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Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications

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Industrial Mathematics and Complex Systems

Part of the book series: Industrial and Applied Mathematics ((INAMA))

Abstract

We consider optimal control problems for the flow of gas or fresh water in pipe networks as well as drainage or sewer systems in open canals. The equations of motion are taken to be represented by the nonlinear isothermal Euler gas equations, the water hammer equations, or the St. Venant equations for flow. We formulate model hierarchies and derive an abstract model for such network flow problems including pipes, junctions, and controllable elements such as valves, weirs, pumps, as well as compressors. We use the abstract model to give an overview of the known results and challenges concerning equilibria, well-posedness, controllability, and optimal control. A major challenge concerning the optimization is to deal with switching on–off states that are inherent to controllable devices in such applications combined with continuous simulation and optimization of the gas flow. We formulate the corresponding mixed-integer nonlinear optimal control problems and outline a decomposition approach as a solution technique.

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References

  1. Abreu, J., Cabrera, E., Izquierdo, J., García-Serra, J.: Flow modeling in pressurized systems revisited. J. Hydraul. Eng. 125(11), 1154–1169 (1999)

    Article  Google Scholar 

  2. Adimurthi, Veerappa Gowda, G.D.: Conservation law with discontinuous flux. J. Math. Kyoto Univ. 43(1), 27–70 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Adimurthi, Mishra, S.: Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2(4), 783–837 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Amin, S., Hante, F.M., Bayen, A.M.: On stability of switched linear hyperbolic conservation laws with reflecting boundaries. In: Hybrid Systems: Computation and Control, vol. 4981. Lecture Notes in Computer Science, pp. 602–605. Springer (2008)

    Google Scholar 

  5. Amin, S., Hante, F.M., Bayen, A.M.: Stability analysis of linear hyperbolic systems with switching parameters and boundary conditions. In: 47th IEEE Conference on Decision and Control, 2008, CDC 2008, pp. 2081–2086. IEEE (2008)

    Google Scholar 

  6. Amin, S., Hante, F.M., Bayen, A.M.: Exponential stability of switched linear hyperbolic initial-boundary value problems. IEEE Trans. Autom. Control 57(2), 291–301 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bastin, G., Coron, J.-M.: On boundary feedback stabilization of non-uniform linear \(2\times 2\) hyperbolic systems over a bounded interval. Syst. Control Lett. 60(11), 900–906 (2011)

    Article  MATH  Google Scholar 

  8. Jacques, F.: Benders. Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4(1), 238–252 (1962)

    Article  MathSciNet  Google Scholar 

  9. Bressan, A., Shen, W.: Optimality conditions for solutions to hyperbolic balance laws. In: Control Methods in PDE-Dynamical Systems, vol. 426. Contemporary Mathematics, pp. 129–152. American Mathematical Society, Providence, RI (2007)

    Google Scholar 

  10. Brouwer, J., Gasser, I., Herty, M.: Gas pipeline models revisited: model hierarchies, nonisothermal models, and simulations of networks. Multiscale Model. Simul. 9(2), 601–623 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Buchheim, C., Meyer, C., Schäfer, R.: Combinatorial optimal control of semilinear elliptic PDEs. Technical report, Fakultät für Mathematik, TU Dortmund (2015)

    Google Scholar 

  12. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert \(W\) function. Adv. Comput. Math. 5(4), 329–359 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Coron, J.-M., Glass, O., Wang, Z.: Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed. SIAM J. Control Optim. 48(5), 3105–3122 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Coron, J.-M., Vazquez, R., Krstic, M., Bastin, G.: Local exponential \(H^2\) stabilization of a \(2\times 2\) quasilinear hyperbolic system using backstepping. SIAM J. Control Optim. 51(3), 2005–2035 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Daafouz, J., Di Benedetto, M.D., Blondel, V.D., Ferrari-Trecate, G., Hetel, L., Johansson, M., Juloski, A.L., Paoletti, S., Pola, G., De Santis, E., Vidal, R.: Switched and piecewise affine systems. In: Handbook of Hybrid Systems Control, pp. 87–137. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  16. Dáger, R., Zuazua, E.: Wave Propagation, Observation and Control in \(1\text{-}d\) Flexible Multi-Structures, vol. 50. Mathématiques & Applications (Berlin). Springer, Berlin (2006)

    Google Scholar 

  17. Dick, M., Gugat, M., Herty, M., Leugering, G., Steffensen, S., Wang, K.: Stabilization of networked hyperbolic systems with boundary feedback. In: Trends in PDE Constrained Optimization, vol. 165. International Series of Numerical Mathematics, pp. 487–504. Birkhäuser/Springer, Cham (2014)

    Google Scholar 

  18. Dick, M., Gugat, M., Leugering, G.: Classical solutions and feedback stabilization for the gas flow in a sequence of pipes. Netw. Heterog. Media 5(4), 691–709 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Dick, M., Gugat, M., Leugering, G.: A strict \(H^1\)-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numer. Algebra Control Optim. 1(2), 225–244 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Domschke, P., Geißler, B., Kolb, O., Lang, J., Martin, A., Morsi, A.: Combination of nonlinear and linear optimization of transient gas networks. INFORMS J. Comput. 23(4), 605–617 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Domschke, P., Kolb, O., Lang, J.: Adjoint-based error control for the simulation and optimization of gas and water supply networks. Appl. Math. Comput. 259, 1003–1018 (2015)

    MathSciNet  Google Scholar 

  22. Dos Santos, V., Bastin, G., Coron, J.-M., d’Andréa Novel, B.: Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments. Autom. J. IFAC 44(5), 1310–1318 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36(3), 307–339 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  24. Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66(1), 327–349 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Fügenschuh, A., Göttlich, S., Herty, M., Klar, A., Martin, A.: A discrete optimization approach to large scale supply networks based on partial differential equations. SIAM J. Sci. Comput. 30(3), 1490–1507 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Fügenschuh, A., Geißler, B., Gollmer, R., Morsi, A., Pfetsch, M.E., Rövekamp, J., Schmidt, M., Spreckelsen, K., Steinbach, M.C.: Physical and technical fundamentals of gas networks. In: Koch, et al. [51], Chap. 2, pp. 17–44

    Google Scholar 

  27. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)

    Article  MATH  Google Scholar 

  28. Geißler, B., Martin, A., Morsi, A., Schewe, L.: Using piecewise linear functions for solving MINLPs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol. 154, pp. 287–314. Springer, New York (2012)

    Chapter  Google Scholar 

  29. Geißler, B., Kolb, O., Lang, J., Leugering, G., Martin, A., Morsi, A.: Mixed integer linear models for the optimization of dynamical transport networks. Math. Methods Oper. Res. 73(3), 339–362 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Geißler, B., Morsi, A., Schewe, L., Schmidt, M.: Solving power-constrained gas transportation problems using an MIP-based alternating direction method. Comput. Chem. Eng. 82, 303–317 (2015)

    Article  Google Scholar 

  31. Geißler, B., Morsi, A., Schewe, L., Schmidt, M.: Solving highly detailed gas transport MINLPs: block separability and penalty alternating direction methods. Technical report, 07 (2016)

    Google Scholar 

  32. Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10(4), 237–260 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  33. Glowinski, R., Marroco, A.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de dirichlet non linéaires. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, 9(R2), 41–76 (1975)

    Google Scholar 

  34. Gugat, M., Leugering, G.: Global boundary controllability of the de St. Venant equations between steady states. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 20(1), 1–11 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Gugat, M., Leugering, G.: Global boundary controllability of the Saint-Venant system for sloped canals with friction. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 26(1), 257–270 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. Gugat, M., Dick, M., Leugering, G.: Stabilization of the gas flow in star-shaped networks by feedback controls with varying delay. In: System Modeling and Optimization, vol. 391. IFIP Advances in Information and Communication Technology, pp. 255–265. Springer, Heidelberg (2013)

    Google Scholar 

  37. Gugat, M., Hante, F.M., Hirsch-Dick, M., Leugering, G.: Stationary states in gas networks. Netw. Heterog. Media 10(2), 295–320 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  38. Gugat, M., Leugering, G., Georg Schmidt, E.J.P.: Global controllability between steady supercritical flows in channel networks. Math. Methods Appl. Sci. 27(7), 781–802 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  39. Gugat, M., Leugering, G., Martin, A., Schmidt, M., Sirvent, M., Wintergerst, D.: Towards simulation based mixed-integer optimization with differential equations. Technical report, Preprint FAU (2016, submitted)

    Google Scholar 

  40. Gugat, M., Leugering, G., Schittkowski, K., Georg Schmidt, E.J.P.: Modelling, stabilization, and control of flow in networks of open channels. In: Online Optimization of Large Scale Systems, pp. 251–270. Springer, Berlin (2001)

    Google Scholar 

  41. Gugat, M., Schultz, R., Wintergerst, D.: Networks of pipelines for gas with nonconstant compressibility factor: stationary states. In: Computational and Applied Mathematics, pp. 1–32 (2016)

    Google Scholar 

  42. Gugat, M., Wintergerst, D.: Finite time blow-up of traveling wave solutions for the flow of real gas through pipeline networks (2016)

    Google Scholar 

  43. Hante, F.M.: Relaxation methods for hyperbolic PDE mixed-integer optimal control problems. ArXiv e-prints, 09 (2015)

    Google Scholar 

  44. Hante, F.M., Leugering, G.: Optimal boundary control of convention-reaction transport systems with binary control functions. Hybrid Systems: Computation and Control. Lecture Notes in Computer Science, vol. 5469, pp. 209–222. Springer, Berlin (2009)

    Chapter  Google Scholar 

  45. Hante, F.M., Leugering, G., Seidman, T.I.: Modeling and analysis of modal switching in networked transport systems. Appl. Math. Optim. 59(2), 275–292 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  46. Hante, F.M., Leugering, G., Seidman, T.I.: An augmented BV setting for feedback switching control. J. Syst. Sci. Complex. 23(3), 456–466 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  47. Hante, F.M., Sager, S.: Relaxation methods for mixed-integer optimal control of partial differential equations. Comput. Optim. Appl. 55(1), 197–225 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  48. Hante, F.M., Sigalotti, M.: Existence of common Lyapunov functions for infinite-dimensional switched linear systems. In: 49th IEEE Conference on Decision and Control (CDC) 2010, CDC 2010, pp. 5668–5673. IEEE (2010)

    Google Scholar 

  49. Hante, F.M., Sigalotti, M.: Converse Lyapunov theorems for switched systems in Banach and Hilbert spaces. SIAM J. Control Optim. 49(2), 752–770 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  50. Holden, H., Risebro, N.H.: Front tracking for hyperbolic conservation laws. In: Applied Mathematical Sciences, vol. 152, second edn. Springer, Heidelberg (2015)

    Google Scholar 

  51. Koch, T., Hiller, B., Pfetsch, M.E., Schewe, L. (eds.): Evaluating Gas Network Capacities. SIAM-MOS series on Optimization. SIAM (2015)

    Google Scholar 

  52. Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: On the analysis and control of hyperbolic systems associated with vibrating networks. Proc. Roy. Soc. Edinburgh Sect. A 124(1), 77–104 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  53. Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modeling, analysis and control of dynamic elastic multi-link structures. In: Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston (1994)

    Google Scholar 

  54. Lagnese, J.E., Leugering, G.: Domain decomposition methods in optimal control of partial differential equations, vol. 148. International Series of Numerical Mathematics. Birkhäuser Verlag (2004)

    Google Scholar 

  55. Lamare, P.-O., Girard, A., Prieur, C.: Switching rules for stabilization of linear systems of conservation laws. SIAM J. Control Optim. 53(3), 1599–1624 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  56. Leugering, G., Georg Schmidt, E.J.P.: On the modelling and stabilization of flows in networks of open canals. SIAM J. Control Optim. 41(1), 164–180 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  57. LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser (1992)

    Google Scholar 

  58. Le Veque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002)

    Google Scholar 

  59. Li, T.: Controllability and observability for quasilinear hyperbolic systems, vol. 3. AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO; Higher Education Press, Beijing (2010)

    Google Scholar 

  60. Mahlke, D., Martin, A., Moritz, S.: A mixed integer approach for time-dependent gas network optimization. Optim. Methods Softw. 25(4–6), 625–644 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  61. Martin, A., Klamroth, K., Lang, J., Leugering, G., Morsi, A., Oberlack, M., Ostrowski, M., Rosen, R., (eds.): Mathematical Optimization of Water Networks. Birkhäuser (2012)

    Google Scholar 

  62. Martin, A., Möller, M., Moritz, S.: Mixed integer models for the stationary case of gas network optimization. Math. Program. 105(2–3, Ser. B), 563–582 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  63. Morsi, A., Geißler, B., Martin, A.: Mixed integer optimization of water supply networks. In: Mathematical Optimization of Water Networks, vol. 162. International Series of Numerical Mathematics, pp. 35–54. Birkhäuser/Springer Basel AG, Basel (2012)

    Google Scholar 

  64. Nicaise, S.: Control and stabilization of 2x2 hyperbolic systems on graphs. Technical report, Université de Valenciennes et du Hainaut Cambrésis (2016). To appear in MCRF 2017

    Google Scholar 

  65. Pfaff, S., Ulbrich, S.: Optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data. SIAM J. Control Optim. 53(3), 1250–1277 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  66. Pfaff, S., Ulbrich, S., Leugering, G.: Optimal control of nonlinear hyperbolic conservation laws with switching. In: Trends in PDE constrained optimization, vol. 165. International Series of Numerical Mathematics, pp. 109–131. Birkhäuser/Springer, Cham (2014)

    Google Scholar 

  67. Pfetsch, M.E., Fügenschuh, A., Geißler, B., Geißler, N., Gollmer, R., Hiller, B., Humpola, J., Koch, T., Lehmann, T., Martin, A., Morsi, A., Rövekamp, J., Schewe, L., Schmidt, M., Schultz, R., Schwarz, R., Schweiger, J., Stangl, C., Steinbach, M.C., Vigerske, S., Willert, B.M.: Validation of nominations in gas network optimization: models, methods, and solutions. Optim. Methods Softw. 30(1), 15–53 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  68. Rüffler, F., Hante, F.M.: Optimal switching for hybrid semilinear evolutions. Nonlinear Anal. Hybrid Syst. 22, 215–227 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  69. Sahinidis, N.V., Grossmann, I.E.: Convergence properties of generalized benders decomposition. Comput. Chem. Eng. 15(7), 481–491 (1991)

    Article  Google Scholar 

  70. Smoller, J.: Shock Waves and Reaction-Diffusion Equations, vol. 258. Grundlehren der mathematischen Wissenschaften. Springer Verlag (1983)

    Google Scholar 

  71. Sontag, E.: From linear to nonlinear: some complexity comparisons. In: 1995 Proceedings of the 34th IEEE Conference on Decision and Control, vol. 3, pp. 2916–2920. IEEE (1995)

    Google Scholar 

  72. Sontag, E.D.: Nonlinear regulation: the piecewise linear approach. IEEE Trans. Autom. Control 26(2), 346–358 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  73. Ulbrich, S.: A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41(3), 740–797 (2002). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the Deutsche Forschungsgemeinschaft for their support within projects A03, A05, B07, and B08 in the Sonderforschungsbereich/Transregio 154 Mathematical Modeling, Simulation and Optimization using the Example of Gas Networks. In addition, parts of this research were performed as part of the Energie Campus Nürnberg and supported by funding through the “Aufbruch Bayern (Bavaria on the move)” initiative of the state of Bavaria.

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Authors and Affiliations

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

    Falk M. Hante & Günter Leugering

  2. Discrete Optimization, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Cauerstr. 11, 91058, Erlangen, Germany

    Alexander Martin, Lars Schewe & Martin Schmidt

  3. Energie Campus Nürnberg, Fürther Str. 250, 90429, Nürnberg, Germany

    Martin Schmidt

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  1. Falk M. Hante
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  2. Günter Leugering
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  4. Lars Schewe
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Correspondence to Falk M. Hante .

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  1. Department of Mathematics, Guru Nanak Dev University, Amritsar, Punjab, India

    Pammy Manchanda

  2. CNRS, Dieudonné Center of Mathematics, University Côte d’Azur, Nice, France

    René Lozi

  3. School of Basic Sciences and Research, Sharda University, Greater Noida, Uttar Pradesh, India

    Abul Hasan Siddiqi

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Hante, F.M., Leugering, G., Martin, A., Schewe, L., Schmidt, M. (2017). Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications. In: Manchanda, P., Lozi, R., Siddiqi, A. (eds) Industrial Mathematics and Complex Systems. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3758-0_5

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