Global Algorithm Based on Inverse Outflow/Head Relationship for Solving Water Distribution Networks with Multiple Nodal Outflows and Improved Convergence

This paper presents developments in the global algorithm based on Newton–Raphson linearization of the equations for water distribution network (WDN) resolution and on the inverse outflow–head relationship, in which the head at demand nodes is expressed as a function of the outflow. These developments concern (1) implementation and application of multiple nodal pressure-driven outflows, including various user outflow and leakage formulations; and (2) algorithm extension to a cubic order of convergence. Unlike algorithms based on the direct outflow–head relationship, in which the outflow is expressed as a function of the head at each node, the implementation of multiple nodal outflows is carried out by considering each kind of outflow as a new unknown vector in the system of equations to be solved, due to analytical difficulties in expressing the head at the single node as a function of the total outflow in the presence of simultaneous nodal outflows. The increase in the order of convergence from quadratic to cubic is obtained by refining the evaluation of some matrixes to account for the curvature of the hyperplane associated with the system in the direction of the Newton–Raphson step. The algorithms developed were applied to five case studies of increasing complexity. The results indicate that algorithm convergence is stable, with no need for relaxation in all case studies, for all kinds of user outflows, including leakage estimated with either the emitter or the fixed and variable area discharge (FAVAD) equation. A reduction in the number of iterations required for convergence is obtained by increasing the order of convergence to cubic.