Monitoring of Three-Phase Signals based on Singular-Value...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSG.2019.2898118, IEEETransactions on Smart Grid

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Monitoring of Three-Phase Signals based onSingular-Value Decomposition

V. Choqueuse, Member, IEEE, P. Granjon, A. Belouchrani, Senior Member, IEEE, F. Auger, SeniorMember, IEEE, and M. Benbouzid, Senior Member, IEEE,

Abstract—In this paper, a new approach for the analysis ofthree-phase electrical signals is considered. While most of theexisting techniques are based on fixed transforms such as theClarke transform, this paper investigates the use of a data-drivenapproach called the Singular Value Decomposition (SVD). Ascompared to other transforms, this study shows that the SVDhas the distinct advantage of clearly separating the contributionsof the phasor configuration and signal instantaneous parameters.Under additive white Gaussian noise, this paper also describesseveral algorithms based on the SVD for signal monitoring. Thefirst algorithm can detect unbalanced systems and classify theminto two categories: system with an off-nominal subspace andsystems with ellipticity. The second algorithm can estimate theangular frequency based on the periodogram of the right singularvectors. As compared to other existing approaches, simulationsshow that the proposed techniques give a good compromisebetween computational complexity and statistical performance.

Index Terms—Three-phase signals, Singular-Value Decompo-sition, Generalized Likelihood Ratio Test, Frequency Estimation.

I. INTRODUCTION

Three-phase electrical signals play a key role in powerelectronic applications at the generation, transmission anddistribution side. Under nominal conditions, each phase signalhas a sinusoidal shape with a constant magnitude and afundamental frequency equal to 60Hz (or 50Hz) [1], [2].Moreover, balanced three-phase signals have equal magnitudeand are ideally 2π/3 apart in terms of phase angles. Inpractice, three-phase signals are seldom stationary and thephasor configuration may deviate significantly from the bal-anced case [3]. In a smart grid, these deviations are monitoredusing synchronized devices called Phasor Measurement Units(PMU) [4]–[6]. In particular, two signal parameters are ofmain interest: the signal fundamental frequency [7]–[10] andthe phasor configuration [11], [12]. This has motivated thedevelopment of new low-complexity algorithms for frequencyestimation and unbalanced detection in PMU devices.

Numerous studies have addressed the frequency estimationproblem for electrical signals [6]. A commonly used approachfor frequency estimation is based on the Discrete Time Fourier

V. Choqueuse is with the ENIB, Lab-STICC, UMR 6285 CNRS,29200 Brest, France (e-mail: [email protected]). P. Granjon is withthe GIPSA-lab, UMR CNRS 5216, Grenoble INP, 38402 Saint MartinD’Heres, France (e-mail:[email protected]). F. Augeris with the IREENA, LUNAM, 44606 Saint-Nazaire, France (e-mail:[email protected]). A. Belouchrani is with the Electrical En-gineering Department., Ecole Nationale Polytechnique, Algiers 16200, Al-geria (e-mail: [email protected]). M. Benbouzid is with theIRDL, UMR CNRS 6027 IRDL, University of Brest, 29238 Brest, France([email protected]).

Transform (DTFT) [13], [14] and its extension such as theInterpolated Discrete Fourier Transform (IpDFT) [15]–[17].Nevertheless, as recently pointed out in [18], these single-phase techniques usually require a large number of cyclesand Signal to Noise Ratio (SNR) to meet the requirementsof the IEEE Standard C37.118-2011 dealing with PMU speci-fications. To improve the estimator performance, one possiblesolution is to exploit the multi-dimensional nature of three-phase signals. For balanced signals, a conventional approachis based on the Clarke transform [19], [20]. After Clarketransform, the so-called αβ components can be further pro-cessed to estimate the signal frequency [7], [8], [21]–[25]and other parameters of interest [23], [25]. For unbalancedsignals, several authors have also investigated the use of theαβ components for frequency estimation [7], [24]. However,this strategy has two major drawbacks. First, under unbalancedconditions, the use of the αβ components usually requires so-phisticated approaches since the signal trajectory depends bothon the phasors configuration and on the signal instantaneousparameters. Then, while the Clarke transform is perfectly well-suited for the analysis of balanced three-phase signals, thistechnique is inherently suboptimal under unbalanced condi-tions [10]. Departing from this approach, some authors havealso investigated the use of a data-driven approach called thePrincipal Component Analysis (PCA) for signal analysis [10],[23]. As compared to the Clarke transform, the PCA has thedistinct advantage of extracting two orthogonal componentsdirectly from the covariance matrix of the multi-dimensionalsignal whatever the unbalance configuration [26].

Regarding the unbalance detection problem, conventionalapproaches are based on the computation of the voltageunbalance factors [27] or the analysis of the αβ compo-nents [28]. To improve the detection performance, someauthors have investigated the use of parametric approaches.These include the Generalized Likelihood Ratio Test (GLRT)[29], the Generalized Locally Most Powerful Test (GLMP)[30], or Information Theoretical Criteria techniques [31], [32].While these techniques are based on the raw three-phasesignal, a GLRT-based approach has also been proposed forthe challenging case where only the native PMU outputsare available for detection purpose [33]. As compared tonon-parametric approaches, parametric techniques have betterstatistical performance but are less robust to model mismatch.In particular, theses parametric detectors assume perfectlysine-wave signals and their performance degrades quickly inthe presence of time-varying amplitude and/or frequency.

In this paper, we propose a new approach for the analysis of



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three-phase signals based on a Singular-Value Decomposition(SVD). As compared to other multi-dimensional tools suchas the Clarke transform or the PCA, the main advantage ofthe SVD is that it gives a better insight into the structure ofthe three-phase signal. Specifically, by decomposing the three-phase signal into a product of three matrices, the SVD allowsto clearly distinguish between the contribution of the phasorsand signal instantaneous parameters. In particular, contraryto the PCA which has been previously used for estimationpurpose [10], [23], the proposed technique can be used both forunbalance detection and frequency estimation. In this context,the contribution of this paper is twofold:• First, we show how the SVD is related to the structure

of the three-phase signal.• Second, under Gaussian noise assumption, we show how

to process the SVD to detect different types of off-nominal conditions and to estimate the signal frequency.

The paper is organized as follows. Section II introducesthe signal model and the assumptions made in the study.Section III presents the expression of the SVD of the three-phase signal. Section IV shows how to detect off-nominalconditions from the SVD and how to estimate the signalfrequency. Finally, Section V reports on the performance ofthe proposed techniques with simulation signals.

II. SIGNAL MODEL

A. Three-phase Signal Model

Unbalanced three-phase signals can be described undernoiseless condition by the following scalar-form model [34]

xk[n] = dka[n] cos(φ[n] + ϕk), (1)

where the quantities a[n] and φ[n] correspond to the instanta-neous amplitude and phase respectively, while dk ≥ 0 andϕk ∈ [0, 2π] correspond to the scaling factor and initialphase-angle on phase k = 0, 1, 2. Using complex notations,the three-phase signal can be expressed more compactly aszk[n] = <e(ckz[n]) where the complex stationary phasors ck(k = 0, 1, 2) and the complex signal z[n] are given by

ck , dkejϕk (2a)

z[n] , a[n]ejφ[n]. (2b)

In practice, note that three-phase signal can also contains otherfrequency components such as harmonics or inter-harmonics.In this study, we assume that these high-frequency componentshave been removed in a pre-processing stage by using a digitallow-pass filter1.

The goal of this study is to analyse the complex phasorsck and the complex signal z[n] from N samples of the three-phase signal xk[n] (k = 0, 1, 2). By using matrix notations,the three-phase signal of Eq. (1) can be expressed as

X = CZT , (3)

where (.)T denotes the matrix transpose, and

1It should be noted that the use of a digital low-pass filter introduces aphase shift and can increase the computation time.

• X is a 3×N matrix which is defined as

X =

x0[0] · · · x0[N − 1]x1[0] · · · x1[N − 1]x2[0] · · · x2[N − 1]

, (4)

• Z is a N × 2 matrix containing the real and imaginaryparts of the complex signal z[n] and is defined as

Z ,[<e(z) =m(z)

], (5)

where z ,[z[0] · · · z[N − 1]

]T,

• C is a 3× 2 matrix which is defined as

C ,[<e(c) =m(c∗)

], (6)

where (.)∗ denotes the complex conjugate and c ,[c0 c1 c2

]T.

B. AssumptionsIn Eq. (1), the signal model cannot by uniquely identified.

Indeed, as xk[n] = <e(ckz[n]), it can be easily checked thatxk[n] = <e(cbkzb[n]) where cbk = βck and zb[n] = z[n]/βwith β ∈ C∗. In this study, we assume a particular normalisa-tion on the stationary phasors and complex signal in order toobtain simple expressions for the SVD.

Assumption 1. The energy of z[n], denoted εz , is equal to

εz , zHz =N−1∑

n=0

|z[n]|2 = 2. (7)

Assumption 2. The sum of the squared phasors, denoted qc,is a positive real number i.e. Arg (qc) = 0 where

qc , cT c =2∑

k=0

c2k. (8)

It is important to note that these two assumptions arenot restrictive. Indeed, it is always possible to preserve thetwo requirements εz = 2 and Arg (qc) = 0 by makingthe substitution ck → βck and z[n] → z[n]/β with β =√εz/2e

−jArg(qc)/2. To obtain simple expressions for the SVD,we also make the classical assumption that z[n] is an analyticsignal, at least asymptotically. Mathematically, a signal is saidto be analytic if its Discrete Time Fourier Transform (DTFT),Z(ω), satisfies [35]

Z(ω) ,N−1∑

n=0

z[n]e−jωn = 0 for − π < ω < 0. (9)

By using the Parseval Identity, this assumption implies that

qz ,N−1∑

n=0

z2[n] =1

2π

∫ π

−πZ(ω)Z(−ω)dω = 0. (10)

By using Assumptions 1 and 2 and Eq. (10), we obtain thefollowing proposition.

Proposition 1. The matrices ZTZ and CTC can be decom-posed as

ZTZ = I2 (11a)

CTC =1

2

[εc + qc 0

0 εc − qc

], (11b)



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where I2 is the identity matrix of size 2× 2, and εc , cHc =∑2k=0 |ck|2.

III. SINGULAR-VALUE DECOMPOSITION OF THETHREE-PHASE SIGNAL

As C in (3) is a 3 × 2 matrix, the Singular-Value Decom-position of X presents at most two non-zero singular values.Therefore, the ”compact” SVD can be expressed as

X = UxSxVTx (12)

• Ux is a 3×2 semi-orthogonal matrix (UTxUx = I2) that

contains the left singular vectors of X,• Sx = diag(σ2, σ1) is a 2×2 diagonal matrix that contains

the singular values of X with σ2 ≥ σ1,• Vx is a N × 2 semi-orthogonal matrix (VT

xVx = I2)that contains the right singular vectors of X.

In the following subsections, we show how the SVD matricesare related to the complex phasors ck and analytic signal z[n].

A. General case

Proposition 2. Using Proposition 1, the SVD of X can beexpressed as X = UxSxV

Tx , where

Ux = CS−1x (13a)

Sx =

√

εc+qc2 0

0√

εc−qc2

(13b)

Vx = Z (13c)

with εc =∑2k=0 |ck|2 and qc =

∑2k=0 c

2k.

Proof. Using Eq. (13a) to (13c), one can check thatUxSxV

Tx = CS−1x SxZ

T = CZT = X. Then, we haveto verify that Ux and Vx are semi-orthogonal matrices.Proposition 1 and Eq. (13b) lead to CTC = S2

x and soUT

xUx = S−1x CTCS−1x = I2. Then, Proposition 1 andEq. (13c) show that VT

xVx = ZTZ = I2.

Remark 1. For the particular case where qc = 0, the SVDof X is not unique since X can also be decomposed as X =UxQSxQ

TVTx where Q is an 2× 2 orthogonal matrix.

The previous proposition gives the structure of the ”com-pact” SVD. From a geometrical point of view, it is alsointeresting to analyse the full SVD, which is given by

X =[Ux u0

]σ2 0 00 σ1 00 0 0

[VT

x

vT0

], (14)

where u0 and v0 are the left and right singular vectorsassociated with the null singular value of X. Regarding the leftsingular vector, u0 can be obtained from the solution of theequation CTu0 = 0 under the unit-norm constraint uT0 u0 = 1.By imposing the unit-norm constraint, u0 can be decomposedas u0 = g/‖g‖ where g is the vector cross-product betweenthe two columns of C, i.e. g = <e(c) ∧ =m(c∗). Therefore,we obtain the following proposition2.

2Note that this proposition constitutes a generalization of a previous resultobtained in [36] for purely stationary three-phase signals.

Proposition 3. The normal vector perpendicular to the signalsubspace is given by

u0 =g

‖g‖ , (15)

where g = =m([c1c∗2 c2c

∗0 c0c

∗1

])T.

B. Nominal conditionUnder nominal condition, the three-phase signal is balanced

and the corresponding complex phasors can be expressed as

ck = c0e−2kjπ/3. (16)

It follows that qc = 0 and εc = 3|c0|2 and we obtain thefollowing decomposition.

Proposition 4. Under nominal condition, the SVD of X canbe expressed as X = UxSxV

Tx where Ux = αC, Sx = 1

αI2,Vx = Z, and α =

√2

3|c0|2 .

Regarding u0, it can be checked that c1c∗2 = c2c∗0 = c0c

∗1 =

|c0|2e2jπ/3. Therefore, we obtain the following result.

Proposition 5. Under nominal condition, the normal vectorperpendicular to the signal subspace i.e.

u0 = n =1√3

[1 1 1

]T. (17)

C. Geometrical interpretationsSeveral studies have shown the advantage of plotting the

three-phase signal in the 3D Euclidean space [36]. As anexample, Figure 1 presents the 3D trajectory for the case ofan off-nominal modulated three-phase signal. The quantitiesdefined through the SVD of Eq. (12) may be interpreted witha geometric point of view, and shed some highlights on the3D trajectory of X.• From Eq. (5) and (13c), it is clear that the right singular

vectors Vx contain the real and imaginary parts ofthe analytic signal z[n], which correlation matrix is theidentity matrix (see Eq. (11a)). This indicates that the realand imaginary parts of z[n] are uncorrelated and have thesame energy, or equivalently that the scatter plot of z[n]in the complex plane has a global circular shape.

• The diagonal matrix Sx containing the two singularvalues of X can be interpreted as a scaling matrix appliedto the previous complex-valued signal, where σ2 is thescaling factor applied to <e(z[n]) and σ1 is appliedto =m(z[n]). Consequently, these two positive factorscontain the information on the global ellipticity of thetrajectory followed by X, σ2 being related to its semi-major axis and σ1 to its semi-minor axis.

• The left singular vectors Ux form an orthonormal basisfor the three-phase signal X, and therefore give theorientation of the plane containing the trajectory of thewhole three-phase signal, as well as the orientation ofits minor and major axes. Equivalently, the 3× 3 matrix[Ux u0

]appearing in the full SVD of X in Eq. (14)

is an orthogonal matrix, and can also be interpreted as a3D rotation matrix applied to the previous plane elliptictrajectory.



4

−5 0 5

·10−2

−5

0

5

·10−2

v2[n]

v 1[n]

(a) Trajectory of the two components of VTx

−2 −1 0 1 2

−1

0

1

×σ2

×σ1

w2[n]

w1[n]

(b) Trajectory of the two components of SxVTx

−2 0 2 −202−2

0

2

u1

u2

u0

n

x0[n]x1[n]

x2[n]

(c) Trajectory of the three-phase signal X = UxSxVTx

Fig. 1: Geometrical analysis of the trajectory of the three-phasesignal (c0 = 1, c1 = 0.8e−2.2jπ/3 and c2 = 1.9e−3.9jπ/3,amplitude-modulated sine wave signals).

IV. ESTIMATION / DETECTION OF OFF-NOMINALCONDITIONS UNDER GAUSSIAN NOISE

In practice, the signal is usually corrupted by an additivenoise i.e. yk[n] = xk[n] + wk[n]. Using a matrix form, thenoisy signal can be expressed as

Y ,

y0[0] · · · y0[N − 1]y1[0] · · · y1[N − 1]y2[0] · · · y2[N − 1]

= X+W, (18)

where X = CZT is the noiseless signal defined in Eq. (3) andW contains the additive noise. In the following, we assumethat the element of W located in the uth row and vth column,denoted w[u, v], has a Gaussian distribution with zero-meanand variance equal to σ2, i.e. w[u, v] ∼ N (0, σ2) [3]. More-over, this noise is assumed to be white in the time domain(regarding v) and uncorrelated from phase to phase (regardingu). Note that because of the additive noise, the SVD of Yis generally different from that of X. The SVD of Y can beexpressed as

Y = USVT , (19)

• U = [u2, u1, u0] is a 3× 3 orthogonal matrix,

• S = diag(σ2, σ1, σ0) is a 3 × 3 diagonal matrix withσ2 ≥ σ1 ≥ σ0 ≥ 0,

• V = [v2, v1, v0] is a N × 3 semi-orthogonal matrix.In this section, we propose to detect unbalance systems

from the structure of U and S, and to estimate the angularfundamental frequency from the right singular vectors V.

A. Off-nominal subspace detection test

As stated in Prop. 5, the noiseless three-phase signal belongsto a plane with normal vector u0 = n under nominal condition.To detect a off-nominal signal subspace, we therefore proposeto address the following hypothesis testing problem:• H0: u0 = n• H1: u0 6= n;

where n is defined in (17). Then, we propose to use aGLRT detector for choosing between the two hypothesis. Thisdetector chooses H1 if

Tn(Y, σ2) = 2ln

(L(Y; Z, σ2,H1)

L(Y; Z, σ2,H0)

)> τ, (20)

where L(Y;Z, σ2,Hk) corresponds to the likelihood functionof Y [37, Section 6.5]. It is demonstrated in appendix A that

Tn(Y, σ2) =1

σ2(‖nTY‖2F − σ2

0). (21)

This detector relies on the difference between two positiveterms normalized by the noise variance. Obviously, Tn(Y, σ2)is close to zero under nominal condition, and positive oth-erwise. Moreover, under assumption H0, it is known thatthe GLRT follows a chi-squared distribution with r degreesof freedom, where r corresponds to the number of freeparameters under H0 [37, Section 6.5]. As u0 is a 3 unit-norm column vector, it follows that the criterion follows underH0 a chi-squared distribution with 2 degrees of freedom,i.e. Tn(Y, σ2) ∼ χ2

2. Note that the evaluation of Tn(Y, σ2)requires the knowledge of the noise variance. When thenoise variance is unknown, we propose to replace it by thefollowing estimator σ2 =

σ20

N . For large N , as the variance ofσ2 is negligible as compared to the variance of Tn(Y, σ2),the distribution of Tn(Y, σ2) can be approximated by thedistribution of Tn(Y, σ2). Using this approximation, we obtainthe following proposition.

Proposition 6. The GLRT detector chooses hypothesis H1 if

Tn(Y) =N

σ20

(‖nTY‖2F − σ20) = N

(‖nTY‖2Fσ20

− 1

)(22)

is greater than a threshold τm. Furthermore, under H0,Tn(Y) follows, asymptotically, a chi-squared distribution with2 degrees of freedom.

In practice, the test threshold τm can be set according to thedesired probability of false alarm, pfa , 1−

∫ τm0

p(Tn(Y) =x|H0)dx. Indeed, by using Proposition 6, we obtain

p(Tn(Y) = x|H0) =

{12e− 1

2x x > 00 x < 0

(23)

and so τm = −2 ln(pfa).



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B. Ellipticity detection test

Under nominal conditions the two largest singular values ofthe three-phase signal are equal. Then, we propose to addressthe following ellipticity detection test:• H0: the signal is globally circular i.e. σ2 = σ1,• H1: the signal is not globally circular i.e. σ2 6= σ1,

The GLRT detector chooses H1 if

Tσ(Y, σ2) = 2ln

(L(Y; X, σ2,H1)

L(Y; X, σ2,H0)

)> τ. (24)

It is demonstrated in the Appendix B that

Tσ(Y, σ2) =1

2σ2(σ2 − σ1)2 . (25)

This equation shows that the GLRT approach leads to asimple intuitive criterion that is equal to the weighted squareddifference of the two largest singular values. This criterion canalso be viewed as the weighted squared difference between thesemi-major and semi-minor axes of the ellipse approximatingthe signal 3D trajectory. Under assumption H0, the two singu-lar values are nearly equal and their difference is close to 0 i.e.Tσ(Y, σ2) ≈ 0. Under assumption H0 and additive Gaussiannoise, thank’s to the properties of the GLRT, the decisionstatistic follows asymptotically a chi-squared distribution with2 degrees of freedom, i.e. Tσ(Y, σ2) ∼ χ2

2. When the noisevariance is unknown, we propose to replace σ2 by σ2 =

σ20

Nusing similar arguments as in section IV-A.

Proposition 7. The GLRT detector chooses hypothesis H1 if

Tσ(Y) =N

2σ20

(σ2 − σ1)2 (26)

is greater than a threshold τm. Furthermore, under hypothesisH0, Tσ(Y) follows, asymptotically, a chi-squared distributionwith 2 degrees of freedom.

C. Estimation of the angular frequency

The analytic signal z can be estimated from the 2 principalright-singular vectors Vx = [v2, v1] as z = v2 + jv1. It isdemonstrated in Appendix C that the estimated analytic signalz = [z[0], · · · , z[N − 1]]T can be approximated by

z[n] ≈ a[n]ej(ρφ[n]+ϕ) + b[n], (27)

where b[n] contains the additive noise, ϕ corresponds to anunknown phase-shift and ρ ∈ {1,−1}. Regarding the additivenoise, as w[u, v] ∼ N (0, σ2), b[n] follows a complex zero-mean noncircular white Gaussian distribution. Note that theadditive noise becomes circular if and only if σ2

1 = σ22 . In

the following, we show how to accurately estimate the signalangular frequency from z[n] for the particular case where theinstantaneous amplitude is constant, i.e. z[n] = cejω0n. Toestimate the unknown signal parameters, a natural approach isto minimize the sum of squared residuals between z[n] andcejω0n. Mathematically, the Least-Squares (LS) estimator isthen given by

{ρ, ω0, c} = arg minr,ω,c

N−1∑

n=0

|z[n]− cejrωn|2, (28)

where r ∈ {1,−1}, ω ∈ [0, π[ and c ∈ C. In the follow-ing, instead of minimizing the LS cost-function directly, wepropose an alternative low-complexity technique based on thedecoupled estimation of ρ, ω0, and c.

1) Estimation of ρ: As z[n] is an analytic signal, itsDiscrete Time Fourier Transform is equal to 0 for −π < ω < 0(modulo 2π). In order to estimate ρ, a simple solution is tocompare the energy of z[n] in the frequency band 0 < ω < πto the one in the band π < ω < 2π. Specifically, let us definethe DTFT of the truncated signal z as

Zm ,M−1∑

n=0

z[n]e−j2πnmM , (29)

where M < N is the truncated signal length. Then, a simpleestimator of ρ is given by

ρ =

1 ifM/2−1∑

m=0

|Zm|2 ≥M−1∑

m=M/2

|Zm|2

−1 elsewhere

. (30)

2) Estimation of c and ω0: In (28), it can be checkedthat the LS cost-function is minimized when c =1N

∑N−1n=0 z[n]e

−jrωn [38]. Then, by replacing c and ρ by theirestimate, the angular frequency estimator can be obtained fromthe maximizer of the direction-corrected Periodogram of z i.e.

ω0 = argmaxω

1

N

∣∣∣∣∣N−1∑

n=0

z[n]e−jρωn

∣∣∣∣∣

2

. (31)

D. Algorithms Summary

Under additive Gaussian noise, we have proposed twotechniques for the detection of unbalanced systems: the firstone is described by the Algorithm 1 and can be used todetect an off-nominal signal subspace, the second one isdescribed by the Algorithm 2 and can be used to detect anelliptic signal trajectory. The proposed frequency estimator isdescribed by the Algorithm 3. Figure 2 shows how to combinethese algorithms for the analysis of three-phase signals. Froma physical viewpoint, it should be mentioned that the proposeddetectors can distinguish between two classes of unbalancedsystems. Specifically, the first class corresponds to the casewhere the zero-sequence is not equal to 0 while the secondone corresponds to the case where the signal is not circularin the 2D plane. If a three-phase signal belongs, at least, toone of these classes, then the system is unbalanced. Note thatthese two classes are not exclusive since some unbalancedsystems may have a non-null zero-sequence component andbe non-circular in the 2D plane.

V. SIMULATION RESULTS

To highlight the interest of this study, this section reports onthe performance of the proposed techniques with Monte Carlosimulations. In each simulation, the three-phase signal is de-scribed by the signal model in (18) where w[u, v] ∼ N (0, σ2).Concerning the analytic signal, z[n] = cejω0n where the(normalized) angular frequency is given by ω0 = 2πf0/Fs =120π/Fs, f0 = 60 Hz corresponds to the signal frequency, and



6

Y

b�0

bS

bV

knT Yk2F Tn(Y)

T�(Y)

bv2 + jbv1

Off-nominal subspace detection

(Algorithm 1)

Ellipticitydetection

(Algorithm 2)SVD

Angular FrequencyEstimation

(Algorithm 3)

Fig. 2: Analysis of three-phase signals using SVD

Algorithm 1 Blind Off-nominal subspace detection test

Input: Three Phase Signal Y, threshold τm.Output: detected (True or False)

1: Compute the SVD of Y = USVT

2: Extract the smallest singular value σ0 from S3: Compute ‖nTY‖2F4: Compute the test criterion Tn(Y) in (22).5: if (Tn(Y) > τm) then6: detected ← True7: else8: detected ← False9: end if

10: return detected

Algorithm 2 Blind Ellipticity detection test

Input: Three Phase Signal Y, threshold τσ .Output: detected (True or False)


2: Extract the three singular values σ2 ≥ σ1 ≥ σ0 from S3: Compute the test criterion Tσ(Y) in (26).4: if (Tσ(Y) > τσ) then5: detected ← True6: else7: detected ← False8: end if9: return detected

Algorithm 3 Angular frequency Estimator

Input: Three Phase Signal Y, truncated signal length MOutput: Angular frequency ω0


2: Estimate the analytical signal with z = v2 + jv1

3: Compute the DTFT of the truncated signal z[n] (n =0, · · · ,M − 1).

4: Estimate the rotating direction ρ using (30).5: Estimate ω0 by maximizing the direction-corrected peri-

odogram of z in (31) using an optimisation algorithm.6: return ω0

Fs = 60 × 24 = 1440 Hz is the sampling rate. Using theseparameters, the cycle duration is equal to N = 24 samples.For the considered signal, it should be mentioned that therequirement in (10) is only satisfied when N = kπ/ω0 = 12k(k ∈ N) or asymptotically when N � π

ω0. The values of ck are

set according to one of the configurations in Table I, where thefour configurations refer respectively to a perfectly balanced

TABLE I: Phasor Configuration

Phasor c0 c1 c2Configuration 1 1 e−j2π/3 e−j4π/3

Configuration 2 1 0.9× e−j2π/3 0.95× e−j4π/3



signal (u0 = n and σ2 = σ1), a signal with off-nominalsubspace (u0 6= n), a non-circular signal (σ2 6= σ1), and a fullunbalanced signal. In the following simulations, the Signal toNoise Ratio (SNR) is defined as SNR = 10 log(‖c‖2/(6σ2))and the performance is estimated from 104 Monte Carlo trials.

A. Off-nominal subspace detection

This subsection presents the performance of several detec-tors for the discrimination of phasor configurations 1 and 2in Table I. In each simulation, three detectors are considered:the clairvoyant detector that requires a perfect knowledge ofσ2 (denoted clair.), the blind detector (denoted blind), andthe GLRT detector proposed by Sun that requires a perfectknowledge of σ2 and ω0 [29]. Mathematically, the clairvoyantand blind detectors are based on the GLRT expressions givenby Eqs.(21) and (22), respectively.

Figure 3a gives the probabilities of detection, Pd, and falsealarm, Pfa, versus the signal length N when the test thresholdis computed as τm = −2 ln(pfa) with pfa = 0.1. Weobserve that the three detectors allow to detect the off-nominalsubspaces with high probability when N > 80 samples.In particular, the proposed blind and clairvoyant detectorsperform well even if the condition qz = 0 is not perfectlyfulfilled. Moreover, we note that the experimental probabilityof false alarm is close to its theoretical value, pfa = 0.1, fromN = 100 samples. Figure 3b shows the Receiver OperatingCharacteristic (ROC) curve i.e. the probability of detectionversus the probability of false alarm for different values of thetest threshold τm. The ROC curves are evaluated for N = 48samples and different SNRs. We observe that the clairvoyantand blind detectors have similar performance whatever theSNR. We also note that the Sun detector clearly outperformsthe two proposed detectors whatever the SNR. Nevertheless, itshould be mentioned that the performance of the Sun detector,which has been designed for perfectly sine-wave signals,significantly degrades in the presence of dynamic conditions.To illustrate this behavior, Fig 3c presents the evolution ofthe experimental probability of false alarm in presence ofamplitude modulation. We see that Sun detector is not able tomaintain the probability of false alarm to pfa = 0.1 when themodulation index increases, while the two proposed detectorsare insensitive to the presence of amplitude modulation.

B. Ellipticity detection

This subsection reports on the performance of several de-tectors for the discrimination of phasor configurations 1 and 3in Table I. In each simulation, three detectors are considered:the proposed clairvoyant detector, the proposed blind detector,and the Sun detector [29]. The two proposed detectors arebased on the equations (25) and (26), respectively. Figure 4a



7

20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

pfa

N (samples)

Prob

abili

ty

Pfa: Clair. (H0)Pfa Blind (H0)Pfa: Sun [29] (H0)Pd Clair. (H1)Pd Blind (H1)Pd Sun [29] (H1)

(a) Probabilities of detection (Pd) and false alarm (Pfa) versusN (SNR = 20 dB, pfa = 0.1).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

SNR = 10 dB

SNR = 15 dB

SNR = 20 dB

Probability of False Alarm

Prob

abili

tyof

Det

ectio

n

ClairvoyantBlind

Sun [29]

(b) ROC curves (N = 48 samples, 2 cycles).

0 5 · 10−2 0.1 0.150

0.2

0.4

0.6

0.8

1

Modulation Index ka

Exp

. Pro

babi

lity

ofFa

lse

Ala

rm

pfa = 0.1

ClairvoyantBlind

Sun [29]

(c) Experimental Probability of False Alarm versus Modula-tion index (SNR = 40 dB, N = 48 samples, pfa = 0.1,a[n] = 1 + ka cos(2π(5/Fs)n)).

Fig. 3: Off-nominal subspace detection (phasor configurations1 and 2 in Table I, ω0 = 120π/Fs)

shows the values of Pd and Pfa versus the signal lengthN when pfa = 0.1. We observe that the probabilities ofdetection and false alarm exhibit strong oscillations for thetwo proposed detectors. These oscillations are caused by thefact that the assumption qz = 0 is not fulfilled for smallvalues of N . We also note that the experimental probabilityof false alarm seems to be exactly equal to its expected valuewhen N = 12k (k ∈ N). Figure 4b presents the ReceiverOperating Characteristic (ROC) for the two detectors when

N = 48 samples. We observe that a large SNR is requiredto guarantee satisfying performance. Figure 4c reports on theability for the three detectors to maintain the probability offalse alarm in presence of amplitude modulation. We see thatthe performance of the considered techniques critically dependon the number of samples. Indeed, we observe that the threedetectors are not able to maintain the probability of false alarmwhen N = 48 samples, but lead to satisfactory results whenN = 288 samples. As compared to the proposed techniques,we observe that the Sun detector clearly outperforms theproposed approaches for ellipticity detection. Nevertheless, itshould be emphasized that this detector requires a perfectknowledge of the fundamental frequency and noise variance,while our techniques can be used without a priori informationabout these parameters.

C. Angular frequency estimation

In this section, we focus on the angular frequency estimationproblem. Specifically, this section compares the performanceof the following techniques for frequency estimation.• The proposed SVD-based estimator described in subsec-

tion IV-C (SVD).• The DTFT algorithm obtained by maximizing the peri-

odogram of the signal on the first phase x0[n] [13].• The Maximum Likelihood Estimator (ML) [10].• The Maximum Likelihood Estimator under balanced as-

sumption (ML-bal) [25],• The Clarke-based approximate Maximum Likelihood Es-

timator (ML-Clarke) [7],For these techniques, the frequency estimate is obtained bymaximizing a cost-function whose evaluation requires thecomputation of several DTFTs. These cost-functions are max-imized by using the Nelder-Mead optimisation algorithm withan initial value ωinit = 120π/Fs [39]. For the proposedestimator, the SVD has been implemented using the svd()subroutine of Python/Numpy and the rotating direction hasbeen estimated from (30) with M = 8. To assess the estimatorperformances, the Mean Square Error MSE , E[(ω0− ω0)

2]has been evaluated using several Monte Carlo trials. Note thatthe MSE can be decomposed as MSE(ω0) = var(ω0) +bias2(ω0, ω0) where var(ω0) is the estimator variance andbias(ω0, ω0) = E(ω0) − ω0 corresponds to the estimatorbias [38]. In each trial, the angular frequency is set toω0 = 2π × 65/Fs (f0 = 65Hz) and the phasors are given bythe configuration 4 in Table I. The approximate Cramer-RaoBound CRB = 24σ2

N3‖c‖2 is also reported for comparison [10].Figure 5a presents the evolution of the MSE versus N . We

observe that the ML estimator achieves optimal performancefor N > 20 samples. Concerning the ML-Clarke and ML-baltechniques, we note that the MSE exhibits a strong oscillationwith period N ≈ 12 samples. To explain this behavior, Fig. 5breports on the estimator bias for the considered techniques. Wesee that the oscillations of the MSE for the DTFT, ML-Clarkeand ML-bal techniques are due to the estimator bias. We alsonote that the contribution of the bias seems to be negligiblefor the proposed and ML techniques. Finally, Figure 5c showsthe influence of the SNR on the MSE. Except for the ML



8

100 200 3000

0.2

0.4

0.6

0.8

1

pfa

N (samples)

Prob

abili

ty

Pfa: Clair. (H0)Pfa Blind (H0)Pfa: Sun [29] (H0)Pd Clair. (H1)Pd Blind (H1)Pd Sun [29] (H1)

(a) Probabilities of detection (Pd) and false alarm (Pfa) versusN (SNR = 20 dB, pfa = 0.1).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

SNR = 10 dB

SNR = 15 dBSNR = 20 dB

Probability of False Alarm

Prob

abili

tyof

Det

ectio

n

ClairvoyantBlind

Sun [29]

(b) ROC curves (N = 48 samples, 2 cycles).

0 5 · 10−2 0.1 0.150

0.2

0.4

0.6

0.8

1

N = 288 samples

N = 48 samples

Modulation Index kx

Exp

. Pro

babi

lity

ofFa

lse

Ala

rm

pfa = 0.1

ClairvoyantBlind

Sun [29]

(c) Experimental Probability of False Alarm versus Mod-ulation index kx (SNR = 40 dB, pfa = 0.1, a[n] =1 + ka cos(2π(5/Fs)n)).

Fig. 4: Ellipticity detection (phasor configurations 1 and 3 inTable I, ω0 = 120π/Fs)

technique, we observe that the estimators exhibit an error floorat SNR > 30 dB. This error floor appears to be caused by theestimator bias for finite N .

To evaluate the estimator performance, the IEEE StandardC37.118.1-2011 has introduced a criterion called the Fre-quency Error (FE) which is defined as FE = |f0 − f0| [40].To be compliant with this standard, the frequency error mustbe bounded by FE < 0.005. As pointed out in [10], whenthe signal is corrupted by an additive noise, this criterion is

10 20 30 40 5010−6

10−5

10−4

10−3

10−2

10−1

N (samples)

MSE(ω

0)

ProposedDTFT [13]

ML-bal [25]ML-Clarke [7]

ML [10]CRB

(a) Angular frequency estimation: Mean Squared Error(SNR = 10 dB)

10 20 30 40 50

10−6

10−4

10−2

N (samples)

bias2(ω

0)

ProposedDTFT [13]


ML [10]

(b) Angular frequency estimation: Estimator bias2 (SNR =10 dB)

10 20 30 40

10−8

10−6

10−4

SNR (dB)

MSE(ω

0)

ProposedDTFT [13]


ML [10]CRB

(c) Angular frequency estimation: Mean Squared Error (N =24 samples)

Fig. 5: Angular Frequency Estimation (phasor configuration 4in Table I, ω0 = 2π × 65/Fs).

not well suited for the analysis of the estimation performancesince the frequency error becomes a random variable. In thiscontext, a natural extension is to evaluate the probability ofFrequency Error compliance, i.e. PFE , P [|f0−f0| < 0.005].Figure 6 presents the evolution of PFE versus N . It showsthat the performance of the proposed estimator is closed tothe one of the ML technique, and that the performance of theother estimators are strongly affected by the presence of theestimator bias.



9

50 100 150 2000

0.2

0.4

0.6

0.8

1

N (samples)

PFE

ProposedDTFT [13]


ML [10]

Fig. 6: Probability of FE Compliance (SNR = 40 dB)

TABLE II: Average Computation time per trial versus N

Technique N = 50 N = 100 N = 150

Proposed 0.71ms 0.83ms 1.05msDTFT [13] 0.58ms 0.67ms 0.83 ms

ML-bal [25] 0.58ms 0.70ms 0.88 msML-Clarke [7] 0.80ms 1.02ms 1.33 ms

ML [10] 1.72ms 2.00ms 2.40ms

Table II reports on the average computational time forthe considered estimators. These estimators have been imple-mented using Python and the simulations have been run onan Linux machine with an Intel CoreTM i7-6700 processor(CPU at 3.4 Ghz). We observe that the proposed technique hasa lower computational complexity than the recently proposedML-Clarke technique. This result is not surprising since theML-Clarke cost-function is composed of 2 DTFTs while theproposed cost-function only requires the computation of 1DTFT.

VI. CONCLUSION

In this paper, we have proposed a new approach for theanalysis of three-phase signals based on the SVD. Instead ofusing a fixed transform for signal analysis, the SVD directlyexploits the content of the three-phase signal for dimensionalreduction and gives a better insight into the signal structure.Under additive white Gaussian noise, we have developed twoGeneralized Likelihood Ratio Tests for the detection of off-nominal condition based on the SVD: the off-nominal sub-space detector aiming at detecting u0 6= n, and the eccentricitydetector aiming at detecting σ2 6= σ1. Furthermore, we havederived a new low-complexity angular frequency estimatorbased on the Periodogram of the right-singular vector. Simu-lation results have shown that the proposed detectors performwell even for a small number of samples and that the proposedangular frequency estimator gives a good compromise betweenstatistical performance and computational complexity.

APPENDIX ADERIVATION OF Tn(Y, σ2)

As w[u, v] ∼ N (0, σ2), we obtain

L(Y;Z, σ2,Hk) = βe−1

2σ2‖Y−CZT ‖2F (32)

where ‖.‖2F corresponds to the Frobenius norm and β =(1

2πσ2

) 3N2 . In practice, the matrix ZT is unknown and can

be replaced by its Maximum Likelihood estimate ZT =(CTC

)−1CTY [38, Chapter 7]. Let us introduce P⊥C ,

I − C(CTC)−1CT , the orthogonal projector into the nullspace of C. As C is a 3 × 2 matrix, this projector can bedecomposed as P⊥C = u0u

T0 [14, Result R17]. Therefore,

L(Y; Z, σ2,Hk) = βe−1

2σ2‖P⊥

CY‖2F = βe−1

2σ2‖uT0 Y‖2F

where u0 = n under assumption H0. Under assumption H1,u0 is given by u0 [41, Section 4.2]. As ‖uT0 Y‖2F = σ2

0 ,Eq. (20) can be simplified as in Eq. (21).

APPENDIX BDERIVATION OF Tσ(Y, σ2)

As w[u, v] ∼ N (0, σ2), we obtain

L(Y;X, σ2,Hk) = βe−1

2σ2‖Y−X‖2F . (33)

To evaluate the likelihood-function, X must be replaced by itsMLE. Two cases must be distinguished. Under H1, the MLEof X is found by minimizing the Frobenius norm ‖Y− X‖2Fwhere X is a rank-2 matrix. The Eckart-Young theorem statesthat this norm is minimized for X = US1V

T where

S1 =

σ2 0 00 σ1 00 0 0

. (34)

Under H0, the MLE of X is found by minimizing theFrobenius norm ‖Y − X‖2F , where X is a rank-2 matrixwith equal singular values. Then, the estimator of X can bedecomposed as X = US0V

T where

S0 = σ ×

1 0 00 1 00 0 0

. (35)

The singular value σ is found by minimizing ‖Y −US0V

T ‖2F = ‖S − S0‖2F with respect to σ. After somecomputations, we obtain σ = 1

2 (σ1 + σ2).By using these two results, Eq. (33) can be expressed as

L(Y; X, σ2,Hk) = βe−1

2σ2‖Y−USkV

T ‖2F = βe−1

2σ2‖S−Sk‖2F .

Then, by using the fact that σ = 12 (σ1 + σ2), we obtain

Tσ(Y, σ2) =1

σ2

(‖S− S0‖2F − ‖S− S1‖2F

)=

1

2σ2(σ2 − σ1)2 .

APPENDIX CEXPRESSION OF z[n]

The analytic signal z is estimated from the 2 principalright-singular vectors Vx = [v2, v1]. These vectors can becomputed as VT

x = S−1x UTxY where Sx and Ux correspond

to the 2 largest singular values of Y and their associated leftsingular vectors.



10

For large N and small σ2, C = UxSx ≈ CQ where Qis an orthogonal matrix containing the SVD indeterminations.By using this property, it follows that

VTx = S−1x UT

xY = S−1x UTxCZT + S−1x UT

xW

≈ QTZT + S−1x UTxW.

As Q is an 2× 2 orthogonal matrix, Q ∈ {Q1,Q2} where

Q1 ,

[cos(θ) − sin(θ)sin(θ) cos(θ)

], Q2 ,

[cos(θ) sin(θ)sin(θ) − cos(θ)

].

Then, the estimator of the analytic signal can be obtained as

z , Vx

[1j

]≈{e−jθz+ b if Q = Q1

ejθz∗ + b if Q = Q2, (36)

where b is an additive white complex noise. Finally, using thedefinition of z, Eq. (36) can be expressed as in Eq. (27).

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11

Vincent Choqueuse (S’08 - M’09) was born inBrest, France, in 1981. He received the Dipl.Ing.and M.Sc. degrees from the Troyes University ofTechnology, France, in 2004 and 2005, respectively,and the Ph.D. degree from the University of Brest,in 2008. From 2008 to 2018, he was an Associateprofessor at the Institut Universitaire de Technologieof Brest, University of Brest, and a member of theInstitut de Recherche Dupuy de Lome (UMR CNRS6027). He is now with the Ecole Nationale desIngenieurs de Brest (ENIB), Brest and a member

of the Lab-STICC (UMR CNRS 6285).His current research interests include signal processing and statistics for

power systems monitoring, smart-grid, digital communication, and digitalaudio.

Pierre Granjon was born in 1971 in France. Francein 2000. He joined the Laboratoire des Images etdes Signaux (LIS) in 2002 and the Gipsa-lab atGrenoble-INP in 2007, where he holds a positionas associate professor.

His research is mainly focused on signal pro-cessing methods for system monitoring and faultdiagnosis. He developed several monitoring methodsfor electrical rotating machines and power networksbased on the analysis of vibrations or electricalquantities. His current research is mainly focused

on developing efficient methods to properly analyze three-component signalssuch as three-axis vibrations or three-phase electrical quantities.

Francois Auger (M’93 - SM’12) was born in Saint-Germain-en-Laye, France, in 1966. He received theEngineer degree and the D.E.A. in signal processingand automatic control from the Ecole NationaleSuperieure de Mecanique de Nantes in 1988, and thePhD from the Ecole Centrale de Nantes in 1991.

From 1993 to 2012, he was “Maıtre deConferences” (assistant professor) at the IUT deSaint-Nazaire (Universite de Nantes). He is now fullprofessor at the same place.

His current research interests include automaticcontrol and diagnosis of electrical power systems, spectral analysis and time-frequency representation of non stationary signals. Dr. Auger is a member ofthe GdR720 CNRS “Information, Signal, Images et ViSion”.

Adel Belouchrani (S’94 - M’96 - SM’14) was bornin Algiers, Algeria, in 1967. He received the StateEngineering degree from the Ecole Nationale Poly-technique (ENP), Algiers, Algeria, in 1991, the M.S.degree in signal processing from the Institut NationalPolytechnique de Grenoble, France, in 1992, andthe Ph.D. degree in signal and image processingfrom Telecom Paris, France, in 1995. He was aPost-Doctoral Fellow with Electrical Engineeringand Computer Sciences Department, the Universityof California, Berkeley, CA, USA, from 1995 to

1996. He was a Research Associate with the Department of Electrical andComputer Engineering, Villanova University, Villanova, PA, USA, from 1996to 1997. From 1998 to 2005, he was an Associate Professor with the ElectricalEngineering Department, ENP. Since 2006, he has been a Full Professor withENP.

His research interests are in statistical signal processing, (blind) arraysignal processing, time-frequency analysis, and time-frequency array signalprocessing with applications in biomedical and communications. He is afounding member of the Algerian Academy of Science and Technology. Heserved as an Associate Editor of the IEEE Transactions on Signal Processingfor two terms from 2013 to 2017. He is currently Senior Area Editor of theIEEE Transactions on Signal Processing.

Mohamed Benbouzid (S’92 - M’95 - SM’98) wasborn in Batna, Algeria, in 1968. He received theB.Sc. degree in electrical engineering from the Uni-versity of Batna, Batna, in 1990, the M.Sc. andPh.D. degrees in electrical and computer engineeringfrom the National Polytechnic Institute of Grenoble,Grenoble, France, in 1991 and 1994, respectively,and the Habilitation Diriger des Recherches degreein electrical engineering from the University of Pi-cardie Jules Verne, Amiens, France, in 2000.

After receiving the Ph.D. degree, he joined theProfessional Institute of Amiens, University of Picardie Jules Verne, wherehe was an Associate Professor of electrical and computer engineering. Since2004, he has been with the Institut Universitaire de Technologie of Brest,University of Brest, Brest, France, where he is currently a Professor ofelectrical engineering. He is also a Distinguished Professor with ShanghaiMaritime University, Shanghai, China.

His current research interests include analysis, design, and control of elec-tric machines, variable-speed drives for traction, propulsion, and renewableenergy applications, and fault diagnosis of electric machines.

Prof. Benbouzid is the Editor-in-Chief of the International Journal onEnergy Conversion. He is also an Associate Editor of the IEEE Transactionson Energy Conversion, the IEEE Transactions on Industrial Electronics, theIEEE Transactions on Sustainable Energy, the IEEE Transactions on VehicularTechnology, and the IET Renewable Power Generation.

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