###### Abstract  
The problem of reactive power compensation in electric distribution networks is addressed in this research paper from the point of view of the combinatorial optimization using a new discrete-continuous version of the vortex search algorithm (DCVSA). To explore and exploit the solution space, a discrete-continuous codification of the solution vector is proposed, where the discrete part determines the nodes where the distribution static compensator (D-STATCOM) will be installed, and the continuous part of the codification determines the optimal sizes of the D-STATCOMs. The main advantage of such codification is that the mixed-integer nonlinear programming model (MINLP) that represents the problem of optimal placement and sizing of the D-STATCOMs in distribution networks only requires a classical power flow method to evaluate the objective function, which implies that it can be implemented in any programming language. The objective function is the total costs of the grid power losses and the annualized investment costs in D-STATCOMs. In addition, to include the impact of the daily load variations, the active and reactive power demand curves are included in the optimization model. Numerical results in two radial test feeders with 33 and 69 buses demonstrate that the proposed DCVSA can solve the MINLP model with best results when compared with the MINLP solvers available in the GAMS software. All the simulations are implemented in MATLAB software using its programming environment. discrete-continuous vortex search algorithm; radial distribution networks; distribution static compensators; annual operational costs minimization; reactive power compensation; daily active and reactive demand curves +
Footnote †: journal: Applied Sciences  
19 January 2021  
5 February 2021  
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**1. Introduction**  
Around the world, electric distribution networks are the channels through which electricity is supplied to millions of end-users at medium- and low-voltage levels [1]. These grids are constantly submitted to increments in the load consumption each year. Therefore, these networks must be prepared to receive new users and guarantee the quality, reliability and security of the public service [2, 3]. Due to the operational voltages, i.e., typically between 10 kV and 15 kV for the Colombian medium-voltage distribution grids, these electrical networks experience higher energy losses as compared to the transmission and sub-transmission systems [4]. In the Colombian context the power system has energy losses of about 1.5% to 2.0% in terms of the total energy transmitted. In the medium-voltage grids energy losses can vary from 5% to 18%, and the lowest losses are found in networks of the grid operators who invested to improve the quality of their grids [5]. The difference between energy losses is principally caused by the radial topology which is typically adopted while building the distribution networks to minimize investment costs. Thistopology is in contrast with the strong meshed structures that are found in power systems. The main problem with the radial topology of the electric distribution grids is the high percentages of energy losses caused by the electricity distribution tasks [6]. To reduce the energy losses in the distribution service, the Utilities can use various methodologies, such as (i) grid reconfiguration [7], (ii) optimal location of shunt capacitors [8] and (iii) optimal location of dispersed power generation plants [9]. These approaches can achieve different levels of reduction of energy losses. Of these methodologies, distributed generation is the best option for energy loss reduction. However, its initial installation costs can be very high as compared to the reconfiguration and shunt capacitor approaches.