## 3 Global Horizontal Irradiance Models
### Clear-Sky Irradiance Formula  
Another essential element of semi-empirical models is the determination of the maximum solar irradiance limit, i.e., the clear-sky irradiance (GHI\({}_{\text{c}}\)). Under a cloudless condition, the solar irradiance attenuation depends on the concentration of absorbers and scatterers, such as aerosols, gases, and water vapor, inside the atmosphere. Most clear-sky models address solar attenuation through a derivation from atmospheric components and geometric position. With regard to the semi-empirical approach, the complexity of the clear-sky irradiance model, which is caused by rapid changes in aerosols, water vapor, and other molecules, is relatively reduced. The atmospheric variables can instead be simplified into a single turbidity variable of \(T_{L}\)[35; 36; 37]. In this study, well-known clear-sky irradiance models (Table 3) were selected to examine GHI\({}_{c}\). The Beyer model applies a straightforward clear-sky model based on only a geometric calculation. This model is called the Bourges clear-sky formula and uses only \(\theta_{z}\) to summarize the attenuation of extraterrestrial solar irradiance during transmission in the atmosphere [10]. The model was purely developed from an empirical correlation between ground measurement, sun position, and site location [38]. The clear-sky irradiance of the Rigollier model, which is known as the ESRA model, is decomposed into direct and diffuse components, considering \(T_{L}\) and the Rayleigh scattering factor (\(\delta_{R}\)). The clear-sky irradiance formula of the Hammer model is similar to that of the Rigollier model. The clear-sky DNI equations in both the Rigollier and Hammer models are identical [39], but the clear-sky DHI equations are different. Owing to the complexities in the derivation of the clear-sky GHI from the Rigollier model, it was preferable to adopt a simpler formula, successfully applied in previous literature. Here, the Kasten clear-sky model was selected to determine the clear-sky GHI of the hybrid model. The model conserved site elevation (z) and atmosphere turbidity for a site. By including z in the equation, the Kasten model demonstrated its advantage in investigating clear-sky GHI at high elevations. In the Perez model, the clear-sky GHI is calculated based on the study by [PERSON] and [PERSON] [36], in which they upgraded the original Kasten model. This new variant also considers \(T_{L}\) and z to enhance its suitability for observational data from the West of the American continent. The most important input parameter in the clear-sky model is the Linke turbidity, which accounts for atmospheric conditions. The Linke turbidity is also a convenient tool for simplifying the degree of atmospheric attenuation due to absorption and scattering. Theoretically, \(T_{L}\) can be evaluated via either a direct or indirect formula [40]. The direct formula, which is called the pyheliometric formula, commonly exploits DNI measurement data, \(AM\), and \(\delta_{R}\)[2]. In contrast, the indirect formula, which is called the parameterized formula, uses atmospheric components, such as \(AM\), precipitable water vapor (\(w\)), angstrom turbidity factor (\(\beta\)), and AOD [41]. After testing various formulas of \(T_{L}\), the model introduced by [PERSON] et al. [41] (Equation (13)) turned out to be optimal and was used throughout this study:  
\[T_{L}=(1.8498+0.2425w-0.0203w^{2})+(15.427+0.3153w-0.0254w^{2})\beta \tag{13}\]