# Identification of Mode Shapes Based on Ambient Signals and the IA-VMD Method
## 5 Simulation
### Analysis of the Test Signal  
As described in Section 2 on the broadband ambient noise model, the simulation signal \(S_{\text{t}}\) is constructed to test the effectiveness of the proposed method. \(S_{\text{t}}\) is constructed by the formula, as follows:  
\[S_{\text{t}}=S_{\text{LFO1}}+S_{\text{LFO2}}+S_{\text{SSO1}}+S_{\text{SSO2}}+S _{\text{SurSO}}+S_{\text{N}} \tag{23}\]  
The test signal \(S_{\text{t}}\) is composed of broadband signals. The order of dominant frequencies is the LFO interval mode \(S_{\text{LFO1}}\), LFO local mode \(S_{\text{LFO2}}\), subsynchronous oscillation \(S_{\text{SSO1}}\), and supersynchronous oscillation signal \(S_{\text{SurSO}}\). Additionally, \(S_{\text{SurSO}}\) is accompanied by \(S_{\text{SSO1}}\); that is, the sum of the frequencies is 120 Hz. \(S_{\text{SSO2}}\) is another independent subsynchronous oscillation signal, and \(S_{\text{N}}\) represents environmental noise. To make the simulation result closer to the actual situation, the test signal is constructed according to the following requirements:  
1. The test signal should include the signals of each frequency band, and the frequency of each oscillation signal should be selected randomly in its frequency band;
2. The oscillation amplitude should be selected according to the actual operation situation, and the oscillation amplitude of each frequency band decreases, in turn, from low to high;
3. Supersynchronous oscillation is accompanied by subsynchronous oscillation. The amplitude of supersynchronous oscillation should be slightly lower than that of subsynchronous oscillation. To satisfy the above conditions, the test signals are selected as follows: \[\left\{ \begin{array}{l}S_{\text{LFO1}}=1.46\times\sin(2\pi t\times 0.68)\\ S_{\text{LFO2}}=2\times\sin(2\pi t\times 1.95)\\ S_{\text{SSO1}}=0.49\times\sin(2\pi t\times 26.58)\\ S_{\text{SSO2}}=0.22\times\sin(2\pi t\times 21.94)\\ S_{\text{SurSO}}=0.32\times\sin(2\pi t\times 93.42)\\ S_{\text{N}}=0.20\times\text{random}[-1,1]\end{array} \right.\] (24)  
Test signal \(S_{\text{t}}\) is composed of five oscillating signals with different frequencies and white noise with an amplitude of 0.2. Test signal \(S_{\text{t}}\) ranges from 0.5 to 100 Hz. The test signal is shown in Figure 3.  
Figure 3: Test signal. According to the broadband signal identification process proposed in this paper, test signal \(S_{t}\) is decomposed into several modes by bandpass filtering and the IA-VMD method, and the dominant oscillation modes of the test signal in different frequency bands are obtained. All oscillation modes of each frequency band of the test signal decomposed by the IA-VMD method are shown in Figure 4. Figure 4 shows that the method proposed in this paper can effectively decompose and extract the oscillation modes in different frequency bands from the test signal. After the oscillation modes of each frequency band are obtained, the information of the modal characteristic parameter can be extracted using the Prony algorithm. The identification results of the proposed method in this paper and the traditional method were compared, and the fitting errors of the different methods are shown in Table 1.  
Compared with the identification results in Table 1, when the test signal is composed of broadband signals, the IA-VMD method combined with the Prony algorithm has the best decomposition and fitting effect among the three methods. Table 1 shows that the maximum fitting error of the oscillation frequency is 1.55%, and the average fitting error is only 0.67%. Thus, the fitting degree of oscillation frequency is good, confirming that the proposed method can meet the requirements of oscillation frequency identification. Table 1 shows that the maximum error of the oscillation amplitude is 9.09%, and the average error is 2.54%. Because the amplitude of subsynchronous oscillation itself is small, the deviation of