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Efficient numerical methods for gas network modeling and simulation

  • * Corresponding author: Yue Qiu

    * Corresponding author: Yue Qiu 
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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  • 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.

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

    Citation:

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  • Figure 1.  Separation of control volume $ C_i $

    Figure 2.  A typical gas network

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

    Figure 7.  Computational diagram for gas network simulation

    Figure 4.  An illustrative network example of a DAG

    Figure 5.  Big benchmark network in [34]

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

    Figure 8.  Pipeline network in [16]

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

    Figure 10.  Medium size network

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

    Figure 12.  Mass flow at supply nodes for case 1

    Figure 13.  Mass flow at supply nodes for case 2

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

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

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

    Figure 17.  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 $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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 -
     | Show Table
    DownLoad: CSV
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