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- Gas networks (2)
- Mixed-integer nonlinear optimization (2)
- Boundary feedback control, feedback stabilization, exponential stability, isothermal Euler equations, second-order quasilinear equation, Lyapunov function, stationary state, non-stationary state, gas pipeline. (1)
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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.
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 stations.
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.
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.
We present an algorithmic approach for the computational solution of optimal control problems with hybrid nature governed by linear parabolic PDEs featuring implicit switches. We propose a stepwise reformulation of the original formulation into a more tractable setting via application of methods from disjunctive programming and a time transformation method.
After removal of the implicit switching rule at the cost of the introduction of explicit switching variables and vanishing constraints, the connection of the resulting formulation to problems with equilibrium constraints is established and studied. The previous steps in combination with smoothening and a Moreau-Yosida type penalty approach allow the derivation of necessary first order optimality conditions to characterize candidates for optimality to the original system. Following the discussion of each individual reformulation step, we introduce the algorithmic framework founded on a semismooth Newton method. Finally, we report on computational of the proposed framework.
Although modern societies strive towards energy systems that are entirely based on renewable energy carriers, natural gas is still one of the most important energy sources. This became even more obvious in Europe with Russia's 2022 war against the Ukraine and the resulting stop of gas supplies from Russia. Besides that it is very important to use this scarce resource efficiently. To this end, it is also of significant relevance that its transport is organized in the most efficient, i.e., cost- or energy-efficient, way. The corresponding mathematical optimization models have gained a lot of attention in the last decades in different optimization communities. These models are highly nonlinear mixed-integer problems that are constrained by algebraic constraints and partial differential equations (PDEs), which usually leads to models that are not tractable. Hence, simplifications have to be made and in this chapter, we present a commonly accepted finite-dimensional stationary model, i.e., a model in which the steady-state solutions of the PDEs are approximated with algebraic constraints. For more details about the involved PDEs and the treatment of transient descriptions we refer to Hante and Schmidt (2023). The presented finite-dimensional as well as mixed-integer nonlinear and nonconvex model is still highly challenging if it needs to be solved for real-world gas transport networks. Hence, we also review some classic solution approaches from the literature.
The optimal control of gas transport networks was and still is a very important topic for modern economies and societies. Accordingly, a lot of research has been carried out on this topic during the last years and decades. Besides mixed-integer aspects in gas transport network optimization, one of the main challenges is that a physically and technically detailed modeling of transient gas dynamics leads to theoretically and computationally highly demanding models involving nonlinear partial differential equations (PDEs). For further background on the application, historical notes and a detailed discussion of mixed-integer aspects for stationary descriptions we refer to Hante and Schmidt (2023). In this chapter, we focus on the most common modeling approaches concerning transient descriptions, point out the challenges, and summarize important contributions concerning the optimization of the most relevant control parameters for this particular class of problems.
The chapter reviews certain computational approaches to solve optimal control problems for evolution-type partial differential equations, where some control functions are limited to switching. The mechanism that enforces switching is modeled as integer restrictions. This brings a combinatorial aspect into the apart from switching already computationally very demanding optimization problems. Recently, great advances have been made to tackle such problems rigorously using relaxation and combinatorial integral approximation. An overview of these methods and the known theoretical results concerning convergence and error estimates are provided in a consistent manner. Further, we point to applications with benchmark character as well as to open problems.