Treffer: Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Natural gas is one of the most important energy sources. Consequently, its transportation through gas networks is an essential task and gives rise to gas transport problems. Such optimization problems involve discrete decisions to switch network elements as valves, control valves, or compressor machines. Moreover, the physical behavior of natural gas is described by differential equations. Thus, when dealing with gas transport optimization, mixed-integer problems constrained by differential equations become relevant. The scientific contribution of this thesis to solve such problems is twofold. First, three new global algorithms are presented. In general, a typical solution approach transforms the differential equations to linear constraints. This is reasonable as mixed-integer linear programming is the most successful instance of mixed-integer programming. The new global algorithms in this thesis do not rely on this transformation and can work with less information about the underlying differential equation constraints. In an iterative process, mixed-integer linear programs and small nonlinear programs are solved alternately and the correct and finite terminations of the algorithms are proven. An extensive theoretical framework that distinguishes the assumptions on the constraints is set up. The developments allow to solve stationary gas transport optimization problems with ordinary differential equations. In this sense, promising numerical results for the Greek natural gas transport network are shown. Furthermore, the way for more general simulation-based algorithms is paved. Second, an instantaneous control algorithm for transient gas network optimization with partial differential equations is presented. A new and specific discretization scheme that allows to use mixed-integer linear programs inside of the instantaneous control algorithm is developed for the example of gas. Again, promising numerical results that illustrate the applicability of the approach are shown. These findings pave the way for more research in the field of transient gas network optimization, which, due to its hardness, is often disregarded in the literature.