Exploiting Partial Convexity of Pump Characteristics in Water Network Design

The design of water networks consists of selecting pipe connections and pumps to ensure a given water demand to minimize investment and operating costs. Of particular importance is the modeling of variable speed pumps, which are usually represented by degree two and three polynomials approximating the characteristic diagrams. In total, this yields complex mixed-integer (non-convex) nonlinear programs. This work investigates a reformulation of these characteristic diagrams, eliminating rotating speed variables and determining power usage in terms of volume flow and pressure increase. We characterize when this formulation is convex in the pressure variables. This structural observation is applied to design the water network of a high-rise building in which the piping is tree-shaped. For these problems, the volume flow can only attain finitely many values. We branch on these flow values, eliminating the non-convexities of the characteristic diagrams. Then we apply perspective cuts to strengthen the formulation. Numerical results demonstrate the advantage of the proposed approach.