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Generic Control Method of Multileg Voltage-Source-Converters for Fast Practical Implementation Philippe Delarue, Alain Bouscayrol, Member, IEEE, and Eric Semail, Member, IEEE

Abstract—A generic and simple control method is suggested for any multileg voltage-source-converter. A specific coding yields an inversion table allowing a fast practical implementation. Phase-tophase voltage references have to be defined for such a table. This original control strategy is validated by experimental results for two-leg, three-leg, four-leg, and five-leg structures supplying balanced and unbalanced multiphase loads. Index Terms—DC–AC converters, digital control, multileg converter, PWM.

I. INTRODUCTION

I

N the last few decades numerous works have been developed to optimize the control of voltage-source-inverters (VSI) [1]–[4]: third harmonic injection, space vector strategy, flat-top modulation… Most of these inverters are three-leg structures for supplying ac machines. Indeed, these drives are more and more used in industrial applications thanks to dynamic machine controls [5]. But other inverter structures are now being studied: four-leg inverters for three-phase four-wire systems [6], [7], four-leg inverters feeding two induction machines [8], [9], five-leg inverters for a two-induction machine drive [10], [11] and for 5-phase reluctance machines [12], [13]… Studies of multimachine multiconverter systems are also being developed for other original solutions [14]. Moreover, power converter manufacturers take a great interest for practical implementations, which make fast modifications possible for fault operating structures. Polyphase ac machines, which need multileg supplies [15]–[17], nowadays have an increasing interest for their reliability. For each of these nonclassical structures, specific and complex controls have been developed. A global control method has already been presented for three-leg, four-leg and five-leg voltage-source-inverters [18]. Only simulation results were provided for open loop operating. Moreover the practical implementation of such a strategy has not often been discussed. In this paper, this generic and simple control method is extended to be applied to any multileg voltage-source-converter with closed-loop current controls. It can be used to supply balanced and unbalanced loads. The aim of this new control Manuscript received February 1, 2002; revised November 25, 2002. Recommended by Associate Editor S. B. Leeb. The authors are with the Laboratory of Electrical Engineering of Lille L2EP, University of Lille, Villeneuve d’Ascq F59655, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TPEL.2003.809349

Fig. 1. Structure of the multileg VSI.

technique is to have a generic algorithm, which can be rapidly implemented and easily modified if the inverter topology changes. Experimental results for different structures validate this practical implementation of the multileg converter control. II. MODELING OF A MULTILEG CONVERTER A. Structure of the Studied Power Converter to ac The studied converter links a dc voltage source . (Fig. 1). It is composed of legs of two current sources power switches, which are assumed to be turn-on and turn-off controlled. As they connect a dc voltage source with ac current sources, each switch is made of a parallel diode-transistor association. Because the leg no. is arbitrarily chosen as potential refmodulated phase-to-phase erence, the converter leads to between the current-sources. The involtages to the dc voltage-source. verter yields a modulated current As this inverter structure is reversible, this control method can also be applied to multileg current source rectifiers. B. Power Converter Modeling is defined for each power switch. A switching function, This function represents the ideal switching order and takes the values 1 when the switch is closed and 0 when it is opened (1) with no. of the leg no. of the switch in the leg Because ideal power switches are considered, the switches of a same leg are in complementary states

0885-8993/03$17.00 © 2003 IEEE

(2)

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m

TABLE I VALUES CALCULATED FROM THE

s

VALUES

Fig. 2. COG of the multileg inverter.

Therefore, only one switching function of a leg has to be defined, because the other is automatically deduced from the first one according to (2). A modulation function (or conversion function) can also be defined from the switching functions of the upper switches (3) So the definition domain of the

functions is (4)

is linked to by the switches The modulated voltage of the leg and of the leg . This modulated voltage has only three levels, the values of which depend on the different combinations of the switches (open or closed). With the assumption of a constant dc voltage, we have (5) So, the modulation function with the dc voltage

links the modulated voltage

(6) can be expressed from Moreover, the modulated current the load currents using the modulation functions (7) A graphical representation of the power converter modeling is obtained by using the causal ordering graph (COG) [19]. The relationships are depicted by balloons. A causal relationship, which depends on time and means that there is a state variable, contains an unidirectional arrow. A noncausal relationship, which has no time dependence, contains a bi-directional arrow. A vector representation simplifies the graph (Fig. 2)

Fig. 3. COG of the average multileg inverter.

The relationship (3) between the switching vector and the can be summarized in a table (Table I). modulation one C. Average Modeling A classical average modeling is used in order to develop the control of the converter. The three-level modulation functions are averaged during a fictive null period

with (9) In consequence, the average voltages uous values

also have contin(10)

This operation allows us to replace discontinuous quantities by their equivalent continuous quantities. The COG is modified by the insertion of an averaging operator and average vectors (Fig. 3). (8) D. Modeling of Multiphase Loads The subscripts are chosen to point out the process that the as inverter, as current sources ). variables come from ( The switching vector is limited to the switching functions of the first switch of legs, because the second one is deduced from a complementary operation (2).

The power converter supplies the current sources through phase-to-phase modulated voltages. Notice, that these phase-to-phase voltages (or their associated modulation functions) can be independent each other. Moreover, this voltage representation is nondependent on the electrical load. So, the

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Fig. 5. Equivalent single-phase electrical load.

Fig. 4.

Multiphase electrical loads.

n-leg inverter can supply unbalanced loads. The multiphase loads are characterized by the following relationship between their currents: (11) Fig. 6. Control COG of the multileg inverter.

But, most multiphase electrical loads are balanced and so are expressed with their phase-to-neutral voltages. Indeed, this mathematical description leads to a symmetric representation of the balanced load. Modern controls of three-phase ac machines are based on this modeling. A global modeling of multiphase electrical loads is thus developed in order to ensure a systematic transposition between phase-to-phase and phase-to neutral voltages. imAll phases of the load are assumed to be identical pedances with an actual or fictive star connection (Fig. 4). The phase no. is chosen as reference for phase-to-phase voltages. For star connected balanced loads (Fig. 4) we have (12) As the potential is taken as reference, the phase-to-phase can be expressed using the phase-to-neutral voltage voltage

(13) According to the balanced voltage equation (12), a transformation matrix can be defined from the phase-to-phase voltages columns to the phase-to-neutral voltages. It is a matrix of lines and

The inverse transformation can be obtained (with number)

the phase

(15)

If this representation is obvious for classical multiphase loads, it can also be extended to be used for a single-phase load. parameters have to be decomposed In this case the initial loads connected in series, with a virtual into two equivalent neutral point (Fig. 5). If this representation of a single-phase load is nonconventional, it allows us a generalization of the ). balanced multiphase load modeling (also for III. CONTROL OF MULTI-LEG CONVERTERS A. Modeling Inversion The control of a power converter can be deduced from an inversion of its modeling [19]. The control structure is deduced from the average modeling (Fig. 6). In the multileg case, the control defines the switching vector from the reference phase-to. phase voltage vector In the first operation the reference voltage vector is divided inversion of (6). It leads by the measured dc voltage, to the average modulation vector (16)

with (14)

Secondly, a specific PWM (comparison with two triangular : inverwaves) (Fig. 7) yields the modulation vector sion of (9). This operation converts the average values into their equivalent three-level values. The last operation is called switching controller and yields the switching vector inversion of (3). The commutation orders of each leg are defined in this way.

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Fig. 7.

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Example of PWM operation.

B. Switching Controller Problem of the inversion—The switching controller corresponds to the inversion of the Table I. which can There are independent switching functions different patterns only have two values. So, there are for the switching vector . modulation functions On the other hand, there are which can take three values. So, the modulation vector owns patterns. The pattern number of the switching vector, , is different . The inversion is not alto that of the modulation vector, ways possible: some modulation patterns have no equivalent switching pattern. Solution to the inversion—First, a specific code is defined for the modulation vector, in order to assign a code number to patterns its

s

TABLE II PATTERNS FOR A 4-LEG VSI (n = 4, S

= 2 = 16 PATTERNS)

(17) with

The same code must used for the switching vector in order to associate equivalent patterns. So the modulation functions are replaced in (17) by their switching functions (3) (18) This relationship can be generalized (19) Since the pattern numbers are different, some of the modulation patterns have no corresponding switching pattern. Many solutions are possible. For example, these modulation patterns can be attributed to some switching patterns in order to minimize the commutation numbers. , which has no In this paper, a modulation pattern equivalent switching pattern, is associated with the switching of the previous line (no change of the commupattern tation orders) according to the suggested numbering. An example is given for a four-leg voltage-source-inverter. The table of the switching patterns (Table II) and the table of the modulation patterns (Table III) are first built. They show the

difference between the modulation and switching pattern numbers. Secondly, the switching patterns are associated with their (Table IV). equivalent modulation patterns thanks to the Finally, the m-patterns with no equivalent s-pattern (dark cells) are associated with the s-pattern of the previous m-pattern acnumbering (Table V). An automatic genercording to the ation of this inversion table is described in the appendix for a Matlab code (see Tables II–V). C. Generation of Phase-to-Phase Voltage References In the case of balanced loads, phase-to-neutral voltage references are generated from current controllers. But the control method for multileg inverters needs phase-to-phase voltage references. They are directly obtained from the transformation matrix (14) of the modeling of multiphase loads

(20)

In the case of unbalanced loads, phase-to-phase voltage references have to be defined directly.

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m

TABLE III PATTERNS FOR A 4-LEG VSI ( = 4,

n

m

= 3 = 27 PATTERNS)

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The control model is firstly described in the Matlab-Simulink environment for simulations (Fig. 9). The VSI control is built (Fig. 10) as explained in the above section (see Fig. 8). The implementation control is then made on a dSPACE 1103 controller board thanks to a direct real-time compilation (RTW). The basic sampling period is fixed to . It permits the definition of the switching functions 25 times every . The refmodulation period erence currents are balanced sine-wave forms: magnitude of 2A . The modulated voltages are and frequency of . thus defined 100 times every reference period These periods have been chosen in function of the studied n-leg VSI and of the controller board. Indeed, we validated two-leg, three-leg, four-leg, and five-leg structures. The last one of theses leads to the greatest computation time, which must be smaller than the chosen sampling period. Moreover, we have chosen to avoid microprocessor coding by using the Simulink toolbox with a direct compilation (greater computation time). Of course, for industrial applications, the modulation frequency can be increased by using, for example, another controller board or FPGAs. B. Experimental Results for Balanced Loads

D. Implementation of the Control Strategy The implementation of such a control is very easy (Fig. 8). It is composed of classical triangular carrier comparisons and of a look-up table. In the case of a structure reconfiguration for fault operating, the look-up table alone has to be modified. Some of the other blocks will not be used, but their operations will be noneffective with a well-adapted modification of the inverse table. IV. APPLICATION TO MULTI-LEG CONVERTERS The generic control method has been validated for several to ). Experimental results are n-leg VSI (from provided in this section. A. Technical Specifications In order to make comparative studies possible for all studied multileg converters, some technical specifications are fixed. All the power structures have the same dc voltage and the same loads (see Fig. 4) multiphase balanced Moreover, the phase-to-neutral reference voltages are obtained from phase current controllers. The reference currents yield balanced ac currents in the load (same rms. value). The small tracking error induced by the PI controller will be negligible in comparison with the modulation ripples.

Five-leg inverter—This structure (Fig. 11) can supply a fivephase ac machine in normal operating [20] or a two-induction machine drive in fault operating [10]. , the number of switching patterns is For and of the modulation pattern one is . So only 32 modulation patterns among 81 have an equivalent switching pattern. The switching controller has been implemented with the general relationship for (21) The five experimental actual currents (Fig. 11) are actually balanced. It can be notice that the current ripples are not too high due to the inductance value and the modulation frequency. The sine reference currents are really obtained: the small tracking errors are absorbed by the current ripples (reference and actual currents are well superposed). Four-leg inverter—Such multileg structures are used to balance three-phase four-wire loads [6]. Moreover, four-leg structures can feed two three-phase induction machines [8], [9]. For a four-leg inverter, the number of switching patterns is and the number of modulation patterns fixed to to (22) The four experimental actual currents (Fig. 12) are actually balanced. Three-leg inverter—Classical three-leg structures are used in order to feed three-phase loads, and in particular ac machines. In this case, the number of switching patterns is fixed to and the number of modulation patterns to (23)

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TABLE IV FIRST CODING TABLE FOR A 4-LEG VSI m

= 1 + (s

0

s

+ 1) + 3(s

0

s

+ 1) + 9(s

0

s

+ 1)

TABLE V INVERSE TABLE FOR 4-LEG VSI m = 1 + (m + 1) + 3(m + 1) + 9(m + 1) EACH NON-CODED M-PATTERN (DARK CELLS) IS ASSOCIATED WITH THE S-PATTERN OF THE PREVIOUS LINE. FOR EXAMPLE, M-PATTERN m = [1; 1; 1] OF m = 3 HAS NO EQUIVALENT S-PATTERN, AND IS ASSOCIATED WITH S-PATTERN OF THE PREVIOUS LINE s = [1; 0; 0; 1] (m = 2)

0 0

The three experimental actual currents (Fig. 13) are actually balanced. It can be notice that a comparison of this multileg control strategy with a more classical one has been made for this

three-leg VSI [18]. The multileg control strategy induces a worse harmonic spectrum than a classical PWM. The number of commutations seems to be equivalent in both control strategies.

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Conclusion for balanced loads—This methodology can be extended to loads with a large number of phase: double star induction machines (six-phase) supplied by six-leg inverters [17], three-phase open-winding induction machines supplied by a six-leg inverters [21], and other classical or original polyphase machine applications. C. Extension for Unbalanced Loads System description—A five-leg VSI has been developed in order to supply two induction machines [10]. The machines are connected with a common potential through a common inverter leg (Fig. 15). This original power structure allows the independent behavior of both machines in a fault operating (breakdown of the leg). Because each machine is a balanced load, there are three bal, , for the machine no. 1 and , anced currents: , for the machine no. 2. As regards to the 5-phase in, verter, the equivalent load induces five currents: , , and . But this structure enables different speeds on the two machines (with different load torque). In this case, the magnitudes and frequencies of the currents can be quite different. So the five currents of the equivalent load are unbalanced, even if each machine is really balanced. The structure is simplified to validate the multileg control strategy for unbalanced loads. It is composed of a five-leg loads (Fig. 16). The VSI, which supplies two three-phase leg no. 5 is shared by a phase of each three-phase load. This common leg is chosen as the reference. In order to take into account this configuration, the transformation matrix must be redefined

Fig. 8. Implementation block diagram for a three-leg VSI.

Fig. 9. Matlab-Simulink model for the five-leg VSI system.

(25)

Experimental results—The same technical specifications in balanced loads are used. In order to impose different behaviors on each three-phase load the current references are defined as with Fig. 10.

Matlab-Simulink model of the five-leg VSI control.

with

Two-leg inverter—The two-leg structure supplies a single load which is decomposed into a 2-phase balanced load with a star connection (see Fig. 5). In this case, the number of switching patterns is fixed to and the number of modulation patterns to (24) The experimental results are presented in Fig. 14. Of course the fictive phase currents are actually inverted (condition of balanced currents in the modeling). Even if this control is not of interest in such a classical structure, this study shows that the suggested method could be applied to any leg number.

The experimental results (Fig. 17) show balanced three-phase currents on each three-phase load. Notice that the current of the has a nonsinusoidal waveform because it is a common leg combination of currents with different magnitudes and frequencies (Fig. 18). Conclusion for unbalanced loads—A classical VSI control is more difficult to implement in this unbalanced configuration. Indeed, the five current references have different waveforms, magnitudes and frequencies. The multileg control strategy is adapted because it requires only phase-to-phase voltage references. It could be relevant to apply this methodology to four-leg VSI for three-phase four-wire systems [6], [7]. Indeed, in this application, the fourth leg has to impose current in order to balance the load.

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Fig. 11.

Five-leg VSI: (a) structure (b) experimental current wave-forms.

Fig. 12.

Four-leg VSI: (a) structure (b) experimental current wave-forms.

Fig. 13.

Three-leg VSI: (a) structure (b) experimental current wave-forms.

Fig. 14.

Two-leg VSI: (a) structure (b) experimental current wave-forms.

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Fig. 15.

Five-leg VSI for two induction machines.

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mental results for different n-leg converters, ( to ), for balanced and unbalanced loads. A specific coding leads to a global strategy for any leg number. It can be easily implemented in a table. Therefore this method yields fast practical implementations and modifications. But, it leads to a worse harmonic spectrum than a classical PWM control. Other methods to define the unused modulation patterns could be found in order to improve the performances. But for a leg number higher than 3, optimum PWM are more complex [6], [7], [10], [16]. This multileg control strategy could be useful in practical cases for which the number of legs may change (modular power structures [22]) or for inverters with a great number of legs. Moreover, this strategy can be applied to unbalanced loads. In particular, supplies of multiphase machines are concerned. Indeed, they are being use more and more in industrial processes thanks to their reliability. In the case of a fault operating, their currents are often unbalanced. APPENDIX GENERIC CODING ALGORITHM OF THE INVERSION TABLE FOR MATLAB CODE (FOUR-LEG VSI)

Fig. 16.

Five-leg VSI for two three-phase loads.

Fig. 17.

Experimental current waveforms of the three-phase loads.

Fig. 18.

Experimental current waveform of the common phase.

V. CONCLUSION A generic control method has been developed for any multileg voltage-source-inverter. It has been validated by experi-

n_leg=4; % leg number m_pat=3^(n_leg-1); % m pattern number s_pat=2^n_leg; % s pattern number % it = inversion table %---- initial coding ------------for i=1:n_leg for j=1:m_pat it(i,j)=-1 end; end; %---- inverse coding ------------for s11=0:1 for s21=0:1 for s31=0:1 for s41=0:1 m_cod=1+(s11-s41+1) +3 (s21-s41+1) +9 (s31-s41+1); it(1,m_cod)=s11; it(2,m_cod)=s21; it(3,m_cod)=s31; it(4,m_cod)=s41; end; end; end; end; %---- final coding ----for i=1:n_leg for j=1:m_pat if it(i,j)==-1 it(i,j)=it(i,j-1); end; end; end;

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REFERENCES [1] J. Houldsworth and D. Grant, “The use of harmonic distortion to increase the output voltage of a three-phase PWM inverter,” IEEE Trans. Ind. Applicat., vol. IA-20, pp. 1224–1228, Sept. 1984. [2] J. Holtz, “Pulse width modulation—a survey,” IEEE Trans. Ind. Electron., vol. 39, pp. 410–419, Dec. 1992. [3] H. van der Broeck, “Analysis of the voltage harmonics of PWM voltage fed inverters using high switching frequencies and different modulations functions,” ETEP J., vol. 2, no. 6, pp. 341–349, November/December 1992. [4] A. Hava, R. Kerkman, and T. Lipo, “Carrier-based PWM VSI overmodulation strategies: analysis, comparison and design,” IEEE Trans. Power Electron., vol. 13, pp. 624–689, July 1998. [5] W. Leonhard, “30 years space vectors, 20 years field orientation, 10 years digital signal processing with controlled ac drives,” EPE J., vol. 1, pp. 13–20, July 1991. [6] S. Ali and M. Kazmierkowski, “Current regulation of four-leg PWMVSI,” in Proc. IECON’98 Conf., vol. 3/4, Aachen, Germany, Aug. 1998, pp. 1853–1858. [7] V. Soares and P. Verdelho, “A comparison of current controllers for three-phase four-wire PWM voltage converter,” in Proc. EPE’99 Conf., Lausanne, Switzerland, Sept. 1999. [8] A. Bouscayrol, M. Pietrzak-David, and B. de Fornel, “Comparative studies of inverter structures for a mobile robot asynchronous motorization,” in Proc. IEEE-ISIE Conf., vol. 1, Warsaw, Poland, July 1996, pp. 447–452. [9] E. Ledzema, B. McGrath, A. Munoz, and T. A. Lipo, “Dual ac-drive system with a reduced switch count,” IEEE Trans. Ind. Applicat., vol. 37, pp. 1325–1333, Sept./Oct. 2001. [10] B. François and A. Bouscayrol, “Design and modeling of a five-phase voltage-source inverter for two induction motors,” in EPE’99 Conference, Lausanne, Switzerland, Sept. 1999. [11] S. Gataric, “A polyphase cartesian vector approach to control of polyphase ac machines,” in Proc. IEEE-IAS Annu. Meeting 2000, Rome, Oct. 2000. [12] H. A. Toliyat, “Analysis and simulation of five-phase variable speed induction motor drives under asymmetrical connections,” IEEE Trans. Power Electron., vol. 13, pp. 748–756, July 1998. [13] T. Gopalarathnam, H. A. Toliyat, and J. C. Moreira, “Multiphase faulttolerant brushless dc motor drives,” in Proc. IEEE-IAS Annu. Meeting 2000, Rome, Italy, Oct. 2000. [14] A. Bouscayrol, B. Davat, B. de Fornel, B. François, J. P. Hautier, F. Meibody-Tabar, and M. Pietrzak-David, “Multimachine multiconverter systems for drives: analysis of couplings by a global modeling,” in Proc. IEEE-IAS Annu. Meeting, Rome, Italy, Oct. 2000. [15] A. Munoz and T. A. Lipo, “Dual stator winding induction machine drive,” IEEE Trans. Ind. Applicat., vol. 36, pp. 1369–1379, Sept./Oct. 2000. [16] E. Semail and C. Rombaut, “New method to calculate the conduction durations of the switches in a n-leg 2-level voltage source,” in Proc. EPE’2001 Conf., Graz, Austria, Aug. 2001. [17] R. Lyra and T. A. Lipo, “Torque density improvement in a six-phase induction motor with third harmonic current injection,” in Proc. IEEE-IAS Annu. Meeting 2001, Chicago, IL, Oct. 2001. [18] P. Delarue, A. Bouscayrol, E. Semail, and B. François, “Control method for multileg voltage source inverter,” in Proc. EPE’2001 Conf., Graz, Austria, Aug. 2001.

[19] X. Guillaud, P.Ph. Degobert, and J. P. Hautier, “Modeling, control and causality: the causal ordering graph,” in Proc. 16th IMACS World Congr., Lausanne, Switzerland, Aug. 2000. [20] H. A. Toliyat, S. Ruhe, and X. Huansheng, “A DSP-based vector control of five-phase synchronous reluctance motor,” in Proc. IEEE-IAS Annu. Meeting 2000, Rome, Oct. 2000. [21] Y. Kawabata, M. Nasu, T. Kawakami, E. Ejiogu, and T. Kawabata, “High efficiency vector control system using open-winding motor and two space vector modulated inverters,” in Proc. EPE’01 Conf., Graz, Austria, Aug. 2001. [22] Z. Ye, D. Boroyevich, and F. C. Lee, “Paralleling nonisolated multiphase PWM converter,” in Proc. IEEE-IAS Annu. Meeting 2000, Rome, Oct. 2000.

Phillipe Delarue received the Ph.D. degree from the University of Sciences and Technologies, Lille, France, in 1989. Since 1991, he has been an Assistant Professor at Ecole Universitaire des Ingénieurs de Lille (EUDIL) and at Laboratory of Electrical Engineering of Lille (L2EP Lille). His main research interests are power electronics and multimachine systems.

Alain Bouscayrol (M’00) received the Ph.D. degree from INP Toulouse, France, in 1995. Since 1996, he has been engaged as assistant Professor at the Laboratory of Electrical Engineering of Lille (L2EP Lille), University of Sciences and Technologies, Lille, France. His research interests include electrical machine controls and multimachine systems. Since 1998, he has managed the Multimachine Multiconverter Systems Project, GdR-SDSE (a national research program of the French CNRS).

Eric Semail (M’00) received the B.S. and M.S. “Agrégation” degrees from the Ecole Normale Supérieure, Cachan, France, in 1986 and the Ph.D. degree from Grande École of Engineering ENSAM, France, in 2000. From 1987 to 2001, he has been Professor (holder of agrégation) with the University of Sciences and Technologies, Lille, France. He became an Associate Professor in Grande École of Engineering ENSAM, in 2001. In Laboratory of Electrical Engineering of Lille (L2EP), his fields of interest include modeling, control, and design of polyphase systems (converters and ac drives).