*2.1. Community Detection: General Overview*

Communities, also named clusters, are dense subgraphs which are well separated from each other. The community structure of complex networks reveals both their organisation and hidden relationships, among other elements [25]. In practice, a simple idea that has attained grea<sup>t</sup> popularity is that a community is a subgraph such that many edges connect nodes within the same group, and comparatively few edges connect nodes in different groups [14].

Many studies in different disciplines have shown that the community structure of complex networks reveals both their organisation and hidden relationships among their elements [25]. In particular, identifying communities can be useful for classifying the nodes in different groups [13]. So, nodes located at a central position in their community may have an important function of control and stability within the cluster, while those nodes located at the proximity of other communities can play a role of mediation or information exchange with these neighbouring communities.

An important consideration to be determined here is the number of communities to be detected. Some algorithms allow one to include a pre-established number of communities to be detected, while other approaches aim to infer the adequate number of communities depending of the characteristics of the networks [14]. A recently published survey paper [26] has reviewed a large number of community detection algorithms in multidisciplinary applications considering both disjoint and overlapping community detection problems. These applications include the study of social

networks [25,27], communication networks [28,29], engineering systems and networks [18,30], biology and ecology [31,32], health sciences [33], scientometrics [34,35], economics [36], etc.

### *2.2. Community Detection in Power Grids*

In recent years, the interest in the development of supergrids has grown remarkably. The supergrid concept was born as a solution to allow large-scale electrical power exchanges over continent-wide areas. This concept has been considered both a potential solution to transmission bottlenecks and an opportunity to trade higher volumes of electricity across longer distances [37]. In particular, we show the complexity of several high-voltage transmission topologies intended to connect two or more subsystems here, and note that supergrids have a meshed form to provide redundancy. In addition to the use of complex control methods [38], the variability of renewable sources [39] at continental scales can be mitigated by using the transmission grid and balancing locally with storage [40]. Some of the future major transmission projects around the world are described in [37]. For example, different projects aim to promote an efficient and reliable transmission grid in North America, including the *Tres Amigas superstation*. This superstation is the first version of this supergrid vision, since it is projected as a high-voltage direct current (HVDC) super-node asynchronously connecting the existing alternating current (AC) networks intended to link the three North American grids: the Eastern Interconnection, the Western Interconnection, and Texas Interconnection. This project involves a three-way alternating current/direct current (AC/DC) transmission superstation with several miles of underground superconducting DC cable, which will eliminate the market separation between the three asynchronous interconnections in the continental U.S. [41]. In the case of Europe, these authors indicate that an important number of major HVDC interconnections are being promoted to establish intercontinental interconnections with neighbouring regions with the aim of integrating regional energy markets into a single European market to achieve the European Union's (EU) renewable energy goals. Some authors have introduced the concept of global grid as the future stage of the electricity network, in which most of the large power plants in the world will be connected [42].

Some recent studies have proposed the analysis of the power grid infrastructure using graph-based network analysis techniques [19]. Usually, the nodes of the network represent the power plants and distribution and transmission substations, while the edges correspond to transmission lines. The application of graph-based analysis techniques has allowed for an analysis of the topological structure of networks representing power grids [43]. As commented above, a typical characteristic of all complex networks is the existence of community structures [13,15], such that detecting those communities can reveal the characteristics or functional relationships in a given network. In the case of power grids, communities represent substations densely connected by high-voltage transmission lines.

The importance of community detection in power grids comes from the fact that it is necessary to maintain grid reliability and enable more efficient restoration from severe disturbances. In particular, it is necessary to prepare a distribution grid for natural disasters (e.g., a storm), by developing switching plans to safely islands or disconnecting portions of the grid, preventing further degradation during incidents and enabling faster restoration after the disturbance. For example, reference [20] applied community detection to island power systems as an emergency response method to isolate failures that could propagate and lead to major disturbances. These authors developed two approaches based on modularity, with the DC power flow model incorporated into them, for islanding in medium and large networks and tested them in networks having 14, 30, 57, 118, and 247 nodes [20]. Other approaches use node similarity indexes to assign each node to the community sharing maximum similarity [22], and have demonstrated the good performance of this method in two IEEE standard power grids (39-bus standard power grid and 118-bus standard power grid). The IEEE 118-bus was also studied in [44]. Other researchers have presented a hierarchical spectral clustering method to reveal the internal connectivity of power transmission, establishing the possibility of islanding systems using a network with nodes and links representing buses and electrical transmission lines, respectively [21]. That approach was evaluated in several test systems of small, medium, and large sizes, including a model of Great Britain's transmission network [21]. Community detection has also been applied to analyse the vulnerability of the power systems under terrorist attacks [45], among other applications. However, none of these previous approaches have analysed supergrids.
