## 2 Materials and Methods
### Empirical Temperature-Based Models
#### 2.3.3 Models Implemented in Tropical Environments  
Knowledge of solar irradiance behavior in tropical zones is a mandatory requirement for this research study. The analyzed cases are from Africa (Nigeria Abuja, Benin City, Katsina, Lagos, Nsukka, and Yola), Brazil (Agua Branca, Pao de Azucar, Santana do Ipanema, Palmeira dos Indios, Arapicara, Maceio, Corcuripe, and Sao Jose da Laje), and Mexico. Due to the fact that Narino is in a tropical area, the results from these three cases constitute relevant inputs for the current research.  
\begin{table}
\begin{tabular}{c c} \hline \hline \(a\) & Region Type \\ \hline
0.16 & Arid and semi-arid \\
0.17 & Interior regions \\
0.19 & Coastal regions \\
0.10\(-\)0.09 & Humid climates \\ \hline \hline \end{tabular} Source: [4,14].  
\end{table}
Table 3: [PERSON] and [PERSON]’s empirical coefficients.  
[PERSON] and [PERSON] (ON) proposed a linear model as follows [37]:  
\[\frac{H}{H_{0}}=a+bT_{R}+cT_{max},\]  
where \(T_{R}\) is the monthly average of the ratio between the daily minimum and maximum temperatures \((T_{min}/T_{max})\), and \(a,b\) and \(c\) are empirical coefficients. The statistical validation of the model yielded a coefficient of determination between 0.809 and 0.952.  
[PERSON] and [PERSON] reviewed empirical models implemented in West Africa in order to compute global solar irradiance. They found that the soft computing models have better accuracy than the empirical models since they can be more easily adapted to several weather conditions. This is because soft computing models allow more inputs, as models or variables, to strengthen their reliability [38].  
[PERSON] et al. studied the performance of ten temperature-based models in Northeastern Brazil. They found that the models did not show significant changes after adjusting them concerning the rainy periods' effects. [PERSON] et al. also concluded that the HS model demonstrates better performance in the hinterlands and interior regions, whereas the BC model showed the best performance in humid and coastal zones [14].  
[PERSON] et al. compared the values of the original empirical coefficients of the HS with new values generated from the model calibrated for Mexico with local data. A notable element of that research was the classification of Mexico's climate zones using the Koppen-Geiger system. Furthermore, the authors developed another classification based on the clearness index, as shown in Table 4, to support the AWS data. It is useful to remember that the clearness index is the ratio between terrestrial solar insolation and extraterrestrial solar insolation. A relevant conclusion in this respect was that regardless of solar irradiance peaks during the day, it is possible to obtain similar clearness index values. Another notable conclusion was that fixed empirical coefficients result in significant error, especially in zones with a temperature difference below 15 \({}^{\circ}\)C. The authors proposed the following equation to overcome the identified issue derived from fixed coefficients [15]:  
\[a_{HS}=a_{1}+a_{2}(\Delta T)\;+\;a_{3}(\Delta T)^{2}\]  
where \(a_{HS}\) is the Hargreaves and Samani equation coefficient.