A two‐phase decomposition method is proposed for the optimal design of new looped water distribution networks as well as for the parallel expansion of existing ones. The main feature of the method is that it generates a sequence of improving local optimal solutions. The first phase of the method takes a gradient approach with the flow distribution and pumping heads as decision variables and is an extension of the linear programming gradient method proposed by Alperovits and Shamir (1977) for nonlinear modeling. The technique is iterative and produces a local optimal solution. In the second phase the link head losses of this local optimal solution are fixed, and the resulting concave program is solved for the link flows and pumping heads; these then serve to restart the first phase to obtain an improved local optimal solution. The whole procedure continues until no further improvement can be achieved. Some applications and extensions of the method are also discussed.

The water supply system studied in this paper consists of a water treatment plant, a ground-level storage, a pumping station, and a distribution network in series. Expected served demand is employed to measure reliability taking into account both insufficient heads and flows at individual nodes in the network since it is the most important service level index provided to individual users. A basic method proposed is to assume that the insufficient nodal head reduces the effectiveness of flow supplied at the node and that the authority provides the maximum service to customers so that the real-time pump and network flow operations maximize the effective served system demand. The average value of the maximum effective served system demand relative to the total system demand over all system states is defined as system reliability, and the nodal reliability for each demand node is similarly defined. The Markov chain method introduced by Beim and Hobbs (1988) is employed to describe the evolution of the storage level over time so that the real-time pump and network flow operations can be accurately implemented by solving a nonlinear programming model. Two example systems are presented to demonstrate numerically the advantage of the method proposed in its consideration of the distribution network and nodal reliabilities.

A modified linear programming gradient (LPG) method is presented for solving looped water distribution network problems, together with a mathematically rigorous derivation of the LPG model. The LPG method of Alperovits and Shamir is modified in terms of both search direction and step size. A quasi‐Newton search direction is proposed instead of the steepest descent direction, and the step size is determined by a backtracking line search method instead of a fixed step size. The modified method is applied to a numerical example, where it provides an improved solution in comparison to the original LPG method.

Abstract. A goal programming model has been developed to analyze the system behavior for the water distribution networks under contingency situations due to failures of pipes and pumps, taking into account three aspects: (1) equity, or sharing inconvenience equally among consumers; (2) redistribution of the network flows to reduce the negative consequences of a failure of one portion on other portions of the network; and (3) consideration of pressure-dependent demand delivery due to insufficient head, namely, if a nodal head falls below a desired level, the flow delivered to that node is reduced. The first priority of the goal program is to maximize the lowest nodal demand supply ratio (or the ratio of actually delivered demand to the required demand at a node). The second priority is to maximize the system demand supply ratio (or the ratio of actually delivered water to the required total system demand). Link flow directions in the model are not fixed but are determined by a set of criteria. The system behaviors with respect to the three aspects of reliability factors are examined through extensive numerical experiments. The impact of equity requirements on redistribution of network flows, link flow directions, nodal demand supply ratio, and system demand supply ratio when failure events become serious is examined in particular detail. It is found that equity requirements can satisfactorily bring about fair sharing of inconvenience among consumers. The model proposed also suggests that network operations should reverse some link flow directions in order to meet equity requirements under severe contingencies.

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