TY - INPR A1 - Grübel, Julia A1 - Huber, Olivier A1 - Hümbs, Lukas A1 - Klimm, Max A1 - Schmidt, Martin A1 - Schwartz, Alexandra T1 - Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium N2 - Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of existence. Three key prerequisites have to be met. First, appropriate bounds on market prices have to be derived from necessary optimality conditions of some players. Second, a technical assumption is required for those prices that are not uniquely determined by the derived bounds. Third, nonconvex optimization problems have to be solved to global optimality. We test the algorithm on well-known instances from the power and gas literature that meet these three prerequisites. There, nonconvexities arise from considering the transmission system operator as an additional player besides producers and consumers who, e.g., switches lines or faces nonlinear physical laws. Our numerical results indicate that equilibria often exist, especially for the case of continuous nonconvexities in the context of gas market problems. KW - Energy markets KW - Nonconvex games KW - Existence KW - Equilibrium computation KW - Perfect competition Y1 - 2021 ER - TY - THES A1 - Nowak, Daniel T1 - Nonconvex Nash Games - Solution Concepts and Algorithms N2 - Game theory is a mathematical approach to model competition between several parties, called players. The goal of each player is to choose a strategy, which solves his optimization problem, i.e. minimizes or maximizes his objective function. Due to the competitive setting, this strategy may influence the optimization problems of other players. In the non-cooperative setting each player acts selfish, meaning he does not care about the objective of his opponents. A solution concept for this problem is a Nash equilibrium, which was introduced by John Forbes Nash in his Ph.D. thesis in 1950. Convexity of the optimization problems is a crucial assumption for the existence of Nash equilibria. This work investigates settings, where this convexity assumption fails to hold. The first part of this thesis extends results of Jong-Shi Pang and Gesualdo Scutari from their paper ``Nonconvex Games with Side Constraints'' published in 2011. In this publication, a game with possibly nonconvex objective functions and nonconvex individual and shared inequality constraints was investigated. We extend these results twofold. Firstly, we generalize the individual and shared polyhedral constraints to general convex constraints and, secondly, we introduce convex and nonconvex, individual and shared equality constraints. After a detailed comparison of solution concepts for the generalized Nash game and a related Nash game, we show that so-called quasi-Nash equilibria exist under similar assumptions than in the original work, provided some additional constraint qualification holds. Subsequently, we prove that the existence of Nash equilibria needs additional assumptions on the gradients of the equality constraints. Furthermore, a special case of a multi-leader multi-follower game is investigated. We show the convergence of epsilon-quasi-Nash equilibria to C-stationary points and prove that these are also Clarke-stationary under reasonable assumptions. In the second part of this thesis, an application in computation offloading is investigated. We consider several mobile users that are able to offload parts of a computation task to a connected server. However, the server has limited computation capacities which leads to competition among the mobile users. If a user decides to offload a part of his computation, he needs to wait for the server to finish before he can assemble the results of his computation. This leads to a vanishing constraint in the optimization problem of the mobile users which is a nonconvex and nonsmooth condition. We show the existence of a unique Nash equilibrium for the computation offloading game and provide an efficient algorithm for its computation. Furthermore, we present two extensions to this game, which inherit similar properties and we also show the limitations of these formulations. The third part investigates a hierarchical constrained Cournot game. In the upper level, several firms decide on capacities which act as constraints for the production variables. In the lower level the same firms engage in a Cournot competition, where they choose production variables to maximize profit. The prior chosen capacities are upper bounds on these production variables. This hierarchical setting induces nonconvexity and nonsmoothness in the upper level objective functions. After a detailed sensitivity analysis of the lower level, we give necessary optimality conditions for the upper level, i.e. for the hierarchical Cournot game. Using these conditions, we construct an algorithm which provably finds all Nash equilibria of the game, provided some assumptions are satisfied. This algorithm is numerically tested on several examples which are motivated by the gas market. KW - Game Theory KW - Nash Games KW - Optimization Y1 - 2021 U6 - https://doi.org/10.26083/tuprints-00017637 PB - E-Publishing-Service der TU Darmstadt CY - Darmstadt ER - TY - INPR A1 - Egerer, Jonas A1 - Grimm, Veronika A1 - Grübel, Julia A1 - Zöttl, Gregor T1 - Long-run market equilibria in coupled energy sectors: A study of uniqueness N2 - We propose an equilibrium model for coupled markets of multiple energy sectors. The agents in our model are operators of sector-specific production and sector-coupling technologies, as well as price-sensitive consumers with varying demand. We analyze long-run investment in production capacity in each sector and investment in coupling capacity between sectors, as well as production decisions determined at repeated spot markets. We show that in our multi-sector model, multiplicity of equilibria may occur, even if all assumptions hold that would be sufficient for uniqueness in a single-sector model. We then contribute to the literature by deriving sufficient conditions for the uniqueness of short- and long-run market equilibrium in coupled markets of multiple energy sectors. We illustrate via simple examples that these conditions are indeed required to guarantee uniqueness in general. The uniqueness result is an important step to be able to incorporate the proposed market equilibrium problem in more complex computational multilevel equilibrium models, in which uniqueness of lower levels is a prerequisite for obtaining meaningful solutions. Our analysis also paves the way to understand and analyze more complex sector coupling models in the future. KW - Energy Markets KW - Sector Coupling KW - Regional Pricing KW - Uniqueness KW - Short- and Long-Run Market Equilibrium Y1 - 2021 ER - TY - INPR A1 - Grimm, Veronika A1 - Nowak, Daniel A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Schwartz, Alexandra A1 - Zöttl, Gregor T1 - A Tractable Multi-Leader Multi-Follower Peak-Load-Pricing Model with Strategic Interaction N2 - While single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. We apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system. Finally, we use our algorithm to provide a publicly available test library for the considered class of multi-leader multi-follower games. This library contains problem instances with different economic and mathematical properties so that other researchers in the field can test and benchmark newly developed methods for this challenging class of problems. KW - Game theory KW - Nash-Cournot equilibria KW - Multi-leader multi-follower game KW - Peak-load pricing Y1 - 2020 U6 - https://doi.org/10.1007/s10107-021-01708-0 ER - TY - INPR A1 - Runge, Philipp A1 - Sölch, Christian A1 - Albert, Jakob A1 - Wasserscheid, Peter A1 - Zöttl, Gregor A1 - Grimm, Veronika T1 - Economic comparison of electric fuels produced at excellent locations for renewable energies: A Scenario for 2035 N2 - The use of electric fuels (e-fuels) enables CO2-neutral mobility and opens therefore an alternative to fossil-fuel-fired engines or battery-powered electric motors. This paper compares the cost-effectiveness of Fischer-Tropsch diesel, methanol, and hydrogen stored as cryogenic liquid (LH2) or in form of liquid organic hydrogen carriers (LOHCs). The production cost of those fuels are to a large extent driven by the energy-intensive electrolytic water splitting. The option of producing e-fuels in Germany competes with international locations with excellent conditions for renewable energy harvesting and thus very low levelized cost of electricity. We developed a mathematical model that covers the entire process chain. Starting with the production of the required resources such as fresh water, hydrogen, carbon dioxide, carbon monoxide, electrical and thermal energy, the subsequent chemical synthesis, the transport to filling stations in Germany and finally the energetic utilization of the fuels in the vehicle. We found that the choice of production site can have a major impact on the mobility cost using the respective fuels. Especially in case of diesel production, the levelized cost of electricity driven by the full load hours of the applied renewable energy source have a huge impact. An LOHC-based system is shown to be less dependent on the kind of electricity source compared to other technologies due to its comparatively low electricity consumption and the low cost for the hydrogenation units. The length of the transportation route and the price of the filling station infrastructure, on the other hand, clearly increase mobility cost for LOHC and LH2. KW - Electric fuels, Hydrogen Utilization, Hydrogen Import, LOHC, Mobility Y1 - 2020 ER - TY - INPR A1 - Bohlayer, Markus A1 - Bürger, Adrian A1 - Fleschutz, Markus A1 - Braun, Marco A1 - Zöttl, Gregor T1 - Multi-period investment pathways - Modeling approaches to design distributed energy systems under uncertainty N2 - Multi-modal distributed energy system planning is applied in the context of smart grids, industrial energy supply,and in the building energy sector. In real-world applications, these systems are commonly characterized by existing system structures of different age where monitoring and investment are conducted in a closed-loop, with the iterative possibility to invest. The literature contains two main approaches to approximate this computationally intensive multiperiod investment problem. The first approach simplifies the temporal decision-making process collapsing the multistage decision to a two-stage decision, considering uncertainty in the second stage decision variables. The second approach considers multi-period investments under the assumption of perfect foresight. In this work, we propose a multi-stage stochastic optimization problem that captures multi-period investment decisions under uncertainty and solves the problem to global optimality, serving as a first-best benchmark to the problem. To evaluate the performance of conventional approaches applied in a multi-year setup and to solve the multi-period problem at lower computational effort, we propose a rolling horizon heuristic that on the one hand reveals the performance of conventional approaches applied in a multi-period set-up and on the other hand enables planners to identify approximate solutions to the original multi-stage stochastic problem. Additionally, we consider an open-loop version of the rolling horizon algorithm to evaluate how single-period investments perform with respect to the entire scenario tree and compared to multi-period investments. We conduct a real-world case study and investigate solution quality as well as the computational performance of the proposed approaches. Our findings indicate that the approximation of multi-period investments by two-stage stochastic approaches yield the best results regarding constraint satisfaction, while deterministic multi-period approximations yield better economic and computational performance. Y1 - 2020 ER - TY - INPR A1 - Biefel, Christian A1 - Liers, Frauke A1 - Rolfes, Jan A1 - Schewe, Lars A1 - Zöttl, Gregor T1 - Robust Market Equilibria under Uncertain Cost N2 - We consider equilibrium problems under uncertainty where firms maximize their profits in a robust way when selling their output. Robust optimization plays an increasingly important role when best guaranteed objective values are to be determined, independently of the specific distributional assumptions regarding uncertainty. In particular, solutions are to be determined that are feasible regardless of how the uncertainty manifests itself within some predefined uncertainty set. Our analysis adopts the robust optimization perspective in the context of equilibrium problems. First, we consider a singlestage, nonadjustable robust setting. We then go one step further and study the more complex two-stage or adjustable case where a part of the variables can adjust to the realization of the uncertainty. We compare equilibrium outcomes with the corresponding centralized robust optimization problem where the sum of all profits are maximized. As we find, the market equilibrium for the perfectly competitive firms differs from the solution of the robust central planner, which is in stark contrast to classical results regarding the efficiency of market equilibria with perfectly competitive firms. For the different scenarios considered, we furthermore are able to determine the resulting price of anarchy. In the case of non-adjustable robustness, for fixed demand in every time step the price of anarchy is bounded whereas it is unbounded if the buyers are modeled by elastic demand functions. For the two-stage adjustable setting, we show how to compute subsidies for the firms that lead to robust welfare optimal equilibria. KW - Continuous Optimization KW - Equilibrium Problems KW - Robust Optimization KW - Adjustable Robustness Y1 - 2021 ER - TY - JOUR A1 - Grimm, Veronika A1 - Rückel, Bastian A1 - Sölch, Christian A1 - Zöttl, Gregor T1 - Regionally differentiated network fees to affect incentives for generation investment JF - Energy N2 - 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. KW - Electricity Markets, Network Expansion, Generation Expansion, Investment Incentives, Computational Equilibrium Models Y1 - 2019 U6 - https://doi.org/10.1016/j.energy.2019.04.035 IS - 177 SP - 487 EP - 502 ER - TY - JOUR A1 - Grimm, Veronika A1 - Grübel, Julia A1 - Rückel, Bastian A1 - Sölch, Christian A1 - Zöttl, Gregor T1 - Storage investment and network expansion in distribution networks: The impact of regulatory frameworks JF - Applied Energy N2 - 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. KW - Computational Equilibrium Models; Electricity Markets; Investment Incentives; Distribution Network Expansion Planning; Storage Investment and Operation; Renewable Energy Production; (Self-) Consumption Y1 - 2019 U6 - https://doi.org/10.1016/j.apenergy.2019.114017 IS - 262 ER - TY - JOUR A1 - Runge, Philipp A1 - Sölch, Christian A1 - Albert, Jakob A1 - Wasserscheid, Peter A1 - Zöttl, Gregor A1 - Grimm, Veronika T1 - Economic comparison of different electric fuels for energy scenarios in 2035 JF - Applied Energy N2 - 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. KW - Sector Coupling, Eletricity Markets, Electric fuels, Hydrogen Utilization, LOHC, Mobility Y1 - 2019 U6 - https://doi.org/10.1016/j.apenergy.2018.10.023 IS - 233-234 SP - 1078 EP - 1093 ER -