Optimal model switching for gas flow in pipe networks

Fabian Rüffler; Volker Mehrmann; Falk M. Hante

doi:10.3934/nhm.2018029

Article Contents

2018, Volume 13, Issue 4: 641-661. Doi: 10.3934/nhm.2018029

This issue Previous Article Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface Next Article Fluvial to torrential phase transition in open canals

Optimal model switching for gas flow in pipe networks

1.
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Lehrstuhl Angewandte Mathematik Ⅱ, Cauerstr. 11, 91058 Erlangen, Germany
2.
Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author
* Corresponding author

Received: January 2018

Revised: August 2018

Published: November 2018

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Abstract

We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach.

Keywords:

Mathematics Subject Classification: Primary: 35L50, 76N25, 93C30; Secondary: 35R02, 49J20.

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Figure 1. A gas network with a supply node $N_1$ and two costumer nodes $N_2$ and $N_3$

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Figure 2. Snapshot of the fully simulated solution showing density (solid, blue) and flux (dashed, red, scaled by $0.05$). On the outer pipes 1 to 5 we see a lot of fluctuation due to the oscillatory boundary flows. The pipes 6 to 9 of the inner circle, however, remain nearly constant

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Figure 3. (A): resulting optimized switching sequence showing, for each time step from $t_0 = 0\ \text{s}$ to $T = 1800\ \text{s}$ and each edge $e_1, \ldots, e_{10}$, if the solution is calculated with the fine model (white) or frozen (black). (B), (C): filtered results with two different filters. (D): $L^2$-error relative to maximum values of the solution $\bar{z}$ corresponding to freezing edges $6$ to $10$ completely

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