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Networks and Heterogeneous Media

2015, Volume 10, Issue 2: 295-320. doi: 10.3934/nhm.2015.10.295
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Stationary states in gas networks

  • 1.
    Lehrstuhl Angewandte Mathematik 2, Cauerstr. 11, 91058 Erlangen, Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU)
  • Received: 01 September 2014 Revised: 01 December 2014

  • 93C20, 93C15.

  • Pipeline networks for gas transportation often contain circles. For such networks it is more difficult to determine the stationary states than for networks without circles. We present a method that allows to compute the stationary states for subsonic pipe flow governed by the isothermal Euler equations for certain pipeline networks that contain circles. We also show that suitably chosen boundary data determine the stationary states uniquely. The construction is based upon novel explicit representations of the stationary states on single pipes for the cases with zero slope and with nonzero slope. In the case with zero slope, the state can be represented using the Lambert--W function.

    Citation: Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering. Stationary states in gas networks[J]. Networks and Heterogeneous Media, 2015, 10(2): 295-320. doi: 10.3934/nhm.2015.10.295

    Related Papers:

    [1] Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering . Stationary states in gas networks. Networks and Heterogeneous Media, 2015, 10(2): 295-320. doi: 10.3934/nhm.2015.10.295
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  • Pipeline networks for gas transportation often contain circles. For such networks it is more difficult to determine the stationary states than for networks without circles. We present a method that allows to compute the stationary states for subsonic pipe flow governed by the isothermal Euler equations for certain pipeline networks that contain circles. We also show that suitably chosen boundary data determine the stationary states uniquely. The construction is based upon novel explicit representations of the stationary states on single pipes for the cases with zero slope and with nonzero slope. In the case with zero slope, the state can be represented using the Lambert--W function.


    [1] M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Networks and Heterogenous Media, 1 (2006), 295-314. doi: 10.3934/nhm.2006.1.295
    [2] A. Bressan, S. Canic, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives, EMS Reviews in Mathematical Sciences, 1 (2014), 47-111. doi: 10.4171/EMSS/2
    [3] R. Carvalho, L. Buzna, F. Bono, M. Masera, D. K. Arrowsmith and D. Helbing, Resilience of natural gas networks during conflicts, crises and disruptions, PLOS ONE, 9 (2014), e0090265. doi: 10.1371/journal.pone.0090265
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    [6] R. M. Colombo, G. Guerra, M. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050. doi: 10.1137/080716372
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    [8] J.-M. Coron, B. d'Andrea-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903
    [9] M. Dick, M. Gugat and G. Leugering, Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Networks and Heterogeneous Media, 5 (2010), 691-709. doi: 10.3934/nhm.2010.5.691
    [10] M. Garavello and B. Piccoli, Conservation laws on complex networks, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 26 (2009), 1925-1951. doi: 10.1016/j.anihpc.2009.04.001
    [11] M. Gugat, M. Dick and G. Leugering, Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM Journal on Control and Optimization, 49 (2011), 2101-2117. doi: 10.1137/100799824
    [12] M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 28-51. doi: 10.1051/cocv/2009035
    [13] M. Gugat, M. Herty and V. Schleper, Flow control in gas networks: Exact controllability to a given demand, Mathematical Methods in the Applied Sciences, 34 (2011), 745-757. doi: 10.1002/mma.1394
    [14] M. Herty, Modeling, simulation and optimization of gas networks with compressors, Networks and Heterogeneous Media, 2 (2007), 81-97. doi: 10.3934/nhm.2007.2.81
    [15] J. H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physico-mathematico-anatomico-botanico-medica, 3 (1758), 128-168.
    [16] T. Li, B. Rao and Z. Wang, Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 243-257. doi: 10.3934/dcds.2010.28.243
    [17] A. Martin, M. Möller and S. Moritz, Mixed integer models for the stationary case of gas network optimization, Mathematical Programming, 105 (2005), 563-582. doi: 10.1007/s10107-005-0665-5
    [18] G. A. Reigstad, Numerical network models and entropy principles for isothermal junction flow, Networks and Heterogeneous Media, 9 (2014), 65-95. doi: 10.3934/nhm.2014.9.65
    [19] V. Schleper, M. Gugat, M. Herty, A. Klar and G. Leugering, Well-posedness of networked hyperbolic systems of balance laws, in Constrained Optimization and Optimal Control for Partial Differential Equations, International Series of Numerical Mathematics, 160, Birkhäuser/Springer Basel AG, Basel, 2012, 123-146. doi: 10.1007/978-3-0348-0133-1_7
    [20] arXiv:1003.1628
    [21] G. P. Zou, N. Cheraghi and F. Taheri, Fluid-induced vibration of composite natural gas pipelines, International Journal of Solids and Structures, 42 (2005), 1253-1268. doi: 10.1016/j.ijsolstr.2004.07.001

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