An Approach for the Modeling of an Autonomous Induction Generator[1]

Volume 4, Issue 1 2005 Article 1052

International Journal of EmergingElectric Power Systems

An Approach for the Modeling of anAutonomous Induction Generator Taking Into

Account the Saturation Effect

Dr. Rekioua Djamila, Department of ElectricalEngineering, University of Bejaia, (Algeria)

Pr. Rekioua Toufik, Department of Electrical Engineering,Univeristy of Bejaia, (Algeria)

Idjdarene Kassa Jr., Department of Electrical Engineering,Univeristy of Bejaia, (Algeria)

Dr. Tounzi Abdelmounaim , LaboratoireD’Electrotechnique et D’Electronique de Puissance de Lille,

L2EP (France)

Recommended Citation:Djamila, Dr. Rekioua; Toufik, Pr. Rekioua; Kassa, Idjdarene Jr.; and Abdelmounaim , Dr.Tounzi (2005) "An Approach for the Modeling of an Autonomous Induction Generator TakingInto Account the Saturation Effect," International Journal of Emerging Electric Power Systems:Vol. 4 : Iss. 1, Article 1052.Available at: http://www.bepress.com/ijeeps/vol4/iss1/art1052DOI: 10.2202/1553-779X.1052

©2005 by the authors. All rights reserved. No part of this publication may be reproduced, storedin a retrieval system, or transmitted, in any form or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the prior written permission of the publisher,bepress, which has been given certain exclusive rights by the author. International Journal ofEmerging Electric Power Systems is produced by Berkeley Electronic Press (bepress).

An Approach for the Modeling of anAutonomous Induction Generator Taking Into

Account the Saturation EffectDr. Rekioua Djamila, Pr. Rekioua Toufik, Idjdarene Kassa Jr., and Dr. Tounzi

Abdelmounaim

Abstract

This paper deals with a model to simulate the operating of an autonomous inductiongenerator. The model used is a diphase one obtained by the application of the Park transform. Thismodel permits, when adopting some simplifying hypothesis, to take account of the saturationeffect. This is achieved using a variable inductance function of the magnetising current. The nonlinearity is then based on the approximation of the magnetising inductance with regards to thecurrent. In our case, we use a polynomial function, of 12th degree to perform it. This approach issimple and very accurate. The developed model has been used to study the operating of aninduction machine when a capacitive bank is connected to the stator windings. The simulationcalculation was achieved using MATLAB®-SIMULINK® package. This paper presents transientanalysis of the self-excited induction generator. In order to simulate the voltage build-up processand the dynamic behaviour of the machine, we first establish the machine's model based on a d-qaxis considering the machine’s saturation effect. Secondly, effect of excitation capacitors or loadimbalances on voltage build-up process is investigated. Simulations results for a 5.5 kW inductiongenerator are presented and discussed. Several experimentations are presented to validatesimulations and verify the effectiveness of the developed model.

KEYWORDS: Autonomous induction generator, Saturation effect, Modelisation, Magnetisinginductance

1. INTRODUCTION

It is well known that induction machines may generate power if sufficient

excitation is provided [1, 2]. The squirrel induction machines are widely used in

the wind energy conversion in the case of isolated or faraway areas from grid

distribution [3, 4, 5]. Theses structures have a lot of advantages. They are robust,

need few maintenance and do not cost so much. When operating as an

autonomous generator, the induction machine has to be magnetised by an external

supply [6]. The simple way to achieve this consists in connecting its stator

windings to a capacitive bank in parallel to the load. Hence, for a given rotation

speed, the remaining magnetic flux yields a low electromotive force. Then when

the capacitances are well designed, the magnetising current through the capacitive

bank yields the built up of the electromotive force and its increase to an useful

value.

A lot of works dealt with the study of the autonomous induction generator.

They concerned the calculation of the required capacitance value or the

performances of the device using the equivalent monophase model [7, 9] taking

into account the saturation effect. These last twenty years, different authors use

models to study also the transient operating of the device [10, 13]. These models

take account of the non linearity of the magnetic material by different approaches

more or less accurate and easy to implement. Hence, in references [12] and [13], a

variable magnetising inductance, using the saturation degree function, permits to

the saturation phenomena to be taken into consideration. However, this method,

which is very accurate, needs the knowledge of the linear and saturated

components of the magnetising flux. Besides, others authors do not use the

approximation of the magnetising inductance but utilise determination techniques

of the parameters (voltage, current…) to achieve the study of the induction

generator [1, 14].

In our approach, the model used is a diphase one obtained by the application of

the Park transform. This model permits, when adopting some simplifying

hypothesis, to take account of the saturation effect. In this case, the non linearity

is based on the approximation of the magnetising inductance with regards to the

current. We use a polynomial function, of 12th

degree [15, 16] to achieve this

approximation. This approach is simple and very accurate. In this case, we can

apply easily control methods in closed loop.

1

Djamila et al.: An Approach for the Modeling of an Autonomous Induction Generator

Published by Berkeley Electronic Press, 2005

In this paper, the developed model is used to study the autonomous generator

running of an induction machine. First, we present the machine which has been

used as an experimental test bench. Then, we perform calculation as no load and

when the generator is loaded. For both cases, we compare the simulation results to

the experiment.

2. PROPOSED STUDIED SYSTEM

Figure 1. shows the three-phase connection diagram of the self-excited induction

generator (SEIG).

Fig. 1: Proposed structure.

To analyse the behaviour of the SEIG under several asymmetrical conditions, the

dynamic equations of generator must be established.

2.1 Induction machine model

The linear model of the induction machine is widely known and used. It yields

results relatively accurate when the operating point studied is not so far from the

conditions of the model parameter identification. This is often the case when the

motor operating, at rated voltage, is studied. As the air gap of induction machines

is generally narrow, the saturation effect is not negligible in this structure. So, to

improve the accuracy of simulation studies, especially when the voltage is

variable, the non linearity of the iron has to be taken into account in the machine

model.

This becomes a necessary condition to study an autonomous induction generator

because the linear model is not able to describe the behaviour of the system. Thus,

only approaches, which take account of the saturation effect, can be utilized. This

effect is not easy to yield with using three phase classical models. So, we usually

SEIG

Ca Cb Cc Load

Capacitive bank

2

International Journal of Emerging Electric Power Systems, Vol. 4 [2005], Iss. 1, Art. 1052

http://www.bepress.com/ijeeps/vol4/iss1/art1052DOI: 10.2202/1553-779X.1052

adopt diphase approaches to take globally account of the magnetic non linearity.

This evidently supposes some simplifying hypotheses. Indeed, the induction is

considered homogenous in the whole structure. Moreover, the use of diphase

model supposes that the saturation effect is considered only on the first harmonics

and does not affect the sinusoidal behaviour of the variables.

In our approach, we adopt the diphase model of the induction machine expressed

in the stator frame. The classical electrical equations are written as follows:

dt

di

dt

di

dt

di

dt

di

i

iLLl

i

iiLl

i

iiL

i

iLLll

i

iLL

i

iiLl

i

iiL

i

iLLl

i

i

i

i

RLlRl

LlRlR

LRl

LlR

v

v

mq

md

sq

sd

m

mq

mmr

m

mqmd

mr

m

mqmd

m

m

md

mmrr

m

mq

mm

m

mqmd

ms

m

mqmd

m

m

md

mms

mq

md

sq

sd

rmrrrrr

mrrrrrr

mssss

mssss

sq

sd

.

..

.0

...0

..

.0

...0

.

).(.

).(.

0..

.0.

0

0

2

''

'

2

'

2

''

'

2

'

(1)

Where Rs, ls, Rr and lr are the stator and rotor phase resistances and leakage

inductances respectively, Lm is the magnetizing inductance and .p

Besides, Vsd, isq , Vsq and isq are the d-q stator voltages and currents respectively,

imd and imq are the magnetizing currents, along the d and q axis, given by:

rdsdmd iii (2)

rqsqmqiii (3)

Where ird and irq are the d-q rotor currents

isd and isq are the d-q stator currents.

Thus, the saturation effect is taken into account by the expression of the

magnetizing inductance Lm with respects to the magnetizing current im defined as:

22

mqmdm iii (4)

To express Lm in function of im, we use a polynomial approximation, of degree

12 [15, 16].

3



n

j

j

mjm

mm

mm

n

j

j

mjmm

iajifid

d

id

dLL

iaifL

0

1'

0

..

.

(5)

2.2 Load model

The stator windings of the induction machine are connected to a star capacitive

bank connected in parallel to a resistive load. Hence, at no load, the diphase stator

voltages and currents are linked by the following expression:

ds

qs

qs

ds

qs

ds

V

V

i

i

C

C

V

V

dt

d.

0

0

.1

0

01

(6)

This takes, when the induction generator is loaded, this other writing:

ds

qs

qchqs

dchds

qs

ds

V

V

ii

ii

C

C

V

V

dt

d.

0

0

.1

0

01

(7)

Where:

idch and iqch are the current through the equivalent diphase resistive load R. They

can be expressed, from the stator voltages (Fig.2.).

Fig.2: Induction machine model.

The dynamic model of a three-phase balanced resistive load in the q-d axis

arbitrary reference frame is given by

qchqs

dchds

iRV

iRV

.

. (8)

Induction

machine

model

ichd, ichd

C Load

R

isd, isq

vsd, vsq

iCd, iCq

4



2.3 The global system

The global differential system to solve is then written as follows:

dt

di

dt

di

dt

di

dt

di

i

iLLl

i

iiLl

i

iiL

i

iLLll

i

iLL

i

iiLl

i

iiL

i

iLLl

i

i

i

i

RR

RR

LpRlp

LplpR

v

v

mq

md

sq

sd

m

mq

mmr

m

mqmd

mr

m

mqmd

m

m

md

mmrr

m

mq

mm

m

mqmd

ms

m

mqmd

m

m

md

mms

mq

md

sq

sd

rr

rr

mss

mss

sq

sd

.

..

.0

...0

..

.0

...0

.

00

00

0..

.0.

0

0

2

''

'

2

'

2

''

'

2

'

(9)

To take into account the non linearity of the resolution, we use Runge Kutta

algorithm to solve the system (7) and system (9) together.

3. RESULTS AND DISCUSSION

The developed model is used to study the autonomous generator running of an

induction machine. First, we present the machine which has been used as an

experimental test bench. Then, we perform calculation at no load and when the

generator is loaded. For both cases, we compare the simulation results to the

experiment.

Fig 3: im versus the phase applied voltage.

Phas

e ap

pli

ed v

olt

age

vs

(V)

Magnetizing current im (A)

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

5



3.1 The experimental machine

Experimental results were obtained from the implementation of the structure

presented in Fig.1. using an induction machine of 5.5 kW (table.1) manufactured

by CEN (Constructions Electriques -Nancy) (figure 4.). [16].

Fig 4: The experimental bench.

Parameter Value Parameter Value

PN 5.5 kW J 0,230 kg.m²

UN 230/400 V f 0,0025 N.m/rads-1

IN 23,8/13,7 Rs 1,07131

f 50 Hz Rr 1,29511

NN 690 rpm p 4

Table1. Machine parameters [16].

Then, from these data and the active power absorbed by the machine, one can

determine the evolution of the magnetising inductance in function of i0. The

obtained curve is drawn in figure 5-a. Lastly, to avoid the problem due to the

absence of experimental inductance values outside the magnetising current range

identification, we drawn a complementary part. This yield the evolution given in

figure 5-b.

6



a: Measurement result b: Simulation result

Fig.5: Magnetizing Curve.

3.2 No load tests.

The experimental device is shown in figure 6. In this section, experimental and

computed results are presented.

Fig.6: The experimental device studied.

The model introduced in the precedent paragraph has been used in the

MATLAB SIMULINK environment to study the performance of the autonomous

Mag

net

izin

g i

nd

uct

ance

Lm

(H

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

0.13

a

b

Magnetizing current im(A)

7



induction generator. The parameter of the model used are the ones of the

experimental machine when the magnetizing inductance is the one given above,

expressed by a proposed polynomial function. To simulate the remaining voltage,

we take a non low initial value for the phase voltages.

3.2.1. Voltage build-up process under balanced conditions

In order to validate the model of the induction generator, we study firstly the

built up process of the stator voltage when the rotor of the induction machine is

driven at 780 rpm under no-load conditions. The value of each self excitation is

fixed to 100 µF [16].

Time [sec].

a: Simulated results.

Time [sec].

b: Experimental results.

Fig. 7: A phase voltage built up process under no-load conditions.

For the same conditions, the evolution of a phase voltage calculated and

measured is shown in the figure 7 (a and b respectively). We can notice the good

agreement between both curves. We can observe that the voltage value before the

Phas

e v

olt

age

Va

[V]

0.14 0.2 0.42 0.56 0.70 0.84 0.980.00

-400

-300

-200

-100

0

100

200

300

400

Phas

e vo

ltag

e V

a [V

]

100 V/div

0.14s/div

8



built-up is different for experimental test and for simulation. This is due to the

initial conditions. The voltage build-up process is due to remaining field in the

machine which can be different after every utilisation of the machine.

We show also the calculation results related to a phase current (Fig.8) and the

magnetizing current (Fig.9.)

Time [sec]

Fig.8: Evolution of a phase current.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

t (s)

i0 (A)

Time [sec]

Fig.9: Evolution of the magnetising current.

3.3 Other operating points.

Other tests have been performed. The first one studies the effect, on the phase

voltage, when the generator is loaded. In the figure 10a and b we show the

simulated and measured evolution of a phase voltage respectively when the

generator is connected to a resistive load of 50 per phase.

Phas

e cu

rren

t ia

[A

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-8

-6

-4

-2

0

2

4

6

8

Mag

net

isin

g c

urr

ent i 0

[A]

9



Time [sec].

a: Computed results.

Time [sec].

b: Measured results.

Fig.10: Evolution of a phase voltage when the generator is loaded (100 F–780

rpm) and R from to 50 .

Once more, the results are in good agreement. The connection of the load yields

a decrease of the phase voltage magnitude and a low variation of its frequency.

Lastly, tests have been carried out to determine the evolutions of the phase

voltage with regards to the capacitance, the speed rotation and the load values.

Hence, for two capacitance values 100 µF and 110 µF, we drawn the curves V(R)

for 3 values of the rotation speed (720, 750 and 780 rpm).

In figure 11a and b, we give the simulation and experimental results

respectively when the capacitance value is 100 µF. Figures 12a and b give the

same evolutions for a capacitance of 110 µF.

Sta

tor

vo

ltag

e V

a [V

]

100 V/div

0.14 s/div

1.42 1.5 1.58 1.66 1.74 1.82 1.9

-400

-300

-200

-100

0

100

200

300

400

Sta

tor

volt

age

Va

[V]

10



R[ ]

a: Simulation results.

R[ ]

b: Experimental results

Fig.11: Evolution of the rms phase voltage with respects to the R for 3 speed

rotation values (100µF).

Ph

ase

volt

age

Vef

f [V

] P

has

e v

olt

age

Vef

f [V

]

0

50

100

150

200

250

300

050100150200250300350400450

780 rpm

750 rpm

720 rpm

50

100

150

200

250

300

050100150200250 300350400450

780 rpm

750 rpm

720 rpm

11



R[ ]

a: Simulation results

R[ ]

b: Experimental results

Fig.12: Evolution of the rms phase voltage with respects to the R for 3 speed

rotation values (110µF).

As we could expect, the magnitude of the phase voltage is an increasing

function of both the capacitance and the speed rotation values. Furthermore, when

the generator is highly loaded, the magnitude voltage decreases quickly. This well

known characteristic, and problem, of the autonomous induction generator when

connected to a simple capacitance bank in parallel.

We drawn the evolution of the phase voltage with regards to the current (Fig.13,

Fig14, Fig.15.) for three rotation speed values (780, 750 and 720 rpm) and two

different values of capacitance (100µF, 110µF).

Ph

ase

vo

ltag

e V

eff

[V]

Phas

e vo

ltag

e V

eff

[V]

0

50

100

150

200

250

300

050100150200250300350400450

0

50

100

150

200

250

300

050100150200250300350400450

780 rpm

750 rpm

720 rpm

780 rpm

750 rpm

720 rpm

12



Current phase ia[A]

C=100µF C=110µF

a: Simulation b: Measurement

Fig. 13: Variation of terminal voltage with current phase at constant speed

(780 rpm).

Current phase ia[A]

C=100µF C=110µF

a: Simulation, b: Measurement

Fig.14: Variation of terminal voltage with current phase at constant speed.

(750 rpm).

0

100

200

300

1 2 3

ab

01 2 3 4 5

ab

01 2 3 4 5 6 7

ab300

200

100

300

200

100

0 1 2 3 4 5 6

ab

100

200

300

Phas

e vo

ltag

e V

eff

[V]

Phas

e volt

age

Vef

f [V

]

13



Current phase ia[A]

C=100µF C=110µF

a: Simulation b: Measurement

Fig. 15: Variation of terminal voltage with current phase at constant speed.

(720 rpm).

As we note, more the capacitance value increases and more the induction machine

provides a constant voltage for a load current. We will find the same results for

the two other speed values N. The curves have all the same shape of hook. We

can add that more the rotor speed increases and more the stator voltage is high.

Finally, all the curves are in the shape of hook. The higher capacity is the more

one has a good behavior in voltage for a higher current. It will be necessary to

choose the excitation capacity adapted for a given use of the induction machine.

3.4. Sudden disconnection of one capacitor

We suddenly disconnected one of the excitation capacitors (C=100 F), the

corresponding simulated transient results are shown respectively on figures 16a, b

and c. This test shows the good accuracy of the machine and the load model.

0

100

200

300

0 1 2 3 4 5 6 7

ab

0 1 2 3 4 5

ab

100

200

300

Phas

e v

olt

age

Vef

f [V

]

14



3 4 5 6 7 8

-200

-150

-100

-50

0

50

100

150

200

Time [sec].

a: Phase a.

3.9 4.1 4.3 4.5

-200

-150

-100

-50

0

50

100

150

200

Time [sec].

b: Phase a, b and c.

3.98 4 4.02 4.04 4.06 4.08 4.1

-250

-200

-150

-100

-50

0

50

100

150

200

250

Time [sec].

c: Zoom on phase a, b and c.

Fig. 16: Effect of a sudden disconnection of one capacitor (C=100 F) on stator

voltage.

Naturally the transient behaviour of the SEIG will depend on the resulting

remaining equivalent excitation capacitor. In fact, if the initial values of the

capacitors are high, the voltage will not fall down under sudden disconnection of

Phas

e vo

ltag

e V

a[V

] P

has

e v

olt

age

Vab

c[V

] P

has

e vo

ltag

e V

abc[

V]

15



one capacitor. It will drops a little during transient and then return to a new

steady-state operation point. The line current variation (Fig.17.) is similar to the

one of voltage (Fig.16.).

3 4 5 6 7 8

-8

-6

-4

-2

0

2

4

6

8

Time [sec].

a: Phase a.

3.9 4.1 4.3 4.5

-8

-6

-4

-2

0

2

4

6

8

Time [sec].


3.98 4 4.02 4.04 4.06 4.08 4.1

-8

-6

-4

-2

0

2

4

6

8

Time [sec].


Fig. 17: Effect of a sudden disconnection of one capacitor (C=100 F) on stator

current.

Cu

rren

t phas

e

ia(A

) C

urr

ent

ph

ase

ia

bc(

A)

Curr

ent

ph

ase

ia

bc(

A)

16



3.4 Influence of capacitor bank imbalance

In order to show the influence of the capacitor bank imbalance (C=160 F), we

present respectively in figure 18. and figure 19. the variations of the stator voltage

and current .

3 4 5 6 7 8

-300

-200

-100

0

100

200

300

Time [sec].

a: Phase a.

3.9 4.1 4.3 4.5

-300

-200

-100

0

100

200

300

Time [sec].


3.98 4 4.02 4.04 4.06 4.08 4.1

-300

-200

-100

0

100

200

300

Time [sec].


Fig. 18: Influence of capacitor bank imbalance (C=160 F) on Stator voltage.

Volt

age

ph

ase

va(

V)

Vo

ltag

e p

has

e v

abc(

V)

Volt

age

ph

ase

v

abc(

V)

17



3 4 5 6 7 8

-15

-10

-5

0

5

10

15

Time [sec].

a: Phase a.

3.9 4.1 4.3 4.5

-15

-10

-5

0

5

10

15

Time [sec].


3.98 4 4.02 4.04 4.06 4.08 4.1

-15

-10

-5

0

5

10

15

Time [sec].


Fig. 19: Influence of capacitor bank imbalance (C=160 F) on Stator current.

3.6. Influence of a sudden disconnection of the load

We can notice that the disconnection of a purely resistive load, involves a voltage

variation but the steady-state is reached after a delay of about 1 s. The same

Cu

rren

t p

has

e ia

(A)

Cu

rren

t p

has

e ia

(A)

Curr

ent

ph

ase

ia

(A)

18



phenomenon can be observed on stator currents. Simulation results show the

validity of the adopted modelling approach.

3 4 5 6 7 8

-300

-200

-100

0

100

200

300

Time [sec].

a: Phase a

3.9 4 4.1 4.2 4.3 4.4 4.5

-300

-200

-100

0

100

200

300

Time [sec].


3.98 4 4.02 4.04 4.06 4.08 4.1

-200

-150

-100

-50

0

50

100

150

200

Time [sec].


Fig. 20: Stator voltages of phases a, b and c under

unbalanced load conditions.

Vo

ltag

e p

has

e

vab

c(A

) V

olt

age

phas

e v

abc(

A)

Vo

ltag

e p

has

e va(

A)

19



3.9 4 4.1 4.2 4.3 4.4 4.5

-10

-5

0

5

10

Time [sec].

a: Phase a, b and c.

3.98 4 4.02 4.04 4.06 4.08 4.1

-10

-8

-6

-4

-2

0

2

4

6

8

10

Time [sec].

b: Zoom on phase a, b and c.

Fig.21: Stator currents of phases a, b and c under

unbalanced load conditions.

3.7. Influence of an unbalanced load.

The aim of this test is to show the resistive load unbalance effect on the behaviour

of the stator voltages and currents. The resistive load parameters are Ra=Rb=50 ,

Rc=80 and each excitation capacitors is equal to 100 µF. The obtained results

are presented respectively on figures 22 and 23.

Cu

rren

t p

has

e ia

bc(

A)

Cu

rren

t p

has

e ia

(bcA

)

20



4 4.02 4.04 4.06 4.08 4.1

-200

-100

0

100

200

Time [sec].

Fig. 22: Stator voltages of phases a, b and c under

unbalanced load condition.

4 4.02 4.04 4.06 4.08 4.1

-8

-6

-4

-2

0

2

4

6

8

Time [sec].

Fig. 23: Stator current of phases a, b and c under

unbalanced load condition.

The influence on stator voltage is negligible while the chosen load imbalance

induces a consequent variation of the peak current value.

4. CONCLUSION

The paper examines the dynamic performances of an autonomous induction

generator, taking the saturation effects into account, by the means of a variable

magnetising inductance, has been presented. This magnetising inductance is

expressed, using a polynomial function, of degree 12, as a function of the

magnetising current. The proposed model has been used, in a MATLAB

SIMULINK simulation environment to study an induction machine in

autonomous generator operating.

Vo

ltag

re p

has

e v

abc(

A)

Curr

ent

ph

ase

ia

bc(

A)

21



Obtained results of the SEIG under voltage build-up process, balanced or

unbalanced network load side conditions are presented and compared. Excessive

conditions like disconnection of one self-excitation capacitor or sudden

disconnection of the load are also analysed.

The analysis presented is validated by experimental results. The comparison of

all these results shows a very good agreement between the experimentation and

simulation. The amplitudes of the signals, their shapes as their duration present

practically the same values for both simulation and experimentation. The

coherence between computed and measured results is very good as well for

dynamic conditions as for steady state. This concordance between the

experimentation and simulation confirms the validity of the developed model.

5. REFERENCES

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An Approach for the Modeling of an Autonomous Induction Generator[1]

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