IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004 505

Performance of Demodulation-Based FrequencyMeasurement Algorithms Used in Typical PMUs

Innocent Kamwa, Senior Member, IEEE, Michel Leclerc, and Danielle McNabb

Abstract—This paper presents a method for evaluating theperformance of demodulation-based frequency measurementalgorithms in the presence of additive interfering sinusoids. Deter-mination of the performance of amplitude measurement schemesunder such conditions is straightforward once the frequencyresponses of the filters involved in the process are known, sincethe error induced by a single interfering tone is easily computedusing the cascade algorithm’s frequency response magnitude. Thispaper presents a similar method for predicting the worst errorof frequency measurement schemes with respect to sinusoidalinterference. Once acquainted with the proposed error predictionformula, the only difficulty in designing effective frequencymeasurement algorithms is the appropriate selection of outputfilters to achieve the specified performance. The method has beenused successfully in designing frequency measurement algorithmscurrently used in Hydro-Québec’s special protection schemes.

Index Terms—Demodulation, frequency measurement, phasormeasurement unit (PMU), recursive discrete Fourier transform,spectral windows, wide-area monitoring.

I. INTRODUCTION

RECENTLY, heavy use has been made of phasor measure-ment unit (PMU) technology in wide-area experiments de-

signed to capture time-tagged normal or forced power systemdynamic responses [1]–[9]. So far, these pioneering investiga-tions, mostly of a research nature, focused on dynamic perfor-mance assessment and information network requirements. Butthey are paving the way to even more sophisticated wide-areacontrol schemes such as those envisaged in [7]–[11], which arebound to remain purely speculative until PMU technology be-comes mature. Although one major acceptance criterion in thiscontext will be the reliability with which the PMU providesits own measurements to a peer-PMU or a data concentratorwithout interruption and in a timely manner [3], [11],1 anotherfundamental maturity issue is related to the PMU’s capabilityof measuring electrical quantities accurately with measurementlags compatible with closed-loop control and special protectionrequirements.

Voltage magnitude measurement algorithms are usually builton state-of-the-art classical filtering methods [12]. In such cases,

Manuscript received November 13, 2002.I. Kamwa is with Hydro-Québec/IREQ, Power System Analysis, Operation

and Control, Varennes, QC J3X 1S1, Canada (e-mail: kamwa@ireq.ca).M. Leclerc was with Cybectec, Inc., Charny, QC, Canada. He is now with

Exfo Inc., Vanier, QC, Canada.D. McNabb is with Hydro-Québec, TransÉnergie/System Studies and Perfor-

mance Criteria, Montréal, QC H5B 1H7, Canada.Digital Object Identifier 10.1109/TPWRD.2004.823185

1SEL-421 is a line protection relay with phasor measurement capabilitiesbased on an external GPS signal. See http://www.selinc.com/sel-421.htm.

knowing which filter has been used and its frequency responseis enough to determine the performances and behavior of the al-gorithm. With frequency measurement algorithms, the story isquite different, since the designer generally relies on time-con-suming simulations of the algorithm to estimate the amountof error induced by each interfering tone. Naturally, this im-plies first coding the frequency algorithm in a software devel-opment environment such as Matlab or Microsoft Visual C++and then performing a dozen simulation runs to draft the algo-rithm response to various additive interfering tones, giving sev-eral values of fundamental frequency.

In the last ten years, many new frequency algorithms basedon iterative error minimization schemes have been presented inthe literature [13], [14]. Generally, these approaches are able toprovide much better frequency accuracy by refining the estimateseveral times for each sample or window of samples, dependingon the application. Their main pitfall is a large computationalburden, which prevents them from implementation on low-costfixed-point digital signal processors such as the Motorola 56 000series. This issue is not only hypothetical, given that frequencyestimation is habitually just a small part of a much larger em-bedded software task, especially in a multichannel PMU that hasto deal simultaneously with, say, six lines, amounting to about20 phasors, in addition to peer-to-peer real-time communica-tions.

Therefore, in PMUs and generally in most digital relays,frequency is estimated using a more computationally efficientscheme based on demodulation [1], [15]–[18]. This is basicallya two-step process where the actual demodulation occursin the first step, in the form of a one-cycle discrete Fouriertransform (DFT) recursively applied to the incoming data, ona sample-by-sample basis. In the second step, the frequency isestimated by finite impulse response (FIR) filtering of the finitedifferences of the phasor angle. Overall, while amplitude accu-racy in such a procedure is dictated only by the DFT frequencyresponse, the frequency error is the result of a complex mixtureof the DFT response and FIR filter used in the second stage.Adding to the challenge, frequency measurements use non-linear mathematical functions such as the arctangent functionto extract the finite difference of the rotating phasor angle.

In this paper, an approximation of the frequency responseof demodulation-based frequency measurements is given. Thisapproximation is used to compute the error associated with asinusoidal interfering signal. This paper is organized as fol-lows: the first section presents the demodulation scheme of fre-quency measurement. The second presents an approximationof the measurement error under sinusoidal interference for thisclass of algorithms. The third and fourth sections provide several

0885-8977/04$20.00 © 2004 IEEE

506 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004

examples using synthetic and actual waveforms, highlightingthe reliability and usefulness of the proposed error approxima-tion formula.

II. DEMODULATION-BASED FREQUENCY

MEASUREMENT ALGORITHMS

Instantaneous frequency estimation is a conventionalsignal-processing problem [19], [20]. In one of the numerousapproaches proposed, the estimation is done by demodulatingthe signal near the fundamental frequency and then usingthe averaged signal phase derivative [15], [18]. Variants ofthis approach have also been studied for tracking changingharmonics in power systems [21]. It is presently used in thenewly commissioned Hydro-Quebec’s under-frequency loadshedding system, which is a major component of the systemdefense plan [22].

First, assume that the measured single-phase signal consistsof a fundamental and noise components (mixture of randomnoise, harmonics, subsynchronous, etc.)

Noise (1)

where refers to time. By definition, the instantaneous fre-quency is

(2)

where is the total phase of . In order to estimate thisquantity , the demodulation technique first brings the signalback to the baseband, using a filter of at least one period of thefundamental frequency

(3)

where is the real signal, is the -point FIR low-passfilter used to demodulate the signal is the demodulatedsignal (or the fundamental phasor associated with ), is thediscrete time index, and is the sampling time period. Denotingby the phase of the demodulated signal, the instantaneousfrequency deviation can be approximated in the discrete-timedomain as follows (replacing the continuous differential in (2)by a finite difference with respect to the discrete time index):

angle

(4a)

where is the frequency deviation and is the complex con-jugate of , with the angle of a complex number defined as

angle (4b)

Since the frequency deviation is inevitably corrupted by noiseas a result of the numerical derivative operation involved in(4), filtering its raw estimate is obviously mandatory whateverthe end use of the frequency measurement. Using an -point

FIR low-pass filter with impulse response , the filtered fre-quency deviation is obtained as follows:

(5)

Tuning this algorithm in practical application is straightfor-ward because at both stages, it relies on FIR filtering. Selectingthe coefficients and therefore completely defines thealgorithm performance. Fig. 1 illustrates some typical choicesof these filters. The most trivial choice for or is theaverage or so-called boxcar filter

(6)

which, when replaced in (3), yields

(7)

The transfer function associated with (7) is

(8)

where and . This is a clear indicationthat the FIR filtering process can be implemented recursively

(9)

which actually defines the well-known recursive discreteFourier transform [23], [24]. Owing to its low computationalburden, this recursive demodulation approach is widely usedin PMU and digital relays, although implementations vary indetail and flavor.

For instance, while (9) gives a nonstationary phasor, the re-cursive DFT introduced many moons ago by Phadke [7] typi-cally provides a stationary phasor, which slips according to thefrequency offset with respect to the fundamental.

As apparent in Fig. 1, while the DFT may be extremely fast,it does not provide good attenuation at high frequency, withside-lobes only 13-dB down the fundamental gain. However,harmonic rejection is perfect in the absence of frequency offset.For better high-frequency rejection, it is possible to use a two-cycle Blackman–Harris filter [25], which pushes the side-lobeswell below 45 dB at the expense of twice more lag. In addition,so-called flat-top windows [25] can help reduce the magnitudemeasurement errors around the fundamental, again at the costof greater measurement lag.

Regarding the second-stage filtering, with a view tosmoothing the raw frequency estimates, the early MacrodynePMU used a simple four-cycle boxcar type filter [3]–[5].However, as noticed by Hauer [4], [26], this is a rather ineffi-cient filter. Fig. 2 demonstrates that a good compromise filter

KAMWA et al.: DEMODULATION-BASED FREQUENCY MEASUREMENT ALGORITHMS IN TYPICAL PMUs 507

Fig. 1. Candidate FIR filters h [ ] for use in phasor demodulation. (a) Fundamental magnitude frequency response and (b) error on the fundamental amplitude.

Fig. 2. Candidate low-pass FIR filters h [ ] used in demodulation frequencymeasurement algorithms.

achieving an acceptable tradeoff between a long measurementtime and effective low-pass filtering is the so-called Kay filter[20], defined as follows:

(10)

Hence, for interferences located at 40 Hz (i.e., at 20 Hz froma 60-Hz fundamental), a Kay filter provides two to three timeslower side-lobes than a boxcar filter with a similar memorylength (four cycles in this case).

III. ERROR APPROXIMATION FOR A SINUSOIDAL INTERFERENCE

Let us assume the following sinusoidal signal with an additiveinterfering component superimposed:

(11)

where and are the frequency and amplitude of the fun-damental component while and are the amplitude and fre-quency of the interfering tone. is the frequency offset whichvanishes in nominal conditions. Given the following additionalnotations:

demodulation of through filter of the previous sectionwill produce the following output baseband signals:

(12a)

508 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004

for the in-phase and

(12b)

for the in-quadrature components, respectively. The corre-sponding rotating phasor therefore takes the form

(12c)

To simplify the process of finding its maximum phase, thefundamental phasor can be made stationary by multiplying itby , leading to

(13)

with

Then, normalizing the fundamental vector in (13) by, its angle can be determined as follows:

angle (14a)

where we have (14b), shown at the bottom of the page.The maximum angle deviation is readily given by

(15)

Since our study is restricted to the measurement of moder-ated frequency deviation around the fundamental (say, 10 Hz),it is obvious that , and are all much smaller thanone. Therefore, the maximum angle deviation (15) is reason-ably small, motivating the use of the arctangent approximationfor small angles, which results in

(16)

Applying the derivative operator to find the instantaneous fre-quency yields

(17)

Low-pass filtering of this signal through results in (18),shown at the bottom of the page.

Since we are looking for the maximum error, let us remove allthe cosine terms, by setting their maximum value to one, shownin (19) at the bottom of the page.

This is the general equation providing the error. However,if the frequency deviation is small or if the first filter is veryefficient, this expression simplifies to (20), shown at the bottomof the next page.

(14b)

(18)

(19)

KAMWA et al.: DEMODULATION-BASED FREQUENCY MEASUREMENT ALGORITHMS IN TYPICAL PMUs 509

Fig. 3. Frequency error for 10% interference: two-cycle Hamming-demodulation followed by a two-cycle Blackman–Harris smoothing. (a) Nominal frequency:60 Hz and (b) offnominal frequency: 57 Hz.

If the first or second filter is very efficient, we can also write

(21)

Finally, if the frequency offset is zero

(22)

It is quite interesting to take a closer look at the general equa-tion providing the error in order to assess the mechanisms bywhich the performance of the measurement is degraded by theinterference and the frequency offset. Let us rewrite the generalequation as

(23)

where we have the equation shown at the bottom of the nextpage.

It is now easy to see that the frequency offset has two maineffects. The first is related to parameter , which increases theerror by the inverse of the frequency response of the first filter.The second (parameter ) is the oscillation added by the term

Fig. 4. Frequency error for a 10% interference: one-cycle phasordemodulation followed by a four-cycle smoothing or averaging.

near the first harmonic. It can be neglected if the filter’s fre-quency response aggressively cuts the first harmonic. The lasttwo parameters involved in (23) are related to the interference

(20)

510 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004

Fig. 5. Actual waveform from Hydro-Québec’s series-compensated transmission system. (a) EMTP simulation and (b) COMTRADE field record.

component. The first one, , is dominant. The second, , can beneglected if the frequency response of the filters aggressivelycuts the first harmonic.

IV. INITIAL VERIFICATION ON SYNTHETIC WAVEFORMS

In this section, a case study is presented to demonstrate howthe relation developed in evaluating frequency measurement al-gorithms is used. Let us first define the filters, signal, and sam-pling rate used in this example:

1) Hz and Hz;2) two-cycle Hamming window for the DFT;3) two-cycle Blackman–Harris window for frequency

smoothing.The response time of the resulting algorithm is four cycles,

meaning a two-cycle delay to a fundamental frequency ramp.The effective magnitude frequency responses of and areillustrated in Figs. 1 and 2, respectively. The actual frequencymeasurement error has been computed numerically by injectinginto the demodulation filters and , a signal of the form (11)with the following parameters:

and Hz

Fig. 6. Frequency error for 10% interference: one-cycle demodulation (DFT)followed by 2.5-cycle Kay smoothing (nominal frequency).

Fig. 3(a) and (b) shows the maximum error as a function of thefrequency for a nominal and off-nominal situation using anidentical filtering scheme. On each graph, one curve shows the

KAMWA et al.: DEMODULATION-BASED FREQUENCY MEASUREMENT ALGORITHMS IN TYPICAL PMUs 511

Fig. 7. Spectral analysis of an EMTP simulation of Hydro-Québec’s eastern transmission system. (a) Estimated frequency deviation and (b) interfering voltage.

error computed using the exact numerical simulation while theother is obtained from the equation derived in Section III. Ac-cording to Fig. 3(a), the maximum error is about 1.5 Hz whenthe frequency of the interfering tone is 45 Hz. This indicates thatthe algorithm is quite robust with respect to interfering tones.The reader will also note that the error falls to zero when theinterfering tones reach 60 Hz. This is because the phase dif-ference between the interfering tone and the fundamental fre-quency varies very slowly at this frequency. The phase is al-most constant and its derivative comes close to zero. Similarcomments apply to the off-nominal scenario in Fig. 3(b), theonly difference being a symmetrical shift in the pattern of error:the maximum error now occurs around 75 Hz instead of 45 Hz.

What is really interesting overall is that the relation developedin Section III is a very good approximation of the performanceof the real algorithm at least for the given set of FIR filters.

This suggests that, within this framework, designing a fre-quency measurement algorithm is a matter of selecting two fil-ters and to achieve some given specifications. If, for thesame response time, the user is more interested in the perfor-mance near the fundamental frequency, the second filter can bechanged for a more selective filter such as a Hamming filter.Then, using (23), the performance in the presence of given in-terfering tones will readily be assessed.

To show how the choice of the filter can influence the fre-quency algorithm performance, Fig. 4 provides the error char-

512 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 2, APRIL 2004

Fig. 8. Spectral analysis of a COMTRADE file recorded in Hydro-Québec’s transmission system. (a) Estimated frequency deviation and (b) interfering voltage.

acteristics of the Kay filter, whose impulse response is definedin (10). Comparing the latter filter with the crude average filterused in the early generation of Macrodyne PMUs [3]–[5] clearlydemonstrates that, under the same one-cycle DFT and four-cycle filtering, the Kay filter is much more effective in rejectinghigh-frequency noise as well as subsynchronous interference inthe usual range of interest (10 to 40 Hz).

V. ANALYSIS OF ACTUAL WAVEFORMS

In a further challenge to the error prediction scheme,two sets of actual power system waveforms were obtained.The first was generated though an EMTP simulation of theHydro-Québec power system, which is characterized by long

735-kV series-compensated transmission lines, giving rise tolow-frequency subsynchronous oscillations (down to 5 Hz).Voltage and current waveforms to be analyzed were recordedon the low-voltage (13.8-kV) side in a large and remote powerplant. The second data set is a COMTRADE file recorded ona 735-kV bus in the Hydro-Québec power system by a digitalfault recorder. Only the phase voltage illustrated in Fig. 5 willbe analyzed in this paper. Notice that the instantaneous mag-nitude of the voltage obtained by applying d-q transformationto the three-phase voltage is heavily corrupted, suggesting thatany frequency measurement will generate significant errorsunder these conditions.

In accordance with current practice in Hydro-Québec’s spe-cial protection systems, we selected to be a one-cycle boxcar

KAMWA et al.: DEMODULATION-BASED FREQUENCY MEASUREMENT ALGORITHMS IN TYPICAL PMUs 513

TABLE IPERFORMANCE OF THE ERROR PREDICTION

METHOD ON ACTUAL SIGNALS

and to be a 2.5-cycle Kay filter. The predicted pattern offrequency measurement error is therefore that in Fig. 6, wheresome discrepancy can be seen between the predicted and com-puted maximum errors, due essentially to the fact that the one-cycle DFT filter is not effective enough, and therefore in vio-lation with one of the recommended applicability conditions of(23). Still, the prediction is sufficiently useful for quick designpurposes.

Our approach for assessing the accuracy and usefulness ofsuch a picture consists of four steps.

1) The frequency and magnitude of the interfering tones onthe voltage are determined

2) The frequency deviation is estimated by applying the de-modulation-based method described in Section II of thispaper.

3) The magnitude of the error superimposed on the fre-quency deviation is determined for each interfering tone.

4) The result in 3) is compared with the predicted pattern oferror in Fig. 6.

Figs. 7 and 8 illustrate the modal decomposition and spec-tral density of the interfering voltage along with the estimatedfrequency deviation. To obtain these results, the original signalsare bandpass filtered between 0.1 and 50 Hz in order to elimi-nate unrelated frequency components. Then a parametric fittingmethod very similar to the well-known Prony analysis [27] isused to extract the magnitude and frequency of the damped si-nusoids constituting the signal. Hence in Figs. 7 and 8, the cap-tion of the upper plot indicates the (amplitude, frequencyand damping ) of the dominant spectral componentsof the signals. In addition, the two superimposed graphs showthe good agreement between the actual signal and its Pronyanalysis-based model. Finally, the lower plot in Figs. 7 and 8presents the spectral density of the analyzed signal, as repre-sented by the upper plot.

For an easy interpretation of these results, a summary is pro-vided in Table I. Consider first the EMTP case. Fig. 7(b) showsthat the dominant interfering tone in the voltage is located at

Hz, which means that the demodulation based fre-quency estimator will yield an error oscillating at

Hz. The predicted pattern of error (Fig. 6) suggests thatthe maximum value of the corresponding error will be about0.056 Hz while the modal decomposition in Fig. 7(a) establishesthe actual maximum magnitude to 0.052, which is acceptablyclose.

A similar reasoning also applies to the COMTRADE case.However, we have here three dominant interfering tones in thevoltage [Fig. 8(b)] leading to the same number of corrupting

tones in the estimated frequency deviation [Fig. 8(a)]. The cor-relation between the predicted magnitude of the frequency errorand the magnitude revealed by the modal analysis of actual sig-nals is acceptable, although the discrepancy is somewhat higherat 21 and 29 Hz.

VI. CONCLUSION

Designing frequency measurement algorithms is generally adifficult task. Many algorithms exist and a great deal of simu-lation is needed to get a feeling of their behavior. In this paper,we have shown that for demodulation-based frequency measure-ment algorithms, a general approximation of the worst measure-ment error can be easily computed for each additive interferingtone. The method developed has proven to be very useful in de-signing frequency measurement for novel protection schemessince, from an engineering point of view, the only problem re-maining is how to select the demodulation and smoothing filters

and so as to fulfill the given requirements. Then, knowingthe frequency response of the selected FIR filters, the behaviorof the algorithm can be determined beforehand. From a practicalpoint of view, the demodulation-based measuring scheme un-derlying this paper is attractive by itself, since it relies on simple,state-of-the-art basic signal-processing techniques. Its currentapplication in the real world, namely, on the Hydro-Québecpower system, has proven a success [22].

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[14] J.-Z. Yang and C. W. Liu, “A precise calculation of power system fre-quency,” IEEE Trans. Power Delivery, vol. 15, pp. 361–366, July 2001.

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Innocent Kamwa (S’83–M’88–SM’98) received the B.Eng. and Ph.D. degreesin electrical engineering from Laval University, Québec City, QC, Canada, in1984 and 1988, respectively.

He has been with the Hydro-Québec research institute, IREQ, since 1988. Atpresent, he is a Senior Researcher in the Power System Analysis, Operation, andControl Department. He is also an Associate Professor of electrical engineeringat Laval University.

Prof. Kamwa is a member of CIGRE. He is a Registered Professional En-gineer. He is a member of the System Dynamic Performance and Electric Ma-chinery Committees of the IEEE Power Engineering Society.

Michel Leclerc received the B.Sc. and M.Sc. degrees in electrical engineeringfrom Laval University, Québec City, QC, Canada, in 1992 and 1994, respec-tively.

After a few years working on algorithms for special network protection atCybectec, he is now with Exfo as a signal processing engineer. His researchinterests include designing signal processing algorithms for measurements.

Danielle McNabb received the B.Sc.A. degree in engineering physics, theM.Ing. degree in nuclear engineering, and the M.Ing. degree in electricalengineering from École Polytechnique, Université de Montréal, QC, Canada,in 1973, 1980, and 1986, respectively.

She joined Hydro-Québec in 1980, where she has been involved with controlmodeling and simulation for the commissioning of Gentilly 2 nuclear powerplant and, since 1986, in control modeling and protection studies for the Hydro-Québec Planning Department.

Dr. McNabb is a member of the Ordre des Ingénieurs du Québec.