Patent Publication Number: US-10312793-B2

Title: Control of electrical converter based on optimized pulse patterns

Description:
FIELD OF THE INVENTION 
     The invention relates to the field of control of high power converters. In particular, the invention relates to a method and controller for operating an electrical converter based on optimized pulse patterns. Furthermore, the invention relates to a converter system controlled by such a method and/or controller. 
     BACKGROUND OF THE INVENTION 
     High power electrical converters are used for converting voltages of different amplitudes and frequencies (such as AC and DC voltages) into one another. In such converters, semiconductor switches are switched according to specific patterns determined by a controller to achieve the voltage conversion. 
     One possibility to control such a converter are pre-computed optimized pulse patterns (OPP), which may be used to shape the spectrum of the currents and voltages of the converter. Traditional OPP-based methods rely on an external control loop to provide a modulation index reference that is used to read offline computed switching angles. These switching angles are then directly applied to the semiconductor switches. 
     Such a method may have several drawbacks. For example, the dynamic performance may be poor because of the slow modulation index control loop. Also, the OPP angles may have to change slowly when changing the modulation index, i.e., almost continuous. The necessity for smooth OPP angles greatly limits the achievable current and voltage spectrums over the operating range. 
     Therefore, in EP 2 469 692 A1, optimized pulse patterns are modified based on model predictive control. Pre-computed OPP switching angles are modified online in order to follow a flux trajectory. This model predictive control method may solve the two issues mentioned above and may be well suited for machine-side applications. 
     However, for grid side applications, it may be preferable to directly control the current. In a flux-based method for the grid side, the current reference may have to be transformed into a voltage reference and then integrated over time to generate a flux reference. In such a way, generating the voltage reference may require good knowledge of the grid voltage and impedance, as well as parameters of a transformer interconnected between the grid and the converter. 
     Another issue with flux-based control on the grid side may be that generating a flux reference by integrating the voltage reference can cause long lasting off-centered flux trajectories when sudden changes in the voltage amplitude occur. This is the case for grid voltage dips, for instance. 
     EP 0 504 449 A1 relates to a control method for an electrical converter, in which OPPs are modified based on a voltage reference. 
     Lacynsky T. et al, “Modulation Error Control for Medium Voltage Drives with LC-Filters and Synchronous Optimal Pulse Width Modulation, INDUSTRY APPLICATIONS SOCIETY ANNUAL MEETING, 2008. IAS &#39;08. IEEE, IEEE, PISCATAWAY, N.J., USA, Oct. 5, 2008 refers to a simulation and describes that precalculated optimized pulse patterns may be optimized by eliminating undesired harmonics (chapter “Definition and Calculation of Modulation Error” on page 3). A modulation error may be corrected, either based on an analytical solution of the optimization problem (chapter “Modulation Error Control by the LQR Controller” on page 4) or directly by minimizing an energy function (chapter “Modulation Error control by the Ljapunow Controller” on page 4). 
     DESCRIPTION OF THE INVENTION 
     It is an object of the invention to provide a simple, fast and reliable control method for an electrical converter. It is a further object of the invention to provide a control method that is especially suited for a grid side controller. 
     These objectives are achieved by the subject-matter of the independent claims. Further exemplary embodiments are evident from the dependent claims and the following description. 
     An aspect of the invention relates to a method for operating an electrical converter. The method may be implemented in hardware and/or software in a controller. 
     According to an embodiment of the invention, the method comprises: determining an optimized pulse pattern from a fundamental voltage reference for the electrical converter, wherein the optimized pulse pattern is determined from a first lookup table and comprises discrete voltage amplitude values changing at predefined switching instants; determining a harmonic content reference from the fundamental voltage reference based on a second lookup table, wherein the harmonic content reference is a harmonic current reference determined from the frequency spectrum of a current of the electrical converter or the harmonic content reference is a filtered voltage reference determined by applying a first order frequency filter to a voltage, which current or voltage is generated, when the optimized pulse pattern is applied to the electrical converter; determining a harmonic content error from the harmonic content reference by subtracting an estimated and/or filtered output voltage and/or estimated output current of the electrical converter from the harmonic content reference; modifying the optimized pulse pattern by timeshifting switching instants such that the fundamental voltage reference and/or a current reference (from which the fundamental voltage reference may be determined) is tracked and the harmonic content error is corrected by the timeshifted switching instants; and applying the modified optimized pulse pattern to semiconductor switches of the electrical converter. 
     The method is based on two lookup tables. 
     In a first lookup table, optimized pulse patterns are stored. These optimized pulse patterns (which may be seen as reference voltage patterns) may have been computed offline. Based on a fundamental voltage reference, which, for example may be provided as a voltage magnitude and a frequency, one voltage reference pattern is selected. 
     In a second lookup table, harmonic content references are stored. These harmonic content references model the behaviour of the electrical converter, when a specific optimized pulse pattern is applied to the converter. Also, the harmonic content references may have been computed offline. For example, a model of the converter may have been used to calculate the harmonic content references for the specific optimized pulse patterns. 
     In such a way, it is not necessary to online evaluate a model of the converter to determine the harmonic content of specific quantities (such as output/input voltages/currents) in the converter. 
     A harmonic content reference may be stored in the lookup table as timepoint/value-pairs in the lookup table. Intermediate points of a trajectory of the harmonic content reference may be determined from this timepoint/value-pairs by interpolation. 
     From the harmonic content reference, a harmonic content error is determined by comparing the harmonic content reference with an estimated output value of the converter such as an estimated and/or filtered output voltage and/or current. The estimated output value may be directly or indirectly calculated from measurements in the electrical system. 
     It has to be noted that the harmonic content reference and/or the harmonic content error may be discrete time trajectories, i.e. may be provided as values at discrete time points. 
     Furthermore, these two quantities may be provided up to a prediction horizon, which for example, may be a specific number of time instants in the future. 
     Furthermore, a harmonic content reference may refer to a fundamental frequency and to frequencies of higher order harmonics of a voltage signal determined from the optimized pulse pattern. It also may be possible that the fundamental frequency is not part of the harmonic content reference. In general, harmonic content in this context may refer to a signal that may be achieved by the converter. The fundamental component of the signal may be provided by the required fundamental voltage/current reference, and the remaining components (and/or harmonics) may be determined based on the optimized pulse pattern. 
     Then, the harmonic content error is used to modify the voltage reference pattern in such a way, that the harmonic content error is corrected. In other words, the voltage reference pattern is modified such that the error of the fundamental component of the filtered voltage and/or current controlled is minimized and the real harmonic content generated by the converter is shaped according to the optimized pulse pattern. The voltage reference pattern is modified in such a way, that time instants indicating, when the output voltage has to be switched to another level, are moved in time. 
     With the method, it may be ensured that the converter operates safely and reliably by maintaining a set of variables within given bounds, that it closely follows arbitrary electrical references, which relate to the active and reactive power, and that it produces a deterministic voltage and current spectrum according to pre-computed voltage reference patterns. Furthermore, such a method may be simple installed and commissioned. I.e. only less work is necessary to customize the method or a corresponding controller to a specific electrical system. 
     According to an embodiment of the invention, the harmonic content error is corrected by minimizing a cost function, which comprises a term quadratic in the harmonic content error and a term quadratic in timeshifts of the switching instants. The harmonic content error, which also depends indirectly on the estimated values from the timeshifts, may be minimized together with a term that penalizes timeshifts which are getting too large. 
     According to an embodiment of the invention, the timeshifts are applied in a time window ending at a prediction horizon. To minimize the harmonic content error, the controller may only modify to timeshifts up to a predefined prediction horizon. The cost function may be based on the harmonic content error at the end of the prediction horizon, i.e. may be evaluated at the end of the prediction horizon. 
     According to an embodiment of the invention, the cost function is minimized subject to the constraint that an order of switching instants does not change. The voltage reference patterns (optimized pulse patterns) may have been optimized with respect to other objectives, such as losses. To maintain these properties of the voltage reference patterns, it may be beneficial also to maintain the order of the switching instants. The voltage reference patterns should not lose its general form after modification. 
     According to an embodiment of the invention, the cost function is minimized by solving a quadratic programming. This quadratic program may be solved online in the controller. It also is possible to solve this quadratic programming online analytically and to implement the resulting equations in the controller. With these equations, the timeshifts may be calculated based on the optimized pulse patterns and the harmonic content error. 
     According to an embodiment of the invention, the method further comprises: determining the fundamental voltage reference from a fundamental current reference (for example provided by an outer control loop) and an estimated grid voltage. With the method, it is possible to directly control an electrical current of the electrical converter. This may be beneficial for an active rectifier, which is usually controlled via the grid current and not via a flux. 
     According to the invention, the harmonic content reference is a harmonic current reference determined from the frequency spectrum of a current of the converter, which is generated, when the voltage reference pattern is applied to the converter. For example, the converter may be simulated offline and the current generated, when a specific voltage reference pattern is applied, may be calculated. From this current, the frequency spectrum may be determined (for example by discrete Fourier transform). The higher order contributions (without the fundamental frequency component) of the current may be saved in the second lookup table indexed by the fundamental component. The higher order contributions may be seen as a ripple on the fundamental component. 
     Online, i.e. during the operation of the converter, the higher order contributions may be read from the second lookup table and added to the fundamental current reference. The error may then be determined by subtracting an estimated current, for example determined from measurements. 
     Summarized, an optimal current trajectory (as harmonic content reference) may be computed online by adding two components: the fundamental component (as current reference) and the ripple or harmonic components. The fundamental component may be provided by some external loop, e.g., the DC link voltage control loop. The ripple component may be provided by the second lookup table and by the grid impedance. The difference between the actual, estimated current and the optimal current trajectory may then be used to correct the switching instants of the optimized pulse patterns. Note that in this case there is no need to have an external controller in the current loop. This may simplify the commissioning work, may increase the robustness of the solution, and may increase the controller bandwidth. 
     Furthermore, in such a way, the control method may directly regulate an electrical current along an optimal trajectory (instead of a flux trajectory). 
     Also note that the fundamental and harmonic content in the current may be treated as a single quantity. This may simplify the implementation, since no special filters and procedures to separate fundamental and harmonic content are required. 
     According to the invention, the harmonic content reference is a filtered voltage reference determined by applying a first order frequency filter to a voltage which is generated, when the optimized pulse pattern is applied to the converter. It also is possible to base the harmonic content analysis on the converter voltage. 
     For example, the converter may be simulated offline and the voltage generated, when a specific voltage reference pattern is applied, may be calculated. This voltage then may be filtered with a first order frequency filter to extract the harmonic content of the voltage. The filtered voltage then may be stored in the second lookup table. 
     According to an embodiment of the invention, the harmonic content error is the difference of the filtered voltage reference and an estimated filtered voltage, wherein the estimated filter voltage is determined by online applying the first order frequency filter to the estimated output voltage of the electrical converter. 
     Online, the harmonic content reference may be read from the second lookup table, for example based on a fundamental voltage reference. Furthermore, an actual, estimated filtered voltage may be determined by applying the same first order filter that was applied offline to the filtered voltage reference, to an actual, estimated voltage, which may be determined from measurements in the converter. The difference of the filtered voltage reference and the estimated filtered voltage may then be used as harmonic content error. 
     According to an embodiment of the invention, the fundamental voltage reference is determined from a fundamental current reference and an error provided by an external controller such as a PI controller. When the harmonic content reference and the harmonic content error are based on a filtered voltage, it may be necessary to correct the fundamental voltage reference with an external controller that determines deviations from the fundamental voltage based on the difference between the fundamental current reference and the estimated output current. 
     Note that the control method with a harmonic content reference based on current may not need an external controller and corresponding control loop for current control. This may be beneficial since the resulting fast control loop allows the DC link voltage to remain within stricter bounds even in the presence of grid voltage disturbances. This results in a safer and more robust operation with less trip events. It also reduces the time and complexity of commissioning because of the lesser number of tuning parameters. 
     According to an embodiment of the invention, the optimized pulse patterns in the first lookup table have been calculated offline with a model of the electrical converter. As already mentioned, the optimized pulse patterns may be (conventional) optimized pulse patterns, which have been calculated offline, subject to a specific optimization objective, such as minimizing losses. 
     According to an embodiment of the invention, the harmonic content references in the second lookup table have been calculated offline from the corresponding optimized pulse patterns applied to a model of the electrical converter. The harmonic content references may be calculated by analysing a harmonic content of the voltage and/or current produced when the corresponding optimized pulse pattern is applied to the converter. This analysis may be done by simulating the converter offline. 
     A further aspect of the invention relates to a controller for an electrical converter, which controller is adapted for controlling the electrical converter according to the method as described above and below. 
     For example, the controller may comprise a processor and a computer program, which, when being executed by the processor, performs the method. It also may be possible that the method is at least partially implemented in hardware, for example in an FPGA and/or a DSP. 
     The first and second lookup table may be stored in a memory of the controller. 
     A further aspect of the invention relates to a converter system, which comprises an electrical converter and such a controller. The electrical converter may be adapted for converting an AC voltage into a DC voltage and/or vice versa. The electrical converter may comprise semiconductor switches that are controlled by the controller for converting the voltages. 
     According to an embodiment of the invention, the electrical converter is an active rectifier for converting an AC input voltage into a DC output voltage. It is also possible to operate the converter in a reverse mode, in which energy from the DC link is supplied into the AC grid. For example, the electrical converter may be connected to an AC grid and may supply a DC link with its DC output voltage. The controller may measure voltages and/or currents in the electrical grid and may control the generation of the DC link voltage based on these measurements. 
     With the method, the current error on the grid side may be controlled based on general optimized pulse patterns by using a fast closed-loop correction mechanism of the switching instants of the optimized pulse patterns. This in turn allows using optimized pulse patterns specifically designed for specific objectives such as shaping the voltage spectrum and/or minimizing switching losses. Shaping the voltage spectrum to fit electrical grid standards may be of particular interest since the electrical converter may meet the standard&#39;s requirements without using a filter, which may reduce cost and size of the electrical converter. 
     Furthermore, with the described control method/controller, the converter system may be more robust to grid events such as voltage dips due to the ability of fast tracking a current reference and in turn ensuring that the voltage levels of the DC link remain within bounds. 
     It may be expected that the number of trips due to disturbances on the grid voltage is reduced. 
     Additionally, the proposed method may reduce costs of drive commissioning, since it does not require tuning of current controller parameters, which has been shown to be troublesome in a number of situations. 
     Also, the converter system may be cheaper to produce, since it may comply with grid standards without requiring a filter. This is possible because the proposed method drops the requirement on smoothness of the voltage reference patterns, allowing the voltage reference patterns to be computed in order to comply with grid standards and/or to fit specific needs. 
     It has to be understood that features of the method as described in the above and in the following may be features of the converter system and/or the controller as described in the above and in the following, as well as vice versa. 
     These and other aspects of the invention will be apparent from and elucidated with reference to the embodiments described hereinafter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The subject-matter of the invention 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 a converter system according to an embodiment of the invention. 
         FIG. 2  shows a diagram illustrating an optimized pulse pattern. 
         FIG. 3  schematically shows a controller according to an embodiment of the invention. 
         FIG. 4  shows a diagram illustrating a harmonic content reference used in the controller of  FIG. 3 . 
         FIG. 5  schematically shows a controller according to a further embodiment of the invention. 
         FIG. 6  shows a diagram illustrating a further harmonic content reference used in the controller of  FIG. 5 . 
         FIG. 7  shows a diagram illustrating a filter function used in the controller of  FIG. 5 . 
     
    
    
     The reference symbols used in the drawings, and their meanings, are listed in summary form in the list of reference symbols. In principle, identical parts are provided with the same reference symbols in the figures. 
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
       FIG. 1  schematically shows a converter system  10  with an electrical converter  12  interconnecting an electrical grid  14  with a DC link  16 . Thus, the converter system  10  may be seen as active rectifier. Three-phases of the electrical grid  14  provide an AC voltage υ g , which is supplied via a filter and/or transformer  18  to the converter  12 , which transforms the three-phase AC voltage υ g  into a DC voltage supplied to the DC link  16 , which, as shown in  FIG. 1 , may be a split DC link with a neutral point between two DC link capacitors C dc . 
       FIG. 1  additionally shows some variables that will be used in the following and that are listed in the following table. 
     
       
         
           
               
               
             
               
                   
               
               
                 variable 
                 description 
               
               
                   
               
             
            
               
                 ν g,ph  (t), ph ∈ {a, b, c} 
                 (estimated) grid voltage of phase ph at time t 
               
               
                 ν ph  (t), ph ∈ {a, b, c} 
                 converter voltage of phase ph at time t 
               
               
                 i ph  (t), ph ∈ {a, b, c} 
                 grid current of phase ph at time t 
               
               
                 L = L g  + L c   
                 (estimated) total inductance seen by the converter 
               
               
                 R = R g  + R c   
                 (estimated) total resistance seen by the converter 
               
               
                 C dc   
                 half DC link capacitance 
               
               
                 Δt 
                 vector of time changes on the nominal switching 
               
               
                   
                 times 
               
               
                 
                   x 
                 
                 reference value of variable x 
               
               
                 {tilde over (x)} 
                 error of variable x with respect to its reference 
               
               
                 x* 
                 optimal value of x 
               
               
                   
               
            
           
         
       
     
     The converter system  10  furthermore comprises a controller  20 , which controls (switches) semiconductor switches of the electrical converter  12 . As described in detail below, the controller  20  is adapted for modifying optimized pulse patterns such that the fundamental component of the voltage and/or current is tracked and a harmonic content of the three-phase AC voltage (which harmonics would be generated by the electrical converter  10 ) is shaped according to the optimized pulse pattern. 
       FIG. 2  illustrates an example of an optimized pulse pattern  22 . The optimized pulse pattern models a voltage level for every phase to be output by the converter  12 . At switching instants  24 , the voltage level of a phase has to be switched to another level. Usually, the switching instants  24  are provided as switching angles, which have to be converted to switching instants based on the period length of the actual frequency. 
     From the optimized pulse pattern  22 , a time dependent voltage reference  v (t) may be determined, which may be translated into switching commands to be applied to the converter  12 . 
       FIG. 3  shows an embodiment of the controller  20 , which is based on current control. The controller  20  receives a fundamental current reference ī 1 (t) and provides a voltage reference/output voltage v(t) (based on a modified optimized pulse pattern  22 ), which may be used for determining switching states to be applied to the electrical converter  12 . 
     In fundamental reference voltage estimation block/step  26 , the controller determines a real-valued reference converter voltage  v   ctr   1 (t) from the desired fundamental current reference ī 1 (t) and the estimated grid voltage v g (t). For example, an external control loop may provide a reference for the desired fundamental current ī 1 (t). 
     The real-valued voltage reference  v   ctr   1 (t) may be obtained from the desired fundamental current ī 1 (t) and the grid voltage (estimate) v g (t) in the following way 
     
       
         
           
             
               
                 
                   
                     
                       
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     where R=R g +R c  and L=L g +L c  are the total series resistance and inductance between the grid voltage source and the converter terminals. Note that it may be assumed that the grid voltage is an ideal fundamental voltage source, i.e., v g (t)=v g   1 (t). 
     In optimal voltage reference determination block/step  28 , an optimal voltage reference  v (t) is determined from the fundamental reference voltage  v   ctr   1 (t). The optimal voltage reference  v (t) is based on an optimal pulse pattern  24  that is read from a first lookup table  30  stored in the controller  20 . 
     With respect to  FIG. 2 , the optimized pulse patterns  22  in the table  30  may be stored in the form of switching instants/angles  24  together with their corresponding voltage levels, which were indexed by the fundamental reference voltage  v   ctr   1 (t). The optimal voltage reference  v (t) then may be reconstructed from these stored values. 
     The optimal voltage reference  v (t) or the corresponding optimized pulse pattern  22  may be computed offline to optimize a specific objective function such as Total Harmonic Distortion (THD) and/or to shape the spectral content of the voltage to fit some requirements such as grid standards. In closed-loop operation, the real-valued voltage reference  v   ctr   1 (t) may be used as an index to read in the table  30  the optimal voltage reference  v (t). 
     In harmonic content determination block  32 , a harmonic current reference ī h (t) is determined from the fundamental reference voltage  v   ctr   1 (t). 
     The optimal voltage reference  v (t) determined from the fundamental reference voltage  v   ctr   1 (t) has discrete amplitude values, which result in higher order contributions to the converter current, when the optimal voltage reference  v (t) would be directly applied to the converter  12 . This harmonic content may be computed offline and may be stored in a second lookup table  34 . For example, a model of the system  10  may be simulated with the optimal voltage reference  v (t) and the generated currents may be analyzed. 
     In particular, when the optimal voltage reference  v (t) is applied over a series resistor-inductor circuit, the resulting current is a low-pass filtered version of the voltage. The magnitude of the fundamental component of the resulting current may be subtracted to obtain the harmonic current reference ī h (t). This is done because the magnitude of the fundamental component of the converter current also depends on the grid voltage v g (t), which is not known a priori and therefore cannot be used to compute the offline trajectory of the harmonic current reference ī(t). 
     The trajectory of the harmonic current reference ī h (t) may be obtained using Fourier series. Let a n  and b n , n=1, 2, . . . , ∞ be the Fourier coefficients that describe  v (t), i.e., 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     where  V   1 (t) and  v   h (t) are the fundamental and the harmonic content of the stored optimal voltage, respectively, and T is the period of the fundamental component. Notice that in (2) the summation for the harmonic content  v   h (t) starts with n=2, thus excluding the fundamental component. 
     The current corresponding to the harmonic content  v   h (t), when applied to an RL series circuit, is given by 
     
       
         
           
             
               
                 
                   
                     
                       
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     where ã n  and {tilde over (b)} n , n=1, 2, . . . , ∞ are the Fourier coefficients modulated through the impedance Z n =R+jwnL, i.e., 
     
       
         
           
             
               
                 
                   
                     
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                             n 
                           
                         
                         } 
                       
                     
                     + 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Im 
                       ⁢ 
                       
                         { 
                         
                           
                             a 
                             n 
                           
                           
                             Z 
                             n 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where 
               j   =       -   1         ,     w   =       2   ⁢   π     T       ,         
and Re(x) and Im(x) are the real and imaginary parts of x, respectively.
 
     The trajectory of ī h (t) may be computed offline and stored in memory of the controller  20  in a lookup table  34  indexed with respect to the fundamental reference voltage  v   ctr   1 (t). In closed-loop operation, the real-valued voltage reference  v   ctr   1 (t) may be used as an index to read in the harmonic current reference ī h (t). 
       FIG. 4  illustrates an example of a trajectory of a harmonic current reference ī h (t) for a specific fundamental reference voltage  v   ctr   1 (t). The harmonic current reference ī h (t) is provided in the stationary αβ-reference frame and may be stored for specific values of the time t. When reading back ī h (t), the missing values may be interpolated. It may be important to store the trajectory at least for the values of t when a switching transition occurs in  v (t), i.e. at least at the switching instants  22  of the corresponding optimized pulse pattern  22 . It may also be important to store the trajectory only during a fraction of a cycle and reconstruct the full trajectory exploring symmetries of the trajectory. 
     After block/step  32 , the fundamental current reference ī 1 (t) is added to the harmonic current reference ī h (t) to determine an optimal current reference ī(t), i.e.,
 
 ī ( t )= ī   1 ( t )+ ī   h ( t )  (6)
 
     A harmonic content error in the form of a current error ĩ(t) is calculated by subtracting a measured/estimated current i(t) from the optimal current reference ī(t), i.e.
 
 ĩ ( t )= ī ( t )− i ( t )  (7)
 
     The measured/estimated current i(t) may be provided from an estimation block  36  of the controller  20 , which based on actual measurements in the system  10  determines the current i(t). 
     For example, the block  38  represents a mapping between voltage level commands based on the voltage reference v(t) and an actual voltage v c (t) on the output of the converter  12 . The block  40  represents a model/modelling of a load. 
     The block  36 ,  38 ,  40  may not be part of the controller  20 . 
     In modification block/step  42 , the current error ĩ(t) is corrected by modifying the switching instants/angles  24  of the optimized pulse pattern  22  and/or the optimal current reference in order to follow the desired current. The switching instants  22  of the optimal voltage reference  v (t) (or the optimized pulse pattern  22 ) may be slightly timeshifted (forward or backward in time t) to correct the current tracking error ĩ(t). 
     Block/step  42  may be performed in a dead beat manner. For example, timeshifts may be applied to the optimal voltage reference  v (t) and the corresponding current corrections may be determined. When the current corrections at least partially equalize the current error ĩ(t), the up to now known determined timeshifts may be used as basis for further corrections with further timeshifts. This iterative optimization may be performed until a predefined time has run out (dead beat). 
     It also is possible that the modification of the optimal voltage reference  v (t) and/or the optimized pulse pattern  22  is performed using the solution of a quadratic programming. Let ĩ(t+W) be the measured/estimated current error at time t+W in the abc frame, W&gt;0 be the desired prediction horizon, N a , N b , and N c  be the number of switching transitions for each phase within the prediction horizon, and Δt a,j , Δt b,j , Δt c,j , j=1, . . . , N be the changes in the planned switching instants within the window t to t+W, i.e.,
 
Δ t   a,j   =t   a,j   *−t   a,j ,
 
     where t a,j * is the offline optimal switching time given by the OPP, and t a,j  is the actual switching time computed by the closed-loop controller. Also, define the vector
 
Δ t=[Δt   a,1    . . . Δt   a,N     a     Δt   b,1    . . . Δt   b,N     b     Δt   c,1    . . . Δt   c,N     c   ]′.  (8)
 
     Define the objective function J as
 
 J=ĩ ( t+W )′ ĩ ( t+W )+Δ t′QΔt,   (9)
 
     where Q is a positive-definite matrix that penalizes changes in the switching instants. 
     Note that ĩ(r+W) also depends on Δt, since the current error is based on the behaviour of the converter  12 , i.e. the estimated current i(t), when the modified optimized pulse pattern is applied. 
     The optimization problem that computes the changes in the switching angles is given by 
     
       
         
           
             
               
                 
                   
                     
                       minimize 
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       J 
                       ⁡ 
                       
                         ( 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         subject 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         to 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           t 
                           
                             a 
                             , 
                             1 
                           
                         
                       
                       &lt; 
                       … 
                       ⁢ 
                       
                           
                       
                       &lt; 
                       
                         t 
                         
                           a 
                           , 
                           
                             N 
                             a 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         t 
                         
                           b 
                           , 
                           1 
                         
                       
                       &lt; 
                       … 
                       ⁢ 
                       
                           
                       
                       &lt; 
                       
                         t 
                         
                           b 
                           , 
                           
                             N 
                             b 
                           
                         
                       
                     
                     , 
                     
                       
                         t 
                         
                           c 
                           , 
                           1 
                         
                       
                       &lt; 
                       … 
                       ⁢ 
                       
                           
                       
                       &lt; 
                       
                         
                           t 
                           
                             c 
                             , 
                             
                               N 
                               c 
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     This quadratic problem may be solved offline and the resulting equations for the solution for Δt may be implemented in the controller  20  in block  42 . 
       FIG. 5  shows an embodiment of the controller  20 , which is also based on current control. Analogously to the controller  20  in  FIG. 3 , the controller in  FIG. 5  receives a fundamental current reference ī 1 (t) and provides a voltage reference v(t) (a modified optimized pulse pattern  22 ), which may be used for determining switching states to be applied to the electrical converter  12 . 
     Contrary to the controller  20  of  FIG. 3 , the fundamental voltage reference  v   ctr   1 (t) is determined according to 
     
       
         
           
             
               
                 
                   
                     
                       
                         v 
                         _ 
                       
                       ctr 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         v 
                         g 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     - 
                     
                       R 
                       ⁢ 
                       
                         
                           
                             i 
                             _ 
                           
                           1 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                     - 
                     
                       L 
                       ⁢ 
                       
                         
                           ∂ 
                           
                             
                               
                                 i 
                                 _ 
                               
                               1 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                     + 
                     
                       
                         C 
                         ⁡ 
                         
                           ( 
                           s 
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 i 
                                 _ 
                               
                               1 
                             
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             i 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     where the last (and additional) term feeds back the current error. The measured/estimated current i(t) is subtracted from the fundamental current reference ī 1 (t) and the result is input to a controller  44 , such as a PI controller, to account for the current error in an outer control loop. The result of the controller is then added to the (basic) fundamental voltage reference provided by block/step  26  to determine the (corrected) fundamental voltage reference  v   ctr   1 (t). 
     Analogously to  FIG. 3 , the optimized pulse pattern  22  and the optimal voltage reference  v (t) are then determined in block/step  28  from the first lookup table  30 . 
     In  FIG. 5 , the harmonic content correction determination and correction in blocks  32  and  42  is based on a filtered voltage. 
     In particular, the harmonic content determination block/step  32  determines a harmonic content reference in the form of a filtered voltage  ψ (t). The filtered voltage  ψ (t) is stored in the second lookup table  34  in the controller  34 . 
       FIG. 6  illustrates an example of a trajectory of a filtered voltage  ψ (t) for a specific fundamental reference voltage  v   1 (t). The filtered voltage  ψ (t) may be provided in the stationary αβ-reference frame and may be stored for specific values of the time t. When reading back  ψ (t), the missing values may be interpolated. The trajectory of the filtered voltage  ψ (t) may be computed offline and stored in a memory of the controller  20  in a lookup table  34  indexed with respect to the fundamental reference voltage  v   ctr   1 (t). 
     For example, the filtered voltage may be computed offline using a first order filter applied to the optimal voltage reference  v (t). 
     
       
         
           
             
               
                 
                   
                     
                       ψ 
                       _ 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       k 
                       
                         s 
                         + 
                         ω 
                       
                     
                     ⁢ 
                     
                       
                         v 
                         _ 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
       FIG. 7  shows a Bode diagram for the first order filter. At low frequencies, the Bode response may be of magnitude one and almost zero phase shift, i.e. the filtered voltage  ψ (t) may be interpreted as a voltage for low frequencies. At high frequencies, the Bode response may be similar to that of an integrator, i.e. the filtered voltage  ψ (t) may be interpreted as a flux for high frequencies. 
     Returning to  FIG. 5 , from the filtered voltage  ψ (t), a filtered voltage error  ψ (t) is calculated as harmonic content error by subtracting a measured/estimate voltage ψ(t) from the filtered voltage  ψ (t). ψ(t) corresponds to the output voltage reference v(t) to which the same first order filter  46  as in (12) is applied, i.e., 
     
       
         
           
             
               
                 ψ 
                 _ 
               
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 k 
                 
                   s 
                   + 
                   ω 
                 
               
               ⁢ 
               
                 v 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
             
           
         
       
     
     The modification block/step  42  then may correct/equalize the voltage error {tilde over (v)}(t) by modifying the switching instants/angles  24  of the optimized pulse pattern  22  analogously to  FIG. 3 . 
     In principle, in the method of  FIG. 5 , the current feedback is moved to an early stage of the control loop, i.e., the reference generation part, and changes the inner loops to manipulate the error {tilde over (ψ)}(t) in the filtered voltage  ψ (t). 
     While the invention 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 invention is not limited to the disclosed embodiments. Other variations to the disclosed embodiments can be understood and effected by those skilled in the art and practising the claimed invention, 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. 
     LIST OF REFERENCE SYMBOLS 
     
         
           10  converter system 
           12  electrical converter 
           14  electrical grid 
           16  DC link 
           18  filter/transformer 
           20  controller 
         υ g  grid voltage 
         L g  grid inductance 
         R g  grid resistance 
         i a (t) grid current 
         i DC   + , i DC   −  DC link current 
         v DC   + , v DC   −  DC link voltage 
         C dc  DC link capacitor 
           22  optimized pulse pattern 
           24  switching instant 
           26  fundamental reference voltage estimation block/step 
           28  optimal reference voltage determination block/step 
           30  lookup table 
           32  harmonic content determination block/step 
           34  lookup table 
         ī 1 (t) fundamental current reference 
           v   ctr   1 (t) fundamental voltage reference 
           v (t) optimal voltage reference 
         v(t) modified optimized pulse pattern, voltage reference 
         ī h (t) harmonic current reference 
         ī(t) optimal current reference 
         ĩ(t) current error 
         i(t) measured/estimated current 
           36  current estimation block/step 
           38  conversion block/step 
           40  load modelling block/step 
           42  modification block/step 
           44  controller 
           46  frequency filter 
           ψ (t) filtered voltage 
         {tilde over (ψ)}(t) filtered voltage error 
         ψ(t) measured/estimated voltage