There is ever increasing commercial and regulatory pressure to minimise the cost of water distribution networks even as the demand for them keeps on growing. But cost minimizing is only one of the demands placed on the network design. Satisfactory networks are required to operate above a minimum level even if they experience failure of components. Reliable hydraulic performance can be achieved if sufficient redundancy is built in the network. This has given rise to various water distribution system optimization methods including genetic algorithms and other evolutionary computing methods. Evolutionary computing approaches frequently assess the suitability of enormous numbers of potential solutions for which the calculation of accurate reliability measures could be computationally prohibitive. Therefore, surrogate reliability measures are frequently used to ease the computational burden. The aim of this paper is to assess the correlation of surrogate reliability measures in relation to more accurate measures. The surrogate measures studied are statistical entropy, network resilience, resilience index and modified resilience index. The networks were simulated with prototype software PRAAWDS that produces more realistic results for pressuredeficient water distribution systems. Statistical entropy outperformed resilience index in this study. The results also demonstrate there is a strong correlation between entropy and failure tolerance.Keywords: Failure tolerance; pressure-deficient water distribution networks; pressure-dependent modelling; redundancy; resilience index; statistical entropy; reliability INTRODUCTIONOne of the major considerations in constructing water distribution networks is the capital cost of the project. To minimise cost of the network modellers endeavour to achieve a balance of smallest possible pipe sizes, tanks and pumps whilst still providing adequate amount of water to meet the demand. However, Walski (2001) questions the benefits of network optimization to achieve minimum cost at the expense of reduced system capacity, and consequently, reliability. He argues that due to uncertainty of future demands and loss in potential project net benefits following cost minimization, water distribution network (WDN) optimization based on cost alone is not viable. Hence another important factor in WDN design is reliability. To guarantee undisrupted water supply even during abnormal conditions, such as fire fighting or network component failure, a WDN has to have some redundancy built in it. But cost and reliability of a water distribution system are at least partly mutually conflicting constraints on design. Studies show that high cost is not always an indicator of high reliability, which means network optimization leading to minimum cost and maximum benefit and efficiency is desirable.

A new multi-objective evolutionary optimization approach for joint topology and pipe size design of water distribution systems is presented. The algorithm proposed considers simultaneously the adequacy of flow and pressure at the demand nodes; the initial construction cost; the network topology; and a measure of hydraulic capacity reliability. The optimization procedure is based on a general measure of hydraulic performance that combines statistical entropy, network connectivity and hydraulic feasibility. The topological properties of the solutions are accounted for and arbitrary assumptions regarding the quality of infeasible solutions are not applied. In other words, both feasible and infeasible solutions participate in the evolutionary processes; solutions survive and reproduce or perish strictly according to their Pareto-optimality. Removing artificial barriers in this way frees the algorithm to evolve optimal solutions quickly. Furthermore, any redundant binary codes that result from crossover or mutation are eliminated gradually in a seamless and generic way that avoids the arbitrary loss of potentially useful genetic material and preserves the quality of the information that is transmitted from one generation to the next. The approach proposed is entirely generic: we have not introduced any additional parameters that require calibration on a case-by-case basis. Detailed and extensive results for two test problems are included that suggest the approach is highly effective. In general, the frontier-optimal solutions achieved include topologies that are fully branched, partially-and fully-looped and, for networks with multiple sources, completely separate sub-networks.

This paper describes a new multi-objective evolutionary optimization approach to the simultaneous layout and pipe size design of water distribution systems. Pressure-deficient and topologically infeasible solutions are fully incorporated in the genetic algorithm without recourse to constraint violation penalties or tournaments. The proposed approach is demonstrated by solving three benchmark problems taken from the literature. New optimal layouts and/or new feasible solutions that are cheaper than the best solutions in the literature were found for both branched and looped network configurations. Specifically, a new best solution was generated for each of the above-mentioned benchmark problems. In addition, the case of the looped design of a hitherto branched network in the literature was considered. Detailed results are included that show that the proposed approach achieves good solutions efficiently and consistently.

A new multi-directional search approach that aims at maximizing the flow entropy of water distribution systems is investigated. The aim is to develop an efficient and practical maximum entropy based approach. The resulting optimization problem has four objectives, and the merits of objective reduction in the computational solution of the problem are investigated also. The relationship between statistical flow entropy and hydraulic reliability/ failure tolerance is not monotonic. Consequently, a large number of maximum flow entropy solutions must be investigated to strike a balance between cost and hydraulic reliability. A multi-objective evolutionary optimization model is developed that generates simultaneously a wide range of maximum entropy values along with clusters of maximum and near-maximum entropy solutions. Results for a benchmark network and a real network in the literature are included that demonstrate the effectiveness of the procedure.Keywords Maximum flow entropy . Hydraulic reliability and redundancy . Water distribution system . Demand driven analysis . Head driven analysis . Penalty-free constrained evolutionary optimization IntroductionIn the context of the design of water distribution systems, the least cost feasible solution is marginally able to satisfy the hydraulic requirements. Thus any failure in any system component can significantly affect the hydraulic performance of such designs. The incorporation of criteria other than cost to distinguish further between feasible solutions would lead to the