Optimal Design-for-Control of Water Distribution Networks via Convex Relaxation

Abstract

This paper considers joint design-for-control problems in water distribution networks (WDNs), where locations and operational settings of control actuators are simultaneously optimized. We study two classes of optimal design-for-control problems, with the objectives of controlling pressure and managing drinking-water quality. First, we formulate the problem of optimal placement and operation of valves in water networks with the objective of minimizing average zone pressure, while satisfying minimum service requirements. The resulting mixed-integer non-linear optimization problem includes binary variables representing the unknown valve locations, and continuous variables modelling the valves’ operational settings. In addition, water utilities aim to maintain optimal target chlorine concentrations, sufficient to prevent microbial contamination, without affecting water taste and odour, or causing growth of disinfectant by-products. We consider the problem of optimal placement and operation of chlorine booster stations, which reapply disinfectant at selected locations within WDNs. The objective is to minimize deviations from target chlorine concentrations, while satisfying lower and upper bounds on the levels of chlorine residuals. The problem formulation includes discretized linear PDEs modelling advective transport of chlorine concentrations along network pipes. Moreover, binary variables model the placement of chlorine boosters, while continuous variables include the boosters’ operational settings. Computing an exact solution for the considered mixed-integer optimization problems can be computationally impractical when large water network models are considered. We investigate scalable heuristic methods to enable the solution of optimal design-for-control problems in large WDNs. As a first step, we solve a convex relaxation of the considered mixed-integer optimization problem. Then, starting from the relaxed solution, we implement randomization and local search to generate candidate design configurations. Each configuration is evaluated by implementing continuous optimization methods to optimize the actuators’ control settings and compute feasible solutions for the mixed-integer optimization problem. Moreover, the solution of the convex relaxation yields a lower bound to the optimal value of the original problem, resulting in worst-case estimates on the level of sub-optimality of the computed solutions We evaluate the considered heuristics to solve problems of optimal placement and operation of valves and chlorine boosters in water networks. As case study, we utilize an operational water network from the UK, with varying sizes and levels of connectivity and complexity. The convex heuristics are shown to generate good-quality feasible solutions in all problem instances with bounds on the optimality gap comparable to the level of uncertainty inherent in hydraulic and water quality models. Future work should investigate the formulation and solution of multiobjective optimization problems for the optimization of pressure and water quality, to evaluate the trade-offs between these two objectives. Moreover, the formulation and solution of robust optimization problems for the design of water networks under uncertainty is the subject of future work.