7 Power Grid Network Analysis for Smart
7.6 TOPOLOGICAL ANALYSIS FOR THE DISTRIBUTION NETWORKS
7.6.3 T he pMf of The n ode d egrees in The s aMple 396-n ode Mv d isTribuTion n eTWork
The average node degree of a power grid transmission network tends to be quite low and does not scale as the network size increases. The topology of a transmission net-work has salient small-world properties, since it features a much shorter average path length (in hops) and a much higher clustering coefficient than that of Erdös–Rényi ran-dom graphs with the same network size and sparsity. While small-world features have been recently confirmed for the HV transmission network, the sample MV network used here implies that a power grid distribution network has a very different kind of topology from that of an HV network, and obviously it is not a small-world topology.
The node degree distribution of the 396-node MV distribution network, shown in Figure 7.10, shows that the maximum node degree in the network equals 4, which is much smaller than that found in the transmission side of the grid, where maximum nodal degrees of 20 or 30 can be found; and about 16% of the nodes connect to only one branch, 60% to two branches, 22% to three branches, and only 2% to four branches.
Figure 7.4e depicts the network’s spectral density, which is a normalized spectral distribution of the eigenvalues of its adjacency matrix. The spectrum of an Erdös–
Rényi random graph network, which has uncorrelated node degrees, converges to a semicircular distribution (see the semicircle dotted line on the background in Figure 7.4e). The spectra of real-world networks have specific features depending on the details of the corresponding models. In particular, scale-free graphs develop
Power Grid Network Analysis for Smart Grid Applications 175
a triangle-like spectral density with a power-law tail, whereas a small-world net-work has a complex spectral density consisting of several sharp peaks. The plot in Figure 7.4e indicates that the sample MV distribution network is neither a scale-free network nor a small-world network.
The branch lengths in the MV distribution network are also analyzed. The cor-responding PMF is shown in Figure 7.11. It indicates that most of the branches are shorter than 1067 m (3500 ft) and the branch length distribution has an exponential tail, with only a very small number of extremely long branches.
1 2 3 4 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
PMF (k)
k-node degree
FIGURE 7.10 PMF of the node degrees in the sample 396-node MV distribution network.
0 1 2 3
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Line length (103 m)
Probability mass function
0 1 2 3
10−3 10−2 10−1 100
Line length (103 m)
FIGURE 7.11 PMF of the line length in the sample 396-node MV distribution network:
(left) probability versus length; (right) log-probability versus length, where the existence of an exponential trend in the tail is clearly visible.
176 Smart Grids
7.6.4 lv disTribuTion neTWork
It is difficult to obtain example data about LV distribution network topologies.
Generally speaking, an LV distribution network is radial, and has a similar network topology to an MV distribution network, except that it may have more nodes with shorter branch length.
7.7 SUMMARY
This chapter presents a comprehensive study on both the topological and electrical characteristics of electric power grids. It finds that the HV transmission network of the power grid is sparsely connected, with a low average node degree that does not scale with the network size; the power grid topology manifests obvious small-world properties, but its small-world rewires are not independent as in the Watts–Strogatz small-world model; instead, long hauls appear among clusters of nodes. This chapter provides evidence that the nodal degrees follow a mixed distribution that comes from the sum of a truncated geometric random variable and an irregular discrete random variable. It also proposes a method to estimate the distribution parameters by analyzing the poles and zeros of the average PGF. The chapter studies the graph spectral density and shows that the power grid network takes on a distinctive pattern for its graph spectral density. Study of the algebraic connectivity of power networks has shown that power networks are exceptionally well connected given their spar-sity, featuring a better scaling law than the Watts–Strogatz small-world graphs. It is shown that the distribution of line impedances is heavy-tailed and can be captured quite accurately by a clipped DPLN distribution. The chapter introduces a novel random topology power grid model, RT-nested-Smallworld, intending to capture the above observed characteristics.
Finally, the chapter extends the same topological analysis to the distribution side of the power grid and reports the results based on a sample 396-node MV distribu-tion network that comes from a real-world US distribudistribu-tion utility mainly located in a rural area.
REFERENCES
1. D. J. Watts and S. H. Strogatz, Collective dynamics of “small-world” networks, Nature, 393, 393–440, 1998.
2. R. Solé, M. Rosas-Casals, B. Corominas-Murtra, and S. Valverde, Robustness of the European power grids under international attack, Physical Review E, 026102, 2008.
3. Z. Wang, A. Scaglione, and R. J. Thomas, The node degree distribution in power grid and its topology robustness under random and selective node removals, in Proceedings of the 1st IEEE International Workshop on Smart Grid Communications (ICC’10SGComm), 23–27 May, Cape Town, South Africa, 2010.
4. R. Albert, I. Albert, and G. L. Nakarado, Structural vulnerability of the North American power grid, Physical Review E, 69, 1–4, 2004.
5. P. Crucittia, V. Latorab, and M. Marchioric, A topological analysis of the Italian electric power grid, Physica A, 338, 92–97, 2004.
6. E. W. Dijkstra, A note on two problems in connexion with graphs, Numerische Mathematik, 1, 269–271, 1959.
Power Grid Network Analysis for Smart Grid Applications 177
7. J. L. Rodgers and W. A. Nicewander, Thirteen ways to look at the correlation coefficient, The American Statistician, 42(1), 59–66, 1988.
8. M. Newman, The structure and function of complex networks, SIAM Review, 45, 167–256, 2003.
9. D. E. Whitney and D. Alderson, Are technological and social networks really differ-ent, in Proceedings of the 6th International Conference on Complex Systems (ICCS06), Boston, MA, 2006.
10. P. Erdös and A. Rényi, On random graphs. I, Publicationes Mathematicae, 6, 290–297, 1959.
11. R. Albert and A. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, 74(1), 47–97, 2002.
12. University of Washington, Electrical Engineering. Power systems test case archive, http://www.ee.washington.edu/research/pstca/.
13. E. J. Kirk, Count loops in a graph, Matlab Central, The Mathworks Inc., http://www.
mathworks.com/matlabcentral/fileexchange/10722, 2007.
14. E. P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, The Annals of Math, 62, 548–564, 1955.
15. E. P. Wigner, On the distribution of the roots of certain symmetric matrices, The Annals of Math, 67, 325–328, 1958.
16. M. Fiedler, Laplacian of graphs and algebraic connectivity, Combinatorics and Graph Theory, 25, 57–70, 1989.
17. W. J. Reed and M. Jorgensen, The double pareto-lognormal distribution: A new para-metric model for size distributions, Communications in Statistics, Theory and Methods, 33(8), 1733–1753, 2004.
18. M. Parashar and J. S. Thorp, Continuum modeling of electromechanical dynam-ics in large-scale power systems, IEEE Transactions on Circuits and Systems, 51(9), 1848–1858, 2004.
19. B. A. Carreras, V. E. Lynch, I. Dobson, and D. E. Newman, Critical points and transi-tions in an electric power transmission model for cascading failure blackouts, Chaos, 12(4), 985–994, 2002.
20. Z. Wang, R. J. Thomas, and A. Scaglione, Generating random topology power grids, in Proceedings of the 41st Annual Hawaii International Conference on System Sciences (HICSS-41), vol. 41, 7–10 January, Waikoloa, HI, 2008.
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