Patent Publication Number: US-2023155481-A1

Title: Optimizing pulse patterns with bernoulli polynomials

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a National Stage Entry of International Patent Application No. PCT/EP2021/052010, filed Jan. 28, 2021, and titled “OPTIMIZING PULSE PATTERNS WITH BERNOULLI POLYNOMIALS”, which claims the benefit of priority to European Patent Application No. 20160674.6, and titled “OPTIMIZING PULSE PATTERNS WITH BERNOULLI POLYNOMIALS”, the entire contents of which are hereby incorporated by reference in their entirety. 
    
    
     BACKGROUND 
     The present disclosure relates to the field of control of electrical converters. In particular, the present disclosure relates to a method, a computer program, a computer-readable medium and a controller for controlling an electrical converter. Additionally, the present disclosure relates to the electrical converter. 
     Optimized pulse patterns (OPPs) minimize harmonic distortions in power electronic systems, such as the total demand distortion (TDD) of the load current. For an electrical converter with an inductive load, a cost function may be formulated with the squared amplitudes of the switching function harmonics divided by their harmonic order. By minimizing this cost function, the optimal switching angles can be computed that minimize the current TDD while providing the desired amplitude and phase of the fundamental voltage component. Traditionally, to make the optimization problem computationally tractable, a finite number of harmonic orders has been considered in the optimization problem, resulting in a finite-sum cost function. 
     US 2018/145579 A1 describes a method for controlling a three-phase electrical converter, wherein a three-phase optimized pulse pattern is selected from a table of pre-computed optimized pulse patterns based. The optimized pulse pattern is optimized by time-shifting switching instants of the optimized pulse pattern such that a cost function being quadratic in the time shifts is minimized. 
     “Bernoulli Polynomials”, pages 29-39, http://www.nhn.ou.edu/‘milton/p5013/chap4.pdf inter alia explicitly describes the fourth order Bernoulli polynomial. 
     BRIEF DESCRIPTION 
     Embodiments of the present disclosure reduce the total demand distortion of an electrical converter. Embodiments of the present disclosure also reduce the computational effort for determining optimized pulse patterns and/or increase the accuracy for determining switching angles of optimized pulse patterns. 
     An aspect of the present disclosure relates to a method for controlling an electrical converter. The method may be automatically performed by a controller of the electrical converter. The method may be performed online, i.e. during the operation of the electrical converter. 
     The electrical converter may include a plurality of semiconductor switches, which may be controlled, such that the electrical converter converts an input voltage into a multi-level output voltage. In general, the output voltage may be a two-level voltage, a three-level voltage or may have a higher number of levels. 
     The electrical converter may be a medium-voltage converter and/or may operate at switching frequencies of a few hundred Hz, such as 100 Hz to 500 Hz. The electrical converter may be a high-power converter, in particular a high-power low-voltage converter, and/or may operate at a few kHz, such as 1 kHz to 5 kHz. 
     According to an embodiment of the present disclosure, the method includes: determining a modulation index for the output voltage from a reference voltage, a flux and/or a current reference. The modulation index may be the ratio between the amplitude of the fundamental component of the output voltage to be generated and a possible maximal amplitude. The modulation index may be determined from another reference quantity, such as a reference voltage, a flux and/or a current reference, which may be provided by a superordinated controller, for example, a current controller. The reference voltage may be provided by its amplitude, its frequency and/or its phase. 
     According to an embodiment of the present disclosure, the method includes: determining an optimized pulse pattern, for example based on the modulation index; and determining switching states from the optimized pulse pattern and applying the switching states to semiconductor switches of the electrical converter, such that the multi-level output voltage is generated. The optimized pulse pattern may be determined from the modulation index. For determining the optimized pattern, the frequency and/or the phase of a reference voltage and/or another reference quantity need not be used, since the optimized pulse pattern may be determined for a 2π period and the time instants of the switching states then later may be determined dependent on the frequency and/or the phase of the reference voltage. However, it may be possible that the number of switching transitions of the optimized pulse pattern is set dependent on the frequency. The lower the frequency, the more switching transitions for the optimized pulse pattern may be used. 
     Switching states of the semiconductor switches of the electrical converter are obtained from the optimized pulse pattern. A switching state defines, when (i.e. at which switching instant) a semiconductor switch of the electrical converter is switched into another state (such as conducting and blocking). For this mapping, the frequency and/or the phase of the reference voltage may be used. The electrical converter then generates an output voltage, which follows the reference voltage. 
     In the optimal case, the output voltage is equal to the reference voltage; because of the discrete nature of the multi-level output voltage, this is, however, not possible. Higher order harmonics are produced. The aim of the optimization procedure of the optimized pulse pattern is to optimize the higher order harmonics with respect to specific goals. Here, the goal may be to reduce the total demand distortion as much as possible. 
     An optimized pulse pattern includes a plurality of switching transitions, each switching transition encoding a transition between two different levels of the multi-level output voltage at a switching angle. 
     A switching transition may be encoded with a natural number, for example, +2 for a transition from the level −1 to 1 and −2 for a transition from the level 1 to 1. For electrical converters with more than two output levels, a switching transition may be encoded by +1 and −1, but there may be more than two types of switching transitions. 
     A switching angle may be a real number encoding an angle between 0 and 2n. For example, the switching angle may be normalized that a value of 1 corresponds to an angle of 2π. Each switching transition may be associated with a switching angle. In the optimization problem, the switching transitions are fixed and the switching angles are varied. 
     The optimized pulse pattern is determined by optimizing a cost function. The switching angles may be varied, such that for the provided switching transitions, the cost function becomes optimal, such as minimal or maximal. As initial input, an optimized pulse pattern, which may have been optimized with respect to another optimization goal, may be selected. Such an initial value optimized pulse pattern may be stored in the controller. The number of switching transitions may be fixed or may be selected with respect to the frequency of the reference voltage. 
     In particular, the cost function J(x)=Δu T  A(x)Δu, with the vector x composed of the plurality of switching angles, is a cost matrix A(x), which is multiplied from both sides with a vector Δu composed of the plurality of switching transitions, wherein each entry of the cost matrix A(x) depends on polynomials, in particular on fourth order Bernoulli polynomials and/or linear combinations of fourth order Bernoulli polynomials, into which the switching angles are input. 
     The switching transition vector Δu may have a length N of the number of switching transitions. The switching transitions may alternate with increasing switching angle and/or may be provided with a specific pattern. As input to the optimization problem, the switching transition vector Δu is fixed. 
     The cost matrix A (x) is an N×N-matrix and/or may be symmetric. Each entry A ij (x) of the cost matrix may be a linear combination of shifted fourth order Bernoulli polynomials. These linear combinations may be further simplified for determining the polynomial form of the entries A ij (x) of the cost matrix. The input values to these polynomials may be sums and/or differences of the switching angles. 
     It will be shown below that such type of cost functions provide an exact solution of the optimization problem to minimize the total demand distortion of the optimized pulse pattern. Due to the polynomial nature of the entries of the cost matrix, the optimization problem is much less computationally demanding than other optimization schemes, which may need trigonometric functions and which only approximate the optimization problem. 
     As will be shown below, an infinite-sum cost function arises in the optimization problem, which however, due to some mathematical tricks can be recasts into polynomial functions in the switching angles. Starting with complex Fourier coefficients and full-wave symmetric single-phase optimized pulse patterns, the cost function may be written in a quadratic form in terms of switching transitions. The coefficients are infinite sums of exponential functions involving pairs of switching angles. It is shown that these infinite sums are fourth-order Bernoulli polynomials. 
     Note that the fourth-order Bernoulli polynomial is 
     
       
         
           
             
               
                 B 
                 4 
               
               ( 
               y 
               ) 
             
             = 
             
               
                 y 
                 4 
               
               - 
               
                 2 
                 ⁢ 
                 
                   y 
                   3 
                 
               
               + 
               
                 y 
                 2 
               
               - 
               
                 
                   1 
                   
                     3 
                     ⁢ 
                     0 
                   
                 
                 . 
               
             
           
         
       
     
     In the cost function and in particular in the cost matrix, shifted Bernoulli polynomials {tilde over (B)} 4 (y)=B 4 (y+s), where s is a constant offset, may be used. 
     The optimization problem may be solved subject to one or more constraints, in particular the ones described below. In general, well known methods are available to solve such a constrained optimization problem, such as gradient-based optimization methods or Newton&#39;s method, for which also reliable implementations in software exists. 
     According to an embodiment of the present disclosure, the cost function is optimized subject to a modulation index constraint, which ensures that the optimized pulse pattern results in the desired modulation index. To this end, the desired modulation index may be added to the optimization problem in the form of a constraint. 
     The modulation index constraint may be 
     
       
         
           
             
               
                 1 
                 
                   2 
                   ⁢ 
                   π 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   g 
                 
                 
                   Δ 
                   ⁢ 
                   
                     u 
                     i 
                   
                   ⁢ 
                   
                     e 
                     
                       
                         - 
                         j 
                       
                       ⁢ 
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                         x 
                         i 
                       
                     
                   
                 
               
             
             = 
             
               m 
               2 
             
           
         
       
     
     where g denotes the number of switching angles and switching transitions, Δu i  are the switching transitions, x i  are the switching angles and m is the modulation index. Here and in the following, j=√{square root over (−1)} is the imaginary unit for complex numbers. 
     According to an embodiment of the present disclosure, the cost function is optimized subject to a DC component constraint, which ensures that a DC component generated by the optimized pulse pattern is 0. However, it also may be possible that a desired DC component is generated. 
     The DC component constraint may be 
     
       
         
           
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 g 
               
               
                 
                   x 
                   i 
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 
                   u 
                   i 
                 
               
             
             = 
             
               u 
               0 
             
           
         
       
     
     where Δu i  are the switching transitions, x i  are the switching angles and u 0  is the first level of the output voltage. 
     According to an embodiment of the present disclosure, the cost function is optimized subject to a transition order constraint, which ensures that an order of the switching transitions is maintained. The transition order constraint may be 
       0≤ x   1   ≤x   2    . . . ≤x   g ≤1
 
     where x i  are the (normalized) switching angles. It may be that the upper bound is further limited to ½ for a half-wave symmetric optimized pulse pattern or to ¼ for a quarter-wave symmetric optimized pulse pattern. 
     According to an embodiment of the present disclosure, the entries A ij (x) of the cost matrix depend on a difference x i −x j  and/or a sum x i +x j  of the switching angles x i . Specifically, x i  and x j  may be the ith and jth component of the vector of switching angles x. 
     According to an embodiment of the present disclosure, the entries A ij (x) of the cost matrix are sums of shifted fourth order Bernoulli polynomials B 4 , i.e. 
     
       
         
           
             
               
                 A 
                 
                   i 
                   ⁢ 
                   j 
                 
               
               ( 
               x 
               ) 
             
             = 
             
               c 
               ⁢ 
               
                 
                   ∑ 
                   k 
                 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     
                       
                         y 
                         k 
                       
                       ( 
                       x 
                       ) 
                     
                     + 
                     
                       s 
                       k 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where c is a constant, y k (x) is either a difference x i −x j  or a sum x i +x j  of the switching angles x i , and where the s k  are shifting constant. The operator “˜” in the claims has to interpreted as “proportional”, i.e. that the left-hand side is equal to the right-hand side multiplied with a constant (here c). 
     According to an embodiment of the present disclosure, the entries A ij (x) of the cost matrix are 
         A   ij ( x )= cB   4 ( x   i   −x   j ) 
     where c is a constant, B 4  is the fourth order Bernoulli polynomial and x i  and x j  are the switching angles. When no further symmetry is imposed on the form of the optimized pulse pattern, the cost matrix may have such a form. This is derived in detail below. 
     To simplify the problem, it is possible to impose a quarter- and/or half-wave symmetry on the optimized pulse pattern. In particular, the number of switching angles, which have to be determined, may then be reduced by the factor of 2 or 4. This may reduce the computational effort significantly. However, these simplifications restrict the solution space and, in general, may lead to suboptimal solutions. In the present approach, using selector variables, the cost functions for half-wave symmetric as well as quarter- and half-symmetric optimized pulse patterns are derived. 
     According to an embodiment of the present disclosure, half-wave symmetry is imposed on the optimized pulse pattern. Half-wave symmetry may mean that the optimized pulse pattern between the switching angles 0 and π is equal to the negated switching pattern between π and 2π. 
     In this case, it can be shown and will be shown below that the entries A ij (x) of the cost matrix are 
     
       
         
           
             
               
                 
                   A 
                   ij 
                 
                 ( 
                 x 
                 ) 
               
               = 
               
                 c 
                 ⁢ 
                    
                 
                   
                     P 
                     3 
                   
                   ( 
                   
                     
                       x 
                       i 
                     
                     - 
                     
                       x 
                       j 
                     
                   
                   ) 
                 
               
             
             , 
                 
             
               
                 where 
                 ⁢ 
                 
                     
                      
                 
                 ⁢ 
                 
                   
                     P 
                     3 
                   
                   ( 
                   y 
                   ) 
                 
               
               = 
               
                 
                   
                     B 
                     4 
                   
                   ( 
                   y 
                   ) 
                 
                 - 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     + 
                     
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                       2 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where c is a constant, B 4  is the fourth order Bernoulli polynomial, and x i  and x j  are the switching angles. The polynomial P 3 (y) may be further simplified to a third order polynomial. 
     According to an embodiment of the present disclosure, quarter-wave and half-wave symmetry is imposed on the optimized pulse pattern. Quarter-wave symmetry may mean that the optimized pulse pattern between the switching angles 0 and π/2 is equal to the mirrored switching pattern between π/2 and π. 
     In this case, it can be shown and will be shown below that the entries A ij (x) of the cost matrix are 
     
       
         
           
             
               
                 
                   A 
                   ij 
                 
                 ( 
                 x 
                 ) 
               
               = 
               
                 
                   c 
                   ⁢ 
                      
                   
                     
                       P 
                       3 
                     
                     ( 
                     
                       
                         x 
                         i 
                       
                       + 
                       
                         x 
                         j 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   c 
                   ⁢ 
                      
                   
                     
                       P 
                       3 
                     
                     ( 
                     
                       
                         x 
                         i 
                       
                       - 
                       
                         x 
                         j 
                       
                     
                     ) 
                   
                 
               
             
             , 
                
             
               
                 where 
                 ⁢ 
                 
                     
                      
                 
                 ⁢ 
                 
                   
                     P 
                     3 
                   
                   ( 
                   y 
                   ) 
                 
               
               = 
               
                 
                   
                     B 
                     4 
                   
                   ( 
                   y 
                   ) 
                 
                 - 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     + 
                     
                       1 
                       2 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where c is a constant, B 4  is the fourth order Bernoulli polynomial, and x i  and x j  are the switching angles. It has to be noted that when the resulting polynomial P 3 (y) of shifted Bernoulli polynomials for quarter- and half-wave symmetry is further simplified, the fourth order terms cancel. P 3 (y) may be a third-order polynomial function. 
     The above cost functions all may be applied to single-phase optimized pulse patterns. These pulse patterns also may be part of a multi-phase optimized pulse pattern. The phases may be optimized independently from each other. 
     For three-phase optimized pulse patterns, further symmetry assumptions may be imposed, which may further simplify the optimization problem. In his case, possible cost functions are again polynomial functions. 
     According to an embodiment of the present disclosure, the multi-level output voltage is a three-phase voltage, and 120° symmetry is imposed between the three phases. The 120° symmetry may mean that triplen-order harmonics of the harmonic order 3, 6, 9, etc can be ignored in the optimization problem. These triplen-order voltage harmonics may incur no current harmonics in a three-phase system with a floating star point. 
     In this case, it can be shown and will be shown below that the entries A ij (x) of the cost matrix are 
     
       
         
           
             
               
                 
                   A 
                   ij 
                 
                 ( 
                 x 
                 ) 
               
               = 
               
                 c 
                 ⁢ 
                    
                 
                   
                     S 
                     
                       3 
                       ⁢ 
                       f 
                     
                   
                   ( 
                   
                     
                       x 
                       i 
                     
                     - 
                     
                       x 
                       j 
                     
                   
                   ) 
                 
               
             
             , 
                
             
               
                 where 
                 ⁢ 
                 
                     
                       
                 
                 ⁢ 
                 
                   
                     S 
                     
                       3 
                       ⁢ 
                       f 
                     
                   
                   ( 
                   y 
                   ) 
                 
               
               = 
               
                 
                   2 
                   ⁢ 
                   
                     
                       B 
                       4 
                     
                     ( 
                     y 
                     ) 
                   
                 
                 - 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     + 
                     
                       1 
                       3 
                     
                   
                   ) 
                 
                 - 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     - 
                     
                       1 
                       3 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where c is a constant, B 4  is the fourth order Bernoulli polynomial, and x i  and x j  are the switching angles. 
     According to an embodiment of the present disclosure, the multi-level output voltage is a three-phase voltage, and a 120° symmetry is imposed between the three phases and half-wave symmetry is imposed on the optimized pulse pattern. 
     In this case, it can be shown and will be shown below that the entries A ij (x) of the cost matrix are 
     
       
         
           
             
               
                 
                   A 
                   
                     i 
                     ⁢ 
                     j 
                   
                 
                 ( 
                 x 
                 ) 
               
               = 
               
                 c 
                 ⁢ 
                    
                 
                   
                     S 
                     
                       3 
                       ⁢ 
                       h 
                     
                   
                   ( 
                   
                     
                       x 
                       i 
                     
                     - 
                     
                       x 
                       j 
                     
                   
                   ) 
                 
               
             
             , 
                
             
               
                 where 
                 ⁢ 
                     
                 
                   
                     S 
                     
                       3 
                       ⁢ 
                       h 
                     
                   
                   ( 
                   y 
                   ) 
                 
               
               = 
                 
               
                 
                   2 
                   ⁢ 
                   
                     
                       B 
                       4 
                     
                     ( 
                     y 
                     ) 
                   
                 
                 - 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     + 
                     
                       1 
                       3 
                     
                   
                   ) 
                 
                 - 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     - 
                     
                       1 
                       3 
                     
                   
                   ) 
                 
                 - 
                 
                   2 
                   ⁢ 
                   
                     
                       B 
                       4 
                     
                     ( 
                     
                       y 
                       + 
                       
                         1 
                         2 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     + 
                     
                       1 
                       6 
                     
                   
                   ) 
                 
                 + 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     - 
                     
                       1 
                       6 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where c is a constant, B 4  is the fourth order Bernoulli polynomial, and x i  and x j  are the switching angles. 
     According to an embodiment of the present disclosure, the multi-level output voltage is a three-phase voltage, and a 120° symmetry is imposed between the three phases and half-wave and quarter-wave symmetry is imposed on the optimized pulse pattern. 
     In this case, it can be shown and will be shown below that the entries A ij (x) of the cost matrix are 
     
       
         
           
             
               
                 
                   A 
                   
                     i 
                     ⁢ 
                     j 
                   
                 
                 ( 
                 x 
                 ) 
               
               = 
               
                 
                   c 
                   ⁢ 
                       
                   
                     
                       S 
                       
                         3 
                         ⁢ 
                         h 
                       
                     
                     ( 
                     
                       
                         x 
                         i 
                       
                       + 
                       
                         x 
                         j 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   c 
                   ⁢ 
                      
                   
                     
                       S 
                       
                         3 
                         ⁢ 
                         h 
                       
                     
                     ( 
                     
                       
                         x 
                         i 
                       
                       - 
                       
                         x 
                         j 
                       
                     
                     ) 
                   
                 
               
             
             , 
                
             
               
                 where 
                 ⁢ 
                     
                 
                   
                     S 
                     
                       3 
                       ⁢ 
                       h 
                     
                   
                   ( 
                   y 
                   ) 
                 
               
               = 
                 
               
                 
                   2 
                   ⁢ 
                   
                     
                       B 
                       4 
                     
                     ( 
                     y 
                     ) 
                   
                 
                 - 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     + 
                     
                       1 
                       3 
                     
                   
                   ) 
                 
                 - 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     - 
                     
                       1 
                       3 
                     
                   
                   ) 
                 
                 - 
                 
                   2 
                   ⁢ 
                   
                     
                       B 
                       4 
                     
                     ( 
                     
                       y 
                       + 
                       
                         1 
                         2 
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     + 
                     
                       1 
                       6 
                     
                   
                   ) 
                 
                 + 
                 
                   
                     B 
                     4 
                   
                   ( 
                   
                     y 
                     - 
                     
                       1 
                       6 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     as defined above, where c is a constant, B 4  is the fourth order Bernoulli polynomial, and x i  and x j  are the switching angles. 
     Further aspects of the present disclosure relate to a computer program, which when executed by a processor is adapted for performing the method as described above and below and to a computer-readable medium, in which such a computer program is stored. The method may be implemented in software and may be run on a controller having a processor and a memory in which the computer program is stored. 
     A computer-readable medium may be a floppy disk, a hard disk, an USB (Universal Serial Bus) storage device, a RAM (Random Access Memory), a ROM (Read Only Memory), an EPROM (Erasable Programmable Read Only Memory) or a FLASH memory. A computer readable medium may also be a data communication network, e.g. the Internet, which allows downloading a program code. In general, the computer-readable medium may be a non-transitory or transitory medium. 
     A further aspect of the present disclosure relates to a controller for an electrical converter adapted for performing the method as described above and below. It has to be noted that the method also may be at least partially implemented in hardware, for example in a DSP or FPGA. 
     A further aspect of the present disclosure relates to an electrical converter, which includes a plurality of semiconductor switches for converting an input voltage into an output voltage and which includes a controller as described above and below. 
     It has to be understood that features of the method as described in the above and in the following may be features of the electrical converter, the computer program, the computer readable medium and the controller, as described in the above and in the following, and vice versa. 
     These and other aspects of the present disclosure will be apparent from and elucidated with reference to the embodiments described hereinafter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The subject matter of the present disclosure will be explained in more detail in the following text with reference to exemplary embodiments which are illustrated in the attached drawings. 
         FIG.  1    schematically shows an electrical converter according to an embodiment of the present disclosure. 
         FIG.  2    shows a flow diagram for a method for controlling an electrical converter according to an embodiment of the present disclosure. 
         FIG.  3    shows a diagram with an optimized pulse pattern used in a method for controlling an electrical converter according to an embodiment of the present disclosure. 
         FIG.  4    shows a diagram comparing the total demand distortion produced with a conventional method with the total demand distortion produced with a method according to an embodiment of the present disclosure. 
         FIG.  5    shows a diagram with switching angles (in degrees) determined with a method according to an embodiment of the present disclosure. 
     
    
    
     In principle, identical parts are provided with the same reference symbols in the figures. 
     DETAILED DESCRIPTION 
     1 The Electrical Converter and the Optimization Problem 
     1.1 The Converter 
       FIG.  1    shows an electrical converter  10 , which includes three phase legs  12 , each of which is composed of two semiconductor switches  14 . A controller  16  of the electrical converter  10  receives a reference voltage v ref  and based on this reference voltage v ref  produces switching states  18 , which are applied to the semiconductor switches  14 . 
     Subject to the control of the controller  16 , the electrical converter  10  converts an it voltage v in , which may be provided by a DC link, to a multi-level output voltage v out . In the shown case, the output voltage v out  has two levels, however, in general, the output voltage v out  may have three, four, five, etc. levels. For example, other types of phase legs  12  may be used, such as neutral point clamped phase legs, which may produce a three-level output voltage v out . 
     As shown, the output voltage v out  may be a three-phase voltage. However, it also is possible that the output voltage v out  is a single-phase output voltage v out , for example such as the one provided by one of the phase legs  12 . 
       FIG.  2    shows a flow diagram illustrating a method that may be performed by the controller  16  for controlling the electrical converter  10 . The controller  16  may include a memory and a processor and the method may be implemented as a computer program in the controller  16 . 
     In step S 10 , the controller  16  receives the reference voltage v ref  and determines a modulation index for the output voltage v out  from the reference voltage V ref . The reference voltage v ref  may be provided by a superordinated controller, which for example is a torque controller for an electrical machine supplied by the electrical converter  10 . 
     In step S 12 , the controller  16  determines an optimized pule pattern u(x) based on the modulation index. One phase of such a pulse pattern u(x) is shown in  FIG.  2   . In general, an initial pulse pattern may be selected, for example dependent on the frequency and/or modulation index of the reference voltage v ref . The initial pulse pattern is then optimized as described above and below. 
     In step S 14 , the controller  16  determines switching states  18  from the optimized pulse pattern u(x) and applies the switching states  18  to the semiconductor switches  14  of the electrical converter  10 , such that the multi-level output voltage v out  is generated. 
     1.2 Pulse Patterns 
       FIG.  2    shows a single-phase two-level pulse pattern u(x) with g+1 switch positions u i , iϵ{0, 1, 2, . . . , g}. The switch positions, which encode the voltage levels of the (single-phase) output voltage v out , are restricted to the integer set u i ϵ{−1, 1}. They switching angles x i , with iϵ{1, 2, . . . , g}, are normalized to the interval 0≤x i ≤1. The g switching transitions Δx i  with iϵ{1, 2, . . . , g} are the differences between two consecutive switch positions 
     The pulse pattern u(x) is a sequence of switch positions; therefore, it may be seen as a piecewise constant signal. In the interval 0≤x≤1, it can be described by 
     
       
         
           
             
               
                 
                   
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                                 u 
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                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 u 
                                 
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                                   - 
                                   1 
                                 
                               
                               , 
                             
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   x 
                                   
                                     g 
                                     - 
                                     1 
                                   
                                 
                               
                               ≤ 
                               x 
                               &lt; 
                               
                                 x 
                                 g 
                               
                             
                           
                         
                         
                           
                             
                               
                                 u 
                                 g 
                               
                               , 
                             
                           
                           
                             
                               
                                 if 
                                 ⁢ 
                                     
                                 
                                   x 
                                   g 
                                 
                               
                               ≤ 
                               x 
                               ≤ 
                               1 
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The pulse pattern defined in Eq. (1) has the initial switch position u 0  and the period 1. Alternatively, the pulse pattern may be defined over the interval [0, 2π] with the switching angles 2πx i . Later, when rewriting the optimization problem below in terms of polynomials, the use of normalized switching angles will lead to a more compact mathematical representation. 
     In this section, no symmetry, other than the fact that the waveform must be periodic with period 1, is imposed on u(x). In order to ensure periodicity, the last switch position is equal to the first one, i.e., u g =u 0 . 
     Given the sequence of switch positions u i , iϵ{1, 2, . . . , g}, the aim is to find the optimal switching angles x i  that minimize the weighted voltage distortion. 
     1.3 Fourier Coefficients 
     Since the pulse pattern u(x) is periodic, it can be expressed as the exponential Fourier series 
     
       
         
           
             
               
                 
                   
                     
                       u 
                       ⁡ 
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         
                           ∑ 
                           ∞ 
                         
                         
                           n 
                           = 
                           
                             - 
                             ∞ 
                           
                         
                       
                       
                         
                           c 
                           n 
                         
                         ⁢ 
                         
                           e 
                           
                             j 
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             nx 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     with the complex Fourier coefficients 
         c   n =∫ 0   1   u ( x ) e   −j2πnx   dx   (3)
 
     where nϵ{0, +1, ±2, . . . , ±∞}. Note that j denotes the imaginary unit. Inserting Eq. (1) into the Fourier coefficient in Eq. (3) leads to 
     
       
         
           
             
               
                 
                   
                     c 
                     n 
                   
                   = 
                   
                     
                       
                         ∫ 
                         0 
                         
                           x 
                           1 
                         
                       
                       
                         
                           u 
                           0 
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             nx 
                           
                         
                         ⁢ 
                         dx 
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           x 
                           1 
                         
                         
                           x 
                           2 
                         
                       
                       
                         
                           u 
                           1 
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             nx 
                           
                         
                         ⁢ 
                         dx 
                       
                     
                     + 
                     … 
                     + 
                     
                       
                         ∫ 
                         
                           x 
                           y 
                         
                         1 
                       
                       
                         
                           u 
                           g 
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             nx 
                           
                         
                         ⁢ 
                         dx 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 
                   
                     c 
                     n 
                   
                   = 
                   
                     j 
                     ⁢ 
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         n 
                       
                     
                     ⁢ 
                     
                       
                         
                           ∑ 
                           g 
                         
                         
                           i 
                           = 
                           1 
                         
                       
                       
                         
                           ( 
                           
                             
                               u 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                             - 
                             
                               u 
                               i 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               nx 
                               i 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where it has been used that u g =u 0 . Defining the ith switching transition Δu i , for iϵ{1, 2, . . . , g}, as 
       Δ u   i   =u   i   −u   i−1   (6)
 
     allows us to write the Fourier coefficient in Eq. (3) in the compact form 
     
       
         
           
             
               
                 
                   
                     c 
                     n 
                   
                   = 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         n 
                       
                     
                     ⁢ 
                     
                       
                         
                           ∑ 
                           g 
                         
                         
                           i 
                           = 
                           1 
                         
                       
                       
                         
                           
                             Δ 
                             ⁢ 
                             u 
                           
                           i 
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               nx 
                               i 
                             
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     for nϵ{±1, ±2, . . . , ±∞}. The DC component of the switching function u(x) is given by 
     
       
         
           
             
               
                 
                   
                     c 
                     0 
                   
                   = 
                   
                     
                       
                         ∫ 
                         0 
                         1 
                       
                       
                         
                           u 
                           ⁡ 
                           ( 
                           x 
                           ) 
                         
                         ⁢ 
                         dx 
                       
                     
                     = 
                     
                       
                         u 
                         0 
                       
                       - 
                       
                         
                           
                             ∑ 
                             g 
                           
                           
                             i 
                             = 
                             1 
                           
                         
                         
                           
                             
                               Δ 
                               ⁢ 
                               u 
                             
                             i 
                           
                           ⁢ 
                           
                             
                               x 
                               i 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     1.4 Total Demand Distortion of the Current 
     The two-level converter  10  with an inductive load is considered. To simplify the analysis, a per unit system is introduced and normalized quantities are adopted. The base quantities are the peak value of the load&#39;s rated phase voltage, the peak value of the rated load current, and the rated fundamental frequency. Unless otherwise stated, all quantities are given in this per unit system. 
     Let V denote the total dc-link voltage, and f 1  is the fundamental frequency. The complex voltage harmonic at the frequency nf 1  is given by 
     
       
         
           
             
               
                 
                   
                     v 
                     n 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       V 
                       dc 
                     
                     ⁢ 
                     
                       
                         c 
                         n 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     n is referred to as the order of the harmonic component. Note that nϵ . 
     It is assumed that the convener  10  supplies a purely inductive load with the load reactance X l . The nth complex current harmonic, with n≠0, follows to 
     
       
         
           
             
               
                 
                   
                     
                       i 
                       n 
                     
                     = 
                     
                       
                         
                           v 
                           n 
                         
                         
                           jn 
                           ⁢ 
                           
                             ω 
                             1 
                           
                           ⁢ 
                           
                             X 
                             l 
                           
                         
                       
                       = 
                       
                         
                           
                             V 
                             dc 
                           
                           
                             2 
                             ⁢ 
                             j 
                             ⁢ 
                             
                               ω 
                               1 
                             
                             ⁢ 
                             
                               X 
                               l 
                             
                           
                         
                         ⁢ 
                         
                           
                             c 
                             n 
                           
                           n 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where the angular fundamental frequency w 1 =2πf 1  has been introduced and Eq. (9) has been used. Thus, the amplitude of the nth current harmonic is the absolute value 
     
       
         
           
             
               
                 
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     
                       i 
                       n 
                     
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                   = 
                   
                     
                       
                         V 
                         dc 
                       
                       
                         2 
                         ⁢ 
                         
                           ω 
                           1 
                         
                         ⁢ 
                         
                           X 
                           l 
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             c 
                             n 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           n 
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The total demand distortion (TDD) of the current in one phase is commonly defined as 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       TDD 
                     
                     = 
                     
                       
                         
                           1 
                           
                             
                               2 
                             
                             ⁢ 
                             
                               I 
                               nom 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 ≠ 
                                 
                                   ± 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   i 
                                   n 
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                               2 
                             
                           
                         
                       
                       = 
                       
                         
                           1 
                           
                             
                               2 
                             
                             ⁢ 
                             
                               I 
                               nom 
                             
                           
                         
                         ⁢ 
                         
                           
                             V 
                             dc 
                           
                           
                             2 
                             ⁢ 
                             
                               ω 
                               1 
                             
                             ⁢ 
                             
                               X 
                               l 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 ≠ 
                                 
                                   ± 
                                   1 
                                 
                               
                             
                             
                               
                                 ( 
                                 
                                   
                                     c 
                                     n 
                                   
                                   n 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where I nom  refers to the nominal rms stator current. Since the per unit system is based on the peak load current, the nominal peak current is one, i.e. √{square root over (2)}l nom =1 pu. Because the Fourier coefficients c n  are defined at nϵ , the sum in Eq. (12) is also defined over nϵ . The positive and negative-sequence fundamental components (with the harmonic order n=±1) are excluded from the sum, 
     1.5 Optimization Problem 
     As a cost function, it is convenient to choose the infinite sum 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           f 
                         
                         ′ 
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             ± 
                             ∞ 
                           
                         
                         
                           n 
                           = 
                           
                             ± 
                             2 
                           
                         
                       
                       
                         
                           ( 
                           
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 c 
                                 n 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                             n 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     which is proportional to the squared current MD in inductive loads, see Eq. (12). The indices 1 and f relate to the single-phase case and full-wave symmetry, respectively. The vector of normalized switching angles is defined as x:=[x 1  x 2  . . . x g ] T . Note that the fundamental component (with the harmonic order n=±1) and the DC component are excluded from the sum. 
     Three constraints may be imposed when formulating the optimization problem. 
     The first constraint may be that the amplitude and phase of the fundamental component of u(x) must be equal to the desired values. For a sine wave, the fundamental component must have an amplitude equal to the modulation index m and a phase of 
     
       
         
           
             - 
             
               π 
               2 
             
           
         
       
     
     rad, i.e., the Fourier coefficient c 1  from Eq. (7) must be equal 
     
       
         
           
             
               - 
               j 
             
             ⁢ 
             
               
                 m 
                 2 
               
               . 
             
           
         
       
     
     According to Eq. (7) this implies that 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         ℊ 
                       
                         
                       
                         Δ 
                         ⁢ 
                         
                           u 
                           i 
                         
                         ⁢ 
                         
                           e 
                           
                             - 
                             j 
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               x 
                               i 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     
                       m 
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The second constraint may result from the fact that the DC component of u(x) usually has to be equal to zero, i.e., 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       ℊ 
                     
                       
                     
                       
                         x 
                         i 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         i 
                       
                     
                   
                   = 
                   
                     u 
                     0 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     according to Eq. (8). 
     The third constraint may ensure that the normalized switching angles are in ascending order and may be restricted to the interval [0, 1], i.e., 
       0≤ x   1   ≤x   2   ≤ . . . ≤x   g ≤1.  (16)
 
     The optimization problem may thus be formulated as 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           x 
                           opt 
                         
                         = 
                         
                           arg 
                           ⁢ 
                               
                           
                             minimize 
                             x 
                           
                           ⁢ 
                               
                           
                             
                               J 
                               
                                 1 
                                 ⁢ 
                                 f 
                               
                               ′ 
                             
                             ( 
                             x 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         
                           
                             subject 
                             ⁢ 
                                 
                             to 
                             ⁢ 
                                 
                             
                               ( 
                               14 
                               ) 
                             
                           
                           , 
                           
                             
                               ( 
                               15 
                               ) 
                             
                             ⁢ 
                                 
                             and 
                             ⁢ 
                                 
                             
                               
                                 ( 
                                 16 
                                 ) 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     2 Single-Phase Formulation 
     As shown in this section, the infinite-sum cost function may be written as a finite sum of quartic polynomials in two variables. The proposed formulation is exact and does not make any approximations. 
     2.1 Full-Wave Symmetry 
     Single-phase optimized pulse patterns are considered. The optimization problem in Eq. (17) is based on the infinite-sum cost function in Eq. (13). 
     2.1.1 Reformulated Cost Function 
     In order to simplify the notation, the slightly modified cost function 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       
                         1 
                         ⁢ 
                         f 
                       
                     
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           ± 
                           1 
                         
                       
                       
                         ± 
                         ∞ 
                       
                     
                       
                     
                       
                         ( 
                         
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               c 
                               n 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           n 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     is introduced. Unlike the original cost function J′ 1f (x) of Eq. (13), J 1f (x) includes the fundamental component with harmonic order n=±1. However, constraint (14) ensures that the amplitude of the fundamental component is equal to the modulation index m. Since the average power in the fundamental component is equal to 
     
       
         
           
             
               
                 m 
                 2 
               
               2 
             
             , 
           
         
       
     
     this leads to the following simple relation between the two cost functions: 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       
                         1 
                         ⁢ 
                         f 
                       
                       ′ 
                     
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           f 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     - 
                     
                       
                         
                           m 
                           2 
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Let Δu denote the vector of switching transitions 
       Δ u :=[Δ u   1   Δu   2    . . . Δu   g ] T .  (20)
 
     For every integer k≥2, let c k  denote the k-dimensional vector of weighted complex Fourier coefficients: 
     
       
         
           
             
               
                 
                   
                     c 
                     k 
                   
                   := 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   c 
                                   
                                     - 
                                     k 
                                   
                                 
                                 
                                   - 
                                   k 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   c 
                                   
                                     - 
                                     2 
                                   
                                 
                                 
                                   - 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   c 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   c 
                                   1 
                                 
                                 1 
                               
                             
                             
                               
                                 
                                   c 
                                   2 
                                 
                                 2 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   c 
                                   k 
                                 
                                 k 
                               
                             
                           
                         
                         ] 
                       
                       T 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Also the binary selector variables δ n , with nϵ{±1, ±2, . . . , ±∞} are introduced, which assume the values 0 or 1. It is assumed that δ n =δ −n  for nϵ{±1, ±2, . . . , ±∞}. These variables will later allow one to remove specific harmonics from the cost function. Multiplying the selector variables with the components of the vector of weighted Fourier coefficients c k  leads to 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   c 
                                   
                                     - 
                                     k 
                                   
                                 
                                 
                                   - 
                                   k 
                                 
                               
                               ⁢ 
                               
                                 δ 
                                 
                                   - 
                                   k 
                                 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 
                                   c 
                                   
                                     - 
                                     2 
                                   
                                 
                                 
                                   - 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 δ 
                                 
                                   - 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   c 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 δ 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   c 
                                   1 
                                 
                                 1 
                               
                               ⁢ 
                               
                                 δ 
                                 1 
                               
                             
                           
                           
                             
                               
                                 
                                   c 
                                   2 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 δ 
                                 2 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 
                                   c 
                                   k 
                                 
                                 k 
                               
                               ⁢ 
                               
                                 δ 
                                 k 
                               
                             
                           
                         
                       
                       ] 
                     
                     T 
                   
                   = 
                   
                     
                       D 
                       k 
                     
                     ⁢ 
                     
                       c 
                       k 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     with the diagonal matrix 
         D   k :=diag(δ −k , . . . , δ −2 , δ −1 , δ 1 , δ 2 , . . . , δ k ).  (23)
 
     By making use of Eq. (7), vector c k  can be factored as: 
     
       
         
           
             
               
                 
                   
                     
                       c 
                       k 
                     
                     = 
                     
                       - 
                       
                         j 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         E 
                         k 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       u 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     where E k  is the matrix of exponential terms: 
     
       
         
           
             
               
                 
                   	 
                   
                     
                       E 
                       k 
                     
                     := 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   2 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   2 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   2 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   2 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   2 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   ? 
                                 
                                 
                                   2 
                                   2 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     The expression 
         J   k ( x ):= c*   k   D*   k   D   k   c   k   (26)
 
     is the finite sum of squared harmonics from harmonic order −k to k, with kϵ{2, 3, . . . }, where * denotes the conjugate transpose. Note that, for every fixed value of x, Eq. (26) is a sequence of numbers that converges to the (modified) cost function in Eq. (18). 
     Owing to the fact that D* k D k =D k , Eq. (26) simplifies to 
         J   k ( x )= c*   k   D   k   c   k   (27)
 
     Eq. (27) is rewritten with the help of Eq. (24), as 
     
       
         
           
             
               
                 
                   	 
                   
                     
                       
                         
                           J 
                           k 
                         
                         ( 
                         x 
                         ) 
                       
                       = 
                       
                         
                           
                             1 
                             
                               4 
                               ⁢ 
                               
                                 π 
                                 2 
                               
                             
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                             u 
                             T 
                           
                           ⁢ 
                           
                             E 
                             k 
                             ★ 
                           
                           ⁢ 
                           
                             D 
                             k 
                           
                           ⁢ 
                           
                             E 
                             k 
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           u 
                         
                         = 
                         
                           
                             1 
                             
                               4 
                               ⁢ 
                               
                                 π 
                                 2 
                               
                             
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                             u 
                             T 
                           
                           ⁢ 
                           
                             A 
                             k 
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           u 
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             where 
           
         
       
       
         
           
             
               
                 
                   
                     	 
                     
                       
                         A 
                         k 
                       
                       := 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       1 
                                       
                                         n 
                                         4 
                                       
                                     
                                       
                                   
                                 
                               
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       
                                         e 
                                         
                                           ? 
                                         
                                       
                                       
                                         n 
                                         4 
                                       
                                     
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     
                                       
                                         ? 
                                       
                                       
                                         n 
                                         4 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       
                                         e 
                                         
                                           ? 
                                         
                                       
                                       
                                         n 
                                         4 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       1 
                                       
                                         n 
                                         4 
                                       
                                     
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     
                                       
                                         ? 
                                       
                                       
                                         n 
                                         4 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                               
                                 ⋮ 
                               
                               
                                 ⋮ 
                               
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     
                                       
                                         ? 
                                       
                                       
                                         n 
                                         4 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     
                                       
                                         ? 
                                       
                                       
                                         n 
                                         4 
                                       
                                     
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       
                                         ± 
                                         1 
                                       
                                     
                                     
                                       ± 
                                       k 
                                     
                                   
                                   
                                     
                                       δ 
                                       n 
                                     
                                     ⁢ 
                                     
                                       1 
                                       
                                         n 
                                         4 
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         . 
                       
                     
                   
                 
                 
                   
                     ( 
                     29 
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     Each of the sums in Eq. (29) is of the form 
     
       
         
           
             
               
                 
                   
                     S 
                     pq 
                   
                   := 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           ± 
                           1 
                         
                       
                       
                         ± 
                         k 
                       
                     
                     
                       
                         δ 
                         n 
                       
                       ⁢ 
                       
                         
                           e 
                           
                             j 
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               n 
                               ⁡ 
                               ( 
                               
                                 
                                   x 
                                   p 
                                 
                                 - 
                                 
                                   x 
                                   q 
                                 
                               
                               ) 
                             
                           
                         
                         
                           n 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     These expressions are even functions, i.e., 
     
       
         
           
             
               
                 
                   	 
                   
                     
                       
                         
                           S 
                           pq 
                         
                         : 
                         
                           + 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   ± 
                                   1 
                                 
                               
                               
                                 ± 
                                 k 
                               
                             
                             
                               
                                 δ 
                                 n 
                               
                               ⁢ 
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   n 
                                   4 
                                 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 ± 
                                 1 
                               
                             
                             
                               ± 
                               k 
                             
                           
                           
                             
                               δ 
                               n 
                             
                             ⁢ 
                             
                               
                                 e 
                                 
                                   ? 
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                         = 
                         
                           S 
                           qp 
                         
                       
                     
                     , 
                     p 
                     , 
                     
                       q 
                       ∈ 
                       
                         { 
                         
                           1 
                           , 
                           2 
                           , 
                           … 
                               
                           , 
                           ℊ 
                         
                         } 
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     which implies that A k  is a symmetric matrix. 
     Now the cost matrix A is defined: 
     
       
         
           
             
               
                 
                   	 
                   
                     A 
                     = 
                     
                       
                         lim 
                         
                           k 
                           
                             → 
                             &#34;\[Rule]&#34; 
                           
                           ∞ 
                         
                       
                         
                       
                         A 
                         k 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   	 
                   
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     1 
                                     
                                       n 
                                       4 
                                     
                                   
                                     
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                       e 
                                       
                                         ? 
                                       
                                     
                                     
                                       n 
                                       4 
                                     
                                   
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                       e 
                                       
                                         ? 
                                       
                                     
                                     
                                       n 
                                       4 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                       e 
                                       
                                         ? 
                                       
                                     
                                     
                                       n 
                                       4 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     1 
                                     
                                       n 
                                       4 
                                     
                                   
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                       e 
                                       
                                         ? 
                                       
                                     
                                     
                                       n 
                                       4 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                       e 
                                       
                                         ? 
                                       
                                     
                                     
                                       n 
                                       4 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                       e 
                                       
                                         ? 
                                       
                                     
                                     
                                       n 
                                       4 
                                     
                                   
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     
                                       ± 
                                       1 
                                     
                                   
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   
                                     δ 
                                     n 
                                   
                                   ⁢ 
                                   
                                     1 
                                     
                                       n 
                                       4 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     For full-wave symmetry all harmonics except the DC component are penalized. To this end, all binary selector variables are set to one, i.e., δ n =1, for all nϵ{±1, ±2, . . . ±k;}. This implies that 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       
                         1 
                         ⁢ 
                         f 
                       
                     
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       lim 
                       
                         k 
                         
                           → 
                           &#34;\[Rule]&#34; 
                         
                         ∞ 
                       
                     
                     
                       
                         
                           J 
                           k 
                         
                         ( 
                         x 
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     With these definitions and according to Eqs. (28) and (34), the cost function in Eq. (18) can be rewritten as 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           f 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         1 
                         
                           4 
                           ⁢ 
                           
                             π 
                             2 
                           
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         T 
                       
                       ⁢ 
                       
                         A 
                         
                           1 
                           ⁢ 
                           f 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       u 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     with the cost matrix 
     
       
         
           
             
               
                 
                   
                     A 
                     
                       1 
                       ⁢ 
                       f 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               1 
                               
                                 n 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         z 
                                       
                                       ⁢ 
                                       ⋯ 
                                       ⁢ 
                                       
                                         x 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                         
                           ⋯ 
                         
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         g 
                                       
                                       ⁢ 
                                       ⋯ 
                                       ⁢ 
                                       
                                         x 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         z 
                                       
                                       ⁢ 
                                       ⋯ 
                                       ⁢ 
                                       
                                         x 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               1 
                               
                                 n 
                                 4 
                               
                             
                           
                         
                         
                           ⋯ 
                         
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         g 
                                       
                                       ⁢ 
                                       ⋯ 
                                       ⁢ 
                                       
                                         x 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         z 
                                       
                                       ⁢ 
                                       ⋯ 
                                       ⁢ 
                                       
                                         x 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         g 
                                       
                                       ⁢ 
                                       ⋯ 
                                       ⁢ 
                                       
                                         x 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                         
                           ⋯ 
                         
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 ± 
                                 ∞ 
                               
                             
                               
                             
                               1 
                               
                                 n 
                                 4 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     Note that A 1f  is a real symmetric matrix. 
     2.1.2 Cost Function Based on Bernoulli Polynomials 
     Recall the constraint (16), which ensures that the normalized switching angles are in ascending order and are bounded by zero and one. This implies that 0≤x p −x q ≤1 holds true for p≥q, with p, qϵ{1, 2, . . . , g}. We define y=x p −x q , with p, qϵ{1, 2, . . . , g} and p≥q. By definition, yϵ[0,1]. To simplify the notation we refrain from adding the indices p and q to y here. 
     Each infinite sum in Eq. (36) is proportional to the Fourier series expansion of a Bernoulli polynomial, i.e., 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           
                             + 
                             1 
                           
                         
                         
                           ± 
                           ∞ 
                         
                       
                       
                         
                           e 
                           
                             j 
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             ny 
                           
                         
                         
                           n 
                           4 
                         
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             2 
                             ⁢ 
                             
                               π 
                               4 
                             
                           
                           3 
                         
                       
                       ⁢ 
                       
                         
                           B 
                           4 
                         
                         ( 
                         y 
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     for 0≤y≤1, where B 4 (y) is the fourth-order Bernoulli polynomial 
     
       
         
           
             
               
                 
                   
                     
                       B 
                       4 
                     
                     ( 
                     y 
                     ) 
                   
                   = 
                   
                     
                       y 
                       4 
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         y 
                         3 
                       
                     
                     + 
                     
                       y 
                       2 
                     
                     - 
                     
                       
                         1 
                         30 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     Eq. (37) is only valid for y in the interval 0≤y≤1. The periodic fourth-order polynomial    4 (y) is introduced as 
           4 ( y )= B   4 ( y−└y ┘),  (39)
 
     where └y┘=max{zϵ |z≤y}. 
     Matrix A 1f , which was defined in Eq. (36), can now be written in terms of fourth-order polynomials, i.e., 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       
                         1 
                         ⁢ 
                         f 
                       
                     
                     = 
                     
                       - 
                       
                         
                           
                             2 
                             ⁢ 
                             
                               π 
                               4 
                             
                           
                           3 
                         
                         [ 
                         
                           
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ( 
                                 0 
                                 ) 
                               
                             
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ( 
                                 
                                   
                                     x 
                                     2 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             
                               ⋯ 
                             
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ( 
                                 
                                   
                                     x 
                                     g 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       2 
                                     
                                     - 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               ⋯ 
                             
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       g 
                                     
                                     - 
                                     
                                       x 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       g 
                                     
                                     - 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       g 
                                     
                                     - 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               ⋯ 
                             
                             
                               
                                 
                                   B 
                                   4 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     and the cost function in Eq. (35) turns into 
     
       
         
           
             
               
                 
                   
                     J 
                     
                       1 
                       ⁢ 
                       f 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           π 
                           2 
                         
                         6 
                       
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       
                         u 
                         T 
                       
                       [ 
                       
                         
                           
                             
                               
                                 B 
                                 4 
                               
                               ( 
                               0 
                               ) 
                             
                           
                           
                             
                               
                                 B 
                                 4 
                               
                               ( 
                               
                                 
                                   x 
                                   2 
                                 
                                 - 
                                 
                                   x 
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             ⋯ 
                           
                           
                             
                               
                                 B 
                                 4 
                               
                               ( 
                               
                                 
                                   x 
                                   g 
                                 
                                 - 
                                 
                                   x 
                                   1 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 B 
                                 4 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     2 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 B 
                                 4 
                               
                               ⁢ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                             ⋯ 
                           
                           
                             
                               
                                 B 
                                 4 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     g 
                                   
                                   - 
                                   
                                     x 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 B 
                                 4 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     g 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 B 
                                 4 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     g 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             ⋯ 
                           
                           
                             
                               
                                 B 
                                 4 
                               
                               ⁢ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       u 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     2.1.3 Optimization Problem Based on Bernoulli Polynomials 
     The optimization problem in Eq. (17) of full-wave symmetric single-phase OPPs can now be stated in terms of Bernoulli polynomials: 
     
       
         
           
             
               
                 
                   
                     x 
                     opt 
                   
                   = 
                   
                     arg 
                     ⁢ 
                         
                     
                       minimize 
                       x 
                     
                     ⁢ 
                         
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           f 
                         
                       
                       ( 
                       x 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 subject 
                 ⁢ 
                     
                 to 
                 ⁢ 
                     
                 
                   1 
                   
                     2 
                     ⁢ 
                     π 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     g 
                   
                     
                   
                     Δ 
                     ⁢ 
                     
                       u 
                       i 
                     
                     ⁢ 
                     
                       e 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                           x 
                           i 
                         
                       
                     
                   
                 
               
               = 
               
                 m 
                 2 
               
             
             ⁢ 
             
 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   g 
                 
                   
                 
                   
                     x 
                     i 
                   
                   ⁢ 
                   Δ 
                   ⁢ 
                   
                     u 
                     i 
                   
                 
               
               = 
               
                 u 
                 0 
               
             
             ⁢ 
             
 
             
               0 
               ≤ 
               
                 x 
                 1 
               
               ≤ 
               
                 x 
                 2 
               
               ≤ 
               ≤ 
               
                 x 
                 g 
               
               ≤ 
               1. 
             
           
         
       
     
     The infinite-sum cost function Ji f  is given in Eq. (41); the vector of normalized switching angles was defined as x:=[x 1  x 2  . . . x g ] T . 
     2.2 Half-Wave Symmetry 
     Half-wave symmetry implies that the second half of the pulse pattern u(x) is the exact opposite of the first half, i.e., 
     
       
         
           
             
               u 
               ⁡ 
               ( 
               
                 x 
                 + 
                 
                   1 
                   2 
                 
               
               ) 
             
             = 
             
               
                 
                   - 
                   
                     u 
                     ⁡ 
                     ( 
                     x 
                     ) 
                   
                 
                 ⁢ 
                     
                 for 
                 ⁢ 
                 
                     
                       
                 
                 ⁢ 
                 0 
               
               ≤ 
               x 
               ≤ 
               
                 
                   1 
                   2 
                 
                 . 
               
             
           
         
       
     
     This condition is only satisfied for a two-level converter if the number of switching transitions h:=g/2 per half cycle is odd. Half-wave symmetry imposes the following two conditions on the normalized switching angles x i  and the switching transitions Δu i : 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       p 
                     
                     = 
                     
                       
                         
                           x 
                           
                             p 
                             - 
                             h 
                           
                         
                         + 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                               
                           and 
                           ⁢ 
                               
                           Δ 
                           ⁢ 
                           
                             u 
                             p 
                           
                         
                       
                       = 
                       
                         
                           - 
                           Δ 
                         
                         ⁢ 
                         
                           u 
                           
                             p 
                             - 
                             h 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               for 
               ⁢ 
                   
               p 
             
             ∈ 
             
               
                 { 
                 
                   
                     h 
                     + 
                     1 
                   
                   , 
                   … 
                   , 
                   g 
                 
                 } 
               
               . 
             
           
         
       
     
     2.2.1 Fourier Coefficients 
     The exponential Fourier series expansion of functions with half-wave symmetry contains only odd-order harmonics, i.e., 
     
       
         
           
             
               
                 
                   
                     c 
                     n 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 0.5 
                               
                               
                                 
                                   u 
                                   ⁡ 
                                   ( 
                                   x 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     nx 
                                   
                                 
                                 ⁢ 
                                   
                                 dx 
                               
                             
                           
                         
                         
                           
                             for 
                             ⁢ 
                                 
                             odd 
                             ⁢ 
                                 
                             n 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             for 
                             ⁢ 
                                 
                             even 
                             ⁢ 
                                 
                             
                               n 
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     This also implies that the DC component is always zero. 
     Similar to the derivation in Sect. 1.3, the Fourier coefficient for odd harmonic orders n follows as 
     
       
         
           
             
               
                 
                   
                     c 
                     n 
                   
                   = 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             ∫ 
                             0 
                             
                               x 
                               1 
                             
                           
                           
                             
                               u 
                               0 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 nx 
                               
                             
                             ⁢ 
                             dx 
                           
                         
                         + 
                         
                           
                             ∫ 
                             
                               x 
                               1 
                             
                             
                               x 
                               2 
                             
                           
                           
                             
                               u 
                               1 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 nx 
                               
                             
                             ⁢ 
                             dx 
                           
                         
                         + 
                         ⋯ 
                         + 
                         
                           
                             ∫ 
                             
                               x 
                               h 
                             
                             0.5 
                           
                           
                             
                               u 
                               h 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 nx 
                               
                             
                             ⁢ 
                             dx 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
             
               
                 
                   = 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                       1 
                       
                         π 
                         ⁢ 
                         n 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         h 
                       
                         
                       
                         Δ 
                         ⁢ 
                         
                           u 
                           i 
                         
                         ⁢ 
                         
                           e 
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               nx 
                               i 
                             
                           
                         
                                                                                                                
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     where it has been used that u h =−u 0  holds for two-level converters. 
     2.2.2 Cost Function 
     The cost function for half-wave symmetric pulse patterns is defined as 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           h 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             
                               + 
                               1 
                             
                           
                           , 
                           
                             + 
                             3 
                           
                           , 
                           … 
                         
                         
                           ± 
                           ∞ 
                         
                       
                         
                       
                         
                           ( 
                           
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 c 
                                 n 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                             n 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     where the vector of normalized switching angles has been redefined as x:=[x 1  x 2  . . . x h ] T . As in the case of the cost function in Eq. (18) for full-wave symmetry, J 1h  includes the fundamental component. Even-order harmonics, however, are not penalized, 
     Following the procedure in Sect. 2.1.1, the cost function can be rewritten in the form 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           h 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         1 
                         
                           π 
                           2 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         T 
                       
                       ⁢ 
                       
                         A 
                         
                           1 
                           ⁢ 
                           h 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       u 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     where the vector of switching transitions has been redefined as Δu:=[Δu 1  Δu 2  . . . Δu h ] T . 
     The matrix A 1h  is equal to A in Eq. (33) with the binary selector variables set to 
     
       
         
           
             
               
                 
                   
                     δ 
                     n 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             e 
                             
                               
                                 - 
                                 j 
                               
                               ⁢ 
                               π 
                               ⁢ 
                               n 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             0 
                           
                           
                             
                               for 
                               ⁢ 
                                   
                               even 
                               ⁢ 
                                   
                               n 
                             
                           
                         
                         
                           
                             1 
                           
                           
                             
                               for 
                               ⁢ 
                                   
                               odd 
                               ⁢ 
                                   
                               
                                 n 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
     Each of the infinite sums in A 1h  has the form 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ∑ 
                           
                             ± 
                             ∞ 
                           
                         
                         
                           n 
                           = 
                           
                             ± 
                             1 
                           
                         
                       
                       
                         
                           δ 
                           n 
                         
                         ⁢ 
                         
                           
                             e 
                             
                               j 
                               ⁢ 
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               ny 
                             
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       [ 
                       
                         
                           
                             
                               ∑ 
                               
                                 ± 
                                 ∞ 
                               
                             
                             
                               n 
                               = 
                               
                                 ± 
                                 1 
                               
                             
                           
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 ny 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                         - 
                         
                           
                             
                               ∑ 
                               
                                 ± 
                                 ∞ 
                               
                             
                             
                               n 
                               = 
                               
                                 ± 
                                 1 
                               
                             
                           
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   n 
                                   ⁡ 
                                   ( 
                                   
                                     y 
                                     + 
                                     
                                       1 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                      
                   
                     
                       where 
                       ⁢ 
                          
                       y 
                     
                     = 
                     
                       
                         x 
                         p 
                       
                       - 
                       
                         x 
                         q 
                       
                     
                   
                   , 
                     
                   
                     with 
                     ⁢ 
                         
                     p 
                   
                   , 
                   
                     q 
                     ∈ 
                     
                       
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             … 
                                
                             , 
                             h 
                           
                           } 
                         
                         ⁢ 
                             
                         and 
                         ⁢ 
                             
                         p 
                       
                       ≥ 
                       
                         q 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
           
         
       
     
     The constraint 
     
       
         
           
             0 
             ≤ 
             
               x 
               1 
             
             &lt; 
             
               x 
               2 
             
             ≤ 
             … 
             ≤ 
             
               x 
               h 
             
             ≤ 
             
               1 
               2 
             
           
         
       
     
     implies that 
     
       
         
           
             
               0 
               ≤ 
               
                 
                   x 
                   p 
                 
                 - 
                 
                   x 
                   q 
                 
               
               ≤ 
               
                 
                   1 
                   2 
                 
                 ⁢ 
                     
                 for 
                 ⁢ 
                     
                 p 
               
               ≥ 
               q 
             
             , 
           
         
       
       
         
           
             
               with 
               ⁢ 
                   
               p 
             
             , 
             
               q 
               ∈ 
               
                 
                   { 
                   
                     1 
                     , 
                     2 
                     , 
                     … 
                     , 
                     h 
                   
                   } 
                 
                 . 
                     
                 Therefore 
               
             
             , 
             
               y 
               ∈ 
               
                 [ 
                 
                   0 
                   , 
                   
                     1 
                     2 
                   
                 
                 ] 
               
             
           
         
       
     
     According to Eq. (37) 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           ± 
                           ∞ 
                         
                       
                       
                         n 
                         = 
                         
                           ± 
                           1 
                         
                       
                     
                     
                       
                         δ 
                         n 
                       
                       ⁢ 
                       
                         
                           e 
                           
                             j 
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             ny 
                           
                         
                         
                           n 
                           4 
                         
                       
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           π 
                           4 
                         
                         3 
                       
                       [ 
                       
                         
                           
                             B 
                             4 
                           
                           ( 
                           y 
                           ) 
                         
                         - 
                         
                           
                             B 
                             4 
                           
                           ( 
                           
                             y 
                             + 
                             
                               1 
                               2 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                                
                   
                     
                       = 
                       
                         
                           - 
                           
                             
                               π 
                               4 
                             
                             3 
                           
                         
                         ⁢ 
                         
                           
                             P 
                             3 
                           
                           ( 
                           y 
                           ) 
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
     where the polynomial P 3 (y) is defined as 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       3 
                     
                     ( 
                     y 
                     ) 
                   
                   = 
                   
                     
                       
                         
                           B 
                           4 
                         
                         ( 
                         y 
                         ) 
                       
                       - 
                       
                         
                           B 
                           4 
                         
                         ( 
                         
                           
                             1 
                             2 
                           
                           + 
                           y 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           - 
                           2 
                         
                         ⁢ 
                         
                           y 
                           3 
                         
                       
                       + 
                       
                         
                           3 
                           ⁢ 
                           
                             y 
                             2 
                           
                         
                         2 
                       
                       - 
                       
                         
                           1 
                           16 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
     Its periodically extended counterpart    3 (y) is defined as 
     
       
         
           
             
               
                 
                   
                     
                       𝒫 
                       3 
                     
                     ( 
                     y 
                     ) 
                   
                   = 
                   
                     
                       
                         ℬ 
                         4 
                       
                       ( 
                       y 
                       ) 
                     
                     - 
                     
                       
                         
                           ℬ 
                           4 
                         
                         ( 
                         
                           
                             1 
                             2 
                           
                           + 
                           y 
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     The cost function in Eq. (48) then reduces to 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       
                         1 
                         ⁢ 
                         h 
                       
                     
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       - 
                       
                         
                           π 
                           2 
                         
                         3 
                       
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       
                         u 
                         T 
                       
                       [ 
                       
                         
                           
                             
                               
                                 P 
                                 3 
                               
                               ( 
                               0 
                               ) 
                             
                           
                           
                             
                               
                                 P 
                                 3 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     2 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 P 
                                 3 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     h 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 P 
                                 3 
                               
                               ( 
                               
                                 
                                   x 
                                   2 
                                 
                                 - 
                                 
                                   x 
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 P 
                                 3 
                               
                               ⁢ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 P 
                                 3 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     h 
                                   
                                   - 
                                   
                                     x 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 P 
                                 3 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     h 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 P 
                                 3 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     h 
                                   
                                   - 
                                   
                                     x 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 P 
                                 3 
                               
                               ⁢ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       u 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   55 
                   ) 
                 
               
             
           
         
       
     
     It is interesting to observe that imposing half-wave symmetry reduces the fourth-order Bernoulli polynomials in Eq. (41) to third-order polynomials. This shows that in the case of half-wave symmetry the cost function is a multivariate cubic polynomial. 
     2.2.3 Optimization Problem 
     Writing the optimization problem of half-wave symmetric single-phase optimized pulse patterns in terms of cubic polynomials leads to 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       opt 
                     
                     = 
                     
                       
                         arg 
                         ⁢ 
                             
                         
                           minimize 
                           x 
                         
                         ⁢ 
                             
                         
                           
                             J 
                             
                               1 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           x 
                           ) 
                         
                         ⁢ 
                             
                         subject 
                         ⁢ 
                             
                         to 
                         ⁢ 
                             
                         
                           1 
                           π 
                         
                         ⁢ 
                         
                           
                             
                               ∑ 
                               h 
                             
                             
                               i 
                               = 
                               1 
                             
                           
                           
                             Δ 
                             ⁢ 
                             
                               u 
                               i 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   x 
                                   ⁢ 
                                   i 
                                 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           m 
                           2 
                         
                         ⁢ 
                             
                         
                           0 
                           ≤ 
                           
                             x 
                             1 
                           
                           ≤ 
                           
                             x 
                             2 
                           
                           ≤ 
                           … 
                           ≤ 
                           
                             x 
                             h 
                           
                           ≤ 
                           
                             1 
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   56 
                   ) 
                 
               
             
           
         
       
     
     with the cost function J 1h  as in Eq. (55). The vector of normalized switching angles was defined as x:=[x 1  x 2  . . . x h ] T . 
     2.3 Quarter- and Half-Wave Symmetry 
     Quarter-wave symmetry implies that the pulse pattern is symmetric around the midpoints of the two half-cycles. In the following, pulse patterns with imposed quarter- and halfwave symmetry are considered, and d:=g/4 is defined as the number of switching angles per quarter cycle. 
     2.3.1 Fourier Coefficients 
     In the case of quarter- and half-wave symmetry, the pulse pattern u(x) is an odd function, i.e., u(x)=−u(−x), and the fundamental component is a sine wave. The trigonometric Fourier series expansion of the switching function u(x) is given by 
     
       
         
           
             
               
                 
                   
                     
                       u 
                       ⁡ 
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         
                           ∑ 
                           ∞ 
                         
                         
                           n 
                           = 
                           0 
                         
                       
                       
                         
                           b 
                           n 
                         
                         ⁢ 
                            
                         sin 
                         ⁢ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             nx 
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   57 
                   ) 
                 
               
             
           
         
       
     
     where b n  is zero for even values of n. For odd n, b n  is given by 
     
       
         
           
             
               
                 
                   
                     b 
                     n 
                   
                   = 
                   
                     8 
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         0.25 
                       
                       
                         
                           u 
                           ⁡ 
                           ( 
                           x 
                           ) 
                         
                         ⁢ 
                         
                           sin 
                           ⁡ 
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             nx 
                           
                           ) 
                         
                         ⁢ 
                            
                         dx 
                       
                     
                   
                 
               
               
                 
                   ( 
                   58 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                          
                   
                     = 
                     
                       
                         4 
                         
                           n 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               u 
                               0 
                             
                             + 
                             
                               
                                 
                                   ∑ 
                                   d 
                                 
                                 
                                   i 
                                   = 
                                   1 
                                 
                               
                               
                                 Δ 
                                 ⁢ 
                                 
                                   u 
                                   i 
                                 
                                 ⁢ 
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       nx 
                                       i 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   59 
                   ) 
                 
               
             
           
         
       
     
     In order to keep the equations symmetric, a dummy variable x 0  is introduced and x 0 =0. The Fourier coefficient b n  for odd n can then be stated as 
     
       
         
           
             
               
                 
                   
                     b 
                     n 
                   
                   = 
                   
                     
                       4 
                       
                         n 
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             
                               u 
                               0 
                             
                             ⁢ 
                                
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   nx 
                                   0 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 ∑ 
                                 d 
                               
                               
                                 i 
                                 = 
                                 1 
                               
                             
                             
                               Δ 
                               ⁢ 
                               
                                 u 
                                 i 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     nx 
                                     i 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
     2.3.2 Cost Function 
     The cost function is now given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           q 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             ∞ 
                           
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                             … 
                           
                         
                         
                           
                             ( 
                             
                               
                                 b 
                                 n 
                               
                               n 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
     where the vector of normalized switching angles has been redefined as x:=[x 1  x 2  . . . x d ] T . The factor ½ accounts for the fact that the Fourier series is single sided. For k≥1, the k-dimensional vector of weighted. Fourier coefficients is 
     
       
         
           
             
               
                 
                   
                     b 
                     k 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   b 
                                   1 
                                 
                                 1 
                               
                             
                             
                               
                                 
                                   b 
                                   3 
                                 
                                 3 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   b 
                                   k 
                                 
                                 k 
                               
                             
                           
                         
                         ] 
                       
                       T 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
     The binary selector variables γ n , with nϵ{1, 3, 5 . . . }, which assume the value 0 or 1, are introduced. These selector variables will later be used to remove specific harmonics from the cost function. Multiplying the selector variables with the components of the vector of weighted. Fourier coefficients b k  leads to 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   
                                     b 
                                     1 
                                   
                                   1 
                                 
                                 ⁢ 
                                 
                                   γ 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   
                                     b 
                                     3 
                                   
                                   3 
                                 
                                 ⁢ 
                                 
                                   γ 
                                   3 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   b 
                                   k 
                                 
                                 k 
                               
                             
                           
                         
                         ⁢ 
                         
                           γ 
                           k 
                         
                       
                       ] 
                     
                     T 
                   
                   = 
                   
                     
                       D 
                       qk 
                     
                     ⁢ 
                     
                       b 
                       k 
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
           
         
       
     
     with the diagonal matrix 
         D   qk :=diag(γ 1 , γ 3 , . . . , γ k ).  (64)
 
     According to Eq. (60), b k  can be factored as 
     
       
         
           
             
               
                 
                   
                     
                       b 
                       k 
                     
                     = 
                     
                       
                         4 
                         π 
                       
                       ⁢ 
                       
                         
                           D 
                           qk 
                         
                         [ 
                         
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       x 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       x 
                                       d 
                                     
                                   
                                   ) 
                                 
                                 
                                   1 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     6 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       x 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 
                                   3 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     6 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 
                                   3 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     6 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       x 
                                       d 
                                     
                                   
                                   ) 
                                 
                                 
                                   3 
                                   2 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               …⋮ 
                             
                             
                                 
                             
                           
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     k 
                                     ⁢ 
                                     
                                       x 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     k 
                                     ⁢ 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     k 
                                     ⁢ 
                                     
                                       x 
                                       d 
                                     
                                   
                                   ) 
                                 
                                 
                                   k 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         q 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       u 
                       q 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               u 
                               0 
                             
                           
                           
                             
                               Δ 
                               ⁢ 
                               
                                 u 
                                 
                                     
                                   1 
                                 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               Δ 
                               ⁢ 
                               
                                 u 
                                 
                                     
                                   d 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     T 
                   
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
     is the augmented vector of switching transitions. For each kϵ{1, 3, 5, . . . }, define 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       
                         1 
                         ⁢ 
                         qk 
                       
                     
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       b 
                       k 
                       T 
                     
                     ⁢ 
                     
                       D 
                       qk 
                       * 
                     
                     ⁢ 
                     
                       D 
                       qk 
                     
                     ⁢ 
                     
                       
                         b 
                         k 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
     Due to the fact that D* qk D qk =D qk , Eq. (67) simplifies to 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           qk 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           b 
                           k 
                           T 
                         
                         ⁢ 
                         
                           D 
                           qk 
                         
                         ⁢ 
                         
                           b 
                           k 
                         
                       
                       = 
                       
                         
                           8 
                           
                             π 
                             2 
                           
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                           u 
                           q 
                           T 
                         
                         ⁢ 
                         
                           A 
                           
                             1 
                             ⁢ 
                             qk 
                           
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                           u 
                           q 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
       
         
           with 
         
       
       
         
           
             
               A 
               
                 1 
                 ⁢ 
                 qk 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               2 
                             
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               n 
                               ⁢ 
                               
                                 x 
                                 g 
                               
                             
                             ) 
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   g 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   g 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   d 
                                 
                               
                               ) 
                             
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   0 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               2 
                             
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               n 
                               ⁢ 
                               
                                 x 
                                 1 
                               
                             
                             ) 
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   1 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   d 
                                 
                               
                               ) 
                             
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     … 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   0 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   d 
                                 
                               
                               ) 
                             
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   1 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 ⁢ 
                                 
                                   x 
                                   d 
                                 
                               
                               ) 
                             
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                         
                         k 
                       
                       
                         
                           γ 
                           n 
                         
                         ⁢ 
                         
                           
                             
                               cos 
                               2 
                             
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               n 
                               ⁢ 
                               
                                 x 
                                 d 
                               
                             
                             ) 
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               
                 Since 
                 ⁢ 
                     
                 
                   
                     J 
                     
                       1 
                       ⁢ 
                       q 
                     
                   
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 
                   lim 
                   
                     k 
                     → 
                     ∞ 
                   
                 
                   
                 
                   
                     J 
                     
                       1 
                       ⁢ 
                       qk 
                     
                   
                   ( 
                   x 
                   ) 
                 
               
             
             , 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           q 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         8 
                         
                           π 
                           2 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         q 
                         T 
                       
                       ⁢ 
                       
                         A 
                         
                           1 
                           ⁢ 
                           q 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         q 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
           
         
       
       
         
           with 
         
       
       
         
           
             
               A 
               
                 1 
                 ⁢ 
                 qk 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 2 
                               
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 
                                   ? 
                                 
                               
                               ) 
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                   ⁢ 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     
                       … 
                     
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 2 
                               
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 
                                   ? 
                                 
                               
                               ) 
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     
                       … 
                     
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                     
                       … 
                     
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                     
                                   
                                     ? 
                                   
                                 
                                 ) 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     
                       … 
                     
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 cos 
                                 2 
                               
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 n 
                                 
                                   ? 
                                 
                               
                               ) 
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     Each of the sums in Matrix A 1q  is of the form 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       1 
                       ⁢ 
                       q 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             1 
                           
                           , 
                           3 
                           , 
                           5 
                           , 
                           … 
                         
                       
                       ∞ 
                     
                     
                       
                         γ 
                         n 
                       
                       ⁢ 
                       
                         
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               n 
                               ⁢ 
                               
                                 x 
                                 p 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                 nx 
                                 q 
                               
                             
                             ) 
                           
                         
                         
                           n 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             ∞ 
                           
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                             … 
                           
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   n 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       x 
                                       p 
                                     
                                     + 
                                     
                                       x 
                                       q 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             ∞ 
                           
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             3 
                             , 
                             5 
                             , 
                             … 
                           
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   n 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       x 
                                       p 
                                     
                                     - 
                                     
                                       x 
                                       q 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     = 
                     
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 , 
                                 3 
                                 , 
                                 5 
                                 , 
                                 … 
                               
                             
                             ∞ 
                           
                           
                             
                               γ 
                               n 
                             
                             ⁢ 
                             
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         q 
                                       
                                       + 
                                       
                                         x 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           
                             2 
                               
                           
                         
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 , 
                                 3 
                                 , 
                                 5 
                                 , 
                                 … 
                               
                             
                             ∞ 
                           
                           
                             
                               γ 
                               n 
                             
                             ⁢ 
                             
                               
                                 cos 
                                 ⁡ 
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         q 
                                       
                                       - 
                                       
                                         x 
                                         p 
                                       
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               with 
               ⁢ 
                  
               p 
             
             , 
             
               q 
               ∈ 
               
                 
                   { 
                   
                     0 
                     , 
                     1 
                     , 
                     … 
                         
                     , 
                     d 
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     Now the fact is used that 
     
       
         
           
             
               
                 
                   
                     cos 
                     ⁡ 
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           e 
                           
                             jx 
                               
                           
                         
                         + 
                         
                           e 
                           
                             - 
                             jx 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   73 
                   ) 
                 
               
             
           
         
       
     
     to show that 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       1 
                       ⁢ 
                       q 
                     
                   
                   = 
                   
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 
                                   ± 
                                   1 
                                 
                               
                               , 
                               
                                 ± 
                                 3 
                               
                               , 
                               
                                 ± 
                                 5 
                               
                               , 
                               … 
                             
                           
                           ∞ 
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   m 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       x 
                                       
                                         p 
                                           
                                       
                                     
                                     + 
                                     
                                       x 
                                       q 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 
                                   ± 
                                   1 
                                 
                               
                               , 
                               
                                 ± 
                                 3 
                               
                               , 
                               
                                 ± 
                                 5 
                               
                               , 
                               … 
                             
                           
                           
                             ± 
                             ∞ 
                           
                         
                         
                           
                             γ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         p 
                                       
                                       - 
                                       
                                         x 
                                         q 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
           
         
       
     
     Up to this point the formulation is general and applies to both single- and three-phase converters with half- and quarter-wave symmetry. According to Eq. (57), γ n =1 for n=1, 3, 5, . . . in the case of a single-phase converter with quarter- and half-wave symmetry. 
     Hence 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       1 
                       ⁢ 
                       q 
                     
                   
                   = 
                   
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 
                                   ± 
                                   1 
                                 
                               
                               , 
                               
                                 ± 
                                 3 
                               
                               , 
                               
                                 ± 
                                 5 
                               
                               , 
                               … 
                             
                           
                           
                             ± 
                             ∞ 
                           
                         
                         
                           
                             e 
                             
                               j 
                               ⁢ 
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                 m 
                                 ⁡ 
                                 ( 
                                 
                                   
                                     x 
                                     
                                       p 
                                         
                                     
                                   
                                   + 
                                   
                                     x 
                                     q 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             n 
                             4 
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 
                                   ± 
                                   1 
                                 
                               
                               , 
                               
                                 ± 
                                 3 
                               
                               , 
                               
                                 ± 
                                 5 
                               
                               , 
                               … 
                             
                           
                           
                             ± 
                             ∞ 
                           
                         
                         
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   n 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       x 
                                       p 
                                     
                                     - 
                                     
                                       x 
                                       q 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               n 
                               4 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
           
         
       
     
     In Eq. (49) binary selector variables δ n  had been defined, which are zero for even harmonic orders n. With their help, Eq. (75) is rewritten as 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       1 
                       ⁢ 
                       q 
                     
                   
                   = 
                   
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 ± 
                                 1 
                               
                             
                           
                           
                             ± 
                             ∞ 
                           
                         
                         
                           
                             δ 
                             n 
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   n 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       x 
                                       p 
                                     
                                     + 
                                     
                                       x 
                                       q 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 ± 
                                 1 
                               
                             
                           
                           
                             ± 
                             ∞ 
                           
                         
                         
                           
                             δ 
                             n 
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   n 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       x 
                                       p 
                                     
                                     + 
                                     
                                       x 
                                       q 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     = 
                     
                       - 
                       
                         
                           
                             π 
                             4 
                           
                           12 
                         
                         [ 
                         
                           
                             
                               𝒫 
                               3 
                             
                             ( 
                             
                               
                                 x 
                                 p 
                               
                               + 
                               
                                 x 
                                 q 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               𝒫 
                               3 
                             
                             ( 
                             
                               
                                 x 
                                 p 
                               
                               - 
                               
                                 x 
                                 q 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
           
         
       
     
     where Eq. (52) has been used. 
     For a single-phase converter with quarter- and half-wave symmetry, the constraint 
     
       
         
           
             0 
             ≤ 
             
               x 
               0 
             
             ≤ 
             
               x 
               1 
             
             ≤ 
             ⋯ 
             ≤ 
             
               x 
               d 
             
             ≤ 
             
               1 
               4 
             
           
         
       
     
     is imposed on the normalized switching angles. This constraint implies that 
     
       
         
           
             
               
                 0 
                 ≤ 
                 
                   
                     x 
                     p 
                   
                   + 
                   
                     x 
                     q 
                   
                 
                 ≤ 
                 
                   
                     1 
                     2 
                   
                   ⁢ 
                       
                   and 
                   ⁢ 
                       
                   0 
                 
                 ≤ 
                 
                   
                     x 
                     p 
                   
                   - 
                   
                     x 
                     q 
                   
                 
                 ≤ 
                 
                   1 
                   4 
                 
               
                   
             
             ⁢ 
             
 
             
               
                 
                   for 
                   ⁢ 
                       
                   p 
                 
                 ≥ 
                 q 
               
               , 
               
                 with 
                 ⁢ 
                     
                 p 
               
               , 
               
                 q 
                 ∈ 
                 
                   
                     { 
                     
                       0 
                       , 
                       1 
                       , 
                       … 
                           
                       , 
                       d 
                     
                     } 
                   
                   . 
                 
               
             
           
         
       
     
     Hence    3 (y) in Eq. (77) is equal to P 3 (y), and the cost function is 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           q 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         8 
                         
                           π 
                           2 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         q 
                         T 
                       
                       ⁢ 
                       
                         A 
                         
                           1 
                           ⁢ 
                           q 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         q 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
           
         
       
       
         
           with 
         
       
       
         
           
             
               A 
               
                 1 
                 ⁢ 
                 q 
               
             
             = 
             
               
                 
                   
                     π 
                     4 
                   
                   12 
                 
                 [ 
                 ⁠ 
                 
                   
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             2 
                             ⁢ 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           0 
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               1 
                             
                             + 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               1 
                             
                             - 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     
                       ⋯ 
                     
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             + 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             - 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               1 
                             
                             + 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               1 
                             
                             - 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             2 
                             ⁢ 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           0 
                           ) 
                         
                       
                     
                     
                       ⋯ 
                     
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               4 
                             
                             + 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             - 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             + 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             - 
                             
                               x 
                               
                                 0 
                                   
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             + 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             - 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                       
                     
                     
                       ⋯ 
                     
                     
                       
                         
                           
                             P 
                             3 
                           
                           ( 
                           
                             2 
                             ⁢ 
                             
                               x 
                               d 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             P 
                             3 
                           
                           ( 
                           0 
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     2.3.3 Optimization Problem 
     For single-phase OPPs with quarter- and half-wave symmetry, the optimization problem in terms of cubic polynomials can now be written as 
     
       
         
           
             
               
                 
                   
                     x 
                     opt 
                   
                   = 
                   
                     arg 
                     ⁢ 
                         
                     
                       minimize 
                       x 
                     
                     ⁢ 
                         
                     
                       
                         J 
                         
                           1 
                           ⁢ 
                           q 
                         
                       
                       ( 
                       x 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               subject 
               ⁢ 
                   
               to 
               ⁢ 
               
                    
                   
               
               ⁢ 
               
                 4 
                 π 
               
               ⁢ 
               
                 ( 
                 
                   
                     u 
                     
                       0 
                         
                     
                   
                   + 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                       
                       d 
                     
                     
                       Δ 
                       ⁢ 
                       
                         u 
                         i 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             x 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
                 ) 
               
             
             = 
             m 
           
         
       
       
         
           
             0 
             ≤ 
             
               x 
               1 
             
             ≤ 
             
               x 
               2 
             
             ≤ 
             ⋯ 
             ≤ 
             
               x 
               d 
             
             ≤ 
             
               
                 1 
                 4 
               
               . 
             
           
         
       
     
     The cost function J 1q  is given in Eq. (78). The vector of normalized switching angles was defined as x:=[x 1  x 2  . . . x d ] T . 
     3 Three-Phase Formulation with Phase Symmetry 
     In this section optimized pulse patterns for a three-phase converter  10  are considered. The main complication with the three-phase formulation is that the cost function may become a piecewise multivariate polynomial function. 
     It is assumed that the three-phase load connected to the converter is balanced. For an active RL load, this implies that the amplitudes of the sinusoidal phase voltages are equal, the voltages are phase shifted by 120° with respect to each other, and the ohmic resistances and inductances are the same in the three phases. 120° symmetry may be imposed on the optimized pulse pattern. This implies that the switching functions in phases b and c are the same as in phase a except for being phase shifted by 120° and 240°, respectively. 
     3.1 Full-Wave Symmetry 
     Consider the single-phase switching function u(x) with period 1 and g switching angles. Neither quarter- nor half-wave symmetry is imposed. The three-phase optimized pulse pattern is constructed from the single-phase switching function by imposing 120° symmetry between the three phases. 
     3.1.1 Cost Function 
     Harmonics of order n=±3, ±6, ±9, . . . are called triplen-order harmonics. Owing to the phase symmetries in the optimized pulse pattern and the load, the triplen-order voltage harmonics are common-mode harmonics, which do not induce harmonic currents. These harmonics may, thus, be removed from the cost function, which is defined as 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         ℋ 
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               
                                 ± 
                                 1 
                               
                             
                             , 
                             
                               ± 
                               2 
                             
                             , 
                             
                               ± 
                               4 
                             
                             , 
                             … 
                           
                         
                         
                           ± 
                           ∞ 
                         
                       
                       
                         
                           ( 
                           
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 c 
                                 n 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                             n 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                   
                     where 
                     ⁢ 
                         
                     x 
                     ⁢ 
                     
                       
                         
                           := 
                           [ 
                           
                             
                               x 
                               1 
                             
                             ⁢ 
                             
                               x 
                               2 
                             
                             ⁢ 
                                 
                             … 
                             ⁢ 
                                 
                             
                               x 
                               g 
                             
                           
                           ] 
                         
                         T 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
           
         
       
     
     The derivations in Sect. 2.1 that led to the matrix A in Eq. (33) are directly applicable also to the three-phase case, provided that harmonics of triplen order are not penalized in the cost function. This is achieved by setting the binary selector variables δ n  of Eq. (22) equal to 
     
       
         
           
             
               
                 
                   
                     δ 
                     
                       n 
                         
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               for 
                               ⁢ 
                                   
                               n 
                             
                             ∈ 
                             
                               { 
                               
                                 0 
                                 , 
                                 
                                   ± 
                                   3 
                                 
                                 , 
                                 
                                   ± 
                                   6 
                                 
                                 , 
                                 … 
                               
                                    
                               } 
                             
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                             
                               for 
                               ⁢ 
                                   
                               n 
                             
                             ∈ 
                             
                               { 
                               
                                 
                                   ± 
                                   1 
                                 
                                 , 
                                 
                                   ± 
                                   2 
                                 
                                 , 
                                 
                                   ± 
                                   4 
                                 
                                 , 
                                 … 
                               
                                    
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
     by defining 
     
       
         
           
             
               
                 
                   
                     δ 
                     n 
                   
                   = 
                   
                     
                       1 
                       3 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           2 
                           - 
                           
                             e 
                             
                               j 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                 
                                 3 
                               
                             
                           
                           - 
                           
                             e 
                             
                               
                                 - 
                                 j 
                               
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   n 
                                 
                                 3 
                               
                             
                           
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
           
         
       
     
     The infinite sums in Eq. (33) are now of the form 
     
       
         
           
             
               
                 
                   
                     
                       
                         S 
                         
                           3 
                           ⁢ 
                           f 
                         
                       
                       ⁢ 
                       
                         ( 
                         y 
                         ) 
                       
                     
                     := 
                     
                       
                         
                           
                             ∑ 
                             
                               ± 
                               ∞ 
                             
                           
                           
                             n 
                             = 
                             
                               ± 
                               1 
                             
                           
                         
                         
                           
                             δ 
                             n 
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 ny 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                       = 
                       
                         
                           1 
                           3 
                         
                         [ 
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   ∑ 
                                   
                                     ± 
                                     ∞ 
                                   
                                 
                                 
                                   n 
                                   = 
                                   
                                     ± 
                                     1 
                                   
                                 
                               
                               
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     ny 
                                   
                                 
                                 
                                   n 
                                   4 
                                 
                               
                             
                           
                           - 
                           
                             
                               
                                 ∑ 
                                 
                                   ± 
                                   ∞ 
                                 
                               
                               
                                 n 
                                 = 
                                 
                                   ± 
                                   1 
                                 
                               
                             
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       y 
                                       + 
                                       
                                         1 
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                           - 
                           
                             
                               
                                 ∑ 
                                 
                                   ± 
                                   ∞ 
                                 
                               
                               
                                 n 
                                 = 
                                 
                                   ± 
                                   1 
                                 
                               
                             
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       y 
                                       - 
                                       
                                         1 
                                         3 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                      
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             
               
                 
                   where 
                   ⁢ 
                       
                   y 
                 
                 = 
                 
                   
                     x 
                     p 
                   
                   - 
                   
                     x 
                     q 
                   
                 
               
               , 
                  
               
                 with 
                 ⁢ 
                    
                 p 
               
               , 
               
                 q 
                 ∈ 
                 
                   
                     
                       { 
                       
                         1 
                         , 
                         2 
                         , 
                            
                         … 
                            
                         , 
                         g 
                       
                       } 
                     
                     ⁢ 
                        
                     and 
                     ⁢ 
                         
                     p 
                   
                   ≥ 
                   
                     q 
                     . 
                   
                 
               
             
           
         
       
     
     The infinite sums can be rewritten with the help of Eq. (37) as 
     
       
         
           
             
               
                 
                   
                     
                       S 
                       
                         3 
                         ⁢ 
                         f 
                       
                     
                     ( 
                     y 
                     ) 
                   
                   = 
                   
                     - 
                     
                       
                         
                           2 
                           ⁢ 
                           
                             π 
                             4 
                           
                         
                         9 
                       
                       [ 
                       
                         
                           2 
                           ⁢ 
                           
                             
                               ℬ 
                               4 
                             
                             ( 
                             y 
                             ) 
                           
                         
                         - 
                         
                           
                             ℬ 
                             4 
                           
                           ( 
                           
                             y 
                             + 
                             
                               1 
                               3 
                             
                           
                           ) 
                         
                         - 
                         
                           
                             ℬ 
                             4 
                           
                           ( 
                           
                             y 
                             - 
                             
                               1 
                               3 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   84 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                  
                   
                     = 
                     
                       
                         - 
                         
                           
                             2 
                             ⁢ 
                             
                               π 
                               4 
                             
                           
                           3 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       
                                         4 
                                         ⁢ 
                                         
                                           y 
                                           3 
                                         
                                       
                                       3 
                                     
                                   
                                   + 
                                   
                                     
                                       8 
                                       ⁢ 
                                       
                                         y 
                                         2 
                                       
                                     
                                     9 
                                   
                                   - 
                                   
                                     8 
                                     243 
                                   
                                 
                               
                               
                                 
                                   
                                     if 
                                     ⁢ 
                                         
                                     0 
                                   
                                   ≤ 
                                   y 
                                   ≤ 
                                   
                                     1 
                                     3 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     
                                       
                                         4 
                                         ⁢ 
                                         
                                           y 
                                           2 
                                         
                                       
                                       9 
                                     
                                   
                                   + 
                                   
                                     
                                       4 
                                       ⁢ 
                                       y 
                                     
                                     9 
                                   
                                   - 
                                   
                                     20 
                                     243 
                                   
                                 
                               
                               
                                 
                                   
                                     if 
                                     ⁢ 
                                         
                                     
                                       1 
                                       3 
                                     
                                   
                                   ≤ 
                                   y 
                                   ≤ 
                                   
                                     2 
                                     3 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       4 
                                       ⁢ 
                                       
                                         y 
                                         3 
                                       
                                     
                                     3 
                                   
                                   - 
                                   
                                     
                                       28 
                                       ⁢ 
                                       
                                         y 
                                         2 
                                       
                                     
                                     9 
                                   
                                   + 
                                   
                                     
                                       20 
                                       ⁢ 
                                       y 
                                     
                                     9 
                                   
                                   - 
                                   
                                     116 
                                     243 
                                   
                                 
                               
                               
                                 
                                   
                                     if 
                                     ⁢ 
                                         
                                     
                                       2 
                                       3 
                                     
                                   
                                   ≤ 
                                   y 
                                   ≤ 
                                   1 
                                 
                               
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   85 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 For 
                 ⁢ 
                     
                 0 
               
               ≤ 
               y 
               ≤ 
               1 
             
             , 
             
               
                 S 
                 
                   3 
                   ⁢ 
                   f 
                 
               
               ( 
               y 
               ) 
             
           
         
       
     
     of Eq. (84) can also be written in terms of absolute value functions, i.e.: 
     
       
         
           
             
               
                 
                   
                     
                       S 
                       
                         3 
                         ⁢ 
                         f 
                       
                     
                     ( 
                     y 
                     ) 
                   
                   = 
                   
                     - 
                     
                       
                         
                           2 
                           ⁢ 
                           
                             π 
                             4 
                           
                         
                         9 
                       
                       [ 
                       
                         
                           2 
                           ⁢ 
                           
                             
                               B 
                               4 
                             
                             ( 
                             y 
                             ) 
                           
                         
                         - 
                         
                           
                             B 
                             4 
                           
                           ⁢ 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               y 
                               - 
                               
                                 2 
                                 3 
                               
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                         
                         - 
                         
                           
                             B 
                             4 
                           
                           ⁢ 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               y 
                               - 
                               
                                 1 
                                 3 
                               
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   86 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                  
                   
                     = 
                     
                       - 
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 4 
                               
                             
                             9 
                           
                           [ 
                           
                             
                               
                                 10 
                                 ⁢ 
                                 y 
                               
                               3 
                             
                             - 
                             
                               
                                 10 
                                 ⁢ 
                                 
                                   y 
                                   2 
                                 
                               
                               3 
                             
                             - 
                             
                               62 
                               81 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     y 
                                     - 
                                     
                                       2 
                                       3 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   y 
                                   - 
                                   
                                     2 
                                     3 
                                   
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     y 
                                     - 
                                     
                                       1 
                                       3 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   y 
                                   - 
                                   
                                     1 
                                     3 
                                   
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                             
                           
                           ] 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   87 
                   ) 
                 
               
             
           
         
       
     
     The resulting cost function is given by 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       
                         3 
                         ⁢ 
                         f 
                       
                     
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       1 
                       
                         4 
                         ⁢ 
                         
                           π 
                           2 
                         
                       
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       
                         u 
                         T 
                       
                       [ 
                       
                         
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ( 
                               0 
                               ) 
                             
                           
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     2 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     g 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ( 
                               
                                 
                                   x 
                                   2 
                                 
                                 - 
                                 
                                   x 
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     g 
                                   
                                   - 
                                   
                                     x 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     g 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     g 
                                   
                                   - 
                                   
                                     x 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 S 
                                 
                                   3 
                                   ⁢ 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     u 
                   
                 
               
               
                 
                   ( 
                   88 
                   ) 
                 
               
             
           
         
       
     
     with S 3f  as defined in Eq. (85) or (87), and the vector of switching transitions Δu:=[Δu 1  Δu 2  . . . Δu g ] T . 
     3.1.2 Optimization Problem 
     The optimization problem of three-phase optimized pulse patterns with full-wave symmetry and balanced three-phase inductive loads may be stated in terms of the single-phase switching function with Bernoulli polynomials: 
     
       
         
           
             
               
                 
                   
                     x 
                     opt 
                   
                   = 
                   
                     arg 
                     ⁢ 
                         
                     
                       minimize 
                       x 
                     
                     ⁢ 
                         
                     
                       
                         J 
                         
                           3 
                           ⁢ 
                           f 
                         
                       
                       ( 
                       x 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   89 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               subject 
               ⁢ 
                   
               to 
               ⁢ 
                   
               
                 1 
                 
                   2 
                   ⁢ 
                   π 
                 
               
               ⁢ 
               
                 
                   
                     ∑ 
                     g 
                   
                   
                     i 
                     = 
                     1 
                   
                 
                 
                   Δ 
                   ⁢ 
                   
                     u 
                     i 
                   
                   ⁢ 
                   
                     e 
                     
                       
                         - 
                         j 
                       
                       ⁢ 
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                         x 
                         i 
                       
                     
                   
                 
               
             
             = 
             
               m 
               2 
             
           
         
       
       
         
           
             
               
                 
                   ∑ 
                   g 
                 
                 
                   i 
                   = 
                   1 
                 
               
               
                 
                   x 
                   i 
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 
                   u 
                   i 
                 
               
             
             = 
             
               u 
               0 
             
           
         
       
       
         
           
             0 
             ≤ 
             
               x 
               1 
             
             ≤ 
             
               x 
               2 
             
             ≤ 
             … 
             ≤ 
             
               x 
               g 
             
             ≤ 
             1. 
           
         
       
     
     The infinite-sum cost function is given in Eq. (88); the vector of normalized switching angles was defined as x:=[x 1  x 2  . . . x g ] T . 
     Recall the function S 3f (y), which was defined in Eq. (85). Its first-order derivative with respect to y is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         dS 
                         
                           3 
                           ⁢ 
                           f 
                         
                       
                       ( 
                       y 
                       ) 
                     
                     dy 
                   
                   = 
                   
                     
                       - 
                       
                         
                           2 
                           ⁢ 
                           
                             π 
                             4 
                           
                         
                         3 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     4 
                                   
                                   ⁢ 
                                   
                                     y 
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     16 
                                     ⁢ 
                                     y 
                                   
                                   9 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                   ⁢ 
                                       
                                   0 
                                 
                                 ≤ 
                                 y 
                                 ≤ 
                                 
                                   1 
                                   3 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     
                                       8 
                                       ⁢ 
                                       y 
                                     
                                     9 
                                   
                                 
                                 + 
                                 
                                   4 
                                   9 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                   ⁢ 
                                       
                                   
                                     1 
                                     3 
                                   
                                 
                                 ≤ 
                                 y 
                                 ≤ 
                                 
                                   2 
                                   3 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   4 
                                   ⁢ 
                                   
                                     y 
                                     2 
                                   
                                 
                                 - 
                                 
                                   
                                     56 
                                     ⁢ 
                                     y 
                                   
                                   9 
                                 
                                 + 
                                 
                                   20 
                                   9 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                   ⁢ 
                                       
                                   
                                     2 
                                     3 
                                   
                                 
                                 ≤ 
                                 y 
                                 ≤ 
                                 1 
                               
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   90 
                   ) 
                 
               
             
           
         
       
     
     whereas the second-order derivative is given by 
     
       
         
           
             
               
                 
                   
                     
                       
                         d 
                         2 
                       
                       ⁢ 
                       
                         
                           S 
                           
                             3 
                             ⁢ 
                             f 
                           
                         
                         ( 
                         y 
                         ) 
                       
                     
                     
                       dy 
                       2 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           2 
                           ⁢ 
                           
                             π 
                             4 
                           
                         
                         3 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     8 
                                   
                                   ⁢ 
                                   y 
                                 
                                 + 
                                 
                                   16 
                                   9 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                   ⁢ 
                                       
                                   0 
                                 
                                 ≤ 
                                 y 
                                 ≤ 
                                 
                                   1 
                                   3 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   8 
                                   9 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                   ⁢ 
                                       
                                   
                                     1 
                                     3 
                                   
                                 
                                 ≤ 
                                 y 
                                 ≤ 
                                 
                                   2 
                                   3 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   8 
                                   ⁢ 
                                   y 
                                 
                                 - 
                                 
                                   
                                     56 
                                     ⁢ 
                                     y 
                                   
                                   9 
                                 
                               
                             
                             
                               
                                 
                                   if 
                                   ⁢ 
                                       
                                   
                                     2 
                                     3 
                                   
                                 
                                 ≤ 
                                 y 
                                 ≤ 
                                 1 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   91 
                   ) 
                 
               
             
           
         
       
     
     The function s 3f (y), is defined piecewise by cubic polynomials, with 0≤y≤1. This so-called cubic spline is continuous and has continuous first and second-order derivatives over the interval (0, 1). 
     Based on these derivatives the gradient and the Hessian of the cost function J 3f (x), in Eq. (88) can be computed. Because the first and second-order derivatives in Eqs. (90) and (91) are continuous over the interval (0, 1), the gradient and the Hessian of the cost function are continuous over the search space. This facilitates the use of standard, e.g. gradient-based, optimization methods. 
     More specifically, first-order derivatives are required for the commonly-used gradient-based optimization methods, whereas Newton&#39;s method also requires second-order derivatives. Newton&#39;s method converges quadratically fast to the optimal solution under certain conditions. 
     3.2 Half-Wave Symmetry 
     In this section half-wave symmetry within a phase and 120° symmetry between the phases is considered. 
     3.2.1 Cost Function 
     Because all harmonics of even order are zero, the cost function simplifies to 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           3 
                           ⁢ 
                           h 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         ∑ 
                         
                           
                             n 
                             = 
                             
                               ± 
                               1 
                             
                           
                           , 
                           
                             ± 
                             5 
                           
                           , 
                           
                             ± 
                             7 
                           
                           , 
                           … 
                         
                         
                           ± 
                           ∞ 
                         
                       
                       
                         
                           ( 
                           
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 c 
                                 n 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                             n 
                           
                           ) 
                         
                         2 
                       
                     
                   
                   , 
                      
                   
                     
                       where 
                       ⁢ 
                           
                       x 
                     
                     := 
                     
                       
                         
                           [ 
                           
                             
                               x 
                               1 
                             
                             ⁢ 
                                
                             
                               x 
                               2 
                             
                             ⁢ 
                                
                             … 
                             ⁢ 
                                
                             
                               x 
                               h 
                             
                           
                           ] 
                         
                         T 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   92 
                   ) 
                 
               
             
           
         
       
     
     The Fourier coefficients c n  are stated in Eq. (46). 
     As shown in Sect. 2.1.1, the cost function can be rewritten as 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       
                         3 
                         ⁢ 
                         h 
                       
                     
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       1 
                       
                         π 
                         2 
                       
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       u 
                       T 
                     
                     ⁢ 
                     
                       A 
                       
                         3 
                         ⁢ 
                         h 
                       
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     u 
                   
                 
               
               
                 
                   ( 
                   93 
                   ) 
                 
               
             
           
         
       
     
     with Δu:=[Δu 1  Δu 2  . . . Δu h ] T . The matrix A 3h  is equal to A in Eq. (33) with the binary selector variables 
     
       
         
           
             
               
                 
                   	 
                   
                     
                       δ 
                       n 
                     
                     = 
                     
                       
                         1 
                         6 
                       
                       ⁢ 
                       
                         ( 
                         
                           2 
                           - 
                           
                             ? 
                           
                           - 
                           
                             ? 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             ? 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   94 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   	     
                   
                     = 
                     
                       
                         1 
                         6 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             2 
                             - 
                             
                               ? 
                             
                             - 
                             
                               ? 
                             
                             - 
                             
                               2 
                               
                                 ? 
                               
                             
                             + 
                             
                               e 
                               
                                 ? 
                               
                             
                             + 
                             
                               e 
                               
                                 ? 
                               
                             
                           
                           ) 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   95 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     The selector variables are equal to one for non-triplen odd harmonic orders, i.e. n=±1, ±5, ±7, . . . ; in all other cases, they are set to zero. Consider y=x p −x q , with p, qϵ{1, 2, . . . , h} and p≥q. The infinite sums in A 3h  have the form 
     
       
         
           
             
               
                 
                   
                     
                       S 
                       
                         3 
                         ⁢ 
                         h 
                       
                     
                     ( 
                     y 
                     ) 
                   
                   := 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           ± 
                           1 
                         
                       
                       
                         ± 
                         ∞ 
                       
                     
                     
                       
                         δ 
                         n 
                       
                       ⁢ 
                       
                         
                           e 
                           
                             j 
                             ⁢ 
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             ny 
                           
                         
                         
                           n 
                           4 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   96 
                   ) 
                 
               
             
           
         
       
       
         
           
             
                          
                      
             
             ⁢ 
             
               = 
               
                 - 
                 
                   
                     
                       
                         π 
                         4 
                       
                       9 
                     
                     [ 
                     
                       
                         2 
                         ⁢ 
                         
                           
                             ℬ 
                             4 
                           
                           ( 
                           y 
                           ) 
                         
                       
                       - 
                       
                         
                           ℬ 
                           4 
                         
                         ( 
                         
                           y 
                           + 
                           
                             1 
                             3 
                           
                         
                         ) 
                       
                       - 
                       
                         
                           ℬ 
                           4 
                         
                         ( 
                         
                           y 
                           - 
                           
                             1 
                             3 
                           
                         
                         ) 
                       
                       - 
                       
                         2 
                         ⁢ 
                         
                           
                             ℬ 
                             4 
                           
                           ( 
                           
                             y 
                             + 
                             
                               1 
                               2 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ℬ 
                           4 
                         
                         ( 
                         
                           y 
                           + 
                           
                             1 
                             6 
                           
                         
                         ) 
                       
                       + 
                       
                         
                           ℬ 
                           4 
                         
                         ( 
                         
                           y 
                           - 
                           
                             1 
                             6 
                           
                         
                         ) 
                       
                     
                     ] 
                   
                   . 
                 
               
             
           
         
       
     
     Owing to the fact that 
     
       
         
           
             
               0 
               ≤ 
               y 
               ≤ 
               
                 1 
                 2 
               
             
             , 
           
         
       
     
     and because the Bernoulli polynomials are symmetric around 
     
       
         
           
             
               y 
               = 
               
                 1 
                 2 
               
             
             , 
           
         
       
     
     we can rewrite this expression as 
     
       
         
           
             
               
                 
                   
                     
                       
                         S 
                         
                           3 
                           ⁢ 
                           h 
                         
                       
                       ( 
                       y 
                       ) 
                     
                     = 
                     
                       - 
                       
                         
                           
                             π 
                             4 
                           
                           9 
                         
                         [ 
                         
                           
                             y 
                             2 
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   y 
                                   - 
                                   
                                     1 
                                     3 
                                   
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                               3 
                             
                           
                           - 
                           
                             2 
                             ⁢ 
                             
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   y 
                                   - 
                                   
                                     1 
                                     6 
                                   
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                               3 
                             
                           
                           + 
                           
                             3 
                             ⁢ 
                             
                               y 
                               2 
                             
                           
                           - 
                           
                             4 
                             ⁢ 
                             
                               y 
                               3 
                             
                           
                           - 
                           
                             1 
                             4 
                           
                         
                         ] 
                       
                     
                   
                   , 
                      
                   
                     
                       for 
                       ⁢ 
                           
                       0 
                     
                     ≤ 
                     y 
                     ≤ 
                     
                       
                         1 
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   97 
                   ) 
                 
               
             
           
         
       
     
     This leads to the cost function 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       
                         3 
                         ⁢ 
                         h 
                       
                     
                     ( 
                     x 
                     ) 
                   
                   = 
                   
                     
                       
                         1 
                         
                           π 
                           2 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         
                           u 
                           T 
                         
                         [ 
                         
                           
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ( 
                                 0 
                                 ) 
                               
                             
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       2 
                                     
                                     - 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       h 
                                     
                                     - 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ( 
                                 
                                   
                                     x 
                                     2 
                                   
                                   - 
                                   
                                     x 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       h 
                                     
                                     - 
                                     
                                       x 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       h 
                                     
                                     - 
                                     
                                       x 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       h 
                                     
                                     - 
                                     
                                       x 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               ⋮ 
                             
                             
                               
                                 
                                   S 
                                   
                                     3 
                                     ⁢ 
                                     h 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       Δ 
                         
                     
                     ⁢ 
                     u 
                   
                 
               
               
                 
                   ( 
                   98 
                   ) 
                 
               
             
           
         
       
     
     with S 3h  defined in Eq. (97), 
     3.2.2 Optimization Problem 
     Writing the optimization problem of three-phase optimized pulse patterns with half-wave symmetry in terms of polynomials leads to 
     
       
         
           
             
               
                 
                   
                     x 
                     opt 
                   
                   = 
                   
                     arg 
                     ⁢ 
                         
                     
                       minimize 
                       x 
                     
                     ⁢ 
                         
                     
                       
                         J 
                         
                           3 
                           ⁢ 
                           h 
                         
                       
                       ( 
                       x 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   99 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               subject 
               ⁢ 
                   
               to 
               ⁢ 
                   
               
                 1 
                 π 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                   
                 
                   Δ 
                   ⁢ 
                   
                     u 
                     i 
                   
                   ⁢ 
                   
                     e 
                     
                       - 
                       j 
                       ⁢ 
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                         x 
                         i 
                       
                     
                   
                 
               
             
             = 
             
               m 
               2 
             
           
         
       
       
         
           
             
               0 
               ≤ 
               
                 x 
                 1 
               
               ≤ 
               
                 x 
                 2 
               
               ≤ 
               … 
               ≤ 
               
                 x 
                 h 
               
               ≤ 
               
                 1 
                 2 
               
             
             , 
           
         
       
     
     with the cost function J 3h  as in Eq. (98). The vector of normalized switching angles was defined as x:=[x 1  x 2  . . . x h ] T . 
     3.3 Quarter- and Half-Wave Symmetry 
     In this section quarter- and half-wave symmetry and 120° symmetry between the phases is considered. 
     3.3.1 Cost Function 
     Due to phase symmetry, triplen harmonics are again not penalized in the cost function. The resulting cost function can be derived from Eq. (61) by removing all triplen harmonics, i.e. 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           3 
                           ⁢ 
                           q 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             5 
                             , 
                             7 
                             , 
                             … 
                           
                           ∞ 
                         
                           
                         
                           
                             ( 
                             
                               
                                 b 
                                 n 
                               
                               n 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   , 
                   
                     
                       where 
                       ⁢ 
                           
                       x 
                     
                     := 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   x 
                                   1 
                                 
                               
                               
                                 
                                   x 
                                   2 
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   x 
                                   d 
                                 
                               
                             
                           
                           ] 
                         
                         T 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   100 
                   ) 
                 
               
             
           
         
       
     
     The Fourier coefficient b n  are given by Eq. (59). 
     Recall the derivation in Sect. 2.3.2 of rewriting the cost function in the quadratic form Eq. (69) with the augmented vector of switching transitions Δu q  as defined in Eq. (66). Removing triplen harmonics from the cost function can be achieved by setting the binary selector variables, γ n  in Eq. (63), equal to one for non-triplen odd harmonic orders, i.e., n=±1, ±5, ±7, . . . and zero for all other values of n. According to Eq. (74) each of the infinite sums in the cost function J 3q  is given by 
     
       
         
           
             
               
                 
                   	 
                   
                     
                       S 
                       
                         3 
                         ⁢ 
                         q 
                       
                     
                     = 
                     
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               ± 
                               ∞ 
                             
                           
                             
                           
                             
                               ? 
                             
                             γ 
                             
                               ? 
                             
                             
                               
                                 e 
                                 
                                   ? 
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           4 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               ± 
                               ∞ 
                             
                           
                             
                           
                             γ 
                             
                               ? 
                             
                             
                               
                                 
                                   e 
                                   
                                     ? 
                                   
                                 
                                 
                                   n 
                                   4 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   101 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ? 
             
             indicates text missing or illegible when filed 
           
         
       
     
     Eq. (101) can be rewritten with the binary selector variable δ n  defined in Eq. (95) as 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       3 
                       ⁢ 
                       q 
                     
                   
                   = 
                   
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             
                               ± 
                               1 
                             
                           
                           
                             ± 
                             ∞ 
                           
                         
                           
                         
                           
                             δ 
                             n 
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                   n 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       x 
                                       p 
                                     
                                     + 
                                     
                                       x 
                                       q 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               n 
                               4 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             
                               ± 
                               1 
                             
                           
                           
                             ± 
                             ∞ 
                           
                         
                           
                         
                           
                             δ 
                             n 
                           
                           ⁢ 
                           
                             
                               
                                 e 
                                 
                                   j 
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                     n 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         x 
                                         p 
                                       
                                       - 
                                       
                                         x 
                                         q 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 n 
                                 4 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   102 
                   ) 
                 
               
             
           
         
       
     
     With Eq. (96) this expression simplifies to 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       3 
                       ⁢ 
                       q 
                     
                   
                   = 
                   
                     
                       
                         1 
                         4 
                       
                       [ 
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               p 
                             
                             + 
                             
                               x 
                               q 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               p 
                             
                             - 
                             
                               x 
                               q 
                             
                           
                           ) 
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   103 
                   ) 
                 
               
             
           
         
       
     
     Recall that the constraint 
     
       
         
           
             0 
             ≤ 
             
               x 
               0 
             
             ≤ 
             
               x 
               1 
             
             ≤ 
             … 
             ≤ 
             
               x 
               d 
             
             ≤ 
             
               1 
               4 
             
           
         
       
     
     implies that 
     
       
         
           
             
               0 
               ≤ 
               
                 
                   x 
                   p 
                 
                 + 
                 
                   x 
                   q 
                 
               
               ≤ 
               
                 
                   1 
                   2 
                 
                 ⁢ 
                     
                 and 
                 ⁢ 
                     
                 0 
               
               ≤ 
               
                 
                   x 
                   p 
                 
                 - 
                 
                   x 
                   q 
                 
               
               ≤ 
               
                 
                   1 
                   4 
                 
                 ⁢ 
                     
                 for 
                 ⁢ 
                     
                 p 
               
               ≥ 
               q 
             
             , 
             
               with 
               ⁢ 
                   
               p 
             
             , 
             
               q 
               ∈ 
               
                 
                   { 
                   
                     0 
                     , 
                     1 
                     , 
                     … 
                         
                     , 
                     d 
                   
                   } 
                 
                 . 
               
             
           
         
       
     
     Hence the simplified expression of S 3q  in Eq. (97) is also applicable here. 
     This leads to the cost function 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         
                           3 
                           ⁢ 
                           q 
                         
                       
                       ( 
                       x 
                       ) 
                     
                     = 
                     
                       
                         2 
                         
                           π 
                           2 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         q 
                         T 
                       
                       ⁢ 
                       
                         A 
                         
                           3 
                           ⁢ 
                           q 
                         
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         u 
                         q 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   104 
                   ) 
                 
               
             
           
         
       
       
         
           with 
         
       
       
         
           
             
               A 
               
                 3 
                 ⁢ 
                 q 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             2 
                             ⁢ 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           0 
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               1 
                             
                             + 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               1 
                             
                             - 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             + 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             - 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               1 
                             
                             + 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               1 
                             
                             - 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             2 
                             ⁢ 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           0 
                           ) 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             + 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             - 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             + 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             - 
                             
                               x 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             + 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             
                               x 
                               d 
                             
                             - 
                             
                               x 
                               1 
                             
                           
                           ) 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           
                             2 
                             ⁢ 
                             
                               x 
                               d 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             S 
                             
                               3 
                               ⁢ 
                               h 
                             
                           
                           ( 
                           0 
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     3.3.2 Optimization Problem 
     For three-phase optimized pulse patterns with quarter- and half-wave symmetry as well as 120° symmetry between the phases the optimization problem in terms of polynomials can now be written as 
     
       
         
           
             
               
                 
                   
                     x 
                     opt 
                   
                   = 
                   
                     arg 
                     ⁢ 
                         
                     
                       
                         minimize 
                             
                       
                       x 
                     
                     ⁢ 
                     
                       
                         J 
                         
                           3 
                           ⁢ 
                           q 
                         
                       
                       ( 
                       x 
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   105 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               subject 
               ⁢ 
                   
               to 
               ⁢ 
                   
               
                 4 
                 π 
               
               ⁢ 
               
                 ( 
                 
                   
                     u 
                     0 
                   
                   + 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       d 
                     
                       
                     
                       Δ 
                       ⁢ 
                       
                         u 
                         i 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             x 
                             i 
                           
                         
                         ) 
                       
                     
                   
                 
                 ) 
               
             
             = 
             m 
           
         
       
       
         
           
             0 
             ≤ 
             
               x 
               1 
             
             ≤ 
             
               x 
               2 
             
             ≤ 
             … 
             ≤ 
             
               x 
               d 
             
             ≤ 
             
               1 
               4 
             
           
         
       
     
     The cost function J 3q  is given in Eq. (104), The vector of normalized switching angles was defined as x:=[x 1  x 2  . . . x d ] T . 
     4 Benefits 
     This section highlights the numerical benefits, when solving the proposed optimization problems. By avoiding the truncation of the Fourier series and by instead considering the infinite-sum cost function, numerical artefacts in the cost function are avoided. These artefacts tend to manifest themselves in the form of artificial local minima, which slow down the convergence rate of the solver, lead to suboptimal solutions and require many initial points to increase the likelihood of finding a close-to-optimal solution. 
     The proposed formulations avoid these numerical issues. In particular, fast convergence to the optimal solution is achieved and optimized pulse patterns with lower cost function values can be found with fewer initial points. The improvements are particularly remarkable at high pulse numbers, which have been hitherto considered numerically intractable. 
     The optimized pulse patterns may be applied to a high-power converter operating at low pulse numbers. The optimized pulse patterns determined with the herein describe method offer advantages at high pulse numbers. A suitable metric is the square root of the cost, √{square root over (J)}, which is proportional to the current total demand distortion, compare Eq. (12) with Eq. (13).  FIG.  4    compares this metric for quarter- and half-wave symmetric optimized pulse patterns (OPP) with that for naturally-sampled carrier-based pulse width modulation (CB-PWM). The comparison is done at the same switching frequency when varying the number of switching angles d between 3 and 100. A double logarithmic graph is used,  FIG.  4    shows √{square root over (J)} on the vertical axis and the number of switching angles d on the horizontal axis. The figure shows that the ratio of √{square root over (J)} for carrier-based pulse width modulation and optimized pulse patterns stays almost constant over the entire range of d, indicating that the relative advantage of optimized pulse patterns in terms of current total demand distortion remains constant. 
       FIG.  5    shows a three-phase optimized pulse patterns with quarter- and half-wave symmetry as well as 120° phase symmetry. The switching angles (in degrees) are depicted on the vertical axis, and the modulation index is depicted on the horizontal axis, Optimized pulse patterns with d=20 switching angles per quarter-wave are depicted, which correspond to the pulse number  41 . Assuming a fundamental frequency of 50 Hz, the corresponding switching frequency is approximately 2 kHz. Such a switching frequency is commonly used for high-power low-voltage converters. It clearly can be seen that the dependence of the switching angles with respect to the modulation index is highly nontrivial. 
     While the present disclosure has been illustrated and described in detail in the drawings and foregoing description, such illustration and description are to be considered illustrative or exemplary and not restrictive; the present disclosure is not limited to the disclosed embodiments. Other variations to the disclosed embodiments can be understood and effected by those skilled in the art, from a study of the drawings, the disclosure, and the appended claims. In the claims, the word “comprising” does not exclude other elements or steps, and the indefinite article “a” or “an” does not exclude a plurality. A single processor or controller or other unit may fulfil the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. Any reference signs in the claims should not be construed as limiting the scope.