Energy Systems

, Volume 5, Issue 1, pp 179–207 | Cite as

Complexity of transmission network expansion planning

NP-hardness of connected networks and MINLP evaluation
  • David Oertel
  • R. Ravi
Original Paper


Transmission network expansion planning in its original formulation is NP-hard due to the subproblem Steiner trees, the minimum cost connection of an initially unconnected network with mandatory and optional nodes. By using electrical network theory we show why NP-hardness still holds when this subproblem of network design from scratch is omitted by considering already (highly) connected networks only. This refers to the case of extending a long working transmission grid for increased future demand. It will be achieved by showing that this case is computationally equivalent to \(3\)-SAT. Additionally, the original mathematical formulation is evaluated by using an appropriate state-of-the-art mixed integer non-linear programming solver in order to see how much effort in computation and implementation is really necessary to solve this problem in practice.


Transmission network expansion planning NP-hard Mixed-integer non-linear programming Electrical network 



The authors would like to thank Prof. D. Wagner and Dr. Ignaz Rutters from KIT for initiation and mentoring of this work, Jay Apt from CMU for valuable references, Arne Lüllmann from Fraunhofer ISI for the topic suggestion, Michael Poss from VUB/ULB for making the test data available, Dr. Matthias Oertel for initial support in electrical engineering, the NEOS project and interACT with the Baden-Württemberg Stipendium for funding this work through a scholarship program with CMU.


  1. 1.
  2. 2.
    Alguacil, N., Motto, A., Conejo, A.: Transmission expansion planning: a mixed-integer lp approach. IEEE Trans. Power Syst. 18(3), 1070–1077 (2003). doi: 10.1109/TPWRS.2003.814891 CrossRefGoogle Scholar
  3. 3.
    Bahiense, L., Oliveira, G.C., Pereira, M., Granville, S.: A mixed integer disjunctive model for transmission network expansion. IEEE Trans. Power Syst. 16(3), 560–565 (2001). doi: 10.1109/59.932295 Google Scholar
  4. 4.
    Bakshi, U., Bakshi, A.: Network analysis & synthesis. Technical Publications (2009).
  5. 5.
    Biggs, N.: Algebraic graph theory. In: Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993).
  6. 6.
    Binato, S.: Optimal expansion of transmission networks by benders decomposition and cutting planes. Ph.D. thesis, Federal University of Rio de Janeiro (2000)Google Scholar
  7. 7.
    Binato, S., Pereira, M.V.F., Granville, S.: A new Benders decomposition approach to solve power transmission network design problems. IEEE Trans. Power Syst. 16(2), 235–240 (2001). doi: 10.1109/59.918292 Google Scholar
  8. 8.
    Bollobás, B.: Modern Graph Theory. Springer, New York (1998)Google Scholar
  9. 9.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2 edn. The MIT Press, Boston (2001)Google Scholar
  10. 10.
    Czyzyk, J., Mesnier, M.P., Moré, J.J.: The neos server. IEEE Comput. Sci. Eng. 5(3), 68–75 (1998). doi: 10.1109/99.714603 Google Scholar
  11. 11.
    Diestel, R.: Graph theory. In: Graduate Texts in Mathematics. Springer, New York (2006).
  12. 12.
    Dolan, E.D.: Neos server 4.0 administrative guide. CoRR cs.DC/0107034 (2001)Google Scholar
  13. 13.
    Dommel, H., Tinney, W.: Optimal power flow solutions. IEEE Trans. Power Appar. Syst. 87(10), 1866–1876 (1968). doi: 10.1109/TPAS.1968.292150 Google Scholar
  14. 14.
    Garver, L.: Transmission network estimation using linear programming. IEEE Trans. Power Appar. Syst. 89(7), 1688–1697 (1970). doi: 10.1109/TPAS.1970.292825 Google Scholar
  15. 15.
    Latorre, G., Cruz, R., Areiza, J., Villegas, A.: Classification of publications and models on transmission expansion planning. IEEE Trans. Power Syst. 18(2), 938–946 (2003). doi:  10.1109/TPWRS.2003.811168 Google Scholar
  16. 16.
    Lee, C., Ng, S., Zhong, J., Wu, F.: Transmission expansion planning from past to future, pp. 257–265 (2006). doi: 10.1109/PSCE.2006.296317
  17. 17.
    MATLAB: version 7.13.0 (R2011b). The MathWorks Inc., Natick, Massachusetts (2011).
  18. 18.
    Moulin, L.S., Poss, M., Sagastizábal, C.: Transmission expansion planning with re-design. Energy Syst. 1(2), 113–139 (2010). doi: 10.1007/s12667-010-0010-9 Google Scholar
  19. 19.
    ONeill, R.P. et al.: A model and approach for optimal power systems planning and investment. Math. Progr. (2011)Google Scholar
  20. 20.
    Powell, M., Buhmann, M., Iserles, A.: Approximation Theory and Optimization: Tributes to M.J.D. Powell. Cambridge University Press, Cambridge (1997).
  21. 21.
    Rider, M., Garcia, A., Romero, R.: Transmission system expansion planning by a branch-and-bound algorithm. Gener. Transm. Distrib. IET 2(1), 90–99 (2008). doi: 10.1049/iet-gtd:20070090 Google Scholar
  22. 22.
    Romero, R., Gallego, R., Monticelli, A.: Transmission system expansion planning by simulated annealing. IEEE Trans. Power Syst. 11(1), 364–369 (1996). doi: 10.1109/59.486119 CrossRefGoogle Scholar
  23. 23.
    Romero, R., Monticelli, A., Garcia, A., Haffner, S.: Test systems and mathematical models for transmission network expansion planning. IEE Proc. Gener. Transm. Distrib. 149(1), 27–36 (2002). doi: 10.1049/ip-gtd:20020026 CrossRefGoogle Scholar
  24. 24.
    Rosas-Casals, M., Corominas, B.: Assessing european power grid reliability by means of topological measures. WIT Trans. Ecol. Environ. 121, 527–537 (2009)CrossRefGoogle Scholar
  25. 25.
    Scilab Consortium: Scilab: Free and open source software for numerical computation (2011).
  26. 26.
    Tawarmalani, M., Sahinidis, N.: Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software, and applications. In: Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Boston (2002).
  27. 27.
    Taylor, J., Hover, F.: Linear relaxations for transmission system planning. IEEE Trans. Power Syst. 26(4), 2533–2538 (2011). doi: 10.1109/TPWRS.2011.2145395 CrossRefGoogle Scholar
  28. 28.
    Torres, S., Castro, C., Pringles, R., Guaman, W.: Comparison of particle swarm based meta-heuristics for the electric transmission network expansion planning problem, pp. 1–7 (2011). doi: 10.1109/PES.2011.6039571
  29. 29.
    von Meier, A.: Electric Power Systems: A Conceptual Introduction. Wiley-Interscience, New York (2006)Google Scholar
  30. 30.
    Wegener, I.: The Complexity of Boolean Functions. Wiley and B.G. Teubner, Stuttgart (1987)Google Scholar
  31. 31.
    Wood, A.J., Wollenberg, B.F.: Power Generation, Operation, and Control. Wiley-Interscience, New York (1996)Google Scholar
  32. 32.
    Zhang, H., Vittal, V., Heydt, G.T., Quintero, J.: A mixed-integer linear programming approach for multi-stage security-constrained transmission expansion planning. IEEE Trans. Power Syst. 27(2), 1125–1133 (2012). doi: 10.1109/TPWRS.2011.2178000 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations