ISBN 978-3-8439-4653-7

978-3-8439-4653-7, Reihe Mathematik

Oliver Habeck
Mixed-Integer Optimization with Ordinary Differential Equations for Gas Networks

215 Seiten, Dissertation Technische Universität Darmstadt (2020), Softcover, B5

Zusammenfassung / Abstract

This thesis deals with the development of a spatial branch-and-bound algorithm for global optimization of a class of mixed-integer nonlinear problems including ordinary differential equation (ODE) constraints on an underlying network structure. The distinguishing feature of this class is that the ODE solutions only need to be known at a finite number of points, that is, the junctions of the underlying network. This particular problem class is motivated by the application on stationary gas transport which is used for computational experiments.

To construct relaxations of ODE constraints for the use in a spatial branch-and-bound algorithm, sufficient conditions under which a discretization method yields a lower or an upper bound are studied and applied to specify the conditions for particular methods. These methods can be used to define convex under- and concave overestimators of the ODE solutions describing the gas flow in pipelines. Moreover, exploiting the underlying network structure allows to incorporate the relaxations defined by discretization methods into the mixed-integer optimization problem without introducing new variables for the discretization. This in turn makes it possible to adaptively refine the discretizations within a spatial branch-and-bound process. With that it is then possible to compute global optimal solutions or decide infeasibility of the optimization problem.

To speed-up the optimization of gas networks combinatorial models for acyclic flows are studied since the gas flow in passive (parts of) networks is necessarily acyclic. In particular one model that captures acyclicity together with the supply and demand behavior of the network is investigated in detail. For example, variable fixing rules and the complexity of linear optimization over the corresponding polytope are provided.