SFB TRR154 Transregio – Mathematical modelling, simulation and optimization using the example of gas networks is focused to provide certified novel answers to these grand challenges, based on mathematical modeling, simulation and optimization. Given the amount of data and the potential of stochastic effects, this is a formidable task all by itself, regardless from the actual process of distributing the proper amount of gas with the required quality to the customer.

In order to achieve this goal new paradigms in the integration of these disciplines and in particular in the interplay between integer and nonlinear programming in the context of stochastic data have to be established and brought to bear. Clearly, without a specified underlying structure of the problems to face, such a breakthrough is rather unlikely. Thus, the particular network structure, the given hierarchical hybrid modeling in terms of switching algebraic, ordinary and partial differential-algebraic equations of hyperbolic type that is present in gas network transportation systems gives rise to the confidence that the challenges can be met by the team of the proposed Transregio-CRC.

Our Publications at TRR154

TRR154 subprojects where we are at

From our team, Enrique Zuazua, Martin Gugat, Michael Schuster and Lukas Wolff are actively involved to the TRR154 through the following current subprojects (Phase 2):

C03: Nodal control and the turnpike phenomenon

Turnpike results provide connections between the solutions of transient and the corresponding stationary optimal control problems that are often used as models in the control of gas transport networks. In this way turnpike results give a theoretical foundation for the approximation of transient optimal controls by the solutions of stationary optimal control problems that have a simpler structure. Turnpike studies can also be considered as investigations of the structure of the transient optimal controls. In the best case the stationary optimal controls approximate the transient optimal controls exponentially fast.

C05: Observer-based data assimilation for time dependent flows on gas networks

This project studies data assimilation methods for models of compressible flows in gas networks. The basic idea of data assimilation is to include measurement data into simulations during runtime in order to make their results more precise and more reliable. This can be achieved by augmenting the original model equations with control terms at nodes and on pipes that steer the solutions towards the measured data. This gives rise to a new system called “observer”. This project is going to explore how much data is needed so that convergence of the observer towards the solution of the original system can be guaranteed, how fast this convergence is and how measurement errors affect the solution.

C08: Random Batch Methods for Optimal Control of Network Dynamics

This Subproject, led by Falk Hante (Humboldt-Universität zu Berlin) and Enrique Zuazua (FAU Erlangen-Nürnberg) focuses on hyperbolic and parabolic dynamics on networks and random batch methods for control. The aim is to restrict the overall network dynamics to subgraphs as a random batch for the computation of a stochastic gradient descent direction. We aim to develop a convergence theory and develop control methodologies of gas networks employing techniques of model predictive control. This approach can then readily be extended to incorporate uncertainties in the model by adapting concepts from the theory of simultaneous control of parameter-dependent systems.

Uncertainty

We analyze the (stationary and transient) gas transport with uncertain boundary data. This leads to optimization problems with probabilistic constraints. Our main methods to work with probabilistic constrained optimization problems are the spheric-radial decomposition and kernel-density estimation.

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