TY - RPRT A1 - Oggioni, Giorgia A1 - Schwartz, Alexandra A1 - Zöttl, Gregor A1 - Wiertz, Ann-Kathrin T1 - Dynamic Pricing and Strategic Retailers in the Energy Sector: A Multi-Leader-Follower Approach N2 - We consider strategic retail pricing in markets, where retail companies buy commodities at fluctuating wholesale prices and resell them to final consumers by applying dynamic retail tariffs. This is of especially large relevance in the context of energy markets where substantial wholesale price fluctuations are observed. Policy makers currently foster the introduction of such dynamic tariff schemes. From a modelling point of view, we propose a multi-leader-follower problem to investigate the implications of strategic retail pricing and we compare the impacts of implementing dynamic tariffs on retailers and final consumers. Our analysis tackles different aspects: first, we formulate the model and provide theoretical results. Second, we develop algorithms, which solve the multi-leader-follower problem and allow us to characterize the resulting market equilibria. Third, we calibrate and solve our framework based on data of the German retail electricity market for the years 2020 and 2021. This allows us to quantitatively assess the impact of introducing real time prices on retailers’ profits and customers’ benefits. As our results show, dynamic real-time pricing on the one hand typically increases market efficiency, which confirms previous results obtained without the explicit consideration of strategic behavior. On the other hand, however, as a novel aspect, dynamic real-time pricing turns out to significantly reduce equilibrium profits in case of strategic firms. This effect is especially large in environments with strongly fluctuating wholesale prices. Y1 - 2022 ER - TY - JOUR A1 - Reuß, Markus A1 - Welder, Lara A1 - Thürauf, Johannes A1 - Linßen, Jochen A1 - Grube, Thomas A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Stolten, Detlef A1 - Robinius, Martin T1 - Modeling Hydrogen Networks for Future Energy Systems: A Comparison of Linear and Nonlinear Approaches JF - International Journal of Hydrogen Energy N2 - Common energy system models that integrate hydrogen transport in pipelines typically simplify fluid flow models and reduce the network size in order to achieve solutions quickly. This contribution analyzes two different types of pipeline network topologies (namely, star and tree networks) and two different fluid flow models (linear and nonlinear) for a given hydrogen capacity scenario of electrical reconversion in Germany to analyze the impact of these simplifications. For each network topology, robust demand and supply scenarios are generated. The results show that a simplified topology, as well as the consideration of detailed fluid flow, could heavily influence the total pipeline investment costs. For the given capacity scenario, an overall cost reduction of the pipeline costs of 37% is observed for the star network with linear cost compared to the tree network with nonlinear fluid flow. The impact of these improvements regarding the total electricity reconversion costs has led to a cost reduction of 1.4%, which is fairly small. Therefore, the integration of nonlinearities into energy system optimization models is not recommended due to their high computational burden. However, the applied method for generating robust demand and supply scenarios improved the credibility and robustness of the network topology, while the simplified fluid flow consideration can lead to infeasibilities. Thus, we suggest the utilization of the nonlinear model for post- processing to prove the feasibility of the results and strengthen their credibility, while retaining the computational performance of linear modeling. Y1 - 2019 U6 - https://doi.org/10.1016/j.ijhydene.2019.10.080 ER - TY - JOUR A1 - Gugat, Martin A1 - Henrion, René A1 - Heitsch, Holger T1 - A turnpike property for optimal control problems with dynamic probabilistic constraints JF - Journal of Convex Analysis N2 - In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give suffcient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. Hence we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics. KW - Probabilistic Constraints KW - Probabilistic Robustness KW - here-and-now decision KW - Turnpike phenomenon KW - Measure turnpike Y1 - 2021 VL - 30 IS - 3 SP - 1025 EP - 1052 PB - Heldermann Verlag 2023 ER - TY - INPR A1 - Geiersbach, Caroline A1 - Hintermüller, Michael T1 - Optimality conditions and Moreau–Yosida regularization for almost sure state constraints N2 - We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau–Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity. Y1 - 2021 ER - TY - INPR A1 - Gahururu, Deborah A1 - Hintermüller, Michael A1 - Surowiec, Thomas T1 - Risk-Neutral PDE-Constrained Generalized Nash Equilibrium Problems N2 - A class of risk-neutral PDE-constrained generalized Nash equilibrium problems is introduced in which the feasible strategy set of each player is subject to a common linear elliptic partial differential equation with random inputs. In addition, each player’s actions are taken from a bounded, closed, and convex set on the individual strategies and a bound constraint on the common state variable. Existence of Nash equilibria and first-order optimality conditions are derived by exploiting higher integrability and regularity of the random field state variables and a specially tailored constraint qualification for GNEPs with the assumed structure. A relaxation scheme based on the Moreau-Yosida approximation of the bound constraint is proposed, which ultimately leads to numerical algorithms for the individual player problems as well as the GNEP as a whole. The relaxation scheme is related to probability constraints and the viability of the proposed numerical algorithms are demonstrated via several examples. Y1 - 2021 ER - TY - JOUR A1 - Kreimeier, Timo A1 - Kuchlbauer, Martina A1 - Liers, Frauke A1 - Stingl, Michael A1 - Walther, Andrea T1 - Towards the Solution of Robust Gas Network Optimization Problems Using the Constrained Active Signature Method N2 - This work studies robust gas network optimization under uncertainties in demand and in the physical parameters. The corresponding optimization problems are nonconvex in node pressures and flows along the pipes. They are thus very difficult to solve for realistic instance sizes. In recent approaches, an adaptive bundle method has been developed, where one solves the occurring adversarial problems via iteratively refined piecewise linear relaxations. These subproblems need to be solved always from scratch using mixed-integer linear programming (MIP). As alternative to the MIP solver, we employ here a nonsmooth optimization approach that allows a warm start strategy such that it can profit from the results obtained for coarser relaxations. We evaluate the approach for realistic gas network topologies and outline possibilities for future research. Y1 - 2022 ER - TY - THES A1 - Huck, Christoph T1 - Perturbation analysis and numerical discretisation of hyperbolic partial differential algebraic equations describing flow networks N2 - This thesis addresses several aspects regarding modelling, analysis and numerical simulation of gas networks. Hereby, our focus lies on (partial) differential-algebraic equations, thus systems of partial and ordinary differential equations which are coupled by algebraic equations. These coupled systems allow an easy approach towards the modelling of dynamic structures on networks. Therefore, they are well suited for gas networks, which have gained a rise of attention in society, politics and science due to the focus towards renewable energies. We give an introduction towards gas network modelling that includes the most common elements that also appear in real gas networks and present two PDAE systems: One for pipe networks and one that includes additional elements like resistors and compressors. Furthermore, we investigate the impact of perturbations onto the pipe network PDAE, where we explicitly allow perturbations to affect the system in the differential as well as in the algebraic components. We conclude that the solution of the PDAE possesses stability properties. In addition, this thesis introduces a new spatial discretisation that is adapted to the net- work topology. This topology-adapted semi-discretisation results in a DAE which possesses the same perturbation behaviour as the space continuous PDAE. Furthermore, we present a topology based decoupling procedure that allows to reformulate the DAE as an ordinary differential equation (ODE), which represents the inherent dynamics of the DAE system. This ODE, together with a decoupled set of algebraic equations, can be derived from the topology and element information directly. We conclude by demonstrating the established results for several benchmark networks. This includes a comparison of numerical solutions for the decoupled ODE and the DAE system. In addition we present the advantages of the topology-adapted spatial discretisation over existing well established methods. Y1 - 2018 U6 - https://doi.org/10.18452/19596 ER - TY - JOUR A1 - Ulbrich, Stefan A1 - Christian, Kirches A1 - Manns, Paul T1 - Compactness and convergence rates in the combinatorial integral approximation decomposition N2 - The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the weak^* topology of L^\infty if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach. Y1 - 2020 ER - TY - JOUR A1 - Ulbrich, Stefan A1 - Manns, Paul T1 - a simplified newton method to generate snapshots for POD models of semilinear optimal controlproblems N2 - n PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however,and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context.In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions. Y1 - 2021 ER - TY - INPR A1 - Veldman, D A1 - Zuazua, E T1 - A framework for randomized time-splitting in linear-quadratic optimal control N2 - Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems. Y1 - ER - TY - INPR A1 - Geshkovski, B A1 - Zuazua, E T1 - Control and Deep Learning: Some connections N2 - This note is an extended abstract for a talk given by the second author during the workshop ”Challenges in Optimization with Complex PDE-Systems”, at Oberwolfach, in February 2021. It is superfluous to state the impact that deep learning has had on modern technology, as it powers many tools of modern society, ranging from web search to content filtering on social networks. A key paradigm of deep learning is that of supervised learning, which may be seen as a compound and high-dimensional simultaneous control problem. This is the viewpoint adopted by our group. And here we present some of our main findings. Y1 - 2021 ER - TY - JOUR A1 - Geshkovski, B A1 - Zuazua, E T1 - Controllability of one-dimensional viscous free boundary flows N2 - In this work, we address the local controllability of a one-dimensional free boundary problem for a fluid governed by the viscous Burgers equation. The free boundary manifests itself as one moving end of the interval, and its evolution is given by the value of the fluid velocity at this endpoint. We prove that, by means of a control actuating along the fixed boundary, we may steer the fluid to constant velocity in addition to prescribing the free boundary’s position, provided the initial velocities and interface positions are close enough. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1137/19M1285354 VL - 59 IS - 3 SP - 1830 EP - 1850 ER - TY - INPR A1 - Esteve, C A1 - Kouhkouh, H A1 - Pighin, D A1 - Zuazua, E 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 a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls. We prove that, when the time horizon TTT tends to infinity, the value function asymptotically behaves as W(x)+c T+λW(x) + c\, T + \lambda W(x)+cT+λ, and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we provide a new turnpike result for the linear-quadratic optimal control problem with constrained control. As a main feature, our turnpike result applies to the case when the steady optimum may saturate the control constraints. This prevented us from proving the turnpike property with an exponential rate, which is well-known to hold for the unconstrained case. Y1 - 2021 ER - TY - JOUR A1 - Joheac, J A1 - Trelat, E A1 - Zuazua, E T1 - Nonnegative control of finite-dimensional linear systems N2 - We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1016/j.anihpc.2020.07.004 VL - 38 IS - 2 SP - 301 EP - 346 ER - TY - INPR A1 - Biccari, U A1 - Warma, M A1 - Zuazua, E T1 - Control and Numerical approximation of Fractional Diffusion Equations N2 - The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain ΩΩΩ. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the L2L^2L2 and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset ω⊂Ωω ⊂ Ωω⊂Ω. Secondly, we move our attention to the exterior control problem, in which the control region O⊂ΩcO ⊂ Ω cO⊂Ωc is located outside ΩΩΩ. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on ∂Ω∂Ω∂Ω. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation. Y1 - 2021 ER - TY - INPR A1 - Heiland, J A1 - Zuazua, E 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 turnpike phenomena 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 phenomena in the non detectable 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 - Riccati equations KW - descriptor systems KW - linear systems KW - long time behavior KW - optimal control Y1 - 2021 ER - TY - INPR A1 - Geshkovski, B A1 - Zuazua, E T1 - Optimal actuator design via Brunovsky’s normal form N2 - In this paper, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear systems with scalar controls. Such systems may be seen as spatially discretized linear partial differential equations with lumped controls. The change of coordinates induced by Brunovsky’s normal form allows us to remove the restriction of having to work with diagonalizable system dynamics, and does not entail a randomization procedure as done in past literature on diffusion equations or waves. Instead, the optimization problem reduces to a minimization of the norm of the inverse of a change of basis matrix, and allows for an easy deduction of existence of solutions, and for a clearer picture of some of the problem’s intrinsic symmetries. Numerical experiments help to visualize these artifacts, indicate further open problems, and also show a possible obstruction of using gradient-based algorithms – this is alleviated by using an evolutionary algorithm. Y1 - 2021 ER - TY - INPR A1 - Biccari, U A1 - Zuazua, E T1 - Multilevel control by duality N2 - We discuss the multilevel control problem for linear dynamical systems, consisting in designing a piece-wise constant control function taking values in a finite-dimensional set. In particular, we provide a complete characterization of multilevel controls through a duality approach, based on the minimization of a suitable cost functional. In this manner we build optimal multi-level controls and characterize the time needed for a given ensemble of levels to assure the controllability of the system. Moreover, this method leads to efficient numerical algorithms for computing multilevel controls. Y1 - 2021 ER - TY - JOUR A1 - Ko, D A1 - Zuazua, E 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. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1142/S0218202521500329 VL - 31 IS - 8 SP - 1569 EP - 1592 ER - TY - INPR A1 - Barcena-Petisco, J.A. A1 - Zuazua, E T1 - Averaged dynamics and control for heat equations with random diffusion N2 - 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. Y1 - 2021 ER -