Mathematical optimization of engineering problems via discretization : pooling, wastewater treatment, and central receiver systems

• Mathematische Optimierung von ingenieurwissenschaftlichen Problemen durch Diskretisierung : Pooling, Abwasserreinigung und zentrale Receiversysteme

Kuhnke, Sascha David; Koster, Arie Marinus (Thesis advisor); Büsing, Christina Maria Katharina (Thesis advisor); Liers, Frauke (Thesis advisor)

Aachen : RWTH Aachen University (2022, 2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

Increasing demand and scarcity of resources require strong and innovative solutions for engineering problems in the energy industry. Such problems can often be formulated as nonconvex optimization problems which require the application of global optimization algorithms to solve them to optimality. As these algorithms struggle to solve real-world instances within reasonable running time, heuristics are a common alternative since they are usually much faster and obtain strong but not necessarily optimal solutions. In this thesis, we develop efficient heuristics based on discretization which approximate the nonconvex problem by a mixed-integer linear program (MILP). This discretized MILP is much easier to solve and may still yield an optimal solution for the original problem if a suitable discretization for the MILP is chosen. The main part of this thesis addresses the selection of a suitable discretization which is often very difficult to find in practice. To this end, we develop adaptive discretization algorithms which iteratively improve the discretization by solving different discretized MILPs. In each iteration, the new discretization is adapted based on the MILP solution of the previous iteration. This yields discretizations that are tailored to the problem structure and thus result in stronger solutions for the original problem. We first apply this approach to the general problem class of quadratically constrained quadratic programs (QCQPs) and perform an extensive computational study to show its effectiveness in comparison to commercial solvers. Then, we develop problem specific adaptive discretization algorithms for the pooling problem and the design of water usage and treatment networks (WUTN design). Again, extensive computational experiments highlight the strength of the adaptive discretization algorithms in comparison to commercial solvers and alternative solution approaches. Since the discretized MILP of WUTN design requires the main computational effort in the above algorithm, we next investigate the polyhedral structure of this MILP from a theoretical point of view. We derive several classes of valid inequalities and prove that some of them are facet-defining for a relaxation of this MILP. In the last part of this thesis, we apply discretization to introduce a robust MILP formulation for the optimization of aiming strategies in central receiver systems (CRS). A case study on real data shows that this formulation obtains solutions with economical benefits over a conventional approach while providing the same degree of safety against material damage.