Adjoint-Based Control of Model and Discretization Errors for Gas and Water Supply Networks

  • Pia Domschke
  • Oliver Kolb
  • Jens Lang
Part of the Studies in Computational Intelligence book series (SCI, volume 359)


We are interested in the simulation and optimization of gas and water transport in networks. Those networks consist of pipes and various other components like compressor/pumping stations and valves. The flow through the pipes can be described by different models based on the Euler equations, including hyperbolic systems of partial differential equations. For the other components, algebraic or ordinary differential equations are used. Depending on the data, different models can be used in different regions of the network. We present a strategy that adaptively applies the models and discretizations, using adjoint-based error estimators to maintain the accuracy of the solution. Finally, we give numerical examples for both types of networks.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abreu, J., Cabrera, E., Izquierdo, J., García-Serra, J.: Flow modeling in pressurized systems revisited. Journal of Hydraulic Engineering 125(11), 1154–1169 (1999)CrossRefGoogle Scholar
  2. 2.
    Bales, P., Kolb, O., Lang, J.: Hierarchical modelling and model adaptivity for gas flow on networks. In: Allen, G., Nabrzyski, J., Seidel, E., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2009. LNCS, vol. 5544, pp. 337–346. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta numerica 10, 1–102 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Braack, M., Ern, A.: A posteriori control of modeling errors and discretization errors. SIAM Journal of Multiscale Modeling and Simulation 1(2), 221–238 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burgschweiger, J., Gnädig, B., Steinbach, M.C.: Optimization models for operative planning in drinking water networks. Optimization and Engineering 10(1), 43–73 (2009)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Domschke, P., Kolb, O., Lang, J.: An adaptive model switching and discretization algorithm for gas flow on networks. In: Procedia Computer Science ICCS 2010, vol. 1(1), pp. 1325–1334 (2010)Google Scholar
  7. 7.
    Domschke, P., Kolb, O., Lang, J.: Adjoint-based control of model and discretisation errors for gas flow in networks. Int. J. Mathematical Modelling and Numerical Optimisation 2(2), 175–193 (2011)zbMATHCrossRefGoogle Scholar
  8. 8.
    Ehrhardt, K., Steinbach, M.C.: Nonlinear optimization in gas networks. In: Bock, H.G., Kostina, E.A., Phu, H.X., Ranacher, R. (eds.) Modeling, Simulation and Optimization of Complex Processes, pp. 139–148. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Hähnlein, C.: Numerische Modellierung zur Betriebsoptimierung von Wasserverteilnetzen. PhD thesis, TU Darmstadt (2008)Google Scholar
  10. 10.
    Herty, M.: Modeling, simulation and optimization of gas networks with compressors. Networks and Heterogeneous Media 2(1), 81–97 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kolb, O., Lang, J., Bales, P.: An implicit box scheme for subsonic compressible flow with dissipative source term. Numerical Algorithms 53(2), 293–307 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Martin, A., Möller, M., Moritz, S.: Mixed integer models for the stationary case of gas network optimization. Mathematical Programming 105, 563–582 (2006)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Moritz, S.: A Mixed Integer Approach for the Transient Case of Gas Network Optimization. PhD thesis, TU Darmstadt (2006)Google Scholar
  14. 14.
    Rossman, L.A.: EPANET 2 users manual. U.S. Environmental Protection Agency, Cincinnati, OH (2000)Google Scholar
  15. 15.
    Sekirnjak, E.: Transiente Technische Optimierung (TTO-Prototyp). Technical report, PSI AG (November 2000)Google Scholar
  16. 16.
    SIMONE Research Group,

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Pia Domschke
    • 1
  • Oliver Kolb
    • 1
  • Jens Lang
    • 1
    • 2
    • 3
  1. 1.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Center of Smart InterfacesTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Graduate School Computational EngineeringTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations