## 4 Case Study and Results Analysis
### KELM for Load Forecasting

Table 3: Partial predicted and actual values of the test set (Unit: kW). Figure 2: Prediction results. The RE ranges [\(-3\%\), \(3\%\)] and [\(-1\%\), \(1\%\)] are usually used as standards to measure the accuracy of forecasting methods. As shown in Figure 4, the forecasting errors at all time points of the KPCA-FWA-KELM model are not only in the [\(-3\%\), \(3\%\)] interval, but also within [\(-2.5\%\), \(2.5\%\)]. The minimum value of the relative error absolute values is \(0.58\%\) and the maximum value is \(2.31\%\), of which only \(8\) sample points have errors outside [\(-2\%\), \(2\%\)], which are sample points with serial numbers 1, 19, 31, 39, 63, 71, 79 and 91, respectively. The relative errors are \(2.25\%\), \(-2.29\%\), \(2.23\%\), \(2.06\%\), \(2.24\%\), \(2.31\%\), \(2.19\%\) and \(-2.28\%\), respectively. In the KELM prediction method, the relative error of \(11\) prediction results is controlled in within [\(-3\%\), \(3\%\)]. The error of the three prediction results is within [\(-1.5\%\), \(1.5\%\)], in which the serial numbers are \(13\), \(31\) and \(45\) respectively. Their relative errors are \(-1.22\%\), \(1.15\%\) and \(1.01\%\) respectively. The minimum absolute value of relative error is \(1.01\%\) and the maximum is \(5.89\%\). In the ELM model, the error of \(4\) sample points is controlled in within [\(-3\%\), \(3\%\)], which are the samples of the sequence number of \(7\), \(68\) and \(84\) respectively. The relative errors are \(2.35\%\), \(-2.26\%\) and \(1.14\%\), respectively, but they are all outside the [\(-1\%\), \(1\%\)] range. Its minimum of the absolute value of the relative error is \(1.14\%\) and the maximum value is \(7.52\%\). The minimum value of absolute relative error of BPNN method is \(1.16\%\), and the maximum is \(10.12\%\). The error of most prediction results is in [\(-9\%\), \(-7\%\)] and [\(7\%\), \(9\%\)], and the fluctuation range is large. From this point of view, the KPCA-FWA-KELM method has the best prediction effect, followed by KELM method and ELM method, and BPNN method has the worst effect. It can be found that KPCA-FWA-KELM modal has the best prediction accuracy and stability, which shows that FWA algorithm improves the ability of model learning, effectively avoids the problem of falling into local optimization, and improves the global search ability of KELM. It is also shown that the prediction results obtained from the KPCA model can achieve satisfactory prediction results and effectively eliminate the interference of redundant data. Furthermore, ILSSVM has better performance than LSSVM, SVM and BPNN. This result shows that LSSVM can achieve better prediction effect after improvement. The RMSE, MAPE and AAE of BPNN, ELM, KELM and KPCA-FWA-KELM are shown in Figure 4. We can find that the RMSE, MAPE and AAE of the proposed methods are \(1.6158\%\), \(1.6079\%\) and \(1.6017\%\) respectively, which are the least error of the above five methods. Furthermore, errors of RMSE, MAPE and AAE of KELM methods are \(4.0873\%\), \(4.0713\%\) and \(4.0649\%\) respectively. The RMSE, MAPE and AAE of ELM method were \(5.4899\%\), \(5.2693\%\) and \(5.0553\%\) respectively. RMSE, MAPE and AAE of the BPNN method are \(7.9917\%\), \(7.8731\%\) and \(7.8878\%\) respectively. Figure 4: RMSE, MAPE and AAE of the prediction results. These indexes reflect the overall error of model forecasting and the degree of error discretization. Therefore, it can be further seen that the overall forecasting effect of the KELM method is superior to the ELM method and the BPNN method, while the overall forecasting effect of the ELM method is superior to the BPNN method, indicating that the overall forecasting performance of the ELM method is significantly improved after the introduction of the kernel function. The forecasting effect of the KPCA-FWA-KELM method is better than that of the KELM method.