Computer modelling of water supply distribution networks using the gradient method
A water distribution network analysis method known as the gradient method, due to Todini (1979), has been generalised and subjected to an extensive program of testing and evaluation. The method has been extended to include pumps and some pressure regulating valves, and an original physically-based method has been proposed for modelling the latter devices. Also, a generalised version of the algorithm which considers the nodal demands as a linear function of the pressures has been introduced. The gradient method has been tested with numerous examples, showing remarkable robustness and convergence speed when compared with the most efficient traditional methods. The gradient algorithm does not break down with disconnected networks. The performance of the gradient algorithm when using seven different linear solvers, including direct and iterative methods, has been investigated. A multifrontal linear solver has been identified as the most efficient method when enough computer memory is available (routine MA27 of the Harwell Library); if storage is limited, a preconditioned (modified) conjugate gradient method is the recommended linear solver. A good compromise between memory and speed is represented by the one-way dissection method of George and Liu (1981). An automatic calibration algorithm has been proposed which estimates the true pipe resistance parameters, based on the estimation of the unmeasured piezometric heads and unmeasured flows. For the piezometric head estimation, three different methods have been proposed and compared: one based on Kriging, another based on bi-cubic splines and a third based on an original deterministic one-dimensional interpolation procedure. The latter producing the closest estimates with respect to the true values. For the estimation of the unmeasured flows, the raw (un-calibrated) network model itself is used, based on initial estimates of the pipe roughnesses, leading to an iterative procedure. The results of using the proposed calibration algorithm with a set of test examples show that the unmeasured flow estimation needs further work and an alternative approach has been suggested, which, hopefully, would lead to improvements both in the flow estimation and in the estimation of the true roughnesses.