# Hybrid GA-SOCP Approach for Placement and
## 1 Introduction  
Recently, the study and analysis of direct current (DC) networks have increased, due to the development of power electronics, advances in renewable generation, and the advantages over alternative current (AC) networks [1; 2]. Initially, the DC networks began to operate at high-level voltages in the power transmission for long-distance between systems, since it allows for reducing transmission power losses, enhancing grid stability systems, easily incorporating off-shore wind systems, and managing power flow both direction simply [3; 4; 5]. Consecutively, the integration of the DC networks has been widened to medium- and low-voltage level applications, generating concepts, such as the microgrids [6; 7]. The main advantages of DC systems as compared to conventional AC systems can be summarized, as follows: (i) fewer power converter devices, (ii) no requirement of synchronizing generators, (iii) better voltage profiles, (iv) less power losses, and (v) do not require frequency and reactive power analysis and controls [8; 9; 10]. Two kinds of analysis are used to study DC networks. The first analysis is dynamic, being performed under the time domain, and integrates concepts as primary and secondary controllers [11; 12]. The second analysis is static, which computes all of the state variables under stationary conditions. This analysis is known as tertiary control [13], which addresses power flow analysis [2; 14], optimal power flow [15], economic dispatch strategies [16], and voltage stability analysis [17; 18]. Furthermore, these analyses are integrated with other problems, such as the placement and sizing of multiple distributed generators (DGs) [19], as well as placement and operation of battery energy storage systems in DC networks [20]. In this study, we focus on the problem of the optimal placement and sizing of DGs in DC networks. Although, in the specialized literature, there are few approaches in this area. In [21], an analytical methodology for sizing energy storage systems in DC networks was proposed. This methodology evaluated the profiles of currents that were injected by energy storage systems in order to minimize power losses. In [22], a feasibility study for the integration of photovoltaic systems in a standalone building was carried out. This study considered strong weather scenarios for off-grid systems and variable electricity prices for the grid-connected system. In [23], a probabilistic optimization methodology for estimating uncertainties in loads, plug-in electric vehicles, and renewable generation in DC distribution networks was analyzed. In [19], a hybrid strategy for optimal location and dimensioning of DGs in DC grids was provided, which mixed a hyperplanes search method (HSM) with semidefinite programming (SDP) model. The HSM relaxed the binary variables, while the SDP computed the optimal power flow. In [24], a similar strategy using a sequential quadratic optimal power flow was presented. Both of the techniques ensure the optimal global solution only for a relaxed problem, where the complexity lies in the number of candidate solutions to recover the binary variables. In [25], a tutorial in the general algebraic modeling system (GAMS) for the optimal placement and dimensioning of DGs in DC grids were presented. In [26], the optimal placement and sizing of photovoltaic generators in DC networks were carried out, when considering an environmental-economic dispatch as an objective function. However, the approach presented in [26] was solved in GAMS, which does not guarantee the optimal global solution, since the problem is nonlinear non-convex mixed-integer. In [27], three metaheuristic methods for optimal location and dimensioning of DGs in DC systems were studied. A classical genetic algorithm was employed in the locations stage, which assigns the location of the DGs. While the black-hole, continuous genetic algorithm, and particle swarm metaheuristic methods were used to dimension the DGs.