Parametric Forecast of Solar Energy over Time by Applying Machine Learning Techniques: Systematic Review

Abstract

To maximize photovoltaic (PV) production, it is necessary to estimate the amount of solar radiation that is available on Earth’s surface, as it can occasionally vary. This study aimed to systematize the parametric forecast (PF) of solar energy over time, adopting the validation of estimates by machine learning models (MLMs), with highly complex analyses as inclusion criteria and studies not validated in the short or long term as exclusion criteria. A total of 145 scholarly sources were examined, with a value of 0.17 for bias risk. Four components were analyzed: atmospheric, temporal, geographic, and spatial components. These quantify dispersed, absorbed, and reflected solar energy, causing energy to fluctuate when it arrives at the surface of a PV plant. The results revealed strong trends towards the adoption of artificial neural network (ANN), random forest (RF), and simple linear regression (SLR) models for a sample taken from the Nipepe station in Niassa, validated by a PF model with errors of 0.10, 0.11, and 0.15. The included studies’ statistically measured parameters showed high trends of dependence on the variability in transmittances. The synthesis of the results, hence, improved the accuracy of the estimations produced by MLMs, making the model applicable to any reality, with a very low margin of error for the calculated energy. Most studies adopted large time intervals of atmospheric parameters. Applying interpolation models can help extrapolate short scales, as their inference and treatment still require a high investment cost. Due to the need to access the forecasted energy over land, this study was funded by CS–OGET.

1. Introduction

Numerous mechanisms that govern life on Earth are primarily driven by solar energy emitted on to the surface of the planet [1,2]. The best way to ensure that life continues as long as possible is to make the best use of this resource and turn it into a non-polluting technology of natural environmental cycles in order to create an endless supply of non-polluting energy [3], also known as renewable or green energy, that is sustainable and has low levels of pollution and energy consumption [4]. The rate of rural electrification is still very low, with approximately 80% of communities lacking access to electricity [5]. This lack is prevalent in many sub-Saharan African communities, particularly the Nigerian region, with Mozambique ranking sixth, with about 20 million people lacking access to electricity [6], many of whom live in rural and suburban areas [7]. The Sustainable Development Goals (SDGs), however, aim to minimize polluting gas emissions by 2050 and mandate electrification by the end of 2030 [5,8,9]. Fossil fuels contribute to global warming by their continued widespread use in many parts of the world today; thus, other energy sources are employed, such as nuclear power plants [10,11], which endanger human health and the environment, and hydroelectric power plants [12], which by themselves cause ecological harm and droughts [4,13]. The SDG aim of lowering net emissions by 2050 [12] is inspired by serious environmental issues [14]. Due to environmental concerns, increased energy needs, fluctuating fuel prices [15], rising costs, and the possibility of fossil fuel depletion, renewable and environmentally friendly green technologies have become more popular [14,16]. In order to make solar energy plants more competitive in the energy market [17] and decrease dependence on fossil fuels [18] for social and economic growth, solar energy forecasting is essential [19,20]. In light of the SDGs for global electrification, PV solar energy is becoming more and more popular as a low-cost, clean, efficient, and sustainable technology [6,15,21] that can be purchased anywhere around the globe [3]. One of the biggest challenges of the modern age is optimizing its operation [22,23]; understanding the many local solar input factors that are taken into account when sizing and installing PV power plants would be the best course of action to enhance these projects’ efficiency [3,15,16,17,18,19,20,21,22,23,24]. An estimated 80% of installed projects globally [25,26] exhibit various anomalies, ranging from solar optimization to appropriate device selection in the sizing model [27,28] and even analyzing energy variability across the regions of interest [2].

In order to achieve grid balance, recent studies have employed MLMs and methodologies to optimize and estimate solar energy utilizing samples of solar, wind, and tidal energy on multiple distinct time scales [29,30]. One study found that their ANN predicted 24 h global horizontal irradiation (GHI) well; on bright days, the correlation coefficient was between 98 and 99%, while on overcast days, it was between 94 and 96% [31,32,33]. In order to estimate GHI and the clear-sky index $\left(K_t^{*}\right)$, researchers used hybrid models to forecast GHI in an island context, and the results showed potential for solar projections [34]. After analyzing aerosol optical thickness (AOT) over irradiance estimated using a normal aerosol scenario, it was concluded that, on average, +9 $\mathrm{W}/{\mathrm{m}}^2$ or $+2\%$ approximated the world mean irradiance values very well, where the errors in direct normal and diffuse horizontal irradiance (DNI and DHI) balanced each other [35,36]. In addition to projecting solar energy and electrical load, it was observed that adjusting wind speed and irradiance from numerical weather prediction presented difficulties [37,38]. Precise energy forecasting from geographically dispersed solar generation resources was defined as crucial to enhance the penetration of solar generation resources into the electrical grid [39]. The researchers based their short-term solar forecasting solely on the measurements of GHI, coming to the conclusion that alternative models, while configurable, require further observations of several atmospheric parameters [40]. MLMs for daily solar energy forecasting using numerical climate models provide a large number of features that can deteriorate learning algorithms’ ability to generalize [41]. Forecasting power generation from renewable energy sources (RESs) like solar panels and wind turbines is becoming crucial for the power grid to run smoothly and continuously. It also aids in making the most use of RESs [42]. Significant sensitivity to variation in the forecast horizon and significance with regard to location selection were indicated by an analysis of the forecast horizon’s sensitivity to solar irradiance and ambient temperature [43]. One study examined the quantification of solar energy fluctuation variability at high frequencies using short-scale measurements and came to the conclusion that the correlation between high fluctuation variability rapidly diminishes at shorter intervals [6]. DNI prediction using multivariate regression (MVR) and MLMs was forecasted, and it was concluded that the lifetime of solar plants was increased when the size of the power plant was known [31,44]. GHI variability was predicted in another study, and the findings of the model tree technique could be used in real time to generate forecasts of solar variability, which can help utilities manage the power grid [45]. They used a gradient-boosted regression tree (GBRT) model, which was demonstrated to be competitive in terms of mean squared error (MSE) across all prediction horizons, to predict solar power at several locations [42,46]. Regression analysis was considered to simulate GHI for any point on Earth [4,47,48]. The models produced by regression analysis outperform existing models in solar irradiance forecast [49].

In areas around a GHI measuring station, where the precise GHI data are unknown even for the future, the primary goal of this research is to forecast solar energy using a parametric model applied to MLM. The same proven MLM parametrically forecasts the GHI sample’s future behavior and its behavior in the regions around it, providing the real variability of partial solar energy and quantifying solar energy availability on a short-scale measurement [2,50,51].

In light of the ePPI review platform’s summaries of 145 bibliographic sources that were accessed between 25 November and 30 November 2023, data from publicly available sources were gathered from Web of Science/of Knowledge, Science Direct, IEEE Xplore, Research for Life, and SCOPUS. These sources also covered knowledge of solar energy behavior in some hard-to-reach locations, like desert regions, paradisiacal high seas, wildlife reserves, and forests, among others, that may still be challenging to master. With this information, the primary issue with power may be resolved as it would enable modeling of the solar energy that is accessible in these locations and be used for a variety of local uses in addition to being injected into the grid [14,52]. Within this framework, the present research primarily focuses on many estimating requirements for solar energy estimates:

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They base their short-term solar projections solely on GHI measurements and conclude that more observations of different atmospheric factors are needed for other models, but they are configurable [40,53,54];

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They studied the quantification of the variability in solar energy fluctuations at high frequencies through short-scale measurements, concluding that the correlation between high fluctuation variability decreases at shorter intervals [4,55,56,57];

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Spatial estimation of sub-second, second, minute, sub-hourly, annual GHI using a combination of special approach, geostatistical interpolation methods, and Regression Kriging (RK) interpolation from in situ imagery to satellite data [21,39,58,59].

Considering that a portion of the population lacks energy and that the study region has restricted access to both comparable research and infrastructure, as well as recommendations for the design of energy usage projects [5,6], in response to the query, what factors go into the accurate forecast of solar energy availability using MLM? The daily GHI population was defined, and the participants were the years of measurements of the GHI sample from the station in the Province of Niassa in Nipepe. The comparisons that are made between the GHI predictions using different MLMs for parametrically forecasting the GHI, and the study of meta-analyses of spatial and temporal variability in solar energy availability and the outcome that the solar energy forecasted obtained, the sample of GHI data collected was validated, through the ePPI reviewer platform that shows some considerations of the objectives of the analysis with the comparison with the STATA Software statistical package platform, version 18.0. The current study’s methodologies seek to both boost forecast accuracy and correct the past forecasting systems’ lack of diversity.

The parametric forecast models were used for theoretical derivation and verification. Regression analysis was also used to compare the model’s estimation accuracy. Ultimately, we identified the best parameters to consider in a parametric forecast. The techniques used in this study introduce a more effective validation process by reducing performance error through the use of new meteorological correction parameters. This leads to improved performance in terms of recommending effective tools for the forecast GHI in station-less locations, as well as for a better evaluation of electricity consumption patterns and the choice of a more precise method to estimate future energy demand. The study of the parametric forecast solar energy and its variability using various MLMs is complemented by this research on solar energy variability.

2. Methods

2.1. Method Used

This systematic scope analysis review adopted to manage all collected data was associated with the bibliometrics method in reference management, a review that supports the evaluation and interpretation of thematic areas, and we used a protocol that included the following: research questions and hypothesis, search process, inclusion, and exclusion criteria, data selection, search methodology, and extraction process and data synthesis (available at https://github.com/Muco-1990/Protocole_Parametric_For_Var.git (accessed on 12 November 2024).

2.2. Inclusion and Exclusion Criteria

The inclusion of the reasoning with logic and the intricacy of the analysis justifies the sample access. According to the inclusion criteria, the articles had to meet the parametric validation of atmospheric components, address solar energy forecasting using MLM, use validation using solar energy samples measured in situ or by remote sensing in short- and long-term measurement scales, and use complexity of reasoning in their titles, abstracts, keywords, and article composition. Studies with samples not in short-scale measurement (0.01 s to 24 h) or long term (up to 1 year) were eliminated based on the exclusion criteria, as were studies that did not validate their findings using MLM.

2.3. Sources of Information and Materials

Utilizing the ePPI reviewer platform available at https://eppi.ioe.ac.uk/eppireviewer-web/home (accessed on 5 November 2023), the bibliographic sources were accessed and included a range of roughly 145 sources of the most recent analysis. The following search terms were used: “forecasting solar energy, solar energy trends, parametric forecast, solar energy availability”. Research for Life available at https://www.sciencedirect.com/ (accessed on 6 November 2023), SCOPUS available at https://www.scopus.com/home.uri (accessed on 16 November 2023), and Web of Science/Knowledge available at https://www.webofscience.com/wos/alldb/basic-search, (accessed on 10 December 2023) were among the other access sources from which data were obtained between 1 November and 25 December 2023. The literature linking articles was carefully selected and examined over 23 years. After that, the chosen terms were input into a reference management system, which generated a file format for keyword analysis in a research information system. Networks of scientific keywords were created using the VOS viewer program available at https://app.vosviewer.com/ (accessed on 12 November 2024) and linked by a co-occurrence link, which is a relationship between two phrases. The quantity and strength of the co-occurrence, which illustrate the viewpoints and patterns of keywords like forecast, spatiotemporal, variability, parametric, GHI, and assessment, were determined by the word’s size.

2.4. Selection and Data Gathering

After examination, the articles were pulled into bibliometric databases so that any potential duplicates could be automatically removed. The online collection was accessed through the flowchart, which made it possible to identify recurrent references and modified versions that shared the same analytical focus. The researchers’ physical examination made it possible to thoroughly examine and remove repeats. However, about six items that were duplicates were found in the databases that were checked during the process. https://github.com/Muco-1990/parametric_for_systematic_review.git (accessed on 22 November 2024) contains their consultation. Data, such as authors, title, method, model, doi, abstract, country, climate, error, and keywords, were already taken out of the identified papers. On average, there were about 145 items that were connected to the literature. The data extraction form that the researchers employed was specifically linked to the seven study topics. As illustrated in Figure 1, which shows the delimitation of articles for the systematic review using the Prisma approach, a peer review validation was carried out to verify the identification of the responses. If there were any differences, agreements were reached for the selection of the responses to achieve full data verification.

Additionally, Figure 1 provides a declaration of the research’s adherence to the PRISMA 2020 guidelines [60]. It was filed on the Shiny platform (https://prisma.shinyapps.io/checklist/) on 13 November 2024, with ID: 12566. As previously mentioned in the preceding paragraphs, Figure 1 illustrates the overall adherence to all writing standards, beginning with identification, screening, retrieval, eligibility, and inclusion. This adds up to the systematic review of the literature on parametric solar energy forecasting with a focus on MLM and validation for in situ measurement stations.

2.5. Informational Data Items

To forecast solar energy variability, the search and extraction of the data sample mainly involved an optimal optimization of the solar energy forecast models. The expected results include a forecast of direct, diffuse, and global solar irradiance; spatial behavior of solar energy; the relationship of solar energy with other renewable energy generating resources; estimation of measurement errors; and variability of atmospheric parameters over time, aerosol concentration, water vapor, ozone layer, uniformly mixed gases, among others. Using solar energy collected in situ or from remote sensing, the studies give various metrics of measurement intervals ranging from 0.0001 s to 24 h, primarily in time intervals of about 1 h, from the set of over 145 sources that were investigated. When these findings from other research for measuring and assessing solar energy fluctuation are included, the current study’s conclusion becomes more evident. Similar to the outcome, the extracted sources likewise specify correlation measurements between local and distant measurement stations, apply MLM to draw a haphazard conclusion, and, most importantly, estimate the origin of several climatic parameters while focusing just on local reasons. This study, however, is anticipated to produce a result in which the generalized reason for each region of operation is already measured by the aggregated parameters alone.

2.6. Measures of Bias and Effect

A variety of models were employed. The majority of studies take into account models like simple linear regression and Kriging regression, but others also tend to use the RF, ANN, and their variants, which estimate lower risks of bias when assessed through peer review with key stakeholders. The measured data deviation for each sample source was caused by the keywords’ relative distance from the previously specified keywords. A summary of the data margin and its categories was provided by the synthesis. Showing an average of 145 sources overall, the majority of which are not estimated, the estimated models’ measurement errors range from 10 to 15 W/m². Considering that the absorbed energy would rise to 1362 W/m², a value near the theoretical solar constant, the results are consistent and demonstrate a massification of energy through the correction of parameters of climatic, atmospheric, and other origins, in an average of 900 W/m² of the measured energy. The sample’s meticulous presentation established a correlational analysis with the production over the years, as well as the cumulative analysis.

2.7. Bias Evaluation

The sample is included in the list of data that have an ambiguity but are applicable; typically, this is in the group of applications of the MLM that illustrate a greater margin of estimation of solar energy. The bibliometric method showed a series of sample references with their references for missing data and presenting a lot of deviation outside the desired characteristic. However, for higher-quality data extraction, physical analysis and exploratory analysis were continued in addition to the bibliometric method. The instrument error plus computational processing of 0.001, the observation error of 0.00001, and the relative error calculated from a 99% confidence level were all taken into consideration when determining the sample error for statistical analysis. Given that the range level is 5 in every scenario, interpolative models were used to address the model’s automatic interpretation errors. Additionally, the RF model yielded a fair interpretation, with an error of about 0.01.

2.8. Search Methodology

This study employed ePPI review techniques for keyword analysis and linkages to 23-year-old literature. By building scientific keyword networks with co-occurrence relationships, a research information system was developed utilizing VOS reader software version 1.6.20. The magnitude of the words affected the co-occurrence strength. Data from many sources, such as Web of Science/Knowledge, Science Direct, IEEE Xplore, Research for Life, and SCOPUS, were compiled utilizing the bibliometrics approach, which employed scoping analysis and systematic review techniques. To predict solar energy trends and availability, this study examined 145 bibliographic sources. Based on the fewest instances of a particular term, 679 relevant keywords with the highest overall link strength were found.

Due to the limited post-processing output over 23 years, the systematic review was examined utilizing a large number of keywords and bibliographic sources as of 2024. The bibliometrics approach in the program’s reference management was combined with the research development method in the information system to improve data management and the ePPI reviewer 6 platform’s search and analysis of bibliographic sources. Data samples, GHI measurements, cloud speed, and solar energy estimates were examples of cross-cutting components.

This study focuses on the variables that affect how solar energy availability is estimated by an MLM. It analyzes the daily GHI and solar radiation behavior and compares parametric models, $K_t^{*}$, and MLM results for stakeholder Niassa. Since the GHI inferred by a pyranometer is complete on Earth’s surface, the analysis sample, collected as part of the FUNAE effort between 2020 and 2021, was selected to validate the range of complete observations from dawn to dark. With a positive correlation between 2001 and 2024, high relative output in 2018 and 2022, and roughly 679 interconnected keywords, the bibliographic source sample was registered in ePPI Reviewer 6 version 6.15 on January 10, under ID 52365. This study used STATA software version 18.0 available at https://www.stata.com/new-in-stata/ (accessed from 1 to 26 November 2023), to examine data and discovered similar meta-analyses, with a focus on parametric energy efficiency calculations. The validated GHI sample must be used to analyze new findings. The research issue, according to the systematic review, is “Parameterization Forecast solar energy over time by applying Machine Learning Techniques,” underscoring its scope and limitations.

2.9. Data Synthesis

A strategy of extracting materials that met the inclusion criteria aided in addressing the research difficulties through database search and analysis. The qualitative method of applied content analysis, which occasionally drew from bibliometric sources, was founded on an exploratory and interpretative component. It developed and extracted objective judgments from the whole reading of the body bibliometric memory for each model. The first phase involved organizing the data in the study region. Next, key data was extracted using the inclusion criteria’s PRISMA 2020 rules and the research questions. The data that was extracted was triangulated, examined, and validated by peer review. The responses were merged with an existing classification to allow for possible analysis and interpretations. Statistical analysis and VOS version 1.6.20, were used to identify interesting intersections of terms, keywords, co-term networks, categories, and subcategories. The data on the quantity of sources and their variation across all years, both geographically and temporally distributed, are analyzed using causal inference approaches. The solar energy sample used to validate the parametric model of solar energy forecasting using MLM is described in the paragraph that follows.

2.10. Solar Energy Data Collection and Processing

The analysis sample was gathered from the National Energy Fund (FUNAE) [61] and consisted of a sample of GHI data obtained during the solar radiation measuring campaign conducted by FUNAE in 2020 and 2021 at the Nipepe station (reference FUNAE–MZ06–Nipepe) in the northern region of Mozambique. To measure the radiation component of GHI and DHI, pyranometer sensors were used, as shown in Figure 2, in conjunction with a system in which a pyrheliometer was inserted to determine the DNI component directly.

The ideal calibration parameters for the pyrheliometer (Eppley–type) that was utilized to gather the data were set to: calibration factor 295–2800 nm, spectral range with 1-min response, linearity ±0.5%, ±1%. A portion of the information was gathered at the National Institute of Meteorology (INAM) [62] and at Eduardo Mondlane University (UEM). This included sample data on solar radiation measured in situ with an automatic Davis station and a pyranometer, as illustrated in Figure 3, as well as measurements of temperature, wind, moisture, greenhouse gases, aerosols, and greenhouse gases.

The meteorological and atmospheric variables were taken from the Aerosol Robotic Network (AERONET) [63] database, which included a sample of aerosol optical thickness (AOT) measured by a sun photometer. These variables were then verified in the same database, producing calibrated results with the help of developed algorithms that, by themselves, already provided some quantities that were relevant to our analysis of the values collected in 1 min measurement intervals as well as daily, hourly, monthly, and annual average intervals. Values of aerosol optical mass, precipitation, and multiple reflectance by gases were precisely collected at the Niassa stations, represented in

Table 1. Coordinates of the measurement stations.

ID	Station	Province	Tower	Code	Long (X)	Lat (Y)		Altitude
MZ25	MZ25_Nipepe	Niassa	TDM	TDM	35°26′12.82″ E	13°54′25.93″ S		60
ID	Station	Province	Tower	λ (nm)	Amplitude	Level	Long (X)	Lat (Y)	Altitude
1_A	Niassa	Niassa	AERONET	400–500	4″, 1/24 H	2.0	37.5665	−12.155	510

, which is available for the years 2020 and 2021.

Samples were taken at short-scale intervals of around four minutes, one minute, one hour, one day, and monthly averages from measurements taken between 19 June 2019 and 31 December 2023. They were measured using data from AERONET, which is obtained through the sun photometer, which gathers light absorbed by aerosols in the upper and lower atmosphere at different wavelengths of identification, AERONET instrument number 1187. The data have multiple levels, including Level 1 and 1.5, but these have been updated and more precisely measured, along with the removal of additional anomalies discovered in Level 2.0 treatment measurements for all stations of the data collected.

Regular seasonal maintenance, including data transfer, battery replacement, cleaning, and realignment, was performed on each device during the measurements.

The sample was collected at the Nissa station, which is situated between the parallels 10°27′ and 26°52′ south latitude and between the meridians 30°12′ and 40°51′ east longitude. The cleaning and orientation periods were recorded during this process, except for data selection periods, such as periods of interference and turbulence. The local topographic section of the measuring station and its latitude and longitude characteristics are displayed in Figure 4.

As seen in Figure 5, in the first step, a sample of bibliographical reference data was collected in the database and registered, after it generated a link of keywords related to the parametric models applied to MLM. This led us to collect a sample of the data, which were next subjected to quality control, spurious value removal, and processing by programs created especially for computing radiation with a 10 min time interval, joined to the AERONET atmospherically, meteorological, spatiotemporal predictor variables. That helped to consider a better choice of different atmospheric parameters considered in the presented parametric model. In the third level, these data were applied to validate the model and registered in the ePPI version 6.15 and STATA version 18.0, which showed a link of possible meta-analysis, as depicted in the last level of Figure 5.

2.11. Evaluation and Effects of the Fuel Sources Currently in Use

According to available data, between 2010 and 2023 [47], the number of individuals not covered by the electrical grid increased by 83% to 91%, from 733 million para 1.2 billion [5,64]. In the interim, to massify renewable actions by more than 30% (net-zero emissions by 2050), dominated by fossil fuels [42,65], electrification efforts are still necessary to achieve the ambitious global electrification outlook by 2030 and to overcome the slowdown of 0.8 to 0.5 percentage points from 2010 to 2018 and 2018 to 2020 [66,67]. In 2020, the percentage of people without access to electricity worldwide who live in rural regions increased by 1% [5,36] annually, with the largest increase occurring in sub-Saharan Africa [24]. Approximately 80% of this population lives in these areas. Approximately 76% of the world’s population lives in the 20 countries with the largest access deficits [26,68]. Mozambique ranks sixth, with roughly 20 million of citizens affected [4,5]. These factors also include interannual oscillations, which are strong variations in cloud cover and rainfall, El Niño and La Niña episodes [3,69], determinants of intermittent temporal fluctuations in solar energy [15,70], etc., which contribute to the existence of life on Earth [64,71].

To fulfill the predicted worldwide demand of 800 GW by 2050 [5,7] and reduce air pollution, the Mozambique Electricity (EDM) declared an expected PV capacity of 50 MW by the end of 2023, with expected expansion in the future [5,25]. The performance of PV systems and the size of battery storage are the primary areas of the 5 MW total PV capacity (stand-alone and grid-connected PV systems) in Mozambique’s Mid–North region that are impacted by solar energy fluctuation.

From seconds and meters [72] to days and thousandths of kilometers [6,66], variability in solar radiation increments can span a wide range of spatiotemporal scales [34,70,71,72,73]. This variability affects generation, load balancing, and power quality maintenance, such as voltage and frequency stability [6].

Generally speaking, primary sources contribute roughly 78% of the energy that keeps fossil fuels strong (4% comes from nuclear power and 74% from fossil fuels). It is significant that these fossil fuels have detrimental effects on the ecosystem. On the other hand, as Figure 6a,b demonstrates, the Sun provides roughly 22% (1% direct solar energy, 21% indirect solar energy, and 0.2% geo, ocean, and environment) to the production of clean, safe, and environmentally healthy energy [5,6,71].

Generally speaking, a variety of instruments can be used to measure solar radiation depending on how well they work. It is significant to note that the radiometer is the fundamental tool used to measure solar radiation [3,26].

2.12. Synthesis Methods

The intervention and assessment of solar energy using atmospheric parameters and other factors determined the eligibility of the studies retrieved for each synthesis; nevertheless, they must meet the established eligibility requirements. Other factors employed in the decision-making process included operationalization using MLM, decreased error estimates, and sample validation using solar energy measured primarily in situ. The cumulative model creates a summative tabulation of the data, the correlative model extracts the data to correlate the studies among themselves, and the inferential model extracts the causal analysis of the tabulated data to tabulate or visually display the results of individual studies and syntheses; all were employed as auxiliary analyses in the data preparation process.

2.13. Research and Data Validation

This study introduces an algorithmic model that offers a structured methodology for analyzing, extrapolating, and forecasting solar radiation at the nuclear station and its associated neighboring stations. In contrast to other algorithms that rely on assumptions based on limited solar irradiance data, this approach is applicable on a global scale. Prior research has indicated that predicted values often deviate significantly from actual measurements, resulting in substantial errors. Here, the application of standard RF and ANN models algorithms in the parametric model resulted in a notably reduced internal error, compared to the reported mean absolute error (MAE) of 10.2 and 7.12 W/m² [50,74]. Although literature models incorporate local meteorological parameters, such as climate, weather, seasons, humidity, latitude, longitude, and altitude, for various applications, they demonstrate commendable performance in testing and predicting energy efficiency.

2.14. Extraction and Correction of Raw Data

A maximum at noon is a characteristic of the data processing. The scripts created especially for computing radiation with a time interval of 10 (ten) minutes were applied to this data sample after it had first undergone quality control and spurious value removal. Regular seasonal maintenance, including data transfer, battery replacement, cleaning, and realignment, was performed on each device during the measurements. Except for data selection events such as periods of interference and turbulence, cleaning and orientation phases were documented during this procedure.

Next, the data process involved treating the GHI sample beforehand to reduce potential losses from errors, also known as uncertainty or margin of error (error), and to acquire data from direct GHI measurements. After that, the information was taken out of the Logger database and made available for confirmation. Forecasting solar energy throughout the hours of sunshine, or from sunrise to sunset, is crucial from the perspective of solar energy consumption. Nevertheless, some days had insufficient data for a variety of reasons, including bird demands and obstructions to measurements, pyranometer shadowing, and a decrease in convection energy for devices that rely on local information.

However, interpolation techniques were used, among other things, to fill in these gaps in the data and make repairs. Data were taken from the original, in situ measurements. Next, models were used to forecast data on solar energy, and the corresponding measurement errors were found. Most of the models under consideration are machine learning models built using cutting-edge artificial intelligence techniques.

2.15. Parametric Estimation of Total Solar Energy

Upon reaching the Earth’s horizontal surface, solar radiation is subject to various factors such as atmospheric and environmental typology [23,44]. These are categorized based on factors, like atmospheric activity, atmospheric properties (such as the ozone layer and greenhouse gases), surface characteristics (such as albedo and vegetation), astronomical and geographic factors (such as altitude and angle of incidence), climatic effects, and climatic events. The troposphere stretches from 8 to 15 km, the stratosphere stretches from 15 km to 50 km [5,34], the mesosphere stretches from 50 km to 85 km, the thermosphere stretches from 85 km to 600 km, and the exosphere stretches from 600 km to 10,000 km; these are the various layers that make up Earth’s atmosphere [7,9,23]. These layers are determined by temperature gradients and other factors [3,6,53].

Various researchers propose different methodologies for predicting and estimating solar energy. However, a common issue arises as each author presents their estimates regarding the percentage of solar energy received and the factors contributing to its attenuation. For instance, the estimative model for solar energy, positing that solar insolation is influenced by wavelength, is represented by the equation $G_{AMK}\left(\lambda \right)=G_{AM0}\left(\lambda \right){\left|{\displaystyle \frac{G_{AM1}\left(\lambda \right)}{G_{AM}\left(\lambda \right)}}\right|}^{{AM}^{0.678}}$ [24,69]. This model acknowledges the solar constant while noting that approximately 30% of sunlight is diminished as it traverses the Earth’s atmosphere before reaching the surface. This reduction is attributed to various effects, including Rayleigh scattering by the atmosphere, particularly at shorter wavelengths, which follows a dependency of $(~{\lambda }^{-4})$, as well as scattering by aerosols and dust particles and absorption by atmospheric gases, such as oxygen, ozone, water vapor, and carbon dioxide [3,6]. Specifically, 7% of sunlight is scattered to the Earth (the diffuse component), 3% is scattered into space, and 18% is absorbed by ozone at altitudes of 20–40 km, upper dust layers at 15–25 km, air molecules from 0 to 30 km [7,9,10,11,12,13,14,15,16,17,18,19], water vapor from 0 to 60 km, and lower dust from 0 to 90 km, with an additional 3% scattered into space [6,24].

Another estimate on numerical models begins by assuming that direct radiation alone can predict 100% of solar energy [11,16]. However, this also takes into account the existence of atmospheric attenuation and considers that, in addition to the direct radiation measured on Earth’s surface, diffuse radiation, which results from diffuse radiation caused by scattering and absorption by aerosols, gases, and water vapor, must be added [3]. This total proves that there is global energy on Earth’s surface.

Other literature provides that only 50% of the 100% of solar radiation that is emitted in short and long waves reaches the Earth’s surface as direct and/or diffuse radiation [3,15]; the remaining 16% is absorbed by particles of ${\mathrm{H}}_2\mathrm{O},{\mathrm{O}}_3,\mathrm{C}{\mathrm{O}}_2$, and other greenhouse gases; 4% is absorbed by clouds [35]; and 30% is fully reflected as solar shortwave, with 6% of the radiation being backscattered by air molecules, 20% reflected in clouds, and 4% reflected on the surface of the Earth [75].

The light can be impacted by the following processes: (1) atmospheric scattering by air, water, and dust molecules; (2) atmospheric absorption by ${\mathrm{H}}_2\mathrm{O},{\mathrm{O}}_3$, and $\mathrm{C}{\mathrm{O}}_2$. These phenomena are detailed in Iqbal [3,45]. Most radiation travels through an air mass (AM) to reach the Earth’s surface throughout the day when the sky is clear. The path that solar radiation takes from its incidence in the atmosphere to the Earth’s surface is known as the air mass $\left(AM={\displaystyle \frac{1}{\cos \theta }}\right)$, where $\theta$ is the angular distance between the solar beam and the vertical that touches the zenith of the incidence site [15].

The radiation is analyzed as being at its minimum in the winter and at its greatest in the summer, so the declination of the radiation at the research site was estimated over the daily course for days of year, given by Equation (1) [7,15].

$$\delta =23.45\sin \left(360{\displaystyle \frac{284+n}{365}}\right)$$

(1)

whose position relative to the sun for horizontal surfaces is given by Equation (2) [16,75],

$$\cos {\theta }_z=\sin \delta \sin \varphi +\cos \delta \cos \varphi \cos {\omega }_s$$

(2)

A typical zenithal angle equal to 90 degrees, resulting in a solar angle ${\omega }_s$, given by Equation (3) as [3,6],

$${\omega }_s={\cos }^{-1} \left(-\tan \varphi \tan \delta \right)$$

(3)

The length of the day $N_d$ is determined according to the rule, which is equal to 2${\omega }_s$, given by Equation (4) [15].

$$N_d={\displaystyle \frac{2}{15}}{\cos }^{-1} \left(-\tan \varphi \tan \delta \right)$$

(4)

The difference in minutes between solar time and standard time is given by Equation (5) [15,75].

$$Solartime-Standardtime=4\times \left(L_{st.}-L_{loc.}\right)+E$$

(5)

where $L_{st.}$ is the Standard Meridian for the local time zone, and $L_{loc.}$ is the longitude of the location in question, measured in degrees $\left(0<L_{loc.}<360°\right)$,

The eccentricity correction factor of the Earth’s orbit in Equation (6) is [3,24],

$$E_0=1+0.033\cos \left(n{\displaystyle \frac{360}{365}}\right)$$

(6)

The equation of time presents variables along the year, presenting positive evaluation in the last days of the end year, in its optimal development, and having the shape as in Figure 7.

The zenith angle is given by Equation (7) [3,24]

$${\theta }_Z=\mathrm{acos}\left(\sin \varphi \sin \delta +\cos \varphi \cos \delta \cos {\omega }_s\right)$$

(7)

The relative optical air mass $m_{\zeta }$ is calculated by Equation (8) [3]

$$m_{\zeta }={\left[\cos {\theta }_z+{\displaystyle \frac{0.15}{{\left(93.885-{\theta }_z\right)}^{1.253}}}\right]}^{-1}$$

(8)

The relative optical mass of the air, considering $p$ the pressure in mbars, is determined by Equation (9) as [69,76],

$$m_a={\displaystyle \frac{p{m}_{\zeta }}{1013.25}}$$

(9)

Reduced precipitable water w, with w′ the precipitable water, is given by Equation (10) [3,77]

$$w={\displaystyle \frac{w^{\prime }}{{\left(\frac{1013.25}{p}\right)}^{\frac{3}{4}}{\left(\frac{T}{273}\right)}^{\frac{1}{2}}}}$$

(10)

The transmittances for the direct insolation of individual atmospheric constituents are compared in detail, assuming the multiplicative total transmittance due to the transmittances for direct irradiance due to absorption by ozone (${\zeta }_0$) [3,27], the transmittances for direct irradiance due to absorption by water vapor (${\zeta }_w$), the transmittance for direct irradiance due to attenuation by aerosols (${\zeta }_a$) [10,31], the transmittances for direct irradiance due to absorption by uniformly mixed gases (${\zeta }_g$) [3], and the transmittances for direct irradiance due to Rayleigh scattering effect of air molecules (${\zeta }_r$) [10,19], so the total transmittance is a multiplicative of all transmittances, as given in Equation (11) [32],

$$\zeta =\prod _{i=1}^{i=j}{\zeta }_i$$

(11)

The multiplicative transmittance for the Nipepe-2 station is displayed in Figure 8, with an average of roughly 0.5281 in 2021 compared to roughly 0.5267 in 2020.

Comparing equations where each absorptance is considered differently is informative. The transmittance via Rayleigh scattering is driven by Equation (12) [15,78],

$${\zeta }_r=e^{\left[-0.0903m_a^{0.84}\left(1.01+m_a-m_a^{1.01}\right)\right]}$$

(12)

The ozone transmittance, influenced by the constant deposition of the emitted gases is given by Equation (13) as follows [3]:

$${\zeta }_0=1-\left[{\displaystyle \frac{0.1611U_3}{{\left(1.01+139.48U_3\right)}^{0.3035}}}-{\displaystyle \frac{0.002715U_3}{1.0+0.044U_3+0.0003U_3^2}}\right]$$

(13)

in which $U_3=l{m}_{\zeta }$ defines $U_3$. Figure 9 illustrates that the numerical values obtained are within approximately 0.071 of each other, even though the test conducted at the same measurement station reveals that there are different concentrations of the ozone layer for different years, with the concentrations rising to 0.9232 in 2020 due to greenhouse gas emissions over the region of consideration.

As can be seen in the previous diagram, the biggest variation in this one is $l=0.89\mathrm{c}\mathrm{m}$, which was established for AERONET average measurement. These variations depend on the zenith angle and the amount of ozone present. Because ozone has a very high transmittance, it has very little effect on the total transmittance. The following yields the transmittance by direct irradiance caused by uniformly mixed gas absorption, given by Equation (14) [3,10,79]:

$${\zeta }_g=e^{\left(-0.0127m_a^{0.26}\right)}$$

(14)

Figure 10 illustrates the reduced attenuation of the transmittance for direct irradiance resulting from absorption by uniformly mixed gases (${\zeta }_g$), with an average of roughly 0.9873 for 2020 and 2021.

In the summer months, there is an increased excitation of these gases due to the effects of temperature, which leads to the formation of systems that exhibit enhanced absorption and transmission [6,75]. Conversely, during the dry and cold seasons, these gases significantly attenuate solar energy, thereby limiting the direct impact of this radiation on the Earth’s surface [25]. This results in a period of instability in solar energy utilization, necessitating careful management of the excess energy generated and careful consideration in the final solar balance calculations to appropriately size the solar system based on the measured energy. Additionally, we examined mathematical models to estimate solar energy availability, assess local grid penetration rates, and identify the variability in cloud cover and atmospheric aerosol concentrations [80,81], as well as the indirect influence of atmospheric gases, to a lesser extent [82]. The transmittance of water vapor is given by Equation (15) as [3,44,56],

$${\zeta }_w=1-2.4959U_1{\left[{\left(1.0+79.034U_1\right)}^{0.6828}+6.385U_1\right]}^{-1}$$

(15)

$U_1=w{m}_{\zeta }$ is defined here. With an average transmittance of 0.9059 in 2020 and growing to 0.9073 in 2021, the variances between the transmittance in those two years are about the same. As Figure 11 indicates, the variations with the zenith angle over the course of the year are related to the amount of water vapor, with $w=1$ cm serving as an example.

An analysis employing precipitable water with a proportion of 2 cm and O₃ with a proportion of 0.85 cm revealed that the GHI spectrum yields higher values [3,83]. No matter how visible it is, the difference is always somewhat less than 25. Considering that there is only a 0.089 difference between the two solar constants, this is an intriguing result.

Provide a different load on populations of particles known as aerosols released into the atmosphere to obtain an expression for the aerosol transmittance of this model. This expression is based on the spectral attenuation at the two wavelengths that the meteorological network typically uses, 400 and 659 nm, which are the wavelengths where ozone molecular absorption is least, as given by Equation (16) [3],

$${\zeta }_a=e^{\left[-{{\delta }_a}^{0.873}\left(1.01+{\delta }_a-{{\delta }_a}^{0.7088}\right){m_a}^{0.9108}\right]}$$

(16)

In this case, ${\delta }_a=0.28{\delta }_{a1}+0.35{\delta }_{a2}$. Different databases and country services routinely measure the thickness of aerosols at these wavelengths. One can disregard a measurement at a wavelength if it is uncertain. An equation for ta in terms of the Angstrom turbidity parameters $\beta$ and $\alpha$, or visibility, will be helpful in places where turbidity or visibility is reported. According to an analysis, the concentration in Nipepe-2 to 0.8547 in 2021 from roughly 0.8538 in 2020 are shown in Figure 12.

For irradiance, transmittance, and absorptance, the direct irradiance, also known as direct normal irradiance that reaches the earth’s surface, is provided and is further generalized by Equation (17) as [32,56].

$$G_n=0.9751G_{sc}\zeta$$

(17)

When the spectral range is taken into account for solar radiation (0.3–3 nm), the factor 0.9751 is added as a result in Equation (18) [24,56],

$$G_b=G_n\cos {\theta }_z$$

(18)

Higher certainty is obtained in the estimation of solar irradiance (at various time scales) through the use of geographic coordinates like latitude and longitude, as well as meteorological parameters, like cloud cover $\left(Cl\right)$, sunshine duration $\left(S\right)$, clearness index $\left(K_t\right)$, pressure $\left(P\right)$, air temperature $\left(T\right)$, relative humidity $\left(RH\right)$, wind speed $\left(W_s\right)$, wind direction $\left(W_d\right)$, etc.

We report results for a clean atmosphere accounting for aerosol attenuation. The spectrally integrated results for a clean atmosphere with $w$ equal to 4 cm and ${\mathrm{O}}_3$ equal to 0.89 cm, are very good at ${\theta }_z<90°$, illustrating a mean of 657.4009 with a deviation of 104.0978 in 2020 and 657.80865 with a deviation of 104.0978 in 2021. Figure 13 shows the direct normal irradiances obtained constituting a spectrally integrated approach.

The impact of turbidity was measured at $150 \mathrm{k}\mathrm{m}$ of visibility, with $w$ equal to 3 cm and ${\mathrm{O}}_3$ equal to 0.89 cm. Nearly equivalent findings are obtained at zenith angles less than $60°$, particularly at greater zenith angles. Nevertheless, for outcomes in broadband computations, the first few millimeters of precipitable water in the atmosphere attenuate.

In Figure 13, whether the atmosphere is empty or laden with $5 \mathrm{c}\mathrm{m}$ of water vapor, the findings will yield an essentially equal degree of error, given an average ozone standard ${\mathrm{O}}_3$ of 0.89 and a visibility of $150 \mathrm{k}\mathrm{m}$ for $0.25 \mathrm{c}\mathrm{m}$ of precipitable water. A modification in the visibility parameter can demonstrate that this statement essentially stays the same. This finding helps create a more straightforward parameterization equation with a constant [15,35].

The method used to determine the irradiance on a horizontal surface is predicated on the findings from two areas of study. Following its initial passage through the atmosphere, Rayleigh scatters diffuse irradiance, which is determined by Equation (19) as [3],

$$G_{dr}=0.79G_{sc}{\displaystyle \frac{\cos {\theta }_z{\zeta }_0{\zeta }_g{\zeta }_w{\zeta }_{aa}.0.5\left(1-{\zeta }_r\right)}{\left(1-m_a+{m_a}^{1.02}\right)}}$$

(19)

Diffuse radiation ($G_d$), which results from a variety of events, interacts with stuff in the atmosphere as depicted in Figure 14.

As can be seen, the ozone layer, aerosol deposition, uniformly mixed gases, and geometric variables all influenced $G_d$ in 2021 (orange line), which was near to the theoretical $G_d$ in 2021 (green line) with average maximums of $435$ and $450 \mathrm{W}/{\mathrm{m}}^2$. On average, however, $G_d$ was higher in 2021 than it was in 2020 (blue line), when it averaged a maximum of $349 \mathrm{W}/{\mathrm{m}}^2$.

Here, we take ${\zeta }_{aa}$ to represent the direct radiation transmittance due to the aerosol absorptance, because of the diffusion, given by Equation (20) [3,24,81],

$${\zeta }_{aa}=1-\left(1-w_0\right)\left(1-m_a+{m_a}^{1.06}\right)\left(1-{\zeta }_a\right)$$

(20)

The diffuse irradiance dispersed by aerosols following their initial passage through the atmosphere is determined by, and should be assumed to be, 0.9 unless a more precise value for w₀ is provided, by Equation (21) as [3,15],

$$G_{da}=0.79G_{sc}{\displaystyle \frac{\cos {\theta }_z{\zeta }_0{\zeta }_g{\zeta }_w{\zeta }_{aa}.F_c\left(1-{\zeta }_{as}\right)}{\left(1-m_a+{m_a}^{1.02}\right)}}$$

(21)

where ${\zeta }_{as}$ was previously defined by Equation (22) [75,84,85]

$${\zeta }_{as}={\displaystyle \frac{{\zeta }_a}{{\zeta }_{aa}}}$$

(22)

Until more precise data on aerosols become available, this model suggests using a constant value of 0.84 for $F_c$. The diffuse irradiance measured an average of 112.8236 in 2020, increased by a percentage to an average of 112.5279 in 2021, and deviated to 38.5645 in 2021, as depicted in Figure 15.

As $\beta$, the quantity of aerosols in the atmosphere, grows, so does the atmospheric albedo. With an increase in $\alpha$, the atmospheric albedo likewise rises [75]. More small particles are represented by an increase in α, because small particles scatter more radiation than large particles [85]. The spectrogram for multi-reflected diffuse radiation looks like this [3,14]. The GHI on a horizontal surface is compared with its behavior in Figure 16.

It yields numbers that are less 2020 than in 2021, mostly because, in the model, the diffuse components differ from one another. Once more, the atmospheric turbidity factors determine how different they are, as shown in Figure 17 [35,86].

The difference between lower and higher values is determined by the characteristics of air turbidity. Before contrasting the diffuse irradiance produced in the two years, the atmosphere’s albedo, or ${\rho }_a^{\prime }$, with α and β, as illustrated in Figure 17, rises with β as predicted. This is a representation of the quantity of aerosols in the atmosphere [6,87,88]. As the value of α grows, so does the atmospheric albedo; since small particles scatter more radiation than large particles, a rise in α indicates a greater concentration of small particles [3,15,56].

The atmospheric albedo has been proposed to have the formula given in Equation (23), to achieve the multiple reflected irradiance [3,75,77]:

$${\rho }_a^{\prime }=0.069+\left(1-F_c\right)\left(1-{\zeta }_{as}\right)$$

(23)

The difference between the two formulations for albedo is not based on the air mass expression $1.66(p/1013.25)$ [3,10,19]. The scattering of light between the Earth and the atmosphere is due to many reflections. This is one way to write the diffuse irradiance [25]: $G_d=G_{dr}+G_{da}+G_{dm}$. The global irradiance (rightmost diffuse) on the horizontal surface can be given by Equation (24) as [69,89]

$$G=G_b +G_d$$

(24)

The global irradiation on a horizontal surface is compared in Figure 18.

The deposition of particles that block the path of solar radiation spreads and diffuses a lot during the hot and rainy season (September to February), as shown in Figure 18. This is primarily caused by the geometric orientation of the earth at this point of analysis (Nipepe), which exposes it to the maximum incidence of solar radiation. This results in maximum transmittances and a consequently greater incidence of solar energy, with a maximum theoretical portion, predicted to be in the range of $870 \mathrm{W}/{\mathrm{m}}^2$ in 2020 and $960 \mathrm{W}/{\mathrm{m}}^2$ in 2021.

However, there is a minimum incidence of solar radiation during the cold and dry season (March to August), which is marked by the earth’s diametrically opposite geometric orientation in the region of analysis. This helps to maintain the stationarity of particles deposited in the atmosphere by a variety of effects and causes, forming even more consistent layers that contribute significantly to maximum attenuation, absorption, and reflectance. There is also a minimum transmittance of solar energy to the earth’s surface, resulting in a minimum incidence of solar energy in the range of $690\mathrm{W}/{\mathrm{m}}^2$ in 2021 and $580\mathrm{W}/{\mathrm{m}}^2$ in 2020.

To put it briefly, the theoretical global energy forecasted for 2021 (orange line) is nearly equal to the theoretical energy for 2021 (green line), which is even higher than that of 2020 with an average difference of $150\mathrm{W}/{\mathrm{m}}^2$ in both years, with the least amount occurring in the cold and dry season and the most during the hot and rainy season. The diffuse components of the two years differ, which is the main cause of the decreased numbers, and the atmospheric turbidity factors determine how different these years are. To rectify the parametric association, MLMs were employed, which may be explained as follows:

Training data for Simple Linear Regression (SLR) with continuous variables ${GHI}_i$ and GHI were obtained for values $\left(GHI,{GHI}_i\right)=1,\dots ,n$, and provided by Equation (25) [74,90,91].

$${\mathrm{G}\mathrm{H}\mathrm{I}}_{\mathrm{i}}={\beta }_0+{\beta }_{1}GHI+{\epsilon }_i,i=1,\dots ,n$$

(25)

With identical independence, the errors ${\epsilon }_1,\dots ,{\epsilon }_n$ have the following values: $E\left({\epsilon }_i\right)=0,Var\left({\epsilon }_i\right)={\sigma }^2$, ${\epsilon }_i~N\left(0,{\sigma }^2\right)$. The regression line represents the estimate $\widehat{f}\left(GHI\right)={\widehat{\beta }}_0+{\widehat{\beta }}_{1}GHI$, which is given as $\widehat{E\left(\left.{GHI}_i\right|GHI\right)}$ for the expected value of the condition $\widehat{{GHI}_i}={\widehat{\beta }}_0+{\widehat{\beta }}_{1}GHI$ [92].

The random forest (RF) model uses three neighborhoods with a search scope of $k_1,k_2\mathrm{and}{k}_3$, with a spatial dataset GHI and a neighborhood of station $N_0^k$. The neighborhood-averaged ${\mathrm{G}\mathrm{H}\mathrm{I}}_{\mathrm{i}}$ is represented by $\overline{GHI}\left(N_0^k\right)$, while the weighted average counterpart is represented by $\overline{{GHI}_i}\left(N_0^k\right)$ based on inverse distance weighting [2,35,93].

$$\overline{{\mathrm{G}\mathrm{H}\mathrm{I}}_{\mathrm{i}}}\left(N_0^k\right)=\sum _{z_i{N}_0^k}{\lambda }_i\mathrm{G}\mathrm{H}\mathrm{I}$$

(26)

The equation ${\lambda }_i={\displaystyle \frac{d_{i0}^{-r}}{\sum _{z_i{N}_0^k}{d}_{i0}^{-r}}}$ measures the relative inverse distance weights between stations, with $r$ being an exponent parameter that amplifies the relevance of closer neighbors while reducing the weights of more distant ones. A larger exponent indicates greater influence of nearby points on predicted values.

We employed the Regression Kriging (RK) model, which combines the expected residuals from Kriging with the projected regression trends as provided by [49,90,93],

$$\widehat{{GHI}_i}\left(x_0\right)=\widehat{m}\left(x_0\right)+\widehat{e}\left({GHI}_0\right)=\sum _{k=0}^p{\widehat{\beta }}_k{q}_k\left({GHI}_0\right)+\sum _{i=1}^m{\widehat{w}}_{i}e\left({GHI}_j\right)$$

(27)

If the number of auxiliary variables is p, the estimated regression coefficients ${\widehat{\beta }}_k,k=0,1,\dots ,p$, a constant term is $q_0\left({GHI}_0\right)=1$, the residual is $\widehat{e}\left({GHI}_0\right)$, and the trend (climatic and geographic variables) is represented by ${\widehat{w}}_i,i=1,\dots ,n$. With each neuron $j$ having an input range of ${\epsilon }_1,\dots ,{\epsilon }_n$, multiplied by the weights $w_{ij}$, and gathered in neuron j specified by Equation (28), the Artificial Neural Network (ANN) model was utilized [33,94].

$${GHI}_i=\sum _{i=1}^n{w}_{ij}{u}_j+w_{0j}$$

(28)

This includes the bias, denoted as $w_{0j}$. For a value ${GHI}_i$, to which an activation function is applied, an expression for this function is $f\left(GHI\right)={\displaystyle \frac{1}{1+e^{-GHI}}}$ [33]. As opposed to a basic linear or exponential regression, whose output has a normal distribution produced by a linear relationship, the Support Vector Machine model (SVM) [95] was employed. It can be regular, binomial, or Poisson.

$${GHI}_i=\sum _{m=1}^M{w}_m\psi \left(GHI,{GHI}_m\right)+c$$

(29)

considering $\psi \left(GHI\right)$ as a function to map a nonlinear input of coefficients using a kernel function, $w_m$ and $c$ are the weight vectors. The ARIMA model is an analytical statistical model that uses time-series data to understand and forecast future trends, as given by Equation (30) [74,96]

$$\left(1+\sum _{i=1}^{p^{,}}{\alpha }_i{L}^i\right){GHI}_t=\left(1+\sum _{i=1}^q{\theta }_i{L}^i\right){\epsilon }_t$$

(30)

The variables ${\epsilon }_t$ are independent variables that are identical to them in the sample to the normal distribution with mean zero; L is the lag operator; α_i is the parameter of the autoregressive section of the module; ${\theta }_i$ is parameters of the moving average in terms of error. Regression classification is conducted using a model called Gradient-Boosting Machines (GBMs). It builds a forecasting model by combining decision trees. When we take into account regression values, the forecast for the initial prediction $F_0\left({GHI}_j\right)$ shown in Equation (31) becomes [90,97],

$${\widehat{GHI}}_i=F_0\left({GHI}_j\right)+\sum _{m=1}^M\eta h_m\left(GHI\right)$$

(31)

A potent nonparametric Bayesian regression method that offers flexibility in modeling intricate relationships in data is Gaussian Process Regression (GPR). Pretend that there is a Gaussian process before functions, given in Equation (32) as [32,43,97],

$$f\left(GHI\right)~GP\left(m\left(GHI\right),k\left(GHI,{GHI}^{\prime }\right)\right)$$

(32)

where $\left({GHI}_i|GHI\right)~N\left(f\left(GHI\right),{\sigma }^{2}I\right)$, the covariance (or kernel), the likelihood of detecting u given GHI for GPR, and the function of the day are denoted by m(GHI), $k\left(GHI,{GHI}^{\prime }\right)$, and so on [32,96].

Networks with Long Short-Term Memory (LSTM): This is a collection of recurrent neural networks (RNNs), which are intended to learn from a sequence of data and provide predictions. $f_t$ is the forget gate (gates determine what position is given the cell state to have); d_t (hidden state) is the output of the LSTM unit and is used for prediction and contains information about constant time weight, and $c_t$ (cell state) represents the memory of the trajectory [96].

$$f_t=\sigma \left(W_f\left[h_{t-1},x_t\right]+b_f\right)$$

(33)

The input gate $\left(i_t\right)$ determines whether part of the data is reduced. The association between a time series and its lag values is measured by the autocorrelation function (ACF). The autocorrelation function $\rho \left(k\right)$ at a lag of $k$ for a time series ${GHI}_t$, and [32,96,97],

$$\rho \left(k\right)={\displaystyle \frac{Cov\left({GHI}_t,{GHI}_{t-1}\right)}{\sqrt{var\left({GHI}_t\right)}Var\left({GHI}_{t-1}\right)}}$$

(34)

where $var$ stands for variance and $C_v$ covariance. The autocorrelation between ${GHI}_t$ and ${GHI}_{t-k}$ that is not explained by the lags between them is represented by PACF, which stands for ${\phi }_{kk}$ [97,98].

3. Results of Forecasting Solar Energy and Analysis of Systematic Bibliographic Sources

3.1. Parametric Forecasting of Solar Energy

3.1.1. Spectral Distribution Histograms for Days

As seen in Figure 19, the anticipated data sample has a high frequency of values with high levels of solar irradiation in the order of 1241.85 W/m².

The analysis conducted over one day, as depicted in Figure 20, reveals a significant correlation between the values predicted by the parametric models, with correlation coefficients nearing 0.989. Conversely, the estimated radiation shows an even higher correlation, with values around 0.9999 when utilizing in situ measurements. Furthermore, Figure 20 also indicates that the radiation on the horizontal surface aligns closely with the theoretical spectrum, thereby enhancing the parametric estimation of solar energy. This observation demonstrates the presence of full Sun conditions with a high incidence of solar energy under clear-sky conditions or skies with minimal cloud cover.

3.1.2. Autocorrelation and Partial-Autocorrelation Function

The energy levels in each model are quantitatively represented using the autocorrelation function (ACF), which has an average order of 0.8956 as shown in Figure 21. Additionally, the parametric analysis is enhanced by the partial-autocorrelation function (PACF) for the years 2020 and 2021, which has an order of 0.9896. The parametric analysis reveals that the solar energy described by the ACF gradually declines, while the one described by the PACF cuts after a few delays, as illustrated in Figure 21, based on the MLM tested with reduced error. But for the other MLM models, including ARIMA, LSTM and others, solar energy has been steadily declining for both ACF and PACF analysis. This demonstrates the extent to which the models used as input in MLM models are adequate, significantly boosting the estimate’s accuracy.

3.1.3. Comparison Between Forecasted, Measured, and Theoretical Solar Energy

The solar energy forecast at the Nipepe station for 2020 and 2021 is very similar, with a margin of 0.01%, and both are very close to the theoretical irradiation spectrum (Figure 22). This observation improves parameterization through MLM, which accepts a set of parameters as input that are corrected using the MLM characteristics.

Considering all atmospheric factors that affect the solar radiation effect, the forecasted 2021 solar energy (orange line) is around the theoretical solar energy (green line). Additionally, the projected solar energy for 2020 (blue line) is around the same magnitude as the projected solar energy for 2021 (orange line). Yet, as expected based on the observation of standard theoretical radiation in Figure 18, the predicted experimental radiation follows the theoretical radiation spectrum, peaking in the months with the highest transmittance because of atmospheric parameters and falling in the months with the lowest transmittance.

Additionally, the forecasted energy in Figure 22, shows relatively little variation between 2020 (blue line) and 2021 (orange line), with 2021 showing a slightly higher quantitative incidence intensity. However, if it is greater in magnitude because it aggregates the energy absorbed by several parameters in its total estimate, compared to the solar energy measured on-site presented in Figure 15, this, in turn, is estimated to be less in 2020 than in 2021 and is mostly because, in the model, the diffuse components differ from one another. In the characteristics of solar energy incidence in the study location (Northern Mozambique), both forms of energy precisely track the development of theoretical irradiation. The behavior in 2020 and 2021 is determined by the previously mentioned spatiotemporal, climatic, and geographic elements of order, which obstruct solar radiation in its route. Figure 22 shows that energy levels are highest from September to March throughout the summer and lowest from April to August, with July serving as a climax.

3.2. Analysis of Systematic Bibliographic Data Sources

3.2.1. Every Technique for Forecasting Worldwide Horizontal Radiation

The process of forecasting GHI entails estimating the quantity of solar radiation that will arrive at a horizontal surface at a given place within a certain time frame [3]. Numerical weather prediction (NWP) models, satellite-based methods, hybrid approaches, machine learning techniques, and physical models (such as simplified radiation models and radiative transfer models) are some of the approaches and methods used for this purpose. Empirical models include regression analysis and artificial neural networks (ANNs) [99].

Solar radiation can be predicted using a variety of machine learning algorithms, such as support vector machines, random forests, and gradient-boosting machines, which are based on historical data and meteorological inputs.

The availability of input data, computational capacity, required forecast accuracy, and application requirements are some of the variables that influence the technique selection.

Additionally, validation datasets should be used to test the accuracy and dependability of the model’s performance because machine learning approaches can capture intricate correlations.

Artificial Neural Network (ANN) Model

These have interconnected nodes, referred to as neurons, arranged in layers, closely resembling the composition and operation of biological brain networks [100]. Feedforward neural networks, recurrent neural networks (RNNs), and convolutional neural networks (CNNs) are the three categories into which they can be separated. The fact that these are frequently used to forecast solar radiation makes them interesting. The model accurately predicted 24 h solar irradiation, and its quality was demonstrated by the correlation coefficient, which ranged from 94–96% for cloudy days to 98–99% for sunny days [32,101,102]. As shown in

Table 2. Artificial neural networks (ANNs) for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error
(Benghanem & Joraid, 2016) [7]	GHI in sitio	1	A multiple correlation between different solar parameters	653,801	0.00028–1	2016	France	Seasoned	Not esteemed
(Ramirez-Vergara et al., 2021) [11]	GHI	24	Ambient temperature and solar irradiance forecasting	9,597,012	0.00028–1	2021	China	Continental	Not estimated
(ousif, et al., 2019) [27]	GHI	1	Analysis and forecasting of weather	522,448	0.00028–1	2019	Oman	Continental	Not estimated
(Mellit & Pavan, 2010) [32]	GHI in sitio	24	A 24-h forecast of solar irradiance using ANN	2,022,175	1	2020	Italy	Continental	Not estimated
(Wen et al., 2023) [35]	GHI in sitio	0.017, 24, 8760	A regional solar forecasting approach	5,557,443	0.00028–1	2023	China	Continental	10.93 ± 2.35%
(Breitkreuz et al., 2009) [36]	AOD, GHI in sitio	0.083	Solar energy forecast basing in different MLM and input	1,504,228	0.00028	2007	Germany	Oceanic temperate	+28 W/m2 (+12%) and (−19 W/m2 or −14%) (on average+9 W/m2 or +2%).
(Blaga et al., 2019) [38]	GHI in sitio	1	Accuracy of solar energy forecasting	426,112	1	2019	Spain	Mediterranean	Not estimated
(Nam & Hur, 2019) [39]	GHI	1	Generating resources for grid integration	1,001,429	0.00028–1	2019	Korea	Continental	Not estimated
(Kumler et al., 2018) [40]	GHI in sitio, atmospheric	1	A Physics-based for Intra-hour forecasting	6,569,201	0.00028–1	2018	USA	Subtropical	Not estimated
(Aler et al., 2015) [41]	GHI	1	MLM for Daily Solar Energy Forecasting	5,809,722	0.00028–1	2015	Netherlands	Continental	Not estimated
(Dhillon et al., 2020) [42]	GHI	1	ANN for Prediction of Power for Wireless	4,422,112	0.00028–1	2020	South Africa	Tropical	Not estimated
(McCandless & Jiménez, 2020) [45]	GHI in sitio	0.25 and 3	A model tree approach to forecasting	5,010,008	1	2020	USA	Subtropical	Not estimated
(Wang et al., 2015) [59]	Wind, GHI	1	Wind, solar, and electrical load forecasting methods	2,489,426	0.00028–1	2022	China	Continental	Not estimated
(Ghayekhloo et al., 2015) [73]	GHI in sitio	24	A novel clustering approach for short-term	2,477,108	24	2015	USA	Subtropical	Not estimated
(Dahmani et al., 2016) [106]	GHI in situ and Satellite	1 and 0.083	Estimating 5-min and hourly GHI	1,244,167	0.083	2015	Algeria	Arid to semi-arid	nRMSE = 13.90%, nRMSE igual a 19.35% e 18.65%.
(Sridharan, 2023) [107]	GHI in sitio	1	Generalized Regression Neural Network Model	3,287,000	1	2023	India	Monsoon tropical	GRNN, fuzzy logic and ANN are 3.55%, 4.64%, and 5.49%
(Flocas, 1980) [108]	GHI in sitio	1	Estimation and prediction GHI Greece	131,957	1	2012	Greece	Mediterranean	Not estimated
(Panamtash et al., 2020b) [109]	GHI in sitio	1	Probabilistic solar power forecasting	1,356,254	0.00028–1	2020	Germany	Oceanic temperate	Not estimated
(H. Sun et al., 2016) [110]	GHI in sitio	1	A Green Energy Application	5,255,033	24	2016	South Africa	Tropical	Not estimated
(Kuo & Huang, 2018) [111]	GHI in sitio	1	MLM in Solar PV Energy	1,106,048	0.00028–1	208	Netherlands	Continental	Not estimated
(Mathiesen et al., 2013) [112]	GHI in sitio	1	A high-resolution, cloud-assimilating	8,146,426	1	2013	USA	Subtropical	Not esteemed
(Devaraj et al., 2021) [113]	GHI in sitio	1	Energy forecasting using big data	3,001,569	24	2021	USA	Subtropical	Not esteemed
(Ramsami & Oree, 2015) [114]	GHI in sitio	24	A hybrid method for forecasting the energy	7,222,109	0.00028–1	2015	China	Continental	Not estimated
(S. Gupta et al., 2020) [115]	GHI	1	Evaluation of solar forecasting technologies	2,459,000	1	2020	India	Subtropical	Not estimated
(Travieso-González et al., 2024) [116]	GHI	1	Solar energy forecasting using machine learning models	2,436,008	24	2024	Romania	Subtropical	Not estimated
(Erdener et al., 2022) [117]	GHI	24	A review of behind-the-meter solar forecasting	6,114,644	0.00028–1	2022	USA	Subtropical	Not estimated
(Sharma & Kakkar, 2020) [118]	GHI	1	Management approaches for energy-harvesting	723,008	1	2020	Germany	Oceanic temperate	Not estimated
(Y. Zhang et al., 2011) [119]	GHI	1	Validation of GFS day-ahead solar irradiance	1,897,445	24	2011	China	Mediterranean	Not estimated

, decision tree and neural network models seem to be reasonable substitutes for the stepwise regression model in terms of comprehending energy consumption patterns and forecasting energy consumption levels, in addition to predicting electricity consumption [103,104]. These models’ ability to learn nonlinear correlations between input meteorological variables (such as temperature, humidity, and cloud cover) and GHI is one of their main features. This ability enables ANNs to be used to represent complex atmospheric processes [94,105].

Support Vector Machine (SVM) Model

Both classification and regression tasks can be completed by this model at the same time. In a nonlinear feature space, it can be effectively used for GHI prediction by mapping meteorological inputs to GHI values [120,121]. Regression was used to estimate solar irradiance using all-sky image characteristics, and a mean absolute error of roughly 22% may be obtained when predicting 5 min solar irradiance [122]. It was demonstrated that predicting electricity consumption is a feasible substitute for analyzing energy consumption trends and estimating energy consumption levels [104]. One benefit of support vector machines (SVMs) is their consideration of an appropriate hyperplane for data point separation in a high-dimensional feature space [48,123]. This hyperplane is optimized to maximize the margin between classes, as further discussed in

Table 3. Support vector machines (SVMs) for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contribution	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error
(Gutierrez-Corea et al., 2014) [58]	GHI in sitio and satellite	0.25	Spatial estimation of GHI	506,030	1	2014	Spain	Mediterranean	67% of stations (average ±2STD.)
(Barhmi et al., 2024) [98]	GHI	1	Solar Forecasting Techniques	9856	0.00028–1	2024	The Netherlands	Continental	Not estimated
(Kühn et al., 2014) [124]	GHI (in sitio), wind	1	Fluctuations in renewable energy supply	357,592	0.00028–1	2014	Germany	Oceanic temperate	Not esteemed
(Zwane et al., 2022) [125]	GHI in sitio	24	Solar Energy Forecasting	8825	24	2022	South Africa	Tropical	Not estimated
(C. Yang & Xie, 2012) [126]	GHI	1	Solar forecasting, its dependence	285	0.00028–1	2012	USA	Subtropical	Not estimated
(Tawn & Browell, 2022) [127]	GHI	1	Short-term wind, solar forecasting	84,789	0.00028–1	2022	France	Seasoned	Not estimated

’s references [104,122].

Random Forest (RF) Model

This involves creating several decision trees during the training process and calculating the average prediction made by each tree [93]. It was applied to phases of testing and development, where the mean absolute percentage error (MAPE) ranged from 3.8% to 13.8% [71]. The tool was employed to enhance the mapping of solar irradiance at high latitudes and produced a new dataset with higher accuracy and precision than the input datasets. The results included a bias of −1.5 W/m² and 4.3 W/m², a root mean square deviation (RMSD) of 17.9 W/m² and 27.1 W/m², a mean absolute deviation (MAD) of 11.9 W/m² and 17.5 W/m², and a bias of 0.01 W/m² [2]. They can handle big datasets with high dimensionality, and they are resistant to overfitting [128,129,130]. As seen in

Table 4. Random forests for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error
(Babar et al., 2020) [2]	GHI in situ and satellite, Cloud, Albedo	0.083	Mapping of solar radiation in high latitudes.	385,207	1	2020	Norway	Oceanic, continental,	RMSD—17.9, 27.1 Wm−2; RMSD—16.2; MAD—10.8,
(Cerralbo et al., 2015) [52]	GHI (in sitio)	0.0000028	Wind variability in a coastal area	643,801	1–24	2015	Alfacs Bay	Seasoned	Not esteemed
(Mei et al., 2014) [76]	GHI (in sitio)	0.0056	Retrieval of aerosol optical depth	4,851,699.4	1–24	2013	USA	Subtropical	Not esteemed
(Zhang et al., 2011) [80]	GHI (in sitio)	1–24	A multi-angle aerosol optical depth retrieval	9,970,610	1–24	2011	Canada	Seasoned	Not esteemed
(Pedro et al., 2012) [81]	GHI (in sitio)	0.0000028	Assessment of forecasting techniques for solar power	2,057,777	1–24	2012	Mexico	Seasoned	Not esteemed
(Krishnan et al., 2023) [84]	GHI (in sitio)	0.17	Solar radiation forecasting impacts	885,156	1–24	2023	Brazil	Tropical	Not esteemed
(Badescu & Budea, 2016) [87]	GHI (in sitio)	0.017	Significant is the stability of the radiative regime	9,597,000	1–24	2016	China	Continental	Not esteemed
(Perpiñán et al., 2013) [131]	GHI (in sitio)	0.017	Urban rooftop PV systems &fluctuation	17,900,000	1–24	2014	Netherlands	Seasoned	Not esteemed
(Klima & Apt, 2015) [132]	GHI (in sitio)	0.0000028	Geographic solar PV smoothing	4,851,699	124	2015	USA	Subtropical	Not esteemed
(Perez et al., 2016) [133]	GHI (in sitio)	0.000028	Spatial and Temporal Variability of GHI	6001.5621	124	2016	USA	Subtropical	Not esteemed
(Marcos et al., 2011) [134]	GHI (in sitio)	0.00028	Power fluctuations: the PV plant/filter	506,030	1–24	2022	Spain	Medi terrain	Not esteemed
(Mills, 2011) [135]	GHI (in sitio)	0.017	Variability GHI Wide-Area Geographic	4,851,699	1–24	2011	USA	Subtropical	Not esteemed
(Arias-Castro et al., 2014) [136]	GHI (in sitio)	0.017	Anisotropic solar ramp rate correlations	4,851,699	1–24	2014	USA	Subtropical	Not esteemed
(Halász & Malachi, 2014) [137]	GHI (in sitio)	0.017	Assessment of PV power fluctuations	20,720	0.017–24	2014	Israel	Temperate a tropical	Not esteemed

, they may be utilized to predict GHI values by identifying patterns in meteorological data and forecasting GHI values based on input features.

Gradient-Boosting Machine (GBM) Model

A collection of ensemble learning techniques is employed to construct models in a sequential manner, where each subsequent model addresses the errors of its predecessors [138,139]. The Pearson correlation coefficient was utilized to determine the most pertinent meteorological data for model training, revealing significant errors and demonstrating the study’s quality and reproducibility [14,20]. The researchers forecasted diffuse solar irradiance through machine learning and multivariate regression approaches. In the context of multivariate regression, the mean absolute error (MAE) for logistic regression, utilizing the aforementioned predictors, was recorded at less than 21.5 W/m² and 30 W/m² [140,141]. This method is particularly advantageous for effectively managing structured data and capturing intricate relationships. It has been successfully applied to global horizontal irradiance (GHI) prediction tasks, achieving high accuracy by progressively refining predictions based on the residuals from earlier models [140,141,142,143,144], as detailed in

Table 5. Gradient-boosting machines (GBMs) for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error
(Tsai et al., 2017) [4]	GHI in sitio	1	Models for forecasting growth trends in renewable energy	9,597,000	1	2017	China	Continental	NGMB(1,1) highest GM(1,1)
(Smith et al., 2013) [128]	GHI (in sitio), wind	1	A comparison of random forest regression	357,592	0.00028–1	2013	Germany	Oceanic temperate	Not esteemed
(Liu & Sun, 2019) [129]	GHI (in sitio)	0.00028, 0.083	Random forest solar power forecast	18	1	2019	USA	Tropical	Not esteemed
(Madanchi et al., 2017) [142]	GHI (in sitio)	0.000028, 0.00028, 0.017	Strong nonlinearity of GHI fluctuations	562,500	0.0000028–0.017	2017	Several countries	Moderate	Not esteemed
(Woyte et al., 2007) [143]	GHI (in sitio)	0.00139	Fluctuations in the instant clarity index	310,500	0.0019–24	2007	Belgium, German, Australia	Moderate maritime, moderate	Not esteemed
(Kumari & D. Toshniwal, 2021) [145]	GHI (in sitio)	0.0167	Long short term memory–convolutional	4,851,699	1–24	2021	USA	Subtropical	Not esteemed

.

Long Short-Term Memory (LSTM) Network Model

A specific kind of recurrent neural network (RNN) is used to represent sequential input and identify long-term relationships [144,145]. First, the structural effect accounts for the largest portion of the total contribution to consumption, reflecting the importance of solar energy growth [88,139,140,141,142]. The factorial decomposition and prediction of solar energy consumption were carried out, and the results show that the proposed approach combined with LSTM has better feasibility [145,146]. Its ability to learn patterns and trends from previous meteorological data, illustrated in

Table 6. Long short-term memory (LSTM) networks for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error
(Sopian & Othman, 1992) [147]	GHI in sitio	1	Estimates of monthly and daily GHI Malaysia	330,803	1	2012	Malaysia	Equatorial	Not estimated
(Kayima et al., 2023) [148]	GHI (in sitio)	1	Socio-economic benefits of on-grid hybrid solar	241,038	1	2023	Uganda	Equatorial, hot and humid	Coef. = 1.73, 1.25, 1.03, increased access ∼1.26–1.77, 1.38
(Stevović et al., 2019) [149]	GHI in sitio	1	Possibilities for wider investment in solar	8,510,000	1	2019	28 EU countries and Serbia	Tropical	Not esteemed
(Alkasassbeh, 2018) [150]	GHI	1	Taxonomy of wind and solar energy forecasting	4,851,699	24	2018	Saudi Arabia	Tropical	Not estimated
(Aslam et al., 2021) [151]	GHI	1	Deep learning methods for power load	4,851,699	0.00028–1	2021	Singapore	Humid tropical	Not estimated
(Kurtgoz & Deniz, 2018) [152]	GHI	1	A virtual sky imager testbed forecasting	5,891,699	8760	2018	USA	Subtropical	Not estimated
(Lee et al., 2018.) [159]	GHI and clear sky	0.083	Assessing variability of wind speed	5,852,589	0.00028–1	2018	Singapore	Humid tropical	Not estimated

, makes it advantageous for application in time-series forecasting, particularly GHI forecasting [142,146,147,148,149,150,151,152,153,154,155,156,157,158].

Gaussian Process Regression (GPR) Model

The link between the input and output variables is modeled as a multivariate Gaussian distribution using this nonparametric Bayesian technique. Investigations into the specific heat capacity of nanofluids for solar energy applications led to the conclusion that GPR $(R=0.99974,RMSE=0.01506\mathrm{J}/\mathrm{k}\mathrm{g})$ performs better than RF $(R=0.99761,RMSE=0.04598\mathrm{J}/\mathrm{k}\mathrm{g})$ and GRNN $(R=0.99563,RMSE=0.06085\mathrm{J}/\mathrm{k}\mathrm{g})$ results [131]. As

Table 7. Gaussian process regression (GPR) for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error
(Ayet & Tandeo, 2018) [97]	GHI Satellite	6	Nowcasting solar irradiance	643,801	1–24	2018	France	Seasoned	Not esteemed
(Nielsen et al., 2012) [100]	GHI (in sitio), wind	0.33	PV Power Fluctuations Neural Networks and Deep Learning	506,030	0.00028–0.33	2012	Spain	Mediterranean	Not esteemed
(Pizarroso et al., 2022) [101]	GHI Satellite	1	Variability Sensitivity Analysis	3,287,263	1–24	2021	Indian	Monsoon tropical	Not esteemed
(D. S. Kumar et al., 2022) [153]	GHI (in sitio)	1	PV Ramp Rate Limiting Strategies	734.3	124	2022	Singapore	Humid tropical	Not esteemed
(Deo et al., 2016) [154]	GHI	1	SVM for forecasting GHI incident	32,527,263	0.00028–1	2016	Australia	Tropical Atlantic	Not estimated
(Said & Alanazi, 2023) [155]	GHI	1	AI-based solar energy forecasting for grid	3,287,263	0.00028–1	2023	Brazil	Tropical	Not esteemed

illustrates, it has the benefit of being able to produce probabilistic GHI projections and quantify forecast uncertainty. It has been used for GHI forecasting assignments, especially when estimating uncertainty is important.

Autoregressive Integrated Moving Average (ARIMA) Model

As demonstrated in

Table 8. Autoregressive integrated moving average (ARIMA) for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error
(Zhang et al., 2015) [23]	GHI	1	A suite of metrics for assessing the performance	5,833,697.1	0.00028–1	2015	Japan	Oceanic temperate	Not estimated
(Ramirez Camargo et al., 2015) [43]	GHI	1	Forecasting prediction horizon sensitivity a	5,859,699	0.00028–1	2015	USA	Mediterranean	Not estimated
(Arslan et al., 2020) [67]	GHI	1	Wind speed variability and wind power potential	5,891,659.4	1	2020	France	Seasoned	Not esteemed
(El-Sebaii & Trabea, 2003) [156]	GHI in sitio	1	Estimation of horizontal diffuse solar radiation	1,002,001	1	2023	Egypt	Tropical	Not esteemed
(Vijayakumar et al., 2005) [157]	GHI in sitio	1	Analysis of short-term solar radiation data	4,851,699	1	2005	USA	Subtropical	Not esteemed
(Awachie & Okeke, 1988) [158]	GHI in sitio and isolation	1	Measurement of solar energy radiation	5545	1	2012	Nigeria, Nsukka	Tropical	0.21 and 0.51 with 0.7%
(J. C. Y. Lee et al., 2018) [159]	GHI	1	Solar Power Generation Forecast Model	7,856,698	0.00028–1	2018	Brazil	Tropical Atlantic	Not esteemed
(Unterberger et al., 2021) [160]	GHI	1	Energy yield of flat-plate solar collector systems	5,871,222.7	0.00028–1	2021	Brazil	Tropical Atlantic	Not estimated
(Alessandrini et al., 2015) [161]	GHI	1	Forecasting daily total solar-radiation	7,468,698.2	0.00028–1	2015	Germany	Oceanic temperate	Not estimated

, they used the autoregressive integrated moving average (ARIMA) method to forecast solar energy consumption after breaking down the factorial. The findings indicate a larger share of total consumption, highlighting the significance of solar energy growth [32,88].

Simple Linear Regression (SLR) Model

However, several satellites provide these raw samples with an emphasis on estimated accuracy [2,162] compared to solar irradiation measured by traditional sensors that offer the real performance of solar energy on the Earth’s surface, used in the present research, with a margin of error of 0.3 to 0.5 [2,3,6]. An analysis and study of solar energy conditions that affect the region that is related to access to horizontal solar irradiation incident on that location were conducted. They used multivariable regression and machine learning to predict diffuse solar irradiation [2,3]. The mean absolute error (MAE) of logistic regression utilizing the aforementioned predictors was less than 21.5 W/m² when the multivariable regression approach was applied, as shown in

Table 9. Simple linear regression (SLR) for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error
(Mucomole et al., 2024) [6]	GHI, wind	1	Regressive and Spatiotemporal Accessibility	506,030	1–24	2024	Mozambique	Tropical	RMSE 21%
(Mucomole et al., 2023a) [25]	GHI in sitio	0.017 and 0.16	Temporal Variability of Solar Energy	801,590	0.167	2023	Mozambique	Tropical	RMSE 30%
(Qing & Niu, 2018) [28]	GHI (in sitio)	1	Hourly day-ahead solar irradiance	281,748	1–24	2088	Brazil	Tropical Atlantic	Not estimated
(Hoff & Perez, 2010) [55]	GHI in sitio	1	Quantifying PV power Output Variability	92,644	1	2010	USA	Subtropical	Not esteemed
(Chen et al., 2023) [88]	GHI (in sitio) wind	24	Big data analysis of PV power fluctuation	9,597,000	1–24	2023	China	Continental	Not estimated
(S. Kumar & Tiwari, 1996) [163]	GHI in sitio	1	Estimation of convective mass transfer in solar	3,287,015	1	2012	India	Monsoon tropical	C = 0.0322; n = 0.4114 for 8 × 106 in an active solar still.
(Teke et al., 2015) [164]	GHI in sitio	1	Estimation of solar radiation	83,562	1	2015	Turkey	Mediterranean, Continental, Semi-arid	Not esteemed
(Park et al., 2001) [165]	GHI (in sitio)	1	Hybrid PV System Operation Control	377,973	1–24	2001	Japan	Oceanic temperate	Not estimated
(Stetz et al., 2015) [166]	GHI	1	The Impact of Solar on Germany’s Energy	357,592	1–24	2015	Germany	Oceanic temperate	Not estimated
(Suri et al., 2007) [167]	GHI	1	Solar electricity GHI prediction fluctuation	442,569	1–24	2007	France	Seasoned	Not estimated
(Jerez et al., 2019) [168]	GHI and wind	1	Future temporal variability PV Europe.	8760	1–24	2019	Spain	Mediterranean	Not estimated
(Trapero et al., 2015) [169]	GHI (in sitio)	1	Short-Term Solar Irradiation Forecast	506,030	1	2015	Spain	Mediterranean	Not estimated
(Akuffo & Brew-Hammond, 1993) [170]	GHI (in sitio)	24	Establishing frequency distribution	238,535	24	2003	Gana	Tropical	Not estimated
(Tiba et al., 2016) [171]	GHI (in sitio)	0.017–3	variability of solar irradiation (minute)	27,843,295	0.0167–3	2016	Brazil	Tropical Atlantic	Not estimated
(Sha & Aiello, 2020) [172]	GHI (in sitio)	0.017	Decentralized Energy Exchange Smart Grid.	17,900,145	1–24	2018	Netherlands	Oceanic temperate	Not estimated
(Koudouris et al., 2018) [173]	GHI (in sitio)	1	GHI process for renewable manage	131,957	1–24	2018	Greece	Mediterranean	Not estimated
(Lam & Li, 1996) [174]	GHI in sitio	1	Regression Analysis of Solar Radiation	9,597,000	131,400	2011	China	Continental	Not esteemed
(Thaker & Höller, 2022) [175]	GHII	1	Time Series Forecasting of Solar Energy	249,951	1	2022	Germany	Oceanic temperate	Not esteemed
(Mucomole et al., 2024b) [176]	GHI	1	Accessibility of Spatial and Temporal Solar	281,948	1	2024	Mozambique	Tropical	Not estimated

[2,3,6].

Regression Kriging (RK) Model

This machine learning model tackles regression in conjunction with correction variables [81]. With an equivalent rise in local network penetration rates, the MLM approaches marginally enhance the results. The worldwide horizontal irradiance at the sub-hourly level was estimated spatially using three geostatistical interpolation techniques, and Regression

Table 10. Regression Kriging (RK) for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Error Assumption
(Funk & Larson, 1998) [13]	GHI, temperature and charge	1	Parametric model of solar cooker	4,851,699.4	1	2012	USA	Subtropical	Coeficient of heat
(Jebli et al., 2021) [20]	DHI	1	Prediction of diffuse solar	9,597,000	1	2016	China	Continental	Not esteemed
(Jung et al., 2016) [21]	DHI	1	Spatiotemporal Characteristics in the Clearness	6,025,321	1	2016	China	Continental	Not esteemed
(Adedeji et al., 2020) [29]	GHI in sitio	1	Resource forecast in site suitability assessment for wind and solar energy	3,287,129	1	2023	India	Monsoon tropical	GRNN, fuzzy logic and ANN are 3.55%, 4.64%, and 5.49%
(Arumugham & Rajendran, 2021) [49]	irradiance, weather and solar angle	0.17	Modeling global solar irradiance	9,597,000	0.17	2021	Several locations	Continental	Not estimated
(Mustafa & Malik, 2023) [90]	GHI in sitio	1	Analysis through a MLR, and Decision Support	881,913	1	2023	Pakistan	Subtropical	Not esteemed
(Hao et al., 2021) [177]	GHI in sitio, temperature	1	Design and prediction method of dual	9,597,000	1	2021	China	Continental	R2 of 0.98 and 0.94
(Nakayama et al., 2020) [178]	GHI in sitio, PV Power	1	Description of short circuit current outdoor	377,975	1	2020	Japan	Temperate oceanic	≥0.5 kW/m2 (on sunny day)
(Younis et al., 2010) [179]	GHI in sitio	1	effect of some factors on water distillation b	1,002,459	1	2010	Egypt.	Egypt	1% significance.
(Fu & Cheng, 2013) [122]	GHI in sitio	0.83	Predicting solar irradiance	8,227,114	0.83	2013	China	Continental	Not estimated
(Gupta, et al., 2016) [82]	GHI,	1	A surface reflectance scheme for retrieving AOT	344,509	1	2016	India	Monsoon tropical	Not estimated
(Vijayakumar et al., 2016) [83]	GHI in sitio	1	Aerosol Optical Depth Retrieval over Bright Areas	4,851,699.4	1	2016	USA	Subtropical	Not esteemed
(Midilli & Kucuk, 2003) [180]	GHI in sitio	1	Mathematical modeling of thin layer drying	83,562	1	2012	Turkey	Mediterranean, Continental, Semi-arid	Rsfsd = 0.9983, χ2sfsd = 2.697 × 10−5
(Lauka et al., 2018) [181]	GHI in sitio	1	First solar power plant in Latvia.	64,589	1	2018	Latvia	cool summers and cold winters	Not esteemed
(Li et al., 2021) [182]	GHI (in sitio)	1	Influencing actual absorption of solar	4057.0496	1	2021	China	Continental	Not esteemed
(Ma et al., 2015) [183]	GHI	1	Description of short circuit current	6,597,230	1	2015	India	Monsoon tropical	Not estimated
(Guta, 2018) [184]	GHI in sitio	1	Solar energy technology in rural	1,112,009	1	2018	Ethiopia	Temperate	Not estimated

indicates that 67% of the volunteer stations investigated present results within the margin of error (mean of ±2 standard deviations) according to Kriging interpolation from satellite photos [131]. Table 10 illustrates the benefits of obtaining results that are near to the estimate, among other things.

Hybrid Machine Learning Models

This consists of combining multiple machine learning models to increase estimation efficiency for comparisons and results [34,184,185,186]. According to [81], there was a comparable rise in local network penetration rates.

Table 11. Hybrid machine learning models for solar energy forecasting worldwide.

Source	Data Source	Resolution (h)	Contributions	Area (km2)	Forecast Horizon (h)	Year	Location	Climate	Erro
(Ohtake et. al., 2019) [8]	GHI in sitio, PV	1	Accuracy of the solar irradiance forecasts	585,144	1	2013	Japan	Tropical	Not estimated
(Mohanty et al., 2017) [9]	GHI in sitio	1	Forecasting growing economy	3,287,596	1	2017	India	Monsoon tropical	Not estimated
(Sahin et al., 2023) [14]	GHI, power solar	1	Predictive ANN and MR	83,562	1	2023	Turkey	Mediterrain, Continental, Semi-arid	Not esteemed
(Pedro et al., 2017) [30]	solar, wind, tidal	1, 2, 3	Mathematical methods for optimized solar forecasting	95,592	1	2017	USA	Subtropical	Not esteemed
(Lauret et al., 2016) [34]	GHI in sitio	1	Solar radiation forecasting in an insular	357,592	1	2015	France	Oceanic temperate	Not esteemed
(Saffaripour et al., 2013) [44]	GHI in sitio	1	Predicting solar radiation fluxes for solar energy	1,648,896	1	2013	Iran	Subtropical and tropical arid high altitude	Not esteemed
(Anuradha et al., 2021a) [46]	GHI (in sitio)	1	Forecasting Using Machine Learning	3,287,263	1	2021	India	Monsoon tropical	Not esteemed
(Ahmad et al., 2018) [50]	GHI in sitio	1	Predictive modeling for solar thermal energy	243,610	1	2018	United Kingdom	Temperate oceanic	Not esteemed
(Klein et al., 1975) [69]	GHI in sitio, thermal	1	Simulation of solar processes and its application	99,452	1	2005	USA	Tropical	Not esteemed
(Charles et al., 2019) [70]	GHI in sitio, PV	0.25	ustainable energy storage for solar home systems in rural	8825	175,200	2019	Saharan Africa	Tropical	7 and 17%
(Puga-Gil et al., 2022) [71]	GHI in sitio	0.017 and 0.167	Global Solar Irradiation Modeling and Prediction	5,227,080	0.167	2022	China	Tropical	Not esteemed
(Fahrmeir et al., 2013) [86]	GHI, power solar	1	Regression: Models, Methods and Applications	710,850	1	2013	Morocco	Temperate	Not esteemed
(Ramedani et al., 2014) [91]	several data tests	1	ANN approach on Type II Regression Analysis	83,562	1	2014	Turkey	Mediterranea, Continental, Semi-arid	ANN based in lower error than the OLS based bisector technique
(Jogunuri et al., 2007) [93]	GHI in sitio	1	Seasonal multi-step ahead short-term solar	6,599,236	1	2007	China	Continental	Not esteemed
(Perveen et al., 2013) [94]	GHI in sitio	1	Global solar energy and PV power forecasting	385,207	1	202	India	Oceanic, continental, subarctic and alpine	RF (MAE) 10.2; STD. 1.5; Interpolation 21.3; STD. 6.4 (Wm−2)
(Wu et al., 2023) [95]	PV data	1	Understanding spatiotemporal energy	7,394,021	1–24	2023	China	Continental	Not estimated
(Alzahrani et al., 2017) [96]	GHI (in sitio)	1–24	Solar Irradiance Forecasting Using Deep Neural Networks	8,514,876	1–24	2017	Brazil	Tropical	Not estimated
(Hill et al., 2021) [120]	GHI Satellite	1	Design-Based Global and Small Area Estimations for Multiphase Forest Inventories	506,030	1–24	2021	Spain	Mediterranean	Not estimated
(Kuipe et al., 2013) [121]	GHI in sitio	1	Generalized Order-Restricted Information Criterion	4,287,289	1	2013	India	Monsoon tropical	GRNN, fuzzy logic and ANN are 3.55%, 4.64%, and 5.49%
(Kurtgoz & Deniz, 2018) [152]	GHI in sitio	1	Estimation of GHI for Different Climatological	83,562	1	2018	Turkey	Mediterranea, Continental and Semi-arid	Not esteemed
(Abdul-Wahab et al., 2005) [185]	wind speed, direction, air temperature, GHI	1	Gground-level ozone and factors affecting	309,501	1	2004	Oman	Arid and dry desert	Not esteemed
(Jamei et al., 2021) [186]	GHI in sitio and nanoparticles	0.0167 and 1	Application of Gaussian process regression	1,648,589	1	2021	Iran	Subtropical and tropical arid high altitude	GPR (R = 0.99974, 0.01506 J/kg); (R = 0.99563, 0.06085 J/kg)
(Verma et al., 2016) [188]	GHI, cloud cover, temperature, wind speed, rainfall	1	Data Analysis to Generate Models Based	2,567,008	1	2016	India	Monsoon tropical	Not estimated
(Alizamir et al., 2020) [189]	GHI (in sitio)	1	Estimating solar radiation	1,648,025	1	2020	Iran	Subtropical and tropical arid high altitude	Not esteemed
(Rapti, 2000) [190]	Climate	1	Atmospheric climatic turbidity, transparency	6,682,023	1–24	2010	Greece	Mediterranean	Not estimated
(Salmanoğlu & ÇetiN, 2022) [191]	Wind	1–24	Harvest wind-Solar PV for Production	5,682,026	1–24	2022	Harvest	Tropical	Not estimated
(Gomes et al., 2020) [192]	GHI (in sitio)	1	Irradiance and power ramp rates	8,514,876	124	2020	India	Monsoon tropical	Not estimated
(Siebentritt, 2011) [193]	GHI (in sitio), wind	0.0167	Efficiency imitates chalcopyrite solar cells	2586	124	2010	Luxembourg	Semi-continental	Not estimated
(Alam et al., 2014) [194]	GHI (in sitio)	0.0167	Solar PV ramp rate control	7,692,023	124	2014	Australia	Hot and dry	Not estimated
(Güçlü et al., 2015) [196]	GHI in sitio	0.00028	Correlational multidimensional and LR	861,591	1	2015	Germany	Oceanic temperate	Not esteemed
(Quoc Hung & Mishra, 2019) [197]	GHI (in sitio)	0.00028, 0.0167	Voltage fluctuation	7,692,023	1–24	2019	Australia	Hot and dry	Not esteemed
(Kawasaki et al., 2006) [198]	GHI (in sitio)	0.0000028	PV fluctuation assessment	377,973	0.00028–0.017	2006	Japan	Oceanic temperate	Not esteemed
(de Freitas Viscondi & Alves-Souza, 2019) [199]	GHI	1	Big data for solar pv electricity	489,595	1	2019	Brasil	Tropical	Not estimated

illustrates how applying machine learning techniques somewhat enhanced the performances reported by [34]. Its benefit is that it can provide more accurate projections based on the situation [29,44,69,186,187,188,189,190,191,192,193,194,195].

Where HDKR is the Hay, Davies, Klucher, and Reindl model; NWM is the numerical weather model; NWP is the numerical weather prediction; the Multiscale Hybrid Forecasting Model (MHFM) USA, UK, MFDFA, tree-based ensemble machine learning models, and the United States of America are terms used to describe the United States and the United Kingdom, respectively. Random Forest (RF); Extra trees (ET) and decision trees (DT) the support vector regression (SVR); Angstrom–Prescott linear regression, or A-PLR for short; forward-looking neural networks, or F-FNNs; multiple linear regression, or MLR for short; CSI: conventional spatial interpolation; PCR: principal component regression; linear regression (LR), Gaussian process regression (GPR), generalized regression neural network (GRNN), multiple regression (MR), geostatistical interpolation (GSID), deterministic and HelioSat2 (GBRT), ARIMA stands for autoregressive integrated moving average in neural networks; S-SLA stands for single-site linear autoregressive; GM(1, 1) for grey prediction models; NGBM(1, 1) for the model and the grey Verhulst model.

By identifying patterns in meteorological data, machine learning algorithms enable precise GHI forecasting, offering trustworthy forecasts for applications in environmental research, renewable energy, and climate monitoring.

Analyzing the Sample Bibliographies and the Data Sources Gathered

A detailed examination of the frequency density reveals that, of the approximately 145 bibliographic sources that were retrieved from different research sources, machine learning models for the estimation and extrapolation of GHI represent the largest relative research output. This trend was more pronounced in several regions in 2015, as shown by the analysis in Figure 23.

With a value of 0.86, the distribution of extracted sources through Figure 24 demonstrates a favorable association between knowledge production in 2001 and 2024.

The most common approach, according to the methods used, is advanced deep learning, which uses machine learning models with roughly 23% SLR, 19% hybrid models, 17% ANNs, 11% RK, 4% SVMs, 9% RF, 5% ARIMA, 4% GBMs, 4% LSTM, and 4% GPR, as shown in Figure 25.

The hybrid model’s method yields a preponderance of 145 bibliographic sources (

Table 12. Comparison of the overall effectiveness and patterns of use of various techniques.

	ANNs	SVMs	RF	GBMs	LSTM	GPR	ARIMA	SLR	RK	Hybrids MLM
Cumulative	28	6	14	6	7	6	9	19	17	33

). This is significantly greater than the estimated six sources utilizing SVM, GBMs, and GPR models, which give a comparatively lower number of explorations.

In a range of 6 and 33 cumulative extracted and processed sources, Figure 26 illustrates how the physical approaches and the traditional methods estimate the cumulative trend of higher use.

4. A Discussion of Machine Learning Techniques for Parameterizing Solar Energy

Utilizing MLM that parametrizes solar radiation, the study enhanced solar irradiance mapping at high latitudes. In most cases, the model produced more accurate and precise results than those found in the input datasets, with a bias of 0.01 W/m² and a lower error of 16.2 W/m² [18]. Moreover, parameterization aids in the more accurate prediction of diffuse solar irradiation through multivariate regression and machine learning. Analogously, one might note that solar generation technologies can achieve significant market expansion by utilizing a specific amount of solar energy that coincides with a horizontal surface, increasing local grid penetration rates [3,200].

The primary ambiguity in the interpretation of the bibliographic sources accessed and analyzed was the inclusion of sources that comprehensively describe the model a priori in terms of parametric relationships to be considered, followed by a somatic analysis. This was primarily to meet the criteria for the summary of results using MLMs that reduce the risk of solar resource misuse. However, the sources use MLM to interpolate and/or anticipate solar energy and provide the parameters in terms of raw samples obtained by remote sensing or in situ measurement instruments (radiometers, photometers, etc.). For a summative analysis, MLMs have the advantage of standardizing the results on the same administrative scale and randomizing the results, which allows the output to be taken into account as a whole. The analysis of generalized parameters (direct and diffuse solar energy, which add up to the predicted global solar energy) brought on by the different atmospheric components (aerosols, water vapor, uniformly mixed gases, ozone layer, among others) is necessary to avoid the ambiguity this introduces into the sample analysis.

There is a significant advancement in estimates using neutral network loops and their variants but interpolated by the RF model, to occasionally fill in measurement gaps and/or measurement data of atmospheric parameters and/or solar energy. The summary and analysis of bibliographic sources reveal a wide range of uses for regression models, including SLR and RK, among others. This presents good analysis bias and lowers the risk of bias towards estimation errors.

The ozone transmittances for the years 2020 and 2021 differ significantly, yet the obtained numerical values are within 0.0098 of one another. The zenith angle and ozone concentration determine the biggest difference. Ozone has very little effect on overall transmittance because of its high transmittance. The water vapor transmittances compare quite well, with values that are around 0.9862 greater at zenithal angle zero. In addition to varying with the zenith angle, the discrepancies also change with the water vapor content.

The expected region data sample has strong variability, with a coefficient of dispersion of roughly 0.52. In contrast to clear and intermediate sky days, roughly 68% of intermediate sky days are mostly represented in the statistical data sample that was generated. This represents a rough statistical analysis of the initial data collected at a station, estimated at 70% of intermediate sky days. While this is in agreement with Kumler et al. (2018) [40], it also shows that short-term solar forecast models that rely only on global horizontal irradiance (GHI) measurements frequently fail to distinguish between factors that affect the GHI and those that can be accurately calculated by atmospheric models.

The time interval under examination and the number of inferred measurement stations shows a decorrelation, which becomes more pronounced for the anticipated values, both for the original GHI data sample that was inferred in situ and for the predicted GHI sample. The output of a solar plant is affected by the variability in solar energy, and this estimate is more pronounced on intermediate sky days, which can be the most problematic. Hoff and Perez (2010) [55] and Mucomole et al. (2024) [6] confirmed this result. These results are similarly consistent with those calculated by Mucomole et al. (2024) [6], but the estimated values tend to exhibit a high frequency of changes in the clear-sky index for their respective increments [15].

For a clean environment, the direct normal irradiances obtained for the two years are relatively similar when using the technique to account for aerosol attenuation; nevertheless, the spectrally integrated findings are very good at zenith angles less than 90°. The biggest difference that can exist between the two years is 0.0893. While the pattern is more configurable than previous models in that it can incorporate new observations to enhance the GHI prediction, if available, it does not require additional observations of different atmospheric parameters [3,40].

Almost identical findings are obtained at zenith angles less than 90° when the influence of turbidity is included, based on a visibility of 150 km. High diffuse multireflected solar energy is produced at bigger zenith angles due to some disagreement caused by albedo and the intensity of solar energy incidence [3,24,176].

The outcomes are more in line with the spectrally integrated approaches. The earlier finding was validated by similar comparisons made using various atmospheric factors. This improved the idealization’s accuracy, and using it is advised.

According to broadband simulations, the initial millimeters of precipitable water in the atmosphere absorb as much radiation as the next centimeters.

Whether the atmosphere is laden with water vapor or is dry, precipitable water produces results with roughly the same degree of error and remains virtually unaltered when the visibility parameter is adjusted. This finding helps to simplify the parameterization equation by assuming a constant volume of water vapor [94,155].

Because aerosol levels and gases can cause significant variations in ground-level solar resources, this study uses both observed and anticipated AOT to investigate the effects of different AOT forecast qualities on solar energy applications [201]. The findings are consistent with research that highlights the necessity of long-term projections for effective control and optimization of solar power systems, such as those published by Breitkreuz et al. (2009) [36] and Lauret et al. (2016) [34]. Local climatic parameters can be used to improve the approximation of the anticipated sample and machine learning models can be used to prepare for future calibrations [3,24,89].

The findings here improve the great PV system dimensioning by optimizing solar energy and solar plant output, hence sizing the PV system based on daily energy use. The evaluation is also a powerful tool for gauging environmental behavior in the context of climate change today. It can provide criteria for making decisions regarding the planning and execution of numerous projects in a variety of fields, including architecture, agriculture, fishing, medicine, aquaculture, and construction. When compared to other models previously mentioned, the results show that the actual quantity of predicted solar energy is around 1241.85 W/m², taking into account all estimating parameters.

The combined data from all sources describe in detail how models are applied utilizing multiple sample sources, providing input of advancements in the use of MLM for parametric forecast analysis of solar energy. The physical effects of radiation scattering in the atmosphere, including the Rayleigh effect, MIE effect, and scattering by small, medium, and large aerosols, are among the parameters that are aggregated and taken into account in the model that is introduced.

However, future research can use a sample of hours of solar brightness and aggregate other physical effects of radiation interaction with matter to perform the same analysis instead of sampling in situ solar energy. This will allow for a more comprehensive understanding of the available solar resources and provide a different perspective on complexity. Nevertheless, an inference estimate of solar energy and the dynamics of the generalized cloud velocity as a whole can also be studied.

5. Summary and Conclusions

The idea of electrifying rural areas is associated with the use of clean sources, such as PV solar energy, which is efficient, clean, and oscillation-free. This type of energy has been gaining popularity worldwide, particularly in the study region of Mozambique, where access to electricity is scarce. Additionally, the installed PV power systems have a limited useful life due to various factors that interfere with their operation; the variability in solar energy is increasing. These factors are crucial for the planning and dependability of future PV electrical networks as well as other solar energy projects. It is feasible to synthesize and make the following deductions from the sample of scientific articles’ bibliographic data analyzed over the past 23 years, which was condensed into solar energy parametric forecast models validated with GHI samples gathered on location in specific areas.

There is a considerable propensity in the literature from the past five years to employ ANN and RF for solar forecasting, with a cumulative order of 28 and 14 sources. However, the SLR model tends to be used by RK, with 19 and 17 sources assessed, respectively. A cumulative analysis of the estimated solar energy using hybrid models from about 33 sources makes several inputs available for the final model summation, increasing the model’s viability and certainty while reducing forecast error.

The evenly mixed gases exhibit the least amount of attenuation based on the atmospheric factors utilized. The uniformly mixed gases, therefore, required the evaluation of a separate absorption term, which is unlimited and is ignored. When compared to gases that are evenly mixed, ozone attenuates slightly more. The amount of ozone in the atmosphere affects this. The three primary attenuators are aerosols, water vapor, and air molecule scattering. Hydrogen vapor and aerosols’ proportional relevance is contingent upon their atmospheric concentration. The simple scattering albedo, or w₀, also affects aerosol absorptance. Regardless of the zenith angle, the transmittances of molecular absorbers are relatively constant. Significant reductions are observed in the Rayleigh and aerosol transmittances.

On clear days, the frequency density of the observation of values near zero is lower than that of values near one; on days with intermediate skies, the frequency density of the observation of values near one is lower than that of values near 0.66; on days with intermediate skies, the frequency density of the observation of values near one is lower than that of values near 0.020. In all cases, the analysis of the sky types shows that the clear sky index is similar to other days in that these values are within the theoretical radiation spectrum, which ranges from −1 to 1.

Owing to several factors that affect solar radiation as it approaches the Earth’s surface, the estimated measurement of solar energy variability offers a practical substitute for the current solar energy fluctuation. As a result, it is a crucial parameter to comprehend the true solar energy behavior from the perspective of future estimation. Still, the following is true: to obtain a more precise estimate, geographical features, atmospheric, climatic, and local influences are needed.

The GHI sample exhibits trends that are comparable to those of the in situ GHI sample, with the majority of the trends differing because cloud movement and meteorological factors will affect future trends. By correcting the real variation through even smaller scale measurements, the GHI’s degree of clarity can be raised.

The majority of the predicted and observed variability in GHI is explained by intermediate sky days, with a larger tendency for lower intermediate sky days. Over the examined spatiotemporal range, solar energy shows a considerable decorrelation, both in the anticipated GHI sample and on intermediate sky days. When machine-readable models are used to examine regression approaches, the prediction error is reduced, with RMSE verifying about 0.30 utilizing the RF model.

The evaluation of radiation in a specific bandwidth can be achieved by simply summing the spectral values. However, radiation over the whole solar spectrum can be calculated using the broadband approach, which is simply integrated over the whole solar spectrum.

6. Patents

This research article appears as one of the first obligations to attend to, in order to obtain a Doctorate degree, the Doctoral Course in Energy Science and Technology, at Eduardo Mondlane University, which the main author is attending.