Field Oriented Control of a Dual Star Induction Machine ... - CiteSeerX

Field Oriented Control of a Dual Star Induction Machine Using Fuzzy Regulators E. Merabet, R. Abdessemed, H. Amimeur and F. Hamoudi LEB–Research Laboratory, Department of Electrical Engineering, University of Batna, Rue Chahid Med El Hadi Boukhlouf, Batna, Algeria. [email protected],

Abstract–In this paper, a space vector control based on a fuzzy regulator applied to the double star induction machine is presented. A DSIM model in the (d-q) frame based on the Park transformation is developed. First, space vector control uses PI regulators for machine intern variables decoupling. After we have replaced PI regulators by fuzzy regulators type PI. Simulation results are finally presented and discussed. Index Terms–Dual star induction machine, field oriented control (FOC), fuzzy regulator.

I.

INTRODUCTION

Since the last of 1920s years, AC double star machines known as six phases machines have been used in many applications (such as pumps, fans, compressors, rolling mills, cement mills, mine hoists [14]) for there advantages in power segmentation, precision, and electromagnetic torque pulsation minimization [1], [7], [15]. Double star machine supplied with non sinusoidal waveforms causes perturbations in the torque (harmonics in torque). Today, with semi conductors development, voltage sources inverters can operates with high commutation frequency (PWM-VSI) and can gives sinusoidal waveforms, add at this multilevel structures which reduce harmonics in output voltages [2]. Nowadays the researchers concentrate their efforts to develop the techniques of control. Several techniques are developed as scalar control or field oriented control. The main difficulty in the asynchronous machine control resides in the fact that complex coupling exists between machine input variables, output variables and machine intern variables as the field, torque or speed. The space vector control assure decoupling between these variables, and the torque is made similar as the one of a DC machine [3], [4]. The origins of the space vector control, contrary to received ideas carry up at the end of the last century to A. Blondel works on the reaction of the two axes theory. In this paper a space vector control is applied to the machine. This technique is based on field orientation, that it is the stator field, the rotating field or by rotor field orientation. The last one will be applied to the DSIM machine using PI regulators and fuzzy regulators type PI. Fuzzy regulators have been successfully used for a few numbers of non linear and complex processes, FLC are robust and their performances are insensible to parameter variations contrary to conventional regulators. Recently several

researchers make efforts to improve the robustness and performances of the FLC by using neuron and genetic algorithms [6], [12]. II. MACHINE MODEL A schematic of the stator and rotor windings for a machine dual three phase is given in Fig. 1. The six stator phases are divided into two wye-connected three phase sets labelled As1 Bs1 Cs1 and As2 Bs2 Cs2 whose magnetic axes are displaced by an arbitrary angle . The windings of each three phase set are uniformly distributed and have axes that are displaced 120° apart. The three phase rotor windings Ar, Br, Cr are also sinusoidally distributed and have axes that are displaced apart by 120° [1], [5]. The following assumptions are made: – Motor windings are sinusoidally distributed; – The two stars have same parameters; – Flux path is linear. The voltage equation is [2]: d [Vabc, s1 ] = [ Rs1 ][ I abc , s1 ] + ϕ abc , s1 dt d (1) [Vabc , s 2 ] = [ Rs 2 ][ I abc , s 2 ] + ϕ abc, s 2 dt d [Vabc, r ] = [ Rr ][ I abc, r ] + ϕ abc, r dt  [ϕ abc , s1 ]  [ Ls1, s1 ] [ Ls1, s 2 ] [ Ls1, r ]   [ I abc , s1 ]       (2) [ϕ abc , s 2 ] = [ Ls 2, s1 ] [ Ls 2, s 2 ] [ Ls 2, r ] [ I abc , s 2 ]  [ϕ abc , r ]   [ Lr , s1 ] [ Lr , s 2 ] [ Lr ,r ]   [ I abc ,r ]       B s1

Ar

B s2

R o to r θ

Br

As2 S ta to r N °2

α

A s1 S ta to r N °1

C

s1

C

s2

Fig. 1. Windings of the DSIM.

C

r

with [Vabc , s1 ] = [vas1 vbs1 vcs1 ]T ; [Vabc , s 2 ] = [vas 2 vbs 2 vcs 2 ]T ;

III. VOLTAGE SOURCE INVERTER MODELLING

[ I abc ,s1 ] = [ias1 ibs1 ics1 ]T ; [ I abc ,s 2 ] = [ias 2 ibs 2 ics 2 ]T ; [Vr ] = [var vbr vcr ]T ; [ I r ] = [iar ibr icr ]T ; [ Rs1 ] = [ Rs 2 ] = Diag [ Rs ] (3×3) ; [ Rr ] = Diag[ Rr ] (3×3) .

The detail of the submatrixes is given at the Appendix. where Rs1=Rs2, Ls1=Ls2 and Lms are the stator resistance, leakage inductance and magnetizing inductance; Rr, L r and Lmr the rotor resistance, leakage inductance and magnetizing inductance; Msr Maximal mutual inductance between stator and rotor. The electromagnetic torque can be expressed [7]: δ δ   Tem = [ I abc , s1 ] [ Ls1, r ][ I abc , r ] + [ I abc, s 2 ] [ Ls 2, r ][ I abc, r ] (3) δθ δθ   The Park model of DSIM is presented below in the references frame at the rotating field (d, q) [5], [16], [17]: vds1 = Rs1ids1 + pϕds1 − ωsϕ qs1 vqs1 = Rs1iqs1 + pϕqs1 + ωsϕds1 vds 2 = Rs 2 ids 2 + pϕ ds 2 − ωsϕqs 2 vqs 2 = Rs 2 iqs 2 + pϕ qs 2 + ωsϕ ds 2

(4)

vdr = Rr idr + pϕdr − (ωs − ωr )ϕqr vqr = Rr iqr + pϕqr + (ωs − ωr )ϕ dr

The three phase voltage source inverter is shown in Fig. 2. Semi conductors can be IGBT, or thyristors for high powers. The operating principle can be expressed by the imposed sequencings at semiconductors which realize modulation of voltages applied to stator windings. Voltages at load neutral point can be given by the following expression [8]:  va   2 −1 −1  f1   v  = E  −1 2 −1  f   b 3   2  vc   −1 −1 2   f 3 

(8)

This modelling for the two converters that feed the DSIM. IV. FIELD O RIENTED CONTROL The objective of space vector control is to assimilate the operating mode of the asynchronous machine at the one of a DC machine with separated excitation, by decoupling the torque and the flux control. With this new technique of control and microprocessor development we can obtain speed and torque control performances comparable at those of DC machine [9]. The field oriented control principle is schematized by the Fig. 3.

The expressions for stator and rotor flux are: ϕ ds1 = Ls1ids1 + Lm (ids1 + ids 2 + idr ) ϕ qs1 = Ls1iqs1 + Lm (iqs1 + iqs 2 + iqr ) ϕ ds 2 = Ls 2 ids 2 + Lm (ids1 + ids 2 + idr ) ϕ qs 2 = Ls 2 iqs 2 + Lm (iqs1 + iqs 2 + iqr )

(5)

ϕ dr = Lr idr + Lm (ids1 + ids 2 + idr ) ϕ qr = Lr iqr + Lm (iqs1 + iqs 2 + iqr )

with p = d/dt; 3Lm/2 =Lms=Lmr=L sr. In the induction machines, rotor windings are short circuited hence, i.e. vdr= 0 and vqr= 0. Fig. 2. VSI scheme.

A. Mechanical Equation

d

The mechanical equation is given by [5], [7]: dΩ J = Tem − Tr − K f Ω dt with Lm ϕ dr (iqs1 + iqs 2 ) − ϕ qr (ids1 + ids 2 )  Tem = p Lm + Lr 

q

(6)

(7)

ϕ d r = ϕ r*

Fig. 3. Field oriented control principle.

By applying this principle (ϕ dr = ϕ r* and ϕ qr = 0) to equations (4) (5) and (7), the final expression of the electromagnetic torque is: Tem* = p

Lm ϕr* ( iqs* 1 + iqs* 2 ) Lr + Lm

(9)

ϕ r*

+

( Lr + Lm ) 2 pLm

* qs1

* ds 2

= Rs 2 ids 2 + Ls 2 pids 2 − ω ( Ls 2iqs 2 + Trϕ ω ) * s

ids1 iqs1

+

* r

* g

g g

(11)

+ vds2 PI

−

( Lr + Lm ) 2 pLm

+ +

+−

vqs 2 PI

−

+

ibs1 ics1 v*ds 2

* s

θ s*

1 S

v*qs 2

+ vqs2c

ias 2 ids 2

Ls 2

iqs 2

+

ias 1

P A R K

vqs 2 c

ω

+

Ls 2

+

ωr

P A R K (−α )

ibs 2 ics 2 Ωr

p

Fig. 4. Decoupling bloc in voltage (FOC).

V. INDIRECT METHOD S PEED REGULATION

vds1 = Rs1ids1 + Ls1

(12)

The equation system (13) shows that stator voltages (vds1, vqs1, vds2, vqs2) are directly related to stator currents (ids1, iqs1, ids2, iqs2). To compensate the error introduced at decoupling time, the voltage references (v*ds1, v*qs1, v*ds2, v* qs2) at constant flux are given by: v*ds1 = vds1 − vds1c

The principle of this method consists to don’t use rotor flux magnitude but simply its position calculated with reference sizes. This method eliminates the need to use flux captor, but only the one of the speed. The speed regulation scheme of the DSIM is given in the following Fig. 5. A. Block of Field Weakening The flux is generally maintained constant at its nominal value ϕ r* for rotor speeds lower or equal to the machine nominal speed nr. For upper speeds, the flux decreases when speed increases in order to limit voltage at machine borne. For this we define reference flux as fellow: ϕ r* = ϕrn ϕ r* =

(13)

Ω rn n ϕr Ωr

For a perfect decoupling, we add stator currents regulation loops (ids1, iqs1, ids2, iqs2) and we obtain at their output stator voltages (vds1, vqs1, vds2, vqs2). The goal of the regulation is to assure a best robustness to intern or extern perturbations. In this work, proportionalintegral (PI) and fuzzy regulators have been used. The decoupling bloc scheme in voltage (Field Oriented control FOC) is given in Fig. 4.

if

Ω r ≤ Ω rn

if

Ω r > Ω rn

(14)

E

v*qs 2 = vqs 2 + vqs 2c

Accepting that i*ds1= i*ds2 and i*qs1= i*qs2.

+

vqs1c

+

The torque expression shows that the reference fluxes and stator currents in quadrate are not perfectly independents, for this, it is necessary to decouple torque and flux control of this machine by introducing new variables.

v*ds 2 = vds 2 − vds 2c

+

PI

−

+

vqs* 2 = Rs1iqs 2 + Ls 2 piqs 2 +ωs* ( Ls 2 ids 2 + ϕr* )

v*qs1 = vqs1 + vqs1c

v*qs 1

vqs1

+

Ls 2

* r

d ids1 dt d vqs1 = Rs1iqs1 + Ls1 iqs1 dt d vds 2 = Rs 2 ids 2 + Ls 2 ids 2 dt d vqs 2 = Rs 2 iqs 2 + Ls 2 iqs 2 dt

+

(10)

= Rs1iqs1 + Ls1 piqs1 +ω ( Ls1ids1 + ϕ )

v

−

i

g g

+

* qs1

v

v*ds1

+

Ls1 * em

v*ds1 = Rs1ids1 + Ls1 pids1 − ωs* ( Ls1iqs1 + Trϕ r*ωg* ) * s

vds1 PI

−

vds1c

and Rr ω = ( iqs* 1 + iqs* 2 ) Lm + Lr

+

Lr Rr

T

* g

i*ds1

Rr + ( Lr + Lm ) S 2 Rr Lm

* ds1

v Ω*

v*qs1

+

−

Ωr

PI

-1

Park

Tem*

ϕr*

F O C

* as1 * bs1 * cs1

v v v

ias1 PWM-VSI N°1

ibs1 ics1

θs*

DSIM E

v*ds 2 v*qs 2

Park-1

Fig. 5. Indirect method speed regulation.

v*as 2 v*bs2 v*cs 2

ias2 PWM-VSI N°2

ibs 2 ics 2

1000 0

0

1

2 3 Time (sec)

4

20 0 -20 0

1

2 3 Time (sec)

4

T o rq u e (N .m )

3000 2000

C u rre n t i q s 1 (A )

C u rre n t i a s 1 (A )

S p e e d (rp m )

0 -1

0

1

2 3 Time (sec)

0 -50 0

1

2 3 Time (sec)

4

0

1

2 3 Time (sec)

4

0

1

2 3 Time (sec)

4

20 0 -20 -40

1 0

4

0

1

2 Time (s)

3

4

20 0 -20 0

1

2 Time (s)

3

4

1 0 -1

T orque (N.m )

0

C urre nt i qs 1 (A )

1000

F lux φ dr (W b)

F lux φ qr (W b)

Current ia s 1 (A )

S peed (rpm )

Fig. 6. Indirect method speed regulation with load torque Cr = 14N.m between [1.5, 2.5] s.

3000 2000

0

1

2 Time (s)

3

4

Torque (N.m)

5

Current iqs1 (A)

0 -2000 0

1

2 3 Time (sec)

4

20 0 -20 0

1

2 3 Time (sec)

4

1

Flux φ dr (Wb)

Speed (rpm)

5

2000

0 -1

0

1

2 3 Time (sec)

4

5

50 0 -50 0

1

2 3 Time (sec)

4

5

0

1

2 3 Time (sec)

4

5

0

1

2 3 Time (sec)

4

5

40 20 0 -20 -40

2 1 0

Fig. 8. DSIM Comportment with inertia variation (J = 2Jn at t = 1s).

VI. FUZZY CONTROL REGULATOR STRUCTURE 50

2

φ d r (W b )

φ q r (W b )

1

Current ias1 (A)

C. Results and Discussions The speed reaches its reference value after 0,.59s with 1.20% overtaking of the reference speed. The perturbation reject is achieved at (0,16s). The electromagnetic torque compensate the load torque and reaches at starting (50N.m) Fig. 6. Simulation results show regulation sensibility with PI for rotor resistance variation; the decoupling is affected. Inertia variation increase the inversion time of rotating sense Fig. 7 and 8.

Flux φ qr (Wb)

B. Robustness Tests The control robustness is its capability to surmount incertitude on the controlled model, we will test the regulation comportment with DSIM parameter variations, by varying rotor resistance Rr and inertia J.

50 0 -50 0

1

2 Time (s)

3

4

A

20 0 -20 -40 0

1

2 Time (s)

3

4

0

1

2 Time (s)

3

4

2 1 0

The fuzzy control with the same objective of regulation as control realized in classical automatic. However, it is possible to happen of an explicit model of the controlled process. This approach is based on two main concepts: The decomposition of a variation domain of a variable under linguistic nuances form: “small, average, big...”, and on rules provided from the expertise which express under linguistic form the control evolution of the system with the observed variables. A fuzzy regulator can be presented under different ways, but in general the adopted presentation is divided in three parts: The fuzzyfication which allows to pass from real to fuzzy, variables, the center of the regulator presented by rules relying inputs and outputs, finally, the inference and defuzzification which allow determination of the real value of the output from the fuzzy sets, Fig. 9. Input set of the process is noted U (set of the calculated actions by fuzzy regulator applied to the controlled process). The set of observed outputs S, the set of references C and the set of inputs of the fuzzy regulator X (for example: the size, the temperature, speed...). The set of gains of inputs normalization GE and the regulator output gains GS allow to adapt the normalized definition domain of different variables.

Fig. 7. DSIM Comportment with rotor resistance variation (R = 2R n at t = 1s).

Input Fuzzyfication The objective of the fuzzyfication is to transform the determinist variables of input to fuzzy variables, i.e. in linguistic variables, with definition of appurtenance functions for these different input variables. The input physical grandeurs X are reduced at normalized grandeurs x in the variation limits, in many time [ 1 1], called universe of speech, which can be discrete or continuous. Always this universe of discourses limited with applying limitation on numerical value of |x| 1, to resolve the problem of huge variations of X. The gains of normalization characterize of input factor of scale x and X. In the case of an continuous universe of discourse, the number of linguistic value (small negative, middle negative, positive...),

GE *

X

+

−

X

TABLE I VARIATION CONTROL CALCULATION

GS Control rule determination

Fuzzyfication

Defuzzyfication

Change of Error

Process

represented by the membership functions, for a variable x1 can vary (for example three, five or seven) [6], [12]. An example of continuous fuzzyfication is illustrated in Fig. 10. for a single variable of x with triangular membership function; corresponding linguistic values are characterized by the symbols likewise:

NM

NS

ZE

PS

PM

PL

NL

NL

NL

NL

NL

NM

NS

ZE

NM

NL

NL

NL

NM

NS

ZE

PS

NS

NL

NL

NM

NS

ZE

PS

PM

ZE

NL

NM

NS

ZE

PS

PM

PL

PS

NM

NS

ZE

PS

PM

PL

PL

PM

NS

ZE

PS

PM

PL

PL

PL

PL

ZE

PS

PM

PL

PL

PL

PL

The defuzzyffication consists to take a decision i.e. obtain a real control from the obtained under fuzzy ensemble form. In the case of reasoning based on the inference of fuzzy rules, several methods exist, we have used in this paper the method of the pondered average [18], [19].

B Rule Bases and Fuzzy Inference The fuzzy rules represent the center of the regulator; it allows to express under linguistic form the input variables of the regulator at control variables of the system [13]. A type of rule for example: if x1 is positive large, x2 is zero equal, then, u is positive large, where x1 and x2 represent two input variables of the regulator likewise: the gap of variable to regulate, its variation and u the control. The experience in elaboration of these rules is important. A graphic representation of the ensemble of rules called inference matrix or table of rules allow to synthesize the center of fuzzy regulator. Tabular (1) represent a table of rules for two linguistic variables of input; the speed error e and its variation e and the output variable u. NS

NL

C Defuzzyfication

NL: Negative Large. NM: Negative Medium. NS: Negative Small. ZE: Zero Equal. PS: Positive Small. PM: Positive Medium. PL: Positive Large.

NM

Error

Fig. 9. Fuzzy regulation principle scheme (FLC).

NL

ZE

PS

VII. CONTROL LAW This law is according to the error and its variation µ= f(e, e) therefore, the activation of the ensemble of rules of decision gives the variation of the control ( µ) necessary, permitting the adjustment of such a control µ. The general form of this control is [19]:

Tem* (k + 1) = Tem* ( k ) + GTem ∆Tem*

−1

−0,5

0

PL

PM

0,5

(15)

Where: GTem is the gain associated to the control T*em (k+1)

VIII.

−1,5

T *em

1

Fig. 10. Continuous fuzzyfication with seven membership functions.

1,5

S PEED REGULATION WITH FUZZY REGULATOR

In Fig. 4. we have changed the classical PI regulators by fuzzy regulators with elimination of the part of compensation voltages (vds1c, vqs1c, vds2c , vqs2c ). In this case the scheme of the decoupling (FFOC: Fuzzy Field Oriented Controller) is presented on Fig. 11. The fuzzy regulators of currents are synthesized with the same way as the speed regulator. The principle scheme of the speed regulation is given in Fig. 12.

g g

i *q s 1

g g

( L r + Lm ) 2 p Lm

+

−

v q*s 1 FLC

( L r + Lm ) 2 p Lm

+

+

ω *s

θ s*

1 s

+

v *q s 2

−

FLC

ia s 2

id s 2

P A R K

iq s 2

i bs 2

(−α ) +

−

v *d s 2

p

2 Time (sec)

3

4

20 0 -20 0

1

2 Time (sec)

3

4

2 1 0

Torque (N.m)

1

0

1

ic s 2

FLC

ωr

0

60 40 20 0 -20

0

1

2 Time (sec)

3

4

0

1

2 Time (sec)

3

4

0

1

2 Time (sec)

3

4

(A)

ics 1

0

qs1

Te*m

ib s 1

1000

Current i

i qs 1

id s 1 P A R K iq s 1

3000 2000

Flux φ (Wb) dr

ia s 1

id s 1

Speed (rpm)

v *ds 1

FLC

(A)

−

as1

+

Current i

i *ds 1

R r + ( L r + Lm ) S 2 R r Lm

Flux φ (Wb) qr

ϕ r*

2 Time (sec)

3

4

20 10 0 -10

1 0 -1

Fig. 13. Speed regulation using fuzzy regulator, with applying resistant torque (Tr=14N.m) during the interval [1,5 2,5]sec.

Ωr

Fig. 11. Decoupling block (FFOC).

E

v*ds1 * qs1

v

Park

Tem*

+

v v v

ias1 ibs1 ics1

PWM-VSI N°1

FLC

Fig. 14. Speed regulator using fuzzy regulator, with inversion of the rotation sense at t=2sec.

Flux φ

The simulation results are effectuated with the output gain of the speed fuzzy regulator GTem = 0.725. The structure of the used fuzzy regulator of current is identical to the one of the speed regulator, with the output gains of the current regulators are: Gvds1 = Gvds2 = 0,15 for the stator currents ids1 and ids2 respectively, Gvqs1 = Gvqs2 = 0,10 for stator currents iqs1 and iqs2 respectively. Figs. 13. and 14 show that, the speed reaches its reference value at 0.55s without overtaking, the electromagnetic torque compensate the resistant torque et and presents at starting a value equal to 70N.m. The Fig. 15 and 16, show respectively the behavior of the DSIM when Rr is 100% increased of its nominal value and J is increased 100% of its nominal value. The simulation results show the insensibility of the control with fuzzy (fuzzy regulator) at Rr variations, only the inertia variation increase the inversion time of the speed but without overtaking.

qr

(Wb)

IX. RESULTS AND DISCUSSIONS

Current i

as1

(A)

Fig. 12. Speed fuzzy regulation scheme using fuzzy regulators.

3000 2000 1000 0

0

1

2 Time (sec)

3

4

20 0 -20 0

1

2 Time (sec)

3

4

2 1 0

Torque (N.m)

PWM-VSI N°1

ias 2 ibs2 ics 2

0

1

2 Time (sec)

60 40 20 0 -20

0

1

2 Time (sec)

3

4

0

1

2 Time (sec)

3

4

0

1

2 Time (sec)

3

4

(A)

vas* 1 vbs* 1 vcs* 1

Speed (rpm)

Park -1 (−α )

qs1

v*ds 2 v*qs 2

Block of Field Weakening

DSIM E

Current i

ϕr*

θs*

(Wb)

Ωm

F O C

dr

−

Flux φ

Ωr*

-1

* as1 * bs1 * cs1

3

4

20 10 0 -10

1 0 -1

Fig. 15. Comportment of the DSIM with Rr variation (Rr = 2×Rnr at t=1sec).

Fig. 16. DSIM comportment with varying J (J = 2 × Jn at t=1sec).

X. CONCLUSION

REFERENCES

In this paper a model of a double star asynchronous machine is presented using Park transformation. Two control methods are compared to regulate stator currents and the speed PIregulator and fuzzy regulator. The simulation results have shown that the PI-regulators present sensitivity to the variation of the DSIM parameters. The fuzzy regulator has a very interesting dynamic performances compared with the conventional PI-regulator. It allows to have fast response without overtaking and minimize in the case of the rotation sense change. Otherwise, the robustness tests have shown that it is insensitive to rotor resistance. In fact, the fuzzy regulator synthesis is realized without take in account the machine model.

[2]

[3]

[4] [5] [6]

[7]

APPENDIX The submatrixes  Ls1 + Lms [ Ls1, s 2 ] =  − Lms / 2  − Lms / 2

[1]

[8]

− Lms / 2 Ls1 + Lms − Lms / 2

− Lms / 2  − Lms / 2  ; Ls1 + Lms 

 Ls 2 + Lms − Lms / 2 − Lms / 2  [ Ls 2, s 2 ] =  − Lms / 2 Ls 2 + Lms − Lms / 2  ;  − Lms / 2 − Lms / 2 Ls 2 + Lms   Lr + Lmr − Lmr / 2 − Lmr / 2  [ Lr , r ] =  − Lmr / 2 Lr + Lmr − Lmr / 2   − Lmr / 2 − Lmr / 2 Lr + Lmr  cos(α + 2π/3) cos(α + 4π/3)   cos(α ) [ Ls1, s 2 ] = Lms  cos(α + 4π/3) cos(α ) cos(α + 2π/3)  ;  cos(α + 2π/3) cos(α + 4π/3) cos(α )  cos(θ + 2π/3) cos(θ + 4π/3)   cos(θ )  [ Ls1, r ] = Lsr cos(θ + 4π/3) cos(θ ) cos(θ + 2π/3)  ;  cos(θ + 2π/3) cos(θ + 4π/3) cos(θ )  cos(γ + 2π/3) cos(γ + 4π/3)   cos(γ )  [ Ls1, r ] = Lsr  cos(γ + 4π/3) cos(γ ) cos(γ + 2π/3)   cos(γ + 2π/3) cos(γ + 4π/3) cos(γ ) 

with γ = θ −α; [ Ls 2, s1 ] = [ Ls1, s 2 ]T ; [ Lr , s1 ] = [ Ls1, r ]T ; [ Lr , s 2 ] = [ Ls 2, r ]T .

[9] [10]

[11] [12] [13]

[14] [15]

[16]

[17]

[18]

[19]

D. Hadiouche, H. Razik and A. Rezzoug, “On the modeling and design of dual-stator windings to minimize circulating harmonic currents for VSI fed AC machines,” IEEE Trans. Ind. Appl., vol. 40, no. 2, pp. 506– 515, March/April 2004. M. Merabtene et E. R. Dehault, “Modélisation en vue de la commande de l’ensemble covertisseur-machine multi-phases fonctionnant en régime dégradé,” Sixième conférence des jeunes chercheurs en génie électrique, JCGE’3, Saint-Nazaire, pp. 193–198, 5 et 6 Juin 2003. G.A. CAPOLINO et Y.Y. FU, “Commande des machines asynchrones par flux orienté : principe, méthode et simulation,” Journée d’études SEE, Lille, 2 Décembre 1992. E. Y. Y. HO and P. C. SEN, “Decoupling control of induction motor drives,” IEEE Trans. Ind. Elect., vol. 35, no 2, pp. 253–262, May 1988. G. K. Singh, K. Nam and S. K. Lim, “A simple indirect field-oriented control scheme for multiphase induction machine,” IEEE Trans. Ind. Elect., vol. 52, no. 4, pp. 1177–1184, August 2005. K. Rajani, N. Mudi and R. Pal, “A Robust Self-Tuning Scheme for PIAnd PD-Type Fuzzy Controllers,” IEEE Trans. Fuzzy. System., vol. 7, no. 1, pp. 2–16, February 1999. D. Hadiouche, “Contribution à l’étude de la machine asynchrone double étoile modélisation, alimentation et structure,” Thèse de doctorat. Université Henri Poincaré, Nancy-1, Soutenue le 20 Décembre 2001. G. Séguier, Electronique de Puissance, Editions Dunod 7 ème édition. Paris, France, 1999. Y. Fu, “Commande découplées et adaptatives des machines asynchrones triphasées,” Thèse de doctorat, Université Montpellier II, 1991. D. Khmessi, “Commande de position des machines asynchrones avec pilotage vectoriel,” Thèse de magister, Ecole militaire polytechnique à Alger, 2000. Gang Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Trans. Fuzzy. System., vol. 14, no. 5, pp. 676– 697, october 2006. J. godjevace, Idées nettes sur la logique floue, Editions Presses Polytechniques et Universitaires Romandes, Suisse, 1980. A. Kalantri, M. Mirsalim and H. Rastegar, “Adjustable speed drive based on fuzzy logic for a dual three-phase induction machine,” Electrics Drives II, Electrimacs. August 18–21, 2002. Y. Zhao and T. A. Lipo, “Space vector PWM control of dual three phase induction machine using vector space decomposition,” IEEE Trans. Ind. Appl., vol. 31, no. 5, pp. 1100–1109, September/October 1995. E. Levi, “Recent developments in high performance variable-speed multiphase induction motor drives,” sixth international symposium nikola tesla October 18–20, 2006, Belgrade, SASA, Serbia. D. Beriber, E. M. Berkouk, A. Talha and M. O. Mahmoudi, “Study and control two two-level PWM rectifiers-clamping bridge-two three-level NPC VSI cascade. Application to double stator induction machine,” 35th Annual IEEE Electronics Specialists Conference, pp. 3894–3899, Aachen, Germany, 2004. V. Pant, G. K. Singh and S. N. Singh, “Modeling of a multi-phase induction machine under fault condition,” IEEE International Conference on Power Electronics and Drive Systems, PEDS’99, pp. 92– 97, July 1999, Hong Kong. C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic control, part II,” IEEE Trans. Systems. Man, and Cybernetics, vol. 20, no. 2, pp. 419– 433, March/April, 1990. M. P. Veeraiah and S. M. Chitralekha Mahanta, “Fuzzy proportional integral–proportional derivative (PI-PD) controller,” Proceeding of the 2004 American control conferences, pp. 4028–4033, Boston Massachusetts June 30 – July 2, 2004.