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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 1, FEBRUARY 2011 323

Robust Detection and Analysis of Power SystemOscillations Using the Teager-Kaiser Energy Operator

Innocent Kamwa, Fellow, IEEE, Ashok Kumar Pradhan, Senior Member, IEEE, and Geza Jos, Fellow, IEEE

AbstractCritical to real-time oscillations monitoring is earlydetection when otherwise dormant natural modes become aserious threat to grid stability. The next urgent issue is to de-termine the frequency and damping of the problematic modeswhen the signal is embedded in noise and the system containsclosely spaced natural modes. The present paper addresses thedetection issue using the Teager-Kaiser energy operator (TKEO)which has shown to be a fast predictor of the instability onsettime when applied to the output signals of an orthogonal filterbank. In the system stability context, linear filter decomposition(LFD) is preferred rather than empirical mode decomposition(EMD), well known for its tendency to generate artificial modes

with no physical meaning. A narrowband LFD with a less than0.2-Hz bandwidth is achieved in the range 0.05 to 3 Hz througha cosine-modulated filter bank design. The effectiveness of thescheme in accurately detecting and tracking the frequency anddamping of oscillatory modes is demonstrated using Monte Carlosimulations of three closely spaced modes and a detailed analysisof an actual event recorded by Hydro-Qubecs WAMS in 2006.

Index TermsFilter bank, interarea oscillation, power systemidentification, power system monitoring, power system oscil-lations, power system stability, Teager-Kaiser energy (TKE),wide-area measurement systems (WAMS).

I. INTRODUCTION

OSCILLATION monitoring [1][5] is a topic of increasingrecent attention, owing to a heightened interest in the

prevention of widespread blackouts, which can result from un-controlled power swings across large geographical areas. Basi-cally, the entity responsible for this task should first determinefrom power system response signals when/if there is an oscil-lation issue and then quantify the underlying threat by meansof an accurate assessment of the frequency and damping of theoscillation. Since the early days of modal analysis of powersystem responses recorded by WAMS [6] or simulated usingpower system studies software [7], the two issues of detecting

and quantifying oscillations have been generally mixed together

Manuscript receivedOctober18,2009; revised December30,2009. Firstpub-lished April 26, 2010; current version published January 21, 2011. Paper no.TPWRS-00823-2009.

I. Kamwa is with Hydro-Qubec/IREQ, Power System and Mathematics,Varennes, QC J3X 1S1, Canada (e-mail:[email protected]).

A. K. Pradhan is with the Department of Electrical Engineering, Indian In-stitute of Technology, Kharagpur 721302, India (e-mail: [email protected])

G. Jos is with the Department of Electrical and Computer Engi-neering, McGill University, Montreal, QC H3A 2A7, Canada (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRS.2010.2046503

with few exceptions [8][10], but detailed analysis obviouslymakes little sense when there is no significant oscillation ac-tivity in the electromechanical frequency range (typically 0.13Hz).

In this paper, a Teager-Kaiser energy operator (TKEO)[11][14] based criterion is proposed as a predictor of poweroscillation problems. To the best of our knowledge, this is thefirst time that the TKEO concept has been used in this context.Since the concept is known to perform poorly on multi-com-ponent signals [13], a multi-band pre-filter is first applied to

the raw input. This allows us to decompose the waveform intoa set of largely orthogonal monochromatic components whichare then subjected to time-frequency analysis using the energyseparation algorithm (ESA). One of the benefits of the linearfilter bank is that, when the prototype is designed with a verynarrow bandwidth, e.g., 0.2 Hz, any analysis method applied tothe output will provide noise-resilient frequency and dampingestimates.

Recent authors have investigated closely related ideas usingthe EMD as filter bank [15] and discrete Hilbert transform(DHT) as the basis for instantaneous frequency and ampli-tude estimation [16], [17]. However, in spite of state-of-art

improvements, even the best implementation of EMD tends togenerate artificial modes due to the frequency-mixing phenom-enon [18][20]. This is especially true at low sampling rates,which are typical of todays WAMS, whose communicationbandwidths are constrained [21].

Although the ESA and Hilbert transform are the shortestpaths to amplitude and frequency information [14], [17], theydo not provide damping information without additional pro-cessing. Furthermore, they are still prone to errors when thesignal is not perfectly monochromatic, which is hard to achievefor systems with closely spaced natural modes. Applicationof parametric methods to the filter bank outputs will, in thiscase, result in a better frequency resolution while providing

damping information [22]. For this reason, the eigensystemrealization algorithm (so-called ERA method) proposed in [23]will be adopted in this research. In addition to its intrinsicallymulti-signal capability, it performs very well for ringdown andstationary ambient noise signals when the SNR ranges frommoderate to high.

The main innovation of the present paper resides in the ap-plication of TKEO to the output signals of a linear filter bankfor robust location in time and frequency of the onset of powersystem oscillations. Locating the oscillation frequency can beuseful for control and identification methods that require a goodinitial value of the natural mode within a certain a priori fre-

quencyrange. It is also a good catastrophe-tracking predictor for0885-8950/$26.00 2010 IEEE

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324 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 26, NO. 1, FEBRUARY 2011

the control room, especially when supplemented with an accu-rate assessment of the underlying modal frequency and dampingusing a high-resolution parametric method like ERA. The pro-posed multi-band filter consists of narrow band-pass cells witha center frequency uniformly distributed from 0.1 to 3 Hz. De-signed using cosine modulation [24] of Taylor spectral win-

dows [25], the linear bandpass prototype FIR-filter possessesattractive attributes like steep stop band (60 dB), fast-decayingside-lobes, and DC rejection. The DC attenuation can be furtherimproved using a 100-sample FIR DHT [1].

The paper is organized as follows. In section two, theessential ideas of the proposed online modal analysis aresummarized. Then in Section III, the Teager-Kaisers energyoperator is reviewed. Its tracking and selectiveness capabilitiesare demonstrated on synthetic multi-sine waveforms signalsincluding a 0.025 Hz/s chirp signal. Section IV is devotedto the design and assessment of a narrow-band linear filterbank for facilitating the application of the TKEO to closelyspaced natural modes. Next, the ERA-based multi-band modal

analysis (MBMA) is developed in Section V for providingfull modal amplitude, frequency, and damping informationin tracking mode. Monte Carlo simulations of a parametricsignal with closely spaced modes [26] demonstrate the higherreliability of the LFD TKEO based modal-analysis approach,compared to a conventional parametric method. Finally inSection VII, the analysis of an actual unstable event recordedby the Hydro-Qubec WAMS [27] in 2006 confirms thatthe MBMA scheme can detect real-life oscillation problems,quantify their severity, and report the results every 5- to 40-sdata blocks, depending on the frequency band with significantoscillation activity, which is much faster that the 120 s required

by typical sliding Fourier or autocorrelation based block-pro-cessing methods [4], [28].

II. OUTLINE OF THE PROPOSED MODAL ANALYSIS METHOD

The power system response signals can generally be de-scribed as the sum of the combined amplitude (AM) andfrequency (FM) modulated primitive signals:

(1)

where , and are, respectively, the amplitude, fre-quency, damping, and phase of the th signal component. Thecorresponding complex signal is of the form

(2)

where is the Hilbert transform of . It should bestressed that all parameters , and can be slowly

time-varying without posing any significant complication tothe proposed algorithms.

Fig. 1. Application of ESA on multi-component AM-FM signals.

Fig. 1 illustrates the overall concept of first decomposing themulti-component power system signal (1) using a filter bank,

and then performing a time-frequency analysis on each channelcomponent using ESA or, alternatively, the Hilbert transform[16]. If the EMD is used for pre-processing, it yields intrinsicmode functions (IMF) [18], which are essentially single-com-ponent sinusoids if and only if the natural modes are well spacedon the frequency axis and the sampling rate is sufficiently high.However, most of the time, such is not the case and the EMDwill create IMFs that contain artificial modes due to the mixingphenomenon [18]. To avoid the EMD pitfalls for closely spacedmodes, the Gabor filter bank could be used, as in [13] and [29],but it is not sharp enough for the present application. We willdesign in Section IV a narrow-band linear filter bank which isfar more suited for power system oscillations signals at the rel-atively low sampling rates of todays WAMS [21].

When accurate damping information is required or the filterbank output signal is not monochromatic, a more detailedmodal analysis, using a parametric method for instance, shouldbe preferred to the ESA. The overall scheme is shown inFig. 2, assuming that ERA [7], [23] is the modal analysis tool,even though any alternative modal estimation method couldequally well be used [4], [22], [30]. An optional DHT [1] isfirst applied to the incoming signal to reject quasi-steady statecomponents more efficiently. The filter bank then splits thepossibly multi-component signal as in (1) into N essentiallyorthogonal components (with in the present paper).

After a linear filter bank decomposition of the incomingsignal, each channels energy is computed using TKEO and the

(typically ) energy-dominant signals are selectedfor parametric modal analysis based on an energy thresholdtest. The modal analysis is then performed on sliding non-over-lapping data blocks whose size can be adjusted per channel,according to the channel center frequency. For instance, a 5-sdata window suffices for filters centered in the range 1 to 3 Hzwhile a 20-s data window could be preferable for analyzingoutput signals of filters in the range 0.1 to 1 Hz.

III. OVERVIEW OF THE TEAGER-KAISERS ENERGY OPERATOR

Teager [11] developed a nonlinear signal operator which wassubsequently shown by Kaiser [12] to be highly effective for

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KAMWA et al.: ROBUST DETECTION AND ANALYSIS OF POWER SYSTEM OSCILLATIONS 325

Fig. 2. Overview of the MBMA scheme with a TKEO-based criterion to detectchannels with the strongest oscillatory activity.

assessing the energy and detecting AM and FM signals in theform (1)

(3)

with, and the first and second derivativesof , while is the instantaneous energy of . In fact, de-noting by the th component of the AM-FM signal in (1)and assuming a constant initial phase , it is proven that [12],[13]

(4)

with a negligible approximation error under quite general andrealistic conditions. This motivated the energy separation algo-rithm (ESA) for tracking the instantaneous amplitude and fre-quency as follows [13], [14]:

(5)

Note that, in the case of a monochromatic signalwith constantand , the relationships (5) are exact. When the continuous

energy operator (3) is sampled with a sampling period T, thefollowing equivalent discrete energy operator is obtained [13]:

(6)

It results in

(7)

Fig. 3. Illustration of the TKEO and ESA concepts on simple signals.

with . By applying to both and itsbackward difference, , the discrete-time ESA was developed in [13], [14]

(8)

The formulation shows that ESA can provide the instanta-neous amplitude and frequency with a single sample delay insharp contrast with the DHT [14], [16], [17] and other block-processing schemes. However, involvement of the signal deriva-tive in (8) also points to a potential sensitivity to noise. Fig. 3illustrates the nearly perfect performance of the ESA on a unitmagnitude, monochromatic damped, and un-damped sinusoidsof the same 1-Hz frequency:

(9)

where , for a pure sine; , for a constantdamping sine and , for a sine with a linearlyvarying damping. The discrete energy in Fig. 3 is divided by Tto match the same energy scale of the continuous operator.

It is clear from (4)(8) that the ESA is defined specifically fora single-component signal. Therefore, its proper application re-quires a pre-processor to decompose any incoming multi-com-ponent signal into monochromatic components using a filterbank, as illustrated in Fig. 1 [29].

IV. LINEAR MULTI-BAND SIGNAL DECOMPOSITION

In order to design a realistic narrow-band filter bank bettersuited than EMD for decomposing power systems signals in

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the form (1) with closely spaced modes, let us consider the fol-lowing definition, which comes from the cosine-modulated filterbank theory [24]:

(10)

where and is the im-pulse response coefficients of a linear-phase low-pass FIR pro-totype filter. Furthermore, the number of filter channels is se-lected so that is the length of the prototype filter. If properlychosen, the scaling factor and the center frequency of the thfilter can be located at a pre-specified frequency. The final im-pulse response data of the th band-pass filter, after a re-scalingfor a unit magnitude and zero phase at the center frequency, is

(11)

where is the response of the th filter cell atits center frequency. The filter bank gain is shown in Fig. 4(a).The corresponding low-pass prototype is based on the Taylorwindow as defined in [25]

(12)

with the filter coefficients being

Given a 40-Hz sampling rate, the prototype window has a400-sample length for filters 1 to 4 and 200 samples for fil-ters 5 to 9. These numbers were selected as a trade-off between

filter delay and narrow-band behavior. It should be noted thatthe component filters are essentially orthogonal as they overlapat their -dB attenuation frequency. For instance, the -dBcrossing frequencies of the third filter are 0.4 Hz and 0.6 Hz onthe low and high sides, respectively. To improve the DC rejec-tion of the filter bank, especially that of the first filter, a DHTpre-filter can be applied to the signal, as in [1]. The effect ofa 100-sample FIR-based DHT designed in Matlab is shown inFig. 4(b), where it is seen that the attenuation at 0.01 Hz has in-creased from 20 dB to 40 dB.

Fig. 5 shows side-by-side the performance of EMD and LFDon the following signal:

(13)

Fig. 4. Example of cosine-modulated filter bank designed from Taylorwindow-based low-pass prototypes. (a) Taylor window based filter bank. (b)

Filter bank with DHT pre-filter.

Fig. 5. Comparison of EMD and LFD on a benchmark signal consisting of twoclosely-spaced modes (13). (a) EMD. (b) Filter bank.

studied in [16] to demonstrate and mitigate the EMD fre-quency mixing effects. Our EMD results were obtained usingstate-of-art software [19], but the masking technique of [16]was not implemented because we were looking for decompo-sition methods with no a priori knowledge about the signal.For both EMD and LFD, only the three dominant componentsare shown with a 40-Hz sampling rate. The periodogram of theLFD signals correctly predicts two components at 0.5 and 0.8Hz while IMF #6 contains four modes, including two modesnot found in the original signal. In addition, it is not monochro-matic, which will diminish the ESA and DHT performances.IMF #4 is monochromatic but its frequency is artificial. All

these artifacts are avoided by the LFD, thanks to its linear andnarrow-band behavior.

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To further demonstrate the tracking capability of the filterbank, let us consider the following unit magnitude chirp signal:

(14)

with

(15)

The input signal and results of its multi-band analysis areshown in Fig. 6(a) and (b), respectively. Each channel signalis represented by a different color. It appears that, during thefrequency sweep, the active filter channel changes accordingto the instantaneous chirp frequency at the given time and thischange is reflected in the energy operator. However, during achannel transition, there is a frequency change from 0 to thesweep signal value. The criterion for detecting the coincidenceof the sweep and channel frequencies is based on the TKEO atthat time, with the detection threshold defined as follows:

(16)

V. TKEO-BASED MULTI-BAND MODAL ANALYSIS

A. ERA-Based Regular Modal Analysis

After a linear filter bank decomposition of the incomingsignal, each channels energy is computed as in (6) and the(typically ) energy-dominant signals are selected forparametric modal analysis based on an energy threshold testsuch as (16). The ERA based modal analysis [7] is then per-formed on data blocks whose size can be adjusted per channel,according to the channel center frequency. Typically, a datawindow of two to four cycles of the center frequency sufficesfor analyzing the corresponding filter output signal.

The ERA method at the core of our modal analysis approachbasically assumed that the digital signal samples are the impulseresponses of a hypothetical proper and linear system, whose dis-crete state-space representation (F, G, C) is determined through

minimal realization of the impulse responses as in [23]. Animplementation of ERA method adapted to single-input mul-tiple-output (SIMO) signals is summarized in the Appendix.Once the continuous time-domain state matrices (A,B,C) are de-rived from (F,G,C), using for instance, the inverse Tustin trans-form, the Prony decompositions of the output signals are ob-tained as follows.

1) Transform (A,B,C) into its diagonal form assuming for pre-sentation simplicity, distinct complex eigenvalues:

(17)

with ,the diagonal matrix of the eigenvalues of A; and

Fig. 6. Multi-band analysis of the chirp signal in (14): each color correspondsto the signal output of a corresponding band-pass filter. Amplitude and fre-quency are computed with ESA. (a) 0.025 Hz/s chirp signal. (b) Multi-bandanalysis.

, the corresponding left-eigenvectormatrix.

2) Compute the vectors of residues, ,for each natural mode

(18)

where designates the th output signal.3) Given the residues , the Prony parameters are defined

for each output signal , according to thefollowing sum of damped sinusoids expression:

(19)

B. Demonstration on Multi-Modes Synthetic Signal

The multi-band modal analysis (MBMA) combining TKEO-based detection with ERA-based parameter estimation will betested first on a noise-free multi-component signal with severalclosely spaced natural modes. The signal is the result of the

Prony analysis of a measured power system response initiallyreported in [6] and later used in [7] as a typical case for assessing

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Fig. 7. Standard ERA-based modal analysis and validation results of the syn-thetic signal in (20).

TABLE IMALIN-ROUND MOUNTAIN MODIFIED NATURAL RESPONSE

LINE CURRENT TO HVDC MODULATION [6], [7]

modal analysis methods. The parameters shown in Table I con-tribute to the signal as follows:

(20)

with obvious relationships between the modal parameters in(19) and (20), giving a single output signal .The results of a conventional ERA analysis of this signal (20)are presented in Fig. 7.

The sampling frequency is 40 Hz, with an assumed systemorder . After Prony decomposition of the state-spacemodel as in (19), a filtering is performed to reject as non-rele-vant all modes with a damping higher than 30% and amplitudelower than 0.1% of the maximum modal amplitude. Under theseassumptions, conventional ERA-based modal analysis is quiteaccurate in spite of the many closely spaced modes present inthe signal (20).

The time and energy signals from the four dominant channelsin the filter bank are shown in Fig. 8. From time-domain plots,

Fig. 8. Multi-band decomposition of the synthetic signal in (20).

Fig. 9. Frequency responses of the ERA models associated with the syntheticsignal in (20). The INPUT plot is the result without LFD.

the red and black signals are not perfectly monochromatic de-spite the narrow bandwidth of the filters. The energy informa-tion points to the fact that a large magnitude in the time do-main does not necessarily translate into large energy, which is

a product of frequency and magnitude. ERA-based modal anal-ysis of the four dominant filter output signals was performedunder conditions similar to those in Fig. 7, but assuming a re-duced model order of 6.

The four frequency responses obtained with these modelsare shown in Fig. 9 along with the overall frequency responsethat resulted from the conventional full-band ERA application(Fig. 7). Again, the red and black signals highlight two peaks,meaning that ESA and DHT cannot accurately extract their am-plitude and frequency. In addition, the full-band ERA yields fivemagnitude peaks, the same number as the MBMA with fourdominant channels but the peaks from the latter are sharper,leading to more accurate frequency location.

Finally, Fig. 10 portrays the damping and frequency estima-tion errors incurred by the multi-band modal analysis. The fre-

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Fig. 10. MBMA frequency and damping errors on the synthetic signal in (20).

quency bias is minor but the error on damping is more signifi-cant when the filter channel used to extract the underlying fre-quency is not monochromatic (i.e., 0.65 Hz and 0.82 Hz). Thisaspect could be further improved by doubling the number of fil-ters in the bank, as this would greatly improve the narrow-band

behavior of the filter channels.C. Monte Carlo Study of MBMA Performance

One claimed attribute of the proposed scheme is its noise-re-siliency, which will now be assessed by the usual means ofMonte Carlo studies. The synthetic signal chosen for this pur-pose is from [26] and is described as follows:

(21)

where

(22)

The above signal was sampled at 40 Hz and white noise wasadded so as to achieve a specified signal-to-noise ratio (SNR)from 20 dB to 100 dB. At each SNR level, 4000 trials weregenerated. Modal analysis of each noisy record was then per-formed using the conventional ERA and the new scheme com-bining multi-band preprocessing, TKEO-based oscillation de-

tection, and ERA. The selected post-filtering criteria and as-sumed model order were the same for both approaches.The first performance metric is the ratio of the number of casesfor which the algorithm obtained exactly three modes (after fil-tering) over the number of trials (4000). When the algorithm isunable to fit the actual number of modes (3), it is assumed tohave failed in that case, even if this is not always true, but sincethe same ERAs implementation is applied with and withoutthe multi-band pre-processing, this assumption allows for a faircomparison.

According to Fig. 11, both methods are sensitive to noise butthe MBMA is more robust. At a 60-dB SNR, it is perfectly reli-able while the conventional ERA shows a 75% reliability level

only. Even at a 100-dB SNR, the latter is still unable to reach a100% reliability target.

Fig. 11. Comparative reliability of the conventional and multi-band modalanalysis of the synthetic signal in (21). There are 4000 trials for each SNRvalue.

Considering only the cases for which the modal analysis wassuccessful (i.e., exactly three modes were obtained after filtering

non-relevant modes), Fig. 12 depicts the dispersion of the es-timated parameters around their mean values, giving a 20-dBSNR. The three plots on the left (12-a-c) present the frequencyagainst damping for the three modes, while the three plots onthe right express (12-d-f) the frequency against the magnitude.It should be noted that the amplitude estimate from the MBMAis compensated according to the frequency response gain of thecorresponding channel filter as follows:

(23)

where is the frequency response magni-tude of the th filter at the natural frequency , and is theERA amplitude estimate based on the th signal. All plots inFig. 12 show a more compact error distribution in the MBMAcase compared with the conventional ERA method, especiallyfor the two lowest frequency modes. In addition, given thehigher reliability index of the former, Fig. 12 contains 2512over 4000 runs which have converged for the multi-band sce-nario in contrast to only 1714 convergent points in the standardERA scenario.

VI. ANALYSIS OF AN ACTUAL EVENT

The field record considered here describes the dynamicprocess preceding and following the trip of a single 350-MWgenerating unit at a 16-unit power plant in the Hydro-Qubecsystem. The event was recorded by the WAMS [27] for about1-min duration on March 11, 2006, starting at 16 h 49 min 20 slocal time. The signals shown in Fig. 13 are very remote fromthe unit in trouble, which explains why the magnitude of theunstable voltage and angle swings is not significant. Selectingthe angle shift signal as target for the MBMA, Fig. 14 showsthe subset of filter-bank output signals to be used in assessingthe event. These are the dominant-energy signals accordingto the TKEO criterion. The energy of all other angle-drivenchannel signals is less than for the entire observation

period. By contrast, the energy of the filters outputs #5 and #6starts to rise immediately at the onset of the oscillation. The

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Fig. 12. Conventional versus multi-band analysis results of the syntheticsignalin (21) at a 20-dB SNR.

Fig. 13. PMU signals on March 11, 2006: single unit trip at a remote powerplant. Response variables from the Churchill Falls PMU, with angle-shift refer-enced near Montreal [27].

ESA is then used for quick visual screening of the amplitude

and frequency of the oscillation.The channel #6 signal was fed to the MBMA tool. A separateanalysis was then performed on a contiguous, non-overlapping6-s buffer of data, for a total of ten snapshots over the approx-imately 60-s period of the record. Fig. 15 shows the frequencyand damping obtained in tracking mode using the ERA. The factthat something is going wrong on the system is clearly identi-fied in the first snapshot by thenegative damping associatedwiththe dominant mode of the channel analyzed. The local mode as-sociated with the unit in trouble has a frequency of about 1.25Hz (same as the value obtained with ESA) and its damping was

%. However, all PMU signals are needed in order to pin-point more precisely the area in which this unit is located. After

growing quickly at first, the oscillation amplitude stabilizes be-tween 10 and 40 s. Around this time, the unit in trouble tripped

Fig. 14. Multi-band analysis of the angle shift shown in Fig. 13. Amplitudeand frequency computed with ESA.

Fig. 15. Tracking of the dominant mode based on a 6-s, frame-by-frame anal-ysis of channel # 5 and 6 signals.

and the damping of the plant local mode was restored immedi-ately to more acceptable values. Throughout this event, the fre-quency of the mode never changed (ERA and ESA concludedthe same), except that the damping moved from negative to pos-itive.

VII. CONCLUSION

In this paper, the TKEO is introduced for the first time as atool for early detection of power system oscillations. In con-trast to other parametric and high-pass filtering-based detec-

tion methods, TKEO makes no assumption about the signal. Itis intrinsically a fast non-parametric detection method with asmooth behavior when applied to the output signals of a linearnarrow-band filter bank. It is found that, although EMD is notappropriate for this task, a bank of cosine-modulated orthog-onal filters appears to be very effective in decomposing the in-coming multi-component power system response in terms of itsmonochromatic components while rejecting both random noiseand DC components. The ESA is used to obtain an initial esti-mate of the frequency and amplitude of the filtered signal oncean oscillation problem is detected.This canbe sufficient in manycircumstances, but it is shown that the ERA provides noise-re-silient damping of the modes over short time frames.

Several multiple-component synthetic signals available fromthe literature areanalyzed in both deterministic andMonte Carlo

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setups to confirm the superiority of multi-band modal analysisover the conventional approach using the same ERA implemen-tation. Analysis of an actual power system event also highlightsthe efficacy of the new detection and MBMA framework inreal-life situations. Overall, the results presented in the paperallow us to believe that the proposed approach could be useful in

analyzing ambient noise, as well as ringdown signals recordedby WAMS or simulated using power system stability analysissoftware.

VIII. APPENDIX: THE EIGENSYSTEMREALIZATION ALGORITHM

The purpose of ERA is to determine numerically the matrices(A, B, C) or (F, G, C) in the following equivalent continuous anddiscrete state-space representations of a single input, -outputlinear system:

(24)

Their sampled impulse response is given by a sequence ofmatrices such as:

(25)

In these expressions, and are, respectively, the state,output, and input vectors of dimensions and 1, while

are the sampling instants. ForMBMA pur-

poses, we can consider that represents a block-data outputfrom the filter bank channel over a time window of dura-tion . The modes found in the signal are exactly theeigenvalues of the state matrix A.

A. Top View of ERA

Fig. 16 recaps the ERA for state-space system identification,which similarly to the N4SID method [6], is based on the con-struction of two Hankel matrices, and .

B. Identification Parameters

sampling period T;

p measured signals, concatenated in ; dimension n of the state representation (A,B,C); dimension of the Hankel matrices and , which in

our case are square;

Fig. 16. Description of the impulse response based ERA.

distribution parameters and of the matrices and, which allow us to change the time span of the window

to be used in the identification procedure.

C. Construction of the Hankel Matrices and

The elements of the Hankel matrices are measurementsthat have been organized in terms of the selected identificationparameters. For a given set of parameters, construction of thetwo matrices is almost identical, the only difference being theposition of is offset by one sample compared to . Forexample, if the first sample used for calculating is ,then the first sample used for will be . For

and with the notations given, the Hankel matrices andare obtained as shown in (26) at the bottom of the page, wherethe suffix is dropped from in order to fit the Hankel matrixmore conveniently in a single column.

D. Remarks

The matrices depend on the parameters T, , and.

If , then and are symmetrical. If , adjacent data are used. The parameters and must be chosen so that

, where represents the number

of samples. For a value of , the parameters and allow the width

of the observation window equal to to beincreased. So, if , and ,

......

......

(26)

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this means that the identification has been done using thefirst 200 samples. If, on the other hand, ,and , this means that one sample out of twois used by the identification procedure and that the lastsample used is the 400th. In both cases, the identificationprocedure uses 200 samples, the difference being that in

the first case, it is the first 200 samples, whereas in thesecond, it considers one sample out of two up to the 400th.

E. State Representation Computation

The discrete state, input and output matrices F, G, and C, re-spectively, of the discrete system are computed using matrices

and . Decomposition of into singular values yields thediagonal matrix S, which contains the singular values of ar-rangedin descending order, andthematrices U andV containingthe left and right singular vectors, respectively. The identifica-tion parameter n is therefore used to select the n highest singularvalues. Thus, we obtain the matrix containing the n highestsingular values and the matrices of the associated singular vec-

tors and from which the matrices (F,G,C) of the staterepresentation of the discrete system are derived as follows:

(27)

Once the matrices F, G, and C have been computed, the contin-uous system (A, B, C) is obtained using the discrete to contin-uous transformation (i.e., the inverse Tustin transform).

ACKNOWLEDGMENT

The authors would like to thank F. Levesque for fruitful dis-cussions on tracking modal analysis and to G. Blais and J. B-land for retrieving and analyzing field data from the Hydro-Qubec WAMS.

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Innocent Kamwa (S83M88SM98F05)received the Ph.D. degree in electrical engineeringfrom Laval University, Laval, QC, Canada, in 1988.

Since then, he has been with the Hydro-QubecResearch Institute (IREQ), Power System Analysis,Operation, and Control, Varennes, QC, where he iscurrently a Principal Researcher in bulk system dy-namic performance. He has been an Associate Pro-

fessor of electrical engineering at Laval Universitysince 1990.Dr. Kamwa has been active for the last 13 years on

the IEEE Electric Machinery Committee, where he is presently the StandardsCoordinator. A member of CIGR and a registered professional engineer, heis a recipient of the 1998, 2003, and 2009 IEEE Power Engineering SocietyPrize Paper Awards and is currently serving on the Adcom of the IEEE SystemDynamic Performance Committee.

Ashok Kumar Pradhan (M94SM09) receivedthe Ph.D. degree in electrical engineering fromSambalpur University, Burla, India, in 2001.

He has been with the Department of Electrical En-gineering, Indian Instituteof Technology, Kharagpur,India, since 2002. Presently, he is an Associate Pro-fessor. His research interests include power systemdynamics and Relaying.

Geza Jos (M82SM89F06) receivedthe M.Eng.and Ph.D.degrees from McGill University,Montreal,QC, Canada.

He has been a Professor with McGill Universitysince 2001 and holds a Canada Research Chair inPower Electronics applied to Power Systems. He isinvolved in fundamental and applied research relatedto the application of high-power electronics to

power conversion, including distributed generationand wind energy, and to power systems. He waspreviously with ABB, the Ecole de technologie

suprieure, and Concordia University. He is involved on a regular basis inconsulting activities in power electronics and power systems.

Dr. Jos is active in a number of IEEE Industry Applications Society com-mittees, IEEE Power Engineering Society working groups, and CIGRE workinggroups. He is a Fellow of the Canadian Academy of Engineering and of the En-gineering Institute of Canada.

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