## 1 Introduction  
The concept of resilience has been increasingly used in the literature related to distribution networks (DN) with high penetration levels of distributed energy resources (DER), with an interest in providing greater autonomy and flexibility in the face of unexpected power generation situations [1]. In this context, microgrids (MG) play a relevant role in the transition process to smart grids because they offer an electrical structure that allows for improving the monitoring and management control of the renewable resources on a small scale [2; 3]. However, following the definition of MGs given by the U.S. Department of Energy [4], a fixed electrical boundary may not be exploiting all the flexibility that an MG provides. The concept of virtual microgrids (VM) with dynamic boundaries has begun to emerge as an alternative for increasing the resilience of conventional DNs [5; 6] based on the IEEE Standard 1547.4 [7], which establishes that the reliability and operation of DNs can be improved by dividing the DN into multiple \"isolated systems\". Discussions regarding a common definition of VMs with dynamic limits continue. However, there are some characteristics in common in the work that has been done; the work by [8] proposes that a VM is developed from a centralized DN, and possesses the same features of an MG but with a flexible structure based on virtual boundaries. The work done by [9] considers a dynamic MG as one with flexible boundaries that expand or shrink to maintain the balance between power generation and power consumption. The work in [10] considers the paradigm of a dynamic MG as a mechanism to continuously partition a DN based on the balance between electricity generation and demand. The key factor in the definitions above is the capability of the DN to find the best partition such that it minimizes the load curtailment, and balances the power system under failure or unbalanced scenarios produced by the stochastic behavior of the DG. Likewise, this responsiveness is based on the autonomy level of the active distribution network (ADN), i.e., the installed capacity of DG and batteries. Thus, the higher the installed capacity is, the higher the autonomy would be. However, to meet 100% of the electricity load with only DERs requires a high investment, which could make the implementation of these technologies in a DN unprofitable [11] and/or bring voltage and frequency issues [12]. Therefore, in most cases, a trade-off occurs between the energy provided by DERs and the utility grid, where the grid helps the DERs fill the energy gap in the critical hours and decreases the installed capacity needed of the renewable resources [13]. Thus, the necessity of partitioning the ADN into VMs under failures of the utility grid increases because, under failure scenarios, the gap cover by the grid, when the DERs do not meet the load, will produce an unbalanced system. That is, unless the ADN faces this issue by dividing itself into virtual \"island systems\". Some clustering techniques [14; 15] and methodologies have been applied to find an optimal partition that allows the network to adopt this dynamic structure. A Genetic Algorithm has been used in [16] to partition a DN to minimize the energy exchange between MGs. The work done by [8] proposes a hierarchical bottom-to-top algorithm based on community detection [17] using an Electrical Coupling Strength as modularity index, which is defined as a composite weighted factor for the analysis of the electrical connection, to size the partition quality. In [18], a Greedy and Particle Swarm Optimization (PSO) algorithm based on community detection is used to find sub-communities through an index based on the voltage sensitivity matrix and the active/reactive power. The Greedy algorithm is used to find the best partition and the PSO to manage the voltage control for each MG. The electrical distance is used in [19; 20] to partition a DN into sub-systems.