1. Summary

2. Data Description

3. Methods

3.3. Clear-Sky Radiation and Its Regression Correlation

By relating GHI and clear sky radiation, the clear-sky index ($K_t^{\ast }$) was obtained [3,5,7]. Using values of$K_t^{\ast }$ for different classes of days (clear, intermediate, and cloudy), a connection was made as a function of the distance of the correlation coefficient or systematic connection of the clear-sky index${\chi }_{ij}^{{∆K}_t^{\ast }}$ [7,14,18],

$${\chi }_{ij}^{{∆K}_t^{\ast }}={\displaystyle \frac{cov\left(∆K_{t,i}^{\ast }\left(t\right),∆K_{t,j}^{\ast }\left(t\right)\right)}{\sqrt{{\sum }_{\tau =1}^{\tau }{\left(∆K_{t,i}^{\ast }\left(t\right)-\overline{∆K_{t,i}^{\ast }}\right)}^2{\sum }_{\tau =1}^{\tau }{\left(∆K_{t,j}^{\ast }\left(t\right)-\overline{∆K_{t,j}^{\ast }}\right)}^2}}}$$

(8)

Figure 3. Forecasted irradiance comparison (scale [1/365]).

Figure 3. Forecasted irradiance comparison (scale [1/365]).

Statistically,${\chi }_{ij}^{{∆K}_t^{\ast }}$ can be$\ge -1$, equal to zero, or$\le 1$ in a space considered for a subspace station between two points x and y. In order to collect data from direct GHI measurements, a priori estimation was used to reduce losses for any errors, also known as uncertainty or the error margin (error). The number of classes (n) was established by classifying the values derived from the$K_t^{\ast }$. Inferring the regression coefficient of$K_t^{\ast }$ and its increments${\beta }_{ij}^{{∆K}_t^{\ast }}$, with${\alpha }_l$ as the intercept constant and${\epsilon }_t$ as the error term, as shown in Equation (9) described the characteristics of the regression inferential analysis of$K_t^{\ast }$ and its increments for various classes and types of days in all years, between the previously indicated seasons [14,33],

$$∆K_{t,j}^{\ast }\left(t\right)={\alpha }_l+{\beta }_{ij}^{{∆K}_t^{\ast }}∆K_{t,i}^{\ast }\left(t\right)+{\epsilon }_t$$

(9)

The correlation between a temporal series and its associated values is measured by the autocorrelation function (ACF). The autocorrelation between$K_t^{\ast }$ and$K_{t-1}^{\ast }$ that cannot be explained by the delays between them is indicated by the Partial Autocorrelation Function (PAF) for a temporal series$K_t^{\ast }$, as shown inFigure 4. This improves the used parametric forecast model by showing that there is a sudden variation and subsequently a homogeneous state for both functions in terms of forecasted energy.

The spatial autocorrelation, as a function of the distance between the pair of pyranometers for acceptable and unacceptable days, modeled for the classes of clear, cloudy, intermediate and all types of days, decreases with the increase in distance between the pair of sensors. This result was also observed in Ahmad et al. (2018) [20] and Mucomole et al. (2024) [7], who evaluated the inference of 50 to 62 pyranometers over hundreds of kilometers, obtaining many correlation points. The rate of decrease is more pronounced on days that record sudden experimental GHI values of more than 1084.04 W/m², which are not common in measurements. These indicate that solar radiation resulting from solar flares and other extreme solar activities may be to blame for these values. However, as detailed in the analysis, the solar energy decreases along the correlative curve. With the increase in distance, the spatial correlation structures of the three sky types, along the analysis section, also differ in their decay rates: for acceptable days, the approximate values of$K_t^{\ast }$ correlation coefficient, are as follows: under cloudy sky conditions 0.2986, clear 0.3586, intermediate 0.3201, and all sky types 0.3844; on unacceptable days under cloudy sky conditions 0.3723, clear 0.3758, intermediate 0.3569, and all sky types 0.4899. The regressive inferred structures of$K_t^{\ast }$ within the different sky types are consistent with the associated cloud patterns, presenting cloudy and clear-sky conditions with greater homogeneity equivalent to the existence or not of cloud layers, sometimes with localized shading factors in pyranometers, as shown inFigure 5.

Both acceptable and unacceptable days have estimated energies near the GHI based on the theoretical model, according to the experimental modeling of the parametric estimation of solar energy classified in terms of day types. Correlative accessibility evaluations regarding heat maps using the$K_t^{\ast }$ heat correlation coefficient have an above-average potential, according to a summative analysis of the correlative-regressive study, as seen inFigure 5a. However, the influence of atmospheric, climatic, geographic, and spatiotemporal parameters related to human-industrial activity, reproductiveness of nature, and other factors that emit gases into the environment, such as aerosols, water vapor, gases that strengthen the ozone layer, and even climate change gases, that absorb, attenuate, and emit solar energy, contributes to significant increases in solar energy that can be quantified increasingly by reducing the observation interval, here quantified in about 1 to 10 min of solar energy measurement interval. The increases in solar energy, both for acceptable and unacceptable days, analyzable in terms of the correlation coefficient of${∆K}_t^{\ast }$, exhibit a decorrelating structure analyzable in the structure of correlative-regression heat maps, as illustrated inFigure 5b.

The$K_t^{\ast }$ correlation of increments${\chi }_{ij}^{{∆K}_t^{\ast }}$ that characterizes the space–time metric of the region taken through the classes of similar spectra of acceptable and unacceptable days presents a decorrelation, as a function of distance; this means that energy accessibility occurs from the beginning of the study period to the end and decreases from cloudy to clear skies and then to intermediate skies. The comparison of the correlation${\chi }_{ij}^{K_t^{\ast }}$ with the model and several different models using a cloud speed of between 4 and 7 m/s shows a decay of the correlation corresponding to the intermediate skies, but after that, the correlations fall much faster, and there are greater energy transmission disturbances in intermediate conditions.

4. Handling and Applicability of the Forecasted Solar Energy Dataset

5. Patents

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