EMP 32(9) #12845

Electric Power Components and Systems, 32:901–912, 2004

Copyright
c Taylor & Francis Inc.
ISSN: 1532-5008 print/1532-5016 online
DOI: 10.1080/15325000490253542

Automatic Reactive Power Control of Isolated

Wind-Diesel Hybrid Power Systems for
Variable Wind Speed/Slip

R. C. BANSAL
Electrical & Electronics Engineering Department
Birla Institute of Technology & Science
Pilani, Rajasthan, India

T. S. BHATTI
D. P. KOTHARI
Centre for Energy Studies
Indian Institute of Technology
Hauz Khas, New Delhi, India

In this article, automatic reactive power control of an isolated wind-diesel

hybrid power system having an induction generator coupled with wind turbine
for power generation is presented. The mathematical model of the system
using reactive power flow equations is developed. The dynamic voltage stability
evaluation is based on small signal analysis considering a typical static VAR
compensator (SVC) and IEEE type-I excitation system. It is shown that a
variable reactive power source, like SVC, is a must to meet the varying demand
of reactive power by induction generator and load and to obtain a very good
voltage regulation of the system with minimum fluctuations. Integral square
error (ISE) criterion is used to evaluate the optimum setting of gain parameters.
Finally, the dynamic responses of the sample power system considered with
optimum gain setting are also presented.

Keywords induction generator, isolated system, static VAR compensator,

wind-diesel hybrid power system

1. Introduction
In recent years, much emphasis has been placed on the squirrel cage induction
machine as the electromechanical energy converter in generation schemes involving
renewable energy sources [1–4]. The advantages of the induction generator over the
synchronous generator are low cost, robustness, no moving contacts, i.e., slip-rings,
no synchronization required, and no need for d.c. excitation. But the induction

Manuscript received in final form on 24 June 2003.

Address correspondence to Prof. R. C. Bansal, Birla Institute of Technology &
Science, Electrical & Electronics Engineering, Pilani, Rajasthan 333031, India. E-mail:
rcbansal@bits-pilani.ac.in

901
902 R. C. Bansal et al.

machine requires a reactive power support for its operation [5–10]. A large number
of articles have appeared in the literature on the subject and a few investigate
the capacitance requirement of self-excited induction generator under steady state
conditions only [1–11]. It has practical significance as it enables the design and
operation engineers to select the proper value of excitation capacitance for a specific
machine. In a stand-alone hybrid power system, the reactive power device has to
fulfill the variable reactive power requirement of the induction generator and of the
load. In the absence of proper reactive power device and controls, the system may
be subjected to large voltage fluctuations, which is not desirable. The device used
for this function in conventional power systems is known as static VAR compensator
(SVC) [12–17] and can also be employed for the hybrid system.
In a conventional power system, the power is exported on transmission lines
to load centers. The reactive power devices are employed in such a way to have
minimum reactive power flow on the transmission lines so that maximum power
can be exported with minimum transmission losses. In hybrid systems, the load is
directly connected to the generator terminals themselves. Therefore, the objective
of the reactive power device in this case is to supply the reactive power required by
the load and the induction machine under varying load conditions. In general, in
any hybrid power system there will be more than one type of electrical generators
[18]. In such circumstances, it is normal, though not essential, for generator(s),
usually on diesel, to be synchronous, and wind/hydro turbine generator(s) to be
asynchronous (induction). An isolated wind-diesel hybrid power system has been
considered for study having induction generator for wind power conversion and
synchronous alternator with automatic voltage regulator (AVR) for diesel unit.
The devices AVR and SVC both operate on the voltage error signal caused by any
disturbance in the system, but have different functions. The main function of the
AVR is to maintain the voltage profile constant at the terminals. The alternator
also provides partial reactive power to the load. Similarly, the main function of
SVC is to eliminate the mismatch of reactive power in the system. The SVC also
partially helps in voltage maintenance at the terminals. A new innovative scheme,
namely, automatic reactive power control, similar to automatic generation control
[19] has been evolved. The system state equations have been derived with transfer
function block diagram representation of the control system. The voltage deviation
signal is used as area reactive power control error to eliminate the reactive power
mismatch in the system. The integral square error (ISE) criterion [20] is used to
evaluate the optimum setting of gain parameters of the controller. Finally transient
responses are shown for different disturbance conditions.

2. Incremental Reactive Power Balance Analysis

A wind-diesel system is considered for mathematical modeling, where a diesel
generator (DG) set acts as a local grid for the wind energy conversion system
connected to it. The system also has a SVC to provide the required reactive power
in addition to the reactive power generated by the synchronous generator. Small
changes in real power are mainly dependent upon the frequency, whereas small
change in reactive power is mainly dependent on voltage [19]. The excitation time
constant is much smaller than the prime mover time constant and its transient
decay much faster and does not affect the load frequency control (LFC) dynamic.
Thus cross coupling between LFC and AVR loop is negligible. The reactive power
 Reactive Power Control of Wind-Diesel Systems 903

balance equation of the system under steady state condition is

QSG + QSVC = QL + QIG (1)

where
QSG = reactive power generated by diesel generator set,
QSVC = reactive power generated by SVC,
QL = reactive power load demand, and
QIG = reactive power required by induction generator.
For the incremental reactive power balance analysis of the hybrid system, let the
hybrid system experience a reactive power load change of magnitude ∆QL . Due to
the action of the AVR and SVC controllers, the system reactive power generation
increases by an amount ∆QSG +∆QSVC . The reactive power required by the system
will also change due to change in voltage by ∆V . The net reactive power surplus
in the system, therefore, equals ∆QSG + ∆QSVC − ∆QL − ∆QIG , and this power
will increase the system voltage in two ways:
• By increasing the electromagnetic energy absorption EM of the induction
generator at the rate d/dt (EM ),
• By an increased reactive load consumption of the system due to increase in
voltage.
This can be expressed mathematically as

∆QSG + ∆QSVC − ∆QL − ∆QIG = d/dt(∆EM ) + DV ∆V (2)

The electromagnetic energy stored in the winding of the induction generator is

given by
1 2 1
EM = LM IM = LM (V /XM )2 (3)
2 2
where XM is the magnetizing reactance of the induction generator. Considering
approximate equivalent circuit model [20], Eq. (3) can be further written as

1 2πf Lm Xm V2
EM = (V /Xm )2 = (V /Xm )2 = (4)
2 2πf 4πf 4πf Xm
For nominal conditions, Eq. (4) can be written as

0 1
EM = (V 0 )2 (5)
4πf Xm
Dividing Eq. (4) by Eq. (5) we get

2
EM V
0 =
EM V0
(6)

Now V = V 0 +∆V and ∆V is small; therefore, Eq. (6) can be written by neglecting
∆V 2 terms as
EM (V 0 + ∆V )2 ∆V
0 = =1+2 o (7)
EM (V 0 )2 V
904 R. C. Bansal et al.

From Eq. (7), ∆EM can be written as

0

0 EM
∆EM = EM − EM =2 ∆V (8)
V0
With increase in voltage all the connected loads experience an increase by DV =
∂QL /∂V p.u. kVAR /pu kV. The parameter DV can be found empirically. The
composite loads are expressed in the exponential voltage form as [22, 23]

QL = C1 V q (9)

where C1 is the constant of the load and the exponent q depends upon the type of
load. For small perturbations Eq. (9) can be written as

∆QL /∆V = q(QoL /V o ) (10)

In Eq. (2), DV can be calculated empirically using Eq. (10). Let QR be the system
reactive power rating. Dividing by QR and using Eq. (5), Eq. (2) can be rewritten
as [24]
o
∆QSG + ∆QSVC − ∆QL − ∆QIG = 2EM /(V o QR )d/dt(∆V ) + DV ∆V (11)

In Eq. (11), QR divides only one term as the other terms are expressed in p.u.
o
kVAR. The term EM /QR can be written as
o
EM /QR = 1/(4πf kR ) = HR (12)

where HR is a constant of the system and its units are sec. and kR is the ratio
of system reactive power rating to rated magnetizing reactive power of induction
o
generator. Substituting the value of EM /QR from Eq. (12) in Eq. (11) we get

∆QSG + ∆QSVC − ∆QL − ∆QIG = 2HR /V o d/dt(∆V ) + DV ∆V (13)

In Laplace form, the above differential equation can be written as

∆V (s) = KV /(1 + sTV )[∆QSG (s) + ∆QSVC (s) − ∆QL (s) − ∆QIG (s)] (14)

where
2Hr
TV = (15)
DV V o
and
1
KV = (16)
DV

2.1. The Synchronous Generator Equation

Under transient condition, QSG is given by [24]

QSG = (Eq V cos δ − V 2 )/Xd (17)

For small perturbation, Eq. (17) can be written as

∆QSG = (V cos δ/Xd )∆Eq + (Eq cos δ − 2V )/Xd ∆V (18)

 Reactive Power Control of Wind-Diesel Systems 905

where ∆Eq = change in the internal armature emf proportional to the change in
the direct axis field flux under transient condition.
Taking Laplace transform of both sides we get

∆QSG (s) = K1 ∆Eq (s) + K2 ∆V (s) (19)

where

K1 = V cos δ/Xd (20)

and

K2 = (Eq cos δ − 2V )/Xd (21)

2.2. SVC Equations

The reactive power supplied by the SVC is given by [16]

QSVC = V 2 BSVC (22)

For small perturbation, Eq. (22) can be written by taking Laplace transform as

∆QSVC (s) = K3 ∆V (s) + K4 ∆BSVC (s) (23)

where

K3 = 2V BSVC and K4 = V 2 (24)

The state equations of a typical SVC having regulator, firing delay and phase
sequence delay, in Laplace transform can be written as
1 
∆BSVC (s) = ∆BSVC (s) (25)
1 + sTd
 Kα
∆BSVC (s) = ∆α(s) (26)
1 + sTα
KR
∆α(s) = [∆Vref (s) − ∆V (s)] (27)
1 + sTR

2.3. The Flux Linkage Equation

The flux linkage equation [25] of the round rotor synchronous machine for small
perturbation is

d/dt(∆Eq ) = (∆Ef d − ∆Eq )/Tdo


(28)

where ∆Eq = change in the internal armature emf proportional to the change in
the direct axis field flux under steady state condition.

Tdo = direct axis open circuit transient time constant

In Eq. (28), Eq is given by

Eq = (Xd /Xd )Eq − (Xd − Xd )/Xd V cos δ (29)

906 R. C. Bansal et al.

For small changes, Eq. (28), using Eq. (29) and taking Laplace transform can be
written as
(1 + sTg )∆Eq (s) = K1 ∆Ef d (s) + K2 ∆V (s) (30)
where
Tg = Xd Tdo

/Xd (31)
K5 = Xd /Xd (32)
K6 = (Xd − Xd ) cos δ/Xd (33)

2.4. The Induction Generator Equation

The real power input, PIW and reactive power QIG , absorbed by the induction
generator can be written in terms of generator terminal voltage, slip, and generator
parameters. These equations can be written for small perturbation, by eliminating
deviation in slip, ∆s as [24, 26]
∆QIG (s) = K7 ∆Pm (s) + K8 ∆V (s) (34)
where
Xeq
K7 = (35)
RP − (RY2 + Xeq
2 )/2R
Y
 
2V RP Xeq
K8 = 2 = X eq − (36)
2
RY + Xeq (RP − (RY2 + Xeq
2 )/2R )
Y

where
r2
RP = (1 − s) (37)
s
RY = RP − Req (38)
Req = r1 + r2 (39)
Xeq = x1 + x2 (40)
r1 , x1 , r2 , x2 are the parameters of the induction generator.

3. Mathematical Modelling of Wind/Diesel System

The block diagram of the system is shown in Figure 1. The state equations in a
standard form can be written as
ẋ = Ax + Bu + Cp (41)
where x, u, and p are state, control and disturbance vectors and A, B, and C are
system, control and disturbances matrices, respectively. The vectors are given by
 T
x = ∆Ef d ∆Va ∆Vf ∆Eq ∆α ∆BSVC 
∆BSVC ∆V (42)
u = [∆Vref ] (43)
 T
p = ∆QL ∆PIW (44)
The elements of the matrices can be obtained from the state equations in the
Laplace form as derived above.
 Reactive Power Control of Wind-Diesel Systems 907

Figure 1. Transfer—function block diagram for reactive power control of wind/diesel

hybrid power system with variable slip/speed.

4. Computer Simulation and Results

The data of the wind-diesel power system considered for simulation is given in Ap-
pendix 1. The gains are optimized using the Liapunov technique [19] for continuous
linear systems with the performance index based upon the integral square error
criterion and is given by

η = [∆V (t)]2 dt (45)
908 R. C. Bansal et al.

Figure 2. Optimization of amplifier gain of wind-diesel isolated hybrid power system

with a typical SVC scheme.

The optimum value of the parameters corresponds to the minimum value of the
performance index. In the studies carried out in this paper η is evaluated over a
time period of 2 seconds. The performance index curve for 1% step increase in
reactive load demand is shown Figure 2. The minimum value of the gain parameter
obtained is KR = 575.
The transient response curves of the system for 1% step increase in reactive
load, and 1% step increase in reactive load with the same increase in wind power for
optimum gain settings are shown in Figure 3. It is observed that the deviation in
both the system voltage and firing angle vanishes in about 0.2 sec. It is observed that
increase in reactive power requirement ∆QL /(∆QL + ∆QIG ) due to disturbances is
met by reactive power ∆QSVC supplied by the SVC with negligible reactive power,
∆QSG supplied by the synchronous generator. It is observed from Figure 3(e) that
the reactive power requirement of the induction generator varies considerably with
change in input wind speed/slip, i.e., real input wind power. The system returns
to steady state conditions in 7 12 cycles of the supply frequency following a step
load disturbance of 1%. It indicates that the AVR controls the system voltage
and the SVC provides the reactive power required by the isolated hybrid power
system.

5. Conclusions
A dynamic voltage stability study has been presented in this article for the hybrid
wind-diesel isolated power system considering transfer function model based on
small signal analysis. The automatic reactive power control model using reactive
power flow equations have been developed for the isolated hybrid systems. The
integral square error criterion has been used to evaluate the optimum gain settings.
It has been shown that SVC is essential for an isolated hybrid power system to
meet the varying demand of reactive power by induction generator and load and to
have minimum voltage fluctuations. Finally, some of the system transient responses
have been shown for optimum gain settings.
 Reactive Power Control of Wind-Diesel Systems 909

(a)

(b)

(c)

Figure 3. (a)–(e) Transient responses of the wind-diesel hybrid power system. ---, for 1%
step increase in reactive power load plus 1% step increase in input wind power, —, for 1%
step increase in reactive power load. (continued )
910 R. C. Bansal et al.

(d)

(e)

Figure 3. (Continued ) (a)–(e) Transient responses of the wind-diesel hybrid power sys-
tem. ---, for 1% step increase in reactive power load plus 1% step increase in input wind
power, —, for 1% step increase in reactive power load.

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Appendix 1
Ratings and data of the typical example of the isolated power system studied

Generation Capacity (kW) Load (kW)

Wind 150.0 150.0

Diesel 150.0 100.0
Total 300.0 250.0

System parameters

Eq = 1.1136 p.u. δ = 21.05◦ Xd = 1.0 p.u. Xd = 0.15 p.u.

Eq = 0.9603 p.u. 
Tdo = 5.0 sec. TE = 0.55 sec. KA = 40.0
TF = 0.715 sec. KF = 0.5 KE = 1.0 KR = 337.0
TA = 0.05 sec. TR = 0.05 sec. Xeq = 1.12 p.u. QL = 0.75 p.u.
RY = 4.06415 p.u. BSVC = 0.73 p.u. α = 2.443985 q = 2.0
XR = 1.0/0.85 p.u. Tα = 0.02/4 sec. Td = 0.02/12 sec.