Formulating the Water Distribution System Equations in Terms of Head and Velocity

The set of equations for solving for pressures and flows in water distribution systems are non-linear due to the head loss-velocity relationship for each of the pipes. The solution of these non-linear equations for the heads and flows is usually based on the Todini and Pilati method. The method is an elegant way of formulating the equations. A Newton solution method is used to solve the equations whereby the special structure of the Jacobian is exploited to minimize the computations and this leads to an extremely fast algorithm. Each iteration firstly solves for the heads and then solves for the flows. In the EPANET implementation of the Todini and Pilati algorithm an initial guess of the flows is based on an assumed velocity of 1.0 fps (0.305 m/s) in each pipe in the network. Each flow is then determined from the continuity equation by multiplying the assumed velocity by the area. Usually velocities in pipes are in the range of 0.5 to 1.5 m/s (and perhaps sometimes higher up to 3 or 4 m/s). Thus the velocities to be solved for are all of the same order of magnitude. In contrast, the range of discharges may be quite large in a system — ranging from below 10 L/s up to above 700 L/s — thus possibly three orders of magnitude of difference. As an alternative to the usual formulation of the Todini and Pilati method in terms of flows and heads, this paper recasts the Todini and Pilati formulation in terms of heads and velocities to attempt to improve the convergence properties. Results are compared for the two formulations for a range of networks from 553 to 10,354 pipes. Convergence criteria for stopping the iterative solution process are discussed. The impact of the initial guess of the velocities in each of the pipes in the network on the convergence behavior is also investigated. Statistics on mean flows and velocities in the network and the minimum and maximum velocities for each of the example networks are given and finally operation counts are also provided for these networks.