## 5 Numerical Experiments and Analysis
### Comparative Analysis of the Results  
To evaluate the performance of different models, four evaluation indices were used to include the root mean square error (RMSE), the mean absolute error (MAE), the mean absolute percent error (MAPE) [23; 36], and the Nash-Sutcliffe Efficiency (NSE) [37] are employed. The four indices are defined as follows. \[\text{RMSE}=\sqrt{\frac{1}{N}{\sum\limits_{i=1}^{N}(f_{\text{s}}(i)-f_{o}(i)) ^{2}}} \tag{32}\]  
\[\text{MAE}=\frac{1}{N}{\sum\limits_{i=1}^{N}}{\left|{f_{\text{s}}(i)-f_{o}(i) }\right|} \tag{33}\]  
\[\text{MAPE}=\frac{1}{N}{\sum\limits_{i=1}^{N}\frac{\left|{f_{\text{s}}(i)-f_{o }(i)}\right|}{f_{o}(i)}}\times 100\% \tag{34}\]  
\[\text{NSE}=1-\frac{{\sum\limits_{i=1}^{N}(f_{\text{s}}(i)-f_{o}(i))^{2}}}{{ \sum\limits_{i=1}^{N}\left({f_{o}(i)-\overline{f_{o}(i)}}\right)^{2}}} \tag{35}\]  
where \(f_{\text{s}}(i)\) and \(f_{o}(i)\) denote the simulated and observed value of the \(i\)th sample point, respectively, \(\overline{f_{o}(i)}\) denotes the mean observed value, and \(N\) denotes the size of the data set. The 3D spatial surfaces for the WH and the WM characteristics based on the AE-PLS-ELM model are shown in Figures 4 and 5, respectively. As can be seen from Figures 4 and 5, the WH and WM spatial surfaces based on the AE-PLS-ELM model are smooth and uniform. In addition, the complete characteristic curves are continuous derivative, which make it easy to ensure the convergence of the water hammer calculation process. The AE-PLS-ELM model can also be used to encrypt and extend the WH and WM characteristic surfaces, according to the practical requirements for research and engineering applications. Therefore, the transition between different opening lines on the surface is smoother and it is convenient for operators to obtain the pump turbine characteristics of different working conditions. Figure 4: 3D surface for WH characteristics based on the proposed AE-PLS-ELM model. The results of the Bagtree, SVR, BP, ELM, and AE-PLS-ELM models for _WH_ characteristics are shown in Table 1. It can be seen from the table that the five models have good training and test accuracy and can model the WH characteristics of the PSU accurately. The AE-PLS-ELM model performs the best in terms of the four indices in both the training and test period among the five models, which indicates that the AE-PLS-ELM model can enhance the modeling accuracy of pump turbine for the WH characteristic curve effectively. Taking the RMSE value of the test period as an example, the RMSE value of ELM is 0.00297, which is lower than that of the Bagtree, SVR, and BP models. The neural network models as BP and ELM perform better than the Bagtree and SVR models. The performance of ELM is slightly better than BP and the Bagtree model performs the worst. Compare the training and test results between the ELM and AE-PLS-ELM models, it can be found that the performance of the AE-PLS-ELM model is significantly better than the ELM model. For the training samples, the RMSE, MAE, MAPE, and NSE values of the AE-PLS-ELM model were 0.00229, 0.00130, 0.00599, and 0.99988, respectively, which were improved by 21.03%, 37.20%, 72.75%, and 0.007% compared with the 0.00290, 0.00207, 0.02198, and 0.99981 obtained by the ELM model. For the test samples, the RMSE, MAE, MAPE, and NSE values of the AE-PLS-ELM model were 0.00217, 0.00142, 0.00697, and 0.99988, respectively, which were improved by 26.93%, 39.32%, 71.75%, and 0.011% when compared with the 0.00297, 0.00234, 0.02467, and 0.99977 obtained by the ELM model. In a word, the proposed AE-PLS-ELM model overcomes the instability and multi-collinearity of the single ELM model, and can improve the generalization ability and fitting accuracy of the ELM for modeling the pump turbine characteristics. The comparison of the residuals for the WH characteristics between the AE-PLS-ELM model and the Bagtree, SVR, BP, and ELM models are shown in Figure 6a-d, respectively.