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 - Kleinert, Thomas A1 - Manns, Julian A1 - Schmidt, Martin A1 - Weninger, Dieter T1 - Presolving Linear Bilevel Optimization Problems JF - EURO Journal on Computational Optimization N2 - Linear bilevel optimization problems are known to be strongly NP-hard and the computational techniques to solve these problems are often motivated by techniques from single-level mixed-integer optimization. Thus, during the last years and decades many branch-and-bound methods, cutting planes, or heuristics have been proposed. On the other hand, there is almost no literature on presolving linear bilevel problems although presolve is a very important ingredient in state-of-the-art mixed-integer optimization solvers. In this paper, we carry over standard presolve techniques from single-level optimization to bilevel problems and show that this needs to be done with great caution since a naive application of well-known techniques does often not lead to correctly presolved bilevel models. Our numerical study shows that presolve can also be very beneficial for bilevel problems but also highlights that these methods have a more heterogeneous effect on the solution process compared to what is known from single-level optimization. As a side result, our numerical experiments reveal that there is an urgent need for better and more heterogeneous test instance libraries to further propel the field of computational bilevel optimization. KW - Linear Bilevel Optimization KW - Presolve KW - Computational Analysis Y1 - 2021 U6 - https://doi.org/10.1016/j.ejco.2021.100020 IS - 9 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 - JOUR A1 - Kleinert, Thomas A1 - Labbé, Martine A1 - Ljubić, Ivana A1 - Schmidt, Martin T1 - A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization JF - EURO Journal on Computational Optimization N2 - Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and practice. The scientific interest in computational bilevel optimization increased a lot over the last decade and is still growing. Independent of whether the bilevel problem itself contains integer variables or not, many state-of-the-art solution approaches for bilevel optimization make use of techniques that originate from mixed-integer programming. These techniques include branch-and-bound methods, cutting planes and, thus, branch-and-cut approaches, or problem-specific decomposition methods. In this survey article, we review bilevel-tailored approaches that exploit these mixed-integer programming techniques to solve bilevel optimization problems. To this end, we first consider bilevel problems with convex or, in particular, linear lower-level problems. The discussed solution methods in this field stem from original works from the 1980's but, on the other hand, are still actively researched today. Second, we review modern algorithmic approaches to solve mixed-integer bilevel problems that contain integrality constraints in the lower level. Moreover, we also briefly discuss the area of mixed-integer nonlinear bilevel problems. Third, we devote some attention to more specific fields such as pricing or interdiction models that genuinely contain bilinear and thus nonconvex aspects. Finally, we sketch a list of open questions from the areas of algorithmic and computational bilevel optimization, which may lead to interesting future research that will further propel this fascinating and active field of research. KW - Bilevel optimization KW - Mixed-integer programming KW - Applications KW - Branch-and-bound KW - Branch-and-cut Y1 - 2021 ER - TY - JOUR A1 - Böttger, Tom A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - The Cost of Decoupling Trade and Transport in the European Entry-Exit Gas Market with Linear Physics Modeling JF - European Journal of Operational Research N2 - Liberalized gas markets in Europe are organized as entry-exit regimes so that gas trade and transport are decoupled. The decoupling is achieved via the announcement of technical capacities by the transmission system operator (TSO) at all entry and exit points of the network. These capacities can be booked by gas suppliers and customers in long-term contracts. Only traders who have booked capacities up-front can "nominate" quantities for injection or withdrawal of gas via a day-ahead market. To ensure feasibility of the nominations for the physical network, the TSO must only announce technical capacities for which all possibly nominated quantities are transportable. In this paper, we use a four-level model of the entry-exit gas market to analyze possible welfare losses associated with the decoupling of gas trade and transport. In addition to the multilevel structure, the model contains robust aspects to cover the conservative nature of the European entry-exit system. We provide several reformulations to obtain a single-level mixed-integer quadratic problem. The overall model of the considered market regime is extremely challenging and we thus have to make the main assumption that gas flows are modeled as potential-based linear flows. Using the derived single-level reformulation of the problem, we show that the feasibility requirements for technical capacities imply significant welfare losses due to unused network capacity. Furthermore, we find that the specific structure of the network has a considerable influence on the optimal choice of technical capacities. Our results thus show that trade and transport are not decoupled in the long term. As a further source of welfare losses and discrimination against individual actors, we identify the minimum prices for booking capacity at the individual nodes. KW - Entry-Exit Gas Market KW - Gas Market Design KW - Multilevel Optimization KW - Robust Optimization Y1 - 2020 ER - TY - JOUR A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - Global Optimization for the Multilevel European Gas Market System with Nonlinear Flow Models on Trees JF - Journal of Global Optimization N2 - 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. Y1 - 2020 U6 - https://doi.org/10.1007/s10898-021-01099-8 ER - TY - JOUR A1 - Beck, Yasmine A1 - Schmidt, Martin T1 - A Robust Approach for Modeling Limited Observability in Bilevel Optimization JF - Operations Research Letters N2 - Many applications of bilevel optimization contain a leader facing a follower whose reaction deviates from the one expected by the leader due to some kind of bounded rationality. We consider bilinear bilevel problems with follower's response uncertainty due to limited observability regarding the leader's decision and exploit robust optimization to model the decision making of the follower. We show that the robust counterpart of the lower level allows to tackle the problem via the lower level's KKT conditions. KW - Bilevel optimization KW - Robust optimization KW - Bounded rationality KW - Limited observability KW - Reformulations Y1 - 2021 IS - 49(5) SP - 752 EP - 758 ER - TY - JOUR A1 - Plein, Fränk A1 - Thürauf, Johannes A1 - Labbé, Martine A1 - Schmidt, Martin T1 - A Bilevel Optimization Approach to Decide the Feasibility of Bookings in the European Gas Market JF - Mathematical Methods of Operations Research N2 - 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, i.e., 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. KW - Gas networks KW - Bilevel optimization KW - European entry-exit market KW - Bookings KW - Active elements Y1 - 2021 U6 - https://doi.org/10.1007/s00186-021-00752-y ER - TY - JOUR A1 - Leugering, Günter T1 - Industrial Applications of Optimal Control for Partial Differential Equations on Networks: Reduction and Decomposition Methods Applied to the Discrete–Continuous Control of Gas Flow in Complex Pipe Systems N2 - This chapter provides an exemplary road map—in a nutshell—from a given industrial application, the control of gas networks, which is far too complex for a direct approach, to a problem that can be actually handled using well-known methods in control theory. It also provides an iterative non-overlapping domain decomposition that can be interpreted as an Uzawa method. The chapter outline two strategies. The first one can be seen as a Jacobi-type approach. In the second approach, fix the integer controls s and decompose the corresponding optimality system for the entire graph into the subgraphs Gk by a another, but very similar, non-overlapping domain decomposition. The problem is the intrinsic coupling of integer controls, continuous controls, and nonlinear dynamics on a metric graph. The idea is to introduce a virtual control that aims at controlling classical in homogeneous Neumann condition including the iteration history at the interface as inhomogeneity to the Robin-type condition that appears in the decomposition. Y1 - 2021 VL - Computational Science and Its Applications IS - 1st Edition SP - 25 EP - 40 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Grimm, Veronika A1 - Schmidt, Martin T1 - Outer Approximation for Global Optimization of Mixed-Integer Quadratic Bilevel Problems JF - Mathematical Programming (Series B) N2 - Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP bilevel problems, i.e., models with a mixed-integer convex-quadratic upper level and a continuous convex-quadratic lower level. This setting allows for a strong-duality-based transformation of the lower level which yields, in general, an equivalent nonconvex single-level reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi- and a single-tree outer-approximation-based cutting-plane algorithm. We show finite termination and correctness of both methods and present extensive numerical results that illustrate the applicability of the approaches. It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches. KW - Bilevel optimization KW - Outer approximation KW - Quadratic programming KW - Convex mixed-integer nonlinear optimization Y1 - 2019 ER - 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 - Kramer, Anja A1 - Krebs, Vanessa A1 - Schmidt, Martin T1 - Strictly and Γ-Robust Counterparts of Electricity Market Models: Perfect Competition and Nash-Cournot Equilibria JF - Operations Research Perspectives N2 - This paper mainly studies two topics: linear complementarity problems for modeling electricity market equilibria and optimization under uncertainty. We consider both perfectly competitive and Nash–Cournot models of electricity markets and study their robustifications using strict robustness and the Γ-approach. For three out of the four combinations of economic competition and robustification, we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition, this result corresponds to the two classical welfare theorems, which also apply in both considered robust cases that again yield convex robustified problems. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of Γ-robustifications of Nash–Cournot models. Thus, an analogue of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling. KW - Robust optimization KW - Linear complementarity problems KW - Electricity market equilibrium models KW - Perfect competition KW - Nash-Cournot competition Y1 - 2018 IS - 89(2) SP - 100197 ER - TY - JOUR A1 - Hojny, Christopher A1 - Joormann, Imke A1 - Lüthen, Hendrik A1 - Schmidt, Martin T1 - Mixed-Integer Programming Techniques for the Connected Max-k-Cut Problem JF - Mathematical Programming Computation N2 - We consider an extended version of the classical Max-k-Cut problem in which we additionally require that the parts of the graph partition are connected. For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics, and symmetry breaking. The main focus of this paper is an extensive numerical study in which we analyze the impact of the different techniques for various test sets. It turns out that the techniques from the existing literature are not sufficient to solve an adequate fraction of the test sets. However, our novel techniques significantly outperform the existing ones both in terms of running times and the overall number of instances that can be solved. KW - Max-cut KW - Connectivity KW - Branch-and-cut KW - Mixed-integer programming Y1 - 2018 ER - TY - JOUR A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - Structural Properties of Feasible Bookings in the European Entry-Exit Gas Market System JF - 4OR N2 - 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 used. KW - Gas networks KW - Booking KW - Entry-exit system KW - Convexity KW - Flow models Y1 - 2018 U6 - https://doi.org/10.1007/s10288-019-00411-3 IS - 18 SP - 197 EP - 218 ER - TY - JOUR A1 - Ambrosius, Mirjam A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Liers, Frauke A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - Endogenous Price Zones and Investment Incentives in Electricity Markets: An Application of Multilevel Optimization with Graph Partitioning JF - Energy Economics N2 - In the course of the energy transition, load and supply centers are growing apart in electricity markets worldwide, rendering regional price signals even more important to provide adequate locational investment incentives. This paper focuses on electricity markets that operate under a zonal pricing market design. For a fixed number of zones, we endogenously derive the optimal configuration of price zones and available transfer capacities on a network in order to optimally govern investment and production decisions in the long run. In a multilevel mixed-integer nonlinear model that contains a graph partitioning problem on the first level, we determine welfare-maximizing price zones and available transfer capacities for a given electricity market and analyze their impact on market outcomes. Using a generalized Benders decomposition approach developed in Grimm et al. (2019) and a problem-tailored scenario clustering for reducing the input data size, we are able to solve the model to global optimality even for large instances. We apply the approach to the German electricity market as an example to examine the impact of optimal zoning on key performance indicators such as welfare, generation mix and locations, or electricity prices. It turns out that even for a small number of price zones, an optimal configuration of zones induces a welfare level that almost approaches the first best. KW - Electricity Markets KW - Price Zones KW - Investment Incentives KW - Multilevel Optimization KW - Graph Partitioning Y1 - 2018 IS - 92 ER - TY - JOUR A1 - Hante, Falk M. A1 - Schmidt, Martin T1 - Convergence of Finite-Dimensional Approximations for Mixed-Integer Optimization with Differential Equations JF - Control and Cybernetics N2 - We consider a direct approach to solve mixed-integer nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach in the sense of the corresponding optimal values. The results are obtained by considering the discretized problem as a parametric mixed-integer nonlinear optimization problem in finite dimensions, where the maximum step size for discretizing the dynamics is the parameter. In this setting, we prove the continuity of the optimal value function under a stability assumption for the integer feasible set and second-order conditions from nonlinear optimization. We address the necessity of the conditions on the example of pipe sizing problems for gas networks. KW - Optimization with differential equations KW - Optimal value function KW - Lipschitz continuity KW - Parametric optimization KW - Mixed-integer nonlinear programming Y1 - 2018 ER - TY - JOUR A1 - Labbé, Martine A1 - Plein, Fränk A1 - Schmidt, Martin T1 - Bookings in the European Gas Market: Characterisation of Feasibility and Computational Complexity Results JF - Optimization and Engineering N2 - As a consequence of the liberalisation of the European gas market in the last decades, gas trading and transport have been decoupled. At the core of this decoupling are so-called bookings and nominations. Bookings are special capacity right contracts that guarantee that a specified amount of gas can be supplied or withdrawn at certain entry or exit nodes of the network. These supplies and withdrawals are nominated at the day-ahead. The special property of bookings then is that they need to be feasible, i.e., every nomination that complies with the given bookings can be transported. While checking the feasibility of a nomination can typically be done by solving a mixed-integer nonlinear feasibility problem, the verification of feasibility of a set of bookings is much harder. The reason is the robust nature of feasibility of bookings - namely that for a set of bookings to be feasible, all compliant nominations, i.e., infinitely many, need to be checked for feasibility. In this paper, we consider the question of how to verify the feasibility of given bookings for a number of special cases. For our physics model we impose a steady-state potential-based flow model and disregard controllable network elements. For this case we derive a characterisation of feasible bookings, which is then used to show that the problem is in coNP for the general case but can be solved in polynomial time for linear potential-based flow models. Moreover, we present a dynamic programming approach for deciding the feasibility of a booking in tree-shaped networks even for nonlinear flow models. It turns out that the hardness of the problem mainly depends on the combination of the chosen physics model as well as the specific network structure under consideration. Thus, we give an overview over all settings for which the hardness of the problem is known and finally present a list of open problems. KW - Gas networks KW - Booking KW - Nomination KW - Computational complexity KW - Trees Y1 - 2018 IS - 21(1) SP - 305 EP - 334 ER - TY - JOUR A1 - Krebs, Vanessa A1 - Schmidt, Martin T1 - Γ-Robust Linear Complementarity Problems JF - Optimization Methods and Software N2 - Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems. Specific examples are game-theoretic settings like the bimatrix game or energy market models like for electricity or natural gas. While optimization under uncertainties is rather well-developed, the field of equilibrium models represented by complementarity problems under uncertainty - especially using the concepts of robust optimization - is still in its infancy. In this paper, we extend the theory of strictly robust linear complementarity problems (LCPs) to Γ-robust settings, where existence of worst-case-hedged equilibria cannot be guaranteed. Thus, we study the minimization of the worst-case gap function of Γ-robust counterparts of LCPs. For box and l1-norm uncertainty sets we derive tractable convex counterparts for monotone LCPs and study their feasibility as well as the existence and uniqueness of solutions. To this end, we consider uncertainties in the vector and in the matrix defining the LCP. We additionally study so-called ρ-robust solutions, i.e., solutions of relaxed uncertain LCPs. Finally, we illustrate the Γ-robust concept applied to LCPs in the light of the above mentioned classical examples of bimatrix games and market equilibrium modeling. KW - Linear complementarity problems KW - Robust optimization KW - Optimization under uncertainty KW - Γ-robustness KW - Tractable counterparts Y1 - 2019 ER - TY - JOUR A1 - Krug, Richard A1 - Mehrmann, Volker A1 - Schmidt, Martin T1 - Nonlinear Optimization of District Heating Networks JF - Optimization and Engineering N2 - We develop a complementarity-constrained nonlinear optimization model for the time-dependent control of district heating networks. The main physical aspects of water and heat flow in these networks are governed by nonlinear and hyperbolic 1d partial differential equations. In addition, a pooling-type mixing model is required at the nodes of the network to treat the mixing of different water temperatures. This mixing model can be recast using suitable complementarity constraints. The resulting problem is a mathematical program with complementarity constraints subject to nonlinear partial differential equations describing the physics. In order to obtain a tractable problem, we apply suitable discretizations in space and time, resulting in a finite-dimensional optimization problem with complementarity constraints for which we develop a suitable reformulation with improved constraint regularity. Moreover, we propose an instantaneous control approach for the discretized problem, discuss practically relevant penalty formulations, and present preprocessing techniques that are used to simplify the mixing model at the nodes of the network. Finally, we use all these techniques to solve realistic instances. Our numerical results show the applicability of our techniques in practice. KW - District heating networks KW - Nonlinear optimization KW - Euler equations KW - Differential-algebraic equations KW - Complementarity constraints Y1 - 2019 IS - 22(2) SP - 783 EP - 819 ER - TY - JOUR A1 - Labbé, Martine A1 - Plein, Fränk A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - Deciding Feasibility of a Booking in the European Gas Market on a Cycle is in P for the Case of Passive Networks JF - Networks N2 - 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. KW - Gas networks KW - European entry-exit market KW - Bookings KW - Potential-based flows KW - Computational complexity Y1 - 2019 U6 - https://doi.org/10.1007/s00186-021-00752-y VL - 78 IS - 2 SP - 128 EP - 152 ER - TY - JOUR A1 - Egerer, Jonas A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - The Impact of Neighboring Markets on Renewable Locations, Transmission Expansion, and Generation Investment JF - European Journal of Operational Research N2 - Many long-term investment planning models for liberalized electricity markets either optimize for the entire electricity system or focus on confined jurisdictions, abstracting from adjacent markets. In this paper, we provide models for analyzing the impact of the interdependencies between a core electricity market and its neighboring markets on key long-run decisions. This we do both for zonal and nodal pricing schemes. The identification of welfare optimal investments in transmission lines and renewable capacity within a core electricity market requires a spatially restricted objective function, which also accounts for benefits from cross-border electricity trading. This leads to mixed-integer nonlinear multilevel optimization problems with bilinear nonconvexities for which we adapt a Benders-like decomposition approach from the literature. In a case study, we use a stylized six-node network to disentangle different effects of optimal regional (as compared to supra-regional) investment planning. Regional planning alters investment in transmission and renewable capacity in the core region, which affects private investment in generation capacity also in adjacent regions and increases welfare in the core region at the cost of system welfare. Depending on the congestion-pricing scheme, the regulator of the core region follows different strategies to increase welfare causing distributional effects among stakeholders. KW - Neighboring Markets KW - Renewables KW - Network Expansion KW - Multilevel Optimization KW - Benders Decomposition Y1 - 2019 ER - TY - JOUR A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - Computing Technical Capacities in the European Entry-Exit Gas Market is NP-Hard JF - Annals of Operations Research N2 - 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 on trees and NP-hard in general. 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. KW - European Entry-Exit Gas Market KW - Technical Capacities KW - Potential-Based Flows KW - Computational Complexity KW - NP-Hardness Y1 - 2020 U6 - https://doi.org/10.1007/s10479-020-03725-2 IS - 295 SP - 337 EP - 362 ER - TY - JOUR A1 - Çelebi, Emre A1 - Krebs, Vanessa A1 - Schmidt, Martin T1 - Γ-Robust Electricity Market Equilibrium Models with Transmission and Generation Investments JF - Energy Systems N2 - We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering Γ-robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a Γ-robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse. KW - Robust optimization KW - Robust market equilibria KW - Electricity market equilibrium models KW - Transmission and generation investment KW - Perfect competition Y1 - 2020 ER - TY - JOUR A1 - Gabriel, Steven A. A1 - Leal, Marina A1 - Schmidt, Martin T1 - Solving Binary-Constrained Mixed Complementarity Problems Using Continuous Reformulations JF - Computers & Operations Research N2 - Mixed complementarity problems are of great importance in practice since they appear in various fields of applications like energy markets, optimal stopping, or traffic equilibrium problems. However, they are also very challenging due to their inherent, nonconvex structure. In addition, recent applications require the incorporation of integrality constraints. Since complementarity problems often model some kind of equilibrium, these recent applications ask for equilibrium points that additionally satisfy certain integer conditions. Obviously, this makes the problem even harder to solve. The solution approach used most frequently in the literature is to recast the complementarity conditions as disjunctive constraints using additional binary variables and big-M constraints. However, both latter aspects create issues regarding the tractability and correctness of the reformulation. In this paper, we follow the opposite route and restate the integrality conditions as complementarity constraints, leading to purely continuous reformulations that can be tackled by local solvers. We study these reformulations theoretically and provide a numerical study that shows that continuous reformulations are useful in practice both in terms of solution times and solution quality. KW - Binary-constrained mixed complementarity problems KW - Mixed-integer optimization KW - Continuous reformulations KW - Spatial price equilibrium problems Y1 - 2020 ER - TY - JOUR A1 - Roland, Marius A1 - Schmidt, Martin T1 - Mixed-Integer Nonlinear Optimization for District Heating Network Expansion JF - at - Automatisierungstechnik N2 - We present a mixed-integer nonlinear optimization model for computing the optimal expansion of an existing tree-shaped district heating network given a number of potential new consumers. To this end, we state a stationary and nonlinear model of all hydraulic and thermal effects in the pipeline network as well as nonlinear models for consumers and the network's depot. For the former, we consider the Euler momentum and the thermal energy equation. The thermal aspects are especially challenging. Here, we develop a novel polynomial approximation that we use in the optimization model. The expansion decisions are modeled by binary variables for which we derive additional valid inequalities that greatly help to solve the highly challenging problem. Finally, we present a case study in which we identify three major aspects that strongly influence investment decisions: the estimated average power demand of potentially new consumers, the distance between the existing network and the new consumers, and thermal losses in the network. KW - District heating networks KW - Network expansion KW - Mixed-integer nonlinear optimization Y1 - 2020 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 - Kleinert, Thomas A1 - Schmidt, Martin T1 - Computing Feasible Points of Bilevel Problems with a Penalty Alternating Direction Method JF - INFORMS Journal on Computing N2 - Bilevel problems are highly challenging optimization problems that appear in many applications of energy market design, critical infrastructure defense, transportation, pricing, etc. Often, these bilevel models are equipped with integer decisions, which makes the problems even harder to solve. Typically, in such a setting in mathematical optimization one develops primal heuristics in order to obtain feasible points of good quality quickly or to enhance the search process of exact global methods. However, there are comparably few heuristics for bilevel problems. In this paper, we develop such a primal heuristic for bilevel problems with mixed-integer linear or quadratic upper level and linear or quadratic lower level. The heuristic is based on a penalty alternating direction method, which allows for a theoretical analysis. We derive a convergence theory stating that the method converges to a stationary point of an equivalent single-level reformulation of the bilevel problem and extensively test the method on a test set of more than 2800 instances - which is one of the largest computational test sets ever used in bilevel programming. The study illustrates the very good performance of the proposed method, both in terms of running times and solution quality. This renders the method a suitable sub-routine in global bilevel solvers as well as a reasonable standalone approach. KW - Bilevel optimization KW - Mixed-integer bilevel optimization KW - Stationary points KW - Penalty methods KW - Alternating direction methods Y1 - 2019 U6 - https://doi.org/10.1287/ijoc.2019.0945 IS - 33 (1) SP - 198 EP - 215 ER - TY - JOUR A1 - Grübel, Julia A1 - Kleinert, Thomas A1 - Krebs, Vanessa A1 - Orlinskaya, Galina A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - On Electricity Market Equilibria with Storage: Modeling, Uniqueness, and a Distributed ADMM JF - Computers & Operations Research N2 - We consider spot-market trading of electricity including storage operators as additional agents besides producers and consumers. Storages 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 storages 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 storages, 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 minutes. Y1 - 2019 U6 - https://doi.org/10.1016/j.cor.2019.104783 IS - 114 ER - TY - JOUR A1 - Gatzert, Nadine A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Seith, Benjamin A1 - Vogl, Nikolai T1 - Portfolio Optimization with Irreversible Long-Term Investments in Renewable Energy under Policy Risk: A Mixed-Integer Multistage Stochastic Model and a Moving-Horizon Approach JF - European Journal of Operational Research N2 - Portfolio optimization is an ongoing hot topic of mathematical optimization and management science. Due to the current financial market environment with low interest rates and volatile stock markets, it is getting more and more important to extend portfolio optimization models by other types of investments than classical assets. In this paper, we present a mixed-integer multistage stochastic model that includes investment opportunities in irreversible and long-term infrastructure projects in the context of renewable energies, which are also subject to policy risk. On realistic time scales for investment problems of this type, the resulting instances are by far too large to be solved with today's most evolved optimization software. Thus, we present a tailored moving-horizon approach together with suitable approximations and simplifications of the model. We evaluate these approximations and simplifications in a computational sensitivity analysis and derive a final model that can be tackled on a realistic instance by our moving-horizon approach. KW - Mixed-integer optimization KW - Multistage stochastic optimization KW - Portfolio optimization KW - Illiquid investments KW - Policy risk Y1 - 2019 ER - TY - JOUR A1 - Grimm, Veronika A1 - Orlinskaya, Galina A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - Optimal Design of Retailer-Prosumer Electricity Tariffs Using Bilevel Optimization JF - Computers & Operations Research N2 - We compare various flexible tariffs that have been proposed to cost-effectively govern a prosumer's electricity management - in particular time-of-use (TOU), critical-peak-pricing (CPP), and a real-time-pricing tariff (RTP). As the outside option, we consider a fixed-price tariff (FP) that restricts the specific characteristics of TOU, CPP, and RTP, so that the flexible tariffs are at least as profitable for the prosumer as the fixed-price tariff. We propose bilevel models to determine the optimal interplay between the retailer's tariff design and the prosumer's decisions on using the storage, on consumption, and on electricity purchases from as well as electricity sales to the grid. The single-level reformulations of the considered bilevel models are computationally highly challenging optimization problems since they, e.g., combine bilinearities and mixed-integer aspects for modeling certain tariff structures. Based on a computational study using real-world data, we find that RTP increases retailer profits, however, leads to the largest price volatility for the prosumer. TOU and CPP only yield mild additional retailer profits and, due to the multiplicity of optimal plans on the part of the prosumer, imply uncertain revenues for the retailer. KW - Electricity tariffs KW - Pricing KW - Bilevel optimization KW - Mixed-integer optimization KW - Tariff design Y1 - 2019 IS - 114 ER - TY - JOUR A1 - Hauschild, Sarah-Alexa A1 - Marheineke, Nicole A1 - Mehrmann, Volker A1 - Mohring, Jan A1 - Badlyan, Arbi Moses A1 - Rein, Markus A1 - Schmidt, Martin T1 - Port-Hamiltonian modeling of district heating networks JF - Progress in Differential Algebraic Equations II (edited by Reis T., Grundel S., and Schöps S). Differential-Algebraic Equations Forum N2 - This paper provides a first contribution to port-Hamiltonian modeling of district heating networks. By introducing a model hierarchy of flow equations on the network, this work aims at a thermodynamically consistent port-Hamiltonian embedding of the partial differential-algebraic systems. We show that a spatially discretized network model describing the advection of the internal energy density with respect to an underlying incompressible stationary Euler-type hydrodynamics can be considered as a parameter-dependent finite-dimensional port-Hamiltonian system. Moreover, we present an infinite-dimensional port-Hamiltonian formulation for a compressible instationary thermodynamic fluid flow in a pipe. Based on these first promising results, we raise open questions and point out research perspectives concerning structure-preserving discretization, model reduction, and optimization. KW - Partial differential equations on networks KW - Port-Hamiltonian model framework KW - Energy-based formulation KW - District heating network KW - Thermodynamic fluid flow Y1 - 2019 ER - TY - JOUR A1 - Schmidt, Martin A1 - Sirvent, Mathias A1 - Wollner, Winnifried T1 - The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities JF - Optimization Letters N2 - It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function---in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute approximate global optimal solutions in finite time and we also provide a worst-case iteration bound. In some first numerical experiments we show that the ``cost of not knowing enough'' is rather high by comparing our approach with the open-source global solver SCIP. This reveals that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research. KW - Mixed-Integer Nonlinear Optimization, Global Optimization, Lipschitz Optimization, Gas Networks Y1 - 2018 ER - TY - JOUR A1 - Schmidt, Martin A1 - Sirvent, Mathias A1 - Wollner, Winnifried T1 - A Decomposition Method for MINLPs with Lipschitz Continuous Nonlinearities JF - Mathematical Programming N2 - Many mixed-integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly. This is, e.g., the case for problems constrained by differential equations or for models that rely on black-box simulation runs. For these problem classes, we present, analyze, and test algorithms that solve mixed-integer problems with only Lipschitz continuous nonlinearities. Our theoretical results depend on the assumptions made on the (in)exactness of function evaluations and on the knowledge of Lipschitz constants. If Lipschitz constants are known, we prove finite termination at approximate globally optimal points both for the case of exact and inexact function evaluations. If only approximate Lipschitz constants are known, we prove finite termination and derive additional conditions under which infeasibility can be detected. A computational study for gas transport problems and an academic case study show the applicability of our algorithms to real-world problems and how different assumptions on the constraint functions up- or downgrade the practical performance of the methods. KW - Mixed-Integer Nonlinear Optimization, Lipschitz Optimization, Inexact Function Evaluations, Decomposition Methods, Gas Networks Y1 - 2017 IS - 178(1) SP - 449 EP - 483 ER - TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter A1 - Wang, Ke ED - Zhang, Xu T1 - Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-Lyapunov function JF - Mathematical Control and Related Fields (MCRF) N2 - For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H2-Lyapunov function and show that the boundary feedback constant can be chosen such that the H2-Lyapunov function and hence also the H2-norm of the difference between the non-stationary and the stationary state decays exponentially with time. KW - Boundary feedback control, feedback stabilization, exponential stability, isothermal Euler equations, second-order quasilinear equation, Lyapunov function, stationary state, non-stationary state, gas pipeline. Y1 - 2017 U6 - https://doi.org/10.3934/mcrf.2017015 VL - 7 IS - 3 SP - 419 EP - 448 ER - TY - JOUR A1 - Gugat, Martin A1 - Keimer, Alexander A1 - Leugering, Günter A1 - Wang, Zhiqiang ED - Piccoli, Benedetto T1 - Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks JF -  Networks and Heterogeneous Media N2 - We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant multi-commodity manufacturing system as encountered in semiconductor production. Every single commodity is mod-eled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress. We illustrate the dynamics for two commodities. In the applications, directed acyclic networks naturally occur, therefore this type of networks is considered. On every edge of the network we have a system of coupled conservation laws with nonlocal velocity. At the junctions the right hand side boundary data of the foregoing edges is passed as left hand side boundary data to the following edges and PDEs. For distributing junctions, where we have more than one outgoing edge, we impose time dependent distribution functions that guarantee conservation of mass. We provide results of regularity, existence and well-posedness of the multi-commodity network model for L p-, BV-and W 1,p-data. Moreover, we define an L 2-tracking type objective and show the existence of minimizers that solve the corresponding optimal control problem. KW - conservation laws on network KW - nonlocal conservation laws KW - optimal nodal control KW - systems of hyperbolic pdes Y1 - 2016 U6 - https://doi.org/DOI: 10.3934/nhm.2015.10.749 VL - 10 IS - 4 SP - 749 EP - 785 ER - TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter ED - Zuazua, Enrique T1 - Time delay in optimal control loops for wave equations JF - ESAIM: COCV N2 - In optimal control loops delays can occur, for example through transmission via digital communication channels. Such delays influence the state that is generated by the implemented control. We study the effect of a delay in the implementation of L 2-norm minimal Neumann boundary controls for the wave equation. The optimal controls are computed as solutions of problems of exact optimal control, that is if they are implemented without delay, they steer the system to a position of rest in a given finite time T. We show that arbitrarily small delays δ > 0 can have a destabilizing effect in the sense that we can find initial states such that if the optimal control u is implemented in the form yx(t, 1) = u(t − δ) for t > δ, the energy of the system state at the terminal time T is almost twice as big as the initial energy. We also show that for more regular initial states, the effect of a delay in the implementation of the optimal control is bounded above in the sense that for initial positions with derivatives of BV-regularity and initial velocities with BV-regularity, the terminal energy is bounded above by the delay δ multiplied with a factor that depends on the BV-norm of the initial data. We show that for more general hyperbolic optimal exact control problems the situation is similar. For systems that have arbitrarily large eigenvalues, we can find terminal times T and arbitrarily small time delays δ, such that at the time T + δ, in the optimal control loop with delay the norm of the state is twice as large as the corresponding norm for the initial state. Moreover, if the initial state satisfies an additional regularity condition, there is an upper bound for the effect of time delay of the order of the delay with a constant that depends on the initial state only. KW - PDE constrained optimization KW - delay KW - wave equation KW - boundary control KW - hyperbolic system Y1 - 2016 U6 - https://doi.org/http://dx.doi.org/10.1051/cocv/2015038 ER - TY - JOUR A1 - Gugat, Martin A1 - Wintergerst, David A1 - Schultz, Rüdiger ED - Iske, Armin T1 - Networks of pipelines for gas with nonconstant compressibility factor: stationary states JF - Computational and Applied Mathematics N2 - For the management of gas transportation networks, it is essential to know how the stationary states of the system are determined by the boundary data. The isothermal Euler equations are an accurate pde-model for the gas flow through each pipe. A compressibility factor is used to model the nonlinear relationship between density and pressure that occurs in real gas in contrast to ideal gas. The gas flow through the nodes is governed by algebraic node conditions that require the conservation of mass and the continuity of the pressure. We examine networks that are described by arbitrary finite graphs and show that for suitably chosen boundary data, subsonic stationary states exist and are uniquely determined by the boundary data. Our construction of the stationary states is based upon explicit representations of the stationary states on each single pipe that can easily be evaluated numerically. We also use the monotonicity properties of these states as functions of the boundary data. Y1 - 2016 U6 - https://doi.org/10.1007/s40314-016-0383-z ER - TY - JOUR A1 - Bärmann, Andreas A1 - Liers, Frauke A1 - Martin, Alexander A1 - Merkert, Maximilian A1 - Thurner, Christoph A1 - Weninger, Dieter T1 - Solving network design problems via iterative aggregation JF - Mathematical Programming Computation N2 - In this work, we present an exact approach for solving network design problems that is based on an iterative graph aggregation procedure. The scheme allows existing preinstalled capacities. Starting with an initial aggregation, we solve a sequence of network design master problems over increasingly fine-grained representations of the original network. In each step, a subproblem is solved that either proves optimality of the solution or gives a directive where to refine the representation of the network in the subsequent iteration. The algorithm terminates with a globally optimal solution to the original problem. Our implementation uses a standard integer programming solver for solving the master problems as well as the subproblems. The computational results on random and realistic instances confirm the profitable use of the iterative aggregation technique. The computing time often reduces drastically when our method is compared to solving the original problem from scratch. KW - Aggregation KW - Network design KW - Combinatorial optimization KW - Mixed-integer programming KW - Branch-and-cut Y1 - 2015 U6 - https://doi.org/10.1007/s12532-015-0079-1 VL - 7 IS - 2 SP - 189 EP - 217 ER - TY - JOUR A1 - Schmidt, Martin A1 - Aßmann, Denis A1 - Burlacu, Robert A1 - Humpola, Jesco A1 - Joormann, Imke A1 - Kanelakis, Nikolaos A1 - Koch, Thorsten A1 - Oucherif, Djamal A1 - Pfetsch, Marc E. A1 - Schewe, Lars A1 - Schwarz, Robert A1 - Sirvent, Mathias T1 - GasLib – A Library of Gas Network Instances JF - Data N2 - 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. KW - Gas Transport KW - Networks KW - Problem Instances KW - Mixed-Integer Nonlinear Optimization KW - GasLib Y1 - 2017 U6 - https://doi.org/10.3390/data2040040 VL - 4 IS - 2 ER - TY - JOUR A1 - Hante, Falk A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications N2 - We consider optimal control problems for the flow of gas or fresh water in pipe networks as well as drainage or sewer systems in open canals. The equations of motion are taken to be represented by the nonlinear isothermal Euler gas equations, the water hammer equations, or the St.~Venant equations for flow. We formulate model hierarchies and derive an abstract model for such network flow problems including pipes, junctions, and controllable elements such as valves, weirs, pumps, as well as compressors. We use the abstract model to give an overview of the known results and challenges concerning equilibria, well-posedness, controllability, and optimal control. A major challenge concerning the optimization is to deal with switching on-off states that are inherent to controllable devices in such applications combined with continuous simulation and optimization of the gas flow. We formulate the corresponding mixed-integer nonlinear optimal control problems and outline a decomposition approach as a solution technique. KW - Networks KW - pipes KW - optimal control KW - Euler and St. Venant equations KW - hierarchy of models Y1 - 2016 ER - TY - JOUR A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Sirvent, Mathias T1 - Nonoverlapping Domain Decomposition for Optimal Control Problems governed by Semilinear Models for Gas Flow in Networks N2 - We consider optimal control problems for gas flow in pipeline networks. The equations of motion are taken to be represented by a first-order system of hyperbolic semilinear equations derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a network and introduce a tailored time discretization thereof. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a nonoverlapping domain decomposition of the optimal control problem on the graph into local problems on smaller sub-graphs - ultimately on single edges. We prove convergence of the domain decomposition method on networks and study the wellposedness of the corresponding time-discrete optimal control problems. The point of the paper is that we establish virtual control problems on the decomposed subgraphs such that the corresponding optimality systems are in fact equal to the systems obtained via the domain decomposition of the entire optimality system. KW - Optimal control, Gas networks, Euler's equation, Semilinear PDE, Nonoverlapping domain decomposition Y1 - 2017 VL - 46 IS - 3 SP - 191 EP - 225 PB - Control and Cybernetics 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 - Sirvent, Mathias A1 - Kanelakis, Nikolaos A1 - Geißler, Björn A1 - Biskas, Pandelis T1 - A Linearized Model for the Optimization of the Coupled Electricity and Natural Gas System JF - Journal of Modern Power Systems and Clean Energy N2 - In the following paper a combined optimization of a coupled electricity and gas system is presented. For the electricity network a unit commitment problem with optimization of energy and reserves under a power pool, considering all system operational and unit technical constraints is solved. The gas network subproblem is a medium-scale mixed-integer nonconvex and nonlinear programming problem. The coupling constraints between the two networks are nonlinear as well. The resulting mixed-integer nonlinear program is linearized with the extended incremental method and an outer approximation technique. The resulting model is evaluated using the Greek power and gas system comprising fourteen gas-fired units under four different approximation accuracy levels. The results indicate the efficiency of the proposed MIP model and the interplay between computational requirements and accuracy. KW - Electricity System KW - Natural Gas System KW - Mixed-Integer (Non)Linear Programming KW - Extended Incremental Method KW - Outer Approximation Y1 - 2017 U6 - https://doi.org/10.1007/s40565-017-0275-2 VL - 5 IS - 3 SP - 364 EP - 374 ER - TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter A1 - Hante, Falk ED - Piccoli, Benedetto T1 - Stationary States in Gas Networks JF - Networks and Heterogeneous Media N2 - Pipeline networks for gas transportation often contain circles. For such networks it is more difficult to determine the stationary states than for networks without circles. We present a method that allows to compute the stationary states for subsonic pipe flow governed by the isothermal Euler equations for certain pipeline networks that contain circles. We also show that suitably chosen boundary data determine the stationary states uniquely. The construction is based upon novel explicit representations of the stationary states on single pipes for the cases with zero slope and with nonzero slope. In the case with zero slope, the state can be represented using the Lambert-W function. KW - Network Y1 - 2016 U6 - https://doi.org/doi:10.3934/nhm.2015.10.295 VL - 10 IS - 2 SP - 295 EP - 320 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 - JOUR A1 - Henrion, René A1 - Schmidt, Martin T1 - Chance-Constrained Linear Complementarity Problems N2 - We study linear complementarity problems (LCPs) under uncer- tainty, which we model using chance constraints. Since the complementarity condition of the LCP is an equality constraint, it is required to consider relax- ations, which naturally leads to optimization problems in which the relaxation parameters are minimized for given probability levels. We focus on these optimization problems and first study the continuity of the related probability functions and the compactness of the feasible sets. This leads to existence results for both types of models: one with a joint chance constraint and one with separate chance constraints for both uncertainty-affected conditions of the LCP. For both, we prove the differentiability of all probability functions and derive respective gradient formulae. For the separate case, we prove con- vexity of the respective optimization problem and use the gradient formulae to derive necessary and sufficient optimality conditions. In a small case study regarding a Cournot oligopoly among energy producers, we finally illustrate the applicability of our theoretical findings. KW - Linear complementarity problems KW - Chance constraints KW - Existence KW - Convexity KW - Optimality conditions Y1 - 2026 ER - TY - JOUR A1 - Aigner, Kevin-Martin A1 - Denzler, Sebastian A1 - Liers, Frauke A1 - Pokutta, Sebastian A1 - Sharma, Kartikey T1 - Scenario Reduction for Distributionally Robust Optimization N2 - Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the oroposed approximation significantly reduces solution time while maintaining high solution quality with only minor errors. KW - distributionally robust optimization KW - scenario reduction KW - scenario clustering KW - approximation bounds KW - mixed-integer programming Y1 - 2025 ER - TY - JOUR A1 - Pokutta, Sebastian A1 - Spiegel, Christoph A1 - Zimmer, Max A1 - Mundinger, Konrad T1 - Extending the Continuum of Six-Colorings JF - Geocombinatorics Quarterly N2 - We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance $d$ in the sixth color. Such colorings have previously been known to exist for $0.41 < \sqrt{2} - 1 \le d \le 1 / \sqrt{5} < 0.45$. Our results significantly expand that range to $0.354 \le d \le 0.657$, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach. Y1 - 2024 U6 - https://doi.org/10.48550 VL - Geocombinatorics Quarterly IS - Volume XXXIV: 2024 ER -