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, Jacobian regularization is employed to improve the condition number of these linear systems and guarantee positive definiteness. After presenting a convergence analysis of the regularized 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.
Highlights
W ATER distribution networks (WDNs) are typically part of an aging infrastructure, which face challenges to efficiently and sufficiently serve a growing population under more stringent economic and environmental constraints
We consider the use of a simple Jacobian regularization technique [23] in the Newton method to prevent the condition numbers from becoming too large or the linear systems becoming semidefinite; we propose appropriate condition number bounds for the regularization that will not negatively affect the convergence properties of the Newton method
The saddle-point structure of the Jacobian in the Newton linear systems is exploited by sparse null-space approaches, which solve the nonlinear hydraulic equations in the kernel space of the mass continuity constraints with less computational resources
Summary
W ATER distribution networks (WDNs) are typically part of an aging infrastructure, which face challenges to efficiently and sufficiently serve a growing population under more stringent economic and environmental constraints. As for many other nonlinear equations, the classical algorithm for solving the hydraulic equations is the Newton method, where a sequence of linear equations is formed and solved at each iteration [12] Since solving these linear systems to high accuracy is the bottleneck of the Newton method, multiple approaches have been proposed to reduce the computational burden [13]. Compared to direct linear solvers, which can produce denser factorizations due to fill-in (i.e., the number of nonzeros in the factors of a matrix A but with corresponding zeros in the matrix A itself [21]), iterative methods, such as the CG algorithm, exploit sparsity better because they involve only matrix-vector multiplication This can result in smaller computational effort along with reduced storage requirements that only depend on the number of nonzero entries of the linear systems [22]. The (right) null space of a matrix A is denoted by ker(A)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
