Abstract
In this paper, a datadriven stochastic subspace identification (SSIDATA) technique is proposed as an advanced stochastic system identification (SSI) to extract the interarea oscillation modes of a power system from widearea measurements. For accurate and robust extraction of the modes’ parameters (frequency, damping and mode shape), SSI has already been verified as an effective identification algorithm for outputonly modal analysis. The new feature of the proposed SSIDATA applied to interarea oscillation modal identification lies in its ability to select the eigenvalue automatically. The effectiveness of the proposed scheme has been fully studied and verified, first using transient stability data generated from the IEEE 16generator 5area test system, and then using recorded data from an actual event using a Chinese widearea measurement system (WAMS) in 2004. The results from the simulated and recorded measurements have validated the reliability and applicability of the SSIDATA technique in power system low frequency oscillation analysis.
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1 Introduction
Extracting and quantifying dynamic behavior from observed oscillations presents a significant challenge in the rapid development of modern power systems [1]. With advances in computing and data communication technologies, a widearea measurement system (WAMS) becomes a powerful tool to provide realtime measurements for analyzing interarea power swing dynamics in large interconnected power systems [2, 3].
So far, published methods in interarea oscillation monitoring and analysis utilizing WAMS data can be divided into two groups. In the first group, spectral analysis approaches such as Prony and Fourier spectral and block processing techniques have been successfully used on complex WAMS data sets to extract modal information [4–7]. Such approaches also include the use of waveletbased spectral analysis [8] and the modified YuleWalker method to estimate lowfrequency modes from simulated data set and actual ambient power system data [7]. However, as the measured data set grows, it becomes difficult to accurately extract the mode and its parameters because of noise and nonstationary phenomena. Recently, techniques of nonlinear and nonstationary analysis based on nonstationary autoregression and empirical mode decomposition (EMD) have been refined to more accurately monitor interarea oscillations [9–12].
In the second group, system identification methods based on the state space have been shown to perform well on ambient noise signals. In [13], the proper orthogonal decomposition (POD) to determine the eigenvalues of the covariance matrix of power system oscillation signals has been shown to be suitable for multiple signals. In [14], the eigensystem realization algorithm (ERA) combining with linear filter decomposition and TeagerKaiser energy (TKE) has been adopted to identify the modal parameters from measured data. An alternative technique for obtaining modal parameters from multiple measured signals is the stochastic subspace identification (SSI) method. In [15] and [16], SSI was used to extract the critical modes of the system from simulation results of test cases. Some wellknown techniques such as QR factorization, least squares and singular value decomposition (SVD) have been utilized in SSI and all are regarded as robust and accurate system identification methods for outputonly modal analysis. The automation of the eigenvalue selection process would provide a lot of advantages for modal identification.
In this paper, the datadriven stochastic subspace identification (SSIDATA) technique combined with automation of the eigenvalue selection process is proposed to extract the interarea oscillation modal parameters from widearea measurements. Power system oscillations and the SSIDATA algorithm will first be reviewed in Sections 2 and 3. Then, the application of SSIDATA to interarea oscillation modal identification is briefly described in Section 4. To validate the effectiveness of the proposed technique, simulated data generated from the IEEE 16generator 5area test system and real data extracted from the postmortem analysis of the record of an actual event in China in 2004 were fully investigated and the identified modes verified in Section 5. The focus of these case studies was on the extraction of interarea power swing dynamics.
2 Power system oscillation
Power system oscillation may be triggered by many incidents in the system. Most power oscillations are suppressed by natural damping of the system, but system collapse can be caused by undamped oscillation. Theoretically, power oscillations are due to the rotor acceleration or deceleration following a change in active transfer from a generator.
It is generally appreciated that a power system may be linearized at an operating point and described in state space form as [17]:
where x is the state vector; y is the output vector; u is the input vector; A is the state matrix; B is the input matrix; C is the output matrix; and D is the feedforward matrix.
The power oscillation properties may be obtained through eigen decomposition of the state matrix A. For smallsignal stability of power system, the i ^{th} eigenvalue is defined as the i ^{th} oscillation mode:
The corresponding oscillation frequency f _{ i } and damping ratio ξ _{ i } are:
The shape of mode λ _{ i } is derived from the elements of the corresponding right eigenvector of the state matrix A, combining with the original state variables.
Therefore, the mode parameters including frequency, damping and the corresponding mode shape may be identified by estimating the state matrix of the linearized power system, and its eigen decomposition, from measured data. With the datadriven SSI technique proposed in Section 3, mode parameters are obtained by identifying the state and output matrices firstly and then performing eigen decomposition.
3 Datadriven stochastic subspace identification
SSIDATA [18, 19] is one of the effective experimental modal analysis methods and has been developed in recent years because of its numerical simplicity, SSIDATA’s statespace form and the robustness of the techniques used in the algorithm.
3.1 Stochastic statespace model
The discrete stochastic state space model can be obtained after sampling at discrete time intervals. Supposing the discrete stochastic statespace model is described as [18]:
where x _{ k } is the discrete time state vector at time step k; y _{ k } is the output vector; w _{ k } and v _{ k } are stochastic terms with zero mean noise E(w _{ k }) = E(v _{ k }) = 0.
In (4), the stochastic terms w _{ k } and v _{ k } are unknown, but they are assumed to have a discrete white noise nature with an expected value of zero and the following covariance matrices:
where E(·) is the expected value operator; δ _{ pq } is the Kronecker delta.
3.2 Datadriven stochastic subspace identification
The measured data y _{ i }, which are generator outputs in this paper, are divided into two parts representing past and future, and formed into an output block Hankel matrix [19], H:
where Y _{ p } = Y _{0,i1}, Y _{ f } = Y _{ i,2i1} are the past and future parts respectively of the block Hankel matrix with i = 2n; n is the system order and j is the number of measured data. Y _{ p } (the past inputs) and Y _{ f } (the future inputs) are defined by splitting Y _{0,2i1} into two equal parts of i rows. By advancing the boundary i in the Hankel matrix, new past and future parts can be obtained, which are Y ^{+}_{ p } = Y _{0,i } and Y ^{−}_{ f } = Y _{ i+1,2i1}.
The projections of the future row spaces to the past row spaces can be obtained:
The SVD of the weighted projection is calculated as:
where W _{1} = ((1/j)Y _{ f } Y ^{T}_{ f } )^{−(1/2)} and W _{2} = [I]_{ jxj } are both weighting matrices; U and V are unitary matrices resulting from the SVD process; S is a diagonal matrix containing the singular values which indicate the rank of the matrix (the order of system). The reduced diagonal matrix S _{1} is obtained by keeping only the part of the matrix S with nonzero diagonal elements.
Extended observables matrices are constructed as:
The Kalman filter state sequences are obtained from (7–10):
where ^{†} denotes the pseudo inverse of the matrix. The state and output matrices can then be determined as:
Now the realizations of the system matrices A _{ d } and C are identified collectively from all measured signals instead of one particular individual signal. Thus, the modal parameters can be obtained through the eigen decomposition of the state matrix A _{ d }:
where ψ is the complex eigenvector matrix; φ is the mode shape matrix; and Λ = diag(η _{1}, η _{2}, …, η _{ n }) is a diagonal matrix composed of the complex poles (eigenvalues) of the system. It shall be noted from (6–14) that a different combination of j and n will give a different state matrix, and furthermore will lead to a different set of modal parameters. Therefore, modal parameters should be derived from a series of combinations, rather than a single combination.
3.3 Determination of the modal parameters and estimation of system order
Mode identification is achieved by adjusting the model order n and the number of data j to achieve convergent eigenvalues. The order of the system is an essential piece of information for modal identification and analysis, but it is often not available, and a good engineering estimate is needed. Here, a simple but effective automatic process is proposed. Firstly, the range of model order to be estimated is specified; then starting from the lower bound, the poles (eigenvalues) corresponding to a given model order n are computed and compared to the poles (eigenvalues) of a oneorderlower model. Stability limits are defined by [20]:
If the frequency and the damping ratio differences are within the desired tolerance, the pole is labelled as a stable one [19]; otherwise, the model order n will be incremented and the procedure repeated until the model order is determined or the upper bound is reached. The modal frequency (f _{ i }) and damping (ξ _{ i }) corresponding to each pole (eigenvalue) can then be calculated by:
where a _{ i } and b _{ i } are real and imaginary parts of the continuous time eigenvalue defined by [19]:
where η _{ i } is the discrete time eigenvalue, corresponding to the ith mode.
The ith mode shape can be computed from:
where ψ _{ i } is the right eigenvector corresponding to the ith mode η _{ i }.
For an interarea oscillation analysis, in addition to the frequency and damping ratio, it’s equally important to extract the corresponding mode shape. As mentioned before, the frequency, damping ratio and mode shape can be identified at the same time by using SSIDATA. This method is well suited to analyze interarea oscillation due to its ability to extract the mode shapes, the oscillation frequencies, and their damping identified from multiple dynamic responses, such as generator speed and active power oscillating signals.
4 Application of SSIDATA in power system analysis
Based on the SSIDATA described in previous section, a scheme for extraction of the interarea oscillation modes from multiple generator oscillating signals is developed as shown in Fig. 1. The three main challenges in its practical implementation are discussed below.
4.1 Selection of measured system outputs
It should be noted that modal parameters including frequency, damping and mode shape are defined in terms of generator states which have historically been difficult to establish in real time. Phasor measurement units (PMUs) are now capable of providing realtime measurements of the operational state of a power system using satellitetriggered time stamps. PMUs installed at generator buses shall be used by operators to measure the system output including the generator states. Other data, such as phase angles and voltages, which are useful for monitoring and controlling, can be obtained.
Hence, modal parameters including frequency, damping and mode shape derived from a group of carefully selected measured system outputs are used to estimate the true mode parameters.
4.2 Placing PMUs for monitoring interarea oscillation
Theoretically, it would be the best to have PMUs placed on all the generator buses, it would be sufficient in practice to have at least one PMU placed in each area as generators in a given area would often swing coherently [11]. In order to improve measurement reliability, it is recommended to have PMUs installed at the terminals of the main generators which have relatively larger rated capacity in each area. For this application of extracting the dominant interarea modes, it is assumed that PMUs will be available at the terminals of two large generators in each area [21].
4.3 Scheme of modal parameters identification
When suitable measured system outputs are selected, interarea oscillation modal parameters can be estimated by using the following scheme.
 Step 1::

Form the output block Hankel matrix, H in (6) with a given set of measured data obtained by PMU
 Step 2::

Determine the projection matrices in (7) with the model order n set to the lower bound value as defined by the user
 Step 3::
 Step 4::

Compute the system matrices A _{ d } and C using (13)
 Step 5::

By employing (17) and (18), estimate the modal parameters through the eigen decomposition of the state matrix A _{ d }
 Step 6::

If the frequency limit l _{ lim,f } and the damping limit l _{ lim,ξ } are below their preset tolerance values, the system order is determined; otherwise, increment the model order n and return to Step 1. Typically, tolerances of 1% for frequencies and 5% for damping [20] have been used for this research
 Step 7::

Report the model order, poles (eigenvalues) and their corresponding modal parameters
5 Simulation and discussion
In this section, case studies based on the simulated IEEE 68bus 5area test system and actual event data are presented to investigate the performance of the proposed scheme.
5.1 IEEE 68bus 5area test system
Transient stability simulation data generated from the IEEE 68bus 5area test system is first used to evaluate the performance of the proposed approach in extracting the features of the interarea oscillation from measurements of interconnected power systems. The details of the system are given in [22].
In Area A, the rated capacity of G6 and G9 are larger than that of other generators, and PMUs are therefore sited on G6 and G9 following the guidelines given in Section 4.2. Likewise, PMUs are sited on G12 and G13 in Area B. Areas C, D and E have only one equivalent generator each, thus PMUs are sited on G14 in Area C, G15 in Area D and G16 in Area E.
Firstly, the theoretical modal parameters (including frequency, damping and mode shape) of interarea oscillation were obtained by the QR method. As observed from Table 1 and Fig. 2, four theoretical interarea oscillation modals exist in this test system. Fig. 2 shows that for Mode 1, generators 1~13 of Areas A and B oscillate against generators 14~16 of Areas C, D and E. In other words, for Mode 1, the oscillation area cluster is Areas A and B against Areas C, D and E. By the same token, the oscillation area clusters of the remaining interarea modes are Areas C and D against Area E, Area A against Area B, and Areas C and E against Area D. In order to normalize mode shapes, each right eigenvector corresponding to the interarea modes is divided by the maximum magnitude among all the right eigenvectors.
The following two contingency scenarios were selected for analysis.
 Case 1::

A threephase fault with duration of 0.05 s is applied at Bus 49
 Case 2::

A 0.15 p.u step disturbance with duration of 0.1 s is applied to the reference voltage of the exciter at G8
Detailed numerical simulations for Cases 1 and 2 were performed with a time step of 0.01 s to generate the data set used in the SSIDATA method. Figure 3 depicts the active power deviation from the steady state operating point of generators measured via PMUs in Cases 1 and 2 over a time window of 10 s after disturbance.
Since interarea oscillation modes are the main focus of this paper, only the modal parameters of interarea oscillations with frequency in the range of 0.2 Hz to 0.7 Hz (in 50 Hz system) were computed. Table 2 shows the identified results obtained with the proposed SSIDATA method applied to the simulated data. Following the eigen decomposition for the estimation of frequency and damping, the mode shapes corresponding to the identified interarea oscillation modes were extracted and shown in Fig. 4 for both Cases 1 and 2.
Modes are excited to different degrees according to the disturbance that causes them. Modes with poor damping are more noticeable. As shown in Table 2, it is clear that all the modes detected from numerical data using the SSIDATA technique are the theoretical interarea modes, but with lower damping. As noted above, G6 and G9 were selected as the representatives of Area A while G12 and G13 were selected as the representatives of Area B. Comparing Figs. 2, 4, it is observed that the estimated swing patterns and mode shapes generally agree with the theoretical ones, in particular for the modes with similar frequency, for example 0.5531 Hz, 0.5642 Hz and 0.5312 Hz.
Compared with traditional methods, the main advantage of the proposed SSIDATA approach is that the mode shape can be extracted following the identification of frequency and damping. This is because the state matrix and the output matrix can also be estimated in the SSIDATA technique and, being derived from multiple inputs, they are likely to reflect the dynamic behavior correctly.
5.2 Northeast China power grid
The event data used in this study describes the dynamic response of a threephase fault on a 500 kV bus in the Northeast China Power Grid. The event was recorded by a WAMS for a duration of about 40 s on March 24, 2004, starting at 13:28:25 local time, and the power swings are plotted in Fig. 5. Selected PMU recordings represented PMU measurements of generators in the study area. The data samples were acquired at a sampling period of 2 ms over a 5 s window period.
Based on the measured data at various locations, the Prony spectral analysis method [23], Matrix Pencil [24], estimation of signal parameters via rotational invariance techniques (ESPRIT) [25] and HHT [10] were firstly used to determine dynamic trends of the selected generators with results shown in Table 3 and Fig. 6. The two identified dominant modes are at about 0.5 Hz and 1.3 Hz respectively.
Further analysis was then conducted with the proposed SSIDATA technique. The frequency and damping of the modes estimated using SSIDATA technique are given in Table 4. As shown in Table 4, two oscillation modes at 1.3201 Hz and 0.5153 Hz were extracted. The results obtained using the SSIDATA technique are consistent with the other mode extraction methods while the corresponded mode shapes as shown in Fig. 7 were also computed using the proposed SSIDATA technique.
It can be seen that the modes extracted from the WAMS data using the proposed SSIDATA technique are the typical local mode and the interarea mode. For mode 1, a local oscillation in the system containing generator YMC is excited. The second mode reveals an interarea mode at 0.51 Hz (lower frequency) between this system and the system containing generators YBC and SZC.
Further study on the measured signals and the extracted modal information reveals that the oscillation died down rapidly due to good damping. This matches with the fact that, in the studied system, a power system stabilizer (PSS) was installed at all generators with capacity over 100 MW. This shows that the provision of mode shape estimation in the proposed SSIDATA technique is particularly useful for the analysis of oscillations in largescale interconnected power system using actual event data.
6 Discussion
The multisignal SSIDATA technique requires a different interpretation paradigm from singlesignal analysis methods, such as the Prony, HHT and Fourier spectral methos. The SSIDATA provides the frequency, damping and the corresponded mode shape through eigen decomposition of the state matrix, and it calculates them at the same time. This is in contrast to the Prony, HHT and Fourier spectral methods, which process the multiple oscillating signals separately. Also, SSIDATA does not require a preprocessing, such as denoising, as measurement noise has already been considered in its formulation.
As shown in the study of an actual event, the SSIDATA technique exhibits a good performance on generator active power oscillating signals, for both local and interarea modes.
7 Conclusion
In this paper, the SSIDATA technique is used to extract modal information from widearea measurements. The advantage of the SSIDATA technique in mode shape identification is that it estimates the state matrix and the output matrix spanning multiple generators in a power system from the measured data. Simulation results based on the IEEE 68bus 5area test system and actual event data showed that one of the prominent advantages of the proposed approach is the systematic determination of mode shapes while the frequency and damping of the modes are also accurately extracted. The proposed technique can be further explored and generalized for different applications and systems such as the design and optimization of lowfrequency interarea damping controller based on widearea system measurements.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 51507028) and the Hong Kong Polytechnic University under Project GUA3Z.
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YANG, D., CAI, G. & CHAN, K. Extracting interarea oscillation modes using local measurements and datadriven stochastic subspace technique. J. Mod. Power Syst. Clean Energy 5, 704–712 (2017). https://doi.org/10.1007/s4056501702716
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DOI: https://doi.org/10.1007/s4056501702716