Electrimacs 2002, August 18 - 21

Electric Drives III

Space Vector Control of 5-phase PMSM supplied by 5 H-bridge VSIs J. P. Martin, E. Semail, S. Pierfederici, A. Bouscayrol, F. Meibody-Tabar, and B. Davat

Abstract—The use of polyphase PMSM supplied by H-bridge VSI allows on one hand to segment the power transferred from the electrical source to the mechanical load and on the other hand to operate in degraded operating mode, with one or several nonsupplied phase. Nevertheless, for an independent current control of the H-bridge VSI the magnetic coupling between each phase winding leads to high phase current ripples. In this paper a global current control method of the H-bridge VSIs, based on an adapted space vector control method is proposed. The proposed method allows a considerable reduction of the current ripple rate in the case of 3-phase and 5-phase nonsalient PMSM supplied by H-bridge VSIs. Index Terms-- AC motor drives, Current control, Permanent magnet motors, Electromagnetic coupling, Pulse width modulation.

I. INTRODUCTION

F

or a given power transferred from the electrical source to the mechanical load the increase of the phase number allows the use of inverters with reduced caliber switches authorizing consequently a higher switching frequency of the inverters components. A structural solution consists in using polyphase machines (fig 1) where each phase is supplied by its own H-bridge Voltage Source Inverter (VSI). By the use of polyphase synchronous machines excited by rotor mounted permanent magnets, the segmented structure of the supply allows the system to operate in degraded mode with one or several non-supplied phases [1]. The structural properties and the modularity of the inverters make this solution attractive for embarked applications, especially for naval propulsion. The Multi-machine Multi-converter System representation of a such system points out energy distribution and magnetic coupling of the electromechanical conversion chain (fig 2) [2]. The sources are depicted by oval forms: the equivalent electrical source (ES) and the mechanical ones (MS). The electrical coupling due to the supply by the same electrical J. P. Martin, S. Pierfederici, F. Meibody-Tabar, B. Davat are with the Group of Research in Electrical engineering and Electronics of Nancy (GREEN), CNRS UMR 7037, Institute National Polytechnic of Lorraine, Vandoeuvre-lès-Nancy, France. (e-mail: [email protected]) E. Semail is with the Laboratory of Electrical engineering and Power Electronics of Lille (L2EP) CNRS UPRES EA 2697, ENSAM Lille, France. A. Bouscayrol is with the Laboratory of Electrical engineering and Power Electronics of Lille (L2EP) CNRS UPRES EA 2697, Lille I University, France.

H-bridge VSI H-bridge VSI

Fig. 1. Segmented supplied structure. Vdc ES

VSI1 SM1

Vdc ES

VSI2

SM2

MS Vdc ES

SMi VSIi SMq

Vdc ES

VSIq

Fig. 2. MMS representation of the segmented structure. source is neglected and the converters are supposed to be supplied by identical independent electrical sources. The electrical converters (H-bridge VSI) are represented by square forms. The odd q-phase PMSM is supposed to be equivalent to q magnetically coupled single-phase machines sharing the same rotor. q circular forms represent this machine and the intersections between them indicates the magnetic coupling. However, the magnetic coupling between the phase windings leads to the fact that the dynamic of one phase current depends not only on its voltage value but also on the applied voltages to the other phases. So, the independent control of each phase current may lead to high current ripple rate, increasing consequently machine and converters losses. By applying the generalized Concordia transformation ([3], [6]) to the variables of a q-phase non-salient PMSM, we show that this machine with an odd phase number is equivalent to (q+1)/2 fictitious machines without magnetic coupling: (q-1)/2 two-phase and one single-phase machines. For a machine with sinusoidal emf waveform, the machine torque depends only on the currents of one of the fictitious 2-phase machine (called main machine). The currents of the other single or 2-phase machines (called secondary machines) generate only additional

losses [6]. Moreover, the voltage equations of the secondary machines show that their current dynamic are only limited by the leakage inductance of the sinusoidal PMSM, which has a quite weak value. The weak value of their inductance leads to a high current ripple rate if the inverter voltages excite the secondary machines. One logical solution to reduce the current ripple rate consists in minimizing the duration of excitation of the secondary fictitious machines. This can be done thanks to a global current controller of the q H-bridge VSIs, which generates the voltage vector reference. This reference voltage vector is reconstituted over each sampling period by an optimal combination of the voltage vectors, which can be imposed by the q H-bridge VSIs. This approach allows to transfer the magnetic coupling between q phase winding to a coupling of the control of the q VSIs control. In this paper, after the presentation of the transformation which allows to obtain the variables of the fictitious machines from those of the real one, we bring out the problem of independent current control of H-bridge VSIs in the case of a 3-phase PMSM. Then, we present a current control method using the properties of the space vector control to choose the convenient voltage vectors to reduce the current ripple rate. Finally, in order to show how to adapt this method to the machines with higher phase number, we apply this method to a 5-phase PMSM.

1 2 1 2 1 2  cos(2 ⋅ π q) cos(4 ⋅ π q)  1  0 sin(2 ⋅ π q) sin(4 ⋅ π q) 2  Tqqt = ⋅ q   1 cos((q −1) ⋅ π q) cos(2 ⋅ (q −1) ⋅ π q)  sin((q −1) ⋅ π q) sin(2 ⋅ (q −1) ⋅ π q)  0  



  cos(2 ⋅ (q −1) ⋅ π q) sin(2 ⋅ (q −1) ⋅ π q)   cos (q − 1)2 ⋅ π q    2   sin (q −1) ⋅ π q    1 2

















Applying this transformation to (1), the voltage equation becomes:

[V0αβ ] = R ⋅ [I0αβ ]+ Tqq t ⋅ [L]⋅ Tqq ⋅ d[I0αβ ] dt + [E 0αβ ] (3) where Tqq t ⋅ [L] ⋅ Tqq is a diagonal matrix for non-salient PMSMs. The second and third diagonal terms of this matrix are the same and have greater value than the proper inductance of each phase winding (cyclic inductance). The other (q-2) diagonal terms are equal or close to the value of the machine leakage inductance. So the equation system (3) contains q independent voltage equations. This property allows to define (q+1)/2 equivalent machines without magnetic coupling [6]: - One single-phase machine called homopolar machine modeled by resistance R, inductance L 0 (leakage inductance) and E 0 the homopolar component of the emf vector. - One 2-phase machine called main machine, modeled by resistance R, equivalent cyclic inductance L c for each phase and E αβ which are the only components of the vector [E ] 1

II. EQUIVALENT FICTITIOUS MACHINES The voltage equation of a non-salient polyphase PMSM can be defined by (1):

containing the fundamental of the emf vector (in steady state). - (q-3)/2 two-phase machines called secondary machines, modeled by their phase resistance R, phase inductance (equal or closed to the leakage inductance) and E αβ (i=2 to (q-1)/2) i

[V] = R ⋅ [I] + [L]⋅ d[I] dt + [E]

(1)

V, I and E are respectively the voltage, the current and the emf vectors of the machine. L is the stator inductance matrix, where the main diagonal values are the self-inductance of each phase winding and the others represent the mutual inductances between phases. R is the stator phase resistance. Using the generalized Concordia transformation (Tqq), one can obtain the component of any vector (voltage, current and emf) in a new reference basis [4]: (2)

with G β1


For sinusoidal q-phase PMSM, the only non-null component of the emf vector in the new reference is E αβ of 1

the main machine. So only the main fictitious machine participates in the torque generation. In order to minimize the machine losses for a given torque amplitude, the current reference vector should have only two non-null components I αβ , in phase with the emf E αβ . 1

1

III. GLOBAL CURRENT CONTROL OF 3-PHASE PMSM

[G 0αβ ] = [Tqq ]t ⋅ [G1,2,..,q ]

[G 0αβ ] = G 0 G α1 [G1,2,..,q ] = [G1 G 2

which are the component of the emf vector containing only some harmonics of the emf vector (in steady state).

]

Gq t

G α (q −1) / 2

G β (q −1) / 2  

t

The 3-phase sinusoidal PMSM can be represented by one homopolar machine and one 2-phase machine, which is the main one. Its MMS representation when the PMSM is supplied by 3 H-bridge VSI is shown on fig. 3. It can be noted that the magnetic coupling between the phase windings (Fig. 2) is transferred to an electrical coupling between the converters (Fig. 3). In fact, the voltage applied to the fictitious machine is a combination of the output voltages of the H-bridge VSIs (2).

H-bridge inverters Basis change u ond3

Equivalent machines

Equivalent torque

shaft

U dc ES

i s1

iond1

u sαβ

Ω am

u ond3

U dc ES U dc

C αβ

is αβ

iond2

u so

ies2 u ond3

ES iond3 Criteria of current undulation rate minimisation

is αβ

i so

co

e so

iso

ies3

e s αβ

Ω am

c em

Ω am

MS cr

Ω am

u s αβ -ref u so-ref

Current control

Fig. 3. MMS representation of a 3-phase machine in the new basis. Ii(t)

Iα(t) Iβ(t)

Fig. 4. Phase current waveform. The figure 4 represents the phase current waveform of the PMSM when the VSI are controlled independently with a fixed switching frequency. It should be noted that output voltage of each H-bridge inverters takes only two values (+Ub, -Ub). The high current ripple rate is due to the excitation of the fictitious homopolar machine having a low inductance. The figure 5 shows that the current of the homopolar fictitious machine contains the major part of the phase current ripples. The current of the main fictitious machine (fig. 6) has a quite low ripple rate thanks to its higher inductance value. Ih(t)

Fig. 6. Current of the fictitious main machine. In order to reduce the current ripple rate, the control of the 3 H-bridge VSIs should be global to minimize the duration of the excitation of the homopolar machine. Taking into account that each H-bridge inverters can impose 3 voltage levels (+Ub, 0, -Ub), the number of possible voltage vectors is 27. The homopolar and α-β components of these 27 voltage vectors are represented on the homopolar axis and on the α-β plane (fig 7). These vectors can be classified in six distinct groups: - The 1st group of 6 vectors M1, M3, M5, M7, M9, M11 have a homopolar component of ± 1 3 ⋅ U b and their projection on α-β plane constitute 6 vectors of norm

2 3 ⋅ U b shifted by

60°. - The 2nd group of 6 vectors M2, M4, M6, M8, M10, M12 have a homopolar component of ± 2 3 ⋅ U b and their projection on α-β plan constitute 6 vectors of norm

Fig. 5. Current of the homopolar fictitious machine.

2 3 ⋅ Ub

shifted by 60°. - The 3rd group of 6 vectors M13 to M18 have null homopolar component and their projection on α-β plane constitute 6 vectors of norm 2 ⋅ U b shifted by 60°. - The 4th group of 6 vectors M19 to M24 have a homopolar component of ± 1 3 ⋅ U b and their projection on α-β plane constitute 6 vectors of norm 2 ⋅ 2 3 ⋅ U b shifted by 60°.

β M21

homopolar M20

M14

M26 M15 M5 , M6 M22

M3 , M4 M26

M7 , M8

M13 M4 , M8 , M12 M19

M1, M2

M0 M25

M16

M1 , M5 , M9 ,

0

M11, M12

M9 , M10

α

M18

M0 ,

M20 , M22, M24

M13 , M14 , M15 , M16 , M17 , M18

M3 , M7 , M11,

M19 , M21, M23

M2 , M6 , M10 M23

M17

M25

M24

Fig. 7. Projection of the 27 voltage vectors on α-β plane and homopolar axis. - The 5th group contains only the vector M0 with a null norm, - The 6th group contains 2 vector M25 and M26, which have

IV. GLOBAL CONTROL OF 5-PHASE PMSM

only non-null homopolar component ( ± 3 ⋅ U b ). We can note that only the 3rd and 5th groups have null homopolar components. So it is suitable to utilize the voltage vectors of these groups to impose the reference voltage vectors given by current controllers. In order to obtain a fixed switching frequency, one can determine the duration of application of the two vectors of the 3rd group, delimiting the projection of reference voltage vector on the α-β plane. The rest of sampling period the zero voltage vector (5th group) is applied. (Classical space vector control) [5]. If the machine emf has a non-zero homopolar component the voltage vector of the 6th group, which are the pure homopolar vectors, are used, in addition to the 3rd and 5th group, to control the homopolar current component. Using this global current control strategy, the phase current ripple rate is considerably attenuated (fig. 8). For a sinusoidal PMSM, the homopolar component of the current vector is instantaneously equal to zero. Iα(t) Iβ(t)

Fig. 8. Phase current waveform of 3-phase PMSM with global current control.

For the PMSM of higher phase number (q>3) supplied by H-bridge VSIs, the number of voltage vectors increases exponentially ( 3q ). Most of these voltage vectors excite the homopolar and secondary fictitious machines. Nevertheless, there is not any voltage vector, which excites the main fictitious machine without exciting the secondary ones. In order to reduce the current ripple rate it is suitable to choose the vectors without homopolar component for which the norm of the projection on the α1-β1 plane being maximal. With these voltage vectors the applied voltage to the secondary fictitious machines are relatively weak. As an example, we present here the results obtained for a 5phase sinusoidal PMSM supplied by 5 H-bridge VSIs. Using the Concordia’s transformation, the voltage equation of the 5phase machine can be diagonalized. Then, one homopolar single–phase, one main 2-phase and one secondary 2-phase fictitious machines are deduced. In the same way, as for the 3phase PMSM, an independent control of the phase current leads to a high ripple rate of the phase current (fig. 9). Ii(t)

Fig. 9. Phase current waveform for independent current control.

β1

β2

M3

M4

M2

M5

M1 M0

M1

α1

M6

M5

M4

M2

M7

M9

α2

M10 M M6 3

M10

M7

M8

M9 M8

Fig. 10. Group of voltage vector with null homopolar component and highest projection norm on α1-β1 plane. a Ih(t)

The figure 11 shows respectively the current of the homopolar, main and secondary machines. We remark again that the high amplitudes of the currents of the homopolar and secondary fictitious machines ( I 0, I α2 , Iβ2 ) are the reason of the high ripple rate of the phase current. In order to apply a global current control strategy of the 5 H-bridge VSIs, the voltage vectors should be chosen between 35 voltage vectors. We choose a group of 10 voltage vectors

Iα2(t) Iβ2(t)

Iα1(t) Iβ1(t)

b

c

Fig. 11. Current waveform of the fictitious machine : homopolar (a), secondary (b) and main (c)

without homopolar component for which the projections on the α1-β1 plane have the highest norm. These vectors are the ones which excite the less the homopolar and the secondary machines. These voltage vectors as well as the zero-value vector are used to control the phase currents. Their projections on the α1-β1 plane and α2-β2 plane are shown on fig. 10. As for the 3-phase PMSM, the current ripple rate of the 5phase sinusoidal PMSM is considerably reduced thanks to the global control of the 5 H-bridge VSIs (fig. 12). The homopolar component of the current vector has effectively a zero-value (sinusoidal PMSM) and the current rms value of the secondary machine is considerably reduced thanks to the use of zerovalue voltage vector (fig. 13). In this case, for each sampling period, the reference voltage vector given by current controller is used to determine the duty cycle of 5 appropriate vectors.

Ii(t)

Fig. 12: Current waveform of a 5-phase PMSM obtained with the global current control of H-bridge VSIs.

[4] J. Huang, "Application of the transformation for a p-pair pole n-phase system to the analysis of 2*3 phase induction motors", International Conference on Electrical Machines (ICEM'94), Paris (France), 1994, pp.591595, vol.2 [5] J. Holtz, "Pulsewidth Modulation – A Survey", IEEE Transaction on Industrial Electronics, vol. 39, N° 5, December 1992, pp. 410-420 [6] E. Semail, Outils et méthodologie d’étude des systèmes électriques polyphasés. Généralisation de la méthode des vecteurs d’espace, Thèse de doctorat, Université des Sciences et Technologies de Lille (USTL), juin 2000.

Iα2(t) Iβ2(t)

VII. BIOGRAPHIES

Fig. 13: Current components of the secondary fictitious machine. Iα1(t) Iβ1(t)

Jean-Philippe Martin is graduate from the University of Nancy, France. He is Ph.D. student at GREENINPL (National Polytechnic Institute of Lorraine). His research interests include electrical machine controls and multi-machine multi-converter systems.

Eric Semail is graduated from the Ecole Normale Supérieure, Cachan, France. He received the teaching degree "Agrégation" in 1986. From 1987 to 2001, he has been professor (holder of agrégation) in University of Lille (USTL). He received Ph.D. degree in 2000 and became associate professor at ENSAM Lille in 2001. In L2EP (Laboratory of Electrical Engineering of Lille) his fields of interest include modeling, control and design of polyphase systems (converters and AC Drives)

Fig. 14: Current components of the main fictitious machine. V. CONCLUSION Thanks to the use of generalized Concordia’s transformation and the diagonalisation of the machine model, it is shown that a q-phase non-salient PMSM supplied by q H-bridge VSI is equivalent to (q+1)/2 fictitious machines without magnetic coupling. Only one of these machines called main 2-phase machine generates the electromagnetic torque for a PMSM with sinusoidal emf waveform. Supplying the others leads only to supplementary losses and to high amplitude ripples in the phase current waveform due to their weak inductance . A global current control of the q H-bridge VSIs based on the application of voltage vectors exciting weakly the homopolar and secondary fictitious machine was proposed. The application of this current control method allows a considerable reduction of the ripple rate of the phase current in the case of 3-phase and 5-phase sinusoidal PMSM supplied by H-bridge VSIs. VI. REFERENCES [1]J-P. Martin, F. Meibody-Tabar, B.Davat, "Multiple-phase permanent magnet synchronous machine supplied by VSIs working under fault conditions", IEEE-IAS annual meeting, Roma, october 2000, CD-ROM [2] A. Bouscayrol, B. Davat, B. de Fornel, B. François, J. P. Hautier, F. Meibody-Tabar, M. Pietrzak-David, "Multi-machine multi-converter systems: applications to electromechanical drives", EPJ Applied Physics, Vol. 10, no. 2, May 2000, pp. 131-147. [3] Semail, E., Rombaut C., " New tools for studying voltage-source inverters", IEEE Power Engineering Review , Volume: 22 Issue: 3 , Mar 2002 p47-48

Serge Pierfederici received the Ph.D. degree from INP Lorraine, France, in 1998. Since 1999, he has been engaged as assistant Professeur at INPL.he work on the area of automatic control applied to power electronic device.

Alain Bouscayrol received the Ph.D. degree from INP Toulouse, France, in 1995. Since 1996, he has been engaged as assistant Professor at University of Lille (USTL), France. In L2EP (Laboratory of Electrical Engineering of Lille), his research interests include electrical machine controls and multi-machine systems. Since 1998, he has managed the Multimachine Multi-converter Systems project of GdRSDSE, a national research program of the French CNRS. Farid Meibody-Tabar received the Engineer degree at ENSEM, Nancy, France, in 1982, the Ph.D. degree in 1986 and the "Habilitation à diriger des recherches" degree in 2000 from the Institut National Polytechnique de Lorraine (INPL, Nancy, France). Since 2000 he has been engaged as Professor at INPL. His research activities in GREEN, UMR/CNRS, deal with architectures and control of electrical machines supplied by static converters. Bernard Davat received the Engineer degree at ENSEEIHT, Toulouse, France, in 1975, the Ph.D. degree in 1979 and the "Docteur d'Etat" degree in 1984, both from the Institut National Polytechnique (INP) de Toulouse. During 1980 - 1988, he was been Researcher at CNRS (Centre National de la Recherche Scientifique) at LEEI (Laboratoire d'Electrotechnique et d'Electronique Industrielle de Toulouse). Since 1988 he has been engaged as Professor at INP de Lorraine. His main research interests deal with architectures of static converters and interactions with electrical machines and new electrical devices (fuel cell and supercapacitors).