Modeling and simulation of gas networks coupled to power grids


A mathematical framework for the coupling of gas networks to electric grids is presented to describe in particular the transition from gas to power. The dynamics of the gas flow are given by the isentropic Euler equations, while the power flow equations are used to model the power grid. We derive pressure laws for the gas flow that allow for the well-posedness of the coupling and a rigorous treatment of solutions. For simulation purposes, we apply appropriate numerical methods and show in an experimental study how gas-to-power might influence the dynamics of the gas and power network, respectively.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


  1. 1.

    Chertkov M, Backhaus S, Lebedev V (2015) Cascading of fluctuations in interdependent energy infrastructures: gas–grid coupling. Appl Energy 160:541–551

    Article  Google Scholar 

  2. 2.

    Zeng Q, Fang J, Li J, Chen Z (2016) Steady-state analysis of the integrated natural gas and electric power system with bi-directional energy conversion. Appl Energy 184:1483–1492

    Article  Google Scholar 

  3. 3.

    Zlotnik A, Roald L, Backhaus S, Chertkov M, Andersson G (2016) Control policies for operational coordination of electric power and natural gas transmission systems. Am Control Conf 2016:7478–7483

    Google Scholar 

  4. 4.

    Herty M, Müller S, Sikstel A (2019) Coupling of compressible Euler equations. Vietnam J Math.

  5. 5.

    Banda M, Herty M, Klar A (2006) Coupling conditions for gas networks governed by the isothermal Euler equations. Netw Heterog Media 1:295–314

    MathSciNet  Article  Google Scholar 

  6. 6.

    Banda M, Herty M, Klar A (2006) Gas flow in pipeline networks. Netw Heterog Media 1:41–56

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bressan A, Canic S, Garavello M, Herty M, Piccoli B (2014) Flow on networks: recent results and perspectives. Eur Math Soc 1:47–111

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Brouwer J, Gasser I, Herty M (2011) Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks. Multiscale Model Simul 9:601–623

    MathSciNet  Article  Google Scholar 

  9. 9.

    Colombo R, Garavello M (2008) On the Cauchy problem for the p-system at a junction. SIAM J Math Anal 39:1456–1471

    MathSciNet  Article  Google Scholar 

  10. 10.

    Reigstad G (2015) Existence and uniqueness of solutions to the generalized Riemann problem for isentropic flow. SIAM J Appl Math 75:679–702

    MathSciNet  Article  Google Scholar 

  11. 11.

    Reigstad G (2014) Numerical network models and entropy principles for isothermal junction flow. Netw Heterog Media 9:65

    MathSciNet  Article  Google Scholar 

  12. 12.

    LeVeque R (2002) Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics. Cambridge University Press, Cambridge

    Google Scholar 

  13. 13.

    Gyrya V, Zlotnik A (2019) An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks. Appl Math Model 65:34–51

    MathSciNet  Article  Google Scholar 

  14. 14.

    Pfetsch ME, Koch T, Schewe L, Hiller B (eds) (2015) Evaluating gas network capacities. Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  15. 15.

    Kolb O, Lang J, Bales P (2010) An implicit box scheme for subsonic compressible flow with dissipative source term. Numer Algorithms 53:293–307

    MathSciNet  Article  Google Scholar 

  16. 16.

    Bienstock D (2015) Electrical transmission system cascades and vulnerability. Society for Industrial and Applied Mathematics, Philadelphia

    Google Scholar 

  17. 17.

    Heuck K, Dettmann K, Schultz D (2013) Elektrische energieversorgung. Springer Vieweg, Berlin

    Google Scholar 

  18. 18.

    Göttlich S, Herty M, Schillen P (2016) Electric transmission lines: control and numerical discretization. Optim Control Appl Methods 37:980–995

    MathSciNet  Article  Google Scholar 

  19. 19.

    Fokken E, Göttlich S, Kolb O (2019) Optimal control of compressor stations in a coupled gas-to-power network. arXiv:1901.00522

  20. 20.

    Egger H (2018) A robust conservative mixed finite element method for isentropic compressible flow on pipe networks. SIAM J Sci Comput 40:A108–A129

    MathSciNet  Article  Google Scholar 

  21. 21.

    Zhou J, Adewumi M (1996) Simulation of transients in natural gas pipelines. SPE Prod Facil 11:202–207

    Article  Google Scholar 

  22. 22.

    Gugat M, Herty M, Klar A, Leugering G, Schleper V (2012) Well-posedness of networked hyperbolic systems of balance laws. In: Leugering G, Engell S, Griewank A, Hinze M, Rannacher R, Schulz V, Ulbrich M, Ulbrich S (eds) Constrained optimization and optimal control for partial differential equations. Springer, Basel, pp 123–146

    Google Scholar 

  23. 23.

    Gugat M, Herty M, Müller S (2017) Coupling conditions for the transition from supersonic to subsonic fluid states. Netw Heterog Media 12:371–380

    MathSciNet  Article  Google Scholar 

  24. 24.

    Grainger J, Stevenson W, Chang G (2016) Power system analysis. McGraw-Hill series in electrical and computer engineering: power and energy. McGraw-Hill Education, New York

    Google Scholar 

  25. 25.

    Kolb O (2014) On the full and global accuracy of a compact third order WENO scheme. SIAM J Numer Anal 52:2335–2355

    MathSciNet  Article  Google Scholar 

  26. 26.

    Naumann A, Kolb O, Semplice M (2018) On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws. Appl Math Comput 325:252–270

    MathSciNet  Google Scholar 

  27. 27.

    Zimmerman RD, Murillo-Sanchez CE, Thomas RJ (2011) MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education. IEEE Trans Power Syst 26:12–19

    Article  Google Scholar 

  28. 28.

    Humpola J, Joormann I, Kanelakis N, Oucherif D, Pfetsch M, Schewe L, Schmidt M, Schwarz R, Sirvent M (2017) GasLib: a library of gas network instances. Technical report

  29. 29.

    Gugat M, Schuster M (2018) Stationary gas networks with compressor control and random loads: optimization with probabilistic constraints. Math Probl Eng 2018:7984079

    MathSciNet  Article  Google Scholar 

  30. 30.

    Göttlich S, Korn R, Lux K (2019) Optimal control of electricity input given an uncertain demand. Math Methods Oper Res.

    Article  Google Scholar 

Download references


The authors gratefully thank the BMBF project ENets (05M18VMA) for the financial support.

Author information



Corresponding author

Correspondence to S. Göttlich.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: A Proof of Lemma 3

Appendix: A Proof of Lemma 3

The following two technical results are the key ingredients to prove Lemma 3.

Lemma 13

Let \(g \in C^1(\mathbb {R}^+, \mathbb {R}^+)\)be a non-negative function, \(g\ge 0\), and let G be given by \(G(\rho ) = \int _\rho ^{\rho _l}g(s)\mathrm{d}s\). Then There holds

  1. 1.

    If \(\rho ^2g(\rho ) \overset{\rho \rightarrow 0}{\rightarrow } 0\), then \(\rho G(\rho ) \overset{\rho \rightarrow 0}{\rightarrow } 0\).

  2. 2.

    If \(\rho G(\rho ) \overset{\rho \rightarrow 0}{\rightarrow } 0\), then \(\liminf _{\rho \rightarrow 0}\rho ^2g(\rho ) =0\).


  1. 1.

    By assumption \(\rho ^2g(\rho ) \overset{\rho \rightarrow 0}{\rightarrow } 0\). For \(m \in \mathbb {N}\) choose \(\rho _m >0\) such that \(g(\rho )\le \tfrac{1}{m\rho ^2}\) for \(\rho <\rho _m\). Now choose \(\rho _{m,0}<\rho _m\) so small that

    $$\begin{aligned} \rho _{m,0} \left( \int _{\rho _m}^{\rho _l}g(s)\mathrm{d}s -\frac{1}{m\rho _m}\right) \le \frac{1}{m} . \end{aligned}$$

    Then, for \(\rho <\rho _{m,0}\), there holds

    $$\begin{aligned} \rho \int _{\rho }^{\rho _l}g(s)\mathrm{d}s&\le \rho \int _{\rho _m}^{\rho _l}g(s) \mathrm{d}s + \frac{1}{m}\rho \int _{\rho }^{\rho _m}\frac{1}{s^2}\mathrm{d}s\nonumber \\&= \rho \left( \int _{\rho _m}^{\rho _l}g(s) \mathrm{d}s-\frac{1}{m\rho _m}\right) +\frac{1}{m}\nonumber \\&\le \frac{2}{m}. \end{aligned}$$

    Therefore \(\lim _{\rho \rightarrow 0}\rho \int _{\rho }^{\rho _l}g(s)\mathrm{d}s \le \tfrac{2}{m}\) for all \(m \in \mathbb {N}\). As \(g\ge 0\), we also have \(0\le \lim _{\rho \rightarrow 0}\rho \int _{\rho }^{\rho _l}g(s)\mathrm{d}s\). Summarizing, we get \(\lim _{\rho \rightarrow 0}\rho G(\rho ) = \lim _{\rho \rightarrow 0}\rho \int _{\rho }^{\rho _l}g(s)\mathrm{d}s = 0\).

  2. 2.

    We prove by contradiction: Assume there are \(\rho _0>0\) and \(a >0\) such that \(\rho ^2g(\rho ) \ge a\) for \(\rho < \rho _0\). Then \(g(\rho )\ge \tfrac{a}{2}\tfrac{1}{\rho ^2}\) for such \(\rho \). Therefore

    $$\begin{aligned} \rho \int _\rho ^{\rho _l}g(s)\mathrm{d}s \ge \rho \int _{\rho _0}^{\rho _l}g(s) \mathrm{d}s + \rho \frac{a}{2}\int _{\rho }^{\rho _0} \frac{1}{s^2} \mathrm{d}s \rightarrow 0 + \frac{a}{2} \rho \left[ -\frac{1}{s} \right] _\rho ^{\rho _0} \rightarrow \frac{a}{2}, \end{aligned}$$

    which contradicts \(\rho G(\rho ) \overset{\rho \rightarrow 0}{\longrightarrow } 0\). \(\square \)

Lemma 14

Let \(g \in C^1(\mathbb {R}^+, \mathbb {R}^+)\), \(g\ge 0\) and \(\liminf _{\rho \rightarrow 0} \rho ^2g(\rho ) = 0\). Let also \(\left( \rho ^2g(\rho ) \right) ^\prime \ge 0\). Then, \(\limsup _{\rho \rightarrow 0} \rho ^2 g(\rho ) = \liminf _{\rho \rightarrow 0} \rho ^2 g(\rho ) = 0\).


We prove by contradiction. Let \(\limsup _{\rho \rightarrow 0} \rho ^2 g(\rho ) > \liminf \rho ^2 g(\rho )\). Then it is easily seen that \(\liminf _{\rho \rightarrow 0} \left( \rho ^2g(\rho ) \right) ^\prime = -\infty < 0\) resulting in a contradiction. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fokken, E., Göttlich, S. & Kolb, O. Modeling and simulation of gas networks coupled to power grids. J Eng Math 119, 217–239 (2019).

Download citation


  • Gas networks
  • Power flow equations
  • Pressure laws
  • Simulation

Mathematics Subject Classification

  • 35L65
  • 65M08