Electrical Power and Energy Systems 33 (2011) 1728–1738

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Electrical Power and Energy Systems

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Comparative seeker and bio-inspired fuzzy logic controllers for powersystem stabilizers

Binod Shaw a, Abhik Banerjee a, S.P. Ghoshal b, V. Mukherjee c,⇑a Department of Electrical Engineering, Asansol Engineering College, Asansol, West Bengal, Indiab Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal, Indiac Department of Electrical Engineering, Indian School of Mines, Dhanbad, Jharkhand, India

a r t i c l e i n f o

Article history:Received 18 May 2010Received in revised form 16 January 2011Accepted 12 August 2011Available online 20 October 2011

Keywords:Bacteria foraging optimizationGenetic algorithmPower system stabilizerSeeker optimization algorithmSugeno fuzzy logic

0142-0615/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijepes.2011.08.015

⇑ Corresponding author. Tel.: +91 0326 2235644; faE-mail addresses: binodshaw2000@gmail.com (

rediffmail.com (A. Banerjee), spghoshalnitdgp@gmaagamani@yahoo.com (V. Mukherjee).

a b s t r a c t

Seeker optimization algorithm (SOA) is a new heuristic population-based search algorithm. In this paper,SOA is utilized to tune the parameters of both single-input and dual-input power system stabilizers(PSSs). In SOA, the act of human searching capability and understandings are exploited for the purposeof optimization. In SOA-based optimization, the search direction is based on empirical gradient by eval-uating the response to the position changes and the step length is based on uncertainty reasoning byusing a simple fuzzy rule. Conventional PSS (CPSS) and the three dual-input IEEE PSSs (namely PSS2B,PSS3B and PSS4B) are optimally tuned to obtain the optimal transient performances. From simulationstudy it is revealed that the transient performance of the dual-input PSS is better than the single-inputPSS. It is further explored that among the dual-input PSSs, PSS3B offers the best optimal transient perfor-mance. While comparing the SOA with recently reported optimization algorithms like bacteria foragingoptimization (BFO) and genetic algorithm (GA), it is revealed that the SOA is more effective than eitherBFO or GA in finding the optimal transient performance. Sugeno fuzzy logic (SFL)-based approach isadopted for on-line, off-nominal operating conditions. On real time measurements of system operatingconditions, SFL adaptively and very fast yields on-line, off-nominal optimal stabilizer parameters.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction tional single-input PSS (machine shaft speed (Dxr) as single

The problem of low frequency electromechanical oscillationsarises from the usage of fast acting, high gain automatic voltageregulator (AVR) in modern generator excitation system [1]. Anyform of disturbances such as sudden change in loads, change intransmission line parameters, fluctuation in the output of the tur-bine and faults, invites the problem of low frequency oscillations(typically, in the range of 0.2–3.0 Hz) under various sorts of systemoperating conditions and configurations. Transfer of bulk poweracross weak transmission lines may also invite this problem oflow frequency oscillation. The usage of power system stabilizer(PSS) is a very common and widely accepted solution, prevailingin the utility houses, to tackle this problem. The PSS adds a stabi-lizing signal to AVR which modulates the generator excitation. 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-

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x: +91 0326 2296563.B. Shaw), abhik_banerjee@

il.com (S.P. Ghoshal), vivek_

input to the PSS) design approach based on a single-machine-infinite-bus (SMIB) linearized model in the normal operatingcondition has some deficiencies.

On the other hand, the two inputs to dual-input PSS are ma-chine shaft speed (Dxr) and the change in electrical torque(DTe). The processed output of the PSS is DVpss that acts as anexcitation modulation signal and the desired damping electricaltorque component is produced. Modeling of IEEE type PSS2B,PSS3B and PSS4B are reported in [2] and those models are takenin the present study.

Pole-placement or eigenvalue assignment for single-inputsingle-output system has been reported in literature [3]. A robustPSS tuning approach [4] based upon lead compensator design hasbeen carried out by drawing the root loci for finite number of ex-treme characteristic polynomials. In [4], 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. [5]. In [5], plant uncertainties are expressedin the form of a linear fractional transformation. Results obtainedin [5] are compared to the results obtained in [6] based on quanti-tative feedback theory.

Linear quadratic control [7] has been applied for coordinatedcontrol design. The problem has been formulated as a standard

B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 1729

LQR and a full feedback control was obtained from the solutionthat retains the dominant modes of the closed-loop system. Struc-tural constraints, such as, simple and decentralized control, feed-back of only measured variables, have been in use in powersystems for many years and cannot be addressed by a standardLQR. Such a structurally constrained optimal control problem hasbeen solved using the generalized Riccati equation [8] and was ap-plied to power systems exploiting sparsity [9].

Fuzzy, GA-fuzzy, neuro-fuzzy are just a few among the othernumerous works reported in the literature to tune PSS. Most ofthese techniques are centered on angular speed deviation (Dxr)as single-input feedback to PSS. Some of these techniques sufferfrom complexity of computational algorithm, heavy computationalburden, memory storage problem and non-adaptive tuning undervarious system operating conditions and configurations. Some suf-fer from robustness because of choice of limited number of controlvariables of PSS, limited number of optimization functions and on-line real time necessity for fast changing 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 GA, simulated annealing suffer fromsettings of algorithm parameters and give rise to repeated revisit-ing of the same suboptimal solutions. The application of particleswarm optimization (PSO) [10] for PSS tuning has attracted morefocus of the researchers for this purpose.

Seeker optimization algorithm (SOA) [11,12] is, essentially, anew population based heuristic search algorithm. It is based on hu-man understanding and searching capability for finding an opti-mum solution. In SOA, optimum solution is regarded as onewhich is searched out by a seeker population. The underlying con-cept of SOA is very easy to model and relatively easier than otheroptimization techniques prevailing in the literature. The highlight-ing characteristic features of this algorithm are the following:

(a) search direction and step length are directly used in thisalgorithm to update the position,

(b) proportional selection rule is applied for the calculation ofsearch direction which can improve the population diversityso as to boost global search ability and decrease the numberof control parameters making it simpler to implement, and

(c) fuzzy reasoning is used to generate the step length becausethe uncertain reasoning of human searching could be bestdescribed by natural linguistic variables, and a simple if–elsecontrol rule.

In the present work, this SOA algorithm is utilized for the pur-pose of optimal tuning of the PSS parameters. The novelty of thepresent work is the study of the performance of the SOA indesigning the PSS parameters and to compare the optimizingperformance of this algorithm with those reported in the state-of-the-art literature for the same purpose.

Bacteria foraging optimization (BFO), a bio-inspired technique,has been reported by Mishra et al. [13] to establish the potentialapplication of the BFO technique as a soft computing intelligencein power system optimization arena. The main focus of this articlewas the tuning of single-input PSS. Ghoshal et al. [14] have ex-plored the possibilities of this algorithm for tuning both the sin-gle-input, as well as, the dual-input PSSs. Is it possible to obtainmore optimal results by exploiting the optimization capability ofthe SOA for this specific engineering optimization application?

Thus, it is very much pertinent to explore a comparative study be-tween the SOA-based PSS tuning and the BFO/GA-based PSS tun-ing. Based on the transient performance, it is also important todraw some comparative logical conclusion between single-inputPSS and dual-input PSSs with the assistance of SOA [11,12,15–18], BFO and GA in line with [14].

A fuzzy logic system-based [14] PSS can adjust its parameterson-line according to the environment in which it works and canprovide good damping over a wide range of operating conditions.The best PSS ultimately derived from this paper proves to be themost robust model in damping all electromechanical modes ofgenerator’s angular speed oscillations for all off-line and on-lineconditions, step changes of mechanical torque inputs (DTm), refer-ence voltage inputs (DVref) and during/after clearing of systemfaults. For the present work, off-line conditions are 34 (=81) setsof nominal system operating conditions which is given in Sugenofuzzy logic (SFL) table. On the other hand, in real time environmentthese input conditions vary dynamically and become off-nominal.And this necessitates the use of very fast acting SFL to determinethe off-nominal PSS parameters for off-nominal input operatingconditions occurring in real time.

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

Major objectives (pertaining to algorithm performance) are:

(a) to study the performance of the SOA for tuning the PSSparameters,

(b) to present the potential benefit of the SOA over bio-inspired technique like BFO [14] and evolutionary tech-nique like GA [14] as optimizing techniques,

(c) to explore the suitability of fuzzy logic-based tuned PSSunder various changes in system operating conditionsincluding occurrence of fault and its subsequent clearing,and

(d) to contrast the convergence profile of the SOA with theBFO, and the GA.

Minor objectives (pertaining to PSS performance) are:

(a) to compare the transient performance of single-inputPSS with dual-input PSSs,

(b) to contrast the generator’s angular speed oscillations fordual-input PSSs (namely PSS2B, PSS3B, and PSS4B)equipped system model, and

(c) to critically examine the best type of PSS for practicalimplementation under any sort of system disturbances.

The rest of the paper is organized as follows. In Section 2, theSMIB system and various PSSs under investigation are presented.Mathematical problem for the present study is formulated in Sec-tion 3. Optimizing algorithms, as implemented to optimal tuning ofPSS parameters, are described in Section 4. Sugeno fuzzy logic asapplied to on-line tuning of PSS parameters is narrated in Section5. Input control parameters for the simulation are given in Section6. Section 7 documents the simulation results. Finally, concludingremarks and the scope for future work are outlined in Section 8.

2. SMIB system and various PSS under investigation

An SMIB [1] model, as considered in the present work, is shownin Fig. 1. As the purpose of PSS is to introduce damping torquecomponent, speed deviation is used as logical signal to control gen-erator excitation for conventional PSS (CPSS). On the other hand,speed deviation and torque deviation are taken as the best pair

Fig. 1. Single-machine-infinite-bus test system.

Fig. 2. Block diagram representation of the SMIB system with AVR, high gainthyristor exciter, synchronous generator and PSS.

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

Fig. 5. Block diagram representation of dual-input PSS3B.

1730 B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738

of inputs for dual-input PSSs [2]. The block diagram of SMIB systemwith AVR, high gain thyristor exciter, synchronous generator andPSS is shown in Fig. 2. The generator including AVR, excitationsystem and transmission-circuit reactance is represented by atwo-axis, fourth order model. IEEE type ST1A model of the staticexcitation system is considered in this work. The block diagramsof different stabilizers under study are shown in Figs. 3–6. Thegenerator with AVR and excitation system along with CPSS/PSS2B/PSS3B/PSS4B is represented by eighth/seventeenth/eighth/eleventh order state matrices, respectively.

Fig. 6. Block diagram representation of dual-input PSS4B.

3. Mathematical problem formulation

The parameters 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 way.

Fig. 3. Block diagram representation of conventional PSS.

OF11() =P

i(r0 � ri)2 if r0 > rq, ri is the real part of the itheigenvalue. The relative stability is determined by �r0. The valueof r0 is taken as 6.0 for the best relative stability and optimal tran-sient performance.

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 ver-tical 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 P 0.0. This means unstableoscillation occurs for the particular parameters of PSS. These par-ticular PSS parameters will be rejected during the optimizationtechnique.

So, first multi-objective optimization function is formulated asin following equation:

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

B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 1731

OF1ðÞ ¼ 10� OF11ðÞ þ 10� OF12ðÞ þ 0:01� OF13ðÞ þ OF14ðÞ ð1Þ

The weighting factors ‘10’ and ‘0.01’ in (1) are chosen to impartmore weights to OF11(), OF12() and to reduce high value of OF13(), tomake them mutually competitive during optimization [19,20]. Byoptimizing OF1(), closed loop system poles are consistently pushedfurther left of jx axis with simultaneous reduction in imaginaryparts also, thus, enhancing the relative stability and increasingthe damping ratio above n0. Finally, all closed loop system polesshould lie within a D-shaped sector (Fig. 7) in the negative halfplane of jx axis for which ri�� r0, ni� n0. Selection of suchlow negative value of r is purposefully chosen. The purpose is topush the closed loop system poles as much left as possible fromthe jx axis to enhance stability to a greater extent.

Thorough computation shows that optimization of OF1() is notsufficient for sharp tuning of PSS parameters. So, it is essential todesign a second multi-objective optimization function for sharptuning of PSS parameters. Thus, the second multi-objective optimi-zation function OF2() is formulated as in following equation:

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

� �2

ð2Þ

In (2); osh, ush, tst, ddt ðDxrÞ are overshoot, undershoot, settling

time and time derivative of change in rotor speed, respectively.All these are referred to the transient response of Dxr. The con-strained optimization problem for the tuning of PSSs is, thus, for-mulated as follows.

Minimize OF1() and OF2() in succession with the help of anyoptimization technique to get optimal PSS parameters, subject tothe limits as given in [1,14].

4. Optimization algorithms as applied to PSS tuning parameters

4.1. Genetic algorithm

Implementation steps of the GA algorithm are given in [14].

4.2. Bacteria foraging optimization algorithm

Implementation steps of the BFO algorithm are shown in [14].

4.3. Seeker optimization algorithm

SOA [11,12] is a population-based, heuristic search algorithm. Itregards optimization process as an optimal solution obtained by aseeker population. Each individual of this population is called see-ker. A neighborhood is defined for each seeker. This neighborhoodrepresents the social component for social sharing of information.The population is randomly categorized into three subpopulations.

These subpopulations search over several different domains of thesearch space. All seekers in the same subpopulation constitute aneighborhood.

4.3.1. Steps of seeker optimization algorithmIn SOA, a search direction dij(t) and a step length aij(t) are com-

puted separately for each seeker i on each dimension j for eachtime step t, where aijðtÞP 0 and dij(t) e {�1, 0, 1}. Here, i repre-sents the population number and j represents the number of vari-ables to be optimized. If the ith seeker goes towards the positivedirection of the coordinate axis on the dimension j, dij(t) is takenas +1. If the ith seeker goes towards the negative direction of thecoordinate axis on the dimension j, dij(t) is assumed as �1. The va-lue of dij(t) is assumed as 0 if the ith seeker stays at the current po-sition. In a population of size S, for of each seeker i (1 6 i 6 S), theposition update on each dimension j is given by the followingequation:

xijðt þ 1Þ ¼ xijðtÞ þ aijðtÞ � dijðtÞ ð3Þ

Each subpopulation is searching for optimal solution using itsown information. It hints that the subpopulation may trap into lo-cal optima yielding a premature convergence. Subpopulationsmust learn from each other about the optimum information sofar they have acquired in their domain. Thus, the positions of theworst seekers of each subpopulation are combined with the bestone in each of the other subpopulations using the following bino-mial crossover operator as expressed in following equation:

xknj;worst ¼xlj;best if randj 6 0:5xknj ;worst else

(ð4Þ

In (4), randj is a uniformly random real number within [0,1],xknj;worst is denoted as the jth dimension of the nth worst positionin the kth subpopulation, xlj,worst is the jth dimension of the best po-sition in the lth subpopulation with and n, k, l = 1, 2, . . . , K � 1 andk – l. In order to increase the diversity in the population, goodinformation acquired by each subpopulation is shared among thesubpopulations.

4.3.2. Calculation of search directionThe natural tendency of the swarm is to reciprocate in a coop-

erative manner while executing needs and deeds. Normally, thereare two extreme types of cooperative behavior prevailing in swarmdynamics. One, egotistic is entirely pro-self and another, altruistic isentirely pro-group [15]. Every seeker, as a single sophisticatedagent, is uniformly egotistic [16]. He believes that he should go to-ward his historical best position according to his own judgment.This attitude of ith seeker may be modeled by an empirical direc-tion vector ~di;egoðtÞ as in following equation:

~di;egoðtÞ ¼ signð~pi;bestðtÞ �~xiðtÞÞ ð5Þ

In (5), sign(.) is a signum function on each dimension of the in-put vector. On the other hand, in altruistic behavior each seekerwants to communicate with each other, cooperate explicitly andadjust their behaviors in response to other seeker in the sameneighborhood region for achieving the desired goal. That meansthe seekers exhibit entirely pro-group behavior. The populationthen exhibits a self-organized aggregation behavior of which thepositive feedback usually takes the form of attraction toward a gi-ven signal source. Two optional altruistic directions may be mod-eled as in following equations:

~di;alt1ðtÞ ¼ signð~gbestðtÞ �~xiðtÞÞ ð6Þ

~di;alt2ðtÞ ¼ signð~lbestðtÞ �~xiðtÞÞ ð7Þ

Fig. 9. The action part of the fuzzy reasoning.

1732 B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738

In (6) and (7), ~gbestðtÞ represents neighbors’ historical best posi-tion,~lbestðtÞ means neighbors’ current best position.

Moreover, seekers enjoy the properties of pro-activeness. Seek-ers do not simply act in response to their environment rather theyare able to exhibit goal-directed behavior [16]. In addition, futurebehavior can be predicted and guided by past behavior [17]. As aresult, the seeker may be pro-active to change his search directionand exhibit goal-directed behavior according to his past behavior.Hence, each seeker is associated with an empirical direction calledas pro-activeness direction as in following equation:

~di;proðtÞ ¼ signð~xiðt1Þ �~xiðt2ÞÞ ð8Þ

In (8), t1, t2 e {t, t � 1, t � 2} and it is assumed that~xiðt1Þ is betterthan~xiðt2Þ: Aforementioned four empirical directions as mentionedin (5)–(8) direct human being to take a rational decision in searchdirection. Every dimension j of~diðtÞ is selected applying the follow-ing proportional selection rule (shown in Fig. 8) as stated in follow-ing equation:

dij ¼

0; if rj 6 pð0Þj

þ1; if pð0Þj 6 rj 6 pð0Þj þ pðþ1Þj

�1; if pð0Þj þ pðþ1Þj 6 rj 6 1

8>><>>: ð9Þ

In (9), rj is a uniform random number in [0,1], pðmÞj

(m e {0, +1 � 1} is the percent of the number of m from the set{dij,ego, dij,alt1, dij,alt2, dij,pro} on each dimension j of all the fourempirical directions, i.e. pðmÞj = (the number of m)/4.

4.3.3. Calculation of step lengthFrom the view point of human searching behavior, it is under-

stood that one may find the near optimal solutions in a narrowerneighborhood of the point with lower fitness value and, on theother hand, in a wider neighborhood of the point with higher fit-ness value. A fuzzy system may be an ideal choice to representthe understanding and linguistic behavioral pattern of humansearching tendency.

Different optimization problems often have different ranges offitness values. To design a fuzzy system to be applicable to a widerange of optimization problems, the fitness values of all the seekersare sorted in descending manner and turned into the sequencenumbers from 1 to S as the inputs of fuzzy reasoning. The linearmembership function is used in the conditional part since the uni-verse of discourse is a given set of numbers, i.e. 1, 2, . . . , S. Theexpression is presented as in the following equation:

li ¼ lmax �S� Ii

S� 1ðlmax � lminÞ ð10Þ

In (10), Ii is the sequence number of~xiðtÞ after sorting the fitnessvalues, lmax is the maximum membership degree value which isequal to or a little less than 1.0. Here, the value of lmax is takenas 0.95.

A fuzzy system works on the principle of control rule as ‘‘If {theconditional part}, then {the action part}. Bell membership functionlðxxÞ ¼ e�x2=2d2

(shown in Fig. 9) is well utilized in the literatureto represent the action part. For the convenience, one dimension

Fig. 10. Flowchart of the seeker optimization algorithm.Fig. 8. The proportional selection rule of search directions.

Table 1Implementation steps of the SOA for the tuning of PSS parameters.

Step 1 Initialization(a) Input operating values of P, Q, Xe, Et. Input fixed SMIB

parameters(b) Setting of limits of variable PSS parameters(c) Maximum population number of PSS parameter strings,

maximum iteration cycles(d) Real value initialization of all the PSS parameter strings

of the population within limits(e) Read the SOA parameters

Step 2 Determine the SMIB parameters like K1, K2, K3, K4, K5, K6

(Fig. 2) [1]Step 3 Initialize the positions of the seekers in the search space

randomly and uniformlyStep 4 Set the time step t = 0Step 5 Compute the objective function of the initial positions.

The initial historical best position among the populationis achieved. Set the personal historical best position ofeach seeker to his current position

Step 6 Let t = t + 1Step 7 Select the neighbor of each seekerStep 8 Determine the search direction and step length for each

seeker and update his positionStep 9 Compute the objective function for the new positionsStep 10 Update the historical best position among the population

and historical best position of each seekerStep 11 Repeat from Step 6 till the end of the maximum iteration

cycles/stopping criterionStep 12 Determine the best string corresponding to minimum

objective function valueStep 13 Determine the optimal PSS parameters string corresponding

to the grand minimum objective function value

Fig. 11. SOA-based comparative transient response profiles of Dxr for CPSS, PSS2B,PSS3B and PSS4B equipped system model with 0.01 pu change in DVref and nochange in DTm.

B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 1733

is considered. Thus, the membership degree values of the inputvariables beyond [�3d + 3d] are less than 0.0111 ðlð�3dÞ ¼0:0111Þ, and the elements beyond ½�3dþ 3d�, in the universe ofdiscourse, can be neglected for a linguistic atom [18]. Thus, theminimum value lmin = 0.0111 is set. Moreover, the parameter ~dof the Bell membership function is determined by the followingequation:

~d ¼ x� absð~xbest �~xrandÞ ð11Þ

In (11), the absolute value of the input vector as the corre-sponding output vector is represented by the symbol abs(.). Theparameter x is used to decrease the step length with time stepincreasing so as to gradually improve the search precision. Inthe present experiments, x is linearly decreased from 0.9 to

Table 2GA–SFL, BFO–SFL and SOA–SFL based comparison of OF2() values for CPSS, PSS2B, PSS3B a

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

Algorithms

1 0.2, �0.2, 0.4752, 1.1 GA–SFL [14]BFO–SFL [14]SOA–SFL

2 0.5, 0.2, 0.4752, 1.0 GA–SFL [14]BFO–SFL [14]SOA–SFL

3 0.75, 0.50, 0.4752, 0.50 GA–SFL [14]BFO–SFL [14]SOA–SFL

4 0.95, 0.30, 0.4752, 0.5 GA–SFL [14]BFO–SFL [14]SOA–SFL

5 1.2, 0.6, 1.08, 0.5 GA–SFL [14]BFO–SFL [14]SOA–SFL

0.1 during a run. The ~xbest and ~xrand are the best seeker and a ran-domly selected seeker, respectively, from the same subpopulationto which the ith seeker belongs. It is to be noted here that ~xrand isdifferent from ~xbest and ~d is shared by all the seekers in the samesubpopulation.

In order to introduce the randomness in each dimension and toimprove local search capability, the following equation is intro-duced to convert li into a vector ~li.

lij ¼ RANDðli;1Þ ð12Þ

In (12), RAND(li, 1) returns a uniformly random real numberwithin [li, 1]. Eq. (13) denotes the action part of the fuzzy reason-ing and gives the step length (aij) for every dimension j.

aij ¼ dj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� lnðlijÞ

qð13Þ

The flow-chart of the SOA is presented in Fig. 10.

4.3.4. Implementation of SOA for PSS tuningThe steps of the SOA, as implemented the tuning of PSS param-

eters of present work, is shown in Table 1.

nd PSS4B equipped system model.

Value of OF2() (�107)

CPSS PSS2B PSS3B PSS4B

7.45 6.23 1.42 2.904.91 4.02 1.01 1.623.61 3.93 0.97 1.42

7.85 7.33 2.47 3.174.87 4.14 1.21 1.524.16 3.99 1.02 1.46

7.16 5.18 2.71 3.883.97 3.17 1.01 1.453.41 2.94 0.95 1.26

7.29 5.85 2.02 4.583.15 2.17 1.42 1.962.97 2.01 1.14 1.45

8.96 8.72 3.74 4.395.16 4.83 2.57 2.434.91 4.16 1.91 2.14

Table 3Comparative BFO- and SOA-based comparison of OF1() values under different operating conditions for CPSS, PSS2B, PSS3B and PSS4B equipped system model.

Operating conditions (P, Q, Xe, Et; all are in pu) Type of PSS Algorithm PSS parameters OF1() tex (s)

1.2, 0.6, 0.4752, 0.5 CPSS BFO [14] 175.00, 0.005, 0.005, 0.001, 0.001, 0.352, 0.001 1455.62 330.65SOA 175.00, 0.088, 0.005, 0.084, 0.001, 0.096, 0.001 629.40 3.90

PSS2B BFO [14] 10.00, 0.097, 0.010, 0.113, 0.01, 0.25 1251.60 639.21SOA 20.59, 0.155, 0.444, 0.603, 0.010, 0.097 1070.91 6.97

PSS3B BFO [14] �10.00, 10.00, 0.271, 0.005, 0.005, 0.334 291.30 344.78SOA �29.10, 34.37, 0.5, 0.005, 0.005, 0.706 212.56 3.94

PSS4B BFO [14] �10.00, 10.00, 0.273, 0.005, 0.138, 0.005 311.52 420.73SOA �10.00, 20.19, 1.963, 0.005, 0.345, 0.005 271.20 4.81

1.0, 0.6, 0.93, 0.5 CPSS BFO [14] 175.00, 0.005, 0.005, 0.001, 0.001, 0.165, 0.001 1448.61 350.09SOA 175.00, 0.070, 0.005, 0.067, 0.001, 0.069, 0.001 630.59 3.96

PSS2B BFO [14] 10.00, 0.01, 0.253, 0.158, 0.01, 0.01 1243.00 632.93SOA 29.98, 0.161, 0.35, 0.408, 0.112, 0.059 1201.12 7.08

PSS3B BFO [14] �10.00, 10.00, 2.00, 2.00, 0.005, 0.099 290.88 334.89SOA �10.00, 37.99, 0.533, 0.005, 0.005, 0.945 212.91 3.89

PSS4B BFO [14] 10.00, 10.00, 0.005, 0.153, 0.249, 0.005 378.87 411.90SOA �10.00, 38.01, 0.005, 0.135, 0.728, 0.005 276.89 5.07

1.0, 0.2, 1.08, 0.5 CPSS BFO [14] 230.00, 0.005, 0.005, 0.029, 0.001, 0.001, 0.001 1491.69 345.16SOA �10.00, 100.00, 0.704, 0.079, 0.005, 0.005 530.52 4.76

PSS2B BFO [14] 10.00, 0.01, 0.126, 0.01, 0.393, 0.204 1444.20 638.75SOA 10.00, 0.182, 0.076, 0.172, 0.142, 0.01 1210.36 6.91

PSS3B BFO [14] �10.00, 10.00, 0.171, 0.005, 0.139, 0.131 267.77 335.89SOA �31.14, 34.79, 0.578, 0.005, 0.031, 0.866 247.79 3.89

PSS4B BFO [14] 10.00, 10.00, 0.005, 0.168, 0.237, 0.005 372.49 415.62SOA �10.00, 100.00, 0.704, 0.079, 0.005, 0.005 4.84

1734 B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738

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

The whole process of SFL [14,19] can be categorized into threesteps viz. Fuzzification of input operating conditions, Sugeno fuzzyinference and Sugeno defuzzification. The detailed of the SFL stepsare given in [14,19].

Fig. 12. SOA-based comparative convergence profiles of OF1() for CPSS, PSS2B,PSS3B and PSS4B equipped system model with 0.01 pu change in DVref and nochange in DTm.

6. Input control parameters

For simulation, step perturbation of 0.01 pu is applied either inreference voltage (DVref) or in mechanical torque (DTm). The simu-lation is implemented in MATLAB 7.1 software on a PC with P-IV3.0 G CPU and 512 M RAM. The following are the different inputcontrol parameters.

(a) For SMIB system: Inertia constant, H = 5, M = 2H, nominal fre-quency, 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 6 1.0, 0.4752 6 Xe 6 1.08, 0.5 6 Et

6 1.1. In the block diagram representation of generator withexciter and AVR [1,14]; Trr = 0.02 s, Ka = 200.0.

(b) For GA: Number of parameters depends on the number ofproblem variables (PSS configuration), number of bits =(number of parameters) � 8 (for binary coded GA, as consid-ered for the present work), population size = 50, maximumnumber of iteration cycles = 200, mutation probability =0.001, crossover rate = 80%.

(c) For BFO: Number of problem variables depends upon the PSSstructure under investigation. All the parameters of the algo-rithm are as in [14].

(d) For SOA: Number of problem variables depends upon the PSSstructure under investigation. All the parameters of the algo-rithm are given in Section 4.3.

7. Simulation results and discussions

Optimized PSS parameters determined by any of the optimiza-tion techniques are substituted in MATLAB–SIMULINK model of

the system to obtain the transient response profiles. Final valuesof OF1() and OF2() are already obtained from the end of optimiza-tion. Final eigenvalues, final undamped and damped frequenciesand final damping ratio are determined by the optimization tech-nique at the end of optimization. Sugeno fuzzy rule base tables(not shown) are obtained by applying each optimization techniquefor distinct 81 number nominal input operating conditions. Theoutputs are 81 distinct nominal optimal PSS parameters sets. Foroptimization, the SOA technique is adopted. BFO [14] and GA[14] are utilized for the sake of comparison. The major observa-tions of the present work are documented below. The results ofinterest are bold faced in the respective tables.

7.1. Analytical transient response characteristics

Table 2 depicts the comparative GA–SFL [14], BFO–SFL [14] andSOA–SFL based optimal transient response characteristics (in terms

B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 1735

of OF2() value) of different PSS equipped system model. From thistable, it may be inferred that the transient stabilization perfor-mance of dual-input PSS equipped system model is better than sin-gle-input counterpart. Comparing dual-input PSSs, it is alsoobserved that the transient stabilization performance of PSS3Bequipped system model is superior to that of others. PSS3Bequipped system model offers lesser values of osh, ush, tst, andddt ðDxrÞ. And thus, lesser value of OF2() is obtained for this PSSequipped system model. It may also be observed from this tablethat the SOA-based optimization technique offers lesser value ofOF2() than BFO- and GA-based ones already reported in [14].Hence, SOA–SFL based optimization technique offers better resultsthan either BFO–SFL and GA–SFL based ones. Fig. 11 depicts the

Fig. 13. Comparative GA-, BFO- and SOA-based transient response profiles of Dxr with 0all are in pu): for (a) CPSS, (b) PSS2B, (c) PSS3B, and (d) PSS4B equipped system model.

Fig. 14. Comparative GA-, BFO- and SOA-based convergence profiles of OF1() for 0.01 pu spu): for (a) CPSS, (b) PSS2B, (c) PSS3B and (d) PSS4B equipped system model.

comparative optimal transient performance of the different PSSequipped power system model corresponding to an operating con-dition of P = 1.2, Q = 0.6, Xe = 0.4752, Et = 0.5 (all are in pu) for0.01 pu change in DVref and no change in DTm. From this figure,it is noticed that the transient stabilization performance of dual-in-put PSS is better than that of the single-input one. Again, amongthe dual-input PSSs, the performance of PSS3B is established tobe the best one.

7.2. Analytical eigenvalue based system performance analysis

SOA-based comparison of OF1() values of CPSS, PSS2B, PSS3Band PSS4B are shown in Table 3 for different system operating

.01 pu simultaneous change in DVref and DTm (P = 0.2, Q = �0.2, Xe = 0.4752, Et = 1.0,

imultaneous change in DVref and DTm (P = 0.2, Q = �0.2, Xe = 0.4752, Et = 1.0, all are in

1736 B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738

conditions. From Table 3, it is observed that the value of OF1() is theleast one for PSS3B, establishing 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(). Thus, for PSS2B equipped systemmodel value of OF1() is very much less for PSS3B equipped systemmodel. On the other hand, majority of the eigenvalues for PSS2B-based system are outside the D-shaped but very close to and rightside of (�r0, j0) point. This yields higher values of OF11(), OF12()and OF13() for PSS2B equipped system model. Thus, the value ofOF1() is more for this system. Hence, from the eigenvalue analysisit may be concluded that a considerable improvement has occurredin the transient performance for the PSS3B-based system model.

7.3. Convergence profiles

The comparative SOA-based convergence profiles of OF1() forCPSS, PSS2B, PSS3B and PSS4B equipped system model are depictedin Fig. 12 corresponding to an operating condition of P = 1.2,Q = 0.6, Xe = 0.4752, Et = 0.5 (all are in pu). From this figure, it is

Table 4Comparative GA–SFL, BFO–SFL, and SOA–SFL based results of eigenvalue analysis correspon

Type of PSS Algorithms-SFL Damping ratio (n) Un-damped natura

Lowest Highest Lowest

CPSS GA–SFL [14] 0.16 0.57 0.38BFO–SFL [14] 0.38 0.65 0.59SOA–SFL 0.41 0.74 0.62

PSS2B GA–SFL [14] 0.26 0.97 0.48BFO–SFL [14] 0.49 1.00 0.63SOA–SFL 0.52 1.23 0.69

PSS3B GA–SFL [14] 0.36 0.98 0.17BFO–SFL [14] 0.72 0.98 0.47SOA–SFL 0.81 1.12 0.59

PSS4B GA–SFL [14] 0.2 0.95 0.55BFO–SFL [14] 0.37 0.95 0.34SOA–SFL 0.45 1.45 0.49

(a) Fault 1

Fig. 15. Comparative GA–SFL, BFO–SFL, and SOA–SFL based transient response profiles o(a) Fault 1, and (b) Fault 2.

notice the value of OF1() corresponding to PSS3B equipped systemmodel is found to converge faster than the others.

7.4. Comparative optimization performance of the optimizationtechniques

With regard to the optimization performances of the optimizingalgorithms, as depicted in Table 2, it may be concluded that theSOA-based approach offers the lower values of OF2() for a particu-lar PSS equipped system model for the same input operatingconditions. Comparative transient performances of Dxr andconvergence profiles of OF1() for SOA-, BFO- [14] and GA- [14]based optimization for all the four PSS modules (CPSS, PSS2B,PSS3B, and PSS4B) are depicted in Figs. 13 and 14, respectively.These figures assist to conclude that the transient stabilization per-formance and the convergence profile of the objective function forthe SOA-based optimization are better than those of BFO-based[14] and GA-based [14] ones. The SOA is offering much better opti-mal transient performance than BFO and GA. Thus, the SOA may beaccepted as a true optimizing algorithm for the power systembased application as considered in the present work.

ding to an input operating condition P = 0.95, Q = 0.30, Xe = 1.08, Et = 0.5; all are in pu.

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

Highest Lowest Highest

2.45 0.42 0.591.45 0.45 4.311.56 0.56 5.21

3.33 0.32 0.7913.42 0.56 7.7914.01 0.62 7.99

1.93 0.15 1.80.96 0.33 0.381.12 0.41 0.56

1.22 0.54 0.362.97 0.28 2.983.15 0.31 3.02

(b) Fault 2

f Dxr for the generator equipped with PSS3B under change in operating conditions:

Table 5Off-nominal operating conditions, simulation of faults, algorithms-SFL, and optimal PSS3B parameters.

Fault no. Operating conditions (P, Q, Xe, Et; all are in pu) Algorithm-SFL PSS3B parameters

Fault 1 0.95, 0.3, 0.4752, 1.0 GA–SFL [14] �59.57, 59.22, 0.558, 0.013, 0.317, 0.169BFO–SFL [14] �13.50, 83.26, 0.529, 0.051, 0.008, 1.566SOA–SFL �14.89, 31.86, 0.38, 0.005, 0.008, 0.544

LT bus fault of duration 220 ms and subsequent clearing GA–SFL [14] No change in parametersBFO–SFL [14]SOA–SFL

0.95, 0.3, 1.08, 1.0 GA–SFL [14] �14.22, 66.25, 0.208, 0.013, 0.161, 0.099BFO–SFL [14] �12.34, 83.67, 1.608, 0.239, 0.326, 0.473SOA–SFL �14.09, 31.46, 1.17, 0.005, 0.008, 0.896

Fault 2 1.0, 0.6, 0.4752, 1.1 GA–SFL [14] �10.00, 53.24, 1.618, 0.192, 0.410, 0.254BFO–SFL [14] �10.88, 26.40, 1.279, 0.369, 0.070, 0.140SOA–SFL �42.56, 33.81, 0.911, 0.005, 0.021, 0.693

LT bus fault of duration 220 ms and subsequent clearing GA–SFL [14] No change in parametersBFO–SFL [14]SOA–SFL

0.2, 0.2, 0.4752, 1.1 GA–SFL [14] �15.27, 12.11, 1.805, 0.893, 0.137, 0.005BFO–SFL [14] �13.07, 39.00, 1.322, 1.816, 1.006, 0.088SOA–SFL �14.05, 35.25, 0.431, 0.005, 0.005, 0.621

B. Shaw et al. / Electrical Power and Energy Systems 33 (2011) 1728–1738 1737

7.5. Comparative eigenvalue analysis

Real parts of some eigenvalues for CPSS/PSS2B equipped systemmodel are always either equal to or greater than r0 in the negativehalf plane of jx axis. A few eigenvalues are always outside the D-shaped sector (Fig. 7) for any operating condition. So, the value ofOF1() are always higher (Table 3) for these two variants of PSSmodel. Much lower negative real parts of eigenvalues for PSS3Band PSS4B (not shown) cause higher relative stability thanCPSS/PSS2B. In case of PSS3B and PSS4B larger reductions in (xn)and (xd) for some electromechanical oscillations are noticed dueto higher damping ratios (ni� n0) for those particular modes(Table 4). It is also noted from Table 4 that as compared to the re-sults published in [14], SOA–SFL simulation offers better dampingratio (n), un-damped natural frequency (xn), and correspondingdamped frequency (xd). These features for PSS3B equipped systemmodel are better than CPSS/PSS2B/PSS4B counterpart. It estab-lishes that the PSS3B damps the oscillation of Dxr very quicklyunder any sort of system perturbations.

7.6. Simulation of fault

It is revealed that PSS3B is offering the best transient perfor-mance in damping all electromechanical modes of generator’sangular speed oscillations for all nominal and off-nominal systemconditions, step changes of mechanical torque inputs (DTm) andreference voltage inputs (DVref). LT bus fault of duration 220 msat the instant of 2.0 s is simulated for PSS3B equipped system mod-el and the corresponding comparative transient response profilesof Dxr for SOA–SFL, GA–SFL and BFO–SFL based responses are plot-ted in Fig. 15 (Fig. 15a) corresponds to Fault 1 and Fig. 15b corre-sponds to Fault 2). A close look into these figures show that afterthe creation of the fault, the SOA–SFL based response recovers fromthis abnormal situation with much lesser fluctuation in angularspeed as compared to those of BFO–SFL [14] and GA–SFL [14] basedones. Table 5 depicts the system model parameters as determinedby SFL. Thus, SOA–SFL based model exhibits better response havinglesser amplitude of angular speed deviation under fault and subse-quent clearing condition yielding better dynamic robust transientperformance than BFO–SFL [14] and GA–SFL [14] based ones.PSS3B equipped system model proves to be much less susceptibleto faults because PSS3B settles all the state deviations to zero muchfaster than any other PSS.

8. Conclusion

The act of human searching ability is simulated in the SOA. It isa newly entrant heuristic stochastic optimization algorithm. Thisalgorithm is easy to understand, simple to implement, very fastand can be used for a wide variety of optimization tasks. In this pa-per, a SOA-based fuzzy logic PSS tuning is presented. The resultsare compared with that of either BFO or GA based ones, reportedin the literature. While comparing the benefits of SOA for PSS tun-ing it is revealed that this algorithm offers true optimal result withfaster convergence ability. Thus, it may be proposed that the SOA iscapable of efficiently and effectively solving PSS tuning problem. Ithas the capability of solving other engineering optimizationproblems.

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