Reducing the error in phasor estimates from phasorlets in fault voltage and current signals

856 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 56, NO. 3, JUNE 2007

Reducing the Error in Phasor Estimates FromPhasorlets in Fault Voltage and Current Signals

José Antonio de la O Serna, Senior Member, IEEE

Abstract—An assessment and reduction of the phasor-errorinfiltration in estimates from phasorlets are proposed in this paperthrough a generalized least-square formulation of the problem.It introduces a computationally simple and quick estimator thatexploits the structure of the sinusoidal model and provides thefinest stable phasor estimate, which is very useful for fault voltagesignals. It also proposes an extended signal model, by including adc component into the signal model, in order to allocate the phasorerror, when an aperiodic component is present in the input signal.The numerical simulations illustrate the improvement in speedand accuracy of the estimates, which are obtained in decicycles,as well as its still wavering nature, which is considered as thepersisting limitation of this technique.

Index Terms—Curve fitting, digital filter, least squares, multi-rate estimation, nonorthogonal basis, normal equations, obliqueplanes, orthogonal decomposition, phasor estimation, phasorlets,signal projection.

I. INTRODUCTION

PHASOR ESTIMATION from phasorlets has been provento be the best technique when the analyzed signal segment

coincides with a pure sinusoidal segment [1]. This methodis also excellent for detecting fast transitions between differ-ent sinusoidal states. However, its phasor estimates are veryvulnerable to transients or noise at the input signal. The firstpublication about this technique appeared in [2] but withoutaddressing its high noise sensitivity. In [1], the focus wasplaced on the frequency response of the filters’ generatorsof phasorlets, which were called phaselets in [2], and only adisclaimer about the huge infiltration of nonsinusoidal signalsinto the phasorlet estimates was raised.

This paper assesses the phasor error in estimates fromphasorlets and proposes several strategies for its reduction,particularly for fault voltage and current signals. This methodis presented as the least-square solution to the sinusoidal fittingto N consecutive components of the input signal. Under thisgeneral formulation, the phasor turns out to be an inverse trans-formation of the N th phasorlet. Under the geometric insightof the linear algebra approach, the phasor-error analysis isstraightforward, and the infiltrations to the Fourier filter [3] arebetter understood.

The main point of this paper is that, in short intervals, thepiecewise sinusoidal fitting works like an all-pass filter, in

Manuscript received April 15, 2005; revised June 26, 2006.The author is with the State University of Nuevo León, Monterrey 66450,

Mexico (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIM.2007.894880

which sinusoidal segments are fitted to the input signal. Inthis process, the nonsinusoidal component is confused withthe sinusoidal, which is producing significant phasor error butwith a small fitting error. As the interval length increases, theprocess becomes a narrower bandpass filter, allocating part ofthe nonsinusoidal component into the fitting error and reducingthe phasor error. But, this introduces a delay in the estimates.Therefore, the key for obtaining accurate and quick (obtainedin short periods of time) phasor estimates resides in using acomplete signal model, containing the signals we want to sep-arate in a very short period of time. Under this time condition,the best way is to use the Taylor expansion of the signal model.The other important idea of this paper is the huge multiplicityof estimates that we can obtain from a piece of signal; forexample, multiple and multirate phasor estimates are proposed,but many more are possible. These ideas will certainly open theway to faster and more accurate phasor estimates for so manyapplications.

In intervals where the nonsinusoidal component is null, thephasorlet estimation technique provides the quickest phasormeasurements. Phasorlets are also very good for detecting fasttransitions between the two different sinusoidal states, in whichthe nonsinusoidal component is present in a very short periodof time, such as those found in fault voltage signals. Thosedisruptions are seen as a fast transition between the two phasormeasurements of the adjacent sinusoidal states. A very efficientand fast phasor-estimation technique for this case is proposed.This technique provides the finest and computationally simplestphasor estimates. However, transients in current signals usuallycontain an aperiodic component. To cope with this kind ofsignal, an extension of the sinusoidal model is proposed by in-cluding a constant component (zero-order Taylor expansion) tothe signal model. In this way, room is allocated into the modelto handle the exponential error, with the advantage that theshorter the segments are, so much the better will be the fittingof the dc model to the exponential. Note that no constant timenor magnitude of the exponential is needed as suggested in [3].

These new strategies are tested and illustrated on real faultcurrent and voltage signals. The obtained results show asignificative reduction on the phasor error in estimates obtainedfrom the decicycle phasorlets.

Most of the works on phasor-estimation algorithms put littleattention to the error generated when the input signal fails tocomply with the assumed underlying model. In several of them,the signal suppositions are not even declared explicitly. Forexample, after a qualitative discussion about the phasor errorin [3], the author discards the filters working in fractionarycycles and recommends the full-cycle Fourier filter as the best

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DE LA O SERNA: REDUCING THE ERROR IN PHASOR ESTIMATES FROM PHASORLETS 857

solution because of its rejection to the full set of harmonics,presupposing a periodic signal at the input. However, beforeaperiodic input signals such as in a power oscillation, this solu-tion has serious shortcomings. In [4], nothing is said about thephasor errors incurred when the signals differ from the assumedmodels; the performance of the algorithms is simply illustratedusing signals that replicate the corresponding underlying signalmodels. The subsequent papers deal with the mitigations ofdifferent error sources such as the exponential infiltration [5]or the errors due to fundamental-frequency deviations [6], [7],or the error incurred when the input signal correspond to apower oscillation [8]. After the recent cascaded blackouts,before serious doubts concerning the accuracy of the phasorestimates under those circumstances, the emphasis is placedon the phasor-error measurement [9]. Even if in this paperphasor-error measurement was euphemistically called qualitymeasure and its proposed algorithm could be in fact measuringharmonic distortion, that paper clearly emphasizes the currentimportance of explicitly addressing the problem of phasor-errormeasurement. Here, again, a strong assumption on periodicinput signal is found, even under a power-oscillation scenario.

In this paper, which was first presented in [10], a quantitativedescription of the phasor-error measurement incurred by partialphasor estimates and a discussion about how to build with thembetter phasor estimates are presented. This paper is writtenunder the scenario of the protection applications, so the inputsignal is assumed to be periodic (the exponential is considereda dc component under short intervals), with a constant funda-mental period. We also assume that digital signals are quantizedwith enough bits, at least 12, with a signal-to-quantization-noiserate superior to 73.76 dB, so quantization noise is not relevant.When additive uncorrelated noise with constant covarianceis present in the input signal, our (Fourier) phasor estimatescoincide with those obtained with the Kalman filter at intervalsmultiples of a half cycle [3, p. 101–105]. But, our scenario ex-cludes noise with time-variant covariance matrix, under whichthe Kalman filter provides the best solutions [3, p. 105]. Finally,for those interested in the Cramér–Rao bound of the estimatedparameters in this paper, [11] is highly recommended.

In the following sections, the formulation of the problem ispresented. The Grammian matrices of the normal equations arefound in closed form for the 2- and 3-D models. The major andminor axes of the elliptical bounds of the phasor error, as wellas the elliptical projections of the vectors of the standard basesof R

N , are illustrated to have an insight of the behavior of thephasor error when N increases. Multiple and multirate phasor-estimation algorithms are presented. Finally, numerical resultsobtained from real voltage and current signals are presented asevidence of the main conclusions.

The suggested strategies show that it is possible to reducethe phasor-error estimates from phasorlets when the model ofthe expected noise is well known. The research remains opento other strategies to reduce the phasor error, such as the use ofwavelets or other weighing functions to shape the Grammianin the normal equations and, of course, to apply these resultsto other applications such as phasor estimation under poweroscillations or in digital transmission receivers, in order toincrease its data rate.

II. PROBLEM APPROACH

Digital current and voltage signals from power systems canbe modeled by the following linear sequence

s(n) = s(n) + e(n), n = 0, 1, . . . , N − 1 (1)

in which s(n) represents a sinusoidal sequence of the forms(n) = A cos(ω0n + θ), with fundamental angular frequencyω0 = 2π/N0 and period N0, and e(n) is a residual signal. In ourperiodic signal scenario for s(n), the residual signal can be anyof the nonfundamental harmonics or any linear combination ofthem. Note that our interest is placed in sequences of lengthN ≤ N0, which correspond to fractions of one cycle. Thissinusoidal sequence can be expressed as a linear combinationof the following N -dimensional vectors

b1 =

cos 0cos ω0

...cos ω0(N − 1)

, b2 = −

sin 0sin ω0

...sinω0(N − 1)

(2)

as it can be seen in the vectorial form

s = Bρ + e (3)

in which b1 and b2 are the first and second columns of theN × 2 matrix B. The coefficients of the linear combination inρ, ρ1 = A cos θ, and ρ2 = A sin θ are constant for all N ’s andcorrespond to the phasor components of the sinusoidal signals(n). Therefore, they span the column space of B, Col(B),which is a plane in R

N , which is referred to as the N th plane.

A. Least-Square Estimates

Least-square phasor estimates can be obtained by projectingthe signals in (3) onto the model subspace Col(B). This issimply done by premultiplying by BT the vectors in (3). Wehave for each N

BT s = BT Bρ + BT e. (4)

This operation equates the projections of the original vectorsin (3) onto the N th plane. It splits R

N into two orthogonalsubspaces: the column space of B, Col(B), and the null spaceof BT , which contains the orthogonal complements of theprojections. In this way, the projections can be completelyexpressed as linear combinations of the vectors in B.

In the case of the e vector, we have

e = e‖ + e⊥ = Bρ(N) + e⊥ (5)

in which e‖ is the projection of e, and e⊥ is its orthogonalcomplement. ρ

(N)contains the coordinates of the projection.

The superscript (N) is used to indicate that the coordinates ofthe projection depend on N , because the planes are changingwith N .

For the input signal, we have a similar composition

s = s‖ + s⊥ = Bρ(N) + s⊥ (6)


and as the signal model does not have orthogonal complement(it defines the N th plane), then (3) becomes

s‖ = s + e‖ (7)

and

s⊥ = e⊥. (8)

In terms of phasors, we have from (7)

ρ(N) = ρ + ρ(N)

. (9)

Therefore, each phasor estimate contains the constant phasor ofthe sinusoidal signal ρ plus the coordinates of the projectionof the exogen signal onto the N th plane. Therefore, the least-square estimate ρ lives in the N th plane and unavoidablycontains the coordinates of the projection of e as a “minimum”error. Of course, if e = 0, then, for all N ’s, the least-squareestimate is exactly equal to the signal phasor ρ. This means that,for a pure sinusoidal signal, its exact phasor ρ can be quicklyobtained from any of the least-square solutions, provided itexists, being inefficient to wait until an orthogonal basis isreached. This seems surprising, but it is not. The sinusoidalsignal changes from plane-to-plane, not because its coordinateschange, but because the basis vectors in B are changing with N .Therefore, the phasor estimate for the sinusoidal signal remainsconstant in all the planes. The main point of this paper is basedon this condition: partial phasor estimates are exact when theinput signal matches the signal model, and for this very reason,they deserve to be taken into account in the phasor-estimationprocess.

The least-square phasor estimate ρ can be obtained from thenormal equations by substituting (9) in (4). We have

BT Bρ(N) = BT s. (10)

This solution provides the best linear fit, in the least mean-square-error sense, to the data set of size N . The vector on theright-hand side of the equation σ(N) = BT s has been calledphasorlet in a previous work [1], because it corresponds toa partial sum of the phasor estimate obtained with the one-cycle Fourier filter. The normal equations lead to interpret thephasorlet as a linear transformation of the phasor estimate. Thistransformation is given by the Gram matrix G = BT B whichdepends only on the geometrical structure of the sinusoidalmodel. Therefore, the phasor estimate can be obtained as aninverse transform of the phasorlet

ρ(N) = G−1σ(N). (11)

The only condition imposed on the data set by the uniquenessof the least-square solution in (11) is that b1 and b2 be linearlyindependent to ensure the inversion of the matrix G. Forcalculation simplicity, the linear algebra theory recommends anorthogonal basis, and the signal processing theory recommendsa “fast” algorithm, exploiting the symmetry and periodicityproperties of the signal model. But, for protection applications,which need estimates in very short periods of time, neitherorthogonality nor calculation fastness are possible, simply

Fig. 1. Elements and determinant of G(N). The entries of its diagonalincrease as N/2, while its determinant increases as N2.

because there is no time for such calculation commodities.Therefore, the proposed real time fast solution must use theoblique basis and the inversion of a set of 2 × 2 matrices.

Equations (3), (4), and (11) are very important in phasorestimation. They define a family of phasor estimators basedon successively available data contained in vectors with moreand more components as the data acquisition time passes. Thesequicker estimates (obtained in fractions of one cycle) are veryattractive for applications in which the speed is the main con-cern. Applications cannot afford to wait until the orthogonalityof the basis is reached, so they start generating estimates as soonas a signal segment of N samples is available. In this case, thevectors bi, i = 1, 2, which are carrying more and more compo-nents, form nonorthogonal bases generating oblique planes, butfrom anyone of them, phasor estimates can be estimated, usingthe corresponding inverse transformation in (11).

The Gram matrices of the sinusoidal model are given by

G =N−1∑n=0

(cos2(nω0) −cos(nω0) sin(nω0)

−cos(nω0) sin(nω0) sin2(nω0)

)(12)

and in closed form by the following geometric series

G(N) =

N

2 + 14

(1+ sinω0(2N−1)

sinω0

)− 1

2sinω0(N−1) sinω0N

sinω0

− 12

sinω0(N−1) sinω0Nsinω0

N2 − 1

4

(1+ sinω0(2N−1)

sinω0

) .

(13)

Fig. 1 shows the evolution of the entries of G(N) as wellas its determinant as a function of N during one cycle, whichis sampled at N0 = 60 samples per cycle. The elements ofthe diagonal correspond to the energy (square of the length)of the columns of B and, therefore, are nonnegative. Theyexhibit a linear trend, increasing in average as N/2. All theentries exhibit an oscillating behavior, at twice the fundamental


Fig. 2. Maximum and minimum principal eigenvalues of the matrixBG−2BT . They bound the minimum error of the least-square solution onthe corresponding projection plane.

frequency, and the determinant follows a parabolic behavior asN2. Note also that for N < 5, the determinant is very smallbecause one of the vectors is very small. For those values,the Grammian matrices are quasi-singular, and their inversionposes numerical stability problems as it was shown in [1]. AtN = N0/2 and N = N0, the bases are orthogonal, and theGram matrices are equal to (N0/4)I and (N0/2)I, respec-tively. These cases correspond to the half-a-cycle and one-cycleFourier filter estimates, for which the matrix inversion on theleast-square solution is reduced to a scalar inversion. The least-square solution rejects the orthogonal component to the N thplane. In the case of the one-cycle Fourier filter (N = N0),it is well known that harmonics are orthogonal to that plane.This is one of the reasons why the one-cycle Fourier filter is sopopular. However, that does not mean that the filter is free fromerrors. Its estimates will contain an unavoidable “minimum”error, which is equal to the coordinates of the projection of thenonsinusoidal signal (the exogenous signal e) onto the phasorplane. There exists a recursive solution [12] for (11), and it isgiven in the Appendix.

B. Phasor Error Bound

It is possible to bound the “minimum error” contained in theestimates in terms of the energy of the exogen signal. For anyvalue of N , we have

ρ = G−1BT e (14)

and therefore

‖ρ‖2 = eT BG−2BT e (15)

which is a quadratic form of e. In that expression, G−2 is thesquare of G−1. Restricting e such that ‖e‖ = 1 with respect to

Fig. 3. Major and minor axes corresponding to the projection of the principaleigenvectors onto the phasor plane, for N = 15 to 60.

the Euclidian inner product on RN , the quadratic form in (15)

is bounded [13] by the maximum and minimum eigenvalues ofthe positive definite N × N matrix BG−2BT . Positive definitematrices have a complete orthogonal set of real eigenvectorsand positive eigenvalues. In our quadratic form, since the errorρ lies in a plane, the rank of BG−2BT is two, and then, it hasonly two nonzero eigenvalues. Fig. 2 shows the two principaleigenvalues of BG−2BT . They bound the phasor error from anexogen signal of unitary Euclidian norm. For N < (N0/2), oneof the eigenvalues is much larger than the other, which meansthat the hypersphere of unitary norm is projected onto theelongated ellipses (big errors). For N ≥ (N0/2), the ellipsesare almost circular. Fig. 3 shows the major and minor axes ofthese ellipses. They were plotted by projecting the principaleigenvectors of the matrix BG−2BT onto the phasor plane.Their lengths correspond to the eigenvalues of the correspond-ing matrices. As it is shown, the error bound shows a specialorientation as N increases from 15 to 60.

Fig. 4 shows the paths traced by the phasor estimates whenthe vectors of the standard bases of R

N from N =15 to N0 =60are projected. Remember that standard bases are the subsetsof vectors of unitary norm. Each line connects the points (thelocus of the phasors) corresponding to each basis and, therefore,represents the complex impulse response of the N th filter. Wecan see the asymmetry of those phasors. While the vectors ofevery basis cover the whole hyperspace (RN ), their phasors arenot omnidirectional, except for N0. Since any error sequencewill be a linear combination of one of those bases, the totalphasor error will also be a linear combination of those phasors.In order to have a better insight of what happens to the standardphasors when N increases, Fig. 5 shows their paths for each Nin ascending levels. They conform the shape of a funnel whichcoincides with the bounds given by the eigenvalues shown inFig. 2. These figures illustrate that the Fourier phasor-estimate


Fig. 4. Projection of the standard basis vectors of RN , from N = 15 to 60.

Fig. 5. Projection of the standard basis vectors of RN , for N = 15 to 60.

N is presented as a vertical level.

family is always susceptible to errors when the signal does notcorrespond to a perfect sinusoidal sequence.

C. Reduction of Normal Equations

Simpler reductions of the normal equations are possible.One of the simplest was suggested in [1]. It consists in takinginto account only the nth and the n + (N0/4) data triplets

(b1n, b2n, sn). For those couples, the Grammian is the identitymatrix, as it can be shown as follows:

G(n) =∑

�∈{n,n+N04 }

(cos2(�ω0) −cos(�ω0) sin(�ω0)

−cos(�ω0) sin(�ω0) sin2(�ω0)

)

= I, ∀n. (16)

This family of phasor estimates is the simplest one. Itsformula can be written as

ρ(n) =(

cos nω0 −sin nω0

−sin nω0 −cos nω0

)(s(n)

s(n + N0

4

)). (17)

It requires only two signal samples a quarter-of-a-cycle apart.The computational efficiency is extraordinary high (it does notrequire matrix inversion at all). It suggests that the best wayof adding the terms of the one-cycle Fourier filter estimator isnot consecutively, but two-by-two a quarter-of-a-cycle apart. Inaddition, averages of those estimates are also phasor estimates.In particular, if ρ(n) is one of them, for the half-a-cycle Fourierestimate, we have

ρ(N0/2) =4

N0

N0/4∑n=1

ρ(n) (18)

and the average of two consecutive half-a-cycle estimates is theone-cycle Fourier estimate. Therefore, those averages convergeto the half-a-cycle or a one-cycle Fourier phasor estimates,when the involved samples in the average cover those intervals.

D. Phase Compensation for Signal Advancement

Now, our interest is placed in generating estimates, while thesignal advances over the fixed fitting interval. In this case, thesinusoidal sequence changes to sk(n) = cos(ω0(n + k) + θ)and depends on the advancement k. In vectorial terms we have

sk = BRkω0

ρ + ek (19)

where Rω0 is the well-known rotation matrix

Rω0 =(

cos ω0 −sin ω0

sin ω0 cos ω0

)(20)

at ω0 radians or 1/N0 of a cycle, which corresponds to thedelay of the sampling period. In cycles, it can also be denotedas R1/N0 , so its N th power will simply be RN/N0 , whichcorresponds to a fraction of a cycle.

By applying (10)–(19), the sequence of estimates will be nowgiven by

ρk = R−kω0

G−1BT sk, (k = 0, 1, . . . ,K). (21)

This is the equation used to generate the estimates in thenumerical simulations shown in the next section.

The rotation matrix has two important properties: symmetry

Rk+

N02

1N0

= −Rk1

N0(22)


Fig. 6. First row elements and determinant of G(N)3 . Orthogonality is reached

until N = N0.

and periodicity

Rk+�N01

N0

= Rk1

N0. (23)

Finally, note that a positive fractional rotation has an equivalentnegative one

RN1

N0= RN−N0

1N0

. (24)

These properties are very useful to interpret the relationshipsbetween the phasorlets in Section IV.

E. Model Extension

Fault current signals usually contain an aperiodic componentor dc offset. In order to cope with this component, a third vectorb0 with ones is necessary in matrix B. In this case, we have a3 × 3 Grammian matrix, which is given in closed form as

G(N)3

=

N 12

(1+ sin

ω02 (2N−1)

sinω02

)−

(sinω0

4 sin2 ω02

+ cosω02 (2N−1)

2 sinω02

)g12

g13 G(N)2

(25)

in which G(N)2 is the former 2 × 2 Grammian matrix. Fig. 6

shows the entries of its first row (and column) and its deter-minant. Now, the determinant grows as N3, and for N < 10,quasi-singular matrices are found. The smaller the dimensionN , the better is the fit to the exponential by a dc component,because under short segments, the exponential looks like aconstant. In the following section, we consider numerical sim-ulations with real fault signals.

Fig. 7. Magnitude response of the phasor estimator with the simplest model.

Fig. 8. Magnitude response of the phasor estimator with the 2 × 2 model.

III. FREQUENCY RESPONSE OF PHASOR ESTIMATORS

Another way to assess the performance of the estimators isby considering the phasor-estimation process as the output ofa filter bank. The frequency response of the filters is given bytheir response to a complex exponential input signal of the formej(θ/θ0)n, n = 0, . . . , N − 1. Figs. 7–9 show the magnituderesponse of the filters corresponding to the simplest filter givenby (17) and those of the 2 × 2 and 3 × 3 models. It is apparentthat the magnitude responses are not symmetric with respect tozero. Many papers in power systems show only the right part ofthe frequency response, which is useful only when the impulseresponse is real. Note that all the filters have unitary gain at thefundamental frequency (u = 1) and zero gain at its negativevalue (u = −1). The one-cycle Fourier filter has, in addition,zero gain at each harmonic, while the half-a-cycle Fourier filterhas zero gain only at the odd harmonics. The inclusion of the cdcomponent in the 3 × 3 model forces a zero at u = 0 in all the


Fig. 9. Magnitude response of the phasor estimator with the 3 × 3 model.

magnitude responses and increases the sensitivity to nonfunda-mental components, particularly for the shorter intervals. Theone-cycle Fourier filter with cd component remains the samebecause this filter had already a zero at u = 0.

As we can see from the frequency responses, all the esti-mates are very sensitive to the nonfundamental components,but despite their huge sensitivity to these components, all ofthem provide an exact solution (measurement) when the inputsignal coincides with the model, which has spectral lines atu = {−1, 1} or u = {−1, 0, 1}. That is why if the exogenoussignal of the application is known in advance, its model can beincluded with the sinusoidal model. In this way, the exogenoussignal will be handled by a new coefficient of the model.

By the dynamic change on the time interval of the phasorlets,this phasor estimation is similar to the wavelet method, even ifno dilation is applied to a mother wavelet. But, the frequencyresponse of the filters changes as it was shown in the figures ofthis section. There are several strategies to validate the phasorestimate from a phasorlet. One is to use a bandpass filter inorder to reject all the frequencies greater than the fundamentalone. The other is to apply a high pass filter in order to detect thefrequencies higher than the fundamental.

IV. MULTIPLE AND MULTIRATE PHASOR ESTIMATION

It is interesting to relate the full-cycle phasorlet σ(N0) withthe shorter phasorlets σ(N). Let us consider the partitions withdyadic intervals inside a cycle p = 0, 1, . . . , log2 N0. For eachpartition p, the length of the intervals is Np = N0/2p samples,so each partition provides a different time resolution level.For simplicity, let us denote the sequence of phasorlets forthe partition p inside the cycle, as σ2−p

� , � = 0, 1, . . . , 2p − 1.Then, the full-cycle phasorlet is given by

σ10 = [BT

0 | BT1 | · · · | BT

2p−1 ]

s0

s1

· · ·s2p−1

(26)

with

BT� = R−�

2−pBT0 , � = 0, 1, . . . , 2p − 1 (27)

and

s� = B0R�2−p ρ, � = 0, 1, . . . , 2p − 1 (28)

so, we have

σ10 =

2p−1∑�=0

R−�2−pσ2−p

� (29)

since the partial phasorlets at each dyadic interval are given by

σ2−p

� = BT0 B0R�

2−p ρ, � = 0, 1, . . . , 2p − 1. (30)

Equation (29) explains the filtering nature of the full-cycle pha-sor estimate. The full-cycle phasorlet σ1

0 is formed by addingtogether the phasorlets in one cycle, but each one is inverserotated at angles corresponding to the phases of the startinginstances of the dyadic intervals inside the cycle. That rotatorynature of the Fourier transform guarantees the suppression ofthe harmonic error in the phasorlets over one cycle.

It is possible to obtain a full-cycle phasorlet (correspondingto the last cycle) every time we advance to the next dyadicinterval through the following recursive equation

σ1k = R2−p

(σ1

k−1 + σ2−p

k − σ2−p

k−2p

)(31)

in which the previous full-cycle phasorlet is renewed by addingthe difference between the partial phasorlets taken at the newand left dyadic interval and, then, rotated. Then, together witheach phasorlet σ2−p

k , it is easy to generate the phasorlet of thelast cycle σ1

k. In fact, it is possible to generate all the inter-mediate phasorlets, at any interval, through recursive equationssimilar to (31).

Phasorlets of lower resolution can also be generated fromthe higher resolution available phasorlets through the followingdescending relationships. At each higher resolution interval k,we have

σ1/4k =σ

1/8k−1 + R−1

18

σ1/8k , k = 2, 3, . . .

σ1/2k =σ

1/4k−2 + R−1

14

σ1/4k , k = 4, 5, . . .

σ1k =σ

1/2k−4 + R−1

12

σ1/2k , k = 8, . . . . (32)

Each one of those phasorlets corresponds to the last dyadicinterval (1/4, 1/2, or 1 cycle), including the kth higher reso-lution interval. Those relationships start at the third resolutionlevel, but they can be easily generalized to any higher order.It is possible to go further and, by decimation, calculate onlythe phasorlets of disjoint dyadic intervals at the correspondingresolution level

σ1/4n = σ

1/82n−1 + R−1

18

σ1/82n

σ1/2n = σ

1/42n−1 + R−1

14

σ1/42n , n = 1, 2, 3, . . . .

σ1n = σ

1/22n−1 + R−1

12

σ1/22n (33)


Fig. 10. Simplest phasor estimates. Successive averages are also phasorestimates.

These phasorlets are calculated at multirates; the index ofeach phasorlet corresponds to the sequence of disjoint dyadicintervals at each resolution level. Finally, multiple or multi-rate phasor estimates are obtained by applying (11) to thephasorlets and by applying the corresponding inverse rotation.The sequences of phasor estimates obtained by this methodcorrespond exactly to the one obtained by applying the disjointwindows of fractionary cycles, but here, we have them as theoutput of a filter bank.

V. NUMERICAL RESULTS

In order to test and illustrate our proposed solutions, threereal fault signals are analyzed: two currents and one voltagesignal. We start considering the simplest estimates of the faultvoltage signal, which is calculated with samples one-quarter-of-a-cycle apart. For those estimates, the Grammian matrix isthe identity, as it was demonstrated in (16). Fig. 10 shows thephasor estimates. As it is shown, the estimates are very good.The signal contains two quick transitions between the threesinusoidal states. Therefore, these phasor estimates are very fastand accurate. The little oscillations in amplitude are due to theconstant change in phase. This sequence of estimates can besmoothed by averaging the successive estimates as it is shownin Fig. 11, in which the sequence of half-a-cycle estimates,which is calculated every quarter, is illustrated.

Now, we consider the performance of the least-squaremethod in fitting the sinusoidal segments to a current signalwith a strong aperiodic component. The idea is to illustratehow good is the phasorlet method to fit the short sinusoidalsegments, and how its performance degrades as the lengthof the fitting interval increases. Fig. 12 shows the sinusoidalpiecewise fitting to the (dotted line) fault current signal overthe intervals of [Fig. 12(a)] one decicycle and [Fig. 12(b)] onecycle. When the length of the fitted interval is one decicycle, the(dashed line) fitting errors are so small that the (continuous line)fitted signal is almost equal to the (dotted line) input signal.

Fig. 11. Half-a-cycle Fourier estimates every quarter-of-a-cycle.

Fig. 12. Parallel and orthonormal projections of a current signal. The sinu-soidal fitting is applied to segments of (a) one tenth and (b) one cycle.

That means that the current signal segments coincide with thesinusoidal model in each one of those small intervals. Theerror signal is generated by the concatenation of the orthogonalprojections of each signal segment. For short intervals, apparenterrors appear when the signal contains high frequency compo-nents. But, when the fitting is performed over one-cycle intervalas in [Fig. 12(b)], the fitting error is significant on intervalswhere the (dotted line) signal does not have a pure sinusoidalsegment over the whole interval, as it can be clearly seen overthe third and fourth cycles. In these intervals, the error consistsof the residual of a sinusoidal approximation to the signal withan aperiodic component, which is not taken into account inthe model; that is why the residuals of sinusoidal pieces areapparent on those intervals. Note also that, as soon as the fittedcycle contains a segment of a pure sinusoidal signal, the fittingerror is again very small, because it coincides with the fittedpattern. In the last cycle but one, a perceptible error appearsbecause of the discontinuity in the current signal. The resultsobtained in this figure suggest a strong relationship between the


Fig. 13. Phasor estimates from decicycle phasorlets. (Continuous line)Magnitude and (dotted line) phase.

phasor estimation and the curve fitting process. For small inter-vals, the multiplicity of phasorlets generates a lot of sinusoidalshapes that are able to fit with precision in any fault signal piece,even those containing an aperiodic component. In this case, thefitting process is acting as an all-pass filter. As the length ofthe interval increases, the fitting process works well only whenthe signal contains a pure sinusoidal segment over the wholefitted interval. Therefore, the phasor error is significant where anonsinusoidal segment appears in the interval.

Fig. 13 shows the magnitude and phase of the phasor es-timates obtained from the phasorlets of one decicycle (sixsamples) that are taken from the same fault current signal. Animportant phasor error (magnitude and phase) in the third andfourth cycles is apparent. It is precisely there where the bestestimates are needed. This high error is due to the all-passfiltering performed by the piecewise sinusoidal fitting process.In the case of the aperiodic component in fault currents, it ismore efficient to include a dc offset in the signal model andhandle the aperiodic with one of the coefficients of the model.In this case, the first coordinate will handle the estimate of thedc in the aperiodic, and the other two, the phasor estimates.The performance of this extended model will be better (moreaccurate and quicker) for smaller values of N .

Fig. 14 shows the dc component, and the phasor magnitudeand phase estimates for the current signal from the decicyclephasorlets. Even if we have a waver (unsteady) estimate, thealgorithm extracts the aperiodic component with high accuracy,and the phasor estimates are also good. Again, the unsteady be-havior of the estimates is reduced when the estimation intervalis increased. Fig. 15 shows the estimates calculated from one-quarter-of-a-cycle phasorlets, which is a huge improvementin the stability of the estimator. Good estimates can also beobtained from one-sixth-of-a-cycle phasorlets.

Finally, Fig. 16 shows the magnitude and phase of the mul-tirate phasor estimates ρ1/4, ρ1/2, and ρ1 corresponding to thecurrent signal in Fig. 15. In those graphs, the phasor estimatesrefer to the center of the steps and cover the corresponding

Fig. 14. Phasor and dc-offset estimates for the fault current signal from thedecicycle phasorlets.

Fig. 15. Phasor and dc-offset estimates for the fault current signal from one-quarter-of-a-cycle phasorlets.

dyadic intervals, but they are available until the (right corners)last quarter of the steps. We can see that the full cycle phasorestimate is very good, but it comes very late as compared withthe higher resolution estimates. Note that for each cycle, wedispose now seven phasor estimates instead of only one.

The former results demonstrate that it is possible to designstrategies for controlling the high sensitivity of the phasorletsto noise, particularly for fault current and voltage signals, evenif the wavering nature of the phasor estimates persists.

VI. DISCUSSION AND FUTURE WORK

The numerical results demonstrate that the inclusion of aconstant into the signal model reduces the phasor error andprovides a way of handling the exponential error by allocatinga room through a defined component.

The main contribution of this paper is to suggest that byimproving the signal model, faster and more accurate phasor


Fig. 16. Multirate phasor estimates from phasorlets σ1/4, σ1/2, and σ1.

estimates can be obtained, and that Taylor series can be appliedto achieve this improvement, with the advantage that the shorterthe interval, the more exact is the Taylor model.

The typical conditions of fault voltages and currents wereconsidered in the input signal. Other types of noisy signals, suchas white noise, are being further developed. This noise is ofparticular interest in digital transmission systems.

Due to the all-pass filtering nature of the phasorlets, futureresearch is needed to reduce the phasor error due to othernonsinusoidal components or its wavering characteristic. Ofparticular interest is the application of phasor measurementunder power oscillations [14], in which the modulating ampli-tude function can be approached by the first- and second-orderTaylor polynomial.

Further research could also be the use of wavelets or win-dows in shaping the normal equations.

Finally, it is worth to mention that oblique bases allow us togenerate quicker estimates without waiting until orthogonalityis reached to elude a 2 × 2 or 3 × 3 matrix inversion.

VII. CONCLUSION

The extended model presented in this paper is a good ap-proach to reduce the error in phasor estimates from phasorletsand improve their performance (speed and accuracy). Providingquicker and more accurate estimates is very important for somany applications. This paper also demonstrates how, from thephasorlets of the higher available level, the phasorlets of lowerresolution level can be generated; this helps to understand muchbetter the phasor-estimation process from phasorlets and breaksthe old paradigm of using only one phasor estimate.

This paper establishes a better relationship between thetheoretical analysis of the estimators and the error bounds,finds the hierarchical relationships between the phasorlets (andphasor estimates) of descending resolution level, and indicatesa general way to use a very well-known mathematical platform(least squares), in order to obtain faster and accurate pha-sor measurements, without waiting to have orthogonal bases.Certainly, this general platform provides a solid foundation

over which further knowledge construction is possible in thisresearch area.

The phasor-estimation error is really reduced by incorpo-rating the exogenous signal into the signal model, and realtime faster estimates can be found when perturbing signals areknown in advance, such as the exponential signal in protectionapplications.

APPENDIX

RECURSIVE SOLUTION

The recursive solution to (10) is given by

ρ(N+1) = ρ(N)+P(N)β(N)

1+β(N)T P(N)β(N)

(s(N)−β(N)T ρ(N)

)(34)

where s(N) is the observed input signal at the instance N ,β(N)T ρ(N) is its predicted value, and the other factor in itsright term is the estimator correction gain. For P = G−1, thefollowing recursion is needed

P(N+1) = P(N) − P(N)β(N)β(N)T P(N)

1 + β(N)T P(N)β(N)(35)

in which β(N)T = (cos ω0N,−sin ω0N).Note that the sequential estimation requires the initialization

of P and ρ at N , but after that, it does not require any furthermatrix inversion. This is the main advantage of the recursivesolution. But, given the high speed and memory capacity of newdigital processors, there exist commercial applications in whichthe inverse of the Grammian matrices is stored in memory andavailable for several values of N , as it was suggested in [3].From its closed form, a simple procedure can be called for eachvalue or N .

ACKNOWLEDGMENT

The author would like to thank W. Premerlani andM. Adamiak for providing the digital signals analyzed inthis paper.

REFERENCES

[1] J. A. de la O Serna, “Phasor estimation from phasorlets,” IEEE Trans.Instrum. Meas., vol. 54, no. 1, pp. 134–143, Feb. 2005.

[2] M. G. Adamiak, G. E. Alexander, and W. Premerlani, Adaptive Algorithmsfor Secure High Speed Distant Protection. Malvern, PA: GE Power Man-agement, 1996.

[3] A. G. Phadke and J. Thorp, Computer Relaying for Power Systems.Baldock, U.K.: Research Studies Press, 1988.

[4] M. Sachdev and M. Nagpal, “A recursive least error squares algorithmfor power system relaying and measurement applications,” IEEE Trans.Power Del., vol. 6, no. 3, pp. 1008–1015, Jul. 1991.

[5] T. Sidhu, X. Zhang, and V. Balamourougan, “A new half-cyclephasor estimation algorithm,” IEEE Trans. Power Del., vol. 20, no. 2,pp. 1299–1305, Apr. 2005.

[6] M. Wang and Y. Sun, “A practical precise method for frequency track-ing and phasor estimation,” IEEE Trans. Power Del., vol. 19, no. 4,pp. 1547–1552, Oct. 2004.

[7] D. Hart, D. Novosel, Y. Hu, B. Smith, and M. Egolf, “A new frequencytracking and phasor estimation algorithm for generator protection,” IEEETrans. Power Del., vol. 12, no. 3, pp. 1064–1073, Oct. 1997.

[8] J. A. de la O Serna and K. Martin, “Improving phasor measurementsunder power system oscillations,” IEEE Trans. Power Syst., vol. 18, no. 1,pp. 160–166, Feb. 2003.


[9] M. Donolo and V. Centeno, “A fast quality assessment algorithmfor phasor measurements,” IEEE Trans. Power Del., vol. 20, no. 4,pp. 2407–2413, Oct. 2005.

[10] J. A. de la O Serna, “Reducing the error in phasor estimates from phasor-lets in fault voltages and current signals,” in Proc. IEEE Power Eng. Soc.General Meeting, San Francisco, CA, Jun. 12–16, 2005, pp. 843–849.

[11] T. Anderson and P. Händel, “IEEE standard 1057, Cramér–Rao boundand the parsimony principle,” IEEE Trans. Instrum. Meas., vol. 55, no. 1,pp. 44–53, Feb. 2006.

[12] L. L. Scharf, Statistical Signal Processing, Detection, Estimation,and Time Series Analysis. Reading, MA: Addison-Wesley, 1991,pp. 384–386.

[13] H. Anton, Elementary Linear Algebra. New York: Wiley, 2003, ch. 9.[14] J. A. de la O Serna, “Dynamic phasor estimates for power system oscil-

lations and transient detection,” in Proc. IEEE Power Eng. Soc. GeneralMeeting, Montréal, QC, Canada, Jun. 18–22, 2006, pp. 2–3.

José Antonio de la O Serna (SM’03) was bornin San Pedro, Coahuila, Mexico, in 1953. He re-ceived the Ph.D. degree from Telecom, Paris, France,in 1982.

In 1987, he was with the Ph.D. program in elec-trical engineering at the State University of NuevoLeón (UANL), where he was a member of the Doc-toral Committee. Currently, he is a Research Profes-sor at the UANL, Monterrey, Mexico. He was also aProfessor with the Monterrey Institute of Technologyfrom 1982 to 1986. From 1988 to 1993, he was with

the Electrical Department, Polytechnic School, Yaoundé, Cameroon.Dr. de la O Serna is a member of the Mexican Research System.

Reducing the error in phasor estimates from phasorlets in fault voltage and current signals

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