## 1 Introduction  
Electrical distribution networks are responsible for transporting energy from power systems to end-users in medium- and low-voltage levels [1]. These are typically fed by substations that keep voltage and frequency variables as constant as possible, in the case of alternative current (AC) networks, or voltage magnitude, as in the case of a direct current (DC) distribution network [2]. Due to the use of voltage levels in distribution networks and their lengths (also its radial configurations), these networks have higher power losses as percentages when compared to transmission networks [1; 3]. Multiple methodologies have been reported to improve the performance of electrical distribution networks in terms of power losses and voltage profiles; some of the classical ones are the following:  
* The optimal location and sizing of capacitor banks [4; 5; 6]. * The optimal location and sizing of distributed generators [7; 8]
* The optimal grid reconfiguration [9; 10; 11]. * The optimal location of voltage regulators and series capacitor banks [12; 13; 14]. Note that in the case of DC networks, the most extended approach is the optimal location and sizing of distributed generators. This is the most reported approach in recent scientific literature where DC technologies are gaining more relevance in distribution scenarios [15]. Some of the main characteristics of the rapid advances in DC distribution are the following: (i) lower power losses and better voltage profiles due to the nonexistence of reactances in distribution lines, and (ii) easy controllability as the concept of frequency does not exist, which implies that controls are only focused on constant voltage support. In addition, multiple distributed energy resources, such as batteries and photovoltaic generators, are typically operated under the DC paradigm, which implies that they are easily integrated into these kinds of grids in contrast to AC networks. This is because a lower power electronic interface is needed to connect them in DC networks, which can reduce costs compared to conventional AC networks [15]. Based on these advantages of DC grids over AC grids, in this study, we focus on the problem of the optimal locations and sizes of distributed generators in DC distribution networks. This problem has been largely studied with regard to AC distribution networks [16]. In the case of DC networks, few approaches have been reported in the literature. The authors of [17] proposed a hybrid approach for the optimal location and sizing of dispatchable distributed generators in DC grids. They combined a semidefinite programming model for an optimal power flow analysis with the hyperplane search method, which relaxes the binary variables. A similar approach based on a sequential quadratic optimal power flow was proposed in [18]. Both optimization models reached the optimal global solution for this problem system with 21 and 69 node test feeders. However, the complexity in their implementations lies in the number of hyperplanes that were needed to be evaluated, as these depend on the number of candidate nodes provided by the relaxed optima power flow (OPF) model that can significantly increase the total processing time required for the solution of the problem. In [19], a tutorial was provided on the optimal location and sizing of distributed generators (DGs) in DC grids using the general algebraic modeling system (GAMS), while in [20], the same software was used for the optimal sizing of photovoltaic sources in DC grids while keeping in consideration a day-ahead environmental dispatch. Both approaches present the main advantages of implementing optimization models in GAMS; however, due to the nonlinear, non-convex mixed-integer structure of the studied problem, these approaches do not ensure the optimum global finding. The authors of [21] presented multiple metaheuristic approaches for the optimal location and sizing of DGs in DC grids.