## 4 Case Studies
### Wind Power, Photovoltaic Power and Load Power Forecasting
#### 4.1.5 The Results of Forecasting  
In order to illustrate the effectiveness of the MIC feature selection method, the results of the forecasting model with feature selection and those of the forecasting model without feature selection are compared, as shown in Table 6. In addition, in order to illustrate the improvement of stacking ensemble learning on the forecasting accuracy of a single model, for each power forecasting problem, this study compares the MAE, RMSE and MAPE of the stacking ensemble learning model with the MAE, RMSE and MAPE of a single algorithm. At the same time, this study is also compared to the forecasting results of a long-short term memory (LSTM) network, a representative method in the forecasting field, to reflect the accuracy of the model. The results of the comparative experiment are as shown in Table 7. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Index**} & \multicolumn{3}{c}{**Without Feature Selection**} & \multicolumn{3}{c}{**With Feature Selection**} \\ \cline{2-7}  & **Wind** & **Photovoltaic** & **Load** & **Wind** & **Photovoltaic** & **Load** \\ \hline MAE (MW) & 0.650 & 0.118 & 0.192 & 0.597 & 0.108 & 0.178 \\ RMSE (MW) & 0.826 & 0.220 & 0.301 & 0.779 & 0.207 & 0.284 \\ MAPE (\%) & 35.369 & 10.394 & 3.923 & 27.418 & 9.977 & 3.005 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Forecasting results comparison of with and without feature selection. Figure 5: The relationship between the number of input features and forecasting results: (**a**) Wind power; (**b**) Photovoltaic power; (**c**) Load power. \begin{table}
\begin{tabular}{c c} \hline \hline
**Type of** **Learners** & **Algorithm** & **Hyper-Parameters Setting** \\ \hline \multirow{4}{*}{
\begin{tabular}{c} base \\ learners \\ \end{tabular} } & LGBM & Boosting tree is GBDT, number of leaves is 100, number of trees is 100, tree depth is 7, learning rate is 0.1, training data ratio is 0.8, feature column sampling rate is 0.9. \\ \cline{2-2}  & KNN & The number of Neighbors is 10. \\ \hline meta- & LR & / \\ learner & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Learners and hyper-parameters of load power forecasting. The results in Table 6 prove the effectiveness of MIC feature selection method, because the MAE, RMSE and MAPE of the forecasting model with feature selection are all lower than those without feature selection. In Table 7, data with * indicate that the corresponding algorithm is not used in the stacking ensemble model of this forecasting problem. The results show that even if the accuracy of some algorithms is not high when predicting independently, the accuracy of the ensemble model can be improved after they join the stacking model. Therefore, they will be also included in the stacking model. For example, the RMSE of the KNN algorithm in photovoltaic and load power forecasting is relatively high, but it is still integrated into the stacking model. At the same time, although the LSTM model can achieve a good forecasting accuracy, its accuracy is still inferior to that of the stacking model. To further understand the forecasting accuracy of the stacking ensemble learning model, 96 time points (4 days) are selected herein to display the forecasting value of the stacking ensemble models of the three forecasting problems and the actual value, respectively. As shown in Figure 6, the forecasting model based on the stacking ensemble learning model achieves a good accuracy in wind power forecasting, photovoltaic power forecasting and load power forecasting, which can provide the accurate wind power, photovoltaic power and load data for RPF calculation. It has been mentioned in Section 2.3 that the forecasting errors of the test set can be statistically obtained to get the probability distribution of the forecasting errors.