MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

  • Martin Gugat
  • Günter Leugering
  • Alexander Martin
  • Martin Schmidt
  • Mathias Sirvent
  • David Wintergerst


We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.


Mixed-integer optimal control Instantaneous control Partial differential equations on graphs Gas networks Mixed-integer linear optimization 

Mathematics Subject Classification

49J15 49J20 76B75 90C11 90C35 



We acknowledge funding through the DFG SFB/Transregio 154, Subprojects A05, B07, B08, and C03. This research has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government. Finally, we thank Marc Steinbach for the provision of some data that we used for our numerical studies.


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Authors and Affiliations

  1. 1.Lehrstuhl für Angewandte Mathematik 2Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)ErlangenGermany
  2. 2.Discrete OptimizationFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)ErlangenGermany
  3. 3.Energie Campus NürnbergNurembergGermany

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