**5. Simulation Results and Data Analysis**

In this section, the present study experiments with the data of IEEE 39-bus power system and establishes the simulation results in detail. The power flow calculation and the isolated island problems are solved using the MATPOWER 6.0 toolkit in MATLAB R2016a [35]. Based on the graph theory, the directed graph gets the average degree *D* = 2.359, cluster coefficient *C* = 0.0385, and average path length *L* = 4.749, while the random network with the same *D*, *Crand* ≈ *D*/*N* = 0.0605 and *Lrand* ≈ ln(*N*)/ ln(*D*) = 4.2687. The graph includes generator nodes ranging from 30 to 39, and it is partitioned into 5 communities according to the Fast–Newman algorithm. The modularity is *Q* = 0.6125, which indicates good community partition of this graph. Each community contains at least one generator node, which is shown as follows.

In Figure 1, communities are labelled by numbers and surrounded by an ellipse. Community 1 is the area of blue solid circles, community 2 the area of red squares, community 3 the area of magenta snowflakes, community 4 the area of green rhombuses, and community 5 the area of black stars.

**Figure 1.** Communities of IEEE 39-bus system.

#### *5.1. Generating Network*

According to the principle of link-addition strategies, IEEE 39-bus system has 9 one-degree nodes ranging from node 30 to node 38. These leaf nodes without heavy loads are unnecessary to connect to each other, because they are all generator nodes. Therefore, the network has to add 9 additional links to get *Dnew* = 2.8205.

#### (1) LDNLAS Network

From Figure 1, the node importance of the original network is obtained to find the most vulnerable node 16 and 2 leaf nodes based on the Equation (9) in the same community. The low-degree nodes are randomly chosen to connect with these leaf nodes to find the average shortest path length. Following the rule, 9 links are added to the original network. In each step, the network can be partitioned into valid communities. The total cost of additional links is 53. See Table 1 for details.


**Table 1.** Connectivity link addition of LDNLAS.

The LDNLAS network detects 4 communities in Figure 2. Community 1 with 9 nodes is the area of blue solid circles, community 2 with 4 nodes is the area of red squares, community 3 with 15 nodes is the area of magenta snowflakes, and community 4 with 11 nodes is the area of green rhombuses.

The modularity of the LDNLAS network is *Q* = 0.5127 ∈ [0.3, 0.7], which indicates the community partition is effective. *C* = 0.0214 is less than that of the original network, and *L* = 3.8475 is reduced to about 19%. Although the LDNLAS network reduces the aggregation degree than that of the original network, it improves the connectivity obviously.

**Figure 2.** Communities of LDNLAS network.

#### (2) NNNLAS Network

Firstly, the neighbors of the leaf nodes are found. Owing to the symmetrical structure, several leaf nodes have the same shortest distance to their neighbors. The total cost of additional links is 22. See Table 2 for details.


**Table 2.** Connectivity link addition of NNNLAS.

In Figure 3, the NNNLAS network detects 5 communities. Community 1 with 7 nodes is the area of blue solid circles, community 2 with 4 nodes is the area of red squares, community 3 with 12 nodes is the area of magenta snowflakes, community 4 with 7 nodes is the area of green rhombuses, and community 5 with 9 nodes is the area of black stars.

**Figure 3.** Communities of NNNLAS network.

The modularity of the NNNLAS network is *Q* = 0.6393 ∈ [0.3, 0.7], which indicates the community partition is highly effective. *C* = 0.2692 is 7 times the original network, and *L* = 4.4872 is reduced to about 5%. Although the NNNLAS network enhances the aggregation degree enormously than that of the original network, it increases the connectivity level slightly.

#### (3) MLNLAS Network

First, the loads of the original network are ordered to select the first 9 load nodes. Then, new links are randomly added to the leaf nodes to satisfy the community partition principle and the average shortest path length. The total cost of additional links is 58. See Table 3 for details.


**Table 3.** Connectivity link addition of MLNLAS.

The MLNLAS network detects 3 communities in Figure 4. Community 1 with 13 nodes is the area of blue solid circles, community 2 with 12 nodes is the area of red squares, and community 3 with 14 nodes is the area of magenta snowflakes.

**Figure 4.** Communities of MLNLAS network.

The modularity of the MLNLAS network is *Q* = 0.4483 ∈ [0.3, 0.7], which indicates the community partition is reasonable. *C* = 0.0342 is close to that of the original network, and *L* = 3.5735 is reduced to about 25%. Although the MLNLAS network decreases the aggregation degree than that of the original network, it increases effectively the connectivity level.

Three networks of the same additional links decrease the APL and increase the connectivity than that of the original network. NNNLAS network significantly improves the aggregation degree at the lowest cost; LDNLAS network effectively increases the connectivity with a higher cost than that of NNNLAS network; MLNLAS network dramatically improves the connectivity and alleviates the burdens of load centers, while the cost is the highest one of three strategies, and the community partition and aggregation degree are relatively weak.

#### *5.2. Network Robustness Analysis*

The robustness of networks is analyzed under three attack scenarios. Random node attacks and high-degree-node-based attacks are regarded as simultaneous attacks, while vulnerability-based attacks are sequential attacks. For reducing the influence of network capacity, this study assumes the universal system tolerance parameter α = 2. Under the simultaneous attack scenarios, the component ratios are graphed with the distribution interval, median, 5%–95% position and mean at various attack ranges. Under the sequential attack scenarios, the component ratio curves are plotted by the number of attacks, and all remaining survival islands are demonstrated as directed graphs.

#### (1) RA Scenario

Random attack groups are *C*<sup>4</sup> 39,*C*<sup>8</sup> 39,*C*<sup>12</sup> 39,*C*<sup>16</sup> 39,*C*<sup>20</sup> 39,*C*<sup>24</sup> 39,*C*<sup>28</sup> 39,*C*<sup>32</sup> 39,*C*<sup>36</sup> <sup>39</sup>, according to the attack ranges respectively. In one attack range, 1000 groups of data are selected to attack 4 networks, which is executed for 50 times to obtain the corresponding results.

From the distribution intervals of Figure 5, the maximum component ratios of the original network are all less than or equal to three new networks of any attack range. The less the range of distribution intervals, the more stable the cascading propagation; the greater the mean value, the better the network robustness. For further comparison, the mean and median values are shown in Figure 6.

**Figure 5.** Component ratio under RAs.

**Figure 6.** Mean and median values under RAs.

Observing the mean histogram and the median curve of Figure 6, the original network lefts fewer nodes when the attack range is up to 60%. The LDNLAS and NNNLAS networks survive up to 70% attack range, while the MLNLAS network can preserve in 80% attack range. Combined with the distribution intervals of Figure 5, the robustness of 4 networks orders is as follows: MLNLAS > LDNLAS > NNNLAS > original.

#### (2) HDNA

The nodes of networks are ordered in degrees. The attack range selects the nodes from the high degrees to the low ones. As the nodes with the same degree have a number of attack groups, the results can be obtained by traversing all attack groups of each attack range.

In Figure 7, when the attack range is up to 50%, the original network totally collapses, and the MLNLAS network lefts a few nodes. In contrast, the LDNLAS and NNNLAS networks remain a large number of nodes. Owing to the impacts of the highest degree nodes on the connectivity, the NNNLAS network losses the maximum nodes at 10% attack range of 4 networks. For further analysis, the mean and median values are shown in Figure 8.

**Figure 7.** Component ratio under HDNAs.

**Figure 8.** Mean and median values under HDNAs.

Combined with Figures 7 and 8, when the attack range reaches 20%, although the mean value of the LDNLAS network is smaller than that of the NNNLAS network, both the maximum value and the median value of the former are larger than the latter, which indicates that the mean value is smaller due to the influence of extreme value. Thus, the overall data should be larger than the latter. Attacking more than 20%, the robustness of the LDNLAS network is obviously superior to other 3 networks. Influenced by the community partition, when the attack range is more than 10%, the robustness of 4 networks orders as follows: LDNLAS > NNNLAS>MLNLAS > original.

#### (3) VSA

Based on the node vulnerability, one node is attacked each time. For comparing with the original, the attack originates from the most vulnerable node 16. The attack sequence of the original network is: 16–26–3–8–6; the attack sequence of the LDNLAS network is: 16–23–7–20–2–9–5–14; the attack sequence of the NNNLAS network is: 16–14–6–26; and the attack sequence of the MLNLAS network is: 16–13–6–8–26–3–22–2.

In Figure 9, the original network sequentially attacks 5 nodes (about 10%) splitting into 4 islands, and *Soriginal* = 2.564. The LDNLAS network sequentially attacks 8 nodes (about 20%) splitting into 3 islands, and *SLDNLAS* = 3.0768. The NNNLAS network sequentially attacks 4 nodes (about 10%) splitting into 3 islands, and *SNNNLAS* = 1.9488. The MLNLAS network sequentially attacks 8 nodes (about 20%) splitting into 4 islands, and *SMLNLAS* = 4.9232.

**Figure 9.** Component ratio under VSAs.

The remaining islands of sequential attacks are shown as follows.

From Figures 9 and 10, it is observed that the MLNLAS network is the most robust one of 4 networks. The LDNLAS network exhibits the difficulty of sequential attacks, while it is weak in islanding operations. The NNNLAS has the worst survivability under sequential attacks. In the sequential attack process, the more the attacks, the more difficult the implementation, and the more robust the network. Moreover, the network with few communities, a small CC and a short APL can resist the sequential attack more efficiently. Therefore, the robustness of 4 networks orders as follows: MLNLAS > LDNLAS > original > NNNLAS.

**Figure 10.** Remaining islands under VSAs (**a**) IEEE 39 system, (**b**) LDNLAS network, (**c**) NNNLAS network, and (**d**) MLNLAS network.

From the above analysis, LDNLAS gets the second largest link-addition cost of the three proposed strategies. The LDNLAS network obtains a shorter APL and smaller CC than the original network, which alleviates the depth of the cascading failure propagation. In fact, this network exhibits the best robustness against HDNAs, and the second best robustness against RAs and VSAs. Although this strategy requires slightly larger investments, it can resist both simultaneous attacks and sequential attacks, and enhance the connectivity of the long-distance transmission structure power system.

MLNLAS obtains the largest link-addition cost of the three proposed strategies. The MLNLAS network with the shortest APL enormously enhances the connectivity than that of the original network. Moreover, this network presents the best performance against RAs and VSAs. Although this strategy requires more investments, it optimizes the electricity supply to greatly alleviate the burdens of load centers. As Ref [36] says, it is difficult to gain the high robustness with the minimal cost simultaneously.

NNNLAS has the smallest link-addition cost of the three proposed strategies. The NNNLAS network with the largest CC improves the centralization of local area management and is robust to the simultaneous attacks. However, it cannot effectively decrease the network vulnerability against VSAs.

#### **6. Conclusions**

Cascading failure propagation can be alleviated by optimizing the network topology. Based on the community partition of the original network, three link-addition strategies are proposed to meet the requirements of engineering practices. It is thus useful to guide the power system planning to improve the network robustness.

From the analysis of simulation results, the three proposed strategies can improve the network connectivity by adding the same number of links. The MLNLAS network exhibits good robustness under RAs; the LDNLAS shows better performances than other networks under HDNAs; the MLNLAS network reveals highly survivability under sequential attacks.

In this study, the proposed strategies are beneficial for improving the robustness of the original network. The focus is on the influence on the power system. In the future work, the authors will continue to study optimal strategies to mitigate cascading failures and improve the robustness of smart grids.

**Author Contributions:** Conceptualization, L.L.; methodology, L.L.; software, L.L.; validation, L.L.; formal analysis, L.L.; investigation, L.L..; resources, L.L. and P.H.; data curation, L.L.; writing—original draft preparation, L.L.; writing—review and editing, L.L., P.H.; visualization, L.L.; supervision, P.H.; project administration, P.H.; funding acquisition, L.L. and P.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**


#### *Processes* **2020**, *8*, 126

#### *Sets and Functions*


#### **References**


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