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Robust optimal discrete arc sizing for tree-shaped potential networks

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Abstract

We consider the problem of discrete arc sizing for tree-shaped potential networks with respect to infinitely many demand scenarios. This means that the arc sizes need to be feasible for an infinite set of scenarios. The problem can be seen as a strictly robust counterpart of a single-scenario network design problem, which is shown to be NP-complete even on trees. In order to obtain a tractable problem, we introduce a method for generating a finite scenario set such that optimality of a sizing for this finite set implies the sizing’s optimality for the originally given infinite set of scenarios. We further prove that the size of the finite scenario set is quadratically bounded above in the number of nodes of the underlying tree and that it can be computed in polynomial time. The resulting problem can then be solved as a standard mixed-integer linear optimization problem. Finally, we show the applicability of our theoretical results by computing globally optimal arc sizes for a realistic hydrogen transport network of Eastern Germany.

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References

  1. Afshar, M.H., Mariño, M.A.: A parameter-free self-adapting boundary genetic search for pipe network optimization. Comput. Optim. Appl. 37(1), 83–102 (2007). https://doi.org/10.1007/s10589-007-9016-1

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall Inc., Upper Saddle River (1993)

    MATH  Google Scholar 

  3. Altın, A., Amaldi, E., Belotti, P., Pınar, M.Ç.: Provisioning virtual private networks under traffic uncertainty. Networks 49(1), 100–115 (2007). https://doi.org/10.1002/net.20145

    Article  MathSciNet  MATH  Google Scholar 

  4. Andre, J., Bonnans, F., Cornibert, L.: Optimization of capacity expansion planning for gas transportation networks. Eur. J. Oper. Res. 197(3), 1019–1027 (2009). https://doi.org/10.1016/j.ejor.2007.12.045

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Ameur, W., Kerivin, H.: Routing of uncertain traffic demands. Optim. Eng. 6(3), 283–313 (2005). https://doi.org/10.1007/s11081-005-1741-7

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton. http://press.princeton.edu/titles/9099.html (2009). Accessed 8 Mar 2019

  7. Bragalli, C., D’Ambrosio, C., Lee, J., Lodi, A., Toth, P.: Water Network Design by MINLP. Technical Report (2008). IBM Research Division, RC24495 (W0802-056)

  8. Bragalli, C., D’Ambrosio, C., Lee, J., Lodi, A., Toth, P.: On the optimal design of water distribution networks: a practical MINLP approach. Optim. Eng. 13(2), 219–246 (2012). https://doi.org/10.1007/s11081-011-9141-7

    Article  MathSciNet  MATH  Google Scholar 

  9. Bragalli, C., D’Ambrosio, C., Lee, J., Lodi, A., Toth, P.: An MINLP solution method for a water network problem. In: Azar, Y., Erlebach, T. (eds.) Algorithms - ESA 2006, Lecture Notes in Computer Science, vol. 4168, pp. 696–707. Springer, Berlin (2006). https://doi.org/10.1007/11841036_62

    Chapter  Google Scholar 

  10. Cacchiani, V., Jünger, M., Liers, F., Lodi, A., Schmidt, D.R.: Single-commodity robust network design with finite and Hose demand sets. Math. Program. 157, 297–342 (2016). https://doi.org/10.1007/s10107-016-0991-9

    Article  MathSciNet  MATH  Google Scholar 

  11. Collins, M., Cooper, L., Helgason, R., Kennington, J., LeBlanc, L.: Solving the pipe network analysis problem using optimization techniques. Manag. Sci. 24(7), 747–760 (1978). https://doi.org/10.1287/mnsc.24.7.747

    Article  MathSciNet  MATH  Google Scholar 

  12. Duffield, N.G., Goyal, P., Greenberg, A., Mishra, P., Ramakrishnan, K.K., van der Merive, J.E.: A flexible model for resource management in virtual private networks. SIGCOMM Comput. Commun. Rev. 29(4), 95–108 (1999). https://doi.org/10.1145/316194.316209

    Article  Google Scholar 

  13. Erlebach, T., Ruegg, M.: Optimal bandwidth reservation in hose-model VPNs with multi-path routing. In: 23rd Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM 2004, vol. 4, pp. 2275–2282. IEEE (2004). https://doi.org/10.1109/INFCOM.2004.1354650

  14. Fingerhut, J., Suri, S., Turner, J.S.: Designing least-cost nonblocking broadband networks. J. Algorithms 24(2), 287–309 (1997). https://doi.org/10.1006/jagm.1997.0866

    Article  MathSciNet  MATH  Google Scholar 

  15. Fügenschuh, A., Geißler, B., Gollmer, R., Morsi, A., Rövekamp, J., Schmidt, M., Spreckelsen, K., Steinbach, M.C.: Chapter 2: physical and Technical Fundamentals of Gas Networks, pp. 17–43. Society for Industrial and Applied Mathematics, Philadelphia (2015). https://doi.org/10.1137/1.9781611973693.ch2

    Google Scholar 

  16. Groß, M., Pfetsch, M.E., Schewe, L., Schmidt, M., Skutella, M.: Algorithmic results for potential-based flows: easy and hard cases. Networks 73(3), 306–324 (2018). https://doi.org/10.1002/net.21865

  17. Gurobi Optimization, Inc.: Gurobi optimizer reference manual. http://www.gurobi.com (2017). Accessed 8 Mar 2019

  18. Hansen, C.T., Madsen, K., Nielsen, H.B.: Optimization of pipe networks. Math. Program. 52(1), 45–58 (1991). https://doi.org/10.1007/BF01582879

    Article  MathSciNet  MATH  Google Scholar 

  19. Hart, W.E., Laird, C., Watson, J.P., Woodruff, D.L.: Pyomo-Optimization Modeling in Python, vol. 67. Springer, Boston, MA (2012). https://doi.org/10.1007/978-1-4614-3226-5

    Book  MATH  Google Scholar 

  20. Hart, W.E., Watson, J.P., Woodruff, D.L.: Pyomo: modeling and solving mathematical programs in python. Math. Program. Comput. 3, 219–260 (2011). https://doi.org/10.1007/s12532-011-0026-8

    Article  MathSciNet  Google Scholar 

  21. Humpola, J., Fügenschuh, A.: Convex reformulations for solving a nonlinear network design problem. Comput. Optim. Appl. 62(3), 717–759 (2015). https://doi.org/10.1007/s10589-015-9756-2

    Article  MathSciNet  MATH  Google Scholar 

  22. Koch, T., Hiller, B., Pfetsch, M., Schewe, L. (eds.): Evaluating Gas Network Capacities. Society for Industrial and Applied Mathematics, Philadelphia (2015). https://doi.org/10.1137/1.9781611973693

    MATH  Google Scholar 

  23. Mischner, J., Fasold, H., Kadner, K.: gas2energy.net: Systemplanung in der Gasversorgung; gaswirtschaftliche Grundlagen. Edition gwf, Gas, Erdgas. Oldenbourg Industrieverl. (2011)

  24. Raghunathan, A.U.: Global optimization of nonlinear network design. SIAM J. Optim. 23(1), 268–295 (2013). https://doi.org/10.1137/110827387

    Article  MathSciNet  MATH  Google Scholar 

  25. Reuss, M., Grube, T., Robinius, M., Preuster, P., Wasserscheid, P., Stolten, D.: Seasonal storage and alternative carriers: a flexible hydrogen supply chain architecture model. Appl. Energy 200, 290–302 (2017). https://doi.org/10.1016/j.apenergy.2017.05.050

    Article  Google Scholar 

  26. Ríos-Mercado, R.Z., Wu, S., Scott, L.R., Boyd, E.A.: A reduction technique for natural gas transmission network optimization problems. Ann. Oper. Res. 117(1), 217–234 (2002). https://doi.org/10.1023/A:1021529709006

    Article  MATH  Google Scholar 

  27. Robinius, M.: Strom- und Gasmarktdesign zur Versorgung des deutschen Straßenverkehrs mit Wasserstoff. Ph. D. Thesis, RWTH Aachen University (2015)

  28. Robinius, M., Otto, A., Syranidis, K., Ryberg, D., Heuser, P., Welder, L., Grube, T., Markewitz, P., Tietze, V., Stolten, D.: Linking the power and transport sectors–part 2: modelling a sector coupling scenario for germany. Energies (2017). https://doi.org/10.3390/en10070957

    Google Scholar 

  29. Ríos-Mercado, R.Z., Borraz-Sánchez, C.: Optimization problems in natural gas transportation systems: a state-of-the-art review. Appl. Energy 147, 536–555 (2015). https://doi.org/10.1016/j.apenergy.2015.03.017

    Article  Google Scholar 

  30. Shiono, N., Suzuki, H.: Optimal pipe-sizing problem of tree-shaped gas distribution networks. Eur. J. Oper. Res. 252, 550–560 (2016). https://doi.org/10.1016/j.ejor.2016.01.008

    Article  MathSciNet  MATH  Google Scholar 

  31. Syranidis, K., Robinius, M., Stolten, D.: Control techniques and the modeling of electrical power flow across transmission networks. Renew. Sustain. Energy Rev. 82, 3452–3467 (2018). https://doi.org/10.1016/j.rser.2017.10.110

    Article  Google Scholar 

  32. Szabó, J.: The set of solutions to nomination validation in passive gas transportation networks with a generalized flow formula. Technical Report, pp. 11–44, ZIB, Takustr.7, 14195 Berlin (2012). www.nbn-resolving.de/urn:nbn:de:0297-zib-15151

  33. Vuffray, M., Misra, S., Chertkov, M.: 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), pp. 4571–4578 (2015). https://doi.org/10.1109/CDC.2015.7402933

  34. Welder, L., Ryberg, D.S., Kotzur, L., Grube, T., Robinius, M., Stolten, D.: Spatio-temporal optimization of a future energy system for power-to-hydrogen applications in Germany. Energy 158, 1130–1149 (2018). https://doi.org/10.1016/j.energy.2018.05.059

  35. Weymouth, T.R.: Problems in natural gas engineering. Trans. Am. Soc. Mech. Eng. 34(1349), 185–231 (1912)

    Google Scholar 

  36. Yates, D., Templeman, A., Boffey, T.: The computational complexity of the problem of determining least capital cost designs for water supply networks. Eng. Optimiz. 7(2), 143–155 (1984). https://doi.org/10.1080/03052158408960635

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the Deutsche Forschungsgemeinschaft for their support within Projects A05, B07, and B08 in the Sonderforschungsbereich/Transregio 154 “Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks”. This research has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government and by the Emerging Field Initiative (EFI) of the Friedrich-Alexander-Universität Erlangen-Nürnberg through the project “Sustainable Business Models in Energy Markets”. Furthermore, this work has been supported by the Helmholtz Association under the Joint Initiative “EnergySystem 2050—A Contribution of the Research Field Energy”.

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Authors and Affiliations

  1. Institute of Electrochemical Process Engineering (IEK-3), Forschungszentrum Jülich GmbH, Wilhelm-Johnen-Str., 52428, Jülich, Germany

    Martin Robinius, Detlef Stolten & Lara Welder

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

    Lars Schewe & Johannes Thürauf

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

    Lars Schewe, Martin Schmidt & Johannes Thürauf

  4. Department of Mathematics, Trier University, Universitätsring 15, 54296, Trier, Germany

    Martin Schmidt

  5. Chair of Fuel Cells, RWTH Aachen University, c/o Institute of Electrochemical Process Engineering (IEK-3), Forschungszentrum Jülich, Wilhelm-Johnen-Str., 52428, Jülich, Germany

    Detlef Stolten

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  1. Martin Robinius
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Robinius, M., Schewe, L., Schmidt, M. et al. Robust optimal discrete arc sizing for tree-shaped potential networks. Comput Optim Appl 73, 791–819 (2019). https://doi.org/10.1007/s10589-019-00085-x

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