A Bilevel Optimization Approach to Decide the Feasibility of Bookings in the European Gas Market - Plein, Fraenk; Thürauf, Johannes; Labbe, Martine; Schmidt, Martin (Mathematical Methods of Operations Research, 2021)
The European gas market is organized as a so-called entry-exit system with the main goal to decouple transport and trading. To this end, gas traders and the transmission system operator (TSO) sign so-called booking contracts that grant capacity rights to traders to inject or withdraw gas at certain nodes up to this capacity. On a day-ahead basis, traders then nominate the actual amount of gas within the previously booked capacities. By signing a booking contract, the TSO guarantees that all nominations within the booking bounds can be transported through the network. This results in a highly challenging mathematical problem. Using potential-based flows to model stationary gas physics, feasible bookings on passive networks, , networks without controllable elements, have been characterized in the recent literature. In this paper, we consider networks with linearly modeled active elements such as compressors or control valves. Since these active elements allow the TSO to control the gas flow, the single-level approaches for passive networks from the literature are no longer applicable. We thus present a bilevel model to decide the feasibility of bookings in networks with active elements. While this model is well-defined for general active networks, we focus on the class of networks for which active elements do not lie on cycles. This assumption allows us to reformulate the original bilevel model such that the lower-level problem is linear for every given upper-level decision. Consequently, we derive several single-level reformulations for this case. Besides the classic Karush–Kuhn–Tucker reformulation, we obtain three problem-specific optimal-value-function reformulations. The latter also lead to novel characterizations of feasible bookings in networks with active elements that do not lie on cycles. We compare the performance of our methods by a case study based on data from the GasLib.<br

Computing Technical Capacities in the European Entry-Exit Gas Market is NP-Hard - Schewe, Lars; Schmidt, Martin; Thürauf, Johannes (Annals of Operations Research, 2020)
As a result of its liberalization, the European gas market is organized as an entry-exit system in order to decouple the trading and transport of natural gas. Roughly summarized, the gas market organization consists of four subsequent stages. First, the transmission system operator (TSO) is obliged to allocate so-called maximal technical capacities for the nodes of the network. Second, the TSO and the gas traders sign mid- to long-term capacity-right contracts, where the capacity is bounded above by the allocated technical capacities. These contracts are called bookings. Third, on a day-ahead basis, gas traders can nominate the amount of gas that they inject or withdraw from the network at entry and exit nodes, where the nominated amount is bounded above by the respective booking. Fourth and finally, the TSO has to operate the network such that the nominated amounts of gas can be transported. By signing the booking contract, the TSO guarantees that all possibly resulting nominations can indeed be transported. Consequently, maximal technical capacities have to satisfy that all nominations that comply with these technical capacities can be transported through the network. This leads to a highly challenging mathematical optimization problem. We consider the specific instantiations of this problem in which we assume capacitated linear as well as potential-based flow models. In this contribution, we formally introduce the problem of Computing Technical Capacities (CTC) and prove that it is NP-complete. To this end, we first reduce the Subset Sum problem to CTC for the case of capacitated linear flows in trees. Afterward, we extend this result to CTC with potential-based flows and show that this problem is also NP-complete on trees by reducing it to the case of capacitated linear flow. Since the hardness results are obtained for the easiest case, i.e., on tree-shaped networks with capacitated linear as well as potential-based flows, this implies the hardness of CTC for more general graph classes.<br
CRIS-Team Scopus Importer:2020-08-21
Keywords: European Entry-Exit Gas Market; Technical Capacities; Potential-Based Flows; Computational Complexity; NP-Hardness

Deciding Feasibility of a Booking in the European Gas Market on a Cycle is in P for the Case of Passive Networks - Labbé, Martine; Plein, Fränk; Schmidt, Martin; Thürauf, Johannes (Networks, 2021)
We show that the feasibility of a booking in the European entry-exit gas market can be decided in polynomial time on single-cycle networks that are passive, i.e., do not contain controllable elements. The feasibility of a booking can be characterized by solving polynomially many nonlinear potential-based flow models for computing so-called potential-difference maximizing load flow scenarios. We thus analyze the structure of these models and exploit both the cyclic graph structure as well as specific properties of potential-based flows. This enables us to solve the decision variant of the nonlinear potential-difference maximization by reducing it to a system of polynomials of constant dimension that is independent of the cycle's size. This system of fixed dimension can be handled with tools from real algebraic geometry to derive a polynomial-time algorithm. The characterization in terms of potential-difference maximizing load flow scenarios then leads to a polynomial-time algorithm for deciding the feasibility of a booking. Our theoretical results extend the existing knowledge about the complexity of deciding the feasibility of bookings from trees to single-cycle networks. <br
Keywords: Gas networks; European entry-exit market; Bookings; Potential-based flows; Computational complexity; Cycle

Global optimization for the multilevel European gas market system with nonlinear flow models on trees - Schewe, Lars; Schmidt, Martin; Thürauf, Johannes (Journal of Global Optimization, 2022)
The European gas market is implemented as an entry-exit system, which aims to decouple transport and trading of gas. It has been modeled in the literature as a multilevel problem, which contains a nonlinear flow model of gas physics. Besides the multilevel structure and the nonlinear flow model, the computation of so-called technical capacities is another major challenge. These lead to nonlinear adjustable robust constraints that are computationally intractable in general. We provide techniques to equivalently reformulate these nonlinear adjustable constraints as finitely many convex constraints including integer variables in the case that the underlying network is tree-shaped. We further derive additional combinatorial constraints that significantly speed up the solution process. Using our results, we can recast the multilevel model as a single-level nonconvex mixed-integer nonlinear problem, which we then solve on a real-world network, namely the Greek gas network, to global optimality. Overall, this is the first time that the considered multilevel entry-exit system can be solved for a real-world sized network and a nonlinear flow model.
CRIS-Team WoS Importer:2022-01-14

Modeling Hydrogen Networks for Future Energy Systems: A Comparison of Linear and Nonlinear Approaches - Reuß, Markus; Welder, Lara; Thürauf, Johannes; Linßen, Jochen; Grube, Thomas; Schewe, Lars; Schmidt, Martin; Stolten, Detlef; Robinius, Martin (International Journal of Hydrogen Energy, 2019)
Common energy system models that integrate hydrogen transport in pipelines typically simplify fluid flow models and reduce the network size in order to achieve solutions quickly. This contribution analyzes two different types of pipeline network topologies (namely, star and tree networks) and two different fluid flow models (linear and nonlinear) for a given hydrogen capacity scenario of electrical reconversion in Germany to analyze the impact of these simplifications. For each network topology, robust demand and supply scenarios are generated. The results show that a simplified topology, as well as the consideration of detailed fluid flow, could heavily influence the total pipeline investment costs. For the given capacity scenario, an overall cost reduction of the pipeline costs of 37% is observed for the star network with linear cost compared to the tree network with nonlinear fluid flow. The impact of these improvements regarding the total electricity reconversion costs has led to a cost reduction of 1.4%, which is fairly small. Therefore, the integration of nonlinearities into energy system optimization models is not recommended due to their high computational burden. However, the applied method for generating robust demand and supply scenarios improved the credibility and robustness of the network topology, while the simplified fluid flow consideration can lead to infeasibilities. Thus, we suggest the utilization of the nonlinear model for post-processing to prove the feasibility of the results and strengthen their credibility, while retaining the computational performance of linear modeling.
Keywords: Hydrogen reconversion; Hydrogen infrastructure; Spatial resolution; Pipeline design optimization; Pressure drop; Robust optimization

On electricity market equilibria with storage: Modeling, uniqueness, and a distributed ADMM - Grübel, Julia; Kleinert, Thomas; Krebs, Vanessa; Orlinskaya, Galina; Schewe, Lars; Schmidt, Martin; Thürauf, Johannes (Computers & Operations Research, 2020)
We consider spot-market trading of electricity including storage operators as additional agents besides producers and consumers. Storage devices allow for shifting produced electricity from one time period to a later one. Due to this, multiple market equilibria may occur even if classical uniqueness assumptions for the case without storage systems are satisfied. For models containing storage operators, we derive sufficient conditions that ensure uniqueness of generation and demand. We also prove uniqueness of the market equilibrium for the special case of a single storage operator. Nevertheless, in case of multiple storage operators, uniqueness fails to hold in general, which we show by illustrative examples. We conclude the theoretical discussion with a general ex-post condition for proving the uniqueness of a given solution. In contrast to classical settings without storage systems, the computation of market equilibria is much more challenging since storage operations couple all trading events over time. For this reason, we propose a tailored parallel and distributed alternating direction method of multipliers (ADMM) for efficiently computing spot-market equilibria over long time horizons. We first analyze the parallel performance of the method itself. Finally, we show that the parallel ADMM clearly outperforms solving the respective problems directly and that it is capable of solving instances with more than 42 million variables in less than 13 min.<br
Keywords: Electricity markets; Storage; Market equilibria; Uniqueness; ADMM; Parallel computing; Large-scale optimization

Radius of Robust Feasibility for Mixed-Integer Problems - Liers, Frauke; Schewe, Lars; Thürauf, Johannes (Informs Journal on Computing, 2021)
For a mixed-integer linear problem (MIP) with uncertain constraints, the radius of robust feasibility (RRF) determines a value for the maximal “size” of the uncertainty set such that robust feasibility of the MIP can be guaranteed. To the best of our knowledge, the approaches for the RRF presented in the literature are restricted to continuous optimization problems. In this work, we first analyze relations between the RRF of a MIP and its continuous linear programming relaxation. In particular, we derive conditions under which a MIP and its continuous relaxation have the same RRF. Afterward, we extend the notion of the RRF such that it can be applied and computed for a large variety of optimization problems and uncertainty sets. In contrast to the setting commonly used in the literature, we consider for every constraint a potentially different uncertainty set that is not necessarily full-dimensional. Thus, we generalize the concept of the RRF to MIPs including “safe” variables and constraints, i.e., where uncertainties do not affect certain variables or constraints. In this extended setting, we again analyze relations between the RRF for a MIP and its continuous relaxation. Afterward, we present methods for computing the RRF for continuous linear as well as mixed-integer problems with safe constraints and variables. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 library for the calculation of the RR
Keywords: Robust Optimization; Mixed-integer programming; Uncertainty sets; Robust feasibility

Robust optimal discrete arc sizing for tree-shaped potential networks - Robinius, Martin; Schewe, Lars; Schmidt, Martin; Stolten, Detlef; Thürauf, Johannes; Welder, Lara (Computational Optimization and Applications, 2019)
We consider the problem of discrete arc sizing for tree-shaped potential networks with respect to infinitely many demand scenarios. This means that the arc sizes need to be feasible for an infinite set of scenarios. The problem can be seen as a strictly robust counterpart of a single-scenario network design problem, which is shown to be NP-complete even on trees. In order to obtain a tractable problem, we introduce a method for generating a finite scenario set such that optimality of a sizing for this finite set implies the sizing's optimality for the originally given infinite set of scenarios. We further prove that the size of the finite scenario set is quadratically bounded above in the number of nodes of the underlying tree and that it can be computed in polynomial time. The resulting problem can then be solved as a standard mixed-integer linear optimization problem. Finally, we show the applicability of our theoretical results by computing globally optimal arc sizes for a realistic hydrogen transport network of Eastern German
Keywords: Discrete arc sizing; Mixed-integer linear optimization; Potential networks; Robust optimization; Scenario generation

Structural Properties of Feasible Bookings in the European Entry-Exit Gas Market System - Schewe, Lars; Schmidt, Martin; Thürauf, Johannes (4OR-A Quarterly Journal of Operations Research, 2019)
In this work we analyze the structural properties of the set of feasible bookings in the European entry-exit gas market system. We present formal definitions of feasible bookings and then analyze properties that are important if one wants to optimize over them. Thus, we study whether the sets of feasible nominations and bookings are bounded, convex, connected, conic, and star-shaped. The results depend on the specific model of gas flow in a network. Here, we discuss a simple linear flow model with arc capacities as well as nonlinear and mixed-integer nonlinear models of passive and active networks, respectively. It turns out that the set of feasible bookings has some unintuitive properties. For instance, we show that the set is nonconvex even though only a simple linear flow model is use
Keywords: Gas networks; Booking; Entry-exit system; Convexity; Flow models