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Closed loop control of gas flow in a pipe: stability for a transient model

Regelung des Gasflusses in einer Pipeline: Stabilität eines transienten Modells
  • Martin Gugat

    Martin Gugat obtained his doctoral degree (summa cum laude) in applied mathematics in 1994 and the Habilitation in Mathematics in 1999 from the University of Trier. From 2000 to 2003 he joined the Department of Mathematics at the Technical University of Darmstadt. Since 2003 he has been a member of the Department of Mathematics at the FAU Erlangen-Nürnberg where he works as a professor with research in optimal control, exact control and stabilization for systems governed by partial differential equations. He was a visiting scholar at IPAM at UCLA, LJLL in Paris and Fudan University in Shanghai.

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    , Falk M. Hante

    Falk M. Hante received his Doctorate degree in mathematics in 2010 from FAU Erlangen-Nürnberg, Germany. He held postdoctoral positions at INRIA Grand Est in Nancy, France, and Universität Heidelberg, Germany, before joining the Department of Mathematics at FAU Erlangen-Nürnberg in 2012 as a Research Associate. Since 2020, he has been Professor at Humboldt-Universität zu Berlin, Germany, in applied mathematics with a specialization in optimization of complex dynamics. His research interests include control, including optimal control; switched and hybrid dynamical systems; network dynamics and optimization.

    and Li Jin

    Li Jin is an Assistant Professor at the New York University Tandon School of Engineering. He received B.Eng. degree in Mechanical Engineering from Shanghai Jiao Tong University in 2011, M.S. degree in Mechanical Engineering from Purdue University in 2012, and Ph.D. degree in Transportation from the Massachusetts Institute of Technology in 2018. He was also a visiting scholar at the University of Erlangen-Nuremberg in 2016. His background is stochastic processes, dynamic control, and optimization. His research focuses on developing resilient control algorithms for cyber-physical systems with guarantees of efficiency in nominal settings, robustness against random perturbations, and survivability under strategic disruptions.

Abstract

This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=32. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow.

We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2-sense. The analysis is done both for the PDE system and an ODE system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions.

Zusammenfassung

Unser Beitrag befasst sich mit einem reibungsdominiertem Modell für den Gasfluss in Pipelines. Das Modell für den Druck ist dabei eine doppelt nichtlineare parabolische p-Laplace Gleichung mit p=32. Solche Gleichungen haben positive Lösungen mit endlicher Ausbreitungsgeschwindigkeit und erfüllen ein Maximumprinzip. Am einen Rohrende wird der Druck und am anderen Rohrende die Flussrate vorgegeben. Durch diese Randbedingungen wird eindeutig ein stationärer Fluss festgelegt.

Wir beschreiben eine Regelung über die Randbedingungen, die Flüsse generiert, welche lokal exponentiell schnell im L2-Sinn gegen den stationären Fluss konvergieren. Die Analyse wird sowohl für das System mit der partiellen Differentialgleichung als auch für eine gewöhnliche Differentialgleichung durchgeführt, die man durch eine Semidiskretisierung im Ort erhält. Die Beweise basieren auf passend gewählten Lyapunovfunktionen. Numerische Beispiele werden präsentiert.

Award Identifier / Grant number: Transregio 154

Award Identifier / Grant number: A03

Award Identifier / Grant number: C03

Award Identifier / Grant number: C05

Award Identifier / Grant number: Z01

Funding statement: This work is supported by DFG Collaborative Research Centre CRC/Transregio 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, projects A03, C03, C05 and Z01; moreover it is supported by NYU Tandon School of Engineering.

About the authors

Martin Gugat

Martin Gugat obtained his doctoral degree (summa cum laude) in applied mathematics in 1994 and the Habilitation in Mathematics in 1999 from the University of Trier. From 2000 to 2003 he joined the Department of Mathematics at the Technical University of Darmstadt. Since 2003 he has been a member of the Department of Mathematics at the FAU Erlangen-Nürnberg where he works as a professor with research in optimal control, exact control and stabilization for systems governed by partial differential equations. He was a visiting scholar at IPAM at UCLA, LJLL in Paris and Fudan University in Shanghai.

Falk M. Hante

Falk M. Hante received his Doctorate degree in mathematics in 2010 from FAU Erlangen-Nürnberg, Germany. He held postdoctoral positions at INRIA Grand Est in Nancy, France, and Universität Heidelberg, Germany, before joining the Department of Mathematics at FAU Erlangen-Nürnberg in 2012 as a Research Associate. Since 2020, he has been Professor at Humboldt-Universität zu Berlin, Germany, in applied mathematics with a specialization in optimization of complex dynamics. His research interests include control, including optimal control; switched and hybrid dynamical systems; network dynamics and optimization.

Li Jin

Li Jin is an Assistant Professor at the New York University Tandon School of Engineering. He received B.Eng. degree in Mechanical Engineering from Shanghai Jiao Tong University in 2011, M.S. degree in Mechanical Engineering from Purdue University in 2012, and Ph.D. degree in Transportation from the Massachusetts Institute of Technology in 2018. He was also a visiting scholar at the University of Erlangen-Nuremberg in 2016. His background is stochastic processes, dynamic control, and optimization. His research focuses on developing resilient control algorithms for cyber-physical systems with guarantees of efficiency in nominal settings, robustness against random perturbations, and survivability under strategic disruptions.

References

1. Martin Gugat. Optimal boundary control and boundary stabilization of hyperbolic systems. SpringerBriefs in Electrical and Computer Engineering. Springer, Cham, 2015. SpringerBriefs in Control, Automation and Robotics.10.1007/978-3-319-18890-4Search in Google Scholar

2. Georges Bastin and Jean-Michel Coron. Stability and boundary stabilization of 1-D hyperbolic systems, volume 88 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser/Springer, Cham, 2016. Subseries in Control.10.1007/978-3-319-32062-5Search in Google Scholar

3. Andrey Smyshlyaev and Miroslav Krstic. Backstepping observers for a class of parabolic PDEs. Systems Control Lett., 54(7):613–625, 2005.10.1016/j.sysconle.2004.11.001Search in Google Scholar

4. Joachim Deutscher. Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs. IEEE Trans. Automat. Control, 61(8):2288–2294, 2016.10.1109/TAC.2015.2491718Search in Google Scholar

5. Martin Gugat and Fredi Tröltzsch. Boundary feedback stabilization of the Schlögl system. Automatica J. IFAC, 51:192–199, 2015.10.1016/j.automatica.2014.10.106Search in Google Scholar

6. Markus Dick, Martin Gugat, Michael Herty, Günter Leugering, Sonja Steffensen and Ke Wang. Stabilization of networked hyperbolic systems with boundary feedback. In Trends in PDE Constrained Optimization, volume 165 of Internat. Ser. Numer. Math., pages 487–504. Birkhäuser/Springer, Cham, 2014.10.1007/978-3-319-05083-6_31Search in Google Scholar

7. Martin Gugat and Rüdiger Schultz. Boundary feedback stabilization of the isothermal Euler equations with uncertain boundary data. SIAM J. Control Optim., 56(2):1491–1507, 2018.10.1137/16M1090156Search in Google Scholar

8. Marc Vuffray, Sidhant Misra and Michael Chertkov. Monotonicity of dissipative flow networks renders robust maximum profit problem tractable: General analysis and application to natural gas flows. In 2015 54th IEEE Conference on Decision and Control (CDC), pages 4571–4578. IEEE, 2015.10.1109/CDC.2015.7402933Search in Google Scholar

9. Sidhant Misra, Marc Vuffray, Anatoly Zlotnik and Michael Chertkov. Monotone order properties for control of nonlinear parabolic PDE on graphs. arXiv preprint arXiv:1601.05102, 2016.Search in Google Scholar

10. Jens Brouwer, Ingenuin Gasser and Michael Herty. Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks. Multiscale Modeling & Simulation, 9(2):601–623, 2011.10.1137/100813580Search in Google Scholar

11. Andrej J. Osiadacz, Different Transient Flow Models – Limitations, Advantages, and Disadvantages. PSIG-9606 Report, 1996. Pipeline Simulation Interest Group, San Francisco, California.Search in Google Scholar

12. Alain Bamberger, Michel Sorine and J. P. Yvon. Analyse et contrôle d’un réseau de transport de gaz. In Computing Methods in Applied Sciences and Engineering (Proc. Third Internat. Sympos., Versailles, 1977), II, volume 91 of Lecture Notes in Phys., pages 347–359. Springer, Berlin-New York, 1979.10.1007/3-540-09119-X_110Search in Google Scholar

13. Terrence W. K. Mak, Pascal van Hentenryck, Anatoly Zlotnik and Russell Bent. Dynamic compressor optimization in natural gas pipeline systems. INFORMS J. on Computing, 31:1–192, 2019.10.1287/ijoc.2018.0821Search in Google Scholar

14. Frédéric Mazenc and Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control and Related Fields, 1(2):231–250, 2011.10.3934/mcrf.2011.1.231Search in Google Scholar

15. Martin Gugat and Fredi Tröltzsch. Boundary feedback stabilization of the Schlögl system. Automatica, 51:192–199, 2015.10.1016/j.automatica.2014.10.106Search in Google Scholar

16. Klaus Ehrhardt and Marc C. Steinbach. Nonlinear optimization in gas networks. In Modeling, Simulation and Optimization of Complex Processes, pages 139–148. Springer, 2005.10.1007/3-540-27170-8_11Search in Google Scholar

17. Marc C. Steinbach. On PDE solution in transient optimization of gas networks. Journal of Computational and Applied Mathematics, 203(2):345–361, 2007.10.1016/j.cam.2006.04.018Search in Google Scholar

18. Morris W Hirsch. Systems of differential equations that are competitive or cooperative ii: Convergence almost everywhere. SIAM Journal on Mathematical Analysis, 16(3):423–439, 1985.10.1137/0516030Search in Google Scholar

19. Shankar S. Sastry. Nonlinear systems: analysis, stability, and control, volume 10. Springer Science & Business Media, 2013.Search in Google Scholar

20. Pia Domschke, Benjamin Hiller, Jens Lang and Caren Tischendorf. Modellierung von Gasnetzwerken: Eine Übersicht. Technische Universität Darmstadt, Preprint 2717, 2017.Search in Google Scholar

21. Michael Herty, Jan Mohring and Veronika Sachers. A new model for gas flow in pipe networks. Mathematical Methods in the Applied Sciences, 33(7):845–855, 2010.10.1002/mma.1197Search in Google Scholar

22. Alain Bamberger. Étude d’une équation doublement non linéaire. J. Functional Analysis, 24(2):148–155, 1977.10.1016/0022-1236(77)90051-9Search in Google Scholar

23. Andrej J. Osiadac. Simulation of transient gas flows in networks. International Journal for Numerical Methods in Fluids, 4(1):13–24, 1984.10.1002/fld.1650040103Search in Google Scholar

24. Kaarthik Sundar and Anatoly Zlotnik. State and parameter estimation for natural gas pipeline networks using transient state data. IEEE Transactions on Control Systems Technology, 27(5):2110–2124, 2019.10.1109/TCST.2018.2851507Search in Google Scholar

25. Martin Gugat, Falk M Hante, Markus Hirsch-Dick and Günter Leugering. Stationary states in gas networks. Networks and Heterogeneous Media, 10(2):295–320, 2015.10.3934/nhm.2015.10.295Search in Google Scholar

26. Erich Kamke. Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II. Acta Mathematica, 58(1):57–85, 1932.10.1007/BF02547774Search in Google Scholar

27. Semyon Aranovich Gershgorin. Über die Abgrenzung der Eigenwerte einer Matrix. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, (6):749–754, 1931.Search in Google Scholar

Received: 2020-05-01
Accepted: 2020-10-13
Published Online: 2020-11-27
Published in Print: 2020-11-18

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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