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Application of Genetic Algorithm For Optimal Load Dispatch With

Non Smooth Cost Equations (CCPP)

Y.Rajendra Babu

Associate Professor, PSCMRCET,Vijayawada,A.P.

Dr.M.Venugopal RaoProfessor&Head of EEE, K.L. University

ABSTRACT

The objective of economic load dispatch(ELD) is to minimize the total fuel cost of

electric power generation in thermal plantswhile ensuring high reliability. In ELDproblems the cost function for each

generator has been approximated by a

single quadratic cost equation. However, it

is more realistic to represent the generationcost function for fossil fired plants (Oil &

natural gas) as a segmented piece wise

quadratic function. As fossil fuel costincreases, it becomes even more important

to have a good model for the production cost

of each generator. A more accurateformulation is obtained for the ELD problemby expressing the generation cost function

as a piece wise quadratic cost function.

However, the solution methods for ELD

problem with piece wise quadratic costfunction require much complicated

algorithms such as the hierarchical

structure approach & Evolutionarycomputations (Ecs). Evolutionary

computations are optimum search methods

using genetics and evolution theory has beenwidely applied to the training of neural

networks, tuning of fuzzy membership

functions, machine learning, system

identification and optimization problems.Obviously applying genetic algorithm to

ELD problems will provide an optimal

solution.A test system comprising of 10 units

with 29 different fuel cost equations areconsidered the system is found to have

minimum and maximum generation

capacities of 1353MWs and 3622MWs

respectively. The load demand is assumed tovary between 2400MWs to 2700MWs insteps

of 100MWs. The applied genetic algorithm

method will provide optimal solution for thegiven load demand.

KEYWORDS: Economic load dispatch

(ELD), Combined cycle Power plant, Piece

wise quadratic cost function, Quadratic

programming Theory, Genetic algorithm

(GA).

1 INTRODUCTION

The operation planning of a power system is

characterized by having to maintain a high

degree of economy and reliability, among

the options that are available in choosing

how to operate the system optimally is

Economic Load Dispatch. The Economic

Dispatch problem is to determine the

optimal combination of power outputs of all

generating units to minimize the total fuel

cost while satisfying the load demand and

operational constraints. It is necessary to

have Input-Output characteristic of a

generator to analyze the ELD problems. In

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existing ELD approaches, the nature of

Input-Output characteristic is approximated

as a single Quadratic variation curve which

gives suboptimal solutions.

Obviously the nature of Input-Outputcharacteristics of modern generating units ishighly non linear because of multifueleffects combined cycle power plants(CCPP)and value loading effects, which may lead tomultiple local minimum points of costfunctions. Hence it is more realistic torepresent the Input-Output characteristic as apiecewise quadratic cost function to avoidhuge revenue loss over time in ELDproblems.

Previous efforts for ELD haveapplied various mathematical programmingmethods and optimization techniques. Theseinclude the lambda-iteration method, thebase point and the gradient methods.Recently a global optimization methodknown as Genetic algorithm has become agood algorithm tool for many optimizationapplications due to its flexibility andefficiency. Optimization is the act of

obtaining the best result under the givencircumstances.

In design, construction andmaintenance of any engineering systemtechnologists have to take many decisions atseveral stages. The ultimate goal of all suchdecisions is either to minimize the effortrequired or to maximize the desired benefit.Since the effort required or the benefitdesired in any practical situation can be

expressed as a function of certain decisionvariables, obviously the optimization can bedefined as the process of finding theconditions that give the maximum orminimum value of a function.

In ELD problems it is necessary tofind the minimum value of a function. Theoptimum seeking methods are also known as

mathematical programming techniques andare generally studied as a part of operationsresearch. It is a branch of mathematicsconcerned with the application of scientificmethods and techniques to decision making

problems providing optimal solutions.Therefore quadratic programming usinggenetic algorithm is more useful in findingthe minimum of a function of severalvariables under a specified set of constraints.

Genetic algorithm (GA) is astochastic searching algorithm; it combinesan artificial survival of the fittest withgenetic operators abstracted from nature toform a surprisingly robust mechanism that is

suitable for a variety of optimizationproblems. GA searches randomly from pointto point, thus allowing it to escape fromlocal optimum in which other algorithmsmight land. Therefore the global optimum ofthe problem can be approached with highprobability. GA has been successfullyapplied in various areas such as feederreconfiguration, hydro generator governortuning, load flow problems and so on.

This project develops geneticalgorithm approach for solving theEconomic dispatch problem considering atest system of 10 plants having 29 fuel costoptions. A salient feature of proposedapproach is that solution time growsapproximately linear with problem sizeother than geometrically. This feature isattractive in large scale problem consideringvarious operational constraints.

2 Economic Load DispatchThe ELD problem is to determine the

optimal combination of power outputs of allgenerating units to minimize the total fuelcost while satisfying the load demand fromthe operational constraints. Minimum fuelcosts are achieved by the economic loadscheduling of the different generating units



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or plants in the power system. By economicload scheduling we mean to find thegeneration of the different generators orplants so that the total fuel cost is minimumand at the same time the total load demand

at any instant must be met by the totalgeneration.

2.1 Economic load dispatch Problem

However, economic load schedulingwas not very important in the beginningwhen there were small power generatingplants for each locality, such as urban powersystem, but now with the growth in thepower demand and at the same timeguarantee regarding the continuity of power

supply to the consumer under normalconditions have forced the power systemengineers to develop grid system. For suchsystem the economic dispatch problem hasbecome increasingly important. Theobjective in the economic dispatch of powersystem is to minimize the cost of meetingthe energy requirements of the system oversome appropriate period of time and in amanner consistent with reliable service. Theappropriate period may be as short as few

minutes or as long as a year or moredepending on the nature of the energysources available to the system.

These sources are frequently referredto as limited energy sources since theycannot continuously maintain full output.Fired combustion turbines depending on theterms of the gas supply contract may fallinto this category. Obviously, the aim in theutilization of limited energy sources is to

realize the greatest possible value, duringthe operating cycle in terms of fuelreplacement at plants where the availablefuel supply is not a limiting factor. Totaloperating cost generally includes only theapplicable fuel cost.

2.2 Problem Statement for ELD withSmooth

Cost Function

Let N be the number of unitsPi power supplied by eachunit to meet load demand

Pd Load demand in MWsThe objective function for each unit can berepresented by a quadratic cost function

Fi = ai Pi2 + bi Pi+ ci Rs/hour

Subject to Pi = Pd i = 1, 2, 3,..nWhere ai , bi and ci are the fuel

consumption cost coefficients of the unit i.

2.3 Equal Incremental cost criteria forEconomic Load Dispatch problem

Using lagrangian multiplier method fuelcost equation can be written as

Fuel Cost = ( ai Pi2 + bi Pi+ci)

+ (Pd Pi)For optimal operation of plants partiallydifferentiating the above fuel cost equationw.r.t. Pi and and equating to zeroF/ Pi = 2 aiPi + bi- = 0where i = 1,2,3.nF/ = Pi = Pd2 aiPi + bi = --------- (1)P1 + b1 / 2 a1 = / 2a1P2 + b2 / 2 a2 = / 2a2...Pn + bn / 2 an = / 2an

Therefore

Pi+ bi/ 2ai = 1/2ai = ( Pd + bi/ 2ai ) / 1/2ai

Once after knowing the value oflambda then generation Pican be calculatedas given below.Pi= ( bi) / 2ai



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Hence the above formulation iscalled the Equal incremental cost procedurefor determination of generation.

3.Piece wise quadratic costfunctions

problem formulation

Recently Combined Cycle Power Plants(CCP) has shown their importance in bothdeveloping and developed countries in orderto improve the efficiency of generation.From the survey, the fuel cost characteristicsof CCP is non-smooth. Thus non-linearity inthe ELD problem is further increased.Numerical Optimization techniques andevolutionary computation methods can playkey role in solving ELD problem.

3.1Combined cycle power plant

In CCP gas and steam turbines areworking in combination to generate electricpower. The exhaust heat produced by thegas turbine is enough to operate the steamturbine. Hence during some range of powergeneration steam turbine is operated without

or with less additional fuel input.In a heat engine (i.e, a device for

converting heat in to work) part of the heattaken up from a source (Ex: burning fuel) ata higher temperature is discharged to a heatsink at a lower temperature.

In a combined cycle system the heatdischarged from one heat engine serves asthe source for the next engine. The net resultis a greater overall operating temperaturerange (i.e, between the initial heat source

and final sink) than is possible with a singleheat engine. The thermal efficiency of thecombined system (i.e., proportion of heatconverted into useful work or electricalenergy) is thus greater than for the two heatengines operating independently.

Less fuel is then required to generatea given amount of electrical energy. As a

general rule, combined cycle systems havetwo heat engine stages, the second stagebeing a conventional condensing steamturbine.

Consequently in a combined cycle

system the steam turbine is preceded by atopping cycle heat engine, which can utilizeheat at high temperatures. The working fluidleaves the topping cycle at a sufficient hightemperature to generate steam for the steamturbine.

3.2 Piecewise quadratic cost function

It is seen from the earlierchapters that the cost function of generatorhas been approximated by a single quadratic

cost function. However, it is more realisticto represent the generation cost function forfossil fired plants as a segmented piecewisequadratic cost function because of multifuelgeneration and value point loading effects.As fossil fuel costs increases, it becomeseven more important to have a good modelfor the production cost of each generator.Therefore a more accurate formulation isobtained for the ELD problem by expressingthe generation cost function as a piecewise

quadratic cost function. However, thesolution methods for ELD problem withpiecewise quadratic cost function require amuch complicated algorithms such ashierarchical structure approach. RecentlyGenetic algorithms which are probabilisticoptimum search methods using genetics andthe evolution theory have been widelyapplied.

Fuel 1

Fuel



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Fig 3.1 Fuel Cost function Of a Thermalgeneration supplied with Multi Fuels

3.2Non smooth cost functions withmultiple

fuelsTraditionally the cost function of

each generator has been approximatelyrepresented by a single quadratic costfunction. In practice the operating conditionsof many generating units require that thegeneration cost function be segmented intopiece wise quadratic cost function.Therefore it is more realistic to representgeneration cost function as segmented intopiece wise quadratic cost function.

Generally a piece wise quadratic function isused to represent the I-O curve of agenerator with multiple fuels. The costfunction of ELD problem is defined asfollows F.c. = Fi(Pi)The piecewise quadratic function is given by

ai,1+ bi,1 Pi+ ci,1Pi2

fuel 1 Pi,min Pi

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quadratic programming problem stated inthe above equation can be obtained by theLagrange multiplier technique.

5 Genetic Algorithm

Genetic algorithm (GA), first introducedby John Holland in early seventies, isbecome a flagship among various techniquesof machine learning and functionoptimization. Algorithm is a set ofsequential steps needs to be executed inorder to achieve a task. A GA is analgorithm with some of the principles ofgenetics included in it. The genetic

principles natural selection andevaluation Theory are main guidingprinciples in the implementation of GA. TheGA combines the adaptive nature of thenatural genetics and search is carried outthrough randomized information exchanged.

There is multitude of search techniques.Among them calculus based, enumerativeand random search techniques are mostlyused. The first two techniques, i.e., Calculusbased and enumerative are capable of

arriving at reasonable good solutions forsearch spaces of smaller size and widevariation from point to point in theirprecinct, like all practical systems, thereefficiency is low in delivering solution forcomplex problems involving huge searchspace due to their lack of ability toovercome these local optimum points andreach the global optimum point. In order toovercome local optimum points we userandom search techniques. Its important to

note that this randomized search is not adirectional search. The search is carried outrandomly and information gained from asearch is utilized in guiding the next search.Genetic algorithm is an example of suchsearch technique.

Genetic algorithmsurpasses all the above limitations ofconventional algorithms by using the basicbuilding blocks that are different from thoseof conventional algorithms. It is different

from them in the following aspects.1. GA works with a coding of the

parameters set and not theparameters themselves.

2. GA searches from a populationof points and not from a singlepoint like conventionalalgorithm.

3. GA uses objective functioninformation, not derivative orother auxiliary data.

4. GA use probabilistic transitionrules by stochastic operands, notdeterministic rules.

The very first step of GA is therandom selection of initial search pointsfrom the total search space. Each and everypoint in the search space corresponds to oneset of values for the parameters of theproblem. Each parameter is coded with astring of bits. The individual bit is calledgene. The content of each gene is called

allele. The total string of such genes of all

parameters written in a sequence is called achromosome. So there exists a

chromosome for each point in the searchspace. The sea of search points selected andused for processing is called a population.That means population is a set ofchromosomes. The no of chromosomes inpopulation is called population size andthe total number of genes in a string is calledstring length. The population is processed

and evaluated through various operators ofGA to generate a new population and thisprocess is carried out till global optimumpoint is reached. The two parts of theprocess are called generation andevaluation.

The evaluation of GA we define afitness function and evaluates the fitness for



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each chromosome of a population. Thisfitness is an indication of the suitability ofthe values of the parameters, as representedby that chromosome, as a solution of theoptimization problem under consideration.

This fitness is used as bias for selecting theparents and generating a new populationfrom the existing one.

6 Simulation results

For a Load demand of 2400MWsMinimum value of fuel cost for 10 units is481.0326 $/hour Generations in MW fromeach unit among 10 units.

Simulation result for 2400Mw loadusing best fitness and score histogram

Generati

ons in

MW

Fuel cost coefficients Incremental fuel cost

in $/Mwhfuel

opti

onsa b C

189.7405 0.0022 -0.3975 26.7600 0.4283 1

202.3427 0.0042 -1.2690 118.400 0.4283 1

253.8953 0.0015 -0.3116 39.7900 0.4283 1

233.0456 0.0059 -2.3380 266.800 0.4283 3

241.8297 0.0011 -0.0873 13.9200 0.4283 1

233.0456 0.0059 -2.3380 266.300 0.4283 3

253.2750 0.0011 -0.1325 18.9500 0.4283 1

233.0456 0.0059 -2.3380 266.800 0.4283 3

320.3832 0.0016 -0.5675 88.5300 0.4283 1

239.3969 .0011 -0.0994 13.9700 0.4283 1



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For a Load demand of 2500MWsMinimum value of fuel cost for 10 units is525.7588 $/hourGeneration

s in MW

Fuel cost coefficients Increment

al fuel cost

in $/Mwh

fuel

option

sa b C

196.0000 0.002

2

-

0.397

5

26.7600 0.4671 2

206.9774 0.004

2

-

1.269

0

118.400

0

0.4671 1

267.2362 0.001

5

-

0.311

6

39.7900 0.4671 1

236.3207 0.005

9

-

2.338

0

266.800

0

0.4671 3

260.0639 0.001

1

-

0.087

3

13.9200 0.4671 1

236.3207 0.005

9

-

2.338

0

266.300

0

0.4671 3

270.8339 0.001

1

-

0.132

5

18.9500 0.4671 1

236.3207 0.005

9

-

2.338

0

266.800

0

0.4671 3

332.8913 0.001

6

-

0.567

5

88.5300 0.4671 1

257.0355 0.001

1

-

0.099

4

13.9700 0.4671 1

Simulation result for 2500Mwload using best fitness and scorehistogram

For a Load demand of 2600MWsMinimum value of fuel cost for 10 unitsis 573.9008$/hour

Generati

ons in

MW

Fuel cost coefficients Incremental fuel cost

in $/Mwhfuel

opti

onsa b C

189.7405 0.0022 -0.3975 26.7600 0.4283 1

202.3427 0.0042 -1.2690 118.400 0.4283 1

253.8953 0.0015 -0.3116 39.7900 0.4283 1

233.0456 0.0059 -2.3380 266.800 0.4283 3

241.8297 0.0011 -0.0873 13.9200 0.4283 1

233.0456 0.0059 -2.3380 266.300 0.4283 3

253.2750 0.0011 -0.1325 18.9500 0.4283 1

233.0456 0.0059 -2.3380 266.800 0.4283 3

320.3832 0.0016 -0.5675 88.5300 0.4283 1

239.3969 .0011 -0.0994 13.9700 0.4283 1



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Simulation result for 2600Mw loadusing best fitness and score histogram

For a Load demand of 2700MWs

Minimum value of fuel cost for 10 units is

623.3292$/hour

Simulation result for 2600Mw load

using best fitness and score histogramAnnual savings in $ for four different loaddemands in MW compared withHierarchical ELD approach[Ref. IEEE transactions on PAS, VOL.20,NO.4, NOVEMBER 2005]

Generations

in MW

Fuel cost coefficients Incremental

fuel cost

in $/Mwh

fuel

options

a b C

218.2499 0.0019 -

0.3059

21.1300 0.5064 2

211.6626 0.0042 -

1.2690

118.4000 0.5064 1

280.7228 0.0015 -

0.3116

39.7900 0.5064 1

239.6315 0.0059 -

2.3380

39.7900 0.5064 3

278.4973 0.0011 -

0.0873

13.9200 0.5064 1

239.6315 0.0059 -

2.3380

266.3000 0.5064 3

288.5845 0.0011 -

0.1325

18.9500 0.5064 1

239.6315 0.0059 -

2.3380

266.8000 0.5064 3

428.5216 0.0006 -

0.0182

14.2300 0.5064 3

274.8667 0.0011 -

0.0994

13.9700 0.5064 1

Generations

in MW

Fuel cost coefficients Incremental

fuel cost

in $/Mwh

fuel

options

a b C

216.5442 0.0019 -0.3059

21.1300 0.5001 2

210.9058 0.0042 -

1.2690

118.4000 0.5001 1

278.5441 0.0015 -

0.3116

39.7900 0.5001 1

239.0967 0.0059 -

2.3380

266.8000 0.5001 3

275.5194 0.0011 -

0.0873

13.9200 0.5001 1

239.0967 0.0059 -

2.3380

266.3000 0.5001 3

285.7170 0.0011 -

0.1325

18.9500 0.5001 1

239.0967 0.0059 -

2.3380

266.8000 0.5001 3

343.4934 0.0016 -

0.5675

88.5300 0.5001 1

271.9861 0.0011 -

0.0994

13.9700 0.5001 1



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S.No

.

Load

Demand inM

W

Fuel costfrom

HierarchicalELD

approach

Fuel cost fromGenetic

AlgorithmELD approach

AnnualSavings

in $$ /

Hour

$ /

year

$ /

Hour

$ /

year

1240

0487.91

4274091.60

481.0326

4213845.58

60246.02

2250

0526.69

4613804.40

525.7588

4605647.09

8157.31

3260

0574.28

5030692.80

573.9008

5027371.01

3321.79

4270

0623.81

5464575.60

623.3292

5460363.79

4211.80

Table 6.1 Annual savings in $ compared withHierarchical ELD approach

SIMULATION PARAMETERS in GAs.

References

[1] RMS Dana raj, Dr F gajendran; Anefficient

algorithm to find optimal economic load

dispatchfor plants having discontinuous fuel costfunctions., Journal of institution ofengineers(India)., Vol 83,pt El dec

2004.

[2] Kwang Y.Lee and Arthit Sode Yome,June Ho

Park; Adaptive Hop field nueralnetworks for

economic load dispatch. IEEE

transactions onpower systems. Vol 13 No 2. May 1998.

[3] Po-Hung Chen, Hong-Chan Chang;Largescale

economic dispatch by genetic algorithm.IEEE

transactions on power systems, Vol10,No 4, Nov

1995.

[4] Derong Liu, Ying Cai; Taguchi methodfor solving

economic dispatch problem with nonsmooth cost

function. IEEE transactions on powersystems,

Vol.20, No. 4, Nov 2005.

[5] N Rama raj, R. RajaRam; Analyticalapproach to

optimize generation schedule of plant

with multiplefuel options. Journal of institution of

engineers(India), Vol68. pt EL Dec 1987.

[6] C E Lin, GL Vivianib; Hierarchicaleconomic

Fitness function @f2

Number of variables 10

Type of the plot Best fitness

Population size 25

String length 88 bits

Initial Population [ ]

Reproduction Elite count 2Cross over Multipoint orscattered

Stopping criteria

Number ofgenerations

120

Stall time limit 50 seconds

Stall generations 105



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dispatch of piece wise quadratic costfunctions.,

IEEE transactions on PAS, Vol PAS -103, No 6

June 1984.

[7] Singiresu S.Rao EngineeringOptimization theory

and practice third edition, New AgeInternational(p)

Limited, Publishers.

[8] Allen J.Wood, Bruce F. WollenBerg,Power

generation operation and control, WileyIndia

edition.

[9] S.Rajasekaran, G.A Vijaya Lakshmi Pai,Nueral

Networks Fuzzy Logic and GeneticAlgorithms,

Prentice Hall of India private Limited.



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