## 4 Case Study and Results Analysis
### KELM for Load Forecasting  
After the dimensionality reduction of input features, input parameters are brought into the KELM model for learning. In this article, the self-programming program is utilized to the operation in Matlab software. It is It is noteworthy that RBF kernel function is selected in the article as the kernel function of KELM method. To ensure its precision and accuracy, the model's important parameters are obtained and optimized by FWA algorithm. The parameters of FWA algorithm are set in Section 2.4, hence they are not repeated here. Through calculation, the KELM model parameters are \(C=8.325\) and \(\sigma=0.0031\). For the sake of proving the forecasting accuracy of the proposed DES load forecasting method, this paper also selects the KELM, ELM and BPNN which are not optimized by FWA to forecast the load data of this sample, and then evaluates and analyzes the prediction results of the four methods. The structure of the BPNN model is 6-3-1, whose hidden layer transfer function is expressed as \"tansig\" function, while the output layer transfer function is expressed as purelin function, with the maximum time of training being 300, the minimum error of training objectives being 0.0001, the rate of training being 0.1, and the original weight and threshold values derived from the model's own learning. In the ELM method, the penalty parameter obtained from training \(C\) is 10.276 and the kernel parameter \(\sigma\) is 0.0013. In the KELM model parameters, \(C\) is 10.108 and \(\sigma\) is 0.0026. Table 3 shows the DES load prediction results of BPNN, ELM, KELM, and the method proposed in the article on the test set. In order to intuitively analyze, the predicted results in Table 3 are drawn into diagrams, as shown in Figure 2. It can be found from Table 3 and Figure 2 that the predicted results of the four methods are not very different from the real data, and the overall trend is consistent. Among them, the prediction results of KPCA-FWA-DIR method are most similar to the actual situation, and the prediction results of other models have relatively large errors. Furthermore, the results show that the prediction curve of KELM method is more accurate than that of single prediction curve, indicating that the introduction of kernel function increases the accuracy of the model to a certain extent. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline The groups of data & 35,424 \\ Maximum load (MW) & 13.57 \\ Minimum load (MW) & 6.15 \\ Maximum temperature(\({}^{\circ}\)C) & 38 \\ Minimum temperature (\({}^{\circ}\)C) & \(-12\) \\ Number of days in spring (day) & 92 \\ Number of days in summer (day) & 97 \\ Number of days in autumn (day) & 91 \\ Number of days in winter (day) & 89 \\ Number of precipitation days (day) & 62 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The main statistical indicators of the collecting data. Figure 3 shows the relative errors of the four methods. By observing Figure 4, we can clearly see the differences in the forecasting results of different models. Figure 3: RE of prediction results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Point** & **Actual Value** & **BPNN** & **ELM** & **KELM** & **KPCA-FWA-KELM** \\ \hline
1 & 9282.68 & 10,008.31 & 9845.31 & 9700.31 & 9491.82 \\
2 & 9522.65 & 1026.208. & 10,100.77 & 9766.33 & 9666.44 \\
3 & 9232.56 & 9938.38 & 9769.71 & 9646.45 & 9388.31 \\
4 & 9333.37 & 10,125.21 & 8826.47 & 8887.60 & 9186.27 \\
5 & 9516.76 & 10,319.69 & 8868.29 & 9979.75 & 9352.31 \\
6 & 9302.71 & 9971.11 & 8750.04 & 9719.28 & 9400.67 \\
7 & & & & & \\
91 & 9942.66 & 10,781.72 & 10,462.36 & 10,354.18 & 9715.57 \\
92 & 9863.11 & 10,732.35 & 10,599.49 & 10,119.45 & 9681.92 \\
93 & 9787.53 & 10,548.22 & 9289.35 & 10,039.95 & 9921.52 \\
94 & 9436.43 & 10,214.84 & 8827.31 & 9875.32 & 9579.87 \\
95 & 9583.09 & 10,364.49 & 9002.35 & 10,039.24 & 9414.91 \\
96 & 9701.47 & 10,477.78 & 10,278.32 & 10,073.82 & 9830.60 \\ \hline \hline \end{tabular}
\end{table}