# American Institute of Mathematical Sciences

December  2020, 15(4): 653-679. doi: 10.3934/nhm.2020018

## 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

Received  August 2019 Revised  August 2020 Published  August 2020

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

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.

Citation: Yue Qiu, Sara Grundel, Martin Stoll, Peter Benner. Efficient numerical methods for gas network modeling and simulation. Networks & Heterogeneous Media, 2020, 15 (4) : 653-679. doi: 10.3934/nhm.2020018
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##### References:
Separation of control volume $C_i$
A typical gas network
Smoothed network of Figure 2 with an ordering of the pipes
Computational diagram for gas network simulation
An illustrative network example of a DAG
Big benchmark network in [34]
Sparsity pattern of $J$ without and with DF ordering
Pipeline network in [16]
Comparison of FVM and FDM for a single pipe network
Medium size network
Comparison of FVM and FDM for a medium network
Mass flow at supply nodes for case 1
Mass flow at supply nodes for case 2
Mass flow for the pipe $31\rightarrow 37$
Nonlinear residual at the first and tenth time step
Number of IDR($4$) iterations at the first time step
Number of IDR($4$) iterations at the $10$-th time step
 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 $x\approx x_m$
 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 $x\approx x_m$
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
 $h$ $\# D_F$ 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
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
 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
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 -
 $h$ $\# D_F$ $t_{S^1}$ 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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