# Synthetical Modal Parameters Identification Method of Damped Oscillation Signals in Power System
## 5 Generalization Ability Test of SMPI with Real-Time Signals  
To test the effectiveness in modal identification of real-time DLFO signals, SMPI is adopted to identify a set of sampling signals from a power system, whose sampling frequency is 100 Hz, the sampling time is 20 s and sampling points is 2000, as shown in Figure 14.  
The comparison of the synthesized signal by EMD with the real-time signals is in Figure 17. The curve of \(c_{EMD}(t)\) fits the oscillating features of the polluted real-time signals smoothly, which illustrates that the EMD method is workable in the signal pre-process.  
Afterward, the SSI and Prony algorithms are utilized to purchase the modal parameters of the synthesized signal after EMD denoising, and their identification results are presented in Tables 10 and 11.  
There are many high-frequency fictitious modes so the parameter matching method is carried out to filter out the true modes, and the matching results are displayed in Table 12. There are two dominant components of the real-time signal, and they can be reconstructed as shown in Figure 18.  
Figure 16: Effective IMFs and a residual of real-time signals.  
Figure 15: All IMFs and residual of real-time signal decomposed by EMD.  
These two dominant components of real-time signal are superimposed to obtain a synthesized signal \(Y\) after SMPI as Equation (27). Subsequently, the comparison between \(Y\) and \(c_{EMD}(t)\) is shown in Figure 19.  
Given that the signal sampling is unstable and the algorithm needs a certain convergence process at the beginning, there are a few errors \(Y\) and \(c_{EMD}(t)\) of the real-time signal. After 0.6 s, the relative errors are controlled well and gradually decrease, owing to the ability of SMPI to fast convergence and the stability of signal sampling. In Figure 19, the curve of signal \(Y\) synthesized with the identified results of SMPI is found in good coincidence with that of \(c_{EMD}(t)\), whose dominant modal parameters have been extracted successfully.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Modes** & **Frequency (Hz)** & **Damping Ratio** \\ \hline
1 & 0.5390 & 0.0089 \\
2 & 0.9811 & 0.0891 \\
3 & 14.7210 & 0.0583 \\
4 & 26.8585 & 0.0607 \\
5 & 35.5469 & 0.0573 \\
6 & 38.2654 & 0.0120 \\
7 & 45.5538 & 0.0124 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Modal parameters of real-time signal identified with SSI approach.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Modes** & **Frequency (Hz)** & **Amplitude (\(\mu\)m)** & **Phase Angle (Rad)** \\ \hline
1 & 0.5397 & 0.0579 & 2.7909 \\
2 & 0.9778 & 0.3053 & 1.8003 \\
3 & 1.6909 & 0.3829 & \(-3.1258\) \\
4 & 2.2648 & 0.0343 & 1.7157 \\
5 & 2.6177 & 0.0217 & 2.0594 \\ ⋮ & ⋮ & ⋮ & ⋮ \\
98 & 49.6196 & 0.0339 & 1.7125 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Modal parameters of real-time signal identified with Prony approach.  
Figure 17: Comparison of the synthesized signal by EMD with the real-time signal.  
In other words, the developed SMPI is a practicable tool, which holds the generalization ability to identify the full modal parameters including frequency, damping ratio, amplitude, and phase angle of DLFO real-time signal in the power system.