# Enhancing Photovoltaic Power Predictions with Deep Physical Chain Model
## 3 Results
### Deep Chain Model  
Table 14 shows that the model trained with BestIpoa and NWPIpoa slightly outperforms the best direct method model in all metrics, although without a significant improvement compared to the direct method. However, this method allows for a modularization of errors, which is crucial for future improvements. The deep chain model presents a more integrated strategy for power prediction.  
Figures 10 and 11 show the graphical representation of the best and worst forecasting results, respectively, using the best model found. The scatter plots in each figure compare the actual versus predicted power values for different stations using the deep chain model method. Each plot corresponds to one station, with the x-axis representing the observed power and the y-axis representing the predicted power. The diagonal line indicates the ideal 1:1 relationship.  
Each row in the figures represents a measurement in one station (or plant), while each column corresponds to a prediction horizon in 15 min increments, starting from 15 min up to 60 min. The main diagonal in each plot indicates the line of identity, where the predicted values perfectly match the observed values.  
\begin{table}
\begin{tabular}{l l c c c c c c c c} \hline \hline \multirow{2}{*}{**Predictors**} & \multirow{2}{*}{**Metric**} & \multicolumn{5}{c}{**Station**} & \multicolumn{5}{c}{**Mean**} \\  & & **0** & **1** & **2** & **3** & **4** & **5** & **8** & \\ \hline CalibrationMAE & MAE & 0.192 & 0.849 & 0.676 & 0.709 & 1.311 & 1.755 & 0.782 & 0.896 \\ Skartveit & RMSE & 0.363 & 1.622 & 1.283 & 1.295 & 2.431 & 3.288 & 1.459 & 1.677 \\  & \(R^{2}\) & 0.931 & 0.924 & 0.906 & 0.940 & 0.877 & 0.870 & 0.909 & 0.908 \\ \hline BestIpoa & MAE & 0.190 & 0.850 & 0.671 & 0.710 & 1.306 & 1.754 & 0.780 & 0.894 \\  & RMSE & 0.363 & 1.634 & 1.280 & 1.308 & 2.449 & 3.305 & 1.454 & 1.685 \\  & \(R^{2}\) & 0.931 & 0.922 & 0.906 & 0.939 & 0.875 & 0.869 & 0.910 & 0.907 \\ \hline NWPIpoa & MAE & 0.191 & 0.857 & 0.668 & 0.709 & 1.322 & 1.759 & 0.789 & 0.899 \\  & RMSE & 0.360 & 1.638 & 1.267 & 1.291 & 2.447 & 3.327 & 1.471 & 1.686 \\  & \(R^{2}\) & 0.932 & 0.922 & 0.908 & 0.941 & 0.875 & 0.867 & 0.908 & 0.908 \\ \hline NWPIpoa & MAE & 0.192 & 0.854 & 0.672 & 0.710 & 1.311 & 1.738 & 0.782 & 0.894 \\  & RMSE & 0.361 & 1.641 & 1.280 & 1.299 & 2.426 & 3.291 & 1.458 & 1.679 \\  & \(R^{2}\) & 0.932 & 0.922 & 0.906 & 0.940 & 0.877 & 0.870 & 0.909 & 0.908 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Results of direct method to power forecast. RMSE and MAE are measured in \([MW]\), and \(R^{2}\) is unitless.  
\begin{table}
\begin{tabular}{l l c c c c c c c c} \hline \hline \multirow{2}{*}{**Predictors**} & \multirow{2}{*}{**Metric**} & \multicolumn{5}{c}{**Station**} & \multicolumn{5}{c}{**Mean**} \\  & & **0** & **1** & **2** & **3** & **4** & **5** & **8** & \\ \hline NWPIpoa & MAE & 0.194 & 0.860 & 0.671 & 0.711 & 1.339 & 1.756 & 0.794 & 0.904 \\  & RMSE & 0.364 & 1.659 & 1.277 & 1.296  
When compared with previous results, it is evident that the deep chain model benefits from each stage of the prediction process, although it shows an inherent error, coming from the first block and propagating through the others, which cannot be eliminated with this strategy.  
It is seen that among the results, there is no significant difference between the different training datasets used, although there is still room for more tests with other variables, whether they are sensor measurements or calculated variables. Despite the above, this study represents a considerable advance in improving the accuracy of photovoltaic power predictions, and given the innovative use of the proposed Transformer Bi-LSTM architecture, it lays a solid foundation for future research in the field.  
Figure 11: Relationship between observed and predicted power output for different stations and prediction horizons for stations with worst results.  
Figure 10: Relationship between observed and predicted power output for different stations and prediction horizons for stations with better results.