TY - JOUR A1 - Geißler, Björn A1 - Morsi, Antonio A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Solving Highly Detailed Gas Transport MINLPs: Block Separability and Penalty Alternating Direction Methods JF - INFORMS Journal on Computing N2 - Detailed modeling of gas transport problems leads to nonlinear and nonconvex mixed-integer optimization or feasibility models (MINLPs) because both the incorporation of discrete controls of the network as well as accurate physical and technical modeling is required in order to achieve practical solutions. Hence, ignoring certain parts of the physics model is not valid for practice. In the present contribution we extend an approach based on linear relaxations of the underlying nonlinearities by tailored model reformulation techniques yielding block-separable MINLPs. This combination of techniques allows us to apply a penalty alternating direction method and thus to solve highly detailed MINLPs for large-scale real-world instances. The practical strength of the proposed method is demonstrated by a computational study in which we apply the method to instances from steady-state gas transport including both pooling effects with respect to the mixing of gases of different composition and a highly detailed compressor station model. Y1 - 2016 IS - 30(2) SP - 309 EP - 323 ER - TY - JOUR A1 - Geißler, Björn A1 - Morsi, Antonio A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Penalty Alternating Direction Methods for Mixed-Integer Optimization: A New View on Feasibility Pumps JF - SIAM Journal on Optimization N2 - Feasibility pumps are highly effective primal heuristics for mixed-integer linear and nonlinear optimization. However, despite their success in practice there are only few works considering their theoretical properties. We show that feasibility pumps can be seen as alternating direction methods applied to special reformulations of the original problem, inheriting the convergence theory of these methods. Moreover, we propose a novel penalty framework that encompasses this alternating direction method, which allows us to refrain from random perturbations that are applied in standard versions of feasibility pumps in case of failure. We present a convergence theory for the new penalty based alternating direction method and compare the new variant of the feasibility pump with existing versions in an extensive numerical study for mixed-integer linear and nonlinear problems. Y1 - 2017 U6 - https://doi.org/10.1137/16M1069687 VL - 27 IS - 3 SP - 1611 EP - 1636 ER - TY - JOUR A1 - Grimm, Veronika A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - Uniqueness of Market Equilibrium on a Network: A Peak-Load Pricing Approach JF - European Journal of Operational Research N2 - In this paper we analyze peak-load pricing in the presence of network constraints. In our setup, firms facing fluctuating demand decide on the size and location of production facilities. They make production decisions constrained by the invested capacities, taking into account that market prices reflect scarce transmission capacities. We state general conditions for existence and uniqueness of the market equilibrium and provide a characterization of equilibrium investment and production. The presented analysis covers the cases of perfect competition and monopoly - the case of strategic firms is approximated by a conjectural variations approach. Our result is a prerequisite for analyzing regulatory policy options with computational multilevel equilibrium models, since uniqueness of the equilibrium at lower levels is of key importance when solving these models. Thus, our paper contributes to an evolving strand of literature that analyzes regulatory policy based on computational multilevel equilibrium models and aims at taking into account individual objectives of various agents, among them not only generators and customers but also, e.g., the regulator deciding on network expansion. KW - Pricing KW - Peak-Load Pricing KW - Networks KW - Uniqueness Y1 - 2017 U6 - https://doi.org/10.1016/j.ejor.2017.03.036 VL - 261 IS - 3 SP - 971 EP - 983 ER - TY - JOUR A1 - Geißler, Björn A1 - Morsi, Antonio A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Solving Power-Constrained Gas Transportation Problems using an MIP-based Alternating Direction Method JF - Computers & Chemical Engineering N2 - We present a solution algorithm for problems from steady-state gas transport optimization. Due to nonlinear and nonconvex physics and engineering models as well as discrete controllability of active network devices, these problems lead to difficult nonconvex mixed-integer nonlinear optimization models. The proposed method is based on mixed-integer linear techniques using piecewise linear relaxations of the nonlinearities and a tailored alternating direction method. Most other publications in the field of gas transport optimization only consider pressure and flow as main physical quantities. In this work, we additionally incorporate heat power supplies and demands as well as a mixing model for different gas qualities. We demonstrate the capabilities of our method on Germany's largest transport networks and hereby present numerical results on the largest instances that were ever reported in the literature for this problem class. Y1 - 2016 U6 - https://doi.org/10.1016/j.compchemeng.2015.07.005 VL - 82 IS - 2 SP - 303 EP - 317 ER - TY - JOUR A1 - Reuß, Markus A1 - Welder, Lara A1 - Thürauf, Johannes A1 - Linßen, Jochen A1 - Grube, Thomas A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Stolten, Detlef A1 - Robinius, Martin T1 - Modeling Hydrogen Networks for Future Energy Systems: A Comparison of Linear and Nonlinear Approaches JF - International Journal of Hydrogen Energy N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1016/j.ijhydene.2019.10.080 ER - TY - INPR A1 - Grimm, Veronika A1 - Hintermüller, Michael A1 - Huber, Olivier A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - A PDE-Constrained Generalized Nash Equilibrium Approach for Modeling Gas Markets with Transport N2 - We investigate a class of generalized Nash equilibrium problems (GNEPs) in which the objectives of the individuals are interdependent and the shared constraint consists of a system of partial differential equations. This setup is motivated by the modeling of strategic interactions of competing firms, which explicitly take into account the dynamics of transporting a commodity, such as natural gas, through a network. We establish the existence of a variational equilibrium of the GNEP. In the case of symmetric firms, we identify an equivalent optimization problem. We use this model to numerically explore the impact of linepacking, that is the use of the network as a temporary storage device. In particular, we study the firms' decisions under various linepacking abilities and analyze which market participants benefit from it. Y1 - ER - TY - INPR A1 - Halbig, Katrin A1 - Hümbs, Lukas A1 - Rösel, Florian A1 - Schewe, Lars A1 - Weninger, Dieter T1 - Computing optimality certificates for convex mixed-integer nonlinear problems N2 - Every optimization problem has a corresponding verification problem which verifies whether a given optimal solution is in fact optimal. In the literature there are a lot of such ways to verify optimality for a given solution, e.g., the branch-and-bound tree. To simplify this task, Baes et al. introduced optimality certificates for convex mixed-integer nonlinear programs and proved that these are bounded in the number of integer variables. We introduce an algorithm to compute the certificates and conduct computational experiments. Through the experiments we show that the optimality certificates can be surprisingly small. Y1 - 2021 ER -