1. Introduction
The power grid system is vulnerable to disruptions caused by manufacturing defects, natural disasters, and human actions, such as terrorist attacks [
1,
2,
3]. Even minor initial disturbances can lead to severe consequences, including economic and social instability [
4,
5,
6]. To reduce the costs associated with the disruptions, one research direction is to enhance the robustness of the power grid to withstand perturbations. The other direction is to propose efficient restoration strategies.
The power grid can be represented as a complex network, where nodes represent generators and loads, and where links represent transmission lines and transformers. Analyzing the power grid’s robustness from a network perspective has attracted significant attention. Network robustness is typically evaluated by assessing changes in network performance due to fluctuations such as node removals or link removals [
7]. Identifying critical nodes and links within the power grid can inform strategies for enhancing robustness by protecting the vital components [
8,
9,
10]. Furthermore, robust failure responses, like partitioning grids into islands, can mitigate the effects of components disruptions [
11,
12]
Numerous studies have focused on strategies for power grid recovery [
13], from black start [
14]—like an optimal generator start-up strategy by solving a mixed integer linear programming (MILP) problem [
15]—to a method that partitions the network to restore parts of the power grid separately and then interconnects them afterward [
16]. Additionally, research has been conducted on recovering power grids after partial component failures. Machine learning methods, specifically reinforcement learning, have been developed for restoring networks after node failures [
17] and link failures [
18]. The machine learning methods demonstrated better performance for node recovery strategies based on node degrees or node loads and link recovery strategies based on link betweenness. Li et al. [
19] developed the Q-learning method, to find an optimal method to recover power grids with link failures. From a network perspective, Wu et al. [
20] developed an effective tool for the sequential recovery graph, to recover nodes in power grids, which performed better than recovery strategies based on node degree and node loads. Forming microgrids can improve the resilience of the system after blackouts. Igder et al. [
21] applied deep reinforcement learning, to establish microgrids from black start after blackouts, so as to restore service in distribution networks. Yeh et al. [
22] developed an enhanced genetic algorithm, to minimize costs while optimizing dispatch in stand-alone microgrid systems.
Despite extensive research on power grid recovery, there is still a lack of investigation into the effectiveness of recovery strategies that rely solely on different network metrics following transmission line failures. From this study, our main contributions are as follows:
We examined recovery strategies based on various network metrics, including degree, betweenness, flow betweenness, eigenvector centrality, weighted eigenvector centrality, closeness, electrical closeness, electrical weighted closeness, zeta vector centrality, and weighted zeta vector centrality. Additionally, we compared these strategies to the random recovery, greedy, and two-step greedy strategies.
To assess the effectiveness of recovery methods, we utilized the general recoverability framework proposed by He et al. [
23], to measure power grid recoverability in the context of random link removals, where recoverability signifies a network’s ability to return to a predefined desired performance level.
Our study did not consider cascading failures after the recovery or removal of a single transmission line. Instead, we used the direct current (DC) power flow model to maximize power flow satisfaction for loads after a link removal or addition.
The paper is structured as follows:
Section 2 provides an overview of network robustness, laying the foundation for the subsequent analyses.
Section 3 details the modeling of power grids, including how to transfer a power grid into a network and optimize the DC power flow model.
Section 4 presents the attack and recovery processes, with a focus on the strategies employed. The results and conclusion are presented in
Section 5 and
Section 6, respectively.
5. Results
To investigate the effectiveness of recovery strategies in power grids, we conducted simulations on three different power grids: the IEEE 30 bus system, the IEEE 39 bus system, and the IEEE 118 bus system. In each realization, we randomly removed links until the
R-value of the system fell below a specified threshold. We then used various strategies to restore the system until all the removed links were added. Finally, we computed the recoverability energy ratio of each recovery strategy, given the same random removal process. To explore the impact of thresholds and tolerance levels, we selected two thresholds (0.8 and 0.5) and four tolerance levels (1, 2, 2.5, and 3) and performed 1000 realizations for a power grid with each setting. In
Figure 5, we demonstrate how the
R-value varied in a realization with different recovery strategies for the IEEE 39 bus system and the abbreviations of the strategies used in the following figures and tables in
Table 2.
We present the recoverability energy ratios of various recovery strategies concerning two thresholds and a tolerance level equal to 1 for the considered power grids, as shown in
Table 3,
Table 4 and
Table 5. The tables illustrate that the greedy two-step method exhibited the highest mean value of recoverability energy ratio and the lowest standard deviation value for all three power grids in both threshold cases. Additionally, the greedy method consistently maintained the second-best performance, in terms of the recoverability energy ratio mean value for all three power grids, albeit its performance was slightly inferior to that of the greedy two-step method. Notably, the difference in performance between the greedy two-step method and the greedy method was more pronounced when the threshold was equal to 0.5, compared to the case when the threshold was 0.8. This observation suggests that the greedy two-step method outperforms the greedy method when the removal link set is relatively large. Another noteworthy finding concerned the results of the random recovery method. The mean recoverability energy ratios of all the power grids with different thresholds of the random recovery method were less than 1, indicating that the cost of recovery outweighed the cost of attacks.
For the recovery strategies utilizing link metrics or the product of node centralities of link end points, we observed noteworthy variations in performance across different power grids. Specifically, the recovery method based on link betweenness exhibited diverse rankings of mean recoverability energy ratio when applied to the three power grids under consideration. In the IEEE 30 bus system, the mean recoverability energy ratio rank of the betweenness recovery method fell within the bottom two. Conversely, within the IEEE 39 bus system, the betweenness recovery method achieved an intermediate rank among all the methods based on the mean recoverability energy ratio. Remarkably, the recovery method based on betweenness centrality yielded the highest mean recoverability energy ratio among all the recovery strategies based on ink metrics or the product of the node centrality values of link end points in the IEEE 118 bus system.
In addition to the betweenness recovery strategy, the zeta vector recovery strategy exhibited a similar performance pattern. The strategy demonstrated effectiveness in the IEEE 30 bus system, with a threshold value of 0.5, as well as in the IEEE 39 bus system. However, the efficiency diminished when applied to the IEEE 118 bus system. The results indicate that the efficacy of recovery methods based on link metrics or the product of node centralities of link end points is contingent upon the underlying network topology and dynamics of the networks.
The observation can be attributed to the distinct allocation of generators in the IEEE 39 bus system, as illustrated in
Figure 6. Specifically, the majority of the generators in the IEEE 39 bus system had only one neighbor, and the proportion of links attached to generators was 22.74%, which was significantly higher than the corresponding proportions in the other two systems. By contrast, the IEEE 30 bus system had only one generator with a neighbor, and the proportion of links connected to generators was 14.63%, while the IEEE 118 bus system had three generators with a neighbor, and the proportion of links connected to generators was 10.62%. Therefore, the location of generators in the IEEE 39 system made the links connected to generators more susceptible to attacks and less likely to be recovered than the other two systems. Based on calculating the average values of the link metrics used for the different recovery strategies of two kinds of links—links connecting generators and loads, and links connecting loads and loads in
Appendix A Table A2—we found that in the IEEE 39 system, the link metrics of the links connecting generators and loads were larger than the link metrics of the links connecting loads and loads in the zeta recovery strategy and weighted zeta recovery strategy, indicating that recovering the links connecting generators and loads matters in the recovery process.
Compared to the IEEE 30 bus system and the IEEE 39 bus system, the recovery methods based on link metrics and the product of node centrality values of link end points in the IEEE 118 system demonstrated substantially better performance. Specifically, most recovery strategies based on link metrics and the product of node centrality values of link end points exhibited higher average recoverability energy ratios than the random recovery strategy for both thresholds.
Link capacity is a crucial factor in causing blackouts in power grids. Although our method did not employ the cascading failure model, we investigated the impact of link capacity on recovery strategies. To this end, we conducted simulations with different tolerance levels (
), and we analyzed the results, which are presented in
Figure 7 for the IEEE 30 bus system, the IEEE 39 bus system, and the IEEE 118 bus system. The simulation results show that the mean values of the recoverability energy ratio with respect to different tolerance levels did not change monotonically for the same recovery method, which could inspire study of the optimal tolerance level. The ranking of recovery strategies based on network metrics may have varied slightly, but the two-step greedy recovery strategy consistently performed the best, with the greedy recovery strategy following closely behind. Moreover, the results indicate that the recovery strategy based on betweenness outperformed the recovery strategy based on flow betweenness, while the electrical weighted closeness recovery strategy was the best among the electrical closeness recovery strategy, closeness recovery strategy, and electrical weighted closeness recovery strategy.
6. Conclusions and Discussion
Based on our study, we observed significant variations in the performances of different recovery strategies. Firstly, the recovery strategies based on link metrics and the product of the node centrality values of link endpoints did not achieve the highest performance. Secondly, the performance of those recovery strategies varied notably across different power systems, emphasizing the significance of network topology and generator placements in power grids. Thirdly, the two-step greedy recovery strategy consistently outperformed all other network metric-based recovery methods across all power systems, thresholds, and tolerance levels. The greedy recovery strategy, which was slightly less effective than the two-step greedy recovery strategy, also showed promising results. The findings confirm that implementing an effective recovery strategy can significantly improve the recoverability of power grids and that the two-step greedy strategy is particularly efficient, while relying solely on one network metric for recovery is less effective.
For future research directions, several avenues are worth exploring. Firstly, we can develop hierarchical recovery strategies, by classifying links into two groups—those connected to generators and those not connected to generators—and then prioritizing restoring links connected to generators. Secondly, we can investigate whether allowing cascading failures after link recovery or removal impacts the conclusions of this study. Thirdly, we can introduce additional constraints—such as generator costs—and incorporate cost minimization as an objective within the DC power flow model. Finally, given that distributed energy systems enhance power system resilience, it is promising to research how to establish and integrate such systems after a blackout, using the discussed strategies.