A modified genetic algorithm (GA) is proposed for water distribution network optimization. ... demands at nodes, the optimal design of a looped network for. 3467 ...
WATER RESOURCES RESEARCH,
VOL. 35, NO. 11, PAGES 3467-3473, NOVEMBER
1999
Water distribution network optimization using a modified genetic algorithm Pilar Montesinos
•
Departmentof Agronomy,Facultyof Agriculturaland ForestryEngineering,CordobaUniversity Cordoba,Spain
Adela
Garcia-Guzman
Departmentof Statistics,Facultyof Agriculturaland ForestryEngineering,CordobaUniversity,Cordoba,Spain
JoseLuis Ayuso Departmentof Rural Engineering,Facultyof Agriculturaland ForestryEngineering,CordobaUniversity Cordoba,Spain
Abstract. A modifiedgeneticalgorithm(GA) is proposedfor water distributionnetwork optimization.Severalchangesare introducedin the selectionand mutationprocesses of a simpleGA. In eachgenerationa constantnumberof solutionsis eliminated,the selected ones are ranked for crossover,and the new solutionsare allowed to undergo at most one mutation.All thesemodificationsgreatly increasethe algorithmconvergence.The modifiedGA is testedon the New York City water supplyexpansionproblem.It obtains the lowest-costfeasiblesolutionreported in the literature in far fewer generationsthan any previousGA. 1.
Introduction
The highinvestmentand maintenancecostsassociated with both new water distributionnetworksand the expansionof existingoneshaveled hydraulicengineersto take greatinterest in mathematicalmethodsto find their optimaldesign,that is, the minimum
cost network.
Over the last 3 decadesa large number of optimization modelshave been developed.Few of them attempt to obtain the optimal layout [Rowelland Barnes, 1982; Morgan and Goulter,1985].Instead,mostmodelsstart from a givenlayout of the hydraulicelements(pipes,valves,tanks,etc.) and a set of specifieddemandpatternsto find the minimalcostnetwork. The latter group of methodsis basedon many different optimizationtechniques(linear and nonlinearprogramming,etc.) combinedin severalways accordingto the type of network (branched or looped)andthemathematical problemformulation (continuous or discretedecisionvariables).Unlike the looped networkproblem,for whicha definitivesolutionmethodto guaranteethe globaloptimumdoesnot yet exist,branchednetwork optimization is a solvedproblem[Martinezet al., 1995]. Recently,severalheuristictechniqueshave been appliedto the pipenetworkoptimizationproblem.The geneticalgorithm (GA) is one of these new methods [Goldberg, 1989; Michalewicz,1994;Martinezet al., 1995;Halhahl et al., 1997]. Basically,a GA is a searchprocedurefor the minimum or maximumof an unconstrained functionusingrandomselection processes which simulatethe followingliving creaturereproductionoperators:selection,crossover, and mutation.The ini-
tial population(set of networks)is randomlygenerated,and from this populationnew setsof solutionsare createdby selectingand modifyingthe lowestcostnetworks.Any selected solutioncan undergotwo kinds of transformations:crossover and mutation.Crossovercreatesnew solutionsby combining partsfrom othersolutions, whilemutationaltersa smallpart of one solution.With efficientproceduresof selectionand crossover,the bestpartial solutionseasilypassfrom one iterationto the next. Justas in naturalpopulations,the fittestindividuals producemore offspring,spreadingtheir genesover the next generation.Although mutationis necessaryto create new solutionsthat cannotbe producedby other meansand to maintain diversityof the genepool, it involvesthe risk of destroying good existingsolutions.GAs have already been applied to obtain optimalwater distributionnetworksand have demonstratedtheir capacityto obtain better solutionsthan classical methods[Simpsonet al., 1994;Dandy et al., 1996;Montesinos, 1995;Savicand Walters,1997]. In this paper we present a new GA that includesseveral changesto a simpleGA in order to optimizeits convergence. The main changesintroducedare the way in which the solutionsare selectedand mutatedto form the nextpopulation.In particular,a constantnumber of solutionsis eliminated,the selected ones are ranked for crossover,and each formed solu-
tion is allowedto undergoat mostone mutation.The penalty terms included in the cost function are also described. The effec-
tivenessof the proposedalgorithmhasbeendemonstrated by its applicationto the New York City water supplyexpansion problem, previouslystudiedby severalauthors[e.g., Morgan and Goulter,1985;Fujiwaraand Khang,1990;Dandyet al., 1996].
•Now at Instituteof EnvironmentalandNatural Sciences, Lancaster University,Lancaster,England.
2.
Copyright1999by the AmericanGeophysicalUnion.
Formulation
Paper number 1999WR900167.
Given the layoutof hydraulicelementsand a set of specified demandsat nodes,the optimaldesignof a loopednetworkfor
0043-1397/99/1999W R900167 $09.00 3467
Looped Network Design Problem
3468
MONTESINOS
ET AL.: WATER DISTRIBUTION
NETWORK
gravitysystems is definedby the setof pipesizeswhichresults
Pipe1 Pipe2 Pipe3 Pipe4 Pipe5
in the minimum investmentcost. The overall optimization
I
problemcanbe statedmathematically in termsof pipeflows
I o0ooI d•
and unit lengthcostsas follows: Minimized
OPTIMIZATION
d2
I 011 I 0oo I d3
d4
d5
Figure 1. Trial network.
cost
c=E E
z) =
i=
ha)
between 50 and 2000 individuals;nevertheless,Dandy et al.
iGD j=l
[1996]suggest N rangingbetween100and 1000individuals. All solutionsare analyzedhydraulicallyto give somemeasure of their fitness,related to their investmentcost and the
subjectto
EQi--EQj--Qextk i•nll•
=
n
(2)
j•n2l•
E gliQ•= AEj
j-
1,''',m
(3)
iGn3j
penaltycostwhichis associated withconstraint violations. The fittestindividualsare selectedto makeup a newpopulationof solutions.This selectionprocessis controlledby the selection probability p• relatedto the fitnessof eachstring.Afterward,
somemembersof the new populationmay undergocrossover in accordance with somespecifiedcrossover probability p c for eachpair of individualsselectedin the previousstep.This (5) probability Prmin • P• --
