*4.3. Results and Discussion*

The accuracies of MIGA and GGA+ were evaluated according to the Q values. Table 2 shows the maxima, means, minima, and standard deviations (SD) of the modularities obtained by MIGA and GGA+ in the European (EA) and the North American (NA) grids considering c = {2, 3, 4, 5, 10, 20, 30, 40, 50} communities. A number of communities within the range of 2 to 50 were used to show how evolutionary algorithms are able to work under different levels of abstraction. However, these algorithms could be applied to obtain a greater number of communities, although their size would decrease considerably. The median runtimes (in minutes) of these 30 independent runs are also provided. Furthermore, the communities detected by Louvain modularity method implemented in Gephi are also shown in this table. To conduct the performance analysis and to statistically compare the quality of the solutions obtained by the two algorithms, a total of 30 independent runs were performed with each algorithm on each network.


**Table 2.** Results obtained by MIGA and GGA+ after 30 independent runs and comparison with Louvain method implemented in Gephi (runtime in minutes).

These results show that GGA+ achieved the best mean and maximum values in both grids, regardless of the number of communities to be detected. These results also indicate that the greater the number of communities, the greater the advantage of GGA+ over MIGA. In addition, the standard deviation obtained from the results of these 30 independent runs was often smaller for GGA+ than for MIGA. The modularity values obtained by both algorithms increased with the number of communities without degradation of the standard deviation, indicating the robustness of these evolutionary approaches.

Table 2 shows that the runtime required by both evolutionary algorithms is of the same order of magnitude in the North American and European networks, while the differences come from the fact that the former has approximately double the number of nodes and edges as the latter (see Table 1). On the other hand, in both cases the runtime tends to decrease when the number of communities is greater. This is due to the crossover and mutation operators moving a given percentage of the nodes between a community and a neighbouring one. Therefore, the bigger the communities are, the higher that number of nodes that are moved between neighbouring communities, and therefore, the runtime increases. It can be concluded that GGA+ is scalable both in terms of network size and in terms of number of communities. Both algorithms require a few hours to complete the search process with these parameter settings, which is not a critical issue since the goal is to find solutions with greater modularity regardless of the execution time. Of course, the execution time could be reduced considerably by modifying the parameter settings or applying parallel processing techniques.

When two algorithms are compared, it is common to determine whether there are significant differences between the solutions they obtain. With this aim, a one-way ANOVA was applied, with the results indicating that the *p*-value was <0.05 in all cases; i.e., the null hypothesis was always rejected, which means that there was a significant difference between at least some of the means of the different groups. Thus, the results obtained by GGA+ were significantly different from those obtained by MIGA, validating the mean values in Table 2.

The analysis of Figure 2 reveals that there are some differences between the results obtained by the two methods when detecting three and twenty communities, especially when the number of communities increases. Each community in these networks is represented with a random colour. Even in the case of detecting only three communities, MIGA has some difficulties in assigning communities in some parts of the graph, while GGA+ obtains clearly differentiated communities. Considering these graphical results and the results provided in Table 2, it can be concluded that GGA+ not only outperforms to MIGA, but also exhibits good performance in these large networks.

(**a**) Louvain modularity (EU, three communities)

(**c**) MIGA (EU, three communities)

(**b**) Louvain modularity (EU, 20 communities)

(**d**) MIGA (EU, 20 communities)

(**e**) GGA+ (EU, three communities)

(**f**) GGA+ (EU, 20 communities)

**Figure 2.** Results obtained by Louvain modularity method, MIGA, and GGA+ for the European grid (three and 20 communities).

The results obtained by GGA+ are analysed in more detail here. Figure 3a–c display the communities detected by GGA+ in the European power grid with 5, 10, and 30 communities. These results reveal that this algorithm is able to obtain good quality solutions even when the number of communities increases. Moreover, Figure 3d–f provide a different layout based on the ForceAtlas2 [61] plugin in Gephi for these three networks. While the results presented in Figure 3a–c correspond to the coordinates of each node, the results displayed in Figure 3d–f cannot be read as a Cartesian projection. Instead, ForceAtlas2 was in a drawing mode that has the specificity of placing each node depending on the other nodes. This visualisation method builds a force directed layout by simulating a physical system in order to accommodate nodes and links in a spatial network. Nodes repel each other like charged particles, while edges attract their nodes like springs. Note that the same colour is used to represent the physical layout and the distribution obtained by ForceAtlas2. Moreover, the number of nodes in each community is often balanced (e.g., the five communities obtained in the European grid have a percentage of nodes between 19.16% and 20.47% of the total of nodes), although there are some significant imbalances between clusters when the number of communities increases (e.g., 30 communities). The analysis of Figure 3 demonstrates the good behaviour of GGA+ independent of the degree of abstraction. Figure 4 shows how geographical structures change with the number of communities.

**Figure 3.** *Cont*.

**Figure 3.** Results obtained by GGA+ for the European power grid: physical layout with (**a**) five communities, (**b**) 10 communities, and (**c**) 30 communities. Distribution obtained by ForceAtlas2 with (**d**) five communities, (**e**) 10 communities, and (**f**) 30 communities.

**Figure 4.** Physical layout of the communities detected by GGA+ in the European network using different degrees of abstraction (the number of communities is indicated in parentheses).

The analysis of the North American network supports similar conclusions. Thus, Figure 5a–c display the results obtained by GGA+ in that network when 5, 10, and 30 communities are detected. These data reveal that this algorithm is able to obtain good quality solutions not only with a few communities, but when the number of communities increases. The results obtained by the layout provided by ForceAtlas2 for these configurations (Figure 5d–f) demonstrate the good behaviour of GGA+ independent of the degree of abstraction. Finally, Figure 5 shows that the algorithm is able to obtain differentiated clusters, even when the number of communities increases significantly. The results obtained here are of particular interest, bearing in mind that the North American electrical grid is made up of three interconnections: the Western Interconnection, the Eastern Interconnection, and the ERCOT (Texas) Interconnection, which are not synchronised, and alternating current (AC) power must be converted to direct current (DC) power for transfer across any of the interconnections. To overcome these limitations, the *Tres Amigas* superstation has been planned in New Mexico (U.S.), a 1.6 billion dollar project that aims to connect these three primary interconnections to facilitate the smooth, reliable, and efficient transfer of green power from region to region while integrating substantial renewable energy sources [62]. Figure 6 shows how geographical structures in the North American grid change with the number of communities.

**Figure 5.** Results obtained by GGA+ for the North American power grid: physical layout with (**a**) five communities, (**b**) 10 communities, and (**c**) 30 communities. Distribution obtained by ForceAtlas2 with (**d**) five communities, (**e**) 10 communities, and (**f**) 30 communities.

**Figure 6.** Physical layout of the communities detected by GGA+ in the North American network using different degrees of abstraction (the number of communities is indicated in parentheses).
