A Dynamic Multilevel Model of the European Gas Market - Gugat, Martin; Schuster, Michael; Steffensen, Sonja (Communications in Optimization Theory, 2023)
The European gas market is governed by rules that are agreed on by the European Union. We present a mathematical market model that takes into account this structure, where the technical system operator (TSO) offers certain transportation capacities that can be booked and later nominated within the previously chosen bookings. The TSO also fixes booking fees and defines an operational control of the gas pipeline system in order to deliver the gas according to the nominations. Since the gas flow is governed by a system of partial differential equations, to realize this control structure partial differential equations (PDEs) should be involved in the model. While the four level gas market model has been discussed previously, in this paper we take into account the flow model by PDEs in the discussion of the model and in the reduction to a single level problem, where we also state the corresponding necessary optimality conditions. We also present some examples for the optimal control problem of the TSO.
Keywords: Gas market; Flow model; Isothermal Euler equations; Partial differential equation; Nash equilibrium; Potential games; Network; Coupling conditions

CONVEXITY AND STARSHAPEDNESS OF FEASIBLE SETS IN STATIONARY FLOW NETWORKS - Gugat, Martin; Schultz, Ruediger; Schuster, Michael (Networks and Heterogeneous Media, 2020)
In this paper, we consider a stationary model for the flow through a network. The flow is determined by the values at the boundary nodes of the network. We call these values the loads of the network. In the applications, the feasible loads must satisfy some box constraints. We analyze the structure of the set of feasible loads. Our analysis is motivated by gas pipeline flows, where the box constraints are pressure bounds.
CRIS-Team WoS Importer:2020-06-09

Max-p Optimal Boundary Control of Gas Flow - Schuster, Michael; Gugat, Martin (Extended Abstracts presented at the 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022, 2022)
In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas. For the operation of gas transport networks it is important to take into account the randomness of the consumers' demand, where often information on the probability distribution is available. Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H^2 norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause embrittlement of the pipeline meta
Keywords: gas pipeline flow; nodal control; boundary control; optimal control; hyperbolic differential equation; random demand; state constraints; pressure bound; classical solutions

Nodal Control and Probabilistic Constrained Optimization using the Example of Gas Networks - Schuster, Michael (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2021)

ON THE CONVERGENCE OF OPTIMIZATION PROBLEMS WITH KERNEL DENSITY ESTIMATED PROBABILISTIC CONSTRAINTS - Schuster, Michael (Applied Set-Valued Analysis and Optimization, 2025)
Uncertainty plays a significant role in applied mathematics and probabilistic constraints are widely used to model uncertainty in various fields even if probabilistic constraints often demand computational challenges. Kernel density estimation (KDE) provides a data-driven approach for properly estimating probability density functions and efficiently evaluating corresponding probabilities. In this paper, we investigate optimization problems with probabilistic constraints, where the probabilities are approximated using a KDE approach. We establish sufficient conditions under which the solution of the KDE approximated optimization problem converges to the solution of the original problem as the sample size goes to infinity. The main results of this paper include three theorems: (1) For sufficiently large sample sizes, the solution of the original problem is also a solution of the approximated problem, if the probabilistic constraint is passive; (2) The limit of a convergent sequence of solutions of the approximated problems is a solution of the original problem, if the KDE uniformly converges; (3) We provide sufficient conditions for the existence of a convergent sequence of solutions of the approximated problems.
CRIS-Team Scopus Importer:2025-04-11
Keywords: Chance constraints; Convergence analysis; Kernel density estimation; Probabilistic constrained optimization; Stochastic optimization

Optimal Control for ODEs and PDEs: The Turnpike Phenomenon - Gugat, Martin; Zuazua Iriondo, Enrique; Schuster, Michael (Extended Abstracts of the 65th Joint Conference on Automatic Control, 2022)
The turnpike phenomenon for dynamic optimal control problems provide insights about the relation between the dynamic optimal control and the solution of the corresponding static optimal control problem. In this talk we give an overview about different turnpike structures for optimal control problems with ordinary differential equations (ODEs) and partial differential equations (PDEs). For optimal control problems with ODEs an exponential turnpike inequality can be shown by basic control theory. These results can be extended to an integral turnpike inequality for optimal control problems with linear hyperbolic systems. For an optimal control problem with non differential tracking term in the objective function, that is exactly controllable, we can show under certain assumptions that the optimal system state is steered exactly to the desired state after finite time. Further we consider an optimal control problem for a hyperbolic system with random boundary data and we show the existence of optimal controls. A turnpike property for hyperbolic systems with random boundary data can be shown numericall
Keywords: Turnpike phenomenon; hyperbolic differential equation; optimal control; boundary control; random boundary data

Optimal Neumann control of the wave equation with L1-control cost: the finite-time turnpike property - Schuster, Michael; Gugat, Martin (Optimization, 2024)
The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an L^1-norm control cost with a weight γ>0" role="presentation" style="display: inline; line-height: normal; font-size: 17.6px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; position: relative;">𝛾>0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared L^L2" role="presentation" style="display: inline; line-height: normal; font-size: 17.6px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; position: relative;">2-norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the L1" role="presentation" style="display: inline; line-height: normal; font-size: 17.6px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; position: relative;">L^1-norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong L^L2" role="presentation" style="display: inline; line-height: normal; font-size: 17.6px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; position: relative;">2-sense.<br
Keywords: finite time turnpike property; non-smooth control cost; smoothing; exact control; Neumann boundary control

Probabilistic constrained optimization on flow networks - Gugat, Martin; Schuster, Michael; Lang, Jens; Strauch, Elisa (Optimization and Engineering, 2022)
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 an approximated problem 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.<br
Keywords: Probabilistic constraints; gas networks; kernel density estimation

Stationary Gas Networks with Compressor Control and Random Loads: Optimization with Probabilistic Constraints - Gugat, Martin; Schuster, Michael (Mathematical Problems in Engineering, 2018)
We introduce a stationary model for gas flow based on simplified isothermal Euler equations in a non-cycled pipeline network. Especially the problem of the feasibility of a random load vector is analyzed. Feasibility in this context means the existence of a flow vector meeting these loads, which satisfies the physical conservation laws with box constraints for the pressure. An important aspect of the model is the support of compressor stations, which counteract the pressure loss caused by friction in the pipes. The network is assumed to have only one influx node; all other nodes are efflux nodes. With these assumptions the set of feasible loads can be characterized analytically. In addition we show the existence of optimal solutions for some optimization problems with probabilistic constraints. A numerical example based on real data completes this paper.<br

The Finite-Time Turnpike Phenomenonfor Optimal Control Problems:Stabilization by Non-smooth TrackingTerms - Gugat, Martin; Zuazua Iriondo, Enrique; Schuster, Michael (Stabilization of Distributed Parameter Systems: Design Methods and Applications, 2021)
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 tim
Keywords: Turnpike phenomenon; finite time; optimal control problems

The location problem for compressor stations in pipeline networks - Gugat, Martin; Schuster, Michael; Sokolowski, Jan (Mathematics and Mechanics of Complex Systems, 2024)
In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related to the gas dynamics that governs the network flow. That results in nonconvex mixed integer stochastic optimization problems with probabilistic constraints. Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, we consider the problem of finding the optimal location for the control on the network such that the control cost is minimal and the gas pressure stays within given bounds. In the deterministic setting, we present explicit bounds for the pipe length and the inlet pressure such that a unique optimal compressor location with minimal control cost exists. In the probabilistic setting, we give an existence result for the optimal compressor location and discuss the uniqueness of the solution depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. We further present the problem of finding optimal compressor locations on networks including the number of compressor stations as a variable. Results for the existence of optimal locations on a graph in both the deterministic and the probabilistic setting are presented, and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example on a diamond graph demonstrating that the minimal number of compressor stations is not necessarily equal to the optimal number of compressor station
Keywords: gas network; compressor control; compressor location; Weber problem; optimal location; uncertain boundary data; nonconvex mixed integer stochastic problem