Research ArticleNonconvex Economic Dispatch Using Particle SwarmOptimization with Time Varying Operators

Vinay Kumar Jadoun Nikhil Gupta K R Niazi and Anil Swarnkar

Malaviya National Institute of Technology Jaipur 302017 India

Correspondence should be addressed to Vinay Kumar Jadoun vjadounmnitgmailcom

Received 23 May 2014 Revised 17 September 2014 Accepted 18 September 2014 Published 12 October 2014

Academic Editor Nikos D Lagaros

Copyright copy 2014 Vinay Kumar Jadoun et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents a particle swarm optimization (PSO) to solve hard combinatorial constrained optimization problems suchas nonconvex and discontinuous economic dispatch (ED) problem of large thermal power plants Several measures have beensuggested in the control equation of the classical PSO bymodifying its operators for better exploration and exploitationThe inertiaoperator of the PSO is modulated by introducing a new truncated sinusoidal function The cognitive and social behaviors aredynamically controlled by suggesting new exponential constriction functions The overall methodology effectively regulates thevelocity of particles during their flight and results in substantial improvement in the classical PSOThe effectiveness of the proposedmethod has been tested for economic load dispatch of three standard test systems considering various operational constraints likevalve-point loading effect prohibited operating zones (POZs) network power loss and so forth The application results show thatthe proposed PSO method is very promising

1 Introduction

The economic dispatch (ED) aims at determining the optimalscheduling of thermal generating units so as to minimizethe fuel cost while satisfying several operational and powersystem network constraintsThe generator fuel cost functionsare invariably nonlinear and also exhibit discontinuities dueto prohibited operating zones (POZs) In addition the valve-point loading effect causes nonconvex characteristic withmultiple minima in the generator fuel cost functions andthus imposes challenges of obtaining the global optima forhigh dimensional ED problems Thus ED is a highly nonlin-ear complex combinatorial nonconvex and multiconstraintoptimization problem with continuous decision variables

The classical mathematical methods like gradient La-grange relaxation methods and so forth except dynamicprogramming are not suitable for such complex optimizationproblemsThemodern metaheuristic search techniques suchas particle swarm optimization (PSO) genetic algorithms(GAs) biogeography-based optimization (BBO) differential

evolution (DE) ant colony optimization (ACO) artificial beecolony (ABC) and hybrid swarm intelligent based harmonysearch algorithm (HHS) [1 2] have shown potential tosolve such complex ED problems due to their ability toobtain global or near global solution but are computationallydemanding especially for modern power systems which arelarge and complex

The PSO has several advantages over other metaheuristictechniques in terms of simplicity convergence speed androbustness [3] It provides convergence to the global or nearglobal optima irrespective of the shape or discontinuities ofthe cost function [4] The potential of PSO to handle nons-mooth and nonconvex ELDproblemwas demonstrated by [56] However the performance of the PSO greatly depends onits parameters and it often suffers from the problems such asbeing trapped in local optima due to premature convergence[6] lack of efficient mechanism to treat the constraints [7]and loss of diversity and performance in optimization process[8] PSO is a population based metaheuristic optimizationtechnique in which themovement of the particles is governed

Hindawi Publishing CorporationAdvances in Electrical EngineeringVolume 2014 Article ID 301615 13 pageshttpdxdoiorg1011552014301615

2 Advances in Electrical Engineering

by the two stochastic acceleration coefficients that is cogni-tive and social components and the inertia component [5] Inorder to enhance the exploration and exploitation capabilitiesof PSO the components affecting velocity of particles shouldbe properly managed and controlled

Several methods have been reported in the recent pastto enhance the computational efficiency of the classical PSOA constriction factor was suggested in the velocity updatingequation to assure convergence of PSO [9ndash11] Howeverthe exact determination of this factor is computationallydemanding Selvakumar and Thanushkodi [12] modifiedcognitive behavior of the swarm by communicating with theworst particle This method provides some additional diver-sity to the particle by the worst experience component butshowing poor local searching ability unless it is hybridizedwith certain other heuristic approaches Roy and Ghoshal[13] proposed crazy PSO (CPSO) where the particle velocityis randomized within predefined limits The idea was torandomize the velocity of some of the particles referred toas ldquocrazy particlesrdquo by applying a predefined probability ofcraziness tomaintain the diversity for global search and betterconvergence However the value of predefined probability ofcraziness can only be achieved after several experimentationsSome attempts [14ndash18] have been made to vary the cognitiveand social behavior of the swarmduring the search process bydynamically controlling the acceleration coefficients withinmaximum and minimum bounds Again the determinationof limiting values of these acceleration coefficients is adifficult task as it required many simulations Coelho andLee [19] randomized cognitive and social behavior of theswarm using chaotic sequences and Gaussian distributionrespectively Selvakumar and Thanushkodi [20] proposedcivilized swarm optimization (CSO) by combining society-civilization algorithm (SCA) with PSO to improve communi-cationTheproposed algorithmprovides clustered search thatresults in better exploration and exploitation of the searchspace but needs several experimentations to determine theoptimum values of the control parameters of CSO Effortshave also been made to suggest a new formulation of thecontrol equation [6 7] Safari and Shayeghi [6] proposediteration PSO (IPSO) where one additional velocity compo-nent pertaining to the best fitness of the current iteration isadded in the control equation of the classical PSO to avoidlocal trap but parameter setting is essential Vlachogiannisand Lee [7] suggested new control equation in improvedcoordinated aggregation PSO (ICAPSO) for better communi-cation among particles to enhance local searchThey allowedparticles to interact with its own best experience along withall other particles have better experience on aggregate basisinstead of the global best experience However the authorsaccepted that the performance of the proposed method isquite sensitive to various parameters setting and their tuningis essential Chaotic PSO (CPSO) of [21] proposed adaptedinertia weight which varies dynamically with fitness value forexploration and chaotic local search was used to determinethe particle position for better exploitation The improvedPSO (IPSO) of [22] suggested chaotic inertia weight whichdecreases and oscillates simultaneously under the decreasingline in a chaotic manner In this way additional diversity

Without valve-point loading effect

With valve-point loading effect

Fuel

cost

($M

W)

Pmin

Output P (MW)Pmax

Figure 1 Fuel cost function with and without valve-point loadingeffect

is introduced but it requires tuning of chaotic controlparameters

This paper attempts to overcome drawbacks of someexisting PSO methods and presents a modified version ofPSO for economic load dispatch of power systems Severalmeasures have been incorporated in the control equationby modifying operators of the classical PSO by introducingnew constriction functionsThe proposed method effectivelycontrols and regulates the components affecting velocityof particles so as to ensure better exploration (searchingnew areas) and exploitation (fine tuning of the currentsolution) A correction algorithm is also suggested to repairinfeasible solutions whenever appeared in the computationalprocess The proposed method is self-adjusting and doesnot require experimentations to obtain the optimal valuesof control parameters and thus overcome the drawbacks ofexisting PSO methods The effectiveness of the proposedmethod has been investigated on three standard test systemsconsidering various operational constraints like valve-pointloading effect prohibited operating zones (POZs) networkpower loss and so forthThe application results show that theproposed PSO method is very promising

2 Problem Formulation

The generator cost function is usually considered as quad-ratic when valve-point loading effects are neglected Thelarge turbine generators usually have a number of fueladmission valves which are operated in sequence to meetout increased generation The opening of a valve increasesthe throttling losses rapidly and thus the incremental heatrate rises suddenlyThis valve-point loading effect introducesripples in the heat-rate curves which introduces nonconvex-ity in the generator fuel cost function as shown in Figure 1The effect of valve-point loading effects can be modeledas sinusoidal function in the cost function Therefore the

Advances in Electrical Engineering 3

objective function for the nonconvex ED problem may bestated as

Minimize119865 (119875119866119894) =

119873119866

sum

119894=1

(119886119894+ 119887119894119875119866119894+ 1198881198941198752

119866119894)

+1003816100381610038161003816119890119894 sin (119891119894 (119875119866119894min minus 119875119866119894))

1003816100381610038161003816

(1)

where 119886119894 119887119894 and 119888119894 are the cost coefficients of the 119894th generator

119890119894 and 119891119894 are the valve-point effect coefficients 119875119866119894 is the realpower output of the 119894th generator and 119873119866 is the number ofgenerating units in the system

Subject to the following constraints

(1) Power Balance ConstraintThe total power generationof all generators must be equal to the sum of totalpower demand plus the network power loss The net-work power loss can be evaluated using 119861-coefficientloss formula [21 23] Therefore the generator powerbalance equation may be stated as follows

119873119866

sum

119894=1

119875119894 = 119875119863 +

119873119866

sum

119894=1

119873119866

sum

119895=1

119875119866119894119861119894119895119875119866119895 +

119873119866

sum

119894=1

1198751198661198941198611198940 + 11986100 (2)

where 119861119894119895 is the transmission loss coefficient 119894 =

1 2 119873119866 and 119895 = 1 2 119873119866 1198611198940 is the 119894thelement of the loss coefficient vector 11986100 is the losscoefficient constant

(2) Generator Constraint For stable operation poweroutput of each generator is restricted within itsminimum andmaximum limitsThe generator powerlimits are expressed as follows

119875min119866119894

le 119875119866119894

le 119875max119866119894

(3)

(3) Prohibited Operating Zones Prohibited operatingzones lead to discontinuities in the input outputrelation of generators Prohibited zones divide theoperating region between minimum and maximumgeneration limits into disjoint convex subregions [1420] The generation limits for the 119894th unit with 119895

number of prohibited zones can be expressed asfollows

119875min119866119894

le 119875119866119894

le 119875119871

1198661198941

119875119880

119866119894119895minus1le 119875119866119894 le 119875

119871

119866119894119895

119875119880

119866119894119873119875119885119894le 119875119866119894

le 119875max119866119894

119894 isin 1 2 119873119866119875119885 119895 isin 2 3 119873119875119885119894

(4)

where superscripts 119871 and 119880 stand for the lowerand upper limit of prohibited operating zones ofgenerators 119873

119866119875119885and 119873

119875119885119894denote the total number

of generators with prohibited zones and the totalnumber of prohibited zones for the 119894th generatorrespectively

3 Proposed PSO

The classical PSO is initialized with a population of randomsolutions and searches for optima by updating particle posi-tions The velocity of the particle is influenced by the threecomponents initial cognitive and the social componentEach particle updates its previous velocity and positionvectors according to the following model [3 24 25]

V119896+1119894

= 119882V119896119894+ 1198881times rand

1 () times119901119887119890119904119905119894 minus 119904119896

119894

Δ119905

+ 1198882times rand

2 () times119892119887119890119904119905119894minus 119904119896

119894

Δ119905

119904119896+1

119895= 119904119896

119895+ V119896+1119895

times Δ119905

(5)

where V119896119894is the velocity of 119894th particle at 119896th iteration rand

1()

and rand2() are random numbers between 0 and 1 119904119896119894is

the position of 119894th particle at 119896th iteration 1198881 1198882 are theacceleration coefficients 119901119887119890119904119905

119894is the best position of 119894th

particle achieved based on its own experience 119892119887119890119904119905119894is the

best particle position based on overall swarm experience Δ119905is the time step usually set to 1 second and 119882 is the inertiaweight which is allowed to decrease linearly as follows

119882 = 119882min +(119882max minus119882min) times (itrmax minus itr)

itrmax (6)

where119882min and119882max are the minimum andmaximum valueof inertia weight respectively itrmax is themaximumnumberof iterations and itr is the current number of iteration

For better performance of PSO the particles must flywith higher velocities during the early flights to enhanceglobal search and should be relatively slow during laterflights of the journey to improve local search Thereforewith appropriate regulation of particlersquos velocity during thejourney the performance of PSO could be improved Initiallythe impact of cognitive component must be high and that ofthe social component be less to ensure global exploration ofthe search space by all particles without trapping into a localminima During later search the impact of social componentmust increase and that of the cognitive component mustdecrease to divert all particles towards global best to improvethe convergence This is essential for a good balance betweenexploration and exploitation as suggested by [15]

In classical PSO only the initial velocity component usinginertia weight is regulated dynamically However the cogni-tive and social behavior of the swarm though randomized toensure diversity is statically controlled by assigning constantvalues to acceleration coefficients These cognitive and socialcomponents of velocity are added in the regulated initialvelocity component to decide themovement of particlesThisprobably results in uncontrolled particle velocities duringthe whole computation process and thus causes insufficientexploration and exploitation of the search space This resultsin poor convergence due to local trapping Therefore amodified control equation (7) is suggested for dynamicallyregulating particlersquos velocity during their whole course of

4 Advances in Electrical Engineering

the flight The modifications suggested in the control equa-tion are explained as follows

V119896+1119894

= 119882 times V119896119894+ 1205771times 1198621119887times rand

1 () times119901119887119890119904119905119894minus 119904119896

119894

Δ119905

+ (1 minus 1205771) times 1198621119901

times rand2 () times

119904119896

119894minus 119901119901119900119900119903

119894

Δ119905

+ 1205772 times 1198622 times rand3 () times

119892119887119890119904119905119894minus 119904119896

119894

Δ119905

(7)

In (7) the inertia weight is modified to regulate the trade-off between the global exploration and the local exploitationof the swarm The poor experience 119901119901119900119900119903

119894has been added

to improve the cognitive component Further dynamic accel-eration coefficients have been introduced using constrictionfunctions 120577

1and 120577

2to regulate the cognitive and social

behaviors of the swarmThese modifications are discussed inthe following sections

31 Inertia Weight Update In [25] Shi and Eberhart sug-gested linear modulation of the inertia weight This trendis followed to solve ELD problems using PSO by manyresearchers till date and some of them can be mentioned as[4 6 8 12 13 15 19 20 25 26] and so forth In the proposedmethod the inertia weight has been allowed to vary inaccordance with a truncated sinusoidal function rather thanto decrease linearlyThemodulations suggested to update theinertia weight is governed by the following relation

119882 = 119882min + (119882max minus119882min) cos2(120579

2)

0 le 120579 le 120587

(8)

where 120579 = 119883 times itr +119884 and the coefficients119883 and 119884 are givenby (9) itr is the iteration countwhich is in general varied fromitrmin to itrmax

119883 =120587

(itrmax minus itrmin)

119884 =minus120587 times itrmin

(itrmax minus itrmin)

(9)

Figure 2 shows a comparison of the conventional linearmodulation and sinusoidal modulation for the inertia weightto be employed in the proposed PSO It can be depicted fromthe figure that using the sinusoidal variations in the inertiaweight the inertia component of the velocity of particlesmaintained always higher during the early half and lowerduring the later half of the search when compared with itslinear variationsTherefore using sinusoidalmodulations thecoarse search is enhanced during the early half by exploringlarger search space with higher values assigned to particlevelocities And during the later half the fine search isenhanced by assigning lower values to particle velocitiesThis facilitates particles to explore the regions in the closeproximity of near global solution

W

Wmax

Wmin

Itrmin Itrmax2 Itrmax Itr

Figure 2 Comparison of linear and sinusoidal modulations ofinertia weight

32 Updating of Poor Experience The cognitive behaviorwas split in [12] by considering the worst experience inaddition to the best experience of the particle Though thismodification provides additional diversity it still demands alocal random search to enhance exploitation potential of thePSOThis occurs as the particlersquos velocity is not well regulatedduring later part of the search Therefore the concept ofpoor experience 119901119901119900119900119903

119894is suggested instead of the worst

experience to improve cognitive behavior of the swarmHerethe current fitness of each particle is compared with its fitnessvalue in the preceding iteration and if it is found less it willbe treated as the poor experience This concept is differentthan that of [12] where the worst particle is determinedby considering the whole past experience of the particlemovement The poor particle produces much less diversitythan the worst particle and thus exploit the region near globaloptima during later iterations in much better way withoutthe support of any local random search

33 Dynamic Control of Acceleration Coefficients In classicalPSO the cognitive and social behaviors are governed byassigning static values to acceleration coefficients Manyresearchers as discussed earlier suggested that these acceler-ation coefficients must be dynamically controlled to regulateparticlersquos velocity during the whole computation process Inthe present work the acceleration coefficients are dynami-cally controlled by suggesting new exponential constrictionfunctions 120577

1and 1205772These constriction functions dynamically

regulate the cognitive and social behaviors of the swarm thuslimiting particlesrsquo velocities during their whole course of theflight and are given by

1205771= eminus1205831120578

1205772= 119896e1205832120578

(10)

Advances in Electrical Engineering 5

0010203040506070809

11

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count12057711205772

Figure 3 Proposed exponential constriction functions

Table 1 Particle encoding for the proposed PSO

1198751198661

1198751198662

sdot sdot sdot 119875119866119894

sdot sdot sdot 119875119866119873

where

120578 =itr

itrmax itrmin le itr le itrmax

119896 =12057711198621119887

12057721198622

(11)

The coefficient of exponent 1205831 has been considered minus55as the term eminus1205831120578 is not perceptible at the end of search With1205832 as 4 the coefficient 119896 is determined for exact match of 1205771and 1205772 at two-third of the search The variation in 1205771 and 1205772

with iterations is shown in Figure 3 for the above mentionedvalue of the exponent coefficients It can be depicted from thefigure that the dominance of cognitive behavior falls sharplyand that of the social behavior rises gradually as the searchprogresses Thus during the early part of the computationalsearch the cognitive behavior is well dominated over thesocial behavior of the swarm to enhance the global searchfor most probable area having the global optima Howeverduring the later part it is the social behavior of the swarm thatdominates over its cognitive counterpart This may enhancelocal exploitation by the swarm to search global or near globaloptima

These alterations in the control equation of the classicalPSO regulate particle velocity within predefined boundswithout any additional formulation as reported in manyimproved versions of PSO [4 6ndash8 10 13ndash16 18 21 26] yetpreserving diversity due to the stochastic nature of cognitiveand social behaviors of the swarm

34 Particle Encoding and Initialization The solution ofan ED problem is the set of most optimal generationsfor the desired objective(s) bounded by certain operationalconstraints In the proposed PSO the particles are encodedin real numbers as the set of current generations in MW asshown in Table 1

Where119875119866119894denotes generation of the 119894th generator inMW

the initial population is randomly created with predefinednumber of particles to maintain diversity Each of these

particles satisfies problem constraints defined by (2)ndash(4)Infeasible particle if appeared is not rejected but correctedusing a correction algorithm as described later in the sectionThis improves the computational efficiency of the PSOThe fitness of each particle is evaluated using (1) and then119901119887119890119904119905 119901119901119900119900119903 and 119892119887119890119904119905 are initialized The initial velocity ofparticles is assumed to be zero

35 Correction Algorithm The velocity and position updatemay create infeasible solutions Infeasible individuals are notrejected but are corrected to feasible individuals by usinga correction algorithm For the purpose the generationsof all generators are adjusted by their respective boundedgeneration limits and then the error is calculated fromthe power balance equation The error in the power isequally distributed among all generators and the procedureis repeated till the error is reduced to a predefined mismatchvalue 120598 In this work themismatch is considered as 0001Thisreduces the computational burden of PSO

36 Elitism and Termination Criterion In stochastic basedalgorithms like PSO the solution with the best fitness in thecurrent iteration may be lost in the next iteration Thereforethe particle with the best fitness is kept preserved for thenext iteration The algorithm is terminated when eitherall particles reach to the best position or the predefinedmaximum iteration number is reached The flow chart of theproposed method is shown in Figure 4

4 Simulation Results

The proposed algorithm is tested on 13-generator system [23]and 40-generator system [23] The control parameters usedfor all these systems to solve the ED problem using classicaland proposed PSO are considered as mentioned in Table 2The proposed algorithm has been developed using MATLABand simulations have been carried on a personal computer ofIntel i5 32 GHz and 4GB RAM

41 Case Study 1 13-Generator System Theproposedmethodis applied on 13 thermal generating units which consist ofvalve-point effect and network power losses The thermalgenerating unitsrsquo data and 119861-coefficient power loss data arereferred from [23] The ED problem is solved for a powerdemand of 2520MW The simulation results obtained forthe best and average fuel cost total power output andpower losses after 100 independent trails using proposedPSO are presented in Table 3 The table shows that theproposed PSO is capable of obtaining better best average andbest fuel costs with smaller power loss than other availableexisting stochastic methods in reasonable CPU time Thusthe proposed method provides good quality solution tosolve complex nonconvex ED problems The best generatingschedule obtained using the proposed PSO is presented inAppendix

42 Case Study 2 40-Generator SystemwithValve-Point EffectThe effectiveness of the proposed method is now investigated

6 Advances in Electrical Engineering

Start

Input cost coefficient data power limit data inertia weight value constriction function valuesset the value of maximum iteration

P = 1

Create one particle randomly

Isparticlefeasible

P = P + 1 Constrained handling

IsP ge popsize

Fitness evaluation initialize pbest ppoor gbest inertia weight constriction function anditeration counter

Itr = 1

P = 1

Velocity and position update

Isparticle feasible Constrained

handling

Fitness evaluation

Iffitness gt old

fitness

Update pbest

Update ppoor

IsP ge popsize

Update gbest

Isstopping criteria satisfied

Stop

Itr = itr + 1update inertia weight (W)and constriction function

Yes

Yes

Yes

Yes

Yes

Yes

No

No

No

No

No

No

P = P + 1

Figure 4 Flow chart of the proposed PSO

Advances in Electrical Engineering 7

Table 2 Various parameters for classical PSO and proposed PSO

Method 119882min119882max 1198621119887

1198621119901

1198622

1205831

1205832

itrmax Population sizeClassical PSO 0109 2 mdash 2 mdash mdash 2500 100Proposed PSO 0109 15 05 2 minus55 4 2500 100

Table 3 Comparison results for case study 1

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

Total power(MW)

Power loss(MW)

CPU time(s)

GA [27] 2463242 2487493 2518859 255987 3987 225DE [27] 2481932 2521764 2565640 256234 4234 258HDE [27] 2459176 2473953 2507490 255916 3916 357STHDE [27] 2456008 2470663 2487244 256433 4433 298ICA-PSO [7] 2454006 2456146 2458945 255905 3905 215SDE [1] 2451488 2451631 mdash 256043 4043 mdashProposed PSO 2451446 2451458 2451526 255807 3807 296

Table 4 Comparison results for case study 2

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

CPU time(s)

SQP [28] 1229044243 1248837692 1265852290 1080EP-SQP [29] 1223239700 1223796300 mdash 99773PSO-SQP [29] 1220946700 1222452500 mdash 73397PSO-LRS [12] 1220357946 1233820000 1257406300 3161NPSO [12] 1217047391 1222213697 1229950976 823NPSO-LRS [12] 1216644308 1229815913 1222093185 2074DEC-SQP [30] 1217419800 1233676500 1253979600 92563DEC(2)-SQP(1) [28] 1217419793 1222951278 1228392941 1426ACO [31] 1215324100 1216064500 1216796400 5245FCASO [32] 1215164700 1220825900 mdash 1452SOH-PSO [14] 1215011400 1218535700 1224463000 mdashTSARGA [33] 1214630700 1229283100 1242965400 6960CPSO-SQP [34] 1214585400 1220281600 mdashGA-PS-SQP [29] 1214580000 1220390000 mdash 4698ABC [35] 1214410300 1219958200 mdash 3002CCPSO [22] 1214125362 1214453269 1215254934 193ICA-PSO [7] 1214221000 mdash mdash 1399DEBBO [36] 1214208948 mdash mdash 12HHS [37] 1214155920 1216158544 mdash 1639IPSO [2] 1214128660 1215095223 1215468420 4289NAPSO [8] 1214125700 mdash mdash 127CSA [38] 1214125355 1215204106 1218102538 303Proposed PSO 1214125355 1214323215 1215643454 999

Table 5 Comparison results for case study 3

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel Cost($hr)

CPU time(s)

PSO [8] 1248758523 1251627011 mdash mdashFAPSO [8] 1222613706 1224710751 1225975196 196NAPSO [8] 1214910662 1214912756 1214915261 127CSA [38] 1214877727 1216113170 1221629295 147Proposed PSO 1214877718 1215113114 1217537157 84

8 Advances in Electrical Engineering

on the most popular test generating system taken from [23]This system consists of 40 thermal units with nonconvexity incost function due to valve-point loading effectsThe expectedpower demand for this test system is 10500MW The resultsobtained after 100 independent trials of the proposed PSOare presented and compared with a variety of other availableexisting deterministic and population based or their hybridtechniques in Table 4 The table validates the effectivenessof the proposed PSO as it generates either comparable orbetter best fuel cost than other several established techniquesincluding hybrid techniques The table also shows that theproposed PSO is less computationally demanding than manyother references including some latest ones Although NPSO[12] and CSA [38] demand less CPU time than the proposedPSO but the proposedmethod is capable of generating betterquality solutionThus the proposed PSO is promising to solvenonconvex ED problems The optimal dispatch of thermalgenerators obtained by the proposed PSO can be referred toin Appendix

43 Case Study 3 40-Generator System with Valve-Point andPOZs Finally the effectiveness of the proposed method isinvestigated on the 40 generators test generating systemwith discontinuities in the cost function due to prohibitedoperating zones The units 10ndash14 have POZs as given in[8] (POZ 2) The expected power demand for this testsystem is 10500 MW The results obtained after 100 trials ofthe proposed PSO are presented and compared with otheravailable existing population based techniques in Table 5Thetable shows that the proposed PSO is capable of generatingcomparable or better result in less computational time thanother established available methods The better value ofaverage fuel cost is obtained by proposed method than othermethods This shows robustness of the proposed PSO Thusthe high dimensional nonconvex discrete ED problems canbe effectively and efficiently solved using the proposed PSOThe optimal dispatch of thermal generators obtained by theproposed PSO can be referred to in Appendix

5 Discussion

In order to appreciate and understand the performance ofthe proposed method a comparison of cognitive and socialbehavior of particle in PSO and the proposed PSO is shown inFigures 5 and 6 respectively Figure 5 shows that in the classi-cal PSO the cognitive and social behaviors of particle velocityvary randomly throughout the computational process withinlimits of 0 to 2 The proposed constriction functions usedto guide the cognitive and social behaviors of the swarm areallowed to vary exponentially as shown in Figure 6The lowerand upper limits of these behaviors are governed by (10)However the sum of the best and poor cognitive behavior ofthe swarm remains constant during the computation processThis plays an important role in providing sufficient diversityby the poor experience during the whole flight of the swarmIt can be seen from Figure 6 that using proposed PSOthe modulations of cognitive (best) cognitive (poor) and

social behaviors though randomly distributed are dynami-cally controlled within exponential bounds of 15 05 and015 respectively This constitutes a marked difference withother versions of existing PSO Thus the particles experienceentirely different cognitive and social behaviors during theirflights and need no additional mechanism to bind theirvelocities

Any stochastic based search technique must be designedto accomplish global exploration and tends to facilitate localexploitation In order to investigate the effectiveness of eachof these modifications a set of convergence characteristicsfor the best and average fuel cost obtained during a sampletrial for 40 generators system is shown in Figures 7 and8 respectively In Figure 7 the characteristic ldquoardquo is for theconventional PSO ldquobrdquo refers to ldquoardquo with sinusoidal mod-ulation in inertia weight ldquocrdquo refers to ldquobrdquo with improvedcognitive behavior due to poor experience and ldquodrdquo refers tothe proposed PSO It can be observed from the figure that theperformance of the PSO is somewhat improved when inertiaweight is sinusoidally modulated and is further improvedwith a good margin when poor experience of particles is alsoconsidered However these two modifications do not seemto be sufficient to exploit the promising region effectively andefficiently This leads to premature convergence due to localtrappings which can be depicted from ldquodrdquo In d the proposedconstriction functions regulate particlesrsquo velocities so thatthey can fly more comprehensively in the search space Infact due to higher initial cognitive component than the socialcomponent the proposed PSO becomes more competentto explore wider search space during the initial phase andthus identify the promising region in about 1000 iterationsHowever particles move with strong communication andthus intensively exploit the region near the global optimaduring later part of the search owing to high values ofsocial component Finally all particles converge towards theglobal minima as can be observed from Figure 8 Thus theproposed PSOprovides better exploration and exploitation ofthe search space and produces better quality solutions Theseresults also highlight that the modifications suggested in thecontrol equation of the classical PSO are very effective as itmakes the proposed PSO perform much better

The proposed method offers better exploration andexploitation of the search space because the velocity ofparticles is regulated throughout their flight The movementof a sample particle in the classical PSOand the proposedPSOis illustrated in Figures 9 and 10 respectively These figuresshow the traces of initial cognitive and social componentsof particlersquos velocity and also the overall velocity imparted toit during a sample trial

The classical PSO searches for about 400 iterations asshown in Figure 9 After this all the three components ofparticlersquos velocity became insignificant and thus the particlegets trapped into local minima Figure 10(b) shows thecognitive component for the best experience which is thensuperimposed by its poor experience as in Figure 10(c) toobtain the overall cognitive component as in Figure 10(d) Itcan be concluded from Figure 10(d) that the poor experienceis contributing to tune the cognitive behavior of the swarmThe social component as shown in Figure 10(e) is providing

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 Advances in Electrical Engineering

by the two stochastic acceleration coefficients that is cogni-tive and social components and the inertia component [5] Inorder to enhance the exploration and exploitation capabilitiesof PSO the components affecting velocity of particles shouldbe properly managed and controlled

Several methods have been reported in the recent pastto enhance the computational efficiency of the classical PSOA constriction factor was suggested in the velocity updatingequation to assure convergence of PSO [9ndash11] Howeverthe exact determination of this factor is computationallydemanding Selvakumar and Thanushkodi [12] modifiedcognitive behavior of the swarm by communicating with theworst particle This method provides some additional diver-sity to the particle by the worst experience component butshowing poor local searching ability unless it is hybridizedwith certain other heuristic approaches Roy and Ghoshal[13] proposed crazy PSO (CPSO) where the particle velocityis randomized within predefined limits The idea was torandomize the velocity of some of the particles referred toas ldquocrazy particlesrdquo by applying a predefined probability ofcraziness tomaintain the diversity for global search and betterconvergence However the value of predefined probability ofcraziness can only be achieved after several experimentationsSome attempts [14ndash18] have been made to vary the cognitiveand social behavior of the swarmduring the search process bydynamically controlling the acceleration coefficients withinmaximum and minimum bounds Again the determinationof limiting values of these acceleration coefficients is adifficult task as it required many simulations Coelho andLee [19] randomized cognitive and social behavior of theswarm using chaotic sequences and Gaussian distributionrespectively Selvakumar and Thanushkodi [20] proposedcivilized swarm optimization (CSO) by combining society-civilization algorithm (SCA) with PSO to improve communi-cationTheproposed algorithmprovides clustered search thatresults in better exploration and exploitation of the searchspace but needs several experimentations to determine theoptimum values of the control parameters of CSO Effortshave also been made to suggest a new formulation of thecontrol equation [6 7] Safari and Shayeghi [6] proposediteration PSO (IPSO) where one additional velocity compo-nent pertaining to the best fitness of the current iteration isadded in the control equation of the classical PSO to avoidlocal trap but parameter setting is essential Vlachogiannisand Lee [7] suggested new control equation in improvedcoordinated aggregation PSO (ICAPSO) for better communi-cation among particles to enhance local searchThey allowedparticles to interact with its own best experience along withall other particles have better experience on aggregate basisinstead of the global best experience However the authorsaccepted that the performance of the proposed method isquite sensitive to various parameters setting and their tuningis essential Chaotic PSO (CPSO) of [21] proposed adaptedinertia weight which varies dynamically with fitness value forexploration and chaotic local search was used to determinethe particle position for better exploitation The improvedPSO (IPSO) of [22] suggested chaotic inertia weight whichdecreases and oscillates simultaneously under the decreasingline in a chaotic manner In this way additional diversity

Without valve-point loading effect

With valve-point loading effect

Fuel

cost

($M

W)

Pmin

Output P (MW)Pmax

Figure 1 Fuel cost function with and without valve-point loadingeffect

is introduced but it requires tuning of chaotic controlparameters

This paper attempts to overcome drawbacks of someexisting PSO methods and presents a modified version ofPSO for economic load dispatch of power systems Severalmeasures have been incorporated in the control equationby modifying operators of the classical PSO by introducingnew constriction functionsThe proposed method effectivelycontrols and regulates the components affecting velocityof particles so as to ensure better exploration (searchingnew areas) and exploitation (fine tuning of the currentsolution) A correction algorithm is also suggested to repairinfeasible solutions whenever appeared in the computationalprocess The proposed method is self-adjusting and doesnot require experimentations to obtain the optimal valuesof control parameters and thus overcome the drawbacks ofexisting PSO methods The effectiveness of the proposedmethod has been investigated on three standard test systemsconsidering various operational constraints like valve-pointloading effect prohibited operating zones (POZs) networkpower loss and so forthThe application results show that theproposed PSO method is very promising

2 Problem Formulation

The generator cost function is usually considered as quad-ratic when valve-point loading effects are neglected Thelarge turbine generators usually have a number of fueladmission valves which are operated in sequence to meetout increased generation The opening of a valve increasesthe throttling losses rapidly and thus the incremental heatrate rises suddenlyThis valve-point loading effect introducesripples in the heat-rate curves which introduces nonconvex-ity in the generator fuel cost function as shown in Figure 1The effect of valve-point loading effects can be modeledas sinusoidal function in the cost function Therefore the

Advances in Electrical Engineering 3

objective function for the nonconvex ED problem may bestated as

Minimize119865 (119875119866119894) =

119873119866

sum

119894=1

(119886119894+ 119887119894119875119866119894+ 1198881198941198752

119866119894)

+1003816100381610038161003816119890119894 sin (119891119894 (119875119866119894min minus 119875119866119894))

1003816100381610038161003816

(1)

where 119886119894 119887119894 and 119888119894 are the cost coefficients of the 119894th generator

119890119894 and 119891119894 are the valve-point effect coefficients 119875119866119894 is the realpower output of the 119894th generator and 119873119866 is the number ofgenerating units in the system

Subject to the following constraints

(1) Power Balance ConstraintThe total power generationof all generators must be equal to the sum of totalpower demand plus the network power loss The net-work power loss can be evaluated using 119861-coefficientloss formula [21 23] Therefore the generator powerbalance equation may be stated as follows

119873119866

sum

119894=1

119875119894 = 119875119863 +

119873119866

sum

119894=1

119873119866

sum

119895=1

119875119866119894119861119894119895119875119866119895 +

119873119866

sum

119894=1

1198751198661198941198611198940 + 11986100 (2)

where 119861119894119895 is the transmission loss coefficient 119894 =

1 2 119873119866 and 119895 = 1 2 119873119866 1198611198940 is the 119894thelement of the loss coefficient vector 11986100 is the losscoefficient constant

(2) Generator Constraint For stable operation poweroutput of each generator is restricted within itsminimum andmaximum limitsThe generator powerlimits are expressed as follows

119875min119866119894

le 119875119866119894

le 119875max119866119894

(3)

(3) Prohibited Operating Zones Prohibited operatingzones lead to discontinuities in the input outputrelation of generators Prohibited zones divide theoperating region between minimum and maximumgeneration limits into disjoint convex subregions [1420] The generation limits for the 119894th unit with 119895

number of prohibited zones can be expressed asfollows

119875min119866119894

le 119875119866119894

le 119875119871

1198661198941

119875119880

119866119894119895minus1le 119875119866119894 le 119875

119871

119866119894119895

119875119880

119866119894119873119875119885119894le 119875119866119894

le 119875max119866119894

119894 isin 1 2 119873119866119875119885 119895 isin 2 3 119873119875119885119894

(4)

where superscripts 119871 and 119880 stand for the lowerand upper limit of prohibited operating zones ofgenerators 119873

119866119875119885and 119873

119875119885119894denote the total number

of generators with prohibited zones and the totalnumber of prohibited zones for the 119894th generatorrespectively

3 Proposed PSO

The classical PSO is initialized with a population of randomsolutions and searches for optima by updating particle posi-tions The velocity of the particle is influenced by the threecomponents initial cognitive and the social componentEach particle updates its previous velocity and positionvectors according to the following model [3 24 25]

V119896+1119894

= 119882V119896119894+ 1198881times rand

1 () times119901119887119890119904119905119894 minus 119904119896

119894

Δ119905

+ 1198882times rand

2 () times119892119887119890119904119905119894minus 119904119896

119894

Δ119905

119904119896+1

119895= 119904119896

119895+ V119896+1119895

times Δ119905

(5)

where V119896119894is the velocity of 119894th particle at 119896th iteration rand

1()

and rand2() are random numbers between 0 and 1 119904119896119894is

the position of 119894th particle at 119896th iteration 1198881 1198882 are theacceleration coefficients 119901119887119890119904119905

119894is the best position of 119894th

particle achieved based on its own experience 119892119887119890119904119905119894is the

best particle position based on overall swarm experience Δ119905is the time step usually set to 1 second and 119882 is the inertiaweight which is allowed to decrease linearly as follows

119882 = 119882min +(119882max minus119882min) times (itrmax minus itr)

itrmax (6)

where119882min and119882max are the minimum andmaximum valueof inertia weight respectively itrmax is themaximumnumberof iterations and itr is the current number of iteration

For better performance of PSO the particles must flywith higher velocities during the early flights to enhanceglobal search and should be relatively slow during laterflights of the journey to improve local search Thereforewith appropriate regulation of particlersquos velocity during thejourney the performance of PSO could be improved Initiallythe impact of cognitive component must be high and that ofthe social component be less to ensure global exploration ofthe search space by all particles without trapping into a localminima During later search the impact of social componentmust increase and that of the cognitive component mustdecrease to divert all particles towards global best to improvethe convergence This is essential for a good balance betweenexploration and exploitation as suggested by [15]

In classical PSO only the initial velocity component usinginertia weight is regulated dynamically However the cogni-tive and social behavior of the swarm though randomized toensure diversity is statically controlled by assigning constantvalues to acceleration coefficients These cognitive and socialcomponents of velocity are added in the regulated initialvelocity component to decide themovement of particlesThisprobably results in uncontrolled particle velocities duringthe whole computation process and thus causes insufficientexploration and exploitation of the search space This resultsin poor convergence due to local trapping Therefore amodified control equation (7) is suggested for dynamicallyregulating particlersquos velocity during their whole course of

4 Advances in Electrical Engineering

the flight The modifications suggested in the control equa-tion are explained as follows

V119896+1119894

= 119882 times V119896119894+ 1205771times 1198621119887times rand

1 () times119901119887119890119904119905119894minus 119904119896

119894

Δ119905

+ (1 minus 1205771) times 1198621119901

times rand2 () times

119904119896

119894minus 119901119901119900119900119903

119894

Δ119905

+ 1205772 times 1198622 times rand3 () times

119892119887119890119904119905119894minus 119904119896

119894

Δ119905

(7)

In (7) the inertia weight is modified to regulate the trade-off between the global exploration and the local exploitationof the swarm The poor experience 119901119901119900119900119903

119894has been added

to improve the cognitive component Further dynamic accel-eration coefficients have been introduced using constrictionfunctions 120577

1and 120577

2to regulate the cognitive and social

behaviors of the swarmThese modifications are discussed inthe following sections

31 Inertia Weight Update In [25] Shi and Eberhart sug-gested linear modulation of the inertia weight This trendis followed to solve ELD problems using PSO by manyresearchers till date and some of them can be mentioned as[4 6 8 12 13 15 19 20 25 26] and so forth In the proposedmethod the inertia weight has been allowed to vary inaccordance with a truncated sinusoidal function rather thanto decrease linearlyThemodulations suggested to update theinertia weight is governed by the following relation

119882 = 119882min + (119882max minus119882min) cos2(120579

2)

0 le 120579 le 120587

(8)

where 120579 = 119883 times itr +119884 and the coefficients119883 and 119884 are givenby (9) itr is the iteration countwhich is in general varied fromitrmin to itrmax

119883 =120587

(itrmax minus itrmin)

119884 =minus120587 times itrmin

(itrmax minus itrmin)

(9)

Figure 2 shows a comparison of the conventional linearmodulation and sinusoidal modulation for the inertia weightto be employed in the proposed PSO It can be depicted fromthe figure that using the sinusoidal variations in the inertiaweight the inertia component of the velocity of particlesmaintained always higher during the early half and lowerduring the later half of the search when compared with itslinear variationsTherefore using sinusoidalmodulations thecoarse search is enhanced during the early half by exploringlarger search space with higher values assigned to particlevelocities And during the later half the fine search isenhanced by assigning lower values to particle velocitiesThis facilitates particles to explore the regions in the closeproximity of near global solution

W

Wmax

Wmin

Itrmin Itrmax2 Itrmax Itr

Figure 2 Comparison of linear and sinusoidal modulations ofinertia weight

32 Updating of Poor Experience The cognitive behaviorwas split in [12] by considering the worst experience inaddition to the best experience of the particle Though thismodification provides additional diversity it still demands alocal random search to enhance exploitation potential of thePSOThis occurs as the particlersquos velocity is not well regulatedduring later part of the search Therefore the concept ofpoor experience 119901119901119900119900119903

119894is suggested instead of the worst

experience to improve cognitive behavior of the swarmHerethe current fitness of each particle is compared with its fitnessvalue in the preceding iteration and if it is found less it willbe treated as the poor experience This concept is differentthan that of [12] where the worst particle is determinedby considering the whole past experience of the particlemovement The poor particle produces much less diversitythan the worst particle and thus exploit the region near globaloptima during later iterations in much better way withoutthe support of any local random search

33 Dynamic Control of Acceleration Coefficients In classicalPSO the cognitive and social behaviors are governed byassigning static values to acceleration coefficients Manyresearchers as discussed earlier suggested that these acceler-ation coefficients must be dynamically controlled to regulateparticlersquos velocity during the whole computation process Inthe present work the acceleration coefficients are dynami-cally controlled by suggesting new exponential constrictionfunctions 120577

1and 1205772These constriction functions dynamically

regulate the cognitive and social behaviors of the swarm thuslimiting particlesrsquo velocities during their whole course of theflight and are given by

1205771= eminus1205831120578

1205772= 119896e1205832120578

(10)

Advances in Electrical Engineering 5

0010203040506070809

11

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count12057711205772

Figure 3 Proposed exponential constriction functions

Table 1 Particle encoding for the proposed PSO

1198751198661

1198751198662

sdot sdot sdot 119875119866119894

sdot sdot sdot 119875119866119873

where

120578 =itr

itrmax itrmin le itr le itrmax

119896 =12057711198621119887

12057721198622

(11)

The coefficient of exponent 1205831 has been considered minus55as the term eminus1205831120578 is not perceptible at the end of search With1205832 as 4 the coefficient 119896 is determined for exact match of 1205771and 1205772 at two-third of the search The variation in 1205771 and 1205772

with iterations is shown in Figure 3 for the above mentionedvalue of the exponent coefficients It can be depicted from thefigure that the dominance of cognitive behavior falls sharplyand that of the social behavior rises gradually as the searchprogresses Thus during the early part of the computationalsearch the cognitive behavior is well dominated over thesocial behavior of the swarm to enhance the global searchfor most probable area having the global optima Howeverduring the later part it is the social behavior of the swarm thatdominates over its cognitive counterpart This may enhancelocal exploitation by the swarm to search global or near globaloptima

These alterations in the control equation of the classicalPSO regulate particle velocity within predefined boundswithout any additional formulation as reported in manyimproved versions of PSO [4 6ndash8 10 13ndash16 18 21 26] yetpreserving diversity due to the stochastic nature of cognitiveand social behaviors of the swarm

34 Particle Encoding and Initialization The solution ofan ED problem is the set of most optimal generationsfor the desired objective(s) bounded by certain operationalconstraints In the proposed PSO the particles are encodedin real numbers as the set of current generations in MW asshown in Table 1

Where119875119866119894denotes generation of the 119894th generator inMW

the initial population is randomly created with predefinednumber of particles to maintain diversity Each of these

particles satisfies problem constraints defined by (2)ndash(4)Infeasible particle if appeared is not rejected but correctedusing a correction algorithm as described later in the sectionThis improves the computational efficiency of the PSOThe fitness of each particle is evaluated using (1) and then119901119887119890119904119905 119901119901119900119900119903 and 119892119887119890119904119905 are initialized The initial velocity ofparticles is assumed to be zero

35 Correction Algorithm The velocity and position updatemay create infeasible solutions Infeasible individuals are notrejected but are corrected to feasible individuals by usinga correction algorithm For the purpose the generationsof all generators are adjusted by their respective boundedgeneration limits and then the error is calculated fromthe power balance equation The error in the power isequally distributed among all generators and the procedureis repeated till the error is reduced to a predefined mismatchvalue 120598 In this work themismatch is considered as 0001Thisreduces the computational burden of PSO

36 Elitism and Termination Criterion In stochastic basedalgorithms like PSO the solution with the best fitness in thecurrent iteration may be lost in the next iteration Thereforethe particle with the best fitness is kept preserved for thenext iteration The algorithm is terminated when eitherall particles reach to the best position or the predefinedmaximum iteration number is reached The flow chart of theproposed method is shown in Figure 4

4 Simulation Results

The proposed algorithm is tested on 13-generator system [23]and 40-generator system [23] The control parameters usedfor all these systems to solve the ED problem using classicaland proposed PSO are considered as mentioned in Table 2The proposed algorithm has been developed using MATLABand simulations have been carried on a personal computer ofIntel i5 32 GHz and 4GB RAM

41 Case Study 1 13-Generator System Theproposedmethodis applied on 13 thermal generating units which consist ofvalve-point effect and network power losses The thermalgenerating unitsrsquo data and 119861-coefficient power loss data arereferred from [23] The ED problem is solved for a powerdemand of 2520MW The simulation results obtained forthe best and average fuel cost total power output andpower losses after 100 independent trails using proposedPSO are presented in Table 3 The table shows that theproposed PSO is capable of obtaining better best average andbest fuel costs with smaller power loss than other availableexisting stochastic methods in reasonable CPU time Thusthe proposed method provides good quality solution tosolve complex nonconvex ED problems The best generatingschedule obtained using the proposed PSO is presented inAppendix

42 Case Study 2 40-Generator SystemwithValve-Point EffectThe effectiveness of the proposed method is now investigated

6 Advances in Electrical Engineering

Start

Input cost coefficient data power limit data inertia weight value constriction function valuesset the value of maximum iteration

P = 1

Create one particle randomly

Isparticlefeasible

P = P + 1 Constrained handling

IsP ge popsize

Fitness evaluation initialize pbest ppoor gbest inertia weight constriction function anditeration counter

Itr = 1

P = 1

Velocity and position update

Isparticle feasible Constrained

handling

Fitness evaluation

Iffitness gt old

fitness

Update pbest

Update ppoor

IsP ge popsize

Update gbest

Isstopping criteria satisfied

Stop

Itr = itr + 1update inertia weight (W)and constriction function

Yes

Yes

Yes

Yes

Yes

Yes

No

No

No

No

No

No

P = P + 1

Figure 4 Flow chart of the proposed PSO

Advances in Electrical Engineering 7

Table 2 Various parameters for classical PSO and proposed PSO

Method 119882min119882max 1198621119887

1198621119901

1198622

1205831

1205832

itrmax Population sizeClassical PSO 0109 2 mdash 2 mdash mdash 2500 100Proposed PSO 0109 15 05 2 minus55 4 2500 100

Table 3 Comparison results for case study 1

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

Total power(MW)

Power loss(MW)

CPU time(s)

GA [27] 2463242 2487493 2518859 255987 3987 225DE [27] 2481932 2521764 2565640 256234 4234 258HDE [27] 2459176 2473953 2507490 255916 3916 357STHDE [27] 2456008 2470663 2487244 256433 4433 298ICA-PSO [7] 2454006 2456146 2458945 255905 3905 215SDE [1] 2451488 2451631 mdash 256043 4043 mdashProposed PSO 2451446 2451458 2451526 255807 3807 296

Table 4 Comparison results for case study 2

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

CPU time(s)

SQP [28] 1229044243 1248837692 1265852290 1080EP-SQP [29] 1223239700 1223796300 mdash 99773PSO-SQP [29] 1220946700 1222452500 mdash 73397PSO-LRS [12] 1220357946 1233820000 1257406300 3161NPSO [12] 1217047391 1222213697 1229950976 823NPSO-LRS [12] 1216644308 1229815913 1222093185 2074DEC-SQP [30] 1217419800 1233676500 1253979600 92563DEC(2)-SQP(1) [28] 1217419793 1222951278 1228392941 1426ACO [31] 1215324100 1216064500 1216796400 5245FCASO [32] 1215164700 1220825900 mdash 1452SOH-PSO [14] 1215011400 1218535700 1224463000 mdashTSARGA [33] 1214630700 1229283100 1242965400 6960CPSO-SQP [34] 1214585400 1220281600 mdashGA-PS-SQP [29] 1214580000 1220390000 mdash 4698ABC [35] 1214410300 1219958200 mdash 3002CCPSO [22] 1214125362 1214453269 1215254934 193ICA-PSO [7] 1214221000 mdash mdash 1399DEBBO [36] 1214208948 mdash mdash 12HHS [37] 1214155920 1216158544 mdash 1639IPSO [2] 1214128660 1215095223 1215468420 4289NAPSO [8] 1214125700 mdash mdash 127CSA [38] 1214125355 1215204106 1218102538 303Proposed PSO 1214125355 1214323215 1215643454 999

Table 5 Comparison results for case study 3

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel Cost($hr)

CPU time(s)

PSO [8] 1248758523 1251627011 mdash mdashFAPSO [8] 1222613706 1224710751 1225975196 196NAPSO [8] 1214910662 1214912756 1214915261 127CSA [38] 1214877727 1216113170 1221629295 147Proposed PSO 1214877718 1215113114 1217537157 84

8 Advances in Electrical Engineering

on the most popular test generating system taken from [23]This system consists of 40 thermal units with nonconvexity incost function due to valve-point loading effectsThe expectedpower demand for this test system is 10500MW The resultsobtained after 100 independent trials of the proposed PSOare presented and compared with a variety of other availableexisting deterministic and population based or their hybridtechniques in Table 4 The table validates the effectivenessof the proposed PSO as it generates either comparable orbetter best fuel cost than other several established techniquesincluding hybrid techniques The table also shows that theproposed PSO is less computationally demanding than manyother references including some latest ones Although NPSO[12] and CSA [38] demand less CPU time than the proposedPSO but the proposedmethod is capable of generating betterquality solutionThus the proposed PSO is promising to solvenonconvex ED problems The optimal dispatch of thermalgenerators obtained by the proposed PSO can be referred toin Appendix

43 Case Study 3 40-Generator System with Valve-Point andPOZs Finally the effectiveness of the proposed method isinvestigated on the 40 generators test generating systemwith discontinuities in the cost function due to prohibitedoperating zones The units 10ndash14 have POZs as given in[8] (POZ 2) The expected power demand for this testsystem is 10500 MW The results obtained after 100 trials ofthe proposed PSO are presented and compared with otheravailable existing population based techniques in Table 5Thetable shows that the proposed PSO is capable of generatingcomparable or better result in less computational time thanother established available methods The better value ofaverage fuel cost is obtained by proposed method than othermethods This shows robustness of the proposed PSO Thusthe high dimensional nonconvex discrete ED problems canbe effectively and efficiently solved using the proposed PSOThe optimal dispatch of thermal generators obtained by theproposed PSO can be referred to in Appendix

5 Discussion

In order to appreciate and understand the performance ofthe proposed method a comparison of cognitive and socialbehavior of particle in PSO and the proposed PSO is shown inFigures 5 and 6 respectively Figure 5 shows that in the classi-cal PSO the cognitive and social behaviors of particle velocityvary randomly throughout the computational process withinlimits of 0 to 2 The proposed constriction functions usedto guide the cognitive and social behaviors of the swarm areallowed to vary exponentially as shown in Figure 6The lowerand upper limits of these behaviors are governed by (10)However the sum of the best and poor cognitive behavior ofthe swarm remains constant during the computation processThis plays an important role in providing sufficient diversityby the poor experience during the whole flight of the swarmIt can be seen from Figure 6 that using proposed PSOthe modulations of cognitive (best) cognitive (poor) and

social behaviors though randomly distributed are dynami-cally controlled within exponential bounds of 15 05 and015 respectively This constitutes a marked difference withother versions of existing PSO Thus the particles experienceentirely different cognitive and social behaviors during theirflights and need no additional mechanism to bind theirvelocities

Any stochastic based search technique must be designedto accomplish global exploration and tends to facilitate localexploitation In order to investigate the effectiveness of eachof these modifications a set of convergence characteristicsfor the best and average fuel cost obtained during a sampletrial for 40 generators system is shown in Figures 7 and8 respectively In Figure 7 the characteristic ldquoardquo is for theconventional PSO ldquobrdquo refers to ldquoardquo with sinusoidal mod-ulation in inertia weight ldquocrdquo refers to ldquobrdquo with improvedcognitive behavior due to poor experience and ldquodrdquo refers tothe proposed PSO It can be observed from the figure that theperformance of the PSO is somewhat improved when inertiaweight is sinusoidally modulated and is further improvedwith a good margin when poor experience of particles is alsoconsidered However these two modifications do not seemto be sufficient to exploit the promising region effectively andefficiently This leads to premature convergence due to localtrappings which can be depicted from ldquodrdquo In d the proposedconstriction functions regulate particlesrsquo velocities so thatthey can fly more comprehensively in the search space Infact due to higher initial cognitive component than the socialcomponent the proposed PSO becomes more competentto explore wider search space during the initial phase andthus identify the promising region in about 1000 iterationsHowever particles move with strong communication andthus intensively exploit the region near the global optimaduring later part of the search owing to high values ofsocial component Finally all particles converge towards theglobal minima as can be observed from Figure 8 Thus theproposed PSOprovides better exploration and exploitation ofthe search space and produces better quality solutions Theseresults also highlight that the modifications suggested in thecontrol equation of the classical PSO are very effective as itmakes the proposed PSO perform much better

The proposed method offers better exploration andexploitation of the search space because the velocity ofparticles is regulated throughout their flight The movementof a sample particle in the classical PSOand the proposedPSOis illustrated in Figures 9 and 10 respectively These figuresshow the traces of initial cognitive and social componentsof particlersquos velocity and also the overall velocity imparted toit during a sample trial

The classical PSO searches for about 400 iterations asshown in Figure 9 After this all the three components ofparticlersquos velocity became insignificant and thus the particlegets trapped into local minima Figure 10(b) shows thecognitive component for the best experience which is thensuperimposed by its poor experience as in Figure 10(c) toobtain the overall cognitive component as in Figure 10(d) Itcan be concluded from Figure 10(d) that the poor experienceis contributing to tune the cognitive behavior of the swarmThe social component as shown in Figure 10(e) is providing

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Electrical Engineering 3

objective function for the nonconvex ED problem may bestated as

Minimize119865 (119875119866119894) =

119873119866

sum

119894=1

(119886119894+ 119887119894119875119866119894+ 1198881198941198752

119866119894)

+1003816100381610038161003816119890119894 sin (119891119894 (119875119866119894min minus 119875119866119894))

1003816100381610038161003816

(1)

where 119886119894 119887119894 and 119888119894 are the cost coefficients of the 119894th generator

119890119894 and 119891119894 are the valve-point effect coefficients 119875119866119894 is the realpower output of the 119894th generator and 119873119866 is the number ofgenerating units in the system

Subject to the following constraints

(1) Power Balance ConstraintThe total power generationof all generators must be equal to the sum of totalpower demand plus the network power loss The net-work power loss can be evaluated using 119861-coefficientloss formula [21 23] Therefore the generator powerbalance equation may be stated as follows

119873119866

sum

119894=1

119875119894 = 119875119863 +

119873119866

sum

119894=1

119873119866

sum

119895=1

119875119866119894119861119894119895119875119866119895 +

119873119866

sum

119894=1

1198751198661198941198611198940 + 11986100 (2)

where 119861119894119895 is the transmission loss coefficient 119894 =

1 2 119873119866 and 119895 = 1 2 119873119866 1198611198940 is the 119894thelement of the loss coefficient vector 11986100 is the losscoefficient constant

(2) Generator Constraint For stable operation poweroutput of each generator is restricted within itsminimum andmaximum limitsThe generator powerlimits are expressed as follows

119875min119866119894

le 119875119866119894

le 119875max119866119894

(3)

(3) Prohibited Operating Zones Prohibited operatingzones lead to discontinuities in the input outputrelation of generators Prohibited zones divide theoperating region between minimum and maximumgeneration limits into disjoint convex subregions [1420] The generation limits for the 119894th unit with 119895

number of prohibited zones can be expressed asfollows

119875min119866119894

le 119875119866119894

le 119875119871

1198661198941

119875119880

119866119894119895minus1le 119875119866119894 le 119875

119871

119866119894119895

119875119880

119866119894119873119875119885119894le 119875119866119894

le 119875max119866119894

119894 isin 1 2 119873119866119875119885 119895 isin 2 3 119873119875119885119894

(4)

where superscripts 119871 and 119880 stand for the lowerand upper limit of prohibited operating zones ofgenerators 119873

119866119875119885and 119873

119875119885119894denote the total number

of generators with prohibited zones and the totalnumber of prohibited zones for the 119894th generatorrespectively

3 Proposed PSO

The classical PSO is initialized with a population of randomsolutions and searches for optima by updating particle posi-tions The velocity of the particle is influenced by the threecomponents initial cognitive and the social componentEach particle updates its previous velocity and positionvectors according to the following model [3 24 25]

V119896+1119894

= 119882V119896119894+ 1198881times rand

1 () times119901119887119890119904119905119894 minus 119904119896

119894

Δ119905

+ 1198882times rand

2 () times119892119887119890119904119905119894minus 119904119896

119894

Δ119905

119904119896+1

119895= 119904119896

119895+ V119896+1119895

times Δ119905

(5)

where V119896119894is the velocity of 119894th particle at 119896th iteration rand

1()

and rand2() are random numbers between 0 and 1 119904119896119894is

the position of 119894th particle at 119896th iteration 1198881 1198882 are theacceleration coefficients 119901119887119890119904119905

119894is the best position of 119894th

particle achieved based on its own experience 119892119887119890119904119905119894is the

best particle position based on overall swarm experience Δ119905is the time step usually set to 1 second and 119882 is the inertiaweight which is allowed to decrease linearly as follows

119882 = 119882min +(119882max minus119882min) times (itrmax minus itr)

itrmax (6)

where119882min and119882max are the minimum andmaximum valueof inertia weight respectively itrmax is themaximumnumberof iterations and itr is the current number of iteration

For better performance of PSO the particles must flywith higher velocities during the early flights to enhanceglobal search and should be relatively slow during laterflights of the journey to improve local search Thereforewith appropriate regulation of particlersquos velocity during thejourney the performance of PSO could be improved Initiallythe impact of cognitive component must be high and that ofthe social component be less to ensure global exploration ofthe search space by all particles without trapping into a localminima During later search the impact of social componentmust increase and that of the cognitive component mustdecrease to divert all particles towards global best to improvethe convergence This is essential for a good balance betweenexploration and exploitation as suggested by [15]

In classical PSO only the initial velocity component usinginertia weight is regulated dynamically However the cogni-tive and social behavior of the swarm though randomized toensure diversity is statically controlled by assigning constantvalues to acceleration coefficients These cognitive and socialcomponents of velocity are added in the regulated initialvelocity component to decide themovement of particlesThisprobably results in uncontrolled particle velocities duringthe whole computation process and thus causes insufficientexploration and exploitation of the search space This resultsin poor convergence due to local trapping Therefore amodified control equation (7) is suggested for dynamicallyregulating particlersquos velocity during their whole course of

4 Advances in Electrical Engineering

the flight The modifications suggested in the control equa-tion are explained as follows

V119896+1119894

= 119882 times V119896119894+ 1205771times 1198621119887times rand

1 () times119901119887119890119904119905119894minus 119904119896

119894

Δ119905

+ (1 minus 1205771) times 1198621119901

times rand2 () times

119904119896

119894minus 119901119901119900119900119903

119894

Δ119905

+ 1205772 times 1198622 times rand3 () times

119892119887119890119904119905119894minus 119904119896

119894

Δ119905

(7)

In (7) the inertia weight is modified to regulate the trade-off between the global exploration and the local exploitationof the swarm The poor experience 119901119901119900119900119903

119894has been added

to improve the cognitive component Further dynamic accel-eration coefficients have been introduced using constrictionfunctions 120577

1and 120577

2to regulate the cognitive and social

behaviors of the swarmThese modifications are discussed inthe following sections

31 Inertia Weight Update In [25] Shi and Eberhart sug-gested linear modulation of the inertia weight This trendis followed to solve ELD problems using PSO by manyresearchers till date and some of them can be mentioned as[4 6 8 12 13 15 19 20 25 26] and so forth In the proposedmethod the inertia weight has been allowed to vary inaccordance with a truncated sinusoidal function rather thanto decrease linearlyThemodulations suggested to update theinertia weight is governed by the following relation

119882 = 119882min + (119882max minus119882min) cos2(120579

2)

0 le 120579 le 120587

(8)

where 120579 = 119883 times itr +119884 and the coefficients119883 and 119884 are givenby (9) itr is the iteration countwhich is in general varied fromitrmin to itrmax

119883 =120587

(itrmax minus itrmin)

119884 =minus120587 times itrmin

(itrmax minus itrmin)

(9)

Figure 2 shows a comparison of the conventional linearmodulation and sinusoidal modulation for the inertia weightto be employed in the proposed PSO It can be depicted fromthe figure that using the sinusoidal variations in the inertiaweight the inertia component of the velocity of particlesmaintained always higher during the early half and lowerduring the later half of the search when compared with itslinear variationsTherefore using sinusoidalmodulations thecoarse search is enhanced during the early half by exploringlarger search space with higher values assigned to particlevelocities And during the later half the fine search isenhanced by assigning lower values to particle velocitiesThis facilitates particles to explore the regions in the closeproximity of near global solution

W

Wmax

Wmin

Itrmin Itrmax2 Itrmax Itr

Figure 2 Comparison of linear and sinusoidal modulations ofinertia weight

32 Updating of Poor Experience The cognitive behaviorwas split in [12] by considering the worst experience inaddition to the best experience of the particle Though thismodification provides additional diversity it still demands alocal random search to enhance exploitation potential of thePSOThis occurs as the particlersquos velocity is not well regulatedduring later part of the search Therefore the concept ofpoor experience 119901119901119900119900119903

119894is suggested instead of the worst

experience to improve cognitive behavior of the swarmHerethe current fitness of each particle is compared with its fitnessvalue in the preceding iteration and if it is found less it willbe treated as the poor experience This concept is differentthan that of [12] where the worst particle is determinedby considering the whole past experience of the particlemovement The poor particle produces much less diversitythan the worst particle and thus exploit the region near globaloptima during later iterations in much better way withoutthe support of any local random search

33 Dynamic Control of Acceleration Coefficients In classicalPSO the cognitive and social behaviors are governed byassigning static values to acceleration coefficients Manyresearchers as discussed earlier suggested that these acceler-ation coefficients must be dynamically controlled to regulateparticlersquos velocity during the whole computation process Inthe present work the acceleration coefficients are dynami-cally controlled by suggesting new exponential constrictionfunctions 120577

1and 1205772These constriction functions dynamically

regulate the cognitive and social behaviors of the swarm thuslimiting particlesrsquo velocities during their whole course of theflight and are given by

1205771= eminus1205831120578

1205772= 119896e1205832120578

(10)

Advances in Electrical Engineering 5

0010203040506070809

11

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count12057711205772

Figure 3 Proposed exponential constriction functions

Table 1 Particle encoding for the proposed PSO

1198751198661

1198751198662

sdot sdot sdot 119875119866119894

sdot sdot sdot 119875119866119873

where

120578 =itr

itrmax itrmin le itr le itrmax

119896 =12057711198621119887

12057721198622

(11)

The coefficient of exponent 1205831 has been considered minus55as the term eminus1205831120578 is not perceptible at the end of search With1205832 as 4 the coefficient 119896 is determined for exact match of 1205771and 1205772 at two-third of the search The variation in 1205771 and 1205772

with iterations is shown in Figure 3 for the above mentionedvalue of the exponent coefficients It can be depicted from thefigure that the dominance of cognitive behavior falls sharplyand that of the social behavior rises gradually as the searchprogresses Thus during the early part of the computationalsearch the cognitive behavior is well dominated over thesocial behavior of the swarm to enhance the global searchfor most probable area having the global optima Howeverduring the later part it is the social behavior of the swarm thatdominates over its cognitive counterpart This may enhancelocal exploitation by the swarm to search global or near globaloptima

These alterations in the control equation of the classicalPSO regulate particle velocity within predefined boundswithout any additional formulation as reported in manyimproved versions of PSO [4 6ndash8 10 13ndash16 18 21 26] yetpreserving diversity due to the stochastic nature of cognitiveand social behaviors of the swarm

34 Particle Encoding and Initialization The solution ofan ED problem is the set of most optimal generationsfor the desired objective(s) bounded by certain operationalconstraints In the proposed PSO the particles are encodedin real numbers as the set of current generations in MW asshown in Table 1

Where119875119866119894denotes generation of the 119894th generator inMW

the initial population is randomly created with predefinednumber of particles to maintain diversity Each of these

particles satisfies problem constraints defined by (2)ndash(4)Infeasible particle if appeared is not rejected but correctedusing a correction algorithm as described later in the sectionThis improves the computational efficiency of the PSOThe fitness of each particle is evaluated using (1) and then119901119887119890119904119905 119901119901119900119900119903 and 119892119887119890119904119905 are initialized The initial velocity ofparticles is assumed to be zero

35 Correction Algorithm The velocity and position updatemay create infeasible solutions Infeasible individuals are notrejected but are corrected to feasible individuals by usinga correction algorithm For the purpose the generationsof all generators are adjusted by their respective boundedgeneration limits and then the error is calculated fromthe power balance equation The error in the power isequally distributed among all generators and the procedureis repeated till the error is reduced to a predefined mismatchvalue 120598 In this work themismatch is considered as 0001Thisreduces the computational burden of PSO

36 Elitism and Termination Criterion In stochastic basedalgorithms like PSO the solution with the best fitness in thecurrent iteration may be lost in the next iteration Thereforethe particle with the best fitness is kept preserved for thenext iteration The algorithm is terminated when eitherall particles reach to the best position or the predefinedmaximum iteration number is reached The flow chart of theproposed method is shown in Figure 4

4 Simulation Results

The proposed algorithm is tested on 13-generator system [23]and 40-generator system [23] The control parameters usedfor all these systems to solve the ED problem using classicaland proposed PSO are considered as mentioned in Table 2The proposed algorithm has been developed using MATLABand simulations have been carried on a personal computer ofIntel i5 32 GHz and 4GB RAM

41 Case Study 1 13-Generator System Theproposedmethodis applied on 13 thermal generating units which consist ofvalve-point effect and network power losses The thermalgenerating unitsrsquo data and 119861-coefficient power loss data arereferred from [23] The ED problem is solved for a powerdemand of 2520MW The simulation results obtained forthe best and average fuel cost total power output andpower losses after 100 independent trails using proposedPSO are presented in Table 3 The table shows that theproposed PSO is capable of obtaining better best average andbest fuel costs with smaller power loss than other availableexisting stochastic methods in reasonable CPU time Thusthe proposed method provides good quality solution tosolve complex nonconvex ED problems The best generatingschedule obtained using the proposed PSO is presented inAppendix

42 Case Study 2 40-Generator SystemwithValve-Point EffectThe effectiveness of the proposed method is now investigated

6 Advances in Electrical Engineering

Start

Input cost coefficient data power limit data inertia weight value constriction function valuesset the value of maximum iteration

P = 1

Create one particle randomly

Isparticlefeasible

P = P + 1 Constrained handling

IsP ge popsize

Fitness evaluation initialize pbest ppoor gbest inertia weight constriction function anditeration counter

Itr = 1

P = 1

Velocity and position update

Isparticle feasible Constrained

handling

Fitness evaluation

Iffitness gt old

fitness

Update pbest

Update ppoor

IsP ge popsize

Update gbest

Isstopping criteria satisfied

Stop

Itr = itr + 1update inertia weight (W)and constriction function

Yes

Yes

Yes

Yes

Yes

Yes

No

No

No

No

No

No

P = P + 1

Figure 4 Flow chart of the proposed PSO

Advances in Electrical Engineering 7

Table 2 Various parameters for classical PSO and proposed PSO

Method 119882min119882max 1198621119887

1198621119901

1198622

1205831

1205832

itrmax Population sizeClassical PSO 0109 2 mdash 2 mdash mdash 2500 100Proposed PSO 0109 15 05 2 minus55 4 2500 100

Table 3 Comparison results for case study 1

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

Total power(MW)

Power loss(MW)

CPU time(s)

GA [27] 2463242 2487493 2518859 255987 3987 225DE [27] 2481932 2521764 2565640 256234 4234 258HDE [27] 2459176 2473953 2507490 255916 3916 357STHDE [27] 2456008 2470663 2487244 256433 4433 298ICA-PSO [7] 2454006 2456146 2458945 255905 3905 215SDE [1] 2451488 2451631 mdash 256043 4043 mdashProposed PSO 2451446 2451458 2451526 255807 3807 296

Table 4 Comparison results for case study 2

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

CPU time(s)

SQP [28] 1229044243 1248837692 1265852290 1080EP-SQP [29] 1223239700 1223796300 mdash 99773PSO-SQP [29] 1220946700 1222452500 mdash 73397PSO-LRS [12] 1220357946 1233820000 1257406300 3161NPSO [12] 1217047391 1222213697 1229950976 823NPSO-LRS [12] 1216644308 1229815913 1222093185 2074DEC-SQP [30] 1217419800 1233676500 1253979600 92563DEC(2)-SQP(1) [28] 1217419793 1222951278 1228392941 1426ACO [31] 1215324100 1216064500 1216796400 5245FCASO [32] 1215164700 1220825900 mdash 1452SOH-PSO [14] 1215011400 1218535700 1224463000 mdashTSARGA [33] 1214630700 1229283100 1242965400 6960CPSO-SQP [34] 1214585400 1220281600 mdashGA-PS-SQP [29] 1214580000 1220390000 mdash 4698ABC [35] 1214410300 1219958200 mdash 3002CCPSO [22] 1214125362 1214453269 1215254934 193ICA-PSO [7] 1214221000 mdash mdash 1399DEBBO [36] 1214208948 mdash mdash 12HHS [37] 1214155920 1216158544 mdash 1639IPSO [2] 1214128660 1215095223 1215468420 4289NAPSO [8] 1214125700 mdash mdash 127CSA [38] 1214125355 1215204106 1218102538 303Proposed PSO 1214125355 1214323215 1215643454 999

Table 5 Comparison results for case study 3

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel Cost($hr)

CPU time(s)

PSO [8] 1248758523 1251627011 mdash mdashFAPSO [8] 1222613706 1224710751 1225975196 196NAPSO [8] 1214910662 1214912756 1214915261 127CSA [38] 1214877727 1216113170 1221629295 147Proposed PSO 1214877718 1215113114 1217537157 84

8 Advances in Electrical Engineering

on the most popular test generating system taken from [23]This system consists of 40 thermal units with nonconvexity incost function due to valve-point loading effectsThe expectedpower demand for this test system is 10500MW The resultsobtained after 100 independent trials of the proposed PSOare presented and compared with a variety of other availableexisting deterministic and population based or their hybridtechniques in Table 4 The table validates the effectivenessof the proposed PSO as it generates either comparable orbetter best fuel cost than other several established techniquesincluding hybrid techniques The table also shows that theproposed PSO is less computationally demanding than manyother references including some latest ones Although NPSO[12] and CSA [38] demand less CPU time than the proposedPSO but the proposedmethod is capable of generating betterquality solutionThus the proposed PSO is promising to solvenonconvex ED problems The optimal dispatch of thermalgenerators obtained by the proposed PSO can be referred toin Appendix

43 Case Study 3 40-Generator System with Valve-Point andPOZs Finally the effectiveness of the proposed method isinvestigated on the 40 generators test generating systemwith discontinuities in the cost function due to prohibitedoperating zones The units 10ndash14 have POZs as given in[8] (POZ 2) The expected power demand for this testsystem is 10500 MW The results obtained after 100 trials ofthe proposed PSO are presented and compared with otheravailable existing population based techniques in Table 5Thetable shows that the proposed PSO is capable of generatingcomparable or better result in less computational time thanother established available methods The better value ofaverage fuel cost is obtained by proposed method than othermethods This shows robustness of the proposed PSO Thusthe high dimensional nonconvex discrete ED problems canbe effectively and efficiently solved using the proposed PSOThe optimal dispatch of thermal generators obtained by theproposed PSO can be referred to in Appendix

5 Discussion

In order to appreciate and understand the performance ofthe proposed method a comparison of cognitive and socialbehavior of particle in PSO and the proposed PSO is shown inFigures 5 and 6 respectively Figure 5 shows that in the classi-cal PSO the cognitive and social behaviors of particle velocityvary randomly throughout the computational process withinlimits of 0 to 2 The proposed constriction functions usedto guide the cognitive and social behaviors of the swarm areallowed to vary exponentially as shown in Figure 6The lowerand upper limits of these behaviors are governed by (10)However the sum of the best and poor cognitive behavior ofthe swarm remains constant during the computation processThis plays an important role in providing sufficient diversityby the poor experience during the whole flight of the swarmIt can be seen from Figure 6 that using proposed PSOthe modulations of cognitive (best) cognitive (poor) and

social behaviors though randomly distributed are dynami-cally controlled within exponential bounds of 15 05 and015 respectively This constitutes a marked difference withother versions of existing PSO Thus the particles experienceentirely different cognitive and social behaviors during theirflights and need no additional mechanism to bind theirvelocities

Any stochastic based search technique must be designedto accomplish global exploration and tends to facilitate localexploitation In order to investigate the effectiveness of eachof these modifications a set of convergence characteristicsfor the best and average fuel cost obtained during a sampletrial for 40 generators system is shown in Figures 7 and8 respectively In Figure 7 the characteristic ldquoardquo is for theconventional PSO ldquobrdquo refers to ldquoardquo with sinusoidal mod-ulation in inertia weight ldquocrdquo refers to ldquobrdquo with improvedcognitive behavior due to poor experience and ldquodrdquo refers tothe proposed PSO It can be observed from the figure that theperformance of the PSO is somewhat improved when inertiaweight is sinusoidally modulated and is further improvedwith a good margin when poor experience of particles is alsoconsidered However these two modifications do not seemto be sufficient to exploit the promising region effectively andefficiently This leads to premature convergence due to localtrappings which can be depicted from ldquodrdquo In d the proposedconstriction functions regulate particlesrsquo velocities so thatthey can fly more comprehensively in the search space Infact due to higher initial cognitive component than the socialcomponent the proposed PSO becomes more competentto explore wider search space during the initial phase andthus identify the promising region in about 1000 iterationsHowever particles move with strong communication andthus intensively exploit the region near the global optimaduring later part of the search owing to high values ofsocial component Finally all particles converge towards theglobal minima as can be observed from Figure 8 Thus theproposed PSOprovides better exploration and exploitation ofthe search space and produces better quality solutions Theseresults also highlight that the modifications suggested in thecontrol equation of the classical PSO are very effective as itmakes the proposed PSO perform much better

The proposed method offers better exploration andexploitation of the search space because the velocity ofparticles is regulated throughout their flight The movementof a sample particle in the classical PSOand the proposedPSOis illustrated in Figures 9 and 10 respectively These figuresshow the traces of initial cognitive and social componentsof particlersquos velocity and also the overall velocity imparted toit during a sample trial

The classical PSO searches for about 400 iterations asshown in Figure 9 After this all the three components ofparticlersquos velocity became insignificant and thus the particlegets trapped into local minima Figure 10(b) shows thecognitive component for the best experience which is thensuperimposed by its poor experience as in Figure 10(c) toobtain the overall cognitive component as in Figure 10(d) Itcan be concluded from Figure 10(d) that the poor experienceis contributing to tune the cognitive behavior of the swarmThe social component as shown in Figure 10(e) is providing

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Advances in Electrical Engineering

the flight The modifications suggested in the control equa-tion are explained as follows

V119896+1119894

= 119882 times V119896119894+ 1205771times 1198621119887times rand

1 () times119901119887119890119904119905119894minus 119904119896

119894

Δ119905

+ (1 minus 1205771) times 1198621119901

times rand2 () times

119904119896

119894minus 119901119901119900119900119903

119894

Δ119905

+ 1205772 times 1198622 times rand3 () times

119892119887119890119904119905119894minus 119904119896

119894

Δ119905

(7)

In (7) the inertia weight is modified to regulate the trade-off between the global exploration and the local exploitationof the swarm The poor experience 119901119901119900119900119903

119894has been added

to improve the cognitive component Further dynamic accel-eration coefficients have been introduced using constrictionfunctions 120577

1and 120577

2to regulate the cognitive and social

behaviors of the swarmThese modifications are discussed inthe following sections

31 Inertia Weight Update In [25] Shi and Eberhart sug-gested linear modulation of the inertia weight This trendis followed to solve ELD problems using PSO by manyresearchers till date and some of them can be mentioned as[4 6 8 12 13 15 19 20 25 26] and so forth In the proposedmethod the inertia weight has been allowed to vary inaccordance with a truncated sinusoidal function rather thanto decrease linearlyThemodulations suggested to update theinertia weight is governed by the following relation

119882 = 119882min + (119882max minus119882min) cos2(120579

2)

0 le 120579 le 120587

(8)

where 120579 = 119883 times itr +119884 and the coefficients119883 and 119884 are givenby (9) itr is the iteration countwhich is in general varied fromitrmin to itrmax

119883 =120587

(itrmax minus itrmin)

119884 =minus120587 times itrmin

(itrmax minus itrmin)

(9)

Figure 2 shows a comparison of the conventional linearmodulation and sinusoidal modulation for the inertia weightto be employed in the proposed PSO It can be depicted fromthe figure that using the sinusoidal variations in the inertiaweight the inertia component of the velocity of particlesmaintained always higher during the early half and lowerduring the later half of the search when compared with itslinear variationsTherefore using sinusoidalmodulations thecoarse search is enhanced during the early half by exploringlarger search space with higher values assigned to particlevelocities And during the later half the fine search isenhanced by assigning lower values to particle velocitiesThis facilitates particles to explore the regions in the closeproximity of near global solution

W

Wmax

Wmin

Itrmin Itrmax2 Itrmax Itr

Figure 2 Comparison of linear and sinusoidal modulations ofinertia weight

32 Updating of Poor Experience The cognitive behaviorwas split in [12] by considering the worst experience inaddition to the best experience of the particle Though thismodification provides additional diversity it still demands alocal random search to enhance exploitation potential of thePSOThis occurs as the particlersquos velocity is not well regulatedduring later part of the search Therefore the concept ofpoor experience 119901119901119900119900119903

119894is suggested instead of the worst

experience to improve cognitive behavior of the swarmHerethe current fitness of each particle is compared with its fitnessvalue in the preceding iteration and if it is found less it willbe treated as the poor experience This concept is differentthan that of [12] where the worst particle is determinedby considering the whole past experience of the particlemovement The poor particle produces much less diversitythan the worst particle and thus exploit the region near globaloptima during later iterations in much better way withoutthe support of any local random search

33 Dynamic Control of Acceleration Coefficients In classicalPSO the cognitive and social behaviors are governed byassigning static values to acceleration coefficients Manyresearchers as discussed earlier suggested that these acceler-ation coefficients must be dynamically controlled to regulateparticlersquos velocity during the whole computation process Inthe present work the acceleration coefficients are dynami-cally controlled by suggesting new exponential constrictionfunctions 120577

1and 1205772These constriction functions dynamically

regulate the cognitive and social behaviors of the swarm thuslimiting particlesrsquo velocities during their whole course of theflight and are given by

1205771= eminus1205831120578

1205772= 119896e1205832120578

(10)

Advances in Electrical Engineering 5

0010203040506070809

11

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count12057711205772

Figure 3 Proposed exponential constriction functions

Table 1 Particle encoding for the proposed PSO

1198751198661

1198751198662

sdot sdot sdot 119875119866119894

sdot sdot sdot 119875119866119873

where

120578 =itr

itrmax itrmin le itr le itrmax

119896 =12057711198621119887

12057721198622

(11)

The coefficient of exponent 1205831 has been considered minus55as the term eminus1205831120578 is not perceptible at the end of search With1205832 as 4 the coefficient 119896 is determined for exact match of 1205771and 1205772 at two-third of the search The variation in 1205771 and 1205772

with iterations is shown in Figure 3 for the above mentionedvalue of the exponent coefficients It can be depicted from thefigure that the dominance of cognitive behavior falls sharplyand that of the social behavior rises gradually as the searchprogresses Thus during the early part of the computationalsearch the cognitive behavior is well dominated over thesocial behavior of the swarm to enhance the global searchfor most probable area having the global optima Howeverduring the later part it is the social behavior of the swarm thatdominates over its cognitive counterpart This may enhancelocal exploitation by the swarm to search global or near globaloptima

These alterations in the control equation of the classicalPSO regulate particle velocity within predefined boundswithout any additional formulation as reported in manyimproved versions of PSO [4 6ndash8 10 13ndash16 18 21 26] yetpreserving diversity due to the stochastic nature of cognitiveand social behaviors of the swarm

34 Particle Encoding and Initialization The solution ofan ED problem is the set of most optimal generationsfor the desired objective(s) bounded by certain operationalconstraints In the proposed PSO the particles are encodedin real numbers as the set of current generations in MW asshown in Table 1

Where119875119866119894denotes generation of the 119894th generator inMW

the initial population is randomly created with predefinednumber of particles to maintain diversity Each of these

particles satisfies problem constraints defined by (2)ndash(4)Infeasible particle if appeared is not rejected but correctedusing a correction algorithm as described later in the sectionThis improves the computational efficiency of the PSOThe fitness of each particle is evaluated using (1) and then119901119887119890119904119905 119901119901119900119900119903 and 119892119887119890119904119905 are initialized The initial velocity ofparticles is assumed to be zero

35 Correction Algorithm The velocity and position updatemay create infeasible solutions Infeasible individuals are notrejected but are corrected to feasible individuals by usinga correction algorithm For the purpose the generationsof all generators are adjusted by their respective boundedgeneration limits and then the error is calculated fromthe power balance equation The error in the power isequally distributed among all generators and the procedureis repeated till the error is reduced to a predefined mismatchvalue 120598 In this work themismatch is considered as 0001Thisreduces the computational burden of PSO

36 Elitism and Termination Criterion In stochastic basedalgorithms like PSO the solution with the best fitness in thecurrent iteration may be lost in the next iteration Thereforethe particle with the best fitness is kept preserved for thenext iteration The algorithm is terminated when eitherall particles reach to the best position or the predefinedmaximum iteration number is reached The flow chart of theproposed method is shown in Figure 4

4 Simulation Results

The proposed algorithm is tested on 13-generator system [23]and 40-generator system [23] The control parameters usedfor all these systems to solve the ED problem using classicaland proposed PSO are considered as mentioned in Table 2The proposed algorithm has been developed using MATLABand simulations have been carried on a personal computer ofIntel i5 32 GHz and 4GB RAM

41 Case Study 1 13-Generator System Theproposedmethodis applied on 13 thermal generating units which consist ofvalve-point effect and network power losses The thermalgenerating unitsrsquo data and 119861-coefficient power loss data arereferred from [23] The ED problem is solved for a powerdemand of 2520MW The simulation results obtained forthe best and average fuel cost total power output andpower losses after 100 independent trails using proposedPSO are presented in Table 3 The table shows that theproposed PSO is capable of obtaining better best average andbest fuel costs with smaller power loss than other availableexisting stochastic methods in reasonable CPU time Thusthe proposed method provides good quality solution tosolve complex nonconvex ED problems The best generatingschedule obtained using the proposed PSO is presented inAppendix

42 Case Study 2 40-Generator SystemwithValve-Point EffectThe effectiveness of the proposed method is now investigated

6 Advances in Electrical Engineering

Start

Input cost coefficient data power limit data inertia weight value constriction function valuesset the value of maximum iteration

P = 1

Create one particle randomly

Isparticlefeasible

P = P + 1 Constrained handling

IsP ge popsize

Fitness evaluation initialize pbest ppoor gbest inertia weight constriction function anditeration counter

Itr = 1

P = 1

Velocity and position update

Isparticle feasible Constrained

handling

Fitness evaluation

Iffitness gt old

fitness

Update pbest

Update ppoor

IsP ge popsize

Update gbest

Isstopping criteria satisfied

Stop

Itr = itr + 1update inertia weight (W)and constriction function

Yes

Yes

Yes

Yes

Yes

Yes

No

No

No

No

No

No

P = P + 1

Figure 4 Flow chart of the proposed PSO

Advances in Electrical Engineering 7

Table 2 Various parameters for classical PSO and proposed PSO

Method 119882min119882max 1198621119887

1198621119901

1198622

1205831

1205832

itrmax Population sizeClassical PSO 0109 2 mdash 2 mdash mdash 2500 100Proposed PSO 0109 15 05 2 minus55 4 2500 100

Table 3 Comparison results for case study 1

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

Total power(MW)

Power loss(MW)

CPU time(s)

GA [27] 2463242 2487493 2518859 255987 3987 225DE [27] 2481932 2521764 2565640 256234 4234 258HDE [27] 2459176 2473953 2507490 255916 3916 357STHDE [27] 2456008 2470663 2487244 256433 4433 298ICA-PSO [7] 2454006 2456146 2458945 255905 3905 215SDE [1] 2451488 2451631 mdash 256043 4043 mdashProposed PSO 2451446 2451458 2451526 255807 3807 296

Table 4 Comparison results for case study 2

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

CPU time(s)

SQP [28] 1229044243 1248837692 1265852290 1080EP-SQP [29] 1223239700 1223796300 mdash 99773PSO-SQP [29] 1220946700 1222452500 mdash 73397PSO-LRS [12] 1220357946 1233820000 1257406300 3161NPSO [12] 1217047391 1222213697 1229950976 823NPSO-LRS [12] 1216644308 1229815913 1222093185 2074DEC-SQP [30] 1217419800 1233676500 1253979600 92563DEC(2)-SQP(1) [28] 1217419793 1222951278 1228392941 1426ACO [31] 1215324100 1216064500 1216796400 5245FCASO [32] 1215164700 1220825900 mdash 1452SOH-PSO [14] 1215011400 1218535700 1224463000 mdashTSARGA [33] 1214630700 1229283100 1242965400 6960CPSO-SQP [34] 1214585400 1220281600 mdashGA-PS-SQP [29] 1214580000 1220390000 mdash 4698ABC [35] 1214410300 1219958200 mdash 3002CCPSO [22] 1214125362 1214453269 1215254934 193ICA-PSO [7] 1214221000 mdash mdash 1399DEBBO [36] 1214208948 mdash mdash 12HHS [37] 1214155920 1216158544 mdash 1639IPSO [2] 1214128660 1215095223 1215468420 4289NAPSO [8] 1214125700 mdash mdash 127CSA [38] 1214125355 1215204106 1218102538 303Proposed PSO 1214125355 1214323215 1215643454 999

Table 5 Comparison results for case study 3

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel Cost($hr)

CPU time(s)

PSO [8] 1248758523 1251627011 mdash mdashFAPSO [8] 1222613706 1224710751 1225975196 196NAPSO [8] 1214910662 1214912756 1214915261 127CSA [38] 1214877727 1216113170 1221629295 147Proposed PSO 1214877718 1215113114 1217537157 84

8 Advances in Electrical Engineering

on the most popular test generating system taken from [23]This system consists of 40 thermal units with nonconvexity incost function due to valve-point loading effectsThe expectedpower demand for this test system is 10500MW The resultsobtained after 100 independent trials of the proposed PSOare presented and compared with a variety of other availableexisting deterministic and population based or their hybridtechniques in Table 4 The table validates the effectivenessof the proposed PSO as it generates either comparable orbetter best fuel cost than other several established techniquesincluding hybrid techniques The table also shows that theproposed PSO is less computationally demanding than manyother references including some latest ones Although NPSO[12] and CSA [38] demand less CPU time than the proposedPSO but the proposedmethod is capable of generating betterquality solutionThus the proposed PSO is promising to solvenonconvex ED problems The optimal dispatch of thermalgenerators obtained by the proposed PSO can be referred toin Appendix

43 Case Study 3 40-Generator System with Valve-Point andPOZs Finally the effectiveness of the proposed method isinvestigated on the 40 generators test generating systemwith discontinuities in the cost function due to prohibitedoperating zones The units 10ndash14 have POZs as given in[8] (POZ 2) The expected power demand for this testsystem is 10500 MW The results obtained after 100 trials ofthe proposed PSO are presented and compared with otheravailable existing population based techniques in Table 5Thetable shows that the proposed PSO is capable of generatingcomparable or better result in less computational time thanother established available methods The better value ofaverage fuel cost is obtained by proposed method than othermethods This shows robustness of the proposed PSO Thusthe high dimensional nonconvex discrete ED problems canbe effectively and efficiently solved using the proposed PSOThe optimal dispatch of thermal generators obtained by theproposed PSO can be referred to in Appendix

5 Discussion

In order to appreciate and understand the performance ofthe proposed method a comparison of cognitive and socialbehavior of particle in PSO and the proposed PSO is shown inFigures 5 and 6 respectively Figure 5 shows that in the classi-cal PSO the cognitive and social behaviors of particle velocityvary randomly throughout the computational process withinlimits of 0 to 2 The proposed constriction functions usedto guide the cognitive and social behaviors of the swarm areallowed to vary exponentially as shown in Figure 6The lowerand upper limits of these behaviors are governed by (10)However the sum of the best and poor cognitive behavior ofthe swarm remains constant during the computation processThis plays an important role in providing sufficient diversityby the poor experience during the whole flight of the swarmIt can be seen from Figure 6 that using proposed PSOthe modulations of cognitive (best) cognitive (poor) and

social behaviors though randomly distributed are dynami-cally controlled within exponential bounds of 15 05 and015 respectively This constitutes a marked difference withother versions of existing PSO Thus the particles experienceentirely different cognitive and social behaviors during theirflights and need no additional mechanism to bind theirvelocities

Any stochastic based search technique must be designedto accomplish global exploration and tends to facilitate localexploitation In order to investigate the effectiveness of eachof these modifications a set of convergence characteristicsfor the best and average fuel cost obtained during a sampletrial for 40 generators system is shown in Figures 7 and8 respectively In Figure 7 the characteristic ldquoardquo is for theconventional PSO ldquobrdquo refers to ldquoardquo with sinusoidal mod-ulation in inertia weight ldquocrdquo refers to ldquobrdquo with improvedcognitive behavior due to poor experience and ldquodrdquo refers tothe proposed PSO It can be observed from the figure that theperformance of the PSO is somewhat improved when inertiaweight is sinusoidally modulated and is further improvedwith a good margin when poor experience of particles is alsoconsidered However these two modifications do not seemto be sufficient to exploit the promising region effectively andefficiently This leads to premature convergence due to localtrappings which can be depicted from ldquodrdquo In d the proposedconstriction functions regulate particlesrsquo velocities so thatthey can fly more comprehensively in the search space Infact due to higher initial cognitive component than the socialcomponent the proposed PSO becomes more competentto explore wider search space during the initial phase andthus identify the promising region in about 1000 iterationsHowever particles move with strong communication andthus intensively exploit the region near the global optimaduring later part of the search owing to high values ofsocial component Finally all particles converge towards theglobal minima as can be observed from Figure 8 Thus theproposed PSOprovides better exploration and exploitation ofthe search space and produces better quality solutions Theseresults also highlight that the modifications suggested in thecontrol equation of the classical PSO are very effective as itmakes the proposed PSO perform much better

The proposed method offers better exploration andexploitation of the search space because the velocity ofparticles is regulated throughout their flight The movementof a sample particle in the classical PSOand the proposedPSOis illustrated in Figures 9 and 10 respectively These figuresshow the traces of initial cognitive and social componentsof particlersquos velocity and also the overall velocity imparted toit during a sample trial

The classical PSO searches for about 400 iterations asshown in Figure 9 After this all the three components ofparticlersquos velocity became insignificant and thus the particlegets trapped into local minima Figure 10(b) shows thecognitive component for the best experience which is thensuperimposed by its poor experience as in Figure 10(c) toobtain the overall cognitive component as in Figure 10(d) Itcan be concluded from Figure 10(d) that the poor experienceis contributing to tune the cognitive behavior of the swarmThe social component as shown in Figure 10(e) is providing

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Electrical Engineering 5

0010203040506070809

11

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count12057711205772

Figure 3 Proposed exponential constriction functions

Table 1 Particle encoding for the proposed PSO

1198751198661

1198751198662

sdot sdot sdot 119875119866119894

sdot sdot sdot 119875119866119873

where

120578 =itr

itrmax itrmin le itr le itrmax

119896 =12057711198621119887

12057721198622

(11)

The coefficient of exponent 1205831 has been considered minus55as the term eminus1205831120578 is not perceptible at the end of search With1205832 as 4 the coefficient 119896 is determined for exact match of 1205771and 1205772 at two-third of the search The variation in 1205771 and 1205772

with iterations is shown in Figure 3 for the above mentionedvalue of the exponent coefficients It can be depicted from thefigure that the dominance of cognitive behavior falls sharplyand that of the social behavior rises gradually as the searchprogresses Thus during the early part of the computationalsearch the cognitive behavior is well dominated over thesocial behavior of the swarm to enhance the global searchfor most probable area having the global optima Howeverduring the later part it is the social behavior of the swarm thatdominates over its cognitive counterpart This may enhancelocal exploitation by the swarm to search global or near globaloptima

These alterations in the control equation of the classicalPSO regulate particle velocity within predefined boundswithout any additional formulation as reported in manyimproved versions of PSO [4 6ndash8 10 13ndash16 18 21 26] yetpreserving diversity due to the stochastic nature of cognitiveand social behaviors of the swarm

34 Particle Encoding and Initialization The solution ofan ED problem is the set of most optimal generationsfor the desired objective(s) bounded by certain operationalconstraints In the proposed PSO the particles are encodedin real numbers as the set of current generations in MW asshown in Table 1

Where119875119866119894denotes generation of the 119894th generator inMW

the initial population is randomly created with predefinednumber of particles to maintain diversity Each of these

particles satisfies problem constraints defined by (2)ndash(4)Infeasible particle if appeared is not rejected but correctedusing a correction algorithm as described later in the sectionThis improves the computational efficiency of the PSOThe fitness of each particle is evaluated using (1) and then119901119887119890119904119905 119901119901119900119900119903 and 119892119887119890119904119905 are initialized The initial velocity ofparticles is assumed to be zero

35 Correction Algorithm The velocity and position updatemay create infeasible solutions Infeasible individuals are notrejected but are corrected to feasible individuals by usinga correction algorithm For the purpose the generationsof all generators are adjusted by their respective boundedgeneration limits and then the error is calculated fromthe power balance equation The error in the power isequally distributed among all generators and the procedureis repeated till the error is reduced to a predefined mismatchvalue 120598 In this work themismatch is considered as 0001Thisreduces the computational burden of PSO

36 Elitism and Termination Criterion In stochastic basedalgorithms like PSO the solution with the best fitness in thecurrent iteration may be lost in the next iteration Thereforethe particle with the best fitness is kept preserved for thenext iteration The algorithm is terminated when eitherall particles reach to the best position or the predefinedmaximum iteration number is reached The flow chart of theproposed method is shown in Figure 4

4 Simulation Results

The proposed algorithm is tested on 13-generator system [23]and 40-generator system [23] The control parameters usedfor all these systems to solve the ED problem using classicaland proposed PSO are considered as mentioned in Table 2The proposed algorithm has been developed using MATLABand simulations have been carried on a personal computer ofIntel i5 32 GHz and 4GB RAM

41 Case Study 1 13-Generator System Theproposedmethodis applied on 13 thermal generating units which consist ofvalve-point effect and network power losses The thermalgenerating unitsrsquo data and 119861-coefficient power loss data arereferred from [23] The ED problem is solved for a powerdemand of 2520MW The simulation results obtained forthe best and average fuel cost total power output andpower losses after 100 independent trails using proposedPSO are presented in Table 3 The table shows that theproposed PSO is capable of obtaining better best average andbest fuel costs with smaller power loss than other availableexisting stochastic methods in reasonable CPU time Thusthe proposed method provides good quality solution tosolve complex nonconvex ED problems The best generatingschedule obtained using the proposed PSO is presented inAppendix

42 Case Study 2 40-Generator SystemwithValve-Point EffectThe effectiveness of the proposed method is now investigated

6 Advances in Electrical Engineering

Start

Input cost coefficient data power limit data inertia weight value constriction function valuesset the value of maximum iteration

P = 1

Create one particle randomly

Isparticlefeasible

P = P + 1 Constrained handling

IsP ge popsize

Fitness evaluation initialize pbest ppoor gbest inertia weight constriction function anditeration counter

Itr = 1

P = 1

Velocity and position update

Isparticle feasible Constrained

handling

Fitness evaluation

Iffitness gt old

fitness

Update pbest

Update ppoor

IsP ge popsize

Update gbest

Isstopping criteria satisfied

Stop

Itr = itr + 1update inertia weight (W)and constriction function

Yes

Yes

Yes

Yes

Yes

Yes

No

No

No

No

No

No

P = P + 1

Figure 4 Flow chart of the proposed PSO

Advances in Electrical Engineering 7

Table 2 Various parameters for classical PSO and proposed PSO

Method 119882min119882max 1198621119887

1198621119901

1198622

1205831

1205832

itrmax Population sizeClassical PSO 0109 2 mdash 2 mdash mdash 2500 100Proposed PSO 0109 15 05 2 minus55 4 2500 100

Table 3 Comparison results for case study 1

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

Total power(MW)

Power loss(MW)

CPU time(s)

GA [27] 2463242 2487493 2518859 255987 3987 225DE [27] 2481932 2521764 2565640 256234 4234 258HDE [27] 2459176 2473953 2507490 255916 3916 357STHDE [27] 2456008 2470663 2487244 256433 4433 298ICA-PSO [7] 2454006 2456146 2458945 255905 3905 215SDE [1] 2451488 2451631 mdash 256043 4043 mdashProposed PSO 2451446 2451458 2451526 255807 3807 296

Table 4 Comparison results for case study 2

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

CPU time(s)

SQP [28] 1229044243 1248837692 1265852290 1080EP-SQP [29] 1223239700 1223796300 mdash 99773PSO-SQP [29] 1220946700 1222452500 mdash 73397PSO-LRS [12] 1220357946 1233820000 1257406300 3161NPSO [12] 1217047391 1222213697 1229950976 823NPSO-LRS [12] 1216644308 1229815913 1222093185 2074DEC-SQP [30] 1217419800 1233676500 1253979600 92563DEC(2)-SQP(1) [28] 1217419793 1222951278 1228392941 1426ACO [31] 1215324100 1216064500 1216796400 5245FCASO [32] 1215164700 1220825900 mdash 1452SOH-PSO [14] 1215011400 1218535700 1224463000 mdashTSARGA [33] 1214630700 1229283100 1242965400 6960CPSO-SQP [34] 1214585400 1220281600 mdashGA-PS-SQP [29] 1214580000 1220390000 mdash 4698ABC [35] 1214410300 1219958200 mdash 3002CCPSO [22] 1214125362 1214453269 1215254934 193ICA-PSO [7] 1214221000 mdash mdash 1399DEBBO [36] 1214208948 mdash mdash 12HHS [37] 1214155920 1216158544 mdash 1639IPSO [2] 1214128660 1215095223 1215468420 4289NAPSO [8] 1214125700 mdash mdash 127CSA [38] 1214125355 1215204106 1218102538 303Proposed PSO 1214125355 1214323215 1215643454 999

Table 5 Comparison results for case study 3

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel Cost($hr)

CPU time(s)

PSO [8] 1248758523 1251627011 mdash mdashFAPSO [8] 1222613706 1224710751 1225975196 196NAPSO [8] 1214910662 1214912756 1214915261 127CSA [38] 1214877727 1216113170 1221629295 147Proposed PSO 1214877718 1215113114 1217537157 84

8 Advances in Electrical Engineering

on the most popular test generating system taken from [23]This system consists of 40 thermal units with nonconvexity incost function due to valve-point loading effectsThe expectedpower demand for this test system is 10500MW The resultsobtained after 100 independent trials of the proposed PSOare presented and compared with a variety of other availableexisting deterministic and population based or their hybridtechniques in Table 4 The table validates the effectivenessof the proposed PSO as it generates either comparable orbetter best fuel cost than other several established techniquesincluding hybrid techniques The table also shows that theproposed PSO is less computationally demanding than manyother references including some latest ones Although NPSO[12] and CSA [38] demand less CPU time than the proposedPSO but the proposedmethod is capable of generating betterquality solutionThus the proposed PSO is promising to solvenonconvex ED problems The optimal dispatch of thermalgenerators obtained by the proposed PSO can be referred toin Appendix

43 Case Study 3 40-Generator System with Valve-Point andPOZs Finally the effectiveness of the proposed method isinvestigated on the 40 generators test generating systemwith discontinuities in the cost function due to prohibitedoperating zones The units 10ndash14 have POZs as given in[8] (POZ 2) The expected power demand for this testsystem is 10500 MW The results obtained after 100 trials ofthe proposed PSO are presented and compared with otheravailable existing population based techniques in Table 5Thetable shows that the proposed PSO is capable of generatingcomparable or better result in less computational time thanother established available methods The better value ofaverage fuel cost is obtained by proposed method than othermethods This shows robustness of the proposed PSO Thusthe high dimensional nonconvex discrete ED problems canbe effectively and efficiently solved using the proposed PSOThe optimal dispatch of thermal generators obtained by theproposed PSO can be referred to in Appendix

5 Discussion

In order to appreciate and understand the performance ofthe proposed method a comparison of cognitive and socialbehavior of particle in PSO and the proposed PSO is shown inFigures 5 and 6 respectively Figure 5 shows that in the classi-cal PSO the cognitive and social behaviors of particle velocityvary randomly throughout the computational process withinlimits of 0 to 2 The proposed constriction functions usedto guide the cognitive and social behaviors of the swarm areallowed to vary exponentially as shown in Figure 6The lowerand upper limits of these behaviors are governed by (10)However the sum of the best and poor cognitive behavior ofthe swarm remains constant during the computation processThis plays an important role in providing sufficient diversityby the poor experience during the whole flight of the swarmIt can be seen from Figure 6 that using proposed PSOthe modulations of cognitive (best) cognitive (poor) and

social behaviors though randomly distributed are dynami-cally controlled within exponential bounds of 15 05 and015 respectively This constitutes a marked difference withother versions of existing PSO Thus the particles experienceentirely different cognitive and social behaviors during theirflights and need no additional mechanism to bind theirvelocities

Any stochastic based search technique must be designedto accomplish global exploration and tends to facilitate localexploitation In order to investigate the effectiveness of eachof these modifications a set of convergence characteristicsfor the best and average fuel cost obtained during a sampletrial for 40 generators system is shown in Figures 7 and8 respectively In Figure 7 the characteristic ldquoardquo is for theconventional PSO ldquobrdquo refers to ldquoardquo with sinusoidal mod-ulation in inertia weight ldquocrdquo refers to ldquobrdquo with improvedcognitive behavior due to poor experience and ldquodrdquo refers tothe proposed PSO It can be observed from the figure that theperformance of the PSO is somewhat improved when inertiaweight is sinusoidally modulated and is further improvedwith a good margin when poor experience of particles is alsoconsidered However these two modifications do not seemto be sufficient to exploit the promising region effectively andefficiently This leads to premature convergence due to localtrappings which can be depicted from ldquodrdquo In d the proposedconstriction functions regulate particlesrsquo velocities so thatthey can fly more comprehensively in the search space Infact due to higher initial cognitive component than the socialcomponent the proposed PSO becomes more competentto explore wider search space during the initial phase andthus identify the promising region in about 1000 iterationsHowever particles move with strong communication andthus intensively exploit the region near the global optimaduring later part of the search owing to high values ofsocial component Finally all particles converge towards theglobal minima as can be observed from Figure 8 Thus theproposed PSOprovides better exploration and exploitation ofthe search space and produces better quality solutions Theseresults also highlight that the modifications suggested in thecontrol equation of the classical PSO are very effective as itmakes the proposed PSO perform much better

The proposed method offers better exploration andexploitation of the search space because the velocity ofparticles is regulated throughout their flight The movementof a sample particle in the classical PSOand the proposedPSOis illustrated in Figures 9 and 10 respectively These figuresshow the traces of initial cognitive and social componentsof particlersquos velocity and also the overall velocity imparted toit during a sample trial

The classical PSO searches for about 400 iterations asshown in Figure 9 After this all the three components ofparticlersquos velocity became insignificant and thus the particlegets trapped into local minima Figure 10(b) shows thecognitive component for the best experience which is thensuperimposed by its poor experience as in Figure 10(c) toobtain the overall cognitive component as in Figure 10(d) Itcan be concluded from Figure 10(d) that the poor experienceis contributing to tune the cognitive behavior of the swarmThe social component as shown in Figure 10(e) is providing

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Advances in Electrical Engineering

Start

Input cost coefficient data power limit data inertia weight value constriction function valuesset the value of maximum iteration

P = 1

Create one particle randomly

Isparticlefeasible

P = P + 1 Constrained handling

IsP ge popsize

Fitness evaluation initialize pbest ppoor gbest inertia weight constriction function anditeration counter

Itr = 1

P = 1

Velocity and position update

Isparticle feasible Constrained

handling

Fitness evaluation

Iffitness gt old

fitness

Update pbest

Update ppoor

IsP ge popsize

Update gbest

Isstopping criteria satisfied

Stop

Itr = itr + 1update inertia weight (W)and constriction function

Yes

Yes

Yes

Yes

Yes

Yes

No

No

No

No

No

No

P = P + 1

Figure 4 Flow chart of the proposed PSO

Advances in Electrical Engineering 7

Table 2 Various parameters for classical PSO and proposed PSO

Method 119882min119882max 1198621119887

1198621119901

1198622

1205831

1205832

itrmax Population sizeClassical PSO 0109 2 mdash 2 mdash mdash 2500 100Proposed PSO 0109 15 05 2 minus55 4 2500 100

Table 3 Comparison results for case study 1

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

Total power(MW)

Power loss(MW)

CPU time(s)

GA [27] 2463242 2487493 2518859 255987 3987 225DE [27] 2481932 2521764 2565640 256234 4234 258HDE [27] 2459176 2473953 2507490 255916 3916 357STHDE [27] 2456008 2470663 2487244 256433 4433 298ICA-PSO [7] 2454006 2456146 2458945 255905 3905 215SDE [1] 2451488 2451631 mdash 256043 4043 mdashProposed PSO 2451446 2451458 2451526 255807 3807 296

Table 4 Comparison results for case study 2

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

CPU time(s)

SQP [28] 1229044243 1248837692 1265852290 1080EP-SQP [29] 1223239700 1223796300 mdash 99773PSO-SQP [29] 1220946700 1222452500 mdash 73397PSO-LRS [12] 1220357946 1233820000 1257406300 3161NPSO [12] 1217047391 1222213697 1229950976 823NPSO-LRS [12] 1216644308 1229815913 1222093185 2074DEC-SQP [30] 1217419800 1233676500 1253979600 92563DEC(2)-SQP(1) [28] 1217419793 1222951278 1228392941 1426ACO [31] 1215324100 1216064500 1216796400 5245FCASO [32] 1215164700 1220825900 mdash 1452SOH-PSO [14] 1215011400 1218535700 1224463000 mdashTSARGA [33] 1214630700 1229283100 1242965400 6960CPSO-SQP [34] 1214585400 1220281600 mdashGA-PS-SQP [29] 1214580000 1220390000 mdash 4698ABC [35] 1214410300 1219958200 mdash 3002CCPSO [22] 1214125362 1214453269 1215254934 193ICA-PSO [7] 1214221000 mdash mdash 1399DEBBO [36] 1214208948 mdash mdash 12HHS [37] 1214155920 1216158544 mdash 1639IPSO [2] 1214128660 1215095223 1215468420 4289NAPSO [8] 1214125700 mdash mdash 127CSA [38] 1214125355 1215204106 1218102538 303Proposed PSO 1214125355 1214323215 1215643454 999

Table 5 Comparison results for case study 3

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel Cost($hr)

CPU time(s)

PSO [8] 1248758523 1251627011 mdash mdashFAPSO [8] 1222613706 1224710751 1225975196 196NAPSO [8] 1214910662 1214912756 1214915261 127CSA [38] 1214877727 1216113170 1221629295 147Proposed PSO 1214877718 1215113114 1217537157 84

8 Advances in Electrical Engineering

on the most popular test generating system taken from [23]This system consists of 40 thermal units with nonconvexity incost function due to valve-point loading effectsThe expectedpower demand for this test system is 10500MW The resultsobtained after 100 independent trials of the proposed PSOare presented and compared with a variety of other availableexisting deterministic and population based or their hybridtechniques in Table 4 The table validates the effectivenessof the proposed PSO as it generates either comparable orbetter best fuel cost than other several established techniquesincluding hybrid techniques The table also shows that theproposed PSO is less computationally demanding than manyother references including some latest ones Although NPSO[12] and CSA [38] demand less CPU time than the proposedPSO but the proposedmethod is capable of generating betterquality solutionThus the proposed PSO is promising to solvenonconvex ED problems The optimal dispatch of thermalgenerators obtained by the proposed PSO can be referred toin Appendix

43 Case Study 3 40-Generator System with Valve-Point andPOZs Finally the effectiveness of the proposed method isinvestigated on the 40 generators test generating systemwith discontinuities in the cost function due to prohibitedoperating zones The units 10ndash14 have POZs as given in[8] (POZ 2) The expected power demand for this testsystem is 10500 MW The results obtained after 100 trials ofthe proposed PSO are presented and compared with otheravailable existing population based techniques in Table 5Thetable shows that the proposed PSO is capable of generatingcomparable or better result in less computational time thanother established available methods The better value ofaverage fuel cost is obtained by proposed method than othermethods This shows robustness of the proposed PSO Thusthe high dimensional nonconvex discrete ED problems canbe effectively and efficiently solved using the proposed PSOThe optimal dispatch of thermal generators obtained by theproposed PSO can be referred to in Appendix

5 Discussion

In order to appreciate and understand the performance ofthe proposed method a comparison of cognitive and socialbehavior of particle in PSO and the proposed PSO is shown inFigures 5 and 6 respectively Figure 5 shows that in the classi-cal PSO the cognitive and social behaviors of particle velocityvary randomly throughout the computational process withinlimits of 0 to 2 The proposed constriction functions usedto guide the cognitive and social behaviors of the swarm areallowed to vary exponentially as shown in Figure 6The lowerand upper limits of these behaviors are governed by (10)However the sum of the best and poor cognitive behavior ofthe swarm remains constant during the computation processThis plays an important role in providing sufficient diversityby the poor experience during the whole flight of the swarmIt can be seen from Figure 6 that using proposed PSOthe modulations of cognitive (best) cognitive (poor) and

social behaviors though randomly distributed are dynami-cally controlled within exponential bounds of 15 05 and015 respectively This constitutes a marked difference withother versions of existing PSO Thus the particles experienceentirely different cognitive and social behaviors during theirflights and need no additional mechanism to bind theirvelocities

Any stochastic based search technique must be designedto accomplish global exploration and tends to facilitate localexploitation In order to investigate the effectiveness of eachof these modifications a set of convergence characteristicsfor the best and average fuel cost obtained during a sampletrial for 40 generators system is shown in Figures 7 and8 respectively In Figure 7 the characteristic ldquoardquo is for theconventional PSO ldquobrdquo refers to ldquoardquo with sinusoidal mod-ulation in inertia weight ldquocrdquo refers to ldquobrdquo with improvedcognitive behavior due to poor experience and ldquodrdquo refers tothe proposed PSO It can be observed from the figure that theperformance of the PSO is somewhat improved when inertiaweight is sinusoidally modulated and is further improvedwith a good margin when poor experience of particles is alsoconsidered However these two modifications do not seemto be sufficient to exploit the promising region effectively andefficiently This leads to premature convergence due to localtrappings which can be depicted from ldquodrdquo In d the proposedconstriction functions regulate particlesrsquo velocities so thatthey can fly more comprehensively in the search space Infact due to higher initial cognitive component than the socialcomponent the proposed PSO becomes more competentto explore wider search space during the initial phase andthus identify the promising region in about 1000 iterationsHowever particles move with strong communication andthus intensively exploit the region near the global optimaduring later part of the search owing to high values ofsocial component Finally all particles converge towards theglobal minima as can be observed from Figure 8 Thus theproposed PSOprovides better exploration and exploitation ofthe search space and produces better quality solutions Theseresults also highlight that the modifications suggested in thecontrol equation of the classical PSO are very effective as itmakes the proposed PSO perform much better

The proposed method offers better exploration andexploitation of the search space because the velocity ofparticles is regulated throughout their flight The movementof a sample particle in the classical PSOand the proposedPSOis illustrated in Figures 9 and 10 respectively These figuresshow the traces of initial cognitive and social componentsof particlersquos velocity and also the overall velocity imparted toit during a sample trial

The classical PSO searches for about 400 iterations asshown in Figure 9 After this all the three components ofparticlersquos velocity became insignificant and thus the particlegets trapped into local minima Figure 10(b) shows thecognitive component for the best experience which is thensuperimposed by its poor experience as in Figure 10(c) toobtain the overall cognitive component as in Figure 10(d) Itcan be concluded from Figure 10(d) that the poor experienceis contributing to tune the cognitive behavior of the swarmThe social component as shown in Figure 10(e) is providing

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Electrical Engineering 7

Table 2 Various parameters for classical PSO and proposed PSO

Method 119882min119882max 1198621119887

1198621119901

1198622

1205831

1205832

itrmax Population sizeClassical PSO 0109 2 mdash 2 mdash mdash 2500 100Proposed PSO 0109 15 05 2 minus55 4 2500 100

Table 3 Comparison results for case study 1

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

Total power(MW)

Power loss(MW)

CPU time(s)

GA [27] 2463242 2487493 2518859 255987 3987 225DE [27] 2481932 2521764 2565640 256234 4234 258HDE [27] 2459176 2473953 2507490 255916 3916 357STHDE [27] 2456008 2470663 2487244 256433 4433 298ICA-PSO [7] 2454006 2456146 2458945 255905 3905 215SDE [1] 2451488 2451631 mdash 256043 4043 mdashProposed PSO 2451446 2451458 2451526 255807 3807 296

Table 4 Comparison results for case study 2

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel cost($hr)

CPU time(s)

SQP [28] 1229044243 1248837692 1265852290 1080EP-SQP [29] 1223239700 1223796300 mdash 99773PSO-SQP [29] 1220946700 1222452500 mdash 73397PSO-LRS [12] 1220357946 1233820000 1257406300 3161NPSO [12] 1217047391 1222213697 1229950976 823NPSO-LRS [12] 1216644308 1229815913 1222093185 2074DEC-SQP [30] 1217419800 1233676500 1253979600 92563DEC(2)-SQP(1) [28] 1217419793 1222951278 1228392941 1426ACO [31] 1215324100 1216064500 1216796400 5245FCASO [32] 1215164700 1220825900 mdash 1452SOH-PSO [14] 1215011400 1218535700 1224463000 mdashTSARGA [33] 1214630700 1229283100 1242965400 6960CPSO-SQP [34] 1214585400 1220281600 mdashGA-PS-SQP [29] 1214580000 1220390000 mdash 4698ABC [35] 1214410300 1219958200 mdash 3002CCPSO [22] 1214125362 1214453269 1215254934 193ICA-PSO [7] 1214221000 mdash mdash 1399DEBBO [36] 1214208948 mdash mdash 12HHS [37] 1214155920 1216158544 mdash 1639IPSO [2] 1214128660 1215095223 1215468420 4289NAPSO [8] 1214125700 mdash mdash 127CSA [38] 1214125355 1215204106 1218102538 303Proposed PSO 1214125355 1214323215 1215643454 999

Table 5 Comparison results for case study 3

Method Best fuel cost($hr)

Average fuel cost($hr)

Worst fuel Cost($hr)

CPU time(s)

PSO [8] 1248758523 1251627011 mdash mdashFAPSO [8] 1222613706 1224710751 1225975196 196NAPSO [8] 1214910662 1214912756 1214915261 127CSA [38] 1214877727 1216113170 1221629295 147Proposed PSO 1214877718 1215113114 1217537157 84

8 Advances in Electrical Engineering

on the most popular test generating system taken from [23]This system consists of 40 thermal units with nonconvexity incost function due to valve-point loading effectsThe expectedpower demand for this test system is 10500MW The resultsobtained after 100 independent trials of the proposed PSOare presented and compared with a variety of other availableexisting deterministic and population based or their hybridtechniques in Table 4 The table validates the effectivenessof the proposed PSO as it generates either comparable orbetter best fuel cost than other several established techniquesincluding hybrid techniques The table also shows that theproposed PSO is less computationally demanding than manyother references including some latest ones Although NPSO[12] and CSA [38] demand less CPU time than the proposedPSO but the proposedmethod is capable of generating betterquality solutionThus the proposed PSO is promising to solvenonconvex ED problems The optimal dispatch of thermalgenerators obtained by the proposed PSO can be referred toin Appendix

43 Case Study 3 40-Generator System with Valve-Point andPOZs Finally the effectiveness of the proposed method isinvestigated on the 40 generators test generating systemwith discontinuities in the cost function due to prohibitedoperating zones The units 10ndash14 have POZs as given in[8] (POZ 2) The expected power demand for this testsystem is 10500 MW The results obtained after 100 trials ofthe proposed PSO are presented and compared with otheravailable existing population based techniques in Table 5Thetable shows that the proposed PSO is capable of generatingcomparable or better result in less computational time thanother established available methods The better value ofaverage fuel cost is obtained by proposed method than othermethods This shows robustness of the proposed PSO Thusthe high dimensional nonconvex discrete ED problems canbe effectively and efficiently solved using the proposed PSOThe optimal dispatch of thermal generators obtained by theproposed PSO can be referred to in Appendix

5 Discussion

In order to appreciate and understand the performance ofthe proposed method a comparison of cognitive and socialbehavior of particle in PSO and the proposed PSO is shown inFigures 5 and 6 respectively Figure 5 shows that in the classi-cal PSO the cognitive and social behaviors of particle velocityvary randomly throughout the computational process withinlimits of 0 to 2 The proposed constriction functions usedto guide the cognitive and social behaviors of the swarm areallowed to vary exponentially as shown in Figure 6The lowerand upper limits of these behaviors are governed by (10)However the sum of the best and poor cognitive behavior ofthe swarm remains constant during the computation processThis plays an important role in providing sufficient diversityby the poor experience during the whole flight of the swarmIt can be seen from Figure 6 that using proposed PSOthe modulations of cognitive (best) cognitive (poor) and

social behaviors though randomly distributed are dynami-cally controlled within exponential bounds of 15 05 and015 respectively This constitutes a marked difference withother versions of existing PSO Thus the particles experienceentirely different cognitive and social behaviors during theirflights and need no additional mechanism to bind theirvelocities

Any stochastic based search technique must be designedto accomplish global exploration and tends to facilitate localexploitation In order to investigate the effectiveness of eachof these modifications a set of convergence characteristicsfor the best and average fuel cost obtained during a sampletrial for 40 generators system is shown in Figures 7 and8 respectively In Figure 7 the characteristic ldquoardquo is for theconventional PSO ldquobrdquo refers to ldquoardquo with sinusoidal mod-ulation in inertia weight ldquocrdquo refers to ldquobrdquo with improvedcognitive behavior due to poor experience and ldquodrdquo refers tothe proposed PSO It can be observed from the figure that theperformance of the PSO is somewhat improved when inertiaweight is sinusoidally modulated and is further improvedwith a good margin when poor experience of particles is alsoconsidered However these two modifications do not seemto be sufficient to exploit the promising region effectively andefficiently This leads to premature convergence due to localtrappings which can be depicted from ldquodrdquo In d the proposedconstriction functions regulate particlesrsquo velocities so thatthey can fly more comprehensively in the search space Infact due to higher initial cognitive component than the socialcomponent the proposed PSO becomes more competentto explore wider search space during the initial phase andthus identify the promising region in about 1000 iterationsHowever particles move with strong communication andthus intensively exploit the region near the global optimaduring later part of the search owing to high values ofsocial component Finally all particles converge towards theglobal minima as can be observed from Figure 8 Thus theproposed PSOprovides better exploration and exploitation ofthe search space and produces better quality solutions Theseresults also highlight that the modifications suggested in thecontrol equation of the classical PSO are very effective as itmakes the proposed PSO perform much better

The proposed method offers better exploration andexploitation of the search space because the velocity ofparticles is regulated throughout their flight The movementof a sample particle in the classical PSOand the proposedPSOis illustrated in Figures 9 and 10 respectively These figuresshow the traces of initial cognitive and social componentsof particlersquos velocity and also the overall velocity imparted toit during a sample trial

The classical PSO searches for about 400 iterations asshown in Figure 9 After this all the three components ofparticlersquos velocity became insignificant and thus the particlegets trapped into local minima Figure 10(b) shows thecognitive component for the best experience which is thensuperimposed by its poor experience as in Figure 10(c) toobtain the overall cognitive component as in Figure 10(d) Itcan be concluded from Figure 10(d) that the poor experienceis contributing to tune the cognitive behavior of the swarmThe social component as shown in Figure 10(e) is providing

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Advances in Electrical Engineering

on the most popular test generating system taken from [23]This system consists of 40 thermal units with nonconvexity incost function due to valve-point loading effectsThe expectedpower demand for this test system is 10500MW The resultsobtained after 100 independent trials of the proposed PSOare presented and compared with a variety of other availableexisting deterministic and population based or their hybridtechniques in Table 4 The table validates the effectivenessof the proposed PSO as it generates either comparable orbetter best fuel cost than other several established techniquesincluding hybrid techniques The table also shows that theproposed PSO is less computationally demanding than manyother references including some latest ones Although NPSO[12] and CSA [38] demand less CPU time than the proposedPSO but the proposedmethod is capable of generating betterquality solutionThus the proposed PSO is promising to solvenonconvex ED problems The optimal dispatch of thermalgenerators obtained by the proposed PSO can be referred toin Appendix

43 Case Study 3 40-Generator System with Valve-Point andPOZs Finally the effectiveness of the proposed method isinvestigated on the 40 generators test generating systemwith discontinuities in the cost function due to prohibitedoperating zones The units 10ndash14 have POZs as given in[8] (POZ 2) The expected power demand for this testsystem is 10500 MW The results obtained after 100 trials ofthe proposed PSO are presented and compared with otheravailable existing population based techniques in Table 5Thetable shows that the proposed PSO is capable of generatingcomparable or better result in less computational time thanother established available methods The better value ofaverage fuel cost is obtained by proposed method than othermethods This shows robustness of the proposed PSO Thusthe high dimensional nonconvex discrete ED problems canbe effectively and efficiently solved using the proposed PSOThe optimal dispatch of thermal generators obtained by theproposed PSO can be referred to in Appendix

5 Discussion

In order to appreciate and understand the performance ofthe proposed method a comparison of cognitive and socialbehavior of particle in PSO and the proposed PSO is shown inFigures 5 and 6 respectively Figure 5 shows that in the classi-cal PSO the cognitive and social behaviors of particle velocityvary randomly throughout the computational process withinlimits of 0 to 2 The proposed constriction functions usedto guide the cognitive and social behaviors of the swarm areallowed to vary exponentially as shown in Figure 6The lowerand upper limits of these behaviors are governed by (10)However the sum of the best and poor cognitive behavior ofthe swarm remains constant during the computation processThis plays an important role in providing sufficient diversityby the poor experience during the whole flight of the swarmIt can be seen from Figure 6 that using proposed PSOthe modulations of cognitive (best) cognitive (poor) and

social behaviors though randomly distributed are dynami-cally controlled within exponential bounds of 15 05 and015 respectively This constitutes a marked difference withother versions of existing PSO Thus the particles experienceentirely different cognitive and social behaviors during theirflights and need no additional mechanism to bind theirvelocities

Any stochastic based search technique must be designedto accomplish global exploration and tends to facilitate localexploitation In order to investigate the effectiveness of eachof these modifications a set of convergence characteristicsfor the best and average fuel cost obtained during a sampletrial for 40 generators system is shown in Figures 7 and8 respectively In Figure 7 the characteristic ldquoardquo is for theconventional PSO ldquobrdquo refers to ldquoardquo with sinusoidal mod-ulation in inertia weight ldquocrdquo refers to ldquobrdquo with improvedcognitive behavior due to poor experience and ldquodrdquo refers tothe proposed PSO It can be observed from the figure that theperformance of the PSO is somewhat improved when inertiaweight is sinusoidally modulated and is further improvedwith a good margin when poor experience of particles is alsoconsidered However these two modifications do not seemto be sufficient to exploit the promising region effectively andefficiently This leads to premature convergence due to localtrappings which can be depicted from ldquodrdquo In d the proposedconstriction functions regulate particlesrsquo velocities so thatthey can fly more comprehensively in the search space Infact due to higher initial cognitive component than the socialcomponent the proposed PSO becomes more competentto explore wider search space during the initial phase andthus identify the promising region in about 1000 iterationsHowever particles move with strong communication andthus intensively exploit the region near the global optimaduring later part of the search owing to high values ofsocial component Finally all particles converge towards theglobal minima as can be observed from Figure 8 Thus theproposed PSOprovides better exploration and exploitation ofthe search space and produces better quality solutions Theseresults also highlight that the modifications suggested in thecontrol equation of the classical PSO are very effective as itmakes the proposed PSO perform much better

The proposed method offers better exploration andexploitation of the search space because the velocity ofparticles is regulated throughout their flight The movementof a sample particle in the classical PSOand the proposedPSOis illustrated in Figures 9 and 10 respectively These figuresshow the traces of initial cognitive and social componentsof particlersquos velocity and also the overall velocity imparted toit during a sample trial

The classical PSO searches for about 400 iterations asshown in Figure 9 After this all the three components ofparticlersquos velocity became insignificant and thus the particlegets trapped into local minima Figure 10(b) shows thecognitive component for the best experience which is thensuperimposed by its poor experience as in Figure 10(c) toobtain the overall cognitive component as in Figure 10(d) Itcan be concluded from Figure 10(d) that the poor experienceis contributing to tune the cognitive behavior of the swarmThe social component as shown in Figure 10(e) is providing

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Electrical Engineering 9

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive behavior

(a)

0

05

1

15

2

1 101 201 301 401 501 601 701 801 901Iteration count

Social behavior

(b)

Figure 5 (a) Cognitive behavior and (b) social behavior in classicalPSO

120

140

160

180

110

0112

0114

0116

0118

0120

0122

0124

01

Iteration count

Cognitive behaviour (best experience)

minus04

01

06

11

16

(a)

0010203040506

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive behaviour (poor experience)

(b)

0

005

01

015

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social behaviour

(c)

Figure 6 (a) Cognitive behavior (best experience) (b) cognitivebehavior (poor experience) and (c) social behavior in proposedPSO

121400

121800

122200

122600

123000

123400

1 501 1001 1501 2001Iteration count

ab

cd

Best

fuel

cost

($h

r)

Figure 7 Effect on the convergence for best fuel cost by suggestedmodifications in the proposed PSO

121400

121900

122400

122900

123400

123900

1 501 1001 1501 2001Iteration count

Aver

age f

uel c

ost (

$hr

)

ab

cd

Figure 8 Effect on the convergence for average fuel cost bysuggested modifications in the proposed PSO

fine tuning as desired in high dimensional optimizationproblem It should be noted that the social component hasbeen kept quite weak in this work as compared to otherpublished literature till date and is one of the keys to obtainhigh quality solutions In addition the proposed modulationin inertia weight intends particles for better explorationand exploitation of the search space by imparting suitablevelocity during the flight as seen from Figure 10(a) Theimpact of improved initial cognitive and social componentsof particlersquos velocity is shown in Figure 10(f) The figureshows a marked improvement in particle movement duringthe whole computation while compared with Figure 9(d)In the proposed PSO during early part of the search theparticles widely travelled in the search space yet their velocityis regulated by the poor experience as the social componentis almost negligible This facilitates the swarm to explore theregion of global optima However in later part of the searchboth poor and the social components are driving the swarmtoward the global optima as the cognitive best experiencehas been made quite weak during this part of the search

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Advances in Electrical Engineering

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Initial component

minus5

minus10

minus15

minus20

(a)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Cognitive component

minus5

minus10

minus15

(b)

05

101520

1 101 201 301 401 501 601 701 801 901Iteration count

Social component

minus5minus10minus15minus20minus25

(c)

05

1015

1 101 201 301 401 501 601 701 801 901Iteration count

Particle velocity

minus5

minus10

minus15

minus20

(d)

Figure 9 Particle velocity and its components in PSO

0102030405060708090

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Initial component

minus10

(a)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (best experience)

minus5minus10minus15

(b)

02

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Cognitive component (poor experience)

minus2minus4minus6minus8minus10minus12

(c)

05

10152025

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Overall cognitive component

minus5minus10minus15

(d)

0005

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Social component

minus005

minus01

minus015

minus02

minus025

(e)

020406080

100

1

201

401

601

801

1001

1201

1401

1601

1801

2001

2201

2401

Iteration count

Particle velocity

(f)

Figure 10 Particle velocity and its components in the proposed PSO

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Electrical Engineering 11

This improves exploitation potential of the PSO for localsearch Thus the proposed PSO provides better explorationand exploitation of the search space and thus produces betterquality solutions than the classical PSO or other existingstochastic based methods

6 Conclusions

The economic dispatch is a highly complex combinatorialconstrained optimization problem with continuous decisionvariables The classical PSO has proven potential to solvesuch hard combinatorial constraints optimization problembut it usually gets trapped into local minima while dealingwith high dimensional ED problems This paper presentsa modified version of PSO to make it suitable for solvinghighly complex EDproblemsTheproposedmethod has beentested to solve ED problems of three different test systems ofdifferent dimensions with a variety of operational and net-work constraints The application results are also comparedwith available existing PSO methods The application resultsshow that the proposed method is efficient and is usuallynot trapped in local minima The comparison shows thatproposed method is capable of giving better results than theexisting PSO and other stochastic based methods This maybe due to the fact that proposed PSO essentially aims toregulate particle velocity during its whole course of flight insuch a fashion so as to enhance exploration and exploitationpotentials of the PSO The operators in the proposed PSOare made to vary dynamically by introducing new truncatedsinusoidal and exponential functions The concept of poorparticle is introduced to improve the cognitive behavior of theswarm and also maintain a good balance between cognitiveand social behavior of the swarm during the whole course ofthe flightThesemodifications guide the swarm to identify thearea where the global optima may exist Thereafter particleshave suitable velocities to wandering within in this area toexplore global or near global solution Further it has beenobserved that in the proposed PSO the particle is acceleratedmore comprehensively during whole of its flight than in theclassical PSO This causes better exploration of the searchspace during the early part and better exploitation during thelater part of the search It is noteworthy that the proposedPSO is free from any mechanism to avoid local trapping anddoes not require any empirical formula to bound particlersquosvelocity Moreover the proposed algorithm is robust as itgenerates better quality solutions irrespective of the initialposition of the particles The proposed PSO can be extendedto solve ED problems with the inclusion of more objectivesand constraints like environmental issues reserve capacitynetwork security network congestion management and soforth

Appendix

See Table 6

Table 6 Optimal generating schedule for case studies 1 2 and 3

Unit Case study 1 Case study 2 Case study 3Power (MW) Power (MW) Power (MW)

1 6283185 110799825 1107997892 2988000 110799825 1107998073 2988000 973999130 9739980804 1597400 179733100 1797330935 1597400 877999050 8779982506 1597400 140000000 1400000007 1597400 259599650 2595996008 1597300 284599650 2845994969 1597400 284599650 28459970010 7620000 130000000 13000000011 1133200 940000000 16879814012 9210000 940000000 16804141913 9210000 214759790 12500000014 mdash 394279370 40000000015 mdash 394279370 39427901816 mdash 394279370 39427920517 mdash 489279370 48927939718 mdash 489279370 48927938019 mdash 511279370 51127937720 mdash 511279370 51127929921 mdash 523279370 52327935422 mdash 523279370 52327937323 mdash 523279370 52327937224 mdash 523279370 52327936525 mdash 523279369 52327937726 mdash 523279370 52327940027 mdash 100000000 10000000028 mdash 100000000 10000000029 mdash 100000000 10000000030 mdash 87799902 87799891031 mdash 190000000 19000000032 mdash 190000000 19000000033 mdash 190000000 19000000034 mdash 164799825 16479976635 mdash 194397782 16479980036 mdash 200000000 16479980337 mdash 110000000 11000000038 mdash 110000000 11000000039 mdash 110000000 10999879840 mdash 511279370 511279348

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the editor and reviewers fortheir valuable comments and recommendations

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Advances in Electrical Engineering

References

[1] A Srinivasa Reddy and K Vaisakh ldquoShuffled differential evolu-tion for large scale economic dispatchrdquo Electric Power SystemsResearch vol 96 pp 237ndash245 2013

[2] A K Barisal ldquoDynamic search space squeezing strategy basedintelligent algorithm solutions to economic dispatch with mul-tiple fuelsrdquo International Journal of Electrical Power amp EnergySystems vol 45 no 1 pp 50ndash59 2013

[3] J Kennedy and R Eberhart Swarm Intelligence Morgan Kauf-mann 2001

[4] D N Jeyakumar T Jayabarathi and T Raghunathan ldquoParticleswarm optimization for various types of economic dispatchproblemsrdquo International Journal of Electrical Power and EnergySystems vol 28 no 1 pp 36ndash42 2006

[5] A Mahor V Prasad and S Rangnekar ldquoEconomic dispatchusing particle swarm optimization a reviewrdquo Renewable andSustainable Energy Reviews vol 13 no 8 pp 2134ndash2141 2009

[6] A Safari and H Shayeghi ldquoIteration particle swarm opti-mization procedure for economic load dispatch with generatorconstraintsrdquo Expert Systems with Applications vol 38 no 5 pp6043ndash6048 2011

[7] J G Vlachogiannis and K Y Lee ldquoEconomic load dispatchmdasha comparative study on heuristic optimization techniques withan improved coordinated aggregation-based PSOrdquo IEEE Trans-actions on Power Systems vol 24 no 2 pp 991ndash1001 2009

[8] T Niknam H DMojarrad andH ZMeymand ldquoNon-smootheconomic dispatch computation by fuzzy and self adaptiveparticle swarm optimizationrdquo Applied Soft Computing Journalvol 11 no 2 pp 2805ndash2817 2011

[9] B Yu X Yuan and J Wang ldquoShort-term hydro-thermalscheduling using particle swarm optimization methodrdquo EnergyConversion andManagement vol 48 no 7 pp 1902ndash1908 2007

[10] G Baskar and M R Mohan ldquoSecurity constrained economicload dispatch using improved particle swarm optimizationsuitable for utility systemrdquo International Journal of ElectricalPower and Energy Systems vol 30 no 10 pp 609ndash613 2008

[11] L Wang and C Singh ldquoStochastic economic emission loaddispatch through a modified particle swarm optimization algo-rithmrdquo Electric Power Systems Research vol 78 no 8 pp 1466ndash1476 2008

[12] A I Selvakumar and K Thanushkodi ldquoA new particle swarmoptimization solution to nonconvex economic dispatch prob-lemsrdquo IEEE Transactions on Power Systems vol 22 no 1 pp42ndash51 2007

[13] R Roy and S P Ghoshal ldquoA novel crazy swarm optimizedeconomic load dispatch for various types of cost functionsrdquoInternational Journal of Electrical Power amp Energy Systems vol30 no 4 pp 242ndash253 2008

[14] K T Chaturvedi M Pandit and L Srivastava ldquoSelf-organizinghierarchical particle swarm optimization for nonconvex eco-nomic dispatchrdquo IEEE Transactions on Power Systems vol 23no 3 pp 1079ndash1087 2008

[15] K T Chaturvedi M Pandit and L Srivastava ldquoParticle swarmoptimization with time varying acceleration coefficients fornon-convex economic power dispatchrdquo International Journal ofElectrical Power and Energy Systems vol 31 no 6 pp 249ndash2572009

[16] K K Mandal and N Chakraborty ldquoDaily combined economicemission scheduling of hydrothermal systems with cascadedreservoirs using self organizing hierarchical particle swarm

optimization techniquerdquo Expert Systems with Applications vol39 no 3 pp 3438ndash3445 2012

[17] Y Wang J Zhou C Zhou Y Wang H Qin and Y LuldquoAn improved self-adaptive PSO technique for short-termhydrothermal schedulingrdquo Expert Systems with Applicationsvol 39 no 3 pp 2288ndash2295 2012

[18] B Mohammadi-Ivatloo ldquoCombined heat and power economicdispatch problem solution using particle swarm optimizationwith time varying acceleration coefficientsrdquo Electric PowerSystems Research vol 95 pp 9ndash18 2013

[19] L D S Coelho and C-S Lee ldquoSolving economic load dispatchproblems in power systems using chaotic and Gaussian particleswarm optimization approachesrdquo International Journal of Elec-trical Power andEnergy Systems vol 30 no 5 pp 297ndash307 2008

[20] A I Selvakumar and K Thanushkodi ldquoOptimization usingcivilized swarm solution to economic dispatch with multipleminimardquo Electric Power Systems Research vol 79 no 1 pp 8ndash16 2009

[21] J Cai X Ma L Li and P Haipeng ldquoChaotic particle swarmoptimization for economic dispatch considering the generatorconstraintsrdquo Energy Conversion andManagement vol 48 no 2pp 645ndash653 2007

[22] J-B Park Y-W Jeong J-R Shin and K Y Lee ldquoAn improvedparticle swarm optimization for nonconvex economic dispatchproblemsrdquo IEEE Transactions on Power Systems vol 25 no 1pp 156ndash166 2010

[23] N Sinha R Chakrabarti and P K Chattopadhyay ldquoEvolution-ary programming techniques for economic load dispatchrdquo IEEETransactions on Evolutionary Computation vol 7 no 1 pp 83ndash94 2003

[24] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks (ICNN rsquo95) pp 1942ndash1948 December 1995

[25] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the Congress on EvolutionaryComputation (CEC rsquo99) pp 1945ndash1950 Piscataway NJ USAJuly 1999

[26] Z-L Gaing ldquoParticle swarm optimization to solving the eco-nomic dispatch considering the generator constraintsrdquo IEEETransactions on Power Systems vol 18 no 3 pp 1187ndash1195 2003

[27] S K Wang J P Chiou and C W Liu ldquoNon-smoothnon-convex economic dispatch by a novel hybrid differential evolu-tion algorithmrdquo IET Generation Transmission and Distributionvol 1 no 5 pp 793ndash803 2007

[28] L dos Santos Coelho and V C Mariani ldquoCombining ofchaotic differential evolution and quadratic programming foreconomic dispatch optimization with valve-point effectrdquo IEEETransactions on Power Systems vol 21 no 2 pp 989ndash996 2006

[29] J S Alsumait J K Sykulski and A K Al-Othman ldquoAhybrid GA-PS-SQP method to solve power system valve-pointeconomic dispatch problemsrdquo Applied Energy vol 87 no 5 pp1773ndash1781 2010

[30] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof Self-adaptive real-coded genetic algorithm using Taguchimethod for Economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[31] S Pothiya I Ngamroo and W Kongprawechnon ldquoAnt colonyoptimisation for economic dispatch problem with non-smoothcost functionsrdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 478ndash487 2010

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Electrical Engineering 13

[32] J CaiQ Li L LiH Peng andYYang ldquoA fuzzy adaptive chaoticant swarm optimization for economic dispatchrdquo InternationalJournal of Electrical Power amp Energy Systems vol 34 no 1 pp154ndash160 2012

[33] P Subbaraj R Rengaraj and S Salivahanan ldquoEnhancementof self-adaptive real-coded genetic algorithm using Taguchimethod for economic dispatch problemrdquo Applied Soft Comput-ing Journal vol 11 no 1 pp 83ndash92 2011

[34] J Cai Q Li L Li H Peng and Y Yang ldquoA hybrid CPSO-SQPmethod for economic dispatch considering the valve-pointeffectsrdquo Energy Conversion and Management vol 53 no 1 pp175ndash181 2012

[35] S Hemamalini and S P Simon ldquoArtificial bee colony algorithmfor economic load dispatch problem with non-smooth costfunctionsrdquo Electric Power Components and Systems vol 38 no7 pp 786ndash803 2010

[36] A Bhattacharya and P K Chattopadhyay ldquoHybrid differentialevolutionwith biogeography-based optimization for solution ofeconomic load dispatchrdquo IEEE Transactions on Power Systemsvol 25 no 4 pp 1955ndash1964 2010

[37] V R Pandi B K Panigrahi R C Bansal S Das and AMohapatra ldquoEconomic load dispatch using hybrid swarmintelligence based harmony search algorithmrdquo Electric PowerComponents and Systems vol 39 no 8 pp 751ndash767 2011

[38] D N Vo P Schegner and W Ongsakul ldquoCuckoo searchalgorithm for non-convex economic dispatchrdquo IET GenerationTransmission and Distribution vol 7 no 6 pp 645ndash654 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of