# Estimation of Direct Normal Irradiance at Antarctica for Concentrated Solar Technology
## 2 Materials and Methods
### Model Validation  
The models, in general, when developed according to the acquired observational data set, need to be compared with the corresponding estimates, and whether the deviation magnitudes do not compromise the quality of the proposed model needs to be evaluated. In order to quantify the quality of the proposed model, some statistical indicators are useful. The first of these indicators, called the Mean Bias Error (MBE), is simply the difference between the mean estimate values and the observation values. In our case, the calculated value is:  
\[\text{MBE}=\frac{1}{N}\sum_{\text{i=1}}^{N}(I_{\text{bn,i,c}}-I_{\text{bn,i,o} })=14.6\ \frac{\text{W}}{\text{m}^{2}} \tag{16}\]  
In the above expression, \(N=10\),641, it is the total number of observation pairs in the data; the subscript \"i\" refers to the i-th pair, while \"o\" and \"c\" refer to the observations and calculated values, respectively. The estimates are calculated using Equation (12) with the values given by Equation (15). The low value of the MBE means that the measurements and estimates are offset on the average value. Figure 10: Dispersion graph of parameter pairs (\(K_{\text{T}}\), \(K_{\text{bn}}\)) obtained from long GHI dataset using the Boland–Ridley model for the European (blue) and Australian (violet) regions [13] and the ENEA model for the Antarctic region (red). Figure 9: Dispersion graph of parameter pairs (\(K_{\text{T}}\), \(k\)) obtained from long GHI dataset using the Boland–Ridley model for the European (blue) and Australian (violet) regions [13] and the ENEA model for the Antarctic region (red). Even though, a low (or, at least, zero) MBE value does not exclude the possibility that there are some significant gaps between the measurements and the estimates, even potentially in a non-negligible number. In order to underline the variance index that takes the above fact into account, it is useful to use the Mean Absolute Error (MAE), which in our case gives:  
\[\text{MAE}=\frac{1}{N}\sum_{\text{i=1}}^{N}\lvert I_{\text{bn,i,c}}-I_{\text{ bn,i,o}}\rvert=117.75\ \frac{\text{W}}{\text{m}^{2}} \tag{17}\]  
or, in relative terms:  
\[\frac{\text{MAE}}{\text{I}_{\text{bn,o}}}=0.227=22.7\% \tag{18}\]  
where, \(\text{I}_{\text{bn,o}}\) is indicated as the average of the DNI measurements, which results in:  
\[\text{I}_{\text{bn,o}}=\frac{1}{N}\sum_{\text{i=1}}^{N}I_{\text{bn,i,o}}=516.7 \ \frac{\text{W}}{\text{m}^{2}} \tag{19}\]  
Another widely used quality deviation index is the Root Mean Squared Error (RMSE) which, compared to the previous evaluated MAE, considers the squares of the discard rather than the absolute values, in order to attribute a greater weight to higher deviations in the summation, such as:  
\[\text{RMSE}\ =\ \sqrt{\frac{1}{N}\sum_{\text{i=i}}^{N}\left(I_{\text{bn,i,c}}-I_{ \text{bn,i,o}}\right)^{2}}=171.8\ \frac{\text{W}}{\text{m}^{2}} \tag{20}\]  
The final quality index considered here can be obtained by comparing the average DNI values of all the measurements and the corresponding estimated values. Since the average value of the DNI estimate is calculated as:  
\[\text{I}_{\text{bn,c}}=\frac{1}{N}\sum_{\text{i=1}}^{N}I_{\text{bn,i,c}}=531.4 \ \frac{\text{W}}{\text{m}^{2}} \tag{21}\]  
we finally get:  
\[\frac{\text{I}_{\text{bn,c}}-\text{I}_{\text{bn,o}}}{\text{I}_{\text{bn,o}}} =\frac{531.4\ \frac{\text{W}}{\text{m}^{2}}\ -\ 516.7\ \frac{\text{W}}{\text{m}^{2}}}{516.7\ \frac{\text{W}}{\text{m}^{2}}}=\ \frac{14.7}{516.7}=\ +0.028=\ +2.8\% \tag{22}\]  
The summary of statistical measures of presented model is compared with Boland-Ridley coefficients for Australia and Europe taken from Reference [13] are presented in Table 2. Within the new calculated \(\alpha\) and \(\beta\) coefficients for Antarctic region from observed data we reduced the model error from 12.9% considering the known coefficients for Australia to 2.8%. The values of the reported quality indicators are affected by the complexity of the phenomenology of solar radiation and its components that reach the ground.