# Potential of Photovoltaic Generation in the Putumayo Department of Colombia
## 2 Materials and Methods
### Annual Irradiation on Inclined Surface \(G_{a}(\alpha,\beta)\)  
Once the irradiation was evaluated \(G_{dm}(0)\), the irradiation on an inclined surface \(G_{a}(\alpha,\beta)\) was evaluated too, where \(\alpha\) is the angle of orientation and \(\beta\) the angle of inclination. For the above, the global irradiation \(G_{dm}(0)\) is decomposed as the sum of the monthly daily diffuse irradiation \(D_{dm}(0)\) and monthly daily direct irradiation \(B_{dm}(0)\). The condition described by [PERSON] and [PERSON] (1960) according to which the relationship between the lightness index \(\frac{G_{dm}(0)}{B_{dm}(0)}\) and the diffuse fraction \(\frac{B_{dm}(0)}{G_{dm}(0)}\) is independent of latitude was considered [28]. Once the daily components of the irradiation \(D_{dm}(0)\) and \(Bo_{dm}(0)\) were obtained, the hourly irradiation on inclined surface \(G_{h}(\alpha,\beta)\) was calculated, for which the model of the three components was taken. This model establishes that the incident irradiation is formed by the direct \(B_{h}(\alpha,\beta)\), diffuse \(D_{h}(\alpha,\beta)\) and reflected \(R_{h}(\alpha,\beta)\) irradiation, and it is expressed as  
\[G_{h}(\alpha,\beta)=B_{h}(\alpha,\beta)+D_{h}(\alpha,\beta)+R_{h}(\alpha,\beta) \tag{8}\]  
where  
\[B_{h}(\alpha,\beta)=\frac{B_{h}(0)}{cos\theta_{\rm zs}}.max(0,cos\theta_{\rm s}) \tag{9}\]  
with \(\theta_{\rm s}\) as the angle of incidence between the solar rays, \(\theta_{\rm zs}\) the normal to the plane considered, and the zenithal solar angle, obtained by means of their respective sinusoidal functions. To calculate the hourly direct \(B_{h}(0)\) and hourly diffuse radiation \(D_{h}(0)\), the following expressions from the Collars-Pereira model were used [29,30]. The diffuse irradiation component on the inclined surface \(D_{h}(\alpha,\beta)\) was calculated by the Hay-Davies isotropic model, because in several studies it stands out for its precision and simplicity [31,32]. This model establishes that the diffuse irradiation on the inclined plane is equal to the sum of two components. The first is the circumsolar component \(D^{c}(\alpha,\beta)\) coming directly from the sun, while the second is the isotropic component \(D^{1}(\alpha,\beta)\) coming from the whole celestial hemisphere. The reflected component \(R_{h}(\alpha,\beta)\) or albedo was calculated considering the reflectivity of the soil, \(\rho_{r}\), as 0.2, assuming that the soil is horizontal, of infinite extension, and reflects light isotropically.  
The hourly components were added to obtain the monthly average daily irradiation on an inclined surface. The value of the annual daily irradiation on inclined surface is approximately equal to the average of the \(G_{dm}(\alpha,\beta)\) values. Finally, to obtain the monthly average of daily irradiation on an inclined surface \(G_{dm}(\alpha,\beta)\), the hourly components were added \(G_{h}(\alpha,\beta)\). The annual mean daily value irradiation on sloped surface \(G_{da}(\alpha,\beta)\) is approximately equal to the average of the values of \(G_{dm}(\alpha,\beta)\).  
To determine the angles and the maximum irradiation, the previous procedure is repeated varying the angles. After knowing the angles, the angular losses were calculated to select an angle that has a suitable inclination for maintenance. With the selected angle, the irradiation \(G_{da}(\alpha,\beta)\) was calculated. These values were averaged by region for each meteorological stations' zone and were taken as the irradiation of each municipality in the region. Finally, the annual irradiation \(G_{a}(\alpha,\beta)\) was calculated as the product of \(G_{da}(\alpha,\beta)\) by the number of days in the year.