TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Sirvent, Mathias A1 - Wintergerst, David T1 - MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems JF - Computational Optimization and Applications N2 - We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network. KW - Mixed-integer optimal control KW - Instantaneous control KW - Partial differential equations on graphs KW - Gas networks KW - Mixed-integer linear optimization Y1 - 2017 U6 - https://doi.org/10.1007/s10589-017-9970-1 VL - 70 IS - 1 SP - 267 EP - 294 ER - TY - JOUR A1 - Burlacu, Robert A1 - Egger, Herbert A1 - Groß, Martin A1 - Martin, Alexander A1 - Pfetsch, Marc A1 - Schewe, Lars A1 - Sirvent, Mathias A1 - Skutella, Martin T1 - Maximizing the storage capacity of gas networks: a global MINLP approach N2 - 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. KW - Mixed-Integer Nonlinear Programming KW - Transient Gas Transport Optimization KW - Storage Capacity Maximization KW - Power-to-Gas KW - First-Discretize-Then-Optimize Y1 - 2019 U6 - https://doi.org/10.1007/s11081-018-9414-5 VL - 20 SP - 543 EP - 573 ET - Optimization and Engineering ER - TY - JOUR A1 - Grimm, Veronika A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Weibelzahl, Martin A1 - Zöttl, Gregor T1 - Transmission and generation investment in electricity markets: The effects of market splitting and network fee regimes JF - European Journal of Operational Research N2 - We propose an equilibrium model that allows to analyze the long-run impact of the electricity market design on transmission line expansion by the regulator and investment in generation capacity by private firms in liberalized electricity markets. The model incorporates investment decisions of the transmission system operator and private firms in expectation of an energy-only market and cost-based redispatch. In different specifications we consider the cases of one vs. multiple price zones (market splitting) and analyze different approaches to recover network cost—in particular lump sum, generation capacity based, and energy based fees. In order to compare the outcomes of our multilevel market model with a first best benchmark, we also solve the corresponding integrated planner problem. Using two test networks we illustrate that energy-only markets can lead to suboptimal locational decisions for generation capacity and thus imply excessive network expansion. Market splitting heals these problems only partially. These results are valid for all considered types of network tariffs, although investment slightly differs across those regimes. KW - Electricity market modeling KW - Mixed-integer nonlinear optimization KW - Multilevel programming KW - Network expansion KW - Transmission management Y1 - 2016 U6 - https://doi.org/10.1016/j.ejor.2016.03.044 VL - 254 IS - 2 SP - 493 EP - 509 ER - TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Sirvent, Mathias A1 - Wintergerst, David T1 - Towards Simulation Based Mixed-Integer Optimization with Differential Equations JF - Networks N2 - We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with “black-box” nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with an approximate feasible global optimal solution of the mixed integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady-state gas transport. For the latter we also present promising numerical results of our method applied to real-world instances that particularly show the effectiveness of our method for problems defined on networks. KW - Mixed-Integer Optimization KW - Simulation Based Optimization KW - Optimization with Differential Equations KW - Decomposition Method KW - Gas Transport Networks Y1 - 2018 U6 - https://doi.org/10.1002/net.21812 ER - TY - JOUR A1 - Krug, Richard A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Weninger, Dieter T1 - Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems with Mixed Two-Point Boundary Conditions JF - Control and Cybernetics N2 - In this article, we continue our work (Krug et al., 2021) on time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations in that we now consider mixed two-point boundary value problems and provide an in-depth well-posedness analysis. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains. KW - Time-domain decomposition KW - Optimal control KW - Semilinear hyperbolic systems KW - Convergence Y1 - 2021 ER - TY - JOUR A1 - Krug, Richard A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Weninger, Dieter T1 - Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems JF - SIAM Journal on Control and Optimization N2 - In this article, we extend the time-domain decomposition method described by Lagnese and Leugering (2003) to semilinear optimal control problems for hyperbolic balance laws with spatio-temporal varying coefficients. We provide the design of the iterative method applied to the global first-order optimality system, prove its convergence, and derive an a posteriori error estimate. The analysis is done entirely on the continuous level. A distinguishing feature of the method is that the decomposed optimality system can be interpreted as an optimality system of a local "virtual" optimal control problem. Thus, the iterative time-domain decomposition of the optimality system can be interpreted as an iterative parallel scheme for virtual optimal control problems on the subintervals. A typical example and further comments are given to show the range of potential applications. Moreover, we provide some numerical experiments to give a first interpretation of the role of the parameters involved in the iterative process. KW - Time-domain decomposition KW - Optimal control KW - Semilinear hyperbolic systems KW - Convergence KW - A posteriori error estimates Y1 - 2020 ER - TY - INPR A1 - Aigner, Kevin-Martin A1 - Burlacu, Robert A1 - Liers, Frauke A1 - Martin, Alexander T1 - Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality N2 - We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions. The latter occur, for instance, in the context of the curtailment of renewables or the switching of power generation units and transmission lines. Our approach delivers globally optimal solutions and is provably convergent. We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs. The solution method starts from a known framework that uses piecewise linear relaxations. These relaxations are modeled as as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled. In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements. In particular, these are mixed-integer second-order cone programs as well as primal and dual cutting planes. For example objective cuts and no-good-cuts help to compute good feasible solutions as where outer approximation constraints tighten the relaxations. We present extensive numerical results for various AC OPF problems where discrete decisions play a major role. Even for hard instances with a large proportion of discrete decisions, the method is able to generate high quality solutions efficiently. Furthermore, we compare our approach with state-of-the-art MINLP. Our method outperforms all other algorithms. KW - Mixed-Integer Nonlinear Programming KW - Second-Order Cone Programming KW - AC Optimal Power Flow KW - Discrete Decisions KW - Piecewise Linear Relaxation Y1 - 2020 ER - TY - INPR A1 - Gugat, Martin A1 - Krug, Richard A1 - Martin, Alexander T1 - Transient gas pipeline flow: Analytical examples, numerical simulation and a comparison to the quasi-static approach N2 - The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a pde model is necessary to handle certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based upon multi-period optimization of steady states does not lead to approximations that converge to the optimal state. Y1 - 2021 U6 - https://doi.org/10.1007/s11081-021-09690-4 ER - TY - INPR A1 - Grimm, Veronika A1 - Martin, Alexander A1 - Sölch, Christian A1 - Weibelzahl, Martin A1 - Zöttl, Gregor T1 - Market-Based Redispatch May Result in Inefficient Dispatch N2 - In this paper we analyze a uniform price electricity spot market that is followed by redispatch in the case of network congestion. We assume that the transmission system operator is incentivized to minimize redispatch cost and compare a cost-based redispatch (CBR) to a market-based redispatch (MBR) mechanism. For networks with at least three nodes we show that in contrast to CBR, in the case of MBR the redispatch cost minimizing allocation may not be short-run efficient. As we demonstrate, in case of MBR the possibility of the transmission system operator to reduce redispatch cost at the expense of a reduced welfare may be driven by the electricity supply side or the electricity demand side. If, however, the transmission system operator is obliged to implement the welfare maximizing (instead of the redispatch cost minimizing) dispatch by regulation, this will result in an efficient dispatch also in case of MBR. KW - Electricity Markets, Redispatch, Congestion Management, Computational Equilibrium Models Y1 - 2019 ER - TY - INPR A1 - Schmidt, Martin A1 - Hiller, Benjamin A1 - Koch, Thorsten A1 - Pfetsch, Marc A1 - Geißler, Björn A1 - Henrion, René A1 - Joormann, Imke A1 - Martin, Alexander A1 - Morsi, Antonio A1 - Römisch, Werner A1 - Schewe, Lars A1 - Schultz, Rüdiger A1 - Steinbach, Marc C. T1 - Capacity Evaluation for Large-Scale Gas Networks N2 - 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. Y1 - 2019 ER -