Therefore, there is a need to develop estimation algorithms that can continuously monitor the stability of the power system. Tj : Mean of the samples for the j-th iteration Sj : Covariance of the samples for the j-th iteration.
General
Synchrophasor-based Wide Area Monitoring Systems (WAMSs)
The voltage and current from the secondary side of the potential and current transformers are converted into corresponding voltage and current signals by voltage and current sensors. These analog voltage and current signals are pre-processed with an anti-distortion filter to remove unwanted high-frequency components.
Power system stability
- Rotor angle stability
The absence of the first component causes non-oscillatory instability, while the latter causes oscillatory instability. Nowadays, rotor oscillations arising due to insufficient damping moment are mostly observed in the power system.
State-of-the-Art
Low frequency critical mode estimation using ringdown data
This method extends the Fourier analysis by directly calculating the attenuation factor, frequency, amplitude and phase of the low frequency oscillations. This algorithm exploits multiple invariance along a single spatial dimension and it is based on a subspace matching formulation of the DOA problem.
Low frequency critical mode estimation for ambient data
An extension of the ESPRIT algorithm, known as multiple invariance (MI) ESPRIT, is presented in [49] for exploiting multiple invariance data. However, these system identification based methods require both the input and output response of the system.
Motivation
To further improve the performance of the above estimator, the MM estimator and modified TLS-ESPRIT are used to mitigate the presence of outliers and high variance noise in the down-call data. To develop an improved estimator which exploits the sparsity of the signals in the surrounding data.
Thesis Organization
A comparative analysis of the estimated modes with the proposed robust TLS-ESPRIT method with the modified TLS-ESPRIT method and the robust modified Prony method is performed using Monte-Carlo simulations. Furthermore, a comparative analysis of the proposed method with improved Prony [38, 40] and modified TLS-ESPRIT and ERA/Prony [44] is performed for real-time data of the Western Electricity Coordinating Council (WECC) system [78, 79].
Improved Prony
Power system signals
Prony estimation and modifications
This also provides robustness to noise as it attempts to fit the noise components with additional exponential terms of the model. The location of these extraneous zeros depends on the choice of g0 which belongs to the null space A0.
Effects of outliers
Proposed Robust Modified Prony
Minimum Covariance Determinant (MCD)
The best samples are iteratively searched for to represent the covariance structure with the lowest possible covariance determinant. This is because with each step the area of the ellipsoids decreases and thus there are more concentrated samples with fewer covariance determinants. Here, the outlier free ellipse and the outlier robust covariance ellipse overlap with an almost similar covariance ellipse, while the classical covariance ellipse becomes circular.
This demonstrates the robust performance of the MCD algorithm in the presence of outliers where the classical technique fails.
Low rank approximation
The three-dimensional scatterplot and the classic and robust ellipsoid are plotted together in Figure 2.5.
Identification of Power System Modes using the Proposed robust modified
Similar simulation is done with matrix A obtained from the signal x(t) = e−0.4tcos(2πt) with an orderL =2 together with a single outlier. The best samples with the lowest possible covariance determinant are iteratively searched using C-steps to represent the covariance structure of the matrix. Thus, the estimated covariance matrix is free from the effect of outlier depicting the robust performance of the MCD algorithm.
SVD is done on the robust matrixRA and polynomial coefficient is calculated from CAi i.e. low rank approximation of RA.
Simulation Results
- Test signal Corresponding to Local Mode
- Test signal Corresponding to Inter-Area Mode
- Mode estimation for different ranges and position of outliers
- Mode estimation for the two-area test system
- Mode estimation utilizing the probing test data of the WECC system
The mean and variance of the estimated mode in terms of attenuation factor and frequency by the improved Prony, modified TLS-ESPRIT, ERA/Prony and the proposed method for outlier placed after the start of the test signal are provided in Table 2.5. It is observed that the mean of the estimated modes by the proposed method is accurate compared to the other methods. It is noted that the proposed method is almost insensitive to the location of the outlier.
Outlier Position Improved Prony Modified TLS-ESPRIT Proposed Method (Proposed No) µ(Mean) σ2(Variance) µ (Mean) σ2 (Variance) µ(Mean) σ2(Variance).
Conclusion
It is noted that when the size of the outliers is smaller, i.e. 1.2 times the peak value of the signal, the performance of all methods is comparable, but as the size of the outliers increases, the proposed method tends to produce more accurate results compared to other methods. with the other methods, as the other methods are not designed to mitigate the effect of the presence of an outlier sample. This chapter motivates the improvement of the existing modified TLS-ESPRIT method so that it can provide a more accurate estimate of the low-frequency oscillatory modes in the presence of outliers in the measured data. Comparison of the modified TLS-ESPRIT [75] with the proposed method is performed for two test signals in the presence of an outlier with different SNR corresponding to local mode and inter-area mode, respectively.
The effectiveness of the proposed method is also demonstrated for the real-time signal of the WECC system [78, 79].
ESPRIT Estimation
Power system signals
Modified TLS-ESPRIT
In TLS-ESPRIT method, the data must be divided into two orthogonal subspaces, namely signal and noise subspace. Singular value decomposition (SVD) technique is used to separate the data into two sub-spaces. Ryy = EsλsE∗s+EnλnE∗n (3.10) Whereλis the singular value of the signal subspace andλis the singular value of the noise subspace.
The basis corresponding to the data vectors p(n), q(n) and r(n) must therefore satisfy the following equations. 3.12) The basis vectors for the signal subspace containing the data vectors p(n), q(n) and r(n) are represented respectively in the columns of the matrices Ep, Eq and Er.
Effects of outliers
Proposed Robust TLS-ESPRIT
Robust Covariance using MM Estimator
MM Estimator
Identification of Power System Modes using the Proposed robust TLS-ESPRIT
To obtain the model ordering required for low ranking of the robust covariance matrix, the index [35] shown below is used by the proposed method. 3.17) where K(i) is a monotonically increasing index and ρi is the i-th singular value.
Simulation Results
- Test signal Corresponding to Local Mode
- Comparison of the Proposed Method With the modified TLS-ESPRIT
- Test signal Corresponding to Inter-Area Mode
- Comparison of the Proposed method With the modified TLS-ESPRIT
- Comparison of the Proposed method With the modified TLS-ESPRIT
- Mode estimation for the two-area test system
- Estimation of modes using the real test signal of the WECC system
The mean and variance of the attenuation factor and frequency of the estimated modes on the test signal with outlier (=10× peak value) after the start are placed in Table 3.1 for the modified TLS-ESPRIT and the proposed method provided. The mean and variance of the attenuation factor and frequency of the estimated modes on the test signal with outlier (=10 × peak value) after the end are placed in Table 3.2 for the modified TLS-ESPRIT and the proposed method provided. The mean and variance of the attenuation factor and frequency of the estimated modes on the test signal with outlier (=10× peak value) after the beginning are placed in Table 3.3 for the modified TLS-ESPRIT and the proposed method provided.
The mean and variance of the attenuation factor and frequency of the estimated modes on the test signal with outlier (=10×peak value) after the end are placed in Table 3.4 for the modified TLS-ESPRIT and the proposed method provided.
Conclusion
The low frequency mode estimation methods as proposed in Chapters 2 & 3 of the thesis are based on the use of ringdown data generated during large power system disturbances. These random disturbances of the system with non-linear trends subtracted appear as noise of very small magnitude and it is difficult to extract the impulse response from it. Extraction of the oscillatory modes from ambient data is difficult as the data has very low SNR and also corrupted with non-linear trends.
The performance of the proposed method, i.e., NwT-RD-TLS-ESPRIT is also performed on ambient data contaminated with nonlinear trends, which are generated by a second-order equivalent system representing a special mode with dynamics that corresponds to modes. present in a two-zone power system [9] and also in data obtained from a real-time digital simulator (RTDS).
Methods for modal parameter estimation
- Nonlinear filter
- Wavelet based signal recovery technique
- Random decrement (RD) technique
- TLS-ESPRIT Estimation
- Identification of Power System Modes using the Proposed NwT-RD-TLS-
After wavelet-based denoising of the measured environmental data, the next task is to obtain the impulse response. The random decrement (RD) technique developed by Cole was previously applied for space structures, aeroelastic systems, large structures [97] and measuring damping [98] of the ground. Estimation of the random reduction signature or impulse response is done by taking the sample average of the selected windows.
To extract modal information from ambient data, the various steps of NwT-RD-TLS-.
Simulation Results
- Test signal corresponding to ambient data
- Estimation of modes for the two-area system
- Mode estimation for the test signal corresponding to two-area data 74
- Estimation of modes using the real PMU data from North Eastern Regional
- Mode estimation for power flow data in line one
- Mode estimation for power flow data in line two
- Estimation of modes using the real test signal of the WECC system
The data obtained from the PMU corresponds to the power flow of two lines connected to the same bus, as shown in Figure 4.8(a) and Figure 4.9(a), respectively. A plot of the preprocessed data, the salient wavelet coefficients, the reconstructed signal, and the estimated free response for the data corresponding to row one are shown in Figures 4.8(b), 4.8(c), 4.8(d), and 4.8(e) , respectively. A plot of preprocessed data, distinct wavelet coefficients, reconstructed signal and estimated free response for the data corresponding to row two is shown in Figures 4.9(b), 4.9(c), 4.9(d) and 4.9(e) , respectively.
Similarly, the plots for the different steps of the proposed method applied to the data corresponding to window 2 and window 3 are given in Figure 4.12 and 4.13 respectively.
Conclusion
The robustness of the proposed method in oscillating mode identification is also demonstrated on a real PMU data obtained from the Northeast Regional Electricity Board and WECC system test data. The results given in Section 4.3 show the efficiency of the proposed method in coping with high noise level and accurate oscillating mode identification. The proposed method, i.e., S-RD-TLS-ESPRIT is compared with the NwT-RD-TLS-ESPRIT method, as proposed in Chapter 4, for simulated data corresponding to ambient data, followed by evaluation of ways to simulate the dynamics of a two-zone system [9].
Further, a comparative analysis of the proposed method with the NwT-RD-TLS-ESPRIT method is performed for real PDC measurements from the Northeast Regional Electricity Board (NEREB) of India and real research data of the Western Electricity Coordination . Council System (WECC) [79] [78].
Methods for modal parameter estimation
- Nonlinear filter
- Sparse Technique for recovery of signal
- Creation of Dictionary (D)
- Determination of sparse vector
- Signal reconstruction
- Random decrement (RD) technique
- TLS-ESPRIT Estimation
- Identification of Power System Modes using the Proposed S-RD-TLS-ESPRIT
The main idea of the sparse technique is to select the best possible combination of columns from the dictionary to represent the input signal. The left singular vectors of the low-rank data matrix form the dictionary for the proposed method. The resultant signal vector y is obtained by concatenating all columns of the reconstructed signal matrix.
A crossing thresholdTth=c was used, which represents a fraction of the standard deviationσof the signal.
Simulation Results
- Test signal corresponding to ambient data
- Estimation of modes for the two-area system
- Mode estimation for the test signal corresponding to two-area data 99
- Estimation of modes using the real PMU data from North Eastern Regional
- Mode estimation for power flow data in line one
- Mode estimation for power flow data in line two
- Estimation of modes using the real test signal of the WECC system
Finally, to get the test signal that matches the environmental data, the output response of the system shown in Figure 5.3(a) is sampled with a sampling frequency. The data measured from the PMU corresponding to the active power flow of the two lines incident on the same bus are provided in Figure 5.8(a) and Figure 5.9(a) respectively. The plot of the preprocessed data, reconstructed signal and the estimated free response for the data corresponding to line one are shown in Figures 5.8(b), 5.8(c) and 5.8(d), respectively.
A plot of the preprocessed data, the reconstructed signal, and the estimated free response for the data corresponding to row two are shown in Figure 5.9(b), 5.9(c), and 5.9(d), respectively.
Conclusion
Lauby, “A comprehensive computer software package for small-signal stability analysis of power systems,” IEEE Trans. Campbell, “An efficient improvement of the AESOPS algorithm for power system eigenvalue computation,” IEEE Trans. Yuan, “Window FFT interpolation algorithm for power system harmonic analysis,” IEEE Trans.
Kumaresan, “On the zeros of the linear prediction-error filter for deterministic signals,” IEEE Trans.
Block diagram of SCADA
Block diagram of WAMS
Block diagram of PMU
Classification of power system stability
Zero plot of signal and noise Prony method without outlier, Prony method with outlier
Covariance ellipsoidal using classical technique for clean data and outlier corrupted
Comparison of the classical and MCD algorithm based covariance ellipse for outlier
Block diagram for on-line estimation using proposed method i.e, robust modified Prony. 28
Single line diagram of the two-area system
Probing data corresponding to the real power flow
Block diagram of the proposed method i.e, robust TLS-ESPRIT for mode estimation. 53
Probing data corresponding to the real power flow
Block diagram of the proposed method i.e, NwT-RD-TLS-ESPRIT for mode estimation. 71
Plots for the generated test signal
SIMULINK model used to generate two-area data
Plots for the generated two-area data
Two-area system for ambient data
Plots for the RTDS data
Plots for the Line 1 data
Plots for the Line 2 data
Probing data corresponding to real power flow
Plots for the window 1 data
Plots for the window 2 data
Plots for the window 3 data
Block diagram of the proposed method i.e, S-RD-TLS-ESPRIT for mode estimation. 94
Plots for the generated test signal
SIMULINK model used to generate two-area data
Plots for the generated two-area data
Two-area system for ambient data
Plots for the RTDS data
Plots for the Line 1 data
Plots for the Line 2 data
Probing data corresponding to real power flow
Plots for the window 1 data
Plots for the window 2 data
Plots for the window 3 data
Single line diagram of the two-area system
