Dynamic Modeling and Control of Three Phase Pulse Width ...

2004 35th Annual IEEE Power Electronics Specialists Conference

Aachen, Germany, 2004

Dynamic Modeling and Control of Three Phase Pulse Width Modulated Power Converters Using Phasors Giri Venkataramanan Bingsen Wang Department of Electrical and Computer Engineering University of Wisconsin – Madison 1415 Engineering Drive Madison, WI 53706 Email: [email protected]

Abstract—Although the application of power electronic converters in ac power systems has been mostly limited to unidirectional loads like motor drives, various evolving applications such as power quality conditioners and distributed generation systems feature complex dynamic interactions affecting the operation of the ac power system. The focus of this paper is to present systematic technologies for modeling switching power converters in conjunction with their controls to determine their dynamic properties and assess their performance in an ac power network. The paper presents a dynamic phasor-oriented modeling technique that is readily compatible with classical power system analysis techniques. A state space model that represents the dynamic properties of the system in the magnitude-angle form is developed. The model can be used for obtaining steady state small signal dynamic properties at various operating conditions, and hence be used for design of appropriate regulators. Application of the technique is illustrated using a current source inverter example.

I.

INTRODUCTION

The application of three phase pulse width modulated power converters is rapidly growing beyond adjustable speed ac motor drives to include distributed generation systems, power quality conditioners, etc. The design of closed loop regulators and controllers for the power converters in such applications have heavily drawn upon the techniques used for ac motor control and dc-dc converters.

basis of recent advances in quasi-stationary phasor dynamic modeling techniques developed for power system analysis [7-10]. The specific focus of the paper is to present systematic techniques for modeling three phase ac power converters in conjunction with their controls to determine their dynamic properties and assess their performance. An example application of the technique for controlling a current source inverter feeding a three-phase ac load is presented in the paper. II.

As an illustrative case for demonstrating the behavior of power converter dynamics in an ac system, a simple example consisting of a first order system at the ac port and a static stiff dc source is being considered. Fig. 1 illustrates the schematic of the power circuit of a three phase current source inverter feeding a balanced R-C load. A nominal application being considered here develops an ac voltage regulator for the inverter system as illustrated in Fig 2. The output voltage (Vac) is measured and compared against a reference value (V*ac) to generate an error. The error drives a voltage regulator Gv(s) that modifies the modulation level of the inverter switches appropriately to regulate the output voltage. In order to design the regulator Gv(s), it is desirable to obtain the transfer function between the modulation input and the output voltage.

The use of ac motor control techniques involves the application of the synchronously rotating D-Q vector coordinates to model the dynamic phenomena of the three phase quantities under time varying excitation [1-3]. On the other hand, the use of dc-dc converter control techniques involves approximating the time varying excitation to be ‘slow’ enough such that they may be considered stationary, while ensuring that the resulting controllers have wide enough bandwidth to faithfully follow the time variations [46]. While both of these approaches provide reasonable solutions to the control problem, they are not readily compatible with the steady state modeling and control techniques that are used to study ac power system dynamics, namely phasors. The paper presents a phasor-oriented modeling technique that is readily compatible with classical power system analysis techniques. It is developed on the

0-7803-8399-0/04/$20.00 ©2004 IEEE.

CASE STUDY EXAMPLE

Idc

Vac R C

Fig. 1. Power circuit schematic of current source inverter feeding a current source inverter with a balanced RC load

The most straightforward technique for approaching this control problem is to develop a scalar transfer function between the control input (modulation) and the output voltage assuming both quantities to be stationary and use it

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2004 35th Annual IEEE Power Electronics Specialists Conference

to design a regulator with adequate controller bandwidth wide enough to faithfully track large signal sinusoidal inputs [4]. Such an approach also assumes that the three phase quantities are decoupled from each other, i.e. varying the modulation level in a particular phase affects the output voltage only in that phase.

Aachen, Germany, 2004

In order to develop these phasor magnitude regulators, it is necessary to develop the transfer function between the magnitude of the modulation phasor input and the magnitude of the ac voltage phasor output, which is discussed further in the following section. 80

-

+

Gain (dB)

V*ac

60

40

Gv(s)

20

1

10

1

10

3

100

1 .10

100

1 .10

Idc 0

Phase (degree)

30

Current+ source MVac inverter -

Vac

R

90

C

M Idc

60

120

3

Frequency (Hz)

Fig. 3. Bode plots of control to output transfer function of current source inverter in the large signal instantaneous domain Fig. 2. Block diagram of the proposed controller to regulate the output voltage of the CSI

200 *

The frequency response of the control to output transfer function (per phase) of the system is illustrated in Fig. 3. The first order plant model may be represented as

a

50

(1),

0

a

v AC ( s ) 1 = I dc R m( s ) 1+ s

100

V * & V (A)

G pL ( s ) =

Va Va

150

ω pL

−50 −100

where ωpL = 1/RC. The first order plant may be controlled using a PI regulator with an appropriately placed zero to cancel the plant pole. The start-up and steady state time domain response of the system with such a controller is illustrated in Fig. 4. As may be observed from Fig. 4, the response features a steady state error in magnitude and phase angle. One of the classical approaches to overcome this performance error is the design of controller in the synchronously rotating reference frame [3]. Alternately, the control problem may be formulated in the phasor domain, wherein the magnitude of the output voltage phasor (VACa) is used as the feedback variable, compared against a corresponding command value to generate a phasor magnitude error. The controller Gv(s) acting upon the error generates a reference modulation phasor magnitude (Ma). The modulation phasor magnitude is then modulated by three phase unit sinusoidal waveforms at the desired power frequency ωe, to generate the switching signals for the inverter. This approach closely follows the point of load regulation of the magnitude of voltage, common in ac power generation, transmission and distribution systems.

−150 −200

0

1

2 3 cycle of 60 Hz

4

5

Fig. 4. Computer simulation results of start-up response of the current source inverter with feedback control implemented in the instantaneous large signal domain

III.

PHASOR DYNAMIC MODELS FOR CIRCUIT ELEMENTS

Let the voltage (vG) across a generic two terminal device with a current (iG) flowing through it be defined as

vG = VGa cos VGθ iG = I Ga cos I Gθ

(2),

following the sign convention shown in Fig. 5. In general, the quantities VGa, VGq, IGa, IGq may be functions of time, thus accommodating complex waveforms, while corresponding to sinusoidal waveforms with arbitrary amplitude and phase as a special case. The superscripts refer

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2004 35th Annual IEEE Power Electronics Specialists Conference

magnitude and phase angle respectively. When the magnitudes are constant and phases are linearly increasing with time, the waveforms are sinusoidal. iL

iR

+

+

vL

-

vR

iC

-

iG

+

+

vC

vG

-

-

The instantaneous power absorbed by the generic device may be determined to be the product of voltage and current as VGa I Ga [cos(VGθ − I Gθ )(1 + cos 2 I Gθ ) − sin(VGθ − I Gθ ) sin 2 I Gθ ] 2 I aV a = G G [cos( I Gθ − VGθ )(1 + cos 2VGθ ) − sin( I Gθ − VGθ ) sin 2VGθ ] 2

(3)

If the generic device is chosen to be a capacitor (C), the energy stored in the capacitor may be expressed by

eC =

1 1 2 CvC = C VCa cosVCθ 2 2

[

]

2

(4).

The instantaneous power absorbed by the capacitor may be obtained by determining the time derivative of (4) as,

 deC CVCa  dVCa dV θ = pC = (1 + cos2VCθ ) − VCa C sin2VCθ  (5)  dt 2  dt dt  Comparing (3) and (5) and substituting the generic component’s subscript (G) with that of a capacitor (C), we obtain, dVCa 1 a = I C cos(I Cθ − VCθ ) dt C dVCθ 1 a I C sin(I Cθ − VCθ ) = dt CVCa

VRa = RI Ra ;

VRθ = RI θR

(8)

I CA = ωCVCA ;

I CΘ − VCΘ = π / 2

(9)

VLA = ωLI LA ;

VLΘ − I LΘ = π / 2

(10)

VRA = RI RA ;

VRΘ − I RΘ = 0

(11)

The uppercase superscripts refer is used to designate steady state quantities. It may be noted that (9-11) are the familiar classical steady state phasor solutions for these elements. In essence, they represent the polar coordinate equivalent of the more common D-Q model in the synchronously rotating reference frame. In general, while developing classical sinusoidal steady state phasor solutions for ac circuits, the sinusoidal voltages and currents are considered to be the projections of rotating vectors of the constant magnitude and constant angular velocity, and the steady state conditions expressed by (9-11) are developed directly, without formulating the dynamic phasor forms (6–8) Therefore, relationships (6-8) may be considered to be extensions of the classical phasor solution (9-11) to dynamic operating conditions, valid also when the magnitude and phase functions of the voltage and current have not yet reached their steady state. Furthermore, in the case of balanced three phase systems, translations of the solution by ±2π/3 in phase readily provide the transient solution for all the three phases simultaneously. IV.

(6)

The pair of equalities in (6) represents the magnitude and phase dynamics of the capacitor voltage related to the magnitude and phase of the capacitor current. They may be construed as generalized phasor dynamic model for time varying excitation of a capacitor. In a similar manner, generalized phasor dynamic model for the inductor may be derived as

dI La 1 a = VL cos(VLθ − I θL ) dt L θ dI L 1 = a VLa sin(VLθ − I θL ) dt LI L

Similarly, an algebraic relationship between the voltage and current for a resistor in the phasor form may be expressed as

For a steady state sinusoidal excitation, the solutions to (6-7) may be readily deduced by setting the time derivative amplitude terms to zero and those of phase by the angular frequency of excitation ω. Thus, we may obtain

Fig. 5. Sign convention of voltages and currents across various two terminal devices.

pG =

Aachen, Germany, 2004

CURRENT SOURCE INVERTER MODEL

The ac dynamic models for the circuit elements in the phasor domain can be coupled with switching power converter models using averaged switching function models. An ideal switch equivalent circuit schematic of the switching circuit of the current source inverter is illustrated in Fig. 6. The inverter consists of two single-pole-triple-throw (SP3T) switches with a stiff dc current connected to the pole terminals and three phase stiff voltages connected across the throws. The switching functions that determine the converter behavior may be defined as 1 H ij (t) =  0

if t ij is closed otherwise

(7)

for j = 1, 2 and i = 1, 2, 3

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(12).

2004 35th Annual IEEE Power Electronics Specialists Conference

Aachen, Germany, 2004

Idc t11

Idc

t31 t21

+ MV ac -

t22 t12

V ac

t32

R M Idc

C

Fig. 6. Ideal switch equivalent circuit of the current source inverter

Furthermore, if the switch throws are operating at repetitive switching frequency considerably higher than the pulse width modulation frequency, the ith phase output current Ii(t) can be expressed as Idc mi(t), where t

mi (t ) =

1 [H i1 (τ ) − H i 2 (τ )]dτ , for i =1, 2, 3 T t −∫T

(13)

Through appropriate pulse width modulation strategy, mi(t) are chosen such that the currents injected into the ac network form a balanced three phase set. In this case, the average output currents and the average dc voltage may be expressed as   θ  cos(ωt + M )   2π  I o = I dc M a  cos(ωt + M θ − ) 3   cos(ωt + M θ + 2π )  3   3 a a θ Vdc = M Vac cos(M θ − I ac ) 2

Fig. 7 illustrates the equivalent circuit of the three phase switching inverter using modulation-dependent voltage and current sources to represent the power transfer mechanism between the ac and dc ports. The dynamic dependence between the ac voltages, dc current and the modulation quantities in the phasor domain may be expressed as follows: (15)

θ dVAC 1 θ = I DC M a sin(Mθ − VAC ) − ωe a dt C VAC

[

d v  dt v

A AC Θ AC

 1 −   RC =   ω  − Ae  V AC

G ps ( s) =

OVERALL SYSTEM MODEL

a dVAC Va  1 θ = I DC M a cos(Mθ − VAC ) − AC  dt C R 

(16)

Once the steady state operating point has been determined, a linearized small-signal state space model for (15) can be determined at the steady state operating point given by (16) as

(14),

where M and M are the modulation amplitude and modulation angle, and Vaca and Vacq are the ac voltage amplitude and phase angle, respectively. V.

A VAC Θ = I DC M A cos(M Θ − VAC ) R A Θ ω e CVAC = I DC M A sin(M ΘAC − VAC )



A ω eV AC 

v 1  v  − RC 

A AC Θ AC

A  V AC   A  M RC  A  +  ω [m ]     MA   

(17)

Using the small signal state space model (17), the small signal transfer function from the input mA to the output vACA may be readily determined as

q

a

Fig. 7. Phasor equivalent circuit model of the CSI feeding an RC load

]

where ωe is the ac excitation frequency. The steady state relationships among quantities in (15) may be determined by setting the time derivatives to be zero (with the uppercase superscripts denoting steady state operating points)

A v AC ( s) = ( sI − A) −1 B A 1 m ( s)

(18),

where the state space matrices are defined from (17) by A  V AC   1 A  − V ω  A  e AC   RC  M RC  . A= ω and B =  1  ω  − Ae −   MA  RC V  AC    The transfer function GpS(s) may be evaluated as

 sω pL  1 + (ω pL + jω e )(ω pL − jω e ) (19) G pS ( s) = G pL ( jω e )    s s 1 + ω pL + jω e  1 + ω pL − jω e     The discrepancies between the transfer function of instantaneous large signal quantities in (1) and the transfer function of the small signal phasor quantities in (19) may be observed as follows: a. The ‘dc gain’ of GpL(s) has been replaced by |GpL(jωe)|. b. The real pole at ωpL has been separated into two conjugate complex poles with imaginary parts given by jωe.

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2004 35th Annual IEEE Power Electronics Specialists Conference

Aachen, Germany, 2004

c. A real zero has been introduced, which is given by the product of the newly introduced complex conjugate poles divided by the original real pole. It is further observed that if the characteristic dynamic frequencies of the network ωpL are considerably higher than the excitation frequency ωe, the transfer functions of the instantaneous large signal model and the small signal phasor model are indistinguishable. In other words, when the imaginary part of the separated pair of complex conjugate poles is much smaller than the real part, one of them essentially appears to be ‘canceled’ by the newly introduced real zero. However, as the characteristic dynamic frequencies of the network are comparable or slower than the ac excitation frequency, the instantaneous large signal dynamic model and the small signal phasor dynamic model are considerably different from each other. Bode plots of the frequency response of the control to output small signal transfer function, GpS(s) for a nominal value R is shown in Fig. 8 (solid red curves). The large signal transfer function of the system GpL(s) is also shown in the figure (dashed blue curves) in order to highlight the variation due to ac excitation. In order to verify the rather preplexing small signal dynamic phasor transfer function as predicted by the analytical model described in (19), a laboratory scale experimental system was built. The small signal transfer function was measured using a frequency response analyzer and the results from the operating condition corresponding to R= 10 kW ; C= 1 mF; and ωe = 257.6 rad/s, are illustrated in Fig. 9. The excellent agreement between the predicted model and the experimental measurement is readily evident from the figure. 80

Fig. 9. Experimental measurement of small signal control (current phasor magnitude) to output (voltage phasor magnitude) transfer function obtained using frequency response analyzer.

Furthermore, in order to examine the effect of the perturbations in the dynamic behavior, the loop gain a closed loop system with a simple PI regulator operating on phasor quantities were developed. The loop gain plots ate nominal (red solid curve), heavy (dashed blue) and light (dash-dot brown) load conditions are shown in Fig. 10. 80 60

40

Gain (dB)

Gain (dB)

60

20

0

1

10

100

40 20 0

3

1 .10

20 0

1 .10

3

1 .10

10

100

1 .10

1

10

100 Frequency (Hz)

1 .10

4

0

30

30 60 Phase (degree)

Phase (degree)

3

1

90 120 150

60 90 120 150

1

10

100

3

1 .10

180

Frequency (Hz)

Small signal Large signal

210

Fig. 8. Bode plot of small signal control (modulation phasor magnitude) to output (voltage phasor magnitude) transfer function of the current source inverter (for C=1 mF, R= 10 kW)

4

Nominal Load Heavy load Light load

Fig. 10. Bode plot of small signal loop gain transfer function of the current source inverter with a PI controller at various loading conditions

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2004 35th Annual IEEE Power Electronics Specialists Conference

It may be observed from the loop gain plots, that the closed loop system would be unstable at light load conditions because of a negative value for phase margin. The results from the time domain computer simulation of the system are illustrated in Fig. 11. The top plots show the output voltage waveforms, and the bottom plots show the averaged ac current waveforms. The step change in load from nominal value to light load is applied at the end of three cycles. The instability in the output upon the step change to a light load is clear from the figure, as predicted by the small signal phasor dynamic transfer function. 150 va vb vc

100

Va,b,c (V)

50 0

Aachen, Germany, 2004

‘system identifier’ operating in real-time to determine the load level using measured quantities. On the other hand, an acceptable controller may be designed based only on the knowledge of range of load levels. If the equivalent load resistance at the heaviest and the lightest anticipated load are known, the controller design may be based on a nominal load level that may be arithmetic or a geometric average of the upper and lower bounds on the load, with adequate margins. Fig. 12 illustrates the frequency response of loop gain of a regulator designed to provide perfect cancellation at the nominal load level (solid red curves). The variation of the curves at heavy load (dashed blue curve) and light load (dotdashed brown curve) are also shown.

−50

40

−100

20 0

1

2

3

4

5

6

7

8

Gain (dB)

−150

8 i a ib ic

6

Ia,b,c (A)

4

0

20

2

40

0 −2

3

1 .10

3

1 .10

1

10

100

1 .10

1

10

100 Frequency (Hz)

1 .10

4

90

−4

45 0

1

2

3

4 t (cycle of 60 Hz)

5

6

7

8

Phase (degree)

−6

Fig. 11. Time domain response of system with a simple PI controller operating in the phasor domain

It has thus clear that the large signal transfer function of the system cannot be used as-is for designing closed loop regulators operating on phasor domain quantities, particularly when the system has dynamic behavior that is ‘slower’ then the ac excitation frequency of the phasors. Having definitively established the validity of the control to output transfer function in the phasor domain through laboratory experiments as well as computer simulations, a suitable controller that accounts for the discrepancies in the phasor domain transfer function may be developed. A candidate control transfer function Gv(s) may be chosen to be of the structure 

G v ( s) =

1+ ω b 

s



ωˆ pL + jω e  1 +

s



ωˆ pL − jω e  (20)

s  sωˆ pL  1 + (ωˆ pL + jω e )(ωˆ pL − jω e ) 

This structure provides pole-zero cancellation for the plant, while featuring a integral loop gain transfer function, when the plant poles and zeros of the small signal phasor domain transfer functions are cancelled exactly by the controller zeros and poles respectively. This approach requires prior knowledge of the location of real pole of the system ωpL, indicated by ‘^’ in the controller transfer function. Such a regulator will have to adapt to changing load conditions, since ωpL depends on the load. It will consist of a

0 45 90 135 180

4

Nominal Load Heavy load Light load

Fig. 12. Bode plot of small signal loop gain transfer function of the current source inverter with the proposed controller at various loading conditions

The operation of the using the proposed controller was verified using computer simulations of the complete system over the entire range of load and transient conditions. Although the design of the regulator is based on the small signal phasor transfer functions, the response of the system to large signal disturbances was found to be satisfactory in the simulations. Fig. 13 provides selected time domain waveforms from the simulations. Fig. 13 (a) shows the traces of three phase load current waveforms and Fig. 13 (b) shows the traces of three phase output voltage waveforms. The set of waveforms on the top illustrates the system response during start up. The set of waveforms in the middle illustrates the system response during a load step change from nominal load to light load. The set of waveforms on the bottom illustrates the system response during a load step change from light load to heavy load. All the traces indicate a fast transient response of the output voltage, while maintaining adequate stability, when compared to the case illustrated in Fig. 11.

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2004 35th Annual IEEE Power Electronics Specialists Conference

loads.

2 i a i b i

abc

I , , (A)

1

c

0 −1 −2

0

1

2

3

4

5

6

abc

I , , (A)

2 i a ib i

1

c

0 −1 −2

0

1

2

3

4

5

6

abc

I , , (A)

20 ia i b i

10

c

0 −10 −20

0

1

2

3 t (cycle of 60 Hz)

4

5

6

(a)

Va,b,c (V)

va vb vc

0

The behavior of the system dynamics in the small signal phasor domain have been compared and contrasted with the large signal instantaneous domain. The small signal phasor domain transfer function has been analytically predicted and subsequently verified using frequency response tests on a laboratory prototype. The pitfalls of using the large signal instantaneous domain transfer functions in the phasor domain has been illustrating using frequency response transfer functions at varying load conditions, and verified using computer simulations. A controller design approach based on pole-zero cancellation at nominal operating conditions has been proposed. Stable operation of the proposed controller has verified using computer simulations over a range of operating conditions. The modeling approach can be conveniently used for controller design and system stability analysis while interfacing power converters with ac power networks.

200 100

Aachen, Germany, 2004

−100 −200

ACKNOWLEDGMENT 0

1

2

3

4

5

6

Va,b,c (V)

200 v a vb vc

100 0 −100 −200

0

1

2

3

4

5

6

Va,b,c (V)

200 v a v b vc

100 0

The authors would like to acknowledge support from Wisconsin Electric Machine and Power Electronics Consortium (WEMPEC) at the University of WisconsinMadison. The authors also thank Paul Van Opens for valuable assistance in developing the laboratory scale hardware for the small signal model frequency response tests. REFERENCES

−100 −200

0

1

2

3 t (cycle of 60 Hz)

4

5

6

(b) Fig. 13. Time domain response of system of the system with the proposed regulator: (a) load current response; (b) load voltage response (from top to bottom: startup, switching to light load, and switching to heavy load)

VI.

CONCLUSIONS

The application of dynamic phasor-oriented modeling technique that is readily compatible with classical power system analysis techniques in the design of controllers for pulse width modulated power converters has been presented in this paper. The technique represents a refinement of the classical scalar approach to controller design for three phase ac power converters. Furthermore, it is an extension of the D-Q synchronously rotating reference frame controller approach to the phasor modeling approach common in practice of ac power systems. The analytical modeling approach has been presented for a case study current source converter, whose results have been verified using experimental and computer simulations. The paper has presented a detailed development for the dynamic phasor model for the current source converters with an RC load. The modeling approach can be systematically extended to study voltage source inverters and to various other classes of

[1]

T.A. Lipo and P.C. Krause, “Stability analysis of a rectifier-inverter induction motor drive,” IEEE Trans. On Power Apparatus and Systems, Vol. PAS-88, No. 1, pp. 55-66, 1965. [2] D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of AC Drives, Oxford, 1996. [3] T. M. Rowan and R. J. Kerkman, “A new synchronous current regulator and an analysis of current-regulated PWM inverters,” IEEE Tran. Ind. App., vol. IA-22, pp. 678-690, Jul./Aug. 1986. [4] R. W. Erickson, S. Cuk, and R. D. Middlebrook, “Large-signal modeling and analysis of switching regulators,” Proc. Power Electron. Specialist Conf., 1982. [5] Mazumder, S.; Alfayyoummi, M.; Nayfeh, A.H.; Borojevic, D, “A theoretical and experimental investigation of the nonlinear dynamics of DC-DC converters,” Power Electronics Specialists Conference, 2000, pp. 729 -734. [6] Venable, H. Dean, and Stephen Foster, “Practical Techniques for Analyzing, Measuring, and Stabilizing Feedback Control Loops in Switching Regulators and Converters,” Proc. Seventh National Power Conversion Conf., PowerCon'7, pp. I2-1-17, 1980 [7] C.L. DeMarco and G.C Verghese, “Bringing phasor dynamics into the power system load flow,” Proc. North America Power Symposium, Oct. 1993, pp. 463-471. [8] V. Venkatasubramanian, H. Schattler, and J. Zaborszky, “Fast time varying phasor analysis in the balanced large electric power system,” IEEE Tran. Automatic Control, vol. AC-40, pp. 1992-2003, 1995. [9] P. W. Sauer, B. C. Lesieutre, and M. A. Pai, “Transient algebraic circuits for power system dynamic modeling,” Int. J. Electric Power Energy Sys. [10] Stankovic, A.M.; Sanders, S.R.; Aydin, T, “Dynamic phasors in modeling and analysis of unbalanced polyphase AC machines,” IEEE Tran. Energy Conversion, vol. 17. No. 1, pp. 107-113, 2002.

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