Power Network Equivalents: A Network Science Based K-Means Clustering Method Integrated with Silhouette Analysis

  • D.  SharmaEmail author
  • K. Thulasiraman
  • D.  Wu
  • J. N. Jiang
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 689)


Network equivalencing is useful for electrical market analysis in large interconnected power networks. Generation shift factor (GSF)-based bus clustering methods have been used for network equivalence to analyze the power transactions across the interconnections. However, the GSF-based methods are sensitive to location changes of slack bus since GSFs depend on the location of slack bus, which may increase the complexity of market analysis. In this paper, we present a new bus clustering method based on Average Electrical Distance (AED), which is independent of location changes of slack bus. AED is based on the concept of electrical distance introduced in [11] and pursued later for applications in social and complex networks analysis [12, 20, 22]. AED represents the average electrical distances from a bus to the nodes of tie-line of interest. In the proposed method, AEDs are used to group buses into clusters for network equivalence. To enhance the efficiency of clustering buses, AED-based improved k-means algorithm is incorporated into the proposed method. Also, silhouette analysis technique is combined with the algorithm to assign appropriate number of clusters to efficiently group buses. The efficacy of the proposed AED-based method is demonstrated on the IEEE 39-bus system by comparing it with the existing GSF-based methods.


  1. 1.
    Cheng, X., Overbye, T.: PTDF-based power system equivalents. IEEE Trans. Power App. Syst. 20(4) (2005)Google Scholar
  2. 2.
    Coppersmith, D., Doyle, P.G., Raghavan, P., Snir, M.: Random walks on a 129 weighted graphs and applications to online algorithms. In: Proceedings of the 22nd symposium on the Theory of Computing, pp. 369–378 (1990)Google Scholar
  3. 3.
    Shi, D., Shawhan, D.L., Li, N., Tylavsky, D.J.: Optimal generation investment planning: Pt. 1: Network equivalents. In: Proceedings of the 44th North American Power Symposium Champaign, IL, USA, pp. 1–6 (2012)Google Scholar
  4. 4.
    Dimo, P.: Nodal Analysis of Power Systems. Kent, UK (1975)Google Scholar
  5. 5.
    Doyle, P., Snell, J.: Random Walks and Electrical Networks. The Mathematical Association of America, Washington (1984)Google Scholar
  6. 6.
    Duran, H., Arvanitidis, N.: Simplification for area security analysis: A new look at equivalencing. IEEE Trans. Power App. Syst. PAS-91(2) (1972)Google Scholar
  7. 7.
    Housos, E.C., Irisarri, G., Porter, R.M., Sasson, A.M.: Steady state network equivalents for power system planning applications. IEEE Trans. Power App. Syst. PAS-99(6) (1980)Google Scholar
  8. 8.
    Faber, V.: Clustering and the continuous k-means algorithm. Los Alamos Sci. 22, 138–144 (1994)Google Scholar
  9. 9.
    Wang, H., Sanchez, C.E.M., Zimmerman, R., Thomas, R.: On computational issues of market-based optimal power flow. IEEE Trans. Power Syst. 22(3), 1185–1193 (2007)Google Scholar
  10. 10.
    Hogan, W.: A market power model with strategic interaction in electricity networks. Energy J. 18, 107–141 (1997)CrossRefGoogle Scholar
  11. 11.
    Klein, D.J., Randic, M.: Resistance distance. J. Math. Chem., 81–95 (1993)Google Scholar
  12. 12.
    Newman, M.: A measure of betweenness centrality based on random walks. Soc. Netw. 27(1) (2005)Google Scholar
  13. 13.
    Oh, H.: A new network reduction methodology for power system planning studies. IEEE Trans. Power App. Syst. 25(2) (2010)Google Scholar
  14. 14.
    Rousseeuw, P.J.: Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math. (1) (1987)Google Scholar
  15. 15.
    Srinivasan, S., Subramanian, V.N., Thulasiraman, K.: Application of equivalence technique in linear graph theory to reduction process in a power system. J. Inst. Eng. (India) (12) (1966)Google Scholar
  16. 16.
    Shi, D., Tylavsky, D.J.: A novel bus-aggregation-based structure preserving power system equivalent. IEEE Trans. Power App. Syst. 30(4) (2015)Google Scholar
  17. 17.
    Srinivasan, S., Sujeer, V.N.: A new equivalence technique in linear graph theory. J. Inst. Eng. (India) (12) (1964)Google Scholar
  18. 18.
    Swamy, M., Thulasiraman, K.: Graphs, Networks and Algorithms. Wiley Interscience (1981)Google Scholar
  19. 19.
    Tinney, W., Bright, J.M.: Adaptive reductions for power flow equivalents. IEEE Trans. Power App. Syst. 2 (1987)Google Scholar
  20. 20.
    Tizghadam, A., Leon-Garcia, A.: Autonomic traffic engineering for network robustness. IEEE J. Sel. Areas Commun. Spec. Issue Auton. Commun. 28(1), 39–50 (2010)Google Scholar
  21. 21.
    US-EIA: U.S. Energy Information Administration—energy explained (1999).
  22. 22.
    Chellappan, V., Rajmohan, K.S., Krithivasan, K.: A centrality entropy maximization problem in shortest path routing networks. Comp. Netw. 104, 1–15 (2016)Google Scholar
  23. 23.
    Ward, J.: Equivalent circuits for power flow studies. AIEE Trans. Power App. Syst. 68, 373–382 (1949)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • D.  Sharma
    • 1
    Email author
  • K. Thulasiraman
    • 1
  • D.  Wu
    • 2
  • J. N. Jiang
    • 1
  1. 1.University of OklahomaNormanUSA
  2. 2.North Dakota State UniversityFargoUSA

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