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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. NONCONVEXECONOMICDISPATCHPROBLEMS

The basic ED becomes a nonconvex optimization problem if

the practical operating conditions are included. In this paper,three different formulations of the ED problems, which reflect

the real-time operating conditions, are used.

A. EDPO

The objective is

(1)

where is the total generation cost ($/hr), is the fuel-cost

function 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 basic

constraints are the real power balance and the real power oper-

ating limits

(2)

(3)

where is the total load in the system (MW), and is

the network loss (MW) that can be calculated by matrix loss

formula. and are the minimum and maximum

power generation limits of generator . The other important con-straints are as follows.

Generator Ramp Rate Limits: If the generator ramp rate

limits are considered, the effective real power operating limits

are 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. The

concept of prohibited operating zones is included as the fol-

lowing constraint in the ED:

(5)

where and are the lower and upper boundaries

of 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 the

rectified sinusoidal component as follows:

(6)

where , and are the fuel-cost coef ficients of gen-

erator . The objective of EDVL is to minimize with the

constraints(2)(4).

C. EDVLMF

Fora power plant with generatorsand fueloptions for

each unit, the cost function of the generator with valve-point

loading is expressed as

if

fuel option (7)

where and are the minimum and maximum power

generation limits of generator with fuel option , respectively;

and , and are the fuel-cost coef ficients of gen-

erator for fuel option . The objective is to minimize sub-

ject to the constraints(2)(4).

III. PROPOSEDNEWPARTICLESWARMOPTIMIZATION(NPSO)

PSO is a population-based, self-adaptive, stochastic op-

timization technique [22]. The basic idea of the PSO is the

mathematical modeling and simulation of the food searching

activities of a swarm of birds (particles). In the multidimen-

sional space where the optimal solution is sought, each particle

in the swarm is moved toward the optimal point by adding a

velocity with its position. The velocity of a particle is influenced

by three components, namely, inertial, cognitive, and social.

The inertial component simulates the inertial behavior of the

bird tofly 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 bird

about the best position among the particles (interaction inside

the swarm). The particles move around the multidimensional

search space until theyfind the food (optimal solution). Based

on the above discussion, the mathematical model for PSO is as

follows.

Velocity update equation is given by

(8)

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

Position update equation is given by

(9)

where

iteration count;

dimension of the velocity of particle at iteration

;

dimension of the position of particle at

iteration ;

inertia weight;

acceleration coefficients;

dimension of the own best position of particle

until iteration ;

dimension of the best particle in the swarm at

iteration ;

dimension of the optimization problem (number

of decision variables);

number of particles in the swarm;

two separately generated uniformly distributed

random numbers in the range [0, 1].

This paper proposes a new variation in the classical PSO by

splitting the cognitive component of the classical PSO into two

different components. Thefirst component can be called good

experience component. That is, the bird has a memory about its

previously visited best position. This component is exactly thesame as the cognitive component of the basic PSO. The second

component is given the name bad experience component. The

bad experience component helps the particle to remember its

previously 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 the

particle toward its best position;

acceleration coefficient, which accelerates the

particle away from its worst position;

dimension of the own worst position of particle

until iteration ;

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

The positions are updated using (9). The inclusion of the

worst experience component in the behavior of the particle gives

additional exploration capacity to the swarm. By using the bad

experience component, the bird (particle) can bypass its pre-

vious worst position and always try to occupy a better position.

IV. LOCAL RANDOMSEARCH(LRS)The metaheuristic algorithms like GA, EP, SA, and PSO are

performing well for small dimensional and less complicated

problems. However, they fail to locate global minima for the

complex multiminima functions. Although they locate the

promising area, they fail to exploit the promising area to get

good quality solutions[6],[9], [11],[14],[15]. With a single al-

gorithm, it is difficult to control and to strike a balance between

exploration of whole search space to locate the promising area

and exploitation of the promising area to get global minima.

Several hybrid methods have been proposed by combining the

metaheuristics 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 LRS

procedure 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 local

area parameter; and are the vectors

of power generation limits; and is the initial local

search 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 generated

as follows:

(14)

where is a random number vector of length

, whose elements are randomly generated be-

tween and 1. If any local search point violates

the limits, it is forced within the boundaries.

Step 3) For each local search point, the objective function

values are calculated. Then the minimum objective

function among all is taken as , and the corre-

sponding is taken as . The optimum values

are updated as follows:

If then and

Otherwise and

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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 by

one and the above procedure is repeated from step

2). Otherwise, and are taken as the op-

timum results found by the LRS algorithm.

V. IMPLEMENTATION OFNPSO-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 Problems

With 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, and

is 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 particles 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 particles 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:

if

if

ifif

(19)

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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 stopping

criterion. The overall NPSO-LRS optimization process is showninFig. 1.

VI. TESTRESULTS ANDANALYSIS

A. Description of the Test Systems

In order to validate the proposed NPSO-LRS, it is tested with

three test systems having nonconvex solution spaces. The first

test system consists of six generators with prohibited operating

zones and has a total load of 1263 MW[9]. All the generatorsare having ramp rate limits. The network losses are calculated

by matrix loss formula. The best generation cost reported

until now is $15 450/h[9]. The previous best generation cost

was $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 minimum

is very difficult to determine. The best generation cost reported

until now is $122 122.16[16].

The third system consists of ten generators with multifuel op-

tions and valve-point effects[21]. Thefirst generator is having

two fuel options, and the remaining generators are having three

fuel 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 the

iterations, and the acceleration coefficient istakenas 2,since

these settings are suitable for many power system problems[9],

[24]. The number of particles is selected as 20. To find suitable

values 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 is

decreased from 1.0 to 0.1 in steps 0.05. For each combi-

nation of and , 100 independent trials have been

made with 1000 iterations per trial.

The EDVL formulation of the 40-generator system is used for

testing. The minimum generation cost for this system reported

so far are $122 624.35 [14], $122 252.265 [15], and $122 122.16

[16]. Hence, for each trial, the minimum generation cost is tested

whether 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 inTable I, and are chosen as 1.6 and

0.4 (case 13), respectively. They give the minimum generationcost more consistently, and the obtained generation cost is also

less 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 be

determined. Since has been taken as 0.05 for most of the prob-

lems[33], it isfixed at 0.05. If the tolerance for range reduction

is 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-determined

NPSO 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) are

tested. For each and combination, 100 independent trialshave been made with 1000 iterations per trial. As before, for

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TABLE IINFLUENCE OFPARAMETERS ONNPSO 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 NPSO

and LRS, it is necessary to find the relative strength of each

constituent. So, three different testing strategies are applied on

the EDPO, EDVL, and EDVLMF problems with a swarm of 20

particles.

1) PSO-LRS: The classical PSO with standard parameters( , and ) is in-

tegrated with LRS. This strategy is selected to analyze the

performance 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 RAM

processor. In order to find the effectiveness and superiority of

the NPSO-LRS algorithm, the test results are compared with the

results obtained by other algorithms available in the literature.

D. Convergence Test

The convergence test is carried out to determine the quickness

of the three PSO strategies in terms of the number of main PSO

iterations. The three PSO strategies are tested with thefirst test

system, and the result is shown inFig. 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 local

searching ability.

The results of convergence test on the 40-generator system

are shown in Fig. 3. For this system also, the NPSO-LRS is

the 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 genetic

algorithm with multiplier updating (CGA_MU)[21]. The same

EDVLMF 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 inFig. 4.

Here also, NPSO-LRS is faster in convergence in terms of

main PSO iterations. The behavior of NPSO falls between

those of PSO-LRS and NPSO-LRS. For all the three test

cases, NPSO-LRS stands first in the performance ladder.The proposed worst experience component in the velocity

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

TABLE IIBESTPOWEROUTPUT FORSIX-GENERATORSYSTEM

update equation and the local search procedure strengthen the

NPSO-LRS.

E. Comparisons of the Best Solutions

The best power output of the six-generator system obtained

by 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, the

losses 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 that

results from the three proposed PSO strategies are listed in

Table III. The number of iteration is taken as 1000 to match

with the previous analysis, and 20 particles are used. The com-parison of generation cost obtained by IFEP[14], MPSO[15],

TABLE IIIBESTPOWEROUTPUT FOR40-GENERATORSYSTEM

ESO[16], and the three PSO strategies is given in Table IV.

All the 40 generators are having valve-point effects, and the

solution space has multiple minima. The optimal generation

cost is difficult to achieve, and the minimum generation cost

reported so far is $122 122.16 [16]. However, the three PSO

strategies have the ability to obtain lower generation cost when

compared to $122 122.16. Among the three PSO strategies, the

NPSO-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, are

given inTable V. The three PSO strategies are able to obtainbetter results compared to IGA_MU[21]. However, NPSO-LRS

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TABLE IVMINIMUMGENERATION COSTOBTAINED BYDIFFERENTMETHODS

TABLE VBESTSOLUTION FORTEN-GENERATORSYSTEM

TABLE VICOMPARISON AMONG DIFFERENT METHODS AFTER

50 TRIALS(SIX-GENERATORSYSTEM)

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 during

all the trials.

The comparison of results after 50 independent trials with the

first test system is shown in Table VI. From the results, the supe-

riority of the PSO_LRS, NPSO, and NPSO-LRS strategies over

GA[9] and PSO[9]can be noticed. Moreover, the maximum

and average values obtained by NPSO-LRS are very close to

the 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 OFCONVERGENCE FOR40-GENERATORSYSTEM

TABLE VIIICOMPARISON AMONG DIFFERENT METHODS AFTER

100 TRIALS(40-GENERATORSYSTEM)

TABLE IXFREQUENCY OFCONVERGENCE FORTEN-GENERATORSYSTEM

frequencies of attainment of a cost within the specific ranges are

presented inTable VII. The minimum, maximum, and average

costs of 100 independent trails are presented inTable VIII.

Tables VIIandVIIIreveal the consistency of NPSO-LRS in

achieving minimum generation cost. NPSO also exhibits similarcharacteristics, but its average cost is slightly greater than the av-

erage cost of NPSO-LRS.Table VIIdiscloses that NPSO-LRS

has the higher probability of attaining quality solution. NPSO

stands next in performance.

Tables IX andX provide the result of robustness test after 100

trials for the EDVLMF problem. Here also, the performances of

the three PSO strategies are superior. They provide good quality

solutions 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

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TABLE XCOMPARISON AMONG DIFFERENT METHODS AFTER

100 TRIALS(TEN-GENERATORSYSTEM)

TABLE XICPU TIMECOMPARISON

the computational efficiency of the NPSO-LRS. The mean CPU

time taken to complete thefixed number of iterations and

the mean CPU time taken to converge into the lower solution

range for 100 trials are shown inTable XI. The

has been calculated only for succeeded trials.

For the 40-generator system, the basic PSO takes an average

CPU time of 4.63 s to complete 1000 iterations. The

increases with the addition of worst experience component and

the LRS. The NPSO-LRS takes 12.1 s more than the NPSO to

complete 1000 iterations. Nevertheless, NPSO-LRS takes only

0.41 s more than the NPSO to converge into the lower solution

range ($120 000$122 500). Similarly, the NPSO-LRS takesonly 0.17 s more than the NPSO to converge into the lower

solution range ($623.5$624.5) of the ten-generator EDVLMF

problem. For both the test cases, the basic PSO is not able to

converge into the lower solution range within the specifiedfixed

iterations, and the PSO-LRS converges into the lower solution

range at the margin of the fixed iterations. It is observed that,

even though the NPSO-LRS contains the burden of LRS, it has

a mean CPU time of converging into the lower solution range

closer to NPSO and has obtained quality solutions

with more robustness than NPSO. If the solution quality and

the robustness of NPSO-LRS are considered, a slight increase

in due to the bad experience component and the LRS

procedure 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 extra

diversification capability provided by the proposed bad expe-

rience component. Since NPSO-LRS has the strength of both

NPSO and LRS, it performs well among the three PSO strate-

gies and outperforms the previous achievements. Hence, the

NPSO-LRS is suggested as a powerful optimization tool for

nonconvex 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 determined

by appropriate experiments. To prove the ability of the proposed

NPSO-LRS in solving nonconvex optimization problems, ED

problems with nonconvex solution spaces are considered and

solved with three different PSO strategies (PSO-LRS, NPSO,

and NPSO-LRS). The three strategies are tested for convergence

and robustness to find the relative strength of NPSO and LRS.With the aid of comparisons of the results obtained by the three

PSO strategies and the results of earlier methods available in the

literature, it is proved that the proposed NPSO-LRS method is

very effective in giving quality solutions consistently for non-

convex ED problems.

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A. Immanuel Selvakumar (M03) 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 in

electrical 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 AnnaUniversity, Coimbatore. His researchinterests include computer modeling and simulation,computer networking, and power systems.

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