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#### Abstract

The goal of this subproject is to apply network flow theory to simplified models for gas pipelines and related transportation networks. Network flow theory has proven itself to be a powerful and valueable tool for solving complex problems in many areas of application, like traffic networks, telecommunication networks, and logistic networks. All of these networks can be designed and operated using network flow algorithms, which exhibit a very efficient runtime behaviour, making them capable of handling the large instances occuring in practice. In this subproject, we strive to employ network flow algorithms for solving problems in gas and related networks. Conventional network flow theory is insufficient to describe gas networks, since it neither accounts for pressures in nodes, nor for the non-linear dependency of the flow from the pressure in nodes. However, there are several problems in gas networks which occur in a similar form in network flow theory and are known and well understood there, for example generalized or length-bounded flows. Furthermore, non-linear relationships in other aspects like for example the flow velocity in traffic networks, have been studied and solved by network flow theory. As a base for our models we use a model based on the Weymouth equations, which can be solved using convex minimum cost flows. This flow model and its solution techniques can be generalized from the stationary to the transient case, analogous to how classic network flows can be generalized to dynamic network flows. For solving dynamic networks (approximately), there are useful techniques known from network flow theory, e.g. adaptive time-expanded networks, which have a great number of applications. These techniques are to be made useable for transient gas flows in this subproject. Gas networks usually contain a number of active elements - valves, compressors, and so on - which require integral decisions which have a significant impact on the gas flow. Therefore, these elements have to be incorporated in the network flow model. This can be handled by Branch & Bound based methods, or by adaption of techniques from network design. For Branch & Bound based techniques it is crucial to identify infeasible solutions as fast as possible. In gas networks, this means identifying partial configurations of active elements which imply infeasiblity. Therefore, analysis of infeasiblity will be a focus of the research done in this subproject. By developing efficient models and solution techniques for gas networks and characterizing infeasibility in these models, this subproject will contribute for Demonstrator 1, with a special focus on large gas networks.

A poster to A07 can be found here.