Parametric Spectral Estimation for Power Quality Assessment

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Parametric Spectral Estimation for Power Quality Assessment Conference Paper · October 2007 DOI: 10.1109/EURCON.2007.4400304 · Source: IEEE Xplore

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EUROCON 2007 The International Conference on “Computer as a Tool”

Warsaw, September 9-12

Parametric Spectral Estimation for Power Quality Assessment ∗ Wroclaw

Zbigniew Leonowicz∗ , Tadeusz Lobos∗

†

University of Technology/Electrical Engineering, Wroclaw, Poland, e-mail: [email protected] † e-mail: [email protected]

Abstract— The authors show that the use of highresolution spectrum estimation methods instead of Fourierbased techniques can improve the accuracy of measurement of spectral parameters of distorted waveforms encountered in power systems, in particular the estimation of the power quality indices (such as harmonic and interharmonic groups and subgroups). The comparison of the frequency and amplitude estimation error, based on numerical simulations is presented. Presentation of selected power quality indices is then followed by comparison of estimation error in the case of application of FFT-based algorithms and parametric methods. Investigated waveforms are typical for dc arc furnace plant. MUSIC and ESPRIT high-resolution methods are used to analyze waveforms in a supply system of a DC arc furnace. Keywords—Power Quality, Power system harmonics, Spectral domain analysis, MUSIC, ESPRIT.

I. I NTRODUCTION The quality of voltage waveforms is nowadays an issue of the utmost importance for power utilities, electric energy consumers and also for the manufactures of electric and electronic equipment. The proliferation of nonlinear loads connected to power systems has triggered a growing concern with power quality issues. The inherent operation characteristics of these loads deteriorate the quality of the delivered energy, and increase the energy losses as well as decrease the reliability of a power system [1], [4], [12]. The methods of power quality assessment in power systems are almost exclusively based on Fourier Transform. The crucial drawback of the Fourier Transform-based methods is that the length of the window is related to the frequency resolution. Moreover, to ensure the accuracy of Discrete Fourier Trans-form, the sampling interval of analysis should be an exact integer multiple of the waveform fundamental period [11]. Parametric spectral methods, such as ESPRIT or MUSIC [11] do not suffer from such inherent limitations of resolution or dependence of estimation error on the window length (phase dependence of the estimation error). The resolution of these methods is to high degree independent on signal-to-noise ratio and on the initial phase of the harmonic components. The author argues that the use of high-resolution spectrum estimation methods instead of Fourier-based techniques can improve the accuracy of measurement of spectral parameters of distorted waveforms encountered in power systems, in particular the estimation of the power quality indices [6]. The paper is composed as follows: After the description

1-4244-0813-X/07/$20.00 2007 IEEE.

of parametric methods (ESPRIT and MUSIC), the comparison of its performance (estimation error), based on numerical simulation is presented. Next part presents basics of selected power quality indices (harmonic sub/groups), followed by comparison of estimation error in the case of application of FFT-based algorithms and parametric methods. II. E RROR OF E STIMATION OF PARAMETRIC S PECTRAL M ETHODS The performance (error of estimation) of the subspace methods has been extensively investigated in the literature, especially in the context of the Direction–of–Arrival (DOA) estimation. Based on [8] and [15], the derivation of variance in the case of frequency component estimation is presented. Comparison of mean square error is useful for theoretical assessment of accuracy of both methods with emphasis to root–MUSIC and ESPRIT. Both methods are similar in the sense that they are both eigendecomposition–based methods which rely on decomposition of the estimated correlation matrix into two subspaces: noise and signal subspace. On the other hand, MUSIC uses the noise subspace to estimate the signal components while ESPRIT uses the signal subspace. In addition, the approach is in many points different. Numerous publications were dedicated to the analysis of the performance of the aforementioned methods (e.g. [14], [5], [18], [19], [15], [8], [9]). Unfortunately, due to many simplifications, different assumptions and the complexity of the problem, published results are often contradictory and sometimes misleading. To the best authors knowledge, the comparison of accuracy to such extent of two different parametric methods based on numerical simulation of real-like signals is for the first time presented in this work. III. P ERFORMANCE A NALYSIS OF MUSIC A. MUSIC From the available N data samples the autocorrelation sequence rx [k] is computed for a chosen number of delays k. The autocorrelation matrix is then formed and then eigen–decomposed as: Rx = UΛU∗T , where U = [u1 , u2 , . . . , uk ]. In one of possible approaches the polynomials are built from eigenvectors spanning the noise subspace. The roots of each of such polynomials

1641

correspond to signal zeros. Now the following expression can be defined [15]: M


D(z) =

[Ui (z)][Ui∗ (1/z ∗ )]

(1)

i=K+1

The idea of MUSIC (Multiple Signal Classification) was developed in [17] where the averaging was proposed for improvement of the performance of Pisarenko estimator [20]. Instead of using only one noise eigenvector, the MUSIC method uses many noise eigenfilters. The number of computed eigenvalues M > K +1. All eigenvalues can be partitioned as follows: λ ≥ λ2 ≥ . . . λK ≥ λK+1 ≥ λK+2 ≥ . . . λM 1      K signal eigenvalues

M −1


SMUSIC =

D(jω)

M −K noise eigenvalues

ui [m]z −m ; i = K + 1, . . . , M

(3)

B. Errors of Estimation The root–MUSIC algorithm uses the estimated covariance matrix to compute the signal zeros from (1). Also from (1) we can obtain the relation between the error of the signal zeros and the estimated D(z) [15]. When analyzing the mean squared error (MSE) of the signal zeros estimates, the relationship between the errors in signal zeros and the estimated D(z) is as follows: (1 − (zl + ∆zl )z −1 )(1 − (zl + ∆zl )∗ z) (4)

2 SMUSIC (L − M )σnoise · · E{|∆zi |2 } = L

M  N
H λk V (ωi )ek 2 · 2 2 (λk − σnoise )

where N is the dimension of the covariance matrix and M is the dimension of signal subspace. In the case of single signal source with following = L · P1 , λ1 = λsignal + parameters: power P1 , λsignal 1 1 2 σnoise , and e1 = V(ω1 ), the sensitivity of root–MUSIC is given by [15] (see (9)): 12L L = (L − 1)(L + 1) V1H (ω1 )Pnoise V1 (ω1 ) (12) Using (11), the expected error of estimation will be [14]: SMUSIC =

E{|∆z1 |2 }

When evaluating the errors of D(z) on the unit circle (D(z)|z=ejω = D(ejω )): D(ejωi ) = c|∆zi |2

|(1 − (zl + ∆zl )zi−1 |2(5)

l=1,l=i

≈ c|∆zi |2

L−1 

|(1 − zl zi−1 )|2

l=1,l=i

Taking the expected value on both sides, we obtain: E{D(ejωi )} = L−1 c l=1,l=i |(1 − zl zi−1 )|2

E{|∆zi |2 } =

=

c

L−1

L

l=1,l=i

= L lim

ω→ωi

|(1 − zl zi−1 )|2

|1 − ejωi e−jω |2 D(ejω )

=

= ·

12L · (13) (L − 1)(L + 1) 2 2 (L − 1) λ1 σnoise 12σnoise ≈ LN (LP1 )2 L2 P1 N

The analysis of more than one sources case is analytically very difficult (see [15]) and demands more arbitrary assumptions about the SNR and other signal parameters. Although reported results of numerical simulations show good correspondence to derived analytical expressions, their usefulness is quite limited. IV. P ERFORMANCE A NALYSIS OF ESPRIT

(6)

A. ESPRIT

E{D(ejωi )} = SMUSIC L where L is the number of samples and SMUSIC can be seen as a sensitivity parameter of the root–MUSIC method and is equal to [15]: SMUSIC

(11)

k=1

l=1

L−1 

= VH (ω)(I − Psignal )V(ω) = (10)

M 
= 1 − VH (ω) el eH V(ω) l

ˆl = el + ηl , where η is the respective and, that estimated e estimation error , it is possible to formulate the MSE of the roots in root–MUSIC [15], as (see (6)):

Every eigenfilter has M − 1 roots, K roots are common for all eigenfilters. The common K roots can be found by averaging.

L−1


(9)

l=1

m=0

D(z) = c

L V
H (ωi )Pnoise V
(ωi )

where: Pnoise = I − Psignal . Considering, that:

(2)

Instead of one annihilating filter (as in Pisarenko’s estimator), MUSIC method uses M −K noise eigenfilters. Ui (z) =

After introduction of the derivative of V(ω):
1  0, jejω , 2je2jω , ..., j(L − 1)e(j(L−1)ω) V T (ω) = √ L (8) and taking into account, that D(jω) = VH (ω)Pnoise V(ω), SMUSIC becomes:

(7)

The original ESPRIT (Estimation of Signal Parameter via Rotational Invariance Technique) was described by Paulraj, Roy and Kailath and later developed, for example, in [16]. It is based on a naturally existing shift invariance between the discrete time series which leads to rotational invariance between the corresponding signal subspaces. The shift invariance is illustrated below. After the eigen–decomposition of the autocorrelation matrix as: (14) Rx = U∗T ΛU

1642

it is possible to partition a matrix by using special selector matrices which select the first and the last (M − 1) columns of a (M × M ) matrix, respectively: Γ1

=

[IM −1 |0(M −1)×1 ](M −1)×M

Γ2

=

[0(M −1)×1 |IM −1 ](M −1)×M

(15)

By using of matrices Γ two subspaces are defined, spanned by two subsets of eigenvectors as follows: S1 S2

= Γ1 U = Γ2 U

(16)

From (18), the approximation of error ∆Φ can be derived using:

0

···

Γ1 (UT)Φ ∗T

Γ1 U (TΦT  

eig. of Φ

) 

= Γ2 (UT) = Γ2 U

In the further considerations the only interesting subspace is the signal subspace, spanned by signal eigenvectors Us . Usually it is assumed that these eigenvectors correspond to the largest eigenvalues of the correlation matrix and Us = [u1 , u2 , . . . , uK ]. ESPRIT algorithm determines the frequencies ejωK as the eigenvalues of the matrix Φ. In theory, the equation (18) is satisfied exactly. In practice, matrices S1 and S2 are derived from an estimated correlation matrix, so this equation does not hold exactly, it means that (18) represents an over–determined set of linear equations. B. Errors of Estimation In the case of ESPRIT algorithm, the main source of errors is the estimate of the matrix Φ. The equation (18) can be solved for Φ using Least Squares or Total Least Squares approach. The choice of approach has no influence on asymptotical performance of ESPRIT as shown in [15]). The error in the matrix Φ, denoted as ∆Φ , causes errors in the eigenvalues of Φ. The error of an eigenvalue (here denoted as ∆zi ), which can be regarded as a performance index of ESPRIT and can be approximated by: ∆zi = pi ∆Φ ei

(20)

where ei is the eigenvector of Φ corresponding to the eigenvalue zi , whereas pi is the corresponding left eigenvector, so that Φei = zi ei and pi Φ = zi pi .

(23)

H

H H +H · (Γ1 − zi∗ Γ2 ) S+H · 1 pi = pi S1   M
  2 H  |eij | (Γ1 − zi∗ Γ2 ) E ζj ζjH (Γ1 − zi∗ Γ2 ]  j=1 +H ·pi S+ = pH (Γ1 − zi∗ Γ2 ) · i S1   1 M L

λ λ j k 2 H · |eij | · 2 Uk Uk N (λ − λ ) j k j=1

(18)

(19)

(22)

E{|∆zi |2 } =   ∗ H H pi S+ 1 (Γ1 − zi Γ2 ) E ∆U ei ei ∆U ·

the following relation can be proven [5]: The matrix Φ contains all information about frequency components. In order to extract this information, it is necessary to solve (18) for Φ. By using a unitary matrix (denoted as T), the following equations can be derived:

+ ∆Φ ≈ S+ 1 ∆S2 − S1 ∆S1 Φ

By substituting (22) in (20) it is possible to obtain expression for MSE of ∆zi , as (Γ1 , Γ2 are defined as in (15), U as in (14) and ζ is the respective eigenvalue estimation error ) [14]:

ejωk

[Γ1 U]Φ = Γ2 U

(21)

as:

For the matrices defined as S1 and S2 in (16), for every ωk ; k ∈ N, representing different frequency components, and matrix Φ, defined as:  jω  e 1 0 ··· 0  0 0  ejω2 0   (17) Φ= . . ..  . . . .  . . . .  0

(S1 + ∆S1 )(Φ + ∆Φ ) ≈ (S2 + ∆S2 )

k=1,k=j

· (Γ1 −

H zi∗ Γ2 )

pi S+ 1

where L is the number of samples, N is the dimension of the covariance matrix and M is the dimension of signal subspace. In the case of single signal source with following parameters: power P1 , λsignal = L · P1 , U1 = V(ω1 ) = 1   jω j(L−1)ω1 T 1 √1 1, e , . . . , e , the dominant term of MSE L of ESPRIT is given by substituting for the parameters in (23) [14]: 2σ 2 (24) E{|∆z1 |2 } ≈ 2 noise L P1 N It can be noted that, approximately, the mean square error of MUSIC (13) is six times higher than the MSE of ESPRIT (24) in the case of a single signal source. V. N UMERICAL P ERFORMANCE C OMPARISON OF MUSIC AND ESPRIT Several experiments with simulated, stochastic signals were performed, in order to compare different performance aspects of both parametric methods MUSIC and ESPRIT, compared to commonly used power spectrum (FFT based method). Testing signals are designed to belong to a class of waveforms often present in power systems. Each run of spectrum and power estimation is repeated many times (Monte Carlo approach) and the mean–square error (MSE) is computed. Parameters of test signals: • one main 50 Hz harmonic with unit frequency and amplitude, • random number of higher odd harmonic components with random amplitudes (lower than 0.5) and random initial phases (from 0 to 8 higher harmonics); if not otherwise specified, • sampling frequency 5000 Hz,

1643

2

4

10

10

MSE freq. ESPRIT MSE freq. MUSIC calc. time ESPRIT calc. time MUSIC

2

10

1

10 0

log MSE; time [s]

log(MSE)

10

−2

10

−4

10

0

10

−1

10

−6

10

−10

10

−2

10

MSE freq. ESPRIT MSE power ESPRIT MSE freq. MUSIC MSE power MUSIC

−8

10

0

20

−3

40

60

80

10

100

150

200

SNR [dB]

Fig. 1: MSE of frequency and power estimation (ESPRIT, MUSIC) depending on SNR. Averaged 1000 independent runs.

250 300 350 400 window length [samples]

450

500

Fig. 3: MSE of frequency and power estimation (ESPRIT, MUSIC) and average calculation time depending on the data window length. Averaged 10000 independent runs.

3

10

0

10 2

10

1

10

−1

10

0

10

MSE

log(MSE); time[s]

MSE amp. MUSIC MSE amp. ESPRIT MSE amp. power spectrum

MSE freq. ESPRIT MSE power ESPRIT MSE freq. MUSIC MSE power MUSIC calculation time [s]

−1

10

−2

10

−2

10

−3

10

−3

10

−4

10

30

40

50 60 70 80 dimension correlation matrix

90

100

−4

10

Fig. 2: MSE of frequency and power estimation (ESPRIT, MUSIC) depending on the size of correlation matrix. Averaged 1000 independent runs. • • • •

each signal generation repeated 1000–100000 times with re–initialization of random number generator, SNR=20 dB if not otherwise specified, size of the correlation matrix = 50 if not otherwise specified, signal length 200 samples if not otherwise specified.

Selected results are presented below: The relation to signal–to-noise ratio (Fig. 1) reveals strong dependence of the accuracy of the frequency estimation on SNR and almost no dependence of amplitude estimation (with exception to MUSIC which shows higher errors for very low and very high noise levels). The size of the correlation matrix must be chosen optimally, as can be seen from Fig. 2. In the case of both methods, there exists an optimum of the size (relative to the data length) which assures the lowest estimation error. Most probably, there exists a trade-off between increasing accuracy of the estimated correlation matrix and increasing numerical errors with the matrix size. The data sequence length influences the accuracy of MUSIC method than ESPRIT stronger(Fig. 3). For shorter

0.1

0.2

0.3 0.4 0.5 0.6 0.7 relative amplitude of higher harmonics

0.8

0.9

Fig. 4: MSE of amplitude estimation (ESPRIT, MUSIC, power spectrum) depending on the relative amplitude of higher harmonics amplitudes. Averaged 10000 independent runs.

data lengths ESPRIT method is faster to calculate; this advantage vanishes with increasing number of data samples taken into calculation. In Fig. 4 the results are shown where the amplitude of higher harmonics was gradually increased from 0.1 to 0.9 of the fundamental 50 Hz component. In such way the problem of masking of the higher low–amplitude harmonics components by a strong fundamental component was investigated. The results show an extremely high masking effect in the case of power spectrum, while MUSIC and ESPRIT methods show very little dependence (almost no dependence in the case of ESPRIT method). This is a very important feature which partially explains excellent performance of parametric methods in the task of calculation of power quality indices. VI. P OWER Q UALITY I NDICES A number of power system applications require an accurate knowledge of the spectral components of current and voltage waveforms. Especially, the power quality field attracts increasing interest. The main application

1644

4

of spectral components in the field of Power Quality refers to the calculation of waveform distortion indices. Several indices are in common use for the characterization of waveform distortions. However, they generally refer to periodic signals which allow an ”exact” definition of harmonic components and require only a numerical value to characterize them. The waveforms obtained from a power supply of a typical DC arc furnace plant are analyzed. The IEC groups and subgroups [7] are estimated by using FFT and the results are compared with advanced methods: the ESPRIT and the root–MUSIC methods.

2

V

A

Fig. 5: Simulated DC arc furnace plant.

V

1.5

B

V

C

1

voltage [V]

0.5 0 −0.5 −1 −1.5

A. Experimental Setup and Preprocessing

−2

The simulated DC arc furnace plant, which is shown in Fig. 5. It consists of a DC arc furnace connected to a medium voltage ac busbar with two parallel thyristor rectifiers that are fed by transformer secondary windings with ∆ and Y connections, respectively. The power supply of arc furnace is modelled using Power System Blockset in Matlab. The electric arc was simulated with a Chua’s circuit, which shows good similarity with real measurements [2]. Exemplary voltage waveforms at the medium voltage AC busbar are shown in Fig. 6. The medium voltage busbar is connected to the high voltage busbar with a HV/MV transformer whose windings are ∆–Y connected. The power of the furnace is 80 MW. The other parameters are: Transformer T1 - 80 MVA, 220kV/21kV; Transformer T2 – 87 MVA, 21kV/0.638kV/0.638kV. The evaluation of harmonic and interharmonic subgroups has been made using the following assumptions: window length – 200 ms non overlapping. For each window, the nth harmonic subgroup includes all spectral components inside the frequency interval [n · f1 − 7.5, n · f1 + 7.5] Hz. The interharmonic subgroup includes all the spectral components inside the frequency interval ]n · f1 + 7.5, (n + 1) · f1 − 7.5[ Hz [3]. When applying parametric methods filters have been applied for pre– processing of data. In particular: a bandstop Butterworth IIR filter blocking the main (50Hz) component; a lowpass (40 Hz) Butterworth IIR filter applied for analyzing interharmonics groupings for n = 0.5 and bandpass Butterworth IIR filters for other subgroups, The amplitudes of the harmonic and interharmonic subgroups Cn−200ms and Cn+0.5−200ms can be evaluated,

x 10

0

0.005

0.01

0.015

0.02 time [s]

0.025

0.03

0.035

0.04

Fig. 6: Voltage waveform of the arc furnace supply – medium voltage AC busbar.

respectively, as: 2 Cn−200ms =

1


2 C10n+k

(25)

k=−1 2 = Cn+0.5−200ms

8


2 C10n+k

(26)

k=−2

where C10n+k are the spectral components (RMS value) of the spectral (DFT) output. According to the cited norms the relations (25) and (26) are computed on 15 successive 200 ms windows in order to obtain values of the progressive average inside a 3 seconds interval. Obtained results were compared to the “Ideal IEC” which is a value of interharmonic or subharmonic subgroups computed over the whole interval of 3 seconds [1] of the waveform under investigation. B. Results and Discussion Selected results of the progressive average of harmonic subgroups calculation of the waveforms of voltage and current are presented in Fig. 7 and 8. From the analysis of other results it can be noted that the results obtained by using ”Ideal IEC” give a very high value of the progressive average in the neighbourhood of the fundamental harmonic referred to the IEC interharmonic subgroups. This phenomenon can be explained by the problem of spectral leakage present in the FFT based algorithms (STFT) and therefore the high energy content leaking into the neighborhood of the fundamental component of the voltage waveform. As shown in the Fig. 7 and 8, the high resolution methods give results closer to the ”Ideal IEC” than the ones obtained with STFT for the evaluation of the progressive average. When analyzing selected current as well as voltage waveforms (not presented in this paper), the poor performance of root–MUSIC can be observed. It can be attributed to spurious roots [20] which in rare cases can ruin the results. STFT and ESPRIT methods are not affected.

1645

TABLE I:

Ideal IEC STFT ESPRIT rootMUSIC

1.15

MSE OF THE P ROGRESSIVE AVERAGE H ARMONICS S UBGROUPS E STIMATION . Method

1.1

Cmean(k) [p.u.]

3rd 1.05

1

OF THE

Harmonic Subgroup No. 5th 7th 11th

C URRENT

13th

STFT

1.23

0.23

0.85

16.00

ESPRIT

1.33

0.22

0.05

2.83

2.23 2.08

MUSIC

1.37

0.22

0.07

1.26

2.24

Ideal IEC [A]

17.00

13.85

23.64

95.50

46.76

0.95

TABLE II:

0.9

2

4

6

8 k

10

12

MSE OF THE PROGRESSIVE AVERAGE OF THE CURRENT INTERHARMONICS SUBGROUPS ESTIMATION .

14

Method

Fig. 7: Progressive average of the fifth harmonic subgroup of

Interharmonic Subgroup No. 1st

2nd

11th

12th

STFT

34.88

52.40

24.93

4.60

ESPRIT

9.22

3.02

2.67

8.14

the voltage. Ideal IEC STFT ESPRIT rootMUSIC

1.08

MUSIC

8.40

6.19

4.57

5.35

Ideal IEC [A]

61.13

43.56

29.26

29.58

1.06

Cmean(k) [p.u.]

1.04

1.02

VII. C ONCLUSION 1

0.98

0.96

0.94 2

4

6

8 k

10

12

14

Fig. 8: Progressive average of the thirteenth harmonic subgroup of the current. In some rare cases parametric methods give less accurate results or almost identical results when comparing to non–parametric STFT. The advantage of using parametric methods becomes evident when analyzing higher harmonic groups of the currents (Fig. 8) and voltages (Fig. 7). In the case of voltage harmonic subgroups estimation (Figures the results are comparable to those obtained using STFT. To summarize obtained results, the errors of indices’ estimation is shown in Tables I–IV which show the value of mean square error (MSE) of the estimation of interharmonic subgroups and allows comparison with the value of Ideal IEC. Values of MSE support excellent performance of parametric methods when computing interharmonic subgroups and slightly decreased accuracy in the case of harmonic subgroups, especially of voltage waveforms. For all results presented previously, it can be seen (Table V) that the use of ESPRIT method for calculation of power quality indices offers reduction of the error of estimation of harmonic subgroups by 53% and the use of MUSIC method reduces the error by 49%, comparing to STFT (FFT–based method).

In practical applications, one of the most important questions concerns the optimal choice of analysis methods when taking into account known parameters of the signal and limitations of the chosen analysis technique. These problems were addressed in the section V. Testing signal were chosen that correspond to mostly often encountered waveforms in power systems. Most important results show that an optimal size of the correlation matrix can be chosen. Further increase of the size of the correlation matrix does not improve the accuracy. In general, parametric methods show similar values of accuracy (with slight advantage of ESPRIT method) which greatly outperform the accuracy of FFT–based non–parametric method. Moreover, parametric methods show almost complete immunity to masking effect (see Figure 4) to variable initial phase of harmonic components and to many other deficiencies off FFT–based techniques, as shown in [10]). Interestingly, when comparing strongly simplified theoretical comparison of performance of ESPRIT and MUSIC

TABLE III:

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MSE OF THE P ROGRESSIVE AVERAGE H ARMONICS S UBGROUPS E STIMATION . Method

STFT

OF THE

VOLTAGE

Harmonic Subgroup No. 3rd

5th

7th

11th

13th

103.37

6.22

2.19

90.93

27.94

ESPRIT

201.01

2.40

6.40

14.56

28.02

MUSIC

210.48

3.02

5.22

11.99

20.90

Ideal IEC [A]

124.99

25.43

19.60

242.57

158.11

TABLE IV:

R EFERENCES

MSE OF THE P ROGRESSIVE AVERAGE OF THE VOLTAGE I NTERHARMONICS S UBGROUPS E STIMATION . Method

Interharmonic Subgroup No. 1st

2nd

11th

12th

13th

357.33

205.34

120.20

27.19

53.40

ESPRIT

110.77

22.77

7.73

11.83

14.92

MUSIC

122.29

9.03

20.39

13.22

17.90

Ideal IEC [A]

70.22

75.16

73.56

82.89

76.17

STFT

TABLE V: R ELATIVE M EAN S QUARE E RROR OF THE P ROGRESSIVE AVERAGE OF H ARMONIC AND I NTERHARMONIC S UBGROUPS E STIMATION (harm. - H ARMONICS , interh. -I NTERHARMONICS ). Method

STFT

Error of current

Error of voltage

harm.

interh.

harm.

interh.

0.067

1.311

1.420

4.475

Total error

1.720

ESPRIT

0.026

0.180

2.173

0.521

0.801

MUSIC

0.024

0.235

2.364

0.563

0.878

(see equations (13) and (24)), the main result is confirmed in numerical simulations (ESPRIT is more accurate than MUSIC), although the difference of performance is not as high as sixfold. Following section VI was devoted to the assessment of the power quality. Most power quality indices use FFT–based techniques. It was shown that application of parametric methods allows approximately 50% reduction of the estimation error. This result was obtained despite the fact that for comparison a procedure was chosen where the minimum error is expected for FFT–based technique (i.e. analysis window length equal to one period of the fundamental harmonic). Even higher gains in accuracy were achieved when analyzing waveforms with high inter/sub–harmonic contents , e.g. [13]. It was shown that the use of high-resolution spectrum estimation methods instead of Fourier-based techniques can significantly improve the accuracy of measurement of spectral parameters of distorted waveforms encountered in power systems, in particular the estimation of the power quality indices, such as inter/harmonic groups and subgroups . ACKNOWLEDGMENT This work was supported in part by the Polish Ministry of Science under Grant No. 3T10A 04030.

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