Medición Parámetros PD Enero 2006

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006 15

Recursive Algorithm for Real-Time Measurement ofElectrical Variables in Power Systems

Tao Lin and Alexander Domijan, Jr.

AbstractIn this paper, a novel complex bandpass filter is pre-sented which overcomes the pitfalls of the techniques in commonuse. This complex bandpass filter can correctly extract the phasorof the fundamental component and symmetrical components involtage or current waveforms and then accurately estimate theirinstantaneous amplitude, phase angle, and frequency, even encoun-tering various power disturbances. Further, a recursive algorithmis also developed for the complex bandpass filtering that updatescurrent filtering output only using several previous sample valuesand filtering outputs. This attribute greatly reduces the computa-tional complexity of complex bandpass filtering, which is the weak-ness of the continuous wavelet transform based on the well-knownMorlet Wavelet. Thus, this recursive algorithm is highly desirablefor real-time applications. The performance of the proposed tech-nique is ascertained by using both simulated and practical powerdisturbance waveforms.

Index TermsDigital filter, frequency estimation, phasormeasurement, power system measurement, recursive algorithm,wavelet transform (WT).

I. INTRODUCTION

ACCURATE and fast measurement of the instantaneousamplitude, phase angle, and frequency of the fundamentalcomponent and/or symmetrical components in three-phasepower systems plays a key role in modern power instru-ments/meters, digital relays, control apparatus, and power-quality (PQ) studies. The performances of the techniquesemployed directly determine the functions of this equipmentand affect their behaviors under various service conditions.Hence, the real-time accurate measurement of the phasor ofthe fundamental component and/or symmetrical componentsis essential and crucial to the safe and economic running ofmodern electric power systems.

A variety of measurement techniques was developed and usedin this field. Among them, the Fourier algorithm (FA), which isbased on the Fourier series, is a conventional method for phasorcomputation by virtue of low computational complexity andgood dynamic response. The FA extracts the phasor of a spe-cific signal component from voltage or current waveform usingan orthogonal bandpass filter bank with a rectangular window[1]. However, due to the undesirable magnitude frequencyproperties (MFPs) of the orthogonal sine and cosine filters

Manuscript received March 1, 2004; revised February 16, 2005. This workwas supported by the National Science Foundation under Grant ECS-014636.Paper no. TPWRD-00102-2004.

T. Lin is with the Department of Electrical and Computer Engineering,University of Florida, Gainesville, FL 32611 USA (e-mail: [email protected];[email protected]).

A. Domijan, Jr. is with the Department of Electrical Engineering, Universityof South Florida, Tampa, FL 33620 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPWRD.2005.858802

in the FA, which are not identical, not symmetrical, and notconcentrated at the nominal frequency, the accuracy of phasormeasurements thus deteriorates, severely encountering variouspower disturbances, such as frequency deviation, decaying dcoffsets, interharmonics, etc. [2]. Recursive algorithms of theFA were also proposed to further reduce the computationalcomplexity [3], [4], but measurement accuracy could not beimproved essentially.

Some amendments have been made to overcome the pitfallsof the FA. As one aspect of the efforts, adaptive attributes wereintroduced: the sampling frequency was adjusted according tothe previous frequency estimate (the window length is fixed)[5]; the window length is an integer multiple of the measuredfundamental cycle (the sampling frequency is unchanged) [6];the central frequency of the orthogonal filter bank is adjusted perprevious frequency measurement [1], [7]; the modified FA doesnot suffer from frequency deviation again but it is still sensitiveto other types of power disturbances.

As the other aspect of the efforts, the rectangular windowin the FA was replaced with other types of windows [2], [8];thus, the modified orthogonal filters achieved satisfactory mea-surement accuracy in all service conditions by virtue of theirdesirable MFPs, which are identical, symmetrical, and concen-trated at the nominal frequency. However, the new windowswere longer than the rectangular window; thus, the dynamic re-sponses of the modified filters were slower; in addition, the com-putational complexities of the corresponding complex bandpassfiltering also increased significantly due to the lack of a recur-sive algorithm. To overcome this contradiction, an optimal com-plex bandpass filter was presented in [9] that the MFPs of thereal and imaginary parts are identical (but still with side lobes)and the computational complexity just increased slightly. Nev-ertheless, the difference between the phase-frequency properties(PFPs) of the filter bank reached 90 , namely the filtering out-puts from the filter bank are orthogonal, only at the nominal fre-quency; thus, the accuracy of phasor computation still sufferedfrom power disturbances in practical applications because theactual frequency cannot always remain unchanged as the nom-inal value.

Recently, the continuous wavelet transform (CWT), basedon the well-known Morlet wavelet, was also used for real-timephasor computation [10]. By the merit of the Gaussian window,the MFPs of the filter bank originating from the (complex)Morlet wavelet are identical and symmetrical over the entirefrequency band without any side lobes. As a result, the CWTcould achieve an accurate estimation of the instantaneousphasor in each sampling interval even during PQ events. Ana-logues to the modified FA, the Morlet wavelet-based CWT

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16 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006

approach also suffered slower dynamic response and muchhigher computational complexity. To simplify the computationprocedure of the CWT, a new complex wavelet and the resultingrecursive algorithm for the CWT were proposed for relay andpower-quality monitoring [11], [12] and that computationalcomplexity was independent of the sampling frequency. How-ever, this recursive algorithm needed future sample values andfuture outputs to update current CWT output, thus it was notsuitable for practical real-time applications.

In order to avoid the sensitivity of real-time phasor compu-tation to frequency deviation, some new methods based on thenonstationary model of a power disturbance waveform wereproposed, which considered the fundamental frequency as avariable. A variety of nonlinear curve fitting and unconstrainedoptimization tools was further employed to realize both highaccuracy and fast convergence of the amplitude and frequencyestimation. These methods included the Kalman Filtering [13],the recursive least squares algorithm [14], the nonrecursiveNewton-type algorithm [15], [16], the recursive Newton-typealgorithm [17], and artificial neural networks [18][20]. Obvi-ously, the measurement accuracy and convergence speed aredetermined by the correctness of the assumed signal model,but the actual signal composition (the number of signal com-ponents) of a voltage or current waveforms is usually unknownin a practical situation; thus, these methods may suffer severelyfrom model mismatch (for example, interharmonics are notconsidered in the signal models).

A novel complex bandpass filter is presented in this paperfor the first time, which replaces the orthogonal filter bankin the FA and CWT, to accurately measure the phasor ofthe fundamental component and symmetrical components inreal time. By virtue of a unique polynomial window functionproposed in Section II, the bandpass filtering can be fulfilledrecursively only using several previous sample values andfiltering outputs; thus, the instantaneous phasor vectors of thefundamental component and symmetrical components can beeasily updated in each sampling interval. This attribute greatlyreduces the computational complexity of phasor computation;hence, this technique is highly desirable for real-time applica-tions. The effectiveness and superiority of this technique areverified in Section III using both simulated and practical powerdisturbance waveforms. At last, the advantages and defects ofthis method are summarized in Section IV.

II. REAL-TIME PHASOR MEASUREMENT BASED ON ANOVEL COMPLEX FILTER

In this section, a novel complex bandpass filter is presentedfor the first time to overcome the weakness of the FA and theCWT approach.

A. A Novel Complex Bandpass FilterProvided function has a bandpass frequency property,

analogous to the construction of a wavelet function, a filter withan adjustable central frequency and radius can be derived from

by introducing a scaling factor

(1)

Its Fourier Transform is

(2)where is the Fourier transform of .

It can be proven that also has a bandpass frequencyproperty; the time domain and frequency domain windows of

are and , respec-tively, where and are the center and radius of , re-spectively; and are those of . Apparently with theaid of a scaling factor , the central frequency of the bandpassfilter can be tuned to any value.

Based on a polynomial-type lowpass window function

(3)

which is proposed for the first time, a complex bandpass filteris easily constructed via the modulation of

(4)where and to ensure have a bandpassfrequency property and .

In order to center the developed bandpass filter on thefundamental component or symmetrical components, the cen-tral frequency of (i.e., ) must be equal to the nominalangular frequency ; hence, the scaling factor is de-termined as , where Hz orHz is the nominal frequency. Obviously, the window functionof is .

As discussed in [2], the bandpass filter bank used for accu-rate phasor measurement must have identical MFP and not haveany side lobes in the frequency domain. The time-domain wave-forms and the MFPs of the real part and imaginary part of(i.e., a bandpass filter bank) are shown in Figs. 1 and 2, re-spectively. In Fig. 2, the thin solid line is the MFP of the realpart, and the thick dashed line represents the imaginary part. Itcan be seen that the MFPs of the filter bank are identical andsymmetrical; they both reach a peak value at the nominal fre-quency (e.g., 50 Hz) and have no side lobes. Thus, this com-plex filter has very good selectivity by suppressing harmonicsand interharmonic components, and the filtering output is notsensitive to frequency deviation. In the meantime, the differ-ence between the PFPs of the filter bank are always 90 at anyfrequency; hence, the bandpass filtering outputs from the filterbank are always orthogonal. In other words, these desirable at-tributes ensure the precise phasor computation as well as thefurther estimation of the instantaneous amplitude, phase angle,and frequency over a wide frequency range and under a compli-cated signal composition. In addition, there is no restriction onthe window width or sampling frequency.

B. Recursive Algorithm for Bandpass FilteringIn computer-based power instruments/meters, the bandpass

filtering is realized discretely as

(5)where is the sampling interval.

LIN AND DOMIJAN: RECURSIVE ALGORITHM FOR REAL-TIME MEASUREMENT 17

Fig. 1. Time-domain waveform of the complex filter (t).

Fig. 2. Identical magnitude frequency properties of the real part and imaginarypart of (t).

Equation (5) can further be interpreted in the Z plane as amultiplication of the Z transform of and

(6)

By virtue of the polynomial form of can be ex-pressed as

(7)

where and are the coefficients related to parametersand . The expressions of these coefficients are

similar to those reported in [11].Equations (6) and (7) lead to

(8)

TABLE ICOMPARISON OF THE COMPUTATIONAL BURDENS

The resulting time-domain expression is

(9)Equation(9)exhibits a recursivealgorithmfor thebandpassfil-

tering that thefilteringoutput isupdatedateachsampleusingonlysix previous sample values and seven previous outputs regardlessof the sampling frequency. In contrast, the FA and the CWT areconducted at each sample using all sample values in their time-domainwindows;hence, theircomputationalcomplexitiesarere-lated to the window lengths and are in proportion to the samplingfrequency. A comparison of the computational burdens of the FA,the CWT based on the Morlet wavelet, and the complex band-pass filtering using recursive algorithm (9) for tracking the funda-mental phasor (60.0 Hz) using a 1000.0-Hz sampling frequencyis exhibited in Table I. Obviously, the computational complexityof (9) is the lowest and constant; on the other hand, the computa-tional burdens of the FA and CWT rise, resulting from the in-crease of sampling frequency. As a result, the proposed recur-sive algorithm is very desirable for real-time applications.

Moreover, it can also be found that the poles of arelocated inside the unit circle in the Z plane; thus, this recursivealgorithm is stable. Locations and the radius of the poles arelisted in Appendix A.

C. Procedure of Accurate Phasor ComputationAnalogous to the FA and the CWT, the complex bandpass

filtering based on extracts the phasor vector of the funda-mental component from a voltage or current waveform. The realpart and imaginary part of the filtering output , namely

and , can be used to estimate the instantaneousphase angle , frequency , and amplitude

(10)where is a gain for calibrating the amplitude. It is related tothe instantaneous angular frequency and ,which is

(11)

where is the nominal frequency.


In three-phase power systems, the instantaneous phasors ofthe symmetrical components can also be measured based on thefiltering outputs of the three-phase voltages or currents

(12)where and

are the instantaneous filtering outputs in phaseA, B, and C, respectively; and are theinstantaneous phasors of the fundamental positive-, negative-,and zero-sequence components, respectively. Similarly, theiramplitudes can be obtained using the respective real and imag-inary parts of these vectors.

By the merit of the recursive computation of via(9), of a fundamental component or a symmet-rical component can be easily and accurately obtained at eachsample, namely the phasor computation can be accomplishedin real time. This feature greatly benefits the development ofor enhances the performance of power instruments/meters, dig-ital relays, and control apparatus. This technique has been suc-cessfully implemented in a digital signal processor (DSP) devel-opment kit and been tested using both simulated and practicalpower disturbance waveforms.

III. PERFORMANCE EVALUATION

The effectiveness and performance of the proposed recursivealgorithm and phasor computation were evaluated using bothsimulated power disturbance waveforms (stationary and nonsta-tionary) and practical PQ event data recorded at an industrial site.The practical data include interruption, sag, swell, harmonics,transients, frequency deviation, system imbalance, and com-binations of these disturbances. The nominal frequencies are50.0 Hz for the simulated cases and 60.0 Hz for the practical data,the sampling frequency is 600.0 Hz (12 samples per cycle) forthe simulated cases, and the initial seven filtering outputs are 0.

The details of three simulated cases and one practical evalu-ation case are provided in the following, in which phasor com-putation based on the recursive algorithm is compared with theFA and the CWT approach. As a result, the effectiveness andsuperiority of the proposed technique are clearly exhibited.

A. Performance Evaluation Under Harmonic DistortionsThe simulated stationary power disturbance waveform can be

expressed as

where is from 45.0 Hz to 55.0 Hz and is 275.5 Hz (fre-quency of an interharmonic component). is a normally dis-tributed random sequence with zero mean and small variance,which can be treated as random noise.

TABLE IIMEAN AND MAXIMUM ERRORS OF FREQUENCY

MEASUREMENT DURING 40 CYCLES

TABLE IIIMEAN AND MAXIMUM RELATIVE ERRORS OF AMPLITUDE

MEASUREMENT DURING 40 CYCLES

A panoramic investigation over a wide frequency range (from45.0 Hz to 55.0 Hz) is illustrated in Tables II and III. Table IIis the mean and maximum errors of the frequency measurementunder severe harmonic distortions (30% third harmonic, 10%fifth harmonic, and 10% interharmonic component). Table III isthe mean and maximum relative (percentage) errors of the mag-nitude measurement in the same conditions. Obviously, the pro-posed technique achieves desirable measurement accuracy overa wide frequency range, even encountering severe harmonic dis-tortions and noise.

B. Performance Evaluation Under Power Disturbances WithTime-Varying Characteristics

1) Case 1: A PQ event, consisting of a decaying dc compo-nent, harmonics, and an interharmonic component.

The waveform of the simulated PQ event is expressed as

where is 50.0 Hz and is 275.5 Hz. is random noise.The instantaneous amplitude and frequency of the funda-

mental component during the PQ event were obtained using theFA, the Morlet wavelet-based CWT, and the proposed recursivealgorithm, respectively. The results are shown and are com-pared in Fig. 3(a) and (b). In these figures, the thick solid linesrepresent the recursive algorithm, the thin solid lines representthe CWT approach, and the thin dashed lines represent the FA.

Obviously, the frequency and amplitude obtained using the re-cursive algorithm match well with the respective target values.


Fig. 3. Phasor computation under a decaying dc component and harmonics:(a) Measured amplitude of the fundamental component. (b) Measuredfundamental frequency.

Performance of the CWT approach is almost equivalent to therecursive algorithm in this case. In contrast, the FA is seriouslyaffected by the power disturbances and it cannot converge to thetrue values during the PQ event. Further, it has also been foundthat the performance of the FA is not significantly improved evenwhen its window length is the same as the novel complex filters.

2) Case 2: A PQ event consisting of frequency deviation,harmonics, and an interharmonic component.

The waveform of the simulated PQ event is expressed as

where Hz when s and Hz whens; is 275.5 Hz; and is random noise.

The instantaneous amplitude and frequency of the funda-mental component obtained using the FA, the CWT, and therecursive algorithm are shown and compared in Fig. 4(a) and

Fig. 4. Phasor computation under frequency deviation and harmonics:(a) Measured amplitude of the fundamental component. (b) The measuredfundamental frequency.

(b), respectively. It can be found that the recursive algorithmconverges to the actual values accurately and rapidly after thedrastic frequency deviation, which is used to examine the dy-namic responses of the techniques although this cannot happenin the real word. The period of the short transient is related tothe implicit window . Performance of the CWT approach isalso equivalent to the recursive algorithm in this case except fora bit slower dynamic response. As a comparison, the FA againcannot converge to a stable value after the frequency deviation.

3) Case 3: A PQ event consisting of system imbalance, fre-quency deviation, and time-varying harmonics.

The waveforms of the simulated PQ event can be expressedas

where is 48.0 Hz, which is deviated from 50-Hz nominal fre-quency, and is random noise.


Before 0.42 s, the amplitudes of the fundamental componentsin phase A, B and C (i.e., , and , are 10.0), which isthe rated amplitude; and no harmonic components appear duringthis period (i.e., the amplitudes and are 0). After0.42 s, decreases to 6.0 while both and increase to12.0; thus, the negative- and zero-sequence components appearwith magnitudes of 2.0, while the positive component remainsunchanged. In addition, the second and third harmonics also ap-pear during this period; their magnitudes are 50% and 20% ofthe fundamental component, respectively (i.e., is 3.0 whileboth and are 6.0 and is 1.2 while both andare 2.4).

The instantaneous frequency and amplitudes of the funda-mental positive-, negative-, and zero-sequence componentsduring the PQ event were obtained using the FA, the CWTapproach, and the recursive algorithm, respectively. The resultsare shown and are compared in Fig. 5(a)(d). As can be seen,the frequency and amplitudes measured using the recursivealgorithm match well with the respective target values regard-less of the effects of frequency deviation or harmonics. Theperformance of the CWT approach is also equivalent to therecursive algorithm in these figures, except that its dynamicresponse is a bit slower.

In contrast, steady-state errors appear in the amplitudes ofboth the positive- and negative-sequence components obtainedusing the FA before 0.42 s, which are caused solely by frequencydeviation, and the more the frequency deviates, the larger theseerrors are. As for the zero-sequence component, although theamplitudes of the real and imaginary parts of the three phaseFA filtering outputs are no longer the same due to the effect offrequency deviation, the sums of both the three real parts andthree imaginary parts are zero at each sample; hence, there is noerror in the amplitude measurement using the FA before 0.42 s.After 0.42 s, due to the effect of both frequency deviation andharmonics, the amplitudes of the three symmetrical componentsobtained using the FA oscillate severely and cannot converge tothe true values.

C. Performance Evaluation Based on Practical PowerDisturbances

The effectiveness and performance of the recursive algorithmwere also evaluated using practical power disturbances. Thesepower disturbances were recorded in the distribution system atPremium Power Park [PPP] in Columbus, OH, during 1999 and2000 [21], which were used as benchmarks for power engi-neering studies. Remote PQ monitors were installed at multiplelocations within the PPP, and all phase voltages and currentswere recorded during a power disturbance. The monitors usedwere BMI 7100 PQ nodes, the system frequency was 60 Hz,the sampling frequency was 7680 Hz (128 samples per cycle),and recording duration was 16 cycles (2048 samples). Extensivepractical cases were selected for the evaluation; however, due tothe page limit, the detailed results of only one case are providedin the following.

The practical PQ event was recorded on September 20, 2000.The voltage waveform in phase C is shown in Fig. 6(a), whichconsists of amplitude variation, frequency deviation, and har-monics. This PQ event might be caused by a maldisconnection

Fig. 5. Phasor computation under frequency deviation and harmonics: (a) Themeasured fundamental frequency. (b) The amplitude of the positive-sequencecomponent. (c) The amplitude of the negative-sequence component. (d) Theamplitude of the zero-sequence component.

of the PPP from the main grid and load shedding did not respondin time; thus leading to voltage collapse and a severe frequencydrop due to the imbalance of power (overload) in the PPP. The


Fig. 6. Amplitude and frequency obtained in the practical case: (a) Originalvoltage waveform. (b) Measured fundamental frequency. (c) The amplitude ofthe fundamental component.

instantaneous frequency and amplitude of the fundamental com-ponent were obtained using the FA, the CWT approach, and therecursive algorithm, respectively. These results are shown andcompared in Fig. 6(b) and (c), which focus only on the sectionof severe waveform distortion.

In Fig. 6(b) and (c), both the frequency and amplitude mea-sured using the FA have a decreasing trend, but oscillate se-verely and irregularly. This phenomenon is caused by the rapidfrequency drop and harmonic interference. Because the actualvalues are unknown, the outputs of the CWT approach and therecursive algorithm are compared. It can be seen that the be-haviors of these two methods are very similar [i.e., they exhibitsmooth drops matching well with the waveform in Fig. 6(a)];the only difference between them is that the recursive algorithm

responds more quickly. These results show that the recursive al-gorithm provides the most reasonable measurements, which canbe considered as true values.

IV. CONCLUSIONA technique based on a novel complex bandpass filter for the

accurate and fast computation of the instantaneous phasor of thefundamental component and symmetrical components is pre-sented in this paper. Being analogous to the modified FA and theCWT, this complex bandpass filter has desirable MFP and PFP;thus, it can achieve high measurement accuracy even during PQevents. Further, by the merit of the unique polynomial windowfunction, a recursive algorithm is derived for the bandpass fil-tering. As a result, the bandpass filtering can be fulfilled recur-sively only using several previous sample values and filteringoutputs and, hence, the instantaneous phasor vectors of the fun-damental component and symmetrical components can be easilyupdated in each sampling interval. This attribute greatly reducesthe computational complexity of phasor computation, which ismuch lower than the modified FA and CWT. Hence, this tech-nique is highly desirable for real-time applications.

The correctness and effectiveness of the proposed techniqueare ascertained using various simulated and practical power dis-turbance waveforms. The evaluation results demonstrate that theproposed technique can be well used in power instruments/me-ters, PQ studies, control apparatus, and backup protections, al-though it is not suitable for primary protection due to a relativelyslower dynamic response (compaed with the FA).

APPENDIX

A. Locations of the PolesLocations of the poles (z1 to z7) in (7) and their individual

radius (r1 to r7) in the Z plane are listed in the following, wherethe nominal frequency is 50 Hz and the sampling frequency is600 Hz

In addition, the locations of the poles related to 60-Hz nom-inal frequency and a various sampling rate have also been ex-amined in that the poles are located in the unit circle.

B. Morlet Wavelet FunctionThe Morlet wavelet used for comparison in Section III is

where and scaling factor are thenominal frequency (60 or 50 Hz).


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Tao Lin received the B.S.E.E., M.E., and Ph.D.degrees from Huazhong University of Science andTechnology, Wuhan, China, in 1991, 1994, and1997, respectively.

Currently, he is a Senior Research Scholar with theDepartment of Electrical and Computer Engineering,University of Florida, Gainesville. He joined CentralChina Power Group Co., China, in 1997. From 2002to 2003, he was a Visiting Research Fellow with theDepartment of Electronic and Electrical Engineering,University of Bath, Bath, U.K. His research interests

include the application of the advanced signal processing techniques and arti-ficial intelligence to electric power systems, power system relaying, metering,and PQ studies.

Dr. Lin was awarded a postdoctoral research fellowship by Japan Society forthe Promotion of Science (JSPS) in 2000 to conduct researchat Nagasaki Uni-versity, Nagasaki, Japan. He is an editorial board member for the InternationalJournal of Power and Energy Systems.

Alexander Domijan, Jr. received the B.S.E.E. de-gree from the University of Miami, Coral Gables, FL,the M.E. degree in electric power engineering fromthe Rensealaer Polytechnic Institute, Troy, NY, andthe Ph.D. degree in electrical engineering from theUniversity of Texas at Arlington.

He was a member of the faculty at the Universityof Florida. Currently, he is a Professor with the De-partment of Electrical Engineering and Director ofthe Power Quality Laboratory and Distributed EnergyLaboratory, University of South Florida, Tampa. He

has been a Consultant with many corporations. His research areas are PQ andelectricity metering, flexible ac transmission systems (FACTS), and customerpower and Flexible, Reliable and Intelligent Electrical eNergy Delivery Sys-tems (FRIENDS).

Dr. Domijan is the Editor-in-Chief for the International Journal of Power andEnergy Systems.

tocRecursive Algorithm for Real-Time Measurement of Electrical VariTao Lin and Alexander Domijan, Jr.I. I NTRODUCTIONII. R EAL -T IME P HASOR M EASUREMENT B ASED ON A N OVEL C OMPLEA. A Novel Complex Bandpass FilterB. Recursive Algorithm for Bandpass Filtering

Fig.1. Time-domain waveform of the complex filter $\eta_{a} (t)Fig.2. Identical magnitude frequency properties of the real parTABLE I C OMPARISON OF THE C OMPUTATIONAL B URDENSC. Procedure of Accurate Phasor ComputationIII. P ERFORMANCE E VALUATIONA. Performance Evaluation Under Harmonic Distortions

TABLE II M EAN AND M AXIMUM E RRORS OF F REQUENCY M EASUREMENT DTABLE III M EAN AND M AXIMUM R ELATIVE E RRORS OF A MPLITUDE M EB. Performance Evaluation Under Power Disturbances With Time-Var1) Case 1: A PQ event, consisting of a decaying dc component, ha

Fig.3. Phasor computation under a decaying dc component and har2) Case 2: A PQ event consisting of frequency deviation, harmoni

Fig.4. Phasor computation under frequency deviation and harmoni3) Case 3: A PQ event consisting of system imbalance, frequency C. Performance Evaluation Based on Practical Power Disturbances

Fig.5. Phasor computation under frequency deviation and harmoniFig.6. Amplitude and frequency obtained in the practical case: IV. C ONCLUSIONA. Locations of the PolesB. Morlet Wavelet Function

T. S. Sidhu and M. S. Sachdev, An iterative technique for fast aJ. A. delaO, H. J. Altuve, and I. Diaz, A new digital filter forK. F. Eichhorn and T. Lobos, Recursive real-time calculation of F. F. Costa, L. A. L. de Almeida, S. R. Naidu, and E. R. Braga-FG. Benmouyal, An adaptive sampling interval generator for digitaD. Hart and D. Novosel, A new frequency tracking and phasor estiH. S. Song and K. Nam, Instantaneous phase angle estimation algoJ. AdelaO, New family of digital filters for phasor computation,J. Suonan, G. Song, J. Zhang, and Y. Song, A novel short-window S. Huang, T. Hsieh, and C. Huang, Application of Morlet wavelet O. Chaari, M. Meunier, and F. Brouaye, Wavelets: A new tool for O. Poisson, P. Rioual, and M. Meunier, Detection and measurementA. A. Girgis and W. L. Peterson, Adaptive estimation of power syI. Kamwa and R. Grondin, Fast adaptive schemes for tracking voltV. V. Terzija and M. B. Djuric, Voltage phasor and local system V. V. Terzija and D. Markovic, Symmetric components estimation tV. V. Terzija, Improved recursive Netwon-type algorithm for freqA. Cichochi and T. Lobos, Artificial neural networks for real tiP. K. Dash, S. K. Panda, B. Mishra, and D. P. Swain, Fast estimaL. L. Lai and W. L. Chan, Real time frequency and harmonic evalu

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