## 3 Results and Discussion  
### Parametric Identification  
The measurements in 2018 from Group 1 in Table 1, which include POA irradiance, air temperature, wind speed, and module temperature, were selected to determine the coefficients of the suggested models. In these locations, the multi-crystalline silicon modules were installed with fixed open cracks; therefore, the coefficients \(\Delta T_{ref}^{*}\) and \(W_{ref}\) in Equation (11) were taken as 28.4 \({}^{\circ}\)C and 13.3 m/s, respectively. These values correspond with \(a=-3.56\) and \(b=-0.075\) s/m in the original Sandia model [3].  
In the next stage, the best fit for the values \(y=\frac{\Delta T_{ref}}{G_{0}}f(\Delta t)\), describing the relevance of the thermal inertia of the module and the interval sampling, were obtained as a function of the time step \(\Delta t\) for both models, based on measured data in three cities in 2018. The measurement data are available for every minute, and to determine \(y\) based on the time step \(\Delta t\), the measurement data of \(T_{m}\) and \(\Delta G_{g}\) are averaged over the duration of the time step \(\Delta t\). The time step is limited to 10 min because otherwise, the time dependency is not described in sufficient detail. The resulting values of \(y\) and the corresponding correlation (\(R^{2}\)) and error (\(RMSE\)) between the simulated values of the temperature \(T_{m}\) and the measurement data for the proposed model with wind speed are presented in Table 4. The mean \(\overline{y}\), averaged over the three locations, and the related standard deviation (SD) were also calculated. The obtained values of \(y\) as a function of \(\Delta t\) for both proposed models are illustrated in Figure 1. The relation between \(\overline{y}\) and \(\Delta t\) is approximately exponential and a least-squares fit yields \(R^{2}\) equal to 0.97 and 0.99 for the model with wind speed and the model without wind speed, respectively: \(\overline{y}=0.0175\exp(-0.061\Delta t)\) and \(\overline{y}=0.0162\exp(-0.056\Delta t)\). These relations can be used to eliminate the function \(f(\Delta t)\) that we have introduced previously.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & \\ \cline{2-10}
**Time Step** & \multicolumn{3}{c}{**Tri An**} & \multicolumn{3}{c}{**Da Nang**} & \multicolumn{3}{c}{**Ha Noi**} & \multicolumn{3}{c}{**Average**} \\ \cline{2-10}
**(Min.)** & \(y\)\({}^{\circ}\)**C m\({}^{2}\)** & **\(R^{2}\)** & **\(RMSE\)** & \(y\)\((^{\circ}\)**C m\({}^{2}\)** & **\(\overline{W}\)** & **(\%)** & **(\(^{\circ}\)**C m\({}^{2}\)** & **\(\overline{W}\)** & **(\%)** & **(\(^{\circ}\)**C m\({}^{2}\)** & **\(\overline{W}\)** & **(\%)** & **(\(^{\circ}\)**C m\({}^{2}\)** & **\(\overline{W}\)** & **SD** \\ \hline
1 & 0.0153 &The Equation (11), therefore, becomes:  
\[T_{m}=T_{a}+\Delta T_{ref}^{*}\frac{G_{g}}{G_{0}}\exp\left(-\frac{W_{s}}{W_{ref}} \right)-\Delta T_{ref}\frac{\Delta G_{g}}{G_{0}}\exp\left(-\frac{\Delta t}{ \tau}\right), \tag{13}\]  
where \(\Delta T_{ref}=18\)\({}^{\circ}\)C and \(\tau=16.7\) min.  
The same method was applied for the second model and then the Equation (12) becomes:  
\[T_{m}=T_{a}+\Delta T_{ref}^{*}\frac{G_{g}}{G_{0}}-\Delta T_{ref}\frac{\Delta G _{g}}{G_{0}}\exp\left(-\frac{\Delta t}{\tau}\right), \tag{14}\]  
with \(\Delta T_{ref}^{*}=T_{0}=25\)\({}^{\circ}\)C, \(\Delta T_{ref}=16\)\({}^{\circ}\)C and \(\tau=16.7\) min.