On the Flow Problem in Water Distribution Networks: Uniqueness and Solvers

Increasing concerns on the security and quality of water distribution systems (WDS), call for computational tools with performance guarantees. To this end, this work revisits the physical laws governing water flow (WF) and provides a hierarchy of solvers of complementary value. Given the water injection or pressure at each WDS node, finding the WFs within pipes and pumps along with the pressures at all WDS nodes, constitutes the WF problem. The latter entails solving a set of (non)-linear equations. We extend uniqueness claims on the solution to the WF equations in setups with multiple fixed-pressure nodes and detailed pump models. For networks without pumps, the WF solution is already known to be the minimizer of a convex function. The latter approach is extended to networks with pumps but not in cycles, through a stitching algorithm. For networks with nonoverlapping cycles, a provably exact convex relaxation of the pressure drop equations yields a mixed-integer quadratically constrained quadratic program (MI-QCQP) solver. A hybrid scheme combining the MI-QCQP with the stitching algorithm can handle a WDS with overlapping cycles, but without pumps on them. Each solver is guaranteed to converge regardless of initialization, as numerically validated on a benchmark WDS.