Dealing with Zero Flows in Solving the Nonlinear Equations for Water Distribution Systems

Abstract

Three issues concerning the iterative solution of the nonlinear equations governing the flows and heads in a water distribution system network are considered. Zero flows cause a computation failure (division by zero) when the Global Gradient Algorithm of Todini and Pilati is used to solve for the steady state of a system in which the head loss is modeled by the Hazen-Williams formula. The proposed regularization technique overcomes this failure as a solution to this first issue. The second issue relates to zero flows in the Darcy-Weisbach formulation. This work explains for the first time why zero flows do not lead to a division by zero where the head loss is modeled by the Darcy-Weisbach formula. In this paper, the authors show how to handle the computation appropriately in the case of laminar flow (the only instance in which zero flows may occur). However, as is shown, a significant loss of accuracy can result if the Jacobian matrix, necessary for the solution process, becomes poorly conditioned, and so it is recommended that the regularization technique be used for the Darcy-Weisbach case also. Only a modest extra computational cost is incurred when the technique is applied. The third issue relates to a new convergence stopping criterion for the iterative process based on the infinity-norm of the vector of nodal head differences between one iteration and the next. This test is recommended because it has a more natural physical interpretation than the relative discharge stopping criterion that is currently used in standard software packages such as EPANET. In addition, it is recommended to check the infinity norms of the residuals once iteration has been stopped. The residuals test ensures that inaccurate solutions are not accepted.