Energy and Power En gi neering, 2011, 3, 87-95
doi:10.4236/epe.2011.32012 Published Online May 2011 (http://www.SciRP.org/journal/epe)
PSS and SVC Controller Design Using Chaos, PSO and
SFL Algorithms to Enhancing the Power System Stability
Saeid Jalilzadeh, Reza Noroozian, Mahdi Sabouri, Saeid Behzadpoor
Electrical Engineering Department, Zanjan University, Zanjan, Iran
Received January 8, 2011; revised March 7, 2011; accepted March 15, 2011
Abstract
In this paper, the Authors present the designing of Power System Stabilizer (PSS) and Static Var Compensa-
tor (SVC) based on Chaos, Particle Swarm Optimization (PSO) and Shuffled Frog Leaping (SFL) Algo-
rithms has been presented to improve the power system stability. Single Machine Infinite Bus (SMIB) sys-
tem with SVC located at the terminal of generator has been considered to evaluate the proposed SVC and
PSS controllers. The coefficients of PSS and SVC controller have been optimized by Chaos, PSO and SFL
algorithms. Finally the system with proposed controllers is simulated for the special disturbance in input
power of generator, and then the dynamic responses of generator have been presented. The simulation results
show that the system composed with recommended controller has outstanding operation in fast damping of
oscillations of power system and describes an application of Chaos, PSO and SFL algorithms to the problem
of designing a Lead-Lag controller used in PSS and SVC in power system.
Keywords: Power System Stabilizer (PSS), Static Var Compensator (SVC), Single Machine Infinite Bus
(SMIB), Chaos, Shuffled Frog Leaping (SFL), Particle Swarm Optimization (PSO)
1. Introduction
Power systems experience low frequency oscillations (in
the range of 0.1 Hz to 2.5Hz) during and after a large or
small disturbance has happened to a system, especially
for middle to heavy loading conditions [1]. These oscil-
lations may sustain and grow to cause system separation
if no adequate damping is available [2]. Power System
Stabilizers (PSSs) are the most cost effective devices
used to damp low frequency oscillations. For many years,
Conventional PSSs (CPSSs) have been widely used in
the industry because of their simplicity [3]. To improve
the performance of CPSSs, numerous techniques have
been proposed for their design, such as using intelligent
optimization methods (simulated annealing, genetic al-
gorithm, tabu search) [4], fuzzy, neural networks and
many other nonlinear control techniques. During some
operating conditions, PSS may not produce adequate
damping, and other effective alternatives are needed in
addition to PSS. Recent development of power electron-
ics introduces the use of Flexible AC Transmission Sys-
tems (FACTS) controllers in power systems [5]. FACTS
utilize high power semiconductor devices to control the
reactive power flow and thus the active power flow of
the transmission system so that the ac power can be
transmitted through a long distance efficiently [6]. The
conception of FACTS as a total network control phi-
losophy was first introduced by N.G. Hingorani [7] from
the Electric Power Research Institute (EPRI) in the USA
in 1988, although the power electronic controlled devices
had been used in the transmission network for many
years before that. The FACTS devices may be connected
so as to provide either series compensation or shunt
compensation depending upon their compensating
strategies. [8]. Nowadays, Static Var Compensator (SVC)
is one of the key elements in the power system that pro-
vides the opportunity to compensate reactive power and
reliability due to its fast response. SVC has the func-
tional capability to handle dynamic conditions, such as
transient stability and power oscillation damping in addi-
tion to providing voltage regulation [6]. Due to the char-
acteristics of power transmission systems, the FACTS
Compensator control algorithm must be designed resort-
ing to control methods capable to deal with system
non-linearities and unknown disturbances [9]. In this
paper the PSS and SVC have the same controller, that
their coefficients have been optimized by PSO, Chaos
and SFL algorithms. Then the system with proposed

88

S. JALILZADEH ET AL.
controller has been simulated for the special disturbance
and the dynamic response of generator has been pre-
sented.
2. Model of Proposed System
A synchronous machine with an IEEE type-ST1 excita-
tion system connected to an infinite bus through a trans-
mission line has been selected to demonstrate the deriva-
tion of simplified linear models of power system for dy-
namic stability analysis [2,10]. Figure 1 shows the
model consists of a generator supplying bulk power to an
infinite bus through a transmission line, with an SVC
located at its terminal. The equations that describe the
generator and excitation system have been represented in
following equations:

01
 
 (1)



1
me
D

   
 (2)
 
qfddd q
EXXid 


do



(3)


f
dAreftpssfd
EKVVU ET

A
(4)
where, m and e are the input and output powers of
the generator, respectively. M and D are the inertia con-
stant and damping coefficient, respectively. 0
 

is the
synchronous speed. δ and ω are the rotor angle and speed,
respectively.
where, is the internal voltage.
q
E
f
d is the field
voltage.
E
d



is the open circuit field time constant.
d and d are the d-axis reactance and the d-axis
transient reactance of the generator, respectively. A


K
and A are the gain and time constant of the excitation
system, respectively. ref is the reference voltage. t
V
is the terminal voltage. Also and can be ex-
pressed as:
T
V
t
Ve 
ttd t
VV jV
q
q

q

q

(5)
tdq q
V   (6)
tqqd d
V 

 (7)
etddtq
VV   (8)
where, Xq is the q-axis reactance of the generator.

12 3
sin
dqb
CC VC  

  (9)

45 6
cos
dqb
CCV C  
   (10)
Solving (9) and (10) simultaneously, d and q


expressions can be obtained. 1 until 6 are constant
and b is the infinite bus voltage. The various parame-
ters of the system have been represented in Table 1.
C C
V
Gen
q




SVC
Inf i n i ti ve
Bus
V
b
< 0
X
e
Re
V
t
I
d
, I
q
Figure 1. Single machine-infinite bus system model with
SVC.
Table 1. System parameters.
Xd = 1.7 Xq = 1.64 d

= 0.245
H = 2.37 τ'do = 5.9 XT = 0.08
Xe = 0.4 Re = 0.02 D = 0
3. Static Var Compensator
A Static Var Compensator (or SVC) is an electrical device
for providing fast-acting reactive power on high-voltage
electricity transmission networks. SVCs are part of the
Flexible AC transmission system device family, regulat-
ing voltage and enhance the transient stability [11] and
provide additional damping to power systems as well
[12]. SVC is mainly operated at load side bus and used
as replacement for existing voltage control devices [10].
A basic topology of SVC consists of a series capacitor
bank C in parallel with a thyristor controlled reactor L, is
shown in Figure 2. The SVC can be seen as an adjust-
able susceptance which is a function of thyristors firing
angle.
4. Power System Linearized Model
A linear dynamic model is obtained by linearizing the
nonlinear model round an operating condition (Pe = 1,
Qe = 0.59). The linearized model of power system as
shown in Figure 1 is given as follows:
o




 (11)
me
D







 (12)


f
dddd
q
do
i
Eq

 

  

 
 (13)




f
dAreftpss fd
EKVVUET

 

A
S VC
(14)
78q
I
ccB


 (15)
910 11dq
Ic ccB
SVC




  (16)

S. JALILZADEH ET AL.
89


11 9dd
cK 
 
(21)
C
L
V
t
5. Chaos Algorithm
Chaos is a general phenomenon in non-line system. It
can get all the states in the search space by the rules of
itself. Moreover, a tiny change of initial values can lead
to a big change of the system. The Chaos search can gen-
erate the neighbourhoods of near-optimal solutions to
maintain solution diversity. It can prevent the search
process from becoming premature. The Chaos optimiza-
tion method based on Chaos Search is proposed to avoid
the local optimal [13]. Chaos variables are usually gen-
erated by the well known logistic map. Figure 4 shows
the flowchart of Chaos algorithm. The logistic map is a
one-dimensional quadratic map defined by following
equation:
Figure 2. Basic SVC topology.



 
11
iii
kk   
 
k
(22)
12 3eqsvc
K
KK 

B
svc
(17)
where,

is a control parameter and


00
i
 
4
1.
Despite the apparent simplicity of the equation, the solu-
tion exhibits a rich variety of behaviours. For


system (22) generates chaotic evolutions. Its output is
like a stochastic output, no value of i is repeated
and the deterministic equation is sensitive to initial con-
ditions. Those are the basic characteristics of Chaos.
Chaos variable

k 


0
i
 is mapped into the variance
ranges of optimisation variables by the following equa-
tions [14]:
45 6tq
VK KKB



  (18)
1
K
until 6
K
are linearization constants. The block
diagram of the linearized power system model is shown
in Figure 3. 7
K
, 8
K
, and 9
K
are constants defined as
follows:

9dd
cK   
7
K
(19)

10 8
1
dd
c 
 
(20)
Figure 3. Block diagram of the linearized model.

S. JALILZADEH ET AL.
90
Figure 4. Flowchart of the Chaos algorithm.
 


2
iiii
xk xk 

 1 (23)
 

0.01, ,
iiiii
ba xab  
i
(24)
where, x is optimization variable,
x
 is the best ex-
periment of variable, and

is the feasible region.
6. PSO Algorithm
The particle swarm optimization (PSO) algorithm was
first proposed by Kennedy and Eberhart [15]. Where is a
novel evolutionary algorithm paradigm which imitates
the movement of birds flocking or fish schooling looking
for food. Each particle has a position and a velocity, rep-
resenting the solution to the optimization problem and
the search direction in the search space the particle ad-
justs the velocity and position according to the best ex-
periences which are called the best found by it and
best
p
g
found by all its neighbors. In PSO algorithms each
particle moves with an adaptable velocity within the re-
gions of decision space and retains a memory of the best
position it ever encountered. The best position ever at-
tained by each particle of the swarm is communicated to
all other particles. Figure 5 shows the flowchart of PSO
algorithm. The updating equations of the velocity and
position are given as follows [16]:
Figure 5. Flowchart of the PSO algorithm.

S. JALILZADEH ET AL.
91


1
 

11
22
1
iiii
gi i
vkwvkrc pxk
rcpx k
 




(25)

1
iii
xkxk vk  (26)
where v is the velocity and x is the position of each par-
ticle. 1 and 2 are positive constants referred to as
acceleration constants and must be , usually
12 . 1 and 2 are random numbers between 0
and 1, w is the inertia weight, refers to the best posi-
tion found by the particle and
c c
r

12 4cc
2cc r
p
g
p refers to the best po-
sition found by its neighbors.
7. SFL Algorithm
The SFL algorithm is a meta heuristic optimization
method that mimic the memetic evolution of a group of
frogs when seeking for the location that has the maxi-
mum amount of available food. The algorithm contains
elements of local search and global information ex-
change ([17,18]). The SFL algorithm involves a popula-
tion of possible solutions defined by a set of virtual frogs
that is partitioned into subsets referred to as memeplexes.
Within each memeplex, the individual frog holds ideas
that can be influenced by the ideas of other frogs, and the
ideas can evolve through a process of memetic evolution.
The SFL algorithm performs simultaneously an inde-
pendent local search in each memeplex using a particle
swarm optimization like method. To ensure global ex-
ploration, after a defined number of memeplex evolution
steps (i.e. local search iterations), the virtual frogs are
shuffled and reorganized into new memeplexes in a
technique similar to that used in the shuffled complex
evolution algorithm. In addition, to provide the opportu-
nity for random generation of improved information,
random virtual frogs are generated and substituted in the
population if the local search cannot find better solutions.
The local searches and the shuffling processes continue
until defined convergence criteria are satisfied. The
flowchart of the SFL algorithm is illustrated in Figure 6.
The SFL algorithm is described in details as follows.
First, an initial population of N frogs

12
,,,
N
PXXX

T
is created randomly. For S-dimensional problems (S
variables), the position of a frog in the search space
is represented as 12
ii
s
th
i
,

,,
X
xx
n
x
m
Nmn
. Afterwards, the
frogs are sorted in a descending order according to their
fitness. Then, the entire population is divided into
memeplexes, each containing frogs (i.e.



1th m
),
in such a way that the first frog goes to the first meme-
plex, the second frog goes to the second memeplex, the
frog goes to the memeplex, and the
frog goes back to the first memeplex, etc. Let k
th
mth m
M
is the
set of frogs in the memeplex, this dividing process
th
k
Figure 6. Flowchart of the SFL algorithm.
can be described by the following expression:




11,1
kkml

.
M
XPknk

m (27)
Within each memeplex, the frogs with the best and the
worst fitness are identified as b
X
and w
X
, respec-
tively. Also, the frog with the global best fitness is iden-
tified as
g
X
. During memeplex evolution, the worst
frog w
X
leaps toward the best frog b
X
. According to
the original frog leaping rule, the position of the worst
frog is updated as follows:


bw
DrX X  (28)



max
,
ww
XnewXDD D 

, (29)

92 S. JALILZADEH ET AL.
where, is a random number between 0 and 1; and
max is the maximum allowed change of frog’s position
in one jump.
r
D
If this leaping produces a better solution, it replaces the
worst frog. Otherwise, the calculations in (28) and (29)
are repeated but respect to the global best frog (i.e. re-
places b
X
). If no improvement becomes possible in this
case, the worst frog is deleted and a new frog is randomly
generated to replace it. The calculations continue for a
predefined number of memetic evolutionary steps within
each memeplex, and then the whole population is mixed
together in the shuffling process. The local evolution and
global shuffling continue until convergence criteria are
satisfied. Figure 6 shows the flowchart of SFL algorithm.
Usually, the convergence criteria can be defined as fol-
lows:
 The relative change in the fitness of the best frog
within a number of consecutive shuffling iterations is
less than a pre-specified tolerance;
 The maximum user-specified number shuffling itera-
tions is reached.
The SFL algorithm will stop when one of the above
criteria is arrived first.
8. Simulation Results
The deviation of speed that obtained from linearization
has been selected for inputs of PSS and SVC controller
which is shown in Figure 7. As shown in this figure,
PSS and SVC have the same Lead-lag controller. The
constant values of Figure 7 have been represented in
Table 2.
The fitness function used in this paper for Chaos, PSO
and SFL algorithms is represented in Equation (30) that
s
im is the simulation time, is the deviation of
speed and is the deviation of terminal voltage of
generator.
tdw
t
dv
0
10*
sim
t
t
f
itnessdwdv dt





50
(30)
The deviation of speed () has been multiplied by
ten to both section of fitness have the same range. Con-
trol parameters and their boundaries are given as follows:
dw
0K (31)
1
0.01 1T (32)
2
0.01 1T (33)
The convergence rate of the fitness function with num-
ber of iterations for SFL, PSO and Chaos algorithms is
shown in Figure 8. As shown in Figure 8, the SFL algo-
rithm is faster than PSO and Chaos algorithm to achieve
the optimum coefficients. Table 3 shows the optimized
1
A
A
K
s
T
1
2
1
11
w
w
sT
s
T
K
s
TsT





1
s
s
K
s
T
1
2
1
11
w
w
sT
s
T
K
s
TsT





Figure 7. PSS and SVC contro lle r.
Table 2. Constant values.
KA [P.U] TA [P.U] Tw [P.U] Ks [P.U] Ts [P.U]
200 0.02 10 10 0.15
Figure 8. Convergence of SFL, PSO and Chaos algorithms.
Table 3. Optimized values.
SFL PSO Chaos
K 4.42 3.84 3.63
T1 0.164 0.18 0.19
T2 0.0015 0.01 0.012

S. JALILZADEH ET AL.
93
(a)
(b)
(c)
Figure 9. System dynamic response for a six cycle fault dis-
turbance. (a) Rotor speed variation; (b) Rotor angle varia-
tion; (c) Terminal voltage variation.
parameters that found by SFL, PSO and Chaos algo-
rithms. The final setting of the optimized parameters
have been given when the input power of generator has
been changed 5% instantaneously and the operating con-
dition was Pe = 1 and Qe = 0.59.
Figure 9 shows the system dynamic response for a six
cycle fault disturbance for rotor speed variation, rotor
angle variation and terminal voltage variation for SFL,
PSO and Chaos controllers, also non-controller.
As shown in these figures, it is clear that the perform-
ance of PSS and SVC controller has good damping
characteristics for low frequency oscillations. However,
this improves greatly the power system stability. Also the
SFL algorithm has pretty faster behavior in convergence
than PSO and Chaos algorithms.
9. Conclusions
In this paper the SMIB system where SVC located at the
terminal of generator has been considered. The SVC and
PSS have the same controller where their optimized co-
efficients have been earned by Chaos, PSO and SFL al-
gorithms. In order to show the excellent operation of
proposed controller, the input power of generator has
been changed 5% instantaneously and the system with
proposed controllers has been simulated, then the dy-
namic response of generator for rotor speed variation,
rotor angle variation and terminal voltage variation have
been represented. The effectiveness of the proposed PSS
and SVC controllers for improving transient stability
performance of a power system are demonstrated under
different operating conditions. The simulation results
shown that the system composed with proposed control-
ler has superior operation in fast damping of oscillations
of power system. Also the results show that SFL algo-
rithm has pretty faster behavior in convergence than PSO
and Chaos algorithms. This procedure can be easily ap-
plied to the systems with similar performances.
Humphreys for English editing. All errors are ours.
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S. JALILZADEH ET AL.
95
Notation
m
Ρ
Ρ
The input power of the generator
e The output power of the generator
M
The inertia constant
D The damping coefficient
0

The synchronous speed

The rotor angle

The rotor speed
q
E The internal voltage
f
d
'd
E The field voltage

The open circuit field time constant
d
 The d-axis reactance of the generator
q
X
The q-axis reactance of the generator
d
X
 The d-axis transient reactance of the generator
A
K
The gain of the excitation system
A
V
T The time constant of the excitation system
ref
V
The reference voltage
t
C
The terminal voltage
1
C
V
6
6
The constants
b The infinite bus voltage
1
K
K The linearization constants
79
K
K
,
The constants defined in (19), (20), (21)
i

The basic characteristics of Chaos
x
The optimization variable of Chaos
x

The best experiment of variable of Chaos

The feasible region of Chaos
V The velocity of PSO
X
The position of each particle of PSO
1
andcc
2
2
The positive constants referred to as accelera-
tion
1
andvv
W
The random numbers between 0 and 1 in PSO
The inertia weight in PSO
and
g
pp The best position found by the particle and the
best position in PSO, respectively
b
X
The frog with the best fitness of SFL
w
X
The frog with the worst fitness of SFL
g
X
The frog with the global best fitness of SFL
r A random number between 0 and 1 in SFL
max
D The maximum allowed change of frog’s posi-
tion in one jump in SFL
s
im
t
dw
The simulation time
The deviation of speed
t
dv The deviation of terminal voltage of generator