Multi-Objective Optimization Models for the Design of Water Distribution Networks by Exploring Scenario-Based Approaches

Demand in a water distribution network (WDN) is an aleatory variable, owing to the unpredictable behaviors of water users. Therefore, it is one of the main reasons for uncertainty in the design of this infrastructure. The increasing number of water demand data sets offers opportunities to improve the traditional deterministic design approaches of WDNs by combining statistical and optimization methods. Robust optimization (RO) takes demand uncertainty into account by studying solutions that perform well under any possible demand scenario, that is, any possible realizations of this variable in the lifetime of a WDN. The right choice of scenario is therefore essential to ensure the reliability of the designed network. This paper presents a statistical methodology for generating scenarios to be used to solve a robust design optimization problem. It involves three steps: (a) descriptive analytics of historical data to derive the marginal distributions of peak hour demand in each node of the WDN, (b) generation of a very large number of snapshots by stratified sampling from the correlated marginal distributions of nodal peak demand, (c) generation of the peak demand scenarios by reducing the number of snapshots. Two heuristic techniques are proposed to reduce the number of snapshots, and for each of them, two different numbers of scenarios are derived. Two multi-objective RO models are solved: the first model includes cost minimization and a mean-variance Generalized Resilience and Failure index maximization objectives, and the second one additionally considers the minimization of the maximum undelivered demand, formalized using a regret function.