Elaboration of the Minimum Capacitor for an Isolated Self Excited

Induction Generator Driven by a Wind Turbine

Fadi Ouaﬁa, Abbou Ahmed

Mohammadia School of Engineers, Mohammed V University, Avenue Ibn Sina, Rabat, Morocco

Department of electrical engineering Mohammed V University, Rabat, Morocco

Keywords:

Wind turbine, Induction generator, self-excitation, excitation capacitance required, wind speed, magnetiz-

ing inductance, steady state, real parts of the roots, voltage build up, no load, prime mover, synchronous speed.

Abstract:

In this paper, a detailed procedure is elaborated, based on evaluation of the roots in the characteristic equation

of the stator current, to determine the proper capacitor bank that will be used to reach the self-excitation. For

that, a d-q model of the self-excited induction generator under no load is presented using Matlab-Simulink

to verify if the self-excitation comes true or not. This study illustrates furthermore, the inﬂuence of the

magnetizing inductance on the voltage buildup.

1 INTRODUCTION

A wind turbine with Self-excited Induction Genera-

tor SEIGs is a subject that is gaining renewed inter-

est with the increasingly frequent use of the asyn-

chronous generator. In the ﬁeld of renewable ener-

gies, in general, and that of wind turbines, in particu-

lar, has largely contributed to the development of the

induction machine as a Self-excited Induction Gen-

erator thanks to its several advantages such as: very

reliable and relatively inexpensive compared to other

types of generators. It also has some mechanical char-

acteristics which makes it very suitable for the conver-

sion of wind energy (sliding of the generator as well

as a certain capacity of overload), very high lifetime;

non-existent maintenance (bearings ...), very simple,

rugged, and produces high power per unit mass (N.

M. Okana and al., 2015), (A.Abbou and al., 2013).

The self excited machine, in its operation in generator

mode, poses a particular problem : it cannot start it-

self as a generator and needs an external source to per-

form this operation which is called self-excitation : it

requires excitation current to magnetize the core and

produce the rotating magnetic ﬁeld. This current is

supplied from an external source, for grid connected

systems; however, this current is supplied from a bat-

tery of capacitors, for isolated system which is our

case. When a charged capacitor is connected and the

generator is driven by a prime mover, a transient ex-

citing current will ﬂow and produce a rotated mag-

netic ﬂux, so as a result, the power will be generated

and supplied to the external source.

Several approaches have been reported in previous

studies to determine the sufﬁcient value of the ca-

pacitance to stimulate a self excited induction gener-

ator (A.K and al., 1990), (N.H.Mal and al., 1986), (

N.H.Malik and al., 1987), (M.Orabi and al., 2000),

(S.S.Murthy al., 1982). Most of these approaches ap-

ply loop equations in the steady state model of per

phase equivalent circuit such us : Nodal Admittance

Approach, Loop Impedance Approach, and LC Res-

onance Principal. These kinds of approaches give the

appropriate capacitance corresponding to the mini-

mum capacitance in steady state analysis, but they are

inapplicable for transient analysis.

The main purpose of this study is to present a new

approach to determine the minimum capacitance re-

quired for excitation in a SEIG. For this, we should,

ﬁrst look behind its process and extract an accurate al-

gorithm that will serve us to achieve the aim. For this,

we have, ﬁrst, recalled the basic concepts for mod-

elling of RLC circuit, since this one is similar to the

self-excited induction generator. Then, a review of the

characteristic stator equation of the SEIG, the roots of

this equation is discussed as well as the choice of the

accurate capacitance.

264

Ouaﬁa, F. and Ahmed, A.

Elaboration of the Minimum Capacitor for an Isolated Self Excited Induction Generator Driven by a Wind Turbine.

DOI: 10.5220/0009774302640270

In Proceedings of the 1st International Conference of Computer Science and Renewable Energies (ICCSRE 2018), pages 264-270

ISBN: 978-989-758-431-2

2 SELF EXCITED INDUCTION

MACHINE

Basically, the mathematical model per phase of the

self-excited induction generator is similar to the clas-

sical induction motor; the only difference is that the

SEIG has a battery of capacitors linked to the stator

terminal.

Figure 1: SEIG with a capacitor connecting across the

stator terminal.

The equivalent circuit representation of an asyn-

chronous machine is proper to use for steady state

analysis. Nonetheless, the Park representation is used

to model the SEIG beneath dynamic conditions.

Figure 2: Park representation of the self induction

generator in stationary frame (a) q-axis, (b) d-axis.

2.1 Modelling of the SEIG under No

Load

Using the d-q representation in Figure 2, the induction

machine can be modelled by equations (1) to (10),

From the stator side :

= L

+ L

(1)

= L

+ L

(2)

= R

dλ

(3)

= R

dλ

(4)

From the rotor side :

= L

+ L

(5)

= L

+ L

(6)

= R

dλ

+ ω

(7)

= R

dλ

− ω

(8)

For the air gap ﬂux linkage side :

= L

+ L

(9)

= L

+ L

(10)

The matrix equation for the d-q model of a self

excited induction generator in the stationary stator

reference frame using the SEIG model is given by

(11).







+ pL

0 pL

0 R

+ pL

0 pL

−ω

+ pL

−ω

+ pL

























cq0

cd0

−K



















(11)

2.2 Analogy between RLC Circuit and

SEIG

Basically, an induction machine can be modelled us-

ing RLC circuit elements. In fact, the behaviour and

analysis of the self-excited induction generator is sim-

ilar to an RLC circuit.

2.2.1 RLC Circuit Approach

Energy can be stored in an inductor as well as in a ca-

pacitor, at t = 0, two initial conditions, current might

have been ﬂowing in an inductor or initial voltage ex-

ist in a capacitor.

Elaboration of the Minimum Capacitor for an Isolated Self Excited Induction Generator Driven by a Wind Turbine

265

Figure 3: RLC circuitl.

Switch S is close : i(t) = 0; the voltage equation :

= Ri(t) + L

di(t)

i(t)d(t) (12)

Introducing the p operator (12) can be written as :

i(t) =

pR + Lp

(13)

The characteristic equation is :

0 = pR + Lp

(14)

The roots of this equation are :

= −

−





−

(15)

= −





−

(16)

If we suppose that :

≤

Then the roots are complex, and can be expressed as :

= ψ + jΩ (17)

= ψ − jΩ (18)

ψ is always negative because the resistance R is posi-

tive, as result, with positive R will be a dampening of

oscillation, the real part ψ represents the rate at which

the transient decays, and Ω the imaginary part repre-

sents oscillation frequency.

In passive circuit like RLC all transient solutions have

a negative ψ meaning that transient is reduced with

the progression of time and ﬁnally will dampen to

zero.

Figure 4: progression of current for a positive resistance

value.

However, if ψ is positive, this implies the transient is

growing with the progression of time.

Figure 5: progression of current for a negative resistance

value.

We can describe the self-excitation in an induction

generator as the growth of current and the associated

increase in the voltage across the capacitor without an

external excitation system.

Transients that grow in magnitude (self-excitation)

with positive real part of the roots can only be happen-

ing if there is an external energy source that is able to

cover all the losses associated with the rising current,

the SEIG is able to have a growing transient because

of the external mechanical energy source.

2.2.2 Projection of the RLC Approach on the

SEIG

We can deal with the circuit of the SEIG by using the

same approach as well as the circuit of the RLC.

The matrix (11) can be written as :







+ pL

0 pL

0 R

+ pL

0 pL

−ω

+ pL

−ω

+ pL

























−V

cq0

−V

cd0

−K







(19)

ICCSRE 2018 - International Conference of Computer Science and Renewable Energies

266

When a balanced three phase system is transformed

into a two-axis system, the stator currents i

and i

have similar waveforms.

numerator

+ k

(20)

The characteristic equation of the current can be ob-

tained from the expression of the current transfer

function (20).

+ k

=0 (21)

With :

= A

= 2AB

= 2ADB

= 2BD +AE

= 2AF + 2BE + D

+ G

= 2BF + 2DE

= 2DF + E

= 2EF

= F

And :

A = C



− L



B = C



+ 2R

− L



D = C(2R

+ (R

+ ω

+ L

− L

) + L

E = CR

+ ω

+ L

) + 2R

F = R

+ ω

+ L

G = CL

This characteristic equation for the current can be

solved using different ways, for our case, the roots are

obtained numerically using the root function in Mat-

lab.

When (21) is factorized, it gives :

(p − ψ

+ jΩ

)(p − ψ

− jΩ

)

(p − ψ

+ jΩ

)(p − ψ

− jΩ

)

(p − ψ

+ jΩ

)(p − ψ

− jΩ

)

(p − ψ

+ jΩ

)(p − ψ

− jΩ

) = 0 (22)

If any of the roots in the equation has a positive real

value, then there is a self-excitation. To determine the

required capacitor for an induction generator running

at the given rotor speed, the roots values are computed

by increasing the capacitor value until one of the real

parts of the roots becomes positive.

2.3 Algorithm to Determine the

Minimum Capacitance

In order to determine the accurate capacitor value,

an algorithm is elaborated and programmed using

Matlab as shown in ﬂow chart ﬁgure 6 :

Figure 6: Flow chart to determine the minimum

capacitance at a given rotor speed at no load.

The minimum capacitance required for a given rotor

speed of induction generator, can be found by ﬁxing

the rotor speed and then increasing the value of the

capacitance until one of the real parts of the roots be-

comes positive. The value of capacitance makes that

happen is the minimum value of capacitance required

for self-excitation.

Figure 7: progression of real part value when capacitance

C increased.

As it is shown in ﬁgure 7 all the real parts of the

roots start with a negative value, so the waveform will

dampen with time and there will be no transients. Af-

ter some iterations, the algorithm returns one of the

real part greater than zero and that by determining

Elaboration of the Minimum Capacitor for an Isolated Self Excited Induction Generator Driven by a Wind Turbine

267

the appropriate capacitance value that makes the self-

excitation achieve.

2.4 Characteristic of Magnetizing

Inductance

In the modelling of an induction machine, it is essen-

tial to determine the magnetizing inductance Lm at

rated voltage and rated frequency. In the SEIG, the

variation of Lm is the major element in voltage build

up and its stabilization. Magnetizing inductance is de-

termined by driving the asynchronous machine at syn-

chronous speed and taking measurements when the

applied voltage was varied from zero to 120% of the

rated voltage. The variation of Lm measured at rated

frequency of the induction machine used in this study

is given by (B.Singh al., 1998) :

= 0.00016I

− 0.002I

+ 0.005I

+ 0.205 (23)

The expression of Lm is elaborated experimentally as

a function of magnetizing current Im, however, in this

investigation, we are looking for the expression of

magnetizing inductance as a function of voltage V

for that, we use equations below to convert from L

f(I

) to L

= f(V

+ ((L

+ L

)2π f )

(24)

= Z

(25)

Figure 8: variation of Lm as function of voltage.

Where the blue dots are experimental results and the

red curve is a fourth order curve ﬁt given by :

= −0.003V

− 0.01V

− 0.013V

− 0.02V

+ 0.19

The magnetizing inductance varies with voltage as

shown in Figure 8. At the start of self-excitation

where the voltage is near to zero, Lm is close to

0.205H. Once self-excitation begins, the generated

voltage will develop and thenL

also increases. When

there is an increase in L

, it increases the value of

the positive real root of the characteristic equation

and consequently the generated voltage grows faster.

Then, L

decreases while the voltage continues to

grow until it reaches its steady state value determined

by : the L

value, capacitance and the rotor speed.

2.5 Impact of Lm on Voltage Build-up

According to the previous ﬁgure, we can divide the

curve in stable an instable area B and A. If the SEIG

starts to generate in region A, a small loss in speed

will cause a drop in voltage and this will bring a de-

crease in L

, which in turn decreases the voltage, and

ﬁnally the voltage will dampen to zero. Once the volt-

age dampens, there is no transient phenomenon and

there will not be voltage build up even if the speed

increases once again to its initial value as shown in

ﬁgure 9.

Figure 9: decrease in rotor speed and its effect on

generation of voltage in the unstable area A.

Area B is a stable operating region. When the speed

of the prime mover decreases, voltage will decrease,

and L

increases, which yield the SEIG to continue to

operate at a lower voltage as shown in Figure 10.

Figure 10: reduction of the rotor speed and its impact on

generation of voltage in the stable area B.

The growth in L

means an increase in the positive

real roots of the characteristic equation.

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3 SIMULATION RESULTE

The induction machine used as the SEIG in this study

is a three-phase induction motor with speciﬁcations

(APPENDIX-table 1). When the induction machine

is driven by a prime mover, the voltage will start to

develop at a corresponding minimum capacitance at

a giving rotor speed. This capacitance is obtained by

solving the roots of the 8th order polynomial equation

given by (21) and then searching if there is a positive

real part of the roots. The curve in ﬁgure 11 gives

the values of minimum capacitance at different rotor

speed at no load.

Figure 11: minimum capacitance as function of rotor speed

at no load.

In this investigation, the minimum capacitance re-

quired has a value of 35.3 µF . The ﬁgure 12 shows

the progression of different waveform of current and

voltage of the SEIG.

Figure 12: progression of current and voltage waveform

with capacitance value C = 35.3 µF.

As we notice the transients are growing with the pro-

gression of time, in theory, these transients will in-

crease to inﬁnity, but in this case, it will increase

until the circuit is saturated so the self-excitation is

achieved. If we reduce the capacitance value from

35.3 µF to 30.5 µF the self-excitation does not come

true as shown in ﬁgure 13, and this value of 30.5

µF yield to a characteristic equation with roots which

have a negative real part.

Figure 13: progression of current and voltage waveform

with capacitance value C = 30.5 µF.

In order to visualize the voltage, a Simulink model of

the SEIG at no load is developed using matrix (19) as

shown in ﬁgure 14, this model gives us the possibility

to visualize the dynamic of our system and the impact

of capacitance value on voltage build up.

Figure 14: model of the SEIG at no load.

The voltage curve is given by the ﬁgure 15. Accord-

ing to this curve, we can notice that we have a voltage

build up at the choosing rotor speed and this validate

the value of capacitance giving by the algorithm de-

veloped in this study.

Figure 15: voltage progression of the SEIG at no load with

capacitance value of C = 35.3 µF.

To conﬁrm that 35.5 µF is the minimum value, we re-

Elaboration of the Minimum Capacitor for an Isolated Self Excited Induction Generator Driven by a Wind Turbine

269

duce another time the capacitance value to 30.5 µF.

The ﬁgure 16 shows that the voltage collapses be-

cause the loss of the self-excitation phenomenon and

that return to the roots of the characteristic equation.

Figure 16: none build-up of the voltage of the SEIG at no

load with capacitance Value of C = 30.5 µF.

4 CONCLUSION

In this study, a simple algorithm is proposed to obtain

the minimum requirement of the capacitance for self

excited induction generator under no-load conditions

for different speeds. This study demonstrates, also,

how excitation capacitance and prime mover speed

affect the steady state performance of SEIG. The min-

imum value of the excitation capacitor must be prop-

erly calculated in order to assure an effective start-

ing of the SEIG. Furthermore, it was concluded that

SEIG has a critical excitation capacitor value at con-

stant rotor speed. We notice that the output voltage

magnitude follows the variation of the rotor speed,

decreasing the rotor speed will lead to decrease the

output voltage. This reveals the voltage regulation is-

sue of the self-excited induction generator when it is

used for wind application. Consequently, if our steady

state point is located in the stable area of Lm curve,

we can have proportionally between voltage and ro-

tor speed, however, if we are located in the instability

area a decrease in rotor speed will lead to a collapse

of the voltage, thereafter, demagnetization of our in-

duction machine.

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APPENDIX

Rated power 15 KW

Voltage 440V Y

Frequency 50 Hz

Pair pole 4

Rated speed 385 rd/s

Stator resistance 0.696 Ω

Rotor resistance 0.743 Ω

Inductance stator 1.1 mH

Inductance rotor 1.1 mH

Table 1: Induction generator parameters.

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