Chaotic ant swarm optimization for fuzzy-based tuning of power system stabilizer

Electrical Power and Energy Systems 33 (2011) 657–672

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Electrical Power and Energy Systems

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

Chaotic ant swarm optimization for fuzzy-based tuning of power system stabilizer

A. Chatterjee a, S.P. Ghoshal b, V. Mukherjee c,⇑a Department of Electrical Engineering, Asansol Engineering College, Asansol, West Bengal 713 305, Indiab Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal 713 209, Indiac Department of Electrical Engineering, Indian School of Mines, Dhanbad, Jharkhand 826 004, India

a r t i c l e i n f o

Article history:Received 30 July 2008Received in revised form 21 December 2009Accepted 9 December 2010Available online 4 February 2011

Keywords:Ant colony optimizationChaotic ant swarm optimizationGenetic algorithmParticle swarm optimizationPower system stabilizerSugeno fuzzy logic

0142-0615/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.12.024

⇑ Corresponding author. Tel.: +91 0326 2235644; faE-mail addresses: [email protected]

nitdgp@ gmail.com (S.P. Ghoshal), vivek_agamani@ya

a b s t r a c t

In this paper, chaotic ant swarm optimization (CASO) is utilized to tune the parameters of both single-input and dual-input power system stabilizers (PSSs). This algorithm explores the chaotic and self-organization behavior of ants in the foraging process. A novel concept, like craziness, is introduced inthe CASO to achieve improved performance of the algorithm. While comparing CASO with either particleswarm optimization or genetic algorithm, it is revealed that CASO is more effective than the others infinding the optimal transient performance of a PSS and automatic voltage regulator equipped single-machine-infinite-bus system. Conventional PSS (CPSS) and the three dual-input IEEE PSSs (PSS2B, PSS3B,and PSS4B) are optimally tuned to obtain the optimal transient performances. It is revealed that thetransient performance of dual-input PSS is better than single-input PSS. It is, further, explored thatamong dual-input PSSs, PSS3B offers superior transient performance. Takagi Sugeno fuzzy logic (SFL)based approach is adopted for on-line, off-nominal operating conditions. On real time measurementsof system operating conditions, SFL adaptively and very fast yields on-line, off-nominal optimal stabilizervariables.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Usage of fast acting, high gain automatic voltage regulator(AVR) in modern generator excitation system invites the problemof low frequency electromechanical oscillation. Transfer of bulkpower across weak transmission lines; any disturbance such assudden change in loads, change in transmission line parameters,fluctuation in the output of the turbine and faults, etc. also invitesthe problem of low frequency oscillations (typically in the range of0.2–3.0 Hz) under various sorts of system operating conditions andconfigurations. The very common and widely accepted solution,prevailing in the utility houses to address this problem, is theusage of power system stabilizer (PSS). The PSS adds a stabilizingsignal to AVR that modulates the generator excitation. Here, itsmain task is to create a damping electrical torque component (inphase with rotor speed deviation) in turbine shaft, which increasesthe generator damping. A practical PSS must be robust over a widerange of operating conditions and capable of damping the oscilla-tion modes in power system. From this perspective, the conven-tional single-input PSS (machine shaft speed (Dxr) as singleinput to PSS) design approach based on a single-machine-

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x: +91 0326 2296563.(A. Chatterjee), spghoshal

hoo.com (V. Mukherjee).

infinite-bus (SMIB) linearlized model in the normal operatingcondition has some deficiencies.

The two inputs to dual-input PSS, unlike the conventional sin-gle-input (Dxr) PSS, are Dxr and DTe. The processed output ofthe PSS is DVpss that acts as an excitation modulation signal andthe desired damping electrical torque component is produced.Modeling of IEEE type PSS2B, PSS3B, and PSS4B are reported in[1] and those models are taken in the present study.

Pole-placement [2] or eigenvalue assignment for single-inputsingle-output system has been reported in the literature. A robustPSS tuning approach [3] based upon lead compensator design hasbeen carried out by drawing the root loci for finite number of ex-treme characteristic polynomials. In [3], such polynomials havebeen obtained by using Kharitonov theorem to reflect wide loadingcondition. An approach based on linear matrix inequalities (LMIs)for mixed H2/H1-design under pole region constraints has been re-ported by Werner et al. [4]. In [4], plant uncertainties are expressedin the form of a linear fractional transformation. Results obtainedin [4] are compared to the results obtained in [5] based on quanti-tative feedback theory.

Linear quadratic control [6] has been applied for coordinatedcontrol design. The problem has been formulated as a standardLQR and a full feedback control was obtained from the solutionthat retains the dominant modes of the closed loop system.Structural constraints, such as, simple and decentralized control,feedback of only measured variables, etc. have been in use in

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Nomenclature

EB infinite bus voltageEt generator’s LT side bus voltageH inertia constant, MW-s/MVAKa thyristor exciter gain,KD damping torque coefficient, pu torque/pu speed devia-

tionosh overshoot of change in rotor speed, puOF1() first objective functionOF2() second objective functionP active power, puQ reactive power, purand() random number in the interval [0, 1]tex execution time, stst settling time of change in rotor speed, sT0 system constant, sTrr terminal voltage transducer time constant, sush undershoot of change in rotor speed, puvcraziness

i a random number chosen uniformly in the interval[vmin

i ; vmaxi ] for the ith ant

vcrazinessmax maximum value of vcraziness

ivcraziness

min minimum value of vcrazinessi

vmaxi ;vmin

i maximum and minimum velocity of the ith ant,respectively

Xe equivalent transmission line reactance, puki ¼ ri þ jbi ith eigenvalueri real part of ith eigenvaluebi imaginary part of ith eigenvaluexd damped frequency, rad/sxn undamped natural frequency, rad/sx0 rated speed = 2pf0, elect, rad/sn damping ratioDTe incremental change in electromagnetic torque, puDTm incremental change in mechanical torque, puDv1 incremental change in terminal voltage, puDVpss incremental change in power system stabilizer output

voltage, puDVmax

pss maximum value of incremental change in power systemstabilizer output voltage, pu

DVminpss minimum value of incremental change in power system

stabilizer output voltage, puDVref incremental change in reference voltage, puDd rotor angle deviation, puDxr speed deviation = xr�x0

x0, pu

658 A. Chatterjee et al. / Electrical Power and Energy Systems 33 (2011) 657–672

power systems for many years and cannot be addressed by a stan-dard LQR. Such a structurally constrained optimal control problemhas been solved using the generalized Riccati equation [7] and wasapplied to a power systems exploiting sparsity [8].

Fuzzy [9], GA-fuzzy [10], neuro-fuzzy [11] are just a few amongthe other numerous works reported in the literature to tune PSS.Most of these techniques are centered on angular speed deviation(Dxr) as single-input feedback to PSS. Some of these techniquessuffer from complexity of computational algorithm, heavy compu-tational burden, memory storage problem and non-adaptive tun-ing under various system operating conditions andconfigurations. Some suffer from robustness because of choice oflimited number of control variables of PSS, limited number of opti-mization functions and on-line real time necessity for fast chang-ing PSS variables.

Recently, evolutionary programming and intelligent controltechniques are being applied to solve many complex optimizationproblems in engineering applications. With high speed computingtools, these search methods are increasingly being applied inpower system planning, design, operation and control problems.The advantage of these methods is that the objective function neednot be explicit or differentiable and nonlinearity or non-convexityis not a problem and optimal damping in the closed loop can be ob-tained. Some algorithms like genetic algorithm (GA), simulatedannealing suffer from settings of algorithm parameters and giverise to repeated revisiting of the same suboptimal solutions.

Chaotic ant swarm optimization (CASO) is, essentially, a searchalgorithm that is based on the chaotic behavior of individual antand the intelligent organization actions of ant colony. This algo-rithm is reported in the literature to obtain the economic dispatchof power systems by Cai et al. [12] for three different power sys-tems. In the present work, this algorithm, (with some inherentmodifications made by the authors of the present work to suitthe present application area), is utilized for the purpose of optimaltuning of the PSS variables. The novelty of the present work is thestudy of the performance of the CASO in designing PSS.

Bacteria foraging optimization (BFO), a bio-inspired technique,has been reported by Mishra et al. [13] to establish the potential

application of the BFO technique as a soft computing intelligencein power system optimization arena. The main focus of the article[14] was the tuning of single-input PSS installed in multi-machinepower system. But what about the tuning of dual-input PSS? Thus,it is very much pertinent to explore a comparative study betweenthese two configurations of PSSs with the assistance of the CASO. Afew variants of particle swarm optimizations (PSOs), and GA maybe taken as some standard algorithms for the sake of comparison.

A fuzzy logic system-based PSS [14] can adjust its variables on-line according to the environment in which it works and can pro-vide good damping over a wide range of operating conditions. Thebest PSS, ultimately, derived from this paper (PSS3B) proves to bethe most robust model compared to either CPSS, or PSS2B, or PSS4Bin damping all electromechanical modes of generator’s angularspeed oscillations for all off-line and on-line conditions, stepchanges of mechanical torque inputs (DTm), reference voltage in-puts (DVref) and during/after clearing of system faults. For the pres-ent work, off-line conditions are 34 (=81) sets of nominal systemoperating conditions which is given in SFL table (not shown inthe paper). On the other hand, in real time environment these in-put conditions vary dynamically and become off-nominal. And thisnecessitates the use of very fast acting SFL to determine the off-nominal PSS variables for off-nominal input conditions occurringin real time.

Thus, the major and minor objectives of this paper may be doc-umented as follow:

Major objectives (pertaining to algorithm performance):

(a) To study the performance of CASO in designing PSS.(b) To present the potential benefit of CASO over other PSOs and

GA as optimizing technique.(c) To explore the suitability of fuzzy logic-based tuned PSS

under various changes in system operating conditions,including occurrence of fault and its subsequent clearing.

Minor objectives (pertaining to PSS performance):

(a) To compare single-input PSS with dual-input PSS.

A. Chatterjee et al. / Electrical Power and Energy Systems 33 (2011) 657–672 659

(b) To contrast the generator’s angular speed oscillations fordual-input PSS (namely PSS2B, PSS3B, and PSS4B) equippedsystem model.

(c) To critically examine the best type of PSS for practical imple-mentation under any sort of system disturbances.

The rest of the paper is organized as follows. In Section 2, SMIBsystem and various PSSs under investigation are presented. Math-ematical problem for the present study is formulated in Section 3.Optimizing algorithms as implemented to optimal PSS tuning aredescribed in Section 4. Sugeno fuzzy logic as applied to on-line tun-ing of PSS variables is narrated in Section 5. Input control parame-ters for the simulation are given in Section 6. Section 7 documentsthe simulation results. Finally, concluding remarks and scope of fu-ture work are outlined in Section 8.

2. SMIB system and various PSSs under investigation

An SMIB [15] model, as considered in the present work, isshown in Fig. 1. As the purpose of PSS is to introduce damping tor-que component, speed deviation is used as logical signal to controlgenerator excitation for conventional power system stabilizer(CPSS). On the other hand, speed deviation and torque deviationare taken as the best pair of inputs for dual-input PSS [16]. The

Fig. 1. Single-machine-infi

Fig. 2. Block diagram representation of SMIB system with AVR,

block diagram of SMIB system with AVR, thyristor high gain exci-ter, synchronous generator and PSS is shown in Fig. 2. The genera-tor including AVR, excitation system and transmission-circuitreactance is represented by a two-axis, fourth order model. IEEEtype ST1A model of the static excitation system is considered.The block diagrams of different stabilizers under study are shownin Figs. 3–6. The generator with AVR and excitation system alongwith CPSS/PSS2B/PSS3B/PSS4B is represented by eighth/seven-teenth/eighth/eleventh order state matrices, respectively.

3. Mathematical problem formulation

The performance specifications of PSS, in terms of damping andspeed of response, may be designed in terms of an admissible poleregion for the linearized small-signal mode. Maintaining stabilityand performance over a range of uncertain parameters may behandled by imposing an upper bound on the H1-norm of theclosed loop transfer function. In LMI-based approach, these maybe handled by designing a single convex optimization problem. Apole region constraint and bound on the H1-norm may be ex-pressed as LMIs.

Minimization of a quadratic performance index under con-straint, provided a solution exists, may be solved by applying anefficient algorithm. But in the context of the present work, two

nite-bus test system.

thyristor high gain exciter, synchronous generator, and PSS.

Fig. 3. Block diagram representation of conventional power system stabilizer.

Fig. 4. Block diagram representation of dual-input PSS2B.



optimization functions, as described below, are formulated to tunethe variables of the PSS.

The variables of the PSS (for CPSS (Fig. 3): Kpss, Td1, Td2, Td3, Td4,Td5, Td6; for PSS2B (Fig. 4): Ks1, T1, T2, T3, T4, T5; for PSS3B (Fig. 5):Ks1, Ks2, Td1, Td2, Td3, Td4; for PSS4B (Fig. 6): Ks1, Ks2, T1, T2, T3, T4)are to be so tuned that some degree of relative stability and damp-ing of electromechanical modes of oscillations, minimized under-shoot (ush), minimized overshoot (osh), and lesser settling time(tst) of transient oscillations of Dxr are achieved. So, to satisfy allthese requirements, two multi-objective optimization functions,OF1() and OF2() which are to be minimized in succession are de-signed in the following ways.

OF11 =P

i(r0 � ri)2 if r0 > ri, ri is the real part of the ith eigen-value. The relative stability is determined by �r0. The value of r0 istaken as 6.0 for the best relative stability and optimal transientperformance.

OF12 =P

i(n0 � ni)2, if (bi, imaginary part of the ith eigen-value) > 0.0, ni is the damping ratio of the ith eigenvalue andni < n0. Minimum damping ratio considered, n0 = 0.3. Minimizationof this objective function will minimize maximum overshoot.

OF13 =P

i(bi)2, if ri P �r0. High value of bi to the right of verti-cal line �r0 is to be prevented. Zeroing of OF13 will increase thedamping further.

OF14 = an arbitrarily chosen very high fixed value (say, 106),which will indicate some ri values P0.0. This means unstableoscillation occurs for the particular variables of PSS. These particu-lar PSS variables will be rejected during the optimizationtechnique.

So, first multi-objective optimization function is formulated asfollows.

OF1ðÞ ¼ 10� OF11 þ 10� OF12 þ 0:01� OF13 þ OF14 ð1Þ

A sensitivity analysis is carried out and is presented in Table 1 todemonstrate the impact of OF11, OF12, OF13, and OF14 on OF1(). Resultsof interest are bold faced. This table helps us to formulate the variousweighting coefficients of (1). The weighting factors ‘10’ and ‘0.01’ in(1) are chosen to impart more weights to OF11, OF12 and to reducehigh value of OF13, to make them mutually competitive duringoptimization. By optimizing OF1(), closed loop system poles areconsistently pushed further left of jx axis with simultaneous


Table 1Sensitivity analysis of OF1().

Operatingconditions

Type ofPSS

OF1() and its components

OF1() OF11 OF12 OF13 OF14

0.2,�0.2, 1.08,1.1

CPSS 1344.23 134.42 0 0 0PSS2B 1295.8 118.6 0 10982.0 0PSS3B 415.63 41.56 0 0 0PSS4B 671.71 55.15 0 12021.37 0

0.5, 0.2,0.4752, 1.0


1.2, 0.6, 0.93,1.0


1.2, 0.6,0.4752, 0.5


Fig. 7. D-shaped sector in the negative half of s plane.


reduction in imaginary parts also, thus, enhancing the relative stabil-ity and increasing the damping ratio above n0.

Finally, all closed loop system poles should lie within a D-shaped sector (Fig. 7) in the negative half plane of jx axis for whichri�� r0, ni� n0.

Selection of such low negative value of r is purposefully chosen.The purpose is to push the closed loop system poles as much left aspossible from the jx axis to enhance stability to a great extent.

Thorough computation shows that optimization of OF1() is notsufficient for sharp tuning of PSS variables. So, it is essential to de-sign second multi-objective optimization function for sharp tuningof PSS variables. Thus, the second multi-objective optimizationfunction OF2() is formulated as follows:

OF2ðÞ ¼ ðosh � 106Þ2 þ ðush � 106Þ2 þ ðtstÞ2

þ ddtðDxrÞ � 106

� �2

ð2Þ

In (2); osh, ush, tst, ddt ðDxrÞ are all referred to the transient re-

sponse of Dxr. A sensitivity analysis is carried out and is presentedin Table 2 to demonstrate the impact of osh, ush, tst, and d

dt ðDxrÞ onOF2(). This table helps us to formulate the various weighting coef-ficients of (2). Results of interest are bold faced. The constrainedoptimization problem for the tuning of PSSs is, thus, formulatedas follows.

Minimize OF1() and OF2() in succession with the help of anyoptimization technique to get optimal PSS variables, subject tothe limits [1,15]:

ðaÞ For CPSS :175:0 � Kpss � 230:00:001 � Tdi � 1:0; i ¼ 1 to 6

�

ðbÞ For PSS2B :

10 � Ks1 � 300:01 6 Ti 6 1:0; i ¼ 1 to 5T10 ¼ 0:0001

8><>:

Other parameters are fixed.

ðcÞ For PSS3B :�100:0 � Ks1 � �10:0; 10:0 � Ks2 � 100:00:005 � Tdi � 2:0; i ¼ 1 to 4

�

No fixed parameters are required.

ðdÞ For PSS4B :�100:0 � Ks1 � �10:0;10:0 � Ks2 � 100:00:005 � Ti � 2:0; i ¼ 1 to 4; T0 ¼ 0:2

�

Table 2Sensitivity analysis of OF2().

Operating conditions Type of PSS OF2() and its components

OF2() � 107 osh � 10�6 ush � 10�6 tst ddt ðDxrÞð�10�6Þ

0.2, �0.2, 1.08, 1.1 CPSS 3.79 896.91 �1089.31 5.999 6.71PSS2B 3.57 830.30 �682.42 5.881 4.87PSS3B 1.46 179.10 �451.48 3.797 4.86PSS4B 1.88 860.31 �1505.31 3.976 7.71

0.5, 0.2, 0.4752, 0.9 CPSS 3.76 727.31 �1245.20 5.965 7.02PSS2B 2.82 657.81 �837.60 5.224 5.01PSS3B 1.21 660.30 �454.30 3.378 4.31PSS4B 1.59 174.76 �1211.12 3.795 7.87




4. Optimization algorithms as applied to PSS tuning

4.1. Genetic algorithm

Implementation steps of the GA algorithm are shown in Fig. 8.

4.2. Chaotic ant swarm optimization

CASO [12] combines the chaotic and self-organization behaviorof ants in the foraging process. It includes both the effects of cha-otic dynamics and swarm-based search. In the present work, thisalgorithm is employed to tune the PSS (both single-input anddual-input) variables with some modifications to suit the presentapplication.

CASO is based on the chaotic behavior of individual ant and theintelligent organization actions of ant colony. Here, the searchbehavior of the single ant is ‘‘chaotic’’ at first and the organizationvariable, ri is introduced to achieve self-organization process of theant colony. Initially, the influence of the organization variable onthe behavior of individual ant is sufficiently small. With the

Step 1 Initialization

a) Input operating values of P, Q, Xe, Et. Inp

b) Setting of limits for PSS variables,

c) Setting of GA parameters like mutation p

d) Maximum population number, maximum

e) Binary value initialization of all the PSS v

f) Decoding of the binary strings within para

Step 2 Determination of SMIB parameters like K1, K

Step 3 Computation of misfitness function/objective f

Step 4 Arraigning the values in increasing order from

Step 5 Copying of 50% selected strings over the rest 5

Step 6 Crossover.

Step 7 Mutation.

Step 8 Repeat from Step 3 till the end of the maximu

Step 9 Determine the optimal PSS variables string co

Fig. 8. Implementation steps of GA alg

continual change of organization variable evolving in time andspace, the chaotic behavior of the individual decreases gradually,via the influence of the organization variable and the communica-tion of previously best positions with neighbors, the individual antalters his position and moves to the best one they can find in thesearch space.

The searching area of ants corresponds to the problem searchspace. In the search space Rl, which is the l-dimensional continuousspace of real numbers, the algorithm searches for optima. A popu-lation of K ants is considered. These ants are located in a searchspace S and they try to minimize a function f: S ? R. Each point sin S is a valid solution to the considered problem. The position ofan ant i is assigned the algebraic variable symbol Si = (zi1, . . . , zil),where i = 1, 2, . . . , K. Naturally, each variable can be of any finitedimension. During its motion, each individual ant is influencedby the organization processes of the swarm. In mathematicalterms, the strategy of movement of a single ant is assumed to bea function of the current position, the best position found by itself.Any member of its neighbors and the organization variable are gi-ven by:

ut fixed SMIB parameters,

robability, crossover ratio,

iteration cycles,

ariables strings of the population within limits,

meters’ limits.

2, K3, T3, K4, K5, K6 (Fig. 2) [12].

unction of each string of the total population.

minimum, and selection of top 50% better strings.

0% inferior strings to form the total population.

m iteration cycle.

rresponding to the grand minimum misfitness.

orithm for tuning of PSS variables.


yiðnÞ¼yiðn�1Þ1þri

zidðnÞ¼ zidðn�1Þþ cwd

Vi

� �exp 1�expð�ayiðnÞÞð Þ 3�wd zidðn�1Þððð

þ cwd

V i

��7:5

wdViþexp �2ayiðnÞþbð Þðpidðn�1Þ�zidðn�1ÞÞ

9>>>=>>>;ð3Þ

wheren is current iteration cycle,n � 1 previous iteration cycle,yi(n) current state of the organization variable (yi(0) = 0.999),a, b, c positive constants,ri e [0, 0.1] a positive constant and is termed as the organization

factor of ant i,zid(n) current state of the dth dimension of the individual ant i,

d = 1, 2, . . . , l,wd determines the selection of the search range of dth ele-

ment of variable in search space, andVi is determines the search region of ith ant and offers the

advantage that ants could search diverse regions of theproblem space (Vi = rand())

The neighbor selection can be defined in the following twoways. The first one is the nearest fixed number of neighbors. Thenearest m ants are defined as the neighbors of single ith ant. Thesecond way of the number of neighbor selection is to considerthe situation in which the number of neighbors increases with iter-ation cycles. This is due to the influence of self-organization behav-iors of ants. The impact of organization will become stronger thanbefore and the neighbors of the ant will increase. That is to say, thenumber of nearest neighbors is dynamically changed as iterationprogresses.

The general CASO is a self-organizing system. When everyindividual trajectory is adjusted toward the successes of neigh-bors, the swarm converges or clusters in optimal regions of thesearch space. The search of some ants will fail if the individualcannot obtain information about the best food source from theirneighbors.

For the present work, the algorithm’s parameters like ri, wd, Vi, a,b, c are different from those reported in [12]. These are, respec-tively, 1 + ri is replaced by (1.02 + 0.04 � rand()), wd = 2.0, Vi = r-and(), a = 1, b = 0.1, c = 3. These values are pre-set after a lot ofexperiments to get the best convergence to optimal solution. Inthis work, finally, craziness is introduced by the authors of thiswork as given in (4), which is not present in [12] but taken from[17,18].

Step 1 Initialization

a) Input operating values of P, Q, X e, Et. Input fix

b) Setting of limits for PSS variables,

c) Setting of CASO parameters,

d) Maximum population number, maximum itera

e) Real value initialization of all the PSS variable

Step 2 Determination of SMIB parameters like K1, K2

Step 3 Computation of misfitness function/objective f

Step 4 Determination of the best string corresponding

Step 5 Updating the strings of the population using C

Step 6 Repeat from Step 3 till the end of the maximum

Step 7 Determine the optimal PSS variables string co

Fig. 9. Implementation steps of CASO al

zidðnÞ ¼ zidðnÞ þ signðr4Þ � vcrazinessi ð4Þ

The values of sign(r4) and vcrazinessi are determined by (5) and (6),

respectively.

signðr4Þ ¼1; ðr4 � 0:5Þ�1; ðr4 < 0:5Þ

�ð5Þ

vcrazzinessi ¼ vcrazziness

min þ ðvcrazzinessmax � vcrazziness

min Þ � randðÞ ð6Þ

Introduction of craziness enhances CASO’s ability of searchingand convergence to a global optimal solution. Variables’ upperand lower bound restrictions are always present. Ultimately, aftera maximum iteration cycles the optimal solution of zid correspondsto global optimal value of fitness function under consideration.

A suite of six benchmark test functions [19,20], broadly catego-rized into three groups viz. (a) unimodal functions, (b) multimodalfunctions with only a few local minima, and (c) multimodal func-tions with many local minima, have been tested using CASO byMukherjee in [21]. In [21], unimodal functions tested are Spheremodel and Generalized Rosenbrock’s function. In the same work,multimodal functions with only a few local minima are MexicanHat function and Six-hump camel back function. Multimodal func-tions with many local minima, as tested in [21], are GeneralizedRastrigin’s function and Generalized Griewank function. Promisingperformances have been obtained in [21] while applying novelCASO on these benchmark test functions. These results motivatethe authors of the present work to apply the same algorithm fortuning the different PSSs equipped with AVR model.

Implementation steps of CASO algorithm are shown in Fig. 9. Aflow chart of the whole algorithm is depicted in Fig. 10.

5. Sugeno fuzzy logic as applied to on-line tuning of PSSvariables

The behavior of a synchronous generator connected to a net-work depends on, among other things, its position in this network,the operating conditions (in particular, the reactive power flow andthe voltage map), the network topology and the generation sche-dule. Usage of the desired optimization technique yields a distinctset of controller variables for different operating conditions. Underdrastic change in operating conditions (e.g. different circuit topol-ogies) the nominal controller is not necessarily going to be tunedenough to yield satisfactory performance. For on-line tuning ofthe variables of PSS, very fast acting SFL is to be adopted.

ed SMIB parameters,

tion cycles,

s strings of the population within limits.

, K3, T3, K4, K5, K6 (Fig. 2) [12].

unction of each string of the total population.

to minimum misfitness/objective function value.

ASO algorithm (described in section 4.2).

iteration cycle.

rresponding to the grand minimum misfitness.

gorithm for tuning of PSS variables.

Initialize input operating parameters ( P , Q , eX and tE ), limits of PSS variables, CASO parameters, maximum population number, maximum iteration number

Compute misfitness function/ objective function of each string

Real value initialization of all the PSS variable strin s of the o ulation

Compute SMIB parameters (K1, K2, K3, T3, K4, K5, K6)

Generation Index

Repeat next generation

Are PSS variables within limits?

Stop

No

Yes

Population Index

Compute new values of PSS variables incorporating velocity vector using CASO algorithm and find pBest , gBest

Compute the best string corresponding to minimum misfitness

Is pBest = gBest ?

Output PSS variables

Move each particle to new position using position updating equation

Repeat next population

Set variables’ values equal to max/min values,

as the case may be

Start

Yes

No

p pg

Fig. 10. Flowchart of CASO algorithm as implemented for PSS tuning.


5.1. Sugeno fuzzy logic for on-line tuning of PSS variables

The whole process of SFL [14] can be categorized into threesteps viz. Fuzzification of input operating conditions, Sugeno fuzzyinference and Sugeno defuzzification. The whole process of SFL[14] involves three steps as discussed below.

5.1.1. FuzzificationThe first step is fuzzification of input operating conditions as ac-

tive power (P), reactive power (Q), equivalent transmission linereactance (Xe) and Generator’s LT side bus voltage (Et) in terms offuzzy subsets (Low, Medium, High). These are associated withoverlapping triangular membership functions. SFL rule base tableis formed, each composed of four nominal inputs and correspond-ing nominal optimal PSS variables as output determined by any ofthe optimizing techniques dealt with. The respective nominal cen-tral values of the input subsets of P are (0.2, 0.7, 1.2), those of Q are(�0.2, 0.6, 1.0), those of Xe are (0.4752, 0.77, 1.08) and those of Et

are (0.5, 0.8, 1.1), respectively, at which membership values areunity (Fig. 11). These are nominal input conditions also. Sugenofuzzy rule base table consists of 34 (=81) logical input conditionsor sets (SFL tables calculated for different PSS structures investi-gated), each composed of four nominal inputs. Each logical inputset corresponds to nominal optimal PSS variables as output.

5.1.2. Sugeno fuzzy inferenceFor on-line imprecise values of input parameters, firstly their

subsets in which the values lie are determined with the help of‘‘IF’’, ‘‘THEN’’ logic and corresponding membership values aredetermined from the membership functions of the subsets. FromSugeno fuzzy rule base table, corresponding input sets and nomi-nal PSS variables are determined. Now, for each input set being sat-isfied, four membership values like lP, lQ, lXe

and lEtand their

minimum lmin are computed. For the input logical sets, whichare not satisfied because variables do not lie in the correspondingfuzzy subsets, lmin will be zero. For the non-zero lmin values only,nominal PSS variables corresponding to fuzzy sets being satisfiedare taken from the Sugeno fuzzy rule base table.

5.1.3. Sugeno defuzzificationSugeno defuzzification yields the defuzzified, crisp output for

each parameter of PSS. Final crisp PSS parameter output is givenby:

Kcrisp ¼P

ilðiÞmin � KiPilðiÞmin

ð7Þ

where i corresponds to input logical sets being satisfied among 81input logical sets, Ki is corresponding nominal PSS parameter. Kcrisp

Fig. 11. Fuzzification of input operating conditions (P, Q, Xe and Et).


is crisp PSS parameter. lðiÞmin is the minimum membership value cor-responding to ith input logical set being satisfied.

5.2. Link between SFL and optimization algorithm

SFL interpolates the variables accurately. Optimizing algorithmis not required to run in real time rather SFL is required to run inreal time. The optimized variables corresponding to nominal modelvariables (P, Q, Xe and Et) as determined by the optimizing algo-rithm, are needed to be stored in memory (in the form of SFL table).SFL only utilizes this SFL table in real time. SFL simply acts as apiecewise linear interpolator within restricted region. SFL workson simple if-then-else logical loops and a very few additions andmultiplications. Hence, the computational burden of SFL is verylow. Its practical implementation is easy. It is to be noted here,in case of SFL application, that more the number of fuzzy subsets,more accurate result will be obtained at the cost of increased com-putational burden. For real time determination of optimal modelvariables fast, adaptive SFL is adopted.

6. Input control parameters

For simulation, step perturbation of 0.01 pu is applied either inreference voltage or in mechanical torque. The simulation is imple-mented in MATLAB 7.1 software on a PC with PIV 3.0 G CPU and512M RAM. The followings are the different input controlparameters.

Table 3GA, HPSOIWA, HPSOCFA, and CASO-based comparison of OF2() values for CPSS, P

Sl. no. Operating conditions (P, Q, Xe, Et; all are in pu) Algor

1 0.2, �0.2, 0.4752, 1.1 GA-SFHPSOHPSOCASO

2 0.5, 0.2, 0.4752, 1.0 GA-SFHPSOHPSOCASO




6.1. Input operating conditions

In the present simulation work, P, Q and Et are varied individu-ally and independently in steps of 0.1 over the respective intervals(0.2, 1.2), (�0.2, 1.0) and (0.5, 1.1). The various values of Xe (referFig. 1) are (Tr + (Xe1//Xe2)), (Tr + Xe1) and (Tr + Xe2), respectively.These values are j0.4752, j0.65 and j1.08, respectively. Thus, 11 Pvalues, 13 Q values, three Xe values and seven Et values are consid-ered. Therefore, a family of 3003 (=11 � 13 � 3 � 7) input operat-ing conditions is considered for the simulation. The main focus ofthis paper is to tune the PSS variables that maintain a damping ra-tio of at least 0.3 and the real part of all eigenvalues more than�6.0 simultaneously for all modes of the family.

6.2. For SMIB system [1, 15]

Inertia constant, H = 5, M = 2H, nominal frequency, f0 = 50 Hz,0.995 6 |Eb| 6 1.0, the angle of Eb = 0�, 0.2 6 P 6 1.2, �0.2 6 Q 61.0, 0.4752 6 Xe 6 1.08, 0.5 6 Et 6 1.1. In the block diagram repre-sentation of generator with exciter and AVR; Trr = 0.02s, Ka = 200.0.

6.3. For GA

Number of parameters depends on problem variables (PSS con-figuration), number of bits = (number of parameters) 8 (for bin-ary coded GA, as considered for the present work), populationsize = 50, maximum number of iteration cycles = 200, mutationprobability = 0.001, crossover rate = 80%.

6.4. For CASO

Number of problem variables depends upon the PSS structureunder investigation. All the parameters of the algorithm are givenin Section 4.

7. Simulation and results

Input system operating conditions are P, Q, Xe, and Et. Each oper-ating condition has three fuzzy logical sets as Low, Medium, andHigh. Thus, total number of nominal input operating condition setsare 34 (=81). Sugeno fuzzy rule base table (not shown) is obtained

SS2B, PSS3B and PSS4B-based systems.

ithms Value of OF2() (�107)

CPSS PSS2B PSS3B PSS4B

L 7.45 6.23 1.42 2.90IWA-SFL [22] 5.91 5.54 1.35 2.64CFA-SFL [22] 5.16 4.42 1.05 1.71-SFL 3.24 2.97 0.99 1.47






for all nominal input operating condition sets by applying eachoptimization technique. Each table contains corresponding 81 setsof distinct nominal optimal PSS variables. Optimized PSS variables,as determined by any of the optimization technique, are substi-tuted in MATLAB-SIMULINK model of the system to obtain thetransient response profiles. Final values of OF1(), and OF2() aredetermined from the end of the optimization. Final eigenvalue, un-damped frequencies, damped frequencies and damping ratio arealso determined by the optimization technique at the end of theoptimization. The major and minor observations of the presentwork are documented below. Results of interest are bold faced inthe respective table.

7.1. Major observations (pertaining to algorithm performance)

7.1.1. Comparative optimization performance of the optimizationtechniques

In [22], Mukherjee and Ghoshal have discussed two basic vari-ants of PSOs like hybrid particle swarm optimization with inertiaweight approach (HPSOIWA), and hybrid particle swarm optimiza-

Table 4GA, and CASO-based comparison of OF1() values under different operating conditions for CPare in pu).

Type of PSS Algorithm-SFL PSS variables

CPSS GA-SFL 181.15, 0.162, 0.080CASO-SFL 175.00, 0.005, 0.005

PSS2B GA-SFL 10.00, 0.174, 0.015,CASO-SFL 10.00, 0.195, 0.306,

PSS3B GA-SFL �10.35, 99.65, 0.854CASO-SFL �10.00, 24.00, 2.000

PSS4B GA-SFL �12.11, 33.20, 0.922CASO-SFL �10.00, 10.00, 0.005

Fig. 12. Comparative GA and CASO-based transient response profiles of Dxr for CPSS, PQ = 0.2, Xe = 1.08;, Et = 0.9, all are in pu).

tion with constriction factor approach (HPSOCFA) and have appliedthese PSOs for the tuning of dual-input PSSs. The same results arereproduced here for the sake of comparison to show the potentialmerits between the proposed approach in this paper using CASOand PSOs [22]. Table 3 depicts the comparative GA, HPSOIWA[22], HPSOCFA [22], and CASO-based optimal transient responsecharacteristics (in terms of OF2() value) of different PSS equippedsystem model. From this table, it is observed that the CASO-basedoptimization technique offer lesser value of OF2() than either GA, orHPSOIWA [22], or HPSOCFA-based one [22]. Thus, CASO-basedoptimization technique offers better results than GA/HPSOIWA/HPSOCFA-based results. With regard to optimization performancesof the optimizing algorithms, as depicted in Table 3 and Table 4, itmay be concluded that the CASO-based approach offers the lowervalues of OF1() and OF2() as compared to GA-based approach forthe same input operating conditions. Comparative transient per-formances of Dxr and convergence profiles of OF1() for GA andCASO-based optimization for all the four PSS modules (CPSS,PSS2B, PSS3B, and PSS4B) are depicted in Fig. 12 and Fig. 13,respectively. From Fig. 12, it may be concluded that the transient

SS, PSS2B, PSS3B, and PSS4B (operating condition: P = 1.0, Q = 0.6, Xe = 0.93, Et = 0.5; all

OF1() tex (s)

, 0.096, 0.053, 0.187, 0.198 1449.69 13.45, 0.001, 0.001, 0.179, 0.001 1428.61 5.20

0.013, 0.191, 0.044 1252.42 20.070.010, 0.010, 0.0100 1228.45 10.56

, 0.098, 0.340, 0.192 535.20 10.42, 1.493, 0.005, 0.093 417.93 5.15

, 0.005, 0.961, 0.005 785.41 17.12, 0.188, 0.206, 0.005 581.03 6.46

SS2B, PSS3B, and PSS4B for 0.01 pu simultaneous change in DTm and DVref (P = 1.0,


stabilization performance of CASO-based optimization is betterthan that of GA-based one. From Fig. 13, it is observed thatCASO-based optimization technique offers lesser OF1() value ascompared to GA-based one, for whatever may be the PSS modelunder consideration. Thus, CASO offers much better optimal per-formance and it may be accepted as a true optimizing algorithmfor the power system-based application as considered in the pres-ent work.

7.1.2. Performance improvement of CASO with crazinessIn the present work, the concept of craziness (a novel concept

proposed by Mukherjee and Ghoshal in the literature [14]) isblended with original CASO algorithm [12] with an attempt to haveimproved performance of CASO algorithm reported in [12]. Table 5shows how improved performance is taking place with the inclu-sion of craziness concept. This table shows for a particular inputoperating condition, improved OF1() value is achieved for all thefour PSS structures. The comparative convergence profiles ofOF1() based on CASO without craziness and CASO with craziness

Fig. 13. Comparative GA and CASO -based transient response profiles of OF1() for CPSS,Q = 0.2, Xe = 1.08, Et = 0.9, all are in pu).

Table 5Comparison of OF1() values based on CASO without craziness, and CASO with craziness forEt = 1.0; all are in pu.

Type of PSS Concept of craziness

CPSS CASO without crazinessCASO with craziness

PSS2B CASO without crazinessCASO with craziness



for CPSS, PSS2B, PSS3B, and PSS4B are depicted in Fig. 14 corre-sponding to a nominal operating condition of P = 0.2, Q = 0.2,Xe = 1.08, Et = 1.0 (all are in pu). From this figure, the objectivefunction value, OF1() corresponding to CASO with craziness isfound to converge to the lesser minimum value faster than theother. It is also noticed from this figure that the OF1() value ofPSS3B corresponding to CASO with craziness is the lowest one.

7.1.3. Fuzzy logic-based tuning of PSS variables under changes insystem operating conditions

LT bus fault of duration 220 ms at the instant of 2.0 s is simu-lated for different PSS equipped system model and the correspond-ing comparative transient response profiles of Dxr for both GA-SFLand CASO-SFL-based approaches are plotted in Fig. 15 for all thePSSs. Fig. 15 reveals that CASO-SFL response exhibits better re-sponse than GA-SFL based one irrespective of the PSS model. Aclose look into this figure also reveals that after the creation ofthe fault, the CASO-SFL-based response for PSS3B equipped systemmodel recovers from this abnormal situation with much lesser

PSS2B, PSS3B and PSS4B for 0.01 pu simultaneous change in DTm and DVref (P = 1.0,

all the four PSSs, nominal input operating conditions being P = 0.2, Q = 0.2, Xe = 1.08,

PSS variables OF1()

175.00, 0.734, 1.00, 0.001, 0.001, 0.001, 0.001 1375.53175.00, 0.082, 0.005, 0.001, 0.001, 0.001, 0.001 1299.39

10.00, 0.375, 0.010, 1.00, 0.010, 0.010 1227.9210.00, 1.00, 0.011, 0.253, 0.010, 0.010 1200.61

�10.00, 10.00, 0.965, 1.510, 0.040, 0.005 449.23�10.00, 10.00, 0.934, 2.00, 0.012, 0.005 442.41

�10.00, 10.00, 0.229, 0.202, 0.191, 0.019 560.42�10.00, 10.00, 0.271, 0.103, 0.005, 0.017 518.35

Fig. 14. Comparative CASO (with craziness and without craziness)-based convergence profiles of OF1() for CPSS, PSS2B, PSS3B and PSS4B (P = 0.2, Q = 0.2, Xe = 1.08, Et = 1.0, allare in pu).

Fig. 15. Comparative GA-SFL, and CASO-SFL-based transient response profiles of Dxr for the generator equipped with different PSS under change in operating conditions.


fluctuation in angular speed as compared to that of GA-SFL-basedcounter part. Table 6 depicts the system model parameters asdetermined by SFL for all the PSS models. Thus, CASO-SFL-based

model exhibits better response having lesser amplitude of angularspeed deviation under fault and subsequent clearing conditionyielding better dynamic transient performance than GA-SFL-based


one for PSS3B equipped system model. Hence, PSS3B proves to bemuch less susceptible to faults because PSS3B settles all the statedeviations to zero much faster than any other PSSs.

Fig. 16. Comparative CASO-based transient response profiles of Dxr for CPSS,PSS2B, PSS3B and PSS4B for 0.01 pu simultaneous change in DTm and DVref (P = 0.5,Q = 0.2, Xe = 0.93, Et = 1.0, all are in pu).

7.2. Minor observations (pertaining to PSS performance)

7.2.1. Analytical eigenvalue-based comparison of PSSsGA-SFL, and CASO-SFL based comparison of OF1() values of CPSS,

PSS2B, PSS3B, and PSS4B are shown in Table 4 for different systemoperating conditions. From Table 4, it is observed that the value ofOF1(), irrespective of the optimization technique adopted, is theleast one for PSS3B. It establishes the performance of PSS3B to bethe best one. For PSS3B equipped system model, majority of theeigenvalues are within D-shaped sector (Fig. 7) which yield lesservalues of OF11(), OF12(), and OF13(). Hence, the value of OF1() is veryless for PSS3B equipped system model. On the other hand, majorityof the eigenvalues for PSS2B-based system are outside the D-shaped but very close to and right side of (�r0, j0) point. Thisyields higher values of OF11(), OF12(), and OF13(). The value ofOF1() is more for this system. Thus, from the eigenvalue analysisit may be concluded that a considerable improvement has occurredin the transient performance for the PSS3B-based system.

Fig. 17. Comparative CASO-based convergence profiles of OF1() for CPSS, PSS2B,PSS3B and PSS4B (P = 1.0, Q = 0.2, Xe = 0.4752, Et = 1.0, all are in pu).

7.2.2. Comparative performances of PSSs in terms of analyticaltransient responses characteristics

From Table 3 and Table 4, it may also be inferred that the tran-sient stabilization performance of dual-input PSS equipped systemmodel is better than single-input counter part. Comparing dual-in-put PSSs, it is also observed that the transient stabilization perfor-mance of PSS3B equipped system model is superior to that ofothers. PSS3B equipped system model offers lesser values of osh,ush, tst, d

dt ðDxrÞ and, thereby, lesser values of OF2(). Fig. 16 depictsthat the comparative optimal transient performance of the differ-ent PSS equipped power system model corresponding to an operat-ing condition of P = 0.5, Q = 0.2, Xe = 0.93, Et = 1.0 (all are in pu) for0.01 pu simultaneous change in DTm and DVref. From this figure, itis noticed that the transient stabilization performance of dual-in-put PSS is better than that of single input one. Among the dual-in-put PSSs, the performance of PSS3B is established to be the bestone for this specific application of SMIB system model under study.From Fig. 16, it is apparently observed that CPSS outperforms theother three PSSs in terms of magnitudes of ush and osh. But theduration of undershoot and tst are very large for CPSS. The secondobjective function OF2(), given in (2), is designed in such a fashion

Table 6PSS variables under change in off-nominal operating conditions.

Off-nominal operating conditions (P, Q, Xe, Et; all are in pu) Type of PSS Algorithms-SFL PSS variables

1.0, 0.5, 0.4752, 1.0 (pre-fault) CPSS GA-SFL 179.68, 0.030, 0.032, 0.245, 0.005, 0.144, 0.401CASO-SFL 230.00, 0.177, 0.005, 0.001, 0.001, 0.001, 0.001

PSS2B GA-SFL 10.00, 0.143, 0.047, 0.089, 0.011, 0.138CASO-SFL 10.00, 0.098, 0.010, 0.010, 0.039, 0.022

PSS3B GA-SFL �10.00, 83.47, 1.992, 0.995, 0.005, 0.036CASO-SFL �10.00, 10.00, 2.00, 1.142, 0.005, 0.077


LT bus fault of duration 220 ms, and subsequent clearing CPSS/PSS2B/PSS3B/PSS4B GA-SFLCASO-SFL No change in variables

0.2, �0.2, 1.08, 1.1 (post-fault) CPSS GA-SFL 182.32, 0.109, 0.090, 0.071, 0.038, 0.088, 0.075CASO-SFL 175.00, 0.065, 0.005, 0.001, 0.001, 0.001, 0.001

PSS2B GA-SFL 10.39, 0.022, 0.150, 0.177, 0.179, 0.077CASO-SFL 10.00, 0.282, 0.010, 0.795, 0.010, 0.010




that it will take care of ush, osh, tstddt ðDxrÞ and as the value of tst is

very large for CPSS, it offers the highest value of OF2() among all thePSSs taken into consideration for this study. Thus, CPSS is exhibitspoor performance for the specific application under study.

7.2.3. Comparative performances of PSSs based on convergenceprofiles

The comparative CASO-based convergence profiles of OF1() forCPSS, PSS2B, PSS3B, and PSS4B are depicted in Fig. 17 correspond-ing to an operating condition of P = 1.0, Q = 0.2, Xe = 0.4752, Et = 1.0(all are in pu). From this figure, the objective function value, OF1()corresponding to PSS3B is found to be the least one and, hence,

Fig. 18. MATLAB-SIMULINK-based transient response profiles of Dv1 and Dxr u

Table 7GA, and CASO-based results of eigenvalue analysis corresponding to operating conditions

Type of PSS Algorithms-SFL Damping ratio (n) Undamped natural

Lowest Highest Lowest

CPSS GA-SFL 0.16 0.57 0.38CASO-SFL 0.41 0.67 0.61

PSS2B GA-SFL 0.26 0.97 0.48CASO-SFL 0.69 1.19 0.78



PSS3B proves to be the best PSS among the PSSs considered forthe specific application under study.

7.2.4. Transient performances of PSSs under different perturbationsFig. 18 displays MATLAB-SIMULINK-based transient response

profiles of Dv1 and Dxr for PSS3B equipped generator. This figurehelps to conclude that PSS3B damps the oscillations of Dv1 andDxr very quickly under the system perturbations considered. Realparts of some eigenvalues for CPSS/PSS2B are always either equalto, or greater than r0 in the negative half plane of jx axis. A feweigenvalues are always outside D-shaped sector (Fig. 7) for anyoperating condition. So, objective function values (OF1()) are

nder different perturbation conditions for generator equipped with PSS3B.

P = 0.95, Q = 0.30, Xe = 1.08, Et = 0.5; all are in pu.

frequency (xn), rad/s Corresponding damped frequency (xd), rad/s

Highest Lowest Highest

2.45 0.42 0.591.49 0.52 4.89

3.33 0.32 0.7913.59 0.63 7.99

1.93 0.15 1.80.99 0.53 0.58

1.22 0.54 0.363.15 0.38 3.25

Fig. 19. Comparative terminal voltage, Dvt (pu) profiles for PSS equipped nonlinearsystem model and linear system model, for 0.01 pu change in DTm and no change inDVref (P = 1.0, Q = 0.2, Xe = 1.08, Et = 1.0;, all are in pu).


always higher (Table 4). Much lower negative real parts ofeigenvalues of PSS3B and PSS4B (not shown) cause higher relativestability than CPSS/PSS2B. Larger reductions of xn and xd for someelectromehanical oscillations are due to higher damping ratiosni� n0 for those particular modes, in case of PSS3B and PSS4B(Table 7).

7.2.5. Comparative performances of nonlinear and linear model forPSS3B

Terminal voltage response profile of an SMIB power systembased on the complete 7th order nonlinear model of the gener-ator system [23,24] is compared to that of the same with two-axis, fourth order linear model [15]. This comparison is carriedout for PSS3B equipped power system model for 0.01 pu changein DTm and no change in DVref, operating conditions beingP = 1.0, Q = 0.2, Xe = 1.08, Et = 1.0, all are in pu. The comparativeresponse profiles are portrayed in Fig. 19. From this figure it isevident that the terminal voltage response under the simulatedinput operating conditions exhibits improved stabilization per-formance for nonlinear system model as compared to that forlinear one.

7.2.6. Optimization timeFor the same iteration cycle, the optimization time taken by

CASO is very much lesser as compared to GA (as shown in Table 4,last column).

8. Conclusion

The present work concludes the followings.

8.1. Major conclusions

i. Among the optimization techniques considered, chaotic antswarm optimization with craziness yields the best optimiza-tion performance.

ii. Under change in operating conditions, faulted or post faultedconditions, the variables of PSS as determined by CASO-SFLis found to be superior than those determined by GA-SFLcounter part.

iii. For on-line, off-nominal system operating conditions, fastacting Takagi Sugeno fuzzy logic is suitable for determina-tion of off-nominal PSS variables. Moreover, the computa-tional burden of the SFL is very low and its practicalimplementation is easy.

8.2. Minor conclusions

i. Dual-input PSS offers better transient performance than sin-gle-input counter part.

ii. Transient performances of dual-input PSS3B among dual-input PSS family (namely PSS2B, PSS3B, and PSS4B) offer lessundershoot, less overshoot and less settling time as com-pared to other dual-input PSSs.

iii. Results obtained from the analysis of PSS3B equipped sys-tem configuration shows that the dynamic stabilization per-formance offered by PSS3B is better with change inoperating conditions and configurations, faulted or post-fault conditions.

iv. PSS3B equipped system model damps out the system pertur-bations very quickly.

v. The seventh order nonlinear model of the system exhibitsbetter performance over linear fourth order model of thesystem.

Thus, CASO with craziness may be appreciated as a powerfuloptimizing technique for power systems application and the prac-tical applicability of dual-input PSS, especially PSS3B, may beestablished for SMIB system. More rigorous testing of PSS3B formulti-generator system with the help of any optimizing tool maybe under consideration for some future works.

Acknowledgement

The comments of the reviewers were instrumental in improvingthis paper from its original version.

References

[1] IEEE Digital Excitation System Subcommittee Report. Computer models forrepresentation of digital-based excitation systems. IEEE Trans EC 1996;11(3):607–15.

[2] El-Sherbiny MK, Hasan MM, El-Saady G, Yousef AM. Optimal pole shifting forpower system stabilization. Electr Power Sys Res 2003;66(3):253–8.

[3] Soliman H, Elshafei AL, Shaltout A, Morsi M. Robust power system stabilizer.IEE Proc Elect Power Appl 2000;147(5):285–91.

[4] Werner H, Korba P, Chen Yang T. Robust tuning of power system stabilizersusing LMI techniques. IEEE Trans CST 2003;11(1):147–52.

[5] Shrikant Rao P, Sen I. Robust tuning of power system stabilizers using QFT.IEEE Trans CST 1999;7(4):478–86.

[6] Hopkins WE, Medanic J, Perkins WR. Output feedback pole placement in thedesign of suboptimal linear quadratic regulators. Int J Control1981;34:593–612.

[7] Geromel JC. Methods and techniques for decentralized control systems. Italy:Clup; 1987.

[8] Costa AS, Freitas FD, Silva AS. Design of decentralized controllers for largepower systems considering sparsity. IEEE Trans PS 1997;12(1):144–52.

[9] Hiyama T, Kugimiya M, Satoh H. Advanced PID type fuzzy logic power systemstabilizer. IEEE Trans EC 1994;9(3):514–20.

[10] Abido MA, Abdel-Magid YL. A genetic based fuzzy logic power systemstabilizer for multimachine power systems. In: Proc IEEE conf; 1997. p. 329–34.

[11] Abido MA, Abdel-Magid YL. A hybrid neuro-fuzzy power system stabilizer formultimachine power systems. IEEE Trans PS 1998;13(4):1323–30.

[12] Cai Jiejin, Ma Xiaoqian, Li Lixiang, Yang Yixian, Peng Haipeng, WangXiangdong. Chaotic ant swarm optimization to economic dispatch. ElectrPower Sys Res 2007;77:1373–80.

[13] Mishra S, Tripathy M, Nanda J. Multi-machine power system stabilizer designby rule based bacteria foraging. Electr Power Sys Res 2007;77(12):1595–607.

[14] Mukherjee V, Ghoshal SP. Intelligent particle swarm optimized fuzzy PIDcontroller for AVR system. Electr Power Sys Res 2007;77(12):1689–98.

[15] Kundur P. Power system stability and control. New York: McGraw-Hill; 1994.[16] Mukherjee V, Ghoshal SP. Towards input selection and fuzzy based optimal

tuning of multi-input power system stabilizer. In: Proceedings of the fourteennational power sys conf (NPSC 2006), paper no. 51, IIT Roorkee, India; 2006.

[17] Mukherjee V, Ghoshal SP. Comparison of intelligent fuzzy based AGCcoordinated PID controlled and PSS controlled AVR system. Int J ElectrPower Energy Syst 2007;29(9):679–89.

[18] Roy Ranjit, Ghoshal SP. A novel crazy swarm optimized economic loaddispatch for various types of cost functions. Int J Electr Power Energy Syst2008;30(4):242–353.


[19] Schwefel HP. Numerical optimization of computer models. Chichester: Wiley;1981.

[20] Yao X, Liu Y. Evolutionary programming made faster. IEEE Trans EC1999;3(2):82–102.

[21] Mukherjee V. Application of evolutionary optimization techniques for someselected power system problems. Ph.D. dissertation, Dept Electrical Engg,National Institute of Technology, Durgapur, India; 2009.

[22] Mukherjee V, Ghoshal SP. Particle swarm optimization-genetic algorithmbased fuzzy logic controller for dual-input power system stabilizers. J Inst Eng(India) 2008;88(pt. EL):36–43.

[23] Anderson PM, Fouad AA. Power system control and stability. IEE Press; 1994.[24] Cheng CH, Hsu YY. Damping of generator oscillation using an adaptive static

VAR compensator. IEEE Trans PS 1992;7(2):718–24.

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Chaotic ant swarm optimization for fuzzy-based tuning of power system stabilizer