Steady state blended gas flow on networks: Existence and uniqueness of solutions

Alena Ulke; Michael Schuster; Simone Göttlich; Alena Ulke; Michael Schuster; Simone Göttlich

doi:10.3934/nhm.2025039

Networks and Heterogeneous Media

2025, Volume 20, Issue 3: 903-937. doi: 10.3934/nhm.2025039

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Research article

Steady state blended gas flow on networks: Existence and uniqueness of solutions

1.
Department of Mathematics, University of Mannheim, B6, 28-29, 68159 Mannheim, Germany
2.
Friedrich-Alexander University Erlangen-Nürnberg, Department of Mathematics, Cauerstraße 11, 91058 Erlangen, Germany

Received: 05 November 2024 Revised: 25 July 2025 Accepted: 05 August 2025 Published: 20 August 2025

We prove an existence result for the steady-state flow of gas mixtures in networks. The model is based on the physical principles of the isothermal Euler equations, coupling conditions for the flow and pressure, and the mixing of incoming flows at the nodes. The state equation is based on a convex combination of the ideal gas equations of state for natural gas and hydrogen. We analyze the mathematical properties of the model, allowing us to prove the existence of solutions for tree-shaped networks and networks with a cycle. Numerical examples illustrate the challenges involved, when extending our approach to general network topologies.
- gas pipeline network,
- gas mixture,
- hydrogen blending,
- isothermal Euler equations,
- stationary states
Citation: Alena Ulke, Michael Schuster, Simone Göttlich. Steady state blended gas flow on networks: Existence and uniqueness of solutions[J]. Networks and Heterogeneous Media, 2025, 20(3): 903-937. doi: 10.3934/nhm.2025039

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Abstract

We prove an existence result for the steady-state flow of gas mixtures in networks. The model is based on the physical principles of the isothermal Euler equations, coupling conditions for the flow and pressure, and the mixing of incoming flows at the nodes. The state equation is based on a convex combination of the ideal gas equations of state for natural gas and hydrogen. We analyze the mathematical properties of the model, allowing us to prove the existence of solutions for tree-shaped networks and networks with a cycle. Numerical examples illustrate the challenges involved, when extending our approach to general network topologies.

References

[1]	Bundesregierung, Energy from climate-friendly gas, 2024. Available from: https://www.bundesregierung.de/breg-en/news/hydrogen-technology-2204238.
[2]	European Commission, A hydrogen strategy for a climate-neutral Europe, 2020. Available from: https://energy.ec.europa.eu/system/files/2020-07/hydrogen_strategy_0.pdf.
[3]	Office of Energy Efficienty & Renewable Energy, Hydrogen: A clean, flexible energy carrier, 2017. Available from: https://shorturl.at/AjQv9.
[4]	European Hydrogen Backbone, Five hydrogen supply corridors for Europe in 2030 executive summary, 2022. Available from: https://ehb.eu/files/downloads/EHB-Supply-corridors-presentation-ExecSum.pdf.
[5]	K. Kumar, G. John, L. Lim, K. Seeger, M. Wakabayashi, G. Kawamura, Green hydrogen in Asia: A brief survey of existing programmes and projects, 2023. Available from: https://www.orrick.com/en/Insights/2023/07/Green-Hydrogen-in-Asia-A-Brief-Survey-of-Existing-Programmes-and-Projects.
[6]	U.S. Department of Energy, OCED, Clean energy demonstrations, mult-year program plan, 2023. Available from: https://www.energy.gov/sites/default/files/2023-08/OCED%202023%20Multi-Year%20Program%20Plan.pdf.
[7]	M. Banda, M. Herty, A. Klar, Gas flow in pipeline networks, Networks Heterogen. Media, 1 (2006), 41–56. https://doi.org/10.3934/nhm.2006.1.41 doi: 10.3934/nhm.2006.1.41
[8]	P. Domschke, B. Hiller, J. Lang, C. Tischendorf, Modellierung von gasnetzwerken: Eine übersicht, 2017. Available from: https://www.mathematik.tu-darmstadt.de/media/mathematik/forschung/preprint/preprints/2717.pdf.
[9]	T. Koch, B. Hiller, M. E. Pfetsch, L. Schewe, Evaluating Gas Network Capacities, MOS-SIAM, 2015.
[10]	M. Schmidt, M. Steinbach, B. Willert, High detail stationary optimization models for gas networks: Validation and results, Optim. Eng., 17 (2016), 437–472. https://doi.org/10.1007/s11081-015-9300-3 doi: 10.1007/s11081-015-9300-3
[11]	M. Gugat, F. M. Hante, M. Hirsch-Dick, G. Leugering, Stationary states in gas networks, Networks Heterogen. Media, 10 (2015), 295–320. https://doi.org/10.3934/nhm.2015.10.295 doi: 10.3934/nhm.2015.10.295
[12]	M. Banda, M. Herty, A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Networks Heterogen. Media, 1 (2006), 295–314. https://doi.org/10.3934/nhm.2006.1.295 doi: 10.3934/nhm.2006.1.295
[13]	M. Gugat, S. Ulbrich, Lipschitz solutions of initial boundary value problems for balance laws, Math. Models Methods Appl. Sci., 28 (2018), 921–951. https://doi.org/10.1142/S0218202518500240 doi: 10.1142/S0218202518500240
[14]	E. S. Baranovskii, Feedback optimal control problem for a network model of viscous fluid flows, Math. Notes, 112 (2022), 26–39. https://doi.org/10.1134/S0001434622070033 doi: 10.1134/S0001434622070033
[15]	M. Herty, Coupling conditions for networked systems of euler equations, SIAM J. Sci. Comput., 30 (2008), 1596–1612. https://doi.org/10.1137/070688535 doi: 10.1137/070688535
[16]	A. Quarteroni, L. Dede', A. Manzoni, C. Vergara, Modelling Blood Flow, Cambridge University Press, 2019, 25–76.
[17]	S. Kazi, K. Sundar, S. Srinivasan, A. Zlotnik, Modeling and optimization of steady flow of natural gas and hydrogen mixtures in pipeline networks, Int. J. Hydrogen Energy, 54 (2024), 14–24. https://doi.org/10.1016/j.ijhydene.2023.12.054 doi: 10.1016/j.ijhydene.2023.12.054
[18]	D. Bothe, W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech., 226 (2015), 1757–1805. https://doi.org/10.1007/s00707-014-1275-1 doi: 10.1007/s00707-014-1275-1
[19]	J. Málek, O. Souček, A thermodynamic framework for heatconducting flows of mixtures of two interacting fluids, ZAMM J. Appl. Math. Mech., 102 (2022), e202100389. https://doi.org/10.1002/zamm.202100389 doi: 10.1002/zamm.202100389
[20]	P. Börner, M. Pfetsch, S. Ulbrich, Modeling and optimization of gas mixtures on networks, 2024.
[21]	A. Bermúdez, M. Shabani, Numerical simulation of gas composition tracking in a gas transportation network, Energy, 247 (2022), 123459. https://doi.org/10.1016/j.energy.2022.123459 doi: 10.1016/j.energy.2022.123459
[22]	M. Gugat, M. Schuster, Stationary gas networks with compressor control and random loads: Optimization with probablistic constraints, Math. Probl. Eng., 2018 (2018), 7984079. https://doi.org/10.1155/2018/7984079 doi: 10.1155/2018/7984079
[23]	M. Schuster, E. Strauch, M. Gugat, J. Lang, Probabilistic constrained optimization on flow networks, Optim. Eng., 23 (2022), 1–50. https://doi.org/10.1007/s11081-021-09619-x doi: 10.1007/s11081-021-09619-x
[24]	S. Srinivasan, K. Sundar, V. Gyrya, A. Zlotnik, Numerical solution to the steady-state network flow equations for a non-ideal gas, IEEE Trans. Control Network Syst., 10 (2022), 1449–1461. https://doi.org/10.1109/TCNS.2022.3232524 doi: 10.1109/TCNS.2022.3232524
[25]	G. Guandalini, P. Colbertaldo, S. Campanari, Dynamic modeling of natural gas quality within transport pipelines in presence of hydrogen injections, Appl. Energy, 185 (2017), 1712–1723. https://doi.org/10.1016/j.apenergy.2016.03.006 doi: 10.1016/j.apenergy.2016.03.006
[26]	A. Witkowski, A. Rusin, M. Majkut, K. Stolecka, Analysis of compression and transport of the methane/hydrogen mixture in existing natural gas pipelines, Int. J. Press. Vessels Pip., 166 (2018), 24–34. https://doi.org/10.1016/j.ijpvp.2018.08.002 doi: 10.1016/j.ijpvp.2018.08.002
[27]	M. Gugat, J. Giesselmann, An observer for pipeline flow with hydrogen blending in gas networks: Exponential synchronization, SIAM J. Control Optim., 62 (2024), 2273–2296. https://doi.org/10.1137/23M1563840 doi: 10.1137/23M1563840
[28]	D. Wintergerst, Optimization on Gas Networks under Stochastic Boundary Conditions, Dissertation, FAU Erlangen-Nürnberg, Germany, 2019.
[29]	M. Gugat, R. Schultz, D. Wintergerst, Networks of pipelines for gas with nonconstant compressibility factor: Stationary states, Comput. Appl. Math., 37 (2018), 1066–1097. https://doi.org/10.1007/s40314-016-0383-z doi: 10.1007/s40314-016-0383-z
[30]	J. Bang-Jensen, G. Gutin, Digraphs: Theory, Algorithms and Applications, Springer London, 2008.
[31]	R. Bapat, Graphs and Matrices, Springer London, 2010.
[32]	C. Gotzes, H. Heitsch, R. Henrion, R. Schultz, On the quantification of nomination feasibility in stationary gas networks with random load, Math. Methods Oper. Res., 84 (2016), 427–457. https://doi.org/10.1007/s00186-016-0564-y doi: 10.1007/s00186-016-0564-y
[33]	S. Srinivasan, N. Panda, K. Sundar, On the existence of steady-state solutions to the equations governing fluid flow in networks, IEEE Control Syst. Lett., 8 (2024), 646–651. https://doi.org/10.1109/LCSYS.2024.3394317 doi: 10.1109/LCSYS.2024.3394317
[34]	S. Srinivasan, K. Sundar, V. Gyrya, A. Zlotnik, Numerical solution of the steady-state network flow equations for a nonideal gas, IEEE Trans. Control Network Syst., 10 (2022), 1449–1461. https://doi.org/10.1109/TCNS.2022.3232524 doi: 10.1109/TCNS.2022.3232524
[35]	L. Baker, S. Kazi, R. Platte, A. Zlotnik, Optimal control of transient flows in pipeline networks with heterogeneous mixtures of hydrogen and natural gas, in 2023 American Control Conference (ACC), 2023, 1221–1228. https://doi.org/10.48550/arXiv.2210.06269
[36]	S. R. Kazi, K. Sundar, A. Zlotnik, Dynamic optimization and optimal control of hydrogen blending operations in natural gas networks, in 2024 Americal Control Conference (ACC), 2024, 5357–5363. https://doi.org/10.23919/ACC60939.2024.10644751
[37]	M. Sodwatana, S. Kazi, K. Sundar, A. Brandt, A. Zlotnik, Locational marginal pricing of energy in pipeline transport of natural gas and hydrogen with carbon offset incentives, Int. J. Hydrogen Energy, 96 (2024), 574–588. https://doi.org/10.1016/j.ijhydene.2024.11.191 doi: 10.1016/j.ijhydene.2024.11.191
[38]	M. Sodwatana, S. Kazi, K. Sundar, A. Zlotnik, Optimization of hydrogen blending in natural gas networks for carbon emissions reduction, in 2023 American Control Conference (ACC), 2023, 1229–1236. https://doi.org/10.48550/arXiv.2210.16385

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