# Synthetical Modal Parameters Identification Method of Damped Oscillation Signals in Power System
## 4 Robustness Test of SMPI under Lower SNR  
To verify the robustness of SMPI, the Gaussian white noise in the ideal simulated signal increased to 20 dB SNR. The new signal will be referred to as signal B following and its curve is illustrated in Figure 9.  
The five effective IMFs and a residual signal are extracted by EMD to eliminate high-frequency noise as Figure 10, and the comparison of the synthesized signal \(c_{EMD}(t)\) with signal B in Figure 11.  
The curve of \(c_{EMD}(t)\) is smooth and extremely coincidence with that of signal B, which indicates that \(c_{EMD}(t)\) has retained the dominant modes of the original DLFO signal. And \(c_{EMD}(t)\) is conducive to modal identification by SSI and Prony following.  
Using the SSI method and Prony algorithm to identify \(c_{EMD}(t)\), the estimated modal parameters are shown in Tables 6 and 7, respectively.  
To extract the true modes from the trivial modes, the parameter matching method is applied to match the full modal parameters in the similar frequency of Tables 6 and 7, and the matching results are displayed in Table 8.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Modal Parameters** & **True Modes** & **True Value** & **Estimated Value** & **Relative Error** \\ \hline \multirow{3}{*}{Frequency (Hz)} & 1 & 0.33 & 0.3318 & 0.0055 \\  & 2 & 0.78 & 0.7807 & 0.0009 \\  & 3 & 1.00 & 1.0072 & 0.0072 \\  & 1 & 2.00 & 2.0785 & 0.0392 \\ \multirow{3}{*}{Amplitude (\(\mu\)m)} & 2 & 4.00 & 3.9976 & 0.0006 \\  & 3 & 6.00 & 5.7695 & 0.0384 \\  & 1 & 0.0579 & 0.0589 & 0.0167 \\ \multirow{3}{*}{Damping ratio} & 2 & 0.0265 & 0.0258 & 0.0252 \\  & 3 & 0.1130 & 0.1119 & 0.0098 \\ \multirow{3}{*}{Phase angle (rad)} & 1 & 1.5708 & 1.5942 & 0.0149 \\  & 2 & 1.5708 & 1.6011 & 0.0193 \\ \cline{1-1}  & 3 & \(-\)1.5708 & \(-\)1.5118 & 0.0375 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of estimated modal parameters of SMPI with true modal parameters.  
Figure 9: Simulated signal with Gaussian white noise at 20 dB SNR.  
Figure 11: Comparison of \(c_{EMD}(t)\) with signal B.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Nodes** & **Frequency (Hz)** & **Damping Ratio** \\ \hline
1 & 0.3306 & 0.0607 \\
2 & 0.7813 & 0.0270 \\
3 & 1.0083 & 0.1124 \\
4 & 2.1442 & 0.0214 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Modal parameters of signal B identified with SSI approach.  
Figure 10: Effective IMFs and a residual signal of signal B.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nodes** & **Frequency (Hz)** & **Amplitude (\(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\bmbm          }}}}}}}}}}}}\))** & **Phase Angle (Rad)** \\ \hline
1 & 0.3275 & 2.0790 & 1.7038 \\
2 & 0.7804 & 4.3000 & 1.6164 \\
3 & 1.0129 & 6.1616 & \(-1.5975\) \\
4 & 1.6755 & 0.0614 & \(-2.9024\) \\
5 & 2.3345 & 0.0218 & 2.8943 \\ ⋮ & ⋮ & ⋮ & ⋮ \\
99 & 49.4658 & 0.0018 & \(-2.7895\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Modal parameters of signal B identified with Prony approach of signal B.  
In light of Table 9, the value of estimated modal parameters is close to that of true modal parameters, and the relative errors between them are comparatively small as well. The frequencies calculated by SMPI are still all within 1% relative error. While the relative errors of amplitude, damping ratio, and phase angle are less than 8%, 5%, and 8.5%, respectively.  
Although the relative errors have increased in a sense, due to the introduction of more noise, the curve fitted by the identified results of SMPI is consistent with the curve of the original simulated signal whose main oscillating features are obtained without omission in Figure 13. Hence, the proposed SMPI method has shown satisfactory robustness under the condition of noise with different SNRs.