Energy Systems

, Volume 10, Issue 1, pp 59–94 | Cite as

Improved spectral clustering for multi-objective controlled islanding of power grid

  • Mikhail GoubkoEmail author
  • Vasily Ginz
Original Paper


We propose a two-step algorithm for optimal controlled islanding that partitions a power grid into islands of limited volume while optimizing several criteria: maximizing generator coherency inside islands, minimizing power flow disruption due to teared lines, and minimizing load shedding. Several spectral clusterings strategies are used in the first step to lower the problem dimension (taking into account coherency and disruption only), and CPLEX tools for the mixed-integer quadratic problem are employed in the second step to choose a balanced partition of the aggregated grid that minimizes a combination of coherency, disruption and load shedding. A greedy heuristics efficiently limits search space by generating the starting solution for the exact algorithm. Dimension of the second-step problem depends only on the desired number of islands K instead of the dimension of the original grid. The algorithm is tested on the standard systems with 118, 2383, and 9241 nodes showing high quality of partitions and competitive computation time.


Emergency control scheme Optimal partitioning of power grid Slow coherency Power flow disruption Load shedding 



The first author would like to thank support from Russian Foundation for Basic Research (Project 16-37-60102).


  1. 1.
    CASE2383WP: Power flow data for Polish system—winter 1999–2000 peak. (2014). Online Accessed 08 July 2016
  2. 2.
    Ahmed, S.S., Sarker, N.C., Khairuddin, A.B., Ghani, M.R.B.A., Ahmad, H.: A scheme for controlled islanding to prevent subsequent blackout. Power Syst. IEEE Trans. 18(1), 136–143 (2003)Google Scholar
  3. 3.
    Ames, B.P.: Guaranteed clustering and biclustering via semidefinite programming. Math. Progr. 147(1–2), 429–465 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ariff, M., Pal, B.C.: Coherency identification in interconnected power system: an independent component analysis approach. Power Syst. IEEE Trans. 28(2), 1747–1755 (2013)Google Scholar
  5. 5.
    Bell, N., Garland, M.: Efficient sparse matrix-vector multiplication on CUDA. Nvidia Technical Report NVR-2008-004, Nvidia Corporation (2008). Accessed 25 May 2017
  6. 6.
    Bie, T., Suykens, J., Moor, B.: Learning from general label constraints. In: Statistical Pattern Recognition, 2004. Proc. 2004 IAPR International Workshop on, Lisbon, Portugal, IAPR (2004)Google Scholar
  7. 7.
    Butyrin, P., Vaskovskaya, T.: Diagnostika elektricheskikh tsepey po chastyam: Teoreticheskiye osnovy i komp’yuterniy praktikum. MPEI Publishing House, Moscow (2003)Google Scholar
  8. 8.
    Chow, J., Kokotovic, P.: Time scale modeling of dynamic networks with sparse and weak connections. In: Singular perturbations and asymptotic analysis in control systems. Lecture Notes in Control and Information Science, vol. 90, pp. 310–353. Springer, New York (1987)Google Scholar
  9. 9.
    Chow, J.H., Galarza, R., Accari, P., Price, W.W.: Inertial and slow coherency aggregation algorithms for power system dynamic model reduction. Power Syst. IEEE Trans. 10(2), 680–685 (1995)Google Scholar
  10. 10.
    Chow, J.H., et al.: A nodal aggregation algorithm for linearized two-time-scale power networks. In: Circuits and Systems, 1988., IEEE International Symposium on, pp. 669–672. IEEE (1988)Google Scholar
  11. 11.
    Chow, J.H., et al.: Aggregation properties of linearized two-time-scale power networks. Circuit Syst. IEEE Trans. 38(7), 720–730 (1991)Google Scholar
  12. 12.
    Christie, R.: 118 Bus Power Flow Test Case. (1993). Online; Accessed 08 July 2016
  13. 13.
    Cotilla-Sanchez, E., Hines, P.D., Barrows, C., Blumsack, S., Patel, M.: Multi-attribute partitioning of power networks based on electrical distance. Power Syst. IEEE Trans. 28(4), 4979–4987 (2013)Google Scholar
  14. 14.
    Defays, D.: An efficient algorithm for a complete link method. Comput. J. 20(4), 364–366 (1977)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Demetriou, P., Quirós-Tortós, J., Kyriakides, E., Terzija, V.: On implementing a spectral clustering controlled islanding algorithm in real power systems. In: PowerTech (POWERTECH), 2013 IEEE Grenoble, pp. 1–6. IEEE (2013)Google Scholar
  16. 16.
    Ding, L., Gonzalez-Longatt, F.M., Wall, P., Terzija, V.: Two-step spectral clustering controlled islanding algorithm. Power Syst. IEEE Trans. 28(1), 75–84 (2013)Google Scholar
  17. 17.
    Ding, L., Wall, P., Terzija, V.: Constrained spectral clustering based controlled islanding. Int. J. Electr. Power Energy Syst. 63, 687–694 (2014)Google Scholar
  18. 18.
    Ding, T., Sun, K., Huang, C., Bie, Z., Li, F.: Mixed-integer linear programming-based splitting strategies for power system islanding operation considering network connectivity IEEE Syst. (99), 1–10 (2015)Google Scholar
  19. 19.
    El-Werfelli, M., Brooks, J., Dunn, R.: Controlled islanding scheme for power systems. In: Universities Power Engineering Conference, 2008. UPEC 2008. 43rd International, pp. 1–6. IEEE (2008)Google Scholar
  20. 20.
    Fan, N., Izraelevitz, D., Pan, F., Pardalos, P.M., Wang, J.: A mixed integer programming approach for optimal power grid intentional islanding. Energy Syst. 3(1), 77–93 (2012)Google Scholar
  21. 21.
    Fan, N., Pardalos, P.M.: Multi-way clustering and biclustering by the ratio cut and normalized cut in graphs. J. Comb. Optim. 23(2), 224–251 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Fliscounakis, S., Panciatici, P., Capitanescu, F., Wehenkel, L.: Contingency ranking with respect to overloads in very large power systems taking into account uncertainty, preventive, and corrective actions. Power Syst. IEEE Trans. 28(4), 4909–4917 (2013)Google Scholar
  23. 23.
    Garey, M., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  24. 24.
    Golari, M., Fan, N., Wang, J.: Two-stage stochastic optimal islanding operations under severe multiple contingencies in power grids. Electr. Power Syst. Res. 114, 68–77 (2014)Google Scholar
  25. 25.
    Golari, M., Fan, N., Wang, J.: Large-scale stochastic power grid islanding operations by line switching and controlled load shedding. Energy Syst. (2016). doi: 10.1007/s12667-016-0215-7
  26. 26.
    Goldschmidt, O., Hochbaum, D.S.: Polynomial algorithm for the k-cut problem. In: Proceedings of the 29th Annual Symposium on Foundations of Computer Science (SFCS ’88). pp. 444–451 (1988)Google Scholar
  27. 27.
    Hamon, C., Shayesteh, E., Amelin, M., Söder, L.: Two partitioning methods for multi-area studies in large power systems. Int. Trans. Electr. Energy Syst. 25(4), 648–660 (2015)Google Scholar
  28. 28.
    Juarez, C., Messina, A., Castellanos, R., Espinosa-Perez, G.: Characterization of multimachine system behavior using a hierarchical trajectory cluster analysis. Power Syst. IEEE Trans. 26(3), 972–981 (2011)Google Scholar
  29. 29.
    Juncheng, S., Jiping, J., Famin, C., Lei, D.: Research on power system controlled islanding. IPSI BgD Trans. 11(2), 1–5 (2015)Google Scholar
  30. 30.
    Kundur, P., Paserba, J., Ajjarapu, V., Andersson, G., Bose, A., Canizares, C., Hatziargyriou, N., Hill, D., Stankovic, A., Taylor, C., et al.: Definition and classification of power system stability ieee/cigre joint task force on stability terms and definitions. Power Syst. IEEE Trans. 19(3), 1387–1401 (2004)Google Scholar
  31. 31.
    Lessig, C.: Eigenvalue computation with CUDA. In: NVIDIA techreport (2007). Accessed 25 May 2017
  32. 32.
    Li, H., Rosenwald, G.W., Jung, J., Liu, C.C.: Strategic power infrastructure defense. Proc. IEEE 93(5), 918–933 (2005)Google Scholar
  33. 33.
    Lütkepohl, H.: Handbook of matrices. Wiley, Amsterdam 1996Google Scholar
  34. 34.
    Ng, A.Y., Jordan, M.I., Weiss, Y., et al.: On spectral clustering: analysis and an algorithm. Adv. Neural Inf. Process. Syst. 2, 849–856 (2002)Google Scholar
  35. 35.
    Pahwa, S., Youssef, M., Schumm, P., Scoglio, C., Schulz, N.: Optimal intentional islanding to enhance the robustness of power grid networks. Physica A 392(17), 3741–3754 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Peiravi, A., Ildarabadi, R.: A fast algorithm for intentional islanding of power systems using the multilevel kernel k-means approach. J. Appl. Sci. 9(12), 2247–2255 (2009)Google Scholar
  37. 37.
    Quirós-Tortós, J., Sánchez-García, R., Brodzki, J., Bialek, J., Terzija, V.: Constrained spectral clustering-based methodology for intentional controlled islanding of large-scale power systems. Gener. Trans. Distrib. IET 9(1), 31–42 (2015)Google Scholar
  38. 38.
    Quirós-Tortós, J., Wall, P., Ding, L., Terzija, V.: Determination of sectionalising strategies for parallel power system restoration: a spectral clustering-based methodology. Electr. Power Syst. Res. 116, 381–390 (2014)Google Scholar
  39. 39.
    Sánchez-García, R.J., Fennelly, M., Norris, S., Wright, N., Niblo, G., Brodzki, J., Bialek, J.W.: Hierarchical spectral clustering of power grids. Power Syst. IEEE Trans. 29(5), 2229–2237 (2014)Google Scholar
  40. 40.
    Song, J., Cotilla-Sanchez, E., Ghanavati, G., Hines, P.D.H.: Dynamic modeling of cascading failure in power systems. Power Syst. IEEE Trans. 31(3), 2085–2095 (2016)Google Scholar
  41. 41.
    Sun, K., Zheng, D.Z., Lu, Q.: Splitting strategies for islanding operation of large-scale power systems using obdd-based methods. Power Syst. IEEE Trans. 18(2), 912–923 (2003)Google Scholar
  42. 42.
    Trodden, P., Bukhsh, W., Grothey, A., McKinnon, K.: Milp formulation for controlled islanding of power networks. Int. J. Electr. Power Energy Syst. 45(1), 501–508 (2013)Google Scholar
  43. 43.
    Trodden, P.A., Bukhsh, W.A., Grothey, A., McKinnon, K.I.: Optimization-based islanding of power networks using piecewise linear ac power flow. Power Syst. IEEE Trans. 29(3), 1212–1220 (2014)Google Scholar
  44. 44.
    Vittal, V., Kliemann, W., Ni, Y., Chapman, D., Silk, A., Sobajic, D.: Determination of generator groupings for an islanding scheme in the manitoba hydro system using the method of normal forms. Power Syst. IEEE Trans. 13(4), 1345–1351 (1998)Google Scholar
  45. 45.
    Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)MathSciNetGoogle Scholar
  46. 46.
    Wang, X., Vittal, V.: System islanding using minimal cutsets with minimum net flow. In: Power Systems Conference and Exposition, 2004. IEEE PES, pp. 379–384. IEEE (2004)Google Scholar
  47. 47.
    Ward Jr., J.H.: Hierarchical grouping to optimize an objective function. J. Am. Stat. Assoc. 58(301), 236–244 (1963)MathSciNetGoogle Scholar
  48. 48.
    Xu, G., Vittal, V.: Slow coherency based cutset determination algorithm for large power systems. Power Syst. IEEE Trans. 25(2), 877–884 (2010)Google Scholar
  49. 49.
    Yang, B., Vittal, V., Heydt, G.T., Sen, A.: A novel slow coherency based graph theoretic islanding strategy. In: Power Engineering Society General Meeting, 2007. IEEE, pp. 1–7. IEEE (2007)Google Scholar
  50. 50.
    You, H., Vittal, V., Wang, X.: Slow coherency-based islanding. Power Syst. IEEE Trans. 19(1), 483–491 (2004)Google Scholar
  51. 51.
    Yusof, S., Rogers, G., Alden, R.: Slow coherency based network partitioning including load buses. Power Syst. IEEE Trans. 8(3), 1375–1382 (1993)Google Scholar
  52. 52.
    Zhao, Q., Sun, K., Zheng, D.Z., Ma, J., Lu, Q.: A study of system splitting strategies for island operation of power system: a two-phase method based on obdds. Power Syst. IEEE Trans. 18(4), 1556–1565 (2003)Google Scholar
  53. 53.
    Zimmerman, R.D., Murillo-Sánchez, C.E., Thomas, R.J.: Matpower: steady-state operations, planning, and analysis tools for power systems research and education. Power Syst. IEEE Trans. 26(1), 12–19 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.V.A. Trapeznikov Institute of Control Sciences of Russian Academy of SciencesMoscowRussia
  2. 2.Skoltech Center for Energy Systems, Skolkovo Innovation CenterMoscowRussia

Personalised recommendations