## 3 Results and Discussion  
### Optimal Tilt and Orientation Angles  
The developed application obtains a flat surface's optimal position to capture solar energy in a given location. The results are based on the climate data introduced for each city (8760 h each year). Figures 4 and 5 illustrate the captured irradiance on a flat surface in Cuenca-Ecuador according to the Klucher model.  
Figure 6 depicts the captured irradiance on a flat surface in Cuenca-Ecuador according to the Liu and Jordan model, and Figure 7 depicts the captured irradiance on a flat surface in Cuenca-Ecuador according to the Temps model. With the mathematical models that have been analyzed through the application, similar results were obtained for the city of Cuenca with the measured data. Note that, for this locality, the more significant capture of solar insolation on a flat surface has an orientation of 34\({}^{\circ}\) concerning the north and an inclination angle of 16\({}^{\circ}\).  
Figure 5: Captured irradiance on a flat surface in Cuenca-Ecuador according to the Klucher model. (0–360\({}^{\circ}\) angle concerning the north, while the radius is the tilt angle in radians).  
Figure 6: Captured irradiance on a flat surface in Cuenca-Ecuador according to the Liu and Jordan model.  
The five mathematical models obtain the optimal inclinations and orientations to capture solar irradiation. However, it is essential to consider that the Klucher and Perez models are the most complex and, at the same time, the ones that best adjust to real conditions since they improve the previously existing models. In the case of the [PERSON] model, it has been selected since it is based on the excellent prediction for clear sky conditions of Temps and Coulson model but with the addition of a \"modulating function\" to correct it under conditions cloudy sky and thus have greater precision in values of this climatic condition. On the other hand, [PERSON]'s model is based on a detailed statistical analysis of the diffuse sky, dividing it into three components: horizon zone, circumsolar zone, and isotropic bottom.  
On the other hand, with the TMY data, different results were obtained for Ecuador's cities, as shown in Tables 4 and 5.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multicolumn{10}{c}{**Irradiance, Inclination, and Orientation Values**} \\ \hline
**Models** & \multicolumn{2}{c}{**Diffuse Isotropic**} & \multicolumn{2}{c}{**Liu and Jordan**} & \multicolumn{2}{c}{**[PERSON] and Coulson**} \\ \hline Cities & (Wh/m\({}^{2}\)) & I (\({}^{\circ}\)) & O (\({}^{\circ}\)) & (Wh/m\({}^{2}\)) & I (\({}^{\circ}\)) & O (\({}^{\circ}\)) & (Wh/m\({}^{2}\)) & I (\({}^{\circ}\)) & O (\({}^{\circ}\)) \\ Guayaquil & 1823.064 & 18 & 50 & 1823.064 & 18 & 50 & 1935.033 & 21 & 51 \\ Quito & 2089.807 & 17 & 35 & 2089.837 & 18 & 35 & 2187.17 & 19 & 35 \\ Cuenca & 2184.402 & 18 & 34 & 2184.402 & 18 & 36 & 2282.8 & 20 & 36 \\ Santo & 1628.135 & 17 & 46 & 1367.136 & 17 & 46 & 1729.643 & 19 & 47 \\ Domingo & & & & & & & & & \\ Machala & 1732.026 & 19 & 51 & 1732.026 & 19 & 51 & 1843.007 & 22 & 52 \\ Portoviejo & 1894.892 & 20 & 53 & 1895.89 & 20 & 53 & 2008.308 & 23 & 53 \\ Loja & 2159.481 & 19 & 40 & 2159.481 & 20 & 40 & 2259.794 & 21 & 40 \\ Ambato & 2120.749 & 19 & 38 & 2120.749 & 19 & 38 & 2218.019 & 20 & 38 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Obtained values for diffuse isotropic, Liu and Jordan, and Temps and Coulson models.  
Figure 7: Captured irradiance on a flat surface in Cuenca-Ecuador according to the Temps model.  
From the mathematical models, each city's variation in Ecuador is clearly shown, taking into account the typical year files' error. Note that these values are taken with a \(-1\%\) margin of the maximum insolation value to obtain tilt and orientation range results closer to the measured data values.