pso

42 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 1, FEBRUARY 2007

A New Particle Swarm Optimization Solutionto Nonconvex Economic Dispatch Problems

A. Immanuel Selvakumar, Member, IEEE, and K. Thanushkodi

Abstract—This paper proposes a new version of the classical par-ticle swarm optimization (PSO), namely, new PSO (NPSO), to solvenonconvex economic dispatch problems. In the classical PSO, themovement of a particle is governed by three behaviors, namely, in-ertial, cognitive, and social. The cognitive behavior helps the par-ticle to remember its previously visited best position. This paperproposes a split-up in the cognitive behavior. That is, the particle ismade to remember its worst position also. This modification helpsto explore the search space very effectively. In order to well exploitthe promising solution region, a simple local random search (LRS)procedure is integrated with NPSO. The resultant NPSO-LRS al-gorithm is very effective in solving the nonconvex economic dis-patch problems. To validate the proposed NPSO-LRS method, itis applied to three test systems having nonconvex solution spaces,and better results are obtained when compared with previous ap-proaches.

Index Terms—Economic dispatch (ED), local search, nonconvexsolution space, particle swarm optimization (PSO).

I. INTRODUCTION

ECONOMIC dispatch (ED) is one of the important opti-mization problems in power systems that has the objec-

tive of dividing the power demand among the online generatorseconomically while satisfying various constraints [1]. Since thecost of the power generation is exorbitant, an optimum dispatchsaves a considerable amount of money. Traditional algorithmslike lambda iteration, base point participation factor, gradientmethod, and Newton method can solve the ED problems effec-tively if and only if the fuel-cost curves of the generating unitsare piece-wise linear and monotonically increasing [2].

The basic ED considers the power balance constraint apartfrom the generating capacity limits. However, a practical EDmust take ramp rate limits, prohibited operating zones, valve-point effects, and multifuel options into consideration to pro-vide the completeness for the ED formulation. The resulting EDis a nonconvex optimization problem, which is a challengingone and cannot be solved by the traditional methods. Dynamicprogramming (DP) [3] can solve such type of problems, but itsuffers from the curse of dimensionality.

This paper considers three types of nonconvex ED problems,namely, ED with prohibited operating zones (EDPO), ED with

Manuscript received February 28, 2006; revised August 30, 2006. Paper no.TPWRS-00115-2006.

A. Immanuel Selvakumar is with the Department of Electrical Sciences,Karunya Deemed University, Coimbatore 641 114, Tamil Nadu, India (e-mail:[email protected]).

K. Thanushkodi is with the Department of Electrical Engineering, Govern-ment College of Technology, Coimbatore 641 114, Tamil Nadu, India (e-mail:[email protected]).

Digital Object Identifier 10.1109/TPWRS.2006.889132

valve-point loading effects (EDVL), and ED with combinedvalve-point loading effects and multifuel options (EDVLMF).A considerable amount of work has been contributed to solvethe EDPO problem. The DP approach [3], decomposition tech-nique [4], advantageous decision spaces approach [5], geneticalgorithm (GA) [6], deterministic crowding genetic algorithm(DCGA)[7], artificial intelligence (AI) technique[8], and PSOapproach [9] are the important contributions to the solution ofEDPO.

The GA[10], GA combined with simulated annealing (SA)[11], evolutionary programming (EP)[12], improved Tabusearch (ITS)[13], improved fast EP (IFEP) [14], modified PSO(MPSO) with a dynamic search-space reduction strategy [15],and evolutionary strategy optimization (ESO) [16] are themodern heuristic techniques that have been used to solve theEDVL.

The solution methodology for ED with multifuel options(EDMF) has been developed by hierarchical method [17],neural networks [18], [19], and EP [20]. Recently, both EDVLand EDMF are combined and solved by an improved GA withmultiplier updating (IGA_MU) [21].

This paper introduces a new PSO (NPSO) and its solutionto the above-mentioned nonconvex ED problems. PSO is oneof the modern heuristic algorithms, which can be used to solvenonlinear and noncontinuous optimization problems [22]. It hasbeen used for many power system problems such as optimaldesign of power system stabilizers [23], distribution state es-timation [24], and optimal reactive power dispatch [25], [26]apart from ED. After the introduction of PSO, many variationshave been proposed for the basic PSO by various researchers[27]–[32].

In the classical PSO, three aspects, namely, inertial, cognitive,and social, govern the movement of a particle. The cognitivebehavior helps the particle to remember its previously visitedbest position. This paper proposes a split-up in the cognitivebehavior. That is, the particle is made to remember its worstposition also. This modification helps in exploring the searchspace very effectively to identify the promising solution region.Moreover, to exploit the promising region well, a simple localrandom search (LRS) procedure, which is a modification of a di-rect search procedure [33], is integrated with NPSO. The resul-tant NPSO-LRS algorithm is very effective in solving the non-convex ED problems.

To validate the proposed NPSO-LRS method, it is tested onthree test systems having nonconvex solution spaces. The re-sults of the proposed NPSO-LRS and those of the previous ap-proaches are compared. The outcome of the comparisons showsthe effectiveness of the proposed NPSO-LRS method in terms

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SELVAKUMAR AND THANUSHKODI: NEW PARTICLE SWARM OPTIMIZATION SOLUTION 43

of solution quality and consistency. For all the test systems con-sidered, NPSO-LRS achieves better results compared to the ex-isting results.

II. NONCONVEX ECONOMIC DISPATCH PROBLEMS

The basic ED becomes a nonconvex optimization problem ifthe practical operating conditions are included. In this paper,three different formulations of the ED problems, which reflectthe real-time operating conditions, are used.

A. EDPO

The objective is

(1)

where is the total generation cost ($/hr), is the fuel-costfunction of generator ($/hr), is the number of generators,

is the real power output of generator (MW), and ,and are the fuel-cost coefficients of generator . The basicconstraints are the real power balance and the real power oper-ating limits

(2)

(3)

where is the total load in the system (MW), and isthe network loss (MW) that can be calculated by matrix lossformula. and are the minimum and maximumpower generation limits of generator . The other important con-straints are as follows.

Generator Ramp Rate Limits: If the generator ramp ratelimits are considered, the effective real power operating limitsare modified as follows:

(4)

where is the previous operating point of generator ,and are the down and up ramp limits of the generator .

Prohibited Operating Zones: A generator with prohibited re-gions (zones) has discontinuous fuel-cost characteristics. Theconcept of prohibited operating zones is included as the fol-lowing constraint in the ED:

(5)

where and are the lower and upper boundariesof prohibited operating zone of generator in (MW), respec-tively; is the number of prohibited operating zones of gen-erator ; and is the number of generators with prohibitedoperating zones.

B. EDVL

The valve opening process of multivalve steam turbines pro-duces a ripple-like effect in the heat rate curve of the generators,and it is taken into consideration in the ED problem by super-imposing the basic quadratic fuel-cost characteristics with therectified sinusoidal component as follows:

(6)

where , and are the fuel-cost coefficients of gen-erator . The objective of EDVL is to minimize with theconstraints (2)–(4).

C. EDVLMF

For a power plant with generators and fuel options foreach unit, the cost function of the generator with valve-pointloading is expressed as

if

fuel option (7)

where and are the minimum and maximum powergeneration limits of generator with fuel option , respectively;and , and are the fuel-cost coefficients of gen-erator for fuel option . The objective is to minimize sub-ject to the constraints (2)–(4).

III. PROPOSED NEW PARTICLE SWARM OPTIMIZATION (NPSO)

PSO is a population-based, self-adaptive, stochastic op-timization technique [22]. The basic idea of the PSO is themathematical modeling and simulation of the food searchingactivities of a swarm of birds (particles). In the multidimen-sional space where the optimal solution is sought, each particlein the swarm is moved toward the optimal point by adding avelocity with its position. The velocity of a particle is influencedby three components, namely, inertial, cognitive, and social.The inertial component simulates the inertial behavior of thebird to fly in the previous direction. The cognitive componentmodels the memory of the bird about its previous best position,and the social component models the memory of the birdabout the best position among the particles (interaction insidethe swarm). The particles move around the multidimensionalsearch space until they find the food (optimal solution). Basedon the above discussion, the mathematical model for PSO is asfollows.

Velocity update equation is given by

(8)


Position update equation is given by

(9)

where

iteration count;

dimension of the velocity of particle at iteration;

dimension of the position of particle atiteration ;

inertia weight;

acceleration coefficients;

dimension of the own best position of particleuntil iteration ;

dimension of the best particle in the swarm atiteration ;

dimension of the optimization problem (numberof decision variables);

number of particles in the swarm;

two separately generated uniformly distributedrandom numbers in the range [0, 1].

This paper proposes a new variation in the classical PSO bysplitting the cognitive component of the classical PSO into twodifferent components. The first component can be called goodexperience component. That is, the bird has a memory about itspreviously visited best position. This component is exactly thesame as the cognitive component of the basic PSO. The secondcomponent is given the name bad experience component. Thebad experience component helps the particle to remember itspreviously visited worst position. To calculate the new velocity,the bad experience of the particle is also taken into considera-tion. This gives the new model of the PSO as below.

The new velocity update equation is given by

(10)

where

acceleration coefficient, which accelerates theparticle toward its best position;

acceleration coefficient, which accelerates theparticle away from its worst position;

dimension of the own worst position of particleuntil iteration ;

three separately generated uniformly distributedrandom numbers in the range [0, 1].

The positions are updated using (9). The inclusion of theworst experience component in the behavior of the particle givesadditional exploration capacity to the swarm. By using the badexperience component, the bird (particle) can bypass its pre-vious worst position and always try to occupy a better position.

IV. LOCAL RANDOM SEARCH (LRS)

The metaheuristic algorithms like GA, EP, SA, and PSO areperforming well for small dimensional and less complicatedproblems. However, they fail to locate global minima for thecomplex multiminima functions. Although they locate thepromising area, they fail to exploit the promising area to getgood quality solutions [6], [9], [11], [14], [15]. With a single al-gorithm, it is difficult to control and to strike a balance betweenexploration of whole search space to locate the promising areaand exploitation of the promising area to get global minima.Several hybrid methods have been proposed by combining themetaheuristics methods with simple local search algorithms.This paper uses a simple LRS procedure, which is a modifi-cation of a direct search technique proposed in [33]. The LRSprocedure is outlined below. The initial search point is taken as

, and the objective function value at is .Step 1) The initial local search range is selected around

as follows:

(11)

(12)

(13)

where and are the lower and upperboundaries of the local search region; is the localarea parameter; and are the vectorsof power generation limits; and is the initial localsearch range. The iteration count is set to 1.(best search point at the beginning of LRS) and(optimum search point) are set to .

Step 2) The local search points are randomly generatedas follows:

(14)

where is a random number vector of length, whose elements are randomly generated be-

tween and 1. If any local search point violatesthe limits, it is forced within the boundaries.

Step 3) For each local search point, the objective functionvalues are calculated. Then the minimum objectivefunction among all is taken as , and the corre-sponding is taken as . The optimum valuesare updated as follows:

If then and

Otherwise and


Step 4) The search range is reduced as

(15)

where is the range reduction parameter.Step 5) If maximum iteration for local search is

not reached, the iteration count is incremented byone and the above procedure is repeated from step2). Otherwise, and are taken as the op-timum results found by the LRS algorithm.

V. IMPLEMENTATION OF NPSO-LRS FOR EDPO,EDVL, AND EDVLMF PROBLEMS

A. Hybridization of NPSO With LRS

In a stochastic optimization algorithm like PSO, the solutionprocess moves through a random trajectory to locate the op-timum point. The trajectory may not be the same for each run.A robust algorithm should locate the optimum solution irrespec-tive of the starting point. Moreover, the stochastic optimizationalgorithms are good enough to locate the promising areas. If apromising area where the global optimum is residing is iden-tified at the end of the optimization process, the algorithm isable to exploit the promising area to get global optimum. On theother hand, if the promising area with global optimum is iden-tified at the earlier stages of the optimization, there is a possi-bility of missing that area without exploitation. Considering theabove fact, the LRS is employed whenever a promising area isobtained by NPSO. The resultant NPSO-LRS hybrid algorithmis robust in finding the global optimum for large dimensionalnonconvex functions.

B. Solution of EDPO, EDVL, and EDVLMF ProblemsWith NPSO-LRS

The main objective of EDPO, EDVL, and EDVLMF is toobtain the amount of real power to be generated by each ofthe committed generators while achieving minimum genera-tion cost within the constraints. This section provides the so-lution methodology for the three types of ED problems withNPSO-LRS.

Representation of the Swarm: Since the decision variablesfor the ED problems are real power generations, they are usedto form the swarm. The real power output of all generatorsis represented as the positions of the particles in the swarm.If there are generators, the particle position is representedas a vector of length . Again if there are particles inthe swarm, the complete swarm is represented as a matrix asfollows:

(16)

where is the position vector of the particle . It representsone of the possible solutions for the optimization problem. Theelement of is the th position component of particle ,and it represents the real power generation of generator of thepossible solution .

Initialization of the Swarm: Each element of the swarm ma-trix is initialized randomly within the effective real power op-erating limits. The initialization is based on (3) for generators

without ramp rate limits and based on (4) for generators withramp rate limits. The velocities of the particles are initialized asfollows:

(17)

where is a small positive number. This velocity initializationscheme always guarantees to produce new particles satisfyingreal power operating limit constraints [15].

Penalized Fuel Cost Function: The nonconvex ED problemsinvolve many constraints. Out of these constraints, limiting themovement of the particles imposes the effective real power op-erating limits. The real power balance and prohibited operatingzone constraints are handled by including penalty terms to theoriginal objective function as follows:

(18)

where is the penalized objective function, is thepenalty factor for real power balance constraint, is thepenalty factor for prohibited operating zone constraint, andis an indicator of falling into the prohibited operating zone. Thepenalty factors and are used to penalize the fuel costproportional to the amount of constraint violations. If there areno prohibited zones, is set to zero.

Initialization of the Best and Worst Positions: In the strategyof PSO, the particle’s best position and global bestposition are the key factors. The best position of aparticle is the position, which gives the minimum , and thebest position out of all the is taken as . In thispaper, the particle’s worst position is introduced. Atthe beginning of the iteration process, the andfor all the particles are taken as the same as the initial positions.The at is taken as .

Moving the Particles: The particles in the swarm are movedto new positions with the help of new velocities. The new veloci-ties are calculated using (10) and the position of the particles areupdated using (9) where is taken as . If any violatesthe effective real power operating limit constraints, its value istaken as the limiting value.

Updating the Best and Worst Positions: The particles areevaluated in the new positions by . Then and

of particle are updated as follows:

ifif

ifif

(19)


Fig. 1. Flowchart for NPSO-LRS.

where is the penalized objective function value of particleat iteration . The best position out of all the new is

taken as , and at is taken as .Employing LRS Procedure: If is better than ,

the LRS subroutine is invoked. The and for the LRSare taken as and , respectively. If obtainedfrom LRS is better than and are re-placed with and , respectively.

Stopping Criterion: There are different criteria available tostop a stochastic optimization algorithm. Tolerance, number offunction evaluations, and maximum number of iterations aresome examples. In this paper, in order to compare with previousresults, maximum number of iterations is taken as the stoppingcriterion. The overall NPSO-LRS optimization process is shownin Fig. 1.

VI. TEST RESULTS AND ANALYSIS

A. Description of the Test Systems

In order to validate the proposed NPSO-LRS, it is tested withthree test systems having nonconvex solution spaces. The firsttest system consists of six generators with prohibited operatingzones and has a total load of 1263 MW [9]. All the generatorsare having ramp rate limits. The network losses are calculatedby matrix loss formula. The best generation cost reporteduntil now is $15 450/h [9]. The previous best generation costwas $15 459/h [9].

The second test system consists of 40 generators with valve-point loading effects and has a total load of 10 500 MW [14].The system has many local minima, and the global minimumis very difficult to determine. The best generation cost reporteduntil now is $122 122.16 [16].

The third system consists of ten generators with multifuel op-tions and valve-point effects [21]. The first generator is havingtwo fuel options, and the remaining generators are having threefuel options. The best generation cost reported so for is $624.5178 [21].

B. Determination of Parameters for NPSO-LRS

To successfully implement the NPSO, the values of the pa-rameters , and have to be determined. The in-ertia weight is linearly decreased from 0.9 to 0.4 over theiterations, and the acceleration coefficient is taken as 2, sincethese settings are suitable for many power system problems [9],[24]. The number of particles is selected as 20. To find suitablevalues for and , the following procedure is used.

1) The , and are fixed at 0.9, 0.4, and 2, re-spectively.

2) is increased from 1.0 to 1.9 in steps 0.05, and isdecreased from 1.0 to 0.1 in steps 0.05. For each combi-nation of and , 100 independent trials have beenmade with 1000 iterations per trial.

The EDVL formulation of the 40-generator system is used fortesting. The minimum generation cost for this system reportedso far are $122 624.35 [14], $122 252.265 [15], and $122 122.16[16]. Hence, for each trial, the minimum generation cost is testedwhether it lies in the region below $122 500 or in the range be-tween $122 500 to $123 000. Based on the outcome of the ex-periments given in Table I, and are chosen as 1.6 and0.4 (case 13), respectively. They give the minimum generationcost more consistently, and the obtained generation cost is alsoless among the remaining cases.

To implement the LRS, the number of iterations for LRS, the number of local search points , local area

parameter , and the range reduction parameter are to bedetermined. Since has been taken as 0.05 for most of the prob-lems [33], it is fixed at 0.05. If the tolerance for range reductionis taken as 1%, the can be calculated as 90. To deter-mine the parameters and , the following experiment is per-formed on the 40-generator system. With the above-determinedNPSO parameters, is varied from 10 to 20 in steps of 5.For each , five different values (0.1 to 0.5 in steps 0.1) aretested. For each and combination, 100 independent trialshave been made with 1000 iterations per trial. As before, for


TABLE IINFLUENCE OF PARAMETERS ON NPSO PERFORMANCE

each trial, and are calculated. Based on the results,and are chosen as 10 and 0.4, respectively.

C. Testing Strategies

Since the proposed NPSO-LRS is the hybridization of NPSOand LRS, it is necessary to find the relative strength of eachconstituent. So, three different testing strategies are applied onthe EDPO, EDVL, and EDVLMF problems with a swarm of 20particles.

1) PSO-LRS: The classical PSO with standard parameters( , and ) is in-tegrated with LRS. This strategy is selected to analyze theperformance of LRS in PSO environment.

2) NPSO: The proposed NPSO is applied without LRS.3) NPSO-LRS: The proposed NPSO is integrated with LRS.The coding is written with MATLAB 6.5 programming lan-

guage and executed in the Pentium IV, 1.5-GHz, 128-MB RAMprocessor. In order to find the effectiveness and superiority ofthe NPSO-LRS algorithm, the test results are compared with theresults obtained by other algorithms available in the literature.

D. Convergence Test

The convergence test is carried out to determine the quicknessof the three PSO strategies in terms of the number of main PSOiterations. The three PSO strategies are tested with the first testsystem, and the result is shown in Fig. 2.

The NPSO and NPSO-LRS are almost similar in conver-gence and show their superiority over the PSO-LRS algorithm.The NPSO algorithm performs well due to the extra diversifi-cation provided by the worst experience component. However,NPSO-LRS is slightly better than NPSO due to the localsearching ability.

The results of convergence test on the 40-generator systemare shown in Fig. 3. For this system also, the NPSO-LRS isthe best performer. It is very fast when compared to the othertwo strategies in terms of main PSO iterations. The PSO-LRS is

Fig. 2. Comparative convergence behaviors of the three PSO strategies for six-generator system.

Fig. 3. Comparative convergence behaviors of the three PSO strategies for40-generator system.

slow in convergence when compared to NPSO and NPSO-LRS,which indicates that the combined strength of PSO and LRS isinferior to those of NPSO and NPSO-LRS.

Recently, the multiple fuel option and valve-point loading for-mulations of ED have been combined and solved by IGA_MU[21]. The IGA_MU could achieve quality solutions with con-siderable amount of speed compared with conventional geneticalgorithm with multiplier updating (CGA_MU) [21]. The sameEDVLMF problem is solved using the proposed PSO, NPSO,and NPSO-LRS strategies with 20 particles, and the conver-gence behaviors of the three PSO strategies are shown in Fig. 4.

Here also, NPSO-LRS is faster in convergence in terms ofmain PSO iterations. The behavior of NPSO falls betweenthose of PSO-LRS and NPSO-LRS. For all the three testcases, NPSO-LRS stands first in the performance ladder.The proposed worst experience component in the velocity


Fig. 4. Comparative convergence behaviors of the three PSO strategies for ten-generator system.

TABLE IIBEST POWER OUTPUT FOR SIX-GENERATOR SYSTEM

update equation and the local search procedure strengthen theNPSO-LRS.

E. Comparisons of the Best Solutions

The best power output of the six-generator system obtainedby the three PSO strategies are compared with those of GA[9] and PSO [9] in Table II. Except GA, all the other algo-rithms give the same minimum generation cost. However, thelosses obtained by PSO_LRS, NPSO, and NPSO-LRS algo-rithms are less when compared to the remaining methods. More-over, NPSO-LRS achieves best generation schedule with min-imum network loss in addition to minimum generation cost.

For the 40-generator system, the best power output thatresults from the three proposed PSO strategies are listed inTable III. The number of iteration is taken as 1000 to matchwith the previous analysis, and 20 particles are used. The com-parison of generation cost obtained by IFEP [14], MPSO [15],

TABLE IIIBEST POWER OUTPUT FOR 40-GENERATOR SYSTEM

ESO [16], and the three PSO strategies is given in Table IV.All the 40 generators are having valve-point effects, and thesolution space has multiple minima. The optimal generationcost is difficult to achieve, and the minimum generation costreported so far is $122 122.16 [16]. However, the three PSOstrategies have the ability to obtain lower generation cost whencompared to $122 122.16. Among the three PSO strategies, theNPSO-LRS algorithm provides minimum generation cost.

The best power output and the fuel options for the ten-gen-erator EDVLMF problem, obtained by different methods, aregiven in Table V. The three PSO strategies are able to obtainbetter results compared to IGA_MU[21]. However, NPSO-LRS


TABLE IVMINIMUM GENERATION COST OBTAINED BY DIFFERENT METHODS

TABLE VBEST SOLUTION FOR TEN-GENERATOR SYSTEM

TABLE VICOMPARISON AMONG DIFFERENT METHODS AFTER

50 TRIALS (SIX-GENERATOR SYSTEM)

proves its superiority among its competitors by providing min-imum generation cost.

F. Robustness Test

Owing to the randomness of the heuristic algorithms, theirperformance cannot be judged by the result of a single run.Many trials with different initializations should be made to ac-quire a useful conclusion about the performance of the algo-rithm. An algorithm is robust, if it gives consistent result duringall the trials.

The comparison of results after 50 independent trials with thefirst test system is shown in Table VI. From the results, the supe-riority of the PSO_LRS, NPSO, and NPSO-LRS strategies overGA [9] and PSO [9] can be noticed. Moreover, the maximumand average values obtained by NPSO-LRS are very close tothe minimum value, which proves that NPSO-LRS is more ro-bust.

For the second test system, 100 independent trials have beenmade. In order to compare the results in a statistical manner, the

TABLE VIIFREQUENCY OF CONVERGENCE FOR 40-GENERATOR SYSTEM

TABLE VIIICOMPARISON AMONG DIFFERENT METHODS AFTER

100 TRIALS (40-GENERATOR SYSTEM)

TABLE IXFREQUENCY OF CONVERGENCE FOR TEN-GENERATOR SYSTEM

frequencies of attainment of a cost within the specific ranges arepresented in Table VII. The minimum, maximum, and averagecosts of 100 independent trails are presented in Table VIII.

Tables VII and VIII reveal the consistency of NPSO-LRS inachieving minimum generation cost. NPSO also exhibits similarcharacteristics, but its average cost is slightly greater than the av-erage cost of NPSO-LRS. Table VII discloses that NPSO-LRShas the higher probability of attaining quality solution. NPSOstands next in performance.

Tables IX and X provide the result of robustness test after 100trials for the EDVLMF problem. Here also, the performances ofthe three PSO strategies are superior. They provide good qualitysolutions when compared to IGA_MU.

G. Computational Efficiency

In comparison to the basic PSO, the NPSO-LRS has two ad-ditional components, i.e., the bad experience component and theLRS procedure. These extra burdens necessitate the analysis of


TABLE XCOMPARISON AMONG DIFFERENT METHODS AFTER

100 TRIALS (TEN-GENERATOR SYSTEM)

TABLE XICPU TIME COMPARISON

the computational efficiency of the NPSO-LRS. The mean CPUtime taken to complete the fixed number of iterations andthe mean CPU time taken to converge into the lower solutionrange for 100 trials are shown in Table XI. Thehas been calculated only for succeeded trials.

For the 40-generator system, the basic PSO takes an averageCPU time of 4.63 s to complete 1000 iterations. Theincreases with the addition of worst experience component andthe LRS. The NPSO-LRS takes 12.1 s more than the NPSO tocomplete 1000 iterations. Nevertheless, NPSO-LRS takes only0.41 s more than the NPSO to converge into the lower solutionrange ($120 000–$122 500). Similarly, the NPSO-LRS takesonly 0.17 s more than the NPSO to converge into the lowersolution range ($623.5–$624.5) of the ten-generator EDVLMFproblem. For both the test cases, the basic PSO is not able toconverge into the lower solution range within the specified fixediterations, and the PSO-LRS converges into the lower solutionrange at the margin of the fixed iterations. It is observed that,even though the NPSO-LRS contains the burden of LRS, it hasa mean CPU time of converging into the lower solution range

closer to NPSO and has obtained quality solutionswith more robustness than NPSO. If the solution quality andthe robustness of NPSO-LRS are considered, a slight increasein due to the bad experience component and the LRSprocedure can be tolerated.

From all the findings, it is concluded that the three PSO strate-gies perform well for the entire test systems selected. The per-formance of NPSO is better than PSO-LRS due to the extradiversification capability provided by the proposed bad expe-rience component. Since NPSO-LRS has the strength of bothNPSO and LRS, it performs well among the three PSO strate-gies and outperforms the previous achievements. Hence, theNPSO-LRS is suggested as a powerful optimization tool fornonconvex ED problems.

VII. CONCLUSION

A NPSO approach is developed and integrated with an LRSprocedure to form a powerful optimization tool called NPSO-

LRS. The suitable parameters for NPSO-LRS are determinedby appropriate experiments. To prove the ability of the proposedNPSO-LRS in solving nonconvex optimization problems, EDproblems with nonconvex solution spaces are considered andsolved with three different PSO strategies (PSO-LRS, NPSO,and NPSO-LRS). The three strategies are tested for convergenceand robustness to find the relative strength of NPSO and LRS.With the aid of comparisons of the results obtained by the threePSO strategies and the results of earlier methods available in theliterature, it is proved that the proposed NPSO-LRS method isvery effective in giving quality solutions consistently for non-convex ED problems.

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A. Immanuel Selvakumar (M’03) was born inTamil Nadu, India, on March 19, 1974. He receivedthe B.E. and M.E. degrees in electrical engineeringform Thiagarajar College of Engineering, Madurai,India. Currently, he is pursuing the Ph.D. degree atAnna University (GCT Campus), Coimbatore, India.

His research topics include power system opera-tion and control.

K. Thanushkodi was born in Theni District, TamilNadu, India, in 1948. He received the B.E. degree inelectrical and electronics engineering and the M.Sc.(Engg) degree from Madras University, Chennai,India, in 1972 and 1976, respectively, and the Ph.D.degree in electrical and electronics engineering fromBharathiar University, Coimbatore, India, in 1991.

He is currently a Professor of electrical Eengi-neering at Anna University, Coimbatore. His researchinterests include computer modeling and simulation,computer networking, and power systems.

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