Optimal Water-Power Flow Solutions Using Polyhedral and Conic Relaxations

Water distribution systems (WDSs) and power grids are vital infrastructures that support daily human activities. Substantial research has focused on optimizing the operation of each system individually. However, the power consumption of WDS creates interdependence between these systems, leading to the need for optimizing their conjunctive operation, also known as the optimal water and power flow (OWPF) problem. Combining the WDS optimal operation problem with the optimal power flow (OPF) problem results in a non-convex mixed integer nonlinear programming (MINLP) problem, presenting significant mathematical and computational challenges. Previous studies have used various approximation methods to make the problem convex and obtain feasible solutions. However, these methods often converge to local optima and lack theoretical guarantees of global optimality. Failing to guarantee global optima or provide an optimality gap may result in inconsistent and untrustworthy results for decision makers. This study introduces a tailored solution method for optimizing the conjunctive operation of WDS and power grids. The method uses polyhedral relaxations of the non-convex hydraulic constraints, along with conic relaxations to address nonlinearities in the OPF problem. By using convex relaxations, the method provides optimality gaps for the computed solutions. To validate its effectiveness, the method is tested on two sample applications and its performance is compared with that of an off-the-shelf nonlinear solver.