Informational entropy : a failure tolerance and reliability surrogate for water distribution networks

Evolutionary algorithms are used widely in optimization studies on water distribution networks. The optimization algorithms use simulation models that analyse the networks under various operating conditions. The solution process typically involves cost minimization along with reliability constraints that ensure reasonably satisfactory performance under abnormal operating conditions also. Flow entropy has been employed previously as a surrogate reliability measure. While a body of work exists for a single operating condition under steady state conditions, the effectiveness of flow entropy for systems with multiple operating conditions has received very little attention. This paper describes a multi-objective genetic algorithm that maximizes the flow entropy under multiple operating conditions for any given network. The new methodology proposed is consistent with the maximum entropy formalism that requires active consideration of all the relevant information. Furthermore, an alternative but equivalent flow entropy model that emphasizes the relative uniformity of the nodal demands is described. The flow entropy of water distribution networks under multiple operating conditions is discussed with reference to the joint entropy of multiple probability spaces, which provides the theoretical foundation for the optimization methodology proposed. Besides the rationale, results are included that show that the most robust or failure-tolerant solutions are achieved by maximizing the sum of the entropies

Assessment of penalty-free multi-objective evolutionary optimization approach for the design and rehabilitation of water distribution systems

This paper describes a penalty-free multi-objective evolutionary optimization approach for the phased whole-life design and rehabilitation of water distribution systems. The optimization model considers the initial construction, rehabilitation and upgrading costs. Repairs and pipe failure costs are included. The model also takes into consideration the deterioration over time of both the structural integrity and hydraulic capacity of every pipe. The fitness of each solution is determined from the trade-off between its lifetime costs and its actual hydraulic properties. The hydraulic analysis approach used, known as pressure-dependent modelling, considers explicitly the pressure dependency of the water supply consumers receive. Results for two sample networks in the literature are included that show the algorithm is stable and finds optimal and near-optimal solutions reliably and efficiently. The results also suggest that the evolutionary sampling efficiency is very high. In other words, the number of solutions evolved and analysed on average before finding a near-optimal solution is small in comparison to the total number of feasible and infeasible solutions. We found better solutions than those reported previously in the literature for the two networks considered. For the Kadu network, for example, the new best solution costs Rs125,460,980 – a significant improvement. Additional statistics that are based on extensive testing are included