## 4 Clustering of Charging Behavior during COVID-19
### Clustering Results  
As mentioned earlier, three clusters were selected using the elbow method. Figure 6 presents the clusters obtained using k-means. The three clusters are almost equal in density and well separated. Cluster one in purple represents the early chargers who usually start between 5 a.m. and 10 a.m. and finish charging between 9 a.m. and 3 p.m. However, it also includes a few users who arrive later during the day and finish charging the next morning. The second cluster in red represents the group that arrives in the afternoon between 12 p.m. and 4 p.m. and leaves between 3 p.m. and 8 p.m. Finally, cluster three in green contains users who arrive after 3 p.m. and finish charging before midnight.  
In Figure 7, the three clusters using agglomerative clustering are presented. The results are similar to k-means clustering in terms of the charging behavior of the three groups. However, cluster two in red is less dense using this algorithm, and therefore, there is a greater overlap between the afternoon chargers, and they belong to all three clusters.  
Figure 5: Elbow method for detecting the number of clusters.  
Figure 6: Cluster visualization for K-means.  
Finally, the three clusters using GMM are presented in Figure 8. The three clusters using this algorithm are very different compared to the aforementioned approaches. In this case, cluster one in purple contains most of the sessions where the arrival time is between 7 a.m. and 8 p.m. and the departure time is between 10 a.m. and midnight. The second cluster in red contains shorter sessions where the charging session lasts for about 2 h. The third and smallest cluster in green contains sessions scattered throughout the day.  
Table 1 summarizes the results using the three internal validation metrics. Silhouette index is between \(-1\) and +1 and higher scores indicate better clustering. Similarly, a higher value for the Calinski-Harabasz index indicates good clustering. Conversely, the lowest possible score for the Davies-Bouldin index is 0, and a lower score represents better clustering. The scores in Table 1 indicate that the Gaussian mixture model produced the worst clustering. This is consistent with the cluster diagram in Figure 8. K-means performed the best in terms of Silhouette and Calinski-Harabasz indices, whereas hierarchical clustering performed the best in terms of the Davies-Bouldin index.  
Figure 8: Cluster visualization for GMM.  
Figure 7: Cluster visualization for hierarchical agglomerative.  
Overall, the results obtained in this work are consistent with the previous work in [19], where three clusters of charging behaviors were identified. However, the authors did not quantify their clustering results. The authors in [20] found four clusters using hierarchical clustering. They also did not present an evaluation of their clustering results. In our work, the quantified results are presented in Table 1. There are a few limitations in the clustering results in the proposed work. First, the number of clusters determined using the elbow method was used for all three algorithms. It is possible to experiment with more methods to determine the optimal number of clusters. Finally, the charging sessions were clustered based on the arrival and departure times. It is likely that using other parameters such as vehicle type and energy consumption could potentially lead to the discovery of more interesting clusters of behaviors.