Energy-optimal pump scheduling and water flow

This paper focuses on the optimal operation of water supply networks. We model water supply networks using hydraulic constraints, and formulate a joint optimal pump scheduling and water flow problem (OWF) using the hydraulic characteristics of variable speed pumps. OWF is a mixed-integer nonlinear program. This problem is nonconvex, and hence NP-hard. To compute an exact solution of OWF, we first focus on the feasibility region of OWF, and propose a mixed-integer second-order cone relaxation for the feasibility region of OWF. We prove that the proposed relaxation is exact for several relevant network topologies. We then focus on the objective function in OWF, and show that for some energy metrics, OWF can be transformed into a mixed-integer second-order cone program. Furthermore, we propose an ADMM-based algorithm to compute suboptimal solutions to OWF and lower bounds on the optimal value of the objective in OWF when the objective function is nonconvex. Finally, we consider a real-world water network, and demonstrate the effectiveness of the proposed relaxation in computing the optimal pump schedules and water flows.