A Novel Subspace Decomposition with Rotational Invariance Technique to Estimate Low-Frequency Oscillatory Modes of the Power Grid

. Tis paper proposes modifed Karhunen–Loeve transform with total least square estimation of signal parameters using rotational in-variance technique (MKLT-TLS-ESPRIT) to approximate the low-frequency oscillatory modes. MKLTdecreases the impact of highly correlated additive colored Gaussian noise (ACGN) from the signal by diferentiating the correlation matrix w.r.t from the fnal time instance. A quantitative study of the suggested method with other estimation methods is used to evaluate the effectiveness of the proposed method. Monte Carlo simulations with 50,000 runs are conducted to test the robustness of the estimation scheme for MKLT-TLS-ESPRIT. Te evaluation of the efciency of the proposed method in real-time perspective, the two-area system, and New England sixty-eight bus test system has been considered. Te analysis shows that the suggested methodology correctly measures the interarea modes and lowers their mean and standard deviation to a minimum value.


Introduction
Wide-area integrated regional power networks are commonly afected by interarea oscillations (IAOs) when they are exposed to shedding of load, line loss, and fault. Such oscillations will lead to instability within the [1] interconnected networks. Small signal stability analysis (SSSA) utilizes state equations to approximate this IAOs by linearizing the power system. But, this technique sufers with heavy numerical burden, inconsistency in parameter estimation, and large computational time. Ten, IAO modes need to be defned directly from the measured signal employing online techniques [2].
Te use of measuring instruments, particularly wide area measurement systems (WAMSs), is drastically increased in the few last decades [3]. Tis allows higher precision IAOs estimates in both of and online. Te online estimation methods are categorized as parametric and nonparametric approaches. Frequency estimation approaches, i.e., wavelet transform (WT) and continuous WT [4,5] whose behavior is the same as a band-pass flter. In WT, changing the window size allows for multiresolution for both slow and fast frequency response. Multidimensional rotating frame measurements for synchronous machines during operations are available for mode estimation in [6]. Although WT can identify both long-and short-duration disturbances, its accuracy is dependent on the decomposition levels used and the mother wavelet used. Tere is currently no defned methodology for selecting the mother wavelet. Approaches addressed in the [7] like robust modifed prony (RMP) and augmented prony (AP) [8] approximate the entire signal parameters by a provided sampled sequence, i.e., duration, direction, intensity, attenuation factor, and damping ratio. However, RMP and AP have a signifcant computing time problem, and AP does have a certain latency in large power system statistical analysis. Data-driven stochastic subspace identifcation (data-SSI) state space technique proposed in [9], which functions well under both ringdown and ambient oscillations in power system. Since data-SSI concentrated on both estimation of the mode and selection of the model order, its computational time period is signifcant, which can contribute to the loss of synchronization. Te authors of [10] proposed a new notch fltering approach along enhanced phase-locked loop to identify the damping factor. Another robust subspace, i.e., signal parameter estimation using rotational in-variance technique (ESPRIT) and its higher variants like modifed TLS-ESPRIT [11,12] are quite effective in terms of time-frequency resolution, fast computational time, and less sensitive to outliers. It is also possible to use the KLT-TLS-ESPRIT subspace-based technique [13], which is used for IAO dynamic phasor estimation.
Tis paper proposed an estimation method that fully denotes the signal as well as maximally compressed the information held by the signal. Tis proposed MKLT-TLS-ESPRIT technique can be extended to probing signals as well as for ringdown and ambient oscillations. It may work for small as well as huge power systems. Te MKLT can be applied to estimate stationary as well as nonstationary signals.
For an efcient and precise measurement of the phase, the signal measured from a PMU must be held noise free. But because of the existence in outliers, the higher order harmonics, as well as unwanted spikes, get devolved in the signal. Employing a Hampel flter (HF), this spikes, and higher-order frequency components can be from a signal without smoothing the data. Te remaining components of the noise are converted into highly correlated ACGN. Figure 1 describes the steps for MKLT implementation.
Te contributions of the paper are as follows: (i) Te proposed technique efectively eliminates the infuence of ACGN and compresses the information carried out by the signal maximally using samples decomposition onto a set of correlation matrix eigenvectors (ii) Among various signal and noise subspace approaches presented over here, this proposed approach is more robust to outliers with the fastest computation (iii) MKLT-TLS-ESPRIT performs more efciently than the above methods when the signal strength buried with noise is very low or at SNRs 30 dB and 10 dB (iv) Te efcacy of the proposed approach for small and midsize power system is verifed using IEEE benchmark two-area and IEEE sixteen-machine sixty-eight bus systems (v) Te proposed technique is evaluated under diferent complex conditions to check its robustness, such as the power system's unexpected detached load and three-phase short-circuit failure Te rest of the paper is structured according to the description. Section 2 discusses with the correct signal model the LFOs, also defnes the impact of outliers in the power system, and also demonstrates the use of HF to eliminate outliers from the power signals. Section 3 discusses the use of MKLT-TLS-ESPRIT for estimation of inter-area mode. Section 4 sets out a step-by-step formulation of MKLT and TLS-ESPRIT for estimation of mode. Section 5 describes the results and discussions. Section 6 presents the functional applicability of the proposed technique using the IEEE benchmark two-area system. Section 7 includes the explanation of the efcacy of the proposed technique for IEEE sixteen-machine sixty-eight bus system, and the paper ends Section 8.

Problem Formulation for LFOs in Power System
Te MKLT-TLS-ESPRIT proposed a new technique, which incorporates the [14] KL transformation theory and TLS-ESPRIT rotational invariance principles to estimate LFO modes.

Modeling of Power Grid Oscillations.
Te oscillations which address the dynamic conditions of the power grid represented using sinusoids which are exponentially damped. We called the active power, i.e., r(n) fows in the IEEE benchmark two-area network tie-lines and also the IEEE sixteen-machine sixty-eight bus system. Te oscillations from the power grid modeled are as follows: Two deterministic quantities ( ω k ) and ( a k ) convey frequencies and amplitudes. Te phases ϕk are uncorrelated random variables, evenly spread over (0, 2π), where attenuation component b k and total sinusoid count K if the AWGN sequence, i.e., w(n), infuences the sinusoids, where E[|w(n)| 2 ] � σ 2 ω of zero mean and variance. Te noise-component signal can be described as where β j � b j + iω j , α j � (a j /2)e iϕ j , and L � 2K.

Efect of Outliers in Power
System. Te building of automated, interconnected regional electric power networks results generation of diverse and complex outlier data in the power system. Major causes of outlier are as follows: (1) Signal acquisition capability: the limited potentiality of the sensors and WAMSs (2) Power system failures: the power system failures such as transmission line outage and faults in the transmission lines (3) Human infuences: the involvement of the human in the signal measurement and control process may lead to the production of outlier data signal measurement For a single out-of-scale measurement, the sample means of the estimation can be afected when the data sequence includes the outliers. Journal of Electrical and Computer Engineering

Removal of Outlier Using Hample Filter.
Te HF is a segment of the group of decision flters which uses the Hample identifer (HI) to remove the outliers from the input data sequence. Te HI works on the principle of the threesigma rule of statistics [15]. Te width of the measurement window also plays a vital role outliers removal. Te steps followed for the removal of outlier are described by the Algorithm 1.

Steps to Implement Proposed Approach to Estimate the Power Grid Oscillations
Let N1 be the recent power measurements received from the tie-line using PMU. It can be expressed by a signal and noise vector using (2).
where ω(n) and r(n) describe noise and signal vectors. Data matrix (Y) can be constructed by using the vector R(n).
Signal vector r(n) can be descried using time-window frequency vector V given as follows: where the time-window of length-M of the frequency vectors holds by the matrix V with L no. of columns.
Te signal vector r(n) can be described with the subwindowing process as Using TLS-ESPRIT, the following matrices can be defned using R M−1 (n).
where unstaggered and staggered windows are expressed as V1 and V2. Frequency vectors of two subtime-window correlate as

Application of Modified KLT for Interarea Oscillating Mode Estimation
Te numerical code implementation of basic KLT approach [16,17] is limited due to its larger computational burden and time. Te idea behind the MKLT is to diferentiate the "dominant eigenvalue" of the autocorrelation matrix with respect to the fnal instant "N" of the basic KLT approach. Te basic KLT approach fxes up the fnal time instant to N � 1, which restricts the basic novelty of KLT that described by the fnite, positive real N as a new continuous variable [12,18]. Following steps need to be carried out to implement the modifed KLT approach, i.e., the fnal variance of the projection coefcients.
Step 2 First, using KLT the autocorrelation of sequence (9) can be expressed at diferent time instants, i.e., Step 3 Te independence property of the random variable C i , i.e., E(C j C i ) � λ i δ ji , where δ ji denotes the Kronecker delta, defned as δ ji � 0 for j ≠ i and δ jj � 1, and λ i is the new positive number sequence which is related to the variance of the corresponding random variable C i as follows:  Journal of Electrical and Computer Engineering 3 Step 4 Using statistical orthogonal property of the random variable C i , i.e., E(C j C i ) � λ i δ ji and taking the summation of both the time instant in one place, i.e., i � j, (10) reduced to where E(C j C i ) � λ i δ jj .
Step 5 As the random function R(n) has zero mean value in the fnite time interval, we can replace the instant n1 � n2 � n to get the variance from the autocorrelation and indexing it by i, and (12) can be expressed by referring (11) Step 6 Using the property of orthogonality, Ψ i (n) should be normalized to one by integrating (13) on both sides with respect to n for 0 to Step 7 For this given zero mean random process R(n), it can introduce the variance σ 2 R(n) of the R(n), i.e., Substituting (15) into (14) gives Step 8 (16) shows that the right side of the summation of eigenvalues λ i must be the few functions of the fnal instant N as left side, i.e., λ i ≡ λ i (N).
Step 9 Now, diferentiating both side of (16) with respect to the fnal instant N yields Te above result σ 2 R(N) is called the fnal variance theorem. It shows that the fnal variance of any random stochastic process depends upon the fnal instant N. Sum of the series of frst-order partial derivatives for eigenvalues λ i (N) provides the fnal variance σ 2 R(N) with respect to the fnal instant N.

Minimization of Complete Error by Utilizing Proposed
MKLT-TLS-ESPRIT Technique. Algorithm 2 can be employed for estimation of modes by minimizing the total error between estimated subspace and actual signal subspace as suggested in [19].

Results and Discussion
Te recent N1 samples provided by the PMU and phasor data concentrator (PDC) are considered for the IAO mode estimation using the proposed approach. Te LFOs with nonlinearity may have approximately a stationary signal to achieve improved estimation. Te MKLT-TLS-ESPRIT minimizes the SD and the bias of estimated modes. Te approximation as an AWGN is regarded as a good analogy for modeling the measurement uncertainty received from the PMU's. However, some ACGN is detected in measurements because of the existence of an antialiasing flter (AAF) (with cut-of frequency between 400 and 1000 Hz) and signal processing algorithms that use convolution to estimate the phasors. At the performance, the ACGN is often ignored. A down-sampler then holds those quantities for further processing.
Additionally, a 10.24 s time window was used, referring to the PMU with the data rate of 512 samples (assuming data rate of 50 phasors/second). Usually, the variance in PMU (1) Consider the N1 samples of signal R(n) from (2).
(2) First, chose a moving window with odd length, which is composed of the current sample of the input signal and make the window centers around the current sample. (3) Next, for the each current window data, its standard deviation σ i and local median m i would be computed. medi where κ is the scale factor which is given by κ � 1/ � 2 √ efrc1/2 ≈ 1.4826. σ i /k describes the median absolute deviation (MAD). For a normally distributed data, the scale factor (k) is equal to 1.4826, which makes the standard deviation estimate a unbiased for Gaussian data.
(4) Recent sample would be compared with t σ σ i , where threshold value is given by t σ . (5) If the flter identifes the current sample, P i , then the HF replaces the median with current samples as follows: we can reduce the efect of HF over the signal. If t is 0, then the HF behaves as a regular median flter. measurements is assumed to be of 10 − 4 p.u. [21]. Te North American SynchroPhasor Initiative (NASPI) ofers standards for PMU and synchroPhasor measurements in different dynamic environments, such as energy variance 0.61 × 10 − 4 p.u.

Test Signal Corresponding to Interarea Mode.
Simulations are conducted with a test signal having frequency � 0.4 Hz, magnitude � 1, and attenuationfactor � −0.07. Figure 2 shows the efect of HF to above-mentioned test signal, i.e., interarea mode to minimize the outliers from the signal and fnal fltered signal further carried out for the process of frequency estimations. Figure 2(a) shows the signal with 10 dB, and Figure 2(b) 30 dB noise afected test signal and fnally fltered out with less variance using HF.
A statistical comparison of the MKLT-TLS-ESPRIT, QR-TLS-ESPRIT, KLT-TLS-ESPRIT, RMP, and data-SSI for SNRs 10 dB and 30 dB is mentioned previously. Te frequency and attenuation factor distribution of a test signal are shown in Figure 3. From Table 1, it is analyzed that the MKLT-TLS-ESPRIT method approximated the standard deviation and mean for the estimated frequency corresponding to interarea oscillating modes are 60.02% and 98.92%, respectively, of that obtained from the data-SSI approach and 60.51% and 99.47% for the KLT-TLS-ESPRIT approach with SNR 10 dB. It is also found that the standard deviation and mean of the estimated attenuation factor for the proposed MKLT-TLS-ESPRIT approach are approximately 76.99% and 54.93%, respectively, of that obtained in case of data-SSI and 63.12% and 70.31% for KLT-TLS-ESPRIT with SNR 10 dB. From Table 2, it is analyzed that the MKLT-TLS-ESPRIT method approximated that the standard deviation and mean for the estimated frequency corresponding to interarea oscillating modes are 59.68% and 99.4%, respectively, of that obtained in case of data-SSI and 55.02% and 99.67% for KLT-TLS-ESPRIT with SNR 30 dB. It is also found that the standard deviation and mean of the estimated attenuation factor for the MKLT-TLS-ESPRIT approach are approximately 56.6% and 88.53%, respectively, of that obtained in case of data-SSI and 36.31% and 78.03% for KLT-TLS-ESPRIT with SNR 30 dB.

Test Signal Corresponding to Local Area Mode.
Simulation is conducted with a test signal having frequency � 1.5 Hz, maginitude � 1, and attenuationfactor � −0.1 Figure 4 shows the efect of HF to test signal, i.e., localarea mode to minimize the outliers of the signal and fnal fltered signal further carried out for the process of frequency estimations. Figure 4(a) shows the signal with 10 dB, and Figure 4(b) shows the signal with 30 dB noise afected test signal and fnally fltered out with less variance using HF.
A statistical comparison of the MKLT-TLS-ESPRIT, QR-TLS-ESPRIT, KLT-TLS-ESPRIT, RMP, and data-SSI for SNRs 10 dB and 30 dB is mentioned as follows.
Te frequency and attenuation factor distribution of a test signal are shown in Figure 5. From Table 3, it is analyzed that the MKLT-TLS-ESPRIT method approximated the standard deviation and mean for the estimated frequency corresponding to interarea oscillating modes are 46.29% and 99.81%, respectively, of the corresponding values obtained with data-SSI and 71.22% and 99.89% with KLT-TLS-ESPRIT for SNR 10 dB. It is also found that the standard deviation and mean of the estimated attenuation factor for the MKLT-TLS-ESPRIT approach are approximately 48.9% and 79.52%, respectively, of the obtained in case of data-SSI and 76.64% and 78.02% with ESPRIR + PM for SNR 10 dB. From Table 4, it is analyzed that the MKLT-TLS-ESPRIT method approximated that the standard deviation and mean for the estimated frequency corresponding to interarea oscillating modes are 61.21% and 99.80%, respectively, of the corresponding values obtained by applying data-SSI and 97.38% and 99.91% with KLT-TLS-ESPRIT for (1) Initially, signal vector R(n) from (3) is used to form a correlation matrix R R ∈ R N×N .
(2) Defnes MKLT of R(n) for a specifed data matrix Y as C i � Ψ T i Y (3) Signal and noise subspaces can be obtained by disintegrating the Ψ i i.e. Ψ i � Ψ s | Ψ n , where Ψ s represents signal subspace.
(5) Tese could be mapped as: V 1 � Ψ 1 T and V 2 � Ψ 2 T. V 1 and V 2 are related as: Te correlation between both the subspaces provided by Algorithm 2 ΨT � TΦ or Ψ � TΦT − 1 Te Ψ eigenvalues are diagonal elements of Φ, where ϕ k k�1,2,3...L . T columns represent the eigenvectors of Ψ. (7) Terefore, the frequency is given by f k � ∠ϕ k /2π where ∠ϕ k is the phase of Φ k .  Figure 6 and Table 5 Figure 4: Test signals for local-area oscillating modes in power system using SNR 10 dB and 30 dB: (a) a test signal corresponds to local-area mode and (b) a test signal corresponds to local-area mode.   Journal of Electrical and Computer Engineering computational time used for IAO modes estimation. It is observed that for 2000 Monte Carlo simulation, the mean computational time taken by the MKLT-TLS-ESPRIT is 0.00501 sec for SNR � 30 dB and 0.00507 sec for SNR � 10 dB, whereas for the data-SSI is 0.0064 sec for SNR � 30 dB and 0.0067 sec for SNR � 10 dB. All tests are performed using Intel(R) Core(TM) i7-4790S CPU @ 3.20 GHz processor with 8 Gb RAM.

Variance Response to Diferent SNRs Using Proposed MKLT-TLS-ESPRIT, KLT-TLS-ESPRIT, RMP, and Data-SSI.
Due to the accurate estimation of the signal frequency, variances of the data-SSI and MKLT-TLS-ESPRIT estimates are somewhat less than the ones for the two different estimates. Figure 7 shows that, for low SNRs, this process well predictable, where the variation by the estimation accuracy is less as compared to the variance by the noise.

LFO Modes Estimation for IEEE Benchmark
Two-Area Grid System Using Proposed Algorithm Figure 8 illustrates a test system [22] refers to IEEE benchmark two-area grid system. It includes two diferent areas, each are having two synchronous machines connected through eleven buses, via bus 7 and bus 9 . Tere are three loads applied to the bus 7 , bus 8 , and bus 9 of the test system. Tere are also two shunt condensers mounted at bus 7 and bus 9 . Te test system operates with fundamental frequency of 60 Hz.
Case 1. IAO mode estimation using data from IEEE benchmark two-area grid system for tie-line shortcircuit fault. Employing the SSSA, the IAO modes of the power system are analyzed. For a period of 1.2 seconds, starting at        Journal of Electrical and Computer Engineering 1 second, a disturbance is provided by applying a short-circuit fault of three phases at tie-line close to bus 8 . Figure 8 shows the real power fow between lines 10 and 9 provided by the PMU located at bus 9 . Te statistical SD, mean, and variance of the IAO modes extracted using Monte Carlo simulation for 50,000 runs are shown in Table 6. Te SD calculated to express the statistical diference of the estimated attenuation factor and frequency from the actual SSSA using MKLT-TLS-ESPRIT by adding a noise of SNR 30 dB is given as 4.155 × 10 − 3 and 3.4867 × 10 − 4 . Te statistical mean extracted by proposed MKLT-TLS-ESPRIT technique related to IAO mode, i.e., 0.6328 Hz, is nearly same that of the obtained in case of SSSA.

LFO Modes Estimation for IEEE Sixteen-
Machine, Sixty-Eight Bus Test System Using Proposed Algorithm Figure 9 shows the IEEE benchmark sixteen-machine sixtyeight bus grid system is considered to verify the efcacy of the proposed approach. Tis system's detailed parameter is listed within [23].
Te statistical behavior of MKLT-TLS-ESPRIT is described in Table 7 using 2000 Monte Carlo runs, i.e., the SD of the estimated frequency and AF is given as (Freq.Std.) and (Attn.fact.-Std), the mean of the Frequency and the AF is given as (Freq mean), and the mean of the frequency and the AF is given as (F (Attn.factor.-mean). Te IAO mode Table 6: Statistical analysis of the IAOs for IEEE benchmark two-area grid system with three-phase line short-circuit fault using RMP,  QR-TLS-ESPRIT, KLT-TLS-ESPRIT, data-SSI, and MKLT-TLS-ESPRIT

Conclusion
In this paper, we proposed a new interarea mode estimation approach based on subspace decomposition, to estimate the power system's IAO modes. Te proposed estimation scheme of the IAOs is found to be more robust and also provides a precise mode estimation when the estimates are slightly afected by the highly correlated ACGN. It has also been examined that the MKLT-TLS-ESPRIT has improved estimation efciency in reference to the QR-TLS-ESPRIT, RMP, KLT-TLS-ESPRIT, and data-SSI in aspects of standard deviation, variance, and the mean of the estimated IAO modes for given frequency and attenuation factor. Moreover, on IEEE benchmark two-area grid system and IEEE sixteen-machine sixty-eight bus system, the estimated performance of the proposed technique is evaluated. Te performance of the MKLT-TLS-ESPRIT at SNR 30 dB with a case of short-circuit failure in three-phase line is analyzed. Te observations made using the MKLT-TLS-ESPRIT technique is much similar to the estimated results obtained with SSSA.

Data Availability
Te data supporting the current study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.