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OPTIMAL TUNING OF POWER SYSTEMS STABILIZERS (PSS) AND AUTOMATIC
VOLTAGE REGULATOR (AVR) GAINS USING PARTICLE SWARM
OPTIMIZATION TECHNIQUE1U. Jafaru,
2G. A. Bakare,
1B. U. Musa and
1M. Abdulkadir
1Department of Electrical and Electronics Engineering, University of Maiduguri,
Borno State.e-mails: jafaru_usman@yahoo.com, musa_bu@yahoo.com, muabdkadir@yahoo.com2
Electrical and Electronics Engineering Programme, Abubakar Tafawa Balewa
University, Bauchi, Bauchi State.
e-mail: bakare_03@yahoo.com
AbstractPower System Stabilizers (PSS) resolves the instability problem which results from the use of
static excitation and long distance transmission line in the power industry. Simultaneous and
coordinated tuning of stabilizer parameters and automatic voltage regulator (AVR) gains in
Single machine infinite bus (SMIB) power systems is considered. This problem is formulated as
an optimization problem. This research focuses on the optimal tuning of the Power SystemStabilizer (PSS) parameters and Automatic Voltage Regulator (AVR) gains using particle swarm
optimization (PSO) technique. It is aimed at minimizing the damping ratio of the power systemby optimally determining the values of the control parameters of the PSS/AVR. The
effectiveness of the PSO technique for tuning PSS/AVR in a large scale power system was tested
by applying it to a SMIB. MATLAB version 7.0 was used as a simulation platform to determinethe effect of the PSS/AVR. The simulation results obtained are the speed deviation, rotor angle
deviation and the terminal voltage with PSO based PSS/AVR, with conventional PSS/AVR and
without PSS/AVR response in per unit value. Simulation results on SMIB revealed that the PSO
based tuning of PSS/AVR damped faster and has less overshoot and settling time (2 sec) ascompared with the conventional PSS/AVR (5 sec) and without PSS/AVR (9 sec). The results
obtained has significantly shown the effectiveness and robustness of the particle swarm
optimization algorithm in solving power system stability problems.
Keywords: Power system stabilizers, Automatic Voltage Regulator, Particle swarm
optimization Single Machine Infinite Bus System.
1. IntroductionThe need for improved power availability and power system quality has been increasing over the
years. This has made a lot of generators to operate continuously and to maintain the electricitysupply within statutory limits of voltage and frequency. Large transmission distances and highly
complex distribution system in Nigeria may lead to dynamic instability, (this instability can lead
to disturbance and frequency oscillation. Power systems experiences low frequency oscillations
due to disturbances, an increase in damping of the system response is desirable, not only becauseit reduces the fluctuations in the controlled variables and hence improving the quality of power
supply, but mainly because damping is translated into an increase in the power transmission
stability limits. Higher stability limits brings significant economic savings as the need for theexpansion of the transmission system can be ignored (Kundur, 1994). The fundamental idea of
Power System Stabilizer is to provide an additional positive damping to the system by
introducing a stabilizing signal of proper phase through the excitation system.
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Over the past four decades, many researches have been published on the design of power
system stabilizers (PSS); to resolve the dynamic instability problem which results from thewidespread use of static excitation systems and long-distance transmission lines in multimachine
power systems. Most of these works are concerned with the tuning of suitable stabilizer
parameters to achieve satisfactory damping characteristics of the system (Gibbard, et-al, 2001).
Low frequency oscillations if sustained may grow to cause system collapse if adequate dampingis not provided. To enhance system damping, the generators are equipped with power system
stabilizers (PSSs) that provide supplementary feedback stabilizing signals in the excitation
system (Feliachi et-al 1988). This is an easy, economical and flexible way to improve powersystem stability in interconnected AC power systems.
The concept of synchronous machine stability as affected by the excitation control is
presented in Do-Bomfim, et-al (2000). Gibbard et-al (2001) used the sensitivity of mechanicalmode damping with respect to stabilizer gain to pick out the generator that is most effective for
stability enhancement using power system stabilizers. Do-Bomfin et-al (2000) proposed a
method that simultaneously optimizes both phase compensation and gain settings for the
stabilizers using Genetic Algorithms (GA). Although GA is sufficient enough in finding global
or near global optimal solution of the problem, it requires a very long run time that may beseveral minutes or even several hours depending on the size of the system under study. That is
why this kind of application could not be applied on-line. The concept of induced torquecoefficients is introduced for the systematic coordination of stabilizers (Gibbard et-al 2001) with
linear programming of a multi-machine system. Techniques for the coordination of stabilizers
based on the calculation of eigenvalue shifts from the residues are developed in Martins andLima (1990). Both techniques, using residues or induced torques are mathematically equivalent
in regard to eigenvalue shifting estimation (Pagola et-al, 1989). Later, El-Zonkoly (2006)
presented an approach using a concept of coherent groups, being a technique which is extremely
useful in transient stability studies.
2.1 Linearized model of a single-machine infinite-bus system for small-signal stability
A single machine-infinite bus (SMIB) system is considered for the present investigations
(Kundur, 1994). A machine connected to a large system through a transmission line may be
reduced to a SMIB system, by using Thevenin’s equivalent of the transmission network external
to the machine. Because of the relative size of the system to which the machine is supplying
power, the dynamics associated with machine will cause virtually no change in the voltage and
frequency of the Thevenin’s voltage (infinite bus voltage). In this section we will study the small
signal performance of a single machine connected to a large system through transmission lines.
A general system configuration is shown in Fig 1. Analysis of system having such simpleconfigurations is extremely useful in understanding basic effects and concepts. After we develop
an appreciation for the physical aspects of the phenomena and gain experience with the
analytical techniques, using simple low-order systems, we will be in a better position to deal with
large complex systems. A block diagram of the system with AVR and PSS is shown in Fig. 2 andthe thyristor excitation system with AVR and PSS is shown in Fig. 3. This study system is
described by the following state space representation:
(1)where
(2)
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The matrices A, B, C and D are as follows:
[
]
(3)
(4)
(5)
(6)
where,
,
, , ,
,
,
Fig.1: Single machine connected to a large power system through transmission lines.
G pss(s)
Fig.2 Block diagram representation SMIB with AVR and PSS
V ref
∆ω
∆T m
∆T e ∆ψ fd ∆δ
Field circuit
-
-+
+
+
+
K 4
K 2
K 1
∑ ∑ ∑
K 6
K 5
∑
∆v1
∆vs
∑ Gex(s) ∆E fd
Exciter-
-
+
++
∆ωr
∆δ
Voltage transducer
∆E 1
Local load
Infinite BusG
E t E B
Z eq=R E + jX E
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2.2 Mathematical Modeling
The synchronous machine is represented by a third-order model comprising the
electromechanical swing equation and the machine’s internal voltage equation. The machineequations are described in Kundur, 1994, Sa’adat, 2002, Panda and Padhy, 2007.
Electrical Equations:Stator winding equations are given by:
qd d ad d I x I R E V '' (7)
d d qaqq I x I R E V '' (8)
Rotor winding equations are given by:
qqd d
d
qoI x x E
dt
dE T )'('
'' (9)
fd d d d q
q
do E I x x E
dt
dE T )'('
''
2'2
'
)()(
)'()()'()(
d eea
oqd eod ea
d x x R R
CosV E x xSinV E R R I
(10)
2'2
'
)()(
)'()()'()(
d eea
oqd eoqea
q x x R R
SinV E x xCosV E R R I
(11)
The torque or electrical power equation is given by;
qqd d e I E I E P'' (12)
The IEEE type-ST1 exciter is used, modeled as:
fd
fe
f
t tr
e
e
fd E sT
sK V V
sT
K E
11(13)
Mechanical Equations:
Rotor swing equation is given by:
dampem T PPdt
d M
(14)
Fi 3: Th ristor excitation s stems with AVR and PSS
∑
transducer Exciter
Com ensationWashoutPhase
V
Gain
(3
(4
(5
(1
(2
-
Power S stem Stabilizer
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Where;
H M 2 DT damp
)(1
em PP D Ms
(15)
so
(16)
The detailed derivations of the equations given above and the notations used are given in
Kundur, 1994.
2.3 Simulink Modelling The derived equations representing the system under study were modeled in the Simulink
environment. The complete model is shown in Fig 4. It contains the subsystem block,
conventional PSS block, differential evolution PSS block and the scopes to indicate the desiredoutputs.
Fig. 4: Complete generator model including DEPSS and CPSS
3.1 Overview of particle swarm optimizationPSO is one of the optimization techniques and belongs to evolutionary computation techniques
(Fukuyama, 1999; Kennedy & Eberhart, 1995; Naka, Genji, Yura, & Fukuyama, 2001). The
method has been developed through a simulation of simplified social models. The features of themethod are as follows:
(1) The method is based on researches on swarms such as fish schooling and bird flocking.
(2) It is based on a simple concept. Therefore, the computation time is short and it requires few
memories.According to the research results for bird flocking, birds are finding food by flocking (not
by each individual). It leaded the assumption that information is owned jointly in flocking.
According to observation of behavior of human groups, behavior pattern on each individual is
based on several behavior patterns authorized by the groups such as customs and the experiencesby each individual (agent). The assumptions are basic concepts of PSO. PSO is basically
developed through simulation of bird flocking in two-dimension space. The position of each
individual (agent) is represented by XY axis position and the velocity is expressed by vx (the
delta
To Workspace3
dw
To Workspace2
t
To Workspace1
Switch
Vpss
Vref
dw
delta
Subsystem
Scope1
Scope
In1Out1
DEPSS
1
Constant
Clock1
In1Out1
CPSS
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velocity of X axis) and vy (the velocity of Y axis). Modification of the agent position is realized
by the position and velocity information. An optimization technique based on the above concept
can be described as follows: namely, bird flocking optimizes a certain objective function. Each
agent knows its best value so far (pbest) and its XY position. Moreover, each agent knows the
best value so far in the group (gbest) among pbests. Each agent tries to modify its position using
the following information:1 the current positions (x,y),
2 the current velocities (vx,vy),3 the distance between the current position, pbest and gbest.
This modification can be represented by the concept of velocity. Velocity of each agent can be
modified by the following equation:
( ) (17)
Using the above equation, a certain velocity, which gradually gets close to pbest and gbest can be
calculated. The current position (searching point in the solution space) can be modified by thefollowing equation:
(18)
Fig. 4 shows a concept of modification of a searching point by PSO and Fig. 5 shows a searchingconcept with agents in a solution space.
For the linearized system model presented in Section 2, the eigenvalues of the total system can
be evaluated. The proposed method is aiming to search for the optimal parameters set of theexciter and the power system stabilizers so that a comprehensive damping index (CDI) ( Cai &
Erlich, 2005) can be minimized:
∑ (19)
where is the damping ratio and n is the total number of the dominant eigenvalues. Theobjective of the optimization is to maximize the damping ratio as much as possible. The control
parameters to be tuned through the optimization algorithm are exciter gain (KA), KSTAB, Tw, T1
and T2 of single machine infinite bus system.
Table 1: Upper and lower limits of the control parameters
Parameter KA KSTAB TW T1 T2
Upper limit 400 50 10 0.5 0.05
Lower limit 50 20 1 0.05 0.005
3.2 Realization of PSO based Optimal Tuning of PSS/AVRStep I: Input system data: the following data are inputted,
Y
X
Vk
Vk+1
Vgbest
Vpbest
Sk+1
Sk X1
X2
X3
X4
Xn
Fig 4. Concept of modification of
a searching point by PSO.
Fig 5. Searching Concept with agents
in a solution space by PSO.
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The network configuration during the operating conditions under study.
The load values at each bus during the same operating conditions.
The generator data and parameters.
The upper and lower limits of the parameters to be optimized.Step II: Initialize the swarm with random position and velocities.
Step III: Evaluate the fitness of each particle (objective value) as described by Eq. (19).Step IV: Determine the personal and global best positions.
Step V: Update the velocity of agents using Eq. (17).
StepVI: Update the position of agents using Eq. (18).
Step VII: Perform the position check (the boundaries of each parameter). If violated then repair
the algorithm then go to step 8. If not violated go to step 8.Step VIII: Check the stopping criterion. If met go to step 9 and if not met go back to step 3.
Step IX: Output the optimal solution, which is the optimal values of the control parameters of the
single machine infinite bus system.
4 Simulation results and discussions
The simulation is carried out using MATLAB version 7.0. Fig 5 shows the characteristics of theDE based system, it can be seen that the algorithm converges in 75 generations. In order toguarantee that the control parameters selected were optimal, the system response under
disturbance conditions, such as increasing or reducing the mechanical torque of the machine was
studied and the system response is given in Fig 6. The Optimal DE values of PSS and AVRParameter Settings for DE Based Tool are as seen in Table 1 below and the DifferentialEvolution Simulation Results are presented in Table 2.
Table 2: Optimal values of PSS/AVR Parameter Settings for PSO Based Tool
Control Parameters Particle Swarm Optimization
Maximum generation, genmax
Number of control devices, DPopulation size, np
Scaling factor for mutation, F
Crossover constant, CR
Objective function scaling constant, a
190
5200.8
0.5
2.0
Fig 5: Convergence characteristics of PSO based system.
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(a) (b) (c)
Figure 14: Speed deviation at different conditions
Figure 14 (a), (b), and (c) are the Speed Deviation at different conditions. The speed deviation’s
amplitude of oscillation in DEPSS/AVR based system is 11 x 10-3
rad/sec which is much morelower than that of CPSS/AVR (15 x 10-3) rad/sec and comes to damp within 3 sec, while
CPSS/AVR converges at about 5 sec. and its amplitude of oscillation is much lower as compared
to the one without PSS/AVR (30 x 10-2
). The response without PSS/AVR converges inapproximately 7 sec. which is lately than even the CPSS/AVR.
(a) (b) (c)
Figure 15: Rotor angle deviation at different conditions
Similarly, in figure 15, the rotor angle deviation’s amplitude of oscillation in DEPSS/AVR basedsystem is 0.6 rad. which is much more lower than that of CPSS/AVR (0.75 rad.) and comes to
converge within 3 sec, while CPSS/AVR converges at about 5 sec. and its amplitude of
oscillation is much lower as compared to the one without PSS/AVR (1.5 rad.) The response
without PSS/AVR converges in approximately 7 sec. which is lately than even the CPSS/AVR.
0 1 2 4 5 6 7 8 9 10 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
t(sec)
Rotor Angle
with DEPSS & AVR
0 1 2 3 4 5 6 7 8 9 10 -0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2 -0.1
0
0.1
t(sec)
Rotor Angle
With CPSS & AVR
0 1 2 3 4 5 6 7 8 9 10 -1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
t(sec)
Rotor Angle
Without PSS & AVR
0 1 2 3 4 5 6 7 8 9 10 -10
-8
-6
-4
-2
0
2
4 x 10 -
t(sec)
Speed Deviation
With DEPSS & AVR
0 1 2 3 4 5 6 7 8 9 10 -8
-6
-4
-2
0
2
4
6 x 10 -3 Speed Deviation
t(sec)
With CPSS & AVR
0 1 2 3 4 5 6 7 8 9 10 -0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
t (sec)
Speed Deviation
Without PSS & AVR
dw(rad/sec)
delta(rad) delta(rad) delta(rad)
dw(rad/sec) dw()rad/sec
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The response before the introduction of fault has less overshoot than that of when fault wasintroduced. Both did not settle until after 10 seconds or more. The disturbance is from altering
the mechanical torque of the subsystem/generator.
Fig 6(b) is the DEPSS/AVR based response of the generator. At this point DEPSS/AVR
was installed. It can be observed from the figure that rotor speed’s overshoots have reduceddrastically and the settling time of the response without disturbance reduced to 3 seconds, while
that with disturbance has reached 5 seconds before it came to settled. This response revealed that
the DE based tuning of PSS and AVR converges faster and have less overshoot and settling timeas compared with the conventional PSS/AVR and without PSS/AVR.
Figure 17: Terminal voltage response
Figure 17 is the terminal voltage response, it represent the voltage at the terminal of the
generator/machine. The DEPSS/AVR based terminal voltage has less overshoot and converges in
lesser time than CPSS/AVR and that without PSS/AVR which makes it convenient fortransmission due to reduction in fluctuation.
(b)
Fig 7 (a) compares the speed deviations of DEPSS/AVR with CPSS/AVR and withoutPSS/AVR at all. It can be seen clearly that the DEPSS/AVR based system has less overshoot and
converges in lesser time than CPSS/AVR and that without PSS/AVR. Fig 7 (b) also compares
the Rotor angle of DEPSS/AVR with CPSS/AVR and without PSS/AVR at all. It can be seen
clearly also that the DEPSS/AVR based system has less overshoot and converges in lesser time
than CPSS/AVR and that without PSS/AVR.
Table 2: DE Simulation Results
Control Parameters Particle Swarm Optimization
0 1 2 3 4 5 6 7 8 9 10 0
0.2
0.4
0.6
0.8
1
1.2
1.4
t(sec)
Without AVR & PSS With CPSS & AVR With DEPSS & AVR
Fig 7: (a) Speed and (b) Rotor angle deviation
Vt(pu)
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Exciter Gain, K A
Stabilizer Gain,K STAB
Washout Time,T W
Compensation Time, T 1 Compensation Time, T2
64.3600
48.5155
1.1811
0.1836
0.0230
In Fig 6(a) the response before the introduction of disturbance has less overshoot than that of when disturbance was introduced. Both did not settle until after 10 seconds or more. The
disturbance is from altering the mechanical torque of the subsystem/generator. In Fig 6(b) the
overshoots have reduced drastically and the settling time of the response without disturbance
reduced to 2 seconds, while that with disturbance has reached 7 seconds before it come to rest.
5. Conclusion
This paper presents an effective technique to maximize the damping ratios of the system byoptimally determining the values of the control parameters of the system generators. The
optimization problem was solved using the particle swarm optimization technique. The proposedtechnique proved to be efficient in determining the optimal values of the control parameters suchthat the system response is satisfactory under different operating conditions. Besides being
effective, the particle swarm optimization technique proved to be fast compared with other
artificial intelligent optimization techniques such as genetic algorithms and compared tomathematical programming optimization approaches such as linear programming and quadratic
programming methods.
References
Byerley, R. T., & Sherman D. E. (1978). Frequency domain analysis of low frequencyoscillations in large electrical power systems. EPRI, Palo Alto, CA, Rep. EL-726, Project 744-1.
Cai, L. J., & Erlich, I. (2005). Simultaneous coordinated tuning of PSS and FACTS damping
controllers in large power systems. IEEE Transactions on Power Systems, 20(1).Do-Bomfim, A. L. B., Taranto, G. N., & Flacao, D. M. (2000). Simultaneous tuning of power
system damping controllers using genetic algorithms. IEEE Transactions on Power Systems,
15(1).
Feliachi, A., Zhang, X., & Sims, C. S. (1988). Power system stabilizers design using optimalreduced order models part II: Design. IEEE Transactions on Power Systems, 3(4).
Fukuyama, Y. (1999). A particle swarm optimization for reactive power and voltage control
considering voltage stability. Proceedings of IEEE international conference on intelligent systemapplications to power systems (ISAP), Rio de Janeiro.
Gibbard, M. J., Martins, N., Sanchez-Gasca, J. J., Uchida, N., Vittal, V., & Wng, L. (2001).
Recent applications of linear analysis techniques. IEEE Transactions on Power Systems, 16(1).
Gibbard, M. J., Vowles, D. J., & Pourbeik, P. (2000). Interactions between, and effectiveness of,power system stabilizers and FACTS devices stabilizers in multimachine systems. IEEE
Transactions on Power Systems, 15(2).
Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. Proceedings of IEEEinternational conference on neural networks (ICNN), IV, Perth, Australia.
Kundur, P. (1994). Power system stability and control. New York, NY: McGraw-Hill. 556 A.M.
El-Zonkoly / Expert Systems with Applications 31 (2006) 551 – 557
8/3/2019 Journal Paper 3
http://slidepdf.com/reader/full/journal-paper-3 11/12
11
Martins, N., & Lima, L. T. G. (1990). Determination of suitable locations for power system
stabilizers and static VAR compensators for damping electromechanical oscillations in largepower systems. IEEE Transactions on Power Systems, 5(4).
Naka, S., Genji, T., Yura, T., & Fukuyama, Y. (2001). Practical distribution state estimation
using hybrid particle swarm optimization. Proceedings of IEEE power engineering socity winter
meeting, Columbus, Ohio, USA.Pagola, F. L., Perez, I. J., & Verghese, G. C. (1989). On sensitivities, residues
and participations — Applications to oscillatory stability analysis and
control. IEEE Transactions on Power Systems, 4(1).Pourbeik, P., & Gibbard, M. J. (1996). Damping and synchronizing torques
induced on generators by FACTS stabilizers in multimachine power
systems. IEEE Transactions on Power Systems, 11(4).Pourbeik, P., & Gibbard, M. J. (1998). Simultaneous coordination of power
system stabilizers and FACTS device stabilizers in a multimachine power
system for enhancing dynamic performance. IEEE Transactions on Power
Systems, 13(2).
Pourbeik, P., & Gibbard, M. J. (2002). Proof of the equivalence of residues andinduced torque coefficients for use in the calculation of eigenvalue shifts.
IEEE Power Engineering Review.Urdaneta, A. J., Bacalao, N. J., Feijoo, B., Flores, L., & Diaz, R. (1991). Tuning
of power system stabilizers using optimization techniques. IEEE
Transactions on Power Systems, 6(1).Zanetta, L. C., & Da Cruz, J. J. (2005). An incremental approach to the
coordinated tuning of power systems stabilizers using mathematical
programming. IEEE Transactions on Power Systems, 20(2).
REFERENCES
Babu, B. V. and Gaurav Chaturvedi, (2000), Evolutionary Computation strategy for
Optimization of an Alkylation Reaction., Proceedings of 53rd Annual Session of IIChE
(CHEMCON-2000), Calcutta.
B. V. Babu, and Rakesh Angira, (2001), Optimization of Nonlinear functions usingEvolutionary Computation., Proceedings of 12th ISME Conference, Chennai, India. Bakare G. A., Venayagamoorthy G. K. and Aliyu U. O. (2007) , Reactive Power and Voltage
Control of the Nigerian Grid System Using Micro-Genetic Algorithm, Proceeding of IEEE
Power Engineering Society General Meeting, San Francisco, California USA.
Do-Bomfim, A. L. B., Taranto, G. N., & Flacao, D. M. (2000), Simultaneous Tuning of Power
System Damping Controllers Using Genetic Algorithms. IEEE Transactions on Power Systems,
15(1).
El-Sayeed MAH, (1998), Ruled Based Approach for Real Time Reactive Power Control in
Interconnected Power System,” Expert System with applications, Vol. 14, pp. 335-360.
El-Zonkoly A.M. (2006), Optimal Tuning of Power Systems Stabilizers and AVR Gains UsingParticle Swarm Optimization.
Feliachi, A., Zhang, X., & Sims, C. S. (1988), Power System Stabilizers Design Using Optimal
Gibbard, M. J., Martins, N., Sanchez-Gasca, J. J., Uchida, N., Vittal, V., and Wng, L.
(2001), Recent Applications of Linear Analysis Techniques. IEEE Transactions on Power
Systems, 16(1).
8/3/2019 Journal Paper 3
http://slidepdf.com/reader/full/journal-paper-3 12/12
12
Gibbard, M. J., Vowles, D. J., & Pourbeik, P. (2001), Interactions between, and Effectiveness
of, Power System Stabilizers and FACTS Devices Stabilizers in Multimachine Systems. IEEE
Transactions on Power Systems, 15(2).
Kit Po Wong and Zhao Yang Dong (2005), Differential Evolution, an Alternative Approach to
Evolutionary Algorithm.
IEE Proc., Pt. C, Gen., Trans.and Distrib.Kundur, P. (1994), Power System Stability and Control. New York, NY: McGraw-Hill.
Pagola, F. L., Perez, I. J., & Verghese, G. C. (1989), On Sensitivities, Residues and
Participations — Applications to Oscillatory Stability Analysis and Control. IEEE Transactions
on Power Systems, 4(1).
Price, K. and Storn, R. (1995), Differential Evolution – A Simple and Efficient Adaptive
Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95-012.
Price, K. and Storn, R. (1997), Differential Evolution - A simple evolution strategy for fast
optimization., Dr. Dobb's Journal, 22 (4).
Price, K. and Storn, R. (2002), Web site of DE, http://www.ICSI.Berkeley.edu/~storn/code.html
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