Modeling Frictional Head Loss in the Redesign of City-Scale Water Networks through Multiobjective Optimization: Strategies to Deal with Nonlinear Terms

Water resource management is a crucial issue today when global warming is advancing, generating water shortages in several world countries. Mathematical and optimization tools have addressed these problems, including new alternative water sources. Water networks are designed to cover consumption. In this context, pumping is critical in modeling water networks through optimization techniques because of the nonlinear terms from the extended Bernoulli equation with the Darcy-Weisbach friction term. This paper compares four strategies to simplify these terms, focusing on the nonlinear constraints generated by the extended Bernoulli equation with the Darcy-Weisbach friction term. This equation has a significant impact on operation costs because of pumping power. The four simplification strategies were compared with a focus on (i) solution, (ii) error, and (iii) execution time. The best results were obtained with a two-stage strategy to address large MINLP multiobjective problems. This strategy is applied to a model with three objective functions (freshwater inlet, global warming potential, and total cost) to illustrate the simplification performance in a city-scale water network. The problem is focused on a case study in Santiago, Chile, and is based on a previous formulation. By solving the multiobjective problem, some results and changes in the network are obtained. When the three objective functions have all the same importance, the results show the following: (i) The current location of water treatment plants is suboptimal. (ii) Water recycling in the city is the best option, with drinking and irrigation qualities. (iii) With the optimal configuration, Santiago can reduce their water consumption by 30%, increasing the economic cost by 108% and the global warming potential by 49%. Finally, this model can be implemented in other contexts to approach nonlinearities by the extended Bernoulli equation with the Darcy-Weisbach friction term in large-scale water networks.