Friedrich-Alexander-Universität Erlangen-Nürnberg
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.
Home energy storage (HES) poses a promising option to foster flexible electricity demand and grid-stabilising self-supply. Understanding the characteristics of potential storage adopters and their motives for adopting is essential to enable targeted measures to promote diffusion. This paper presents an in-depth analysis of motivational and psychological factors as well as product characteristics that affect the willingness to adopt domestic energy storage. Based on a selective sample of 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 characteristics of home energy storage. Using correlation-based average linkage clustering, we identify four distinct types of storage adopters: finance-oriented, independence-oriented, idealistic and multilaterally oriented households. Our findings emphasise that segment-specific policy strategies and support mechanisms are needed to stimulate HES adoption to pave the way for a sustainable energy system.
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.
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.
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.
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.
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.
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.
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.
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 maximum possible welfare gain of the first-best benchmark.
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.
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.
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.
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.
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.
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.
In optimal control problems, often initial data are required
that are not known exactly in practice.
In order to take into account this uncertainty,
we consider optimal control problems for a system with an uncertain initial
state. A finite terminal time is given. On account of the uncertainty of the
initial state, it is not possible to prescribe an exact terminal state.
Instead, we are looking for controls that steer the system into a given
neighborhood of the desired terminal state with sufficiently high
probability. This neighborhood is described in terms of an inequality for
the terminal energy. The probabilistic constraint in the considered optimal
control problem leads to optimal controls that are robust against the
inevitable uncertainties of the initial state.
We show the existence of such optimal controls.
Numerical examples with
optimal Neumann control of the wave equation are presented.
Exploiting complete linear descriptions for decentralized power market problems with integralities
(2019)
It is well known that linear prices supporting a competitive equilibrium exist in the case of convex markets, however, in the presence of integralities this is open and hard to decide in general. We present necessary and sufficient conditions for the existence of such prices for decentralized market problems where market participants have integral decision variables and their feasible sets are given in complete linear description. We utilize total unimodularity and the aforementioned conditions to show that such linear prices exist and present some applications. Furthermore, we compute competitive equilibria for two classes of decentralized market problems arising in energy markets and show that competitive equilibria may exist regardless of integralities.
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.
Portfolio optimization is an ongoing hot topic of mathematical optimization and management science. Due to the current financial market environment with low interest rates and volatile stock markets, it is getting more and more important to extend portfolio optimization models by other types of investments than classical assets. In this paper, we present a mixed-integer multistage stochastic model that includes investment opportunities in irreversible and long-term infrastructure projects in the context of renewable energies, which are also subject to policy risk. On realistic time scales for investment problems of this type, the resulting instances are by far too large to be solved with today's most evolved optimization software. Thus, we present a tailored moving-horizon approach together with suitable approximations and simplifications of the model. We evaluate these approximations and simplifications in a computational sensitivity analysis and derive a final model that can be tackled on a realistic instance by our moving-horizon approach.