Article Contents
Article Contents

# Optimal model switching for gas flow in pipe networks

• * Corresponding author
• 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.

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

 Citation:

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

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

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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