Efficient numerical methods for gas network modeling and simulation

Yue Qiu; Sara Grundel; Martin Stoll; Peter Benner

doi:10.3934/nhm.2020018

Article Contents

2020, Volume 15, Issue 4: 653-679. Doi: 10.3934/nhm.2020018

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Efficient numerical methods for gas network modeling and simulation

1.
School of Information Science and Technology, ShanghaiTech University, 393 Middle Huaxia Road, 201210, Shanghai, China
2.
Computational Methods for Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106, Magdeburg, Germany
3.
Faculty of Mathematics, Technische Universität Chemnitz, Reichenhainer Straße 41, 09126, Chemnitz, Germany

^* Corresponding author: Yue Qiu
^* Corresponding author: Yue Qiu

Received: July 31, 2019

Revised: July 31, 2020

Early access: August 2020

Published: December 2020

This work is partially funded by the European Regional Development Fund (ERDF/EFRE: ZS/2016/04/78156) within the Center Dynamic Systems (CDS)

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Abstract

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a system of nonlinear differential algebraic equations (DAEs). With our modeling approach, we reduce the number of algebraic constraints, which correspond to the $ (2,2) $ block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the $ (1, 1) $ block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

Keywords:

Mathematics Subject Classification: Primary: 65F08, 37M05, 37N30, 94C15.

Citation:

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Figure 1. Separation of control volume $ C_i $

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Figure 2. A typical gas network

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Figure 3. Smoothed network of Figure 2 with an ordering of the pipes

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Figure 7. Computational diagram for gas network simulation

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Figure 4. An illustrative network example of a DAG

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Figure 5. Big benchmark network in [34]

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Figure 6. Sparsity pattern of $ J $ without and with DF ordering

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Figure 8. Pipeline network in [16]

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Figure 9. Comparison of FVM and FDM for a single pipe network

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Figure 10. Medium size network

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Figure 11. Comparison of FVM and FDM for a medium network

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Figure 12. Mass flow at supply nodes for case 1

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Figure 13. Mass flow at supply nodes for case 2

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Figure 14. Mass flow for the pipe $ 31\rightarrow 37 $

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Figure 15. Nonlinear residual at the first and tenth time step

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Figure 16. Number of IDR($ 4 $) iterations at the first time step

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Figure 17. Number of IDR($ 4 $) iterations at the $ 10 $-th time step

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Algorithm 1: Newton's method to solve (16)

1: Input: maximal number of Newton steps $ n_{\max} $, stop tolerance $ \varepsilon_0 $, initial guess $ x_0 $
2: $ m=0 $
3: while

$ m\leq n_{\max}\& \ \|F(x)\|\geq \varepsilon_0 $

do
4:     Compute the Jacobian matrix $ D_F(x_m)=\frac{\partial}{\partial x}F|_{x=x_m} $
5:     Solve $ F(x_m) + D_F(x_m)(x-x_m)=0 $
6:     $ m\gets m+1 $, $ x_m\gets x $
7: end while
8: Output: solution $ x\approx x_m $

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Table 1. Computational time (seconds) for Schur complement $ S^1 $

$ h $	$ \# D_F $	with DF	without DF
20	2.01e+05	$ 8.12 $	8.75
10	3.97e+05	$ 17.84 $	19.14
5	7.91e+05	$ 38.44 $	41.75
2.5	1.58e+06	$ 81.42 $	87.77

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Table 2. Condition number of the Jacobian matrix $ D_{F} $ from FVM and FDM, 1st time step, $ h = 60 $

Newton iter.	1	2	3	4
FVM	1.56e+07	1.57e+07	1.57e+07	1.57e+07
FDM	1.24e+08	1.25e+08	1.25e+08	1.25e+08

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Table 3. Computational time for the 1st Newton iteration

$ h $	$ \# D_F $	$ t_{S^1} $	IDR($ 4 $)	backslash
40	1.03e+05	$ 3.85 $	0.25	0.13
20	2.01e+05	$ 8.12 $	0.52	0.36
10	3.97e+05	$ 17.84 $	1.06	1.18
5	7.91e+05	$ 38.44 $	2.13	1054.62
2.5	1.58e+06	$ 81.42 $	4.34	-

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