Genetics and Replacing Pipes

Computer-Aided Civil and Infrastructure Engineering 23 (2008) 625635

Using Evolutionary Optimization Techniquesfor Scheduling Water Pipe Renewal Considering

a Short Planning Horizon

Leila Dridi

National Research Council Canada, Institute for Research in Construction, 1200 Montreal Road,M20, Ottawa, Ontario, Canada

Marc Parizeau

Departement de genie electrique et de genie informatique, Universite Laval, Pavillon Adrien-Pouliot,Quebec (Quebec), Canada

Alain Mailhot & Jean-Pierre Villeneuve

Institut national de la recherche scientique, Centre Eau, Terre et Environnement (INRS-ETE),490 rue de la Couronne, Quebec (Quebec), G1K 9A9, Canada

Abstract: The maintenance and management of under-ground infrastructures is a growing problem for a ma-jority of municipalities. The maintenance costs are in-creasing while the financial resources of municipalitiesremain limited. Water distribution system (WDS) man-agers therefore need tools to assist them in the elabora-tion of pipe renewal management plans. In this article, re-sults of a newly developed strategy for pipe renewal basedon a cost function are presented. The strategy allows theminimization of a cost function while also consideringhydraulic criterion. This strategy was tested on a shortplanning horizon of five years. The pipe number to bereplaced and the optimal moment for renewal are identi-fied using three different optimization techniques: IGA(Island Genetic Algorithm), NPGA-2 (Niched ParetoGenetic Algorithm 2), and NSGA-II (Non-dominatedSorting Genetic Algorithm-II). The proposed approachhas five distinctive features: (1) it is coupled with a flexi-ble evolutionary framework that allows the user to select

To whom correspondence should be addressed. E-mail: [email protected].

any type of operator for IGA or any kind of multiob-jective genetic algorithm; (2) it uses the hydraulic sim-ulator Epanet2.0 which allows steady state or dynamicsimulations; (3) it considers a probabilistic break modelto evaluate the structural deterioration of pipes; (4) it in-tegrates a Bayesian approach for the estimation of thepipe break model parameters that take into account theinfluence of inherent uncertainties related to the qualityof data during the decision-making process; and (5) itsimulates the variation of the pipes roughness over theyears. The developed strategy/model is explained usingan example that allows us to elucidate its most importantcomponents. Simulation experiments on a small network(100 pipes) are presented. A comparison of three evolu-tionary algorithm results is provided. Tests showed thatIGA performs well, but for networks of larger sizes, werecommend increasing the number of demes to reach bet-ter solutions. Higher quality results were achieved withNSGA-II than NPGA-2 on differently sized networks.We recommend the use the NSGA-II to optimize largeWDS. Future developments for this strategy are alsodiscussed.

C 2008 Computer-Aided Civil and Infrastructure Engineering. Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA,and 9600 Garsington Road, Oxford OX4 2DQ, UK.

626 Dridi, Parizeau, Mailhot & Villeneuve

1 INTRODUCTION

Distribution networks are constructed to satisfy theneeds of users and to ensure/provide fire protection.The distributed water must be of good quality and sup-ply a sufficient pressure. However, over the years, pipesdeteriorate, breaks become more and more frequent,and certain networks fail to satisfy the required pres-sure. To improve a networks state, managers need toolsthat allow them to make good decisions regarding whenit is essential to replace or rehabilitate network pipes,taking into account the financial resources, which inmost cases are limited.

The objective of this article is to present and dis-cuss optimization results of a new strategy/model forpipe renewal that was tested to minimize total break re-pairs and replacement costs and to identify when andwhich pipes need renewal. This strategy was tested us-ing three evolutionary optimization algorithms to finda solution to the problem. The first algorithm is IGA(island genetic algorithm), which considers the fitnessvalue to be optimized to find the optimal solution (orclose to optimum). The advantage of the considered ge-netic algorithm (GA) is that the search process for thebest solution is realized on various subspaces. The sec-ond algorithm is the NPGA-2 (niched Pareto geneticalgorithm), a multiobjective technique which is an im-proved version of the NPGA first developed by Hornet al. (1994) and tested by Erickson et al. (2001) inground water quality management. The last algorithmis named NSGA-II (non-dominated sorting genetic al-gorithm), a multiobjective technique that is a revisedversion of the NSGA developed by Srinivas and Deb(1994). This version is more efficient computationallythan its predecessor.

The article is organized as follows. The followingsection presents a brief literature review of the mainpublished works related to rehabilitation/replacementproblem. Section 3 presents the proposed renewal strat-egy applied on a short planning horizon. Section 4presents the mathematical formulation of the problem.Three evolutionary optimization techniques are pre-sented in Section 5. Finally, the results are discussedand the different techniques are analyzed and comparedconsidering a 100-pipe network.

2 LITERATUREREVIEW

The rehabilitation/replacement problem was previouslyaddressed at the pipe level by several authors. For ex-ample, Shamir and Howard (1979) were among the firstwho presented a method to determine the optimal re-

placement time for a pipe. Kleiner et al. (1998a, 1998b)also proposed an approach of long-term renewal plan-ning applicable to small WDS (less than 20 pipes). De-terioration of structural integrity and hydraulic capac-ity of pipes were both considered. The study of Kimand Mays (1994) differs from previous works becauseit includes, in addition to replacement, rehabilitationand break repair costs, a fourth cost function associatedwith pumping, with the objective of minimizing the sumof the four costs. In Europe, the CARE-W (computer-aided rehabilitation of water networks) model was cre-ated within a European research project framework,with the association of 10 research centers (Le Gauf-fre and Eisenbeis, 2004). This model includes five mainmodules, linked with a database and data managementtools. The first module is for the evaluation of per-formance indicators related to network rehabilitation(performance indicators). The second module includesfailure prediction tools for breaks and leaks (failureforecasting). The third module is used to assess the hy-draulic reliability of pipes in the network (water supplyreliability). The fourth module helps during the con-struction and comparison of different long-term man-agement strategies (long-term rehabilitation planning).Finally, the last module is dedicated to the multicri-teria decision-making method for the construction ofan annual rehabilitation program (annual rehabilitationplanning).

The applications of GAs in science and engineer-ing are growing. The variety of application is numer-ous and includes civil engineering-structural optimiza-tion, environmental engineering (Erickson et al., 2001),and many other fields. In water distribution networks,GAs are used to solve design problems (Dandy et al.,1996; Walters et al., 1999; Vairavamoorthy and Ali,2000; Savic and Walters, 2000; Wu et al., 2001), renewalplanning problems (Dandy and Engelhardt, 2001), andpump scheduling problems (Savic et al., 1997).

The standard GA is an algorithm that first requiresthe coding of all of the parameters of the optimizationproblem in a finite length string. The principle of GA issimple. It consists of the evolution of a population of in-dividuals until a stop criterion. At first, an initial popula-tion of individuals (or solutions) is generated randomly.Then, in each generation, individuals are selected; thisselection is based on an objective function called fitness.Then, two other operators, crossover and mutation, areapplied to generate a new population of solutions. Thisprocess is repeated until a stop criterion is satisfied. Thecriterion most often used is a maximal number of gen-erations. Generational GA first generates an initial pop-ulation and then evaluates the fitness of all individuals.Individuals are then randomly selected for reproduction

Using evolutionary optimization techniques 627

according to the survival principle of the most adapted.The individuals children (or offspring) are created byapplying the following two genetic operators: crossoverand mutation. Those children are placed in a new popu-lation P(t) and will replace, in all or in part, the popula-tion of the previous generation. New populations of in-dividuals are then going to succeed iteratively from onegeneration (t) to the next generation (t + 1). The GAdescribed above is said to be generational because allthe children of the individuals generated are placed in apopulation and will replace completely the populationof parents.

Dandy and Engelhardt (2001) used a GA with a singleobjective to schedule the optimal pipe replacement inAdelaides metropolitan network (Australia) composedof 488 pipes. The only costs considered were replace-ment and break repair costs (direct and indirect costs).Rahman et al. (2005) have provided an overview of theexisting social costs related to municipal infrastructureconstruction, maintenance, repair, rehabilitation, andrenewal. The objective of their report was to stress theimportance of considering social costs in the calculationof total construction and maintenance costs of munici-pal infrastructure projects. Their study has shown thatsocial costs can account for up to 400% of constructioncosts on certain projects.

Multiobjective approaches have also been suggested.Halhal et al. (1997, 1999) used the structured messy ge-netic algorithm to solve this problem. Their approach isbased on the maximization of the benefit resulting fromthe adoption of a solution and minimization of solutioncosts, under the constraint of available budgets for theimprovement of the networks state. The total benefitof one solution is the sum of four benefits weighted andchosen by the user: hydraulic benefit, physical integritybenefit, flexibility benefit, and quality benefit.

3 PROPOSEDRENEWALSTRATEGY

The flowchart (Figure 1) summarizes various steps ofthe proposed renewal strategy. The principle of therenewal methodology developed consists of supposingthat one break model represents the break data avail-able for the network considered (Weibul-Exponentialin our case). In reality, a preliminary stage is requiredand consists of identifying the break model, which bestfits the available real data. The reader who is inter-ested in this stage can consult an example of the workdone by Mailhot et al. (2000). The researchers devel-oped a methodology based on survival analysis to modelthe pipe structural deterioration for the municipality ofChicoutimi (Quebec), having a short-break history and

Real Data in water distribution system

Break model X

Break model Weibul-Expo Break model Y

Estimation of BBM parameters with

Bayesian approach

Network objective function

Optimization with: Island GA, NPGA-2 and NSGA-II

Best solutions : 1st, 2nd, etc.

Results visualization

Identification of the Best Break Model (BBM)

Posterior Distribution of parameters

Prior Distribution of parameters

Fig. 1. Flowchart of a short-term renewal strategy.

testing four various break models. Mailhot et al. (2000)explained how it was possible to compare between twobreak models, with the goal of identifying the most suit-able and statistically significant model.

The following step consists of using a Bayesian ap-proach to estimate parameters of a supposed breakmodel. The Bayesian approach is a theory, which isbased on Bayess theorem. This theorem allows pass-ing from a prior distribution, to a new posterior distri-bution, which integrates all information included in theobservation data. The result is a distribution illustrat-ing occurrence probability of various values for an un-known parameter, considering a prior distribution forthis unknown parameter and information contained inreal observations. Uncertainty in the modeling of thepipes structural deterioration is explicitly introduced bybuilding this distribution. Then, the value, which corre-sponds to the maximum of the posterior distribution, isused as an estimator for the unknown parameter. Thedetailed description of the probabilistic break modelused in this work, developed by Mailhot et al. (2003),and how parameters of this break model are estimatedby a Bayesian approach are outside the scope of this ar-ticle, but can be found in Dridi (2006) or Dridi et al.(2005). Once the parameters of the break model are


identified for each break order, a relation between pa-rameters and break order is used to estimate other pa-rameters with a higher order. All the parameters esti-mate the evolution of the average number of breaksfor every pipe in the network. This average number isa function of the number of breaks recorded by eachpipe. The third stage shows whether it is economical torenew a pipe or not. An objective function consideringrepair costs and future replacement cost is used. The re-pair break function considered for each pipe depends onthe recorded breaks in the same pipe. At this stage, thehydraulic component, which is also used for deteriora-tion capacity, is also considered. Then, an optimization,based on a 5-year interval to minimize cost function, isrun and best solutions are provided. The supposed plan-ning horizon used in this study was equal 5 years to re-flect a short horizon planning level. However, this inter-val can be modified easily.

To resolve the optimization problem, an evolutionarytechnique was selected. The choice of genetic algorithmvariety as the technique of optimization was forced bythe fact that there were a large number of combinationsto estimate. For example, let us consider a case whereonly two options are available: to replace or not to re-place pipe. These two options give a total number ofcombinations for a simple network with 100 pipes as2100, which is roughly 1.26 1030 combinations, whichis extremely large, especially if we expect to add a timeinfluence (replace first year, second year, third year,etc.). Besides the fact that GA can resolve large prob-lems, they are well adapted to resolve discrete, nonlin-ear problems, with or without constraint and they do notrequire auxiliary calculations as derivative calculations.

4 MATHEMATICALFORMULATIONOFTHEPROBLEM

The proposed algorithm has the goal of minimiz-ing a cost function, including a break repair termand a replacement term for all pipes in the network(Equation (1)). We supposed that the present time is t,with an annual step and a planning horizon of 5 years.Thus, each pipe in the network can be replaced be-tween (t + 1) and (t + t), for the first and lastyears, respectively. We supposed in this model that thewhole break history in the network is known. By con-sidering a discrete notation, the optimization problem(minimization) of cost is formulated with Equations (1)to (11):

MinNj=1

[CTot( j ; Tf ( j) |k( j))] (1)

Where: N: Total number of pipes in the network; j: Pipeindex; CTot( j ; Tf ( j) |k( j)): Cost function for the pipe j;Tf (j): Replacement time for pipe j; k(j): Total numberof breaks recorded by pipe j, at present time;

CTot( j ; Tf ( j) |k( j))

= Cr ( j)( j)(1 + R)(Tf ( j)ta) +

k( j)i=1

Cb( j)(1 + R)(Ti ( j)ta)

+Tf ( j)t=t +1

Cb( j)M(t 1, t |k( j); t )(1 + R)(tta) if k( j) = 0 (2)

Cr ( j) = 1.1D( j)1.15 (3)

Cb( j) = 16.5D( j)1.15 (4)Where: Cr(j): Replacement unit cost for the pipe j($/m);Cb(j): Unit repair cost for the pipe j ($/break); D(j): Di-ameter of the pipe j(mm); (j): Length of the pipe j(m);R: Annual discount rate; ta: Discounted time for costs;Ti(j): Break occurrence time for ith break of pipe j; t:Present time; M(t 1, t |k( j); t ): Average number ofbreaks for the pipe j during year (t 1);

Cj (t) = 18 37.2 log10[ke0 + ak[t t0( j)]

D( j)

]t > t (5)

Cj (t) = 18 37.2 log10[ke0 + ak[t Tf ( j)]

D( j)

]t > Tf ( j)

(6)Pipesj=1

Q( j, t)in Pipesk=1

Q(k, t)out = Demnode,

node {1, . . . , MaxNode} (7)mLoop j

Hm = 0, j {1, . . . , MaxLoop} (8)

H( j, t) = 10.67( j)D( j)4.87Cj (t)1.852

Q( j, t)1.852;

j {1, . . . , N} (9)H(node,t) Hmin(node, t) node {1, . . . , MaxNode}

(10)Penalty = max

t(t=t +1,t +5)

[

maxnode

(node=1,MaxNode)[max(0, Hmin(node) H(node, t))]]

(11)

Where: Cj(t): HazenWilliams coefficient for pipe j; ak:Roughness rate growth (m/year); ke0: Initial roughnessof pipe j(mm); Q(j, t): Flow in pipe j at time t; Demnode:Demand node; H(node, t): Pressure head; Hmin(node,t): Minimal required pressure; H(j, t): Head losses;Penalty: Pressure penalty function.


The cost function (Equation (2)) for a pipe havingrecorded breaks (k(j) = 0), contains a first term for re-placement during year Tf (j), a second term for breakrepair k(j) undergone by the pipe at present time t,and a third term for future break repairs estimatedfrom the average number of breaks between presenttime and replacement time M(t 1, t |k( j); t ). Calcu-lation details of this last term can be found in Mailhotet al. (2003) or Dridi (2006). For a pipe having recordedno breaks (k(j) = 0), the average number of breaks,M(t 1, t |k( j) = 0; t ) for the annual interval, is sup-posed to be zero and the cost function for this pipe hasonly one term for replacement at Tf (j). Equations (3)and (4) identified, respectively, the unit replacementcost and the unit repair cost of each break. Theunit repair cost is supposed to be a function of pipediameter.

To take into account the variation of the HazenWillians coefficient over the years, one of the roughnessprediction equations developed by Sharp and Walski(1988), and used by Kleiner et al. (1998a), is considered.Equation (5) gives the calculation example of this coef-ficient for a pipe installed at t0(j). If the pipe is replacedat Tf (j), Equation (6) is used to calculate the HazenWilliams coefficient after replacement. As any tempo-ral variation of this coefficient can create a pressurevariation at network nodes, Equations (7) to (9) (mass,energy conservation, and head losses) are used eachyear for verification. The minimal pressure constraint(Equation (10)) must be checked for all nodes of thenetwork except the source node. A penalty function de-fined by Equation (11) is used to check the solution andcalculations of this function are clarified in the follow-ing section. Notice that all tests were realized by con-sidering as a hydraulic constraint, the maximum deficitpressure, but the user can specify any other constraintwithout disturbing the structure of this algorithm.

5 EVOLUTIONARYOPTIMIZATIONTECHNIQUESTESTED

The evolutionary optimization techniques used are allintegrated to Openbeagle framework (Beagle Engine isan Advanced Genetic Learning Environment) writtenin C ++ by Gagne and Parizeau (2004). This frame-work was linked to the hydraulic simulator Epanet2.0(Rossman, 2000) for hydraulic steady-state analysis.This framework was used because it presents severaladvantages. For example, it allows the integration ofthe operator to minimize or maximize the objectivefunction without modifying the form of the function tobe optimized. Another advantage is that the user canchoose various operators (selection, crossover, etc.) as

well as the evolution process of generations. During ourtests, we used the generational IGA and the multiob-jective genetic algorithms NPGA-2 and NSGA-II. Inboth multiobjective applications (NPGA-2 and NSGA-II), the hydraulic penalty function (Equation (11)) isthen considered as the second objective.

5.1 Island genetic algorithm

Instead of using a single population, IGA uses a set ofsmall subpopulations (called demes), which evolve sep-arately. With this isolation, every island can evolve withits own parameters, in different directions and towarddifferent solutions. In this kind of GA, a certain num-ber of individuals can migrate from one subpopulationto the nearest subpopulation. The island that evolves to-ward a local optimum or which converged prematurelycan be helped by arrival of one or several individual mi-grants. The individuals selected for migration can stayin their island and only a copy is sent to the nearest is-land, or those individuals are sent directly to the nearbyisland. According to Ryan (1995), selection of individ-ual migrants can be made in two ways. The first one isstochastic; the advantage of this method is the highervariety of individuals who can result from it. The sec-ond method consists of selecting individuals who, ac-cording to their fitness values, are the most successful, tocopy them to other islands, and this method can createa more direct evolution than the first method. Cantu`-Paz (2000) tested other configurations, for example: thebest migrants replace the worst, the migrants selectedrandomly replace the worst, and the best migrants re-place migrants selected randomly. The choice of indi-vidual migrants transferred from one deme to anotherand the individuals replaced in every deme can influ-ence the selection pressure. Configurations that selectmigrant individuals or replace them according to theirfitness, direct them towards an accelerated convergence.The advantage of this GA is that search processing forthe best solution is realized in parallel, in various sub-research spaces. This characteristic results in the gen-eration of several solutions, which can be very useful,especially in the case of multimodal functions. The sec-ond advantage is that a premature convergence in eachisland can be avoided by copying individuals from oneisland to another rather than sending them to the near-est island. The effect is that there is no loss in the qual-ity of individuals, even when those individuals will haveto mate with other individuals with less quality. IslandGA requires specifying, in addition to the parametersof a standard GA, the following parameters: numberof demes, migration frequency of individuals, and thenumber of individuals who migrate each time. In ourcase, we have considered a generational IGA with a


tournament selection and one point crossover becauseall results obtained were acceptable. In every deme, theGA is carried out sequentially. Demes can exchange in-formation from time to time by allowing certain individ-uals to migrate from one subpopulation to another oneaccording to certain topology.

5.2 NPGA-2

A solution is called dominated if there are other solu-tions, which can be identified and can give better valuesfor each objective. In the NPGA, two individuals ran-domly chosen are compared against a subset from theentire population. If one of them is dominated (by theindividuals randomly chosen from the population) andthe other is not, then the nondominated individual wins.When both competitors are either dominated or non-dominated, so that there is a tie, the result of the tourna-ment is decided through fitness sharing (Coello Coelloet al., 2002). Although this algorithm produced good re-sults in some applications, it was improved by Ericksonet al. (2001). The authors proposed the NPGA-2, whichuses the Pareto ranking principle: a group of individu-als is randomly selected from the population. If thereis a single candidate with the lowest rank (less domi-nated), then this candidate will be the chosen individ-ual of the tournament and is selected for reproduction.If there is no winner, none of the candidates are pre-ferred and a tie is realized. As in the previous version,the continuously updated fitness sharing is carried out.The niche radius value is chosen by the user at an es-timation of the minimal separation expected betweenthe goal solutions. The niche count, which indicates thenumber of individuals belonging to a particular niche,is also used. The niche counts are calculated using indi-viduals in the partially filled next generation populationrather than from the current generation population. Todetermine a good niche count, the values of objectivefunctions need to be scaled to equal ranges; the follow-ing function was used in our test to realize comparisonwith the second multiobjective algorithm (NSGA-II).The cost function selected was reduced between 0 and 1,which will also help to choose an adequate niche radiusvalue:

F 1 =F1

500000(12)

where F 1 is the adjusted value of the first objective costfunction F1.

5.3 NSGA-II

In NSGA-II, an initial population of parents of size Nand a population of children of the same size are assem-

bled to compose a new population (Deb et al., 2002).Then, the new population of size (2 N) is sorted ac-cording to nondomination criterion to identify the var-ious fronts F1, F2, and so on. Best individuals will bemembers of the first fronts. The new parent population(Pt+1) is formed, by adding solutions from the first front(first front F1, second front F2, etc.) until the size ex-ceeds N. If the number of individuals in (Pt+1) is lowerthan (N), a crowding procedure is applied to the firstfollowing front, (Fi), not included in (Pt+1). The pur-pose of this operator is to insert the best (N |Pt+1|)individuals missing in the population (Pt+1). These in-dividuals of the front are used to calculate the crowd-ing distance between two close solutions. Once individ-uals belonging to the population (Pt+1) are identified,a new child population (Qt+1) is created. The processcontinues, from one generation to the next, until thetermination criterion is reached. The crowding distanceof solution (i) measures the current density of solutionsaround this solution. This distance is calculated accord-ing to the perimeter formed by points closest to (i) foreach objective. Once this distance is calculated, it willthen proceed to choose missing individuals to fill thepopulation of size N. Between two solutions with differ-ing nondomination fronts, the solution with the lowerfront is preferred. Otherwise, if both solutions belongto the same front, the solution that is located in the areawhere the solution density is lowest (less crowded) ispreferred. This algorithm is considered very effectivebecause it uses an elitist approach, which allows record-ing of the best-founded solutions during previous gen-erations. It also uses a selection procedure based on afaster nondominance process, which does not requireany adjustment parameter or sharing factor, and it usesa comparison operator based on crowding distancecalculation.

5.4 Solution encoded

For all the applications, the coding used has a fixedlength; all pipes in the network are represented. The so-lution representation uses two numbers for all pipes inthe network. The first number represents the decisionvariable for replacement coded with 1 bit equal to 1if a pipe is replaced, and 0 if no replacement is made.The second number represents a replacement time, ifthe time varies between 1 and 5, a minimum of 3 bitsis necessary to code all possibilities. This coding allowsconsidering in future works, intervals of 7 years. This so-lution representation was chosen because it allows theidentification, at each interval tested, a variable numberof replaced pipes. The Gray code is used for decodingof all the solutions.


5.5 Fitness evaluation

The cost of each decision is estimated by using the costfunction identified by Equation (1). For the evaluationof a penalty function (Equation (11)), hydraulic charac-teristics of the pipe selected for replacement are mod-ified for the year identified by the solution. For thefirst year, if a decision is to replace a pipe, the HazenWilliams coefficient of this pipe is modified by that ofa new pipe, having the same diameter, and Equation(6) is used. On the other hand, if a decision does notidentify replacement for this pipe, the roughness coef-ficient for this year is calculated by using Equation (5).A steady-state hydraulic simulation is then run to esti-mate the pressure at network nodes. If the pressure cal-culated is lower than the pressure required at any node,a pressure deficit is calculated for this node. Then, themaximal pressure deficit for this year, representing themaximum deficit on all nodes, is estimated. This pro-cess is realized for all 5 years, and then a penalty func-tion representing the maximum amplitude of pressuredeficit for all years is evaluated. Once a penalty func-tion of a solution is estimated, the cost function for asolution proposed by the GA can be estimated consid-ering the cost of the network and the penalty function.The cost function of each solution in the population isestimated. The process of the GA is repeated; the newpopulations replace the older ones, until the maximalnumber of generations is reached. In the end, the bestsolutions of all generations are identified: 1st, 2nd, andso on. The best solution is chosen and the number ofpipes replaced and the corresponding replacement timeis identified. Because the GA does not guarantee the lo-cation of the global minimum of fitness function, severaltests are realized with various initial population initial-izations and sensibility analyses to the GAs parametersare undertaken.

6 APPLICATIONS

The application of the developed strategy is illustratedwith a hypothetical WDS. The complete data set can beconsulted in the study by Dridi (2006). This network hasone single reservoir and 100 pipes; the minimal requiredpressure at each node is fixed to 20 m.

6.1 IGA results and influence of the penaltyfunction coefficient

The first test was realized with a break history over aperiod of 40 years and, regardless of a penalty function,with the goal being to check the influence of the param-eters of IGA. By using value parameters of Table 1, the

Table 1Generational island GA parameters

Deme number 5Deme size 200Migration interval 10Migrants size (5%) 10Tournament size 4Crossover probability 0.30Individual mutation probability 1.00Bit mutation probability 0.01Maximum of generations 500

best fit obtained was $202,785. To validate the qualityof an optimal solution and the influence of a numberof parameters on convergence toward the best solution,a sensibility analysis was realized. For example, tour-nament size was modified while other parameters weremaintained at their initial values (see Table 1). Figure 2illustrates the obtained results when considering a tour-nament of 2, 3, 4, and 10 individuals. In every case,we observed that the fitness function value decreasedwhen the generation number increased. Figure 2 indi-cates that for the same maximal number of generations,the test realized with a tournament selection of two in-dividuals gave a fitness value of $202,838 while the otherthree tests carried out with, respectively three, four, and10 individuals gave the same result ($202,785), with thebest function also obtained. As could be expected, ifthe number of individuals who participate in the tour-nament is small, the selection pressure is small and theconvergence is slow. To have the best solution whilemaintaining a minimal value of tournament size (2), itwill be necessary to increase the size of demes, or num-ber of demes, or the maximal number of generations.Increasing one of those cited parameters has however, aconsequence, an increased calculation time. With breakdata over this period, the best GA solution suggested re-placing four pipes between the 41st and 45th year: pipes16, 28, and 63 at the 41st year and pipe 96 at the 44th

202 500

204 000

205 500

207 000

0 100 200 300 400 500Generation

Fit

ness

fun

ctio

n

10

4

3

2

Fig. 2. Tournament size influence on the convergence towarda best solution.


year in this example. To validate the results obtainedwith the IGA, another numerical method was used tocalculate, for each pipe of the network, the cost func-tion for the 5 years. The minimum of the cost func-tion, if it exists, over the 5 years indicates the optimalyear for replacement. Then the number of pipes identi-fied for replacement is calculated to indicate how manypipes are replaced during this interval. For the valida-tion, different tests were realized considering differentbreak histories (30 to 75 years) and two different net-works. The comparison was satisfactory. Both methodsidentified the same number of pipes and times for thereplacement. However, this validation method did notwork when a hydraulic constraint was considered.

The second test was realized on another interval witha break history of 30 years and considering the influ-ence of a minimal pressure constraint. Each solutionproposed by the GA is now checked with the hydraulicsimulator to make sure that pressures in all nodes arehigher than the required minimal pressure. A solutionthat does not satisfy this condition is not discarded fromthe population of individuals, but its fitness is penal-ized. During all the tests, the coefficient of the penaltyfunction was chosen to keep the same order betweencost and penalty functions. By considering a minimalpressure of 20 m, simulation results showed that thebest fitness was $281,220, this value corresponds to re-placement of pipe number 94 at the 32nd year. This re-sult was obtained by considering parameters of Table 1and a maximal number of 1,000 generations. To increasepressure at the most critical node, results showed thatit was essential to replace the 94th pipe even if it didnot record any breaks. From a practical point of view,the option to replace a pipe must be considered onlywhen its structural state is in a poor condition, and theoption for rehabilitation needs to be considered beforethis, because, in most cases, municipalities do not havesufficient financial resources to renew pipes and there-fore take action only when there is an emergency. Infurther work, integration of different options for re-habilitation will need to be considered as a decisionvariable.

Other sensibility analyses were realized to evaluatethe influence of the penalty function coefficient on thefinal solution. The results showed that for a small coef-ficient (1,000), the solution obtained is not feasible be-cause the pressure obtained at the most critical node islower than 20 m during 2 years. This solution suggestedreplacing pipe number 83 at the 33rd year. For a highercoefficient (100,000), pressures at each node for all yearsare more than the 20 m head specified. It seems clearthat the penalty function coefficient has a large impacton the solution obtained by the GA as illustrated byFigure 3.

18

19

20

21

22

23

24

25

30 31 32 33 34 35Year

Pre

ssu

re a

t n

ode

58 (m

) Solution without hydraulic constraintSolution with hydaulic constraint (kp=100,000)Required minimum pressure (20m)Solution with hydraulic constraint (kp=1,000)

Fig. 3. Weight influence of the hydraulic penalty with anisland genetic algorithm.

6.2 NSGA-II

Parameters of NSGA-II used are: population size =1,000; crossover probability = 0.50; mutation probabil-ity = 0.50; bit mutation probability = 0.01; and maxi-mum of generation = 500. Figure 4 illustrates the to-tal number of nondominated solutions, as well as thecost value of the network (Equation (1)) and the max-imal deficit pressure value over years (Equation (11)).It is interesting to note that the first solution is identi-cal to that found with IGA with no hydraulic constraintand the 6th solution is like that obtained with the ap-proach considering the hydraulic constraint, by impos-ing a minimal pressure of 20 m. The results indicate thatsix solutions were identified, five proposed to replaceone pipe to increase the pressures at the most criticalnode. The first solution had the cheapest cost and thelast one had the most expensive cost function but in-sures that all nodes had the required minimal pressureof 20 m. Because all solutions suggest the replacementof one 100 mm pipe, the total replacement cost of eachsolution is equal to $21,947.

6.3 Comparison NPGA-2 and NSGA-II

Before beginning, a sensibility analysis of maximumgeneration number had been made. Then, this number

0,0

0,2

0,4

0,6

0,8

1,0

280 800 281 000 281 200 281 400

Total cost ($)

Max

imum

pre

ssur

e de

fici

t (m

)

NSGA-II (500generations)

NPGA 2 (radius = 0,50, 500 generations)



Fig. 4. NPGA-2 and NSGA-II nondominated solutions.


was fixed to 500 generations. For the NPGA-2, we usedEquation (12) for the first objective and for the secondobjective; the same equation was used because the max-imum pressure deficit value was bounded between 0 and1. Figure 4 presents, for a maximal number of 500 gen-erations, the nondominated solutions obtained with theNSGA-II as well as the results of three tests made withNPGA-2, considering a tournament with four individ-uals and niche radius value of 0.05, 0.10, and 0.50. Itwas noticed that solutions of NPGA-2 depend stronglyon the niche radius value. The advantage of this multi-objective approach is to allow for the identification ofboth solutions found with the mono-objective approach(solution with and without hydraulic criteria), as well asseveral other solutions in a single test. The superiorityof the NSGA-II was demonstrated with tests realized onother networks of different sizes. With this network size(100 pipes), run time did not exceed 15 minutes. For an-other network size (250 pipes), run time was evaluatedto 4 seconds per generation (4 hours). This time is negli-gible compared to the 5 years horizon planning. Noticethat each network size has its own parameters (maxi-mum of generations, population size, required minimalpressure, etc.). However, other improvements can beeasily done to reduce the run time.

6.4 Discussion

The first aim of this application is to present three dif-ferent methods to generate a renewal planning in WDSand to test which of them generates the best result andcould be recommended to optimize a real system oflarge size. Our tests showed that in term of solutionidentification (replacement time and pipe number), thefirst algorithm (IGA) performs well when no penaltyfunction coefficient is considered. As the size of thenetwork grows (250 pipes), the performance decreases.The user (decision maker for example) needs to in-crease the number of demes to reach a good solution.Additionally, the results obtained using IGA, consider-ing the penalty coefficient, are highly dependent on thechoice of this coefficient. This algorithm will necessitatemore time to calibrate this coefficient when real distri-bution systems are considered. Letting the user choosethis factor is not an entirely suitable process from a prac-tical point of view. The second (NPGA-2) and third(NSGA-II) algorithms are more advantageous than thefirst because they provide several different solutions.For a large system, this procedure cannot be enumer-ated manually due to the very large amount of possi-ble solutions. These two algorithms simplify the task ofthe decision maker. The results showed that both algo-rithms gave slightly different Pareto fronts (Figure 4).However, the third algorithm (NSGA-II) is recom-

mended for larger systems because the decision makerhas few parameters to adjust compared to NPGA-2.

The developed strategy showed the efficiency of us-ing evolutionary algorithms to resolve problems of piperenewal planning in WDS while considering short hori-zon planning. This strategy was developed consideringthat we have a complete break data history of 30 years(and up). For other real systems, different break mod-els could be integrated and then the user can selectthe most suitable model that fits best with the avail-able real data. Different random break histories havebeen used to test this strategy. With each history, thenumber of pipes replaced, the corresponding time ofreplacement and the number of pipe breaks recordedprior to replacement are specified. In all cases tested,the results showed that replaced pipes were those withthe largest number of breaks. For a longer break his-tory, replacement occurs when the pipes have reacheda critical number of breaks (after 8 to 10 pipe breaksfor the network tested). This strategy allows the consid-eration of the specified break model, a variable num-ber of replaced pipes that could reduce costs but alsothe estimated cost of new pipes. From a practical pointof view, this strategy can be useful and helpful to anydecision maker. Such strategies answer to the followingquestions: Which pipe needs renewal? When shouldthe renewal be done? and How much will each solu-tion cost? After identifying a Pareto front, the decisionmaker can easily evaluate each solution and proposethe most suitable considering, for example, other prac-tical aspects of the network. This strategy was limitedto replacement; however, the coding used can be testedto consider the option of rehabilitation as a decisionvariable.

Perhaps the test problem seems simple. However, itis very important to test the behavior of such strategieson a hypothetical system. This step will help to eval-uate the performance of each evolutionary algorithm,and the best algorithm can be recommended for a largersystem. If the test results with a simple benchmark aresatisfactory, it can be simple to jump to a real and morecomplex system. We are conscious that real problemswill need more validation tests.

6.5 Future development

The developed strategy was limited within this frame-work to only two criteria. However, the modular struc-ture of this algorithm allows us to easily include otherindicators. Indeed, three other components can be inte-grated. A first module representing node demand vari-ation and uncertainty related to estimation of those val-ues in the long-term planning, especially for networksthat are expected to new supply areas. A second module


that takes into account the diameter variation for thereplaced pipe because it is supposed that all replacedpipes have the same diameter and that the material isthe same as the previous one. However, this modulealso has to take into account the practical aspects ofmaximum velocity. Finally, a third module allows thecomparison, from a practical point of view, of two al-most equal solutions. For example, suppose we havetwo adjacent solutions, the first optimal solution sug-gests replacing two pipes several kilometers apart at acost C1 and the second best solution suggests replac-ing two nearby pipes with a slightly higher cost C2. Al-though the cost of the optimum solution is lower thanthe second solution, it is more advantageous to choosethe second solution because, from a practical point ofview, having replacement work on nearby sites reducesthe indirect costs. The choice of an appropriate solutioncan also depend on several other criteria, which dependfor example on road repair. Other criteria must there-fore be considered, for example, the possibility of re-alizing other work at the same time, such as work onother sewer systems or during road repairs. The algo-rithm was developed so that other rehabilitation optionsfor renewal or a function to minimize cost energy can beincluded.

Finally, the developed algorithm was tested on a sim-ple hypothetical network, which includes only pipes andone reservoir. In future, it would be interesting to studya more complex network with storage tanks and pumpsthat are closer to real networks. Testing other categoriesof evolutionary algorithms as evolution strategies andanalyzing their performance on other distribution net-works (larger in size), is also suggested.

7 CONCLUSION

The objective of this article was to present results ofa new strategy developed on a short planning hori-zon that allows the identification of which pipes needto be replaced and the optimal time for replacementby testing three different evolutionary algorithms: anIGA, NPGA-2, and NSGA-II. The generational IGAused gave satisfying results for identification of best so-lutions. However, the results depend on coefficient ofthe penalty function. During sensibility analysis, it wasestablished that the number of individuals who par-ticipate in the tournament selection can also have alarge influence on the convergence toward the best so-lution. Tests conducted on other networks (less than250 pipes) showed also that the IGA used performswell, but for larger sized networks we recommend in-creasing the number of demes (more than 10 demeswith 500 individuals in each deme) to reach better so-

lutions. The NPGA-2 results showed that the niche ra-dius value could affect the quality of the nondominatedsolutions. The comparison of NPGA-2 with NSGA-IIshowed that the latter could give better nondominatedsolutions, thus a better quality of Pareto front. We rec-ommend the use of NSGA-II to optimize a large WDS,because few parameters need to be adjusted with thisalgorithm and its performance is better.

The results confirm that using optimization tech-niques based on evolutionary algorithms as genetic al-gorithm could be useful to solve scheduling pipe re-newal problems in WDS and that such a renewal modelcan be useful to assist any decision maker when eval-uating different options to improve their systems. Fu-ture work is needed to apply this approach to largerand more realistic systems. This algorithm can be eas-ily expanded to integrate other modules. Future appli-cations will, however, take time because the most im-portant factor in such strategies is the process of datacollection, which takes a lot of time, energy, and moneyto organize all the information into a format that is easyto use.

ACKNOWLEDGMENTS

The work presented in this article reflects the results ofthe research work of the first author for his Ph.D. the-sis. The authors are grateful to reviewer Karen Cossittfor the suggestions in improving this article. The authorswish to thank the five reviewers and the editor for pro-viding valuable comments.

REFERENCES

Cantu`-Paz, E. (2000), Efficient and Accurate Parallel GeneticAlgorithms, Kluwer Academic Publishers, Boston, 162 p.

Coello Coello, Carlos A., Van Veldhuizen, D. A. & Lamont,G. B. (2002), Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, NewYork, 576 p.

Dandy, G. C. & Engelhardt, M. (2001), Optimal scheduling ofwater pipe replacement using genetic algorithms, Journalof Water Resources Planning andManagement, 127(4), 21423.

Dandy, G. C., Simpson, A. R. & Murphy, L. J. (1996), Animproved genetic algorithm for pipe network optimization,Water Resources Research, 32(2), 44958.

Deb, K., Pratap, A., Agarwal, S. & Meyarivan, T. (2002), Afast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6(2),18297.

Dridi, L. (2006), Developpement et validation dune approchede remplacement des conduites pour les reseaux deaupotable, Ph.D. thesis, Institut National de la Recherche Sci-entifique, Centre Eau Terre et Environnement, Quebec,Canada (in French).


Dridi, L., Mailhot, A., Parizeau, M. & Villeneuve, J.-P. (2005),A strategy for optimal replacement of water pipes integrat-ing structural and hydraulic indicators based on a statisticalwater pipe break model, in Proceeding, Eight Intern Conf.on Computing and Control for the Water Industry, D. A.Savic, G. A. Walters, R. King, and S.-T. Khu (eds.), Uni-versity of Exeter, UK, 6570.

Erickson, M., Mayer, A. & Horn, J. (2001), The Niched ParetoGenetic Algorithm 2 Applied to the Design of Ground-water Remediation Systems, in Proceedings, First Inter-national Conference on Evolutionary Multi-Criterion Opti-mization, E. Zitzler, K. Deb, L. Thiele, C. A. Coello Coello,and D. Corne (eds.), Lecture Notes in Computer Science,Springer-Verlag, 68195.

Gagne, C. & Parizeau, M. (2004), Open BEAGLE, a VersatileEC Framework (Open BEAGLE, Beagle Engine Is an Ad-vanced Genetic Learning Environment), Report, UniversityLaval, Quebec, Canada.

Halhal, D., Walters, G. A., Ouazar, D. & Savic, D. A. (1997),Water network rehabilitation with structured messy geneticalgorithm, Journal of Water Resources Planning and Man-agement, 123(3), 13746.

Halhal, D., Walters, G. A., Savic, D. A. & Ouazar, D. (1999),Scheduling of water distribution system rehabilitation usingstructured messy genetic algorithm, Evolutionary Compu-tation, 7(3), 31129.

Horn, J., Nafpliotis, N. & Goldberg, D. E. (1994), A nichedPareto genetic algorithm for multiobjective optimization, inProceedings, First IEEE Conference on Evolutionary Com-putation, IEEE World Congress on Computational Intelli-gence, IEEE Service Center, Piscataway, NJ, 8287.

Kim, J. H. & Mays, L. W. (1994), Optimal rehabilitation modelfor water distribution systems, Journal of Water ResourcesPlanning and Management, 120(5), 67492.

Kleiner, Y., Adams, B. J. & Rogers, J. S. (1998a), Long-termplanning methodology for water distribution system reha-bilitation, Water Resources Research, 34(8), 203951.

Kleiner, Y., Adams, B. J. & Rogers, J. S. (1998b), Selectionand scheduling of rehabilitation alternatives for water dis-tribution systems, Water Resources Research, 34(8), 205361.

Le Gauffre, P. & Eisenbeis, P. (2004), Gestion patrimoni-ale des infrastructures : de nouveaux outils daide a` ladecision, Dix-septie`mes entretiens Jacques Cartier. ColloqueEau potable et technologies environnementales : enjeuxinternationaux, Centre Eau, Terre et Environnement Institut national de la recherche scientifique, Quebec,Canada, 115.

Mailhot, A., Pelletier, G., Noel, J.-F. & Villeneuve, J.-P.(2000), Modeling the evolution of the structural state of wa-ter pipe networks with brief recorded pipe break histories:Methodology and application, Water Resources Research,36(10), 305362.

Mailhot, A., Poulin, A. & Villeneuve, J.-P. (2003), Optimal re-placement of water pipes, Water Resources Research, 39(5),1136, doi:10.1029/2002WR001904.

Rahman, S., Vanier, D. J. & Newton, L. (2005), Munici-pal Infrastructure Investment PlanningSocial Cost Con-siderations forMunicipal InfrastructureManagement. NRC-CNRC, report B-5123.8, 20 p.

Rossman, L. A. (2000), EPANET 2 users manual, U.S. Envi-ronmental Protection Agency, Cincinnati, Ohio.

Ryan, C. (1995), Niche and species formation in ge-netic algorithms, in Practical Handbook of Genetic Algo-rithms: Applications, vol. 1, Lance Chambers, editor, 5774.

Savic, D. A. & Walters, G. A. (2000), Genetic algorithm op-timization for design and management of water systems,in Proceedings, the 2nd ASMO/ISSMO Conference on En-gineering Design Optimization, J. Sienz Swansea (eds.),Wales, UK, 20510.

Savic, D. A., Walters, G. A. & Schwab, M. (1997), Multiobjec-tive genetic algorithms for pump scheduling in water sup-ply, in AISB InternationalWorkshop on Evolutionary Com-puting. Lecture Notes in Computer Science 1305, Berlin,Springer-Verlag, pp. 22736.

Shamir, U. & Howard, C. D. D. (1979), An analytic approachto scheduling pipe replacement, Journal of American WaterWorks Association, 71, 24858.

Sharp, W. W. & Walski, T. M. (1988), Predicting internalroughness in water mains, Journal of AWWA, 80, 3440.

Srinivas, N. & Deb, K. (1994), Multiobjective optimizationusing non-dominated sorting in genetic algorithms, Evolu-tionary Computation, 2(3), 22148.

Vairavamoorthy, K. & Ali, M. (2000), Optimal design of wa-ter distribution systems using genetic algorithm, Computer-Aided Civil and Infrastructure Engineering, 15(5), 37482.

Walters, G. A., Halhal, D., Savic, D. A. & Ouazar, D. (1999),Improved design of Anytown distribution network usingstructured messy genetic algorithms, Urban Water, 1, 2338.

Wu, Z. Y., Boulos, P. F., Orr, C. H. & Ro, J. J. (2001), Us-ing genetic algorithms to rehabilitate distribution systems,Journal of American Water Works Association, 93(11), 7485.

Genetics and Replacing Pipes

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