Probability Density Function of Three-Phase Ellipse Parameters for the Characterization of Noisy Voltage Sags

This work deals with the statistical analysis of additive noise impact on the space-vector ellipse parameters used to detect and classify three-phase voltage sags. In fact, since voltage waveforms are always corrupted by additive noise and harmonics, the space vector is pre-processed through the Discrete Fourier Transform to extract the power frequency components. Thus, harmonics can be readily discarded, but additive noise can still have significant impact on the elliptical trajectory of the space vector on the complex plane. Therefore, by modeling the ellipse parameters (i.e., the shape index and the inclination angle) as random variables, the related statistical characterization is derived in the paper. In particular, the main results and the novelty of the paper are given by the analytical derivation in closed-form of the probability density function, cumulative distribution function, mean value, and variance of the ellipse parameters as functions of the additive noise variance. Since the ellipse shape index and inclination angle are commonly used to detect and classify voltage sags, the results derived in the paper are useful for both uncertainty propagation analysis, and assessment of detection capability in case of voltage dips close to the minimum value defined in the IEEE Standard 1159. Analytical results are validated through numerical simulation of noisy voltage sags.


I. INTRODUCTION A. MOTIVATION
Among all the power quality issues regarding three-phase power systems, voltage sags, also called voltage dips, represent a major concern together with supply interruptions [1], [2]. Indeed, according to the IEEE Standard 1159, a voltage sag is a decrease in the RMS voltage of 10-90% of the nominal value for durations from 0.5 cycles to 1 minute [3]. As the most frequent power quality disturbance, voltage sags can have severe consequences spanning from malfunctioning of control system equipment, to disconnection or loss of efficiency in electric machines. For this reason, many researchers have focused on the analysis, classification and characterization of voltage sags (e.g., [4]- [10]). Not much attention, however, has been devoted to measurement uncertainty due to the presence of harmonics The associate editor coordinating the review of this manuscript and approving it for publication was Arash Asrari . and additive noise. Nevertheless, this point is crucial since harmonics and noise can affect proper classification and characterization of voltage sags.

B. LITERATURE REVIEW
One of the most effective techniques for real-time detection and classification of voltage sags is the space vector approach [11]- [20]. Indeed, starting from the pioneering work [11], the space vector approach has become popular for its straightforward geometrical properties. In fact, under normal operation, the power-frequency component (i.e., the 50/60 Hz component) of the space vector corresponding to balanced three-phase voltages describes a circular trajectory on the complex plane. When an unbalanced voltage sag occurs (i.e., when the voltage drop is not the same for all the phases), however, the space vector trajectory becomes elliptical [11]- [18]. In this case, the geometrical parameters of the ellipse, i.e., the semi-major axis, the semi-minor axis, and the inclination angle, allow a fast identification of the VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ type of voltage sag. In particular, the ratio between the ellipse axes (i.e., the so-called shape index) detects the onset of an unbalanced voltage sag, whereas the ellipse inclination angle identifies the dropped phases in both single-phase and double-phase dips. An interesting generalization was proposed in [19], [20] where a 3-D polarization ellipse was introduced. The five parameters of the 3-D ellipse were exploited for both classification and localization of fault voltage sags.

C. CONTRIBUTION AND PAPER ORGANIZATION
As pointed out in several papers, the main drawback of the space vector approach is that the space vector is normally corrupted by additive noise and harmonics [11], [14]. In fact, the ellipse mentioned above is related to the power-frequency component only. Thus, the space vector must be properly preprocessed in order to extract relevant information at power frequency. This can be readily accomplished through the Discrete Fourier Transform (DFT) which allows efficient evaluation of the space vector spectrum. Notice that other interesting transforms could be used for extraction of relevant information from noisy and distorted space vectors (e.g., orthogonal polynomials in [21], [22]). However, the derivations proposed in this paper are based on the DFT for two main reasons. First, the DFT is a common and wellknown tool within the context of power system analysis. Second, in Section II it will be shown that the statistical effects of additive noise on DFT coefficients is already a wellknown issue in the relevant literature. Thus, while harmonics can be effectively discarded through the DFT, the impact of additive noise at power frequency can be still significant. This is a crucial point since additive noise at power frequency affects the estimates of the ellipse parameters for voltage sag detection and classification. Despite this issue was pointed out in several papers, a specific analytical investigation of additive noise effects on ellipse parameters is still missing in the relevant literature. Therefore, the statistical analysis of additive noise effects on ellipse parameters for voltage sag detection and classification is the main objective of this work. Possible impact of multiplicative noise is not investigated in this paper. In particular, the main contribution of this paper is the analytical derivation of the probability density function, cumulative distribution function, mean value, and variance of the ellipse parameters as functions of the additive noise level. The importance of the presented results is twofold. First, complete uncertainty analysis of the ellipse parameters allows proper classification of voltage sags by improving the decisionmaking process. Second, the impact of additive noise on detection of voltage sags is described in probabilistic terms.
The paper is organized as follows. In Section II the main properties of the DFT of a noisy and distorted space vector are described. In Section III the ellipse parameters for voltage sag detection and classification are defined in terms of the powerfrequency DFT coefficients of the space vector. In Section IV the statistical analysis of the additive noise impact on ellipse parameters is presented. In particular, the statistical properties of the ellipse shape index and inclination angle are derived in closed form. In Section V the analytical results are validated through numerical simulations of noisy space vectors with elliptical trajectory, whereas in Section VI real data recorded by DOE/EPRI are used for further validation. Finally, the main results of the paper are summarized and discussed in Section VII.

II. FOURIER ANALYSIS OF VOLTAGE SPACE VECTOR
Space vector definition is based on the time-domain Clarke transformation of the phase variables in a three-phase circuit.
T is given by [23]: where T is an orthogonal matrix, i.e., T −1 = T t . The corresponding space vector is defined as the complexvalued time-domain function given by [24], [25]: where a = e j2π / 3 . Under sinusoidal steady-state conditions with angular frequency ω 0 it can be shown that the space vector can be written as: where V p and V * n are the positive-sequence and the negativesequence (complex conjugate) phasors, according to the wellknown symmetrical component transformation [26]: where S is a unitary matrix, i.e., S −1 = S * t . Notice that (3) can be advantageously interpreted as a double decomposition of the space vector, i.e., the positive/negative-sequence decomposition, and the frequency decomposition in two spectral lines located in ±ω 0 . Thus, V p and V * n can be seen as the complex Fourier coefficients corresponding to the sinusoidal steady-state, i.e., V 1 = V p and V −1 = V * n . Contrary to the case of real-valued timedomain functions, however, the two Fourier coefficients corresponding to ±ω 0 are not related each other by complex conjugation (i.e., V −1 = V * 1 ). In fact, since the space vector is a complex-valued function the two Fourier coefficients V 1 = V p and V −1 = V * n , corresponding to ±ω 0 , are independent each other.
According to the interpretation mentioned above, in the more general case of distorted steady-state conditions the space vector can be written in the complex series form [27]: where V k and V −k are the positive-sequence and negativesequence (complex conjugate) phasors at kω 0 , respectively. It can be readily shown that the complex coefficients in (5) are given by: where T = 2π /ω 0 = 1/f 0 . The complex coefficients (6) can be evaluated through the well-known Discrete Fourier Transform (DFT) of the space vector samples. By taking N s samples in one period of v(t) (i.e., N s T s = T where T s = 1/f s is the sampling interval), the complex coefficients (6) are given by: Notice that, according to (5), the complex coefficients V k given by (7) must be evaluated for both positive and negative k values, i.e., the bilateral frequency spectrum of v(t) must be evaluated.
Let us consider now the impact of additive noise on the evaluation of the coefficients V k through the DFT (7). In case the time-domain phase voltages v a , v b , v c are corrupted by additive zero-mean identically-distributed white noise: with variance σ 2 n , from (2) we obtain: where Thus, n α and n β are zero-mean uncorrelated white noise with variance [28]: Therefore, the space vector is corrupted by additive noise with the same characteristics as the additive noise corrupting the phase voltages. This point allows the use of wellestablished results concerning the impact of additive noise on the DFT coefficients provided by (7). Thus, a noisy space vector corrupted by additive zero-mean white noise with variance σ 2 n results in random DFT coefficients where, from the Central Limit Theorem [28], the real and the imaginary parts are Gaussian, uncorrelated, unbiased random variables with equal variance given by [29], [30]: This is the fundamental result that will be exploited in the next Sections to derive the probability density functions of the ellipse parameters generated by a noisy three-phase system.

III. ELLIPSE PARAMETERS FOR VOLTAGE SAG ANALYSIS
Voltage sags in three-phase systems can be effectively characterized in terms of the parameters of the elliptical trajectory described by the fundamental frequency component (i.e., the component located at f 0 ) of the voltage space vector. In fact, under sinusoidal steady-state conditions the space vector (3) describes an ellipse on the complex plane, characterized by the following parameters [11]: where |·| denotes the magnitude, r m and r M are the semiminor and semi-major ellipse axes, respectively, and ϕ is the ellipse inclination angle. Notice that in (16) the difference between the two phasor arguments would be considered, instead of the sum, if the negative-sequence phasor was not conjugated. Under ideal conditions, the negative-sequence phasor is equal to zero, and the corresponding space-vector trajectory is a circle with radius V p . A circle, with reduced radius with respect to the nominal voltage, is obtained also in case of balanced faults of the three phases.
In case of unbalanced faults, however (i.e., faults involving only one or two phases), the corresponding negativesequence component |V n | > 0 yields an elliptical trajectory of the space vector (see Fig. 1). A proper parameter able to detect the elliptical shape of the trajectory is the so-called Shape Index (SI) defined as [11]- [20]: By taking into account the types of possible voltage sags, and by considering that a voltage sag is defined as a decrease between 10% and 90% of the nominal voltage, in [11]- [20] the threshold level SI = 0.933 was calculated. Thus, the condition SI < 0.933 detects an unbalanced fault, and the ellipse inclination angle ϕ provides the type of unbalanced fault according to Fig. 2 where single-phase sags are denoted with S, double-phase sags with D, and the subscripts denote the dropped phases [11]. Inclination angles in Fig. 2 are integer multiples of 30 • . Thus, each type of sag is characterized by one of the angles in Fig. 2 with uncertainty ±15 • . For example, a single-phase voltage dip involving phase a (i.e., S a ) is characterized by ϕ = 90 • ± 15 • .  Ellipse inclination angles for each type of unbalanced dip (i.e., S for single-phase and D for double-phase dips) [11]. The angles are located at integer multiples of 30 • . Therefore, each type of dip is characterized by an inclination angle within the range n30 • ±15 • . For example, single-phase dip on phase a (i.e., S a ) is characterized by inclination angle within 90 • ±15 • .
As far as noisy and distorted voltage waveforms are considered, pre-processing is needed in order to extract the positive and negative-sequence phasors at the fundamental frequency. To this aim, the procedure outlined in Section II based on the DFT can be effectively used to discard harmonic waveform components, but additive noise has still impact on magnitude and phase of V p and V * n at ±f 0 . Of course, as a general principle, noise impact is larger for spectral components with smaller magnitude. In the present case, it is expected that additive noise has large impact around the threshold condition for SI where the negative-sequence phasor is much smaller in magnitude with respect to the positive-sequence phasor. In fact, at the threshold level SI = 0.933, from (17) we have V * n V p ∼ = 0.035. Thus, it is expected that additive noise has significant impact in the detection of unbalanced faults (i.e., the condition SI < 0.933), since a large impact is expected on the small-magnitude negative-sequence phasor. Moreover, once an unbalanced fault is detected, the inclination angle (16) of the ellipse can be greatly affected by additive noise since a significant impact is expected also on the argument of the small-magnitude negative-sequence phasor.
In the next Section the impact of additive noise on ellipse shape index and inclination angle will be analyzed by deriving the probability density function (PDF) and the main statistical moments of such parameters in terms of the noise variance.

IV. STATISTICAL ANALYSIS OF ADDITIVE NOISE EFFECTS
The ellipse shape index SI (17) and inclination angle ϕ (16) can be evaluated from the positive-sequence and the negativesequence phasors V p and V * n through the DFT of the space vector as shown in Section II. Since the space vector is corrupted by additive zero-mean white noise with variance σ 2 n (see (9)- (12)) the sequence phasors V p and V * n can be treated as complex unbiased random variables, whose real and imaginary parts have Gaussian distribution with variance (13). For the sake of simplicity, a more compact notation will be used: where the subscripts denote the real and the imaginary parts.
In (17) the absolute values of (18)- (19) are required, i.e.: Therefore, the objective of the statistical analysis can be reformulated as the PDF of the two functions of random variables: As mentioned above, the largest noise impact is expected for small values of y, i.e., for SI around its threshold limit 0.933 where y x. Thus, a reasonable approximation is treating only y as a random variable, while x is treated as a noise-free variable x = x 0 = X 2 0r + X 2 0i (where the subscript 0 denotes the deterministic values corresponding to zero additive noise). Therefore, (22)-(23) can be rewritten as: The validity of the approximations in (24)-(25) will be numerically assessed in Section V.

A. ELLIPSE SHAPE INDEX 1) PROBABILITY DENSITY FUNCTION
The PDF of the ellipse shape index given by (24) can be obtained by resorting to the theorem on transformation of random variables [28]. The starting point is the Gaussian distribution with variance (13) of the unbiased random variables Y r and Y i . From (21) we obtain that the magnitude y has a Rice PDF [28]- [30]: where y 0 is the noise-free value of y, and I 0 is the modified Bessel function of the first kind. In (24) the random variable y is divided by the constant x 0 . Thus, the PDF of the new random variable u = y/x 0 is given by: Therefore, the transformation (24) can be written as: where z denotes the random variable SI. To the aim of obtaining the PDF of z, the first derivative and the inversion formula of (28) are needed: Therefore, from the theorem on transformation of random variables the PDF of the ellipse shape index can be obtained: The Cumulative Distribution Function (CDF) of the ellipse shape index can be also obtained with a similar procedure. Starting from the CDF of the Rice random variable y: where Q 1 is the first-order Marcum Q-function, the CDF of the scaled random variable u = y/x 0 can be readily obtained: By considering the transformation (28) we have that Therefore, the CDF of the ellipse shape index is given by: This result is crucial since, by definition, (36) provides the probability that the ellipse shape index is smaller than a given level z (e.g., the SI threshold level). By denoting as SI 0 the noise-free value of the shape index (24), i.e.: x 0 in (36) can be expressed as: Thus, by considering the threshold value z T = 0.933 for voltage sag detection, from (36) we obtain the detection probability: where the signal-to-noise ratio (SNR) is defined as: Finally, for the sake of completeness, the more general case where the assumption y x is not met can be considered. In this case, the random variable u = y/x is not a simple scaled version of y, but it is the ratio between two random variables with Rice distribution and equal variance. The PDF of such ratio is given by [31]: The numerical convergence of the PDF (41) is more critical than (27) as the ratio x 0 y 0 /σ 2 increases.
The CDF of the random variable u = y/x is given by [31]: where: Equations (41) and (42) can be used in (31) and (36) to obtain the general expressions for the PDF and CDF of the shape index SI without the restriction y x.

2) MEAN VALUE AND VARIANCE
Approximate expressions for the mean value and the variance of the ellipse shape index can be obtained by resorting to the Taylor expansion approach [28]. To this aim, the first and second order derivatives of the transformation (28) are needed. The first order derivative is given by (29), whereas the second order derivative is given by: 3 (44) VOLUME 8, 2020 Thus, the approximate mean value of the ellipse shape index is given by: where µ u and σ 2 u can be readily expressed in terms of µ y and σ 2 y (i.e., the well-known mean value and variance of a Rice distribution) [32]: where L is the Laguerre polynomial. The approximate variance of the ellipse shape index is given by [28]: where µ u and σ 2 u are given by (46)-(47).

B. ELLIPSE INCLINATION ANGLE 1) PROBABILITY DENSITY FUNCTION
From (25) the first step in order to obtain the PDF of the inclination angle ϕ is the analysis of the random variable w = Y i Y r . The PDF of the ratio of two Gaussian random variables with equal variance is given by [31]: where Y 0r and Y 0i are the noise-free values of Y r and Y i , respectively, and erfc is the complementary error function. According to (25) the following transformation of random variable must be solved: Similarly to the shape index case, the first derivative and the inversion formula of (50) are needed: Thus, according to the theorem on the transformation of random variables the PDF of the ellipse inclination angle is given by: Notice that (53) provides the PDF of the inclination angles expressed in radians. The PDF for inclination angles in degrees is given by: where g d = g · 180/π. The CDF of the inclination angle can be obtained by numerical integration of the PDF (53)-(54).

2) MEAN VALUE AND VARIANCE
Approximate expressions for the mean value and the variance of the ellipse inclination angle can be obtained by using the Taylor expansion of (25) with respect to the two Gaussian random variables Y r and Y i [28]. Thus, the mean value of ϕ is given by: where the correction term related to the second-order partial derivatives is not present since it is null. Therefore, the inclination angle is a random variable with negligible bias. The variance of the inclination angle can be estimated through the first-order partial derivatives: Notice that since σ 2 ϕ is a function of the SNR only, it is not dependent on the nominal shape index SI 0 .
Finally, it is worth noticing that by removing the assumption y x, i.e., by treating also X i /X r as a random variable in (23), the inclination angle ϕ would be given by the weighted sum of the two random variables arctan(X i /X r ) and arctan(Y i /Y r ). Thus, the PDF of ϕ would be given by the convolution of two PDFs [28]. A closed-form analytical result is not available in the literature. Therefore, the analytical results derived in this Subsection are limited to the assumption y x. Numerical simulations, however, will prove that such assumption leads to reasonably accurate results.

V. NUMERICAL VALIDATION
The analytical results derived in Section IV were validated by means of numerical simulations in Matlab. The simulation process was implemented according to the following steps. First, the noise-free values of the positive and negativesequence phasors V p and V n were selected in order to set the noise-free value SI 0 of the ellipse shape index (37), and the noise-free value ϕ 0 of the ellipse inclination angle (55). Second, by inversion of the symmetrical component transformation (4) (with null zero-sequence component) the phasors of the phase voltages V a , V b , V c were calculated. Then, the corresponding time-domain voltages v a (t), v b (t) , v c (t) were obtained by assuming fundamental frequency f 0 = 50Hz. Each time-domain waveform was corrupted by additive zeromean Gaussian noise with variance σ 2 n . Such variance was selected according to the desired value of the SNR (40), where σ is related to σ n through (13). The three noisy waveforms were sampled by taking N s = 256 samples per period, i.e., with sampling frequency f s = N s f 0 = 12.8kHz. Then, according to (2), the three waveforms where combined to define the voltage space vector, and the time-to-frequency transformation of the space vector (i.e., the DFT (7)) was calculated through the Fast Fourier Transform. The time window was one period in length (i.e., the number of processed samples was N s ). Therefore, the DFT frequency resolution was f = f s /N s = 50Hz. Thus, according to (7), the obtained frequency coefficients V 1 and V −1 were the estimates of V p and V * n , respectively. The ratio |V −1 | / |V 1 |, and the arguments of V 1 and V −1 were used to evaluate the shape index (17) and the inclination angle (16). Such evaluations were repeated N times (with N ≥ 10 4 ) by generating new sequences of random noise added to the time-domain waveforms v a (t), v b (t) , v c (t). Thus, for each selection of the parameters SI 0 , ϕ 0 , and SNR, a set of N realizations of the random variables SI and ϕ were calculated. Therefore, the statistical properties of SI and ϕ, i.e., PDF, CDF, mean value and variance, were numerically evaluated and compared with the corresponding analytical results obtained in Section IV.  Four different values of the noise-free shape index SI 0 were selected, i.e., 0.5, 0.6, 0.7, and 0.8. Numerical results obtained through the simulation procedure outlined above are represented by the red lines. The approximate analytical PDF given by (27) substituted into (31) is represented by blue lines, whereas the general analytical PDF given by (41) substituted into (31) is represented by black lines. The objective of this figure is showing that by increasing SI 0 the approximate analytical PDF (blue lines) provides satisfactory results. Since we are interested in the case of SI 0 approaching the limiting value 0.933, the approximate analytical PDF can be effectively used. Thus, in the next figures, only the approximate analytical PDF will be represented.   4 shows the behavior of the PDF of the shape index with SI 0 = 0.933, and for three different SNR values, i.e., 3, 6, and 9. By increasing the SNR (i.e., by decreasing the noise level) the spread of the PDF decreases. Notice that even for SNR = 9 the shape index can take values around 0.96, i.e., well above the threshold value 0.933 used to detect a voltage sag. For SNR = 3 the shape index can take values even around 1, that is the case where the ellipse becomes a circle. This is consistent with the fact that SNR = 3 means y 0 = 3σ . Thus, the noisy y can take values even close to zero.   5 shows the CDF of the shape index corresponding to the PDF shown in Fig. 4. The blue lines represent the analytical CDF given by (36). The slope of the central part of the CDF increases with SNR. In fact, by decreasing the noise level to zero the CDF approaches a discontinuous behavior at 0.933. VOLUME 8, 2020   6 shows the detection probability (39) of a voltage sag, i.e., the probability that the shape index is lower than the threshold 0.933, for three different values of SI 0 , i.e., 0.933, 0.93, and 0.92. For SI 0 = 0.933, by increasing the SNR the PDF peak approaches the location 0.933 (see Fig. 4). Thus, the detection probability approaches 0.50 for increasing SNR. For SI 0 = 0.93 and SI 0 = 0.92, however, the location of the PDF peak is lower than 0.933. Thus, by increasing SNR the PDF shrinks and the detection probability approaches 1.   7 shows the mean value (45) of the shape index, with SI 0 = 0.933, as a function of the SNR. The mean value approaches SI 0 as SNR increases. This behavior corresponds to the shift of the PDF peak in Fig. 4 as SNR increases. Similar curves can be obtained for different values of SI 0 . Fig. 8 shows the behavior of the standard deviation of the shape index (i.e., the square root of (48)) as a function of  SNR, for three different values of SI 0 , i.e., 0.933, 0.9, 0.8. This figure can be compared with Fig. 3 where it is shown that, for a given SNR value, the standard deviation increases for decreasing SI 0 . This is because both y 0 and σ increase to keep SNR constant. Fig. 9 shows the behavior of the PDF (54) of the ellipse inclination angle with SI 0 = 0.933 and for three different values of SNR, i.e., 3, 6, and 9. The selected phases for V p and V * n were such that the noise-free inclination angle was ϕ 0 = 90 • . Notice that even for SNR = 9 the spread of the PDF is about ±10 • . This is a crucial point since from Fig. 2 each type of voltage sag can be identified within an angular range ±15 • due to circuit parameters. Therefore, the impact of noise could result in a wrong identification of the type of voltage sag. Of course, this potential problem is emphasized for smaller SNR values. For example, SNR = 6 corresponds to an approximate PDF spread ±15 • , i.e., the same angular range of each sector in Fig. 2.
Finally, Fig. 10 shows the behavior of the standard deviation of the inclination angle (i.e., the square root of (56)). As already pointed out, σ ϕ is a function of the SNR only (in the approximate result (56)). Thus, the behavior shown in Fig. 10 is valid independently of SI 0 . Notice that, since the PDF in Fig. 9 are approximately Gaussian, the PDF spread is approximately ±3σ ϕ , where σ ϕ can be read in Fig. 10.

VI. VALIDATION WITH REAL DATA
Real data recorded by DOE/EPRI [33] were used for further validation of the analytical results derived in the paper. In particular, the event number 0243 was considered. According to [33] the fault was due to a tree fallen on phase a.    of samples per period can be advantageous since real-time continuous monitoring is needed.
The procedure based on the DFT described in this paper was used to evaluate the voltage ellipse shape index SI and inclination angle ϕ in each period. The shape index takes its minimum value 0.9364 at the third period, and the corresponding inclination angle is 77.5 • (i.e., within the range 90 • ± 15 • characteristic of a grounded phase a). Notice that, according to the choice of a threshold level equal to 0.933, the voltage sag would not be detected. Additive noise, however, is always present in measured waveforms. Unfortunately, estimate of additive noise level in Fig. 11 is a hard task since the underlying signal is non-stationary. For comparison purposes only, we can for example consider the case of additive noise with SNR = 9, already considered in Sections IV and V. From the DFT analysis of the third period of the space vector, we obtain that |V −1 | = V * n ∼ = 750V . Thus, from (40) the corresponding frequency-domain standard deviation is given by σ ∼ = 80V . From (13), for VOLUME 8, 2020 kV ∼ = 14kV . Therefore, since the ratio V rms /σ n ∼ = 44, the effect of time-domain additive noise cannot be clearly observed on the waveform amplitude. Fig. 12 shows an example of the three voltage waveforms corrupted by additive zero-mean Gaussian noise with σ n = σ √ N s (dashed lines). Thus, the time-domain effect of such noise level seems negligible in the time-domain. However, in Section IV and V it was shown that such noise level can produce significant deviations of the shape index and the inclination angle. To illustrate this point, and for further validation of the analytical results, a repeated run analysis (with N = 10 5 ) was performed by adding zeromean Gaussian noise with variance σ 2 n to the three voltage waveforms in Fig. 11. The distribution of the shape index and the inclination angle are represented by the histograms in Figs. 13 and 14, respectively. The range of the histogram in Fig. 13 (i.e., 0.91 ÷ 0.96) is approximately the same range of the PDF with SNR = 9 in Fig. 4. Thus, noise effect can result in either detection or no-detection of the voltage sag. The range of the histogram in Fig. 14 (i.e., ϕ ∼ = 20 • ) is approximately the same range of the PDF with SNR = 9 in Fig. 9, with different mean value because this is a function of circuit parameters. Notice that noise effect can result in inclination angle outside the characteristic range 90 • ± 15 • , producing a wrong identification of the voltage sag type.

VII. CONCLUSION AND DISCUSSION
In this Section, the main achievements obtained in the paper are summarized and discussed.
The parameters of the elliptical trajectory (i.e., shape index and inclination angle) of a voltage space vector on the complex plane at power frequency are widely used to detect and classify three-phase voltage sags. In real applications, however, the voltage space vector is corrupted by harmonics and additive noise. Therefore, since the ellipse parameters refer to the power frequency component only, pre-processing of space vector is required.
Within the context of power system analysis, the most common technique for harmonic analysis is the well-known DFT. It is worth noticing that, for the present analysis, the DFT was used only to measure the power frequency components of the voltage space vector, i.e., the positive-sequence and the negative-sequence components at power frequency. Thus, since the ellipse parameters depend only on the power frequency components, the DFT is used only to isolate the power frequency components with respect to the harmonic content. Moreover, in the paper it was assumed that sampling was synchronized with voltage fundamental frequency. In practice this condition can be achieved by using a closedloop measuring system. Indeed, in case of lack of synchronization, windowing of voltage samples (e.g., Hann window) is required to minimize spectral leakage. In this case the analytical results derived in the paper hold provided that the variance (13) is multiplied by the equivalent noise bandwidth (ENBW) of the selected window (e.g., ENBW = 1.50 for the Hann window) [29].
Once the power frequency components of the voltage space vector are measured, the impact of additive noise on the corresponding DFT coefficients must be evaluated since such coefficients are used to calculate the ellipse parameters. Under weak assumptions, regardless the specific distribution of additive noise, the Central Limit Theorem guarantees that the real and imaginary parts of DFT coefficients can be approximated as uncorrelated and unbiased Gaussian random variables. This is the fundamental starting point of the analytical derivations presented in the paper. In fact, from this point, the statistical properties of the ellipse parameters were derived in analytical form. In particular, the PDF, the CDF, the mean value, and the variance of the ellipse shape index and inclination angle were derived as functions of the additive noise variance. These results allow a complete statistical characterization of the ellipse parameters. To the Author's knowledge, such statistical characterization was still lacking in the relevant literature, and therefore a straightforward comparison with similar results cannot be accomplished. In Section V the analytical results were validated through numerical simulation of the whole measurement process.
The obtained statistical characterization of the ellipse parameters can be useful for two reasons. First, measured ellipse parameters can be characterized in terms of uncertainty. In fact, the analysis proposed in the paper can be regarded as a study of uncertainty propagation. Since ellipse parameters are used to distinguish and classify voltage sags, the corresponding uncertainty levels provide the required information for a proper identification of voltage sag types. Second, voltage sag detection is conventionally characterized by shape index SI < 0.933. In case of noisy measurements, the comparison between the measured shape index and the threshold level 0.933 can be evaluated in statistical terms, i.e., in terms of detection probability. To this aim, it is worth noticing that, according to (17), the threshold SI = 0.933 corresponds to voltage unbalance V * n / V p ∼ = 0.035. Therefore, if a system is working in unbalanced mode (e.g., a distribution system), then the whole methodology based on the space vector can be still used provided that the voltage unbalance is lower that 3.5% [34]. For higher voltage unbalance the whole space-vector approach investigated in [11]- [20] and in this paper requires further investigation.