TY - GEN A1 - Hiller, Benjamin A1 - Walther, Tom T1 - Improving branching for disjunctive polyhedral models using approximate convex decompositions N2 - Disjunctive sets arise in a variety of optimization models and much esearch has been devoted to obtain strong relaxations for them. This paper focuses on the evaluation of the relaxation during the branch-and-bound search process. We argue that the branching possibilities (\ie binary variables) of the usual formulations are unsuitable to obtain strong bounds early in the search process as they do not capture the overall shape of the the entire disjunctive set. To analyze and exploit the shape of the disjunctive set we propose to compute a hierarchy of approximate convex decompositions and show how to extend the known formulations to obtain improved branching behavior. T3 - ZIB-Report - 17-68 KW - approximate convex decomposition KW - branch-and-bound KW - convex hull reformulation KW - disjunctive set Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-67462 SN - 1438-0064 ER - TY - GEN A1 - Hiller, Benjamin A1 - Walther, Tom T1 - Modelling compressor stations in gas networks N2 - Gas networks are an important application area for optimization. When considering long-range transmission, compressor stations play a crucial role in these applications. The purpose of this report is to collect and systematize the models used for compressor stations in the literature. The emphasis is on recent work on simple yet accurate polyhedral models that may replace more simplified traditional models without increasing model complexity. The report also describes an extension of the compressor station data available in GasLib (http://gaslib.zib.de/) with the parameters of these models. T3 - ZIB-Report - 17-67 KW - MINLP model KW - compressor station KW - gas compressor modeling KW - nonlinear model KW - polyhedral model Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-67443 SN - 1438-0064 ER - TY - GEN A1 - Hiller, Benjamin A1 - Saitenmacher, René A1 - Walther, Tom T1 - Analysis of operating modes of complex compressor stations N2 - We consider the modeling of operation modes for complex compressor stations (i.e., ones with several in- or outlets) in gas networks. In particular, we propose a refined model that allows to precompute tighter relaxations for each operation mode. These relaxations may be used to strengthen the compressor station submodels in gas network optimization problems. We provide a procedure to obtain the refined model from the input data for the original model. This procedure is based on a nontrivial reduction of the graph representing the gas flow through the compressor station in an operation mode. T3 - ZIB-Report - 17-65 KW - bound tightening KW - flow patterns KW - model reformulation KW - multi-way compressor station Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-68179 SN - 1438-0064 ER - TY - GEN A1 - Hiller, Benjamin A1 - Koch, Thorsten A1 - Schewe, Lars A1 - Schwarz, Robert A1 - Schweiger, Jonas T1 - A System to Evaluate Gas Network Capacities: Concepts and Implementation N2 - In 2005 the European Union liberalized the gas market with a disruptive change and decoupled trading of natural gas from its transport. The gas is now transported by independent so-called transmissions system operators or TSOs. The market model established by the European Union views the gas transmission network as a black box, providing shippers (gas traders and consumers) the opportunity to transport gas from any entry to any exit. TSOs are required to offer the maximum possible capacities at each entry and exit such that any resulting gas flow can be realized by the network. The revenue from selling these capacities more than one billion Euro in Germany alone, but overestimating the capacity might compromise the security of supply. Therefore, evaluating the available transport capacities is extremely important to the TSOs. This is a report on a large project in mathematical optimization, set out to develop a new toolset for evaluating gas network capacities. The goals and the challenges as they occurred in the project are described, as well as the developments and design decisions taken to meet the requirements. T3 - ZIB-Report - 18-11 Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-67438 SN - 1438-0064 N1 - An earlier version of this report is available as ZR 17-03 at https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/6193. ER - TY - JOUR A1 - Hiller, Benjamin A1 - Koch, Thorsten A1 - Schewe, Lars A1 - Schwarz, Robert A1 - Schweiger, Jonas T1 - A System to Evaluate Gas Network Capacities: Concepts and Implementation JF - European Journal of Operational Research N2 - In 2005 the European Union liberalized the gas market with a disruptive change and decoupled trading of natural gas from its transport. The gas is now trans- ported by independent so-called transmissions system operators or TSOs. The market model established by the European Union views the gas transmission network as a black box, providing shippers (gas traders and consumers) the opportunity to transport gas from any entry to any exit. TSOs are required to offer the maximum possible capacities at each entry and exit such that any resulting gas flow can be realized by the network. The revenue from selling these capacities more than one billion Euro in Germany alone, but overestimating the capacity might compromise the security of supply. Therefore, evaluating the available transport capacities is extremely important to the TSOs. This is a report on a large project in mathematical optimization, set out to develop a new toolset for evaluating gas network capacities. The goals and the challenges as they occurred in the project are described, as well as the developments and design decisions taken to meet the requirements. Y1 - 2018 VL - 270 IS - 3 SP - 797 EP - 808 ER - TY - GEN A1 - Hiller, Benjamin A1 - Becker, Kai-Helge T1 - Improving relaxations for potential-driven network flow problems via acyclic flow orientations N2 - The class of potential-driven network flow problems provides important models for a range of infrastructure networks. For real-world applications, they need to be combined with integer models for switching certain network elements, giving rise to hard-to-solve MINLPs. We observe that on large-scale real-world meshed networks the usually employed relaxations are rather weak due to cycles in the network. We propose acyclic flow orientations as a combinatorial relaxation of feasible solutions of potential-driven flow problems and show how they can be used to strengthen existing relaxations. First computational results indicate that the strengthend model is much tighter than the original relaxation, thus promising a computational advantage. T3 - ZIB-Report - 18-30 KW - potential-driven network flows KW - acyclic orientations KW - flow orientations KW - MINLPs Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-69622 SN - 1438-0064 ER - TY - GEN A1 - Becker, Kai-Helge A1 - Hiller, Benjamin T1 - ASTS Orientations on Undirected Graphs: Structural analysis and enumeration N2 - All feasible flows in potential-driven networks induce an orientation on the undirected graph underlying the network. Clearly, these orientations must satisfy two conditions: they are acyclic and there are no "dead ends" in the network, i.e. each source requires outgoing flows, each sink requires incoming flows, and each transhipment vertex requires both an incoming and an outgoing flow. In this paper we will call orientations that satisfy these conditions acyclic source-transhipment-sink orientations (ASTS-orientation) and study their structure. In particular, we characterize graphs that allow for such an orientation, describe a way to enumerate all possible ASTS-orientations of a given graph, present an algorithm to simplify and decompose a graph before such an enumeration and shed light on the role of zero flows in the context of ASTS-orientations. T3 - ZIB-Report - 18-31 Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-69632 SN - 1438-0064 ER - TY - GEN A1 - Anderson, Lovis A1 - Hiller, Benjamin T1 - A Sweep-Plane Algorithm for the Computation of the Volume of a Union of Polytopes N2 - Optimization models often feature disjunctions of polytopes as submodels. Such a disjunctive set is initially (at best) relaxed to its convex hull, which is then refined by branching. To measure the error of the convex relaxation, the (relative) difference between the volume of the convex hull and the volume of the disjunctive set may be used. This requires a method to compute the volume of the disjunctive set. Naively, this can be done via inclusion/exclusion and leveraging the existing code for the volume of polytopes. However, this is often inefficient. We propose a revised variant of an old algorithm by Bieri and Nef (1983) for this purpose. The algorithm uses a sweep-plane to incrementally calculate the volume of the disjunctive set as a function of the offset parameter of the sweep-plane. T3 - ZIB-Report - 18-37 KW - Disjunctive Programming KW - Volume Algorithm KW - Union of Polytopes Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-69489 SN - 1438-0064 ER -