Explicit demand uncertainty formulation for robust design of water distribution systems

The objective of the least cost design 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 that guarantee its feasibility independent of the data, i.e., under model uncertainty. Typically the uncertainty of the model is assumed to be in the consumers' demands, as opposed to its reliability being computed based on violation of hydraulic heads, resulting in an implicit inclusion of the uncertainty in the optimization model. In this work, uncertainty in the demand is accounted for through explicit formulation of the demands in the mass, head-loos, and minimum head constraints. A robust equivalent (Ben-Tal and Nemirovski, 1998, 1999) incorporating the uncertainty is formulated and solved to optimize the design or rehabilitation of water distribution systems. Explicit uncertainty formulation and tractability of the problem is accomplished through linearization of the head-loss equations for the description of the robust equivalent approach. The uncertain data is described by deterministic ellipsoidal uncertainty sets with a predefined size determined by the decision maker reflecting risk aversion and providing a trade-off between robustness and performance. This work demonstrates the structure, tractability, and flexibility of robust optimization to water distribution systems least cost design.