solution of economic dispatch problems by seeker optimization algorithm

Expert Systems with Applications 39 (2012) 508–519

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Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Solution of economic dispatch problems by seeker optimization algorithm

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

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

Keywords:Economic dispatchMultiple fuel optionsSeeker optimization algorithmTransmission lossValve point loading

0957-4174/$ - see front matter � 2011 Published bydoi:10.1016/j.eswa.2011.07.041

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

yahoo.com (V. Mukherjee), [email protected]

Seeker optimization algorithm (SOA), a novel heuristic population-based search algorithm, is utilized inthis paper to solve different economic dispatch (ED) problems of thermal power units. In the SOA, the actof human searching capability and understanding are exploited for the purpose of optimization. In thisalgorithm, the search direction is based on empirical gradient by evaluating the response to the positionchanges and the step length is based on uncertainty reasoning by using a simple fuzzy rule. The effective-ness of the algorithm has been tested on four different small, as well as, large scale test power systems tosolve the ED problems. The outcome of the present work is to establish the SOA as a promising alternativeapproach to solve the ED problems in practical power systems. Both the near-optimality of the solutionand the convergence speed of the algorithm are promising. The results obtained are compared with thosepublished in the recent literatures.

� 2011 Published by Elsevier Ltd.

1. Introduction

Economic dispatch (ED) allocates generations among the com-mitted generating units in the most economical manner subjectto different operational constraints (Wood & Wollenberg, 1984).To date, various investigations on ED problems have been under-taken as better solutions would result in more saving in the oper-ating cost.

Increased interests are being paid by the researchers towardsthe application of artificial intelligence technology for the solutionof ED problems. Genetic algorithm (GA) (Chen & Chang, 1995;Gaing, 2003), artificial neural networks (Su & Lin, 2000), simulatedannealing (SA) (Wong & Fung, 1993), Tabu search (Lin, Cheng, &Tsay, 2002), evolutionary programing (Jayabarathi, Sadasivam, &Ramachandran, 1999; Yang, Yang, & Huang, 1996), particle swarmoptimization (PSO) (Kuo, 2008), ant colony optimization (ACO)(Pothiya, Ngamroo, & Kongprawechnon, 2010), differential evolu-tion (Coelho & Mariani, 2006) are just a few among the numeroustechniques adopted for this specific end.

PSO is a global search technique originally introduced by Ken-nedy and Eberhart. It simulates the social evolvement knowledge,probing the optimum by evolving the population which includescandidate solutions. Compared with the other evolutionary algo-rithms, PSO shows incomparable advantages in searching speedand precision. In order to improve the global search ability, reduc-ing the chance of getting trapped into local optimum in solving

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x: +91 0326 2296563.(B. Shaw), vivek_agamani@m (S.P. Ghoshal).

multimodal problems, many revised versions of PSO have ap-peared, mainly concentrating in improving the evolution imple-mentations and exploring best parameters’ combinations. Theserevised and hybrid PSOs have successfully been applied in solvingdifferent ED problems by the researchers’ pool and have been re-ported in the literature time to time.

Inspired by the mechanism of the survival of bacteria, e.g.,Escherichia coli, an optimization algorithm, called bacterial foragingoptimization (BFO) (Passino, 2002) has been developed. The BFOhas been applied successfully in several kinds of power systemoptimization problems including ED (Panigrahi & Pandi, 2008).But this algorithm requires the proper setting of many parametersfor its optimal performance. Also, the algorithm executes many dif-ferent loops, and hence, takes tremendous time to converge tonear-global solutions.

In 2008, a new optimization algorithm based on biogeographyhas been proposed by Simon (2008). Biogeography is modeled afterthe process of natural immigration and emigration of species be-tween islands in search of more friendly habitats. This methodhas been termed as biogeography-based optimization (BBO). In or-der to exploit the exploration and the exploitation capabilities ofdifferential evolution (DE) and BBO, a hybrid technique combiningDE with BBO, referred to as DE/BBO, has been proposed in the lit-erature. In this method, the migration operator of the BBO alongwith mutation, crossover and selection operators of DE have beencombined together to effectively utilize the goodness of both DEand BBO to enhance the convergence property and to improvethe quality of the solution. Both BBO and DE/BBO have beensuccessfully applied in solving ED problems (Bhattacharya &Chattopadhyay, 2010).

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B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519 509

Seeker optimization algorithm (SOA) (Dai, Chen, Zhu, & Zhang,2009a, 2009b) is, essentially, a novel population based heuristicsearch algorithm. It is based on human understanding and search-ing capability for finding an optimum solution. In the SOA, opti-mum solution is regarded as one which is searched out by aseeker population. The underlying concept of the SOA is very easyto model and relatively easier than other optimization techniquesprevailing in the literature. The highlighting characteristic featuresof 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 the global search ability and decrease the num-ber of control parameters making it simpler to implement.

(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.

The present work focuses on the performance of the SOA as anoptimizing tool in solving the ED problems. In view of the above,the main contribution of this paper can be summarized as follows:

(1) Four test systems of the ED problem are solved with the SOAand the best results obtained are presented in this paper.

(2) The best results obtained by adopting the SOA are comparedwith those published in the recent papers.

(3) Considering the quality and the improved convergencespeed of the solution as obtained, application of the SOA insolving the ED seems to be a promising alternative approachfor practical power systems.

Table 1 displays the brief descriptions of the four test systemsadopted in this paper vis-à-vis different algorithms compared withSOA.

The rest of the paper is organized as follows. In Section 2, math-ematical modeling of ED problem is done. In Section 3, an objectivefunction is formulated which requires to be optimized. The SOA isnarrated in Section 4. Numerical example and simulation resultsare presented in Section 5 to demonstrate the performance of thealgorithm for the ED problem. Section 6 focuses on conclusionsof the present work and future research directions.

2. Mathematical modeling of the ED problem

The prime objective of the ED problem is to minimize the totalgeneration cost in power system (with an aim to deliver power tothe end user at minimal cost) for a given load demand with due re-gard to the system equality and inequality constraints (Wood &Wollenberg, 1984).

2.1. ED with quadratic cost function and transmission loss

The problem of ED is multimodal, non-differentiable and highlynon-linear. Mathematically, the problem can be stated as in (1):

Min FTðPÞ ¼Xm

i¼1

FiðPiÞ $=h; i ¼ 1; . . . ;m; ð1Þ

subject to,

(i) Real power balance constraint.The power balance operation can be modeled as in (2):
Xm
i¼1

Pi � PD � PL ¼ 0; i ¼ 1; . . . ;m: ð2Þ

The transmission loss (PL) is a function of active power generation ofeach generating unit for a given load demand (PD). It may be ex-pressed as a quadratic function of generations (using B coefficientmatrix) as given by (3):

PL ¼Xm

i¼1

Xm

j¼1

PiBijPj þXm

i¼1

B0iPi þ B00;

i ¼ 1; . . . ;m; and j ¼ 1; . . . ;m; ð3Þ

where Bij, (i � j)th element of loss coefficient symmetric matrix, B;Bi0, ith element of the loss coefficient vector; and B00, constant losscoefficient.

(ii) Generation capacity constraints.The generating capacity constraints are written as in (4):

Pmini 6 Pi 6 Pmax

i ; i ¼ 1; . . . ;m: ð4Þ

2.2. ED problem with valve point loading

For simplicity, the generator cost function was mostly approxi-mated by a single quadratic function (Lin & Viviani, 1984). How-ever, because the cost function of a fossil fired plant, containinginflexions owing to valve point loading, is highly non-linear, andhence, the cost function is realistically denoted as a recurring rec-tified sinusoidal function (Sinha, Chakrabarti, & Chattopadhyay,2003). Each generator has multi-valve steam turbines and costfunctions comprising of very different input–output curves. Toconsider the valve-point loadings, a sinusoidal function is intro-duced into the quadratic function as given in (5):

FiðPiÞ ¼ ai þ biPi þ ciP2i þ ei � sin fi � Pmin

i � Pi

� �� $=h;

i ¼ 1; . . . ;m: ð5Þ

Ripples are introduced in the input–output curves due to thevalve-point loadings, and thereby, the number of local optima is in-creased. It is to be noted here that the fuel cost coefficients ei and fi

are introduced in (5) to model valve point loadings. Variation offuel cost Fi(Pi) due to valve-point loading with the change of gener-ation value Pi is shown in Fig. 1.

2.3. ED problem with valve point loading and multiple fuel options

To model an accurate and practical ED problem, both valvepoint loading effect and multiple fuel options are also taken intoaccount in Chiang (2005). Units with multiple fuels option utilize‘‘hybrid cost function’’. Each segment of the hybrid cost functionbears some information about the fuel being burnt for the unit’soperation. The single continuous quadratic function of each unitis replaced by several piecewise quadratic functions that reflectthe effects of fuel type changes and the generators must identifythe most economic fuel to be burnt.

To frame both valve point loading effect and multiple fuels, thecost function (Chiang, 2005) may be represented as in (6):

FiðPiÞ ¼

ai1 þ bi1Pi þ ci1P2i þ ei1 � sin fi1 � Pmin

i1� Pi1

� �� ;for fuel 1; Pmin

i 6 Pi 6 Pi1;

ai2 þ bi2Pi þ ci2P2i þ ei2 � sin fi2 � Pmin

i2� Pi2

� �� ;for fuel 2; Pi1 < Pi 6 Pi2 ;

..

.

aik þ bikPi þ cikP2i þ eik � sin fik � Pmin

ik� Pik

� �� ;for fuel k; Pik�1 < Pi 6 Pmax

i :

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð6Þ

Table 1Brief descriptions of the four test systems vis-à-vis algorithms considered.

Test system System description Sl.no.

Name of the algorithm Abbreviationused

Reference taken from

Test system 1 20-generating units without valvepoint loading

1 Biogeography-based optimization BBO Bhattacharya and Chattopadhyay (2010a)2 Lambda iteration method Li Su and Lin (2000)3 Hopfield model based approach HM Su and Lin (2000)4 Chaotic and Gaussian particle swarm

optimizationCGPSO Coelho and Lee (2008)

Test system 2 38-Generating units without valvepoint loading

1 Hybrid differential evolution withbiogeography-based optimization

DE/BBO Bhattacharya and Chattopadhyay (2010b)

2 Biogeography-based optimization BBO Bhattacharya and Chattopadhyay (2010a)3 Particle swarm optimization with

time varying acceleration coefficientsPSO-TVAC Chaturvedi et al. (2009)

4 New particle swarm optimization New PSO Chaturvedi et al. (2009)5 Particle swarm optimization with

crazyPSO-Crazy Chaturvedi et al. (2009)

6 Simple particle swarm optimization SPSO Chaturvedi et al. (2009)

Test system 3 40-Generating units with valve pointloading



2 Biogeography-based optimization BBO Bhattacharya and Chattopadhyay (2010a)3 PSO with both chaotic sequences and

crossover operationCCPSO Park et al. (2010)

4 Quantum-inspired particle swarmoptimization

QPSO Meng et al. (2010)

5 Improved coordinated aggregation-based particle swarm optimization

ICA-PSO Vlachogiannis and Lee (2009)

6 Self-organizing hierarchical particleswarm optimization

SOH-PSO Chaturvedi et al. (2008)

7 New particle swarm optimizationwith local random search

NPSO-LRS Selvakumar and Thanushkodi (2007)

8 Particle swarm optimization withlocal random search

PSO-LRS Selvakumar and Thanushkodi (2007)

9 Combined particle swarmoptimization with real-valuedmutation

CBPSO-RVM Lu et al. (2010)

10 Real coded genetic algorithm RCGA Amjady and Nasiri-Rad (2009)11 Improved fast evolutionary

programingIFEP Sinha et al. (2003)

12 Hybrid evolutionary programing, andsequential quadratic programing

EP-SQP Alsumait et al. (2010)

13 Hybrid genetic algorithm, patternsearch, and sequential quadraticprograming

GA-PS-SQP Alsumait et al. (2010)

14 Bacterial foraging with Nelder–Meadalgorithm

BF-NM Panigrahi and Pandi (2008)

15 Chaotic differential evolution CDE Coelho and Mariani (2006)16 Ant colony optimization ACO Pothiya et al. (2010)

Test system 4 10-Generating units with valve pointloading and multiple fuel options



2 Biogeography-based optimization BBO Bhattacharya and Chattopadhyay (2010a)3 New particle swarm optimization

with local random searchNPSO-LRS Selvakumar and Thanushkodi (2007)

4 New particle swarm optimization NPSO Selvakumar and Thanushkodi (2007)5 Particle swarm optimization with

local random searchPSO-LRS Selvakumar and Thanushkodi (2007)

6 Ant colony optimization ACO Pothiya et al. (2010)7 Real coded genetic algorithm RCGA Amjady and Nasiri-Rad (2009)8 Improved genetic algorithm with

multiplier updatingIGA-GA Chiang (2005)

9 Conventional genetic algorithm withmultiplier updating

CGA-MU Chiang (2005)

510 B. Shaw et al. / Expert Systems with Applications 39 (2012) 508–519

3. Formulation of the objective function

We must define the objective function OF() for evaluating thefitness of each individual in the population (Gaing, 2003). In orderto emphasize the ‘‘best’’ solution and the speed up convergence ofthe iteration procedure, the evaluation value is normalized into therange between 0 and 1. The objective function is as given in (7). Itis the reciprocal of the sum of generation cost function and powerbalance constraint. This implies if the values of Fi(Pi) and Ppbc (Pi) ofindividual Pi were small, then its evaluation value would be large:

OFðÞ ¼ 1Fcost þ Ppbc

; ð7Þ

where

Fcost ¼ 1þ absPm

i¼1FiðPiÞ � Fmin� �

Fmax � Fminð Þ

� �; ð8Þ

Ppbc ¼ 1þXm

i¼1

Pi � PD � PL

!2

; ð9Þ

Fig. 1. Input–output curve with valve-point loadings. a, b, c, d, e – valve points.


where Fmax, the maximum generation cost among all individuals inthe initial population; and Fmin, the minimum generation costamong all individuals in the initial population.

4. Seeker optimization algorithm and its application to the EDproblem

4.1. Seeker optimization algorithm

SOA (Dai et al., 2009a, Dai, Chen, Zhu, & Zhang, 2009b) is a pop-ulation-based heuristic search algorithm. It regards optimizationprocess as an optimal solution obtained by a seeker population.Each individual of this population is called a seeker. The total pop-ulation is randomly categorized into three subpopulations. Thesesubpopulations search over several different domains of the searchspace. All the seekers in the same subpopulation constitute aneighborhood. This neighborhood represents the social componentfor the social sharing of information.

4.2. Steps of seeker optimization algorithm

In SOA, a search direction dij(t) and a step length aij(t) are com-puted separately for each ith seeker on each jth variable at eachtime step t, where aij(t) P 0 and dij(t) 2 {�1,0,1}. Here, i representsthe population number and j represents the variable number to beoptimized.

(1) Calculation of search direction, dij(t): It is the natural tendencyof the swarms to reciprocate in a cooperative manner whileexecuting their needs and goals. Normally, there are twoextreme types of cooperative behavior prevailing in swarmdynamics. One, egotistic, is entirely pro-self and another,altruistic, is entirely pro-group. Every seeker, as a singlesophisticated agent, is uniformly egotistic (Eustace, Barnes,& Gray, 1993). He believes that he should go towards his his-torical best position according to his own judgment. Thisattitude of ith seeker may be modeled by an empirical direc-tion vector ~di;egoðtÞ as shown in (10):

~di;egoðtÞ ¼ sign ~pi;bestðtÞ �~xiðtÞ� �

: ð10Þ

Fig. 2. The proportional selection rule of search directions.

In (10), sign(�) is a signum function on each variable of the inputvector. On the other hand, in altruistic behavior, seekers want tocommunicate with each other, cooperate explicitly, and adjust theirbehaviors in response to the other seeker in the same neighborhoodregion for achieving the desired goal. That means the seekers exhi-

bit entirely pro-group behavior. The population then exhibits a self-organized aggregation behavior of which the positive feedback usu-ally takes the form of attraction towards a given signal source. Twooptional altruistic directions may be modeled as in (11),(12):

~di;alt1ðtÞ ¼ sign ~gbestðtÞ �~xiðtÞð Þ; ð11Þ~di;alt2ðtÞ ¼ sign ~lbestðtÞ �~xiðtÞ

� �: ð12Þ

In (11),(12), ~gbestðtÞ represents neighbors’ historical best position,~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; they areable to exhibit goal-directed behavior (Wooldridge & Jennings,1995). In addition, the future behavior can be predicted and guidedby the past behavior (Ajzen, 2002). As a result, the seeker may bepro-active to change his search direction and exhibit goal-directedbehavior according to his past behavior. Hence, each seeker is asso-ciated with an empirical direction called as pro-activeness directionas given in (13):

~di;proðtÞ ¼ sign ~xiðt1Þ � ~xiðt2Þð Þ; ð13Þ

In (13), t1, t2 2 {t, t �1, t �2} and it is assumed that ~xiðt1Þ is betterthan ~xiðt2Þ. Aforementioned four empirical directions as presentedin (10)–(13) direct human being to take a rational decision in hissearch direction.

If the jth variable of the ith seeker goes towards the positivedirection of the coordinate axis, dij(t) is taken as +1. If the jth vari-able of the ith seeker goes towards the negative direction of thecoordinate axis, dij(t) is assumed as �1. The value of dij(t) is assumedas 0 if the ith seeker stays at the current position. Every variable j of~diðtÞ is selected by applying the following proportional selectionrule (shown in Fig. 2) as stated in (14):

dij ¼

0; if rj 6 pð0Þj ;

þ1; if pð0Þj 6 rj 6 pð0Þj þ pðþ1Þj ;

�1; if pð0Þj þ pðþ1Þj < rj 6 1;

8>><>>: ð14Þ

In (14), rj is a uniform random number in [0,1], pðmÞj ðm 2 f0;þ1;�1gis the percent of the numbers of ‘‘m’’ from the set {dij,ego,dij,alt1,dij,alt2,dij,pro} on each variable j of all the four empirical directions,

i.e., pðmÞj ¼ ðthe number of mÞ=4.

(2) Calculation of step length, aij(t): from the view point ofhuman searching behavior, it is understood that one mayfind the near-optimal solutions in a narrower neighborhoodof the point with lower fitness, value and on the other hand,in a wider neighborhood of the point with higher fitnessvalue. A fuzzy system may be an ideal choice to representthe understanding and linguistic behavioral pattern ofhuman searching tendency.Different optimization problems often have different rangesof fitness values. To design a fuzzy system to be applicable toa wide range of optimization problems, the fitness values of


all the seekers are sorted in descending manner (for minimi-zation problem)/in ascending manner (for maximizationproblem) and turned into the sequence numbers from 1 toS as the inputs of fuzzy reasoning. The linear membershipfunction is used in the conditional part since the universeof discourse is a given set of numbers, i.e., 1,2, . . . ,S. Theexpression is presented as in (15):

Comeach

li ¼ lmax �S� Ii

S� 1lmax � lmin

� �: ð15Þ

Fig. 3. The action part of the fuzzy reasoning.

Real coded initialization of S seekers

Divide the population into K subpopulations randomly

Calculate the objective function value for each seeker

Calculate the personal best position, neighborhood best position and population best position

Compute step length for each seeker by using (18)

Update the position of each seeker

Subpopulations learn from each other by using (20)

Meet stopping criterion?

Display the optimal fitness value and optimal solution

Start

Stop

No

Yes

pute search direction for seeker by using (14)

Calculate the objective function value for each seeker

Update the personal best position, neighborhood best position and population best position

Set t = 0

Increment t = t + 1

Fig. 4. Flowchart of the seeker optimization algorithm.

In (15), 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 {the con-ditional part}, then {the action part}. Bell membership functionlðxÞ ¼ e�x2=2d2

(shown in Fig. 3) is well utilized in the literature torepresent the action part. For the convenience, one variable is con-sidered. Thus, the membership degree values of the input variablesbeyond [� 3d,+3d] are less than 0.0111 (l(±3d) = 0.0111), and theelements beyond [�3d, +3d] in the universe of discourse can be ne-glected for a linguistic atom (Li, 1998). Thus, the minimum valuelmin = 0.0111 is set. Moreover, the parameter,~d of the Bell member-ship function is determined by (16):

Table 2Implem

~d ¼ x� abs ~xbest �~xrandð Þ: ð16Þ

In (16), the absolute value of the input vector as the correspondingoutput vector is represented by the symbol abs(�). The parameter xis used to decrease the step length with increasing time step so as togradually improve the search precision. In the present experiments,x is linearly decreased from 0.9 to 0.1 during a run. The ~xbest and~xrand are the best seeker and a randomly selected seeker respectivelyfrom the same subpopulation to which the ith seeker belongs. It is tobe noted here that~xrand is different from ~xbest and ~d is shared by allthe seekers in the same subpopulation.In order to introduce the randomness in each variable and to im-prove local search capability, the following equation is introducedto convert li into a vector ~li with elements as given by (17):

lij ¼ RANDðli;1Þ: ð17Þ

In (17), RAND(li,1) returns a uniformly random real number within[li,1]. Eq. (18) denotes the action part of the fuzzy reasoning andgives the step length (aij) for every variable j:

entation steps of the SOA algorithm for ED problems.

Step 1 Initialization(a) Read input data(b) Read cost curves of machines and B coefficients(c) Set number of run counter(d) Set maximum population number of generator output parame-

ter strings(e) Set lower and upper limits of each generator output(f) Read SOA parameters(g) Set termination criteria (i.e., maximum iteration cycles)

Step 2 Initialize the positions of the seekers in the search space ran-domly and uniformly

Step 3 Set the time step t = 0Step 4 Compute the objective function of the initial positions. The ini-

tial historical best position among the population is achieved.Set the personal historical best position of each seeker to hiscurrent position

Step 5 Let t = t + 1Step 6 Select the neighbor of each seekerStep 7 Determine the search direction and step length for each seeker,

and update his positionStep 8 Update the position of each seekerStep 9 Compute the objective function for each seeker

Step 10 Update the historical best position among the population andhistorical best position of each seeker

Step 11 Subpopulation learns from each otherStep 12 Repeat from Step 6 till the end of the maximum iteration

cycles/stopping criterionStep 13 Determine the best string corresponding to optimum objective

function valueStep 14 Determine the optimal generation string corresponding to the

grand optimum objective function value

Table 3Best results for the test system 1 with PD = 2500 MW.

Unit SOA BBO (Bhattacharya &Chattopadhyay, 2010a)

Li (Su & Lin, 2000) HM (Su & Lin, 2000) CGPSO (Coelho & Lee, 2008)

P1 471.6036 513.0892 512.7805 512.7804 563.3155P2 100.9020 173.3533 169.1033 169.1035 106.5639P3 123.2770 126.9231 126.8898 126.8897 98.7093P4 82.7784 103.3292 102.8657 102.8656 117.3171P5 133.2635 113.7741 113.6386 113.6836 67.0781P6 43.2814 73.06694 73.5710 73.5709 51.4702P7 72.1406 114.9843 115.2878 115.2876 47.7261P8 119.1722 116.4238 116.3994 116.3994 82.4271P9 153.1067 100.6948 100.4062 100.4063 52.0884P10 80.5750 99.99979 106.0267 106.0267 106.5097P11 198.8873 148.9770 150.2394 150.2395 197.9428P12 444.3575 294.0207 292.7648 292.7647 488.3315P13 111.9436 119.5754 119.1154 119.1155 99.9464P14 59.1284 30.54786 30.8340 30.8342 79.8941P15 77.5241 116.4546 115.8057 115.8056 101.525P16 61.9256 36.22787 36.2545 36.2545 25.8380P17 45.2780 66.85943 66.8590 66.8590 70.0153P18 58.4592 88.54701 87.9720 87.9720 53.9530P19 60.2692 100.9802 100.8033 100.8033 65.4271P20 41.3794 54.2725 54.3050 54.3050 36.2552

Total power output (MW) 2539.2530 2592.1011 2591.9670 2591.9670 2512.3343Total transmission loss (MW) 39.1177 92.1011 91.9670 91.9669 12.3343Power mismatch (MW) 0.1348 0 �0.000187 0.000021 NRa

Total generation cost ($/h) 59291 62456.77926 62456.6391 62456.6341 59804.0500Time/iteration(s) 0.0234 0.29282 0.033757 0.006355 0.44

a NR means not reported in the referred literature.


aij ¼ dj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ln lij

� �r: ð18Þ

(3) Updating of seekers’ position: in a population of sizeS, for eachseeker i(1 6 i 6 S), the position update on each variable j isgiven by the following equation:

Table 4Convergence results (50 trial runs) for the test system 1 with PD = 2500 MW.
xijðt þ 1Þ ¼ xijðtÞ þ aijðtÞ � dijðtÞ; ð19Þ
Algorithms Total generation cost ($/h)

Minimum Maximum Average

BBO (Bhattacharya &Chattopadhyay, 2010a)

62456.7926 62456.7935 62456.7928

Li (Su & Lin, 2000) 62456.6391 NRa NRa

HM (Su & Lin, 2000) 62456.6341 NRa NRa

where xij(t + 1), the position of the jth variable of the ith see-ker at time step t + 1; xij(t), the position of the jth variable ofthe ith seeker at time step t; aij(t), the step length of the jthvariable of the ith seeker at time step t; and dij(t), the searchdirection of the jth variable of the ith seeker at time step t.

CGPSO (Coelho & Lee, 2008) 59804.05 63184.63 61171.84SOA 59291 59520 59900


(4) Subpopulations learn from each other: each subpopulation issearching for the optimal solution using its own information.It hints that the subpopulation may trap into local optimayielding a premature convergence. Subpopulations mustlearn from each other about the optimum information sofar they have acquired in their respective domain. Thus,the position of the worst seeker of each subpopulation iscombined with the best one in each of the other subpopula-tions using the following binomial crossover operator asexpressed in (20):

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

: ð20Þ

Fig. 5. Convergence profile of the total generation cost for 20-generating units.

In (20), randj is a uniformly random real number within [0,1],xknj;worst is denoted as the jth variable of the nth worst position inthe kth subpopulation, xlj,best is the jth variable of the best posi-tion in the lth subpopulation. Here, 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 amongthe other subpopulations. The flowchart of the algorithm isdepicted in Fig. 4.

4.3. Implementation of the SOA for the ED problem

The steps of the SOA, as implemented for the solution of EDproblem of this work, are shown in Table 2.


Unit SOA DE/BBO (Bhattacharya &Chattopadhyay, 2010b)


PSO-TVAC (Chaturvediet al., 2009)

New-PSO (Chaturvediet al., 2009)

P1 398.4807 426.606060 422.230586 443.659 550.000P2 384.9040 426.606054 422.117933 342.956 512.263P3 338.4664 429.663164 435.779411 433.117 485.733P4 316.8407 429.663181 445.481950 500.00 391.083P5 271.2609 429.663193 428.475752 410.539 443.846P6 270.3114 429.663164 428.649254 492.864 358.398P7 259.7542 429.663185 428.115368 409.483 415.729P8 441.9667 429.663168 429.900663 446.079 320.816P9 193.5191 114.000000 115.904947 119.566 115.347P10 276.9221 114.000000 114.115368 137.274 204.422P11 203.3973 119.768032 115.418662 138.933 114.000P12 170.1541 127.072817 127.511404 155.401 249.197P13 229.5496 110.000000 110.000948 121.719 118.886P14 145.4604 90.0000000 90.0217671 90.924 102.802P15 253.1107 82.0000000 82.0000000 97.941 89.039P16 239.3751 120.000000 120.038496 128.106 120.000P17 100.5346 159.598036 160.303835 189.108 156.562P18 123.2491 65.0000000 65.0001141 65.00 84.265P19 109.7293 65.0000000 65.0001370 65.00 65.041P20 202.3414 272.000000 271.999591 267.422 151.104P21 188.1623 272.000000 271.872680 221.383 226.344P22 177.4071 260.000000 259.732054 130.804 209.298P23 104.8227 130.648618 125.993076 124.269 85.719P24 62.1276 10.0000000 10.4134771 11.535 10.000P25 79.1108 113.305034 109.417723 77.103 60.000P26 86.7702 88.0669159 89.3772664 55.018 90.489P27 46.7746 37.5051018 36.4110655 75.000 39.670P28 58.3767 20.0000000 20.0098880 21.682 20.000P29 39.5597 20.0000000 20.0089554 29.829 20.995P30 48.9387 20.0000000 20.0000000 20.326 22.810P31 58.8491 20.0000000 20.0000000 20.000 20.000P32 53.4548 20.0000000 20.0033959 21.840 20.416P33 45.2953 25.0000000 25.0066586 25.620 25.000P34 33.6909 18.0000000 18.0222107 24.261 21.319P35 18.6810 8.00000000 8.00004260 9.667 9.122P36 49.7400 25.0000000 25.0060660 25.000 25.184P37 28.3711 21.7820891 22.0005641 31.642 20.000P38 20.0464 21.0621792 20.6076309 29.935 25.104

TGa 6129.507 NRf NRf NRf NRf

TTLb 129.497 NRf NRf NRf NRf

PMc 0.01 NRf NRf NRf NRf

TGCd 9.0940e + 06 9417235.78639 9417633.637644 9500448.307 9516448.312TIe 0.19 NRf NRf NRf NRf

a TG means total generation (MW).b TTL means total transmission loss (MW).c PM means power mismatch (MW).d TGC means total generation cost ($/h).e TI means time/iteration.f NR means not reported in the referred literature.

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5. Test systems and solution results

SOA has been applied to solve ED problems of four different testsystems for investigating its optimization capability. The softwarehas been written in MATLAB-7.3 language and executed on a 3.0-GHz Pentium IV personal computer with 512-MB RAM.

5.1. Description of the test systems

The following four test systems are considered in this work.

(A) Test system 1: 20-generating units without valve point loading:A system with 20 generators is taken as the test system 1.The system input data are available in Su & Lin (2000); Coel-ho & Lee (2008). The valve point loading effect is not consid-ered for this system but transmission loss is considered. Forthis test system load demand is 2500 MW. The resultsreported in the literature like BBO (Bhattacharya & Chatto-padhyay, 2010a), Li (Su & Lin, 2000), HM (Su & Lin, 2000),and CGPSO (Coelho & Lee, 2008) are compared with theSOA-based results and the potential benefit of the SOA asan optimizing algorithm for this specific application is estab-lished. The best solutions of the generation schedules, thetotal optimal generation cost etc. as obtained from 50 trialruns of the SOA and the other afore-mentioned algorithmsare presented in Table 3. Convergence results for the differ-ent algorithms with the same PD are also shown in Table 4.The convergence profile of the cost function is depicted inFig. 5.

(B) Test system 2: 38-generating units without valve point loading:A system with 38 generators is taken as the test system 2.Fuel cost characteristics are quadratic. Transmission loss isconsidered for this test system. The input data of the systemare taken from Sydulu (1999). The load demand is 6000 MW.The best results obtained by using the SOA have been com-pared with those by using DE/BBO (Bhattacharya & Chatto-padhyay, 2010b), BBO (Bhattacharya & Chattopadhyay,2010a), PSO-TVAC (Chaturvedi, Pandit, & Srivastava, 2009),New PSO (Chaturvedi et al., 2009), PSO-Crazy (Chaturvediet al., 2009), and SPSO (Chaturvedi et al., 2009). The bestsolutions of the generation schedules and the total genera-tion cost etc. as obtained from 100 trial runs of the differentalgorithms are shown in Table 5. Convergence results for thealgorithms with the same PD are also presented in Table 6.The convergence profile of the cost function is depicted inFig. 6.




SPSO (Chaturvedi et al.,2009)

9543984.777 NRa NRa

PSO-Crazy (Chaturvediet al., 2009)

9520024.601 NRa NRa

New PSO (Chaturvediet al., 2009)

9516448.312 NRa NRa

PSO-TVAC (Chaturvediet al., 2009)

9500448.307 NRa NRa


9417633.637 NRa NRa

DE/BBO (Bhattacharya &Chattopadhyay, 2010b)

9417235.786 NRa NRa

SOA 9.0940e + 006 9.1812e + 006 9.0912e + 006


(C) Test system 3: 40-generating units with valve point loading: Asystem with 40 generators with valve point loadings is con-sidered as the test system 3. The input data are given in Sin-ha et al. (2003). The load demand is 10500 MW. The bestresults obtained from the SOA are compared to thoseobtained by using DE/BBO (Bhattacharya & Chattopadhyay,2010b), BBO (Bhattacharya & Chattopadhyay, 2010a), CCPSO(Park, Jeong, Shin, & Lee, 2010), QPSO (Meng, Wang, Dong, &Wong, 2010), ICA-PSO (Vlachogiannis & Lee, 2009), SOH-PSO(Chaturvedi, Pandit, & Srivastava, 2008), NPSO-LRS (Selvaku-mar & Thanushkodi, 2007), NPSO (Selvakumar & Thanushko-di, 2007), PSO-LRS (Selvakumar & Thanushkodi, 2007),CBPSO-RVM (Lu, Sriyanyong, Song, & Dillon, 2010), RCGA(Amjady & Nasiri-Rad, 2009), IFEP (Sinha et al., 2003), EP-SQP (Alsumait, Sykulski, & Al-Othman, 2010), GA-PS-SQP(Alsumait et al., 2010), BF-NM (Panigrahi & Pandi, 2008),CDE (Coelho & Mariani, 2006), and ACO (Pothiya et al.,2010). The best solutions of the generation schedules andthe total generation cost etc. as obtained from 50 trial runsare presented in Table 7. Convergence results for the differ-ent algorithms with the same PD are also presented inTable 8. Table 9 shows the frequency of attaining minimumcost within different ranges for this test system out of 50independent trials. The convergence profile of the cost func-tion is depicted in Fig. 7.

(D) Test system 4: 10-generating units with valve point loading andmultiple fuel options: A system comprising of 10 thermalunits with valve point loading and multiple fuels option isconsidered as the test system 4. The input data are takenfrom Chiang (2005). The load demand is 2700 MW. Trans-mission loss is not considered for this case. The best resultsobtained by the SOA are compared to those obtained by thecombined DE-BBO (Bhattacharya & Chattopadhyay, 2010b);BBO (Bhattacharya & Chattopadhyay, 2010a); different PSOtechniques namely NPSO-LRS (Selvakumar & Thanushkodi,2007), NPSO (Selvakumar & Thanushkodi, 2007), PSO-LRS(Selvakumar & Thanushkodi, 2007); ACO (Pothiya et al.,2010), and different GA methods like are RCGA (Amjady &Nasiri-Rad, 2009), IGA-MU (Chiang, 2005), CGA-MU (Chiang,2005). The best solutions of the generation schedules andthe total generation cost etc. as obtained from 100 trial runsof the algorithms are shown in Table 10. Convergence resultsfor the algorithms with the same PD are presented in




CCPSO (Parket al., 2010)

QPSO (Menget al., 2010)

ICA-PSO (Vlachogiannis &Lee, 2009)

SOH-PSO (Chaturvediet al., 2008)

NPSO-LRS (Selvakumar &Thanushkodi, 2007)

P1 104.5681 110.7998 111.0465 110.7998 111.20 110.80 110.80 113.9761P2 89.5109 110.7998 111.5915 110.7999 111.70 110.80 110.80 113.9986P3 102.6168 97.3999 97.60077 97.3999 97.40 97.41 97.40 97.4241P4 170.8105 179.7331 179.7095 179.7331 179.73 179.74 179.73 179.7327P5 81.9546 87.9576 88.30605 87.7999 90.14 88.52 87.80 89.6511P6 127.5261 140.00 139.9992 140.0000 140.00 140.00 140.00 105.0444P7 245.8990 259.5997 259.6313 259.5997 259.60 259.60 259.60 259.7502P8 256.1526 284.5997 284.7366 284.5997 284.80 284.60 284.60 288.4534P9 259.9806 284.5997 284.7801 284.5997 284.84 284.60 284.60 284.6460P10 258.5663 130.00 130.2484 130.0000 130.00 130.00 130.00 204.8120

P11 279.0242 168.7998 168.8461 94.0000 168.80 168.80 94.00 168.8311P12 295.6307 94.00 168.8239 94.0000 168.8 94.00 94.00 94.0000P13 414.2454 214.7598 214.7038 214.7598 214.76 214.76 304.52 214.7663P14 418.8108 394.2794 304.5894 394.2794 304.53 394.28 304.52 394.2852P15 399.6508 394.2794 394.2461 394.2794 394.28 394.28 394.28 304.5187P16 422.0261 304.5196 394.2409 394.2794 394.28 304.52 398.28 394.2811P17 407.6866 489.2794 489.2919 489.2794 489.28 498.28 489.28 489.2807P18 446.4102 489.2794 489.4188 489.2794 489.28 489.28 489.28 489.2832P19 486.8343 511.2794 511.2997 511.2794 511.28 511.28 511.28 511.2845P20 424.1161 511.2794 511.3073 511.2794 511.28 511.28 511.27 511.3049

P21 514.9760 523.2794 523.417 523.2794 523.28 523.28 523.28 523.2916P22 520.6876 523.2794 523.2795 523.2794 523.28 523.28 523.28 523.2853P23 500.4430 523.2794 523.3793 523.2794 523.29 523.28 523.28 523.2797P24 485.1688 523.2794 523.3225 523.2794 523.28 523.28 523.28 523.2994P25 514.0632 523.2794 523.3661 523.2794 523.29 523.28 523.28 523.2865P26 512.7867 523.2794 523.4262 523.2794 523.28 523.28 523.28 523.2936P27 112.0454 10.00 10.05316 10.0000 10.01 10.00 10.00 10.0000P28 135.0547 10.00 10.01135 10.0000 10.01 10.00 10.00 10.0001P29 103.1277 10.00 10.00302 10.0000 10.00 10.00 10.00 10.0000P30 81.5614 97.00 88.47754 87.8000 88.47 96.39 97.00 89.0139

P31 160.4972 190.00 189.9983 190.0000 190.00 190.00 190.00 190.0000P32 149.1974 190.00 189.9881 190.0000 190.00 190.00 190.00 190.0000P33 147.2893 190.00 189.9663 190.0000 190.00 190.00 190.00 190.0000P34 171.3613 164.7998 164.8054 164.7998 164.91 164.82 185.20 199.9998P35 179.2939 200.00 165.1267 194.3976 165.36 200.00 164.80 165.1397P36 176.9985 200.00 165.7695 200.0000 167.19 200.00 200.00 172.0275P37 85.5451 110.00 109.9059 110.0000 110.00 110.00 110.00 110.0000P38 85.4917 110.00 109.9971 110.0000 107.01 110.00 110.00 110.0000P39 81.6582 110.00 109.9695 110.0000 110.00 110.00 110.00 93.0962P40 347.7701 511.2794 511.2794 511.2794 511.36 511.28 511.28 511.2996

TGa 10757.04 NRf NRf NRf NRf NRf NRf NRf

TTLb 257.03 NRf NRf NRf NRf NRf NRf NRf

PMc 0.01 NRf NRf NRf NRf NRf NRf NRf

TGCd 114060 121420.89 121426.95 121403.5362 121448.21 121413.2 121501.14 121664.4308TIe 0.05 0.06 0.11 NRf NRf 0.22 NRf NRf

a TG means total generation (MW).b TTL means total transmission loss (MW).c PM means power mismatch (MW).d TGC means total generation cost ($/h).e TI means time/iteration.f NR means not reported in the referred literature.

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IFEP (Sinha et al., 2003) 122624.3500 125740.6300 123382.0000EP-SQP (Alsumait et al., 2010) 122324 NRa 122379PSO-LRS (Selvakumar & Thanushkodi, 2007) 122035.7946 123461.6794 122558.4565CDE (Coelho & Mariani, 2006) 121741.9793 NRa 121814.9465NPSO (Selvakumar & Thanushkodi, 2007) 121704.7391 122995.0976 122221.3697NPSO-LRS (Selvakumar & Thanushkodi, 2007) 121664.4308 122981.5913 122209.3186CBPSO-RVM (Lu et al., 2010) 121555.32 123094.98 122281.14ACO (Pothiya et al., 2010) 121532.41 121679.64 121606.45SOH-PSO (Chaturvedi et al., 2008) 121501.14 122446.30 121853.57GA-PS-SQP (Alsumait et al., 2010) 121458.14 NRa 122039QPSO (Meng et al., 2010) 121448.21 NRa 122225.07BBO (Bhattacharya & Chattopadhyay, 2010a) 121426.953 121688.6634 121508.0325BF-NM (Panigrahi & Pandi, 2008) 121423.63792 NRa 122295.1278DE/BBO (Bhattacharya & Chattopadhyay, 2010b) 121420.8948 121420.8968 121420.8952RCGA (Amjady & Nasiri-Rad, 2009) 121418.5425 121628.5987 121504.1169ICA-PSO (Vlachogiannis & Lee, 2009) 121413.20 121453.56 121428.14CCPSO (Park et al., 2010) 121403.5362 121525.4934 121445.3269SOA 114060 116000 114850


Fig. 7. Convergence profile of the total generation cost for the 40-generating units.

Table 9Frequency of convergence in 50 trial runs for 40-generating units with PD = 10500 MW.

Algorithms Range of total generation cost (�103, $/h)

<120.0 120.0–121.5

121.5–122.5

122.5–123.0

123.0–123.5

123.5–124.0

124.0–124.5

124.5–125.0

125.0–125.5

125.5–126.0

>126.0

SOA 50 0 0 0 0 0 0 0 0 0 0DE/BBO (Bhattacharya &

Chattopadhyay, 2010b)0 50 0 0 0 0 0 0 0 0 0

BBO (Bhattacharya & Chattopadhyay,2010a)

0 38 12 0 0 0 0 0 0 0 0

QPSO (Meng et al., 2010) 0 2 27 20 1 0 0 0 0 0 0SOH-PSO (Chaturvedi et al., 2008) 0 0 50 0 0 0 0 0 0 0NPSO-LRS (Selvakumar &

Thanushkodi, 2007)0 0 40 10 0 0 0 0 0 0 0

NPSO (Selvakumar & Thanushkodi,2007)

0 0 37 13 0 0 0 0 0 0 0

PSO-LRS (Selvakumar &Thanushkodi, 2007)

0 0 26 17 7 0 0 0 0 0 0

CBPSO-RVM (Lu et al., 2010) 0 41 8 1 0 0 0 0 0 0 0IFEP (Sinha et al., 2003) 0 0 0 11 25 9 2 2 0 0 0


Table 11. Table 12 shows the frequency of attaining mini-mum cost within different ranges for this test system out

of 100 independent trials. The convergence profile of thecost function is depicted in Fig. 8. The results of interestare bold faced in the respective tables.Discussions on the results of the different test systems aremade in the following subsection.

5.2. Discussions on the results of the test systems

(A) Solution quality: It is noticed from Tables 3, 5, 7, and 10 thatthe minimum cost achieved by applying the SOA is the leastone as compared to those achieved by earlier reported algo-rithms as mentioned in the respective tables. It may also benoted from Tables 4, 6, 8, and 11 that the minimum, themaximum, and as well as, the average costs achieved bythe SOA are the least once among all the other methodstaken into consideration. It emphasizes on the fact that theSOA offers the best near-optimal solution for the ED prob-lems considered.

(B) Comparison of best generation costs: It may be observed fromTables 3, 5, 7, and 10 that the minimum costs achieved bythe SOA based method for test systems 1–4, are 59291 $/h,9.0940e + 06 $/h, 114060 $/h, and 598.1652 $/h, respec-tively. Again, power mismatches are the least ones in the




NPSO-LRS (Selvakumar& Thanushkodi, 2007)

IGA-MU (Chiang, 2005)

Generation Fuel type Generation Fuel type Generation Fuel type Generation Fuel type

P1 174.5461 1 213.4589 2 212.9 2 223.335 2 219.126 2P2 122.3390 2 209.4836 1 209.4 1 212.195 1 211.164 1P3 408.7926 1 332.0000 3 332.0 3 276.216 1 280.657 1P4 200.2129 1 238.0269 3 238.3 3 239.418 3 238.477 3P5 330.2403 1 269.1423 1 269.2 1 274.647 1 276.417 1

P6 235.0207 2 238.0269 3 237.6 3 239.797 3 240.467 3P7 384.7870 1 280.6144 1 280.6 1 285.538 1 287.739 1P8 206.2955 1 238.1613 3 238.4 3 240.632 3 240.761 3P9 343.1772 3 414.7001 3 414.8 3 429.263 3 429.337 3P10 294.5888 1 266.3850 1 266.3 1 278.954 1 275.851 1

Total generation (MW) 2700 2700 2700 2700 2700Total transmission loss (MW) 0 0 0 0 0Power mismatch (MW) 0 0 0 0 0Total generation cost ($/h) 598.1652 605.6230127 605.6387 624.127 624.517Time/iteration(s) 0.15 0.48 0.80 0.52 7.25

NR⁄ means not reported in the referred literature.




IGA-GA (Chiang, 2005) 627.5178 630.8705 625.8692CGA-MU (Chiang, 2005) 624.7193 633.8652 627.6087PSO-LRS (Selvakumar & Thanushkodi, 2007) 624.2297 628.3214 625.7887NPSO (Selvakumar & Thanushkodi, 2007) 624.1624 627.4237 625.2180NPSO-LRS (Selvakumar & Thanushkodi, 2007) 624.1273 626.9981 624.9985RCGA (Amjady & Nasiri-Rad, 2009) 623.8281 623.8814 623.8495ACO (Pothiya et al., 2010) 623.70 624.09 623.90BBO (Bhattacharya & Chattopadhyay, 2010a) 605.6387 605.9103 605.8622DE/BBO (Bhattacharya & Chattopadhyay, 2010b) 605.6230 605.6231 605.6252SOA 598.1652 601.2624 600.0214

Table 12Frequency of convergence in 100 trial runs for 10-generating units with PD = 2700 MW.

Algorithms Range of total generation cost ($/h)

<605.5 605.5–623.5

623.5–624.5

624.5–625.5

625.5–626.5

626.5–627.5

627.5–628.5

628.5–629.5

629.5–630.5

630.5–631.5

631.5–632.5

632.5–633.5

633.5–634.5

SOA 100 0 0 0 0 0 0 0 0 0 0 0 0DE/BBO (Bhattacharya &

Chattopadhyay, 2010b)0 100 0 0 0 0 0 0 0 0 0 0 0


0 100 0 0 0 0 0 0 0 0 0 0 0

NPSO-LRS (Selvakumar &Thanushkodi, 2007)

0 0 20 58 17 5 0 0 0 0 0 0 0

NPSO (Selvakumar &Thanushkodi, 2007)

0 0 18 54 16 12 0 0 0 0 0 0 0

PSO-LRS (Selvakumar &Thanushkodi, 2007)

0 0 5 37 36 17 5 0 0 0 0 0 0

IGA-MU (Chiang, 2005) 0 0 0 39 45 11 2 2 0 1 0 0 0CGA-MU (Chiang, 2005) 0 0 0 5 20 31 21 10 7 3 2 0 1


SOA as compared to those offered by the other algorithms.Hence, it can be concluded that for all the four test systemsthe performance of the SOA is found to be the best one.

(C) Testing of robustness: The performance of any heuristicsearch based optimization algorithm is best judged throughrepetitive trial runs so as to compare the robustness/consis-tency of the algorithm. For this specific goal, the frequency ofconvergence to the minimum cost at different ranges of gen-eration cost with fixed load demand is recorded and pre-

sented in Table 9 for the test system 3 and in Table 12those for the test system 4, respectively. The same for thetest systems 1 and 2 are not included in the referred litera-tures, and hence, are not reported in the present work. But itis experimented by the authors of the present work with theSOA for the test systems 1 and 2 and it is noticed that theconvergence to the minimum value of the cost function witha minimal variation is achieved. Tables 9 and 12 show thatthe frequency of converging to a better solution is higher



in the SOA as compared to the other methods. Thus, it maybe inferred that the SOA method is the most consistent inachieving the lowest cost in all the runs.

(D) Computational efficiency: Apart from yielding minimum costby the SOA, it may also be noted that the SOA yields mini-mum cost at comparatively lesser time of execution of theprogram. Thus, this approach is also efficient as far as thecomputational time is concerned.

6. Conclusion

In this paper, the SOA has been successfully implemented tosolve different ED problems. It has been observed that the SOAhas the ability to converge to a better quality near-optimal solutionand possesses better convergence characteristics and robustnessthan other prevailing techniques reported in the recent literatures.It is also clear from the results obtained by different trials that theSOA is free from the shortcoming of premature convergence exhib-ited by the other optimization algorithms. Thus, this algorithmmay become very promising for solving some more complex engi-neering optimization problems for future researchers.

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solution of economic dispatch problems by seeker optimization algorithm

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