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We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decomposition approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sum-up-rounding strategies and mixed-integer linear programming techniques. The coupling is handled using a penalty-approach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands.
We prove an existence result for the steady state flow of gas mixtures
on networks. The basis of the model are the physical principles of the isothermal
Euler equation, coupling conditions for the flow and pressure, and the mixing of
incoming flow at nodes. The state equation is based on a convex combination of
the ideal gas equations of state for natural gas and hydrogen. We analyze mathematical
properties of the model allowing us to prove the existence of solutions in
particular for tree-shaped networks and networks with exactly one cycle. Numerical
examples illustrate the results and explore the applicability of our approach
to different network topologies.
In this paper, we study hydrogen-natural gas mixtures transported through pipeline networks. The flow is modeled by the isothermal Euler equations with a pressure law involving a non-constant, composition-dependent compressibility factor. For a broad class of such compressibility models, we prove the existence of steady-state solutions on networks containing compressor stations. The analysis is based on an implicit representation of the pressure profiles and a continuity argument that overcomes the discontinuous dependence of the gas composition on the flow direction. Numerical examples illustrate the influence of different compressibility models on the resulting states.
