Constraint preconditioned inexact Newton method for hydraulic simulation of large-scale water distribution networks

Abstract

Many sequential mathematical optimization methods and simulation-based heuristics for optimal control and design of water distribution networks rely on a large number of hydraulic simulations. In this paper, we propose an efficient inexact subspace Newton method for hydraulic analysis of water distribution networks. By using sparse and well-conditioned fundamental null space bases, we solve the nonlinear system of hydraulic equations in a lower-dimensional kernel space of the network incidence matrix. In the inexact framework, the Newton steps are determined by solving the Newton equations only approximately using an iterative linear solver. Since large water network models are inherently badly scaled, a Jacobian regularization is employed to improve the condition number of these linear systems and guarantee positive definiteness. After presenting a convergence analysis of the regularised inexact Newton method, we use the conjugate gradient (CG) method to solve the sparse reduced Newton linear systems. Since CG is not effective without good preconditioners, we propose tailored constraint preconditioners that are computationally cheap because they are based only on invariant properties of the null space linear systems and do not change with flows and pressures. The preconditioners are shown to improve the distribution of eigenvalues of the linear systems and so enable a more efficient use of the CG solver. Since contiguous Newton iterates can have similar solutions, each CG call is warm-started with the solution for a previous Newton iterate to accelerate its convergence rate. Operational network models are used to show the efficacy of the proposed preconditioners and the warm-starting strategy in reducing computational effort.