1.1. General Context
Due to the growing concern about global warming, mitigating the emission of polluting gases has become an important issue in recent years [1
]. According to [4
], for 2018, the amount of
emissions caused by electricity generation increased by 1.7%. It is for this reason that, in many countries, the integration of photovoltaic (PV) systems into the grid is being supported and carried out by both utilities and government agencies, which has led to a structural change for energy suppliers, opening up competition in the generation and distribution markets [5
Additionally, the ability to collect solar energy and supply it in the form of electricity has become a reality thanks to the development of solar panels and power electronics-based converters [6
]. Therefore, companies are looking for new technologies to provide their customers with energy quality and reliability, so that the consumer, as a competitive decentralized generator, can use PV systems for various purposes, as is the case of residential or commercial use, seeking to supply a generated demand [6
]. This allows consumers to take advantage of the solar resource to supply their own demand and thereby save money.
One of the main advantages of implementing PV generators in distribution power systems is that they can improve the network voltage profiles, reduce the branch load level, and provide environmental benefits by reducing pollutant gas emissions if adequate planning and operation strategies are implemented [8
]. Otherwise, the high integration of PV systems in distribution networks can generate problems with regard to the network’s operating limits, as mentioned in [9
], which causes a poor energy dispatch and may imply economic penalties for not complying with operation restrictions. Therefore, one of the main challenges that arise when integrating PV systems into electrical systems is to correctly determine their best location and optimal size, as the technical-operating status of the electrical system can be seriously compromised with an inadequate integration of these devices [10
]. Consequently, the PV generator integration problem is part of the mixed-integer nonlinear linear programming (MINLP) family of models, which includes other planning problems for the expansion of distribution networks, such as the optimal reconfiguration of distribution networks, the selection of the optimal conductor size, and the location and sizing of existing and new substations, among others.
1.2. Literature Review
Some works have addressed the problem of integrating distributed energy resources into distribution networks for multi-period scenarios. The authors of [11
] used multi-period planning for the integration of multiple energy resources in microgrids, aiming for the minimization of investment and grid operation costs for all periods. The authors of [10
] proposed the integration of batteries and distributed generators based mainly on renewable energy sources at medium and low voltages. The problem was decoupled into two stages. First, a heuristic simulation-based algorithm defined the placement of the distributed energy resources, and the solution of the mixed-integer linear programming model defined their optimal daily outputs, which, together with test systems consisting of 11, 135, and 230 buses, demonstrated the effectiveness of the proposed approach in comparison with conic programming models. The authors of [12
] studied the problem regarding the optimal siting and sizing of air-based energy sources in distribution and transmission networks. This problem was represented through an MINLP model, which was solved through the GAMS specialized software. Their main contribution was the discovery of the capabilities of the reactive component of wind generator power to minimize the grid energy losses. The main issue with this study is that the authors did not consider any costs in the objective function, which implies that the systems may be oversized. The same concept was extended to high-voltage transmission networks in [13
] by considering PV generators with dynamic apparent power capacities. The exact MINLP model was also solved in the GAMS environment. However, the investment and operating costs of renewable energies were not taken into account. The study by [14
] proposed a convex optimization model based on second-order conic programming to minimize, through the integration of PV sources, the total greenhouse gas emissions of distribution networks in rural areas. With the proposed optimization approach, the search for the global optimum could be ensured. However, the authors did not include economics in the optimal sizing problem, and, as a consequence, it was not possible to apply this approach to real life. In light of the above, the main contributions of this research are the formulation of a MINLP model that formulates the optimal siting problem by minimizing the total costs of energy acquisition at the compensator node, as well as those associated with PV generator investment and operating costs, for a planning horizon of 20 years.
Similarly, PV panels, given their characteristics, can improve the voltage profiles of the power supply and reduce the load level of the branches. In addition, thanks to the use of the solar resource as a source of energy, no greenhouse gas emissions are generated when producing electricity [8
]. Therefore, a large number of PV generators have been installed in distribution networks around the world, implying that the number of PV systems in distribution networks should increase and be comparable to the output of the main source. In this case, PV systems could have serious implications for important technical aspects, such as the quality of power supplied to customers by power utilities, operational protection systems, and the islanding of PV systems [15
]. In practice, utility regulations dictate that, if the output is greater than 10% of power, PV systems must operate with a power factor greater than 0.85 (supply or return). In some cases, voltage compensation systems may be operated outside this limit with the consent of the utility company. To determine if the PV array has a significant impact on the supply voltage, the size and location of the PV array must be considered. This means that the voltage quality, influenced by the large number and location of grid-connected PV systems, becomes an important issue. This paper summarizes the results of a large number of random PV array installations in order to provide useful guidance for the future implementations of PV arrays in distribution systems. Therefore, within the framework of research on distribution networks, there are multiple methods and/or possible solutions to the problem regarding the optimal placement and sizing of photovoltaic sources with different approaches, such as reducing power losses, improving voltage regulation, and minimizing operating costs, among others.
In 2012, ref. [17
] presented the combination of genetic algorithms and particle swarm optimization with the objective of minimizing grid power losses, improving voltage regulation, and enhancing voltage stability with regard to system operation and safety constraints in radial distribution systems. In 2017, ref. [18
] presented an approach towards symbiotic organism search in order to minimize active power loss. A year later, ref. [19
] proposed the implementation of the population-based incremental learning algorithm, along with the use of particle swarm optimization to optimally locate distribution generators, and ref. [20
] suggested the use of the artificial bee colony algorithm to minimize power distribution losses and improve voltage profiles. In 2019, ref. [21
] presented a heuristic algorithmic approach aimed at the optimal allocation of distributed generation and capacitor banks, and ref. [22
] followed a master–slave optimization approach with the constructive heuristic vortex search algorithm, which works upon the basis of a Gaussian distribution and a variable radius function in order to explore and exploit the solution space.
Then, in 2022, the authors of ref. [23
] proposed a mixed-integer conic model (MIC), whose main contribution was to find the optimal global solution through the combination of the branch-and-cut method, which does not yield local optimal solutions, as is the case of other types of algorithms. In ref. [24
], the annual operating costs of distribution networks were reduced by means of the discrete-continuous vortex search algorithm. Finally, the authors of ref. [25
] located and sized PV systems by means of a generalized normal distribution optimization approach, and, in ref. [26
], the location and sizing of PVs was carried out using a modified gradient-based metaheuristic optimizer.
Finally, the objective function of many of the aforementioned studies aimed at minimizing the total electrical energy losses, which can be easily modified into a multi-objective problem when the optimal location and sizing of dispersed generation systems is required. Therefore, many optimization algorithms were used to meet the multiple design objectives. Furthermore, multi-objective algorithms can be divided into classical [27
] and artificial intelligence algorithms. Classical algorithms include several approaches, such as the analytical one [28
], which uses the iterative Newton–Raphson method for solving the load flow problem; linear programming [29
] in order to minimize the cost of energy sales according to the existing demand and maximize the benefit of customers who have distributed generation; dynamic programming [30
], which seeks to minimize distributed generation curtailment and load disconnection, considering the current and future cost of energy, the uncertainty of distributed generation, the reconfiguration of the grid, and the reduction of distributed generation; and the exhaustive search technique [31
], with which all possible inputs of an objective function are evaluated in order to determine the best input; even though the process is not efficient, it is reliable when it comes to arriving at the result. However, artificial and hybrid intelligence algorithms perform better than classical ones. The superiority of the former stems from their ability to find better optimal solutions at the cost of complexity. Many proposals have addressed dispersed generation planning using various artificial intelligence algorithms, such as ant colony search [32
], the Monte Carlo algorithm [33
], artificial bee colonies [34
], and hybrid techniques [35
]. The Jaya algorithm is among the intelligent algorithms proposed by Rao and published in 2016 [36
]. This algorithm aims for victory in finding the best solution, hence the name Jaya
(a Sanskrit word meaning victory
). The authors of [37
] used a modified version of the crow search algorithm, adapting it into a discrete–continuous form for the location and sizing of PV systems in distribution networks. So far, this has been the study that has yielded the best results regarding the reduction of power purchasing, investment, and operation and maintenance costs associated with PV systems. Here, the proposed methodology is modified and implemented in order to estimate the optimal capacities and the best locations for very high-penetration PV systems.
A summary of the algorithms that have been implemented, along with their year of publication, can be found in Table 1
1.3. Motivation, Contributions, and Scope
In order to find a strategy for reducing the time and computational resources required to perform an optimization process, this study proposes the communication between two specialized software applications in order to achieve the best locally optimal solution. As evidenced above, different optimization methodologies have been implemented which allow obtaining a solution to the problem regarding the location and sizing of PV generators in distribution systems. However, most of these solutions can get stuck in local optima, which is why this paper proposes an optimization methodology that interconnects the interfaces of MATLAB and GAMS through a master–slave methodology, with the aim to solve the problem under study. The main advantage of this methodology is that it allows said problem to be divided by stages in order to find a high-quality solution. The master stage was programmed in MATLAB, where the discrete version of the sine–cosine algorithm was used to determine the location of the PV generators. In addition, in the GAMS software, using the BONMIN solver and the locations proposed in the master stage, the mathematical model that represents the problem is solved, thus finding the optimal size of the PV generators and an objective function value that respects all the technical-operating conditions of the system. The main contributions of this research are:
A complete description of the mathematical formulation representing the problem of the siting and sizing of PV generators in distribution networks while considering the deactivation of maximum power point tracking.
A new optimization methodology based on the interconnection of MATLAB and GAMS that allows finding the best locally optimal solution to the problem under study.
A new master–slave methodology to solve the mathematical model representing the problem under study. In the master stage, the MATLAB software is used as a tool to develop the discrete version of the sine–cosine algorithm, with the aim to determine the locations of the PV generators. Then, in the slave stage, GAMS is used to solve the MINLP model that represents the studied problem, thus yielding the objective function value and the necessary nominal power to be generated by the PV systems.
It is worth mentioning that this research identifies, as an opportunity for research, the possibility of exploiting the advantages of combinatorial optimization methods in order to solve optimization problems with discrete variables, as is the case of the sine–cosine algorithm and its implementation in the MATLAB software, with the effectiveness and robustness exhibited by the nonlinear programming tools available in the GAMS software. This, in order to obtain an efficient solution methodology that allows the improvement of the reported results regarding the optimal placement and sizing of PV generation units in AC distribution grids. Note that the numerical results of this research will be regarded as the new reference for the studied problem for future research focusing on the area of renewable generation and its applications for electrical distribution networks.