Adaptive Field-Oriented Control with PD Regulator for the Induction Machine

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Adaptive Field-Oriented Control with PD Regulator for the Induction Machine

AHMED CHOUYA Department of Genie Electrical University of Djilali Bounaama Khemis-Miliana City, 44 225

ALGERIA ahmedchouya@gmail.com

MOHAMED CHENAFA & ABDELLAH MANSOURI Department of Genie Electrical

ENPO, BP 1523 El Mnaouer Oran City

ALGERIA mchenafa@yahoo.fr ammansouri@yahoo.fr

Abstract: In all the research tasks, the researchers use regulator PI. In this work, one treats the Field-Oriented Control (FOC) where the regulator is PD. Simulation is made by the software MATLAB/SIMULINK.

Key–Words: Field-Oriented Control (FOC); Reference Adaptative Control (MRAC); Proportional-Derive(PD); Induction Ma- chine.

Nomenclature

: Stator inductance cyclic ,

: Rotor inductance cyclic ,

: Cyclic mutual inductance between stator and rotor

: Stator resistance,

: Rotor resistance,

: Scattering coefficient,

: Time constant of the rotor dynamics,

: Rotor inertia,

: Resistive torque,

: Pole pair motor,

¾ : is the 2-dimensional identity matrix,

¾ : is a skew - symmetric matrix.

1 Synthesis of Field-Oriented Con- trol

For an asynchronous machine supplied with tension, ten- sions stator and are the variables of control, and we consider rotor flux, the currents stator and the mechanical speed like variables of state [1].

Ö

½

Ö

½

Ö

½

×

½

×

(1)

The parameters are defined as follows:

¾

½

Ö

¾

The stator pulsation is not exploitable sinceis null with the starting of the machine. We will use for establish- ment, the following equation:

(2) The vectorial field-oriented control is based on an orienta- tion of the turning reference mark of axessuch as the axisthat is to say confused with the direction of[1].

The fluxbeing directed on the axis, the equation of state (1) us allows to express , ,and with

and

. The following equations are ob- tained [2] :

Ö

½

×

½

×

Ö

½

Ö

Ú

Ð

(3)

Closely connected of uncoupled the two first equations from the system (3). We define two new variables of order

(2)

and :

(4) Where and are the terms of coupling given by :

Ö

(5) and orders it uncoupling

(6)

Transfer functions of this system uncoupled while taking as in-puts

and as out-puts

and :

(7)

We will present the synthesis of each regulator sepa- rately closely connected to clarify the methodology of synthesis of each one of them.

Flux regulator :

The combination enters the third equation of the system (3) and the first of the system (7), we will have

½

Ö

(8)

We wish to obtain in closed loop a response of the type

order. To achieve this objective, one takes an adaptive proportional-derive regulator with MRAC of the type:

½

¾ (9)

We can represent the system in closed loop by the figure (FIG.1)

Adaptation Law

vsd

e_\r

\r

\rm

U_\c

) ( s G_\_m

) 1)(

( J

V

s

s T T L

M

r r s

¸¸

¹

·

¨¨

©

§

\

\ D P

D k

s k k

Figure 1: Diagram block in closed loop of PD adaptive regulator with model reference of flux.

The reference model of the system in closed loop is selected with a first-order transfer function:

That is to say the optimality criterionof the ad- justment loop is expressed by the quadratic integral [3]:

(10)

Its derivative is :

!" (11) The out-put is written:

½

¾

½

Ö

½

¾

#

(12) Let us compensate for the slowest pole by the numerator of the transfer function of our regulator, which is translated by the condition:

¾

½

(13)

In open loop, the transfer function is written:

½

with

(14) And in closed loop:

½

(15)

If the condition (13) is not considered, therefore one will have

½

¾

½

Ö

½

¾

(16)

One calculates the adjustable parameters½and¾:

½

¾

(17)

¾

½

¾

(18)

Where

½

and

½

.

(3)

Taking into account (11), (17) and (18), one can write the equation of gradient½and¾:

½

$

½

(19)

½

$

½

!"

½

#

½

$

½

!"

½

#

½

$

½

!"

½

(20) And

¾

$

¾

(21)

¾

$

¾

!"

¾

#

¾

$

½

!"

½

¾

#

¾

$

¾

!"

½

¾

(22) Speed regulator:

According to the mechanical equation of the machine (3); we have :

(23)

From where the expression of the electromechanical torque is given by the formula :

(24)

While replacing, the system (7) in the torque (24)

(25)

Therefore, equation (23) becomes :

×

Ö

(26) For closed loop speed it was proposed regulator PD with MRAC of the form :

ª

%

½

%

¾ (27)

The functional diagram is given by the figure (FIG.2)

Adaptation law

vsq

:m

e:

) (s G

_:_m

:

U_:c T^l

fv

Js 1

J V

\

s

L L pM

r s

rd :

: _P

Ds k k

Figure 2: Diagram block in loop closed of PD adaptive regulator to model reference speed .

The reference model of the loop system closed is selected with a second-order transfer function:

ª

Let us compensate for the slowest pole by the numerator of the transfer function of our regulator, which is translated by the condition:

ª

%

¾

%

½

(28) In open loop, the transfer function is written:

ª

%

½

ª

with ª

(29) And in closed loop:

ª

%

½

ª

%

½

ª

(30) If the condition (28) is not considered, therefore one will have

ª

½

%

¾

ª

Ú

ª

%

½

%

¾

(31) The adjustable parameter is calculated^%½and^%¾:

ª

%

½

ª

%

½

ª

(32)

ª

%

¾

%

½

%

¾

ª

(33)

(4)

Whereª

%

½

ª

andª

%

½

ª

. Taking into account (11), (32) and (33), one can write the gradient equation^%½and^%¾:

%

½

$

ª½

%

½

(34)

%

½

$

ª½

!"

ª

%

½

#

ª

%

½

$

ª½

!"

%

½

ª

#

ª

$

ª½

!"

%

½

ª

(35)

And

%

¾

$

ª¾

%

¾

(36)

%

¾

$

ª¾

!"

ª

%

¾

#

ª

&

¾

$

ª¾

!"

%

½

%

¾

ª

#

ª

$

ª¾

!"

ª

%

½

%

¾

(37)

2 Results and Simulations

We conceived simulation by carrying out the diagram general in blocks as the figure shows it (Fig.3).

2.1 Simulation Block Diagrams, Machine Data and a Benchmark

We ride a detailed scheme SIMULINKof the MRAC regulator in figure (FIG.4).

In addition to that, we perform a simulation with MATLAB-SIMULINKby using the benchmark in (FIG. 3) and the asynchronous machine parameters given in Table 1.

Parameters Notation Value Unit

Poles pairs

Stator resistance

Rotor resistance

Stator inductance ^'

Rotor inductance

'

Mutual Inductance ^'

Rotor inertia ^!(^¾

viscous damping

)(*+

coefficient

Resistive torque

)(

Table 1: Parameters of the asynchronous machine.[4]

2.2 Powerful of MRAC based FOC

The vector of machine state is initial- ized whit

, and the results are given for the machine of which a direct starting.

The figure (FIG.5) shows the pace of desired flux and flux reference. One notices desired flux follows very well his reference with a respect to the transient state and the moment where speed changes its direction. The means flux error is^(,with a variance ^¿ to see the figure (FIG.6)

Figure 3: Diagram general of FOC with MRAC regulator.

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Figure 4: SIMULINKof FOC with MRAC regulator.

0 0.5 1 1.5 2 2.5 3 3.5

0 0.5 1 1.5 2 2.5

Flux [Wb]

Time [s]

ψr ψ_m

Figure 5: Flux performance.

0 1 2 3

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3

Time [s]

Error flux [Wb]

−0.2 0 0.2

0 0.5 1 1.5 2

x 10⁴

Gaussian Histogram

Figure 6: Flux errors of control.

The curves of the figure (FIG.7) give desired speed and controlled speed; there are differences right in the transient states at moment and , with a very weak means of error which is+*and variance

to go up to the figure (FIG.8).

0 0.5 1 1.5 2 2.5 3 3.5

−100

−80

−60

−40

−20 0 20 40 60 80 100

Time [s]

Speed [rad/s]

ΩΩ_m

Figure 7: Speed performance.

0 1 2 3

−80

−60

−40

−20 0 20 40

Time [s]

Error speed [rad/s]

−20 0 20

0 2 4 6 8 10 12

x 10⁴

Gaussian Histogram

Figure 8: Speed errors of control.

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The evolutions of the self-adjustable parameters½and

¾gave by the figure (FIG.9) and of the parameters^%½and

%

¾by the figure (FIG.10)

0 0.5 1 1.5 2 2.5 3 3.5 4

0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13

ρ1

0 0.5 1 1.5 2 2.5 3 3.5 4

0 100 200 300 400 500 600 700

Time [s]

ρ2

Figure 9: Parameters½and¾.

0 0.5 1 1.5 2 2.5 3 3.5

−10

−5 0 5 10

ρ1

0 0.5 1 1.5 2 2.5 3 3.5

0 2 4 6 8 10x 10⁵

Time [s]

ρ2

Figure 10: Parameters^%½and^%¾.

3 Conclusion

We clarified FOC control. The uncoupling control, us made it possible to use adaptive regulators and to have an effect uncoupled on the regulation from rotor flux and rotating speed.

The two regulators used are PD adaptive in the control loops of the rotor flux and of rotating speed. The results are very well. Simulations show the effectiveness of adaptation.

References:

[1] Carlos Canudas de Wit. Mod´elisation controle vec- toriel et DTC. Commande des moteurs asynchrones.

HERMES Science Publications. Volume 1. 2000 [2] Carlos Canudas de Wit. Optimisation discr´etisation et

observateurs. Commande des moteurs asynchrones 2.

HERMES Science Publications. 2000

[3] Mimoun Zelmat. Automatisation des processus in- dustriels : Commande modale et adaptative. Office des Publications Universitaires. Tome 2. 2000 [4] A. Chouya, A. Mansouri and M. Chenafa. Sen-

sorless speed nonlinear control of induction motor, Recent Advances in Circuits, Systems, Signal and Telecommunications, 5th WSEAS International Con- ference on Circuit, Systems and Telecommunications (CISST’12), Puerto Morelos, Mexico, Junary 29- 31,2011, pp:69-75.