## 3 Results and Discussion
### Evaluation of the Proposed Network Models  
Before starting the simulation, the dataset was divided into three parts: testing (1680 cases), validation (840 cases), and training (5880 cases). The training and validation sets are used during the training stage. The training dataset part is used to find the neural network bias and weights. The over fitting was minimized using the validation set. Figure 7, shows the results throughout the training and validation stages.  
As shown in Figure 7, the measured error with the NARX model is much lower compared with the DFF model. Moreover, Figure 8 illustrates the function of error autocorrelation (for the NARX network) which describes the time correlation of the prediction errors. It can be seen from Figure 8 that  
Figure 7: Regression of error for both the NARX and DFF neural networks.  
there were significant prediction errors correlation, the only autocorrelation value outside the interval occurs at lag 0.  
After the training stage is done, the testing dataset part is used to evaluate the model and test its performance. Figures 9-12, shows real time comparison between the real data of the networks models and the simulated results for two different manufacturing PV panels.  
The results of these figures are obtained from the average hourly solar radiation and temperature distributions during April 2018 (see Figure 6). The comparison of NARX and DFF results demonstrate that NARX is more efficiently track the real time data compared with DFF.  
To further verify the effectiveness and significance of the proposed models, we employed the Diebold-Mariano (DM) statistic [51]. The DM compares the forecast accuracy of two models. The MSE is adopted as the loss function to build the loss differential. The DM test results are exhibited in the following Table 5.  
According to the DM test results presented in Table 5, the first two sample windows indicates that there is no significant different between the two forecasting models (\(p\)-value \(>\) 0.05); however, in the third test-sample window, the forecasting performance of the NARX model is better than the DFF model (\(p\)-value \(<\) 0.05).  
Figure 8: NARX error autocorrelation (**a**) zoom out view (**b**) zoom-in view.  
Figure 9: Real time comparison using the Astronergy-CHSM6610P PV panel (**a**) between the actual power and simulated DFF network (**b**) DFF voltage and current errors.  
Figure 10: Real time comparison using Sharp’s-NUS0E3E PV panel (**a**) between the actual power and simulated DFF network (**b**) DFF voltage and current errors.  
Figure 11: Real time comparison using the Astronergy-CHSM6610P PV panel (**a**) between the actual power and simulated NARX network (**b**) NARX voltage and current errors.  
Figure 12: Real time comparison using Sharp’s-NUS0E3E PV panel (**a**) between the actual power and simulated NARX network (**b**) NARX voltage and current errors.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Average Value**} \\ \hline  & **First Test-Sample** & **Second Test-Sample** & **Third Test-Sample** \\  & **Window (406)** & **Window (912)** & **Window (1680)** \\ \hline DM-test \({}^{\mathrm{c}}\) & 1.6873 & 1.3783 & \(-6.5539\) \\ \(p\)-value & 0.0915 & 0.1681 & \(<\)0.00001 \\ \hline \hline \end{tabular} \({}^{\mathrm{c}}\) Significance level is 5%.  
\end{table}
Table 5: Results of the Diebold-Mariano (DM) test.