Pressure Flow–Based Algorithms for Pressure-Driven Analysis of Water Distribution Networks

Ezio Todini,Simone Santopietro,Rudy Gargano,Lewis A. Rossman

doi:10.1061/(asce)wr.1943-5452.0001401

Abstract

In recent years, pressure-driven analyses of water distribution systems using pressure-flow relationships (PFRs) were shown to be more computationally efficient than those based on flow-pressure relationships (FPRs), such as the widely used Wagner equation and various spline approximations. Using a PFR enhances the convergence properties of the Newton–Raphson method used by most water distribution network (WDN) solvers. This paper derives a new way of incorporating a PFR into the classical Todini and Pilati global gradient algorithm (GGA). The convergence properties of the resulting solution algorithm are compared with those from two other existing PFR algorithms, EPANET 2.2 and the active-set method (ASM), as well as from a conventional flow-pressure–based algorithm on a number of different networks of varying size.

Highlights

Solvers for water distribution network (WDN) models, from the original Cross (1936) method to the widely used Todini and Pilati global gradient algorithm (GGA) (Todini and Pilati 1988), were essentially developed for design purposes and run by fixing demands while modifying pipe diameters until the desired pressures were reached at all network nodes
This work proposes an approach using an inverse flow-pressure relationship (FPR) in conjunction with the GGA, which has the advantage of leaving in demanddriven mode all the nodes for which the power is equal or higher than the power required to deliver at the node the desired demand at the required head, as well as all the nodes for which the estimated operating pressure and demand result at the same time smaller than or equal to zero, failing to deliver water
The derived solution equations, based on an extension of the demanddriven approaches (DDAs) GGA, are practically identical to the ones used in EPANET 2.2, which were derived in analogy with the use of a virtual pipe and a virtual reservoir connected to each node

Summary

Introduction

Solvers for water distribution network (WDN) models, from the original Cross (1936) method to the widely used Todini and Pilati global gradient algorithm (GGA) (Todini and Pilati 1988), were essentially developed for design purposes and run by fixing demands while modifying pipe diameters until the desired pressures were reached at all network nodes. DDAs fail to realistically represent the actual WDN behavior and, when pressure falls below the desired value, one needs to insert a pressure-based condition at the relevant nodes This alternative pressure-driven approach (PDA) requires defining a flow-pressure relationship (FPR) to describe the reduced nodal flow delivered as a function of available pressure and demand. In 2020, the US Environmental Protection Agency (USEPA) released version 2.2 of EPANET (USEPA 2020) which uses an inverse formulation of the FPR by adding a virtual one-way link between a node and a virtual reservoir (fixed head node), to represent an inverse flow-pressure–type relation (FPR) in the form of H 1⁄4 HðdÞ, which will be referred to as a pressure-flow relation (PFR) This approach, already advocated and tested by Rossman several years before for describing emitters, resulted into a noticeable improvement in the convergence properties without increasing the size or the density of the system matrix. The EPANET 2.2 PDA algorithm can be derived as follows

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Conclusions

Data Availability Statement