An adaptive discretization algorithm for the design of water usage and treatment networks

Abstract

In this paper, we consider the design of water usage and treatment systems in industrial plants. In such a system, the demand of water using units as well as environmental regulations for wastewater have to be met. To this end, water treatment units have to be installed and operated to remove contaminants from the water. The objective of the design problem is to simultaneously optimize the network structure and water allocation of the system at minimum total cost. Due to many bilinear mass balance constraints, this water allocation problem is a nonconvex mixed integer nonlinear program (MINLP) where nonlinear solvers have difficulties to find feasible solutions for real world instances. Therefore, we present a problem specific algorithm to iteratively solve this MINLP. In each iteration, this algorithm deals with an interplay of a mixed integer linear program (MILP) and a quadratically constrained program (QCP). First, an MILP approximates the original problem via discretization and provides a suitable network structure. Then, by fixing this structure, the original MINLP turns into a QCP which yields feasible solutions to the original problem. To improve the accuracy of the generated structure, the discretization of the MILP is adapted after each iteration based on the previous MILP solution. In many cases where nonlinear solvers fail, this approach leads to feasible solutions with good solution quality in short running time.