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IEEE Transactions on Power Systems, Vol. 9, No. 4 November 1994 927GENE1'1C ALGORITHMS IN OPTIMAL MULTISTAGE DISTRIBUTION NETWORK PLANNINGVladimiro Muanda J V.Ranito L. M. PmenqtINESC Inst. de Engenharia de Sistemase Computadoresand FEUP/DFdiC Faculdade deEngenharia da Univ. doPort0L. Mompilher 4000 Porn PORTUGALFax:351.2.318692 a n : vmiranda@porto.inescn.ptA 4 This paper describes a genetic algorithm

approach to ttrc optimal multistage planning of distributionnetworks. Tlw authors describe a mathematical and algorithmicmodel that thcy havc developed and experimented with success.The paper a lw prcsents application examples, with real sizesystems. Thc advantages of adopting this new approach arediscussed i n Lhc planning context, namely in conjunction withthe adoption o f rr~ullicriteriaecision making methods.Ior opt imol distribution network planning, manymathematical wodcls have been proposed in the past forinstance, scc rcl'u 11 - 91. Many of them [l, ,4,5,71 are staticmodels, giving on optimal solution for a fixed set of data and asingle time prrid Essays in dynamic planning, consideringthe evolution i n power demand through time and consequenttopological changcs in the networks (new branches, new sub-stations, rcinforccmcnts, etc.) have never been a definitesuccess, whcn applicd to real sized networks. The mathematicalmodels behind thosc proposals either were heavy or neglectedseveral projcct Ccatures that engineers take as important on dis-uibution systcm dcsign; for example, economies gained fromcables in the same trench, reuse of disclassified lines, reshapingthe whole network by introducing a new substation, etc.Furthermorc, the vast majority of the model proposals,on the last 15 years, was aimed at a so called Optimalsolution . Howcver, during the last few years the objective ofrcaching this optimal concept has been challenged more andmore, namely within the US, with the acceptance of principlesof the least cost planning approach .This evolution favors the option for multi-criteriamethods and for algorithms giving, as an answer, a large set ofpossibly good solutions, instead of a single optimum.Genetic algorithms share precisely this property, andtherefore an invcstigation has been made on their behavior inthe distribution network design or distribution planningproblem (DPP). his research has been directed into answeringthe following questions :

94 WM 229-5 PWRSby the IEEE Power System Engineering Committee of theIEEE Power Engineering Society for presentation at theIEEE/PES 1994 Winter Meeting, New York, New York,January 30 February 3, 1994. Manuscript submittedJuly 12, 1993; made available for printingDecember 6, 1993.

A paper recommended and approved

Can genetic algorithms adequately model the DPPproblem?Is it possible to develop efficient computing algorithmsfor real size networks?Has the genetic approach an easy interface withmulticriteria decision making procedures ?Do genetic algorithms show any advantages overprevious proposed approaches ?As the following sections report, the answer is positiveto all these four questions. In fact, differently from manymodels previously proposed, this new model is flexible enoughso that many realistic features and conditions of practical naturemay be taken care of. For instance:the inclusion of multiple feeders in the same trench,with therelated savings achieved;the possibility of disclassifying lines at some stage, andof re-using them at a later stage;the possibility of generating solutions with some openloops, which may be valuable from a reliability point ofview;dealing naturally with load diversity factors;no need to specify, in advance, at which year should asubstation be built or put into service.

considering multi-objectives;

C AL-Genetic Algorithms (GA) are search and optimizationmethods based on natural evolution [Golberg89]. They consiston a population of bit strings transformed by three geneticoperators: selection, crossover and mutation. Each string(chromosome) represents a possible solution for the problembeing optimized and each bit (or group of bits), represents avalue for some variable of the problem (gene). These solutionsare classified by a evaluarion function, giving better values, orfitness, to better solutions.Although there are many forms [Grefestette91] forGenetic Algorithms, we will only refer to the canonicalalgorithm. This means that we will be dealing with threegenetic operators (selection, crossover and mutation) and linear,binary, fixed-size chromosomes. Canonical GA use a fixed-size,non-overlapping population scheme and each new generation iscreated by the selection operator and altered by crossover andmutation. The first population is generated at random.

Genetic Algorithm comDonentSEach chromosome represents a potential solution for theproblem to solve and must be expressed in binary form. For

instance, if we want to maximize the function f ( X ) = X Z , inthe integer interval I =[8 3 1 we could simply code X in

0885-8950j94/$04.00 1994 IEEE

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1928binary base, using 5 bits. Each solution must be evaluated bythe fitness function to produce a value. In our example, thechromoscjmc 1101 1 would receive the fitness value 272=729,while the chromosome00111 would receive the value 72=49.The pair (chromosome, fitness) represents anindividual.

The selection operator creates a new population (orgeneration) by sclccting individuals from the old population,biased towards thc best. This means that there will be morecopies of thc k s t individuals, although there may be somecopies of thc worst . This operator can be implemented in avariety of ways, although we use here a technique known asStochastic Tournanic'nt [Goldberg91]. This implementationissuited to a luturc distributed implementation and is verysimple: evcry tinic we want to select an individual forrcproduction, w e chtwse two, at random, nd the best wins withsome fixed probability, typically 0.8. This scheme can beenhanced by us ing more individuals on the competition[Goldberg9 c or cvcn by considering evolving winningprobability, cvcnlually leading to Bolttman Tournament[Goldberg91],gi'ncralizing the Simulated Annealing paradigm[Kirkpatrick83

C r o s s o w r i s the main genetic operator and consists inswapping chroinosome parts between individuals. The simplestcrossover opcralor is implemented by selecting a randomcrossover point in thc chromosome, and swapping the genesthat reside bclwecn the crossover point and the end of thechromosornc. For example, if we have two individuals:and choose a crossover point C=3 (indicated by 1) the resultingindividuals aftcr crossover would be:

A=O 10I00 B=010 11

A'=010 11 B'=010 IO0Crossovcr is not performed on every pair of individuals,its frequency bcing controlled by a crossoverprobability. Thisprobability should have a large value, typically Pc=0.8.The last genetic operator is mutation and consists intoggling a random bit in an individual. This operator should be

used with some care, with low probability, typicallyPm=O.OOl, for normal populations.How does a Gcnclic Algorithm work?

A canonical G A is a very simple process: we firstgenerate a random initial population, evaluate it and startcreating new populations by applying genetic operators. Thishigh-level behavior can be depicted on the following piece ofpseudo-C:m i n )int gen;generate (oldpop);f o r (gen = 0; gen < MAXGEN; gen++)I eval uate (oldpop)

newpop = s e le t (oldpop);cro sso ve r (newpop)mutation (oldpop)oldpop = newpop;1

Obviously, there is the need for some bookkeepingfunctions, for statistics and so on, but they are not central tothis explanation.This very simple behavior hides a powerful processing,done by the GA. In fact, the combination of selection andcrossover leads to a proliferation of individuals that possesssmall, tightly coupled blocks of bits leading to goodperformance. These blocks, usually called s c h e m a t a

[Holland75], are replicated through selection and combined orseparated by crossover.And mutation, what is its job? Mutation works as akind of life insurance . Some important bit values (genes)may be lost during selection; mutation c n bring them back, ifnecessary. Nevertheless, too much mutation c n be harmful: amutation probability of 0.5 always leads to random search[Goldberg89], independently of crossover probability.So, GA tends to select individuals with goodperformance and recombine some of their building blocks,creating more and more copies of good schemata, simply by theuse of selection and crossover. This hidden processing is calledimplicit parallelism because the number of schemata processed

in each generation is typically O N3) ,beingN the populationsize [Holland75]. This compares very well with the number offitness function evaluations, N. This characteristic is distinctiveof Genetic Algorithms rGrefenstette9 11.Genetic algorithms in Power Svstems

There have been some attempts to apply GeneticAlgorithms to solve problems in the Power System area, butso far, to our knowledge, not in multi-stage distribution(planning. In references [16] to [22] one may find the reports ofattempted approaches to problems such as:optimal capacitor placement [17];voltage optimization [181;harmonics [19];- system observability [20];reactive power control [21];load flow analysis [22];

clustering and network reduction [161;

This section presents a model to solve the problems ofthe optimal sizing, timing and location of distributionsubstation and feeder expansion, using genetic algorithms. Themodel allows the inclusion of constraints related to networkradiallity, voltage drops and reliability assessment.The objectives for distribution system planning that will

be discussed are related to providing the designs and associatedimplementation plans necessary for an orderly expansion offacilities, minimizing new facility installation costs andoperation costs, as well as achieving an acceptable level ofreliability, under the following constraints:a) Operation of the networks under radial configuration(although some open loops may exist);

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b) Voltagcdrop constraints;c) Powcr dctnand specifications;9 Power llow availability, namely constrained by linethermal 1 ll1 its;e) Possiblc site location for substation and lines pre-specilicul.

Facilily installation costs will be divided in threeelements : s i h l i h n cost, substation capacity expansion costand new fecclc-r cost. Power losses in the network will be takenas operation cc)>is.Already existing elements (substationsorfeeders) arc iiiiludcd at no investment cost in the model;however, thcir Iwwcr losses are taken in account.The following assumptions are made:a) A pcak load is considered for each stage expansionplanning load lorwasting is out of the scope of this paper).Some extra inlormation about load curves (suchas load factor)is required, i n ordc-r to evaluate losses and assess reliabilityindices such a.\ avcragc load disconnectedor average annualenergy not supplic'd.b) NCH nslallation facility candidates are knownbeforehand, arid lhcir location and installationcosts estimated.An cxpansion slrategy will be driven by load growth.The planning pcriod will be divided into several stages (oneyear, for institncc). One aims at having, as a result, a list ofinvestments to bc madc at each stage.The gcnctic algorithm approach to the DPP is drawn

1 . A sct or variables is chosen to represent a multi-stagenetwork solution; these variables are encoded in achromosome.2. A genctic algorithm is applied to a family of solutions,giving birth U new generations.3. Each solution in the new generation is evaluatedthrough a imess function, that includes investmentcosts, power loss costs, reliability, voltage dropdeviations; non compliance with other constraints isdealt with by the fitness function an unfeasiblesolution, with a low fitness value, will hardly survive.4. At the end of the process, a family of well fitted plans isavailable.

under the following general lines:

Variablcs:For a m-stage planning problem, the following (0-1)integer decision variables couldbedefined:Fis : Fis = 1 : if feeder i is used at stagesFis=O : otherwiseSis : Sis = 1 : if substation i is used at stage sSis = 0 : otherwiseEis : Eis = 1 : if subst. expansion i is used at stage sEis = 0 : otherwiseExisting facilities may be considered by fixing theirrespective valucs LO 1.

-me codingThe direct coding of the above variables into achromosome has been tried and tested. Although the resultsobtained were satisfactory, the process was not very efficient,because the extremely large number of unfeasible solutionsappearing at each generation led to a large computing timebefore reachingan acceptable stability.Therefore, we devised a new coding process. Itsrequirements were: it should lead to a minimum amount ofunfeasible solutions genera^, and it should provide very fastdecoding, as this operation is required at every fitnessevaluation.Its characteristics are:

a) at one time stage, some nodes matbe assigned with loadvalues, while other may have a forecastedzero load;b) a node with a positive load must be connected to, atleast, one feeder, while nodes with zero load may eitherhave no connections or have one or more connections inorder to allow power flow to other nodes;c) each node is represented in the chromosome by anumber of bits needed to encode the number of possibleconnections to it e.g., if four lines connect to thenode, one needs two bits, if it is a positive load node,ora three bit string, if it is a zero load node (to include thec sewhere no line goes through that node);0 a substation is assumed as a special type of line, thatmay be connected to the nodes chosen as possiblebuilding sites;This encoding strategy is such that it gives, as a result,only solutions with a number of lines=(nodes- ). a radiallitycondition. A few other details were included in the model,which are not relevant to its understanding but helped ingaining computer efficiency.Fitness Fu nctionThe itness function must reflect both the desired and theunwanted propertiesof a solution, rewarding the former and

strongly penalizing the latter. In the DPP, esired propertiesare, for inktance, low cost and high reliability, while unwantedfeaturesare non-radial configurations (open loops are accepted,but not closed loops), violations of thermal cable limits or ofvoltage drop constraints.Fitness is evaluated a posteriori; therefore it may benon-linear, non-continuous, non-convex, whatever. This is veryadvantageous over strict mathematical programmingapproaches.The general trend is to maximize fitness. Figure 1presents the general scheme of evaluation of the fitness of asolution, represented by a chromosome, at any generation. Thefunctions g ), h() and v(), referred to in this figure, must bechosen so that, for no matter what solutions x are evaluated,

me always obtains

In function f, we have included the following:

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1930

clzromosomc decoding

chcck coniicclivity andradiality

corn. tosources)

uc h i i d Flow

h(PNS)

FITNESS = f (power losscost, volt. drop quality,

Fig.1 Fitncss function evaluation schemeinvestment costs IC are added for all the planningstages, consideringa fixed discount rate;power loss costs PL are treated in the same way;voltage quality VQ may be assessed in many ways wehave used the following scheme:1) if thc volbge drop at any node exceeds some threshold(e.g., 8 ), the solution has been taken as unfeasibleand its fitness evaluated by function v;2) else, if the voltage drops at some nodes lie within agivcn inlcrval (e.g., [5%,8%]), he voltage qualityindex of Lhe solution is assessed by the sum of Lhesquarc dcviations to the lower limit of this interval;

3) else, if at all nodes the voltage drop stays below thatlower limit, the voltage quality index receives 0value.the reliability RB of a solution is evaluated through theapproximate calculation of the expectedannual energynot supplied this calculation takes in account theexistence of open loops in the network; we have usedthe following scheme:1) an upper bound U, in reliability level is calculatedassuming that no switching devices are included inthe network therefore, disconnections take placed y t the substation;is calculated assuming that allbranches are equipped with switching devices,allowing the isolation of failed branches and servicerestoration (taking in account the line capacities);

2) a lower bound

3) reliability fitness is given byRI3 =uu,+(1ct))L,where a E 0,1] is an improvement coefficient thataims at simulating the effect of a compromiseodution in switching device location policy asthis, in itself, is a very complex problem [23].The fitness value f of a solution x is given by

f(X) =M Cl(IC+PL) c ~ V Qc ~ R Bwhere M - Large (enough) constant value;

Ci constants externally fixed.Otherswial featuresa) the DC oad flow calculations areused as a first filterfor unfeasibilities, and are performed at the same time as thestructure of the network is recognized this strategy provedvery efficient;b) for AC load flow calculations, we used the methoddescribed in [24], applied to balanced systems, as it wasspecially developed forradialnetworks;c) before load flow calculations, loads valuesto be usedmay be affected by coefficients to take in account the diversityin demand load curves this featurealsodistinguishes this GA

approach romother models.

m The GA approach was applied to the system shown inFig. 2, where solid lines represent existing cables in the initialradial system, and dotted lines represent possible sites for theexpansion of the system. A complete data listing may beobtained from the authors, by request. The proposed substationsizes and other complementary system data are shown belowNumber of time stagesDiscount RateNominal voltageVoltage thresholdsNo. of nodesNo. of branches totalNo. of potential branchesTotal load for each time stg . (MVA)Feeder cost (PTE)

3 (+ initial)10%15kV5%,8%3 x 503 x 6 43 x4845; 63; 9.54*106/km

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s2o Km

Fig. 2 Initial system

s2Fig. 4 Solution for Stage 2

Fig. 3 Solution for Stage I

Fig. 5Solution for Stage 3

Run Several runs were made. The results presented wereobtained for a population of 40, and for 300 generations (avalue for which a robust stability in the best-so-far solutionhas been consistently observed). On average, user run timeshave been of around 300 sec., in a NeXTStation (33 MHz), butwe are now aware that these times may still be largelyimproved by carefully revising the computer code.

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ON LUSIONS1932330 -ENSMWh)300270240210

Pi\?

850 9(X) 950 lo00 1050COST ( EX o6)

Fig. 6 PIOI or a I-aIuily of solutions in a bi-attribute space' (invcstment vs. reliability).

ResultsIn I jprcs 3-4-5, a substation symbol within a squaremeans that hcrc has been a reinforcement in capacity of thatsubstation; shadowcd lines represent feeders already installedthat are not uwd n a given stage, remaining, however, as openloops. See, for instance, branches 33-34-35-36, built in stage1; in stage 2,3435-36 iu unused, but in stage 3 they are putin active scrvicc whilc 33-34 is flow open.Notice that node 35has no forccnsicd load until stage 3; nevertheless, a solutionwas built through it in d e r o reach node S3, ight in stage 1,which had a positive load value attached. Notice also therearrangemcnl i n the network when a new subs t a t i on gets in.Figure 6presents a family of solutions, obtained at theend of the evolution process, in a bi-attribute space(investment, in Portuguese Escudos, vs. reliability, in averageannual energy not supplied, both to be minimized). Solutions

for which thcrc exists an alternative with less investment and,at least, no incrcase in EiNS, or a more reliable alternative with,ai least, no incrcase in cost are saidto be dominated.We have drawn a curve representing a convexapproximation to the non-dominated border; the black dot,pointed by the arrow, stands for the solution depicted in figs. 2-5, and the thrcc shadowed dots represent some special non-dominated soluiions: they would not be found by classicalmathematical programming methods such as the weightedaddition of thc functions representing the objectivesnevertheless, thcy may be thought of as good compromises, byh e planner.Onc may clearly see how few the non-dominatedsolutions (also callcd Pareto optimal) are in this example

which, in principle, are the interesting solutions to look at.This clearly indicates that the combination of geneticalgorithms Ibllowcd by multi-criteria screening is likely to be apowerful tool in dccision aid for network planning. We believethat there is ficld hcre for further research.

The research reported in this paper clearly demonstratesthat a GA pproachb dynamic multi-stage planning problemis both feasible and advantageous:It provides the planner with a set of time-orderedinvestment decisions which is not obtained from static sub-optimizations, but directly from the considemtion of the timedimension of the problem. In this respect, it is more completethan many publisha apprmhes which claim feasibility underconstrained computing environments.Furthermore, i t allows the representation of non-linearitks which are hard to include in pure mathematicalprogrammingmethods, in fact, the existence of non-linearitiesenhancesthe advantages of using GA against pure mathematicalprogramming. These non-linqities arise not only from thenon-linear character of objective functions and constraints butalso from the discrete nature of many aspectsof thedistributionplanning problem. These in some cases could lead to a non-convex domain, perhaps in some cases not even connex butGA are able to deal with such environments and can detect localminima or even islands of solutions.The results of a GA are a generation of solutions,filtered through the struggle for survival. Therefore, manyinteresting and valuable exercises on comparisons and trade offsmay be executed, helping the planner to gain insight on theproblem he is faced with and Jlowing field for better decisionsto be taken.The fact that many solutions will be available alsoenhances the opportunity for multicriteria methods to beexplicitly applied, which is, in our point of view, a steptowards an adequate direction in distribution planning.

REFERF,NCESY. Baklund, J.A. Bubenko, I' Computer-aided distributionsystem phning , Electrical Power Energy Sy ste m,vol.1, no.1, Apr 1979M.F. Oliveira, V. Miranda, Etudes d'optimisation dansles rdseaux de distribution comprenant des calculs defiabilitd , Proceedings of CIRED'79, s.6,Li&ge,Belgium,Apr 1979R.N. Adams et al., A methodology for distributionsystem planning , Proceedings of the 8th PSCC,Helsinki, Finland, Aug. 1984M.A el-Kady. Computer aided planning of distributionsubstation and primary feeders , IEEE Transactions onP A S , vol. 103, Jun 1984T. Burkhardt et al., Decision making including forecastuncertainties and optimal routing in distributionnetworks , Proceedings of CIRED'85, s.6, Brighton,U.K., Apr 1985

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1933[19] H. Yang, Worst case analysis of distribution systemharmonics using genetic algorithms , Proc. IEEESOUTHEASTCON '92 New York, NY, USA, IEEE1992

[6] T. Gijncn, .J. Ramirez-Rosado, Review of distributionplanning modcls: a model for optimal multistageplanning , TEEE Proceedings, Vol. 133, Pt. C, NO. 7,Nov 1986[7] K.Aoki cl ; i l . , Ncw pproximate optimization method fordistrihut on systcm planning , IEEE Transactions onPWRS vo l 5 no.1, Feb 1990181 N. Kagan, K.N. Adams, Application of Bendersdecompo4 lion tcchnique to the distribution planning

problcni , I'rocccdings of the 10th. PSCC, Graz, Austria,Aug 1990, cd. Butterworths, London[9] K.Nara ct al., Multi-year expansion planning fordistribulion systcms ,IEEE Transactions on PWRS, ol

6 no.3, A u g 1991[lo] D. E. Goldhc rg , Genetic algorithms in Search,Optiini;/aiion and Machine Learning , 1989 Addison-Wesley[ I 11 J. Grclcnwttc, Conditions for Implicit Parallelism ,Navy Ccnlcr lor Applied Research in Artificial

Intelligcncc, lnlcrnal Report, 1991 Washington[12] D. E Goldbcrg, A Note on Boltzman TournamentSelection for Gcnctic Algorithms and Population-OrientedSiniulatcd Annealing , 1991, Complex Systems 3[13] D. E Goldbcrg, K. Deb, A Comparative Analysis ofSelection SchcnicsUsed in Genetic Algorithms ,GeneticAlgorilhms i n Search, Optimization and MachineLearning Suinincr School, 1991 Stanford[141 S . Kirkpatrick et al. Optimization by SimulatedAnnealing 1983 Science1151 J . H. Holland, Adaption in Natural and ArtificialSystcnis , The University of Michigan Press, 1975 Ann

Arbor1161 H. Ding et al, Optimal clustering of power networksusing genclic algorithms , Proc. 3rd Biennial Symp.Indust. Elcct. Applications, Ruston, LA, USA, LA Tech.Univ., 19921171 V. Ajjarapu et al., Application of genetic basedalgorilhins LO oplimal capacitor placement , Proc. FirstInternationalForum on Applications of Neural Networksto Power Systems, New York, NY, USA, IEEE 1991[181 T. Haida et al., Genetic algorithms approachto voltageoptimization , Proc. First International Forum onApplicationsof Neural Networks to Power Systems, NewYork, NY, USA, IEEE 1991

[20] H. Mori, A genetic approach to power systemtopological observability , Proc. IEEE InternationalSymp. on Circuits and Systems, New York, NY,USA,IEEE 1991[21] K. Iba, Reactive Power Optimization by GeneticAlgorithm , PICA93, Phoenix, AR USA, May 1993[22]X.Yin, N. Germany, Investigations on Solving the LoadFlow Problem by Genetic Algorithms , Electric PowerSystems research,22 (1991). 1991 Elsevier[23] V. Miranda , Using fuzzy reliability indices in a decisionaid environment for establishing interconnection andswitching location policies , Proc. CIREDP1, s.6, Libge,Belgium, Apr 1991[ I V. Miranda, F.M. Barbosa, Three phase load flow forradial networks', Proc. of MELECON'83, rp D4.07,

Athens, Greece, 1983

tVladimiro Miranda was born in Oporto, Portugal, on March11, 1955. He received his Licenciado, Ph.D. and Agregadodegrees from the Faculty of Eng. of the University of OportoFEUP)n 1977, 1982 and 1991, in Electrical Engineering. In1981 he joined FEUP and currently holds the position ofProfessor Associado. In 1985 he joined also INESC ResearchEngineering Institute for Systems and Computers and holdspresently the position of Project Manager Head ofInformation and Decision in Energy Systems area.J o b Vasco Ranito was born in Oporto, Portugal, on January23, 1965. He received his Licenciado and M.Sc. degrees fromFEUP in 1988 and 1993, in Electrical Engineering andComputers. In 1988 he joined INESC as a researcher in theInformation Systems area. He presently holds there theposition of Team Leader in the Software EngineeringDevelopment Center and is a Ph.D. student.Luis Mirruel Proenca was born in Lisbon, Portugal, on May10, 1966. He received his Licenciado degree from FEUP in1989 in Electrical Engineering and Computers. He is now aM.Sc. student at INESC, in the Information and Decision inEnergy Systems group.