Polynomial optimization for water networks: Global solutions for the valve setting problem

This paper explores polynomial optimization techniques for two formulations of the energy conservation constraint for the valve setting problem in water networks. The sparse hierarchy of semidefinite programing relaxations is used to derive globally optimal bounds for an existing cubic and a new quadratic problem formulation. Both formulations use an approximation for friction loss that has an accuracy consistent with the experimental error of the classical equations. Solutions using the proposed approach are reported on four water networks ranging in size from 4 to 2000 nodes and are compared against a local solver, Ipopt and a global solver, Couenne. Computational results found global solutions using both formulations with the quadratic formulation having better time efficiency due to the reduced degree of the polynomial optimization problem and the sparsity of the constraint matrix. The approaches presented in this paper may also allow global solutions to other water network steady-state optimization problems formulated with continuous variables.