Electrical Power and Energy Systems 33 (2011) 8–16

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Automatic generation control application with craziness based particle swarmoptimization in a thermal power system

Haluk Gozde ⇑, M. Cengiz TaplamaciogluGazi University, Faculty of Engineering, Department of Electrical & Electronics Engineering, 06750 Maltepe, Ankara, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 March 2009Received in revised form 13 June 2010Accepted 13 August 2010

Keywords:Automatic generation controlCRAZYPSOThermal power systemPI-controller

0142-0615/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.08.010

⇑ Corresponding author. Tel.: +90 312 2311340; faxE-mail addresses: halukgozde@gmail.com (H. Goz

Taplamacioglu).

In this study, a novel gain scheduling Proportional-plus-Integral (PI) control strategy is suggested forautomatic generation control (AGC) of the two area thermal power system with governor dead-band non-linearity. In this strategy, the control is evaluated as an optimization problem, and two different costfunctions with tuned weight coefficients are derived in order to increase the performance of convergenceto the global optima. One of the cost functions is derived through the frequency deviations of the controlareas and tie-line power changes. On the other hand, the other one includes the rate of changes which canbe variable depends on the time in these deviations. These weight coefficients of the cost functions arealso optimized as the controller gains have been done. The craziness based particle swarm optimization(CRAZYPSO) algorithm is preferred to optimize the parameters, because of convergence superiority. Atthe end of the study, the performance of the control system is compared with the performance whichis obtained with classical integral of the squared error (ISE) and the integral of time weighted squarederror (ITSE) cost functions through transient response analysis method. The results show that theobtained optimal PI-controller improves the dynamic performance of the power system as expected asmentioned in literature.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

An interconnected electric power system generates, transportsand distributes electric energy. The aim of such these systems isto supply electric energy with nominal system frequency and ter-minal voltage, values and tolerances of those are defined by somepower quality standards. According to power system control the-ory, a nominal system frequency depends on the balance betweengenerated and consumed real powers [1]. The difference betweengenerated power and instant load demand causes changing ofnominal system frequency at the normal state. If the amount ofgenerated power is less than the demanded amount, speed and fre-quency of the generator units begin to decrease, and vice versa.Hence, the amount of production of the synchronous generatorsis made sense for frequency deviations occurred in the power sys-tem in order to maintain that balance. For this purpose, an auto-matic generation control concept is used. The aim of automaticgeneration control is that the steady state error of the system fre-quency deviations following a step load demand is made zeroerror.

ll rights reserved.

: +90 312 2308434.de), taplam@gazi.edu.tr (M.C.

When the literature is investigated, it can be seen that earlyworks on AGC was initiated by Cohn [2]. However, a modern opti-mal control concept for AGC designs of interconnected systems isput forward by Elgerd and Fosha for the first time [3]. They sug-gested a proportional controller and different feedback form to de-velop optimal controller. Until the present day, lots of differentcontrol strategies such as conventional, adaptive, variable struc-ture, robust and some based on artificial intelligence have been re-ported [4]. However, gain scheduling adaptive control can bedistinguished from the other control techniques because it makesthe process which is under control less sensitive to changes in pro-cess parameters and in particular, it is also simpler to implementthan the other modern control techniques. For these reasons, it iscarried out to AGC system, frequently.

The first gain scheduling control method for AGC of intercon-nected power system was proposed by Lee and coworkers in1991 [5]. Their controller provided better control performancefor a wide range of operating conditions than the performances ob-tained so far. Later on, Rubaai and Udo presented a multi-variablegain scheduling controller by defining a cost function with a termrepresenting the constraints on the control effort and then mini-mizing that with respect to the control vector [6]. Since the con-ventional gain scheduling methods may be unsuitable in someoperating conditions due to the complexity of the power systemssuch as nonlinear load characteristics and variable operating

Nomenclature

Ki integral gain constantKp proportional gain constantwi cost function weight coefficientRi regulation constantTg speed governor time constantTt turbine time constant

Tp power system time constantDfi frequency deviation from nominal value in area-iDPtie change in tie-line power between two areaDPLi load demand incrementBi frequency bias factorT12 synchronization coefficient

H. Gozde, M.C. Taplamacioglu / Electrical Power and Energy Systems 33 (2011) 8–16 9

points, a usage of artificial intelligence based methods were pre-ferred by researchers from the beginning of these dates. In 1997,Chang reported the usage of fuzzy logic based gain schedulingmethod for power system AGC [7]. As different from the usage oftwo fuzzy rules for integral and proportional gains in PI-controllerby Chang, Çam and Kocaarslan who improved the performance ofthis approach in 2005 by which the rules for the gains are chosento be identical [8]. Talaq suggested an adaptive fuzzy gain schedul-ing method for conventional PI-controller in 1999 [9]. Then, Pingk-ang optimized the gains of PI and PID controllers through realcoded genetic algorithm in a two area power system in 2002[10]. After 1 year from Pinkang’s study, Abdel-Magid and Abidoproposed a usage of PSO for the same purpose [11]. In 2004, Yes�ilsuggested the self tuning fuzzy PID type controller for AGC [12].One year after, Juang put forward the genetic algorithm based fuz-zy gain scheduling method for PI-controller [13]. In this study, toreduce both the fuzzy system design effort and the number of fuz-zy rules, the fuzzy system used for gain scheduling is automaticallydesigned by genetic algorithms. In 2006, Massiala used two layeredfuzzy gain scheduling controller in order to improve the dynamicperformance of AGC in a two area reheat thermal power systemwith generation rate constraints [14]. Taher composed a hybridPSO algorithm for the gain scheduling of PI-controller in a two areathermal power system in 2008 [15]. The hybrid PSO algorithm con-tained evolutionary operators like selection, crossover and muta-tion as in genetic algorithms or DE algorithm. In the year 2009,Nanda suggested a maiden application of bacterial foraging optimi-zation based gain scheduling PI-controller in a multi area AGC [16].Rao and coworkers studied an automatic generation control onTCPS based hydrothermal system [17].

The aim of this study which is different from the above litera-ture is that a novel gain scheduling PI-control strategy is proposedfor automatic generation control (AGC) of a two area thermalpower system with governor dead-band nonlinearity. In this strat-egy, the control is evaluated as an optimization problem, and twodifferent cost functions with tuned weight coefficients are derivedto increase the performance of convergence to the global optima.The weight coefficients of the cost functions are also optimizedas the controller gains have been done. Then, the craziness basedparticle swarm optimization (CRAZYPSO) algorithm is preferredin order to optimize the parameters, because of its convergencesuperiority and also its relatively simple codes. At the result of thisapplication, it is seen that the response of AGC of the power systemis improved.

2. Materials and methods

2.1. Power system model

A two area interconnected thermal power system is consideredfor application of optimal automatic generation control, because ofbeing the simplest model of interconnected power system. A trans-fer function model of this system is depicted in Fig. 1, and system

parameters are also given in Table 1. At the simulation study, it isassumed that there is a step load changing in the control area-1.

In the above model, u1 and u2 are the control inputs from pro-posed gain scheduling PI-controllers. DPL1 is 1% of step load pertur-bations of nominal loading in the control area-1. Df1 and Df2 arefrequency deviations of each control area, and DPtie is the changingof tie-line power between control areas.

In the model, a governor dead-band effect is also added to allcontrol areas to simulate nonlinearity. A governor dead-band is de-fined as the total magnitude of a sustained speed change wherethere is no change in valve position of the turbine. Describing func-tion approach is used to represent the governor dead-band in theareas. The governor dead-band nonlinearity tends to produce acontinuous sinusoidal oscillation of natural period of aboutT0 = 2 s. This approach is being used to linearize the governordead-band in terms of change and rate of change in the speed[18]. The nonlinearity of the hysteresis is defined as,

y ¼ Fðx; _xÞ ð1Þ

In this function, x is taken as a sinusoidal oscillation withf0 = 0.5 Hz.

x ¼ A sin w0t ð2Þ

Since the dead-band nonlinearity tends to give continuouslysinusoidal oscillation, such an assumption is quite realistic. Then,the F function can be evaluated as a Fourier series as follows,

Fðx; _xÞ ¼ F0 þ N1xþ N2

w0dx=dt þ � � � ð3Þ

For an approximation, it is enough to consider the first threeterms in (3). As the dead-band nonlinearity is symmetrical aboutthe origin, and then F0 is equal to zero,

Fðx; _xÞ ¼ N1xþ N2

w0dx=dt ¼ DBx ð4Þ

where DB denotes the dead-band. In this work, the backlash ofapproximately 0.5% is chosen and the Fourier coefficients are ob-tained as N1 = 0.8 and N2 = �0.2. At the result of the analysis, thetransfer function of the governor with dead-band nonlinearity canbe expressed in (5) [19]. It is used as this form in the power systemmodel.

Gg ¼0:8� 0:2

p s1þ Tgs

ð5Þ

2.2. Gain scheduling control

A gain scheduling control is an adaptive control technique thatchanges some control parameters of the controller according toscheduling variables related to different operating regions whichthe plant works. This control technique deals with particularlynonlinear processes, processes with time variations or situationswhere the requirements on the control that change with the oper-ating conditions. The main advantage of this control is that the

Fig. 1. Block diagram of the two area interconnected thermal power system with governor dead-band nonlinearity.

Table 1System parameters.

Parameter Quantity

Tg1,2 0.2 sTt1,2 0.3 sKp1,2 120 Hz/puMWTp1,2 20 sT12 0.0707 MW/radB1,2 0.425 puMW/HzR1,2 2.4 Hz/puMW

10 H. Gozde, M.C. Taplamacioglu / Electrical Power and Energy Systems 33 (2011) 8–16

controller parameters can be adjusted very quickly in response tochanges in the plant dynamics. It is also simpler to implement thanthe other adaptive control techniques. A typically block diagram ofgain scheduling control system is depicted in Fig. 2.

In this figure, the scheduling variables can be the measured sig-nal, the control signal or an external signal. The control parametersof the controller are determined with a gain scheduling algorithm.The algorithm is run for all different operating conditions by auto-matic tuning. However the classical tuning methods have beenused as a gain scheduling algorithm, nowadays the swarm intelli-gence based methods are carried out increasingly due to their sim-pler implementing and better performance of converging and lessrun times [20]. As an optimization algorithm based on swarmintelligence, craziness based particle swarm optimization methodis preferred in order to its superiority in respect of the standardPSO and the other algorithms, and also its short and simple codes.From this point of view, a microprocessor program of this algo-rithm can be easily prepared in reality. Nowadays, this algorithmis applied to the lots of power system control applications in order

Fig. 2. Block diagram of a gain scheduling control system.

to tune the parameters of the controller or the system such as eco-nomic dispatch control [21].

2.3. Craziness based particle swarm optimization

Particle swarm optimization is a population based optimizationalgorithm which is first introduced by Kennedy and Eberhart in1995 [22]. It can be obtained high quality solutions within shortercalculation time and stable convergence characteristics by PSOthan other stochastic methods such as genetic algorithm.

PSO uses particles which represent potential solutions of theproblem. Each particles fly in search space at a certain velocitywhich can be adjusted in light of preceding flight experiences.The projected position of ith particle of the swarm xi, and the veloc-ity of this particle vi at (t + 1)th iteration are defined and updatedas the following two equations:

v tþ1i ¼ v t

i þ c1r1ðpti � xt

i Þ þ c2r2ðgti � xt

i Þ ð6Þ

xtþ1i ¼ xt

i þ v tþ1i ð7Þ

where i = 1, � � �, n and n is the size of the swarm, c1 and c2 are posi-tive constants, r1 and r2 are random numbers which are uniformlydistributed in [0, 1], t determines the iteration number, pi repre-sents the best previous position (the position giving the best fitnessvalue) of the ith particle, and g represents the best particle amongall the particles in the swarm. At the end of the iterations, the bestposition of the swarm will be the solution of the problem. It cannotbe always possible to get an optimum result of the problem, but theobtained solution will be an optimal one. The flowchart of PSO isdepicted in Fig. 3.

Since the standard PSO algorithm can fall into premature con-vergence especially for complex problems with many local optimaand optimization parameters, the craziness based PSO algorithmwhich is particularly effective in finding out the global optimumin very complex search spaces is developed. The main differencebetween PSO and CRAZYPSO is the propagation mechanism todetermine new velocity for a particle as follows:

v tþ1i ¼ r2signðr3Þv t

i þ ð1� r3Þc1r1ðpti � xt

i Þþ ð1� r2Þc2ð1� r1Þðgt � xt

i Þ ð8Þ

xtþ1i ¼ xt

i þ v tþ1i þ Pðr4Þsignðr4ÞVcr ð9Þ

where pi is the local best position of particle i, and gi is the globalbest position of the whole swarm. r1, r2, r3 and r4 are random

Initialize the randomvelocities and

positions of particles

Evaluate the fitness valuesfor each particles

Compare particle’sfitness with its best

previous one

Best previous positionequal to the current value

Best previous positionequal to itself

Currentvalue isbetter

Previousvalue isbetter

Best of the best position isappointed to the global

best

Change velocities and positionsaccording to (6) and (7)

Is the criterion metor end of iterations?

YesNo

StopGlobal best position is the

solution

Fig. 3. The flowchart of the standard PSO algorithm.

Fig. 4. The novel control strategy of AGC.

Table 2Tuned parameters with ISE, ITSE and proposed cost functions.

Cost function Kp Ki w1 w2 w3

Proposed – 1 �0.5762 0.1962 0.7300 0.6848 0.7879Proposed – 2 �0.4000 0.3000 0.5000 0.5000 0.5001ITSE �0.2647 0.3317 – – –ISE �0.2514 0.2491 – – –

Table 3Settling times with ISE, ITSE and proposed cost functions.

Cost function Df1 (s) Df2 (s) DPtie (s)

Proposed – 1 9.65 10.98 13.16Proposed – 2 11.64 12.42 17.55ITSE 21.66 21.67 30.80ISE 20.45 20.47 29.45

H. Gozde, M.C. Taplamacioglu / Electrical Power and Energy Systems 33 (2011) 8–16 11

parameters distributed uniformly in [0, 1], and c1, c2 are named stepconstants and are taken 2.05 generally. The sign is a function de-fined as follows for r3 and r4,

signðr3Þ ¼�1( r3 6 0:051( r3 > 0:05

�ð10Þ

signðr4Þ ¼�1( r4 6 0:51( r4 > 0:5

�ð11Þ

In birds flocking or fish schooling, since a bird or a fish oftenchanges directions suddenly, in the position updating formula, acraziness factor, Vcr, is used to describing this behavior. In thisstudy, it is decreased linearly from 10 to 1. P(r4) is defined as

Pðr4Þ ¼1( r4 6 Pcr

0( r4 > Pcr

�ð12Þ

where Pcr is a predefined probability of craziness and is introducedto maintain the diversity of the particles. It is taken 0.3 in this study.The CRAZYPSO algorithm can prevent the swarm from beingtrapped in local minimum, which would cause a premature conver-gence and lead to fail in finding the global optimum [23].

3. Control strategy

As a control strategy, the new control configuration which is de-picted in Fig. 4 is suggested. In this configuration, to achieve thecontrol inputs, the optimal PI-controllers are used together witharea control errors, ACE1 and ACE2, in (13) and (14) respectively.

ACE1 ¼ B1Df1 þ DPtie1 ð13Þ

ACE2 ¼ B2Df2 � DPtie2 ð14Þ

In the control strategy, control inputs of the power system, u1

and u2, are obtained with PI-controllers as below.

u1 ¼ Kp1ACE1 þ Ki1

ZACE1dt ð15Þ

u2 ¼ Kp2ACE2 þ Ki2

ZACE2dt ð16Þ

The object of the obtaining optimal solutions of control inputs istaken as an optimization problem, and CRAZYPSO algorithm isbeing used to tune the gains of the controllers and cost functionweights w1, w2 and w3 as a novel control approach. In AGC system,in order to convergence to optimal solution, two different costfunctions in (17) and (18) are derived. While one of the cost func-tions is derived through the frequency deviations of the controlareas and tie-line power changes, the rates of changes in thesedeviations according to time are used in the other one.

J ¼Z t

0t w1

dDf1

dt

� �2

þ w2dDf2

dt

� �2

þ w3dDPtie

dt

� �2" #

� dt ð17Þ

J ¼Z t

0t w1Df1ð Þ2 þ ðw2Df2Þ2 þ ðw3DPtieÞ2h i

� dt ð18Þ

12 H. Gozde, M.C. Taplamacioglu / Electrical Power and Energy Systems 33 (2011) 8–16

4. Results and discussion

The results are obtained by MATLAB 6.5 software run on Core2of 2 GHz, and RAM of 1 GB. Forty particles are used, and 100

5 10-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

Ti

delta

f1 (H

z)

Fig. 5. The deviations of Df1 with ISE, ITSE and propo

6 8 10 12 14-1.5

-1

-0.5

0

0.5

1

1.5x 10-3

Ti

delta

f1 (H

z)

9.65 s 11.64 s

Fig. 6. The zoom of deviations of Df1 with

5 10-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

T

delta

f2 (H

z)

Fig. 7. The deviations of Df2 with ISE, ITSE and propo

iterations are chosen for converging to solution in the crazinessbased PSO algorithm. The simulations are realized in case ofDPL1 = 0.01 puMW. At the end of the simulations, the tuned para-meters of the control system are shown in Table 2, and the settling

15 20 25

me (s)

Proposed cost function 1Proposed cost function 2ITSE cost functionISE cost function

sed cost functions in case of DPL1 = 0.01 puMW.

16 18 20 22 24me (s)

Proposed cost function 1Proposed cost function 2ITSE cost functionISE cost function

20.45 s 21.66 s

ISE, ITSE and proposed cost functions.

15 20 25ime (s)

Proposed cost function 1Proposed cost function 2ITSE cost functionISE cost function

sed cost functions in case of DPL1 = 0.01 puMW.

5 10 15 20 25

-1.5

-1

-0.5

0

0.5

1

x 10-3

Time (s)

delta

f2 (H

z)

Proposed cost function 1Proposed cost function 2ITSE cost functionISE cost function

10.98 s 12.42 s 20.47 s 21.67 s

Fig. 8. The zoom of deviations of Df2 with ISE, ITSE and proposed cost functions.

5 10 15 20 25 30-10

-8

-6

-4

-2

0

x 10-3

Time (s)

delta

Ptie

(puM

W)

Proposed cost function 1Proposed cost function 2ITSE cost functionISE cost function

Fig. 9. The deviations of DPtie with ISE, ITSE and proposed cost functions in case of DPL1 = 0.01 puMW.

15 20 25 30 35 40-1.5

-1

-0.5

0

0.5

1

1.5x 10-4

Time (s)

delta

Ptie

(puM

W)

Proposed cost function 1Proposed cost function 2ITSE cost functionISE cost function

13.16 s 17.55 s 29.45 s 30.80 s

Fig. 10. The zoom of deviations of DPtie with ISE, ITSE and proposed cost functions.

H. Gozde, M.C. Taplamacioglu / Electrical Power and Energy Systems 33 (2011) 8–16 13

0

5

10

15

20

25

30

35

Δf1 Δf2 ΔPtie

Proposed – 1

Proposed – 2

ITSE

ISE

Fig. 11. The comparison of settling times according to cost functions.

14 H. Gozde, M.C. Taplamacioglu / Electrical Power and Energy Systems 33 (2011) 8–16

times of the frequency and tie-line power deviations are repre-sented in Table 3. These results are compared to the results ob-tained with the integral of the squared error (ISE) and theintegral of time weighted squared error (ITSE) cost functions in(19) and (20) respectively.

0 5 10-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

T

delta

f1 (H

z)

Fig. 12. Curves of Df1 for di

0 5 10-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

T

delta

f2 (H

z)

Fig. 13. Curves of Df2 for di

ISE ¼Z t

0ðACEiÞ2dt ð19Þ

ITSE ¼Z t

0t � ðACEiÞ2dt ð20Þ

The results show that the proposed control configuration withtuned cost function achieves good dynamic performance for thepower system. Especially, the proposed cost functions expose bet-ter solutions than the standard cost functions. In addition to this,the proposed cost function-1 depicted in (17) with the rates ofchanges in the deviations shows the best solutions for the powersystem. It can be clearly seen that the CRAZYPSO based gain sched-uling PI-controller with this cost function improved to control ofthe power system in order to minimize the frequency and tie-linepower deviations. These deviations and their zooms determiningthe settling times are also depicted in Figs. 5–10.

These figures and Table 3 show that the settling time of Df1

obtaining with proposed cost function-1 is better than that of Df1

obtaining with proposed cost function-2, ISE and ITSE of 17.09%,

15 20 25ime (s)

nominal load+25% load+50% load- 25% load- 50% load

fferent changes of DPL1.

15 20 25ime (s)

nominal load+25% load+50% load- 25% load- 50% load

fferent changes of DPL1.

0 5 10 15 20 25-16

-14

-12

-10

-8

-6

-4

-2

0

2x 10-3

Time (s)

delta

Ptie

(puM

W)

nominal load+25% load+50% load- 25% load- 50% load

Fig. 14. Curves of DPtie for different changes of DPL1.

H. Gozde, M.C. Taplamacioglu / Electrical Power and Energy Systems 33 (2011) 8–16 15

55.44% and 52.81%, respectively. And then, the settling time of Df2

obtaining with proposed cost function-1 is better than that of Df2

obtaining with proposed cost function-2, ISE and ITSE of 11.59%,49.33% and 46.36%, respectively. In addition to these, the settlingtime of DPtie obtaining with proposed cost function-1 is better thanthat of DPtie obtaining with proposed cost function-2, ISE and ITSEof 25.01%, 57.27% and 55.31%, respectively. Furthermore, the com-parison of settling times according to cost functions is depicted inFig. 11. According to this figure, the power change between controlareas is minimized by the proposed control strategy faster than theother control strategies.

On the other hand, some different values of DPL1 are applied tothe power system in order to evidence the robustness of the con-trol strategy optimized by CRAZYPSO algorithm. For this purpose,a step load change of the control area-1 is decreased the step of25% and 50%, and increased the step of 25% and 50%, respectively.In this case, the proposed cost function-1 is used due to its conver-gence superiority. As a result, the obtained frequency deviationcurves of these cases are depicted in Figs. 12–14. The tunedparameters are shown in Table 4, and the maximum overshootsof these curves are shown in Table 5, respectively. It can be seenfrom these figures and tables that the deviation of the frequenciesand the tie-line power change are obtained less than half accordingto nominal values. Then, the proposed control strategy provides a

Table 4Tuned parameters for different loads.

+50% Load +25% Load Nominal load �25% Load �50% Load

Kp �0.5389 �0.4000 �0.5762 �0.4428 �0.5478Ki 0.2098 0.2464 0.1962 0.2000 0.2162w1 0.5473 0.8603 0.7300 0.7259 0.9882w2 1.2365 0.6533 0.6848 0.8438 0.9454w3 0.9877 1.1000 0.7879 0.6694 0.9781

Table 5Maximum overshoots for different loads.

+50% Load +25% Load Nominal load �25% Load �50% Load

Df1 �0.04726 �0.03764 �0.03194 �0.02293 �0.01580Df2 �0.05429 �0.04171 �0.03710 �0.02570 �0.01817DPtie �0.01411 �0.01098 �0.00958 �0.00673 �0.00472

robust control in the range of ±50% of the step load change,sufficiently.

5. Conclusion

In this article, the new gain scheduling PI-controller strategy isproposed for automatic generation control (AGC). In this strategy,the control is evaluated as an optimization problem, and theweight coefficients of the cost function are also optimized as thecontroller gains have been done. CRAZYPSO algorithm which isone of the recent population based optimization algorithms is usedbecause of its convergence superiority in order to optimize theparameters and also its short and simple codes. The performanceof the proposed controller is compared with the performanceswhich are obtained with standard ISE and ITSE cost functions.The results obtained from the simulations show that the proposedcontrol strategy optimized with new cost functions achieves betterdynamic performances than the standard cost functions as sum-marized in Fig. 11. On the other hand, the robustness of the controlstrategy with the proposed cost function-1 is also investigated.Then, it can be said that this control approach is the effectiveand the relatively robust strategy in order to provide optimal auto-matic generation control to the power system, and the choosingsuitable cost function is also quite important for performance ofthe convergence to the best solution. Finally, due to its superiori-ties, the proposed control strategy can be applied to the differentcontrol system applications, successfully.

Acknowledgment

The authors wish to thank to Prof. _Ilhan KOCAARSLAN for hiscontributions to this paper.

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