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Robustness-Based Design of Water Distribution SystemsDonghwi Jung1; Doosun Kang2; Joong Hoon Kim3; and Kevin Lansey, A.M.ASCE4
Abstract: Robustness is generally defined as an ability of a system to maintain its function under a defined set of disturbances. To introducerobustness to the water distribution systems (WDSs) design, chance constrained, or so-called reliability-based models have been formulated.Under variations in system parameters, such as nodal demands and pipe roughness, system reliability is generally measured as the probabilitythat the stochastic nodal pressures will be higher than an allowable minimum pressure limit. However, chance constraints may not be the bestformulation to improve system robustness because it focuses on the likelihood of failure under a specified set of conditions rather thandeveloping a solution that consistently provides adequate service. In addition, the reliability-based design requires defining the demandcondition, its probability distribution and its statistics, which are not straight forward in practice. To address these difficulties, a robustnessindex that limits the range of the system function variability is posed here and incorporated in a two objective optimization problem. Resultingdesigns are compared with those from the reliability constraint formulation. The authors demonstrate that the robustness-based designimproves resilience relative to the reliability-based design. DOI: 10.1061/(ASCE)WR.1943-5452.0000421. © 2014 American Societyof Civil Engineers.
Author keywords: WDS design; Reliability; Robustness; Resilience; System hydraulic availability (SHA).
Introduction
Early studies of optimal water distribution systems (WDSs) designminimized system costs while satisfying the minimum allowablepressure requirement at nodes; so-called least-cost design (Schaakeand Lai 1969; Alperovits and Shamir 1977; Lansey and Mays1989; Simpson et al. 1994; Savic and Walters 1997). However,these least-cost design formulations do not provide redundancyor robustness in the system design when system demands vary fromthe design conditions. To overcome this shortcoming, later optimi-zation work considered broader indicators of system performancethat addressed uncertainties to system parameters (Lansey et al.1989; Xu and Goulter 1999; Todini 2000).
Reliability has been the most popular measure for systemperformance. System reliability is generally defined as the abilityof the network to provide an adequate service to customers underuncertain system conditions (Goulter 1995) and measured by theprobability that the stochastic nodal pressures are greater than orequal to a prescribed minimum pressure. Reliability indices havebeen included in an optimization framework: (1) as a constraintin the least-cost design problem to meet an allowable minimumreliability value (Lansey et al. 1989; Xu and Goulter 1999;
Babayan et al. 2005) or (2) as a second objective competing withcost in a multiobjective optimization formulation (e.g., Kapelanet al. 2005).
Lansey et al. (1989) were the first to develop a least-cost designmethodology by assuming that the nodal demands, pressure headrequirements, and pipe roughness coefficients are variable in achance constraint model formulation. Xu and Goulter (1999) intro-duced least-cost design with reliability constraint and used thefirst-order reliability method to quantify the uncertainties in the no-dal pressure. Babayan et al. (2005) suggested a chance constraint-based least cost design of WDS under demand uncertainty usingthe integration-based approach to overcome the time requirementsof Monte Carlo simulation (MCS).
The transition to multiobjective frameworks followed the devel-opment of efficient multiobjective optimization algorithms suchas non-dominated sorting genetic algorithm-II (NSGA-II) (Debet al. 2002). The trade-off between total cost and reliability wasinvestigated in Pareto optimal solutions. Kapelan et al. (2005) firstimplemented a multiobjective optimal model using the robustNSGA-II (RNSGA-II) to minimize total cost and maximize reli-ability considering uncertainties in water demand and pipe rough-ness coefficients. Latin hypercube sampling (LHS) was used toquantify uncertainties in nodal pressure.
Robustness is generally defined as an ability of the systems tomaintain its function under a defined set of disturbance (Jen 2003;Lansey 2012) therefore, can be measured by the variation of systemperformance. In this context, reliability-based design fits within thedomain of robustness. Although improving reliability will improvesystem robustness, the question is how the reliability index willimprove system robustness. Robustness incorporates the variationof system performance; an additional aspect of system performancethat reliability does not encompass. Thus, the reliability index maynot be the best formulation for obtaining a robust design. For ex-ample, the two systems of the same level of reliability can havedifferent variations in system performance (Table 1). Althoughthe reliability level in each design is the same, the degree of under-performance in the failure case is significantly more severe inNetwork 1. Thus, a reliability formulation does not possess all
1Postdoctoral Research Associate, Dept. of Civil Engineering andEngineering Mechanics, Univ. of Arizona, Tucson, AZ 85721. E-mail:[email protected]
2Assistant Professor, Dept. of Civil Engineering, Kyung Hee Univ.,446-701, 1 Seocheon-Dong, Giheung-Gu, Yongin-Si, Kyunggi-do, SouthKorea. E-mail: [email protected]
3Professor, School of Civil, Environmental and Architectural Engineer-ing, Korea Univ., 136-713, Anam-ro 145, Seongbuk-gu, Seoul, SouthKorea. E-mail: [email protected]
4Professor, Dept. of Civil Engineering and Engineering Mechanics,Univ. of Arizona, Tucson, AZ 85721 (corresponding author). E-mail:[email protected]
Note. This manuscript was submitted on June 27, 2013; approved onNovember 8, 2013; published online on November 11, 2013. Discussion per-iod open until October 20, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal ofWater Resources Plan-ning and Management, © ASCE, ISSN 0733-9496/04014033(14)/$25.00.
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desired explanatory characteristics and an alternative (robustness-based) formulation may be useful for effective WDS design.
Giustolisi et al. (2009) proposed such an alternative in multipleobjective formulations to improve WDS robustness. While mini-mizing total cost, they also maximized the parameter α, that isequivalent to the standard normal deviate relating the mean pres-sure, its standard deviation, and the minimum pressure require-ment. The measure is an indicator of the difference between themean pressure and the allowable minimum pressure (i.e., it has neg-ative value if mean pressure is below the minimum requirement).Theoretically, to maximize this robustness index, the mean pressureshould be increased while its standard deviation is decreased, re-sulting in the network with low relative pressure variations. Here, arobustness alternative indicator is introduced.
Robustness-based design avoids or minimizes failure severityrather than focusing on the probability of successful operation.Reducing severity is advantageous when pressure constraintscan be considered as soft limits and pressure slightly below theminimum are acceptable in extreme conditions. It also will providesolutions that function properly under a broader set of conditionsthat may not be explicitly defined as design conditions.
The reliability-based design requires the modeler to know orassume the probability distribution of demand and pipe roughnesscoefficients. These probability distributions are generally notknown (e.g., Distribution A in Fig. 1 is unknown) and little datahas been published to support their definition. In the previous
reliability-based model formulations, the normal distribution wasoften assumed for demand, roughness, and the resulting pressures.However, the probability distribution can be an atypical shape(e.g., a bimodal distribution), making reliability estimation diffi-cult. Predicting the input parameter distributions for some futurestate becomes more problematic.
Finally and most importantly for practical applications, the def-inition of the demand conditions used in a chance constraint for-mulation have not been defined explicitly. For example, typically itis not clearly stated if the demand is the peak demand with its as-sociated variability (Distribution B in Fig. 1) or the average demandwith its variability (Distribution A in Fig. 1). As shown, the dis-tributions for the demand are related but they imply very differentconditions. A peak demand distribution is particularly difficult toassess since they occur quite rarely.
In this study, the authors propose a robustness index to take intoaccount the variation of WDS performance to uncertain conditions.The proposed index uses the coefficient of variation (CV) of sto-chastic pressures; avoiding the need to define the probability dis-tribution. First-order second-moment (FOSM) approach is usedfor uncertainty quantification and NSGA-II is used to solve themultiobjective optimization problem. Structural differences areidentified in optimal designs using this new index and those deter-mined from the conventional reliability-based design formulationfor the Anytown network. Postoptimization analyses were per-formed to examine the WDS resilience of the optimal systems bycalculating system hydraulic availability (SHA) considering pipebreaks and fire flow conditions.
Water Distribution System Reliability andRobustness
Reliability
WDS reliability is generally defined as the system’s ability to pro-vide an acceptable supply of water to customers under varying sys-tem conditions including normal and abnormal conditions (Goulter1995). Reliability can be measured in terms of the quantity andquality of delivered water. Water quality reliability is beyond thescope of this study. In terms of water quantity, reliability is gener-ally related to the flow to be supplied (Su et al. 1987; Goulter andCoals 1986; Cullinane et al. 1992) and the range of pressures atwhich those flows must be provided (Lansey et al. 1989; Goulter1995; Xu and Goulter 1999).
Hydraulic reliability (Xu and Goulter 1999) is used here anddefined as the probability that the nodal demand is met with pres-sure greater than or equal to the allowable minimum pressure. Thepipe network has a fixed configuration and uncertainties are intro-duced in the system parameters (demands and pipe roughness). Theprobability of hydraulic failure at node i can be expressed by
PFi ¼ ProbðPi < PminÞ ð1Þ
where PFi = the probability of nodal hydraulic failure at node i;Pi = the random nodal pressure at node i; and Pmin = the allowableminimum pressure.
The hydraulic reliability at node i is then
Reli ¼ 1 − PFi ð2Þ
An exact hydraulic reliability can be obtained by integrating theprobability density function (PDF) or
Demand Factor
Probability
1.8
Fig 1. Distribution of demand, where A indicates distribution of aver-age demand and B indicates distribution of the peak
Table 1. Critical Node’s Pressures of Two Systems under FiveDisturbances (Each Disturbance has Difference Type and Magnitude)
Disturbances
Pressure (m)
Network 1 Network 2
1 34 312 28 283 31 294 22 275 29 28Probability of meeting a 28 mpressure requirement (%)
80 80
Variance of pressure 16 2
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Reli ¼Z ∞Pmin
fiðPÞdP ð3Þ
where fiðPÞ = the probability density function of pressure P atnode i.
The system reliability of a given network can be an averageor flow weighted average of the nodal reliabilities (Su et al. 1987;Bao and Mays 1990) or the minimum value of the nodal hydraulicreliability as used here
sysRel ¼ minðReliÞ i ¼ 1; : : : ; nn ð4Þwhere nn = the number of the nodes in the network. Assuming thatthe nodal pressure is normally distributed, the PDF is defined by themean and standard deviation of the nodal pressure.
Robustness
Robustness is the ability of a fixed system to avoid or minimize theseverity of a failure when under a defined set of stresses. The sys-tem robustness can be assessed using the variation of nodal pressureunder system parameter uncertainties. Here, a robustness index isproposed using the coefficient of variation (CV) of the stochasticnodal pressure.
The CV of a node’s pressure is
CVi ¼σPi
Pavgi
ð5Þ
where CVi = the coefficient of variation of pressure at node i;Pavgi = mean of stochastic nodal pressures at node i; and σPi
=standard deviation of random nodal pressure at node i.
The robustness index for node i is defined as
Robi ¼ 1 − CVi ð6ÞThe system robustness is defined as the minimum nodal
robustness
sysRob ¼ minðRobiÞ i ¼ 1; : : : ; nn ð7Þ
Methodology
Two optimization model formulations, reliability-based androbustness-based, are posed and resulting designs are compared.The objectives of the former model are to minimize total cost andto maximize system reliability while the latter one’s second objec-tive is to maximize the system robustness. The reliability androbustness indices [Eqs. (4) and (7), respectively] are estimatedusing mean and standard deviation of stochastic nodal pressuresat nodes and the statistics are obtained using the FOSM uncertaintyapproximation method. NSGA-II is used for system optimization.Postoptimization analyses using SHA compare the solutions fromtwo design approaches. The following subsections describe thedetails of the objective functions, optimization approach, and post-optimization analyses.
Economic Cost Function
Total system cost consists of capital costs of pipe and pump con-struction and operation and management (O&M) costs for pumpingstations. For pipeline construction, pipe material and installationcosts are included. All costs in this paper are updated using theEngineering News-Record (ENR) construction cost index for2005 and expressed in US dollars. The O&M costs are calculated
on annual basis and converted to a present-worth for the systemplanning time and annual discount rate.
Pipe Construction CostClark et al. (2002) fit general pipe construction cost (PipeCC)functions accounting for pipe material and installation
y ¼ aþ bðxcÞ þ dðueÞ þ fðxuÞ ð8Þwhere y = unit pipe cost ($=ft; 1 ft ¼ 0.3048 m); x = pipe diameterin inches (1 in: ¼ 2.54 cm); u = indicator variable; and a, b, c, d, e,and f are component-specific parameter values estimated using re-gression analyses. Pipe construction costs are the sum of the baseinstallation cost, trenching and excavation, embedment, backfill,and compaction costs, and as well as valves, fittings, and hydrantscosts. Detailed information for the pipe cost function can be foundin Clark et al. (2002).
Pump Construction CostThe pump construction cost (PumpCC) (Walski et al. 1987) is
PumpCC ¼ 500 ×Q0.7 ×H0.4 ×maxðNPa;NPpÞ ð9Þ
where Q = rated discharge in gpm (gallons per minute,1 gal: ¼ 0.003785 m3); H = rated head in ft (1 ft ¼ 0.3048 m);and NPa and NPp are the numbers of pumps operating for averageand peak demand conditions, respectively. For the sake of simplic-ity, it is assumed that individual pumps in a pump station areidentical.
The rated head, H, in ft is formulated as
H ¼ 550 × HP × ηmotor
γ�QPNP
� ð10Þ
where γ = specific weight of water in lb=ft3 (62.4 lb=ft3 ¼1,000 kg=m3); HP = pump horsepower (hp); ηmotor = motorefficiency; QP = pump discharge in ft3=s (1 ft3=s ¼0.0283 m3=s); and NP = the number of pumps that are operating(average or peak demand condition). HP and NP are decision var-iables. The rated head, H, is calculated based on HP, NP and QPdepending on demand condition.
Pump Operation CostThe present value of the pump operation cost (PumpOC) is givenby
PumpOC ¼ NPa ×0.746ηpump
× HP × EC × 24 × 365 × PVF ð11Þ
where ηpump = pump efficiency; EC = pump energy cost ($=kWh);and PVF = the present value factor. The energy consumption isestimated from the average demand condition for the entire plan-ning period assuming constant pump energy cost.
The present value factor is
PVF ¼ ð1þ AIÞn − 1
AIð1þ AIÞn ð12Þ
where AI = annual discount rate and n = planning period durationin years.
Multiobjective Optimization
System reliability and robustness are inversely related to systemcost. Understanding the tradeoff between system indices and costswill allow decision makers to make better decisions. To that end,
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two formulations, reliability and robustness based design, are posedand solved.
Objectives and ConstraintsIn both models, the first objective function is to minimize the totalsystem cost (F1). The second objective (F2) is to maximize thesystem reliability index (sysRel) or the robustness index (sysRob),which can be expressed as
Minimize F1 ¼ PipeCCþ PumpCCþ PumpOC ð13Þ
Maximize F2 ¼ sysRel or sysRob ð14ÞPressure constraints are included in both design problems, that
is, the nodal pressure should be within the allowable pressure limitsfor each demand conditions
Pk ≤ Pi;k ≤ Pk ð15Þwhere Pk and Pk = the lower and upper allowable pressure limitsfor demand condition k, respectively, and Pi;k = the pressure head atnode i for demand condition k.
Uncertainty Quantification Method
WDS uncertainty has been assessed using MCS, FOSM analysis,and LHS. MCS is a random enumeration technique in which a largeset of realizations are developed and evaluated. MCS is assumedto be correct if a sufficient sample size is used but it is oftentoo time consuming to be used in an optimization framework.FOSM is a variance approximation method based on a Taylor-series linear approximation. LHS is a quasi-MCS using stratifiedsampling. FOSM and LHS are often used as alternative approachesto MCS to reduce the computational effort. In this study, FOSM isemployed since it significantly reduces the number of functionevaluation and has been proved to be acceptably accurate forWDS uncertainty estimates (Kang et al. 2009).
First-Order Second-MomentFOSM, also called the variance propagation method (Berthouex1975), estimates the variance by approximating a function witha Taylor series expansion around the mean parameter value anddropping the higher order terms (Tung and Yen 2005). If a modeloutputW is related to K model parameters, the covariance matrix ofthe model predictions, σ2, can be shown to be
σ2 ¼ sTDxs ð16Þwhere Dx ¼ diagðσ2
1;σ22; : : : ; σ
2KÞ is a K × K diagonal matrix
of variance of the model parameters and s ¼ ∇xWðμxÞ is aK-dimensional vector of the sensitivity coefficients evaluated atmean of parameters, μx. The sensitivity matrix is the gradient ofthe model output with respect to a model parameter. Therefore,for a system with K parameters using a finite difference approxi-mation of the derivatives, FOSM requires K þ 1 function evalua-tions for the K perturbations plus the analysis of the base condition.
Postoptimization Analysis
To compare the impact of the two objectives on the network design,SHA of their optimal solutions are calculated for single pipe breakconditions and for an extreme fire flow condition in postoptimiza-tion analyses. Resilience is a system’s ability to “gracefully degradeand subsequently recover from” a failure event (Lansey 2012) andthus, is a function of failure severity and recovery time. SHA isdefined as the availability of the desired quantity of water and isa first estimate for system resilience. It is influenced by the nodal
pressure under pipe break and fire flow conditions. This analysisfollows the Scholz et al. (2012) definition of specified resilience. Ifthe nodal pressure does not meet allowable minimum pressure re-quirement, a partial demand is considered to be delivered to cus-tomers. SHA does not address a system’s ability to respond to theevent (Zhuang et al. 2013) nor does it assess extreme conditionssuch as earthquake events described by Scholz et al. as generalizedresilience.
SHA during Pipe BreaksThe minimum cut-set method (Su et al. 1987) is employed toestimate the SHA. Following Su et al. (1987), only single pipe fail-ures are considered in calculating the SHA. For a given optimaldesign, one of the pipes is closed in the hydraulic model represent-ing a pipe failure and nodal pressures are calculated. If a nodalpressure does not meet the minimum pressure requirement, thehydraulic availability (HA) for that node is calculated using thefunction given in Fig. 2 (Cullinane et al. 1992). System hydraulicunavailability (SHUA) under a pipe j failure is computed using ademand weighted average of nodal unavailabilities (1 − HAi;j) or
SHUAj ¼Pnn
i¼1ð1 − HAi;jÞ × qiPnni¼1 qi
ð17Þ
where SUHAj = system hydraulic unavailability during pipe j’s(j ¼ 1; : : : ; nl) break condition; HAi;j = hydraulic availabilityfor node i (i ¼ 1; : : : ; nn) in pipe j’s break condition; qi = nodaldemand for node i; nn = the number of nodes; and nl = the numberof pipes.
The closed pipe is then opened and another one is closed and theabove calculations are repeated. This process continues until breaksin all pipes have been evaluated. The SHA is then calculated as
SHAp ¼ 1 −Pnl
j¼1 SHUAj
nlð18Þ
where SHAp is system hydraulic availability of the given optimaldesign.
SHA under Fire Flow ConditionFires result in unexpected and abnormal system stress conditions.The procedure for estimating SHA under fire flow conditions usesthe availability function in Fig. 2. For a given optimal design, alocation near one of the nodes is assumed to have a fire and acorresponding increase in flow is required. Nodal pressures are cal-culated for the network under that condition. As in the minimumcut-set method, if the nodal pressure does not meet the minimum
Fig. 2. Hydraulic availability (HA) versus pressure for pipe break andfire flow conditions
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pressure requirement, the HA at the node is calculated using theavailability function. SHUA under fire flow for node i is calculatedas a demand weighted nodal unavailabilities (1 − HAi;i) whereHAi;i is hydraulic availability for node i (i ¼ 1; : : : ; nn) withthe fire flow occurring at node i. The fire flow is then moved fromnode i to node iþ 1 and the SHA procedure is repeated. This pro-cess continues until fire flows are simulated at all the nodes. Thesystem fire flow SHAf is then calculated by
SHAf ¼ 1 −Pnn
i¼1 SHUAi
nnð19Þ
where SHUAi = the SHUA with the fire flow at nodei (i ¼ 1; : : : ; nn).
Study Network
The analyses outlined above are completed for the Anytown net-work (Walski et al. 1987) with several modifications to the originalsystem. Optimal reliability-based and robustness-based designs aredetermined for (1) pipes only and (2) pipes and pumps as decisionvariables. For the pipe only design, the elevated source reservoirhead is assigned an elevation of 73.2 m (240 ft) while for the pipeand pump design, the original Anytown reservoir head of 3.0 m(10 ft) is applied. Peak and average demand conditions are includedin the optimal design model. The average demand condition is as-sumed to continue constantly during planning periods and drivepump operations. The peak demand condition is the dominant fac-tor determining pipe sizes and the number of pumping units andtheir sizes.
Several other modifications to the original Anytown systemwere also introduced. First, this study optimizes new pipe and
pump station sizing assuming that there are no existing pipesand pumps in the system. Next, two tanks and two riser pipes areeliminated from the network. Therefore, the modified system con-sists of 19 nodes, one source, and 38 pipes as shown in Fig. 3. TheHazen-Williams roughness coefficient is 130 for all pipes. The min-imum pressure requirements are 28.1 m (40 psi) for the average andpeak demand conditions and 14.1 m (20 psi) for the fire flow con-dition. The maximum allowable pressure is 77.3 m (110 psi) for alldemand conditions. Cost parameters are listed in Tables 2 and 3.
A peaking factor of 1.8 is applied to given average base demand(2005 average daily use) to create the daily peak condition and thefire flow condition is assumed to occur at the peak plus the0.252 m3=s (4,000 gpm, 22.7% of total system peak demand) nodalfire demand.
Nodal demands and pipe roughness coefficients are consideredas uncertain and are assumed to follow the normal distribution witha CV of 0.1 (i.e., σ ¼ 0.1 × μ) for both parameters. The same op-timization parameters and functions applied to the NSGA-II areused for both formulations. The population of 100 was evolvedover 10,000 generations with crossover rate of 95% and mutationrate of 10%. The solutions that enhance their non-domination rankthrough selection, crossover, and mutation were survived to nextgeneration. Hydraulic simulations are performed using EPANet(Rossman 2000) for both design and SHA evaluation. A pressure-driven model can be used for SHA evaluation (Wu et al. 2009;Giustolisi and Walski 2012).
A number of assumptions and simplification are made in thisapplication: (1) the base nodal demand and pipe roughness coef-ficients are known with certainty; (2) average base demand is with-drawn constantly during planning periods; (3) peak demands andpipe roughness variations are considered for the robustness indexestimation although multiple demand conditions can be considered;
20
30
40
50
55
60
70
75
80
90
100
110
115
120
130
140
150
160
170
10
Zone A
Zone B
Zone C
Pipe 2
Pipe 6
Pipe 4
Reservoir : 3.0 m
Pump Station
Pipe 50
Pipe 34
Pipe 52Pipe 56
Pipe
62
Pipe
68
Pipe
70
Pipe 74
Fig. 3. Network layout and three elevation zones; zone A are at 36.6 m (120 ft) and zone B are the most distal nodes at an elevation of 24.4 m (80 ft);elevation zone C includes node 20 at 6.1 m (20 ft) and the rest of the nodes at 15.2 m (50 ft); the network’s EPANet input file is found at the linkhttps://www.dropbox.com/sh/6idcjhfkekhfflp/W1Nd_uY_CG
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(4) pump and motor efficiency are constant and independent offlow; (5) the likelihood of pipe breaks and fire are the same at everylocation; (6) valves are located at each end of all pipes allowing thepipe to be isolated without affecting the attached nodes; and(7) pipes are installed now and the improvement action such ascleaning and rehabilitation does not occur.
These assumptions above are consistently applied to both reli-ability and robustness-based design to maintain consistency thecomparison between methods. All can be relaxed in further studiesusing the techniques adopted here.
Results
Verification of Statistical Assumptions
Nodal and system reliabilities were estimated assuming that nodalpressures followed the normal distribution. This assumption wasconfirmed to be true for pipe and pump designs by visual inspection[Fig. 4(b)], Chi-squared goodness-of-fit test, and Kolmogorov-Smirnov test (results not presented).
However, the hypothesis of normality was rejected for pipe onlydesigns by the two statistical tests. The coefficients of skewness and
Table 2. Pipe Construction Cost Parameters Based on Clark et al. (2002) Used in Eq. (8)
Description Type
Parameter values
Indicator variable (u)a b c d e f
Base installation PVC pressure pipe −1.0 0.0008 3.59 0.011 1.0 0.0067 150 (Pressure class rating)Trenching and excavation Sandy gravel soil with 1∶1 side slope
(4–8 in.)−24 0.32 0.67 16.7 0.38 0.0 4 (Trenching depth)
Sandy gravel soil with 1∶1 side slope(8–144 in.)
2.9 0.0018 1.9 0.13 1.77 0.0 4 (Trenching depth)
Embedment Ordinary embedment 1.6 0.0062 1.83 −0.2 1.0 0.07 0 (Ordinary bedding)Backfill and compaction Sandy gravel soil with 1∶1 side slope −0.094 −0.062 0.73 0.18 2.03 0.02 4 (Trenching depth)Valve, fitting and hydrant Medium spacing 9.8 0.02 1.8 0.0 0.0 0.0 No indicator variables
Table 3. Pump Construction and Operation Cost Parameters Used inEqs. (10)–(12)
Parameter Value
Pump efficiency (ηpump) 0.75Motor efficiency (ηmotor) 0.75Pump energy tariff ($=kWh) 0.12Amortization period (n, year) 20Interest rate (AI, %) 3
Node 55 (Skewness: -0.295, Kurtosis: 0.173) Node 55 (Skewness: 0.035, Kurtosis: 0.047)
Node 170 (Skewness: -0.414, Kurtosis: 0.356) Node 170 (Skewness: 0.048, Kurtosis: 0.128)
Pressure (m)
0
100
200
300
400
Fre
qu
ency
Pressure (m)
0
100
200
300
400
Fre
qu
ency
32 34 36 38 40 42 22 24 26 28 30 32 34 36 38 40 42 44 46
24 26 28 30 32Pressure (m)
0
100
200
300
400
Fre
qu
ency
18 20 22 24 26 28 30 32 34 36 38 40Pressure (m)
0
100
200
300
400
Fre
qu
ency
(a) (b)
Fig. 4. Nodal pressure histograms from 10,000 randomly generated conditions for robustness-based design Solution 3: (a) pipe only design; (b) pipe/pump design
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kurtosis of the stochastic pressures were very close to zero and thehistogram of pressures visually appears to be a normal distribution[Fig. 4(a)]. The slightly negatively skewed distributions results fromthe fixed supply reservoir elevation that bounds the tail of the high
pressures and causes the failure of the hypotheses test. The lowpressure tail is of primary concern here and visually appears tobe consistent with the normal distribution so the normal distributionassumption for pressures is also maintained for the pipe only design.
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Fig. 5. Alternative robustness/reliability measures for Pareto optimal solutions from pipe only design [robustness-based design (open circle) versusreliability-based design (open diamond)]
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Optimization Results
Optimal results for the chance constraint and robustness problemsare presented in a series of figures and tables. Figs. 5(a and b) showrobustness and reliability values, respectively, of optimal solutionsfrom the two formulations for the pipe only design. Figs. 6(a and b)
show results for the pipe/pump design. Minimum mean pressureand the maximum standard deviations of pressure of the optimalsolutions are plotted in Figs. 5(c and d), 6(c and d) to show howthe two design approaches change the minimum pressure and maxi-mum standard deviation with increasing total cost. Solutions with
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Fig. 6. Alternative robustness/reliability measures for Pareto optimal solutions from pipe/pump design [robustness-based design (open circle) versusreliability-based design (open diamond)]
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similar costs from the two design approaches were selected and aresummarized in Tables 4 and 5. Least cost solutions for each designproblem with reliability equal to 1 (by a single precision expres-sion) are also summarized in the tables. PDFs of the minimum pres-sures node are shown in Figs. 7 and 8 using statistics given inTables 4 and 5 assuming pressures are normally distributed.
As expected, the robustness-based designs have higher sys-tem robustness than comparable costing reliability-based designs[Figs. 5(a) and 6(a)]. Similarly, reliability-based designs havehigher system reliability than robustness-based designs [Figs. 5(b)and 6(b)]. The Pareto fronts in Figs. 5(b) and 6(b) are quite steepslope indicating a rapid increase of system reliability with a smallincrease of total cost; whereas, relatively, system robustness in-creases more gradually [Figs. 5(a) and 6(a)].
The reliability increases result from higher mean minimum pres-sure rather than reductions in the pressure head’s standard deviation[Figs. 5(c and d), 6(c and d)]. This result is in contrast to the robust-ness based design that has lower minimum pressures and smallermaximum variances [Figs. 5(c and d), 6(c and d)]. Although thereliability-based design decreases the maximum standard deviationof pressure, reductions were not as significant as in the robustness-based design.
The difference between the two design approaches is moreclearly seen in the PDFs of the mean pressure at the minimumpressure node and its standard deviation (Figs. 7 and 8). Bothreliability-based and robustness-based designs begin from Solution
1 that are similar in PDF shape and location. As the total solutioncost increases (from Solution 1 to 4 in Fig. 7 and to 5 in Fig. 8), therobustness-based design reduces the variability as seen by narrowerand more peaked PDFs while the reliability-based designs tend toshift the distribution to the right by increasing the mean pressure.
The robustness of reliability-based designs increases with reli-ability nearly linearly [Figs. 5(e) and 6(e)]. The change in reliabilityof the robustness-based designs, on the other hand, does not changeuniformly. For example, in the pipe/pump design, until the totaloptimal design cost reaches $30.58M (Solution 2 in the robustness-based design in Table 5), the reliability values remain at 0.5 as seenin the vertical lines of points in Fig. 6(e) [the vertical lines of pointsare more apparently seen in Fig. 5(e)]. Until this cost is reached, themean pressure at critical nodes does not change but the standarddeviation decreases as shown in the PDF’s transition from Solution1 to 2 [Fig. 8(a)]. This pattern is quite different from the reliability-based design in which the mean pressure increases at the criticalnode without a corresponding decrease in the standard deviation[Solution 1 to 2 in Fig. 8(b)].
For costs above $30.58M, the reliability increases with robust-ness until the reliability value reaches 1. The standard deviation ofpressure decreases with increasing cost in the robustness-based de-sign until it becomes too costly relative to changes in the meanpressure then the mean pressure at the critical node is raised.
Note that the range of robustness values in the pipe only design[Fig. 5(a)] and that in pipe/pump design [Fig. 6(a)] are different
Table 4. Selected Pipe Only Design Solutions
Solution# Total cost Robustness Reliability SHAp SHAf
Min nodal pressure(m) (stdv)
Max nodalstdv (m) Remarks
Rob design1 15.08M 0.936 0.507 0.946 0.958 28.13 (0.81) 1.87 Least cost Rob design2 15.62M 0.971 0.507 0.971 0.992 28.13 (0.81) 1.05 Most expensive solution with Rel ≈ 0.53 16.28M 0.975 0.749 0.981 0.994 28.59 (0.70) 0.93 —4 17.31M 0.980 0.908 0.990 0.996 28.90 (0.59) 0.78 Comparable to Rel design sol #45 18.76M 0.984 1.000 0.993 1.000 30.97 (0.51) 0.66 Least cost design w=Rel ¼ 1
Rel design0 14.76M 0.934 0.504 0.938 0.961 28.14 (1.86) 1.86 Least cost Rel design1 15.08M 0.941 0.718 0.950 0.971 28.58 (0.80) 1.73 Comparable to Rob Design sol #12 15.61M 0.949 0.972 0.953 0.975 29.49 (0.70) 1.61 Comparable to Rob Design sol #23 16.29M 0.956 1.000 0.966 0.988 30.24 (0.63) 1.44 Comparable to Rob Design sol #34 17.34M 0.966 1.000 0.972 0.993 31.28 (0.57) 1.19 Least cost design w=Rel ¼ 1
Note: Rob = robustness, Rel = reliability.
Table 5. Selected Pipe/Pump Design Solutions
Solution# Total cost Robustness Reliability SHAp SHAf
Min nodal pressure(m) (stdv)
Max nodalstdv (m) Pump size (HP), NPp;NPa, remarks
Rob design1 30.13M 0.882 0.503 0.932 0.827 28.14 (2.87) 3.50 750,2,1, Least cost Rob design2 30.58M 0.911 0.505 0.949 0.909 28.16 (2.50) 3.06 720,2,1, Most expensive solution with Rel ≈ 0.53 31.17M 0.920 0.738 0.961 0.950 29.64 (2.38) 2.91 720,2,14 32.45M 0.932 0.995 0.970 0.970 34.20 (2.33) 2.71 750,2,15 34.71M 0.941 1.000 0.990 0.990 38.96 (2.28) 2.63 780,2,1, Comparable to Rel Design sol #56 35.36M 0.944 1.000 0.992 0.991 40.56 (2.31) 2.75 790,2,1, Least cost design w=Rel ¼ 1
Rel design0 30.08M 0.871 0.505 0.921 0.776 28.16 (3.03) 3.75 770,2,1, Least cost Rel design1 30.13M 0.871 0.562 0.922 0.786 28.58 (2.97) 3.73 770,2,1, Comparable to Rob Design sol #12 30.59M 0.883 0.830 0.934 0.840 31.00 (2.99) 3.74 790,2,1, Comparable to Rob Design sol #23 31.18M 0.896 0.953 0.946 0.894 32.97 (2.89) 3.56 790,2,1, Comparable to Rob Design sol #34 32.46M 0.919 1.000 0.962 0.957 37.18 (2.43) 3.23 790,2,1, Comparable to Rob Design sol #45 34.80M 0.935 1.000 0.974 0.973 40.56 (2.28) 2.85 790,2,1, Least cost design w=Rel ¼ 1
Note: Rob = robustness; Rel = reliability.
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while the ranges of reliability values are same [Figs. 5(b) and 6(b)].The robustness values are between 0.936 and 0.991 in the pipe onlydesigns and between 0.882 and 0.953 in pipe/pump designs, indi-cating that the proposed robustness index reflects the componentsin the system. Because the pump station in pipe/pump design haslarger pressure variations and lower pressures at increasing de-mands, the pipe/pump designs are less robust compared to the pipeonly design that is supplied by a fixed elevated reservoir. On theother hand, the reliability values are between 0.5 and 1.0 for bothpipe only and pipe/pump designs because the reliability value wasmeasured by the probability that the stochastic nodal pressures areequal to or greater than the minimum required pressure, 28.1 mrather than the consistency in the pressure. For the solutions withsimilar costs given in Tables 4 and 5, the reliability-based designshas at maximum 8.2% higher mean pressure at the critical node inpipe only design and 11.2% in pipe/pump design, compared to therobustness-based designs. Thus, the robustness index is moreindicative of different configurations although it does not have clearengineering interpretation. This contrast was not identified earliersince no other application considered pipe/pump designs.
Design Differences using Alternative Measures
Both formulations recognize that robustness is improved by in-creasing a node’s pressure or reducing its variance. Higher meanpressures allow for more variability in stresses as they are fartherfrom the failure condition while designs with mean pressures nearerthe threshold with low variability can also be considered robustagainst failure since a consistent pressure is delivered. However,the resulting designs from the two models in terms of the spatialpipe size distributions and pump/pipe sizes differ suggesting thediffering influence of the two measures.
Fig. 9 shows pipe diameters of the selected solutions sum-marized in Table 4. Regardless of total cost of the designs, therobustness-based design always has larger and more uniformlysized pipes along Pipe 6, 50, 62, and 68 or 70 (Route 1) and alongRoute 2 (Pipe 4, 34, and 74). Pipes were increased on these routesto obtain robust pressures in the zones A and B [Fig. 9(a)] becausethe critical node is shifted between the two zones as the robustnessrequirement increased.
The reliability-based design, on the other hand, primarily im-proved Route 2 and links 52 and 56 that branch off Route 1 to sup-ply water to the zone A [Fig. 9(b)]. These pipes insure reliability inthe zone A that is at the highest elevations in the system. The differ-ent spatial distribution of larger pipes in the two design approacheswas similar in the pipe/pump solutions. Consistent with the behav-ior of selecting pipes to increase the mean pressure, pump sizesselected in the reliability-based design were larger than those inrobustness-based design (Table 5).
Postoptimization Analysis Results
The robustness and chance-constraint based models applied hereadd robustness to the system against extreme demand conditionsin different ways. A question of interest that follows is: what isthe robustness of those systems under peak demand and pipe failureconditions outside of the design states? To that end, a postoptim-ization analysis was performed to compute each solution’s SHAunder two failure conditions, pipe break and fire flow conditions(Figs. 10 and 11).
SHAp and SHAf measure the system’s ability to provide ad-equate service during the abnormal conditions. Overall, the optimalsolutions obtained from robustness-based design had higherSHAs and are more robust against the two abnormal conditions.
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Fig. 8. Transitions in PDF for most critical node’s pressure for selectedpipe/pump designs: (a) robustness-based design; (b) reliability-baseddesign
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Fig. 7. Transitions in PDF for most critical node’s pressure for selectedpipe only designs: (a) robustness-based design; (b) reliability-baseddesign
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As expected, the source pipe failure (e.g., pipe 4 and 6) is the mostcritical in determining SHAp while fire flow at the nodes in thezone B and node 40 is severe. As noted, in the example considered,chance constraints tend to focus design decisions on one path
whereas robustness appears to have a broader system-wide influ-ence. Thus, larger pipes are sized for multiple locations reducingthe impact of pipe failures and fires at locations that were not con-sidered as design loads. For example, when fire flow occurred at
Solution#1
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Fig. 9. Pipe layout comparison for four pipe only design solutions (Solution 1 to 4 in Table 4); pipe diameters are in mm; thicker and darker pipe hasbigger size than the one that is not: (a) robustness-based design; (b) reliability-based design
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node 75 [Fig. 12(b)], the most critical node’s pressure was 15.3 m(21.7 psi) in the robustness-based design while that of the reliability-based design was computed in the hydraulic model as −9.8 m(−14.0 psi). Thus, this preliminary result suggests that the hydrauli-cally based robustness design provides a broader system robustnesscompared to the reliability based design.
Summary and Conclusions
Early optimal design research on WDS focused on minimizingeconomic costs while meeting the pressure requirement. In lasttwo decades, studies have moved to multiobjective problems thatmaximize the system robustness while minimizing total cost. Thereliability/chance constrained formulation has been used for iden-tifying robust designs. However, the authors’ contention is that ro-bustness is not effectively introduced by the reliability-based designmethods because they focus on the probability of successful systemperformance rather than minimizing failure severity. In addition
and equally important, reliability-based design methodologies re-quire defining the demand condition, its probability distributionand its statistics that are not generally available.
An alternative robustness index is proposed to consider systemrobustness. As a constraint in an optimal design model, the robust-ness index limits the range of variability of system function by con-straining the coefficient of variation (CV) of stochastic pressuresdue to demand and pipe roughness variability. The authors com-pared the robustness-based and reliability-based designs. The au-thors also demonstrated that the robustness-based design with arobustness formulation improves resilience relative to the reliabilityformulation. The main drawback of the robustness measure is de-fining an acceptable robustness level since, unlike reliability in achance constraint, it does not have clear engineering interpretation.
From numerical results, the reliability-based design emphasizedincreasing the mean pressure to increase system reliability whilethe robustness-based design simultaneously increases standard de-viation of pressure and the mean pressure. As a result, robustness-based design is more effective in introducing the system robustness
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Fig. 10. System hydraulic availability under pipe break conditions [robustness-based design (open circle) versus reliability-based design (opendiamond)]: (a) only pipe design; (b) pipe and pump design
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Fig. 11. System hydraulic availability under fire flow conditions: (a) only pipe design; (b) pipe and pump design [robustness-based design (opencircle) versus reliability-based design (open diamond)]
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than the reliability-based method. In the example considered, ro-bustness constrained solutions contain larger pipes in the main linesto the multiple pressure zones while reliability-based design isdominated by large pipes along the path leading to the node withlowest pressure.
Postoptimization analysis was completed to compute the SHAfor the optimal solution obtained from two design approaches.SHA is a resilience indicator that considers the system’s ability tosupply adequate service during the abnormal conditions, i.e., pipebreak and fire flow conditions. Overall, robustness-based optimaldesigns have higher hydraulic availability and are less vulnerable tothe two failure conditions. Therefore, the robustness-based designsappear preferable to reliability-based design with respect to systemresilience in a multiobjective WDS design model.
This study has several limitations that future research mustaddress. First, large networks with varying configurations shouldbe examined to confirm this study’s conclusions. Here, for a net-work with a single source, the robustness-based design includedrelatively large and consistent pipes in the main lines to the pressurezones. Thus, the robustness-based design performed better than thereliability-based design during failure events. However, if the net-work has multiple sources, the superiority of the robustness-baseddesign could diminish due to the proximity of the source to demand
points. Second, the risk type robustness index can be considered asa design objective by weighting the proposed robustness index bythe proportion of demand at a failed node (Kapelan et al. 2006).Third, reliability and robustness indices should be extended tounsteady conditions and results for two design approaches com-pared when designing valves, pumps, and tanks. Finally, rigorousanalysis should be completed to provide guidance on selectingthreshold robustness value to support engineering decision making.
Acknowledgments
This material is based in part upon work supported by the NationalScience Foundation under Grant No. 083590. Any opinions, find-ings, and conclusions or recommendations expressed in this mate-rial are those of the author(s) and do not necessarily reflect theviews of the National Science Foundation.
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