**1. Introduction**

Distributed generation (DG) has been used to produce energy in remote and isolated places, where the distance between the demand and the producer is short [1]. At present, this trend is changing as it has been proven that DG provides technical, economic, and environmental improvements [2]. The benefits of using DG are based on the reduction of network losses, the voltage level improvement, or the dependence reduction of energy, fuel prices, and traditional generation. All these advantages cause CO2 emission reduction [3,4]. On the other hand, DG has also disadvantages, mainly related to technical aspects due to the fact that the existing networks have not been designed to incorporate this type of generation [2]. Some drawbacks are reverse current flows, the need for network redesign, or frequency instability [5].

In addition, planning of distribution networks must account for renewable generation uncertainty to meet the future demand in any possible future scenario. In that case, the decisions to be made could be to invest in the network, in substations, in DG generation, or in any combination of them [6]. When the chosen option is DG, it is necessary to determine the type of location, by solving the Distribution Generation Planning (DGP) problem. An example of this sort of problems can be seen in [7], where a linear model that optimizes size and location of DG is used. The objective function maximizes DG real power. In [6], a model is proposed for minimizing the investment and operation DG costs, the cost of the electricity bought from the substation, and the cost of network losses of a distribution company. The model is formulated as a mixed-integer-nonlinear one. Other authors have worked with particle swarm optimization methods [8], where a multi-objective model considers investment and operating costs of new generation, the cost of the energy purchased, and CO2 emission cost.

A number of references using Benders' decomposition to solve computational-complex problems in distribution [9–12] and transmission systems [13–17] can be found in the technical literature.

The DGP is addressed by a multi-objective mixed-integer linear problem in [9], using Benders' decomposition with an implicit enumeration algorithm where cost and reliability, among others, are included in the objectives' set. No renewable technology is modeled, just feeders and substations. Another multi-objective operation approach is developed in [10]. In this case, the aim is to minimize the total operational costs and emissions, as well as to generate Pareto-optimal solutions for the energy and reserve scheduling problem, using fuzzy decision-making processes. The scenario combination merges wind generation and forecasted demand. The problem is formulated as mixed-integer non-linear problem. Reference [11] studies the point of view of a local distribution company to maximize its profit, using nodal hourly prices within a smart grid. DG, including fossil fuel and renewable units, is taken into account. Reference [12] deals with the day-ahead unit commitment problem in a microgrid system. The problem is formulated as a stochastic mixed integer program that takes into account the uncertainty in PV generation.

A generation expansion planning model is proposed in [13] where the network is disregarded. It includes a generic model for renewable units (wind and solar) and hydro units modeled as storage units. Another generation expansion planning in generalized networks is studied in [14] considering production costs and system reliability in the lower level and the expansion plan in the upper level. A transmission expansion planning model is proposed in [15] incorporating the uncertainty of wind units via scenarios, including the cost of the added lines, and wind curtailment and using a DC power flow. Studies [16,17] propose complementarity models to determine the optimal investment decisions of a profit-oriented private investor interested in building new conventional and wind-power generating units, respectively.

The basis of this paper is a two-stage stochastic mixed-integer linear programming problem (MILP) that is used to determine where and when renewable energy sources should be allocated as part of generation planning in distribution systems [18]. The model proposed in [18] has an important limitation; namely, its computational burden is very high if a large number of scenarios and planning periods is considered. In the worst case, the problem may be even intractable.

Despite this relevant issue, the problem described in [18] has an interesting property: if investment variables are fixed, the problem can be decomposed per scenario and planning period. The proposed Benders' decomposition algorithm takes advantage of this decomposable problem structure to reduce the computational burden of the problem. This is the main benefit of the proposed approach.

Note that, to the best of our knowledge, there is no reference in the technical literature that considers a Benders' decomposition approach for a stochastic multi-stage DGP problem such as the one considered in this paper. The existing literature has not taken into account the benefits of Benders' decomposition for solve the DGP problem with a large number of scenarios of renewable energy. This problem is generally intractable for realistic case studies since it is necessary to consider a large number of scenarios and time periods to obtain informed expansion decisions. Moreover, it includes binary variables that further complicate the problem. Therefore, the traditional resolution methods, as a MILP, for stochastic multi-stage DGP problems are not viable. The Benders' decomposition model presented, although it has greater computational complexity, dramatically reduces resolution times. This model can solve problems with a large number of scenarios and planning periods. In addition, it is able to solve problems that are typically intractable with traditional methods.

Given the above, the contributions of this work are:

