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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 3, AUGUST 2002879Application of Evolutionary Algorithmsfor the Planning of Urban DistributionNetworks of Medium VoltageEloy Daz-Dorado, Jos Cidrs, Member, IEEE, and Edelmiro Mguez, Associate Member, IEEEAbstractCurrently,animportantissueinpowerdistributionisthe need to optimize medium voltage (mv) networks serving urbanareas.Thispapershowshowanevolutionaryalgorithmcanbeusedasthebasisforthetypeofefficientalgorithmsuchoptimizationde-mands. The search for optimal network solution will be restrictedto a graph defined from the urban map, so each graph branch rep-resents a trench. The solution space (networks) is assumed with“loop feeder circuits”: networks with two electrical paths from thehighvoltage/mediumvoltage(hv/mv)substationstothecustomers.In the optimization process, the investment and loss load costs areconsidered taking into account the constraints of conductor capac-ities and voltage drop. The investment costs will take into accountthat some cables can be lying in the same trench. The process wasapplied for a Spanish city of 200000 inhabitants.IndexTermsEvolutionaryalgorithm,mediumvoltagenetworkplanning, network design, urban distribution network.I. INTRODUCTIONURBANmediumvoltage(mv)distributionnetworksarede-signed with two criteria in mind: 1) minimal cost and 2)reliability of supply. To obtain a reliable supply, the techniquemost employed consists of searching for configurations withloop feeders, so that all mv/lv substations have two possiblepaths from (high voltage/medium voltage (hv/mv) substations,in which the system is operated as radial configuration 13.The mostcommonlyusedurban mvnetworkconfigurations2,3 are: ring, interconnective, and clasp (see Fig. 1). The ex-tremes of the feeders of ring and interconnective configurationsare hv/mv substations. The clasp configuration has switchingstations toresupplycustomersafter afault has beenisolated anduses a reserve feeder 2, 3 from the switching station to thehv/mv substations that is normally opened.The search for minimal cost network configuration is doneassuming certains constraints, costs, and switching needs.The bibliography related to planning mv urban networks isscarce 210 and is based on heuristic methods. Particularly,some of the models are expansion plans based on the resolutionof the multiple traveling salesman problem (m-TSP) or ofmultiple vehicle routing problem (m-VRP) 27. In theseheuristic optimization models the losses and voltage dropare treated only with post-optimization processes, and theManuscript received March 12, 2001; revised February 18, 2002. This workwas supported in part by the Union Fenosa S.A. Electric Utility.The authors are with the Departamento de Enxeeria Electrica, Universi-dade de Vigo, Vigo 36289, Spain (e-mail: ediazuvigo.es; jcidrasuvigo.es;edelmirouvigo.es).Publisher Item Identifier 10.1109/TPWRS.2002.800975.Fig. 1.Urban distribution network mon trench problem is not considered. Only Freund 8and Burkhardt 9, 10 consider the possibility of laying twocables in the same trench.In this paper, a technique based on evolutionary algorithms11 is proposed to solve mv urban networks with ring, clasp,and interconnective configurations, taking into account the in-vestment and losses costs, the constraints of conductor capac-ities, and voltage drops, and trenches with more that one con-ductor. The position of the switch that must be open is deter-mined to obtain the lowest losses for each open loop.This paper is organized as follows. In Section II, a completedescriptionoftheproposedevolutionaryalgorithm ispresented.SectionIIIpresentsanexamplewiththemvnetworkforacityinSpain, the curves for the evolution of the optimal solution, anda table of results with different configurations. The conclusionsare summed up in Section IV.II. EVOLUTIONARYALGORITHM FORURBANDISTRIBUTIONMV NETWORKSThe design of the mv urban network can be defined with agraphcontaining all possible routes. This graphis obtained from the city map, whereis the set of branches(trenches) andis the set of nodes. The nodes include thehv/mv and mv/lv substations, loads, switching stations, and theintersections of branches. The coordinates of the hv/mv andmv/lvsubstations,mvcustomersandswitchingstations,andtheload of the mv/lv substations and mv customers are known.Evolutionary algorithms work with a population of individ-uals (codified solutions), which is able to evolve in a given en-vironment by application of the selection, crossover, and muta-tionoperators.The“elite”orbestindividuals(solutions)surviveduring the optimization process.Each individual or “chromosome,” represents a complete so-lution of the mv network. The chromosomes are integer stringsthatcodifytheconnectionbetweenthenodesofthemvnetwork.0885-8950/02$17.00 2002 IEEE880IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 3, AUGUST 2002Fig. 2.Example of an mv network and the codification.The first step of an evolutionary algorithm is to generate theinitial population. The initial population consists ofdifferentchromosomesgeneratedrandomly,thatrepresentdifferentso-lutions for the configurations in Fig. 1.A random sampling of this initial population is selected tocreate an intermediate population, and the crossover and muta-tion operatorsareappliedtothemin ordertoobtain anew popu-lation withchromosomes. This process is called a generationand it is repeated until a stop function is decided.The population of the next generation must havechromo-somes. Thesechromosomes will be the best (minimal costfunction) of the set composed by theandchromosomes ofthe previous generation.The crossover and mutation operators must obtain new chro-mosomes that represent urban distribution networks in Fig. 1(with loop feeders).A. CodificationThe codification used to represent a network is a variable-length integer string. The chromosomeon generation , isdefined by(1)whereare the genes of the chromosome and contain nodenumbers of the graph. The string represents the sequence ofnodes that shape the network loops and the values can be pos-itive (cross) or negative (connection). When the network hasmore than one loop, the string represents the consecutive se-quence of the nodes of all loops (see Fig. 2). When there aresome loops going through a node, the node number will be re-peated on the chromosome (e.g., nodes 4, 7, 8, and 12 in Fig. 2).The negative value of a gene means that the hv/mv or mv/lvsubstation or the switching station located at this loop node isconnected.Thesenodeswillbecalledprincipalnodes.Thepos-itive values represent the sequence of nodes of the path betweentwo consecutive principal nodes across the graph.In the example of Fig. 2, the sets of principal nodes of thethree loops are 4, 1, 3, 7, 4, 5, 13, 7, and 4, 10, 8, 7,respectively. Segments (8, 12) and (7, 9) belong to loops 2 and3. The mv/lv substation of node 8 is connected on loop 3 and,therefore,node8isaprincipalnodeofloop3andanormalnodeof loop 2.Fig. 3.Reconnection of two segments of the same loop.B. Initial PopulationThe initial population, randomly generated, consists ofchromosomes. The network codified by each chromosomemust have, for all the mv/hv substations, two possible pathsfrom hv/mv substations or switching stations. These pathsare selected by applying the heuristic algorithm proposed inSection II-F.C. Mutation OperatorThemutationoperatorisappliedindividuallytoeachselectedchromosomeofthepopulation,andtheresultisanewindividualof the intermediate population. The mutation is the most im-portant operator of the algorithm, and the modification implieschanges of connections between the customers and the numberof loops. This operator is based on a topological transformationof the codified network. The method consists of selecting twosegments of the network associated to the chromosome, definedby:1) both extremes of the segments are principal nodes(normal nodes can be between the principal nodes); or2) fictitious segments between hv/mv substations and/orswitching stations.The probability of selecting two segments is a function of therelationbetweenthecurrentandthenewlengthafterthechange.The pairs of segments with greater length reduction have moreprobability of beingselected. Thetwo selected segmentswillbeeliminated and replaced by two new segments reconnecting theprincipal nodes with two paths crossed with the previous. In thenext examples the different possibilities when the branches areselected are represented. To simplify the examples, they haveonly principal nodes.1) The segments can belong to the same loop, and the resultis a new loop (see Fig. 3).2) When the segments belong to different loops, there aretwo possible solutions see Fig. 4(a) and (b). When theloops have a switching station on an extreme, only thesolution with a hv/mv substation on both loops can beselected.3) Whenthetwo selectedsegmentsbelongtodifferentloopsandbothareincidentwithhv/mvsubstationsorswitchingstations, the result is one loop and a fictitious branch (seeFig. 5).4) Likewise,afictitioussegmentandasegmentofaloopcanbe selected and the result is two loops (see Fig. 6).5) When the extreme principal nodes of a new segment arenot adjacent in the graph, it is necessary to determine aDAZ-DORADO et al.: APPLICATION OF EVOLUTIONARY ALGORITHMS881(a)(b)Fig. 4.(a) Reconnection type 1 of segments of different loops. (b)Reconnection type 2 of segments of different loops.Fig. 5.Reconnection resulting in a fictitious segment.Fig. 6.Connection with eliminated fictitious segment.path that connects these nodes. The employed method isdeveloped in Section II-F.D. Crossover OperatorThe crossover operator belongs to the designated group “bi-sexual,” because it is applied to two chromosomes (parents),obtaining two new chromosomes (children), with mixed char-acteristics of both parents. The steps are as follows (see Fig. 7):Step 1) select two chromosomes of thepopulation (parents);Step 2) obtain the set of pairs of principal nodes that areeliminated (each pair separately), make discon-nected the graph corresponding to the union of thenetworks associated to both individuals (taking intoFig. 7.Example of crossover operation with two changes.account that the hv/mv substations and switchingstations are connected with fictitious branches);Step 3) for each pair of nodes, the decision is taken to ex-change the existing isolated subnetworks betweenboth individuals, with a probability of 50%.The example Fig. 7 has pairs of nodes (3, 6) and (11, 17) that,if one of them is eliminated, the graph is disconnected. In theexample all the nodes of the graph are principal nodes.E. Selection OperatorThe selection operator belongs to the type called elitist, be-cause it is used for the selection of thebest chromosomes.The elements of the population will be selected from among thechromosomes with minor costs from the set formed by thechromosomes of the previous population, plus theindividualsobtained with the mutation and crossover operators.F. Search for the Path Between Two Principal NodesWhen two principal nodes must be connected (crossover op-erator and initial population), it is necessary to obtain a path onthe graph, and this does not have to be the shortest.The process begins from the two principal nodes at the sametime, until both paths are connected. Let node , be the extremeof a calculated section of the path from node, andtheset of adjacent nodes to ; the probability of the following nodeselected being nodeis(2)882IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 3, AUGUST 2002Fig. 8.Weight ellipses defined by the points?and?.Fig. 9.Resultant path from?to?.whereis the weight of the node:(3)To obtain the weights for the different adjacent nodes, theellipses with the focus on pointsandwill be considered (seeFigs. 8 and 9)G. Objective FunctionTheobjectivefunctiontominimizeisthesumofthetotalcostfor the loops, trenches, and hv/mv substations plus a penaltyfunction for unfeasible networks. It is important to allow unfea-sible networks into the population, because good solutions canbe the result of operating with unfeasible networks and becausethe operators do not ensure feasible descendants when the par-ents are feasible.The optimal solution of the mv network must fulfill the con-straints of conductor capacities and voltage drops. Each oneof the loops of the network must be able to supply all theirloads from both extremes (emergency operation). The cost ofthe trenches is calculated separately, because it cannot be con-sidered a term of the cost of the loops.When one of the extremes of the loop is a switching stationthe backup line is open, and only the conductor type of the loopmust be determined. If the loop has two hv/mv substations onthe extremes, it is necessary to obtain the open branch of theloop to minimize the losses.Fig. 10.The mv network of Vigo with three hv/mv substations and sixswitching stations.Given a loop whose extremes are hv/mv substations (bothdifferent or the same), the obtaining of the minimal cost of theloop, implies determining: the optimal conductor of each loopand thelocationoftheopen branch.The costof aloop fora typeconductor will be(4)whereandare the investment and losses unitarycosts, respectively, for the typeconductor.In the proposed algorithm, different types of conductor areadmitted, but each loop will be formed by only one type. Themost unfavorable situations are given when all the power of theloop is supplied from one extreme (minimal section).When the cost of a loop with branchopened isknown, the cost with the branchopened can be ob-tained as(5)whereis the cost variation of the losses, and its value is(6)withwhereis the set of branches located between nodeandthe hv/mv substation that is supplying it power.Only whenis negative is it possible to reduce the cost.To prove when an open branch reduces the cost, (7) must beverified(7)DAZ-DORADO et al.: APPLICATION OF EVOLUTIONARY ALGORITHMS883(a)(b)(c)Fig. 11.(a) Cost evolution curve. (b) Cables and trenches lengths evolution curve. (c) Number of loops evolution curve.Through the recursive checking of the adjacent branches, thebranch that must be opened can be determined, taking into ac-count that the cost curve of the losses is convex with respect tothe optimal branch.Once the optimal open branch is obtained, it is necessary todetermine the optimal conductor type. When the cost of a loopis known, with branchopened and the conductor type, the cost with the branchopened and the conductortypecan be obtained as(8)The cost variation, when the conductor typeis changed fromtoand the open branch isis(9)Equations (6) and(9) are functions only of thedata of thetwoimplicated branches in the change and their adjacent nodes.The values ofandof each principal node andand the minimal cost of each loop can be associated tothe chromosomes and must only be recalculated by the loopsmodified with the mutation and crossover operators.H. Stop FunctionThe number of generations (iterations) of the algorithm isvariable. The process is stopped when the population is homo-geneous and there are not variations of the population duringenough generations.III.III. EXAMPLEThe example of Fig. 10 represents the mv network of Vigocity (Spain) with 200000 inhabitants, 242 mv/lv substationsTABLE IRESULTS FORVIGOCITY(195 MW installed), three hv/mv substations, and six switchingstations. The graph has 1852 nodes and 2996 branches.Some of the initial networks are unfeasible. The evolution ofthepopulationtowardtofeasiblesolutionsisveryfast,duetothemutation operator. Later evolution is slower, and it correspondswith a fine fitting, in which the crossover operation has moreeffect. The total cost, after 60000 iterations of a (6, 10)-ES, is7.84 MC (1C0.9$). The network is composed of 19 loops,with 56.9 km of trenches and 71.8 km of cables. The curves ofFig. 11(a)(c) represent the evolution of the cost, the cable, andtrench lengths, and the number of loops for the best chromo-some of the population of each generation. Table I presents theoptimal results with different numbers of hv/mv substations (2or 3), with or without switching stations.IV. CONCLUSIONSThe proposed method makes it possible to obtain the mv net-work from a large city, knowing the position of the mv/lv sub-stations, hv/mv substations, and switching stations, by consid-ering investment and losses costs and constraints of conductorcapacities and voltage drop. The investment costs will take intoaccount that some cables can be lying in the same trench. Thecost of the losses are calculated considering the switch whichmust be opened in each loop, for optimal radial operation. The884IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 3, AUGUST 2002Fig. 12.Costs for typical Spanish commercial conductors.TABLE IICOSTS FORCOMMERCIALSUBSTATIONARRANGEMENTSTABLE IIICOSTS FORCOMMERCIAL HV/MVTRANSFORMERSresolutionforlargecitiesisfeasibleinreasonabletimes,withoutmaking simplifications in the objective function or in the graph.The proposed method can be modified for application to anexpansion planning model. In this case, the objective functionmust consider the existing lines.APPENDIXThe conductor costs are represented by Fig. 12.The trench costs are 48000 C/km under the sidewalks and84000 C/km under the road.The hv/mv substation costs are shown in Tables II and III.The economic parameters are: 0.04 C/kW-losses, 25-yearplanning, 25% overload factor, 1% annual inflation and 5%annual interest.REFERENCES1 H.L. Willis,Power DistributionPlanning ReferenceBook.NewYork:Marcel Dekker, 1997.2 Z. Bozic and E. Hobson, “Urban underground network expansion plan-ning,” IEE Proc., Gener. Trans. Distrib., vol. 144, pp. 118124, Mar.1997.3 Z. Bozic, “Software system for hv network expansion planning,” Ph.Ddissertation, University of South Australia, June 1996.4 Y. Backlund and J. A. Bubenko, “Computer aided distribution systemplanning,” Electr. Power Energy Syst., vol. 1, no. 1, pp. 3545, 1979.5, “Computer aided distribution system planning: Part 2Primaryand se
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