Least-cost design of water distribution networks under demand uncertainty

Due to inherent variability in instantaneous water consumption levels, values of demands at nodes in a water distribution system remain one of the major sources of uncertainty in the network design process. Uncertainty in demand leads to uncertainty in head at the nodes, which, in turn, affects the system performance and has to be taken into account when designing new water distribution systems or extending/rehabilitating existing ones. One approach to dealing with this difficulty is to formulate and solve the stochastic optimization problem providing a robust, cost-effective solution. However, stochastic formulation usually requires Monte Carlo simulation, which involves calculation of a large number of state estimates, even for relatively simple networks. This renders the approach intractable when combined with heuristic adaptive search techniques, such as genetic algorithms (GAs) or simulated annealing. These methodologies require the fitness function to be evaluated for thousands of possible network configurations in the course of the search process. In this paper a new approach to quantifying the influence of demand uncertainty on nodal heads is proposed. The original stochastic model is reformulated as a deterministic one, which uses standard deviation as a natural measure of variability. Such an approach allows the use of effective numerical methods to quantify the influence of uncertainty on the robustness of water distribution system solutions. The deterministic equivalent is then coupled with an efficient GA solver to find robust and economic solutions. The proposed methodology was tested on the New York tunnels and Anytown problems. A number of low cost network solutions were found for different levels of reliability and different forms of probability distribution function for demands. The robustness of the solutions found was compared to known solutions for deterministic formulations, whose results were postprocessed using full Monte Carlo simulation.