Robust optimization for water distribution systems least cost design
Lina Perelman,1 Mashor Housh,2 and Avi Ostfeld1
Received 12 March 2013; revised 1 August 2013; accepted 16 September 2013; published 21 October 2013.
[1] The objective of the least cost design problem of a water distribution system is to findits minimum cost with discrete diameters as decision variables and hydraulic controls asconstraints. The goal of a robust least cost design is to find solutions which guarantee itsfeasibility independent of the data (i.e., under model uncertainty). A robust counterpartapproach for linear uncertain problems is adopted in this study, which represents theuncertain stochastic problem as its deterministic equivalent. Robustness is controlled by asingle parameter providing a trade-off between the probability of constraint violation andthe objective cost. Two principal models are developed: uncorrelated uncertainty modelwith implicit design reliability, and correlated uncertainty model with explicit designreliability. The models are tested on three example applications and compared foruncertainty in consumers’ demands. The main contribution of this study is the inclusion ofthe ability to explicitly account for different correlations between water distribution systemdemand nodes. In particular, it is shown that including correlation information in the designphase has a substantial advantage in seeking more efficient robust solutions.
Citation: Perelman, L., M. Housh, and A. Ostfeld (2013), Robust optimization for water distribution systems least cost design, WaterResour. Res., 49, 6795–6809, doi:10.1002/wrcr.20539.
1. Introduction
[2] Optimal design of water distribution systems hasbeen studied extensively starting from the late 1960s[Schaake and Lai, 1969], up to this date [Kang and Lansey,2012]. The vast majority of these studies have formulatedthe problem assuming perfectly known parameters result-ing in deterministic optimization models. The resultsobtained by such models may perform poorly when imple-mented in the real world, when the problem parameters arerevealed and are different from those assumed in the deter-ministic model. Hence, it is necessary to find more ‘‘ro-bust’’ solutions than the classical deterministicoptimization designs, which assume that all model parame-ters are known with certainty. Typically, water distributionsystem robustness is concerned with the system’s capabilityto supply consumers’ demands with adequate heads orpressures. This is generally evaluated as the probabilitythat the minimum allowable nodal pressures are met underdemand variability. However, such an approach assumesthat nodal demand flows are satisfied by perfectly known
probability density functions (PDFs) [Xu and Goutler,1999].
[3] More recently, new methodologies for optimaldesign/rehabilitation of water distribution systems underuncertainty have been developed. Various uncertainty han-dling techniques have been integrated with different opti-mization models for both single and multiobjectiveformulations. Typically, uncertainty quantification can beclassified as:
[4] 1. Surrogate approach—The uncertainty isexpressed using a surrogate reliability index (e.g., resil-iency, network resiliency, flow entropy) and combined withthe minimum cost design objective of water distributionsystems. A reliability index, representing head surplus, isdefined (i.e., increasing the reliability index increases sys-tem reliability). The optimization problem is then formu-lated and solved using multiobjective optimization forminimizing cost and maximizing the reliability index, usingevolutionary techniques which form trade-off Pareto fronts[Todini, 2000; Prasad and Park, 2004; Farmani et al.,2005; Prasad and Tanyimboh, 2008; Raad et al., 2009,2010; Jung et al., 2012].
[5] 2. Stochastic approach—System robustness is nor-mally expressed in terms of system probability to maintainadequate pressures. This probability is added to the optimi-zation problem either as chance constraints or as a secondobjective function. The optimization problem is then solvedby analytical (e.g., through the general reduced gradientGRG2) [Lasdon and Waren, 1986] or sampling-based opti-mization techniques [Lansey et al., 1989; Xu and Goutler,1999; Tolson et al., 2004; Kapelan et al., 2005; Babayanet al., 2005, 2007; Giustolisi et al., 2009].
[6] 3. Fuzzy logic—The uncertainty is represented usingfuzzy theory with membership functions describing theuncertainty in demands. The design problem is formulated
Additional supporting information may be found in the online version ofthis article.
1Faculty of Civil and Environmental Engineering, Technion—IsraelInstitute of Technology, Haifa, Israel.
2Faculty of Civil and Environmental Engineering, University of Illinoisat Urbana-Champaign, Urbana, Illinois, USA.
Corresponding author: L. Perelman, Faculty of Civil and EnvironmentalEngineering, Technion—Israel Institute of Technology, Technion City,Haifa 32000, Israel. ([email protected])
WATER RESOURCES RESEARCH, VOL. 49, 6795–6809, doi:10.1002/wrcr.20539, 2013
as a two-objective optimization problem of minimizingcost and maximizing system reliability [Fu and Kapelan,2011] and solved using the multiobjective nondominatedsorted genetic algorithm II (NSGAII) of Deb et al. [2002].
[7] 4. Deterministic equivalent—Adding safety margins(redundancy) to the constraints or to the uncertain variablesresulting in a robust deterministic equivalent formulationfor the uncertain problem. The deterministic equivalent isthen solved by an evolutionary optimization technique,such as a genetic algorithm (GA) [Babayan et al., 2005,2007].
[8] The main drawbacks of most previously applied sto-chastic oriented approaches are: (a) The uncertain data areassumed to have a known PDF, solutions based on assumeddistributions may be unjustified. A drawback is that thePDF itself (i.e., the shape of the PDF) must be recognizedas being uncertain. (b) The size of the optimization problemis substantially increased when using stochastic orientedapproaches. This is because the PDFs, which represent theuncertain demands, are assimilated through a large numberof sampled scenarios. This drawback is further emphasizedwhen simulation-based optimization techniques, known fortheir computational complexity, are utilized.
[9] This study proposes formulating a deterministicequivalent of the uncertain problem of optimal design ofwater distribution systems, namely, a nonprobabilistic ro-bust counterpart (RC) [Ben-Tal and Nemirovski, 1999].The data uncertainty is quantified by a deterministic ellip-soidal uncertainty set with the decision maker seeking a so-lution that is robust optimal for all possible scenarios in theuncertainty set. The robust counterpart uses some charac-teristics of data distribution instead of using full distribu-tion information of the uncertain variables. The methoddoes not require the construction of a representative sampleof scenarios, and has the same size as the original model.RC thus copes with the drawbacks of the stochastic ori-ented approaches discussed above.
[10] RC for the optimal design of water distribution sys-tems under demand uncertainty was suggested by Perelmanet al. [2013]. Perelman et al. [2013] noted some drawbacksof their proposed RC model which are further addressed inthis study.
[11] Typically, the optimization problem is solvedassuming that nodal demands are satisfied for a givenpipes’ diameters selection. The robustness of the design, onthe other hand, is being computed thereafter the optimiza-tion stage based on violation of the hydraulic heads throughhydraulic simulations [Tolson et al., 2004; Kapelan et al.,2005; Babayan et al., 2005, 2007; Giustolisi et al., 2009;Fu and Kapelan, 2011; Perelman et al., 2013].
[12] These results in an implicit demand uncertaintyeffect system reliability. In addition, although the RCmodel [Ben-Tal and Nemirovski, 1999] incorporates corre-lated uncertainty, its straightforward application to waterdistribution system optimal design introduces a degeneratecase in which the dependency is not apparent, as discussedin Perelman et al. [2013].
[13] This work explores both the straight forward applica-tion of the RC approach and develops an adaptive approachto the water distribution systems least cost design problem.
[14] As the RC approach was developed for linear uncer-tain optimization problems, correlated data uncertainty
model, and explicit head constraints formulation areincluded through linearization of the headloss equation.The resulting equivalent deterministic problem is solvedusing the cross-entropy (CE) optimization method [Rubin-stein, 1999]. CE was successfully applied to deterministicsingle and multiobjective optimal design of water distribu-tion systems [Perelman et al., 2008].
[15] The following sections describe the RC formulationfor linear uncertain problems, formulation of the uncorre-lated and correlated models of data uncertainty for the leastcost design of water distribution systems, and applicationand comparison of the proposed approaches.
2. Robust Formulation of Linear ProgrammingProblems
[16] The uncertain linear optimization problem in aconstraint-wise form can be formulated as:
Minimize cT x ð1Þ
Subject to : eaTi x � 0 8eai 2 Ui 8i ð2Þ
where x ¼ x1; . . . ; xn;�1½ �T is a vector of decision variablesincluding the right-hand side, c is a vector of objectivefunction parameters, eai ¼ ½eai1; . . . ;eain;ebi� is a vector ofuncertain parameters for the ith constraint, Ui are uncer-tainty sets, and ebi is the right-hand side. The overall uncer-tainty set of all uncertain parameters in the problem isdefined as U where each Ui is taken as the projection of Ualong its corresponding dimensions [Ben-Tal and Nemirov-ski, 1999].
[17] For a particular constraint i, let J represent the set ofcoefficients that are subject to uncertainty, then each eaij; j 2J lays in the interval aij � aij; aijþaij
� �, where aij is the
deviation from the nominal value aij. Ben-Tal andNemirovski [1999] proposed to model the uncertainty withellipsoidal uncertainty. The advantages of the ellipsoidaluncertainty are: (a) it incorporates correlations between pa-rameters while controlling the distance from their nominalvalues, and (b) it is less conservative than the box modeluncertainty, which assumes that all parameters are at theirworst values simultaneously [Soyster, 1973]. The reader isrefereed to Housh et al. [2011] for further details on the useof ellipsoidal uncertainty sets.
[18] For uncertain parameter eaij, with nominal value aij
and covariance matrixP
, the ellipsoidal uncertainty set iswritten using the Mahalanobis distance in the form:
U �ð Þ ¼ eaij j eaij � aij
� �TX�1 eaij � aij
� �� �2
n oð3Þ
[19] The ellipsoidal uncertainty set can be described asan affine mapping of the random uncertain variable by set-ting eaij ¼ aij þ Pu, � ¼ P � PT and kuk � �. Equation (3)can be rewritten as:
U �ð Þ ¼ eaij j aij þ Pu; kuk � �� �
ð4Þ
where � ¼ P � PT , P is the mapping matrix which can becomputed by Cholesky decomposition of the covariance
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matrix, u is the perturbation vector, k � k is the Euclideannorm, and � is a value controlling the size of the ellipsoidand is also referred to as the protection level.
[20] To derive the robust equivalent the lower bound ofthe uncertain constraint in equation (2) needs to be nonneg-ative with the mapping derived from equation (4). For theith uncertain constraint :
minu:kuk��
eaTi uð Þx
� �� 0) min
u:kuk��ai þ Piuð ÞT x
n o¼aT
i x
þ minu:kuk��
uT PTi x
� � ð5Þ
where Pi is the ith mapping matrix, corresponding to the ithconstraint.
[21] The analytical solution to the problem can beattained trough Lagrangian duality [Ben-Tal and Nemirov-ski, 1999] and the robust equivalent of the uncertain con-straint (equation (5)) equals :
aTi x� �kPT
i xk � 0 ð6Þ
[22] Finally, the deterministic equivalent problem of theuncertain linear problem in equations (1) and (2) can beformulated as:
Minimize cT x ð7Þ
Subject to : aTi x� �kPT
i xk � 0 8i ð8Þ
[23] Although the deterministic equivalent is of asecond-order conic form, as opposed to the linear originalproblem, this still a convex tractable formulation whichcould be solved efficiently. Moreover, this formulationdoes not require sampling of multiple scenarios as requiredby a stochastic uncertainty model.
2.1. Probability Bounds
[24] The price of robustness is controlled by the parame-ter � (i.e., it controls the trade-off between the probabilityof violation of the uncertain constraint and the cost of theobjective function). Under the model uncertainty kuk � �for eaij, the probability that the uncertain constraint is vio-lated, is bounded by a wide family of PDFs [Ben-Tal et al.,2009]:
Pr eaTij xj � 0
� � e
��2=2 ð9Þ
[25] In other words, to guarantee a probability 1�" ofmaintaining the constraint, the parameter of protectionlevel should be selected as � �
. For example,for a 0.95 probability guarantee (i.e., 1� " ¼ 1� 0:05),select � � 2:45.
3. Water Distribution System Least Cost Design
[26] The least cost design problem of a water distributionsystem is to find its minimum cost as a function of pipes’discrete diameters as decision variables and lengths (equa-tion (10)) subject to linear mass conservation constraints(equation (11)), nonlinear energy conservation constraints
(equation (12)), and head bounds constraints (equation(13)). Todini and Pilati [1987] generalized the mass andenergy constraints in a matrix form:
Minimize fC D; Lð Þ ð10Þ
Subject to : fQ Q;Hð Þ ¼ A21Q� q ¼ 0 ð11Þ
fP Q;Hð Þ ¼ A11Qþ A12H ¼ 0 ð12Þ
H � Hmin ð13Þ
where fC is the cost function, D pipes’ diameters, L pipes’lengths, A21 ¼ AT
12is a topological matrix with elements of
the ith row equal to 1;�1; 0f g depending on the networkconnectivity, A11 is a diagonal matrix with nonlinear ele-ments representing pipe’s resistance, Q and H are unknownflows and heads, q are the consumer’s demands, and Hmin
are minimum desired nodal heads.[27] The term A11Q in equation (12) represents the head-
loss equation of the form:
DH ¼ DH Qð Þ ¼ RQe1 ð14Þ
where R ¼ e3LCe1 De2
, C pipe’s friction coefficient, e1 ¼ 1:852,e2 ¼ 4:87, and e3 is a unit-dependent coefficient for theHazen-Williams equation.
4. Data Uncertainty and Robust Formulation
[28] In this work, the uncertainty is assumed to be in theconsumers’ demands and the optimization problem issolved to find the least cost design. The water supply sys-tem includes a nonlinear objective function, and nonlinearand linear constraints. However, all constraints withdemand uncertainty are linear. Two principal models areexplored and compared: the uncorrelated uncertaintymodel and implicit design reliability, and the correlateduncertainty model and explicit design reliability.
4.1. Uncorrelated Model of Data Uncertainty
[29] Considering uncertainty in the demands eq 2 U , thelinear mass balance constraint (equation (11)) is uncertainwith only the right-hand side parameters being uncertain.Then, the deterministic equivalent of the mass balanceequation according to equation (8) is:
A21Q� q� �v ¼ 0 ð15Þ
vi ¼ kPTi k8i ð16Þ
[30] Equation (11) is replaced by equation (15) in the RCformulation. The mapping matrix P can be computed byCholesky decomposition of the covariance matrix � ¼P � PT of the demands. Each mass balance constraint con-tains only one demand. Hence, for each ith constraint Pi
compromises one vector and kPTi k ¼ �i, where �i is the
standard deviation of the ith uncertain demand. Thus, therobust equivalent optimization problem is:
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Minimize fC D; Lð Þ ð17Þ
Subject to : fQ Q;Hð Þ ¼ A21Q� q� � � � ¼ 0 ð18Þ
fP Q;Hð Þ ¼ A11Qþ A12H ¼ 0 ð19Þ
H � Hmin ð20Þ
where � is a vector of all standard deviations.[31] The resulting robust equivalent problem resembles
the original problem (equations (10)–(14)) with a safetyfactor added to all uncertain demands. This formulation,previously suggested by Perelman et al. [2013], presentstwo main drawbacks: (a) the correlation between differentdemand nodes does not influence the optimal design (i.e.,this is evident, since the optimization problem above isonly a function of the standard deviations and the meandemands), and (b) only the linear water balance equationsare replaced with their deterministic equivalents.
[32] In this study, the robust model is further developedto include correlated uncertain data through explicit formu-lation of the demands in head constraints. The subsequentsections describe the correlated model of data uncertaintyfollowed by a comparative study of the uncorrelated versusthe correlated model.
4.2. Correlated Model of Data Uncertainty
[33] To explicitly formulate demand uncertainty in therobust formulation, the hydraulic head loss function (equa-tion (14)) is expressed in a linear form. Two linearizationforms are considered herein:
[34] 1. Linearization around an operating point Q0 suchthat the line is tangent to the headloss curve at the operatingpoint:
fDH Qð Þ ¼ 1� e1ð ÞRQe10 þ e1RQe1�1
0 Q ¼ B0 þ B1Q ð21Þ
[35] 2. Linearization of the operating domain [Q1, Q2] :
fDH Qð Þ ¼ DH Q1ð ÞQ2 � DH Q2ð ÞQ1
Q2 � Q1þ DH Q2ð Þ � DH Q1ð Þ
Q2 � Q1
; Q ¼ B0 þ B1Q
ð22Þ
[36] Figure 1 graphically demonstrates the nonlinear, lin-ear with a single point, and linear with two operating pointsheadloss as a function of flow. It can be seen that modelLinear I (1 point) underestimates headloss in pipe and con-sequently overestimates nodal heads. Model Linear II (2points) overestimates headloss in pipe within the specifieddomain and thus underestimates nodal heads, and contrary-outside the specified domain.
[37] Next equation (11) is replaced by its linear version(equation (21) or (22)) and a full-rank linear system ofequations can be solved instead of equations (11) and (12):
A12 B1
0 A21
� �HQ
� �¼ G
HQ
� �¼ �B0 þ h
q
� �)
HQ
� �¼ K
B�0q
� �¼ K11 K12
K21 K22
� �B�0q
� � ð23Þ
where G�1 ¼ K ¼�
K11 K12
K21 K22
�is the inverse of the block
matrix in equation (23), K11 is of size nnodes� nlinks½ �,and K12 of size nnodes� nnodes½ � and B�0 ¼ �B0 þ h, h isa given vector corresponding to the fixed head nodes.
[38] The nodal heads H can be computed as:
H ¼ K11B�0 þ K12q ð24Þ
[39] Following this formulation, the minimum head con-straints in equation (13) can be explicitly expressed as a lin-ear function of the demands q.
[40] Next, considering uncertainty in the demandseq 2 U , the linear optimization problem with uncertainty indemands takes the form:
Minimize fC D; Lð Þ ð25Þ
Subject to : K11B�0 þ K12eq � Hmin ð26Þ
eq 2 U ð27Þ
where K Q 0f g 1;2f g;D� �
and B�0 Q 0f g 1;2f g;D� �
are a functionof linearization model and pipe diameters.
[41] The attained optimization problem (equations (25)–(27)) follows the formulation of equations (1) and (2) andits robust equivalent problem can be formulated accordingto equation (8). The minimum head constraint for node i inthe system is:
K11;iB�0 þ eqT K12;i
T � Hmin;i ð28Þ
where K11,i and K12,i are the ith row in the matrices K11 andK12, respectively. Hmin,i is the minimum head constraint atnode i.
[42] The deterministic equivalent is :
K11;iB�0 þ qT K12;i
T � �kPT K12;iTk � Hmin;i ð29Þ
[43] Finally, the robust design of a water distribution sys-tem is the solution of the following optimization problem:
Minimize fC D; Lð Þ ð30ÞSubject to :
K11;iB�0 þ qT K12;i
T � �kPT K12;iTk � Hmin;i � 0 8i ð31Þ
Figure 1. Head loss models—nonlinear (Hazen Wil-liams), Linear I (1 point), and Linear II (2 points).
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[44] The formulation of the optimization problem for theleast cost design of water distribution system under demanduncertainty derived in equations (30) and (31) incorporatesboth the correlations between the consumers’ demands, andit explicitly relates uncertainty in demands to the resultednodal heads.
4.3. Overconservativeness of the Uncorrelated Model
[45] In the framework presented in Perlman et al. [2013]and summarized in the previous section, the robustfication(i.e., replacing uncertain elements by their robust counter-parts) is performed on the water balance equations (Kirch-off’s first low) resulting in the deterministic equivalentformulation in equation (18). It is evident that the optimiza-tion problem in equations (17)–(20) is only a function of thestandard deviations and the mean demands and thus does notcapture the correlation between different demand nodes.
[46] Correlation assimilation in the design phase is mostimportant to achieve an efficient and reliable solution aswill be demonstrated next in the examples. Another consid-erable drawback is that the robustification is performedonly on the water balance and not on the energy balance.The demand uncertainty propagates through the systemequations (i.e., through Kirchoff laws 1 and 2) resulting inan uncertain heads in the system. The minimum head con-straint is then uncertain by itself and need to be replaced byits robust counterpart as suggested by the methodology pre-sented in the previous section. Next, it will be shown thatomitting the explicit dependency between the demand andthe head results in overconservative solutions.
[47] Consider the robust counterpart of the uncertainhead requirement constraint (with explicit demand correla-tion) derived in equation (29). To show that formulation(29) is less conservative robust counterpart than formula-tion (18), we need to prove that the left-hand side of equa-tion (29) is larger than the one obtained in formulation (18)even if no-correlation is present in the problem. Thus, thecorrelation effect is filtered out and the difference isentirely attributed to the lack of explicit dependencybetween the demand and the head.
[48] The linearization of formulation (18) consideringthe robust counterpart of the water balance results in:
K11;i B�0 þ qT K12;iT þ ��T K12;i
T � Hmin;i ð32Þ
[49] It needs to be shown that:
�kPT K12;iTk � �T K12;i
T ð33Þ
when no-correlation is present [i.e., when PT¼ diag(�)].[50] All element of K12;i 8 i are negative. This could be seen
by noting that K12,i is multiplied by the demand in the head equa-tion, and increasing the demand results in decease in head (seealso Appendix B for an example of the negativity of K12,i).
[51] Relying on the above, inequality (33) could berewritten as:
kdiagð�ÞK12;iTk � jdiagð�ÞK12;i
T j ð34Þ
where jxj is the L1 norm defined as jxj ¼P
j jxjj. Askxk � jxj, it is shown (Appendix B) that the robust counter-
part presented herein is less conservative than the one sug-gested in Perelman et al. [2013].
5. Applications
[52] The proposed method was applied on three growingcomplexity networks: (1) an illustrative example [Bouloset al., 2006], (2) Hanoi system [Fujiwara and Khang[1990], and (3) large network (adapted from Alperovits andShamir [1977]). The following notions were used (metadataand all utilized source codes are attached as supportinginformation):
5.1. Robust Optimization Models
[53] Three uncertainty models were explored andapplied:
[54] 1. Uncorrelated implicit nonlinear—based on theproblem formulated in equations (17)–(20). This modeldoes not consider correlation between uncertain data andassigns all uncertain demands a safety factor. This modelcan thus be considered as a redundancy based formulation.
[55] 2. Correlated explicit Linear I (1 point)—based onthe problem formulated in equations (30) and (31) with asingle point linearization of the head loss equation (21).This model underestimates headloss in individual pipes andthus typically overestimates nodal heads (depending on thecombination of all network pipes). Thus, this model isexpected to be less conservative than the two points linearmodel (below). One solution of the system’s nonlinearequations is required to compute model parameters at oper-ating point [Q0].
[56] 3. Correlated explicit Linear II (2 points)—thismodel is as well based on the formulation in equations (30)and (31) with operating range considered for the headlosslinearization in equation (22). It overestimates headloss inindividual pipes and underestimates nodal heads. Thus, thismodel is expected to be more conservative than the onepoint linear model. Two solutions of the system’s nonlinearequations are required to compute model parameters at thetwo operating points [Q1, Q2].
5.2. Optimization Technique
[57] The nonlinear discrete optimization problem wassolved using the cross-entropy (CE) method for combinato-rial optimization [Rubinstein, 1999; Rubinstein and Kro-ese, 2004].
[58] The CE algorithm is a two-stage iterative procedureinvolving: (1) generation of random data solutions, and (2)updating of the parameters of the problem on the basis ofthe sampled data in the direction of solution improvements.The cross-entropy scheme seeks to find optimal probabilitysuch that the Kullback-Leibler distance [Kullback and Lei-bler, 1951] between the sampling probability and the theo-retical optimal probability is minimal. The CE algorithmparameters are the sample size N defining the number ofsamples in each iteration, the elite sample percentage �defining the best set of solutions used for updating sam-pling probability, and the smoothing parameter � used toprevent the algorithm from converging prematurely. Pa-rameter values are generally set through model calibration.
[59] A full description of the CE method and its applica-tion to single and multiobjective optimal design of waterdistribution systems can be found in Perelman et al.
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[2008]. Appendix A provides a small illustrative exampleof the cross-entropy method.
5.3. Example 1—Illustrative
[60] The illustrative example was adopted from Bouloset al. [2006] and is used for demonstrating purposes. Thenetwork consists of a single constant head source, threedemand nodes, and four pipes. The length and friction coef-ficient of all links and elevation of all nodes are 304.8 (m),100, and 0 (m), respectively. The available diameters con-sidered in this example are (20.32, 25.4, 30.48, 35.56,40.64) (cm) (i.e., 54¼ 625 solution space) with correspond-ing costs (23, 32, 50, 60, 90) ($/m). The layout of the net-work and base demands are shown in Figure 2. For the easeof representation the demand at node 2 is considered to beconstant and equals to zero.
[61] The demands at nodes 3 and 4 are considered to beuncertain with nominal demands, standard deviations, andcovariance matrix correlation coefficients :
q ¼203:88
254:85
� �cmh½ �; � ¼
58:855
73:569
� �;
� ¼�3�3 ��3�4
��3�4 �4�4
� �; where� 1 � � � 1 is a
ð35Þ
[62] Given the small problem size, all possible solutionswere enumerated, without the need for using the CEmethod. The optimal designs were computed for differentvalues of protection level � ¼ 0; 0:1; . . . ; 2½ � for the threemodels: (1) uncorrelated implicit nonlinear, (2) correlatedexplicit Linear I (1 point) with Qi0½ � ¼ Qi qð Þ½ � for each pipei, and (3) correlated explicit Linear II (2 points) withQi1;Qi2½ � ¼ Qi qð Þ;Qi qþ 2�q
� �� �for each pipe i.
[63] The robust designs were tested on three types ofdata uncertainty: (a) positive correlation (PC) between con-sumers’ demands, (b) zero correlation (ZC) between con-sumers’ demands, and (c) negative correlation (NC)between consumers’ demands. For example, positively cor-related representing similar types of consumers (e.g.,domestic), zero correlated—random demands, and nega-tively correlated—different types of consumers (e.g.,domestic and industry). Particularly, the correlation was setto � 2 0:8; 0;�0:8f g representing positively, zero, andnegatively correlated consumers, respectively. Solutionrobustness was evaluated as the probability of head con-straint violation based on 1000 Monte Carlo samples
assuming uniform PDF with given above mean and stand-ard deviation.
[64] Figures 3–5 demonstrate the price of robustness as afunction of protection level � for all models. Figures 3–5ashow the theoretical and simulated probability of head con-straint violation as a function of protection level �. Figures3–5b show the cost of designs as a function of �. From thefigures, it can be seen that as protection level increases,probability of head constraint violation decreases anddesign cost increases.
[65] The robust design of the two linear models is signifi-cantly affected by the relationship between the demandsadapting the optimal design and its cost to the model uncer-tainty. It is also evident, as expected, that the cost of thedesigns for negative correlation are lower than for zero cor-relation, and those in turn are lower than positive correla-tion. This performance is not evident in the nonlinearmodel which does not account for correlation of the uncer-tain data and results in most costly designs. In all threecases (PC, ZC, and NC) uncorrelated implicit Linear I (1point) model results in less conservative designs than the
Figure 2. Example 1—layout.
Figure 3. Example 1—(a) head constraint violation prob-ability and (b) cost versus protection level � for positivelycorrelated demands �¼ 0.8.
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correlated explicit Linear II (2 points) model. This wasexplained previously by the overestimation of nodal headsof the 2 points model. In case of positively correlateddemands, as shown in Figure 3a, the correlated model iseven more conservative than the uncorrelated model forseveral values of �. Additionally, it can be seen from Fig-ures 3–5 that the probability of head constraint violation islower than the theoretical bound. However, the slacknessof the probability bound can be attributed to the discretenature of the problem, the given available diameters, andthe intrinsic robustness of the design for reduced demands.
[66] To estimate the overall error Herror resulting fromthe linearization in the formulation of the deterministicequivalent, the robust head at each node was compared tothe nonlinear headloss given the robust demand by:
Herror;i ¼ HR;i � H qR;i
� �ð36Þ
where HR;i ¼ qT K12;iT � �kPT K12;i
Tk þ K11;iB�0 is the ro-
bust equivalent head at node i, qR;i ¼ q � �PPT K12;i
T
kPT K12;iTk is the
robust equivalent demands to seek robust equivalent head atnode i, and H(qR) is the nonlinear head loss given demand qR.
[67] The proximity of the optimal solution to minimumhead constraint, Hslack, was evaluated as:
Hslack;i ¼ HR;i � Hmin;i ð37Þ
[68] Table 1 lists the errors in nodal heads Herror and thedistance of the optimal solution from the minimum headconstraint Hslack for the three correlation values and maxi-mum protection level �¼ 2. Positive values of Herror indi-cate that the linear model overestimates nodal heads andnegative values—underestimates. Low positive values ofHslack indicate that the optimal design is close to the mini-mum head constraint. From the results, it can be seen that,in this example, the nodal heads calculated by correlatedexplicit Linear II (2 points) model are always slightlyunderestimated, thus the solution may be overconservative,and contrary for the correlated explicit Linear I (1 point)model. The optimal design is closest to the minimum headconstraint for negative correlation between demands, andfarther for the other two cases. This can be attributed to the
Figure 4. Example 1—(a) head constraint violation prob-ability and (b) cost versus protection level � for zero corre-lated demands �¼ 0.
Figure 5. Example 1—(a) head constraint violation prob-ability and (b) cost versus protection level � for negativelycorrelated demands �¼�0.8.
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discrete nature of the problem and the given availablediameters.
[69] The reliability of the designs was further evaluatedagainst all three test sets and the results for correlatedexplicit Linear II (2 points) model are shown in Table 2. Asexpected, design based on positive correlation has the high-est reliability in all cases at the expense of its cost, asopposed to design based on negative correlation.
5.4. Example 2—Hanoi Water Distribution System
[70] The Hanoi network is a relatively large gravitationalsystem introduced by Fujiwara and Khang [1990]. The net-work (Figure 6) is subject to one demand loading condition,and consists of 34 links and 32 demand nodes supplied by asingle reservoir at a constant head of þ100 (m). All nodes
are at zero elevation and the minimum pressure headrequirement is 30 (m) at all nodes. Six candidate pipe diam-eters (30.48, 40.64, 50.8, 60.96, 76.2, 101.6) (cm) with aHazen-Williams coefficient of 130 (�) are considered foreach pipe. The full data for this example can be found inCentre for Water Systems (CWS) [2001]. The cost ($) ofinstalling a pipe of diameter D (mm) and length L (m) is :
fC D; Lð Þ ¼ 8:593� 10�3D1:5L ð38Þ
[71] To model uncertainty in consumers’ demands, sys-tem nodes were partitioned to three demand zones: zone1—nodes 1:15, zone 2—16:24, zone 3—25:32 (Figure 6).Demands in zone 2 were assumed to be certain and in zones1 and 3 uncertain with standard deviation of 12% frommean demand of each zone [i.e., 80 and 50 (m3/h)], respec-tively. The problem was again formulated and solved fordifferent values of protection level � ¼ 0; 0:2; . . . ; 2½ � forthe three models: (1) uncorrelated implicit nonlinear, (2)correlated explicit Linear I (1 point) with Qi0½ � ¼ Qi qð Þ½ �for each pipe i, and (3) correlated explicit Linear II (2points) with Qi1;Qi2½ � ¼ Qi qð Þ;Qi qþ 2�q
� �� �for each
pipe i. Additionally, intrazone correlation was set to�¼ 0.8 meaning that consumers in the same zone followsimilar demand pattern. Interzone correlation was set to
Figure 6. Hanoi water distribution system layout.
Table 1. Example 1—Robust Design Analysis for �¼ 2
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� ¼ 0:6; 0;�0:6½ � again to represent positive, zero, andnegative correlation between different zones in thenetwork.
[72] The cross-entropy optimization method was used tofind the optimal design for each model and each value of�. The CE parameters set for all runs were: sample sizeN¼ 10,000, elite sample percentage �¼ 0.005, andsmoothing parameter �¼ 0.6. The algorithm converged af-ter 16 iterations on average with average running times of:(1) uncorrelated implicit nonlinear model 160 (s), (2) cor-related explicit Linear I (1 point) 200 (s), and (3) corre-lated explicit Linear II (2 points) 400 (s), all on Intel(R)2Core 2.80 GHz machine.
[73] Figures 7–9 demonstrate the price of robustness as afunction of protection level � for all models. Figures 7–9ashow the theoretical and simulated probability of head con-straint violation as a function of protection level �. Theo-retical probability of head constraint violation wascomputed based on equation (9) and the actual probabilitywas calculated based on 1000 Monte Carlo samples assum-ing uniform PDF with given above means and standard
deviations. Again, it can be seen that the probability ofhead constraint violation are lower than the theoreticalbound.
[74] Figures 7–9b show the cost of designs as a functionof �. It can be seen that the nonlinear model not consider-ing the correlations between demands resulted in a moreconservative design than taking correlations under consid-eration. This becomes more evident for the zero and nega-tive correlation uncertainty models. As in the previousexample, the linear model based on a single operating pointis less conservative than the two point linear model.
[75] Table 3 lists the errors in nodal heads Herror (equa-tion (36)) and the distance of the optimal solution from theminimum head constraint Hslack (equation (37)) for themaximum protection level �¼ 2 and positive correlation�¼ 0.6. The results demonstrate that the nodal heads calcu-lated by correlated explicit Linear II (2 points) model arealways slightly underestimated and always slightly overes-timated by correlated explicit Linear I (1 point) model.Thus, the two point linear model is always more expensivethan the one point linear model. This can be observed forall runs (Figures 7–9a).
Figure 8. Hanoi—(a) head constraint violation probabil-ity and (b) cost versus protection level � for zero correlateddemands �¼ 0.
Figure 7. Hanoi—(a) head constraint violation probabil-ity and (b) cost versus protection level � for positively cor-related demands �¼ 0.6.
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5.5. Example 3—Large Network
[76] In this example, a more challenging water distribu-tion system is introduced based on a real water distributionsystem example application described in Alperovits and
Shamir [1977]. Uncertainty is incorporated in all demandnodes considering a fully heterogeneous covariance matrix.
[77] The water distribution system layout is shown inFigure 10. The network consists of 65 links, and 52 demandnodes. The two pumping stations were replaced with a con-stant head reservoir of þ408 (m) representing pump headduring the loading condition. The elevation varies acrossthe nodes while the minimum pressure head requirement is30 (m) at all nodes [Alperovits and Shamir, 1977, Table9d]. Eleven candidate pipe diameters are considered allwith a Hazen-Williams coefficient of 130 (�) and capitalcosts as described in Alperovits and Shamir [1977, Table
Figure 9. Hanoi—(a) head constraint violation probabil-ity and (b) cost versus protection level � for negatively cor-related demands �¼�0.6.
Figure 10. Network layout for Example 3 based on Alperovits and Shamir [1977].
Table 3. Hanoi—Robust Design Analysis for �¼ 2 and �¼ 0.6
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9c]. The EPANET [2012] input data file for this example isattached as supporting information.
[78] Demands at all nodes are considered uncertain. Themean of the demand is equal to the base demand as givenin Table 9d in Alperovits and Shamir [1977]. The standarddeviation of the demand is set to 20% of the mean demandfor each node. Hence, unlike the second example in whichthe standard deviation of the demand is 50 or 80 (m3/h), inthis example each demand node has different standarddeviations. Moreover, the covariance between the demandnodes is fully heterogeneous with correlation rangingbetween �0.3 and 0.8, as graphically depicted in Figure 11.
[79] The network is solved using the correlated explicitLinear I model and the deterministic equivalent is solvedusing the cross-entropy optimization method with the sameparameters described in Example 2, except for the samplesize which is set to N¼ 50,000 due to the increased size ofthe problem.
[80] As the RC methodology requires only meandemands, standard deviations, and correlation matrix tosolve the uncertain optimization problem, it was interestingto test the reliability of the attained designs on differentPDF’s. The reliability was empirically calculated by MonteCarlo simulation based on three different distributions: (a)marginal uniform, (b) multivariate normal, and (c) trans-formed uniform (i.e., independent uniform distributionswhich are linearly transformed to give the desired meanvector and covariance matrix). Each of these three distribu-tions has the same mean and standard deviation vectors andcorrelation matrices, as indicated in the problemdescription.
[81] Figure 12 shows the robust design cost versus reli-ability trade-off using 1000 Monte Carlo samples for thethree distributions above. The close trade-off curves indi-cate that the design suggested by the model is almost indif-ferent (i.e., robust) to different PDFs. Thus, no matter whatprobability distribution is realized (out of the three above),the design is still satisfactory as it gives very similar reli-ability outcomes.
5.6. Importance of Demand Correlation Assimilation
[82] The main contribution of the present study over therobust design approach developed previously by the
authors in Perlman et al. [2013] is the ability to account fordifferent correlations between demand nodes. Next it isshown that including correlation information in the designphase has a substantial advantage in seeking more efficientrobust solutions.
[83] The same means and standard deviations are consid-ered, as described in Example 3, but with different correla-tion matrices. The problem is first solved with a zero-correlation assumption and then compared to correlationbased design in terms of reliability and design costs.
[84] Figure 13 presents the correlation matrices for: (a)zero-correlation demands, (b) positively correlateddemands, (c) mixed positively and negatively correlateddemands with a correlation range of �0.6 to 0.6, and (d)two zones homogeneous demands.
[85] In the two zones correlation matrix (Figure 13d) thelargest four water consumers (nodes 5, 7, 13, and 22) areconsidered in one zone which is negatively correlated (cor-relation of �0.6) with all others nodes (second zone). Theinterzone correlation is positive and is set to 0.6 for allnodes.
[86] To show the relative advantage of the correlationassimilation in the robust design, the performances betweenzero-correlated and correlated designs are compared. Set-ting a reliability target of 95% the correlated explicit LinearI model formulation is then solved to achieve the reliabilitytarget under each of the correlation cases in Figure 13, andunder the base run correlation in Figure 11.
[87] Figure 14 shows the performance comparisonbetween the zero-correlated and the correlated designs.Design that uses zero-correlation information has the samedesign cost for all correlations, while the design cost forcorrelation based design is changing as a function of thecorrelation.
[88] For example, the first two groups of columns in Fig-ure 14 show that the design cost to reach 95% reliability inthe base run correlation (Figure 11) is 4.38 (M$), whilst ifthe correlation in Figure 13a is assumed, then the designcost is 4.43 (M$). The design cost based on zero-correlation is the same regardless of the correlation, and it
Figure 11. Correlation matrix for the base run of Exam-ple 3.
Figure 12. Reliability versus cost trade-off curve of therobust design under different PDF’s, for Example 3.
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is equal to 4.44 (M$). The reliability however, is differentas shown in Figure 14. A slightly higher reliability isobtained if the design is based on zero-correlation, for thecases of the base run and mixed correlation.
[89] On the one hand, the correlation based design in thebase run and mixed correlation reduces the cost whilemeeting the targeted reliability of 95%. However, on theother hand, the correlation based design increases the costin the two zones and the positive correlation cases com-pared to the zero-correlation based design. Nevertheless,this cost difference is well justified since the zero-correlation design will perform poorly if these two cases ofcorrelation are realized, as reliability will be substantiallyreduced (Figure 14).
[90] The above results highlight the importance of theconsumer’s demand correlation incorporation in the designphase of water distribution systems. Omitting the correla-tion information may result in unnecessarily more expen-sive design or even more hazardous situations in whichsignificantly lower than expected reliability is obtained.
6. Conclusions
[91] The proposed work suggests using a deterministicequivalent to the uncertain problem of optimal designunder demand uncertainty. Three robust optimization mod-els were explored—one not considering dependence in datauncertainty and two-dependent uncertainty models. Allthree models were further tested and compared on different
Figure 13. Correlation matrices for Example 3 for (a) zero-correlation, (b) positive correlation, (c)mixed correlation, and (d) two zones.
Figure 14. Cost and reliability comparison between zeroand different correlation based designs for Example 3.
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data uncertainty models representing different type of con-sumers in a water distribution system, including positive,zero, negatively, and fully heterogeneous correlationsbetween consumers.
[92] The presented method shows four main strengths: (1)Inclusion of correlated uncertain data. The results showed thatnot considering uncertainty dependency leads to more con-servative and thus costly designs. (2) Explicit head constraintformulation by means of linearization of the nonlinear head-loss equation. The two linear models explored showed consis-tency in their results and performances. (3) The explicit headconstraint formulation allowed to provide theoretical probabil-ity bounds for violating this constraint. The slackness of theprobability bound can be attributed to the intrinsic robustnessof the design due to the discrete nature of the problem, thegiven available diameters, and that uncertainty in demandscan also result in reduced demands, in which case designrobustness remains intact. (4) System robustness is monitoredthrough a single parameter � controlling the trade-offbetween the probability of violation of the uncertain constraintand the cost of the objective function.
[93] Further research will seek to incorporate into robustoptimization real sized networks with pumping stations,multiple loading conditions, water quality considerations,and multiobjective problems.
Appendix A
[94] The cross-entropy (CE) algorithm is a heuristicsearch technique which utilizes probabilities of possibleoutcomes of decision variables instead of the actual values.
[95] For example, if a decision variable x can receive thevalues x¼ {1, 2, 3}, then an associate probability ofp(x)¼ {p(1), p(2), p(3)} can be defined where each elementdefines the probability of receiving the actual associatedvalue. For example, p¼ {1=2, 0, 1=2} means that there is 50%chance of the decision variable to get the values of one orthree, and zero for receiving two. Obviously, this does nothelp much in getting the optimal value x�, so the optimalprobability should be degenerated or close to it. For example,p(x�)¼ {0, 0, 1} means that the optimal solution is x� ¼ 3.
[96] The CE algorithm starts with some user defined ini-tial probability piter¼ 0 (it can be from any family of PDFsassociated with the underlying decision space) and is evolv-ing iteratively until convergence (i.e., until p � {0, 1}).
[97] In each iteration a new sample of solutions is gener-ated based on the current probability piter, and the probabil-ity is updated based on counting the frequencies of eachdecision value in a user defined elite set of solutions. Addi-tionally, to prevent premature convergence the probabilityis smoothed by taking a weighted average of the probabil-ities from the last two iterations:
piter ¼ �piter þ ð1� �Þpiter�1: ðA1Þ
[98] For example (Table A1), consider six possible(N¼ 6) values for a decision variable x, p0 as a uniform ini-tial probability, and S(x) as the performance function of thedecision variable x. The goal is to find x that maximizesS(x). In each iteration three solutions are sampled and theprobability is updated based on the top two solutions. F(x)counts the number of times that a solution was sampled
(i.e., in the first iterations each solution x¼ 1, 2, 4 weresampled once). The probability is updated based on theelite two solutions (i.e., those that were sampled, S(x¼ 2)¼ 4 and S (x¼ 4)¼ 16). The updated probability(without smoothing) is 1=2 for these two solutions, x¼ 2 andx¼ 4 and zero for the remaining values of x¼ {1, 3, 5, 6}.Without the smoothing, these solutions can never besampled again. With the smoothing, for example �¼ 0.6,their probability is lower than the initial, but is not zero andthose solutions can be sampled again. In the next iterationthe process is repeated with the new probability p¼ {0.067,0.367, 0.067, 0.367, 0.067, 0.067}, and so on.
Appendix B
[99] Proof that kxk � jxj for any x. We use the defini-tions of the L1 and the L2 norms to write the problem inindex form as:
ffiffiffiffiffiffiffiffiffiffiffiXn
j¼1
x2j
s�Xn
j¼1
jxjj
Xn
j¼1
x2j �
Xn
j¼1
jxjj !2 ðB1Þ
[100] We then prove the above by induction as follows:[101] Step 1. Base case for n¼ 2
x21 þ x2
2 � jx1j þ jx2j� �2
0 � 2jx1jjx2jðB2Þ
[102] Step 2. Assume that (B1) is true for n¼m
M Xm
j¼1
jxjj !2
�Xm
j¼1
x2j � 0 ðB3Þ
[103] Step 3. Show that the inequality holds forn¼mþ 1
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[104] Since both terms in the right-hand side are positivethe inequality holds.
B1. Negativity of the K12 Elements
[105] Here the matrix K12 is demonstrated for the illustra-tive example in Figure 2. It is shown that all elements ofthe matrix are negative as assumed in the proof of theoverconservativeness.
[108] Recalling that the elements Bi correspond to theslope of the linear approximation for the Hazen-Williamsequation, and recalling that this equation is convex, thusBi � 08i which implies that all elements in K12 above arenegative.
[109] Acknowledgment. This research was supported by the Techn-ion Grand Water Research Institute.
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