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Genetic Algorithm SolutiontoEconomic Dispatch Problemsx. Lei, E. Lerch, D. Povh

Abstract

This paper presents an economic dispatch algorithm based on the genetic algorithm (GA) for the determina­tion of the global or quasi-global optimum dispatch solution under consideration of transmission losses. !npractice, the efficiency ofGA is sometimes hindered by a poor performance in a localized search or by the dif­ficulty offinding and maintaining feasibility for a constrained problem. To deal with these problems, an ad­vanced genetic algorithm is developed. With the help ofa local improvement operator, combined with a con­jugate gradient method, the performance of the GA is enhanced. By means of heuristic scaling offitness fun­ctions and adaptation ofpenalty factors. the genetic search in the feasible region can be improved. The algo­rithm is implemented in the binary search space and tested with multi-modal non-linear functions. The appli­cation ofthe algorithm to a test system is also demonstrated. where transmission losses are represented by theB-matrix loss formula,

Fig. 1. Single search and multiple search by GAa) Single search b) Multiple search

rithms search for various states in every iteration usingmultiple paths, so that they can jump out of a local min­imum. However, this kind of algorithm is usually time­intensive. A genetic algorithm (GA) also is such an al­gorithm (Fig. 1) and has been developed using the "sur­vival-of-the-fittest" concept [15] in searching for bettersolutions. The GA has attractive features such as: simplic­ity, its ability to handle all sorts of functional representa­tions of problems, including problems with very complexinter-functional and intra-functional relationships, and itsrobustness; etc. Due to a number of advantages over otherconventional optimization and search techniques, GAhave also been applied to optimization problems in powerengineering such as in the design of power distributionsystems [5], optimal power flow [6,7], analysis of power­system topologies [8] and also reactive power optimiza­tion [9]. In this paper, the application of GA to solve theeconomic dispatch problem under consideration of trans­mission losses is presented.

1 Introduction

The economic dispatch problem is to allocate gen­eration from a number of dispatchable units in such away that power demands are met and fuel costs are min­imized, so that the system load will be supplied entirelyand most economically. This procedure, which is funda­mental to power system planning and control, has beenstudied extensively. The best solution is achieved if thefuel cost function reaches the global minimum and alltransmission constraints are satisfied. In general, the fuelcost function may contain a set of local minima becausethe fuel cost characteristic of a thermal generator is usu­ally approximated by piecewise quadratic functions orby a polynomial function of a high order.

The economic dispatch problem has traditionallybeen solved ignoring transmission constraints. Due tothe fact that transmission constraints in production costanalysis are very important issues for power utilities inplaning and operation of power system, various ap­proaches to include these constraints have been investi­gated and applied, even though the procedures then be­come much more complicated.!rving et al. [I] proposeda dual revised simplex algorithm and Liang et al. [2] de­veloped a dynamic programming approach. An algo­rithm based on simulated annealing has been presentedby Wong et al. [3] while a solution using artificial neuralnetworks has been proposed by Yalcinoz et al. [4].Among all approaches, linear or non-linear program­ming are single-path search algorithms. Starting from aninitial operating point. the approximation to the solutionis improved in every iteration along a single path untilreaching a convergent solution. These methods, howev­er, can be caught or arrested in a local minimum solu­tion, from which it may be difficult to escape. On theother hand, to find the global or a quasi-global optimumof a dispatch function, stochastic sampling and simulat­ed annealing algorithms have been used. These algo-

o Initial operating point• Intermediate operating point• Final operating point

o Initial generation• Intermediate generation• Final generation

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2 Economic Dispatch Problem

The achievement of economic dispatch in power­system operation consists of minimizing the operatingcosts depending on demand and subject to certain con­straints, i. e. how to allocate the required load demandamong the available generation units. Considering thetransmission losses PL , the total fuel cost F for runningn generators to meet the system load PD is given by

Minimize F = J; (PI) + Iz (pz) + ... + 1;, (p,,),

"subject to: L Pi = PD - PL ,

i=1

Pi. min :::; Pi :::; Pi. max '

(I)

No.7

Fig. 2. Wheel roulette in stochastic selection

where

Pi. min' Pi. max

Fig. 3. Crossover and mutation operators

r--- Crossover point

Individual I

Reproduction (selection) is an operator that produc­es a fixed number of copies of solutions according totheir fitness values [10, 14, IS]. If the fitness value ishigh then the number of copies is also large. That meansindividuals with higher fitness have a higher probabilityto be crossed with other strings to produce news stringsfor the next generation. Fig. 2 illustrates a stochastic se­lection strategy using a wheel roulette according to thevalue of the fitness function. Here, the individual no. 4has the highest probability to be selected for reproduc­ing the next generation.

The crossover and mutation operators have been in­spired by the mechanisms of chromosome recombina­tion and gene mutation found in biological genetics. Thecrossover operator leads to a mixture of solutions asshown in Fig. 3. By crossing Individual I and Individu­al II, two new individuals are generated. The mixture isperformed by randomly choosing a point within thestrings. The crossover rate (C) is a factor that determinesthe number of crossed strings in a generation. The mostcommonly used crossover methods are single-point,two-point and uniform crossover [I I, 14].

The mutation operator is defined by a random bit ma­nipulation in a chosen string (shown in Fig. 3) with a lowprobability of such change. The mutation adds a randomsearch character to the genetic algori thm and is necessaryto prevent that, after some generations, all possible solu­tions become alike. All strings and bits have the sameprobability of mutation. The mutation rate (M) is thenumber of mutations that are carried out in a generation.

Individual

Individual II '-----'-_--'--_---'-_-'---_..L.---J'------'-_---'

Individual I

Individual II

Newindividual

real power output of the i-th generator,operating cost of unit i,total transmission losses,minimum and maximum outputs of thisunit.

The transmission losses PL can be either given froma load-flow study or approximated by the B-matrix lossformula [2] (see Appendix, Tab. 3) given by

pL=pTBP+pTBo+Boo, (2)

A genetic algorithm is an evolutionary computationtechnique that works with a population of potential so­lutions to a problem. Individuals in the GA populationmate and reproduce as in nature. Different populationmembers are assigned reproduction rates proportional totheir fitness. The fitness is deri ved from the problem ob­jective function. A GA is frequently based a structuresimilar to a biological gene and searches for an optimalsolution in a coded parameter space (e. g. a binary cod­ification) by application of selection, reproduction,crossover and mutation process. The strongest individ­uals (solutions) survive during the optimization.

The first step of a GA is to generate an initial popu­lation. A binary string of length L is associated with eachindividual of the population. This string is usuallyknown as a chromosome and represents a solution of theproblem. A sample of this initial population creates anintermediate population by applying some operations(reproduction, crossover and mutation). The process,that starts from a present population and leads to a newpopulation, is named a generation when executing a ge­netic algorithm.

where

pT vector for generator loading (p I' P» ... , p,,),B loss-coefficient matrix,Bo loss-coefficient vector,B(x) loss constant.

Obviously, this economic dispatch problem is a con­strained non-linear optimization problem. A solution tothis problem is called "feasible" if all constraints are sat­isfied.

3 Genetic Algorithm

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3.2 Genetic Operators

With traditional GA. whose number of crossoverpoints (see above) is fixed at a very low constant valueof I or 2, premature convergence may occur in some op-

boundary by placing penalties into the fitness function topenalize strings that violate the constraints.

With the penalty function included, the objectivefunction can be written in the form

(3)

(4)

(5)

(6)if k = I,

otherwise.

I [ i-I](J's = In 17max - (17ma, - 17min) In _ I '

Z(P) =f(P) + aljJ(P),

wheref(P) is the operating cost function, and a is a pen­alty coefficient intended to control the degree of explo­ration in infeasible regions. For the problem presentedin this paper a quadratic penalty function IjJ is adopted,which is given by

After evaluating the randomly generated initialpopulation, the GA begins the creation of a new gener­ation of solutions. Genotypes with higher fitness valuesare selected to produce new generations. In the use ofpenalty functions, constraint violations could becometoo harsh and the majority of the population would beoutside of the feasible region. In this case, a proportion­al selection scheme [10] depending on the absolute fit­ness value of individuals becomes unreliable. There­fore, a rank-based se lection method [ 14] is used to over­come this problem. Using this mechanism, the numberof copies allocated to each individual is determinedonly by its relative ordering, i. e. its index i in the sort­ed population, and not by the absolute fitness value ofeach individual. Thus, the magnitude used for mappingis no longer important. The selection probabilities ofthis mechanism are given by

The coefficient </J is set to zero if all constraints aresatisfied, otherwise, it is set to the value of the distancebetween the feasible and infeasible region. The choiceof the penalty term can be significant as discussed in thenext section.

where m is the number of individuals in the population.The two constants TJmin = 0.75 and 17max = 1.1 are main­tained during the entire search procedure.

Because the economic dispatch function is con­strained by boundaries, the feasibility of the outputs ofeach unit must be taken into consideration. Therefore, aboundary mutation concept is used to manipulate eachunit output within its upper and lower bounds; it random­ly selects one unit output I and sets it equal to a uniformrandom number within the upper and lower boundsE (Pmin.k' Pmax.k)·

Hence,

I jE (Pmin.k' Pma,.k )Pk

Pk

3 Implementation of the Algorithm

If L is the string length, then M· N L bits change theirvalue in each generation. Generally, M is a small number.

Comparing traditional optimization procedures, theiteration count, the number of the operating points and theobjective function in genetic algorithms are called "gen­eration", "population" and "fitness", respectively. As­suming that a state variable vector Pi (PI' ... , p,,) (n gen­erator loading) is defined as the i-th individual in a popu­lation T, a typical structure of a GA is then as follows:

T(O): = (PI' ... , P,,) Initial Population.

Fitness Evaluation of each Individual in the currentPopulation.

While Not (End Condition) do:

T'(t + I): = Reproduction (T(t»),

T"(t + I): = Crossover (T'(t + I)},

T"'(t + I): = Mutation (T"(t + I)},

T(t + I): = T"'(r + I),

t:= t + I,

where T(t) indicates the population in the z-th gener­ation, and the generator loadings PI (p I, ... , PII)' ...Pm (PI' ... , p,,) designate each of the In individuals in thepopulation, respectively. Starting from a randomly gen­erated initial population T(O) the genetic operators mod­ify individuals in the population T(t) and generate a pop­ulation Ttt + I) with a higher probability of reproducingindividuals with a higher fitness. In the search proce­dure, the selected structures are copied into the new pop­ulation. The search procedure is stopped either when agiven maximum number of generations is reached orwhen the present solution cannot be improved.

3.1 Fitness Function

For the application of GA to the economic dispatchproblem, a simple binary codification was chosen to rep­resent a solution. The use of binary encoding has advan­tages in solving scheduling problems containing both in­teger and continuous numbers. If N represents the num­ber of units and H the scheduling period in hours, thenthe assumption that, at every hour a certain unit can beeither On or Off, an H-bit string is needed to describe theoperation schedule of a single unit. In such a string, a "I"at a certain location indicates that the unit is On where­as "0" indicates that the unit is Off. By concatenating thestrings of N units, a solution for a certain time period isformed.

To start the GA, a number of initial binary-coded so­lutions (genotypes) are produced at random to form theinitial population. Then, each genotype is evaluated andits fitness value calculated. The solution of the con­strained economic dispatch problem involves a mini­mization of the operating cost function, taking transmis­sion loss into account. For finding a solution that satisfiesall the constraints described in eq. (1) penalty functionsare used by the algorithm. The penalty functions try toforce the unconstrained optimum towards the feasibility

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3.4 GA with a Local Hill-Climbing Method

wise, Z(P) will be greater than one, and, hence, Z(P) isre-written as:

It is obvious that the scaling of Z(P) guarantees thatall feasible strings are being always awarded higher fit­ness values than infeasible ones. Thus, this method al­ways gives preference to searching feasible strings andto finding feasible solutions.

In general, GA are capable of quickly finding regionsof interest in the search space and perform well in a glo­bal search, while they perform poorly in a localized searchand may take a relatively long time to reach the optimalsolution. Furthermore, the most frequently used stoppingcriterion is a specified maximum number of generations.GA are forced to reach the specified number of genera­tions without considering whether a desired solution hasalready been obtained. To overcome this weakness, a hy­brid method [12] has been used instead. This method com­bines the genetic algorithm with a local hill-climbingmethod - a conjugate gradient method which evaluatesonly a narrow range of solutions to quickly climb to oneof the local maxima. Here, the search begins with the ge­netic algorithm locating a global hill with the search spaceand continues with the conjugate gradient method climb­ing the global maximum. This combination provides theGA with an efficient local improvement. As an addition­al advantage, the stopping criteria of the conjugate gradi­ent method can be used to manage the search process.

The switch from the GA to the hill-climbing meth­od depends on the improvement rate of the fitness val­ues.lfthe last generations of the genetic search could notachieve any improvement on the fitness values, the con­jugate gradient search is evoked. Currently ongoing ex­periments indicate that the resulting number of genera­tions lies in the range from 10 to 20.

(9)k it ( k /I k J2Z = I + t:=T Po - PL - I Pi~arge 1=1

3.3 Penalty Function Approach

timization problems due to a high homogeneity of thepopulation. This is especially true for large dimensionalproblems with many local minima. In order to avoid pre­mature convergence, a uniform crossover is appliedwhich has a strong ability to create new individuals fromparents with nearly identical genetic material than thetwo-point crossover method, especially as the popula­tion loses diversity [11].

An important motivation for this research work is todeal with problems subject to constraints. The applica­tion of GA to constrained optimization problems has fre­quently been hindered by the inefficiency of reproduc­tion, mutation and crossover when the feasibility ofgen­erated solutions is no longer guaranteed or when feasiblesolutions are difficult to find. By using the penalty func­tion given in eq. (3), therefore, the choice of the penaltycoefficient it becomes significant for infeasible stringsthat carry useful information for the optimal solution,but lie outside the feasible region. Small values of itallow greater exploration outside the feasible region,while large values of it essentially restrict the search tothe feasible region. In order to control the search in fea­sible regions, a large it is expected. However, if the fac­tor it is monotonically increased, it may become infiniteand the genetic search may also fail. Sometimes, infea­sible strings shared with strong genotypic similarity tothe optimal solution are more useful in an intermediatepopulation than feasible strings with weaker genotypicaffinity to the optimum. If a search is restricted only tofeasible regions, it will be more difficult to find a path tothe optimum, especially when the feasible region is notconvex, or the solution lies on the boundary. Converse­ly, if the search is expanded to include near-feasible re­gions - search paths will be allowed to wander outsideof(but not too far from) the feasible region - the frequen­cy of occurrence ofschemata with optimal genotype willbe increased.

In this work it is continuously updated with

(7)

(8)

where itk - I is the value from the previous generation, andZ~; 1 and Z~;2 are the average values of Z(P) of the pre­vious generation and the generation before that, respec­tively. The key advantage of this method is to choose italways in accordance with constraint violations and thefeasibility of solutions found so far, and to avoid an in­finite increase of it.

Similar to [13], a heuristic scaling of the functionZ(P) is adopted, to avoid an dominate search in infea­sible regains. Thus,

k Zk

Z = t:=T'fi:Jrge

wherefl:~.~ is the largest value of the cost function with­out violating the constraints of the last generation. In thecase of all constraints being satisfied, the penalty termequals zero and Z(P) is equal to or less than one. Other-

3.5 Numerical Tests

The proposed genetic algorithm has been tested onvarious problems with respect to efficiency and reliabil­ity. As examples, the optimization of a family of multi­minima, non-linear test problems given in [16] was per­formed in comparison with the simulated annealing al­gorithm. The results of simulated annealing (SA) aretaken from the ten replications of the test problems fromthis reference. In testing, the GA ran ten times with dif­ferent random seeds. The found solutions and the num­ber of function evaluations needed are given in Tab.T,

Dim. Method Av-Res Min-Res Av-No. Min-No.

4GA 1.84.10- 10 4.50. 10- 12 2.24·10-1 1.25·10-1

SA 6.18·10--1 8.70·10-x 1.38.106 1.18· 106

10GA 2.23. 10- 10 1.43.10- 12 5.23·10-1 3.42·10-1

SA 5.40· 10--1 5.40. 10--1 1.62.106 1.55· 106

Tab. 1. Results and evaluation numbers of the testing

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where Av-Res and Min-Res represent the average andminimum value of the solutions, and Av-No. and Min­No. are the average and the minimum number of evalu­ations of the function, respectively. Note that the globalminima of the test function occur at the origin (0, ... ,0).

The comparison in Tab. I shows that the GA is ca­pable of finding better solutions with less function evalu­ations than simulated annealing. With the value I· 10-6 forepsilon, as used in [16], GA obtained the optimal solutionin each of the two cases for all replications, while SAfailed twice in finding the optimal solution for the four-di­mensional case, and it always failed for the ten-dimen­sional case. The test results of the testing show that theproposed GA is efficient and accurate. The use of a localimprovement operator (a conj ugate gradient method) cansignificantly enhance the local search of the algorithm.

4 Application Example

The test system with three generators described inthe Appendix is taken to illustrate the solution to eco­nomic dispatch with transmission losses. The third­order polynomial fuel cost functions of the generators inthe test system and the B-matrix coefficients are alsogiven in the Appendix. The incremental fuel cost func­tions of the generators are non-monotonic.

The initial values of the generator loading are arbi­trarily set for testing the proposed algorithm, even withinfeasible values. Unlike most local search procedures.in which obtained solutions are strongly dependent onthe initial variables, the proposed algorithm is less sen­sitive to the initial variables. The GA proposed ran tentimes with different random initial settings and, in allcases led to almost the same solution. i. e. a low cost of6639.49 $/h. The population size and generation num­ber were set to 80 individuals and 100 generations foreach program run during the calculations, it was ob­served that the fuel cost functions of the first 30 to 40generations had already reached the optimum solution,while the reduction offuel costs of the remaining gener­ations was very small.

Tab. 2 summarizes the dispatch solutions obtainedby the proposed algorithm as well as by dynamic pro­gramming [2] and by simulated annealing [3]. Fromthis table, it can be seen that the solution found by thegenetic algorithm is almost the same as the solutionwith the scale factor y= 0.1 found by simulated anneal­ing, while the dynamic programming and the simulat­ed annealing with y= 0.01 converged to 6642.45 $/hand 6642.657 $/h, respectively. The best solution at6639.502 $/h was determined by the genetic algorithm.Although the transmission loss is now 62.446 MW in­stead of 43.4 MW, the total operating costs for thesystem are reduced. All of these confirm that the algo­rithm proposed is capable of determining the global orquasi-global optimum solution.

The proposed GA has been applied to some realis­tic systems with a large number of units in order to min­imize daily fuel costs. This involves the determinationof the start-up and shut-down schedules of thermal uni tsto meet a forecast demand over a future short-term (24 h)period. This kind of problem is a complex mathematical

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a) Loading Fuel cost(inMW) (in $/h)

Gen. I 360.2 1661.95Gen. 2 406.4 1843.42Gen. 3 676.8 3137.08Gen. total 1443.4 6642.45Loads 1400.0Losses 43.4

b) Loading Fuel cost(inMW) (in $/h)

Gen. 1 359.5459 1659.016Gen. 2 406.7342 1844.985Gen. 3 677.1525 3138.656Gen. total 1443.4339 6642.657Loads 1400.0Losses 43.4339

c) Loading Fuel cost(inMW) (in $/h)

Gen. I 376.1226 1733.799Gen. 2 100.0521 397.030Gen.3 986.2728 4508.675Gen. total 1462.448 6639.504Loads 1400.0Losses 62.448

d) Loading Fuel cost(inMW) (in $/h)

Gen. I 376.122 1733.798Gen.2 100.052 397.029Gen.3 986.272 4508.675Gen. total 1462.446 6639.502Loads 1400.0Losses 62.446

Tab. 2. Economic dispatch solutionsa) Solution by dynamic programming [31b) Solution by simulated annealing [4] (y= 0.0 I)c) Solution by simulated annealing [41 (y= 0.1)d) Solution by genetic algorithm

optimization problem with both integer and continuousvariables. The desired results were achieved which alsovalidated the efficiency of the algorithm. The applica­tion of the algorithm to unit commitment problems willbe subject of another publication.

5 Conclusions

A genetic algorithm solution to the economic dis­patch problem with consideration of transmission losshas been presented. It was necessary to enhance a stan­dard GA implementation with problem-specific modifi­cations to obtain satisfactory solutions. Two major mod­ifications have been proposed: The first focuses on han­dling constraint violations by introducing a penaltyfunction in the evaluation of the fitness function. Bymeans of heuristic scaling of the fitness function andcontinuous updating of the penalty factor. the search inthe feasible regions has been improved. The second im­provement enhances the local search performance of theGA with the help of a local hill-climbing algorithm.Once the GA achieves no improvement in a localized

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a) Unit 2 3

Ao 11.20 -632.0 147.144

AI 5.10238 13.01 4.28997A2 - 2.6429· 10-3 -3.0571-10- 2 3.0845 . 10-4

A 3 3.3333· 10-6 3.3333· 10-" -1.7677.10-7

Pmin 100MW 100MW 200MW

Pma, 500MW 500MW 1000MW

search, the conjugate gradient method will continue thesearch and climb quickly to the located peak. The algo­rithm is implemented in the binary search space and test­ed with multi-modal non-linear functions. The efficien­cy of the algorithm has been demonstrated by its appli­cation to a three-generator test system, for which a de­sired solution of the economic dispatch is obtained. Theproposed GA has also been successful to solve realisticproblems. A main disadvantage of the GA is its highcomputation time. However, with parallel processingthe speed of computation can be significantly enhanced.

b) Row\CoI.

1

23

7.5.10- 5

2

5.0.10- 6

1.5. 10-5

3

7.5.10- 6

1.0. 10-5

4.5.10- 5

Appendix: Data of the Test System

6 List of Symbols and Abbreviations

A three-generator example is taken from [2] to il­lustrate the new algorithm for solving an economic dis­patch problem, considering transmission line losses.The case of a load demand of 1400 MW is considered.The operating cost of each generator is represented bythe following polynomial

f(p;>=A o+A IP;+A-:.p,2+A1P;1, i= 1,2,3, (AI)

where the polynomial coefficients are listed inTab. 2. along with generator minimum and maximum

operating limits. The transmission line losses are rep­resented by:

(A2).1 .1

PL = I I PjBjkPk'j=1 k

where the B coefficients are also listed in Tab. 3.

Tab. 3. Input data for three-generator examplea) Generator datab) B- coefficients

References

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[21 Liang, Z. X; Glover. 1. D.: A Zoom Feature for A Dy­namic Programming Solution to Economic Dispatch In­cluding Transmission Losses. IEEE Trans. on PowerSyst. PS-7 (1992) no. 2. pp. 544-550

[3] I#lIlg. K. P.; Fung, C. C: Simulated annealing based ec­onomic dispatch algorithm. lEE Proc. C-140 (1993)no. 6, pp. 509-515

[4J Yalcinoz: T.; Short. M. J.: Neural Network Approach forSolving Economic Dispatch Problem with TransmissionCapacity Constraints. IEEE Trans. on Power Syst, PS-13(1998) no. 2, pp. 307-313

[5J Ramirez-Rosado, I. J.; Bernal-Augstin, J. R.: Genetic Al­gorithm Applied to Design of Large Power DistributionSystems. IEEE Trans. on Power Syst. PS-13 (1998)no. 2, pp. 696 - 703

[6] Wong. K. P.; Li, A.; Law, M. Y.: Development of Con­strained-Genetic-Algorithm Load Flow Method. lEEProc. Gen. Trans. and Distr. 144 (1997) no. 2, pp, 91-99

[7] Nara. K.et. aI.: Implementation ofGenetic Algorithm forDistribution System Loss Minimum. IEEE/PES SM1991. 9ISM467-IPWSR, 1991

[8] Mori, H. et. al.: A genetic Approach to Power SystemTopological Observability. Proc. IEEE 199 I ISCAS,pp. 1141-1147

[9] Iba Kenji: Reactive power optimization by genetic al­gorithm. IEEE Trans. on Power Syst. PS-9 (1994) no.2. pp. 685-692

[IOJ Goldberg, D. E.: Genetic Algorithms in Search, Optimi­zation. and Machine Learning. New York/USA: Addi­son-Wesley. 1989

[III Spears, W; De Jong, K. A.: An Analysis of Multi-PointCrossover. Found. of Genet. Algorithms Workshop, In­diana/USA 1990, Proc. pp. 865 -976

[121 Lei. X; Stuib. L. H.: A Genetic Algorithm for Con­strained Optimization: Application to 3D Medical ImageRegistration. IEEE Workshop on Biomedical ImageAnul., Washington/USA 1994. Proc. pp. 225-234

dimensionaverage value of the solutionsminimum value of the solutionsaverage number of evaluationsminimum number of evaluationsgenetic algorithmsimulated annealing

objective function with the penalty termunit output, lower and upper limitsof generator ioperating cost of unit itotal system load demandtotal transmission lossesloss coefficient matrix, coefficient vec­tor and constantpopulationindividual in a population (vector ofthe output of n-unit)number of unitsnumber of individuals in a populationgenetic generation indexrandom numberpenalty functionscalar as penalty coefficientscalar factor of the simulated annealingmethodelement ofselection probabilityconstants for determination of selec­tion probabilitytransposeiteration indexelement index of a vector

E

r

(J,

17l1lax' '7min

Z(P), Z

TP

nmf

f3I/JA.

Pi' Pi. min' Pi.llla,

DimAv-ResMin-ResAv-No.Min-No.GASA

/;(Pi)PDPLB,Bo,Boo

(. )T

(fOi, (.);

352 ETEP Vol. 9. No.6. November/December 1999

[13] Powell, D.; Skolnick, M. M: Using Genetic Algorithmsin Engineering Design Optimization with Non-linearConstraints. 5th lnt. Cant". on Genet. Algorithms,Town/Country 1993, Proc. pp. 424-430

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[15] Holland, J.: Adaptation in natural and artificial system.Ann Arbor/USA: The University of Michigan Press,1975

[16] Houck, C. R.; Joines, J. A; Kay, M. G.: A Genetic Algo­rithm for Function Optimization: A Matlab Implementa­tion. Matlab Toolbox of Genet. Algorithm Optimization,1996

Manuscript received on Apri/27, /999

The Authors

Xianzhang Lei (1958) received theBSc degree from Zhejiang Univer­sity/China, and his MSc and PhD de­grees in electrical engineering fromthe Technical University of Berlin/Gerrnany, in 1982, 1987 and 1992, re­spectively. From 1987 to 1993 heworked as a research fellow in theDept. of Electrical Engineering at theTechnical University of Berlin. He iscurrently working as a senior project

manager in the Power Transmission and Distribution Groupat Siemens in Erlangen/Germany. His areas of interest arepower-system stability, simulation, optimization and controlof power systems, as well as system planning. (Siemens AG,EV NP2, Paul-Gossen-Str. 100. 91052 Erlangen/Germany,Phone: +499131734010. Fax: +499131735159, E-mail:xianzhang.lei@erls04.siemens.de)·

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Edwin N. Lerch (1953) received hisDipl.-Ing. degree from the UniversityofWuppertal/Germany in 1979, wherehe also completed his PhD in electricalengineering in 1984. Since 1985 he hasbeen a member of the systems planningdepartment at Siemens in the industri­al power system and machines group.He is currently working in the areas ofpower system stability, dynamics ofmulti machine systems, control, opti­

mization and identification problems in electrical powersystems and is a deputy director of the Systems Planning de­partment. (Siemens AG, EV NP2, Paul-Gossen-Str, 100,91052 Erlangen/Germany, Phone: + 49 91 3173 40 52, Fax:+499131735159, E-mail: edwin.lerch@erls04.siemens.de)

Dusan Povh ( 1935) received his Dipl.Ing. degree from the University ofLiubljana/Slovenia in 1959, and hisDr. lng. degree from the TechnicalUniversity in Darmstadt/Germany in1972. Since 1989 he has been Profes­sor at the University of Ljubljana. Heis active in a number of committeesand working groups of ClGRE andIEEE. He is the chairman of ClGRESC 14 on HVDC and FACTS. His

areas of interest are systems analysis, network planning, in­sulation coordination of EHV and HVDC transmissionsystems and development of HVDC and FACTS techniques.He is the president of the Department of System Planning inthe Siemens Power Transmission and Distribution Group.(Siemens AG, EV NP, Paul-Gossen-Str, 100,91052 Erlan­gen/Germany, Phone: +499131734443, Fax: +499131734445, E-mail: dusan.povh@erls04.siemens.de)

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