PSO_for_Economic Dispatch

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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost Functions

A Comparative Study of Particle Swarm Optimization and Differential

Evolution on the Economic Dispatch with Non-smooth Cost Functions

ohammad Nawaz Edoo and !o"ert #$ F$ Ah %in&'

Department of Electrical and Electronic EnineerinFaculty of Enineerin!niversity of "auritius

#eduit$ "auritiusEmail% r&ah'in(uom&ac&mu

A"stract

Over the last few decades$ an impressive num)er of methods have )een used to solve the Economic Dispatch Pro)lem

*EDP+ that considers such enerator characteristics as prohi)ited operatin zones$ multiple fuel options and valve-point

effects& ,n relatively recent times$ two alorithms$ namely Particle Swarm Optimization *PSO+ and Differential

Evolution *DE+ have )een introduced and have shown superior performance& ,n this paper$ usin four widely used test

systems with non-smooth cost functions an attempt is made to compare the performance of PSO and DE on this type of

pro)lem& .he simulation results show that DE is the )etter alorithm$ outperformin PSO and other alorithms in the

literature )oth in terms of /uality of solution and ro)ustness&

%eywords

Economic dispatch$ differential evolution$ multiple fuels$ particle swarm optimization$ prohi)ited operatin zones$

valve-point loadin effects&

*For correspondences and reprints

0


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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost Functions($ )ntroduction

.he cost of the fuel used for the eneration of electrical enery forms the most sinificant portion of the operatin costs

of a power system& .his is truer as the prices of fossil fuels have reached record-hih levels in the recent past althouh

they are now on the low side& Nevertheless$ the Economic Dispatch Pro)lem *EDP+ is assumin reater importance in

power systems operation&

Proper modelin in the EDP is vital to achieve ood results& "oreover$ any improvement in the model which could lead

to )etter solutions is$ o)viously$ hihly desira)le& .raditionally$ the input-output relationship of the fossil-fired

enerator has )een modeled as a /uadratic curve which is continuous and has a monotonically increasin derivative&

Such a curve is conve1 and conse/uently traditional mathematical techni/ues have )een used to solve this classical

EDP&

!nfortunately$ not all units can )e represented )y the /uadratic model& For e1ample$ lare steam-tur)ine enerators

have input-output relationships that are realistically modeled )y superimposin the )asic /uadratic function with a

rectified sinusoidal component *2ood 3 2ollen)er 0445+& "oreover$ fossil-fired units with the capa)ility to )urn

multiple types of fuels have input-output curves which consist of piecewise /uadratics *6in 3 7iviani 0489+& And

lastly$ some units have lare discontinuities in their cost curves so as to represent their prohi)ited operatin zones

*PO:+ *;ain +& 2hile all these additions to the )asic /uadratic model reatly increase the accuracy of the latter$

the pro)lem that arises is that the EDP )ecomes discontinuous$ hihly nonlinear or even non-conve1& .hus$ traditional

optimization techni/ues such as 6arane "ultipliers can no loner )e applied&

.o tac'le the non-conve1 EDP$ researchers are increasinly ma'in use of nature-inspired metaheuristics& !nli'e

traditional methods$ the metaheuristics do not impose any constraint on the function to )e optimized& Also$ the main

idea )ehind the respective metaheuristic is enerally simple to understand and$ its application is also /uite

straihtforward& ,t is mainly for these reasons that these methods have )ecome e1tremely popular amon researchers& ,n

Sinha et al& *+$ for instance$ several types of Evolutionary Prorammin *EP+ techni/ues are applied to the EDP

with 7alve-point 6oadin effects *ED76+& ,n ?im et al& *


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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost FunctionsBuite recently$ two metaheuristics$ namely Particle Swarm Optimization *PSO+ *?ennedy 3 E)erhart *044@++ and

Differential Evolution *DE+ *Storn 3 Price *044@++ have )een introduced and have rown very popular with researchers

tac'lin the non-conve1 EDP& ,n that respect$ the o)ective of this paper is to provide a comparative study of PSO and

DE on these non-conventional EDPs& .o that effect$ four test systems with 5$ 0=$ 0> and 9= units$ all e1hi)itin non-

smooth cost or non-conve1 cost functions$ have )een used& Simulation results are presented and comparisons are made

)etween PSO and DE as well as with other alorithms in the literature&

.he rest of the paper is oranized as follows& Section < formulates the different 'inds of EDPs& An overview of the PSO

and DE alorithms are iven in Section >& .he constraint handlin methods used in this wor' are descri)ed in Section 9&

Section @ presents the implementations of the alorithms while the results and discussions are iven in Section 5&

Finally$ some conclusions are drawn in Section &

*$ Pro"lem Formulation

.he inclusion of certain enerator characteristics transforms the classical EDP into a discontinuous andor non-conve1

pro)lem& ,n this paper$ three different Economic Dispatch formulations are considered$ namely$ Economic Dispatch

with Prohi)ited Operatin :ones$ Economic Dispatch with 7alve-point 6oadin effects and Economic Dispatch with

7alve-point effects and "ultiple Fuels&

*$($ Economic Dispatch with Prohi"ited Operatin& +ones*EDPO,

.he o)ective is

= =++==

G GN

i

N

i

iGiiGiiGii cPbPaPFF

0 0

+

1$*$ E3uality Constraint

.he e/uality constraint is$ pro)a)ly$ what distinuishes the EDP from other optimization pro)lems& ,t is also what

ma'es the EDP a very difficult optimization pro)lem$ especially when considerin the non-linear enerator

characteristics& A num)er of schemes have )een used to handle the power )alance constraint& Perhaps the most common

is the penalty function method& !nfortunately$ this method has serious draw)ac's which ma'e it very difficult for

metaheuristics to stri'e the riht )alance )etween e1ploration and e1ploitation& .herefore$ in this paper$ a new

alternative method is used$ where)y all particlesindividuals are made to satisfy the e/uality constraint at initialization

and durin evolution&

8


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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost Functions.he procedure is descri)ed as follows& Firstly$ for any particleindividual$ a se/uence of num)ers$ of lenth e/ual to the

num)er of enerators is randomly permuted& .he random se/uence specifies the order in which the eneratorsM outputs

will )e modified in order to satisfy the power )alance constraint& .o illustrate this approach$ consider for e1ample a 5-

unit system& Suppose that for a certain individual$ the randomly permuted se/uence is @$ & ,n this wor'$

this means that the last enerator specified )y the se/uence *in this case$ enerator >+ is used as a standard slac'

enerator in a first attempt to satisfy the power )alance constraint& ,f$ after this$ the e/uality constraint is still not

satisfied$ the output of enerator @ is then modified& .his process will continue as lon as the power )alance constraint

is not satisfied and with the se/uence of the enerator output to )e modified specified )y the randomly permuted

se/uence&

4$ PSO and DE )mplementations for Economic Dispatch

4$($ PSO-"ased Economic Dispatch

,n addition to the particlesM positions themselves$ PSO also re/uires the initialization of the particlesM velocities& ,n this

paper$ each dimension of the particlesM velocities is initialized randomly )etween the followin limits *Par' et al&

*


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,n this wor'$ the stoppin criterion is a ma1imum num)er of iterations&

0=


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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost Functions4$*$ DE-"ased Economic Dispatch

.he pseudo code for the DE-)ased Economic Dispatch is iven )elow&

)nitialize each individual of the population Each individual is forced to satisfy the e/ualityconstraint$ usin the procedure e1plained earlier

whilestopping conditions- not tre do

for each indi/ida+doCreate trial vector and ensure that it satisfies the power )alanceif tria+ /ector is better than crrent indi/ida+ do

#eplace current individual )y trial vector end end

end

.he stoppin criterion for DE is a ma1imum num)er of iterations as for PSO&

6$ Simulation !esults and Analysis

6$($ Description of the #est Systems

.o test the effectiveness of the two alorithms and compare their performances$ four test systems with non-conventional

cost functions were chosen& .he first system consists of 5 enerators with prohi)ited operatin zones and has a total

load of 0 "2 *;ain *++& .ransmission losses are determined usin B-coefficients& #amp rate limits are also

ta'en into account in this system&

.he second test system$ from Chian *


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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost Functions6$*$ #est System ( /6-unit system,7 EDPO

For this case study$ the followin parameters were used for PSO&

Num)er of particles% 0==

Num)er of iterations%


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#a"le ( 8est power output for 6-unit system

!nitPSO *;ain

*++

NPSO-6#S*Selva'umar

3.hanush'odi

*9@5

99&94=

0>&>49950>4&@0&>09&954&=55>05@&9=8&0>@@

.otal;eneration

*"2+0


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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost Functions6$0$ #est System * /(= while the ma1imum num)er of iterations is @==&

One hundred runs were performed for each alorithm&

.a)le 9 shows the )est solutions out of 0== runs for the PSO and DE alorithms as well as those from previous wor's

in the literature& .he DE alorithm provides the )est solution followed )y the PSO alorithm& ,n fact$ the alorithms

from the literature do not provide solutions lower than 5


16/28

A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost Functionsconfirmed in .a)le 5 where the mean num)er of iterations for PSO to reach a value of 5


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#a"le > Fre3uency of conver&ence for (&@

P

5094&


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0 100 200 300 400 500 6001.79

1.8

1.81

1.82

1.83

1.84

1.85

1.86x 10

4

Iterations

ObjectiveFunctionValue/$

DE

PSO

Fi&ure 0 Comparative conver&ence characteristics on (0-unit system for PSO and DE

,t should )e emphasized that it is only throuh the contri)ution of the chaotic se/uences that DE succeeds in findin the

minimum of 0$45>&8


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0 100 200 300 400 500 6001.79

1.8

1.81

1.82

1.83

1.84

1.85x 10

4

Iterations

Objectivefunctionvalue/$

Fi&ure 1 Conver&ence characteristics showin& the influence of the chaotic se3uences

#a"le : Statistical comparison of al&orithms for (; -

PSO-SBP*7ictoire 3

Qeya'umar * - 08$=


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5&9&


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#a"le (< 8est solution of (0-unit system with *4*< ? loadin&

;eneratorPSO-SBP *7ictoire$

3 Qeya'umar *

9@584

0=000

5>=0@4&>>=0@4&>>=0@4&>>=&>444&>44=8&58554445

.otal powerenerated *"2+


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An averae of 8&>0s was ta'en )y a run for PSO whereas that of DE was 0>&< s &

#a"le (( Statistical comparison of al&orithms

6$4 #est System 1 /1


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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost FunctionsAs it can )e seen in Fiure 5$ PSO convered e1tremely fast )ut aain at the e1pense of a more thorouh e1ploitation

phase& At the end of a)out


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0 500 1000 1500 2000 2500 3000 3500 4000 4500 50001.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34x 10

5

Iteration

Objectivefunctionvalue/

$

PSO

DE

Fi&ure 6 Comparative conver&ence characteristics of DE and PSO on 1


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#a"le (* 8est solution of 1


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#a"le (0 Statistical comparison of al&orithms /"ased on 4< runs,

"ethod.otal eneration cost *hr+ Standard

deviation"inimum "a1imum "ean

,FEP *Sinha et al&

*+ 0


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A Comparative Study of Particle Swarm Optimization and Differential Evolution on the Economic Dispatch with Non-smooth Cost Functionsfor PSO has improved the results reported in the literature for the four test systems e1cept for the one with the 08==

"2 loadin of 0>-unit system&

!eferences

CH,AN;$ C&-6& *

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PSO_for_Economic Dispatch

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