## 4 Simulation Results
### Preprocessing of Data and Parameters
#### 4.2.2 Analysis of Predictive Performance
Part of the reason is that the parameter optimization of the mechanism model can make it close to the true value, which significantly reduces the predictive error, but still cannot be consistent with the true value completely. Another part of the reason may be that there are errors which cannot be eliminated due to the simplification of mechanism model. Because of the differences in the optimization results of parameter \(w_{ho11}^{2}\) that cannot be ignored, there are large predictive errors in dynamic processes of HMPIO for \(T_{gs}\), which leads to the dynamic error of \(Q_{sl}\) indirectly. Therefore, the ECM is added to compensate the error of HMPIO. The green line in the figures represents the hybrid model with ECM. The hybrid model (HM) proposed in this paper reduces the predictive residual of HMPIO effectively. The results of GDTA in Table 11 demonstrate that the HM has the highest accuracy, only one tenth of EM. Due to the ECM, the response time of HM is slower than that of MMNP and HMPIO, but only half of that of EM. Although the improvement of HM accuracy comes at the cost of sacrificing some response time, it can still meet the requirements for the boiler system. Similarly, the predictive results of drum, including \(M_{dl}\), \(\rho_{v}\), \(H_{w}\) and \(H_{r}\), are shown in Figures 17 and 18. The reason why the curve of MMNP for output \(M_{dl}\) has been rising is that the initial value of parameter \(w_{cp14}^{2}\) is inaccurate, resulting in the mass imbalance of liquid in drum. Due to the mutual conversion and calculation between vapor and liquid, the drum is a complex non-linear block, which leads to poor predictive performance of the simplified mechanism model MMNP, even for HMPIO. For outputs \(M_{dl}\) and \(H_{w}\), even though HMPIO has better predictive effect than MMNP, it still has obvious predictive errors, especially in the dynamic processes. Meanwhile, as to output \(M_{dl}\), the predictive result of EM is also not very satisfactory. Comparing the results of these models, we can see that the hybrid model has relatively excellent predictive performance. From the results of GDTA in Table 12, we can derive that the accuracy of HM has been improved by 66% and 76% compared to EM-LSTM and EM-GRU, respectively. Finally, we provide the flexibility comparison in Table 13 according to reference [59]. From the total score, EM has the highest flexibility, followed by HMPIO and HM, and MMNP has the lowest. HM, MMNP and HMPIO are not as flexible as EM due to their mechanism model limiting the number and relationship of input and output parameters. Although the interpretability of HM comes at the cost of sacrificing some flexibility, compared to EM, HM improves the accuracy and response time of digital twin models effectively. In conclusion, the hybrid model established in this paper has relatively excellent predictive performance, which ensures the high fidelity of digital twin model for boiler system. However, the datasets used for training and verifying the model in this paper is limited and may not cover all working conditions. Therefore, the generalization performance of hybrid model, especially ECM, needs to be further studied in future work. Figure 14: The predictive results of pulverizing system based on different models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Indexes** & **Output** & **MMNP** & **EM-LSTM** & **EM-GRU** & **HMPIO** \\ \hline \multirow{2}{*}{AOP (\%)} & \(W_{cf}\) & 85.51 & 2.66 & 1.41 & 0.06 \\  & \(T_{o}\) & 75.01 & 0.2 & 0.5 & 0.17 \\  & \(W_{cf}\) & 0.0438 & 0.0391 & 0.0394 & 0.0363 \\ \multirow{2}{*}{ACVAR} & \(T_{o}\) & 0.6771 & 0.0147 & 0.0149 & 0.0148 \\  & global & 80.26 & 1.43 & 0.95 & 0.12 \\ \multirow{2}{*}{GIVAR} & global & 0.3605 & 0.0269 & 0.0271 & 0.0256 \\  & global & \(2.84\times 10^{-5}\) & 0.0044 & 0.0041 & \(7.43\times 10^{-6}\) \\ \multirow{2}{*}{RTVAR} & global & \(2.20\times 10^{-7}\) & \(2.27\times 10^{-5}\) & \(9.29\times 10^{-6}\) & \(5.23\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table}