4.1. Case Study
Part of the Guangdong power system in China (
Figure 3), including 155 transmission lines and 125 nodes, is utilized to verify the effectiveness of the proposed multi-attribute evaluation method based on the modified VIKOR method (MMVIK) for critical lines identification. In an actual regional power system of China, its subsystems with different voltage levels are regulated by different levels of power grid corporations. The lines with 500 kV voltage level in the power system are managed by the provincial power grid corporation, so these lines are ignored by the regional power grid corporations which are the subsidiaries of the provincial one. The results of critical lines identification are determined by the lines’ importance degree evaluation results and the actual demands of power grid corporations. In this paper, the top ten lines obtained by using the MMVIK method are identified as critical lines for easy demonstration. It is assumed that
,
vT = 5.6 m/s,
vmin = 35 m/s,
,
Kλ = 0.1,
αcid = 0.5,
= 0.8,
and
[
19].
The correlation between the evaluation indexes is needed to be analyzed first to verify whether the index selection is reasonable. Spearman correlation coefficient is adopted to analyze the correlation between the six indexes because it is not very strict on data qualification and can analyze the correlation by the ranking position of indexes’ values, which is suitable for the critical lines identification. Spearman correlation coefficients between any two indexes are shown in
Table 2. It can be seen in
Table 2 that most of the correlations between the evaluation indexes are weak except for the pair of indexes
Ircn and
Icid, because lines’ position in network topology is considered in
Ircn and nodes’ position in network topology is considered in
Icid. However, the objective of
Ircn is to identify critical lines which locate in the center of the network, and the objective of
Icid is to include critical nodes which are important for both network topology and operation state of power systems. Thus, these two indexes are different with each other actually. It can be concluded that there is not much overlap and cross information among the selected indexes and the selection of these indexes is reasonable.
After identification, the top ten lines are determined as the lines with bold dotted lines in
Figure 3, as also shown in
Table 3. It can be seen in
Table 3 that the selected evaluation indexes have similarities and complementarity. For example, the values of
Ircn and
Irlc for
LTF-LFA are 1, which indicates that some critical lines locate at the center of the power network topology; and the nodes voltage might be out of limits or the reactive power regulation of generators might be declined after removing line
k.
LHZ-ZK is with a smaller
Ircn (i.e., 0.034) and a relatively bigger
Iwfb (i.e., 0.531), indicating that
LHZ-ZK plays a less important role in topology but is important for power transmission. In addition, if a single index (e.g.,
Ircn) is utilized to evaluate line importance degree, some critical lines would be ignored for critical lines identification. For instance, the value of
Ircn for
LHZ-SD is 0.038, which ranks in the 35th based on
Ircn only,
LHZ-SD could not be identified as a critical line because it is not in the center of network topology. However, the value of
Iwfb for
LHZ-SD is 0.572, which ranks 4th based on
Iwfb only, indicating
LHZ-SD has a great influence on the operation state of the power system; the value of
Ifil for
LHZ-SD is 1, indicating removing
LHZ-SD might cause a bigger cascading failure. Thus,
LHZ-SD should be identified as a critical line. As a result, the key characteristic parameters of the power systems can be considered more reasonably and comprehensively by integrating the six indexes, reducing the probability of missing some critical lines to a certain extent.
Table 4 shows the results of indexes weights calculated by single weighting methods and the proposed combination weighting method. It can be seen in
Table 4 that the weights of
Ifil and
Icid determined by the entropy weighting method are 0.0549 and 0.0616, respectively, and then
Ifil and
Icid are neglected in critical lines identification. The objective weights of these two indexes are relatively small because the difference of the indexes values is small and their weights obtained by the objective weighting method are based on data characteristics. Thus, the objective weighting results might be unreasonable because the objective weighting method ignores the experts’ experience. On the other hand, with the subjective impacts caused by the experts with different levels of knowledge and experience considered, it is not scientific to rely on the subjective weights given by the experts results only. In the proposed combination weighting method, the synthetic weights of
Ircn,
Ifil and
Icid are 0.1163, 0.0950 and 0.1146, respectively. These three indexes’ weights are relatively small, which is reasonable because
Ircn and
Ifil are utilized to evaluate the importance degrees of lines from the pure topology, and the difference between the values of
Ifil and
Icid is small. Therefore, the proposed combination weighting method that can consider both the data characteristic and experts’ experience is more suitable for determining the indexes’ weights for the critical lines identification.
For the top ten lines, three parameters of the MMVIK method, i.e., group utility
Sk, individual performance
Rk and comprehensive evaluation
Qk, are shown in
Table 5. It can be seen in
Table 5 that the values of
Sk for both
LHZ-ZK and
LHZ-SD are bigger than those for
LYY-SD and
LBL-JY, indicating the overall performances of
LHZ-ZK and
LHZ-SD are not as good as those of
LYY-SD and
LBL-JY. However, the values of
Qk for
LHZ-ZK and
LHZ-SD are smaller than those for
LYY-SD and
LBL-JY because the values of
Rk for
LHZ-ZK and
LHZ-SD are smaller than those for
LYY-SD and
LBL-JY. It can be seen in
Table 3 that both
LHZ-ZK and
LHZ-SD are with big values of
Ifil (i.e., 1), which indicates that removal of them might cause a bigger cascading failure of the power system. In other words,
LHZ-ZK and
LHZ-SD play important roles in a particular aspect of the power system. It also can be seen from
Table 3 that most of the values of the six indexes for both
LYY-SD and
LBL-JY is not large. Thus, it is more reasonable that
LHZ-ZK and
LHZ-SD rank slightly higher than
LYY-SD and
LBL-JY in the MMVIK method. Therefore, the MMVIK method is effective in identifying critical lines for the skeleton-network which wants to contain a variety of critical lines.
4.2. Comparisons with Other Methods
To further demonstrate the effectiveness of the MMVIK method for critical lines identification, five methods, i.e., single-attribute evaluation method based on the rate of change of network efficient (SRCNE) [
8], single-attribute evaluation method based on hybrid flow betweenness (SHFB) [
11], multi-attribute evaluation method based on utility theory and cooperative game (MUTCG) [
15], SWA [
28] and TOPSIS [
29], are employed for comparisons. The final line importance evaluation results and the top ten lines of the actual regional Guangdong power system obtained by the six methods are shown in
Figure 4 and
Table 6, respectively.
It can be seen in
Figure 4 that most of the critical lines identified by single-attribute evaluation methods are included in the identification results obtained by the proposed MMVIK method, such as line 27 (i.e.,
LHZ-TF), line 50 (i.e.,
LTF-LFA), line 60 (i.e.,
LBL-BL2) and line 61 (i.e.,
LHZ-HZ1) identified by SRCNE method. However, some critical lines are ignored in single-attribute evaluation methods (i.e., SRCNE and SHFB), such as line 24 (i.e.,
LHZ-ZK). Line 24 connects central node
HZ with node
ZK, and it is the main channel to transmit power from the generator node
HZ to load nodes. Thus, it is more reasonable to identify line 24 as a critical line with higher importance degree by the MMVIK method. In the MUTCG method, only network topology and the power transmitted on the lines are considered, and the lines’ failure probability and the impact of line removal are ignored. For example, line 31 (i.e.,
LZK-WT2) with a big value of
Ifpl (i.e., 0.868), which is easy to fail under typhoon, should be involved in reinforcement work to reduce lost caused by extreme weather. Line 28 (i.e.,
LYY-SD) with a big value of
Ifil (i.e., 0.7), which has a great impact on other lines and is the main channel to transmit power from the generator node HZ to load nodes, should have a relatively high importance degree. The nodes voltages might be out of limits or the reactive power regulations of generators might be declined obviously after removing line 2 (i.e.,
LBL-JY) with a big value of
Irlc (i.e., 1). However, these three lines rank 27th, 29th and 18th with a relatively low importance degree when using the MUTCG method. Therefore, it can be concluded that the proposed MMVIK method is better than these three methods for critical lines identification in the skeleton-network.
It can be seen in
Table 6 that the top ten lines identified by the MMVIK, SWA and TOPSIS methods are similar, but their ranking order is slightly different.
LTF-LFA ranks first by using the proposed MMVIK while
LBL-BL2 and
LHZ-HZ1 rank first by using the SWA and TOPSIS methods, respectively. It can be seen in
Figure 3 that
LTF-LFA, which locates at the center of the network topology, is the sole interconnected line between two subsystems, and it has relatively high values of
Icid and
Irlc (i.e., 1). The power system could be split into two isolated islands and the voltage and the reactive power could also be affected after the removal of
LTF-LFA. It can also be seen in
Figure 3 that both the location in the network topology and the function for power transmission of
LBL-BL2 and
LHZ-HZ1 are similar. If any of them failed, the power flow can be transmitted through the other line. Thus, the impact of removing
LTF-LFA is bigger than removing
LBL-BL2 or
LHZ-HZ1 and it is more reasonable to rank
LTF-LFA as the first one. In addition,
LHZ-SD and
LZK-WT2 are identified as the critical lines by the MMVIK method. It can be seen in
Table 3 that removing
LHZ-SD from the network could lead to a cascading failure for the power system because the value of
Ifil for
LHZ-SD is equal to 1, and
LZK-WT2 is easy to fail under typhoon because the value of
Ifpl for
LHZ-SD is equal to 0.868, so these two lines should be identified as the critical lines with high ranking orders. However,
LHZ-SD and
LZK-WT2 are not identified as critical lines by the SWA method. It can be concluded that the MMVIK method not only takes the comprehensive performance of the lines under various indexes into account, but also considers the outstanding contribution of some important lines on a particular aspect of the power system, which indicates that it is a better method to identify critical lines for the skeleton-network.