Hydraulic modelling of closed pipes in loop equations of water distribution networks

Abstract

Zero flows in closed pipes in pressurised water supply networks require additional computational boundary constraints, which can result from two different operational states for pipe appurtenances. Static closed pipes are produced by turning off the isolation valves to undertake inspections/repairs of related pipe segments or to separate adjacent pressure zones. In addition, closed pipes can occur dynamically due to shutdowns of check valves and pressure/flow regulating facilities equipped with non-return valves. In this study, a generalised solution approach is developed to solve the nonlinear loop equations of the network flow problem involving closed pipes, which is integrated directly into hydraulic simulations of water distribution systems. The iterative approach uses the Newton–Raphson method based on the energy equations. The head loss across closed pipes is estimated using a novel computational technique called the unknown hydraulic function to avoid the singularity of the Jacobian matrix. The flow corrections of the related loops are performed locally using the Hardy-Cross technique, which allows small amounts of the flow rates to vary within lower (εmin) and upper (εmax) bounds during the iteration process. As an additional convergence criterion, the upper limit (εmax ≤ 0.020 l/s) is allowed to vary in both flow directions for static closed pipes, whereas this is permitted only under reverse flow conditions for closed pipes of unidirectional control devices. The reliability and efficiency of the proposed algorithm are demonstrated using some example network applications. The results obtained after many computer runs demonstrated that the iteration process is robust and efficient, thereby ensuring consistent convergence in a rapid manner. The flow correction magnitude was reduced to negligible small values for the related loops after executing a Hardy-Cross step, which significantly helped to stabilise the direction of the solution matrix.