Since contingency analysis needs to solve the system for as many as |E| times to rank all the edges, it becomes impractical in the real-world system. Our goal is to develop an efficient algorithm which identifies the important edges (high PI) without carrying out the full contingency analysis.
To find the risky edges, we approximate the significance of an edge by comparing its power flow to its expected value if the network is completely randomized but with the same degree distribution. In this paper, we call our measureQij modularity score for the convenience of notation. Specifically, for each edge(i, j)the scoreQij = Aij−kikj/m is computed and compared to its performance index(PI), which serves as a measure for
the impact of removing the edge. Let fij be the current on an edge (i, j) and Cij be capacity of the edge. The performance index is defined as P I =P
(fij/Cij)4.
(a) N-1 for power system size 57 (b) N-1 for power system size 118
Figure 7.2. x label indicates each transmission line, while the y label
shows the comparision between our score and the performance index
Figures 7.2a and 7.2b show that the two measures agree with each other well, espe- cially for risky edges for the 57-node test case and 118-node test case.
(a) reduction case:57 (b) reduction case:118
Figure 7.3. We firstly apply reduction to this power grid network system.
And then we sort the transmission lines according to our contingency anal- ysis method. The x direction represents the performance index computed from exact calculation, while the y axis shows the score defined in this section. From the figure, only few points appear in the fourth quadrant, which indicates the most significant transmission lines are well captured by the our score
In figure 7.4, we showed N-1 analysis for size 2383 power system. After the network reduction, we sort the transmission lines according to our contingency analysis method.
(a) all the transmission lines (b) zoom in to study the top 70
Figure 7.4. The x direction represents the performance index computed
from exact calculation, while the y axis shows the score defined in this sec- tion. The power grid network is composed of several components and our method works well one component. However this component captures the most important edges and indicates that he most significant transmission lines are well captured by the our score
For a large system, the computation of time of N −x analysis grows exponentially with x, and so it is important to find an efficient contingency selection method. In
N −x analysis, one wants to study the impact of various combinations of individual transmission lines failing at the same time. When x = 2, for each pair of edges (i, j) and (s, t), we define the significant score: (Aij − kikj/m) ×(Ast −kskt/m) and compare it with the performance index. Figures 7.5a and 7.5b show the significant score captures most of the top-ranking edges in the performance index list. We filter our the non-severe transmission lines and apply N-2 contingency analysis to power system size 57 and 118 as shown in figure 7.5a and 7.5b.
(a) N-2 for power system size 57 (b) N-2 for power system size 118
Figure 7.5. N-2 contingency analysis: The x axis shows the index of
each pair of transmission lines, while the vertical axis shows the score comparison our definition and the performance index [4]
7.4. Results and Discussion
In this section, we will compare our results with the algorithm described in [45] and utilize the performance indices (PI) [48][49] to quantify our comparison. Table 1 shows the rank performance of three algorithms: PI represents the performance index algorithm,
bus bus PI CD BC 8 9 1 1 29 1 15 2 2 26 7 29 3 11 38 1 17 4 6 70 1 16 5 9 79 8 7 6 19 52 3 4 7 5 41 1 2 8 3 59 15 45 9 10 8 45 44 10 25 5 bus bus PI CD BC 8 5 1 1 15 10 9 2 43 18 9 8 3 42 3 66 49 4 3 45 89 92 5 2 39 89 90 6 8 173 26 30 7 6 49 65 64 8 111 51 63 59 9 107 44 64 63 10 108 58
Table 1. System size 57 and size 118
CD indicts the community detection based algorithm while BC shows the betweenness centrality related method in [45]
7.5. Conclusion
Stability is the key issue of power systems. Previously power grids are well studied as a physical system, but as all kinds of new energy are integrated into power grids most recently, the system becomes very complicated and its dynamics become intractable. This chapter aims to bring in novel perspectives from network science and build tight connections between the two fields. Power grids are essentially complex networks, but they distinguish themselves from other network models in computer science or social science by the high correlation between components and heterogeneity across them. We study the stability of power networks, analyze their stability via the idea of modularity and compare our results with those of contingency analysis.
In conclusion, we reduced the size of the power system through an inverse prob- lem of the modularity-based community detection. Then we further introduced a novel transmission line score, which aims to understand how the network topology affects the severeness of the transmission lines. Our result is compared with other network-based algorithms, which is very promising for size-57 and size-118 system.
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