TY - JOUR A1 - Gontran, Lance A1 - Trélat, Emmanuel A1 - Zuazua, Enrique T1 - Shape turnpike for linear parabolic PDE models N2 - We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations. Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1016/j.sysconle.2020.104733 VL - 142 PB - Syst. Control. Lett. ER - TY - JOUR A1 - Biccari, Umberto A1 - Navarro-Quiles, Ana A1 - Zuazua, Enrique T1 - Stochastic optimization methods for the simultaneous control of parameter-dependent systems N2 - We address the application of stochastic optimization methods for the simultaneous control of parameter-dependent systems. In particular, we focus on the classical Stochastic Gradient Descent (SGD) approach of Robbins and Monro, and on the recently developed Continuous Stochastic Gradient (CSG) algorithm. We consider the problem of computing simultaneous controls through the minimization of a cost functional defined as the superposition of individual costs for each realization of the system. We compare the performances of these stochastic approaches, in terms of their computational complexity, with those of the more classical Gradient Descent (GD) and Conjugate Gradient (CG) algorithms, and we discuss the advantages and disadvantages of each methodology. In agreement with well-established results in the machine learning context, we show how the SGD and CSG algorithms can significantly reduce the computational burden when treating control problems depending on a large amount of parameters. This is corroborated by numerical experiments. KW - Parameter-dependent systems KW - simultaneous controllability KW - stochastic optimization KW - computational cost Y1 - 2020 ER - TY - JOUR A1 - Esteve, Carlos A1 - Kouhkouh, H A1 - Pighin, Dario A1 - Zuazua, Enrique T1 - The Turnpike property and the long-time behavior of the Hamilton-Jacobi equation N2 - In this work, we analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to an optimal control problem. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation. In order to carry out our study, we use the setting of a finite-dimensional linear-quadratic optimal control problem, for which the turnpike property is well understood. We prove that, when the time horizon T tends to infinity, the value function converges to a travelling-front like solution of the form W(x) + c T + λ. In addition, we provide a control interpretation of each of these three terms in the spirit of the turnpike theory. Finally, we compare this asymptotic decomposition with the existing results on long-time behavior for Hamilton-Jacobi equations. We stress that in our case, the Hamiltonian is not coercive in the momentum variable, a case rarely considered in the classical literature about Hamilton-Jacobi equations. KW - Optimal control problems KW - long-time behavior KW - the turnpike property KW - Hamilton-Jacobi-Bellman equations KW - linear-quadratic Y1 - 2020 ER - TY - JOUR A1 - Ruiz-Balet, Domenec A1 - Zuazua, Enrique T1 - Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations N2 - Dynamic phenomena in social and biological sciences can often be modeled employing reaction diffusion equations. Frequently in applications, their control plays an important role when avoiding population extinction or propagation of infectious diseases, enhancing multicultural features, etc. When addressing these issues from a mathematical viewpoint one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in [0; 1])). Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady states of spatially homogeneous semilinear heat equations, with constraints in the state, and using boundary controls, which is indeed a natural way of acting on the system in the present context. The nonlinearities considered are among the most frequent: monostable and bistable ones. We prove that controlling the system to a constant steadystate may become impossible when the diffusivity is too small (or when the domain is large), due to the existence of barrier functions. When such an obstruction does not arise, we build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves, some connectivity of the set of steady states. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady states. These techniques fail in the present multidimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain. The results are illustrated by numerical simulations of these models that find several applications, such as the extinction of minority languages or the survival of rare species in sufficiently large reserved areas. KW - Constraints KW - Controllability KW - Mathematical biology KW - Reaction-diffusion Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1016/j.matpur.2020.08.006 VL - 143 SP - 345 EP - 375 ER - TY - JOUR A1 - Ko, Dongnam A1 - Zuazua, Enrique T1 - Model predictive control with random batch methods for a guiding problem N2 - We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space. The numerical simulation of such models quickly becomes unfeasible for a large number of interacting agents. To reduce the computational cost, we use the Random Batch Method (RBM), which provides a computationally feasible approximation of the dynamics. At each time step, the RBM randomly divides the set of particles into small subsets (batches), considering only the interactions inside each batch. Due to the averaging effect, the RBM approximation converges to the exact dynamics as the time discretization gets finer. We propose an algorithm that leads to the optimal control of a fixed RBM approximated trajectory using a classical gradient descent. The resulting control is not optimal for the original complete system, but rather for the reduced RBM model. We then adopt a Model Predictive Control (MPC) strategy to handle the error in the dynamics. While the system evolves in time, the MPC strategy consists in periodically updating the state and computing the optimal control over a long-time horizon, which is implemented recursively in a shorter time-horizon. This leads to a semi-feedback control strategy. Through numerical experiments we show that the combination of RBM and MPC leads to a significant reduction of the computational cost, preserving the capacity of controlling the overall dynamics. KW - Agent-based models KW - Guiding problem KW - large scale complex systems KW - Random Batch Method KW - Model Predictive Control Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1142/S0218202521500329 N1 - Ko Dongnam, Zuazua Enrique. Model predictive control with random batch methods for a guiding problem (2021). Math. Models Methods Appl. Sci., Vol. 31, No. 8, pp. 1569-1592. (2021) DOI: https://doi.org/10.1142/S0218202521500329 VL - 31 IS - 8 SP - 1569 EP - 1592 PB - Math. Models Methods Appl. Sci. ER - TY - JOUR A1 - Zuazua, Enrique T1 - Asymptotic behavior of scalar convection-diffusion equations N2 - In these lecture notes, we address the problem of large-time asymptotic behaviour of the solutions to scalar convection-diffusion equations set in [katex]\mathbb{R}^N[/katex]. The large-time asymptotic behaviour of the solutions to many convection-diffusion equations is strongly linked with the behavior of the initial data at infinity. In fact, when the initial datum is integrable and of mass [katex]M[/katex], the solutions to the equations under consideration oftentimes behave like the associated self-similar profile of mass [katex]M[/katex], thus emphasising the role of scaling variables in these scenarios. However, these equations can also manifest other asymptotic behaviors, including weakly non-linear, linear or strongly non-linear behavior depending on the form of the convective term. We give an exhaustive presentation of several results and techniques, where we clearly distinguish the role of the spatial dimension and the form of the nonlinear convective term. Y1 - 2020 ER - TY - JOUR A1 - Biccari, Umberto A1 - Zuazua, Enrique T1 - A Stochastic Approach to the Synchronization of Coupled Oscillators N2 - This paper deals with an optimal control problem associated with the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation. KW - coupled oscillators KW - gradient descent KW - Kuramoto model KW - optimal control KW - random batch method Y1 - 2020 U6 - https://doi.org/10.3389/fenrg.2020.00115 VL - 8 ER - TY - JOUR A1 - Heiland, Jan A1 - Zuazua, Enrique T1 - Classical system theory revisited for Turnpike in standard state space systems and impulse controllable descriptor systems N2 - The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of the turnpike is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike in the nondetectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems, we establish existence and convergence of solutions to a generalized differential Riccati equation. KW - descriptor systems KW - linear systems KW - long time behavior KW - optimal control KW - Riccati equations Y1 - 2020 ER - TY - JOUR A1 - Esteve, Carlos A1 - Geshkovski, Borjan A1 - Pighin, Dario A1 - Zuazua, Enrique T1 - Large-time asymptotics in deep learning N2 - It is by now well-known that practical deep supervised learning may roughly be cast as an optimal control problem for a specific discrete-time, nonlinear dynamical system called an artificial neural network. In this work, we consider the continuous-time formulation of the deep supervised learning problem, and study the latter’s behavior when the final time horizon increases, a fact that can be interpreted as increasing the number of layers in the neural network setting. When considering the classical regularized empirical risk minimization problem, we show that, in long time, the optimal states converge to zero training error, namely approach the zero training error regime, whilst the optimal control parameters approach, on an appropriate scale, minimal norm parameters with corresponding states precisely in the zero training error regime. This result provides an alternative theoretical underpinning to the notion that neural networks learn best in the overparametrized regime, when seen from the large layer perspective. We also propose a learning problem consisting of minimizing a cost with a state tracking term, and establish the well-known turnpike property, which indicates that the solutions of the learning problem in long time intervals consist of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal solution of an associated static learning problem. This property in fact stipulates a quantitative estimate for the number of layers required to reach the zero training error regime. Both of the aforementioned asymptotic regimes are addressed in the context of continuous-time and continuous space-time neural networks, the latter taking the form of nonlinear, integro-differential equations, hence covering residual neural networks with both fixed and possibly variable depths. KW - deep learning KW - Neural ODEs KW - optimal control KW - Residual Neural Networks KW - Supervised Learning Y1 - 2020 ER - TY - JOUR A1 - Esteve, C A1 - Zuazua, Enrique T1 - The Inverse Problem for Hamilton-Jacobi equations and Semiconcave Envelopes N2 - We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function [katex]u_T[/katex] and a time horizon [katex]T > 0[/katex], we aim to construct all the initial conditions for which the viscosity solution coincides with [katex]u_T[/katex] at time [katex]T[/katex]. As it is common in this kind of nonlinear equations, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of [katex]u_T[/katex] , is the one obtained by reversing the direction of time in the equation, considering [katex]u_T[/katex] as terminal condition. In this case, we use the notion of backward viscosity solution, that provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, that relates the reachability of the target with its semiconcavity properties. Then, for the case when [katex]u_T[/katex] is reachable, we construct the set of all initial conditions for which the solution coincides with [katex]u_T[/katex] at time [katex]T[/katex]. Note that in general, such initial conditions are not unique. Finally, for the case when the target [katex]u_T[/katex] is not necessarily reachable, we study the projection of [katex]u_T[/katex] on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem, and can be interpreted as the semiconcave envelope of [katex]u_T[/katex] , i.e. the smallest reachable target bounded from below by [katex]u_T[/katex] . KW - Remove term: Hamilton-Jacobi equation Hamilton-Jacobi equation KW - inverse design problem KW - obstacle problems KW - semiconcave envelopes Y1 - 2020 ER - TY - JOUR A1 - Bárcena, J.A. A1 - Zuazua, Enrique T1 - Averaged dynamics and control for heat equations with random diffusion N2 - Abstract. This paper deals with the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the $\frac{1}{2}$-fractional Laplacian, which is wellknown to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime. KW - Averaged controllability, averaged observability, observability, random heat equation Y1 - 2020 ER - TY - INPR A1 - Gabriel, Steven A. A1 - Leal, Marina A1 - Schmidt, Martin T1 - On Linear Bilevel Optimization Problems with Complementarity-Constrained Lower Levels N2 - We consider a novel class of linear bilevel optimization models with a lower level that is a linear program with complementarity constraints (LPCC). We present different single-level reformulations depending on whether the linear complementarity problem (LCP) as part of the lower-level constraint set depends on the upper-level decisions or not as well as on whether the LCP matrix is positive definite or positive semidefinite. Moreover, we illustrate the connection to linear trilevel models that can be reduced to bilevel problems with LPCC lower levels having positive (semi)definite matrices. Finally, we provide two generic and illustrative bilevel models from the fields of transportation and energy to show the practical relevance of the newly introduced class of bilevel problems and show related theoretical results. KW - Bilevel optimization KW - Linear programs with complementarity constraints KW - Linear complementarity problems KW - Reformulations KW - Spatial price equilibria Y1 - 2020 ER - TY - INPR A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - Why there is no need to use a big-M in linear bilevel optimization: A computational study of two ready-to-use approaches N2 - Linear bilevel optimization problems have gained increasing attention both in theory as well as in practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to the ability of this class of problems to model hierarchical decision processes. However, this ability makes bilevel problems also very hard to solve. Since no general-purpose solvers are available, a "best-practice" has developed in the applied OR community, in which not all people want to develop tailored algorithms but "just use" bilevel optimization as a modeling tool for practice. This best-practice is the big-M reformulation of the Karush-Kuhn-Tucker (KKT) conditions of the lower-level problem - an approach that has been shown to be highly problematic by Pineda and Morales (2019). Choosing invalid values for M yields solutions that may be arbitrarily bad. Checking the validity of the big-Ms is however shown to be as hard as solving the original bilevel problem in Kleinert et al. (2019). Nevertheless, due to its appealing simplicity, especially w.r.t. the required implementation effort, this ready-to-use approach still is the most popular method. Until now, there has been a lack of approaches that are competitive both in terms of implementation effort and computational cost. In this note we demonstrate that there is indeed another competitive ready-to-use approach: If the SOS-1 technique is applied to the KKT complementarity conditions, adding the simple additional root-node inequality developed by Kleinert et al. (2020) leads to a competitive performance - without having all the possible theoretical disadvantages of the big-M approach. KW - Bilevel optimization KW - Big-M KW - SOS-1 KW - Valid inequalities KW - Computational analysis Y1 - 2020 ER - TY - JOUR A1 - Egger, Herbert A1 - Philippi, Nora T1 - On the transport limit of singularly perturbed convection-diffusion problems on networks N2 - We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order O(sqrt(eps)) in the L1(L2)- norm with diffusion eps -> 0. In this paper, we prove a corresponding result for problems on one-dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a-priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers. Y1 - 2020 ER - TY - JOUR A1 - Egger, Herbert A1 - Philippi, Nora T1 - A hybrid discontinuous Galerkin method for transport equations on networks JF - Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples N2 - We discuss the mathematical modeling and numerical discretization of 5 transport problems on one-dimensional networks. Suitable coupling conditions are derived that guarantee conservation of mass across network junctions and dissipation of a mathematical energy which allows us to prove existence of unique solutions. We then consider the space discretization by a hybrid discontinuous Galerkin method which provides a suitable upwind mechanism to handle the transport prob10 lem and allows to incorporate the coupling conditions in a natural manner. In addition, the method inherits mass conservation and stability of the continuous problem. Order optimal convergence rates are established and illustrated by numerical tests. Y1 - 2020 ER - TY - INPR A1 - Gugat, Martin A1 - Herty, Michael T1 - Limits of stabilizabilizy for a semilinear model for gas pipeline flow N2 - We present a positive and a negative stabilization result for a semilinear model of gas flow in pipelines. For feedback boundary conditions we obtain an unconditional stabilization result in the absence and conditional instability in the presence of the source term. We also obtain unconditional instability for the corresponding quasilinear model given by the isothermal Euler equations Y1 - 2020 ER - TY - INPR A1 - Gugat, Martin A1 - Herty, Michael T1 - Modeling, Control and Numerics of Gas Networks N2 - In this article we survey recent progress on mathematical results on gas flow in pipe networks with a special focus on questions of control and stabilization. We briefly present the modeling of gas flow and coupling conditions for flow through vertices of a network. Our main focus is on gas models for spatially one-dimensional flow governed by hyperbolic balance laws. We survey results on classical solutions as well as weak solutions. We present results on well–posedness, controllability, feedback stabilization, the inclusion of uncertainty in the models and numerical methods. KW - Hyperbolic Balance Laws, Stabilization, Exact Controllability, Modeling of Gas Flow, Finite-Volume Schemes, Optimal control, Uncertainty Y1 - 2020 ER - TY - JOUR A1 - Branda, Martin A1 - Henrion, René A1 - Pistek, Miroslav T1 - Producer’s Best Response in Pay-as-clear Day-ahead Electricity Market with Uncertain Demand N2 - We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer’s best response given a historical distribution of both estimated and real electricity demand is provided. KW - electricity market KW - multi-leader-common-follower game KW - stochastic demand KW - day-ahead bidding KW - chance constraints Y1 - 2020 ER - TY - JOUR A1 - Adam, Lukas A1 - Branda, Martin A1 - Heitsch, Holger A1 - Henrion, René T1 - Solving joint chance constrained problems using regularization and Benders' decomposition JF - Annals of Operations Research N2 - In this paper we investigate stochastic programs with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution. KW - chance constrained programming KW - optimality conditions KW - regularization KW - Benders cuts KW - gas networks Y1 - U6 - https://doi.org/10.1007/s10479-018-3091-9 VL - 292 SP - 683 EP - 709 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 - 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 - JOUR A1 - Burger, Martin T1 - Network structured kinetic models of social interactions N2 - The aim of this paper is to study the derivation of appropriate meso- and macroscopic models for interactions as appearing in social processes. There are two main characteristics the models take into account, namely a network structure of interactions, which we treat by an appropriate mesoscopic description, and a different role of interacting agents. The latter differs from interactions treated in classical statistical mechanics in the sense that the agents do not have symmetric roles, but there is rather an active and a passive agent. We will demonstrate how a certain form of kinetic equations can be obtained to describe such interactions at a mesoscopic level and moreover obtain macroscopic models from monokinetics solutions of those. The derivation naturally leads to systems of nonlocal reaction-diffusion equations (or in a suitable limit local versions thereof), which can explain spatial phase separation phenomena found to emerge from the microscopic interactions. We will highlight the approach in three examples, namely the evolution and coarsening of dialects in human language, the construction of social norms, and the spread of an epidemic. Y1 - 2021 U6 - https://doi.org/10.1007/s10013-021-00505-8 ET - Vietnam Journal of Mathematics ER - TY - JOUR A1 - Gugat, Martin A1 - Giesselmann, Jan T1 - Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks N2 - The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. In this system, the influence of certain source terms that model friction effects is essential. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. In this paper, we analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by 1 or -1 respectively. Thus in the corresponding eigenvalues the influence of the gas velocity is neglected, which is justified in the applications since it is much smaller than the sound speed in the gas. For a star-shaped network of horizontal pipes for suitable coupling conditions we present boundary feedback laws that stabilize the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the $H^1$-norm for arbitrarily long pipes. This is remarkable since in general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon observability inequalities for the $L^2$ and the $H^1$-norm. Y1 - 2020 U6 - https://doi.org/10.1051/cocv/2021061 CY - ESAIM:COCV ER - TY - JOUR A1 - Gugat, Martin A1 - Schuster, Michael A1 - Zuazua, Enrique T1 - M. Gugat, M. Schuster, E. Zuazua. The Finite-Time Turnpike Phenomenon for Optimal Control Problems: Stabilization by Non-Smooth Tracking Terms, in “Stabilization of Distributed Parameter Systems: Design Methods and Applications”. Grigory Sklyar Alexander Zuyev Eds., ICIAM 2019 SEMA SIMAI Springer Series 2, p. 17-42. ISSN 2199-3041 N2 - In this paper, problems of optimal control are considered where in the objective function, in addition to the control cost, there is a tracking term that measures the distance to a desired stationary state. The tracking term is given by some norm, and therefore it is in general not differentiable. In the optimal control problem, the initial state is prescribed. We assume that the system is either exactly controllable in the classical sense or nodal profile controllable. We show that both for systems that are governed by ordinary differential equations and for infinite-dimensional systems, for example, for boundary control systems governed by the wave equation, under certain assumptions, the optimal system state is steered exactly to the desired state after finite time. Y1 - 2020 VL - SEMA SIMAI Springer Series 2 SP - 17 EP - 42 PB - Springer International Publishing ET - Grigory Sklyar Alexander Zuyev Eds., ICIAM 2019 ER - TY - INPR A1 - Sarna, Neeraj A1 - Giesselmann, Jan A1 - Benner, Peter T1 - Data-Driven Snapshot Calibration via Monotonic Feature Matching N2 - Snapshot matrices of hyperbolic equations have a slow singular value decay, resulting in inefficient reduced-order models. We develop on the idea of inducing a faster singular value decay by computing snapshots on a transformed spatial domain, or the so-called snapshot calibration/transformation. We are particularly interested in problems involving shock collision, shock rarefaction-fan collision, shock formation, etc. For such problems, we propose a realizable algorithm to compute the spatial transform using monotonic feature matching. We consider discontinuities and kinks as features, and by carefully partitioning the parameter domain, we ensure that the spatial transform has properties that are desirable both from a theoretical and an implementation standpoint. We use these properties to prove that our method results in a fast $m$-width decay of a so-called calibrated manifold. A crucial observation we make is that due to calibration, the $m$-width does not only depend on $m$ but also on the accuracy of the full order model, which is in contrast to elliptic and parabolic problems that do not need calibration. The method we propose only requires the solution snapshots and not the underlying partial differential equation (PDE) and is therefore, data-driven. We perform several numerical experiments to demonstrate the effectiveness of our method. Y1 - 2020 ER - TY - JOUR A1 - Schuster, Michael A1 - Strauch, Elisa A1 - Gugat, Martin A1 - Lang, Jens T1 - Probabilistic Constrained Optimization on Flow Networks N2 - Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimation. In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for the stationary and the dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability. KW - Probabilistic Constraints KW - Flow Networks KW - Gas Networks KW - Spheric Radial Decomposition KW - Kernel Density Estimator Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1007/s11081-021-09619-x VL - Optimization and Engineering ER - TY - INPR A1 - Aigner, Kevin-Martin A1 - Burlacu, Robert A1 - Liers, Frauke A1 - Martin, Alexander T1 - Solving AC Optimal Power Flow with Discrete Decisions to Global Optimality N2 - We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions. The latter occur, for instance, in the context of the curtailment of renewables or the switching of power generation units and transmission lines. Our approach delivers globally optimal solutions and is provably convergent. We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs. The solution method starts from a known framework that uses piecewise linear relaxations. These relaxations are modeled as as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled. In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements. In particular, these are mixed-integer second-order cone programs as well as primal and dual cutting planes. For example objective cuts and no-good-cuts help to compute good feasible solutions as where outer approximation constraints tighten the relaxations. We present extensive numerical results for various AC OPF problems where discrete decisions play a major role. Even for hard instances with a large proportion of discrete decisions, the method is able to generate high quality solutions efficiently. Furthermore, we compare our approach with state-of-the-art MINLP. Our method outperforms all other algorithms. KW - Mixed-Integer Nonlinear Programming KW - Second-Order Cone Programming KW - AC Optimal Power Flow KW - Discrete Decisions KW - Piecewise Linear Relaxation Y1 - 2020 ER - TY - JOUR A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Thürauf, Johannes T1 - Global Optimization for the Multilevel European Gas Market System with Nonlinear Flow Models on Trees JF - Journal of Global Optimization N2 - The European gas market is implemented as an entry-exit system, which aims to decouple transport and trading of gas. It has been modeled in the literature as a multilevel problem, which contains a nonlinear flow model of gas physics. Besides the multilevel structure and the nonlinear flow model, the computation of so-called technical capacities is another major challenge. These lead to nonlinear adjustable robust constraints that are computationally intractable in general. We provide techniques to equivalently reformulate these nonlinear adjustable constraints as finitely many convex constraints including integer variables in the case that the underlying network is tree-shaped. We further derive additional combinatorial constraints that significantly speed up the solution process. Using our results, we can recast the multilevel model as a single-level nonconvex mixed-integer nonlinear problem, which we then solve on a real-world network, namely the Greek gas network, to global optimality. Overall, this is the first time that the considered multilevel entry-exit system can be solved for a real-world sized network and a nonlinear flow model. Y1 - 2020 U6 - https://doi.org/10.1007/s10898-021-01099-8 ER - TY - THES A1 - Manns, Julian T1 - Presolve of Linear Bilevel Programs N2 - Bilevel programs are complex optimization problems that can be used to model hierarchical decision processes, which occur e.g. in energy markets, critical infrastructure defense or pricing models. Even the most simple bilevel programs, where only linear objective functions and constraints appear, are non-convex optimization problems and equivalent single level formulations replace the lower level problem by its non-convex optimality constraints. This makes linear bilevel programs inherently difficult so solve. The simplification of mixed-integer linear programs before solving them, called presolve, significantly accelerated the solving of these problems. However, there is only very few literature on the topic of presolve of bilevel programs. In this thesis we review said literature on presolve of bilevel programs in the context of linear bilevel programming, derive new theoretical foundations for presolve of linear bilevel programs and then apply these results to analyze how common presolve techniques for linear and mixed integer programs can be used to presolve linear bilevel programs. Y1 - 2020 ER - TY - INPR A1 - Biefel, Christian A1 - Liers, Frauke A1 - Rolfes, Jan A1 - Schmidt, Martin T1 - Affinely Adjustable Robust Linear Complementarity Problems N2 - Linear complementarity problems are a powerful tool for modeling many practically relevant situations such as market equilibria. They also connect many sub-areas of mathematics like game theory, optimization, and matrix theory. Despite their close relation to optimization, the protection of LCPs against uncertainties - especially in the sense of robust optimization - is still in its infancy. During the last years, robust LCPs have only been studied using the notions of strict and Γ-robustness. Unfortunately, both concepts lead to the problem that the existence of robust solutions cannot be guaranteed. In this paper, we consider affinely adjustable robust LCPs. In the latter, a part of the LCP solution is allowed to adjust via a function that is affine in the uncertainty. We show that this notion of robustness allows to establish strong characterizations of solutions for the cases of uncertain matrix and vector, separately, from which existence results can be derived. Our main results are valid for the case of an uncertain LCP vector. Here, we additionally provide sufficient conditions on the LCP matrix for the uniqueness of a solution. Moreover, based on characterizations of the affinely adjustable robust solutions, we derive a mixed-integer programming formulation that allows to solve the corresponding robust counterpart. If, in addition, the certain LCP matrix is positive semidefinite, we prove polynomial-time solvability and uniqueness of robust solutions. If the LCP matrix is uncertain, characterizations of solutions are developed for every nominal matrix, i.e., these characterizations are, in particular, independent of the definiteness of the nominal matrix. Robust solutions are also shown to be unique for positive definite LCP matrix but both uniqueness and mixed-integer programming formulations still remain open problems if the nominal LCP matrix is not positive definite. KW - Linear Complementarity Problems KW - Adjustable Robustness KW - Robust Optimization KW - Existence KW - Uniqueness Y1 - 2020 ER - TY - JOUR A1 - Gugat, Martin A1 - Hante, Falk A1 - Jin, Li T1 - Closed loop control of gas flow in a pipe: Stability for a transient model JF - at - Automatisierungstechnik N2 - This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=2/3. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow. We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2-sense. The analysis is done both for the pde system and an ode system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions. Y1 - 2020 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 - JOUR A1 - Böttger, Tom A1 - Grimm, Veronika A1 - Kleinert, Thomas A1 - Schmidt, Martin T1 - The Cost of Decoupling Trade and Transport in the European Entry-Exit Gas Market with Linear Physics Modeling JF - European Journal of Operational Research N2 - Liberalized gas markets in Europe are organized as entry-exit regimes so that gas trade and transport are decoupled. The decoupling is achieved via the announcement of technical capacities by the transmission system operator (TSO) at all entry and exit points of the network. These capacities can be booked by gas suppliers and customers in long-term contracts. Only traders who have booked capacities up-front can "nominate" quantities for injection or withdrawal of gas via a day-ahead market. To ensure feasibility of the nominations for the physical network, the TSO must only announce technical capacities for which all possibly nominated quantities are transportable. In this paper, we use a four-level model of the entry-exit gas market to analyze possible welfare losses associated with the decoupling of gas trade and transport. In addition to the multilevel structure, the model contains robust aspects to cover the conservative nature of the European entry-exit system. We provide several reformulations to obtain a single-level mixed-integer quadratic problem. The overall model of the considered market regime is extremely challenging and we thus have to make the main assumption that gas flows are modeled as potential-based linear flows. Using the derived single-level reformulation of the problem, we show that the feasibility requirements for technical capacities imply significant welfare losses due to unused network capacity. Furthermore, we find that the specific structure of the network has a considerable influence on the optimal choice of technical capacities. Our results thus show that trade and transport are not decoupled in the long term. As a further source of welfare losses and discrimination against individual actors, we identify the minimum prices for booking capacity at the individual nodes. KW - Entry-Exit Gas Market KW - Gas Market Design KW - Multilevel Optimization KW - Robust Optimization Y1 - 2020 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Labbé, Martine A1 - Schmidt, Martin A1 - Plein, Fränk T1 - Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality JF - Optimization Letters N2 - Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker (KKT) conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances. Y1 - 2020 IS - 15 SP - 1027 EP - 1040 ER - TY - INPR A1 - Thürauf, Johannes T1 - Deciding the Feasibility of a Booking in the European Gas Market is coNP-hard N2 - We show that deciding the feasibility of a booking (FB) in the European entry-exit gas market is coNP-hard if a nonlinear potential-based flow model is used. The feasibility of a booking can be characterized by polynomially many load flow scenarios with maximum potential-difference, which are computed by solving nonlinear potential-based flow models. We use this existing characterization of the literature to prove that FB is coNP-hard by reducing Partition to the infeasibility of a booking. We further prove that computing a potential-difference maximizing load flow scenario is NP-hard even if we can determine the flow direction a priori. From the literature, it is known that FB can be decided in polynomial time on trees and a single cycle. Thus, our hardness result draws the first line that separates the easy from the hard variants of FB and finally answers that FB is hard in general. KW - Potential-based flows, Gas networks, Computational complexity, European entry-exit market, Bookings Y1 - 2020 ER - TY - INPR A1 - Aigner, Kevin-Martin A1 - Clarner, Jan-Patrick A1 - Liers, Frauke A1 - Martin, Alexander T1 - Robust Approximation of Chance Constrained DC Optimal Power Flow under Decision-Dependent Uncertainty N2 - We propose a mathematical optimization model and its solution for joint chance constrained DC Optimal Power Flow. In this application, it is particularly important that there is a high probability of transmission limits being satisfied, even in the case of uncertain or fluctuating feed-in from renewable energy sources. In critical network situations where the network risks overload, renewable energy feed-in has to be curtailed by the transmission system operator (TSO). The TSO can reduce the feed-in in discrete steps at each network node. The proposed optimization model minimizes curtailment while ensuring that there is a high probability of transmission limits being maintained. The latter is modeled via (joint) chance constraints that are computationally challenging. Thus, we propose a solution approach based on the robust safe approximation of these constraints. Hereby, probabilistic constraints are replaced by robust constraints with suitably defined uncertainty sets constructed from historical data. The uncertainty sets are calculated by encompassing randomly drawn scenarios using the scenario approach proposed by Margellos et al. (IEEE Transactions on Automatic Control, 59 (2014)). The ability to discretely control the power feed-in then leads to a robust optimization problem with decision-dependent uncertainties, i.e. the uncertainty sets depend on decision variables. We propose an equivalent mixed-integer linear reformulation for box uncertainties with the exact linearization of bilinear terms. Finally, we present numerical results for different test cases from the Nesta archive, as well as for a real network. We consider the discrete curtailment of solar feed-in, for which we use real-world weather and network data. The experimental tests demonstrate the effectiveness of this method and run times are very fast. Moreover, on average the calculated robust solutions lead only to a small increase in curtailment, when compared to nominal solutions. KW - OR in energy KW - optimal power flow KW - chance constrained programming KW - robust optimization KW - decision-dependent uncertainty Y1 - 2020 ER - TY - INPR A1 - Habeck, Oliver A1 - Pfetsch, Marc E. T1 - Combinatorial Acyclicity Models for Potential-based Flows N2 - Potential-based flows constitute a basic model to represent physical behavior in networks. Under natural assumptions, the flow in such networks must be acyclic. The goal of this paper is to exploit this property for the solution of corresponding optimization problems. To this end, we introduce several combinatorial models for acyclic flows, based on binary variables for flow directions. We compare these models and introduce a particular model that tries to capture acyclicity together with the supply/demand behavior. We analyze properties of this model, including variable fixing rules. Our computational results show that the usage of the corresponding constraints speeds up solution times by about a factor of 3 on average and a speed-up of a factor of almost 5 for the time to prove optimality. KW - Network Optimization KW - Potential networks KW - Potential-based flows KW - acyclic flows 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 - INPR A1 - Klimm, Max A1 - Pfetsch, Marc A1 - Raber, Rico A1 - Skutella, Martin T1 - On the Robustness of Potential-Based Flow Networks N2 - Potential-based flows provide a simple yet realistic mathematical model of transport in many real-world infrastructure networks such as, e.g., electricity, gas, or water networks, where the flow along each edge is controlled via the (difference of) potentials at its end nodes. A potential-based flow network is robust if the maximal difference of node potentials needed to satisfy a set of demands cannot increase if demands are decreased. This notion of robustness is motivated by infrastructure networks where users first make reservations for certain demands that may be larger than the actual amounts sent later on. Here node potentials correspond to physical quantities such as the pressures or the voltages and must be guaranteed to lie within a fixed range, even if the actual amounts are smaller than the previously reserved demands. Our main results are a precise characterization of such robust networks for the case of point-to-point demands via forbidden node-labeled graph minors, as well as an efficient algorithm for testing robustness. Y1 - 2019 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 - INPR A1 - Kuchlbauer, Martina A1 - Liers, Frauke A1 - Stingl, Michael T1 - Adaptive bundle methods for nonlinear robust optimization N2 - Currently, there are few theoretical or practical approaches available for general nonlinear robust optimization. Moreover, the approaches that do exist impose restrictive assumptions on the problem structure. We present an adaptive bundle method for nonlinear and non-convex robust optimization problems with a suitable notion of inexactness in function values and subgradients. As the worst case evaluation requires a global solution to the adversarial problem, it is a main challenge in a general non-convex nonlinear setting. Moreover, computing elements of an epsilon-perturbation of the Clarke subdifferential in the l2-norm sense is in general prohibitive for this class of problems. In this article, instead of developing an entirely new bundle concept, we demonstrate how existing approaches, such as Noll's bundle method for non-convex minimization with inexact information (Computational and analytical mathematics 50: 555-592, 2013) can be modified to be able to cope with this situation. Extending the non-convex bundle concept to the case of robust optimization in this way, we prove convergence under two assumptions: Firstly, that the objective function is lower C1 and secondly, that approximately optimal solutions to the adversarial maximization problem are available. The proposed method is hence applicable to a rather general setting of nonlinear robust optimization problems. In particular, we do not rely on a specific structure of the adversary's constraints. The considered class of robust optimization problems covers the case that the worst-case adversary only needs to be evaluated up to a certain precision. One possibility to evaluate the worst case with the desired degree of precision is the use of techniques from mixed-integer linear programming (MIP). We investigate the procedure on some analytic examples. As applications, we study the gas transport problem under uncertainties in demand and in physical parameters that affect pressure losses in the pipes. Computational results for examples in large realistic gas network instances demonstrate the applicability as well as the efficiency of the method. Y1 - 2020 ER - TY - THES A1 - Philippi, Nora T1 - Analysis and Numerical Approximation of Transport Equations on Networks N2 - 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. Y1 - 2020 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 - 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 - INPR A1 - Liers, Frauke A1 - Martin, Alexander A1 - Merkert, Maximilian A1 - Mertens, Nick A1 - Michaels, Dennis T1 - Solving Mixed-Integer Nonlinear Optimization Problems using Simultaneous Convexification - a Case Study for Gas Networks N2 - 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 function 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 that relies on determining the convex envelope of linear combinations of the constraint functions 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. KW - Mixed-Integer Nonlinear Programming KW - Simultaneous Convexification KW - Convex Envelope KW - Gas Network Optimization Y1 - 2020 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Grimm, Veronika A1 - Schmidt, Martin T1 - Outer Approximation for Global Optimization of Mixed-Integer Quadratic Bilevel Problems JF - Mathematical Programming (Series B) N2 - Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP bilevel problems, i.e., models with a mixed-integer convex-quadratic upper level and a continuous convex-quadratic lower level. This setting allows for a strong-duality-based transformation of the lower level which yields, in general, an equivalent nonconvex single-level reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi- and a single-tree outer-approximation-based cutting-plane algorithm. We show finite termination and correctness of both methods and present extensive numerical results that illustrate the applicability of the approaches. It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches. KW - Bilevel optimization KW - Outer approximation KW - Quadratic programming KW - Convex mixed-integer nonlinear optimization Y1 - 2019 ER - TY - INPR A1 - Veronika, Grimm A1 - Kretschmer, Sandra A1 - Mehl, Simon T1 - Home energy storage: A characterization of potential adopters in Germany N2 - 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. KW - Energy storage, Consumer segmentation, Adoption decision, Innovation diffusion Y1 - 2019 ER - TY - JOUR A1 - Grimm, Veronika A1 - Kretschmer, Sandra A1 - Mehl, Simon T1 - Green innovations: The organizational setup of pilot projects and its influence on consumer perceptions JF - Energy Policy N2 - 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. KW - Pilot projects, Organizational setup, Adoption decision, Quadruple Helix, Innovation diffusion Y1 - 2019 U6 - https://doi.org/10.1016/j.enpol.2020.111474 IS - 142 ER - TY - JOUR A1 - Bichler, Martin A1 - Grimm, Veronika A1 - Kretschmer, Sandra A1 - Sutterer, Paul T1 - Market Design for Renewable Energy Auctions: An Analysis of Alternative Auction Formats JF - Energy Economics N2 - 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. KW - renewable energy auction, combinatorial auction, market design Y1 - 2019 U6 - https://doi.org/10.1016/j.eneco.2020.104904 IS - 92 ER - TY - JOUR A1 - Ambrosius, Mirjam A1 - Egerer, Jonas A1 - Grimm, Veronika A1 - van der Weijde, Adriaan H. T1 - Uncertain bidding zone configurations: The role of expectations for transmission and generation capacity expansion JF - European Journal of Operational Research N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1016/j.ejor.2020.01.024 IS - 285 (1) SP - 343 EP - 359 ER -