# American Institute of Mathematical Sciences

June  2015, 10(2): 295-320. doi: 10.3934/nhm.2015.10.295

## Stationary states in gas networks

 1 Lehrstuhl Angewandte Mathematik 2, Cauerstr. 11, 91058 Erlangen, Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Germany, Germany, Germany, Germany

Received  September 2014 Revised  December 2014 Published  April 2015

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. Networks & Heterogeneous Media, 2015, 10 (2) : 295-320. doi: 10.3934/nhm.2015.10.295
##### References:
 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] F. Chapeau-Blondeau, Numerical evaluation of the lambert W function and application to generation of generalized Gaussian noise with exponent 1/2, IEEE Transactions on Signal Processing, 50 (2002), 2160-2165. doi: 10.1109/TSP.2002.801912.  Google Scholar [5] R. M. Colombo and F. Marcellini, Smooth and discontinuous junctions in the p-system, Journal of Mathematical Analysis and Applications, 361 (2010), 440-456. doi: 10.1016/j.jmaa.2009.07.022.  Google Scholar [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.  Google Scholar [7] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comp. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physico-mathematico-anatomico-botanico-medica, 3 (1758), 128-168. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] D. Veberic, Having fun with Lambert W(x) function,, , ().   Google Scholar [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.  Google Scholar

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##### References:
 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] F. Chapeau-Blondeau, Numerical evaluation of the lambert W function and application to generation of generalized Gaussian noise with exponent 1/2, IEEE Transactions on Signal Processing, 50 (2002), 2160-2165. doi: 10.1109/TSP.2002.801912.  Google Scholar [5] R. M. Colombo and F. Marcellini, Smooth and discontinuous junctions in the p-system, Journal of Mathematical Analysis and Applications, 361 (2010), 440-456. doi: 10.1016/j.jmaa.2009.07.022.  Google Scholar [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.  Google Scholar [7] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comp. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physico-mathematico-anatomico-botanico-medica, 3 (1758), 128-168. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] D. Veberic, Having fun with Lambert W(x) function,, , ().   Google Scholar [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.  Google Scholar
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