## 1 Introduction  
Electric power systems are composed of four sectors: generation, transmission/sub-transmission, distribution, and commercialization; the transmission and distribution networks are in charge of connecting the generation and consumption points over hundreds or thousands of kilometers [1]. In general, electric power transmission and distribution networks differ in the voltage levels at which they operate, which implies transmission voltages greater than 220 kV, between 57 and 220 kV for sub-transmission, and less than 57 kV for distribution [2]. However, the predominant voltage energy distribution level in Colombia ranges between 11.4 and 13.8 kV, respectively [3]. These electric networks present differences in their characteristics since the transmission usually uses meshed systems, and the distribution uses radial characteristics [4, 5]. Due to these differences in electrical energy losses, the transmission levels are between 1.5% and 2% of the energy generated in peak hours, while the energy distribution can vary between 6% and 18%. The above-mentioned implies that, in the worst case, 18% of the distribution level energy is transformed into heat in the resistance lines and transformers in the distribution mainly [3; 6]. Given the high levels of losses in the energy distribution, it is necessary to quantify them to determine an optimal solution to this problem [7]. There are different methods to reduce losses; for example, by reconfiguring the network using the binary gravitational search algorithm based on the algebraic graph theory [8]. Other examples include, through energy storage systems [9], and using particle swarm meta-heuristic optimization and genetic algorithms, with the inclusion of a static distribution compensator D-STATCOM that provides reactive power [7], and through the location and optimal size of distributed generation [8]. This article aims to reduce losses in distribution networks using the localization and the optimal size of PV systems, i.e., PV generators, in distribution systems with 33 and 69 nodes [6; 10]. For this analysis, the generators inject active power according to their typical behavior during hours of solar radiation. Furthermore, the characteristic demand curve of the distribution networks is presented in this study. Due to the growing interest in solar energy for its benefits to the clear contribution of the trimming of CO2, in recent years, there has been an increase in research regarding this renewable energy, which focuses on issues associated with optimization, performance, and solar systems cells, among others, as mentioned in [11]. It means that the field of application of solar energy includes a wide variety of energy solutions, such as carrying out studies to reduce power and energy losses that integrate energy sources of renewable generation. For this, it is of vital importance to resort to metaheuristic techniques through algorithms. These algorithms use advantages that vary according to the implemented numerical optimization methods, which are adaptable algorithms to any problem. This variety allows finding a space solution that reduces losses [12] and costs of the distributed generation [13], which is because the algorithm provides the information necessary to determine the optimal solution. The study of losses is a complex mixed integer nonlinear programming problem (MINLP). The MINLP structure of the optimization problem complicates the possibility of finding a globally optimal solution due to the non-convex shape of the solution space [14]. Therefore, a hybrid discrete-continuous modification of the vortex search algorithm (DCVSA) is proposed as a solution. So, it is necessary to use the unconventional method known as successive approximations (SA) [15] to evaluate the power flow problem. The DCVSA algorithm works through an optimization strategy that consists of a master stage responsible for determining the optimal location and size of the PV systems.