Stochastic optimization for feasibility determination: an application to water pump operation in water distribution network

The energy consumption by public drinking water and wastewater utilities represent up to 30%-40% of a municipality energy bill. The largest energy consumption is used to operate motors for pumping. As a result, the engineering and control community develop the Variable Speed Pumps (VSPs) which allow for regulating valves in the network instead of the traditional binary ON/OFF pumps. Potentially, VSPs save up to 90% of annual energy cost compared to the binary pump. The control problem has been tackled in the literature as “Pump Scheduling Optimization” (PSO) with a main focus on the cost minimization. Nonetheless, engineering literature is mostly concerned with the problem of understanding “healthy working conditions” (e.g., leakages, breakages) for a water infrastructure rather than the costs. This is very critical because if we operate a network under stress, it may satisfy the demand at present but will likely hinder network functionality in the future.

This research addresses the problem of analyzing working conditions of large water systems by means of a detailed hydraulic simulation model (e.g., EPANet) to gain insights into feasibility with respect to pressure, tank level, etc. This work presents a new framework called Feasible Set Approximation – Probabilistic Branch and Bound (FSA-PBnB) for the definition and determination of feasible solutions in terms of pumps regulation. We propose the concept of feasibility distance, which is measured as the distance of the current solution from the feasibility frontier to estimate the distribution of the feasibility values across the solution space. Based on this estimate, pruning the infeasible regions and maintaining the feasible regions are proposed to identify the desired feasible solutions. We test the proposed algorithm with both theoretical and real water networks. The results demonstrate that FSA-PBnB has the capability to identify the feasibility profile in an efficient way. Additionally, with the feasibility distance, we can understand the quality of sub-region in terms of feasibility.

The present work provides a basic feasibility determination framework on the low dimension problems. When FSA-PBnB extends to large scale constraint optimization problems, a more intelligent sampling method may be developed to further reduce the computational effort.