Genetic Algorithms for Reliability-Based Optimization of Water Distribution Systems

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Genetic Algorithms for Reliability-Based Optimizationof Water Distribution Systems

Bryan A. Tolson1; Holger R. Maier2; Angus R. Simpson, M.ASCE3; and Barbara J. Lence, A.M.ASCE4

Abstract: A new approach for reliability-based optimization of water distribution networks is presented. The approach links a gealgorithm ~GA! as the optimization tool with the first-order reliability method~FORM! for estimating network capacity reliability.Network capacity reliability in this case study refers to the probability of meeting minimum allowable pressure constraints acronetwork under uncertain nodal demands and uncertain pipe roughness conditions. The critical node capacity reliability approximanetwork capacity reliability is closely examined and new methods for estimating the critical nodal and overall network capacity reliausing FORM are presented. FORM approximates Monte Carlo simulation reliabilities accurately and efficiently. In addition, FORMbe used to automatically determine the critical node location and corresponding capacity reliability. Network capacity reliability apmations using FORM are improved by considering two failure modes. This research demonstrates the novel combination of a GFORM as an effective approach for reliability-based optimization of water distribution networks. Correlations between random vaare shown to significantly increase optimal network costs.

DOI: 10.1061/~ASCE!0733-9496~2004!130:1~63!

CE Database subject headings: Algorithms; Water distribution; Optimization; Hydraulic networks; Network reliability.

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Introduction

Water distribution networks~WDNs! are essential and costly in-frastructure in every modern community. As WDNs continue tage and cities continue to grow, the design of new WDNs and trehabilitation or upgrade of existing WDNs will continue to be aimportant problem. The general WDN design problem involveminimizing whole of life network costs~e.g., pipe and construc-tion! subject to meeting minimum allowable pressure and/maximum allowable velocity constraints under design demalevels. Traditionally, WDN design, upgrade, or rehabilitation habeen based on engineering judgment. More recently, a significamount of research has focused on the optimal design or upgrof WDNs. Some of the first studies utilized linear programmin~LP! ~Alperovits and Shamir 1977; Quindry et al. 1981! whilelater studies applied nonlinear programming~NLP! ~Su et al.

1Graduate Student, School of Civil and Environmental EngineerinCornell Univ., Ithaca, NY, 14853-3501; and Research Associate, Schof Civil and Environmental Engineering, The Univ. of Adelaide, Adelaide, S.A., 5005, Australia. E-mail: [email protected]

2Senior Lecturer, Centre for Applied Modelling in Water EngineeringSchool of Civil and Environmental Engineering, The Univ. of AdelaideAdelaide, S.A., 5005, Australia.

3Associate Professor, Centre for Applied Modelling in Water Engneering, School of Civil and Environmental Engineering, The Univ. oAdelaide, S.A., 5005, Australia.

4Associate Professor, Dept. of Civil Engineering, Univ. of British Columbia, Vancouver, B.C., Canada, V6T 1Z4.

Note. Discussion open until June 1, 2004. Separate discussions mbe submitted for individual papers. To extend the closing date by omonth, a written request must be filed with the ASCE Managing EditoThe manuscript for this paper was submitted for review and possibpublication on June 19, 2001; approved on January 14, 2003. This pais part of theJournal of Water Resources Planning and Management,Vol. 130, No. 1, January 1, 2004. ©ASCE, ISSN 0733-9496/2004/63–72/$18.00.

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1987; Lansey and Mays 1989; Xu and Goulter 1999!, or chanceconstrained approaches~Lansey et al. 1989! to the pipe networkoptimization problem. Much of the recent literature has utilizedgenetic algorithms~GAs! for the determination of low cost WDNdesigns and they have been shown to have several advantaover more traditional optimization methods~Simpson et al. 1994;Savic and Walters 1997!.

The focus of a number of the above studies is the least-codesign of reliable WDNs. Goulter~1995! states that reliability isgenerally concerned with the ability of WDNs to provide an adequate level of service to consumers, under both normal and anormal conditions. For example, failure to meet an adequate levof service would occur due to nodal flow demands being supplieat inadequate pressures or flow rates. In general, WDN reliabilitbased optimization is focused on either the mechanical failurecomponents~e.g., Goulter and Coals 1986; Su et al. 1987! such aspipe or pump failure, or the hydraulic failure of the system due tdegraded pipe capacities and/or uncertain nodal demand flo~Lansey and Mays 1989; Xu and Goulter 1999!. Goulter ~1995!and Xu and Goulter~1999! provide reviews of reliability analysismethods.

Reliability-based optimization of WDNs requires the combination of an optimization algorithm with a method for estimatingWDN reliability. In this paper, a novel approach to reliability-based optimization of WDNs is proposed and tested on a 14-picase study. This approach combines a GA as the optimization towith improved methods for estimating different WDN reliabilitymeasures based on the first-order reliability method~FORM!. Itshould be noted that the approach presented in this paper isstricted to ‘‘capacity reliability’’ estimation as in Xu and Goulter~1999!, which refers to the probability that the minimum allow-able nodal pressures are met under the assumption that thequired nodal demand flows are satisfied, and is a function of thuncertain nodal demands and the uncertain degree to which phydraulic capacities will be reduced over the design period.

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Proposed Approach

Optimization

Although GAs have been found to be a robust technique foroptimization of deterministic WDNs~Simpson et al. 1994; Saviand Walters 1997!, their application to reliability-based optimization of WDNs is scarce and they appear to have only been usestudies where a surrogate of reliability is considered in the omization framework~Halhal et al. 1997; Shin and Park 1999!.This is despite the fact that GAs have a number of advantaover the mathematical programming techniques traditionally ufor WDN optimization ~e.g., NLP!, including ~1! decision vari-ables are represented as a discrete set of possible values,~2! theability to find near globally optimum solutions, and~3! the gen-eration of a range of good solutions in addition to one leadsolution. Consequently, GAs are used as the optimization tnique as part of the proposed approach.

GAs are robust heuristic iterative search methods thatbased on Darwinian evolution and survival of the fittest~Holland1975; Goldberg 1989!. The GA techniques and mechanisms slected as part of the proposed approach are binary coding odecision variables, tournament selection, uniform crossover, cmutation, elitism, and the MicroGA technique. These combinGA procedures produce a relatively new and efficient GA cathe small-elitist-creeping-uniform-restart GA or ‘‘securGA’’ thatis built on theMicroGA technique~see Krishnakumar 1989! andis first introduced by Yang et al.~1998!. Computational efficiencyis important in the context of reliability-based optimization,GAs generally require many more function evaluations compawith traditional optimization methods.

Briefly, thesecurGAworks by using a smaller population siz~relative to that which would be used in a traditional GA! thatevolves like a traditional GA using uniform crossover, elitisand creep mutation. When convergence is reached, howevnew random population is generated and combined with theindividual from the previous generation and the evolution procrepeats itself using a new pool of genetic information. This cycontinues until a maximum generation limit is reached.

Reliability Estimation

The performance of any engineered system can be expressterms of its load and resistance. IfX5(X1 ,X2 ,...,Xn)T is thevector of random variables that influences a system’s load~L!and/or resistance~R!, the performance function,G(X), is com-monly written as

G~X!5R2L (1)

The failure ~limit state! surfaceG50 separates all combinations of X that lie in the failure domain~F! from those in thesurvival domain~S!. Consequently, the probability of failure,pf ,is given as~Sitar et al. 1987!

pf5Pr$XPF%5Pr$G~X!,0%5EG~X!,0

f X~X!dx (2)

where f X(x) is the joint probability density function~PDF! of X.In most realistic applications, the integral in Eq.~2! is difficult

to compute. Approximate solutions can be obtained by usinvariety of techniques including Monte Carlo simulation~MCS!and the first-order reliability method~FORM! ~Madsen et al.1986!. The most accurate reliability estimation method is MCwith a large number of realizations. Since all other reliabil

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estimation techniques are generally developed to be more computationally efficient than MCS, their accuracy should always beassessed in comparison with MCS benchmark solutions.

MCS approximates the integral in Eq.~2! by repeatedly gen-erating random realizations of the variables inX and then evalu-ating the performance function in Eq.~1! for each realization. Thereliability measure as given by MCS is then the ratio of the num-ber of realizations whereG(X).0 to the total number of MCSrealizations evaluated. The MCS reliability estimate approachesthe actual reliability as the number of MCS realizations used in-creases.

The objective of FORM is to compute the reliability indexb,which is then used to obtain the reliabilitya, using

a512pf512F~2b!5F~b! (3)

where F~ !5standard normal cumulative distribution function~CDF!. In the n-dimensional space of then random variables,bcan be interpreted as the minimum distance between the poindefined by the values of then variable means~mean point! andthe failure surface. The point on the failure surface closest to themean point is generally referred to as the design point, which maybe thought of as the most likely failure point. The reliability ob-tained using FORM is only an approximation, unless the perfor-mance function is linear. The degree of non-linearity in the per-formance function, and hence the accuracy of FORM, is problemdependent~see Madsen et al.~1986! and Xu and Goulter~1999!for a more complete description of FORM!.

Much of the research in WDN reliability studies has utilizedMonte Carlo simulation~MCS! for WDN reliability estimation~Bao and Mays 1990; Gargano and Pianese 2000!. However, asMCS is relatively computationally inefficient, Xu and Goulter~1998, 1999! pioneered the use of the first-order reliabilitymethod ~FORM! for WDN capacity reliability estimation. Thisresearch builds on the work by Xu and Goulter~1999! on the useof FORM for approximating MCS predictions of WDN capacityreliability within an optimization framework by introducing newformulations for using FORM to estimate critical node and over-all network capacity reliability.

Both MCS and FORM can be used to estimate nodal, criticalnode, and network capacity reliability. Nodal and critical nodecapacity reliabilities are defined with respect to a specified nodeand the node in the network that has the worst nodal capacityreliability, respectively. In contrast, network capacity reliabilitycan be defined as the probability that the minimum allowablepressures are met atall nodes. The way MCS and FORM can beused to estimate these different measures of reliability is dis-cussed below.

Nodal Capacity ReliabilityThe performance function used to evaluate the capacity reliabilityof nodei, by MCS or FORM is

Gi~X!5Hi~X!2Himin (4)

whereHi(X)5head predicted at nodei as a function of the vectorof both random nodal demands and pipe hydraulic capacities,X,andHi

min5minimum allowable specified head required at nodei.In order to assess the reliability at all the nodes in the networkusing FORM, the performance function in Eq.~4! must be speci-fied for each node and a separate FORM computational proceduris then required. In contrast, MCS can be used to estimate thereliability at all nodes in the network in the same computationalprocedure since the performance functions at all nodes can beevaluated for each MCS realization. Therefore, the relative com-

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putational advantage of FORM over MCS diminishes quicklythe reliability at many or all nodes in the network is of interes

Critical Node Capacity ReliabilityThe most basic way that FORM or MCS can be used to determthe critical node capacity reliability is to evaluate Eq.~4! for eachnode in the network and then select the smallest nodal reliabWhen MCS is used, this approach is no less efficient than a sinodal capacity reliability estimate. However, as recognized byand Goulter~1999!, the increased computational burden imposby FORM for estimating multiple nodal capacity reliabilitiemakes this basic approach for critical node determination unsirable when FORM is the reliability estimation technique. Cosequently, Xu and Goulter~1999! employed Eq.~4! to find onemeasure of nodal capacity reliability at the most critical nodethe network. Xu and Goulter~1999! identified the most criticalnode in the network by analyzing the intermediate resultsFORM and therefore observed the increase in computationalto be minimal. A detailed description of this approach is givenXu and Goulter~1998!.

Depending on the reliability analysis program utilizedimplement FORM, the intermediate results of FORM may notavailable to the program user. Therefore, an alternative apprthat does not require intermediate FORM results is proposedtested here for identification of the critical node in the netwoThe following performance function can be defined for use wFORM to identify the critical node in the network:

Gc~X!5min~Hi~X!2Himin!,1,2, . . . ,I (5)

where Gc(X)5performance function at the node that is mocritical with respect to meeting its corresponding minimum prsure requirement andI5number of nodes considered. In this peformance function, the location of the critical node is not fixand can change locations with each FORM evaluation of theformance function. Thus, FORM is left to converge to the critinode location at the design point. This performance functiondesigned so that the critical node location and reliability atcritical node are determined without accessing the intermedresults of the FORM computational procedure.

Network Capacity ReliabilityIt is often convenient and sometimes more meaningful to estima single reliability measure that characterizes overall networkformance. Previous approaches have proposed heuristic meaof network reliability such as the arithmetic mean or weighaverage of all nodal reliabilities~e.g., Bao and Mays 1990!. An-other common heuristic measure is to use the critical nodeability as an approximation of network reliability~e.g., Bao andMays 1990; Xu and Goulter 1999!. While these heuristics caprovide reasonable estimates of network reliability, it is importto realize that MCS can be used to directly estimate netwreliability by treating WDNs as a series system in which failureprovide adequate heads atany one or more nodes with a minimum pressure requirement constitutes a network failure.

A direct estimate of the above definition of network capacreliability can be found using MCS to evaluate the performafunction in Eq.~5!. Even though both FORM and MCS can bused to evaluate the performance function defined in Eq.~5!, thereliability measures estimated by each method are not the sWhen Eq.~5! is used as the FORM performance function, FORsearches for and converges to the critical node at the design~i.e., the single most likely event to cause failure! and thereforeestimates the capacity reliability only for that event at that no


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If the same performance function is defined for MCS reliabilityestimation, failure is defined with respect to meeting the mini-mum pressure requirement at the most critical node in the systefor eachMCS realization. Thus, the MCS technique measures thereliability with respect to all failure events simultaneously, insteadof just measuring the reliability with respect to the most likelyfailure event.

When FORM is used for reliability estimation in WDNs, aheuristic measure of network reliability must be employed. Atpresent, the only proposed approach for approximating networreliability using FORM is to assume it is approximately equal tothe critical node reliability~Xu and Goulter 1999!. Relying on aheuristic measure of network reliability such as the critical nodereliability is a significant disadvantage of FORM with respect toMCS for network reliability estimation unless the heuristic mea-sure accurately approximates the MCS measure of network relability. A common assumption in the literature is that the criticalnode reliability does in fact closely approximate the true networkreliability. Although this may be the case and could even beevaluated in reliability studies dealing with a limited number ofpredefined network configurations, it is probably unreasonable textend this assumption to WDN reliability-based design studiesfor two reasons. First, the critical node reliability approximationis non-conservative as it only considers failure at a single node ithe network. Events leading to failure at other nodes in the network that are independent of failure at the critical node are noconsidered by this heuristic. Therefore, when the approximation inot accurate, the reliability of the network is overestimated. Theother difficulty with this assumption is that in many or even allWDN design situations it is generally impossible to determine theappropriateness of the assumption for all or even a significanproportion of the vast number of possible network designs~oftengreater than a million!. The combined effect that these two short-comings have on the critical node approximation to network reli-ability is that when the approximation is employed in networkoptimization trials, there will most likely be a number of candi-date network designs that are judged to have a significantly highenetwork reliability than their true network reliabilities. Conse-quently, if FORM is to be used to approximate network reliabilityin reliability-based optimization models, the FORM network re-liability estimate should be improved.

The FORM measure of network capacity reliability can beenhanced if an additional failure mode is considered. For example, two failure modes could represent the capacity reliabilityat two nodes of interest in the network. In any system with twofailure modes, the probability of system failure,pf s , is

pf s5pf 11pf 22pf 125Pr$G1,0%1Pr$G2,0%

2Pr$G1,0 and G2,0% (6)

wherepf 1 andpf 25probabilities of failure due to failure Modes 1and 2, respectively;pf 125joint probability of failure for failureModes 1 and 2 andG15G(X1) andG25G(X2) are5the perfor-mance functions for failure Modes 1 and 2, respectively. Thefailure probabilities for the individual failure modes (pf 1 andpf 2)can be obtained using Eq.~3! while the joint probability of fail-ure, pf 12, is given by Madsen et al.~1986! as

pf 125F~2b1 ,2b2 ;r12!5F~2b1!F~2b2!

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c~2b1 ,2b2 ;y!dy (7)


Fig. 1. Layout of two-reservoir network upgrade problem under uncertainty

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where F~ , ;r!5CDF for a bivariate normal vector with zeromean values and unit variances and correlation coefficientr andc~ , ;r!5corresponding PDF. The integral in Eq.~7! is generallyobtained numerically. The approximate correlation coefficieneeded to evaluate this integral,r12, is calculated using~Madsenet al. 1986!

r125V1*

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where V1* and V2* 5design points in standard normal space fofailure Modes 1 and 2, respectively.

Although Eq.~6! can be evaluated exactly when there are ontwo failure modes, only the bounds on the system probabilityfailure can be calculated when there are more than two modesfailure. In addition, consideration of each additional mode of faiure adds one more FORM computational procedure. Thereforeis proposed that a more accurate point estimate of network capity reliability for a reasonable increase in computational cost cbe evaluated using FORM for reliability estimation in the twmost critical failure modes~i.e., at two nodes in the network! inconjunction with Eqs.~6!, ~7!, and~8! to estimate the series sys-tem probability of failure. The basic idea underlying this approacis to identify the second most critical failure node, in addition tthe most critical failure node in the network, to attempt to accoufor a significant fraction of the network failure events that manot lead to failure at the most critical node. The two most criticnodes in the network are determined by first estimating the mcritical nodal capacity reliability using the performance functiodefined in Eq.~5! and then estimating the capacity reliability forthe next most critical node in the network according to the folowing performance function:

G2c~X!5min@Hi~X!2Himin#, 1,2, . . . ,I , and iÞ l (9)

where the performance function in the second mode of failureEq. ~9! and is defined as in Eq.~5! except that nodel, the criticalnode determined in the first failure mode, is not considered a
possible location of failure in the second mode of failure. .
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Based on Eq.~6!, the new system reliability measure,as, asestimated by FORM becomes

as512pf 12pf 21pf 12512pf s (10)

where the individual probabilities of failure at the first and secondmost critical nodes in the network arepf 1 andpf 2 , respectively,and are calculated from Eq.~3!, and the joint probability of failurebetween the two most critical nodes (pf 12) is calculated from Eq.~7!.

Case Study

The novel combination of a GA with FORM for reliability-basedoptimization of WDNs is demonstrated for a case study that hasbeen previously optimized under deterministic conditions~Simp-son et al. 1994!. The original design problem was to determinethe required pipe sizes for an expansion to an existing WDN inorder to minimize total pipe costs while still meeting the mini-mum nodal head requirements under three different flow demandpatterns. The three demand patterns considered represent thpeak-hour demand pattern and two fire-loading demand patternsthat occur for a demand equal to the average peak-day demandpattern.

The layout of the case study WDN and the proposed expan-sion, other network characteristics, and the head requirements anmean nodal flows for each demand pattern, are summarized inFig. 1. There are five new pipes to be sized~Pipes 6, 8, 11, 13,and 14! and three existing pipes~Pipes 1, 4, and 5! that can be leftas is, cleaned, or duplicated with another pipe in parallel. Newpipes and cleaned pipes are assumed to have a Hazen-WilliamcoefficientC value of 12̇0. For the five new pipes, eight possiblediameters from which to choose, in millimeters, are 152, 203,254, 305, 356, 406, 457, and 508. The available pipe diametersconsidered in the duplication of the existing pipes are, in milli-meters 152, 203, 254, 305, 356, and 406. Therefore, there are alseight possible decisions for Pipes 1, 4, and 5. The pipe costs permeter associated with each possible decision are listed in Table 1

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Modifications to the Original Case Study

The original deterministic problem in Simpson et al.~1994! mustbe converted to a design problem under uncertain conditions.objectives of the expansion of the WDN in Fig. 1 are to minimitotal pipe costs and maximize network capacity reliability durithe three critical demand patterns. In order to combine the nwork reliability values for each demand pattern, the three dempatterns are assumed to be of equal importance with respemeeting the required nodal pressures. Therefore, the minimnetwork capacity reliability of the three demand patterns is uto represent the value of the reliability objective. In other wordthe network reliability of each demand pattern is evaluated inpendently and then the minimum reliability is taken as the vaof the reliability objective. This is just one way to generatecomposite reliability measure characterizing multiple desevents. It should be noted that the network capacity reliabimeasures outlined earlier are meant to estimate network reliabduring a single design condition—not to combine reliabilitiacross multiple design conditions.

The sources of uncertainty considered are the uncertain vaof roughness for each pipe and the uncertain nodal demand flfor each demand pattern. Estimation of WDN reliability durinthe critical design demand levels is complicated by the combition of the pipe hydraulic capacities and the nodal demand floas uncertain variables because the time scales at which tsources of uncertainty vary are different. For example, pipedraulic capacities degrade slowly over time due to corrosiondeposition instead of varying randomly over time, as could becase with the critical demand patterns. Therefore, the reliabilitythis study is calculated with respect to the uncertain conditionthe end of the design period under consideration and will thgenerally, be the worst-case network reliability during any poin time during the design period. It is assumed that a randamount of degradation in the pipe hydraulic capacities~i.e., ran-dom reductions in theC values from present day values! willoccur over the design period and that the variability in the annnodal demand patterns will be constant throughout the entiresign period.

The reduction inC values of each pipe is assumed to be reresented by a normal random variable with an average degrtion of 10% over the design period and a coefficient of variatiof the reduction inC (COVC) of 40%. For example, a pipe thahas a present dayC value of 120 is reduced on average b0.1*120512, with a standard deviation of 0.4*1254.8, to C51202N(12,4.8) at the end of the design period. The randoamount of degradation in each pipe is assumed to be indepenand is bounded such that the minimum and maximumC values atthe end of the design period are 60 and the present dayC value,

Table 1. Available Pipe Sizes and Associated Costs from Simpset al. ~1994!

Diameter~mm!

Cost of newpipe ~dollars/m!

Cost of cleaningexisting pipe~dollars/m!

152 49.5 47.6203 63.3 51.5254 94.8 55.1305 132.9 58.1356 170.9 60.7406 194.9 63.0457 231.3 66.3508 262.5 69.2


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respectively. It should be noted that the reliability analysis pro-gram used in this work for the implementation of MCS andFORM automatically adjusts the PDF for each bounded randomvariable so that the total probability is equal to one. The nodaldemand flows in all three critical demand patterns are assumed tobe normally distributed, with means as listed in Fig. 1 and a COVof 40% for all nonfire demand nodes. The fire demand nodesconsidered are Node 7, in Demand Pattern 2, and Node 12, inDemand Pattern 3, and both are assumed to have a reduced COof 5%. All nodal demands are bounded to be greater than 0 andare assumed uncorrelated unless stated otherwise.

Model Formulation and Implementation

The optimization formulation presented in this section is definedin general terms such that the reliability objective can be based onany measure of reliability, estimated by either MCS or FORM,and the cost objective can be based on any network cost characterization. The reliability-constrained cost minimization model in-volves minimizing the total pipe costs of the WDN while meetinga specified minimum reliability constraint for the demand patternwith the lowest reliability and is given as follows:

Maximize: $k~Y!1A@100~ht~Y,Hmint ,Xpar

t !2a* !#B%21

(11)

where k(Y)5total cost of the network as a function ofY, thevector of the selected decision variables in the system;ht(Y,Hmin

t ,Xpart )5estimated value of the reliability measure of in-

terest for demand patternt, which has the lowest reliability of thethree demand patterns considered, and is a function of the vectoof decision variablesY, the vector of minimum specified nodalhead requirements for demand patternt Hmin

t , and the vector ofprobability distribution parameters for demand patternt Xpar

t thatdescribe the random variables in the WDN model;a*5minimumdesired reliability level for the WDN under the least reliable de-mand pattern andA and B are the penalty coefficient and expo-nent, respectively, in the GA penalty function. A penalty is onlyimposed, per reliability unit~i.e., per 1%! in which the reliabilityconstraint is violated, if the estimated reliability is less than thespecified minimum reliability level. Values ofA andB need to beselected such that the penalty term in Eq.~11! drives the objectivefunction value to very small values for unacceptable designs. Theobjective is inverted to become a maximization objective becausethe GA utilized in this work, as with most GAs, is coded as amaximization algorithm. Reliability-cost trade-off curves can beconstructed by solving this model for a range of reliability con-straints.

The general reliability-constrained cost minimization model isapplied to the WDN case study. The mathematical formulationconsiders the objectives of minimizing total pipe costs and maxi-mizing the minimum reliability of the three critical demand pat-terns. For each trial WDN design, the total pipe costs and thereliability under all three critical demand patterns must be esti-mated. The model is implemented in FORTRAN by linking ahydraulic network solver, a reliability analysis program, and a GAprogram together. All three of these programs, as well as allsupplemental subroutines, such as that needed for the calculationof the WDN pipe costs, are also written inFORTRAN. The Wad-iso hydraulic network simulation program~Gessler and Walski1985! is used to simulate the hydraulic system for each set ofrandom variables and pipe network configurations generated dur-ing the optimization trials. Wadiso assumes that the nodal demandflows are met and solves for the resultant nodal heads across th

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Table 2. Water Distribution Network Designs Selected for Evaluation of Proposed Water Distribution Network Reliability Measures

Design

Pipe diameter sizes~mm!

Total cost(106 dollars/yr)

Pipe@1#

Pipe@4#

Pipe@5#

Pipe@6#

Pipe@8#

Pipe@11#

Pipe@13#

Pipe@14#

A dup356 dup356 dup254 406 356 254 254 305 3.1860

B dup254 dup356 dup305 254 508 305 203 356 2.9378

C leave dup406 clean 356 305 305 254 305 2.4035

D dup406 dup305 dup152 254 305 254 203 254 2.6585

E dup152 dup356 clean 305 203 254 152 254 2.1738

F dup356 dup305 dup152 305 254 203 203 254 2.4922

Note: dup254means the pipe is duplicated with a pipe 254 mm in diameter in parallel.

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network. The original version of Wadiso is modified so that it cabe repeatedly executed without the use of an input file andaccuracy is verified against a standard hydraulic simulation paage~EPANET2!. A slightly modified version of the general reli-ability analysis programRELAN ~Foschi et al. 1993! is used toimplement the FORM and MCS reliability estimation techniqueRELANhas been previously used and described in many studsuch as Maier et al.~2001! and is generally robust for high reli-ability estimation given that it was developed to analyze very loprobability structural failures. FORM and MCS as implementein RELANallow random variables to be drawn from a numberprobability distributions and allow the user to specify correlatiobetween any combination of the random variables. TheRELANimplementation of FORM uses the Rackwitz-Fiessler meth~Madsen et al. 1986! to find the minimumb.

The GA source code used in this study~FORTRAN GA version1.7.1! after minor modifications is written by Dr. David Carroland is available at http://cuaerospace.com/carroll/ga.html~alsosee Yang et al. 1998!. An efficient set of GA parameters, as determined by Tolson~2000! for use in a different case study, isused. ThesecurGAparameter values used here are a populatisize of 5, a maximum generation limit of 1,000, a uniform crosover probability of 0.5, a creep mutation probability of 0.1, andsingle offspring per pair of parents. The binary coded valuesthe decision variables that are used by Simpson et al.~1994! arealso adopted in this study. The deterministic network optimizatiproblem solved by Simpson et al.~1994! and Simpson and Gold-berg~1994! is solved again with the above GA parameter set witen random seeds to ensure that thesecurGAwith the above pa-rameter settings is comparable in efficiency to the GAs usedthese previous studies. Results show approximately the sambetter performance in comparison with the previous GAs usedsolve this problem.

Analyses Conducted

The first set of analyses conducted involves testing the varioFORM and MCS reliability definitions on six example networdesigns and then further analyses demonstrating the applicatiothe reliability-constrained cost minimization model follow. Thsix example designs are selected so as to cover network desthat span a range of reliability values. The pipe sizes and tocost of the six designs selected are summarized in Table 2.each network design, the three demand patterns are analyConsequently, the first sets of analyses are carried out on 18ferent design conditions.1. Node Capacity Reliability Estimation. To test whether

FORM provides a good approximation to MCS reliabilitiefor the case study considered, FORM and MCS estimates

68 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT

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nodal reliability are obtained for each node across all 1design conditions in accordance with Eq.~4!. Initial testingis undertaken to determine the number of MCS realizationrequired to generate accurate MCS benchmark reliabilitieMCS nodal and network reliability estimates using 100,00realizations converge to within approximately 0.001 of thMCS reliabilities found using 500,000 realizations. Therefore, all benchmark MCS reliabilities are generated usin100,000 realizations;

2. Critical Node Capacity Reliability Identification. To testwhether FORM can be used to automatically determine thcritical node, FORM is used to estimate the critical nodreliabilities in accordance with Eq.~5! for all 18 design con-ditions. The results obtained are compared with the knowFORM reliability predictions at the critical node obtained apart of the complete enumeration of the nodal capacity reabilities in ~1! above;

3. Network Capacity Reliability Estimation. To quantify the dif-ference between critical node and network capacity reliabity estimates for the case study under consideration, netwocapacity reliability is calculated using MCS in accordancwith Eq. ~5! for the 18 design cases and the results obtaineare compared with the critical node capacity reliabilities obtained using MCS in~1! above.

4. Network Capacity Reliability Approximation Using FORM.To quantify the improvement associated with considerintwo failure modes when using FORM to approximate nework capacity reliability, rather than using a single failuremode, estimates of network capacity reliability are obtainefor the 18 design cases in accordance with Eq.~10! usingFORM, and the results obtained are compared with thFORM results obtained using a single failure mode in~2!above and the true network capacity reliabilities obtaineusing MCS in~3! above;

5. Reliability Constrained Cost Minimization. The reliability-constrained cost minimization model is solved using these-curGA to obtain minimum cost solutions for reliability con-straints of 0.7, 0.8, 0.9, 0.95, 0.99, and 0.999. The FORapproximation of network capacity reliability@obtainedusing two failure modes—see~4! above# is utilized as thereliability constraint. The penalty exponent in Eq.~11! is setat 1.0 for all model evaluations to remain consistent witSimpson et al.~1994!. However, the penalty coefficient isdetermined by trial and error for different reliability con-straint levels. The penalty coefficients found to be reasonabare 0.1 million dollars for reliability constraints under 0.90.5 million for constraints between 0.9 and 0.99, and 1.million for reliability constraints to 0.999. Since the opera

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tion of the GA is probabilistic in nature, at least two optimzation trials, using different random GA seeds, are usegenerate all model solutions; and

6. Correlated Nodal Demands. To investigate the impact noddemand correlations have on the reliability-constrainedminimization model solutions, the analyses in~5! above arerepeated with a correlation coefficient of 0.5~rather than 0.0!between the nodal demands. Correlations betweenroughness coefficients are not considered.

Results and Discussion

Nodal Capacity Reliability Estimation

The comparison of 180 MCS and FORM nodal capacity reliabties for ~1! above shows that, at worst, the absolute differencthe FORM and MCS estimates of nodal reliability is 0.026 areliability level of approximately 0.5. The magnitude of the ablute differences decreases as the reliability level increasesexample, above a reliability level of 0.95, the absolute differenbetween FORM and MCS are most often less than 0.001 amaximum of 0.006. For nodal reliabilities between 0.8 and 0the absolute differences in FORM and MCS reliabilities rabetween 0.000 and 0.017 with an average absolute differen0.007. Therefore, the case study is deemed appropriate for eating the new FORM reliability measures.

In 17 of the 18 design cases considered for~2! above, the useof Eq. ~5! leads to the correct identification of the critical noand the correct reliability estimate for that node, indicatingFORM can be used to automatically determine the critical nIn the one case where the correct critical node is not identithe third most critical node is identified as critical and the diffence in reliabilities between the critical and third most critinode is less than 0.002. The average and worst observed incin computational cost associated with using Eq.~5! instead ofspecifically identifying the critical node as in Eq.~4! is observedto be 11.4 and 31.0%, respectively. This small average increacomputational cost is deemed a reasonable trade-off for theplicity, consistency, and accuracy with which the proposedformance function definition allows FORM to determine the crcal node and its corresponding capacity reliability withaccessing intermediate FORM results.

Network Capacity Reliability Estimation

The comparison of the MCS critical node and critical netwcapacity reliabilities for~3! above shows that for 11 of the 1design cases investigated, the approximation to network capreliability given by the critical node capacity reliability is quireasonable, as the critical node capacity reliability approximatfall within 0.001 of the actual network capacity reliability. Evthough the results for the remaining seven cases remainapproximations, the differences in reliabilities in excess of 0.show that these two quantities can be dissimilar. The largestcrepancy in the approximation occurs for a case where the anetwork capacity reliability of 0.943 is overestimated by 0.0with a critical node capacity reliability of 0.958. Although therror may seem small, such a difference could have a signifiimpact on network cost. Therefore, FORM approximations ofwork reliability are necessary that go beyond the critical ncapacity reliability.

Table 3 presents the FORM approximations to network caity reliability for ~4! above obtained using two failure mod


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~Column 3!, as well as the MCS estimates of network capacityreliability ~Column 2!, and the FORM critical node reliabilityestimates~Column 4!. Many cases show little or no absolute im-provement by using the FORM estimate of network capacity re-liability as can be seen by comparing Columns~3! and ~4! inTable 3. However, if the relative reduction in the FORM approxi-mation errors is considered~last column in Table 3 which com-pares Columns 3 and 4!, then it becomes clear that for severaldesigns the novel FORM approximation of network capacity re-liability proposed here results in more accurate estimates of theMCS network capacity reliability. In fact, the average reduction inerror for the eight cases where the reduction is greater than 0% i35.2%.

Design B~2! in Table 3 shows the largest observed error reduc-tion and can be analyzed further to show that the improvement issignificant with respect to the approximate 95% confidence limitson the MCS reliability~a binomial proportion!. Since the approxi-mate confidence limits are calculated as~0.942,0.944! and do notcontain the FORM critical node capacity reliability~0.958!, theapproximation using FORM with two failure modes~0.946! thatis now closer to the upper bound of the confidence limits demon-strates that the improvement is significant. More evidence that theimprovement in network reliability estimation by FORM is note-worthy is that in 8 of 13 design cases where improvement ispossible~e.g., Column 4ÞColumn 2 in Table 3!, at least someerror reduction is observed. The increase in computational timerequired for FORM evaluation of two failure modes over thecritical node approximation to network reliability is approxi-mately 100% since this new measure of network reliability re-quires two FORM computational procedures instead of one.

Table 3. Comparison of Alternative Measures of Network CapacityReliability

Design~demandpattern!

Monte Carlosimulationnetworkcapacity

reliability

First-orderreliabilitymethodnetworkcapacity

reliability

First-orderreliabilitymethod

critical nodecapacity

reliability

Percentreduction inFirst-order

reliability methodapproximation

errora ~%!

A~1! 1.000 1.000 1.000 —A~2! 0.981 0.982 0.982 0.0A~3! 1.000 1.000 1.000 —B~1! 0.995 0.996 0.998 66.7B~2! 0.943 0.946 0.958 80.0B~3! 0.995 0.996 0.996 0.0C~1! 0.999 0.999 0.999 —C~2! 0.924 0.924 0.924 —C~3! 0.996 0.996 0.996 —D~1! 0.954 0.957 0.957 0.0D~2! 0.804 0.822 0.826 18.2D~3! 0.818 0.835 0.835 0.0E~1! 0.982 0.985 0.987 40.0E~2! 0.689 0.709 0.719 33.3E~3! 0.948 0.954 0.957 33.3F~1! 0.911 0.919 0.919 0.0F~2! 0.605 0.621 0.622 5.9F~3! 0.524 0.549 0.55 3.8aReduction in first-order reliability method network capacityreliability error5100%@1.02$~Column 3!2~Column 2!%/$~Column 4!2~Column 2!%#.


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Reliability-Constrained Cost Minimization

Reliability-cost trade-off curves obtained for~5! above by solvingthe reliability-constrained cost minimization model withFORM network capacity reliability as estimated from Eq.~10! areshown in Fig. 2. The best solution of the two or more optimtion trials used to solve the model at each reliability constraiplotted. Fig. 2 shows that the model solutions found do nooccur adjacent to their respective reliability constraints. Thisgests that the total number of noninferior solutions in this plem may be fairly limited. For each model solution in Fig. 2,FORM network capacity reliabilities were checked using 100MCS realizations. The magnitude of the absolute differencetween FORM and MCS network reliability estimates are a mmum of 0.017 at approximately the 0.80 reliability level andmain below 0.004 for all solutions with reliability levels greathan or equal to 0.90. Therefore, FORM approximates MCSwork reliability quite well for the trade-off solutions in the ranwhere most realistic designs would be considered~e.g., above 0.reliability!.

The typical execution time for one optimization model t~i.e., the generation of one trade-off curve point! with 5,000 GAevaluations is approximately 75 min on a Pentium III, 666 Mprocessor although some trials required in excess of 120 mexecution. The variance in the processor time is due to vaconvergence rates of FORM for the trial network configuratiIn comparison, if 2,000 MCS realizations were used to estimnetwork capacity reliability in the model instead of FORM,execution time is increased approximately six times over thecal FORM execution time.

Comparison of the trade-off curves with and without demcorrelations for~6! above in Fig. 2 shows that if correlation btween nodal demands is considered, higher cost network deare generally required to achieve a given desired level of netcapacity reliability. Furthermore, the difference in cost betwthe trade-off curves increases as more reliable designs aquired. Consequently, it is important that correlations betweerandom variables are taken into account.

Limitations and Future Work

There are some limitations to the ideas presented in thisstudy that are noteworthy. First of all, the accuracy of FO

Fig. 2. Reliability-cost trade-off curves generated by reliabilconstrained cost minimization model with correlation coefficientstween random nodal demands of 0 and 0.5

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relative to MCS is generally application specific and initial testingis required to determine if FORM is appropriate for otherreliability-based optimization studies. Other limitations relate tothe application of the ideas presented here to larger, more coplex WDNs. Future work in this area is needed to extend thapproaches outlined here to be applicable to large networks whundreds and perhaps thousands of pipes. These limitationsdiscussed in more detail below to help guide future work.

Although convergence to the critical node is observed in thicase study, future applications using Eq.~5! should initially betested to ensure that FORM does converge to the correct criticnode and critical node reliability. In practice, large networks arlikely to require a modified approach to that proposed here. Iparticular, the new FORM methods proposed here could likely bapplied to subsections of large networks such that FORM waconstrained to find the critical node in a subsection of a networand then further constrained to find the second most critical nodin another subsection of the network. This approach would function to find nonadjacent nodes as the two most critical nodes awould thus improve the accuracy of the FORM network reliability measure.

As network size increases, the number of failure events causby independent failures at multiple nodes also increases. Thefore, the errors in any approximation of network reliability thatdoes not capture some of the possible failure events will alsgenerally increase. Studies that are focused on managing for nwork reliability should consider this since the result is that thecritical node reliability approximation of network reliability, andeven the approximation using the FORM network reliability measure with two failure modes, may not be sufficiently accurateConsequently, when FORM is to be utilized, more than two failure modes may need to be considered. However, there is a traoff between the accuracy of the network reliability approximationand required computational time using FORM.

Conclusions

This paper has introduced a number of new and useful techniqufor consideration in future reliability-based optimization studiesof WDNs. Some techniques proposed are specific to studies eploying FORM for WDN reliability estimation while others aremore general and should be applicable in future WDN reliabilitybased optimization studies regardless of the type of reliabilitmeasure considered. FORM has been applied to accurately emate WDN nodal capacity reliability and a new FORM approacthat is both accurate and efficient is proposed which automaticaidentifies the most critical node in the network. For this casstudy, it is demonstrated that the MCS critical node capacity reliability approximation can significantly underestimate the trueMCS network capacity reliability. Considering that the criticalnode capacity reliability approximation to network capacity reli-ability estimation may not be reasonable in all WDN reliability-based optimization studies, a new and more accurate FORM aproximation to network capacity reliability is developed thatconsiders failure events at the two most critical nodes in the nework. This work also demonstrates the novel combination ofGA with FORM as a reasonably efficient approach for reliability-based optimization of WDNs. Last, correlations between noddemands are shown to significantly increase WDN costs designto meet a specific reliability target.

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Acknowledgments

Financial support for this project was provided by the Universof Adelaide Small Grants Scheme. This support is gratefullyknowledged. The writers would also like to acknowledge tanonymous reviewers for their insightful comments that improvthe manuscript.

Notation

The following symbols are used in this paper:A 5 penalty multiplier;B 5 penalty exponent;C 5 Hazen-Williams coefficient;F 5 failure domain;

f X(x) 5 is joint probability density function~PDF!of X;

G(X) 5 performance function, where failure~limitstate! surfaceG50 separates all combinationsof X that lie in failure domain from those insurvival domain;

Gc(X), G2c(X)5 performance function at node that is most critica

with respect to meeting its correspondingminimum allowable pressure requirement, 2refers to second most critical node;

Hi(X) 5 head predicted at nodei;Hi

min 5 minimum allowable specified head required atnodei;

ht(Y,Hmint ,Xpar

t )5 estimated value of reliability measure for demand

pattern with lowest reliability of three demandpatterns considered as function of vector of decisivariablesY, vector of minimum specified nodalhead requirementsHmin

t and vector of probabilitydistribution parametersXpar

t ,I 5 number of nodes in network with minimum

pressure requirements;i 5 index for specific node in network;

k(Y) 5 total cost of networkL 5 system’s load;l 5 most critical node in network;

pf ,pf 1 5 probability of failure, 1 refers to Mode 1;pf s 5 probability of system failure;

pf 12 5 joint probability of failure for failure Modes 1and 2;

R 5 system’s resistance;S 5 survival domain;

V1* 5 design point in standard normal space forfailure Mode 1;

X 5 vector of random variables that influencessystem’s load and resistance;

Y 5 vector of decision variable values~pipe sizes!;a 5 reliability;

as 5 system reliability measure;a* 5 is minimum desired reliability level for WDN

under any critical demand pattern;b 5 reliability index;

F~ ! 5 standard normal cumulative density function;F~ , ;r! 5 CDF for bivariate normal vector with zero

mean values, unit variances, and correlationcoefficientr; and


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c~ , ;r! 5 PDF for bivariate normal vector with zeromean values, unit variances, and correlationcoefficientr.

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Genetic Algorithms for Reliability-Based Optimization of Water Distribution Systems