Humboldt-Universität zu Berlin
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This document aims to provide a concise and clear introduction to the topic of gas flow modeling. We present several models for gas flow, organized into hierarchies based on complexity. We discuss in detail the modeling of individual components such as valves and compressors. Network model classes based on purely algebraic relations and energy-based port-Hamiltonian models are included, along with a brief overview of basic numerical methods for hyperbolic balance laws and port-Hamiltonian systems.
We do not claim completeness and refer in many places to the existing literature.
We study model predictive control (MPC) schemes
for non-cooperative dynamic games regarding stabilization. The
dynamic games are modelled as generalized Nash equilibrium
problems (GNEPs), in which a shared constraint is given as a
jointly controlled time-discrete (linear) dynamics. Furthermore,
the players’ objectives are interdependent. We present recent
results concerning their stabilizing properties using α-quasi-
GENP-approximation and terminal conditions in the form of
equilibrium endpoint constraints. Moreover, we extend the result
towards Lyapunov terminal costs, which is a more general type
of terminal condition. Furthermore, we show that a suitable
Lyapunov terminal cost can be obtained from a non-game-
based MPC scheme. This non-game-based MPC scheme relies
on a classical optimal control problem for the aggregated cost.
Hence, known results for determining the Lyapunov cost can
be applied and carried over to the game-based setting. The
theoretical results are complemented by numerical experiments.
This paper addresses the critical challenge of hydrogen embrittlement in the context of Germany’s transition to a sustainable, hydrogen-inclusive energy system. As hydrogen infrastructure expands, estimating and pricing embrittlement become paramount due to safety, operational, and economic concerns. We present a twofold contribution: (1) We discuss hydrogen embrittlement modeling using both continuum models and simplified approximations. (2) Based on these models, we propose optimization-based pricing schemes for market makers, considering simplified cyclic loading and more complex digital twin models. Our approaches leverage widely-used subcritical crack growth models in steel pipelines, with parameters derived from experiments. The study highlights the challenges and potential solutions for incorporating hydrogen embrittlement into gas transportation planning and pricing, ultimately aiming to enhance the safety and economic viability of Germany’s future energy infrastructure.
We consider an optimal control problem for the heat equation as a prototypical parabolic partial differential equation with a non-convex control mechanism of the form continuous-or-off. We model this fundamental switching mechanism as the product of a classically continuous and a binary control both in the control term of the dynamics and in the objective. A total variation regularization is added to the cost in order to restrict the number of switching times. This renders the problem as a mixed-integer non-linear PDE-constrained problem. We discuss well-posedness of the problem and present an exact relaxation result for a linearized and a trust-region type penalized problem. The exactness result is constructive and provides a way to numerically compute mixed-integer optimal solutions from the optimality conditions of an associated PDE-constrained problem without integer restrictions. It lays a foundation for a new class of sequential relaxation algorithms to solve the considered class of mixed-integer control problems. This is demonstrated numerically by showcasing a descent step in the presence of binary restrictions.
We present a feedback scheme for non-cooperative dynamic games and investigate its stabilizing properties. The dynamic games are modeled as generalized Nash equilibrium problems (GNEP), in which the shared constraint consists of linear time-discrete dynamic equations (e.g., sampled from a partial or ordinary differential equation), which are jointly controlled by the players’ actions. Further, the individual objectives of the players are interdependent and defined over a fixed time horizon. The feedback law is synthesized by moving-horizon model predictive control (MPC). We investigate the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce α-quasi GNEPs, a family of auxiliary problems based on a modification of the Nikaido–Isoda function, which approximate the original games. Basing the MPC scheme on these auxiliary problems, we derive conditions on the players’ objectives, which guarantee asymptotic stability of the closed-loop if stabilizing end constraints are enforced. This analysis is based on showing that the associated optimal-value function is a Lyapunov function. Additionally, we identify a suitable Lyapunov function for the MPC scheme based on the original GNEP, whose solution fulfills the stabilizing end constraints. The theoretical results are complemented by numerical experiments.
With this overview we want to provide a compilation of different models for
the description of gas flow in networks in order to facilitate the introduction
to the topic. Special attention is paid to the hierarchical structure inherent
to the modeling, and the detailed description of individual components such
as valves and compressors. Also included are network model classes based
on purely algebraic relations, and energy-based port-Hamiltonian models. A
short overview of basic numerical methods and concepts for the treatment
of hyperbolic balance equations is also given. We do not claim completeness
and refer in many places to the existing literature.
We propose a method to solve linear generalized Nash equilibrium problems (LGNEPs). For this purpose, a reformulation of the LGNEPs as piecewise linear problems is considered. This requires the calculation of all vertices for a special kind of unbounded convex polyhedra. Then the active signature method for constrained abs-linear problems can be used to determine the Nash equilibria. We analyse the computational effort for the resulting solution procedure. This includes also the verification of suitable optimality conditions. Finally, we present and analyse numerical results for some test problems.
We propose a novel algorithmic approach to computationally solve optimal control problems governed by linear parabolic partial differential equations (PDEs) including a state-dependent control-regime switching mechanism. We state an equivalent mixed-integer formulation featuring vanishing constraints (VCs) arising from methods of disjunctive programming. We embed the problem into the class of equilibrium constraints (ECs) by introduction of an additional slack variable. Based on theoretical results associated with Sum-Up-Rounding (SUR) strategies, we proceed with the solution of the related relaxed formulation by an indirect approach. In order to obtain a computationally tractable optimality system, we apply a Moreau-Yosida type penalty approach for the VCs. After a theoretical discussion, we introduce and exert the algorithmic framework founded on a semismooth Newton method. Finally, we communicate computational experiments based on the proposed approach.
Design of foundations on an elastic base is carried out using the solution of three-dimensional problems of contact interaction. Improving the accuracy of engineering calculations is necessary to ensure economic efficiency and increase energy savings in green building. The problems of indentation of punches with a flat base bounded by doubly connected close to polygonal contact areas are researched in the present work. Small parameter method is used to obtain explicit analytical expressions for the contact pressure distribution and the punch displacement dependence in a simplified form, which is convenient for engineering practice. The found load-displacement dependence satisfies the known inequalities that are valid for an arbitrary contact domain. Also a numerical-analytical method is in consideration. It uses the simple layer potential expansion and successive approximations for the problems accounting roughness of the elastic half-space. Roughness coefficient is considered as a parameter of regularization of the integral equation for the smooth contact problem. The results of both methods coincide with sufficient accuracy.
The objective is to optimize the pressure distribution under a rigid punch having a doubly connected contact domain close to a circular ring and interacting with an elastic half-space. The required design variable is the punch shape. The functional to be minimized is the root-mean-square deviation of the pressure distribution from some given distribution. An analytical technique is developed for solving the problem for the punches with doubly connected shape, by reducing to a sequence of similar problems for the circular ring punches using expansions of the simple layer potential. The method of expansion in terms of a small parameter is used. The simple layer potential expansion is proposed when mapping a doubly connected integration domain onto a circular ring by transforming the integration variables and transforming the coordinates of the pole of the kernel. As a result, a sequence of similar problems was obtained for a circular ring to determine the functions characterizing the distribution of normal pressure under the punch in the form of a non-circular ring, as well as the normal displacements, from where the optimal punch shape is determined.