An Improved MP Algorithm for Power System Mode Identification

An improved matrix pencil (MP) algorithm for power system mode identification has been proposed in this paper. Aiming at the problem of inaccurate order determined by the singular value decomposition (SVD) step in traditional MP algorithms and the lack of Hankel structure in the reconstructed matrix, which leads to inaccurate identification under low signal-to-noise ratio (SNR) condition, singular entropy order determination is introduced. An iterative algorithm is proposed to calculate a reconstructed matrix with both Hankel structure and rank-deficient attributes. This reconstruction matrix has a better approximation effect and improves the identification accuracy of the MP algorithm under low SNR. The effectiveness of the proposed algorithm was verified by modal parameter identification of numerical signals and oscillation signals of the 4-machine and 2-area systems. The algorithm proposed in low SNR has higher accuracy than the traditional MP algorithm, and the reconstructed signal has a better fitting effect.


Introduction
Low-frequency oscillation of the power system is the phenomenon of relative power angle swing of the generator set caused by a slight disturbance.This oscillation behavior may cause out-of-step splitting of the generator units and seriously threaten the safe and stable operation of the power system.Researching the dynamic stability characteristics of the power system and identifying the potential oscillation risk is a significant means to ensure safety and stable operation.There are usually two methods: one is to directly establish the model of large systems and solve the eigenvalues of the characteristic matrix to determine the oscillation mode of the system [1][2][3], and the other is to identify modal parameters based on actual measurement data, which can bypass the modeling process of complex systems and obtain the proper system oscillation mode without any system and equipment parameters.Meanwhile, the mode identification algorithm should have sufficient anti-noise ability and identification accuracy.
The matrix pencil (MP) algorithm is a mode identification algorithm based on observed signals previously proposed by [4][5].The frequency and damping ratio of each modal component can be solved by constructing a matrix pencil and solving the signal poles in the MP algorithm.The traditional MP algorithm needs to use the measurement sequence to construct the Hankel matrix (i.e., observation matrix) and then use the singular value decomposition (SVD) to determine the modal order by filtering the singular value through the threshold.However, this method may not be accurate in order determination under low signal-to-noise ratio (SNR), which may result in false modes or ignoring some modes.In order to solve the problem of order determination, singular entropy was introduced in [6][7], and exact mode order (EMO) was introduced in [8].Besides, the Hankel matrix constructed by the ideal signal without noise has the rank-deficient attribute.The observation matrix reconstructed by SVD in the traditional MP algorithm only has the rank-deficient attribute but lacks the Hankel structure, which means the observation matrix is biased and may cause a significant error in the estimation of modal parameters [9].
Aiming at the order determination problem of the traditional MP algorithm and the problem that the observation matrix reconstructed by SVD does not have a Hankel structure, the singular entropy order determination was introduced.A rank-deficient Hankel matrix approximation iterative method was proposed in this paper, which achieved the goal that the observation matrix has both Hankel structure and rank-deficient and improves the accuracy of modal parameter identifications under low SNR.The simulation results of the numerical signals and the 4-generator and 2-area simulation models verified the effectiveness of the algorithm proposed in this paper.

Rank-deficient Hankel matrix approximation
An important pre-step for implementing the MP algorithm is to use SVD decomposition to remove small singular values to determine the number of signal components (i.e., the modal order, the value of M) and achieve the goals of the noise reduction effect.This method will produce significant errors under the condition of low SNR, and the value of M will be inaccurate, reducing the estimated accuracy of the MP algorithm.This paper introduces the method of singular entropy to estimate the M value and rankdeficient Hankel matrix approximation to improve the parameter estimation accuracy of the MP algorithm under low SNR.
The value of M is determined by using the SVD method to process the noisy discrete sequence and k is an integer) obtained by actual sampling.Hankel matrix is constructed: and SVD is performed on Y: where U is an (N-L)×(N-L) orthogonal matrix, Σ is a (N-L)×(L+1) diagonal matrix and V is an (L+1)×(L+1) orthogonal matrix.The main diagonal elements of Σ are the singular values of Y and are arranged from large to small: The dominant components of the signal cause the first 2M singular values and those from the 2M+1 to the end are caused by noise.The singular value corresponding to the noise is significantly smaller than that corresponding to the effective components of the signal under the condition of high SNR.If the i-th singular value meets: (4) where  is the threshold.The accuracy of M depends on the appropriate selection of  .When the signal SNR is low, and the noise component is not negligible compared with the effective signal component, it is difficult to determine the singular value caused by the noise through threshold screening.It has been proven in [6] that introducing singular entropy and utilizing its rapid convergence characteristics is an effective way to obtain M in strong noise backgrounds accurately.The j-th singular entropy increment is defined as E j : Moreover, the change rate of E j can be expressed as: ,1 1 Since there is an order of magnitude difference in the singular entropy increment corresponding to the effective component of the signal and the noise, the value of Dj will jump significantly.Thus, when Dj achieves the maximum value, j equals 2M.The diagonal matrix ' Σ is constructed as: .

Correspondingly, '
V can be constructed by taking the first 2M columns from V and ' Y can be calculated by ' V does not contain the singular value caused by noise, ' Y can be considered as noiseless.The calculation process of ' Y is defined as: ' Y is rank-deficient and rank( ') rank( ) Y does not have a Hankel structure.The Hankel matrix constructed by an ideal signal sequence must be rank-deficient.The critical point of estimating the modal parameters of the noisy sequence is to use the rank-deficient Hankel matrix to approximate that constructed by the ideal signal sequence.In the traditional MP algorithm, only the rank-deficient property of ' Y is used.For efficient identification, ' Y should be a rank-deficient Hankel matrix.
A rank-deficient Hankel matrix approximation method is introduced.H operation is defined as: YH is a rank-deficient Hankel matrix.The p-th row, q-th column element of YH can be calculated by: where ', ' ' p q y is the element in row ' p and column ' q of ' Y , and 1  represents the set of all elements of the anti-diagonal where the p-th row, q-th column element of ' Y is located, which can be expressed as: . After the H operation, YH has a Hankel structure.An iterative operation J is defined as: Operation J iterates the combination of L and H. Theoretically, the result YJ is a rank-deficient Hankel matrix with a better approximation to the Hankel matrix constructed by the ideal signal sequence.It has been proved in [10] that the iteration of Equation ( 12) is convergent.Termination conditions of the iteration in Equation ( 12) should be set in practical applications.

The improved MP algorithm based on rank-deficient Hankel matrix approximation
The LFO signal can be described as a series of cosine functions with different frequencies and damped amplitudes: Equation ( 13) is the system response when a slight disturbance occurs, which contains the actual system response ( ) x t and noise ( ) n t .i A , i  , i  , and i  are the amplitude, angular frequency, phase of the i-th component, and damping factor, respectively.The corresponding complex signal function of Equation ( 13) is: Equation ( 14) is discretized: Ts is the sampling interval, k is an integer and 0 1 k N    .N is the length of the sampling sequence.The complex frequency domain operator is expressed as ( ) , which contains the information of i  and i  .Parameter Pi is expressed as , and the relationship between x(t), Pi and zi is: In [11], the definition of matrix pencil is given: where f(t, λ) is the pencil function of g(t) and h(t), λ is the constraint parameter.g(t) and h(t) are nonlinear related.For the signal function s(t), the parameters of s(t) can be estimated by choosing appropriate g(t), h(t), and λ.The basic idea of the MP algorithm is to transform the problem of solving mode parameters into the solution of matrix pencil generalized eigenvalues.By substituting the Hankel matrix Y into Equation ( 12), the rank-deficient Hankel matrix YJ can be obtained.The Frobenius norm is introduced to describe the convergence of Equation ( 12): where X is the Hankel matrix constructed by the ideal signal sequence, i is the number of iterations and F  is the Frobenius norm.Equation ( 17) indicates that the i-th iterative result ( ) i J Y is closer to the X than the previous iteration result ( 1)   i J  Y .The criterion for iteration termination can be set as: 0.001 The calculation process may converge slowly, so it is also necessary to set an upper limit on the number of iterations.Based on experience, the maximum number of iterations can be set to 100.YJ can be regarded as the Hankel matrix constructed by discretized x(t).The last and first columns of YJ are deleted separately to form Y1H and Y2H : where L is the matrix pencil parameter and N/4≤ L≤ N/3.The value of M can be calculated through singular entropy.Y1H and Y2H can be decomposed into: where: , , , M diag P P P  P  (26).
The matrix pencil is constructed by: where the matrix I is a 2M×2M identity matrix.When Then, the frequency and damping factor can be obtained by: The coefficient matrix P can be obtained by solving the least square problem shown in overdetermined Equation (31): and it can be obtained from YJ. ZM can be expressed as: ( ) cos(2 ) The closer the value of R2 is to 1, the higher the approximation degree between the reconstructed sequence and the original data signal sequence is, the better the fitting effect is, and the more accurate the identification parameters are.The values of R2 in Table 1 are close to 1, indicating that the identification results are accurate and reliable.0.9997 0.9997 In order to verify the anti-noise performance of the proposed algorithm, the white noise is superimposed on the original signal, and the SNR is within the range of 6 to 15 dB.The test is repeated 200 times value at each SNR, and the average values of the identification results are used to reconstruct the signals.The R 2 -SNR curves are presented in Figure 4.In the low SNR region corresponding to the left side of the curves, the R 2 values of the proposed algorithm are closer to 1 than those of the traditional MP algorithm, which indicates that the identification results of the proposed algorithm are more credible.With the SNR values increasing, the R 2 values of the two algorithms are getting closer, which confirms that the identification results of the two algorithms for noiseless signals are almost the same.

Conclusions
An improved MP algorithm based on rank-deficient Hankel matrix approximation was proposed in this paper.The simulation results of numerical signals and the 4-machine and 2-region models indicated that the proposed algorithm has higher identification accuracy than the traditional MP algorithms under low SNR conditions.In practical applications, termination conditions for the iteration should be set in the process, and its convergence characteristics should be further studied.The proposed algorithm can be extended to other algorithms, including SVD computation, improving the noise resistance performance.
the elements of the i-th row of 0

.Figure 1 .
Figure 1.The flowchart of the improved MP algorithm based on rank-deficient Hankel matrix approximation.4. Case study 4.1.Numerical signal simulation Consider the signal shown in Equation (33), which contains three components.The parameters in Equation (33) are set as: A1=0.8,A2=1.6, A3=2.4,α1=-0.15,α2=-0.20,α3=-0.25,f1=0.5, f2=1.3,f3=1.87.φ1, φ2 and φ3 are the random values within the range of [0, 2π].The white noise on the signal is superimposed.The SNR range is [8, 15] dB.The sampling frequency is set as 100 Hz.The traditional and improved MP algorithms proposed in this paper are used to identify the parameters.The test is repeated 500 times at each SNR value ( in dB), and the mean square error (MSE) value (in dB) of the identification error of each algorithm is counted.The results are shown in Figure 2. It can be observed that with the increase of SNR, the identification accuracy of each algorithm also improves.The calculation accuracy of the two algorithms under high SNR (right end of the curve) is almost the same,and they can all achieve higher identification accuracy.However, under low SNR conditions, the identification error curves of the algorithm proposed in this paper are all located below the traditional MP algorithm, and the error is significantly smaller than the traditional MP algorithm.Therefore, the accuracy advantage of the algorithm proposed in this paper is obvious under low SNR conditions.3 1 1

Figure 3 .Figure 4 .
Figure 3.The single line drawing of the 4-machine and 2-area system.