## 1 Introduction
A cost minimization is used in [24] to find the optimal battery size in active distribution networks. Based on a multi-temporal constrained optimization problem, in [25] the sizing and placement of storage devices are addressed by minimizing investment and operation costs. In [26; 27; 28; 29; 30; 31] the planning problem has been treated with specific reference to LV grids. Particularly, these papers aim to mitigate the impact of renewable generation [26; 27; 28], to improve reliability [29] and to reduce costs [30; 31]. In [27; 28; 29; 30] the problem of the unbalances is also discussed within the framework of the storage planning issue. In this paper, an unbalanced LV microgrid is considered and the allocation (siting and sizing) problem is formulated as a mixed, non-linear, constrained optimization problem whose objective function deals with economic issues and whose constraints take into account technical requirements of both network and distributed resources. This planning problem is quite complex and involves mathematical models of high dimension. As discussed in [10], in fact, the use of storage devices requires to face with problems of intertemporal nature: the operation of the storage at a time step will affect its operation at the following time steps. Moreover, computational complexity is particularly cumbersome in case of LV unbalanced grids, for which three phase representations are required for all the grid components, and when load patterns during the years have to be taken into account in the planning problem. This implies that simplified approaches or approximations in the models have to be considered in order to make the study possible even in realistic applications where the problem is further complicated by the presence of large number of buses. Analysis of new methods that can limit the computational burden of the optimization problem is of great importance. In our opinion, a possible solution to this problem can be to reduce the computational burden by containing the solution's search space of the planning problem into a limited number of buses of the grid, which are considered as the \"best\" for allocating DESSs (\"allocation bus set\"). Therefore, in this paper the computational complexity is overcome by solving the mixed, non-linear,constrained optimization problem of DESS allocation, through a two-step hybrid procedure: (step 1) determine the limited subset of buses to be included in the reduced search space (allocation bus set); and (step 2) find the optimal solution in this reduced region by applying a genetic algorithm (GA). The buses of the allocation bus set (step 1) are obtained by selecting among all the network buses suitable for the DESS siting (i.e., theoretically all the network buses) a subset of buses that includes only the buses whose injection has the expected better favorable impact on the network constraints satisfaction. These buses are selected by applying the Inherent Structure Theory of Networks (ISTN) and the Loading Constraints Criterion (LCC). The ISTN is based on the reformulation of network admittance and impedance matrices with respect to their eigenvalues and eigenvectors and was first proposed by [PERSON] in [32; 33] and successively extended to solve other important problems in the field of power systems [34; 35; 36; 37; 38; 39]. The LCC was proposed in the frame of the European project ADDRESS to identify different sub-networks of distribution systems, referred to as Load Areas, by evaluating the influence of prosumers' injections on the relevant network constraints [40; 41]. These techniques have never been applied to the DESS allocation problem so far. The GA (step 2) includes an inner algorithm that performs an optimization aimed at finding the optimal daily charge/discharge cycles of storage batteries. The main original contribution of this paper is the proposal of a complex and detailed planning problem for DESS allocation in an unbalanced LV microgrid which drastically reduces the computational burden.