Patent Publication Number: US-8976554-B2

Title: Control for fault-bypass of cascaded multi-level inverter

Description:
BACKGROUND 
     The present embodiments relate to cascaded multi-level inverters. In particular, the embodiments relate to fault-bypass operation of such inverters. 
     Cascaded multilevel inverters are used in industrial control systems. For example, cascaded multilevel inverters are used to control medium voltage motor drives (e.g., 4.1 kV-13.8 kV motor drives) and/or static voltage compensators. 
     Cascaded multilevel inverters are modular. The high power inverter is made-up of a number of smaller series connected single-phase inverters or cells. Any number (e.g., 3-8) of cells may be used for a given leg of a three-phase system. When one or more of these cells fail, the inverter can still operate and produce balanced line-line voltages by computing different voltage references. Although the fault-bypass maximum line-line voltage is less than in normal conditions, continued operation at the reduced power level may be preferable to a complete shut-down. 
     To provide balance, each leg may be operated with the same number of cells. Where a number of cells fail in one leg, the other legs may bypass a corresponding number of cells. However this approach does not provide maximum available voltage to the motor since not all available cells are used. In another approach, a feedback mechanism is used to generate the voltage references. However, the accuracy of the method is influenced by the feedback gain, which has to be fairly high for good results. This approach makes it more challenging when the output frequency of the cascaded multilevel inverter is high as the gain cannot be increased beyond certain limits. 
     BRIEF SUMMARY 
     By way of introduction, the preferred embodiments described below include methods, systems, instructions, computer readable media, and digital electronic circuitry for control in fault-bypass of a cascaded multi-level inverter. The reference voltages are generated as an analytic solution based on the number of operating or active cells. 
     In a first aspect, a system is provided for control in fault-bypass of a cascaded multi-level inverter. A first plurality of first inverter cells leg connects in series for a first phase. A second plurality of second inverter cells leg connects in series for a second phase. A third plurality of third inverter cells leg connects in series for a third phase. The fault-bypass results in a different number of the inverter cells in the first plurality than of the second inverter cells in the second plurality being active. A processor is configured to generate reference voltages, free of feedback, for active ones of the first, second, and third inverter cell legs. All of the active first inverter cells are operated with a first common one of the reference voltages, all of the active second inverter cells being operated with a second common one of the reference voltages, and all of the active third inverter cells being operated with a third common one of the reference voltages. 
     In a second aspect, a method is provided for control in fault-bypass of a cascaded multi-level inverter. The cascaded multi-level inverter is operated in the fault-bypass. A first leg is identified as having fewer operating cells of the multi-lever inverter than a second leg of the multi-level inverter during the operating. The operating of cells of the second leg is controlled as a function of a number of the operating cells of the first leg. 
     In a third aspect, a system is provided for control in fault-bypass of a cascaded multi-level inverter. First, second, and third legs of the cascaded multi-level inverter are provided. At least the third leg has a fewer number of cells operating than a number the second leg and a number of the first leg. A processor is configured to control a pulse width modulation of the cells of the first, second, and third legs as a function of first, second, and third reference voltages, respectively. The first, second and third reference voltages each include a term for a balanced number of cells in the first, second and third legs, and the second and third reference voltages including a term for a difference between the number of the first leg and the number of the second leg without a difference between the number of the first leg and the number of the second leg or the number of the third leg. 
     The present invention is defined by the following claims, and nothing in this section should be taken as a limitation on those claims. Further aspects and advantages of the invention are discussed below in conjunction with the preferred embodiments and may be later claimed independently or in combination. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The components and the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. Moreover, in the figures, like reference numerals designate corresponding parts throughout the different views. 
         FIG. 1  is a block diagram of one embodiment of a system for control in fault-bypass of a cascaded multi-level inverter; 
         FIG. 2  is a circuit diagram of one embodiment of a cell for a cascaded multilevel inverter; 
         FIGS. 3A-C  are graphic representations of balanced line-line voltage in situations with different bypassed cells in different phases; 
         FIG. 4  is a flow chart diagram of one embodiment of a method for control in fault-bypass of a cascaded multi-level inverter; 
         FIG. 5  is an example simulation of phase references with one cell bypassed in a phase; 
         FIG. 6  is an example simulation of phase currents with one cell bypassed in a phase; 
         FIG. 7  is an example simulation of the line-line voltage between phases A and B with one cell bypassed in phase C; 
         FIG. 8  is an example simulation of the line-line voltage between phases B and C with one cell bypassed in phase C; 
         FIG. 9  is an example simulation of phase references with two cell bypassed in a phase; 
         FIG. 10  is an example simulation of phase currents with two cell bypassed in a phase; 
         FIG. 11  is an example simulation of the line-line voltage between phases A and B with two cells bypassed in phase C; 
         FIG. 12  is an example simulation of the line-line voltage between phases B and C with two cells bypassed in phase C; 
         FIG. 13  is an example simulation of phase references with two cells bypassed in phase C and one cell bypassed in phase B; 
         FIG. 14  is an example simulation of phase currents with two cells bypassed in phase C and one cell bypassed in phase B; 
         FIG. 15  is an example simulation of the line-line voltage between phases A and B with two cells bypassed in phase C and one cell bypassed in phase B; 
         FIG. 16  is an example simulation of the line-line voltage between phases B and C with two cells bypassed in phase C and one cell bypassed in phase B; 
         FIG. 17  is a spectrum of line-line voltage between phases A and B of one example with two cells bypassed on phase C and one cell bypassed on phase B; 
         FIG. 18  is a spectrum of line-line voltage between phases B and C of one example with two cells bypassed on phase C and one cell bypassed on phase B; 
         FIG. 19  is a simulated DC-link current through one cell of phase A with two cells bypassed on phase C and one cell bypassed on phase B; 
         FIG. 20  is an example simulation of phase references with two cells bypassed in phase C and one cell bypassed in each of phases A and B; 
         FIG. 21  is an example simulation of phase currents with two cells bypassed in phase C and one cell bypassed in each of phases A and B; 
         FIG. 22  is an example simulation of the line-line voltage between phases A and B with two cells bypassed in phase C and one cell bypassed in each of phases A and B; 
         FIG. 23  is an example simulation of the line-line voltage between phases B and C with two cells bypassed in phase C and one cell bypassed in each of phases A and B; 
         FIG. 24  is a spectrum of line-line voltage between phases A and B of one example with two cells bypassed on phase C and one cell bypassed on each of phases A and B; 
         FIG. 25  is a spectrum of line-line voltage between phases B and C of one example with two cells bypassed on phase C and one cell bypassed on each of phases A and B; and 
         FIG. 26  is a simulated DC-link current through one cell of phase A with two cells bypassed on phase C and one cell bypassed on each of phases A and B. 
     
    
    
     DETAILED DESCRIPTION OF THE DRAWINGS AND PRESENTLY PREFERRED EMBODIMENTS 
     The voltage references for a cascaded multilevel drive in fault bypass are generated analytically. Compared to other approaches, this approach determines exact values from mathematical formulas for the phase voltage references in order to obtain maximum possible output line-line voltage. This approach does not rely on any approximation or feedback loop. 
     The cascaded multilevel inverter is a three-phase device. The three-phase system is modeled under bypass operation where one or more cells in one or more of the phase legs is inoperable, but each leg still includes operable cells. 
     The three phase system has legs labeled as A, B, and C (i.e., an ABC orientation). The voltage output by each leg is given in equations (1)-(3): 
                     V   a     =     V   ·     cos   ⁡     (     ω   ·   t     )                 (   1   )                 V   b     =     V   ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )                 (   2   )                 V   c     =     V   ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )                 (   3   )               
When all the cells are active or participating in the generation of the three phase system, the reference voltages used to control each cell are written as equations (4)-(6):
 
                     V     ref_a   ⁢           ⁢   1       =       m   ·     2     3       ·     cos   ⁡     (     ω   ·   t     )         +   CMO             (   4   )                 V     ref_b   ⁢           ⁢   1       =       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO             (   5   )                 V     ref_c   ⁢           ⁢   1       =       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +     CMO   .               (   6   )               
where m is a modulation index for pulse width modulation and can have any value from 0 to 1, ω is the frequency, t is the time, and CMO is the common mode offset. The term
 
             2     3           
accounts for the 15% increase in the maximum modulation index due to the introduction of a common mode offset as given by equation (7):
 
     
       
         
           
             
               
                 
                   CMO 
                   = 
                   
                     - 
                     
                       
                         
                           
                             
                               
                                 Max 
                                 ( 
                                 
                                   
                                     V 
                                     
                                       ref_a 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   , 
                                   
                                     V 
                                     
                                       ref_b 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   , 
                                   
                                     
                                       V 
                                       
                                         ref_c 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     + 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 Min 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       V 
                                       
                                         ref_a 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     , 
                                     
                                       V 
                                       
                                         ref_b 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     , 
                                     
                                       V 
                                       
                                         ref_c 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     When a certain number of cells are bypassed due to a failure, the drive cannot deliver equal voltages in each phase. To maintain equal line-line voltages and thus achieve balanced phase currents, the amplitude and phase displacement for one or more of the phase voltages are adjusted from their normal values. To achieve this, the reference voltages are adjusted accordingly for each leg. 
     Assuming that the number of active cells on each phase during the bypass operation is N a , N b  and N c , respectively, the per unit phase voltages generated by the drive during a fault bypass operation are written in a general form as:
 
 V   a   =M   a ·cos(ω· t )+CMO  (8)
 
 V   b   =M   b ·cos(ω· t−φ   1 )+CMO  (9)
 
 V   c   =M   c ·cos(ω· t−φ   2 )+CMO  (10)
 
 M   a   =N   a   ·m   1   (11)
 
 M   b   =N   b   ·m   1   (12)
 
 M   c   =N   c   ·m   1   (13)
 
In equations (8)-(10), the common mode offset (CMO) is no longer assumed to be as given in equation (7), while m 1  is the maximum modulation index, under bypass operation.
 
     The amplitudes of the line-line voltages are written as:
 
| V   ab |=√{square root over ( M   a   2 −2 ·M   a   ·M   b ·cos φ 1   +M   b   2 )}≦ M   a   +M   b   (14)
 
| V   ac |=√{square root over ( M   a   2 −2 ·M   a   ·M   c ·cos φ 2   +M   c   2 )}≦ M   a   +M   c   (15)
 
| V   bc |=√{square root over ( M   b   2 −2 ·M   b   ·M   c ·cos(φ 1 −φ 2 ) +M   c   2 )}≦ M   b   +M   c   (16)
 
     From equations (14) and (16), the two unknowns, φ 1  and φ 2  can be determined in order to maintain all three line-line voltages equal:
 
 M   b   2   −M   c   2 =2 ·M   a   ·M   b ·cos φ 1 −2 ·M   a   ·M   c ·cos φ 2   (17)
 
 M   a   2 −2 ·M   a   ·M   c ·cos φ 2   =M   b   2 −2 ·M   b   ·M   c ·cos(φ 1 −φ 2 )  (18)
 
     From equations (17) and (18), the two angles displacement may, in theory, be determined, but obtaining an analytical form is not possible. In addition, this approach cannot determine the expression for the common mode offset in equations (8)-(10) needed in order to obtain maximum line-line voltage. As a result, the motor will operate but will not be able to deliver maximum speed, power, and/or torque. 
     A different approach is to split the three-phase system into two systems: one which is a balanced in the number of cells on each leg that can be operated as any normal three-phase system and a second one which operates with different phase shifts in order to maintain balanced line-line voltages. However, that approach has the drawback of having different phase references to the cells belonging to the same phase. This means that when using a phase shifted PWM modulation, the waveform generated contains additional harmonics. 
     By using an analytical approach, the same reference voltage may be provided for each operating cell of a phase. Identical voltage references are provided to all cells belonging to the same phase, thus making sure that the use of a standard phase shifted PWM modulation does not lead to additional harmonics. 
       FIG. 1  shows a system for control in fault-bypass of a cascaded multi-level inverter. In the example of  FIG. 1 , the system is a cascaded multilevel medium voltage motor drive. The voltage frequency and/or amplitude are controlled to operate the alternating current (AC) motor  18 , such as for industrial process control. 
     The AC motor  18  is any AC-type motor: synchronous, asynchronous, permanent magnet, and may be rated for low voltage, medium voltage or high-voltage. For example, medium-voltage AC motors may operate in the 4.1 kV to 13.8 kV range. Greater or lesser voltage may be used. More than one AC motor  18  may be connected. Other loads may be used instead of or in addition to the AC motor  18 . The AC motor  18  responds to the voltage applied by the multilevel inverter on the three phases to increase, decrease or maintain a speed or position. 
     The cascaded multilevel inverter includes three phases or legs  14 , each connected with the AC motor  18  and controlled by a processor  20  operating with a memory  22 . Additional, different, or fewer components may be used. Other control structures than a processor  20  and memory  22  may be used. 
     Each of the legs  14  is formed from a plurality of cells  16 . Any number of cells  16  may be used in each leg  14 , such as 3-8 cells. In the example of  FIG. 1 , three legs  14  are each formed from a same number, N, of cells  16 . Other inverters, connection arrangements, or combinations thereof may be used. 
     The cells  16  are single-phase inverters. The high power inverter for the AC motor  18  is made up of a cascade of cells  16  connected in series in different legs  14 . Each cell  16  is responsive to control signals to alter the voltage level and/or frequency output, resulting in multilevel voltage waveform for each leg  14 . 
     The cells  16  include power semiconductor switching devices, passive components (inductors, capacitors), control circuits, processors, interfaces, and other components for communicating with the processor  20 . The cells  16  operate based on signals from the processor  20 . For example, power levels are established by the processor  20 . The control circuit or control board in an inverter cell  16  receives the voltage reference and generates the gating pulses for power switching devices using appropriate vector controls and pulse-width modulation. Alternatively, the processor  20  outputs the gating pulses provided to the cells  16  based on the voltage references. 
       FIG. 2  shows one example circuit schematic for one of the cells  16 . Four insulated gate bipolar transistors Q 1 - 4  connect in line one (L 1 ) and line two (L 2 ) output groups. Diodes and a capacitor connect with a three phase input AC signal and across the transistors Q 1 - 4 . The transistors Q 1 - 4  have gate input signals controlled by the processor  20  ( FIG. 1 ). By pulse-width modulating the voltage reference, the processor  20  controls each of the cells  16  and thus, the voltage and frequency of the output, which is the voltage between lines L 1  and L 2 . Other inverter circuits may be used with the same or different components (e.g., different types and/or numbers of transistors). The other inverter circuits may have a differential output voltage and may be connected in series with the output voltage of a different inverter. 
     Referring again to  FIG. 1 , one or more of the cells  16  may fail or not operate correctly. To continue operation of the cascaded multilevel inverter, the failed or inoperable cell or cells  16  are bypassed, such as by connecting L 1  and L 2  together in  FIG. 2 . A mechanical switch connects across each of the inoperable cells  16 , allowing bypass of the selected cell  16 . The mechanical switch is activated by applying a DC or AC voltage across a solenoid in response to a signal from processor  20 . Alternatively, operable transistors Q 1 - 4  of the cell  16  are used for bypass. For example, Q 1  and Q 3  are controlled to be always “on” to bypass, or Q 2  and Q 4  are controlled to be always “on” to bypass. Other bypass mechanisms may be used, where for instance an electronic device is used instead of a mechanical switch. 
     By bypassing a cell  16  in one leg  14 , a different number of active cells  16  results in different legs  14 . For example, the legs  14  of the three phases are labeled as A, B, and C. In the description below, the number of active cells  16  in leg A is Na, the number of active cells  16  in leg B is Nb, and the number of active cells  16  in leg C is Nc. If a cell  16  fails in leg C, then legs A and B have a greater number of operable cells  16 . A cell  16  may fail in leg B, resulting in leg A having more operable cells  16  than legs B and C. Another cell  16  may fail in leg C, resulting in leg C having fewer operable cells  16  than leg B, which has fewer operable cells  16  than leg A. Any combination of different numbers (e.g., Na=N, Nb=N, and Nc=N−2) of operable or failed cells  16  in the different legs  14  may be provided. The bypass operation may continue until an insufficient number of cells  16  is provided in one or more of the legs  14 , such as fewer than a threshold number (e.g., fewer than one or two). 
     When one or more of the cells  16  fail, the inverter may still operate and produce balanced line-line voltages by using different voltage references compared to the case where all cells  16  are available. The processor  20  generates different voltage references for controlling the transistors Q 1 - 4  of the active cells  16 . Although the available maximum line-line voltage provided to the AC motor  18  is less than in normal conditions (all cells operable), in many industrial processes, it is preferable to continue to operate at a reduced power level without a complete shut-down. 
     The processor  20  is a general processor, central processing unit, control processor, digital signal processor, application specific integrated circuit, field programmable gate array, digital circuit, analog circuit, combinations thereof, or other now known or later developed device for controlling inverter cells  16 . The processor  20  is a single device or multiple devices operating in serial, parallel, or separately. The processor  20  may be a main processor of a computer, or may be a processor for handling some tasks in a larger system, such as a controller in a panel or programmable logic controller. 
     The processor  20  is configured by instructions, design, hardware, and/or software to be able to perform the acts discussed herein. For example, the processor  20  is configured to generate the reference voltages by accessing a look-up table in the memory  22  by the combination of operable or bypassed cells  16  per leg  14 . The look-up table provides the analytical solution in the form of stored reference voltages for the appropriate bypass situation. In other embodiments, the processor  20  performs one or more calculations to analytically solve for the reference voltages as needed. 
     The processor  20  generates the control reference voltages or signals necessary to control inverter cells  16 . The reference voltages are generated for active ones of the inverter cells  16 . Reference voltages are not generated for the bypassed or inactive cells  16 . 
     The reference voltage is compared to a triangle wave. The result of the comparison is a pulse width modulation signal which is amplified by a gate driver circuit and then applied to each IGBT&#39;s Q 1 -Q 4  of a cell  16 . When the reference voltage is higher than the triangle waveform, Q 1  is turned on. When the reference voltage is lower than the triangle waveform Q 1  is turned off. Q 1  and Q 2  are always switched in opposite manner: when one is turned on the other one is turned off and vice-versa. Similar operation occurs for Q 3  and Q 4 , the only difference being that the triangle waveform used is in opposite phase with respect to the triangle waveform used for Q 1  and Q 2 . The pulse wave modulation is responsive to the reference voltage. Control of the reference voltage controls the operation of the transistors Q 1 - 4  of the active cells  16 . The processor  20  performs the comparison. Alternatively, circuitry of the cells  16  performs the comparison. 
     One reference voltages is generated for each of the legs  14 . For example, all of the operable cells  16  of a given leg  14  operate using the same reference voltage, but different reference voltages are provided to different legs  14 . The pulse width modulation of one leg  14  is different than for another leg  14 . 
     The processor  20  is configured to generate the reference voltages for the cells  16  of the legs  14  such that an equal magnitude voltage is provided between each two phases, therefore a balanced three-phase line-line voltage is applied on the motor. The reference voltages are altered or controlled to provide the desired voltage, frequency and phase relationship for controlling the AC motor  18 . Other voltage, frequency and/or phase limitations may be used, depending on the operation of the AC motor  18  or other load. 
     The processor  20  is configured to generate the reference voltages based on the number of available cells on each leg. According to equations (19)-(21), three numbers defined as Min, Mid and Max represent the minimum, medium and maximum number of available cells per phase, respectively, once the cascaded multilevel inverter has entered bypass operation:
 
Min=min{ N   a   ,N   b   ,N   c }  (19)
 
Mid=mid{ N   a   ,N   b   ,N   c }  (20)
 
Max=max{ N   a   ,N   b   ,N   c }  (21)
 
     The following matrix notations represent the original inverter structure (normal), having N cells per phase and the inverter operating in bypass operation, having N a , N b  and N c  cells per phase, respectively: 
     
       
         
           
             
               
                 
                   
                     I 
                     Normal 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           N 
                         
                       
                       
                         
                           N 
                         
                       
                       
                         
                           N 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     Bypass 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             N 
                             a 
                           
                         
                       
                       
                         
                           
                             N 
                             b 
                           
                         
                       
                       
                         
                           
                             N 
                             c 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     In one example, the greatest number of bypassed cells  16  is assigned to the leg C, a middle number to leg B, and the least number to leg A. This is represented as:
 
Min= N   c Mid= N   b Max= N   a   (24).
 
Different legs may be associated with the minimum, middle, or maximum numbers of active cells  16 .
 
     Following the assumption from equation (23), the inverter under bypass operation may be represented in the form: 
                           I   Bypass     =       ⁢     [         Max           Mid           Min         ]                 =       ⁢       [         Min           Min           Min         ]     +     [           Mid   -   Min               Mid   -   Min             0         ]     +     [           Max   -   Mid             0           0         ]                   =       ⁢       I   1     +     I   2     +     I   3                     (   25   )               
From equation (25), the inverter under bypass operation may be thought of as a combination of three inverters, I 1 , I 2  and I 3 . The first component, I 1 , corresponds to a regular inverter operated with a balanced set of voltage references. An equal number of cells  16  are provided in each leg for this first set of cells, or inverter I 1 . The known common mode offset may be applied to the reference voltage of the first component or set I 1  to increase the modulation index. The second set of cells or inverter I 2  corresponds to an inverter in which only two phases may be modulated. The third set of cells  16  or inverter I 3  corresponds to an inverter where only one phase may be modulated.
 
     For the analytic solution, the processor  20  generates the reference voltages based on the numbers of active inverter cells  16  from the legs with the minimum and medium numbers of active cells  16 . The third component, I 3 , cannot bring any contribution to the maximum achievable line-line voltage, but may influence the harmonic spectrum. The analytic solution may use the first and second sets of cells, I 1  and I 2  and not use third set I 3  for generating the reference voltages. The maximum line-to-line voltage of the phases uses the first and second components (I 1  and I 2 ) and the corresponding minimum and medium numbers, but not the maximum term which only occurs in the third component, I 3 . The maximum line-to-line voltage of the overall inverter is a sum of voltages from legs C and B of the active inverters cells. The reference voltages for each leg include terms for a balanced number of cells and a term for two legs based on a difference between the medium and minimum numbers of cells  16  without being based on the maximum number of cells  16  (or a corresponding difference from the maximum). 
       FIGS. 3A-C  show how a three-phase line-line balanced system is obtained, when the number of minimum operating cells is on phases C, B and A, respectively. Since the phase or leg labels A, B, and C are arbitrarily assigned, the procedure is similar for all situations. Accordingly, the explanation below is for  FIG. 3A , which corresponds to the assumption in equation (24). The results may be extended to the situations of  FIGS. 3B  and C. 
     From  FIG. 3A , the inverter I 1  from decomposition of equation (25) is considered as formed by the vectors V a1 , V b1 , and V c . V a1 , V b1  and V c  represent a balanced inverter having the minimum number of cells  16  per phase. In this way, all the available cells on phase C are utilized but there are more cells available on phases A and B. To maintain a balanced line-line voltage, the remaining cells on these two phases may be used in such a way that the vector produced by the extra difference between the middle and the minimum numbers (Mid−Min) of cells  16  in phase A, V a2 , is aligned with the vector CA produced by inverter I 1  while at the same time, the vector produced by the extra difference between the middle and the minimum numbers (Mid−Min) of cells in phase B, V b2 , is aligned with the vector CB produced by inverter I 1 . This vector system of V a2  and V b2  is the second term in equation (25). From geometry, the triangle A s B s C is still equilateral, just as ABC, meaning that the line-line system produced in this manner is still balanced. In essence, a number of (Min) cells have the regular voltage references of a balanced system as given by equations (4)-(7) while the cells of the difference in the minimum from the middle number (Mid−Min) of cells have the references as given below by equations (26) and (27): 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       ref 
                       ⁢ 
                       _ 
                       ⁢ 
                       a 
                       ⁢ 
                       2 
                     
                   
                   = 
                   
                     m 
                     · 
                     
                       cos 
                       ( 
                       
                         
                           ω 
                           · 
                           t 
                         
                         - 
                         
                           π 
                           6 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     
                       ref 
                       ⁢ 
                       _ 
                       ⁢ 
                       b 
                       ⁢ 
                       2 
                     
                   
                   = 
                   
                     m 
                     · 
                     
                       cos 
                       ( 
                       
                         
                           ω 
                           · 
                           t 
                         
                         - 
                         
                           π 
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Also from  FIG. 3A , the maximum line-line voltage produced is written as the sum of the line-line voltage of the inverter I 1  plus the voltage produced by the inverter I 2 :
 
 V   LINE-LINE     —     BYPASS   =CA   s   =CB   s   =A   s   B   s   =V   LINE-LINE     —     I     1     +V   a2   =+V   LINE-LINE     —     I     1     +V   b2   (28a)
 
Recall that inverter I 1  is a balanced system. A standard CMO is added as expressed in equations (4)-(7), and with a maximum modulation index of
 
               2     3       ,         
considering equations (11)-(13), it follows that, in PER UNIT terms:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                           
                             LINE 
                             - 
                             
                               
                                 LINE 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 I 
                               
                               1 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           CA 
                           s 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           CB 
                           s 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             A 
                             s 
                           
                           ⁢ 
                           
                             B 
                             s 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           Min 
                           · 
                           
                             2 
                             
                               3 
                             
                           
                           · 
                           
                             3 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           2 
                           · 
                           Min 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           2 
                           · 
                           
                             N 
                             c 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     28 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Since there are Nb−Nc (Mid−Min) cells to which references of equations (26) and (27) apply, and with a maximum modulation index of 1, the maximum voltage that can be obtained by system I 2  is given by:
 
 V   a2   =V   b2 =(Mid−Min)= N   b   −N   c   (28)
 
     From equations (28a)-(28b), the maximum PER UNIT line-line voltage under bypass is:
 
 V   LINE-LINE     —     BYPASS =2 ·N   c   +N   b   −N   c   =N   b   +N   c   (29)
 
     The relationship of equation (29) was developed under the assumptions of equation (24) that the minimum number of cells are on phase C and maximum number of cells are on phase A. In a more general form, equation (29) is rewritten such that the maximum line-line voltage during bypass operation is written in PER UNIT as:
 
 V   LINE-LINE     —     BYPASS =Min+Mid  (30)
 
Comparing the general expressions of equations (14)-(16) with equation (30), this analytical solution is able to achieve maximum line-line voltage under bypass operation.
 
     The solution obtained so far uses two different sets of waveforms for cells that belong to the same phase. A number Nc (Min) of cells  16  in each of the three phases have the references given in equations (4)-(7), and a number, Nb−Nc (Mid−Min), of cells  16  on two phases, A and B, have the references given by equations (26)-(27) under the assumptions of equation (24). This is represented as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                           
                             
                               ref 
                               ⁢ 
                               _ 
                               ⁢ 
                               A 
                             
                             i 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           V 
                           
                             ref 
                             ⁢ 
                             _ 
                             ⁢ 
                             a 
                             ⁢ 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               m 
                               · 
                               
                                 2 
                                 
                                   3 
                                 
                               
                               · 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     ω 
                                     · 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             CMO 
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             for 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             each 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             of 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             the 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               Min 
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cells 
                           
                           , 
                           
                             
                               i 
                               . 
                               e 
                               . 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                             &lt; 
                             i 
                             &lt; 
                             
                               ( 
                               Min 
                               ) 
                             
                           
                           , 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 and 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           V 
                           
                             
                               ref 
                               ⁢ 
                               _ 
                               ⁢ 
                               A 
                             
                             j 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           V 
                           
                             ref 
                             ⁢ 
                             _ 
                             ⁢ 
                             a 
                             ⁢ 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             m 
                             · 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     ω 
                                     · 
                                     t 
                                   
                                   - 
                                   
                                     π 
                                     6 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         , 
                         
                           for 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           each 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           of 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           the 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               Mid 
                               ⁢ 
                               
                                 - 
                               
                               ⁢ 
                               Min 
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cells 
                         
                         , 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             i 
                             . 
                             e 
                             . 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                           &lt; 
                           j 
                           &lt; 
                           
                             
                               ( 
                               
                                 Mid 
                                 ⁢ 
                                 
                                   - 
                                 
                                 ⁢ 
                                 Min 
                               
                               ) 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     The introduction of different phase shifts as well as magnitudes to the cells  16  for the same phase has a detrimental effect when a phase shifted PWM modulation method is used. The use of different references in controlling different cells which belong to the same phase introduces additional harmonics in each phase voltage. 
     In the proposed analytic solution, all cells  16  pertaining to one phase are operated with the same voltage reference. The reference voltages for different phases are different, but all of the cells of each phase use the same reference voltage. Adding equations (31) and (32) provides a single reference as if one single cell is present on phase A. Since there are a number Nb (Mid) of cells  16  per phase in leg A, the single reference is averaged by all cells as: 
                     V     ref_A   ⁢   _Bypass       =       ⁢       (         ∑   1   Min     ⁢     V     ref_A   i         +       ∑   1     (     Mid   -   Min     )       ⁢     V     ref_A   j           )     Mid                 =       ⁢         Min   Mid     ·     V     ref_a   ⁢           ⁢   1         +         (     Mid   -   Min     )     Mid     ·     V     ref_a   ⁢           ⁢   2                       
In a similar way, the reference for all cells  16  in phase B is written as:
 
                     V     ref_B   ⁢   _Bypass       =         Min   Mid     ·     V     ref_b   ⁢           ⁢   1         +         (     Mid   -   Min     )     Mid     ·     V     ref_b   ⁢           ⁢   2                   (   34   )               
Finally, the reference on phase C, the leg with the lowest number Nc of working cells  16 , is unchanged. As a result, equation (6) provides the reference:
 
 V   ref     —     C     —     Bypass   =V   ref     —     c1   (35)
 
     From equations (33) and (34), the reference voltage for phases A and B are functions of the number of active cells  16  in phases C and B, but not phase A (i.e., the phase with the largest number of active cells  16 ). The reference voltages of phases A and B are ratios of the reference voltages from the first component I 1  and the second component I 2 . The ratio for the reference voltage from the balanced first component is of Nc/Nb. The ratio for the reference voltage from the second component is (Nb−Nc)/Nb. 
     Since phase A has more active cells  16 , all or only enough cells  16  to equal the middle number Nb of cells  16  may be used. When all of the active inverter cells  16  of phase A are used, the reference voltage is: 
                     N   c       N   a       ·     (         2     3       ·     cos   ⁡     (     ω   ·   t     )         +     C   ⁢           ⁢   M   ⁢           ⁢   O       )       +         (       N   b     -     N   c       )       N   a       ·     cos   ⁡     (       ω   ·   t     -     π   6       )           ,         
where Na is the number of the active inverter cells  16  in leg A. When Na−Nb of the active inverter cells  16  of leg A are not operated, the reference voltage for the active inverter cells  16  is:
 
     
       
         
           
             
               
                 
                   N 
                   c 
                 
                 
                   N 
                   b 
                 
               
               · 
               
                 ( 
                 
                   
                     
                       2 
                       
                         3 
                       
                     
                     · 
                     
                       cos 
                       ⁡ 
                       
                         ( 
                         
                           ω 
                           · 
                           t 
                         
                         ) 
                       
                     
                   
                   + 
                   
                     C 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     O 
                   
                 
                 ) 
               
             
             + 
             
               
                 
                   ( 
                   
                     
                       N 
                       b 
                     
                     - 
                     
                       N 
                       c 
                     
                   
                   ) 
                 
                 
                   N 
                   b 
                 
               
               · 
               
                 cos 
                 ⁡ 
                 
                   ( 
                   
                     
                       ω 
                       · 
                       t 
                     
                     - 
                     
                       π 
                       6 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The reference voltages obtained with equations (33) and (34) do not exceed unity. The cells  16  are not forced to operate in over-modulation. Over modulation is not produced, regardless of the bypass configuration as described in the following paragraph. 
     Assuming Min/Mid=α, where 0&lt;α&lt;1, the two references (33) and (34) are re-written as: 
                     V     ref_A   ⁢   _Bypass       =       α   ·     (       V     ref_a   ⁢           ⁢   1       -     V     ref_a   ⁢           ⁢   2         )       +     V     ref_a   ⁢           ⁢   2                 (   A1   )                 V     ref_B   ⁢   _Bypass       =       α   ·     (       V     ref_b   ⁢           ⁢   1       -     V     ref_b   ⁢           ⁢   2         )       +     V     ref_b   ⁢           ⁢   2                 (   A2   )               
Looking at the definition of α, it follows that the highest amplitudes for the voltage references occur when Min=Mid−1, that is when only one cell  16  is bypassed. By plotting for V ref     —     A     —     Bypass  at maximum modulation index for various values of a, the modulation waveforms never exceed unity. This assumes that on the phase with maximum number of cells  16 , a number of (Max−Mid) healthy cells  16  are also bypassed. If, however all cells  16  are used on that particular phase, the reference will be smaller, by a factor of Mid/Max, so the plot represents the worst case scenario. Similarly, a family of plots for V ref     —     B     —     Bypass  at maximum modulation for various values of α show that the modulation waveforms never exceed unity. Not surprisingly, the two waveforms appear to be symmetric as suggested by the vector representation depicted in  FIGS. 3A-C .
 
     From equation (25), the inverter under bypass operation is split in three fictitious inverters I 1 , I 2  and I 3  but all the analysis has ignored the last component, I 3 . The result obtained by ignoring this third component however has demonstrated that the maximum line-line voltage is achieved without the use of this last component. This is physically explained by the fact that the line-line voltage cannot be bigger than the maximum voltage achievable by the two phases with the lowest number of cells. With the assumptions from equation (24), the extra (Max−Mid) cells  16  which are available on one phase (e.g., A in this case) cannot contribute anymore to the maximum available line-line voltage. 
     A benefit may be gained if all available switches are used in the leg  14  with the most operable cells  16 . The line-neutral voltage of that leg  14  may have the dominant harmonics pushed higher into the frequency spectrum. Because of that, two line-line voltages will also be slightly improved, (Max−Min) and (Max−Mid), but (Mid−Min) voltage is not influenced. 
     If all cells are to be used in the phase with the maximum number of operable cells  16  (e.g., A in this case), the reference on leg A is to be changed to: 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       ref_A 
                       ⁢ 
                       _Bypass 
                     
                   
                   = 
                   
                     
                       
                         Min 
                         Max 
                       
                       · 
                       
                         V 
                         
                           ref_a 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             Mid 
                             - 
                             Min 
                           
                           ) 
                         
                         Max 
                       
                       · 
                       
                         V 
                         
                           ref_a 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     Depending on the particular number of cells  16 , and depending on the application, the additional benefit of using all cells  16  may not be justified. Since the (Max−Mid) cells  16  on the phase with the highest number of cells  16  cannot further increase the line-line voltage, these extra cells  16  may be also bypassed, either in the same way as the faulty cells  16  or by keeping in the either top or bottom switches in the cells  16  in the on position, in order to reduce the losses. 
     The above discussion deals with one possibility of the example inverter of  FIG. 1  where leg C has the fewest number of active cells  16 , with leg B having a middle number, and leg A having the most number of active cells  16 . In other situations, other arrangements or distributions of the active cells are provided  16 . Below are analytical reference voltage calculations for other distributions: 
             u   =     m   ·     1     3       ·     cos   ⁡     (     ω   ·   t     )                     v   =     m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )                     w   =     m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )                     CMO   =     -           Max   ⁡     (     u   ,   v   ,   w     )       +     Min   ⁡     (     u   ,   v   ,   w     )         )     2                     Case   ⁢           ⁢   1.   ⁢           ⁢     N   a       ≥     N   b     ≥     N   c           
If (N a −N b ) cells are not used on phase A:
 
               V     ref_A   ⁢   _Bypass       =           N   c       N   b       ·     (       m   ·     2     3       ·     cos   ⁡     (     ω   ·   t     )         +   CMO     )       +         (       N   b     -     N   c       )       N   b       ·   m   ·     cos   ⁡     (       ω   ·   t     -     π   6       )                 
If all N a  cells are used on phase A:
 
               V     ref_A   ⁢   _Bypass       =           N   c       N   a       ·     (       m   ·     2     3       ·     cos   ⁡     (     ω   ·   t     )         +   CMO     )       +         (       N   b     -     N   c       )       N   a       ·   m   ·     cos   ⁡     (       ω   ·   t     -     π   6       )                         V     ref_B   ⁢   _Bypass       =           N   c       N   b       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO     )       +         (       N   b     -     N   c       )       N   b       ·   m   ·     cos   ⁡     (       ω   ·   t     -     π   2       )                               ⁢       V     ref_C   ⁢   _Bypass       =       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +   CMO                           ⁢       Case   ⁢           ⁢   2.   ⁢           ⁢     N   a       ≥     N   c     ≥     N   b             
If (N a −N c ) cells are not used on phase A:
 
               V     ref_A   ⁢   _Bypass       =           N   b       N   c       ·     (       m   ·     2     3       ·     cos   ⁡     (     ω   ·   t     )         +   CMO     )       +         (       N   c     -     N   b       )       N   c       ·   m   ·     cos   ⁡     (       ω   ·   t     +     π   6       )                 
If all N a  cells are used on phase A:
 
               V     ref_A   ⁢   _Bypass       =           N   b       N   a       ·     (       m   ·     2     3       ·     cos   ⁡     (     ω   ·   t     )         +   CMO     )       +         (       N   c     -     N   b       )       N   c       ·   m   ·     cos   ⁡     (       ω   ·   t     +     π   6       )                               ⁢       V     ref_B   ⁢   _Bypass       =       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO                     V     ref_C   ⁢   _Bypass       =           N   b       N   c       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +   CMO     )       +         (       N   c     -     N   b       )       N   c       ·   m   ·     cos   ⁡     (       ω   ·   t     +     π   2       )                               ⁢       Case   ⁢           ⁢   3.   ⁢           ⁢     N   b       ≥     N   a     ≥     N   c                     V     ref_A   ⁢   _Bypass       =           N   c       N   a       ·     (       m   ·     2     3       ·     cos   ⁡     (     ω   ·   t     )         +   CMO     )       +         (       N   a     -     N   c       )       N   a       ·   m   ·     cos   ⁡     (       ω   ·   t     -     π   6       )                 
If (N b −N a ) cells are not used on phase B:
 
               V     ref_B   ⁢   _Bypass       =           N   c       N   a       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO     )       +         (       N   a     -     N   c       )       N   a       ·   m   ·     cos   ⁡     (       ω   ·   t     -     π   2       )                 
If all N b  cells are used on phase B:
 
               V     ref_B   ⁢   _Bypass       =           N   c       N   b       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO     )       +         (       N   a     -     N   c       )       N   b       ·   m   ·     cos   ⁡     (       ω   ·   t     -     π   2       )                               ⁢       V     ref_C   ⁢   _Bypass       =       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +   CMO                           ⁢       Case   ⁢           ⁢   4.   ⁢           ⁢     N   b       ≥     N   c     ≥     N   a                           ⁢       V     ref_A   ⁢   _Bypass       =         m   ·     2     3         ⁢     cos   ⁡     (     ω   ·   t     )         +   CMO             
If (N b −N c ) cells are not used on phase B:
 
               V     ref_B   ⁢   _Bypass       =           N   a       N   c       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO     )       +         (       N   c     -     N   a       )       N   c       ·   m   ·     cos   ⁡     (       ω   ·   t     -       5   ·   π     6       )                 
If all N b  cells are used on phase B:
 
               V     ref_B   ⁢   _Bypass       =           N   a       N   b       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO     )       +         (       N   c     -     N   a       )       N   b       ·   m   ·     cos   ⁡     (       ω   ·   t     -       5   ·   π     6       )                         V     ref_C   ⁢   _Bypass       =           N   a       N   c       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +   CMO     )       +         (       N   c     -     N   a       )       N   c       ·   m   ·     cos   ⁡     (       ω   ·   t     +       5   ·   π     6       )                               ⁢       Case   ⁢           ⁢   5.   ⁢           ⁢     N   c       ≥     N   b     ≥     N   a                           ⁢       V     ref_A   ⁢   _Bypass       =         m   ·     2     3         ⁢     cos   ⁡     (     ω   ·   t     )         +   CMO                     V     ref_B   ⁢   _Bypass       =           N   a       N   b       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO     )       +         (       N   b     -     N   a       )       N   b       ·   m   ·     cos   ⁡     (       ω   ·   t     -       5   ·   π     6       )                 
If (N c −N b ) cells are not used on phase C:
 
               V     ref_C   ⁢   _Bypass       =           N   a       N   b       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +   CMO     )       +         (       N   b     -     N   a       )       N   b       ·   m   ·     cos   ⁡     (       ω   ·   t     +       5   ·   π     6       )                 
If all N c  cells are used on phase C:
 
               V     ref_C   ⁢   _Bypass       =           N   a       N   c       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +   CMO     )       +         (       N   b     -     N   a       )       N   c       ·   m   ·     cos   ⁡     (       ω   ·   t     +       5   ·   π     6       )                               ⁢       Case   ⁢           ⁢   6.   ⁢           ⁢     N   c       ≥     N   a     ≥     N   b                     V     ref_A   ⁢   _Bypass       =           N   b       N   a       ·     (       m   ·     2     3       ·     cos   ⁡     (     ω   ·   t     )         +   CMO     )       +         (       N   a     -     N   b       )       N   a       ·   m   ·     cos   ⁡     (       ω   ·   t     +     π   6       )                               ⁢       V     ref_B   ⁢   _Bypass       =       m   ·     cos   ⁡     (       ω   ·   t     -       2   ·   π     3       )         +   CMO             
If only (N c −N a ) cells are used on phase C:
 
               V     ref_C   ⁢   _Bypass       =           N   b       N   a       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +   CMO     )       +         (       N   a     -     N   b       )       N   a       ·   m   ·     cos   ⁡     (       ω   ·   t     +     π   2       )                 
If all N c  cells are used on phase C:
 
               V     ref_C   ⁢   _Bypass       =           N   b       N   c       ·     (       m   ·     2     3       ·     cos   ⁡     (       ω   ·   t     +       2   ·   π     3       )         +   CMO     )       +         (       N   a     -     N   b       )       N   c       ·   m   ·     cos   ⁡     (       ω   ·   t     +     π   2       )                 
The term m represents the modulation index and it is assumed to have values between 0 and 1, as the term
 
             2     3           
accounts for the 15% increase in the maximum modulation index due to the introduction of the common mode offset, where applicable.
 
     From  FIGS. 3A-C , when the load power factor exceeds certain limits, then some of the cells  16  are forced to operate in regeneration mode due to the shift in the neutral position. Operating some cells  16  in regeneration mode increases the DC-link voltage and eventually leads to a drive overvoltage trip if the drive is only uni-directional. The analytic solution for the reference voltage may be used to determine the safe operating range of the cells when the load power factor varies. The cells  16  located on the phase with the minimum (Min) number of cells  16  are not affected by the load power factor because the vector orientation of that particular phase is unchanged. These cells  16  will not experience any potential overvoltage as long as the load power angle is within −90 to +90 degrees. 
     With respect to  FIG. 3A , for the cells on phases A or B, assuming that δ is the angle between the vector A s N and the D axis, the maximum load angle is limited to: 
                       -     π   2       +   δ     &lt;     ϕ   LOAD     &lt;       π   2     -   δ             (   37   )               
where φ LOAD  is the load power factor angle. The angle δ is determined as follows:
 
     
       
         
           
             
               
                 AA 
                 s 
               
               
                 sin 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 δ 
               
             
             = 
             
               
                 
                   
                     A 
                     s 
                   
                   ⁢ 
                   N 
                 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       
                         5 
                         · 
                         π 
                       
                       6 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   2 
                   · 
                   
                     A 
                     s 
                   
                 
                 ⁢ 
                 N 
               
             
           
         
       
       
         
           
             
               AA 
               s 
             
             = 
             
               m 
               · 
               
                 ( 
                 
                   Mid 
                   - 
                   Min 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 A 
                 s 
               
               ⁢ 
               N 
             
             = 
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             m 
                             · 
                             
                               2 
                               
                                 3 
                               
                             
                             · 
                             Min 
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         
                           m 
                           2 
                         
                         · 
                         
                           
                             ( 
                             
                               Mid 
                               - 
                               Min 
                             
                             ) 
                           
                           2 
                         
                       
                       - 
                     
                   
                 
                 
                   
                     
                       
                         2 
                         · 
                         
                           m 
                           2 
                         
                         · 
                         
                           2 
                           
                             3 
                           
                         
                         · 
                         Min 
                         · 
                         
                           ( 
                           
                             Mid 
                             - 
                             Min 
                           
                           ) 
                         
                         · 
                         cos 
                       
                       ⁢ 
                       
                         
                           5 
                           · 
                           π 
                         
                         6 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     δ 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       Mid 
                       - 
                       Min 
                     
                     
                       2 
                       · 
                       
                         
                           
                             
                               4 
                               3 
                             
                             · 
                             
                               Min 
                               2 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 Mid 
                                 - 
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                               ) 
                             
                             2 
                           
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                             2 
                             · 
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                             · 
                             
                               ( 
                               
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                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         3 
                       
                       · 
                       
                         ( 
                         
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                         ) 
                       
                     
                     
                       2 
                       · 
                       
                         
                           
                             3 
                             · 
                             
                               Mid 
                               2 
                             
                           
                           + 
                           
                             Min 
                             2 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     To guarantee that no cells  16  in any phase are in danger of operating in regeneration mode, the load angle is restricted by the following condition: 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         π 
                         2 
                       
                     
                     + 
                     
                       arcsin 
                       ( 
                       
                         
                           
                             3 
                           
                           · 
                           
                             ( 
                             
                               Mid 
                               - 
                               Min 
                             
                             ) 
                           
                         
                         
                           2 
                           · 
                           
                             
                               
                                 3 
                                 · 
                                 
                                   Mid 
                                   2 
                                 
                               
                               + 
                               
                                 Min 
                                 2 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   &lt; 
                   
                     ϕ 
                     LOAD 
                   
                   ≤ 
                   
                     
                       π 
                       2 
                     
                     - 
                     
                       arcsin 
                       ( 
                       
                         
                           
                             3 
                           
                           · 
                           
                             ( 
                             
                               Mid 
                               - 
                               Min 
                             
                             ) 
                           
                         
                         
                           2 
                           · 
                           
                             
                               
                                 3 
                                 · 
                                 
                                   Mid 
                                   2 
                                 
                               
                               + 
                               
                                 Min 
                                 2 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     With respect to  FIG. 3A  (i.e., the minimum number of cells  16  on phase C), the first part of the inequality is the condition that no cells  16  are in regeneration mode on phase B, while the second part is the condition that no cells  16  are in regeneration mode on phase A. With respect to  FIG. 3B  (i.e., the minimum number of cells  16  on phase B), the first part of the inequality is the condition that no cells  16  are in regeneration mode on phase A, while the second part is the condition that no cells  16  are in regeneration mode on phase C. With respect to  FIG. 3C  (i.e., the minimum number of cells  16  on phase A), the first part of the inequality is the condition that no cells  16  are in regeneration mode on phase C, while the second part is the condition that no cells  16  are in regeneration mode on phase B. 
     During a bypass following a normal operation where the load angle is limited to between −π/2 and π/2, there can be only one phase where the cells  16  may be subjected to a regeneration mode. The phase in question may be determined if the load power factor is known. Using this method with equation (38), if the same number of cells  16  fails on any two phases, there is no need for additional restriction on the load power factor. This is a natural effect given the decomposition into different components (see equation 25). Since in practice only a certain number of cells  16  may be bypassed before the whole drive is stopped, equation (38) is used to tabulate and store the minimum power factor for a given drive and a given bypass configuration, thus reducing the overhead associated with the actual calculation of equation (38). 
     The processor  20  generates the reference voltages. The processor  20  uses the memory  22  for looking up the reference voltages. A look up table outputs the reference voltages as a function of input of numbers of active inverter cells in the different legs  14  (e.g., input of Na, Nb, and Nc). The corresponding reference voltages for each of the legs  14  and/or cells  16  are output. In alternative embodiments, the memory  22  stores instructions for calculating the reference voltages and performing any checks. 
     The memory  22  is a random access memory, system memory, cache memory, hard drive, buffer, database, combinations thereof, or other now known or later developed memory device for storing a look-up table of reference voltages as a function of distributions of operable cells for bypass fault operation of the cascaded multilevel inverter. 
     The memory  22  or other memory is alternatively or additionally a processor readable storage medium storing data representing instructions executable by the programmed processor  20  for control in fault-bypass of a cascaded multi-level inverter. The instructions for implementing the processes, methods and/or techniques discussed herein are provided on non-transitory processor-readable storage media processors, microcontrollers, digital signal processors or memories, such as a cache, buffer, RAM, removable media. The functions, acts or tasks illustrated in the figures or described herein are executed in response to one or more sets of instructions stored in or on the memory or the processor used. The functions, acts or tasks are independent of the particular type of instructions set, storage media, processor or processing strategy and may be performed by software, hardware, integrated circuits, firmware, micro code and the like, operating alone, or in combination. Likewise, processing strategies may include multiprocessing, multitasking, parallel processing, and the like. 
       FIG. 4  shows one embodiment of a method for control in fault-bypass of a cascaded multi-level inverter. The method is implemented by the system of  FIG. 1 , a plurality of the cells  16  of  FIG. 2 , and/or another system or cells. The acts are performed in the order shown and/or, additional, different, or fewer acts may be provided. 
     In act  62 , initially the cascaded multi-level inverter is operated in a normal mode where no cells are faulted. In act  64 , one or more cells in one or more phases become inoperable or operate incorrectly. For example, one or more transistors or other circuit components burn out, short, or open. 
     These faulty cells are bypassed. The result is a different number of cells contributing to the output of the inverter. The numbers of cells for each phase are equal or different. 
     A check may be performed to determine whether sufficient numbers of cells are operable in each phase. Some combinations of operable cells per phase may result in the drive overvoltage trip if the power factor exceeds certain limits. 
     For fault bypass operation, the cells are controlled differently to account for the bypassed cells. For pulse width modulation control, the reference voltage for establishing the pulse widths is altered. In act  66  where one or more cells are bypassed, the reference voltages change to account for the bypass. 
     The operable cells respond to pulse width modulation signals. The same reference voltage is used by all cells of each leg to establish the pulse width modulation. In response to the control, the cells of the different legs output equal magnitude voltages between the three phases. A balanced phase relationship is provided in the line-to-line output voltage. 
     In act  64  the bypassed cells are identified for bypass operation. The specific cells that are active or inactive are determined. 
     The cells are identified from the control of the cells. Where a cell is no longer responsive or where other processes have determined a cell is to be bypassed, the determination is used to identify the cells. Alternatively, an electronic circuit detects a malfunction, and the faulty cell is identified from the output of this circuit. In another embodiment, proper operation is sensed. The faulty or bypassed cells are identified by not being included in a list of properly operating cells. Any approach may be used to identify the cells for bypass or being bypassed. 
     The numbers of bypassed and/or operable cells for each phase are determined. One phase may have more or fewer operable cells with a corresponding more or fewer faulty cells than another phase. 
     In act  66 , the operation of the operable cells is controlled. The reference voltage used to establish pulse width modulation is set based on the number of active cells. The reference voltages are calculated. Alternatively, the pre-calculated reference voltages for a given bypass situation are looked up. 
     The reference voltages are calculated by modeling the inverter as a combination of three inverters. The first inverter has the same number of cells on each leg, equal to the minimum number of available cells on each leg. The second inverter has two legs with a number of cells equal to the difference between the middle number of cells in one leg and the minimum number of cells in another leg. The third leg of the second inverter has no cells. The third inverter has one leg with a number of cells equal to the difference between the maximum number of cells in one leg and the medium number of cells in another leg. The second and third legs of the third inverter have no cells. 
     The leg with the most operable cells may be operated in one of two ways. In a first way, all of the operable cells are controlled. In a second way, only a number of cells equal to the number of cells on the leg with the middle number are operated. The reference voltage and/or pulse width modulation for the leg with the greatest number of cells is set based on the selected number of cells to use. 
     The reference voltages represent an analytic solution. The control of the operation is performed without feedback from voltage output of the cascaded multilevel inverter. The processor  20  generates the reference voltages without feedback. The voltage output by one or more cells  16 , by one or more legs  14 , or to the AC motor  18  is not used in a feedback path to set the reference voltage. No feedback gain accuracy is needed. The frequency of the cascaded multilevel inverter drive may be high since no feedback gain limits high frequency operation. The reference voltages are generated without approximation based on the number of active to inactive cells for a given phase. An approximation based on the number of active to inactive cells in phase A is not used; an approximation based on the number of active to inactive cells in phase B is not used; and an approximation based on the number of active to inactive cells in phase C is not used. Different ratios are used, based on the relationship of active cells between phases. 
       FIGS. 5-26  relate to a computer simulation using the analytically calculated voltage references. The simulated drive is a PERFECT HARMONY drive from SIEMENS with nine cells, three cells per phase. Nine carriers under normal conditions when no cells are bypassed are used. The carrier is a triangle wave compared with the reference voltage for establishing the pulse wave modulation. The switching frequency is assumed to be 600 Hz for each transistor (e.g., IGBT), and the output frequency is 60 Hz. A star connected R-L load is considered with the values of 2 ohms and 5 mH, respectively. Each DC-link voltage of the H-bridge cell (see  FIG. 2 ) is assumed as 800 Vdc. Unless otherwise specified, the waveforms are depicted at maximum modulation index. 
       FIGS. 5-8  depict the voltage references on each phase, the phase currents, and line-to-line voltages from A to B and B to C, respectively, for the case where one cell is bypassed in phase C.  FIGS. 9-12  depict the voltage references on each phase, the phase currents, and line-to-line voltages from A to B and B to C, respectively, for the case where two cells are bypassed in phase C.  FIGS. 13-16  depict the voltage references on each phase, the phase currents, and line-to-line voltages from A to B and B to C, respectively, for the case where two cells are bypassed in phase C and one cell on phase B. The harmonic spectrum of two line-line voltages (A to B) and (B to C) are depicted in  FIGS. 17 and 18 , respectively.  FIG. 19  is a simulated DC-link current through one cell of phase A with two cells bypassed on phase C and one cell bypassed on phase B.  FIGS. 20-22  depict the voltage references on each phase, the phase currents, and line-to-line voltages from A to B and B to C, respectively, for the case where two cells are bypassed in phase C and one cell each, on phases A and B.  FIG. 23  is an example simulation of the line-line voltage between phases B and C with two cells bypassed in phase C and one cell bypassed in each of phases A and B. The harmonic spectrum of two line-line voltages (A to B) and (B to C) are depicted in  FIGS. 24 and 25 , respectively.  FIG. 26  is a simulated DC-link current through one cell of phase A with two cells bypassed on phase C and one cell bypassed on each of phases A and B. 
     Where phase A has all three cells functioning and one more than either of the other two phases, either three or two cells may be used on phase A. One drive configuration (Na=3, Nb=2, Nc=1) may be compared with the other drive configuration (Na=2, Nb=2, Nc=1) configuration. This other configuration is one in which one healthy cell has been intentionally bypassed on phase A. Table I compares the two configurations, showing that the differences are rather minor in terms of waveforms quality. 
     
       
         
           
               
             
               
                 TABLE I 
               
             
            
               
                   
               
               
                 Comparison between two bypass configurations 
               
            
           
           
               
               
            
               
                   
                 Total Harmonic 
               
               
                   
                 Distortion (THD %) 
               
            
           
           
               
               
               
               
               
               
            
               
                 Configuration 
                 I a   
                 I b   
                 I c   
                 V ab   
                 V bc   
               
               
                   
               
               
                 321 (Two faulty cells in phase C and 
                 0.72 
                 0.81 
                 1.04 
                 25.55 
                 23.07 
               
               
                 one in phase B) 
               
               
                 221(Two faulty cells in phase C and 
                 0.83 
                 0.83 
                 1.03 
                 26.03 
                 22.13 
               
               
                 one in phase B plus one healthy cell 
               
               
                 bypassed on phase A) 
               
               
                   
               
            
           
         
       
     
     Table II confirms that both configurations output the same maximum output voltage, a result which is in accordance with equation (30). 
                     TABLE II                  Maximum voltage comparison between two bypass configurations                             Maximum               Line-Line           Voltage           (Fundamental)                                 Configuration   V ab     V bc                         321 (Two faulty cells in phase C and   2388   2394           one in phase B)           221(Two faulty cells in phase C and one   2388   2394           in phase B plus one healthy cell           bypassed on phase A)                        
The simulated fundamental voltage in Table II are very close to the ideal maximum voltage according to equation (30), which is (Mid+Min)*Vdc=3*800=2400 Vac.
 
     While the invention has been described above by reference to various embodiments, it should be understood that many changes and modifications can be made without departing from the scope of the invention. It is therefore intended that the foregoing detailed description be regarded as illustrative rather than limiting, and that it be understood that it is the following claims, including all equivalents, that are intended to define the spirit and scope of this invention.