application of staggered undersdampling to power quality monitoring

864 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 15, NO. 3, JULY 2000

Application of Staggered Undersampling to PowerQuality Monitoring

Hanoch Lev-Ari, Senior Member, IEEE, Aleksandar M. Stankovic, and S. Lin

AbstractIn this paper we address the issue of monitoringpower quality with limited sensing and computational re-sources. The proposed staggered sampling methodology utilizesclose-to-periodic nature of waveforms of interest. The sparsesamples obtained in our scheme are reordered (shuffled) beforeFFT is used for spectral calculations. We provide a completecharacterization of the shuffling process in time domain and of thedual inverse shuffling in the frequency domain. The methodologyis applied to a number of cases of interest in power quality,demonstrating that harmonics of the fundamental frequency canbe recovered without error, and that effective descriptions arepossible for flicker, sag, interharmonics, and noisy measurements.

Index TermsComputational algorithms, power quality, signalsampling and reconstruction.

I. INTRODUCTION

T HE IMPORTANCE of power quality is increasing with thepower market deregulation and with the emergence of newelectric loads that are sensitive to variations in power supplywaveforms. The ability to adequately measure and documentpower quality is thus critical to the unimpeded functioning ofthe market. Existing sensing and digital signal processing tech-nologies are capable performing the required measurements inreal time, but at significant signal conversion (AID) and com-putational costs. In this paper we propose a sampling and signalprocessing scheme that reduces these penalties by exploiting theparticular close-to-periodic nature of most variables of interest.This makes our scheme relevant for even very cost-sensitiveequipment, such as digital wattmeters and similar apparatus.

Problems related to power quality are becoming prevalent in amajority of power utilities, and are a rising concern to customers[1]. Signal processing technology is of primary importance inassessment and classification of disturbances in a power net-work. Many digital signal processing (DSP) schemes have beenapplied successfully in power systems, and a number of special-ized algorithms has been developed over the last three decades.A comprehensive summary of power quality issues and relatedcomputational requirements is presented in [2]. A number ofpotential sources of errors (such as finite word length) that wastypical in the early DSP technologies is described in [3]. Whilethe impact of some of the difficulties has been mitigated byadvances in microcontroller technology, certain problems (likesynchronization errors) persist [4]. Recent developments in DSP

Manuscript received January 28, 1999. The work of A. M. Stankovic wassupported by the National Science Foundation under Grant ECS-9502636 andthe Office of Naval Research under Grant N14-97-1-0704.

The authors are with Northeastern University, Boston, MA.Publisher Item Identifier S 0885-8977(00)07200-9.

technology, like multirate algorithms, have been fruitfully ap-plied to power quality problems [5].

The rest of the paper is organized as follows: in Section IIwe describe the algorithm and present a motivating example;the analysis of the algorithm is presented in Section III, andfollowed by applications to power quality monitoring in Sec-tion IV; the paper is briefly summarized in Section V, while theanalytical details are explained in the Appendix.

II. DESCRIPTION OF THEALGORITHM

The main idea in staggered sampling is to manipulate sparsesamples of a signal to obtain a set equivalent to the one obtainedby fast sampling. This equivalence is of course possible onlyfor some classes of waveforms such as periodic signals. Recallthat the Nyquist theorem states that a band-limited signal can beexactly reconstructed from samples provided that the samplingrate is greater or equal to twice the bandwidth of the signal.Slower sampling would in principle introduce deviations in thefrequency content (aliasing); this can, however, be avoided if thesignal belongs to a restricted class (e.g., periodic) and samplesare 1) taken at instants that are carefully chosen, and 2) preciselymanipulated before applying the fast Fourier transform (FFT).

Assume that a signal is periodic with period, and that a totalof samples is taken (e.g., in our examples). Also as-sume that harmonics higher than have been com-pletely eliminated before sampling by the anti-aliasing filter.Samples are taken every seconds, where and aremutually coprime ( in our examples). It will thus take

periods of the signal to acquire samples, but all sampleswill be taken at distinct relative positions within a period (as weprove later). If the samples are then reordered (shuffled), theycan equal the set of samples taken every samples withinoneperiod. Given such a set of samples, we can recover theharmonic content up to the ( )st harmonicexactlyviaFFT. In a sense, one is using the periodicity of the signal to re-duce the sampling frequencytimeswithoutcausing aliasing[recall that the signal is assumed to be bandlimited toby anti-aliasing filtering]. The analysis presented here builds onresults from [6].

We show in the next section that the required shuffling mapsthe th sample into the th position according to the followingformula

(1)

where denotes the reminder of after division by .In particular, when we obtain

08858977/00$10.00 2000 IEEE

LEV-ARI et al.: APPLICATION OF STAGGERED UNDERSAMPLING TO POWER QUALITY MONITORING 865

Fig. 1. Unordered samples of a 3 tone signal.

Fig. 2. Ordered samples of a 3 tone signal and harmonic content.

so that no shuffling is required. This special case of the al-gorithm presented here was used forsampling oscilloscopes[7, pp. 463465].

Next, we show an example that illustrate the capabilities ofthe method. We consider a sinusoidal waveform with 10% fifthharmonic and 1% seventh harmonic added, and ,

and . Fig. 1 shows unordered samples, while Fig. 2shows ordered (shuffled) samples and the correctly determinedharmonic content (expressed in dB of the fundamental).

III. A NALYSIS OF THE ALGORITHM

In this section we introduce a multirate signal processingcharacterization of the staggered sampling algorithm, andwe use it to analyze the performance of this algorithm in the

presence of nonharmonic components, including interhar-monics, modulated harmonics, and white or colored noise.

As described in the previous section, the staggered undersam-pling algorithm forms a sequence of sparse samples , viz.,

where denotes the continuous-time (analog) inputsignal. In contrast, conventional (dense) sampling of thesame analog signal would have produced a sequence,where

Notice that the sparse sequence can also be interpreted asa subsampled version of because

(2)

This relation between and is highlighted in a block-diagram representation of the staggered undersampling algo-rithm, in which we represent the sparse sampler as a cascade of adense sampler followed by a factor-of-downsampler (Fig. 3).

As explained in Section II, the harmonic content of a periodicanalog input signal can be determined without error from a fi-nite segment of the densely sampled sequence, providedthat the dense sampler is synchronized with the periodof theanalog signal (see Fig. 3). Consequently, is periodic withperiod , i.e., .

Our first objective in this section is to demonstrate that, for aperiodic input signal , we can reconstruct a single periodof without errorby applying the shuffling operation toconsecutive samples of the subsampled sequence. Com-bining the periodicity of with (2) we find that

(3)

which establishes a correspondence between the finite set ofsparse samples and the finite setof dense samples . We now show thatthis correspondence is onto and one-to-one, so that the latter setcan be obtained by shuffling the elements of the former, usingthe index mapping (1).

Result 1: If , are mutually coprime, than the index map-ping (1) is onto and one-to-one.

Proof: See Appendix.Thus shuffling the finite set of sparse samples

reproduces one complete period of the denselysampled sequence . The harmonic content of the analoginput signal is then obtained by applying a scaled FFT to theshuffled set of samples, as explained in Section II.

Our next objective in this section is to determine the effectof nonharmonic components of on the harmonic content

as determined by the FFT block inFig. 3. Since the system described by Fig. 3 is linear (albeittime-varying) we can analyze the effect of each component in-dividually. For instance, a white noise component in the analoginput signal gives rise to an independent identically distributed(i.i.d.) sequence component in both the densely sampled signal


Fig. 3. Block-diagram representation of the staggered undersampling algorithm.

Fig. 4. Equivalent block-diagram representation of the staggered undersampling algorithm.

and its undersampled version . The power (=vari-ance) of this component is , where is the cutoff fre-quency of the anti-aliasing lowpass filter, and denotes the(one-sided) power spectral density of the analog white noisecomponent. Since the FFT is a (scaled) orthogonal transforma-tion, the end result is a random contribution to every harmoniccomponent . The power of that contribution is the same forall harmonics, and equals .

While the time-domain shuffle leaves the frequency contentof white noise unaltered, the same is not true for other typesof signals, such as colored noise or interharmonic sinusoldalsignals. As we shall presently show, the application of the al-gorithm of Fig. 3 to such signals results in mixing up theirfrequency content. Our next result provides an explicit charac-terization of this frequency-domain effect.

Result 2: A shuffle operation [as given by (1)] followed byan FFT is equivalent to an FFT operation followed by aninverseshuffle.

Proof: See Appendix.This result implies that,for any analog input signal, the

block-diagram of Fig. 3 is equivalent to the block-diagramshown in Fig. 4. We can now use standard multirate analysistechniques (see, e.g., [8]) to determine the discrete-time Fouriertransform (DTFT) of the sequence , viz.,

The application Of an FFT to a finite segment of the sequenceresults in a windowed (and frequency-sampled) version

of the DTFT, which is then mixed up by the inverse shuffleoperation.

The harmonic content after the inverse shuffle can be deter-mined by using the following explicit expression.

Result 3: The inverse to the shuffle operation (1) is given by

where is the smallest positive integer that satisfies the equa-tion .

Proof: See Appendix.

To demonstrate the utility of the equivalent block-diagramrepresentation of Fig. 4, consider a single (complex) interhar-monic component at frequency , viz.,

where . The DTFT of is, where denotes the Dirac delta function. The

FFT block uses a windowed version (= a finite segment) of, namely the FFT input is , where

else

The DTFT of this windowed signal is a convolution betweenand , which equals . The

output of the FFT operation is a sampled version of this DTFT,namely , where . Thus a singleinterharmonic complex sinusoid contributes a single smearedpeak, centered at , to the signal at the input of the inverseshuffle block in Fig. 4, while a real interharmonic sinusoid con-tributes two smeared peaks, one at , and the other at .

For instance, a 30 Hz analog interharmonic component at theinput of a system designed to determine the harmonics of 60Hz signals (with , and ) results insmeared peaks centered at . Thismeans that the strongest harmonics before inverse shuffling arelocated at . Since in this case , the inverseshuffle moves the strongest harmonics to

, which gets folded back to , and. Fig. 5 shows the ordered samples and

spectral content of a signal comprising the 60 Hz fundamentaland 10% of the 30 Hz interharmonic. Note how prominent are13th and 14th harmonic in the lower panel.

IV. FURTHERAPPLICATIONS INPOWERQUALITY MONITORING

We first consider the case offlicker [2], i.e., a 60-Hz sinu-soid whose magnitude is modulated (10% in this case) with thefrequency of (set to Hz). In that case the analogsignal has the main component at Hz, and smaller (5%)components at sidebands . The analysis from the previoussection establishes that the main peak before inverse shuffling


Fig. 5. Ordered samples of a 30 Hz interharmonic signal and harmoniccontent.

Fig. 6. Ordered samples of a flicker signal and harmonic content.

will be at , which will be mapped to after the in-verse shuffling. The side components are at and ,and they are mapped by the inverse shuffling to and

, respectively. Fig. 6 shows ordered samples and the har-monic content in the corresponding numerical simulation; notethe magnitude of components at and which is

dB. A family of spectral signatures (dependenton ) can be used for detection of flicker in power system ap-paratus that uses staggered sampling.

Next, we consider the case ofsagi.e., a 60-Hz sinusoid whosemagnitude is modulated (20% in this case) over several cycles(3 in our case). Fig. 7 shows unordered samples, while Fig. 8shows ordered samples and the harmonic content. In the caseof a sag, more effective detection methods are likely to be in

Fig. 7. Unordered samples of a sag.

Fig. 8. Ordered samples of a sag signal and harmonic content.

time domaine.g., in the top panel of Fig. 8 samples obtainedduring sag clearly differ from values obtained by a simple (evenlinear) interpolation of the left and right neighboring values.These simple calculations can be used in for fast on-line detec-tion of sag.

Finally, we consider the case ofnoisy measurements, i.e., a60-Hz sinusoid with added white noise (standard deviation is5% of the magnitude of the fundamental); Fig. 9 shows orderedsamples and the harmonic content. As explained in the previoussection, we expect the noise to influence all harmonics, which isthe case in Fig. 9. In the case of white (or colored) noise detec-tion, frequency domain criteria are likely to be most effective.For example, a flat platform is expected in the case of additivewhite noise, as explained in the previous section.


Fig. 9. Ordered samples of a noisy sinusoid and its harmonic content.

Fig. 10. Ordered samples of a measured voltage and its harmonic content.

The results for the case of (simulated) noisy measurementsare quite comparable with results obtainedexperimentally fordistribution voltage in our laboratory using a digitizing storageoscilloscope (with 50 kHz sampling frequency, and2% digi-tizing error, normalized by nominal voltage amplitude170 V);Fig. 10 shows ordered samples and the harmonic content.

Detection ofbrown-outs (i.e., sustained reductions in mag-nitude of the fundamental) is easy in frequency domain, as onlythe first harmonic needs to be monitored.

In practical implementations the choice ofis a very impor-tant issue: if is chosen too small, then we do not gain enoughin terms of reduced processing requirements from down sam-pling; if is too large, then it takes too long to collectsam-ples, making our information about the sampled signal outdated.

While the final choice depends on the application and charac-teristics of the employed hardware, is reasonable withpresent technology.

V. CONCLUSIONS

This paper introduces a signal processing methodology thatreduces the sampling frequency and data conversion require-ments in power quality monitoring. The methodology utilizesclose-to-periodic nature of waveforms of interest. The sparsesamples obtained in our scheme have to be reordered (shuffled)before FFT can be used for spectral characterization. A com-plete characterization of the shuffling process in time domainand of the dual inverse shuffling in the frequency domain is acentral theme in the paper. Applications to a number of cases ofinterest in power quality monitoring demonstrate that harmonicsof the fundamental frequency are recovered without error, andthat effective descriptions are possible for flicker, sag, interhar-monics and noisy measurements (the last case was explored ex-perimentally as well). Thus the staggered sampling procedureallows equipment designers to extract more power quality in-formation with limited signal processing resources.

APPENDIXPROOF OFRESULTS13

Proof of Result 1:To establish the one-to-one natureof the index map (1) we need to show that two distinctvalues of cannot be mapped into the same valueof . Indeed, implies that

, and since are coprime, wededuce that . The only solution to thisequation with both and restricted to the rangeis , which establishes the one-to-one property.

Since distinct values of map into distinct values of , theentire range is covered by the mapping (1) as

varies in the same range, namely the map is indeed onto.Proof of Result 2:The shuffling relation (3) can also be

written in the form

where , and is the Kronecker delta. Thismeans that the mapping of the setinto the set is a permutation, and thatthe corresponding permutation matrix is . Thus,the result we need to establish is , where isthe DFT matrix, viz.,

We shall prove, instead, the equivalent statement .Indeed, the ( )th element of is


and, consequently, the ( )th element of is

which establishes the result.Proof of Result 3:From (1) we have

Since , it follows thatand, therefore, , which is the

inverse of the map (1).

REFERENCES

[1] E. W. Gunther and H. Mehta, A survey of distribution system powerquality, IEEE Trans. Power Delivery, vol. 10, pp. 322329, January1995.

[2] G. T. Heydt,Electric Power Quality: Stars in a Circle Publications, 1991.[3] K. Srinivasan, Errors of digital measurement of voltage, active and re-

active powers and an on-line correction of frequency drift,IEEE Trans.Power Delivery, vol. 2, pp. 7276, January 1987.

[4] X. Dai and R. Gretsch, Quasisynchronous sampling algorithm and itsapplications,IEEE Trans. Instrument. Measure., vol. 43, pp. 204209,April 1994.

[5] L. Toivonen and J. Morsky, Digital multirate algorithms for measure-ment of voltage, current, power and flicker,IEEE Trans. Power De-livery, vol. 10, pp. 116126, January 1995.

[6] S. Lin, Undersampling method and its application in developing adigital spectral wattmeter with power quality monitoring, M.S. thesis,Northeastern University, Boston, MA, May 1998.

[7] W. McC. Siebert,Circuits, Signals and Systems: The MIT Press, 1986.

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Hanoch Lev-Ari (S78M84SM93) received the B.S., Summa Cum Laude,in 1971, and the M.S. in 1978, both in electrical engineering from the Technion,Israel Institute of Technology, Haifa, Israel; and the Ph.D. in electrical engi-neering from Stanford University, Stanford, CA, in 1984. During 1985 he helda joint appointment as an Adjunct Research Professor of electrical engineeringwith the Naval Postgraduate School, Monterey, CA and as a Research Associatewith the Information Systems Laboratory at Stanford; he stayed at Stanford asa Senior Research Associate until 1990. He is currently an Associate Professorwith the Department of Electrical and Computer Engineering at NortheasternUniversity. During 19941996 he was also the Director of the Commununica-tions and Digital Signal Processing (CDSP) Center at Northeastern University.His present areas of interest include model-based spectrum analysis and estima-tion for nonstationary signals, scale-recursive (multirate) detection and estima-tion of random signals, and adaptive linear and nonlinear filtering techniques,with applications to channel equalization, over-the horizon (OTH) radar, auto-matic target detection and recognition, and identification of time-variant sys-tems. Dr. Lev-Ari served as an Associate Editor of Circuits, Systems and SignalProcessing, and of Integration, The VLSI Journal. He is a member of SIAM,and a senior member of IEEE.

Aleksandar M. Stankovicobtained the Dipl.Ing. degree from the University ofBelgrade, Yugoslavia in 1982, the M.S. degree from the same institution in 1986,and the Ph.D. degree from Massachusetts Institute of Technology in 1993, all inelectrical engineering. He has been with the Department of Electrical and Com-puter Engineering at Northeastern University, Boston since 1993, presently asan Associate Professor. Dr. Stankovic is a member of IEEE Power Engineering,Power Electronics, Control Systems, Industry Applications and Industrial Elec-tronics Societies. He serves as an Associate Editor for the IEEE TRANSACTIONSFOR CONTROL SYSTEM TECHNOLOGY.

application of staggered undersdampling to power quality monitoring

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