@unpublished{SchmidtHillerKochetal.2019, author = {Schmidt, Martin and Hiller, Benjamin and Koch, Thorsten and Pfetsch, Marc and Geißler, Bj{\"o}rn and Henrion, Ren{\´e} and Joormann, Imke and Martin, Alexander and Morsi, Antonio and R{\"o}misch, Werner and Schewe, Lars and Schultz, R{\"u}diger and Steinbach, Marc C.}, title = {Capacity Evaluation for Large-Scale Gas Networks}, pages = {8}, year = {2019}, abstract = {Natural gas is important for the energy turnaround in many countries like in Germany, where it serves as a "bridging energy" towards a fossil-free energy supply in the future. About 20\% of the total German energy demand is provided by natural gas, which is transported through a complex pipeline network with a total length of about 30000 km and the efficient use of the given transport infrastructure for natural gas is of political, economic, and societal importance. As a consequence of the liberalization of the European gas market in the last decades, gas trading and transport have been decoupled. This has led to new challenges for gas transport companies, and mathematical optimization is perfectly suited for tackling many of these challenges. However, the underlying mathematical problems are by far too hard to be solved by today's general-purpose software so that novel mathematical theory and algorithms are needed. The industrial research project "ForNe: Research Cooperation Network Optimization" has been initiated and funded by Open Grid Europe in 2009 and brought together experts in mathematical optimization from seven German universities and research institutes, which cover almost the entire range of mathematical optimization: integer and nonlinear optimization as well as optimization under uncertainty. The mathematical research results have been put together in a software package that has been delivered to Open Grid Europe at the end of the project. Moreover, the research is still continuing - e.g., in the Collaborative Research Center/Transregio 154 "Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks" funded by the German Research Foundation.}, language = {en} } @article{PfetschSchmitt2021, author = {Pfetsch, Marc E. and Schmitt, Andreas}, title = {A Generic Optimization Framework for Resilient Systems}, year = {2021}, abstract = {This paper addresses the optimal design of resilient systems, in which components can fail. The system can react to failures and its behavior is described by general mixed integer nonlinear programs, which allows for applications to many (technical) systems. This then leads to a three-level optimization problem. The upper level designs the system minimizing a cost function, the middle level represents worst-case failures of components, i.e., interdicts the system, and the lowest level operates the remaining system. We describe new inequalities that characterize the set of resilient solutions and allow to reformulate the problem. The reformulation can then be solved using a nested branch-and-cut approach. We discuss several improvements, for instance, by taking symmetry into account and strengthening cuts. We demonstrate the effectiveness of our implementation on the optimal design of water networks, robust trusses, and gas networks, in comparison to an approach in which the failure scenarios are directly included into the model.}, language = {en} } @article{JoormannPfetsch2018, author = {Joormann, Imke and Pfetsch, Marc E.}, title = {Complexity of Minimum Irreducible Infeasible Subsystem Covers for Flow Networks}, series = {Discrete Applied Mathematics}, volume = {244}, journal = {Discrete Applied Mathematics}, doi = {10.1016/j.dam.2018.02.025}, pages = {124 -- 142}, year = {2018}, abstract = {For an infeasible network flow system with supplies and demands, we consider the problem of finding a minimum irreducible infeasible subsystem cover, i.e., a smallest set of constraints that must be dropped to obtain a feasible system. The special cases of covers which only contain flow balance constraints (node cover) or only flow bounds (arc cover) are investigated as well. We show strong NP-hardness of all three variants. Furthermore, we show that finding minimum arc covers for assignment problems is still hard and as hard to approximate as the set covering problem. However, the minimum arc cover problem is polynomially solvable for networks on cactus graphs. This leads to the development of two different fixed parameter algorithms with respect to the number of elementary cycles connected at arcs and the treewidth, respectively. The latter can be adapted for node covers and the general case.}, language = {en} } @article{GrossMarcEScheweetal.2019, author = {Groß, Martin and Marc E., Pfetsch and Schewe, Lars and Schmidt, Martin and Skutella, Martin}, title = {Algorithmic Results for Potential-Based Flows: Easy and Hard Cases}, volume = {73}, number = {3}, edition = {Networks}, doi = {10.1002/net.21865}, pages = {303 -- 324}, year = {2019}, abstract = {Potential-based flows are an extension of classical network flows in which the flow on an arc is determined by the difference of the potentials of its incident nodes. Such flows are unique and arise, for example, in energy networks. Two important algorithmic problems are to determine whether there exists a feasible flow and to maximize the flow between two designated nodes. We show that these problems can be solved for the single source and sink case by reducing the network to a single arc. However, if we additionally consider switches that allow to force the flow to 0 and decouple the potentials, these problems are NP-hard. Nevertheless, for particular series-parallel networks, one can use algorithms for the subset sum problem. Moreover, applying network presolving based on generalized series-parallel structures allows to significantly reduce the size of realistic energy networks.}, language = {en} } @article{GrossPfetschSkutella2018, author = {Groß, Martin and Pfetsch, Marc E. and Skutella, Martin}, title = {On the Complexity of Instationary Gas Flows}, volume = {46}, number = {3}, edition = {Operations Research Letters}, doi = {10.1016/j.orl.2018.01.007}, pages = {286 -- 290}, year = {2018}, abstract = {We study a simplistic model of instationary gas flows consisting of a sequence of k stationary gas flows. We present efficiently solvable cases and NP-hardness results, establishing complexity gaps between stationary and instationary gas flows (already for k=2) as well as between instationary gas s-t-flows and instationary gas b-flows.}, language = {en} } @article{HabeckPfetschUlbrich, author = {Habeck, Oliver and Pfetsch, Marc E. and Ulbrich, Stefan}, title = {Global optimization of mixed-integer ODE constrained network problems using the example of stationary gas transport}, volume = {29}, number = {4}, edition = {SIAM Journal of Optimization}, doi = {10.1137/17M1152668}, pages = {2949 -- 2985}, abstract = {In this paper we propose a new approach for finding global solutions of mixed-integer nonlinear optimization problems with ordinary differential equation constraints on networks. Instead of using a first discretize then optimize approach, we combine spatial and variable branching with appropriate discretizations of the differential equations to derive relaxations of the original problem. To construct the relaxations we derive convex under- and concave over-estimators for the ODE solution operators using numerical discretization schemes. Thereby, we make use of the underlying network structure, where the solutions of the ODEs only need to be known at a finite number of points. This property enables us to adaptively refine the discretization and relaxation without introducing new variables. The incorporation into a spatial branch-and-bound process allows to compute global epsilon-optimal solutions or decide infeasibility. We prove that this algorithm terminates finitely under some natural assumptions. We then show how this approach works for the example of stationary gas transport and provide some illustrative computational examples.}, language = {en} } @article{SchmidtAssmannBurlacuetal.2017, author = {Schmidt, Martin and Aßmann, Denis and Burlacu, Robert and Humpola, Jesco and Joormann, Imke and Kanelakis, Nikolaos and Koch, Thorsten and Oucherif, Djamal and Pfetsch, Marc E. and Schewe, Lars and Schwarz, Robert and Sirvent, Mathias}, title = {GasLib - A Library of Gas Network Instances}, series = {Data}, volume = {4}, journal = {Data}, number = {2}, doi = {10.3390/data2040040}, pages = {18}, year = {2017}, abstract = {The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML format. In this paper, we explain this format and present the instances that are available in the library.}, language = {en} } @unpublished{KlimmPfetschRaberetal.2021, author = {Klimm, Max and Pfetsch, Marc and Raber, Rico and Skutella, Martin}, title = {Approximation of Binary Second Order Cone Programs of Packing Type}, year = {2021}, abstract = {This paper considers binary second order cone programs of packing type where a linear objective is optimized under m second order cone packing constraints and all decision variables are binary. We show that when m is part of the input, these problems cannot be approximated within a factor of 1/(m + 1)^(1-ε) for any ε > 0, unless P = NP. We then propose approximation algorithms based on different algorithmic principles that almost match this approximation factor: a pipage rounding technique that solves fractional relaxations of the problems and modifies the solutions so that few fractional variables remain, a greedy approach, and a randomized rounding technique. While all algorithms have similar theoretical approximation guarantees in the order of 1/m, we also test the algorithms on realistic instances that arise in the context of gas transportation networks. This empirical study reveals in particular that taking the best of the proposed algorithms produces highly competitive solutions that yield on average 96 \% of the value of an optimal solution.}, language = {en} } @unpublished{KlimmPfetschRaberetal.2021, author = {Klimm, Max and Pfetsch, Marc and Raber, Rico and Skutella, Martin}, title = {Reduction of Potential-Based Flow Networks}, year = {2021}, abstract = {We consider potential-based flow networks with terminal nodes at which flow can enter or leave the network and physical properties such as voltages or pressures are measured and controlled. We study conditions under which such a network can be reduced to a smaller, equivalent network with the same behavior at the terminal nodes. Potential-based flow networks are widely used to model infrastructure networks such as electricity, gas, or water networks. In contrast to Kron's reduction for electrical networks, we prove that, in general, potential-based flow networks with at least three terminals cannot be reduced to smaller networks whose size only depends on the number of terminals. On the other hand, we show that it is possible to represent a special class of potential-based flow networks by a complete graph on the terminals, and we establish a characterization of networks that can be reduced to a path network. Our results build on fundamental properties of effective resistances proved in this paper, including explicit formulae for their dependence on edge resistances of the network and their metric properties.}, language = {en} } @unpublished{KlimmPfetschRaberetal.2019, author = {Klimm, Max and Pfetsch, Marc E. and Raber, Rico and Skutella, Martin}, title = {Packing under Convex Quadratic Constraints}, year = {2019}, abstract = {We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are APX-hard to approximate and present constant-factor approximation algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the golden ratio; (2) a greedy strategy; (3) a randomized rounding method leading to an approximation algorithm for the more general case with multiple convex quadratic constraints. We further show that a combination of the first two strategies can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of the three algorithms for problem instances arising in the context of real-world gas transport networks.}, language = {en} } @unpublished{KlimmPfetschRaberetal.2019, author = {Klimm, Max and Pfetsch, Marc and Raber, Rico and Skutella, Martin}, title = {On the Robustness of Potential-Based Flow Networks}, year = {2019}, abstract = {Potential-based flows provide a simple yet realistic mathematical model of transport in many real-world infrastructure networks such as, e.g., electricity, gas, or water networks, where the flow along each edge is controlled via the (difference of) potentials at its end nodes. A potential-based flow network is robust if the maximal difference of node potentials needed to satisfy a set of demands cannot increase if demands are decreased. This notion of robustness is motivated by infrastructure networks where users first make reservations for certain demands that may be larger than the actual amounts sent later on. Here node potentials correspond to physical quantities such as the pressures or the voltages and must be guaranteed to lie within a fixed range, even if the actual amounts are smaller than the previously reserved demands. Our main results are a precise characterization of such robust networks for the case of point-to-point demands via forbidden node-labeled graph minors, as well as an efficient algorithm for testing robustness.}, language = {en} } @unpublished{HabeckPfetsch2020, author = {Habeck, Oliver and Pfetsch, Marc E.}, title = {Combinatorial Acyclicity Models for Potential-based Flows}, year = {2020}, abstract = {Potential-based flows constitute a basic model to represent physical behavior in networks. Under natural assumptions, the flow in such networks must be acyclic. The goal of this paper is to exploit this property for the solution of corresponding optimization problems. To this end, we introduce several combinatorial models for acyclic flows, based on binary variables for flow directions. We compare these models and introduce a particular model that tries to capture acyclicity together with the supply/demand behavior. We analyze properties of this model, including variable fixing rules. Our computational results show that the usage of the corresponding constraints speeds up solution times by about a factor of 3 on average and a speed-up of a factor of almost 5 for the time to prove optimality.}, language = {en} } @article{BurlacuEggerGrossetal.2019, author = {Burlacu, Robert and Egger, Herbert and Groß, Martin and Martin, Alexander and Pfetsch, Marc and Schewe, Lars and Sirvent, Mathias and Skutella, Martin}, title = {Maximizing the storage capacity of gas networks: a global MINLP approach}, volume = {20}, edition = {Optimization and Engineering}, doi = {10.1007/s11081-018-9414-5}, pages = {543 -- 573}, year = {2019}, abstract = {In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations (PDEs). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic PDEs and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem (MINLP). For the numerical solution of the MINLP, we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems (MIPs). Finally, we present two case studies that illustrate the applicability of our approach.}, language = {en} } @article{StrelowGerischLangetal., author = {Strelow, Erik Laurin and Gerisch, Alf and Lang, Jens and Pfetsch, Marc E.}, title = {Physics-Informed Neural Networks: A Case Study for Gas Transport Problems}, volume = {Journal of Computational Physics}, number = {Vol. 481}, pages = {112041}, abstract = {Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method.}, language = {en} }