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Perspective Cuts for the ACOPF with Generators

  • Esteban Salgado
  • Claudio Gentile
  • Leo LibertiEmail author
Chapter
Part of the AIRO Springer Series book series (AIROSS, volume 1)

Abstract

The alternating current optimal power flow problem is a fundamental problem in the management of smart grids. In this paper we consider a variant which includes activation/deactivation of generators at some of the grid sites. We formulate the problem as a mathematical program, prove its NP-hardness w.r.t. activation/deactivation, and derive two perspective reformulations.

Keywords

Power flow Reformulation NP-hardness 

Notes

Acknowledgements

This paper has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement n. 764759. The second author is partially supported by MIUR PRIN2015 project no. 2015B5F27W. The last author (LL) is grateful for financial support by CNR STM Program prot. AMMCNT-CNR n. 80058 dated 05/12/2017.

References

  1. 1.
    Bacher, R.: Power system models, objectives and constraints in optimal power flow calculations. In: Frauendorfer, K., Glavitsch, H., Bacher, R. (eds.) Optimization in Planning and Operation of Electric Power Systems, pp. 217–264. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  2. 2.
    Bienstock, D.: Electrical Transmission System Cascades and Vulnerability: an Operations Research Viewpoint. Number 22 in MOS-SIAM Optimization. SIAM, Philadelphia (2016)Google Scholar
  3. 3.
    Bienstock, D., Verma, A.: Strong NP-hardness of AC power flows feasibility. Technical report (2015). arXiv:1512.07315
  4. 4.
    Chen, C., Atamtürk, A., Oren, S.: Bound tightening for the alternating current optimal power flow problem. IEEE Trans. Power Syst. 31(5), 3729–3736 (2016)CrossRefGoogle Scholar
  5. 5.
    Fourer, R., Gay, D., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. AMPL Optimization LLC (2003)Google Scholar
  6. 6.
    Frangioni, A., Furini, F., Gentile, C.: Approximated perspective relaxations: a project&lift approach. Comput. Optim. Appl. 63(3), 705–735 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Frangioni, A., Furini, F., Gentile, C.: Improving the approximated projected perspective reformulation by dual information. Oper. Res. Lett. 45, 519–524 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Frangioni, A., Gentile, C.: Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program. 106(2), 225–236 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    IBM. ILOG CPLEX 12.6 User’s Manual. IBM (2014)Google Scholar
  10. 10.
    Josz, C.: Application of polynomial optimization to electricity transmission networks. Ph.D. thesis, Univ. Paris VI (2016)Google Scholar
  11. 11.
    Kuang, X., Ghaddar, B., Naoum-Sawaya, J., Zuluaga, L.: Alternative LP and SOCP hierarchies for ACOPF problems. IEEE Trans. Power Syst. 32(4), 2828–2836 (2017)CrossRefGoogle Scholar
  12. 12.
    Lavaei, J., Low, S.: Zero duality gap in optimal power flow problem. IEEE Trans. Power Syst. 27(1), 92–107 (2012)CrossRefGoogle Scholar
  13. 13.
    Lehmann, K., Grastien, A., van Hentenryck, P.: AC-Feasibility on tree networks is NP-hard. IEEE Trans. Power Syst. 13(1), 798–801 (2016)CrossRefGoogle Scholar
  14. 14.
    Panciatici, P.: Private Communication (2016)Google Scholar
  15. 15.
    Ruiz, M., Maeght, J., Marié, A., Panciatici, P., Renaud, A.: A progressive method to solve large-scale AC optimal power flow with discrete variables and control of the feasibility. In: Proceedings of the Power Systems Computation Conference, PSCC, Piscataway, vol. 18. IEEE (2014)Google Scholar
  16. 16.
    Sahinidis, N.: BARON user Manual v. 17.8.9. The Optimization Firm LLC (2017)Google Scholar
  17. 17.
    Salgado, E.: Fast relaxations for alternating current optimal power flow. Master’s thesis, LIX, Ecole Polytechnique (2017)Google Scholar
  18. 18.
    Salgado, E., Scozzari, A., Tardella, F., Liberti, L.: Alternating current optimal power flow with generator selection. In: Proceedings of ISCO 2018, vol. 10856, pp. 364–375. LNCS (2018)Google Scholar
  19. 19.
    Vavasis, S.: Quadratic programming is in NP. Inf. Process. Lett. 36, 73–77 (1990)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wolsey, L.A.: Integer Programming. Wiley, New York (1998)zbMATHGoogle Scholar
  21. 21.
    Zimmermann, R., Murillo-Sanchez, C., Thomas, R.: MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26(1), 12–19 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Ecole PolytechniquePalaiseauFrance
  2. 2.IASI-CNRRomeItaly
  3. 3.CNRS LIX Ecole PolytechniquePalaiseauFrance

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