## 1 Introduction  
Distributed energy resources (DERs) are an attractive technology to install in power distribution networks due to their technical, economic, and environmental benefits [1]. In general, a DER is a small power plant that can employ different kinds of energy source such as small-hydro, micro-turbine, fuel cell, methane-generating bioreactors, gas-turbine, and renewable distributed generation (wind and solar power). Typically, DERs are located close to the customers in a distribution network, which has allowed decreasing power losses associated with transmission lines and improving the voltage profiles [2, 3]. However, DER can generate adverse effects in electrical distribution networks, such as dis-coordination of protection schemes, increased current fault level, bidirectional power flows, transmission line overload, and voltage profile deterioration [4] if these are not correctly placed and sized. An optimization model for the optimal location and sizing of DERs is required to avoid these possible problems. This problem has been extensively studied in the literature with several mathematical formulations and methodologies. The problem is complicated to solve since its formulation combines discrete and continuous variables with nonlinear/non-convex constraints. Therefore, the global optimum cannot be guaranteed [5]. The problem has been typically solved using master-slave strategies [6]. The master strategy chooses the DERs placement and size, while the slave strategy solves an optimal power flow problem. For the master strategy, metaheuristic optimization techniques have been commonly implemented, such as Chu-Beasley genetic algorithms [7], tabu search algorithms [8], simulated annealing methods [9], ant-lion optimizers [10], whale optimization algorithm [11], krill herd algorithms [12], teaching-based learning optimizers [13], population-based incremental learning [6], imperialist competitive algorithms [14], harmonic search algorithms [15], and bat and firefly algorithms [16], among others. Although these optimization techniques ensure good solutions, none of them can guarantee the optimal solution, and some are not well justified mathematically [17]. These techniques include some degree of randomness, which leads to slightly different solutions from one execution to another. In addition, they contain many adjustment parameters that make them highly dependent on the programmer, adding to the fact that statistical analysis must be required to determine their reproducible capacities [18; 19; 20]. Recent approaches have combined heuristic methods with second-order cone programming (SOCP) models to solve the problem of the optimal location and sizing of DERs in distribution networks as the case of the discrete version of the sine-cosine algorithm reported in [21] and the genetic algorithm proposed in [22] however, these approaches only ensures the optimal solution for the continuous part of the problem, i.e., the DERs' sizing problem, while for the integer part, i.e., the DERs location problem, it is not possible due to the heuristic nature of the discrete algorithms. The heuristic nature of the the integer part does not allow for ensuring the global optimum reaching, which implies that statistical test are required to verify their efficiency [19]. Unlike heuristics, this paper proposes an exact formulation based on recent advances in convex optimization for power systems applications [23]. The optimal placement and sizing of DERs are transformed into a mixed-integer second-order cone programming (MI-SOCP) model. The problem is solved via the Branch & Bound method, which guarantees optimal global optimization if each sub-problem of optimization is convex. For this purpose, the optimal power flow becomes a convex problem through a relation. Thus, the main contributions of this work are summarized below:  
* A reformulation of the classical model represents the problem of the optimal location and sizing of DERs in power distribution networks by using an MI-SOCP model.