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 - INPR A1 - Krebs, Vanessa A1 - Müller, Michael A1 - Schmidt, Martin T1 - Γ-Robust Linear Complementarity Problems with Ellipsoidal Uncertainty Sets T2 - International Transactions in Operational Research N2 - We study uncertain linear complementarity problems (LCPs), i.e., problems in which the LCP vector q or the LCP matrix M may contain uncertain parameters. To this end, we use the concept of Γ-robust optimization applied to the gap function formulation of the LCP. Thus, this work builds upon [16]. There, we studied Γ-robustified LCPs for l1- and box-uncertainty sets, whereas we now focus on ellipsoidal uncertainty set. For uncertainty in q or M, we derive conditions for the tractability of the robust counterparts. For these counterparts, we also give conditions for the existence and uniqueness of their solutions. Finally, a case study for the uncertain traffic equilibrium problem is considered, which illustrates the effects of the values of Γ on the feasibility and quality of the respective robustified solutions. KW - Robust optimization KW - Linear complementarity problems KW - Ellipsoidal uncertainty sets KW - Traffic equilibrium problems Y1 - 2019 IS - 29(1) SP - 417 EP - 441 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 - Grimm, Veronika A1 - Grübel, Julia A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Zöttl, Gregor T1 - Nonconvex Equilibrium Models for Gas Market Analysis: Failure of Standard Techniques and Alternative Modeling Approaches JF - European Journal on Operational Research N2 - This paper provides a first approach to assess gas market interaction on a network with nonconvex flow models. In the simplest possible setup that adequately reflects gas transport and market interaction, we elaborate on the relation of the solution of a simultaneous competitive gas market game, its corresponding mixed nonlinear complementarity problem (MNCP), and a first-best benchmark. We provide conditions under which the solution of the simultaneous game is also the solution of the corresponding MNCP. However, equilibria cannot be determined by the MNCP as the transmission system operator's (TSO’s) first-order conditions are insufficient, which goes back to nonconvexities of the gas flow model. This also implies that the welfare maximization problem may have multiple solutions that sometimes do not even coincide with any of the market equilibria. Our analysis shows that, even in the absence of strategic firms, market interaction fails to implement desirable outcomes from a welfare perspective due to the TSO’s incentive structure. We conclude that the technical environment calls for a market design that commits the TSO to a welfare objective through regulation and propose a design where the market solution corresponds to a welfare maximum and vice versa. KW - Natural Gas Markets KW - Nonconvex Equilibrium Models KW - Uniqueness KW - Multiplicity KW - Fundamental Welfare Theorems Y1 - 2017 IS - 273(3) SP - 1097 EP - 1108 ER - TY - JOUR A1 - Krebs, Vanessa A1 - Schmidt, Martin T1 - Uniqueness of Market Equilibria on Networks with Transport Costs JF - Operations Research Perspectives N2 - We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a welfare maximization problem. Using the latter we prove uniqueness of the resulting equilibrium for piecewise linear and symmetric transport costs under additional mild assumptions. Moreover, we show the necessity of these assumptions by illustrating examples that possess multiple solutions if our assumptions are violated. KW - Market Equilibria KW - Networks KW - Transport Costs KW - Uniqueness KW - Perfect Competition Y1 - 2017 IS - 5 SP - 169 EP - 173 ER - TY - JOUR A1 - Mehrmann, Volker A1 - Schmidt, Martin A1 - Stolwijk, Jeroen J. T1 - Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization JF - Vietnam Journal of Mathematics N2 - The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples. KW - Gas network optimization KW - Isothermal stationary Euler equations KW - Model hierarchy KW - Adaptive error control KW - Marking strategy Y1 - 2017 IS - 46(4) SP - 779 EP - 801 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 - INPR A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Weninger, Dieter T1 - A Decomposition Heuristic for Mixed-Integer Supply Chain Problems T2 - Operations Research Letters N2 - Mixed-integer supply chain models typically are very large but are also very sparse and can be decomposed into loosely coupled blocks. In this paper, we use general-purpose techniques to obtain a block decomposition of supply chain instances and apply a tailored penalty alternating direction method, which exploits the structural properties of the decomposed instances. We further describe problem-specific enhancements of the algorithm and present numerical results on real-world instances that illustrate the applicability of the approach. KW - Supply chain KW - Mixed-integer optimization KW - Decomposition KW - Penalty method KW - Alternating direction methods Y1 - 2019 IS - 48(3) SP - 225 EP - 232 ER - TY - INPR A1 - Kleinert, Thomas A1 - Labbé, Martine A1 - Plein, Fränk A1 - Schmidt, Martin T1 - There's No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization T2 - Operations Research N2 - One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity conditions using techniques from mixed-integer linear optimization. The latter step requires to determine some big-M constant in order to bound the lower level's dual feasible set such that no bilevel-optimal solution is cut off. In practice, heuristics are often used to find a big-M although it is known that these approaches may fail. In this paper, we consider the hardness of two proxies for the above mentioned concept of a bilevel-correct big-M. First, we prove that verifying that a given big-M does not cut off any feasible vertex of the lower level's dual polyhedron cannot be done in polynomial time unless P=NP. Second, we show that verifying that a given big-M does not cut off any optimal point of the lower level's dual problem (for any point in the projection of the high-point relaxation onto the leader's decision space) is as hard as solving the original bilevel problem. KW - Bilevel optimization KW - Mathematical programs with complementarity constraints (MPCC) KW - Bounding polyhedra KW - Big-M KW - Hardness Y1 - 2019 IS - 68(6) SP - 1716 EP - 1721 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 - Kleinert, Thomas A1 - Schmidt, Martin T1 - Global Optimization of Multilevel Electricity Market Models Including Network Design and Graph Partitioning JF - Discrete Optimization N2 - We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in [25] that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of [25] by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model. KW - Network design KW - Graph partitioning KW - Multilevel optimization KW - Mixed-integer optimization KW - Electricity market design Y1 - 2018 IS - 33 SP - 43 EP - 69 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 - Schewe, Lars A1 - Schmidt, Martin T1 - The Impact of Potential-Based Physics Models on Pricing in Energy Networks JF - Central European Journal of Operations Research N2 - Pricing of access to energy networks is an important issue in liberalized energy sectors because of the natural monopoly character of the underlying transport infrastructures. We introduce a general pricing framework for potential-based energy flows in arbitrarily structured transport networks. In different specifications of our general pricing model we discuss first- and second-best pricing results and compare different pricing outcomes of potential-free and potential-based energy flow models. Our results show that considering nonlinear laws of physics leads to significantly different pricing results on networks and that these differences can only be seen in sufficiently complex, e.g., cyclic, networks as they can be found in real-world situations. KW - Energy Networks KW - Pricing KW - Gas Networks KW - Electricity Networks Y1 - 2018 IS - 28(3) SP - 1003 EP - 1029 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 - 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 - Schmidt, Martin T1 - Complementarity-Based Nonlinear Programming Techniques for Optimal Mixing in Gas Networks JF - EURO Journal on Computational Optimization N2 - We consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems. As mixed-integer reformulations of pooling-type mixing models already render small-size instances computationally intractable, we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based on recent results for remodeling piecewise affine constraints using an inverse parametric quadratic programming approach, we show that classical stationarity concepts are meaningful for the resulting complementarity-based reformulation of the mixing equations. Further, we investigate in a numerical study the performance of this reformulation compared to a more compact complementarity-based one that does not feature such beneficial regularity properties. All computations are performed on publicly available data of real-world size problem instances from steady-state gas transport. KW - Gas transport networks KW - Mixing KW - Inverse parametric quadratic programming KW - Complementarity constraints KW - MPCC Y1 - 2017 IS - 7(3) SP - 299 EP - 323 ER -