Your browser does not support JavaScript!
http://iet.metastore.ingenta.com
1887

access icon free Locating all real solutions of power flow equations: a convex optimisation-based method

  • Author(s): Bin Liu 1, 2 ; Wei Wei 1 ; Feng Liu 1
    • View affiliations
  • Source: Volume 12, Issue 10, 29 May 2018, p. 2273 – 2279
    DOI:  10.1049/iet-gtd.2017.1870 , Print ISSN 1751-8687, Online ISSN 1751-8695
© The Institution of Engineering and Technology
Received 28/11/2017, Accepted 13/02/2018, Revised 06/02/2018, Published 15/02/2018

This study proposes a convex optimisation-based method that either locates all real roots of a set of power flow (PF) equations or declares no real solution exists in the given area. In the proposed method, solving the PF equations is reformulated as a global optimisation problem (GPF for short) that minimises the sum of slack variables. All the global minima of GPF with a zero objective value have a one-to-one correspondence to the real roots of PF equations. By solving a relaxed version of GPF over a hypercube, if the optimal value is strictly positive, there is no solution in this area and the hypercube is discarded. Otherwise the hypercube is further divided into smaller ones. This procedure repeats recursively until all the real roots are located in small enough hypercubes through the successive refinement of the feasible region embedded in a bisection paradigm. This method is desired in a number of power system security assessment applications, for instance, the transient stability analysis as well as voltage stability analysis, where the closest unstable equilibrium and all Type I unstable equilibrium is required, respectively. The effectiveness of the proposed method is verified by analysing several test systems.

Inspec keywords: convex programming; load flow; power system transient stability; power system security

Other keywords: transient stability analysis; Type I unstable equilibrium; PF equation; GPF; bisection paradigm; voltage stability analysis; power flow equation; global optimisation problem; convex optimisation-based method; power system security assessment application

Subjects: Optimisation techniques; Power system control

References

    1. 1)
      • 13. Lee, J., Chiang, H.D.: ‘A singular fixed-point homotopy method to locate the closest unstable equilibrium point for transient stability region estimate’, IEEE Trans. Circuits Syst. II Express Briefs, 2004, 51, (4), pp. 185–189.
    2. 2)
      • 21. Molzahn, D., Lesieutre, B., Chen, H.: ‘Counterexample to a continuation-based algorithm for finding all power flow solutions’, IEEE Trans. Power Syst., 2013, 28, (1), pp. 564–565.
    3. 3)
      • 22. Metha, D., Nguyen, H., Turitsyn, K.: ‘Numerical polynomial homotopy continuation method to locate all the power flow solutions’, IET Gener. Transm. Distrib., 2016, 10, (12), pp. 2972–2980.
    4. 4)
      • 15. Iba, K., Suzuki, H., Egawa, M., et al: ‘A method for finding a pair of multiple load flow solutions in bulk power systems’, IEEE Trans. Power Syst., 1990, 5, (2), pp. 582–591.
    5. 5)
      • 32. Sherali, H., Adams, W.: ‘A reformulation-linearization technique for solving discrete and continuous nonconvex problems’ (Springer Science & Business Media, Berlin, Germany, 2013).
    6. 6)
      • 5. Wannoi, C., Khumdee, A., Wannoi, N., et al: ‘An optimum technique for renewable power generations integration to power system using repeated power flow technique considering voltage stability limit’, Procedia Comput. Sci., 2016, 86, pp. 357–360.
    7. 7)
      • 11. Liu, C.W., Thorp, J.S.: ‘A novel method to compute the closest unstable equilibrium point for transient stability region estimate in power systems’, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 1997, 44, (7), pp. 630–635.
    8. 8)
      • 27. Lavaei, J., Tse, D., Zhang, B.: ‘Geometry of power flows and optimization in distribution networks’, IEEE Trans. Power Syst., 2014, 29, (2), pp. 572–583.
    9. 9)
      • 17. Liu, C., Chang, C., Jiang, J., et al: ‘Toward a CPFLOW-based algorithm to compute all the type-1 load-flow solutions in electric power systems’, IEEE Trans. Circuits Syst. I Regul. Pap., 2005, 52, (3), pp. 625–630.
    10. 10)
      • 30. Maranas, C., Floudas, C.: ‘Finding all solutions of nonlinearly constrained systems of equations’, J. Global Optimiz., 1995, 7, (2), pp. 143–182.
    11. 11)
      • 34. Kocuk, B., Dey, S., Sun, X.: ‘Strong SOCP relaxations for the optimal power flow problem’. School of Industrial and Systems Engineering, Georgia Institute of Technology, 2013.
    12. 12)
      • 33. Burer, S., Saxena, A.: ‘The MILP road to MIQCP’, in ‘Mixed integer nonlinear programming’ (Springer, New York, NY, USA, 2012), pp. 373–405.
    13. 13)
      • 7. Tinney, W.F., Hart, C.E.: ‘Power flow solution by newton's method’, IEEE Trans. Power Appar. Syst., 1967, 86, (11), pp. 1449–1460.
    14. 14)
      • 20. Salam, F., Ni, L., Guo, S., et al: ‘Parallel processing for the load flow of power systems: the approach and applications’. 28th IEEE Conf. Decision and Control, 1989.
    15. 15)
      • 19. Ma, W., Thorp, J.S.: ‘An efficient algorithm to locate all the load flow solutions’, IEEE Trans. Power Syst., 1993, 8, (3), pp. 1077–1083.
    16. 16)
      • 28. Madani, R., Sojoudi, S., Lavaei, J.: ‘Convex relaxation for optimal power flow problem: mesh networks’, IEEE Trans. Power Syst., 2015, 30, (1), pp. 199–211.
    17. 17)
      • 31. Hiskens, I., Davy, R.J.: ‘Exploring the power flow solution space boundary’, IEEE Trans. Power Syst., 2001, 16, (3), pp. 389–395.
    18. 18)
      • 25. Low, S.H.: ‘Convex relaxation of optimal power flow part I: formulations and equivalence’, IEEE Trans. Control Syst. Technol., 2014, 1, (1), pp. 15–27.
    19. 19)
      • 26. Low, S.H.: ‘Convex relaxation of optimal power flow part II: exactness’, IEEE Trans. Control Syst. Technol., 2014, 1, (2), pp. 177–189.
    20. 20)
      • 1. Shakarami, M.R., Beiranvand, H., Beiranvand, A., et al: ‘A recursive power flow method for radial distribution networks: analysis, solvability and convergence’, Int. J. Electr. Power Energy Syst., 2017, 86, pp. 71–80.
    21. 21)
      • 9. Stott, B., Alsac, O.: ‘Fast decoupled load flow’, IEEE Trans. Power Appar. Syst., 1974, 93, (3), pp. 859–869.
    22. 22)
      • 35. 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., 2011, 26, (1), pp. 12–19.
    23. 23)
      • 14. Tamura, Y., Mori, H., Iwamoto, S.: ‘Relationship between voltage instability and multiple load flow solutions in electric power systems’, IEEE Trans. Power Appar. Syst., 1983, 102, (5), pp. 1115–1125.
    24. 24)
      • 6. Yang, X., Zhou, X.: ‘Application of asymptotic numerical method with homotopy techniques to power flow problems’. Int. J. Electr. Power Energy Syst., 2014, 57, pp. 375–383.
    25. 25)
      • 23. Chow, S.N., Mallet-Paret, J., Yorke, J.A.: ‘Finding zeroes of maps: homotopy methods that are constructive with probability one’, Math. Comput., 1978, 32, (143), pp. 887–899.
    26. 26)
      • 18. Pereira, L.E.S., da Costa, V.M.: ‘An efficient starting process for calculating interval power flow solutions at maximum loading point under load and line data uncertainties’, Int. J. Electr. Power Energy Syst., 2016, 80, pp. 91–95.
    27. 27)
      • 4. Yang, N.-C., Tseng, W.-C.: ‘Adaptive three-phase power-flow solutions for smart grids with plug-in hybrid electric vehicles’, Int. J. Electr. Power Energy Syst., 2015, 64, pp. 1166–1175.
    28. 28)
      • 8. Kumar, R.S., Chandrasekharan, E.: ‘A parallel distributed computing framework for Newton-Raphson load flow analysis of large interconnected power systems’, Int. J. Electr. Power Energy Syst., 2015, 73, pp. 1–6.
    29. 29)
      • 12. Lee, J.: ‘Dynamic gradient approaches to compute the closest unstable equilibrium point for stability region estimate and their computational limitations’, IEEE Trans. Autom. Control, 2003, 48, (2), pp. 321–324.
    30. 30)
      • 2. Nagrial, M.H., Solimant, H.M.: ‘Power flow solution using the modified quasilinearization method’, Comput. Electr. Eng., 1984, 11, (4), pp. 213–217.
    31. 31)
      • 24. Lavaei, J., Low, S.H.: ‘Zero duality gap in optimal power flow problem’, IEEE Trans. Power Syst., 2012, 27, (1), pp. 92–107.
    32. 32)
      • 3. Ahmed, S.S., Rahman, M.M.: ‘Use of base case jacobians in power flow solution under topological changes’, Comput. Electr. Eng., 2002, 26, (1), pp. 17–23.
    33. 33)
      • 16. Chiang, H.D., Dobson, I., Thomax, R.J., et al: ‘On voltage collapse in electric power systems’, IEEE trans. Power Syst., 1990, 5, (2), pp. 601–611.
    34. 34)
      • 10. Chiang, H.D., Thorp, J. S.: ‘The closest unstable equilibrium point method for power system dynamic security assessment’, IEEE Trans. Circuits Syst., 1989, 36, (9), pp. 1187–1200.
    35. 35)
      • 29. Jubril, A., Olaniyan, O., Komolafe, O., et al: ‘Economic-emission dispatch problem: a semi-definite programming approach’, Appl. Energy, 2014, 134, pp. 446–455.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-gtd.2017.1870
Loading

Related content

content/journals/10.1049/iet-gtd.2017.1870
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading
Print this page
This is a required field
Please enter a valid email address