1. Introduction
Medium-voltage distribution networks cover big areas, with hundreds of kilometers in both urban and rural zones, to attend to the electricity end-users [
1]. A distribution grid essentially interfaces the power systems in transformation nodes (i.e., substations) with consumers at medium- and low-voltage magnitudes to provide a quality, reliable, and secure service [
2]. The main feature of an electrical network is mainly associated with its topology, as these grids are normally structured in radial form to simplify the protection schemes and reduce investment costs in all infrastructures related to conductors [
3]. The radial characteristics of these networks result in considerable energy losses when compared with the transmission/subtransmission grids [
4].
On the other hand, the harmful effects of global warming have significantly increased the interest in the reduction of greenhouse gas emissions from fossil fuels [
5], that is combustion of coal, diesel, natural gas, and in general, petroleum-derived products [
6]. Renewable energy resources are essential elements in the evolution of the electrical sector to achieve this goal, as this is the third emitter of greenhouse gas emissions after beef and pork production [
7] and transportation systems [
8], respectively. In the context of renewable generation, wind turbines and solar photovoltaic (PV) sources have been widely developed in recent decades, which have allowed essential reductions in the investment and maintenance costs of small-scale installations, making these technologies suitable for installation in medium- and low-voltage distribution grids [
9].
For tropical countries, that is countries located near the Equatorial Line (as in the case of Colombia), the most suitable technology to produce clean energy and replace thermal generation from fossil fuels is PV generation since solar radiation suffers minor variations [
10]. Instead, wind power is only suitable in coastal areas [
11]; thus, large continental territories are relegated to other types of renewable energy resources, that is mainly PV sources [
12]. An important aspect that must be considered in the optimal integration of PV generation units is the high variability of the energy production due to the cloud-induced fluctuations, which can produce important changes in the total power generation in intervals on the order of seconds [
13]. A complete analysis regarding these fast energy generation fluctuations (due to partially shaded conditions) and their forms was reported in [
14].
On the other hand, the low cost and high expansion of PV sources along the distribution grids makes the optimal design and inclusion of this technology not an easy task. This is due to bad planning, which can cause over-voltages and over-currents in nodes and distribution lines, energy loss increments, misoperation of protective devices, and deterioration of the quality of service, among other problems [
15] (a complete review regarding problems derived from renewable energies in power systems can be found in [
16]). To avoid those problems, distribution companies need to efficiently plan these grids, which requires an optimal integration of PV generation sources. Such a process must be performed by solving the mixed-integer nonlinear programming (MINLP) model that represents the problem of the optimal siting and sizing of the generation sources in distribution grids [
17].
The current literature has addressed the problem of the optimally sizing and integration of renewable energy resources from two different perspectives. The first approach considers only the technical improvement of the grid, that is power loss reduction [
18], voltage profile improvement [
19], and voltage stability improvement [
20]; however, the main problem of those approaches is related to the economic feasibility of those projects. The second approach deals with the optimal integration of renewable generation technologies, which considers a planning period based on an economic indicator as the objective function. This approach has the main advantage of ensuring that the final solution is technically and economically feasible [
17].
Some approaches, recently reported in the scientific literature, consider economical objective functions in the problem of the optimal siting and dimensioning of renewable energy resources. For example, the application of the Chu and Beasley genetic algorithm (CBGA) using discrete–continuous codification was proposed in [
21], which addresses the problem of the optimal siting and dimensioning of PV sources in medium-voltage distribution grids. Computational validations in test feeders formed by 33 and 69 buses demonstrated the efficiency when numerical results were compared with the exact solution of the MINLP model in the general algebraic modeling system (GAMS) and the BONMIN solver. Valencia et al. [
9] proposed a linear approximated model to size and site batteries and renewable energy resources in distribution networks. The optimal location of those devices (integer problem) was left to a classical simulated annealing algorithm, and the operation problem (sizing problem) was solved using a linear programming model. Test feeders from 11 to 230 nodes were used to validate the effectiveness of the proposed model; nevertheless, no comparisons with other optimization methods were provided, which is the main problem of that work since it is impossible to ensure that the solution provides global optimization. In [
22], the application of a new metaheuristic optimizer called the Newton-metaheuristic algorithm (NMA) for the same problem was presented. Numerical results in test feeders with 34 and 85 nodes demonstrated the effectiveness of this optimization method through comparisons with the GAMS and CBGA, respectively. In [
23], the authors reported the application of the particle swarm optimizer to locate and size PV sources and energy storage systems simultaneously. The main contribution of this research was the complete economic analysis made by the authors, which included the installation, maintenance, and operating costs of the devices. However, the authors simplified the optimization problem by considering a single nodal model for the distribution grid, which reduced the exact MINLP model to an operative nonlinear programming model. Thus, the component associated with the optimal location of the renewables and batteries remains unsolved. Cortes-Caicedo et al. [
17] presented a discrete–continuous version of the vortex-search algorithm (DCVSA) for the PV location and sizing problem. Simulations in the IEEE 33-bus and IEEE 69-bus grids demonstrated the effectiveness of the proposed approach through comparisons with the exact solution in the GAMS software and the application of the CBGA, respectively. A two-stage methodology based on the combination of a mixed-integer quadratic convex model to decide the location of the PV sources and the optimal power flow (PF) solution via the interior point method to determine the PV sizes was proposed in [
12]. Numerical results obtained in this work demonstrated that the method reached similar results to the CBGA and the NMA approaches for the IEEE 33-bus and IEEE 85-bus grids.
In addition, optimization methodologies for the locating and sizing of PV generation units in distribution networks, considering technical or economic objective functions, include the Jaya optimization algorithm [
24], the heuristic-based local search algorithm [
25], the modified gradient-based metaheuristic optimizer [
26], the mixed-integer linear approximation [
27], the multi-criteria decision system [
28], and recursive simulations using multiple PF evaluations [
29], among other methods. The main characteristic of those approaches is that they overcome the problem of location from the problem of sizing. The former problem is solved with sensitivity analyses or heuristic algorithms, and the latter problem is solved using multiple PF evaluations.
Considering the previous revision of the literature, the main contribution of the article is the following: the application of a recently developed generalized normal distribution optimizer (i.e., GNDO), with a discrete–continuous codification, to solve the problem of the PV location (selection of nodes) and optimal sizing in the master stage. This solution allows transforming the MINLP model into a simple PF problem for distribution networks, which is solved using the successive approximation power flow (SAPF) method in the slave stage. Simulations in the IEEE 33-bus and IEEE 69-bus grids confirmed the effectiveness of the proposed optimization method, since the final objective function values were better than those provided by current literature approaches. In this way, the satisfactory results reported by the vortex-search algorithm in [
17] were improved by approximately USD 89.95 and USD 341.18 per year of operation, respectively.
It is important to remark that the GNDO has not been previously applied to problems of distribution system planning; thus, it is as an opportunity for research addressed in this work. The GNDO algorithm was proposed in 2020 by Zhang et al. [
30] to extract the parameters of PV modules with different numbers of diodes. However, that is an optimization problem defined in the domain of the real variables (i.e., continuous variables). Therefore, the work reported in this paper adapted the GNDO method to deal with a mixed codification, which combines integer variables related to the buses where the PV sources will be assigned and continuous variables describing their optimal sizes.
The rest of the paper is organized as follows:
Section 2 shows the exact MINLP model for the studied problem using a mathematical representation in the complex domain. Then,
Section 3 describes the main features of the proposed MS optimization approach, which combines the GNDO in the master stage with the PF method based on the successive approximations in the slave stage.
Section 4 presents the test feeder characteristics, including their peak load behavior, demand, and generation curves; moreover, this section also describes the parametrization of the objective function.
Section 5 presents the complete analysis and discussion of the results obtained by the proposed MS optimizer and provides comparisons with approaches recently reported in the scientific literature. Finally,
Section 6 lists the main conclusions of this work.