An Energy-Based Centrality for Electrical Networks

Ruiyuan Kong, Congying Han, Tiande Guo, Wei Pei

College of Humanities and Social Sciences, University of Chinese Academy of Sciences, Beijing, China.
Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing, China.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China.

DOI: 10.4236/epe.2013.54B115                

Abstract

We present an energy-based method to estimate centrality in electrical networks. Here the energy between a pair of vertices denotes by the effective resistance between them. If there is only one generation and one load, then the centrality of an edge in our method is the difference between the energy of network after deleting the edge and that of the original network. Compared with the local current-flow betweenness on the IEEE 14-bus system, we have an interesting discovery that our proposed centrality is closely related to it in the sense of that the significance of edges under the two measures are very similar.

Keywords

Centrality; Energy; Effective Resistance; Current-flow Betweenness

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. Albert, I. Albert and G. L. Nakarado, “Structural Vulnerability of the North American Power Grid,” Physical Review E, Vol. 69, No. 2, 2004, pp. 1-4. doi:10.1103/PhysRevE.69.025103
[2]	B. A. Carreras, V. E. Lynch, I. Dobson and D. E. Newman, “Critical Points and Transitions in an Electric Power Transmission Model for Cascading Failure Blackouts,” Chaos, Vol. 12, No. 4, 2002, pp. 985-994. doi: 10.1063/1.1505810
[3]	P. Crucittia, V. Latorab and M. Marchioric, “A Topological Analysis of the Italian Electric Power Grid,” Physica A: Statistical Me-chanics and its Applications, Vol. 338, No. 1-2, 2004, pp. 92-97. doi:10.1016/j.bbr.2011.03.031
[4]	M. Rosas-Casals, S. Valverde and R. V. Solé, “Topological Vulnerability of the European Power Grid Under Errors and Attacks,” International Journal of Bifurcation and Chaos in Ap-plied Sciences and Engineering, Vol. 17, No. 7, 2007, pp. 2465-2475. doi: 10.1142/S0218127407018531
[5]	M. E. J. Newman, “A Measure of Betweenness Centrality Based on Random Walks,” Social Networks, Vol. 27, No. 1, 2005, pp. 39-54. doi:10.1016/j.socnet.2004.11.009
[6]	U. Brandes and D. Fleischer, “Centrality Measures Based on Current Flow,” Proceedings of the 22nd Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Springer, Berlin, Vol. 3404, 2005, pp. 533-544. doi:10.1007/978-3-540-31856-9_44
[7]	K. A. Stephenson and M. Zelen, “Rethinking Centrality: Methods and Examples,” Social Networks, Vol. 11, No. 1, 1989, pp. 1-37. doi:10.1016/0378-8733(89)90016-6
[8]	P. Hines and S. Blumsack, “A Centrality Measure for Electrical Networks,” Proceedings of the 41st Hawaii International Conference on System Sciences, Hawaii, 7-10 January 2008, p.185. doi:10.1109/HICSS.2008.5
[9]	Z. H. Wang, A. Scaglione and R. J. Thomas, “Electrical Cen-trality Measures for Power Grids,” Control and Optimi-zation Methods for Electric Smart Grids, Power Elec-tronics and Power System, Springer, New York, Vol.3, 2012, pp. 239-255. doi:10.1007/978-1-4614-1605-0_12
[10]	E. Bompard, D. Wu and F. Xue, “The Concept of Betweenness in the Analysis of Power Grid Vulnerability,” Complexity in Engineering, Rome, 22-24 February, 2010, pp. 52-54. doi:10.1109/COMPENG.2010.10
[11]	D. J. Klein and M. Randic, “Resistance Distance,” Journal of Mathematical Chemistry, Vol. 12, No. 1, 1993, pp.81-95. doi:10.1007/BF01164627
[12]	K. C. Das, A. D. Gungor and A. S. Cevik, “ On Kirchhoff Index and Resistance-Distance Energy of a Graph,” MATCH Communications in Mathematical and in Computer Chemistry, Vol. 67, No. 2, 2012, pp. 541-556.
[13]	B. Zhou and N. Trinajstic, “On Resistance-Distance and Kirchhoff Index,” Journal of Mathematical Chemistry, Vol. 46, No. 1, 2009, pp. 283-289. doi:10.1007/s10910-008-9459-3
[14]	L. C. Freeman , “A Set of Measures of Centrality Based On Betweenness,” Sociometry, Vol. 40, No. 1, 1977, pp. 35-41.
[15]	M. Girvan and M. E. J. Newman, “Community Structure in Social and Biological Networks,” Proceedings of the National Academy of Sciences, Vol. 99, No. 12, 2002, pp. 7821-7826. doi:10.1073/pnas.122653799
[16]	B. Bollobás, “Modern Graph Theory,”3rd Edition, Springer-Verlag, New York, 2003.
[17]	R. Bapat, I. Gutman and W. Xiao, “A Simple Method for Computing Resistance Distance,” Zeitschrift für Naturforschung, Vol. 58a, No. 9-10, 2003, pp. 494-498.
[18]	E. Zio , R. Piccinelli and M. Delfanti , “Application of the Load Flow and Random Flow Models for the Analysis of Power Transmission Networks,” Reliability Engineering and System Safety, Vol. 103, 2012, pp. 102-109. doi:10.1016/j.ress.2012.02.005