Robust optimization for water distribution systems least cost design

Authors

  • Lina Perelman,

    Corresponding author
    1. Faculty of Civil and Environmental Engineering, Technion—Israel Institute of Technology, Haifa, Israel
    • Corresponding author: L. Perelman, Faculty of Civil and Environmental Engineering, Technion—Israel Institute of Technology, Technion City, Haifa 32000, Israel. (lina@tx.technion.ac.il)

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  • Mashor Housh,

    1. Faculty of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
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  • Avi Ostfeld

    1. Faculty of Civil and Environmental Engineering, Technion—Israel Institute of Technology, Haifa, Israel
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Abstract

[1] The objective of the least cost design problem of a water distribution system is to find its minimum cost with discrete diameters as decision variables and hydraulic controls as constraints. The goal of a robust least cost design is to find solutions which guarantee its feasibility independent of the data (i.e., under model uncertainty). A robust counterpart approach for linear uncertain problems is adopted in this study, which represents the uncertain stochastic problem as its deterministic equivalent. Robustness is controlled by a single parameter providing a trade-off between the probability of constraint violation and the objective cost. Two principal models are developed: uncorrelated uncertainty model with implicit design reliability, and correlated uncertainty model with explicit design reliability. The models are tested on three example applications and compared for uncertainty in consumers' demands. The main contribution of this study is the inclusion of the ability to explicitly account for different correlations between water distribution system demand nodes. In particular, it is shown that including correlation information in the design phase has a substantial advantage in seeking more efficient robust solutions.

1. Introduction

[2] Optimal design of water distribution systems has been 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 formulated the problem assuming perfectly known parameters resulting in deterministic optimization models. The results obtained by such models may perform poorly when implemented in the real world, when the problem parameters are revealed and are different from those assumed in the deterministic model. Hence, it is necessary to find more “robust” solutions than the classical deterministic optimization designs, which assume that all model parameters are known with certainty. Typically, water distribution system robustness is concerned with the system's capability to supply consumers' demands with adequate heads or pressures. This is generally evaluated as the probability that the minimum allowable nodal pressures are met under demand variability. However, such an approach assumes that nodal demand flows are satisfied by perfectly known probability density functions (PDFs) [Xu and Goutler, 1999].

[3] More recently, new methodologies for optimal design/rehabilitation of water distribution systems under uncertainty have been developed. Various uncertainty handling techniques have been integrated with different optimization models for both single and multiobjective formulations. Typically, uncertainty quantification can be classified as:

[4] 1. Surrogate approach—The uncertainty is expressed using a surrogate reliability index (e.g., resiliency, network resiliency, flow entropy) and combined with the minimum cost design objective of water distribution systems. A reliability index, representing head surplus, is defined (i.e., increasing the reliability index increases system reliability). The optimization problem is then formulated and solved using multiobjective optimization for minimizing cost and maximizing the reliability index, using evolutionary 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 normally expressed in terms of system probability to maintain adequate pressures. This probability is added to the optimization problem either as chance constraints or as a second objective function. The optimization problem is then solved by analytical (e.g., through the general reduced gradient GRG2) [Lasdon and Waren, 1986] or sampling-based optimization techniques [Lansey et al., 1989; Xu and Goutler, 1999; Tolson et al., 2004; Kapelan et al., 2005; Babayan et al., 2005, 2007; Giustolisi et al., 2009].

[6] 3. Fuzzy logic—The uncertainty is represented using fuzzy theory with membership functions describing the uncertainty in demands. The design problem is formulated as a two-objective optimization problem of minimizing cost and maximizing system reliability [Fu and Kapelan, 2011] and solved using the multiobjective nondominated sorted genetic algorithm II (NSGAII) of Deb et al. [2002].

[7] 4. Deterministic equivalent—Adding safety margins (redundancy) to the constraints or to the uncertain variables resulting in a robust deterministic equivalent formulation for the uncertain problem. The deterministic equivalent is then 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 stochastic oriented approaches are: (a) The uncertain data are assumed to have a known PDF, solutions based on assumed distributions may be unjustified. A drawback is that the PDF itself (i.e., the shape of the PDF) must be recognized as being uncertain. (b) The size of the optimization problem is substantially increased when using stochastic oriented approaches. This is because the PDFs, which represent the uncertain demands, are assimilated through a large number of sampled scenarios. This drawback is further emphasized when simulation-based optimization techniques, known for their computational complexity, are utilized.

[9] This study proposes formulating a deterministic equivalent of the uncertain problem of optimal design of water distribution systems, namely, a nonprobabilistic robust counterpart (RC) [Ben-Tal and Nemirovski, 1999]. The data uncertainty is quantified by a deterministic ellipsoidal uncertainty set with the decision maker seeking a solution that is robust optimal for all possible scenarios in the uncertainty set. The robust counterpart uses some characteristics of data distribution instead of using full distribution information of the uncertain variables. The method does not require the construction of a representative sample of scenarios, and has the same size as the original model. RC thus copes with the drawbacks of the stochastic oriented approaches discussed above.

[10] RC for the optimal design of water distribution systems under demand uncertainty was suggested by Perelman et al. [2013]. Perelman et al. [2013] noted some drawbacks of their proposed RC model which are further addressed in this study.

[11] Typically, the optimization problem is solved assuming that nodal demands are satisfied for a given pipes' diameters selection. The robustness of the design, on the other hand, is being computed thereafter the optimization stage based on violation of the hydraulic heads through hydraulic 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 uncertainty effect system reliability. In addition, although the RC model [Ben-Tal and Nemirovski, 1999] incorporates correlated uncertainty, its straightforward application to water distribution system optimal design introduces a degenerate case in which the dependency is not apparent, as discussed in Perelman et al. [2013].

[13] This work explores both the straight forward application of the RC approach and develops an adaptive approach to the water distribution systems least cost design problem.

[14] As the RC approach was developed for linear uncertain optimization problems, correlated data uncertainty model, and explicit head constraints formulation are included through linearization of the headloss equation. The resulting equivalent deterministic problem is solved using the cross-entropy (CE) optimization method [Rubinstein, 1999]. CE was successfully applied to deterministic single and multiobjective optimal design of water distribution systems [Perelman et al., 2008].

[15] The following sections describe the RC formulation for linear uncertain problems, formulation of the uncorrelated and correlated models of data uncertainty for the least cost design of water distribution systems, and application and comparison of the proposed approaches.

2. Robust Formulation of Linear Programming Problems

[16] The uncertain linear optimization problem in a constraint-wise form can be formulated as:

display math(1)
display math(2)

where inline image is a vector of decision variables including the right-hand side, c is a vector of objective function parameters, inline image is a vector of uncertain parameters for the ith constraint, Ui are uncertainty sets, and inline image is the right-hand side. The overall uncertainty set of all uncertain parameters in the problem is defined as U where each Ui is taken as the projection of U along its corresponding dimensions [Ben-Tal and Nemirovski, 1999].

[17] For a particular constraint i, let J represent the set of coefficients that are subject to uncertainty, then each inline image lays in the interval inline image, where inline image is the deviation from the nominal value aij. Ben-Tal and Nemirovski [1999] proposed to model the uncertainty with ellipsoidal uncertainty. The advantages of the ellipsoidal uncertainty are: (a) it incorporates correlations between parameters while controlling the distance from their nominal values, and (b) it is less conservative than the box model uncertainty, which assumes that all parameters are at their worst values simultaneously [Soyster, 1973]. The reader is refereed to Housh et al. [2011] for further details on the use of ellipsoidal uncertainty sets.

[18] For uncertain parameter inline image, with nominal value aij and covariance matrix ∑, the ellipsoidal uncertainty set is written using the Mahalanobis distance in the form:

display math(3)

[19] The ellipsoidal uncertainty set can be described as an affine mapping of the random uncertain variable by setting inline image, inline image and inline image. Equation (3) can be rewritten as:

display math(4)

where inline image, P is the mapping matrix which can be computed by Cholesky decomposition of the covariance matrix, u is the perturbation vector, inline image is the Euclidean norm, and Ω is a value controlling the size of the ellipsoid and is also referred to as the protection level.

[20] To derive the robust equivalent the lower bound of the uncertain constraint in equation (2) needs to be nonnegative with the mapping derived from equation (4). For the ith uncertain constraint:

display math(5)

where Pi is the ith mapping matrix, corresponding to the ith constraint.

[21] The analytical solution to the problem can be attained trough Lagrangian duality [Ben-Tal and Nemirovski, 1999] and the robust equivalent of the uncertain constraint (equation (5)) equals:

display math(6)

[22] Finally, the deterministic equivalent problem of the uncertain linear problem in equations (1) and (2) can be formulated as:

display math(7)
display math(8)

[23] Although the deterministic equivalent is of a second-order conic form, as opposed to the linear original problem, this still a convex tractable formulation which could be solved efficiently. Moreover, this formulation does not require sampling of multiple scenarios as required by a stochastic uncertainty model.

2.1. Probability Bounds

[24] The price of robustness is controlled by the parameter Ω (i.e., it controls the trade-off between the probability of violation of the uncertain constraint and the cost of the objective function). Under the model uncertainty inline image for inline image, the probability that the uncertain constraint is violated, is bounded by a wide family of PDFs [Ben-Tal et al., 2009]:

display math(9)

[25] In other words, to guarantee a probability 1−ε of maintaining the constraint, the parameter of protection level should be selected as inline image. For example, for a 0.95 probability guarantee (i.e., inline image), select inline image.

3. Water Distribution System Least Cost Design

[26] The least cost design problem of a water distribution system is to find its minimum cost as a function of pipes' discrete diameters as decision variables and lengths (equation (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 and energy constraints in a matrix form:

display math(10)
display math(11)
display math(12)
display math(13)

where fC is the cost function, D pipes' diameters, L pipes' lengths, inline image is a topological matrix with elements of the ith row equal to inline image depending on the network connectivity, A11 is a diagonal matrix with nonlinear elements representing pipe's resistance, Q and H are unknown flows and heads, q are the consumer's demands, and Hmin are minimum desired nodal heads.

[27] The term A11Q in equation (12) represents the headloss equation of the form:

display math(14)

where inline image, C pipe's friction coefficient, inline image, inline image, and e3 is a unit-dependent coefficient for the Hazen-Williams equation.

4. Data Uncertainty and Robust Formulation

[28] In this work, the uncertainty is assumed to be in the consumers' demands and the optimization problem is solved to find the least cost design. The water supply system includes a nonlinear objective function, and nonlinear and linear constraints. However, all constraints with demand uncertainty are linear. Two principal models are explored and compared: the uncorrelated uncertainty model and implicit design reliability, and the correlated uncertainty model and explicit design reliability.

4.1. Uncorrelated Model of Data Uncertainty

[29] Considering uncertainty in the demands inline image, the linear mass balance constraint (equation (11)) is uncertain with only the right-hand side parameters being uncertain. Then, the deterministic equivalent of the mass balance equation according to equation (8) is:

display math(15)
display math(16)

[30] Equation (11) is replaced by equation (15) in the RC formulation. The mapping matrix P can be computed by Cholesky decomposition of the covariance matrix inline image of the demands. Each mass balance constraint contains only one demand. Hence, for each ith constraint Pi compromises one vector and inline image, where σi is the standard deviation of the ith uncertain demand. Thus, the robust equivalent optimization problem is:

display math(17)
display math(18)
display math(19)
display math(20)

where σ is a vector of all standard deviations.

[31] The resulting robust equivalent problem resembles the original problem (equations (10)-(14)) with a safety factor added to all uncertain demands. This formulation, previously suggested by Perelman et al. [2013], presents two main drawbacks: (a) the correlation between different demand nodes does not influence the optimal design (i.e., this is evident, since the optimization problem above is only a function of the standard deviations and the mean demands), and (b) only the linear water balance equations are replaced with their deterministic equivalents.

[32] In this study, the robust model is further developed to include correlated uncertain data through explicit formulation of the demands in head constraints. The subsequent sections describe the correlated model of data uncertainty followed by a comparative study of the uncorrelated versus the correlated model.

4.2. Correlated Model of Data Uncertainty

[33] To explicitly formulate demand uncertainty in the robust formulation, the hydraulic head loss function (equation (14)) is expressed in a linear form. Two linearization forms are considered herein:

[34] 1. Linearization around an operating point Q0 such that the line is tangent to the headloss curve at the operating point:

display math(21)

[35] 2. Linearization of the operating domain [Q1, Q2]:

display math(22)

[36] Figure 1 graphically demonstrates the nonlinear, linear with a single point, and linear with two operating points headloss as a function of flow. It can be seen that model Linear I (1 point) underestimates headloss in pipe and consequently overestimates nodal heads. Model Linear II (2 points) overestimates headloss in pipe within the specified domain and thus underestimates nodal heads, and contrary-outside the specified domain.

Figure 1.

Head loss models—nonlinear (Hazen Williams), Linear I (1 point), and Linear II (2 points).

[37] Next equation (11) is replaced by its linear version (equation (21) or (22)) and a full-rank linear system of equations can be solved instead of equations (11) and (12):

display math(23)

where inline image is the inverse of the block matrix in equation (23), K11 is of size inline image, and K12 of size inline image and inline image, h is a given vector corresponding to the fixed head nodes.

[38] The nodal heads H can be computed as:

display math(24)

[39] Following this formulation, the minimum head constraints in equation (13) can be explicitly expressed as a linear function of the demands q.

[40] Next, considering uncertainty in the demands inline image, the linear optimization problem with uncertainty in demands takes the form:

display math(25)
display math(26)
display math(27)

where inline image and inline image are a function of linearization model and pipe diameters.

[41] The attained optimization problem (equations (25)-(27)) follows the formulation of equations (1) and (2) and its robust equivalent problem can be formulated according to equation (8). The minimum head constraint for node i in the system is:

display math(28)

where K11,i and K12,i are the ith row in the matrices K11 and K12, respectively. Hmin,i is the minimum head constraint at node i.

[42] The deterministic equivalent is:

display math(29)

[43] Finally, the robust design of a water distribution system is the solution of the following optimization problem:

display math(30)
display math(31)

[44] The formulation of the optimization problem for the least cost design of water distribution system under demand uncertainty derived in equations (30) and (31) incorporates both the correlations between the consumers' demands, and it explicitly relates uncertainty in demands to the resulted nodal 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 counterparts) is performed on the water balance equations (Kirchoff's first low) resulting in the deterministic equivalent formulation in equation (18). It is evident that the optimization problem in equations (17)-(20) is only a function of the standard deviations and the mean demands and thus does not capture the correlation between different demand nodes.

[46] Correlation assimilation in the design phase is most important to achieve an efficient and reliable solution as will be demonstrated next in the examples. Another considerable drawback is that the robustification is performed only on the water balance and not on the energy balance. The demand uncertainty propagates through the system equations (i.e., through Kirchoff laws 1 and 2) resulting in an uncertain heads in the system. The minimum head constraint is then uncertain by itself and need to be replaced by its robust counterpart as suggested by the methodology presented in the previous section. Next, it will be shown that omitting the explicit dependency between the demand and the head results in overconservative solutions.

[47] Consider the robust counterpart of the uncertain head requirement constraint (with explicit demand correlation) derived in equation (29). To show that formulation (29) is less conservative robust counterpart than formulation (18), we need to prove that the left-hand side of equation (29) is larger than the one obtained in formulation (18) even if no-correlation is present in the problem. Thus, the correlation effect is filtered out and the difference is entirely attributed to the lack of explicit dependency between the demand and the head.

[48] The linearization of formulation (18) considering the robust counterpart of the water balance results in:

display math(32)

[49] It needs to be shown that:

display math(33)

when no-correlation is present [i.e., when PT = diag(σ)].

[50] All element of inline image are negative. This could be seen by noting that K12,i is multiplied by the demand in the head equation, and increasing the demand results in decease in head (see also Appendix Appendix for an example of the negativity of K12,i).

[51] Relying on the above, inequality (33) could be rewritten as:

display math(34)

where inline image is the L1 norm defined as inline image. As inline image, it is shown (Appendix Appendix) that the robust counterpart presented herein is less conservative than the one suggested in Perelman et al. [2013].

5. Applications

[52] The proposed method was applied on three growing complexity networks: (1) an illustrative example [Boulos et al., 2006], (2) Hanoi system [Fujiwara and Khang [1990], and (3) large network (adapted from Alperovits and Shamir [1977]). The following notions were used (metadata and all utilized source codes are attached as supporting information):

5.1. Robust Optimization Models

[53] Three uncertainty models were explored and applied:

[54] 1. Uncorrelated implicit nonlinear—based on the problem formulated in equations (17)-(20). This model does not consider correlation between uncertain data and assigns all uncertain demands a safety factor. This model can thus be considered as a redundancy based formulation.

[55] 2. Correlated explicit Linear I (1 point)—based on the problem formulated in equations (30) and (31) with a single point linearization of the head loss equation (21). This model underestimates headloss in individual pipes and thus typically overestimates nodal heads (depending on the combination of all network pipes). Thus, this model is expected to be less conservative than the two points linear model (below). One solution of the system's nonlinear equations is required to compute model parameters at operating point [Q0].

[56] 3. Correlated explicit Linear II (2 points)—this model is as well based on the formulation in equations (30) and (31) with operating range considered for the headloss linearization in equation (22). It overestimates headloss in individual pipes and underestimates nodal heads. Thus, this model is expected to be more conservative than the one point linear model. Two solutions of the system's nonlinear equations are required to compute model parameters at the two operating points [Q1, Q2].

5.2. Optimization Technique

[57] The nonlinear discrete optimization problem was solved using the cross-entropy (CE) method for combinatorial optimization [Rubinstein, 1999; Rubinstein and Kroese, 2004].

[58] The CE algorithm is a two-stage iterative procedure involving: (1) generation of random data solutions, and (2) updating of the parameters of the problem on the basis of the sampled data in the direction of solution improvements. The cross-entropy scheme seeks to find optimal probability such that the Kullback-Leibler distance [Kullback and Leibler, 1951] between the sampling probability and the theoretical optimal probability is minimal. The CE algorithm parameters are the sample size N defining the number of samples in each iteration, the elite sample percentage β defining the best set of solutions used for updating sampling probability, and the smoothing parameter α used to prevent the algorithm from converging prematurely. Parameter values are generally set through model calibration.

[59] A full description of the CE method and its application to single and multiobjective optimal design of water distribution systems can be found in Perelman et al. [2008]. Appendix Appendix provides a small illustrative example of the cross-entropy method.

5.3. Example 1—Illustrative

[60] The illustrative example was adopted from Boulos et al. [2006] and is used for demonstrating purposes. The network consists of a single constant head source, three demand nodes, and four pipes. The length and friction coefficient of all links and elevation of all nodes are 304.8 (m), 100, and 0 (m), respectively. The available diameters considered in this example are (20.32, 25.4, 30.48, 35.56, 40.64) (cm) (i.e., 54 = 625 solution space) with corresponding costs (23, 32, 50, 60, 90) ($/m). The layout of the network and base demands are shown in Figure 2. For the ease of representation the demand at node 2 is considered to be constant and equals to zero.

Figure 2.

Example 1—layout.

[61] The demands at nodes 3 and 4 are considered to be uncertain with nominal demands, standard deviations, and covariance matrix correlation coefficients:

display math(35)

[62] Given the small problem size, all possible solutions were enumerated, without the need for using the CE method. The optimal designs were computed for different values of protection level inline image for the three models: (1) uncorrelated implicit nonlinear, (2) correlated explicit Linear I (1 point) with inline image for each pipe i, and (3) correlated explicit Linear II (2 points) with inline image for each pipe i.

[63] The robust designs were tested on three types of data uncertainty: (a) positive correlation (PC) between consumers' demands, (b) zero correlation (ZC) between consumers' demands, and (c) negative correlation (NC) between consumers' demands. For example, positively correlated representing similar types of consumers (e.g., domestic), zero correlated—random demands, and negatively correlated—different types of consumers (e.g., domestic and industry). Particularly, the correlation was set to inline image representing positively, zero, and negatively correlated consumers, respectively. Solution robustness was evaluated as the probability of head constraint violation based on 1000 Monte Carlo samples assuming uniform PDF with given above mean and standard deviation.

[64] Figures 3-5 demonstrate the price of robustness as a function of protection level Ω for all models. Figures 3-5a show the theoretical and simulated probability of head constraint violation as a function of protection level Ω. Figures 3-5b show the cost of designs as a function of Ω. From the figures, it can be seen that as protection level increases, probability of head constraint violation decreases and design cost increases.

Figure 3.

Example 1—(a) head constraint violation probability and (b) cost versus protection level Ω for positively correlated demands ρ = 0.8.

Figure 4.

Example 1—(a) head constraint violation probability and (b) cost versus protection level Ω for zero correlated demands ρ = 0.

Figure 5.

Example 1—(a) head constraint violation probability and (b) cost versus protection level Ω for negatively correlated demands ρ = −0.8.

[65] The robust design of the two linear models is significantly affected by the relationship between the demands adapting the optimal design and its cost to the model uncertainty. It is also evident, as expected, that the cost of the designs for negative correlation are lower than for zero correlation, and those in turn are lower than positive correlation. This performance is not evident in the nonlinear model which does not account for correlation of the uncertain data and results in most costly designs. In all three cases (PC, ZC, and NC) uncorrelated implicit Linear I (1 point) model results in less conservative designs than the correlated explicit Linear II (2 points) model. This was explained previously by the overestimation of nodal heads of the 2 points model. In case of positively correlated demands, as shown in Figure 3a, the correlated model is even more conservative than the uncorrelated model for several values of Ω. Additionally, it can be seen from Figures 3-5 that the probability of head constraint violation is lower than the theoretical bound. However, the slackness of the probability bound can be attributed to the discrete nature of the problem, the given available diameters, and the intrinsic robustness of the design for reduced demands.

[66] To estimate the overall error Herror resulting from the linearization in the formulation of the deterministic equivalent, the robust head at each node was compared to the nonlinear headloss given the robust demand by:

display math(36)

where inline image is the robust equivalent head at node i, inline image is the robust equivalent demands to seek robust equivalent head at node i, and H(qR) is the nonlinear head loss given demand qR.

[67] The proximity of the optimal solution to minimum head constraint, Hslack, was evaluated as:

display math(37)

[68] Table 1 lists the errors in nodal heads Herror and the distance of the optimal solution from the minimum head constraint Hslack for the three correlation values and maximum protection level Ω = 2. Positive values of Herror indicate that the linear model overestimates nodal heads and negative values—underestimates. Low positive values of Hslack indicate that the optimal design is close to the minimum head constraint. From the results, it can be seen that, in this example, the nodal heads calculated by correlated explicit Linear II (2 points) model are always slightly underestimated, 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 head constraint for negative correlation between demands, and farther for the other two cases. This can be attributed to the discrete nature of the problem and the given available diameters.

Table 1. Example 1—Robust Design Analysis for Ω = 2
ModelNodesCorrelation Between Demand Zones
ρ = 0.8ρ = 0ρ = −0.8
Herror (m)Hslack (m)Herror (m)Hslack (m)Herror (m)Hslack (m)
Linear I (1 point)10.0543.1330.0543.1330.0543.133
20.0602.4150.0681.9860.1020.509
30.1422.1340.0551.9180.0320.472
Linear II (2 points)1−0.0644.458−0.1232.957−0.1232.957
2−0.0482.029−0.1681.750−0.2200.188
3−0.1972.795−0.2101.652−0.3320.110

[69] The reliability of the designs was further evaluated against all three test sets and the results for correlated explicit Linear II (2 points) model are shown in Table 2. As expected, design based on positive correlation has the highest reliability in all cases at the expense of its cost, as opposed to design based on negative correlation.

Table 2. Head Constraint Reliability for Ω = 2
 Design Based on CorrelationRealized
Linear II (2 points)ρ = 0.8ρ = 0ρ = −0.8Correlation
Cost ($)62.48458.826450.292 
Reliability10.9650.748ρ = 0.8
10.9940.845ρ = 0
110.989ρ = −0.8

5.4. Example 2—Hanoi Water Distribution System

[70] The Hanoi network is a relatively large gravitational system introduced by Fujiwara and Khang [1990]. The network (Figure 6) is subject to one demand loading condition, and consists of 34 links and 32 demand nodes supplied by a single reservoir at a constant head of +100 (m). All nodes are at zero elevation and the minimum pressure head requirement is 30 (m) at all nodes. Six candidate pipe diameters (30.48, 40.64, 50.8, 60.96, 76.2, 101.6) (cm) with a Hazen-Williams coefficient of 130 (−) are considered for each pipe. The full data for this example can be found in Centre for Water Systems (CWS) [2001]. The cost ($) of installing a pipe of diameter D (mm) and length L (m) is:

display math(38)
Figure 6.

Hanoi water distribution system layout.

[71] To model uncertainty in consumers' demands, system nodes were partitioned to three demand zones: zone 1—nodes 1:15, zone 2—16:24, zone 3—25:32 (Figure 6). Demands in zone 2 were assumed to be certain and in zones 1 and 3 uncertain with standard deviation of 12% from mean demand of each zone [i.e., 80 and 50 (m3/h)], respectively. The problem was again formulated and solved for different values of protection level inline image for the three models: (1) uncorrelated implicit nonlinear, (2) correlated explicit Linear I (1 point) with inline image for each pipe i, and (3) correlated explicit Linear II (2 points) with inline image for each pipe i. Additionally, intrazone correlation was set to ρ = 0.8 meaning that consumers in the same zone follow similar demand pattern. Interzone correlation was set to inline image again to represent positive, zero, and negative correlation between different zones in the network.

[72] The cross-entropy optimization method was used to find the optimal design for each model and each value of Ω. The CE parameters set for all runs were: sample size N = 10,000, elite sample percentage β = 0.005, and smoothing parameter α = 0.6. The algorithm converged after 16 iterations on average with average running times of: (1) uncorrelated implicit nonlinear model ∼160 (s), (2) correlated explicit Linear I (1 point) ∼ 200 (s), and (3) correlated 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 a function of protection level Ω for all models. Figures 7-9a show the theoretical and simulated probability of head constraint violation as a function of protection level Ω. Theoretical probability of head constraint violation was computed based on equation (9) and the actual probability was calculated based on 1000 Monte Carlo samples assuming uniform PDF with given above means and standard deviations. Again, it can be seen that the probability of head constraint violation are lower than the theoretical bound.

Figure 7.

Hanoi—(a) head constraint violation probability and (b) cost versus protection level Ω for positively correlated demands ρ = 0.6.

Figure 8.

Hanoi—(a) head constraint violation probability and (b) cost versus protection level Ω for zero correlated demands ρ = 0.

Figure 9.

Hanoi—(a) head constraint violation probability and (b) cost versus protection level Ω for negatively correlated demands ρ = −0.6.

[74] Figures 7-9b show the cost of designs as a function of Ω. It can be seen that the nonlinear model not considering the correlations between demands resulted in a more conservative design than taking correlations under consideration. This becomes more evident for the zero and negative correlation uncertainty models. As in the previous example, the linear model based on a single operating point is less conservative than the two point linear model.

[75] Table 3 lists the errors in nodal heads Herror (equation (36)) and the distance of the optimal solution from the minimum head constraint Hslack (equation (37)) for the maximum protection level Ω = 2 and positive correlation ρ = 0.6. The results demonstrate that the nodal heads calculated by correlated explicit Linear II (2 points) model are always slightly underestimated and always slightly overestimated by correlated explicit Linear I (1 point) model. Thus, the two point linear model is always more expensive than the one point linear model. This can be observed for all runs (Figures 7-9a).

Table 3. Hanoi—Robust Design Analysis for Ω = 2 and ρ = 0.6
 Linear I (1 point)Linear II (2 points)
NodeHerror (m)Hslack (m)Herror (m)Hslack (m)
10.0166.72−0.0166.70
20.1726.30−0.1725.95
30.2223.14−0.1922.39
40.2619.38−0.2118.16
50.2915.78−0.2314.04
60.2915.06−0.2413.19
70.3212.62−0.2412.42
80.3211.19−0.2610.45
90.309.24−0.279.44
100.328.71−0.278.89
110.357.15−0.287.30
120.375.33−0.285.45
130.2410.17−0.3111.29
140.1811.31−0.3112.18
150.0812.98−0.3113.54
160.1217.67−0.2517.86
170.1421.87−0.2021.77
180.1624.80−0.1824.54
190.2318.64−0.1418.59
200.239.30−0.149.25
210.238.03−0.147.98
220.2315.19−0.1415.40
230.2313.19−0.1413.52
240.2110.76−0.1711.36
250.0510.72−0.2511.66
260.0611.54−0.2611.81
270.1612.09−0.1712.99
280.1410.19−0.259.07
290.158.56−0.248.84
300.158.66−0.248.91
310.169.09−0.219.52

5.5. Example 3—Large Network

[76] In this example, a more challenging water distribution system is introduced based on a real water distribution system example application described in Alperovits and Shamir [1977]. Uncertainty is incorporated in all demand nodes considering a fully heterogeneous covariance matrix.

[77] The water distribution system layout is shown in Figure 10. The network consists of 65 links, and 52 demand nodes. The two pumping stations were replaced with a constant head reservoir of +408 (m) representing pump head during the loading condition. The elevation varies across the nodes while the minimum pressure head requirement is 30 (m) at all nodes [Alperovits and Shamir, 1977, Table 9d]. Eleven candidate pipe diameters are considered all with a Hazen-Williams coefficient of 130 (−) and capital costs as described in Alperovits and Shamir [1977, Table 9c]. The EPANET [2012] input data file for this example is attached as supporting information.

Figure 10.

Network layout for Example 3 based on Alperovits and Shamir [1977].

[78] Demands at all nodes are considered uncertain. The mean of the demand is equal to the base demand as given in Table 9d in Alperovits and Shamir [1977]. The standard deviation of the demand is set to 20% of the mean demand for each node. Hence, unlike the second example in which the standard deviation of the demand is 50 or 80 (m3/h), in this example each demand node has different standard deviations. Moreover, the covariance between the demand nodes is fully heterogeneous with correlation ranging between −0.3 and 0.8, as graphically depicted in Figure 11.

Figure 11.

Correlation matrix for the base run of Example 3.

[79] The network is solved using the correlated explicit Linear I model and the deterministic equivalent is solved using the cross-entropy optimization method with the same parameters described in Example 2, except for the sample size which is set to N = 50,000 due to the increased size of the problem.

[80] As the RC methodology requires only mean demands, standard deviations, and correlation matrix to solve the uncertain optimization problem, it was interesting to test the reliability of the attained designs on different PDF's. The reliability was empirically calculated by Monte Carlo simulation based on three different distributions: (a) marginal uniform, (b) multivariate normal, and (c) transformed uniform (i.e., independent uniform distributions which are linearly transformed to give the desired mean vector and covariance matrix). Each of these three distributions has the same mean and standard deviation vectors and correlation matrices, as indicated in the problem description.

[81] Figure 12 shows the robust design cost versus reliability trade-off using 1000 Monte Carlo samples for the three distributions above. The close trade-off curves indicate that the design suggested by the model is almost indifferent (i.e., robust) to different PDFs. Thus, no matter what probability distribution is realized (out of the three above), the design is still satisfactory as it gives very similar reliability outcomes.

Figure 12.

Reliability versus cost trade-off curve of the robust design under different PDF's, for Example 3.

5.6. Importance of Demand Correlation Assimilation

[82] The main contribution of the present study over the robust design approach developed previously by the authors in Perlman et al. [2013] is the ability to account for different correlations between demand nodes. Next it is shown that including correlation information in the design phase has a substantial advantage in seeking more efficient robust solutions.

[83] The same means and standard deviations are considered, as described in Example 3, but with different correlation matrices. The problem is first solved with a zero-correlation assumption and then compared to correlation based design in terms of reliability and design costs.

[84] Figure 13 presents the correlation matrices for: (a) zero-correlation demands, (b) positively correlated demands, (c) mixed positively and negatively correlated demands with a correlation range of −0.6 to 0.6, and (d) two zones homogeneous demands.

Figure 13.

Correlation matrices for Example 3 for (a) zero-correlation, (b) positive correlation, (c) mixed correlation, and (d) two zones.

[85] In the two zones correlation matrix (Figure 13d) the largest four water consumers (nodes 5, 7, 13, and 22) are considered in one zone which is negatively correlated (correlation of −0.6) with all others nodes (second zone). The interzone correlation is positive and is set to 0.6 for all nodes.

[86] To show the relative advantage of the correlation assimilation in the robust design, the performances between zero-correlated and correlated designs are compared. Setting a reliability target of 95% the correlated explicit Linear I model formulation is then solved to achieve the reliability target under each of the correlation cases in Figure 13, and under the base run correlation in Figure 11.

[87] Figure 14 shows the performance comparison between the zero-correlated and the correlated designs. Design that uses zero-correlation information has the same design cost for all correlations, while the design cost for correlation based design is changing as a function of the correlation.

Figure 14.

Cost and reliability comparison between zero and different correlation based designs for Example 3.

[88] For example, the first two groups of columns in Figure 14 show that the design cost to reach 95% reliability in the base run correlation (Figure 11) is 4.38 (M$), whilst if the correlation in Figure 13a is assumed, then the design cost is 4.43 (M$). The design cost based on zero-correlation is the same regardless of the correlation, and it is equal to 4.44 (M$). The reliability however, is different as shown in Figure 14. A slightly higher reliability is obtained if the design is based on zero-correlation, for the cases of the base run and mixed correlation.

[89] On the one hand, the correlation based design in the base run and mixed correlation reduces the cost while meeting the targeted reliability of 95%. However, on the other hand, the correlation based design increases the cost in the two zones and the positive correlation cases compared 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 of correlation are realized, as reliability will be substantially reduced (Figure 14).

[90] The above results highlight the importance of the consumer's demand correlation incorporation in the design phase of water distribution systems. Omitting the correlation information may result in unnecessarily more expensive design or even more hazardous situations in which significantly lower than expected reliability is obtained.

6. Conclusions

[91] The proposed work suggests using a deterministic equivalent to the uncertain problem of optimal design under demand uncertainty. Three robust optimization models were explored—one not considering dependence in data uncertainty and two-dependent uncertainty models. All three models were further tested and compared on different data uncertainty models representing different type of consumers in a water distribution system, including positive, zero, negatively, and fully heterogeneous correlations between consumers.

[92] The presented method shows four main strengths: (1) Inclusion of correlated uncertain data. The results showed that not considering uncertainty dependency leads to more conservative and thus costly designs. (2) Explicit head constraint formulation by means of linearization of the nonlinear headloss equation. The two linear models explored showed consistency in their results and performances. (3) The explicit head constraint formulation allowed to provide theoretical probability bounds for violating this constraint. The slackness of the probability bound can be attributed to the intrinsic robustness of the design due to the discrete nature of the problem, the given available diameters, and that uncertainty in demands can also result in reduced demands, in which case design robustness remains intact. (4) System robustness is monitored through a single parameter Ω controlling the trade-off between the probability of violation of the uncertain constraint and the cost of the objective function.

[93] Further research will seek to incorporate into robust optimization 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 heuristic search technique which utilizes probabilities of possible outcomes of decision variables instead of the actual values.

[95] For example, if a decision variable x can receive the values x = {1, 2, 3}, then an associate probability of p(x) = {p(1), p(2), p(3)} can be defined where each element defines the probability of receiving the actual associated value. For example, p = {½, 0, ½} means that there is 50% chance of the decision variable to get the values of one or three, and zero for receiving two. Obviously, this does not help much in getting the optimal value x*, so the optimal probability should be degenerated or close to it. For example, p(x*) = {0, 0, 1} means that the optimal solution is x* = 3.

[96] The CE algorithm starts with some user defined initial probability piter = 0 (it can be from any family of PDFs associated with the underlying decision space) and is evolving iteratively until convergence (i.e., until p ∈ {0, 1}).

[97] In each iteration a new sample of solutions is generated based on the current probability piter, and the probability is updated based on counting the frequencies of each decision value in a user defined elite set of solutions. Additionally, to prevent premature convergence the probability is smoothed by taking a weighted average of the probabilities from the last two iterations:

display math(A1)

[98] For example (Table A1), consider six possible (N = 6) values for a decision variable x, p0 as a uniform initial probability, and S(x) as the performance function of the decision variable x. The goal is to find x that maximizes S(x). In each iteration three solutions are sampled and the probability 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 x = 1, 2, 4 were sampled once). The probability is updated based on the elite two solutions (i.e., those that were sampled, S (x = 2) = 4 and S (x = 4) = 16). The updated probability (without smoothing) is ½ for these two solutions, x = 2 and x = 4 and zero for the remaining values of x = {1, 3, 5, 6}. Without the smoothing, these solutions can never be sampled again. With the smoothing, for example α = 0.6, their probability is lower than the initial, but is not zero and those solutions can be sampled again. In the next iteration the process is repeated with the new probability p = {0.067, 0.367, 0.067, 0.367, 0.067, 0.067}, and so on.

Table A1. Simple Cross-Entropy Example
Xp0(x)S(X)F(X)p1(x) inline image
11/6110 inline image
21/6411/2 inline image
31/6900 inline image
41/61611/2 inline image
51/62500 inline image
61/63600 inline image

Appendix B

[99] Proof that inline image for any x. We use the definitions of the L1 and the L2 norms to write the problem in index form as:

display math(B1)

[100] We then prove the above by induction as follows:

[101] Step 1. Base case for n = 2

display math(B2)

[102] Step 2. Assume that (B1) is true for n = m

display math(B3)

[103] Step 3. Show that the inequality holds for n = m + 1

display math(B4)

[104] Since both terms in the right-hand side are positive the inequality holds.

B1. Negativity of the K12 Elements

[105] Here the matrix K12 is demonstrated for the illustrative example in Figure 2. It is shown that all elements of the matrix are negative as assumed in the proof of the overconservativeness.

[106] The matrix inline image is the inverse matrix of inline image. For the illustrative examples the G matrix reads:

display math

[107] After inversing the matrix the block matrix inline image in the upper right corner of the inverse matrix K reads:

display math

[108] Recalling that the elements Bi correspond to the slope of the linear approximation for the Hazen-Williams equation, and recalling that this equation is convex, thus inline image which implies that all elements in K12 above are negative.

Acknowledgment

[109] This research was supported by the Technion Grand Water Research Institute.

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