A New Derivative‐Free Linear Approximation for Solving the Network Water Flow Problem With Convergence Guarantees

Abstract

Addressing challenges in urban water infrastructure systems, including aging infrastructure, supply uncertainty, extreme events, and security threats, depends highly on water distribution networks modeling emphasizing the importance of realistic assumptions, modeling complexities, and scalable solutions. In this study, we propose a derivative‐free, linear approximation for solving the network water flow problem. The proposed approach takes advantage of the special form of the nonlinear head loss equations, and, after the transformation of variables and constraints, the water flow problem reduces to a linear optimization problem that can be efficiently solved by modern linear solvers. Ultimately, the proposed approach amounts to solving a series of linear optimization problems. We demonstrate the proposed approach through several case studies and show that the approach can model arbitrary network topologies and various types of valves and pumps, thus providing modeling flexibility. Under mild conditions, we show that the proposed linear approximation converges. We provide sensitivity analysis and discuss in detail the current limitations of our approach and suggest solutions to overcome these. All the codes, tested networks, and results are freely available on Github for research reproducibility.

Probabilistic State Estimation in Water Networks

“…where n t × n t matrix A h , n t × n q matrix B q , n j × n q matrix E J q , and n q × n h matrix E h are constant matrices that depend on the topology and hydraulic properties of the underlying WDN. Equation (2a) collects the dynamic equations of tanks (22); Equation (2b) collects the mass balance equations for all junctions (21); the nonlinear function Φ(•) in Equation (2c) includes the nonlinear pipe (23), pump (24), and valve (27), (28) models in Appendix B. Additional details are also given in our recent work [23].…”

“…Equation (2a) collects the dynamic equations of tanks (22); Equation (2b) collects the mass balance equations for all junctions (21); the nonlinear function Φ(•) in Equation (2c) includes the nonlinear pipe (23), pump (24), and valve (27), (28) models in Appendix B. Additional details are also given in our recent work [23]. The roughness coefficients for pipes are collected in vector c(k).…”

“…2) Conservation of energy at pipes, pumps, and valves: The major head loss of a pipe is determined by Hazen-Williams, and can be expressed as (23) in Tab. V, where resistance coefficient R ij = 4.727L P (C HW ) −1.852 (D P ) −4.871 is a function of the corresponding roughness coefficient C HW , pipe diameter D P , and pipe length L P .…”

Computing Lipschitz Constants for Hydraulic Models of Water Distribution Networks

“…The aim of this section is to investigate how the analytical derivation of the Lipschitz constants (Section IV) for various networks compare with the numerical computations (Section VI). To that end, six WDN templates (Three-Node [22], Eight-node [13], Anytown [23], Net2, Net3, and OBCL networks [24]) are used to assess our methods in finding Lipschitz constants K via analytical and numerical methods. We do not compute OSL since L has the same value as K; see Preposition 4.…”

A Mixed‐Integer Linear Programming Framework for Optimization of Water Network Operations Problems

“…On the other hand, mathematical programming methods allow to explicitly model the governing hydraulic equations within the optimization problem as constraints, and represent the performance as objective functions (Coelho & Andrade‐Campos, 2014; Collins et al., 1978; Sela Perelman & Amin, 2015). Comprehensive and efficiently designed mathematical programming models can provide optimal or near‐optimal solutions even for large networks without relying on a multitude of hydraulic simulations, thus typically requiring shorter running times and lower computational effort (Bragalli et al., 2012; Wang et al., 2020). Therefore, due to advantages with regards to accuracy and computational complexity, we examine only mathematical programming techniques in this paper.…”