Tightened Polyhedral Relaxations of a Non-Convex Mixed Integer Program for Optimal Placement and Control of Valves and Chlorine Boosters in Water Networks

In this paper, a new mixed integer nonlinear programming formulation is proposed for optimally placing and operating pressure reducing valves and chlorine booster stations in water distribution networks. The objective is the minimisation of average zone pressure, while penalising deviations from a target chlorine concentration. We propose a tailored solution method based on tightened polyhedral relaxations and a heuristic method to compute good quality feasible solutions, with bounds on their level of sub-optimality. This is because off-the-shelf global optimisation solvers failed to compute feasible solutions for the considered non-convex mixed integer nonlinear program. The implemented methods are evaluated using three benchmarking water networks, and they are shown to outperform off-the-shelf solvers, for these case studies. The proposed heuristic has enabled the computation of good quality feasible solutions in the vast majority of the tested problem instances.