## 4 Simulation of Parameter Identification for Envelope Current Signal
### Simulation of Oblique Envelope Signal with 3 Times Harmonic
#### 4.1.3 The Analysis of the Improved Method Based on WIFFT and DE Algorithms  
The differential evolution algorithm is introduced to improve the precision in envelope parameter identification. It has no need to peroform optimization for frequency, since the frequency estimation error calculated by WIFFT is very small. Therefore, this section only considers the optimization of envelope parameters \(a\) and \(b\), amplitude parameters \(A_{\text{m}}\) and phase parameters \(\theta_{\text{m}}\). Set the aforementioned estimated value as the initial value of iteration, and DE algorithm parameters are as follows. The upper and lower bounds of the independent variables are 1.5 and 0.5 times of the initial value, respectively, the population size is 60, crossover factor is 0.4, crossover rate is 0.9 and the number of evolution generations is 100. Because the amplitude and envelope parameters are involved in the iteration, and they meet Formula (11) at the same time, it will result in different descriptions (\(a\), \(b\), \(A_{\text{m}}\)) of the same signal. Therefore, the following Formula (13) is proposed to calculate the envelope parameter error when analyzing the parameter identification.  
\[Err_{\_a}=\frac{a_{\_\_\_\text{cal}}\cdot norm(A_{\_\_\text{cal}})-a_{\_\_ \text{exact}}\cdot norm(A_{\_\_\text{exact}})}{a_{\_\_\text{exact}}\cdot norm (A_{\_\_\text{exact}})} \tag{13}\]  
where \(a_{\_\_\text{cal}}\) is the optimal value given by the differential evolution algorithm, \(a_{\_\_\text{exact}}\) is the set value, \(norm(A_{\_\_\text{cal}})\) is the amplitude norm of the optimal estimation which obtained by the DE algorithm, and \(norm(A_{\_\text{exact}})\) is the norm of the set harmonic amplitude value. The dynamic parameter identification in this example is shown in Table 3.  
The signal reconstructed by the above identification parameters and errors is shown in Figure 5.  
Figure 5: Simulation results of the improved method based on WIFFT and DE algorithm: (**a**) Comparison of the original signal and reconstructed signal; (**b**) Absolute error of original signal and reconstructed signal.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Parameter** & \(a^{*}\) & \(b^{*}\) & \(A_{\_\_}1\) & \(A_{\_}2\) & \(A_{\_}3\) & \(\theta_{\_}1\) & \(\theta_{\_}2\) & \(\theta_{\_}3\) \\ \hline \hline \multirow{2}{*}{Valuation} & 0.3829 & 0.7656 & 130.6214 & 6.5348 & 13.0577 & 1.0490 & 0.7874 & 0.5246 \\  & 0.0176 & 7.1951 \(\times\) 10\({}^{-4}\) & / & / & / & 0.1723 & 0.2580 & 0.1865 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameter identification and errors of the oblique envelope signal.  
According to Figure 5, because of the DE algorithm, the error of envelope parameters is less than 0.0176% and the precision of phase calculation increases a lot--the maximum error decreases from 1.4589% to 0.258%. Furthermore, the approximation ability of the reconstructed signal is improved greatly. To be specific, the RMSE is only 0.0735, and the maximum absolute error is reduced from about 10 A to about 0.2 A, which means the precision is raised about 50 times. In addition, the computation time of WIFFT is 0.0492 s and the computation time of the proposed method is 1.8401s which is about 37 times of WIFFT. The computation time is the average of 30 times. The analysis result indicates that it is feasible to analyze the envelope signal through the combination method of WIFFT and DE algorithm.  
Furthermore, the adaptability of this improved method in the case of noise is investigated. The difference of harmonic amplitudes has reached 26 dB, hence 40 dB, 50 dB and 60 dB noises are added to the signal for consideration. The results of parameter identification are as follows (Table 4):  
The results show that the method has good adaptability to noise, and it has a good precision even in these noisy cases.