A Lagrangian decomposition approach for the pump scheduling problem in water networks

• Published in: European journal of operational research Vol. 241; no. 2; pp. 490 - 501
• Main Authors: Ghaddar, Bissan, Naoum-Sawaya, Joe, Kishimoto, Akihiro, Taheri, Nicole, Eck, Bradley
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.2015; Elsevier Sequoia S.A
• Subjects: Decomposition; Electricity; Electricity pricing; Integer programming; Lagrange multiplier; Lagrangian relaxation; Large scale optimization; Linear programming; Mathematical models; Network flow problem; Networks; Nonlinearity; Optimization; Pump scheduling; Pumps; Schedules; Scheduling algorithms; Studies; Supply & demand; Water utilities; Large scale optimization; Integer programming; Lagrangian relaxation; Pump scheduling
• DOI: 10.1016/j.ejor.2014.08.033
• ISSN: 0377-2217

•We propose a Lagrangian decomposition approach for the pump scheduling problem in water network.•The proposed approach exploits the structure of the problem leading to smaller problems that are solved independently.•A simulation based limited discrepancy search heuristic is proposed to provide high quality feasible solutions.•The Lagrangian decomposition approach provides solutions with bound guarantees.•Numerical testing is conducted on two real water networks. Dynamic pricing has become a common form of electricity tariff, where the price of electricity varies in real time based on the realized electricity supply and demand. Hence, optimizing industrial operations to benefit from periods with low electricity prices is vital to maximizing the benefits of dynamic pricing. In the case of water networks, energy consumed by pumping is a substantial cost for water utilities, and optimizing pump schedules to accommodate for the changing price of energy while ensuring a continuous supply of water is essential. In this paper, a Mixed-Integer Non-linear Programming (MINLP) formulation of the optimal pump scheduling problem is presented. Due to the non-linearities, the typical size of water networks, and the discretization of the planning horizon, the problem is not solvable within reasonable time using standard optimization software. We present a Lagrangian decomposition approach that exploits the structure of the problem leading to smaller problems that are solved independently. The Lagrangian decomposition is coupled with a simulation-based, improved limited discrepancy search algorithm that is capable of finding high quality feasible solutions. The proposed approach finds solutions with guaranteed upper and lower bounds. These solutions are compared to those found by a mixed-integer linear programming approach, which uses a piecewise-linearization of the non-linear constraints to find a global optimal solution of the relaxation. Numerical testing is conducted on two real water networks and the results illustrate the significant costs savings due to optimizing pump schedules.