Evaluating Flexibility in Water Distribution System Design under Future Demand Uncertainty

Evaluating Flexibility in Water Distribution System Designunder Future Demand Uncertainty

Innocent Basupi1 and Zoran Kapelan2

Abstract: Performance of water distribution systems (WDSs) is highly dependent on consumer water demand that has to be met withadequate pressure. With uncertain future water demand due to the effect of rapid urbanization and climate change, WDSs may underperformor end up being overdesigned because of the long-term unforeseen future conditions. Long-term planning of WDS therefore requires stra-tegic, cost-effective, and sustainable design intervention investment across the entire planning horizon that is uncertain in nature. However,making the most appropriate decisions on such intervention measures that keep up the performance of WDS under uncertainty is a challenge.This paper illustrates the importance of flexible WDS design under uncertain future water demand. The methodology has been tested on theNew York Tunnels and the Anytown network interventions in the long-term planning. The approach is compared with the traditional deter-ministic intervention plans. The analysis results demonstrate that there is potential WDS performance value obtained when flexible designapproach is adopted rather than the conventional deterministic practice. DOI: 10.1061/(ASCE)IS.1943-555X.0000199. © 2014 AmericanSociety of Civil Engineers.

Author keywords: Uncertainty; Flexible design; Deterministic design; Water distribution systems.

Introduction

Water demand is one of the major factors that determine thehydraulic performance of a water distribution system (WDS).The effect of rapid urbanization and climate change render waterdemand highly uncertain. Redesign of WDSs is necessary to keepup with the water service requirements. However, the extent of de-sign has to be done now to supply adequate water to customers intothe unknown future. Making appropriate and long-term decisionsbecomes a challenge that requires strategic methodologies that less-ens the susceptibility of overdesigned or underperforming WDS asan attempt to better the traditional (deterministic) approach.

The aim of this paper is to evaluate the flexibility inherent in theflexible WDS design approach as compared with the rigid anddeterministic method that can lead to overdesigned or underper-forming WDSs because of the uncertain nature of the long-termwater demand.

The paper is organized as follows: after this introduction, thebackground of flexible designs and application in engineering sys-tems under uncertainty is briefly discussed, and the WDS flexibledesign evaluation methodology under water demand uncertainty isexplained. The methodology is then demonstrated on two literaturecase studies and the results are compared with both precautionaryand staged deterministic approaches. Finally, conclusions aredrawn.

Background

Uncertainty in engineering system parameters requires long-termplanning, designing, and management. Creaco et al. (2013) foundout that long-term designs (i.e., sequencing of construction)perform better than the deterministic designs, but the uncertaintyof design parameters was not considered. The WDS design is tradi-tionally based on the deterministic future water demand pro-jections. The deterministic approach makes WDS susceptible topoor performance because of the actual demand that is most likelyto differ from the projections. Building redundancy in WDS is oneway of achieving robustness. Kapelan et al. (2005) and Babayanet al. (2005) worked on robustness-based solutions to WDS designproblems under water demand uncertainty. Other researchersformulated and solved similar robust WDS design problems underuncertainty (Lansey et al. 1989; Xu and Goulter 1999; Giustolisiet al. 2009). However, in all these approaches, the future demanduncertainty has been addressed only passively by building in addi-tional system redundancy via suitably sized interventions that arefixed over some prespecified long-term planning horizon. Otherresearchers, e.g., Kang and Lansey (2012), considered scenario-based multistage construction of water supply infrastructure underdemand uncertainty. In their method, scenarios (plausible futureswith assumed probabilities of occurrence) are simultaneouslyconsidered to identify a solution that minimizes the expected costs(including overpayment and supplementary costs needed to meetthe requirements) as a single objective. The method presented byKang and Lansey (2012) adds robustness in system design by iden-tifying solutions that allow for adaptive modifications with mini-mized overpayment and supplementary costs. However, optionalpaths for relevant levels of uncertain demands that are accountedfor in the current study were not considered. The method presentedalso simulates uncertain demands around the mean demand(without focus on different spatial distributions) at each stage asexplained in the “Methodology” section of this paper. Furthermore,the intervention plans are evaluated over a large number of possiblefutures compared with the scenario-based method.

1Ph.D. Student, Centre for Water Systems, College of Engineering,Mathematics and Physical Sciences, Univ. of Exeter, North Park Rd.,Harrison Building, Exeter EX4 4QF, U.K. (corresponding author). E-mail:[email protected]

2Professor, Centre for Water Systems, College of Engineering, Mathe-matics and Physical Sciences, Univ. of Exeter, North Park Rd., HarrisonBuilding, Exeter EX4 4QF, U.K. E-mail: [email protected]

Note. This manuscript was submitted on November 19, 2012; approvedon November 5, 2013; published online on May 19, 2014. Discussion per-iod open until October 19, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Infrastructure Sys-tems, © ASCE, ISSN 1076-0342/04014034(14)/$25.00.

© ASCE 04014034-1 J. Infrastruct. Syst.

J. Infrastruct. Syst.

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http://dx.doi.org/10.1061/(ASCE)IS.1943-555X.0000199




As pointed out by De Neufville (2004), alternative ways exist tomore proactively manage future uncertainties by creating and main-taining flexibility in the engineering design. Huang et al. (2010) arethe authors who also suggested using the concept of flexible designin the context of long-term planning of WDSs. However, the meth-odology presented in this paper differs from the methodologyshown in the previous conference paper as follows:• Demand uncertainty is represented differently: In this meth-

odology, uncertain future demands are represented using the(Gaussian, can be any other) probability density functions withincreasing mean (denoting increase in most likely future de-mands) and increasing variance (i.e., level of uncertainty) overthe planning horizon, whereas in the Huang et al. (2010)approach, uncertain future demands are represented using a sce-nario tree denoting a number of predefined demand scenarioswith arbitrarily chosen probabilities for different paths on thetree. The approach adopted in this paper seems more compatiblewith the way demand projections are normally made in theengineering practice.

• Intervention plans are represented and evaluated differently: Inthis methodology, flexible intervention plan is represented as adecision tree with threshold demand values defined at branching(i.e., decision) points. The associated intervention path probabil-ities are then estimated indirectly by using the Monte Carlo(MC) sampling method. This setup allows for decisions to bemade according to how the uncertain water demands wouldactually evolve. In the Huang et al. (2010) approach, flexibleintervention plan is represented as a set of time-staged interven-tions that are evaluated using the enumeration method over asmall number of predefined demand scenarios over the analyzedplanning horizon (see previous discussion).

• The designed WDS performance is evaluated differently: Inthis methodology, this is evaluated in terms of total cost and

resilience as the average of all sampled demand profiles,whereas in the approach by Huang et al. (2010), this is evaluatedin terms of cost and pressure deficiency by summing up the pro-ducts of individual WDS design costs and the assumed probabil-ity of each demand scenario analyzed.

Flexible WDS Design Methodology

Uncertain Demands

The future water demand is considered as the only source of un-certainty in this analysis, i.e., to demonstrate the value of flexibleWDS design approach. The nodal water demands at given points intime over some long-term planning horizon are assumed to follow anormal distribution function (Babayan et al. 2005; Kapelan et al.2005) with mean values equal to the traditional, deterministic pro-jection of future demands and the increasing standard deviation.The design methodology shown in this paper is not limiting in thissense and any other probability density function(s) (PDF/PDFs) (orscenario-based approach) can be used to describe uncertain futurewater demand.

A single demand profile (or scenario) over the planning horizonis generated by randomly sampling system level demands (Dt1;Dt2; : : : ;Dtend shown in Fig. 1) from respective, prespecified de-mand PDFs at the end of each planning horizon stage (i.e., at timet1; t2; : : : ; tend). A large number of demand profiles (or scenarios) isgenerated for the purpose of evaluation of flexible (and determin-istic) WDS designs (i.e., 10,000 samples in the case studies shownin this paper).

Finally, the randomly generated system level demands areallocated to demand nodes proportionately, according to eachnode’s respective contribution to (i.e., percentage of) the system

Water Demand

[l/s]

Time [years]

Intervention Paths

1st

EPANET RUN2nd

EPANET RUN3rd

EPANET RUN

TA

TB-1

TB-2

TC-1

TC-2

TC-3

TC-4

Demand Forecast(Deterministic)

t0 t1 t2 tend

H-D

L-D

LL-D

LH-D

HL-D

HH-D

HHH-D

HHL-D

HLH-D

HLL-D

LHH-D

LHL-D

LLH-D

LLL-D

Fig. 1. Illustration of uncertain water demand with increasing variance over time



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level (mean) demand. This indicates that an absolute increase(i.e., in l=s) for a node with larger demand will be greater thanthe increase of the node with smaller demand.

Representation of Flexible WDS Design

The flexible design, i.e., intervention plan, is represented using a de-cision tree (Fig. 1). Each branch on the tree represents a set of indi-vidual intervention measures (i.e., interventions), and each path on thetree (from the root of the tree to its leaves) represents one possible(i.e., optional) intervention path across the analyzed planning horizon(a total of eight paths shown in Fig. 1). The branching of the tree cor-responds to the design stages of the planning horizon (e.g., t0–t1 andt1–t2). The decision tree used is not a full tree, i.e., it does not havecombinations of tree nodes (opposite of binomial lattices). The deci-sion tree explained in this section has optional intervention paths at theend of each time step (Fig. 2) of the planning horizon. These featuresof the decision tree enable it to deal with path-dependent, interdepend-ent, and irreversible flexible interventions. In this case, path depend-ence indicates that the extent of future design interventions or the stateof the system at any point in time depends on the previous interventionpath undertaken (given changes in uncertain water demand). This fea-ture is as opposed to the financial options (of which this methodologyis analogous to) that have valuewhich only depends on the price at anyparticular time. Interdependence refers to the interaction of interven-tions that influence each other’s performance, hence the overall per-formance of the whole system. The flexible designs are nonreversiblebecause once they have been exercised, the interventions (e.g., struc-tures) implemented are not going to be demolished.

As shown in Fig. 1, a decision point exists at each tree node. Thedecision to be made is which tree branch to follow (i.e., which set ofoptional interventions to implement) in the next planning horizon(i.e., design stage). This decision is taken based on the level of de-mand at that decision point relative to the prespecified demandthreshold (see intervention paths, e.g., L-D, LH-D, and LHH-Dand the corresponding threshold demands TA, TB−2, and TC−3shown in Fig. 1). L-D and H-D refer to Lower and Higher designinterventions, respectively (i.e., in the first design stage). In the sub-sequent stages, this design notation (LL-D, LH-D, HL-D, HH-D,and so on) also shows information about the previous stage(s). Forexample, LH-D is a Higher design in the second design stage butcomes after a Lower design in the first stage. If the generated sys-tem demand Dt1 at the end of first stage (i.e., at time t1) is largerthan the corresponding threshold demand (TA), then path H-D isfollowed. If, however, demand Dt1 is smaller than the thresholddemand TA, then the L-D intervention path is followed. Further-more, if Dt1 results in the selection of the H-D path, then Dt2can only be used to select either HL-D or HH-D path (dependingon whetherDt2 is above or below TB−1). The subscripts A, B, and Crepresent the three design stages that are used in this paper to dem-onstrate the evaluation of design plans. Therefore, as shown inFig. 1, for a given random demand profile, the demands generatedat t1, t2, and tend are used to select (and eventually evaluate, see nextsection) the WDS designs at stages 1, 2, and 3, respectively. Thedemand threshold value is not linked directly to any pipe/tank/pump capacity. Also, when the flexible designs are evaluated,no further pipe and/or tank sizing are performed, i.e., the flexibledesign plans are not modified in any sense. The do more/design anddo less/nothing interventions are represented with solid and brokenlines, respectively. The threshold values essentially define indi-rectly the probabilities of optional paths on the intervention deci-sion tree (given random future demands). Given a certain systemwater demand level (say at t1) and assumed demand PDFs acrossthe planning horizon, there is a probability associated with the

likelihood of sampling system demand that is above or below thatcertain water demand level. This certain water demand level iswhat is referred to as the threshold demand in this paper.

Evaluation of Flexible WDS Designs

The WDS design performance at each stage of the planning horizonis evaluated by calculating the cumulative cost and the WDSresilience, as defined subsequently. For this purpose, a number ofrandom demand profiles (i.e., scenarios) are generated first, as de-scribed in the demand section previously. Once this is done, for eachdemand profile generated, the corresponding intervention path on thedecision tree (used to represent the evaluated flexible design) is de-termined, as explained in the previous section. Now, for each inter-vention path selected, the EPANET network hydraulic solver(Rossman 2000) is run at the end of each design stage (i.e., att1; t2; : : : ; tend) with the aim of estimating the system resilience atthese points in time (runs are not performed at t0 as initial resilienceis known, i.e., needs to be estimated only once, prior to samplingdemands). At the same time, the corresponding intervention (andif necessary operational) costs are evaluated and accumulated. Eachtime the hydraulic solver is run (e.g., at time t1), both the networkconfiguration (and, if necessary, system operation) and system de-mands are updated to take into account the interventions imple-mented during the analyzed stage (t0–t1) and the modified end ofstage demands (at t1). The above is repeated for all design stagesresulting in two planning horizon profiles (resilience and cost) builtfor each analyzed demand profile. At the end, the cumulative costand system resilience profiles generated this way are averaged over aprespecified, typically large number of Monte Carlo simulations(i.e., demand samples). The procedure for the evaluation of the flex-ible WDS design represented using a decision tree in Fig. 1 is shownin Fig. 3. The approach presented in this section is generic in terms ofthe number of time-steps and the intervention paths considered.

As discussed previously, to demonstrate the value embedded in theflexible intervention under the proposed methodology, the followingtwo indicators were used: (1) the total intervention cost incurred in theinterventions that include duplicating existing or adding new pipes,cleaning and lining of existing pipes, addition of tanks and their lo-cations, tank parameters (diameter, bottom elevation, minimum, andmaximum operating levels), and the pump schedule, i.e., the numberof operating pumps at a given time; and (2) the WDS resilience index(RI), which serves as a measure of WDS’s intrinsic capability to en-sure continuity of supply to users after sudden failure conditions.

Total system intervention cost is estimated as follows:

Average Total Cost

¼PNs

k¼1

PSi¼1

1ð1þrÞt ð

Pnj¼1 Cj;cap þ

Pnpuj¼1 Cj;opÞ

Ns

ð1Þ

Redesign (or 'do more')

Threshold Value

Intervention Option Selection

Mean Demand

Do nothing (or 'do less')

Fig. 2. Illustration of a decision criterion for implementing interven-tions at each decision point



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where Ns is the total number of samples (demand scenarios); r isthe cost discount rate; t is the time elapsed (years) in the planninghorizon; S is the total number of design stages; Cj;cap is the capitalcost of the jth rehabilitation intervention option in the ith stage; n isthe number of rehabilitation options implemented; Cj;op is thewhole stage interval operating energy consumption cost becauseof the jth pump; and npu is the number of pumps in the systemfor a particular ith stage. The total cost is cumulative across theentire planning horizon for every kth sample.

The WDS resilience was introduced by Todini (2000) as a mea-sure of WDS’s intrinsic capability to ensure continuity of supply tousers after sudden failure conditions. It is estimated as follows:

Resilience Index ¼Pnn

i¼1 qiðhav − hreqÞPnrr¼1 QrHr þ

Pnpuj¼1ðPj

γ Þ −Pnn

i¼1 qihreqð2Þ

Because of the multistaged designs and a large number ofdemand scenarios considered in this methodology, the averageresilience index (RIav) at each time-step is calculated as follows:

RIav ¼PNs

k¼1 RIkNs

ð3Þ

where qi corresponds to the flow of the ith node; hav is the availablepiezometric head; hreq is the minimum required head; Qr is the res-ervoir flow;Hr is the reservoir head; Pj is the pump power; γ is thespecific weight of water; nn is the number of demand nodes; nr isthe number of reservoirs; and npu is the number of pumps. Otherresilience (and/or reliability/risk) measures could be used as well, ifdesired, e.g., the network resilience and the modified resilience in-dex by Prasad and Park (2004) and Jayaram and Srinivasan (2008),

respectively. For more information on the performance of theseresilience indices, see Banos et al. (2011). The focus of this paperis on the flexible design concepts, not on the most appropriate resil-ience measure. Eq. (1) uses the present value analysis for discount-ing future costs. System resilience improvement is used as ameasure of benefit achieved by design interventions. This benefitis not expressed in monetary units, which prevents the use of moreconventional net present value (NPV) analysis.

The intervention plan analysis is subject to hydraulic model(i.e., mass and energy balance) equations

Xni

m¼1

qm − qd;i ¼ 0 ði ¼ 1; : : : ; nnÞ ð4Þ

hi;u − hi;d −Δhi ¼ 0 ði ¼ 1; : : : ; nlÞ ð5Þwhere qm are the flows in all ni pipes; qd;i is the water demand atthe ith node; hi;u is the head at upstream node of the ith pipe; hi;d isthe head at downstream node of the ith pipe; Δhi is the differencebetween the ith pipe’s total head loss and pumping head; nn is thenumber of network nodes; and nl is the number of network links.

Solutions whose performance is evaluated are kept within thepractical constraints as shown in Figs. 4–7. These constraints meanthat a potential solution should have path-dependent, interdepend-ent, and irreversible interventions that can be implemented at mostonce in any particular uncertain water demand scenario across theplanning horizon.

Generation of Flexible WDS Designs

The flexible and other WDS designs analyzed in the case studiesshown in this paper have been derived from the correspondingoptimized WDS designs reported previously in the literature. Thesedesigns were generated manually by using engineering judgment only.The relevant details of the designs generated this way can be found inthe next section. Although this is not ideal (flexible designs could beoptimized in the same way as the more conventional, deterministicones), it is fit for the objective of this paper which is to comparethe deterministic and flexible WDS designs and demonstrate that flex-ible WDS designs can outperform the more conventional (i.e., deter-ministic) ones under conditions of uncertainty. Therefore, if the lattercan be shown with manually generated flexible WDS designs (as it isdone in the next section), then things can only be improved further byusing some more formal methodology (e.g., optimization based) togenerate further improved flexible WDS designs.

Case Studies

The flexibility evaluation methodology presented in this paper hasbeen illustrated on the New York Tunnels (Dandy et al. 1996) andthe Anytown network (Walski et al. 1987) redesign problems.Figs. 4–8 and 6–10 show results for both networks that demonstratethe performance differences between deterministic and flexible de-sign plans over the WDS planning horizon. The literature solutionsfor both networks were used in the current study as the basis foranalysis because they were optimized for the deterministic waterdemand at the end of the planning horizon which is also usedas the mean value in the current study. Two deterministic planswere considered, namely, the precautionary and the staged deter-ministic approaches. The precautionary implementation of theleast-cost solution indicates that all the interventions of the designare carried out at the beginning of the planning horizon. As forthe staged approach, interventions were distributed across the plan-ning horizon based on the critical demand nodes of the networks.

Design 1 (at t0):

1. Generation of a specified number of Random Water Demand Profiles - i.e., random demand values at t1 (Dt1), t2 (Dt2) and tend

(Dt,end)

2. Evaluation of the Flexible Design Plan for all the number of Sampled Scenarios (Demand Profiles)

Design 3 (at t2):

Design 2 (at t1):

Output:

Average Cumulative Cost at each Design stage Average Resilience Index at each Design stage

If Dt,end >TC-1

Evaluate HHH-DElse

Evaluate HHL-D

If Dt,end <TC-2

Evaluate HLL-DElse

Evaluate HLH-D

If Dt1 > TA

Evaluate H-DIf Dt1 < TA

Evaluate L-D

If Dt2 >TB-1

Evaluate HH-DIf Dt2 < TB-1

Evaluate HL-D

If Dt,end <TC-4

Evaluate LLL-DElse

Evaluate LLH-D

If Dt,end >TC-3

Evaluate LHH-DElse

Evaluate LHL-D

If Dt2 >TB-2

Evaluate LH-DIf Dt2 <TB-2

Evaluate LL-D

using MC simulations

Fig. 3. The flowchart illustration of the flexible WDS intervention planperformance evaluation



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Interventions (e.g., pipes) that lead to the critical demand nodeswere meant to be implemented at earlier stages. Also, note that flex-ible plans considered in this paper are cost equivalent to the stageddeterministic plans only.

Data Analyses and Assumptions

For both of the networks considered in this study (defined in the nextsection), a planning horizon of 60 years with three 20-year decision-making intervals was used for analysis. The planning period is con-sidered to be suitable for the analysis of long-term WDS design andthe associated climate change/urbanization effects.

Kapelan et al. (2005) assumed nodal demands as uncertain var-iables that follow a Gaussian PDF with a mean value equal to thedeterministic value and a standard deviation equal to 10% of themean value of the New York Tunnels network problem. Burovskiyet al. (2011) made similar assumption to Kapelan et al. (2005) butwith 30% of the deterministic demands on the Anytown networkproblem. In a similar manner, Gaussian PDFs are assumed withstandard deviations of 5, 10, and 20% of the mean water demandvalues (referred to as Case 2) at years 20, 40, and 60, respectively,for both the New York Tunnels and the Anytown network solutionsconsidered in the present study. These standard deviations reflectthe fact that demands are more uncertain if projected for longer intothe future. The initial water demand for each network was assumedto be known, and it was fixed at a value which after a 30% incre-ment results in the end of the planning horizon mean demand(the latter value being defined in the literature). The initial systemwater demands are 43.9 m3=s (1,552 ft3=s) and 0.48 m3=s(7,538 gal:=min) for New York Tunnels and Anytown networks,respectively. The initial water demand is assumed to increase ex-ponentially until the end of the planning horizon demand. Also,with regard to spatial distribution, all the nodal demands in the sys-tem are assumed to increase at the same rate (percentage). Becauseof the standard deviation assumptions made in this study, two morecases (þ25% and −25% of the Case 2 standard deviations acrossthe planning horizon) were also investigated for sensitivity analy-sis. These two more cases are referred to as Cases 1 and 3, respec-tively (Fig. 11).

As for the cost, time preference is accounted for by using presentvalue analysis. Many water utilities adopt a discount rate close to thecapital cost which is around 6–8% (Wu et al. 2010). In this study, adiscount rate for the cost of implementing interventions at different

times as depicted by the solution plans across the planning horizon is6%. Because of the assumption made in this section, other discountrates used for sensitivity analysis are 4.5 and 7.5%. The study con-sidered the capital costs of pipes, tanks, and pump operation only asthey have a major contribution in the WDS design expenses. Theadditional costs components such as maintenance, labor, and waterloss will unnecessarily obscure the intended purpose of the approachpresented in this section. These additional aspects are common fac-tors that apply to both staged deterministic and the flexible methods,which are more important in the comparison made in this study. Atotal of 10,000 Monte Carlo simulations of uncertain future waterdemand scenarios were used in the analysis of all the design planspresented in this study.

The New York Tunnels Problem

DescriptionThe network has a single source (i.e., reservoir)—19 demand nodesand 21 pipes (Fig. 12). There is a minimum head requirement of77.72 m (255 ft) at all demand nodes except Nodes 16 and 17 thathave head requirements of 79.25 m (260 ft) and 83.15 m (272.8 ft),respectively. The reservoir at Node 1 has a fixed head of 91.44 m(300 ft). In the current study, the flexibility evaluation indicators arethe total cost and the WDS resilience. The means of rehabilitation isto duplicate existing pipes with new ones at different design decisionpoints as water demand evolves. An existing pipe can only be du-plicated once across the entire planning horizon. In the present analy-sis, deterministic (precautionary and staged) and flexible designplans are evaluated separately under uncertain water demands.

The precautionary intervention plan is based on the least-costsolution identified by Maier et al. (2003) by assuming that allinterventions are implemented in the first stage of the planning hori-zon. The corresponding staged deterministic intervention plan wascreated by distributing the individual interventions of the aforemen-tioned least-cost design across the planning horizon, in a way which is

Fig. 4. Comparative (cost equivalent) flexible and deterministic de-signs for the New York Tunnels network (Case 2)

Fig. 5. Comparative (resilience equivalent) flexible and deterministicdesigns for the New York Tunnels network (Case 2)



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compatible with the forecasted demand increase and critical nodes.Finally, the flexible plan was created by adding or keeping the pre-vious interventions on the decision tree paths that correspond to thepossible higher and lower future demand forecasts. The flexible plan,which is intended to envelope uncertain demands in each stage, wascreated by including all and also adding more interventions (i.e., alonghigher demand paths) to those that were selected for the staged deter-ministic plan. It is important that the flexible designs created are pathdependent and irreversible as explained in the methodology section.Along the lower demand paths, interventions from the previous stagesare kept in the consecutive stages without any additional systemreinforcement (i.e., do nothing). The threshold values were set man-ually in the example designs evaluated in this paper. For example, inFig. 4, in Design Stage 1, if a threshold demand of 47.1 m3=s(1,665 ft3=s) was placed, this indicates that of the 10, 000MC systemdemands simulated around an average deterministic value of48.8 m3=s (1,723 ft3=s), in approximately 75% cases, the demandvalue will be higher than 47.1 m3=s (1,665 ft3=s). The percentageis determined by the relevant demand PDF defined at the end of Stage1. The higher demands are used to evaluate the more reinforced sys-tems, whereas those that are equal to or lower than the threshold de-mand are used for less reinforced systems. The same evaluationprocedure applies to the consecutive stages. The system performanceat each design stage is averaged across all 10,000 samples.

The three intervention plans generated this way are shown inFigs. 4 and 5. The fact that the staged and flexible intervention plansgenerated may not be optimal (hence potentially disadvantaged inthat sense when compared with the precautionary plan) but are stillcomparable is acknowledged. The reason for this is that if stagedand/or flexible plan show improvement over the precautionaryone (in terms of average cost and/or resilience), this improvementcan only be further increased if these two plans are optimized.

The cost of pipe duplication intervention for the analyzed plansin US dollars ($) is given by

Average Total Cost ¼PNs

k¼1

PSi¼1

1ð1þrÞti

Pnpj¼1 1.1D

1.24j Lj

Nsð6Þ

where pipe diameter Dj and length Lj are in inches and feet, re-spectively; ti is the time elapsed at the beginning of each designstage (design point). This is an original cost model for the NewYork Tunnels network (CWS 2013), which has been extendedto account for staged designs.

Results and DiscussionThe intervention plans’ evaluation results based on the previouslyexplained methodology are presented in Figs. 4–8. The flexible,precautionary, and staged deterministic design plans are comparedin terms of average cost and average resilience index.

The precautionary plan of the New York Tunnels problem inFigs. 4 and 5 shows that all the interventions (six duplication pipes)are carried out in the first design stage, as it is normally done in thistype of approach. The duplication pipes selected are shifted towardsimproving available head at high demand and far downstream nodesbecause of high head loss from source to critical nodes (16, 18, 19,and 20) in the network. Also, it is shown that the solution duplicatesPipe 7 that has a relatively small (132 in.) existing diameter. The restof the duplicated pipes (16, 17, 18, 19, and 21) have smaller diametersizes of 1829, 1829, 1524, 1524 and 1829 mm (i.e., 72, 72, 60, 60and 72 in., respectively). For the corresponding staged deterministicplan, the interventions were selected based on critical nodes. For ex-ample, Pipe 19 is duplicated with a 1829-mm (72-in.) pipe in the firststage of the staged deterministic plan. Pipe 19 conveys water to Node20 which has a high mean demand at the end of the first design stage.

Pipe No. 1,..6 7 8,..15 16 17 18 19 20 21 1,2 3 4,..6 7 8,..15 16 17 18 19 20 21 1 2 3 4,..6 7 8,..14 15 16 17 18 19 20 21

Av. total Cost Av. End RI

t0 t1 t2 tend ($M) (-)

0 144 0 96 96 84 72 0 72 0 0 0 144 0 96 96 84 72 0 72 0 0 0 0 144 0 0 96 96 84 72 0 72 38.64 0.402

0 0 0 0 0 0 72 0 0 0 0 0 0 0 96 0 0 72 0 72 0 0 0 0 144 0 0 96 96 84 72 0 72 9.67 0.402

120 0 36 0 144 0 192 96 96 84 72 0 72

0 36 0 0 0 96 0 84 72 0 72

0 0 36 0 0 0 0 96 0 84 72 0 72

0 0 0 0 0 0 72 0 0

120 0 36 0 144 0 192 96 96 84 72 0 72

0 0 0 0 0 0 0 0 72 0 0

0 0 0 0 0 0 0 0 0 0 72 0 0

9.62 0.441

120 0 36 0 144 0 192 96 96 84 72 0 72

0 36 0 0 0 96 0 84 72 0 72

0 0 36 0 0 0 0 96 0 84 72 0 72

0 0 0 0 0 0 0 0 0

120 0 36 0 144 0 192 96 96 84 72 0 72

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

Current Stage Pipe Duplication

Threshold Demand: 1678 ft3/s, Samples (HLH-D = 24%, HLL-D = 6%)

Threhold Demand: 1678 ft3/s, Samples (HHH-D = 36%, HHL-D = 9%)

Threshold Demand: 1678 ft3/s, Samples (LHH-D = 12%, LHL-D = 3%)

Threshold Demand: 1678 ft3/s, Samples (LLH-D = 8%, LLL-D = 2%)

No Pipe Duplication / Previously Duplicated Pipe

Samples (HH-D) = 45%

Threshold Demand: 1866 ft3/s


Samples (H-D) = 75%

Samples (L-D) = 25%

Fle

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Pipe Diameters (inches): Design 1 Pipe Diameters (inches): Design 2

Det

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ns mean demand = 1723 ft3/s mean demand = 1914 ft3/s

Staged

Samples: LL-D =10%


Precautionary

Performance IndicatorsPipe Diameters (inches): Design 3

Samples (HL-D) = 30%

Samples: LH-D = 15%

mean demand = 2017.5 ft3/s

Fig. 6. Comparative (cost equivalent) flexible and deterministic designs for the Anytown network (Case 2); note: pipes that are not reinforced/rehabilitated in any of the three designs presented are not shown



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It is also known that Node 20 is one of the highest demand nodes inthe network with 4.814m3=s (170 ft3=s) at the end of the planninghorizon. The corresponding flexible design interventions were se-lected as an attempt to add more reinforcement on the interventionsthat were selected for the staged deterministic interventions. Forexample, in Figs. 4 and 5 Design Stage 2, Pipes 16, 19, and 21 thathave been selected in the staged deterministic plan are also part of thealternative routes’ interventions in the flexible plan.

The solution evaluation results obtained indicate that there isvalue associated with flexibility in the implementation of interven-tions across the WDS planning horizon. This is evidently demon-strated in Fig. 4 by the flexible intervention plan that shows asimilar average total cost (slightly lower) to the staged deterministicsolution but has clearly higher end resilience. A flexible plan whichhas an average total cost of $9.62 million results to an average resil-ience index of 0.441, whereas the deterministic plan has an endaverage resilience of 0.402 with costs of $38.64 million and$9.67 million for precautionary and staged design, respectively.

Fig. 5 shows a flexible plan that outperforms the resilience equiv-alent deterministic plan in terms of average total cost. They all havesimilar resilience index of 0.402 but cost $7.79 million, $38.64 mil-lion, and $9.67 million for flexible, precautionary, and staged deter-ministic plans, respectively. The total cost or the end resiliencedifference between the deterministic and the flexible interventionplans is the engineering value added by flexibility. For example,the $1.88 million difference between the staged deterministic andthe flexible approach reflects the cost reduction that is introducedby the flexible plan. This reduction results because of the capabilityof the flexible design plan to adapt to uncertain water demand. Thiscapability indicates that the flexible approach can effectively avoidimplementing further interventions altogether if the future waterdemand turns out not to increase (much) further. The deterministic

Pipe No. 1,..6 7 8,..15 16 17 18 19 20 21 1,2 3 4,..6 7 8,..15 16 17 18 19 20 21 1 2 3 4,..6 7 8,..14 15 16 17 18 19 20 21

Av. total Cost Av. End RI

t0 t1 t2 ($M) (-)

Precautionary

0 144 0 96 96 84 72 0 72 0 0 0 144 0 96 96 84 72 0 72 0 0 0 0 144 0 0 96 96 84 72 0 72 38.64 0.402

Staged

0 0 0 0 0 0 72 0 0 0 0 0 0 0 96 0 0 72 0 72 0 0 0 0 144 0 0 96 96 84 72 0 72 9.67 0.402

120 0 36 0 144 0 192 96 96 84 72 0 72

0 36 0 0 0 96 0 84 72 0 720 0 36 0 0 0 0 96 0 84 72 0 72

0 0 0 0 0 0 72 0 0

120 0 36 0 144 0 192 96 96 84 72 0 72

0 0 0 0 0 0 0 0 72 0 00 0 0 0 0 0 0 0 0 0 72 0 0

7.79 0.402

120 0 36 0 144 0 192 96 96 84 72 0 72

0 36 0 0 0 96 0 84 72 0 720 0 36 0 0 0 0 96 0 84 72 0 72

0 0 0 0 0 0 0 0 0

120 0 36 0 144 0 192 96 96 84 72 0 72

0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0

Current Stage Pipe Duplication

No Pipe Duplication / Previously Duplicated Pipe

Threshold Demand: 1806 ft3/s, Samples (HLH-D = 9.45%, HLL-D = 4.05%)

Threshold Demand: 1806 ft3/s, Samples (LHH-D = 26.95%, LHL-D = 11.55%) Fle

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Samples (H-D) = 30%

Samples (L-D) = 70%

Threshold Demand: 1806 ft3/s, Samples (HHH-D = 11.55%, HHL-D = 4.95%)

Threshold Demand: 1806 ft3/s, Samples (LLH -D = 22.05%, LLL-D = 9.45%)



Samples (H-D) = 16.5%

Samples (L-D) = 13.5%

Samples (LH-D) = 38.5%

Samples (H-D) = 31.5%

Performance Indicators

Det

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Pipe Diameters (inches): Design 1 Pipe Diameters (inches): Design 2 Pipe Diameters (inches): Design 3

mean demand = 1723 ft3/s tendmean demand = 1914 ft3/s mean demand = 2017.5 ft3/s

Fig. 7. Comparative (resilience equivalent) flexible and deterministic designs for the Anytown network (Case 2); note: pipes that are not reinforced/rehabilitated in any of the three designs presented are not shown

Fig. 8. New York Tunnels average cumulative cost profiles of threedesign plans (Case 2)



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intervention plans lack this attribute which indicates that consider-able economic and WDS resilience values can be left unexploited.The capability to adapt the WDS configuration when and where nec-essary allows postponing (or avoiding) design intervention measureswhich, in turn, results in lower average present value cost of the flex-ible design. Intervention measures implemented in future is an attrib-ute of both the staged deterministic and flexible plans, but the latterallows for more interventions to be possibly implemented in a certainroute. This feature raises the average resilience but still maintain acomparable average total cost.

Figs. 13 and 8 show the profiles of average resiliencies and thecumulative costs of the precautionary, staged, and the flexibleintervention plans for Case 2. Table 1 provides additional informa-tion of intervention plans and their respective stage costs and resil-ience performances. End average resilience information resultingfrom all scenarios considered in this study is shown in Table 2.

The least-cost design solution for New York Tunnels problem is$38.64 million (Maier et al. 2003). The precautionary plan for thissolution has a higher cost than both the staged deterministic and theflexible plan that explains its higher initial average resilience(0.564). However, its average resilience reduces because of the ris-ing future demands until it intersects with the average resilience(0.402) of the developmental deterministic plan at the end of theplanning horizon. This fact happens despite that the precautionaryplan has a higher average total cost. The staged deterministic designshows lower average total cost because of the discounted cost offuture interventions. The cost equivalent flexible plan’s averageresilience starts off at a lower level (0.243) with a lower initial cost($2.40 million) than the staged deterministic plan ($3.18 million).This happens due to the fact that the interventions are the same atthat stage but the flexible plan allows for doing nothing which ex-plains the lower average total costs and average resilience values.

In the second stage (at 40 years into the future), the cost equiv-alent flexible plan outperforms the staged deterministic approachbecause of the possibility to take intervention paths with additionalsystem reinforcement if the water demand is high. In this secondstage, the cost equivalent flexible solution has an average resilienceof 0.167 and an average total cost of $4.08 million compared withthe average resilience of 0.149 and an average total cost of$4.42 million for the staged deterministic approach.

More importantly, the cost equivalent flexible design plan outper-forms both the precautionary and the staged deterministic interventionplans in terms of the average resilience index at the end of the planninghorizon even though it has a similar or lower average total cost. Theend resilience equivalent solution also stresses that the same averageresilience to the staged deterministic approach can be achieved atlower average total cost. All these results reveal the consequenceof the built-in flexibility to adapt to changes in the future demand.

Table 2 provides the sensitivity analysis results of the perfor-mance of WDSs in terms of average resilience with varying level

Fig. 9. Anytown network average resilience index profiles of three de-sign plans (Case 2)

Table 1. Design Plan, End-of-Stage Average Costs, Average Resiliencies and Average Total Costs (Cumulative) for New York Tunnels and AnytownNetworks in Case 2

Solution plan Design time

New York tunnels Anytown network

Average cost($ million)

Averageresilience index

Average total cost(cumulative)

Average cost($ million)

Averageresilience index

Average total cost(cumulative)

Precautionary t0 0 0.310 0 0 0 0t1 38.64 0.564 38.64 16.23 0.164 16.23t2 0 0.468 38.64 3.35 0.172 19.57tend 0 0.402 38.64 1.11 0.117 20.68

Staged deterministic t0 0 0.310 0 0 0 0t1 3.18 0.266 3.18 13.59 0.147 13.59t2 4.42 0.149 7.60 3.53 0.169 17.12tend 2.07 0.402 9.67 1.31 0.117 18.43

Flexible (cost equivalent) t0 0 0.310 0 0 0 0t1 2.40 0.243 2.40 13.12 0.026 13.12t2 4.08 0.167 6.47 4.24 0.169 17.36tend 3.15 0.441 9.62 1.06 0.164 18.42

Flexible (resilience equivalent) t0 0 0.310 0 0 0 0t1 0.95 0.195 0.95 13.12 0.026 13.12t2 3.96 0.137 4.91 3.55 0.035 16.67tend 2.88 0.402 7.79 1.12 0.117 17.79

Note: Average cost at any given time refers to the cost incurred before the corresponding design time. Bold values that indicate the most important (end ofplanning horizon) values that are considered in respective Figs. 6, 7, 11, and 12.



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of demand uncertainty. The results indicate that the relative (i.e., per-centage) increase of average resilience from the deterministic to thecost equivalent flexible plan increases with more water demand un-certainty. For example, an increase of an average resilience at the endof the planning horizon for a comparable cost equivalent flexibleintervention to the deterministic plans is from 7.3% (Case 1) to9.7% (Case 3). These average resilience increment differences con-firm that uncertainty does not always need to be avoided but it can bean opportunity to be exploited (De Neufville 2003).

The sensitivity analysis results for varying discount rates inTable 3 show that reducing cost discount rate from 6 to 4.5% leadsto a 31.5% increase of the average total cost in the case of the stageddeterministic solution. The increase is lower than 39.2% obtainedfrom the flexible plan. Again, the flexible approach is more sensi-tive than the staged deterministic plan when the discount rate isincreased to 7.5%. The average total cost of the flexible designis reduced by 24.4%, whereas in the case of staged deterministicplan, it is 20.4%. This is a consequence of optional interventionpaths that result in a wider range of possible costs.

The Anytown Network Problem

DescriptionThe Anytown WDS (Fig. 14) was originally set up by Walski et al.(1987) as a realistic example of a more challenging WDS rehabili-tation problem even though it does not have all features of real

systems (e.g., multiple pressure zones, seasonal and local demandfluctuations, fiscal constraints, uncertainty of future demands andpipe roughness, and complicated staging of construction).

The network consists of existing pipes in the central city (thicksolid lines) which are difficult to access, making cleaning or pipe du-plication more expensive. In the residential region (thin lines), pipesare easier to access and therefore cheaper to clean or duplicate. Formore details of the different costs involved in this problem, seeWalskiet al. (1987). The dashed lines indicate the new pipes for the plannedextension to the north of the city which is also part of the networkrehabilitation. The network has two existing tanks. The treatmentworks is maintained at a fixed level of 3.05 m (10 ft), and thetwo existing tanks operate with levels between 68.58 m (225 ft)and 76.20 m (250 ft). Water is pumped into the system from a nearbytreatment plant by three parallel identical pumps.

The objective of the problem is to determine the most economi-cally effective solution to reinforce the existing network to meet thefuture demands considering pumping (operational) and capitalcosts. Rehabilitation options for each existing pipe include dupli-cation, cleaning and lining, or do nothing. A pipe that has beencleaned and lined has a Hazen-Williams coefficient of C ¼ 125compared with C ¼ 130 for new pipes. New pipes can be chosenfrom a range of 10 possible diameters (6, 8, 10, 12, 14, 16, 18, 20,24, and 30 in.). Any node (except Node 1) which is not alreadyconnected to the existing tank is considered as a potential locationsite of a new tank. Each tank has an emergency volume and anormal operating volume. A maximum of two new tanks each with

Table 2. Cost Equivalent Design Plan Analysis for Varying Standard Deviations for New York Tunnels and Anytown Networks

Solution planStandard

deviation (case)


Average total cost($ million)

Average endresilience index



Precautionary 1 38.64 0.409 20.67 0.1202 38.64 0.402 20.68 0.1173 38.64 0.392 20.69 0.112

Staged deterministic 1 9.67 0.409 18.43 0.1202 9.67 0.402 18.43 0.1173 9.67 0.402 18.45 0.112

Flexible 1 9.62 0.439 18.41 0.1662 9.62 0.441 18.42 0.1643 9.62 0.441 18.43 0.157

Percent increase from a staged deterministicto a flexible approach

1 — 7.3 — 38.32 — 9.7 — 40.23 — 9.7 — 40.2

Table 3. Cost Equivalent Design Plan Analysis for Varying Cost Discount Rates for New York Tunnels and Anytown Networks in Case 2

Solution planDiscountrate (%)






Precautionary 4.5 38.64 0.402 24.76 0.1176 38.64 0.402 20.68 0.1177.5 38.64 0.402 17.97 0.117

Staged deterministic 4.5 12.72 0.402 22.73 0.1176 9.67 0.402 18.43 0.1177.5 7.70 0.402 15.59 0.117

Flexible 4.5 13.39 0.441 22.75 0.1646 9.62 0.441 18.42 0.1647.5 7.27 0.441 15.51 0.164

Percent increase (+)/reduction (−)Staged deterministic 4.5 þ31.5 — þ23.3 —

7.5 −20.4 — −15.4 —Flexible approach 4.5 þ39.2 — þ23.5 —

7.5 −24.4 — −15.8 —



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its location, overflow elevation, normal day elevation, diameter,and the bottom elevation as decision variables are considered.The tank is connected to the demand nodes by a riser pipe that alsohas to be sized. In addition to the rehabilitation of the network, theoperation schedule of the pumps for a typical day is to be selected.To optimize the pumping schedule, the design of new tanks that filland empty over average daily flows and allow for emergency flowsmake it difficult to choose between solutions because a number ofsolutions can satisfy pressure requirements under average dailyflows, but the end-of-day tank levels may differ from the start-of-day levels. Some of the solutions may satisfy the start- andend-of-day levels under average day flows but fail to satisfy theminimum required pressures under instantaneous peak flows.The network solutions have to satisfy the minimum pressureand the tanks level requirements to be feasible solutions.

To demonstrate the value of considering water demand uncer-tainty in the flexible design approach, a 24-h simulation [i.e., ex-tended period simulation (EPS)] with 1-h hydraulic time step foraverage day flows in each design stage considered is performed inthe present study. The WDS resilience is calculated based on theminimum pressure across the 24-h simulation. The total cost of anintervention plan is the cumulative capital cost of pipes, tanks, andthe present value of pump operation over 60 years. The annual en-ergy cost for pumping is calculated by multiplying the energy usedin a day (obtained from the EPANET 24-h simulation) by the unitcost of energy ($0.12=kWh) and the number of days per year. Theseannual costs are then discounted for each year of the design stage(i.e., time interval; e.g., 20 years) and these are then added up toestimate the total cost of energy at that stage. Under the currentstudy, the original Anytown network redesign problem describedpreviously has also been considered for the deterministic (precau-tionary and staged) and flexible designs across the planning hori-zon. Furthermore, the Anytown network problem is normallysolved for five operating conditions (average day flow, instantane-ous peak hour flow, and three fire flows) (Walski et al. 1987), but inthis paper, only the normal operation condition was analyzed forsimplicity in both the deterministic and the flexible design plans.

As it was done in the case of New York Tunnels, the pre-cautionary intervention plan was created by assuming that all

interventions of the optimal solution identified by Farmani et al.(2005) are occurring in the first stage. The staged deterministic in-tervention plan was generated by staging, i.e., distributing afore-mentioned optimal interventions across the planning horizon andby leaving out only Pipe Number 4 clean and line interventionbecause it is not necessary (the pipe has an adequate roughnesscoefficient 130). However, cleaning and lining or doing nothingon Pipe 4 has a negligible contribution in the solution objectivesbecause of its short length. Also, three new pipes (10, 14, and 16)were assumed to be already in place at the beginning of planninghorizon to avoid demand node isolation in the analysis. All threepumps were switched on for every hour and a tank was set up atNode 8. Finally, in each stage, the flexible plan was generated byimplementing all the staged interventions and adding more inter-ventions in the higher demand paths, all by using engineeringjudgment. Along the lower demand paths, interventions from theprevious stages are kept in the consecutive stages without any addi-tional interventions. The threshold values were also set manually inthe flexible plan analyses by using engineering judgment. The threeintervention plans generated this way are shown in Figs. 6 and 7.Even though the design plans generated this way may not be opti-mal, the comparison still makes sense. This is because of the factthat if the flexible plan shows improvement over deterministicplans in terms of average cost and/or resilience, there is only apotential to further enhance the improvement if these two plansare optimized. The staged deterministic plan has less possiblesolutions compared with the flexible plan which makes much betterflexible solutions difficult to achieve.

Results and DiscussionThe design plan analysis results based on the new approach arepresented in Figs. 6–10 and Tables 1–3. The results comparethe two deterministic (precautionary and staged) and the flexibleapproaches in terms of WDS average total cost and the averageresilience index over the planning horizon. Figs. 6 and 7 showthe precautionary, staged deterministic, and the flexible design in-terventions. In Figs. 6 and 7, a cross through a pipe diameter (i.e., inthe first time step) shows the diameter that is already in place (op-posed to the original network problem as explained in the networkdescription). The shaded cells show design interventions that arenewly implemented in the time step they are in. The precautionaryapproach indicates that all the design interventions are imple-mented at the beginning of the planning horizon as in the previouscase study. The solution duplicates critical pipes. For example,Pipes 1 and 2 convey pumped water to the rest of the network.The selected developmental interventions (staged deterministicand flexible) were meant to duplicate critical pipes in the earlierstages. For example, Pipes 1 and 2 are duplicated in the first designstage. The interventions in the staged deterministic provide basisfor the alternative routes in the flexible plan. This indicates thatfor any given alternative design path, the interventions consistof at least the same intervention(s) as in the staged deterministicapproach. It can be observed that Pipes 1, 2, 6, 26, 27, 29, and30 have all been selected in the first stage of both the staged deter-ministic and the flexible design plans.

Fig. 6 confirms that a flexible intervention plan outperforms thedeterministic design plans in terms of end average resilience eventhough it has a similar or less cost than the latter. A flexible plan,which has an average total cost of $18.42 million results in an endaverage resilience index of 0.164, whereas the two deterministicplans have an average end resilience of 0.117 with the costs of$20.68 million and $18.43 million for the precautionary and stageddesigns, respectively. Fig. 7 also shows a flexible plan withless cost than the deterministic plans at equivalent end average

Fig. 10. Anytown network average cumulative cost profiles of threedesign plans (Case 2)



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resilience. For an end average resilience of 0.117, a flexible planhas a cost of $17.79 million as compared with the $20.68 millionand $18.43 million shown by the precautionary and the stageddeterministic plans, respectively. The differences indicate that thereis value that has been derived from flexibility as an opportunitypresented by uncertainty. For example, the $0.64-million differencebetween the staged deterministic and the flexible approach showsthe potential cost reduction by the latter. This reduction results dueto the same reason of the previous case study that the flexible de-sign plan allows for postponement of design interventions up tosuch time that they would be needed. In this case, the system avoidsless desirable design for some water demands and captures themore favorable ones. Flexible or conditional implementation ofintervention measures in the future can drastically increase WDSaverage resilience with an average total cost which is similar or lessthan the deterministic intervention plans. In addition, designflexibility provides for the possibility to implement additional in-terventions but still maintain the comparable average total cost.This finding is attributable to the low initial cost and the futureintervention measures that have a lower present value cost.

Table 1 and Figs. 9 and 10 display the average resilience and thecumulative average cost performance profiles of the Anytown net-work design plans. A similar trend to the New York network resultsindicates that in the initial stage, the deterministic approachesoutperform the flexible intervention plans in terms of average resil-ience which is explained by the fact that flexible design approachhas a do nothing option which reduces the average system resil-ience and cost. For example, in the initial stage, the staged deter-ministic has an average resilience of 0.147 and an average totalcost of $13.59 million compared with the average resilience of0.026 and the average of $13.12 million for the cost equivalent

flexible plan. With cumulative design interventions that respondto uncertain demands as time passes on, the WDS average resil-ience at the end of the planning horizon for a cost equivalent flex-ible intervention plan clearly outperforms both the precautionaryand the staged deterministic intervention plans in terms of averageresilience. Table 1 shows that the staged deterministic plan has anend average resilience of 0.117, whereas the cost equivalent flex-ible approach has 0.164. The end resilience equivalent solutionalso stresses the economic value that can be achieved by havingthe same resilience to the deterministic approach in the flexibledesigns. Both deterministic plans (staged and precautionary) showthe potential of performing highly earlier on when water demandand uncertainty is lower.

The sensitivity analysis in this case study confirms that the in-crease in uncertainty represented by the standard deviation in waterconsumption (Table 2) leads to higher percentage increment of theaverage resilience index from the deterministic to the flexible plan.For example, an increase of an average resilience at the end of theplanning horizon for a comparable flexible intervention to thestaged deterministic plan is from 38.3% (Case 1) to 40.2% (Case3). These results also stress the point that there is value inherent inflexibility, and the more uncertain the future water demand is, thehigher the relative value increase of flexible design. These also sug-gest that uncertainty presents an opportunity that can be exploited.

In Table 3, the flexible design shows an increase of 23.5%,whereas the staged deterministic design has 23.3% when a lowercost discount rate of 4.5% is used. A similar trend is shown by ahigher discount rate of 7.5%, which shows a 15.8 and 15.4% re-duction in the total cost of flexible and staged deterministic designs,respectively. These cost increases’ differences mean that the flex-ible design plan is more sensitive to cost discount rate than the

Pipe No. 1 2 6 10 13 14 15 16 17,18 20 24 25 26 27 29 30 31 33 34 38 39 Tank Node 1 2 3 6 10 13 14 15 16 17,18 20 23 24 25 26 27 29 30 31 33 34 38 39 Tank Node 1 2 3 6 10 13 14 15 16 17 18 20 23 24 25 26 27 29 30 31 33 34 38 39 Tank NodeAv. Total Cost Av. End RI

t0 t1 Demand Mean value = 9300 gallons/minute t2 tend ($M) (-)

20 30 8 12 6 10 8 14 12 16 CL 12 24 12 8 12 14 CL 10 CL 6 8 20 30 0 8 12 6 10 8 14 12 16 0 CL 12 24 12 8 12 14 CL 10 CL 6 8 20 30 0 8 12 6 10 8 14 12 12 16 0 CL 12 24 12 8 12 14 CL 10 CL 6 8 20.68 0.117

20 30 8 12 0 10 0 14 0 0 0 0 24 12 8 12 0 0 0 0 0 no tank 20 30 0 8 12 6 10 8 14 12 16 0 0 0 24 12 8 12 0 0 0 0 6 no tank 20 30 0 8 12 6 10 8 14 12 12 16 0 CL 12 24 12 8 12 14 CL 10 CL 6 8 18.43 0.117

20 30 20 8 12 6 10 8 14 12 12 16 20 CL 12 24 12 8 12 14 CL 10 CL 6 820 30 20 8 12 6 10 8 14 12 16 20 0 12 24 12 8 12 14 0 10 0 6 no tank

20 30 20 8 12 6 10 8 14 12 12 16 20 0 12 24 12 8 12 14 0 10 0 6 no tank

20 30 8 12 0 10 0 14 0 0 0 0 24 12 8 12 0 0 0 0 0 no tank

20 30 20 8 12 6 10 8 14 12 12 16 20 CL 12 24 12 8 12 14 CL 10 CL 6 820 30 0 8 12 0 10 0 14 0 0 0 0 0 24 12 8 12 0 0 0 0 0 no tank

20 30 0 8 12 0 10 0 14 0 0 0 0 0 0 24 12 8 12 0 0 0 0 0 no tank

18.42 0.164

20 30 20 8 12 6 10 8 14 12 12 16 20 CL 12 24 12 8 12 14 CL 10 CL 6 820 30 20 8 12 6 10 8 14 12 16 20 0 12 24 12 8 12 14 0 10 0 6 no tank

20 30 20 8 12 6 10 8 14 12 12 16 20 0 12 24 12 8 12 14 0 10 0 6 no tank

0 0 0 12 0 10 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 no tank

20 30 20 8 12 6 10 8 14 12 12 16 20 CL 12 24 12 8 12 14 CL 10 CL 6 80 0 0 0 12 0 10 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 no tank

0 0 0 0 12 0 10 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 no tank

New Pipes Already in place

New Design Intervention

Threshold Demand: 13025 Gallons/minute, Samples (HHH-D = 4.29%, HHL-D = 81.51%)

Threshold Demand: 13025 Gallons/minute, Samples (HLH-D = 0.11%, HLL-D = 2.09%)

Threshold Demand: 9304 Gallons/minute, Samples (LHH-D = 7.02%, LHL-D = 4.68%)

Threshold Demand: 9304 Gallons/minute, Samples (LLH-D = 0.18%, LLL-D = 0.12%)

Samples (LH-D) = 11.7%

Threshold Demand: 7477 gallons/minute

Samples (LL-D) = 0.3% Tank riser diameter: 14

No Intervention / Previously Rehabilitated

Samples: (H-D) = 88%


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Tank riser diameter: 14

Samples (HH-D) = 85.8%


Samples (HL-D) = 2.2% Tank riser diameter: 14

Samples: (L-D) = 12% Tank riser diameter: 14


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Design 1 Interventions: Pipe Diameters (inches) / Clean and Line (CL) Design 2 Interventions: Pipe Diameters (inches) / Clean and Line (CL) Design 3 Interventions: Pipe Diameters (inches) / Clean and Line (CL) Demand Mean value = 8369 gallons/minute Demand Mean value = 9800 gallons/minute

Precautionary

Tank riser diameter: 14 Tank riser diameter: 14 Tank riser diameter: 14

Staged


Fig. 11. Three demand standard deviation cases for intervention plan evaluation



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Pipe No. 1 2 6 10 13 14 15 16 17 18 20 24 25 26 27 29 30 31 33 34 38 39 Tank Node 1 2 3 6 10 13 14 15 16 17 18 20 23 24 25 26 27 29 30 31 33 34 38 39 Tank Node 1 2 3 6 10 13 14 15 16 17 18 20 23 24 25 26 27 29 30 31 33 34 38 39 Tank NodeAv. Total Cost Av. End RI

t0 t1 t2 tend ($M) (-)

20 30 8 12 6 10 8 14 12 12 16 CL 12 24 12 8 12 14 CL 10 CL 6 8 20 30 0 8 12 6 10 8 14 12 12 16 0 CL 12 24 12 8 12 14 CL 10 CL 6 8 20 30 0 8 12 6 10 8 14 12 12 16 0 CL 12 24 12 8 12 14 CL 10 CL 6 8 20.68 0.117

20 30 8 12 0 10 0 14 0 0 0 0 0 24 12 8 12 0 0 0 0 0 no tank 20 30 0 8 12 6 10 8 14 12 12 16 0 0 0 24 12 8 12 0 0 0 0 6 no tank 20 30 0 8 12 6 10 8 14 12 12 16 0 CL 12 24 12 8 12 14 CL 10 CL 6 8 18.43 0.117

20 30 20 8 12 6 10 8 14 12 12 16 20 CL 12 24 12 8 12 14 CL 10 CL 6 820 30 20 8 12 6 10 8 14 12 12 16 20 0 12 24 12 8 12 14 0 10 0 6 no tank

20 30 20 8 12 6 10 8 14 12 12 16 20 0 12 24 12 8 12 14 0 10 0 6 no tank

20 30 8 12 0 10 0 14 0 0 0 0 0 24 12 8 12 0 0 0 0 0 no tank

20 30 20 8 12 6 10 8 14 12 12 16 20 CL 12 24 12 8 12 14 CL 10 CL 6 820 30 0 8 12 0 10 0 14 0 0 0 0 0 0 24 12 8 12 0 0 0 0 0 no tank

20 30 0 8 12 0 10 0 14 0 0 0 0 0 0 24 12 8 12 0 0 0 0 0 no tank

17.79 0.117

20 30 20 8 12 6 10 8 14 12 12 16 20 CL 12 24 12 8 12 14 CL 10 CL 6 820 30 20 8 12 6 10 8 14 12 12 16 20 0 12 24 12 8 12 14 0 10 0 6 no tank

20 30 20 8 12 6 10 8 14 12 12 16 20 0 12 24 12 8 12 14 0 10 0 6 no tank

0 0 0 12 0 10 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 no tank

20 30 20 8 12 6 10 8 14 12 12 16 20 CL 12 24 12 8 12 14 CL 10 CL 6 80 0 0 0 12 0 10 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 no tank

0 0 0 0 12 0 10 0 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 no tank

New Pipes Already in place

New Design Intervention

Samples: LH-D = 4.2%

Demand Threshold: 9658 gallons/minute

Samples: LL-D = 7.8%

No Intervention Measure / Previously Rehabilitated

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Samples: HH-D = 30.8%


Samples: HL-D = 57.2%

Threshold Demand: 12103 Gallons/minute, Samples: (HHH-D = 3.696%, HHL-D = 27.104%)

Threshold Demand: 12204 Gallons/minute, Samples (HLH-D = 6.292%, HLL-D = 50.908%)

Threshold Demand: 8773 Gallons/minute, Samples (LHH-D = 2.94%, LHL-D = 1.26%)

Threshold Demand: 8773 Gallons/minute, Samples (LLH-D = 5.46%, LLL-D = 2.34%)



Samples (H-D) = 88%

Samples (L-D) = 12% Tank riser diameter: 14



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s Design 1 Interventions: Pipe Diameters (inches) / Clean and Line (CL) Design 2 Interventions: Pipe Diameters (inches) / Clean and Line (CL) Design 3 Interventions: Pipe Diameters (inches) / Clean and Line (CL) Demand Mean value = 8369 gallons/minute Demand Mean value = 9300 gallons/minute Demand Mean value = 9800 gallons/minute

Precautionary

Tank riser diameter: 14 Tank riser diameter: 14 Tank riser diameter: 14

Staged


Fig. 12. The New York Tunnels network (Babayan et al. 2005, © ASCE)

Fig. 13. New York Tunnels average resilience index profiles of threedesign plans (Case 2) Fig. 14. The Anytown network (Farmani et al. 2005, © ASCE)



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staged deterministic approach. This is a consequence of the built-inflexibility, i.e., optional intervention paths result in wider range ofpossible costs. This finding implies that flexible WDS designs maynot always be the best choice, i.e., discount rate for the long-termplanning of WDS should be carefully chosen.

The authors acknowledge that practicing engineers have beenintuitively dealing with the issues of staged and adaptive long-termplanning of distribution systems based on multiple demand scenar-ios. This study introduces formal concepts and techniques used inthis context (e.g., decision trees to represent flexible plans and MCsimulation to evaluate flexible plans). The reason why the authorsare representing intervention plans using decision trees definedover the full length of the planning horizon is to take the long-termview, i.e., to make sure that what is proposed for implementation inthe near future (the first stage in this paper or next year for practi-tioners) is compatible with different possible demand and otherfutures based on the current best information available. This is con-sistent with Walski (2013) who also suggested that decisions mustbe made in the short term but also fit into a long-term plan. Deci-sion trees can (and should) be updated as frequently as desired inthe future by using engineering judgment or some other approach.

Conclusions

Methodologies that can identify design interventions that are adapt-able to future climate and urbanization changes in the long-termplanning of WDSs are essential. De Neufville (2004) classifiedmanagement of uncertainty in three ways as controlling uncertaintyby demand management, protecting passively by building in ro-bustness, and lastly by protecting actively by creating flexibilitythat managers can use to react to uncertainties. In this study, a meth-odology that analyzes the potential value of flexibility that is cre-ated in WDS design interventions under uncertain water demandhas been explored.

The new methodology was tested on two case studies based onthe WDS redesign problem of New York Tunnels and Anytown.Three alternative intervention plans, two deterministic (staged andprecautionary), and one flexible, were evaluated and comparedagainst one another by assuming uncertain future demands oversome long-term planning horizon. The methodology recognizesuncertainty for flexibility purpose in the long-term planning ofWDSs. The flexible methodology gives managers opportunities toexploit the uncertain nature of water demand as a design parameter.The approach allows the WDS to cope with the changes in waterdemand by changing the actual design elements of the system when(and if) necessary.

The results obtained lead to the following conclusions:• It has been demonstrated that flexible WDS design (i.e., inter-

vention plan) can have lower economic cost and/or improvedhydraulic performance (i.e., higher resilience) when comparedwith the corresponding deterministic precautionary and stageddesigns (i.e., intervention plans) under uncertain future waterdemand conditions. The value of flexibility can be estimatedas the difference in respective expected design costs (giventhe same/similar design resiliencies) or as the difference inrespective expected resiliencies (given the same/similar designcosts).

• The value of flexibility comes from the ability of the flexibleWDS design approach to adapt the water distribution systemto uncertain future water demands in a cost-effective, resilient,and timely manner. This is a consequence of the fact that flex-ible WDS design approach allows both postponing (i.e., delay-ing in time) and implementing (or not implementing) optional

interventions that are compatible with future demands. Thestaged deterministic WDS design allows only delaying interven-tions in time, whereas the precautionary approach does notallow either.

• The flexible WDS design seems more sensitive to changes in thecost discount rate than the staged deterministic plan. This is aconsequence of the built-in flexibility, i.e., optional interventionpaths resulting in wider range of possible costs. This findingimplies that flexible WDS designs may not always be the bestchoice depending on the discount rate used.These conclusions are based on the assumptions, data, and cost

models used in the two-case studies presented in this paper. Futurework on larger, more complex WDSs is required to further analyzeand quantify the benefits of flexible WDS designs. Future work isalso required to identify the optimal staged and flexible designplans by formulating and solving the relevant WDS optimizationproblems.

Acknowledgments

This research work has been fully supported by a University ofExeter Ph.D. scholarship, which is gratefully acknowledged.

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Evaluating Flexibility in Water Distribution System Design under Future Demand Uncertainty