Advertisement

Mathematical Programming Computation

, Volume 10, Issue 4, pp 557–596 | Cite as

Matrix minor reformulation and SOCP-based spatial branch-and-cut method for the AC optimal power flow problem

  • Burak Kocuk
  • Santanu S. Dey
  • X. Andy Sun
Full Length Paper
  • 56 Downloads

Abstract

Alternating current optimal power flow (AC OPF) is one of the most fundamental optimization problems in electrical power systems. It can be formulated as a semidefinite program (SDP) with rank constraints. Solving AC OPF, that is, obtaining near optimal primal solutions as well as high quality dual bounds for this non-convex program, presents a major computational challenge to today’s power industry for the real-time operation of large-scale power grids. In this paper, we propose a new technique for reformulation of the rank constraints using both principal and non-principal 2-by-2 minors of the involved Hermitian matrix variable and characterize all such minors into three types. We show the equivalence of these minor constraints to the physical constraints of voltage angle differences summing to zero over three- and four-cycles in the power network. We study second-order conic programming (SOCP) relaxations of this minor reformulation and propose strong cutting planes, convex envelopes, and bound tightening techniques to strengthen the resulting SOCP relaxations. We then propose an SOCP-based spatial branch-and-cut method to obtain the global optimum of AC OPF. Extensive computational experiments show that the proposed algorithm significantly outperforms the state-of-the-art SDP-based OPF solver and on a simple personal computer is able to obtain on average a \(0.71\%\) optimality gap in no more than 720 s for the most challenging power system instances in the literature.

Mathematics Subject Classification

90C20 90C26 90C57 90C90 

References

  1. 1.
    Andersen, M.S., Hansson, A., Vandenberghe, L.: Reduced-complexity semidefinite relaxations of optimal power flow problems. IEEE Trans. Power Syst. 29(4), 1855–1863 (2014)CrossRefGoogle Scholar
  2. 2.
    Bai, X., Wei, H.: Semi-definite programming-based method for security-constrained unit commitment with operational and optimal power flow constraints. IET Gener. Transm. Distrib. 3(2), 182–197 (2009)CrossRefGoogle Scholar
  3. 3.
    Bai, X., Wei, H., Fujisawa, K., Wang, Y.: Semidefinite programming for optimal power flow problems. Electr. Power Energy Syst. 30, 383–392 (2008)CrossRefGoogle Scholar
  4. 4.
    Bienstock, D., Chen, C., Muñoz, G.: Outer-product-free sets for polynomial optimization and oracle-based cuts. arXiv preprint arXiv:1610.04604 (2016)
  5. 5.
    Bienstock, D., Munoz, G.: On linear relaxations of OPF problems. arXiv preprint arXiv:1411.1120 (2014)
  6. 6.
    Bose, S., Gayme, D.F., Chandy, K.M., Low, S.H.: Quadratically constrained quadratic programs on acyclic graphs with application to power flow. IEEE Trans. Control Netw. Syst. 2(3), 278–287 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bose, S., Gayme, D.F., Low, S., Chandy, K.M.: Optimal power flow over tree networks. In: 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1342–1348 (2011)Google Scholar
  8. 8.
    Bukhsh, W.A., Grothey, A., McKinnon, K., Trodden, P.: Local solutions of optimal power flow. IEEE Trans. Power Syst. 28(4), 4780–4788 (2013)CrossRefGoogle Scholar
  9. 9.
    Cain, M.B., O’Neill, R.P., Castillo, A.: History of optimal power flow and formulations. http://www.ferc.gov/industries/electric/indus-act/market-planning/opf-papers/acopf-1-history-formulation-testing.pdf (2012)
  10. 10.
    Carpentier, J.: Contributions to the economic dispatch problem. Bull. Soc. Fr. Electr. 8(3), 431–447 (1962)Google Scholar
  11. 11.
    Chen, C., Atamtürk, A., Oren, S.S.: Bound tightening for the alternating current optimal power flow problem. IEEE Trans. Power Syst. PP(99), 1–8 (2015)Google Scholar
  12. 12.
    Chen, C., Atamtürk, A., Oren, S.S.: A spatial branch-and-cut method for nonconvex QCQP with bounded complex variables. Math. Program. 165, 549–577 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Coffrin, C., Gordon, D., Scott, P.: NESTA, The NICTA energy system test case archive. arXiv preprint arXiv:1411.0359 (2014)
  14. 14.
    Coffrin, C., Van Hentenryck, P.: A linear-programming approximation of AC power flows. INFORMS J. Comput. 26(4), 718–734 (2014)CrossRefGoogle Scholar
  15. 15.
    Coffrin, C., Hijazi, H.L., Van Hentenryck, P.: The QC relaxation: a theoretical and computational study on optimal power flow. IEEE Trans. Power Syst. 31(4), 3008–3018 (2016)CrossRefGoogle Scholar
  16. 16.
    Coffrin, C., Hijazi, H.L., Van Hentenryck, P.: Strengthening the SDP relaxation of AC power flows with convex envelopes, bound tightening, and valid inequalities. IEEE Trans. Power Syst. 32(5), 3549–3558 (2017)CrossRefGoogle Scholar
  17. 17.
    Coffrin, C., Van Hentenryck, P.: A linear-programming approximation of AC power flows. INFORMS J. Comput. 26(4), 718–734 (2014)CrossRefGoogle Scholar
  18. 18.
    Dey, S.S., Gupte, A.: Analysis of MILP techniques for the pooling problem. Oper. Res. 63(2), 412–427 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Frank, S., Steponavice, I., Rebennack, S.: Optimal power flow: a bibliographic survey I—formulations and deterministic methods. Energy Syst. 3(3), 221–258 (2012)CrossRefGoogle Scholar
  20. 20.
    Frank, S., Steponavice, I., Rebennack, S.: Optimal power flow: a bibliographic survey II—nondeterministic and hybrid methods. Energy Syst. 3(3), 259–289 (2012)CrossRefGoogle Scholar
  21. 21.
    Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion I: general framework. SIAM J. Optim. 11(3), 647–674 (2001)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gupte, A., Ahmed, S., Dey, S.S., Cheon, M.-S.: Relaxations and discretizations for the pooling problem. J. Glob. Optim. 67(3), 631–669 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hijazi, H., Coffrin, C., Van Hentenryck, P.: Polynomial SDP cuts for optimal power flow. In: 2016 Power Systems Computation Conference (PSCC), pp. 1–7 (June 2016)Google Scholar
  24. 24.
    Hijazi, H.L., Coffrin, C., Van Hentenryck, P.: Convex quadratic relaxations of mixed-integer nonlinear programs in power systems. Technical Report, NICTA, Canberra, ACT Australia (2013)Google Scholar
  25. 25.
    Hillestad, R.J., Jacobsen, S.E.: Linear programs with an additional reverse convex constraint. Appl. Math. Optim. 6(1), 257–269 (1980)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  27. 27.
    Jabr, R.A.: Radial distribution load flow using conic programming. IEEE Trans. Power Syst. 21(3), 1458–1459 (2006)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Jabr, R.A.: Optimal power flow using an extended conic quadratic formulation. IEEE Trans. Power Syst. 23(3), 1000–1008 (2008)CrossRefGoogle Scholar
  29. 29.
    Jabr, R.A.: Exploiting sparsity in SDP relaxations of the OPF problem. IEEE Trans. Power Syst. 27(2), 1138–1139 (2012)CrossRefGoogle Scholar
  30. 30.
    Jabr, R.A., Coonick, A.H., Cory, B.J.: A primal-dual interior point method for optimal power flow dispatching. IEEE Trans. Power Syst. 17(3), 654–662 (2002)CrossRefGoogle Scholar
  31. 31.
    Josz, C., Maeght, J., Panciatici, P., Gilbert, J.C.: Application of the moment-sos approach to global optimization of the OPF problem. IEEE Trans. Power Syst. 30(1), 463–470 (2015)CrossRefGoogle Scholar
  32. 32.
    Kocuk, B.: Global Optimization Methods for Optimal Power Flow and Transmission Switching Problems in Electric Power Systems. PhD thesis, Georgia Institute of Technology (2016)Google Scholar
  33. 33.
    Kocuk, B., Dey, S.S., Sun, X.A.: Strong SOCP relaxations for the optimal power flow problem. Oper. Res. 64(6), 1176–1196 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Kocuk, B., Dey, S.S., Sun, X.A.: Inexactness of SDP relaxation and valid inequalities for optimal power flow. IEEE Trans. Power Syst. 31(1), 642–651 (2016)CrossRefGoogle Scholar
  35. 35.
    Lavaei, J., Low, S.H.: Zero duality gap in optimal power flow problem. IEEE Trans. Power Syst. 27(1), 92–107 (2012)CrossRefGoogle Scholar
  36. 36.
    Madani, R., Ashraphijuo, M., Lavaei, J.: OPF Solver Guide (2014). http://ieor.berkeley.edu/~lavaei/Software.html
  37. 37.
    Madani, R., Ashraphijuo, M., Lavaei, J.: Promises of conic relaxation for contingency-constrained optimal power flow problem. Allerton (2014)Google Scholar
  38. 38.
    Madani, R., Sojoudi, S., Lavaei, J.: Convex relaxation for optimal power flowproblem: Mesh networks. In: Asilomar Conference on Signals, Systems, and Computers (ACSSC), pp. 1375–1382 (2013)Google Scholar
  39. 39.
    Madani, R., Sojoudi, S., Lavaei, J.: Convex relaxation for optimal power flow problem: Mesh networks. IEEE Trans. Power Syst. 30(1), 199–211 (2015)CrossRefGoogle Scholar
  40. 40.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems. Math. Program. 10(1), 147–175 (1976)CrossRefGoogle Scholar
  41. 41.
    Misener, R., Thompson, J.P., Floudas, C.A.: Apogee: global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes. Comput. Chem. Eng. 35(5), 876–892 (2011)CrossRefGoogle Scholar
  42. 42.
    Molzahn, D.K., Hiskens, I.A.: Sparsity-exploiting moment-based relaxations of the optimal power flow problem. IEEE Trans. Power Syst. 30(6), 3168–3180 (2015)CrossRefGoogle Scholar
  43. 43.
    Molzahn, D.K., Holzer, J.T., Lesieutre, B.C., DeMarco, C.L.: Implementation of a large-scale optimal power flow solver based on semidefinite programming. IEEE Trans. Power Syst. 28(4), 3987–3998 (2013)CrossRefGoogle Scholar
  44. 44.
    Momoh, J.A., El-Hawary, M.E., Adapa, R.: A review of selected optimal power flow literature to 1993 part I: nonlinear and quadratic programming approaches. IEEE Trans. Power Syst. 14(1), 96–104 (1999)CrossRefGoogle Scholar
  45. 45.
    Momoh, J.A., El-Hawary, M.E., Adapa, R.: A review of selected optimal power flow literature to 1993 part II: Newton, linear programming and interior point methods. IEEE Trans. Power Syst. 14(1), 105–111 (1999)CrossRefGoogle Scholar
  46. 46.
    MOSEK ApS. MOSEK Optimizer API for .NET manual. Version 8.1 (2017)Google Scholar
  47. 47.
    Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results. Math. Program. 95(2), 303–327 (2003)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Nesterov, Y., Wolkowicz, H., Ye, Y.: Semidefinite programming relaxations of nonconvex quadratic optimization. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming. International Series in Operations Research & Management Science, vol. 27, pp. 361–419. Springer, Boston (2000)CrossRefGoogle Scholar
  49. 49.
    Phan, D.T.: Lagrangian duality and branch-and-bound algorithms for optimal power flow. Oper. Res. 60(2), 275–285 (2012)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Qualizza, A., Belotti, P., Margot, F.: Linear programming relaxations of quadratically constrained quadratic programs. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, pp. 407–426. Springer, New York (2012)CrossRefGoogle Scholar
  51. 51.
    Sojoudi, S., Lavaei, J.: Physics of power networks makes hard optimization problems easy to solve. In: IEEE Power and Energy Society General Meeting, pp. 1–8 (2012)Google Scholar
  52. 52.
    Tawarmalani, M., Richard, JP.P.: Decomposition Techniques in Convexification of Inequalities. Technical Report. Working paper (2013)Google Scholar
  53. 53.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, vol. 65. Springer, New York (2002)zbMATHGoogle Scholar
  54. 54.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Taylor, J.A.: Convex Optimization of Power Systems. Cambridge University Press, Cambridge (2015)CrossRefGoogle Scholar
  56. 56.
    Torres, G.L., Quintana, V.H.: An interior-point method for nonlinear optimal power flow using voltage rectangular coordinates. IEEE Trans. Power Syst. 13(4), 1211–1218 (1998)CrossRefGoogle Scholar
  57. 57.
    Wang, H., Murillo-Sánchez, C.E., Zimmerman, R.D., Thomas, R.J.: On computational issues of market based optimal power flow. IEEE Trans. Power Syst. 22(3), 1185–1193 (2007)CrossRefGoogle Scholar
  58. 58.
    Wu, Y., Debs, A.S., Marsten, R.E.: A direct nonlinear predictor–corrector primal-dual interior point algorithm for optimal power flows. IEEE Trans. Power Syst. 9(2), 876–883 (1994)CrossRefGoogle Scholar
  59. 59.
    Zhang, B., Tse, D.: Geometry of feasible injection region of power networks. In: 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1508–1515 (Sept 2011)Google Scholar
  60. 60.
    Zimmerman, R.D., Murillo-Sanchez, C.E., Thomas, R.J.: MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26(1), 12–19 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  1. 1.Industrial Engineering ProgramSabancı UniversityIstanbulTurkey
  2. 2.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations