ESTIMATION OF DIFFUSE SOLAR RADIATION IN THE CITY OF NATAL – RN BASED ON THE KT INDEX

Objective: This study aimed to validate and compare three models for estimating diffuse solar radiation in the city of Natal - RN, using data from the SONDA Project, with a focus on developing and validating the Ubial model. Theoretical Framework: The research is based on solar radiation concepts, which are essential for optimizing photovoltaic systems. Accurate models for estimating diffuse radiation are necessary for the efficient performance of these systems, especially in locations like Natal - RN. Method: The methodology adopted a quantitative and comparative approach, analyzing the Erbs, Bourges, and Ubial models. Data from the SONDA Project were used, and the statistical analysis included evaluating the Mean Bias Error (MBE) and correlating the data with the clearness index (Kt). Results and Discussion: The Ubial model showed superior performance, with a negative MBE, indicating greater accuracy without overestimating the data. These results were discussed in light of the theoretical framework, highlighting the suitability of the Ubial model for the climatic conditions of Natal - RN. Research Implications: The implications of this research include the potential application of the Ubial model in other regions with similar climatic characteristics, contributing to the optimization of photovoltaic systems. Originality/Value: This study introduces an innovative model that incorporates additional variables, resulting in more accurate estimates of diffuse solar radiation, with a direct impact on the solar energy field.


INTRODUCTION
All electromagnetic radiation coming from the Sun is called solar radiation and is responsible for hitting the planet (Querino et al., 2006).This radiation plays a crucial role in the main physical, chemical and biological processes, both in animal and plant organisms.Furthermore, it is directly responsible for the disposition of primary energy in all terrestrial processes, from photosynthesis to the development of storms, which can cause adverse weather conditions (Souza et al., 2005).Therefore, knowledge of the behavior of Global Solar Radiation (GR), composed of the direct and diffuse components, both above and within forests, is essential to understand the availability of energy for the various processes of this system.
According to Crotti (2018), the frequent droughts that have occurred in recent years in Brazil have been compromising the credibility of the system of generating electricity through hydroelectric power plants, given that the majority of the energy supply in the country comes from this source (EPE, 2020).Faced with this situation, diversifying the electricity matrix, giving priority to renewable energy sources, seems to be the best strategy for guaranteeing the energy supply and, at the same time, reducing the environmental impact.In this context, solar energy stands out as a promising option (MCTIC, 2017).Therefore, an accurate analysis of radiation at the installation site is fundamental for the performance of the photovoltaic modules and for the evaluation of the so-called generation of electrical energy.
For the development of solar energy applications, knowledge of the direct and diffuse fractions of solar radiation is fundamental (CROTTI, 2018).In most weather stations, only global radiation is measured, which corresponds to the sum of direct and diffuse radiation.
However, the specific understanding of the contribution of each form of radiation can significantly influence the development of power generation projects for these purposes.
Therefore, for the development of advanced projects in the generation of photovoltaic energy, a comprehensive knowledge of solar radiation at the installation site is required, including global, diffuse and direct radiation.The latter is especially crucial for the energy production of photovoltaic modules (LEALEA, 2013).According to Gomes (2007), one of the most widely used statistical methodologies for data validation is the model developed by Liu & Jordan (1960).This methodology has been widely applied in the estimates of horizontal and diffuse direct radiation (Orgill & Hollands, 1977;Painter, 1981;Iqbal, 1983;Dehne, 1984;Stanhill, 1985;Skartveit & Olseth, 1987;Sirén, 1987;Soler, 1990;Battles et al., 1995;Jacovides et al., 196;González & Calden bó, 1999;Oliveira et al., 2002a).One advantage of the Liu & Jordan method is the elimination of the locality dependency.Furthermore, the model is capable of relating the fraction of direct radiation on incidence (Kb) to the clarity index (Kt), as demonstrated in the studies by Bartoli

METHODOLOGY
A comparative methodology was adopted for the analysis, in which the models Erbs, Bourges and the proposed model (Ubial) were compared.The latter was developed by the authors for the treatment of solar radiation data specific to Natal, Rio Grande do Norte, using information provided by the SONDA project.The Ubial model considers, in addition to Kt and Kd, other relevant variables.

LOCATION
Natal is a coastal city located at latitude 5º 45' 54" South and longitude 35º 12' 05" West.
Its climate is hot and humid, with average temperature around 26°C and moderate winds predominant from the Southeast.Due to its proximity to the equator, the city has about 2,184.80The intensity of local solar radiation is high, with a solar trajectory ranging from 65° to 90°, with diffuse radiation predominating and high luminosity (Araújo, 1996).

WEATHER STATION
The Solarimetric Station of the Laboratory of Tropical Environmental Variables -LAVAT belonging to the National Institute for Space Research / Northeast Regional Center, is located 58 meters above sea level, at coordinates 5.8367° S and 35.2065°W, mounted on top of a water tank.The main characteristics of the equipment of the Station of interest of the present work are specified in Table 1 (SONDA, 2024).Among the available equipment, the station has two pyranometers for measuring global and diffuse radiation, in W/m², equipped with ventilation units and a shading ring (diffuse radiation sensor) (SILVA, 2008).The last calculation step will be to find the direct solar radiation in which Equation 4was used.
=   ×   (4) where: Hd = direct solar radiation incident on the Christmas surface -RN (W/m2) (2010), the use of a logistic equation is a coherent approach to modeling the variation of the direct solar fraction with respect to the clarity index, especially in places where estimates based on polynomial models are not adequate.However, since it is not possible to directly extract the empirical coefficients from logistic regression, a mathematical adaptation is necessary to express it in the form  =  +.This adaptation was performed as described in equations 5 to 7.

MODEL BUILD
Normally the logistic equation is as follows: To obtain the coefficients a and b it is necessary to replace the linear equation with an auxiliary logarithmic function (Equations 6 and 7, respectively): In this way:

RESULTS AND DISCUSSIONS
Diffuse radiation results from the dispersion of solar radiation, a physical process that involves particulate matter, in which gas molecules, water droplets and aerosols remain suspended in the atmosphere.This phenomenon occurs due to the interaction of sunlight with the particles present, which disperses the energy in all directions on the Earth's surface.The amount of diffuse radiation is influenced by several factors, such as the thickness and composition of the atmosphere, the presence of clouds and aerosols, and the solar angle (Fiorin et al., 2011).
In order to visualize the daily distribution of Kd as a function of Kt, a graph was drawn that incorporates the polynomial equation of 4th order and the coefficient of determination R2, representing the characteristic curve of the data behavior.The distribution obtained is consistent with studies conducted after 2020 (Lopes Junior et al., 2021;Medeiros et al., 2022) and is congruent with the results presented by Galdino et al. (2016), which conducted a similar study in the region using data from 2004 to 2014.It is important to note that as Kt tends to 1, Kd also exhibits the same behavior, as shown in Figure 2.   2. This indicates a significant relationship between the two DHI measurements, which is clearly illustrated by the dispersion of the points around the adjustment line in Figure 3.Although these results are not identical to those obtained by Barros et al. (2018), due to the differences in the periods of the analyzed data, they are quite close and do not indicate changes in diffuse radiation patterns for the region.(Ubial), it should be noted, that the models reproduce moments of cloudy, partially cloudy, and clear skies (Medeiros et al., 2022), it can be seen that most of the data analyzed is concentrated in the center -for the three models -indicating a higher prevalence of clear skies.The proposed model (Ubial) showed greater difference, due to the adjustment for the Christmas/RN climate.This is because the local climate is predominantly dry which reduces diffuse radiation.The results in Table 3 corroborate what is observed in the graphs presented.Although the results of the proposed method (Ubial) show a greater difference, the three evaluated methods are considered to have yielded results within expectations.This is evidenced by the proximity of the results of the Erbs and Bourges models, as well as by the trend curve of the Ubial method, which maintains the same format as the others, even though their results are different.

Correlation and representation values of the sky cover
The Ubial model has an advantage compared to the other two models.The negative MBE value indicates that there is no overestimation in the estimated measurements of Kd (Ricieri et al., 1999), while the other methods tend to overestimate these values.Thus, the results indicate that the proposed model (Ubial) has positive aspects compared to the models of Erbs and Bourges.Based on these findings, it is considered that this study contributes both as material for future research and in a practical way, offering an improved model for the analysis of diffuse radiation.

CONCLUSION
The data obtained by the meteorological station and analyzed using the methodologies described in this study allowed a detailed comparison of the three models of estimation of diffuse solar radiation: Erbs, Bourges and the proposed model (Ubial).The results obtained confirm the validity of the Ubial model, highlighting its significant advantages in relation to the other two traditional models.In addition, it is recommended that future studies explore the application of the Ubial model in different locations and climatic conditions.Replicating the model across multiple regions will allow you to assess its robustness and adaptability, providing a more comprehensive understanding of its global applicability.
This study contributes significantly to the existing literature by presenting a new model for the assessment of diffuse solar radiation, offering a valuable tool both for the academic community and for professionals involved in solar energy projects.The academic value of this work lies in the proposition of a model that incorporates additional variables, resulting in more accurate and adjusted estimates, and the provision of grants for future research that may further explore and improve this approach.The validation of the Ubial model in different contexts could reinforce its importance and usefulness in the optimization of photovoltaic systems, promoting greater efficiency and sustainability in the generation of solar energy.
Among the meteorological stations that supply the most complete information about local solar radiation, one can highlight the project developed by INPE (National Institute for Estimation of Diffuse Solar Radiation in the City of Natal -RN Based on the KT Index ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.9 | p.1-15 | e08342 | 2024.4 Space Research), known as the SONDA network (National Organization System for Environmental Data).This system is aimed at the energy sector and aims to provide meteorological information to support the energy sector in Brazil (MARTINS, 2005).The meteorological stations of SONDA were strategically installed in different regions of the country to measure solar energy in the most varied climates characteristic of the Brazilian territory.The data collected by SONDA is publicly accessible and is used in the development of research and projects in several areas (PES, 2020).

Figure 2
Figure 2 shows the daily distribution of Kb as a function of Kt together with the polynomial equation of 2° and After data processing, a logistic function was used to represent the actual dispersion of  relative to .According to Boland et al. (2013) and Ridley et al.
fraction of solar radiation (Kd) observed in the solar station Ln = natural logarithm x = local brightness index (Kt) a and b = empirical coefficients estimated by linear regression analysis.Based on the mathematical correlations described, a graph of Ye per Kt was constructed in Microsoft Excel 2016 software.By means of linear regression, coefficients a and b were obtained.In the calculation of Ye, only values of Kd ≤ 0.9999 were considered, in order to avoid extremes in the equation.From these empirical coefficients and the local clarity index, it was possible to apply them in the logistic equation to generate a second graph, which demonstrates the behavior of the estimated direct fraction (Kbe) as a function of Kt, in comparison with the observed direct fraction (Kb) as a function of Kt.Finally, a third graph was created, showing the correlation between the Estimated Direct Solar Radiation, obtained Estimation of Diffuse Solar Radiation in the City of Natal -RN Based on the KT Index ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.9 | p.1-15 | e08342 | 2024.9 by the model developed in this work, and the Direct Solar Radiation Observed by the Natal/RN Solarimetric Station.

Figure 2
Figure 2Direct radiation as a function of brightness

Figure 3
Figure 3 Correlation of Kt by Ln adjustment

Figure 4
Figure 4Direct radiation estimated as a function of the brightness index et al. (2009) For the purpose of comparison between the proposed models (Ubial), Erbs and Bourges elaborated the graph observed in Figure 4.In the Figure, it is already evident that the results obtained by the Erbs and Bourge models are close to each other and that the proposed model

First
, the Ubial model proved superior to the Erbs model by incorporating additional variables beyond the clarity index (Kt) and diffuse fraction (Kd).This broader approach has resulted in more accurate and adjusted estimates of diffuse solar radiation.The inclusion of these extra variables allows a better understanding and modeling of local atmospheric conditions, reflecting directly on the accuracy of the results.Compared to the Bourges model, the Ubial model demonstrated a noticeably superior performance, evidenced by the negative Mean Absolute Error (MBE).This indicator suggests that the Ubial model does not overestimate diffuse radiation data, a crucial feature for practical Estimation of Diffuse Solar Radiation in the City of Natal -RN Based on the KT Index ___________________________________________________________________________ Rev. Gest.Soc.Ambient.| Miami | v.18.n.9 | p.1-15 | e08342 | 2024.13 applications in solar power generation.The absence of overestimation of data is particularly important to ensure the reliability and efficiency of photovoltaic systems, since inflated estimates can lead to sub-optimal design decisions.The time restriction of data, which only covers the period from January 1, 2015 to December 31, 2016, limits the generalization of results.Future studies should consider a longer time span to further validate the Ubial model and investigate possible seasonal and interannual variations.Another limitation concerns the comparative analysis restricted to only two additional models.Although Erbs and Bourges are widely used and recognized in literature, they are relatively old models.Future research should expand the comparison to include newer, innovative models that can offer new perspectives and further improvements in the estimation of diffuse solar radiation.

Table 1
Constituents of the Natal-RN Solarimetric Station.

Table 3
Comparison of statistical indices between the analyzed models