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Sun, 16 Feb 2020 18:19:05 +0100Sun, 16 Feb 2020 18:19:05 +0100Solving Binary-Constrained Mixed Complementarity Problems Using Continuous Reformulations
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/308
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.Steven A. Gabriel; Marina Leal; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/308Sun, 16 Feb 2020 18:19:05 +0100An adaptive bundle method for nonlinear robust optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/307
We present an adaptive bundle method to minimize non-convex nonsmooth marginal functions when only inexact function values and subgradients are available.
This class of problems covers nonlinear robust optimization problems written in minimax form with an inexact evaluation of the worst case.
Currently, there are few theoretical or practical approaches available for general nonlinear robust optimization. Moreover, the approaches that do exist impose further assumptions on the problem structure.
Our method only requires availability of approximate worst case evaluations, and does not rely on a specific structure of the worst case constraints.
To evaluate the worst case with the desired degree of precision, one possibility is the use of techniques from mixed-integer linear programming (MIP).
The proposed algorithm and convergence proof techniques are inspired by Noll's bundle method for non-convex minimization with inexact information.
By extending his approach, we prove convergence to approximate critical points of the non-convex minimax problem under two assumptions:
Firstly, that the objective function is lower C1 for any realization of the
uncertainties and secondly, that approximately optimal solutions to the inner maximization problem are available up to any requested error.
As an example application, we investigate the gas transport problem under
uncertainties in demand and in physical parameters that affect pressure losses in the pipes.
Approximating the worst case problem by a piecewise linearization strategy and solving the resulting problem up to an arbitrarily small error using MIP techniques, the required assumptions are met.
Computational results for examples in large realistic gas network instances demonstrate the efficiency of the method.Martina Kuchlbauer; Frauke Liers; Michael Stinglpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/307Tue, 11 Feb 2020 17:44:08 +0100Analysis and Numerical Approximation of Transport Equations on Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/306
This work deals with the analysis and numerical approximation of transport problems on networks. Appropriate coupling conditions are proposed that allow to establish well-posedness of the continuous problem by semigroup theory. A discontinuous Galerkin method is proposed for the space discretization and its well-posedness and order optimal convergence rates are proven. In addition, the time discretization by the implicit Euler method is investigated.Nora Philippimasterthesishttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/306Tue, 28 Jan 2020 11:31:42 +0100Γ-Robust Electricity Market Equilibrium Models with Transmission and Generation Investments
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/305
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 a detailed case study on an academic example to gain insights into the general effects of robustification on electricity market models. In particular, our case study reveals that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse.Emre Çelebi; Vanessa Krebs; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/305Mon, 27 Jan 2020 17:45:44 +0100Computing Technical Capacities in the European Entry-Exit Gas Market is NP-Hard
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/304
As a result of its liberalization, the European gas market is organized as an entry-exit system in order to decouple the trading and transport of natural gas. Roughly summarized, the gas market organization consists of four subsequent stages. First, the transmission system operator (TSO) is obliged to allocate so-called maximal technical capacities for the nodes of the network. Second, the TSO and the gas traders sign mid- to long-term capacity-right contracts, where the capacity is bounded above by the allocated technical capacities. These contracts are called bookings. Third, on a day-ahead basis, gas traders can nominate the amount of gas that they inject or withdraw from the network at entry and exit nodes, where the nominated amount is bounded above by the respective booking. Fourth and finally, the TSO has to operate the network such that the nominated amounts of gas can be transported. By signing the booking contract, the TSO guarantees that all possibly resulting nominations can indeed be transported. Consequently, maximal technical capacities have to satisfy that all nominations that comply with these technical capacities can be transported through the network. This leads to a highly challenging mathematical optimization problem. We consider the specific instantiations of this problem in which we assume capacitated linear as well as potential-based flow models. In this contribution, we formally introduce the problem of Computing Technical Capacities (CTC) and prove that it is NP-complete. To this end, we first reduce the Subset Sum problem to CTC for the case of capacitated linear flows in trees. Afterward, we extend this result to CTC with potential-based flows and show that this problem is also NP-complete on trees by reducing it to the case of capacitated linear flow. Since the hardness results are obtained for the easiest case, i.e., on tree-shaped networks with capacitated linear as well as potential-based flows, this implies the hardness of CTC for more general graph classes.Lars Schewe; Martin Schmidt; Johannes Thüraufpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/304Tue, 21 Jan 2020 10:11:19 +0100Towards the Solution of Mixed-Integer Nonlinear Optimization Problems using Simultaneous Convexification
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/303
Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial
branch-and-bound become more effective the tighter the used
relaxations are. Relaxations are commonly established by convex underestimators, where each constraint function is considered separately. Instead, a considerably tighter relaxation can be found via so-called simultaneous convexification, where convex underestimators are derived for more than one constraint at a time.
In this work, we present a global solution approach for solving mixed-integer nonlinear problems that uses simultaneous convexification. We introduce a separation method for the convex hull of constrained sets. It relies on determining the convex envelope of linear combinations of the constraints and on solving a nonsmooth convex problem. In particular, we apply the method to quadratic absolute value functions and derive their convex envelopes. The practicality of the proposed solution approach is demonstrated on several test instances from gas network optimization, where the method outperforms standard approaches that use separate convex relaxations.Frauke Liers; Alexander Martin; Maximilian Merkert; Nick Mertens; Dennis Michaelspreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/303Wed, 08 Jan 2020 22:23:37 +0100Outer Approximation for Global Optimization of Mixed-Integer Quadratic Bilevel Problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/302
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.Thomas Kleinert; Veronika Grimm; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/302Mon, 23 Dec 2019 13:58:17 +0100Diffusion of green innovations: The role of consumer characteristics for domestic energy storage adoption
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/301
Against the background of a growing share of renewable energies and the ensuing fluctuations in electricity supply, domestic energy storage poses a promising option to foster flexible electricity demand and grid-stabilising self-supply. Understanding the determinants of adoption decisions regarding energy storage is therefore essential to enable targeted measures to promote their diffusion. This paper presents an in-depth analysis of motivational and psychological factors as well as product characteristics that affect the will-ingness to adopt domestic energy storage based on a selective sample focused on solar panel owners as the main energy storage target group. We find that potential adopters differ systematically in their evaluation of economic and non-economic utility aspects of domestic energy storage. Using correlation-based average linkage clustering, we identify four distinct types of storage adopters, namely finance-oriented, security-oriented, idealistic and multilaterally oriented households. The results of random forest analysis further reveal that adoption motives and the willingness to adopt vary substantially among the different types. We show that the most promising future adopters can be predicted based solely on observable characteristics. Our findings emphasise that segment-specific policy strategies and support mechanisms are needed to stimulate energy storage adoption to pave the way for a sustainable energy system.Grimm Veronika; Sandra Kretschmer; Simon Mehlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/301Sun, 22 Dec 2019 22:57:06 +0100Green innovations: The organizational setup of pilot projects and its influence on consumer perceptions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/299
Pilot-, test- and demonstration-projects (PTDs) are a prominent policy tool to promote the adoption of smart, green technologies. However, as technology adoption is heavily dependent on the individual attributes and beliefs of potential adopters, it is important to understand the influence of a PTD’s organizational setup on technology perception. By varying the information about a PTD’s organizational setup in a survey experiment among a selected sample of potential PTD-participants, we gather first experimental evidence for the effect of different setups on the perception of green technologies. We show that the organizational setup has a significant impact on a product’s perceived contribution to the energy transition, its establishment in the market, cost-reduction potential, innovativeness and environmental friendliness. In particular, full organizational cooperation between government, university and industry consistently improves perceptions compared to a partial setup. Regarding the willingness to participate in a PTD, we find that communication and support are the most imperative aspects and even more important than economic benefits. Our findings provide policy-makers with a more ample foundation on how PTDs should be designed to successfully transfer technologies to the market.Veronika Grimm; Sandra Kretschmer; Simon Mehlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/299Thu, 19 Dec 2019 13:34:23 +0100Market Design for Renewable Energy Auctions: An Analysis of Alternative Auction Formats
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/298
Auctions are widely used to determine the remuneration for renewable energies. They typically induce a high concentration of renewable energy plants at very productive sites far-off the main load centres, leading to an inefficient allocation as transmission line capacities are restricted but not considered in the allocation, resulting in an inefficient
system configuration in the long run. To counteract these tendencies effectively, we propose a combinatorial auction design that allows to implement regional target capacities, provides a simple pricing rule and maintains a high level of competition between bidders by permitting package bids. By means of extensive numerical experiments we evaluate
the combinatorial auction as compared to three further RES auction designs, the current
German nationwide auction design, a simple nationwide auction, and regional auctions. We find that if bidders benefit from high enough economies of scale, the combinatorial auction design implements system-optimal target capacities without increasing the average remuneration per kWh as compared to the current German auction design. The prices resulting from the combinatorial auction are linear and anonymous for each region whenever possible, while minimal personalised markups on the linear prices are applied only when necessary. We show that realistic problem sizes can be solved in seconds, even though the problem is computationally hard.Martin Bichler; Veronika Grimm; Sandra Kretschmer; Paul Suttererpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/298Wed, 18 Dec 2019 16:41:23 +0100Uncertain bidding zone configurations: the role of expectations for transmission and generation capacity expansion
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/297
Ongoing policy discussions on the reconfiguration of bidding zones in European electricity markets induce uncertainty about the future market
design. This paper deals with the question of how this uncertainty affects market participants and their long-run investment decisions in generation and transmission capacity. Generalizing the literature on pro-active network expansion planning, we propose a stochastic multilevel model which incorporates generation capacity investment, network expansion, and market operation, taking into account uncertainty about the future bidding zone configuration. Using a stylized two-node network, we disentangle different effects that uncertainty has on market outcomes. If there is a possibility that future bidding zone configurations provide improved regional price signals, welfare gains materialize even if the change does not actually take place. As a consequence, welfare gains of an actual change of the bidding zone configuration are substantially lower due to those anticipatory effects. Additionally, we show substantial distributional effects in terms of both expected gains and risks, between producers and consumers and between different generation technologies.Mirjam Ambrosius; Jonas Egerer; Veronika Grimm; Adriaan H. van der Weijdepreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/297Wed, 18 Dec 2019 16:41:21 +0100An enumerative formula for the spherical cap discrepancy
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/296
The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed.
Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing
optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy
to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial
nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes.
Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its
explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy
with respect to a sample of given size.Holger Heitsch; René Henrionarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/296Tue, 17 Dec 2019 16:18:45 +0100MARKET-BASED REDISPATCH MAY RESULT IN INEFFICIENT DISPATCH
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/295
In this paper we analyze a uniform price electricity spot market that is followed by redispatch in the case of network congestion. We assume that the transmission system operator is incentivized to minimize redispatch cost and compare a cost-based redispatch (CBR) to a market-based redispatch (MBR) mechanism. For networks with at least three nodes we show that in contrast to CBR, in the case of MBR the redispatch cost minimizing allocation may not be short-run efficient. As we demonstrate, in case of MBR the possibility of the transmission system operator to reduce redispatch cost at the expense of a reduced welfare may be driven by the electricity supply side or the electricity demand side. If, however, the transmission system operator is obliged to implement the welfare maximizing (instead of the redispatch cost minimizing) dispatch by regulation, this will result in an efficient dispatch also in case of MBR.Veronika Grimm; Alexander Martin; Christian Sölch; Martin Weibelzahl; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/295Tue, 17 Dec 2019 15:13:55 +0100Economic comparison of different electric fuels for energy scenarios in 2035
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/294
Electric fuels (e-fuels) enable CO2-neutral mobility and are therefore an alternative to battery-powered electric vehicles. This paper compares the cost-effectiveness of Fischer-Tropsch diesel, methanol and Liquid Organic Hydrogen Carriers. The production costs of those fuels are to a large part driven by the energy-intensive electrolytic hydrogen production. In this paper, we apply a multi-level electricity market model to calculate future hourly electricity prices for various electricity market designs in Germany for the year 2035. We then assess the economic efficiency of the different fuels under various future market conditions. In particular, we use the electricity price vectors derived from an electricity market model calibrated for 2035 as an input for a mathematical model of the entire process chain from hydrogen production and chemical bonding to the energetic utilization of the fuels in a vehicle. Within this model, we perform a sensitivity analysis, which quantifies the impact of various parameters on the fuel production cost. Most importantly, we consider prices resulting from own model calculations for different energy market designs, the investment cost for the electrolysis systems and the carbon dioxide purchase price. The results suggest that the use of hydrogen, which is temporarily bound to Liquid Organic Hydrogen Carriers, is a favorable alternative to the more widely discussed synthetic diesel and methanol.Philipp Runge; Christian Sölch; Jakob Albert; Peter Wasserscheid; Gregor Zöttl; Veronika Grimmpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/294Mon, 16 Dec 2019 14:50:17 +0100STORAGE INVESTMENT AND NETWORK EXPANSION IN DISTRIBUTION NETWORKS: THE IMPACT OF REGULATORY FRAMEWORKS
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/293
In this paper we propose a bi-level equilibrium model that allows to analyze the impact of different regulatory frameworks on storage and network investment in distribution networks. In our model, a regulated distribution system operator decides on network investment and operation while he anticipates the decisions of private agents on storage investment and operation. Since, especially in distribution networks, voltage stability and network losses have a decisive influence on network expansion and operation, we use a linearized AC power flow formulation to adequately account for these aspects. As adjustments of the current regulatory framework, we consider curtailment of renewable production, the introduction of a network fee based on the maximum renewable feed-in, and a subsidy scheme for storage investment. The performance of the different alternative frameworks is compared to the performance under rules that are commonly applied in various countries today, as well as to a system-optimal (first-best) benchmark. To illustrate the economic effects, we calibrate our model with data from the field project Smart Grid Solar. Our results reveal that curtailment and a redesign of network fees both have the potential to significantly reduce total system costs. On the contrary, investment subsidization of storage capacity has only a limited impact as long as the distribution system operator is not allowed to intervene in storage operation.Veronika Grimm; Julia Grübel; Bastian Rückel; Christian Sölch; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/293Mon, 16 Dec 2019 14:46:46 +0100REGIONALLY DIFFERENTIATED NETWORK FEES TO AFFECT INCENTIVES FOR GENERATION INVESTMENT
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/292
In this paper we propose an equilibrium model that allows to analyze subsidization schemes to affect locational choices for generation investment in electricity markets. Our framework takes into account generation investment decided by private investors and redispatch as well as network expansion decided by a regulated transmission system operator. In order to take into account the different objectives and decision variables of those agents, our approach uses a bi-level structure. We focus on the case of regionally differentiated network fees which have to be paid by generators (a so called g-component). The resulting investment and production decisions are compared to the outcome of an equilibrium model in the absence of such regionally differentiated investment incentives and to an overall optimal (first-best) benchmark. To illustrate possible economic effects, we calibrate our framework with data from the German electricity market. Our results reveal that while regionally differentiated network fees do have a significant impact on locational choice of generation capacities, we do not find significant effects on either welfare or
network expansion.Veronika Grimm; Bastian Rückel; Christian Sölch; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/292Mon, 16 Dec 2019 14:46:45 +0100THE IMPACT OF MARKET DESIGN ON TRANSMISSION AND GENERATION INVESTMENT IN ELECTRICITY MARKETS
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/291
In this paper we propose an equilibrium model in order to analyze the impact of electricity market design on generation and transmission expansion in liberalized electricity markets. In a multi-level structure, our framework takes into account that generation investment and operation is decided by private investors, while network expansion and redispatch is decided by a regulated transmission system operator — as well as the different objectives of firms (profit maximization) and the regulator (welfare maximization). In order to illustrate the possibilities to quantify long term economic effects with our framework, we calibrate our model for the German electricity market. We consider various moderate adjustments of the market design: (i) the division of the market area into two price zones, (ii) the efficient curtailment of renewable production and (iii) a cost-benefit-driven balance between network expansion and network management measures. We then analyze the impact of these market designs on generation and transmission investment in case those design elements are anticipated upon network development planning. The resulting investment and production decisions are compared to a benchmark that reflects the current German electricity market design and to an overall optimal first-best benchmark. Our results reveal that price zones do have a significant impact on locational choice of generators and result in a reduced need for network expansion, but lead to only moderate annual welfare gains of approximately 0.9% of annual total system costs. Anticipation of optimal curtailment of renewables and a cost-benefit-driven use of redispatch operations upon network expansion planning, however, implies a welfare gain of over 4.9% of annual total system costs per year as compared to the existing market design, which equals 85% of the maximal possible welfare gain of the first-best benchmark.Veronika Grimm; Bastian Rückel; Christian Sölch; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/291Mon, 16 Dec 2019 14:46:44 +0100Energy-intense production-inventory planning with participation in sequential energy markets
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/290
To support the uprise of demand response, especially in the context of industrial processes, we propose a new approach to integrally determine the production-inventory plan and the cost-minimizing bids to participate in sequential reserve and energy-only markets. In particular, our approach considers time-coupling constraints which occur in the context of a production-inventory planning problem. We extend this problem with a comprehensive bidding formulation, which allows evaluating revenues and potential cost from the market participation, considering price uncertainties and uncertain activations of committed reserve capacity. This results in a multistage stochastic mixed-integer linear program, which explicitly considers the stage-wise revelation of information in our setup. To illustrate the capabilities of our approach, we apply our model to a real-world case study in which we investigate the participation of a cement plant in the German energy-only and reserve markets. The results of our case study indicate significant revenues for flexible industrial processes when participating in German spot and reserve markets.Markus Bohlayer; Markus Fleschutz; Marco Braun; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/290Mon, 16 Dec 2019 14:46:43 +0100The Impact of Neighboring Markets on Renewable Locations, Transmission Expansion, and Generation Investment
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/289
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.Jonas Egerer; Veronika Grimm; Thomas Kleinert; Martin Schmidt; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/289Tue, 03 Dec 2019 22:05:57 +0100On the numerical discretization of optimal control problems for conservation laws
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/288
We analyze the convergence of discretization schemes for the adjoint equation arising in the adjoint-based derivative computation for optimal control problems governed by entropy solutions of conservation laws. The difficulties arise from the fact that the correct adjoint state is the reversible solution of a transport equation with discontinuous coefficient and discontinuous end data. We derive the discrete adjoint scheme for monotone difference schemes in conservation form. It is known that convergence of the discrete adjoint can only be expected if the numerical scheme has viscosity of order O(h^\alpha) with appropriate 0 < \alpha < 1, which leads to quite viscous shock profiles. We show that by a slight modification of the end data of the discrete adjoint scheme convergence to the correct reversible solution can be obtained also for numerical schemes with viscosity of order O(h) and with sharp shock resolution. The theoretical findings are confirmed by numerical results.Stefan Ulbrich; Johann Michael Schmitt; Paloma Schäfer Aguilar; Michael Moosarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/288Tue, 03 Dec 2019 15:19:27 +0100Packing under Convex Quadratic Constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/287
We consider a general class of binary packing problems with a convex quadratic knapsack constraint.
We prove that these problems are APX-hard to approximate and present constant-factor approximation
algorithms based upon three different algorithmic techniques: (1) a rounding technique tailored
to a convex relaxation in conjunction with a non-convex relaxation whose approximation ratio equals the
golden ratio; (2) a greedy strategy; (3) a randomized rounding method
leading to an approximation algorithm for the more general case with multiple convex quadratic
constraints. We further show that a combination of the first two strategies can be used to yield a monotone algorithm leading to a strategyproof mechanism for a game-theoretic variant of the problem. Finally, we present a computational study of the empirical approximation of the three
algorithms for problem instances arising in the context of real-world gas transport networks.Max Klimm; Marc E. Pfetsch; Rico Raber; Martin Skutellapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/287Fri, 29 Nov 2019 17:22:07 +0100Deciding Feasibility of a Booking in the European Gas Market on a Cycle is in P
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/286
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. 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.Martine Labbé; Fränk Plein; Martin Schmidt; Johannes Thüraufpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/286Tue, 12 Nov 2019 16:31:05 +0100Capacity Evaluation for Large-Scale Gas Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/285
Natural gas is important for the energy turnaround in many countries like in Germany, where it serves as a "bridging energy" towards a fossil-free energy supply in the future. About 20% of the total German energy demand is provided by natural gas, which is transported through a complex pipeline network with a total length of about 30000 km and the efficient use of the given transport infrastructure for natural gas is of political, economic, and societal importance.
As a consequence of the liberalization of the European gas market in the last decades, gas trading and transport have been decoupled. This has led to new challenges for gas transport companies, and mathematical optimization is perfectly suited for tackling many of these challenges. However, the underlying mathematical problems are by far too hard to be solved by today's general-purpose software so that novel mathematical theory and algorithms are needed. The industrial research project "ForNe: Research Cooperation Network Optimization" has been initiated and funded by Open Grid Europe in 2009 and brought together experts in mathematical optimization from seven German universities and research institutes, which cover almost the entire range of mathematical optimization: integer and nonlinear optimization as well as optimization under uncertainty.
The mathematical research results have been put together in a software package that has been delivered to Open Grid Europe at the end of the project. Moreover, the research is still continuing - e.g., in the Collaborative Research Center/Transregio 154 "Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks" funded by the German Research Foundation.Martin Schmidt; Benjamin Hiller; Thorsten Koch; Marc Pfetsch; Björn Geißler; René Henrion; Imke Joormann; Alexander Martin; Antonio Morsi; Werner Römisch; Lars Schewe; Rüdiger Schultz; Marc C. Steinbachpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/285Fri, 08 Nov 2019 11:13:52 +0100Γ-Robust Linear Complementarity Problems with Ellipsoidal Uncertainty Sets
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/284
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.Vanessa Krebs; Michael Müller; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/284Thu, 10 Oct 2019 16:33:04 +0200Nonlinear Optimization of District Heating Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/283
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.Richard Krug; Volker Mehrmann; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/283Fri, 04 Oct 2019 16:49:19 +0200