Global Mixed-Integer Nonlinear Optimization Revisits Optimal Valve Placement in Water Distribution Networks

Abstract

Pressure management through pressure-reducing valves (PRVs) is among the most cost-effective leakage-control measures in water distribution networks, and the underlying placement problem is a nonconvex mixed-integer nonlinear programme. This paper revisits the classical 25-node benchmark of Sterling and Bargiela with two independent tools—a spatial branch-and-bound solver applied to the quadratic-friction MINLP of Eck and Mevissen, and a documented reimplementation of a published Harmony Search method—validating every candidate solution by exact EPANET resimulation. Both find placements better than the one published for this benchmark; the best-known placement attains 705.7 m of total nodal pressure against 722.8 m, and supplying the metaheuristic incumbent to the exact solver tightens the optimality gap from 6.4% to 4.5%, so the best-known solution is provably within 4.5% of the global optimum of the quadratic model, with the residual gap traced to the relaxation of the nonconvex energy constraints. The study also isolates, through a dedicated control experiment, a tolerance effect of broader relevance: placements whose settings are optimal for the approximate model violate the minimum-pressure requirement by centimetres in the exact hydraulics, and rankings can shift by metres; sensitivity analyses on tolerances, setting discretization and demands show that all reported rankings are nevertheless stable. On the 445-node Balerma network, re-optimization with a 2 m safety margin yields a strictly feasible three-valve solution reducing the sum of nodal pressures by 19.6%, with pressures reproduced by EPANET within 3.8%. These results imply that exact resimulation and explicit safety margins on the pressure requirement should become standard steps of model-based valve placement, and that pairing metaheuristic incumbents with global solvers offers a practical route to certified pressure-management decisions.