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
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 


  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