One premise to perform DMD is that the rank of the measurement matrix needs to be no less than the number of the dominant modes [22,23]. In the case that the measurement matrix is a series of temporal synchrophasors, its rank would be one, which is insufficient for SSRPI [22]. To address this issue, the concept of Hankel matrices is used here. For the Nth measured synchrophasor Xc(N), its N time-shifting historic data (Xc(0),…,Xc(N−1)) are collected to construct two Hankel matrices H1 and H2 as follows:

(13)H1=[Xc(0)Xc(1)…Xc(N−L−1)Xc(1)Xc(2)…Xc(N−L)…………Xc(L−1)Xc(L)…Xc(N−1)]and

(14)H2=[Xc(1)Xc(2)…Xc(N−L)Xc(2)Xc(3)…Xc(N−L+1)…………Xc(L)Xc(L+1)…Xc(N)]where H1,H2∈CL×(N−L) with min(L,N−L)>r, and L is generally set smaller than N−L.

Actually, H1, stacked from overlapping elements between adjacent rows or adjacent columns with high coherence, has the property of a low-dimensional structure, as shown by the singular value (i.e., σi i = 1,2,…,r) curve of noise-free H1 in Fig. 1a, where the singular values are obtained by using singular value decomposition (SVD) on noise-free H1. The number of nonzero singular values depends on the number of dominant modes in H1 (or Xc). Note that Fig. 1a represents the noise-free condition. In practice, the noises in Xc would perturb small singular values in H1 and H2. In other words, the number of nonzero singular values could be more than that of the dominant mode in Xc, as shown in Fig. 1b. It is definite that H2 has the same property as H1.

With the derivation in the Appendix, the relationship of H2 and H1 is determined as

(15)H2=AH1where A∈CL×L is an unknown map from H1 to H2, given by

(16)A=QΛQ†

where ()† is the pseudo inverse of the matrix, and Q∈CL×r is given by

Q=[a1a2⋯ara1λ1a2λ2⋯arλra1λ12a2λ22⋯arλr2⋯⋯⋯⋯a1λ1L−1a2λ2L−1…arλrL−1]and Λ∈Cr×r is given by

(17)Λ=diag(λj),j=1,…,rwhere diag(.) constructs a diagonal matrix of λj in (11). Eqs. (16) and (17) imply that the frequency and damping of the SSR can be identified from the eigenvalues λj of the unknown map A. Since Hankel matrices H1 and H2 span the dominant modes of Xc, the unknown map A can be approached by computationally efficient DMD.

To reduce the impact of noise, reduced SVD is performed to seek the low-dimensional representation of H1. The SVD of H1 is given as

(18)H1=UΣV*where U∈CL×L and V∈C(N−L)×L are the left and right singular vectors satisfying UU∗=I and VV∗=I, ∑∈CL×L is a diagonal matrix, i.e., Σ=diag(Σ1,…Σi,…ΣL), and Σi is the singular value of H1.

By retaining the first r columns of U and V, the reduced SVD gives a low-dimensional approximation of H1 as follows:

(19)H1≈UrΣrVr∗where Ur=U(:,1:r), Σr=Σ(1:r,1:r), and Vr=V(:,1:r) in MATLAB notation. As shown in Fig. 1, in the noise-free case, Σr+1=…=ΣL=0 holds, whereas in the noisy case, Σr+1>…>ΣL>0 is caused by noise. Eq. (19) thus reduces the influence of noise.

The matrix Ur, as the proper orthogonal modes of H1, can project A into reduced dimensional space. Then, the reduced order model A¯∈Cr×r of A is given as

(20)A¯≈Ur*AUr

Since H2=AH1, A¯ can be determined by solving the optimization problem in (20):

(21)minA¯‖H2−UrA¯ΣrVr*‖Fwhere ‖⋅‖F is the Frobenius norm, and the least-square solution of (21) is given as

(22)A=Ur∗H2VrΣr−1where (⋅)−1 is the inverse of the matrix. Note that the eigenvalues for A and A¯ are identical [20]; thus, an eigenvalue analysis of this matrix A¯ generates the eigenvalues and dynamic modes of the system.

The modal parameters are obtained from the eigenvalues of A¯ by using eigenvalue decomposition as follows:

(23)A¯=WΛW−1,where W∈Cr×r is the matrix of eigenvectors, and Λ is constructed by the eigenvalues λi, given in (17). Based on the relation between λi and (fi,αi) in (8) and (10), fi and αi can be determined as

(24)fi=Imag(log⁡(λi))/(2π∗Tr)αi=Re(log⁡(λi))/Tri=1,2,…,r,where Imag(⋅) and Re(⋅) represent the imaginary and real parts of a complex number, respectively. In this paper, we consider that the frequency range of the SSR phasor is different from that of the fundamental phasor. Moreover, according to (10), a pair of conjugated frequencies come from fs. Based on the two rules, fs and λs of SSR can be determined.

Once λi is estimated, the amplitudes of SSR can be obtained by first solving αi in (12). Rewriting (12) in the matrix form gives

(25)X¯c=Ea¯,with

(26)a¯=[a1,a2,…,ar]T,X¯c=[Xc(0),Xc(1),…,Xc(N)]T,E=[11…1ejλ1ejλ2…ejλr…………ejλ1(N)ejλ2(N)…ejλr(N))]where (⋅)T is the matrix transpose, and a¯ is solved in the least-square sense as follows:

(27)a¯=(E∗E)−1E∗X¯cwhere (⋅)−1 is the inverse of the matrix.

Finally, the amplitude As of the SSR component is calculated according to (10):

(28)As=|αs|/Cswhere |⋅| is the absolute value, αs(1≤s≤r) corresponds to that of fs, and Cs is calculated by (5) with the obtained fs and αs.

The proposed Hankel-DMD method provides a good dynamic performance, as it uses a very short data window for SSRPI. Under noise-free conditions, the proposed method can perform well as long as the dimensions of the constructed H1 and H2 satisfy min(L,N−L)≥r (r=4 in our problem). In practice, a longer data window is desired to reduce the impact of noise, but it also degrades the dynamic performance of DMD. Through sensitive studies using both simulation and field data, this paper suggests using N=50 or N=100 synchrophasors Xc, i.e., a 0.5 or 1 s data window for a 50 Hz system, and L is set to be L=N/3.

Another parameter that affects the performance of DMD is the selection of the number of dominant modes, i.e., r. Theoretically, r should be set as 2 when SSR does not occur and 4 after SSR occurs. The cumulative percentage sj of σi is used here to automatically determine the value of r, as shown in (29).

(29)sj=∑i=1jσi∑i=1Jσiwhere J=min(L,N−L) and 1≤j≤J. Then, r is determined by a predefined threshold Th, given as

(30)r=⌈j/2⌉×2,whensj≥Th,where ⌈⋅⌉ is the round up operator. Due to the impact of measurement noises, σi is not equal to zero when i>r. However, their eigenvalues are much smaller than those of the fundamental and SSR components. This indicates that j in (30) would be 1 or 2 when SSR does not occur and 3 or 4 after SSR occurs. Thus, (30) eliminates the impact of measurement noise and guarantees r to be 2 or 4 depending on the occurrence of SSR.

The whole procedure of the Hankel-DMD method to identify three key parameters of the SSR component, i.e., fs, αs and As, from the reported synchrophasors Xc can be summarized as follows:

Construct two Hankel matrices H1 and H2 by using (Xc(0),…,Xc(N−1),Xc(N)) according to (13) and (14);

A synthetic SSR current data was constructed as (31), where an off-nominal frequency f1=49.7 Hz is considered

(31)x(t)=1.0×cos⁡(2πf1t+π/2)+xs(t)

The parameters of the SSR components, i.e., fs, αs and As, are set as

(32){xs(t)=0t∈[0,2)xs(t)=0.2×e0.08tcos⁡(2π×35.2t+π/6)t∈[2,4)xs(t)=0.35×e−0.1tcos⁡(2π×34.5t+π/3)t∈[4,10]

Fig. 2a shows the waveform of i(t) with a sampling rate of 10 kHz. It can be seen that SSR starts at t=2s and its modal parameters change at t=4s. The DFT algorithm was first performed on x(t) with Np=200 (i.e., one cycle) to obtain synchrophasors Xp, and then the reported synchrophasors Xc were generated by resampling Xp with reporting frequency fr=100 Hz. Fig. 2b shows the magnitude of Xc.

The proposed method applies a sliding window to identify the parameters of SSR, and each window contains N=50 synchrophasor data (i.e., 0.5 s data). The window is shifted N=2 each time, so N=2 synchrophasor data overlap between two successive windows. For the synchrophasors in Fig. 2b, the estimated fs, αs and As are shown as black curves in Figs. 2c–2e, while the true values are depicted as red dot curves. To compare the proposed method with the InpDFT method and Prony method, the same sliding window scheme and window data with N=50 were used in the three methods. Figs. 2c–2e also show the results obtained from the InpDFT and Prony methods, depicted as blue dot curves and magenta dot curves, respectively. As seen from the figure, the curves of the proposed method match better with the true values than those of the two comparative methods.

In another test, Gaussian noise was also added to the signal. According to our field data and those reported in the literature [24], the noises in practical PMU data are generally around a signal-to-noise ratio SNR = 45 dB. Thus, the paper considers noise with SNR = 40 dB. The results obtained from the proposed and two comparative methods are shown in Figs. 3b–3d. Since Prony is sensitive to noise, the results of Prony become worse, showing a large deviation from the ground truth under noisy conditions. The proposed method also coincides better with the true values than the InpDFT and Prony methods. Comparatively, the estimation of fs is more accurate than that of αs and As.

Furthermore, the mean errors of the three methods are displayed in Table 1 for two model periods, i.e., [2,4) s and (4,6] s, as shown in Fig. 2a. The mean errors Eb of each window were calculated as

(33)Eb=mean(|b(t)−b~(t)|b(t)×100%)b:fs,αs,Ast∈[2,4)or(4,6]where mean(⋅) is the mean operator and b(t) and b~(t) are the true and estimated parameter values at time t, respectively. In both noise-free and noisy conditions, the proposed method achieves a higher accuracy for fs and αs. It is clear that the estimation of αs from InpDFT is unacceptable. The reason is that [13] relies on DFT; thus, a longer window is required to overcome the spectral leakage issue due to the off-nominal f1. In the case that spectral leakage does not occur, InpDFT can achieve acceptable estimation of the three parameters by using short window data.

The proposed method was further tested by simulated SSR data. For this purpose, a series-compensated wind farm system was modeled in MATLAB/Simulink software, as shown in Fig. 4a. A sixth-order model of the induction machine is used, with a two-mass drive train model to represent the generator shaft. Figs. 4b and 4c show the control strategies of the grid-side converter (GSC) and the rotor-side converter (RSC), respectively. Table 2 provides the key system parameters, and further details of the simulation model can be found in [25].

An SSR event is initiated at t=1 s when the series compensation level of the line is increased from 0.2 to 0.3. The wind speed is initially set as 8 m/s and gradually changes to 9 m/s. The waveform of the phase A current flowing through the line and the magnitude of the corresponding synchrophasors Xc are shown in Figs. 5a and 5b, respectively. Due to the change in wind speed, the SSR mode changes, which can be clearly observed from Fig. 5a.

After the proposed method was performed on each window with N=50 synchrophasors, the estimated SSR parameters are shown as black lines in Figs. 5c–5e. Meanwhile, the results obtained from the InpDFT and Prony methods are also shown in Figs. 5c–5e. To evaluate the accuracy of the three methods, Figs. 5c–5e show the reference values obtained from the waveform-based analysis. For the waveform-based analysis, the frequency and amplitude were directly calculated from DFT with a long-length window, while the damping was computed by the amplitude difference of two successive data windows. From Figs. 5d and 5e, it can be seen that the damping and amplitude trends obtained from the proposed method are in good agreement with the waveform-based analysis. However, the Prony method provides an unsatisfactory estimation of damping and amplitude, and the InpDFT method exhibits a poor estimation of damping due to the spectral leakage of f1 shown in Fig. 5f.

This subsection investigates the performance of the proposed method using practical SSR incidents that occurred in North China. Two sets of field data at different periods are used. Figs. 6a and 7a show the waveform data provided by the fault recorder with a sampling rate of 1000 Hz, while the magnitude of the corresponding synchrophasors is shown in Figs. 6b and 7b. It can be seen that the SSR mode varies over time. The possible reasons could be the tripping of wind generators and the change in the wind speed.

The estimation results of the first case are shown in Figs. 6c–6e. Similarly, the InpDFT and Prony methods are considered for comparison, and waveform-based analysis is used as a reference. According to the results, the estimation of the proposed method matches well with the reference value, while the damping results estimated by the two comparative methods deviate from those of the proposed method and waveform analysis.

The estimation results of the second case are shown in Figs. 7c–7e. Different from the first case, the off-nominal condition in the second period is not severe. As a result, the curves of the estimated parameters from the proposed and InpDFT methods match well, whereas the Prony method still cannot achieve satisfactory estimation and cannot effectively capture the variation in the damping and amplitude. The estimated fs varies between 7 and 8 Hz in Fig. 7c, which coincides with the results in [3], and the damping curves in Fig. 7d clearly show the rising and falling variation, which is consistent with the variation in the waveform amplitude. The two sets of experimental results demonstrate that the proposed method can capture the dynamic variation of the three parameters of the SSR component regardless of the spectral leakage effect.