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Mathematical Problems in Engineering
Volume 2018, Article ID 7984079, 17 pages
Research Article

Stationary Gas Networks with Compressor Control and Random Loads: Optimization with Probabilistic Constraints

Lehrstuhl Angewandte Mathematik 2, Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Cauerstr. 11, 91058 Erlangen, Germany

Correspondence should be addressed to Michael Schuster; ed.uaf@retsuhcs.ihcim

Received 27 April 2018; Revised 29 July 2018; Accepted 4 September 2018; Published 26 September 2018

Guest Editor: Gerhard-Wilhelm Weber

Copyright © 2018 Martin Gugat and Michael Schuster. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We introduce a stationary model for gas flow based on simplified isothermal Euler equations in a non-cycled pipeline network. Especially the problem of the feasibility of a random load vector is analyzed. Feasibility in this context means the existence of a flow vector meeting these loads, which satisfies the physical conservation laws with box constraints for the pressure. An important aspect of the model is the support of compressor stations, which counteract the pressure loss caused by friction in the pipes. The network is assumed to have only one influx node; all other nodes are efflux nodes. With these assumptions the set of feasible loads can be characterized analytically. In addition we show the existence of optimal solutions for some optimization problems with probabilistic constraints. A numerical example based on real data completes this paper.