Comparison of second and third order algorithms for steady state resolution of water distribution networks

This paper presents the comparison of second and third order algorithms for steady state resolution of water distribution networks (WDNs). The algorithms are obtained by using the direct outflow/pressure relationship and linearizing the global equations using the Newton Raphson method. The increase in the order of convergence from quadratic to cubic is obtained by refining system matrices at half Newton Raphson step. Two variants are considered for the third order algorithm, differing in the evaluation of the matrix expressing the derivative of the outflow/pressure relationship at WDN nodes: the derivative is evaluated analytically and numerically for the first and second versions, respectively. Specifically, the numerical evaluation is obtained by using outflow and head values that are available at the half Newton Raphson step. The results of applications to five case studies of increasing complexity point out that the third order algorithm converges in a smaller number of iterations than the second order algorithm. The third order algorithm with numerical evaluation of the derivative of the outflow/pressure relationship gives significant benefits in terms of convergence performance when the service pressure range for passing from no outflow to full outflow conditions at WDN nodes is small. All the algorithms developed in this work will be considered for implementation inside the SWANP version 4.0 software.