## 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.