A new joint sliding-window ESPRIT and DFT scheme for waveform distortion assessment in power systems

Electric Power Systems Research 88 (2012) 112–120

Contents lists available at SciVerse ScienceDirect

Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

A new joint sliding-window ESPRIT and DFT scheme for waveform distortion assessment in power systems Antonio Bracale a,∗ , Guido Carpinelli b , Irene Yu-Hua Gu c , Math H.J. Bollen d,e a

Department for Technologies, University of Naples Parthenope, Centro Dir. di Napoli Is. C4, 80143 Napoli, Italy Department of Electrical Engineering, University of Naples Federico II, Via Claudio, 21, 80125 Napoli, Italy Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden d STRI AB, 421 31 Gothenburg, Sweden e Electric Power Engineering, Luleå University of Technology, 731 81, Skellefteå, Sweden b c

a r t i c l e

i n f o

Article history: Received 16 June 2011 Received in revised form 23 November 2011 Accepted 1 February 2012 Available online 17 March 2012 Keywords: Power system measurements Power system disturbances Waveform distortion assessment Sliding-window ESPRIT Sliding-window DFT Harmonics Interharmonics

a b s t r a c t This paper proposes a novel scheme that jointly employs a sliding-window ESPRIT and DFT for estimating harmonic and interharmonic components in power system disturbance data. In the proposed scheme, separate stages are utilized to estimate the voltage fundamental component, harmonics and interharmonics. This includes the estimation of the fundamental component from lowpass filtered data using a sliding-window ESPRIT, of harmonics from a sliding-window DFT with a synchronized window, and of interharmonics from the residuals by applying the sliding-window ESPRIT. Main advantages of the approach include high resolution and accuracy in parameter estimation and significantly reduced computational cost. Experiments and comparisons are made on both synthetic and measurement data. Results have shown the effectiveness and efficiency of the proposed scheme. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Distortion of voltage and current waveforms remains one of the major power-quality disturbances. Its origins are discussed in many literatures [1–3]. Waveform distortion is mainly assessed in terms of harmonic and interharmonic components of the power system (simplified as harmonics and interharmonics in the remaining text). In power system community, the terminology “harmonics” refer to voltage (or current) components in the data whose frequencies fk are integer-multiples of the power system nominal frequency f0 , (i.e. fk = kf0 , k = integer, f0 = 50 Hz or 60 Hz), whereas “interharmonics” relate to components whose frequencies are non-integer multiples of f0 , (i.e. fm = mf0 , m = / integer). For periodic data analysis, Fourier transform is a very effective tool. Several international standards and recommendations exist, e.g. on indices characterizing voltage and current distortions in a power system, and on methods measuring and interpreting the results [2]. In particular, IEC standards contain specific signal processing recommendations and definitions where DFT is applied over successive rectangular

∗ Corresponding author. Tel.: +39 0815476757; fax: +39 0815476777. E-mail addresses: [email protected], [email protected] (A. Bracale). 0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2012.02.002

time windows (or, “Short Time Fourier Transform” in signal processing terminology) of ten/twelve cycles (in 50/60 Hz systems) of fundamental period to obtain spectral components [4,5]. While the standard method is important to quantify the waveform distortion, it is of limited use for extracting detailed information and for studying individual components. Difficulties arise, such as the occurrence of spectral leakage when the time window length is not exactly an integer-multiple of power system fundamental period. The spectral leakage is primarily due to errors in synchronizing harmonics as the result of deviation in fundamental frequency, and secondary due to the presence of interharmonics [6]. Interharmonics are likely to grow with the increased use of new type generators and equipments. A direct approach to obtain good synchronization is to use an analog Phase Locked Loop (PLL), though problems for interharmonics remain. Other reasons of de-synchronization are related to the instrument used. For instance, the frequency resolution of sampling clock is finite, and it is practically difficult to synchronize the clock with the fundamental frequency. Another important factor causing de-synchronization is the presence of non-stationary waveform distortion. Many efforts are made to improve the DFT-based distortion analysis, e.g. using a Hanning instead of rectangular window to reduce the spectral leakage [6], using self-tuning that synchronizes the window as integer-multiple of actual fundamental period [7],

A. Bracale et al. / Electric Power Systems Research 88 (2012) 112–120

using a desynchronized two-stage algorithm that estimates harmonics followed by interharmonics from the harmonic-subtracted data [8], using parametric methods, e.g. Prony and ESPRIT, for estimating spectral components [2,3,9–15]. In [12], a sliding-window ESPRIT is introduced to estimate harmonics and interharmonics from non-stationary distortion recordings. In [13–15], slidingwindow Prony and ESPRIT-based methods are proposed for solving de-synchronization problems. These parametric methods provide improved estimation of harmonics and interharmonics with high frequency resolution, usually at the cost of significantly increased computations. Motivated by the above, this paper proposes a novel scheme that may provide accurate estimation of parameters (mainly, frequencies, amplitudes and initial phases) of distortion components meanwhile keep a relatively low computational cost. The proposed method applies a sliding-window ESPRIT, a slidingwindow DFT, and a sliding-window ESPRIT in different passbands for separately estimating the power system fundamental, harmonics and interharmonics. Using the strategy of divide and conquer, each step focuses on estimating different data components. To increase the efficiency and accuracy, the sliding-window ESPRIT is employed that focus on estimating the power system fundamental, whereas harmonics are estimated using the sliding-window DFT with a synchronized window to mitigate spectral leakage. In such a way, the proposed method may significantly reduce the inaccuracy caused by the spectral leakage in DFT-based methods. It may be utilized as a reference method in laboratory measurements for research purposes. In addition, the proposed method significantly reduces the computational cost as compared with conventional parametric methods. This is more attractive for industrial applications where evaluation of large amount of data requires heavy computations in quantifying the distortions. The remainder of the paper is organized as follows. In Section 2, sliding-window DFT and sliding-window ESPRIT are briefly reviewed. Section 3 describes the proposed scheme in detail. In Section 4, several case studies are described with results included. Performance evaluations and comparisons to the conventional methods are also made. Finally, the conclusion is given in Section 5. 2. Sliding window DFT and sliding-window ESPRIT: brief review This section briefly reviews the sliding-window DFT, ESPRIT and sliding-window ESPRIT [2,3,12,16–18] for the sake of mathematical convenience in the next section.

The sliding-window DFT, commonly referred to as windowed DFT in power engineering literatures or STFT (Short Time Fourier Transform) in signal processing literatures, is a standard tool for time-dependent spectral analysis [1,2]. Given a waveform sequence x(n), an L-point DFT is given by: X(k) =

L−1 

x(n)e−j2(k/L)n ,

k = 0, 1, · · ·L − 1

(1)

n=0

If x(n) is non-stationary, a sliding-window DFT is required. This is done by using a window that slides forward successively over time. Rectangular windows are commonly adopted, e.g. in IEC 61000-4-7 [5] and IEC 61000-4-30 [4]. Given a waveform sequence x(n) of N samples (N > L), the sliding-window DFT is defined as: Xm (k) =

L−1  n=0

x(n) w(n − m)e−j2(k/L)n ,

where w(n) is a window function of size L, and m is the starting time instant. The time duration of sliding window Tw = LTs (Ts is the sampling interval) determines the frequency resolution f = 1/Tw of the spectrum. The choice of L is often a compromise between the time and frequency resolution. IEC standards recommend a window duration Tw of 10 cycles (50-Hz systems) and 12 cycles (60-Hz systems) of fundamental frequency [4,5]. It is worth noting that a rectangular window size in the sliding-window DFT may impact the analysis result. In particular, a synchronized window, where Tw , equals to an integer-multiple of power system fundamental period, is desirable to avoid the spectral leakage. 2.2. The ESPRIT and sliding-window ESPRIT Estimation of Signal Parameters by Rotational Invariance Technique (ESPRIT) [16] is based on the shift invariance between discrete time series, which leads to rotational invariance between the corresponding signal subspaces. It models the data as the sum of M complex exponentials in white noise [3,12]. For a given block of measurement data x(n) of size L, the following model is employed: x(n) =

M 

hk ek(˛k +j2fk )n + r(n),

k = 0, 1, · · ·, L − 1

(2)

n = 0, 1· · ·L − 1

(3)

k=1

where hk = Ak ej k , Ak is the amplitude, k the initial phase, fk the frequency, ␣k the damping factor of the kth exponential, and r(n) is the white noise. Eq. (3) may be equivalently written in a matrix form: n

x(n) = V˚ H + r(n)

⎢ ⎢ ⎣



x(n) = x(n) · · ·

where

 T h1 · · · hM , ⎡



r(n) = r(n)

1

1

e˛1 +j2f1 .. .

e˛2 +j2f2 .. .

x(n + N1 − 1) ··· ··· ··· .. . ···

T

(4) ,

r(n + N1 − 1) 1 e˛M +j2fM .. .

T

H= ,

⎤

V=

⎥ ⎥,  = ⎦

e(˛1 +j2f1 )(N1 −1) e(˛2 +j2f2 )(N1 −1) e(˛M +j2fM )(N1 −1) ⎤ (˛ +j2f ) 1 e 1 0 ··· 0 0 e(˛2 +j2f2 ) · · · 0 ⎢ ⎥ ⎢ ⎥and N1 < L is the sliding. .. .. .. ⎣ ⎦ .. . . . (˛ +j2f ) M M 0 ··· e 0 window size. The matrix  contains all information on the damping factors and frequencies. Using rotational invariance, estimating ˛k and fk in  can be equivalently done from using  satisfying [18]:

⎡

S2 = S1 

2.1. The DFT and sliding-window DFT

113

(5)

where S1 and S2 are two suitable matrices. One of the most common methods to estimate  is the least-squares (LS) approach ˆ = (Sˆ ∗ Sˆ 1 )  1

−1 ∗ Sˆ 1 Sˆ 2

(6)

Frequencies and damping factors are then calculated from the ˆ . Once the frequencies and damping faceigen-decomposition of  tors are estimated, amplitudes and initial phases can be obtained from another LS estimation [12]. It is worth noting that ESPRIT is particularly suitable for estimating frequencies when high frequency resolution is required; further, the computation increases with the data block size N1 and the number of exponentials M. Several criteria may be used to estimate M as this is a prespecified parameter in the ESPRIT. Commonly used criteria, among many others, include the Final Prediction Error (FPE), Akaike Information Criterion (AIC), Minimum Description Length (MDL), auto-regressive transfer criterion (CAT), and criteria based on eigen decomposition of sample autocorrelation matrix [19]. The MDL criterion is shown to be suitable for estimating M for waveform

114

A. Bracale et al. / Electric Power Systems Research 88 (2012) 112–120

Fig. 1. The block diagram of the proposed joint SW ESPRIT and DFT scheme.

distortions in power systems [19]. For non-stationary waveform distortion sequences, the SW (sliding-window) ESPRIT should be used [12], resulting in time-dependent estimates of frequency components.

3. Joint sliding-window ESPRIT and DFT scheme This section describes the proposed scheme in details. As mentioned previously, the sliding-window size in the DFT should be synchronized to be exactly an integer-multiple of the power system fundamental period, to mitigate the spectral leakage from the harmonics, since non-synchronized window may severely impact the accuracy of estimated harmonic components due to spectral leakage, especially for a small window size. ESPRIT-based methods may result in accurate estimation for both harmonics and interharmonics, however, the computation is high if the number of exponentials M is large. We propose a novel estimation scheme that jointly using SW ESPRIT and SW DFT maintains high estimation accuracy with relatively low computation. Initially, the SW ESPRIT is applied only to a lowpass filtered signal for estimating the power system fundamental frequency in time. This is because the fundamental frequency often deviates from its nominal value in real world scenarios. Once the fundamental frequency is estimated, a SW DFT with a synchronized window size is then applied to the remaining band for estimating power system harmonics. Interharmonic components are then estimated by employing the SW ESPRIT again, however, on the remaining signal with fundamental and harmonic components subtracted. Using SW DFT for estimating harmonic components results in a reduced number of remaining spectral components that needs to be estimated by the second SW ESPRIT (for the first SW ESPRIT, only the fundamental and a few discrete interharmonics generally exist within the lowpass signal band; for the second SW ESPRIT, only discrete interharmonics are estimated). Since the number of discrete interharmonics is significantly smaller than the total number of components M in x(n), a significant lower computational cost is required without sacrificing the estimation accuracy. The joint scheme, as depicted in the block diagram of Fig. 1, consists of three separate stages.

Table 1 The pseudo code for the proposed scheme. Step 1: Estimate power system fundamental (1.1) Select a starting time; set the window size = 10 (12) cycles of the fundamental period for a 50 Hz (60 Hz) system; and apply the LPF to x(n) followed by down-sampling; (1.2) Apply the SW ESPRIT to the waveform from Step 1.1; estimate the components (frequencies, amplitudes, initial phases); estimate Tˆfund and set the window size Tw = 10 Tˆfund (12 Tˆfund ) for a 50 Hz (60 Hz) system. Reconstruct the fundamental xˆ fund (n). Step 2: Estimate power system harmonics (2.1) Form the fundamental subtracted signal as the input of this step, x1 (n) = x(n) − xˆ fund (n); Apply the SW DFT with a synchronized window size Tw ; estimate the parameters of harmonics; reconstruct the sum of harmonics xˆ harm (n). Step 3: Estimate power system interharmonics Form the residual signal as the input of this step, x2 (n) = x(n) − xˆ fund (n) − xˆ harm (n) Apply the SW ESPRIT, and estimate the parameters of interharmonic components.

3.2. Estimate parameters of harmonic components In this step, the SW DFT is applied to estimate power system harmonics from the remaining signal, x1 (n) = x(n) − xˆ fund (n), where the reconstructed fundamental component is subtracted. The window size of SW DFT is set to be an integer-multiple of the estimated fundamental period Tˆfund to minimize the spectral leakage from other harmonics. The time-window size can be set to 10 cycles (or, 12 cycles) of Tˆfund if the power system frequency is 50 Hz (or, 60 Hz), following the IEC Standards.1 Using the synchronized SW DFT, harmonic components may be accurately estimated. The output from this step includes the estimated parameters (i.e. frequencies, amplitudes, initial phases and damping factors) of all harmonics, with which the sum of reconstructed harmonic components xˆ harm (n) is obtained. Main advantages of synchronized SW DFT are the accurate harmonic estimation with very low computation. One could observe that the presence of interharmonics in the signal x1 (n) could introduce some spectral leakage in the SW DFT, and nonsynchronized interharmonics also cause errors but relatively small [6].

3.3. Estimate parameters of interharmonic components

3.1. Estimate parameters of power system fundamental component In the first step, a SW ESPRIT is applied to estimate the power system fundamental component in the lowpass band f ∈ [0, 100] Hz. The lowpass filtered signal is then down-sampled, and the SW ESPRIT is applied to each window (size L) of lowpass filtered data with a pre-specified small number of exponentials M1 . This results in the estimated parameters of power system components in the lowpass band, including the fundamental component. Using the estimated parameters associated with the fundamental component (including the estimated fundamental period Tˆfund ), the reconstructed fundamental component xˆ fund (n) can be obtained. Since M1 is significantly smaller than M, the computation in this step is significantly less as compared with applying the SW ESPRIT to the full band signal x(n).

In this step, the SW ESPRIT method is again employed, however, to the residuals x2 (n) = x(n) − xˆ fund (n) − xˆ harm (n). This step is designed to obtain an accurate estimation of all interharmonics in the highpass band, i.e. frequency f > 100 Hz. The output from this step includes the estimated parameters (i.e. frequencies, amplitudes, initial phases and damping factors) of all interharmonics at frequencies f > 100 Hz. Since harmonic components are already subtracted from x2 (n), the SW ESPRIT in this step requires much less computation as compared with the SW ESPRIT applying to signal with all harmonics included in the band. Table 1 summarizes the pseudo code of the proposed scheme.

1 It is worth mentioning that resampling is applied to the signal before applying the synchronized SW DFT, so that the resulting time-window size is an integermultiple of Tfund .

A. Bracale et al. / Electric Power Systems Research 88 (2012) 112–120

3.4. Discussions (a) Window size in the SW DFT: the window-synchronized SW DFT in this scheme uses a time window of exactly 10 or 12 cycles (as suggested by the IEC standards). However, one may reduce the window size, e.g. to two cycles, in order to obtain a higher time resolution for harmonic estimates. Since harmonics can be time-varying, a shorter window (with an integer-multiple of fundamental) should give a better estimation. However, using a shorter time window reduces the frequency resolution, which may lead to more spectral leakage of interharmonics and hence less accurate estimates of harmonics, as the improved estimate of Tˆfund only prevents the spectral leakage for harmonics. Hence, when applying SW DFT for estimating harmonics from nonstationary disturbance sequences, selecting time window size should be a tradeoff between the high time resolution and low spectral leakage if interharmonics are present. (b) Computational cost: while the proposed scheme is shown to provide high estimation accuracy, its computational cost is larger than that in SW DFT, and smaller than SW ESPRIT applied to a full band data x(n). Our experiments indicated that the proposed scheme requires no more than 40% of computations as compared with the full-band SW ESPRIT.

4. Experiments and results To evaluate the effectiveness and the performance of the proposed scheme, tests were performed on both waveform recordings measured in the power system as well as synthetically generated data sequences. As examples, five case studies, two from the synthetic waveforms and three from the measured voltage and current waveforms, are included in this section for the demonstration and the evaluation in terms of effectiveness, estimation accuracy, and computational speed. In the first two cases, the distorted waveforms were generated synthetically where comparisons between the proposed scheme, the SW DFT, and the SW ESPRIT are made using the ‘ground truth’ parameter values of frequency components. Using synthetically generated data sequences in the first two case studies allows us to compare the estimated parameters with their ground truth values, and to evaluate and compare the performance of the proposed scheme. For all cases, lowpass filter is formed by a linear-phase FIR filter with a cutoff frequency of 200 Hz. Matlab programs were made and tested on a Windows PC with an Intel® CoreTM 2 Duo 2.66 GHz processor and 2048 MB of RAM memory. 4.1. Case studies and results Case study-1: The synthetic signal of interest in this case study is an “acid test”, consisting of a 50.02 Hz component of amplitude 100, and two interharmonic components of amplitude 0.1 at frequencies of 58 and 63 Hz. The fundamental frequency value is set to introduce desynchronization. Such desynchronization results are comparable with the maximum permissible error of IEC instruments shown in [5]. The additive noise in (3) is white, Gaussian distributed with zero-mean and a standard deviation of 0.01. The duration of data sequence is 10 s, and the sampling rate is 5000 Hz. Since interharmonics are present only in [0,100] Hz band, only the first step in the proposed scheme is required in this case study. In the tests, neighboring data blocks are chosen with 0.16 s overlap in order to yield more smooth results through windows. Table 2 shows the estimated mean values and standard deviations of the estimated harmonic and interharmonic amplitudes, frequencies and initial phases from the proposed scheme. The mean frequency and

115

magnitude values and their standard deviations are calculated using the results obtained from all data blocks as defined in [12]; observing Table 2, it shows that the proposed scheme has yielded an accurate estimation for all spectral components in terms of amplitudes, frequencies and initial phases. Case study-2: This signal is a synthetic, consisting of a fundamental component of magnitude 100 at frequency 50 Hz, and two interharmonics of amplitude 3 and 5 at frequencies of 82 Hz and 182 Hz, respectively; the signal also includes 2nd, 3rd, 4th, 5th, and 7th harmonics of magnitudes 0.5, 3.0, 1.0, 4.0, and 2.5, respectively. The additive noise in (3) is white zero-mean Gaussian distributed with a standard deviation of 0.05. The data sequence x(n) is 10 s long, and the sampling rate is 5000 Hz. Table 3 shows the results obtained from the proposed scheme, in terms of mean estimates and standard deviations of amplitudes, frequencies and initial phases for the harmonics and interharmonics. From Table 3, one may observe that the proposed scheme has yielded accurate parameter estimation for all frequency components including harmonics and interharmonics. The standard deviations of the harmonic amplitude estimates are somewhat increased however within an acceptable level. The largest standard deviations in the 3rd and 4th harmonic amplitude estimates are due to the spectral leakage in the SW DFT caused by the large and non-synchronized 182 Hz interharmonic in x1 (n). Case study-3: A measured voltage recording (sampling rate 7200 Hz) is used in this test. For all measured data recordings, an anti-aliasing filter is present in the measurement device before analog-to-digital conversion. Fig. 2 shows the time-dependent parameters of signal components from the proposed scheme. A same number marked on the curve in Fig. 2a and b implies that these two curves correspond to a same signal component. Observing Fig. 2a, one can see that the spectral line splits when estimating 50 Hz fundamental frequency. One interharmonic magnitude in [0,100] Hz band is relatively large, about 2% of the fundamental amplitude (about 231 V). It also shows the presence of a 2nd harmonic whose magnitude does not reach 0.1% of the fundamental. Fig. 2b shows that there are two interharmonics: one is at the frequency range between 183.240 and 183.438 Hz with a slow varying magnitude between 3 and 4 V; another is at the frequency range between 283.325 and 283.541 Hz with a magnitude of about 1.5 V. Further, Fig. 2a shows that 2nd, 3rd, 4th, 5th, 7th, 9th, 11th, 13th and 15th harmonics are present; all amplitudes of harmonics fluctuate in time. Further, 5th and 7th harmonics have relatively large amplitudes however below 3 V (1.2% of the fundamental magnitude). Case study-4: In this case study, a measured voltage recording of 40 s, obtained close to a large arc furnace, is used. Fig. 3 shows the time dependent parameters of frequency components (frequencies and magnitudes) estimated from the proposed scheme. Observing the first two rows in Fig. 3a, one can see that the most significant interharmonics are in the frequency range of [0,200] Hz. Two interharmonics in the third and fourth rows are stationary in terms of frequency; however, their magnitudes contain small fluctuations (note the scale of the vertical axis). One can also see that there is a relatively strong interharmonic and relatively large magnitude fluctuation in time at a frequency around 125 Hz. Case study-5: In this case study, measured waveform sequences of both the current and voltage, obtained during the soft-starting of a wind turbine, are used. The sampling rate is 2048 Hz. An overlapped window of 0.04 s is used in the proposed scheme. Fig. 4 shows the time variations of estimated spectral components (frequencies and magnitudes) from the proposed scheme. Observing Fig. 4a, one can see that all components are at harmonic frequencies (and also stationary). Observing Fig. 4b shows that the magnitudes are non-stationary containing significant variation in time during the soft-starting of the wind turbine (between the time interval of

116

A. Bracale et al. / Electric Power Systems Research 88 (2012) 112–120

Table 2 Case study-1: mean and standard deviation (std) of estimated amplitudes, frequencies and initial phases of components over data blocks, from the proposed scheme. Proposed scheme Frequency

Magnitude

Initial phase

True

Mean

Std

True

Mean

Std

50.02 58.00 63.00

50.02 58.00 63.00

1.98e−05 0.04 0.03

100.00 0.10 0.10

100.01 0.10 0.10

0.01 5.90e−04 1.01e−03

True 0,00 0.00 0.00

Mean

Std

0.00 −0.04 0.03

1.30e−05 0.38 0.54

Table 3 Case study-2: mean and standard deviation (std) of estimated amplitudes, frequencies and initial phases of components over data blocks, from the proposed scheme. Proposed scheme Frequency

Magnitude

Initial phase

True

Mean

Std

True

Mean

Std

True

Mean

Std

50.00 82.00 100.00 150.00 182.00 200.00 250.00 350.00

50.00 82.00 100.00 150.00 182.00 200.00 250.00 350.00

5.64e−05 1.00e−03 0.00 0.00 5.91e−04 0.00 0.00 0.00

100.0 3.00 0.50 3.00 5.00 1.00 4.00 2.50

100.0 3.00 0.49 3.00 5.00 1.05 4.00 2.50

0.06 2.50e−03 0.03 0.14 2.32e−03 0.24 0.06 0.02

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

3.00e−04 0.81 1.97e−01 5.05e−02 0.81 2.80e−01 2.27e−02 1.66e−02

a

b

110

90 80 70

8 7 6 5 4 3

60

2 1

50 0.1

10 9

Magnitude [V ]

Frequency [Hz]

100

11

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.1

0.5

0.15

0.2

0.25

t [s]

0.3

0.35

0.4

0.45

0.5

t [s]

300

8

5 4

250

Magnitude [V ]

Frequency [Hz]

7

3 200

150 0.1

2

0.2

0.25

0.3

0.35

0.4

0.45

5

2

4

4

3

5

2

0

0.5

0.1

0.15

0.2

0.25

t [s]

0.35

0.4

0.45

0.5

8

5

750

7

650

Magnitude [V ]

4

700

Frequency [Hz]

0.3

t [s]

800

3

600 550

2

500

1

400

6 5 4

1

3

2

2

450

350 0.1

1

3

1

1 0.15

6

1

4

3

5

0 0.15

0.2

0.25

0.3

t [s]

0.35

0.4

0.45

0.5

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t [s]

Fig. 2. Case study-3: estimated signal components versus time without the fundamental. (a) Estimated frequencies in [0,100], [150,300], [350,800] Hz; (b) corresponding amplitudes.

A. Bracale et al. / Electric Power Systems Research 88 (2012) 112–120

a

b

110 100

117

1.5

80

Magnitude [kV]

Frequency [Hz]

90 70 60 50 40

1

0.5

30 20 10 0

0 0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

t [s]

t [s] 200

1.5

190

Magnitude [kV]

Frequency [Hz]

180 170 160 150 140 130

1

0.5

120 110 100

0

5

10

15

20 t [s]

25

30

35

0

40

460

5

10

15

20 t [s]

25

30

35

40

5

10

15

20 t [s]

25

30

35

40

5

10

15

20

25

30

35

40

0.5 0.45

440

0.4

Magnitude [kV]

Frequency [Hz]

0

420 400 380

0.35 0.3 0.25 0.2 0.15 0.1

360

0.05

340 0

5

10

15

20 t [s]

25

30

35

00

40

700

0.5 0.45

Magnitude [kV]

Frequency [Hz]

650 600 550 500

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

450

0 0

5

10

15

20

25

30

35

40

t [s]

0

t [s]

Fig. 3. Case study-4: estimated signal components versus time without the fundamental. (a) Estimated frequencies in [0,100], [100,200], [340,460], [440,700] Hz; (b) corresponding amplitudes.

1 s and 2.5 s). Also, the 5th and 7th harmonics are significant for both the current and the voltage. 4.2. Comparisons To further evaluate the performance, comparisons are made with two existing methods: • “Method-1” is the SW DFT that is directly applied to the full band signal x(n) using a 200 ms sliding window.

• “Method-2” is the conventional SW ESPRIT that is directly applied to the full band signal x(n), where the window size is set to Tw = 10 Tˆfund in 50 Hz systems (or, 12 Tˆfund in 60 Hz systems), and Tˆfund is the estimated fundamental period from the SW ESPRIT in the first stage. Comparisons are made in terms of both estimation accuracy (mean and standard deviation) and computational time. Since the ground truths from the measured recordings (case studies 3–5) are unknown, only case studies 1–2 are compared.

118

A. Bracale et al. / Electric Power Systems Research 88 (2012) 112–120

b

100 90

700 600

80

Magnitude [A]

Frequency [Hz]

a

70 60 50 40 30

500 400 300 200

20 100

10 0

0

1

2

3

4

5

600

0 0

7

4

70

400

Magnitude [A]

450

3

350 300

2

250 200

2

3

4

5

6

7

4

5

4

5

6

7

4

5

6

7

5

6

7

2

60

3

50 40 30 20

1

150 1

3

90 80

0

2

100

500

100

1

t [s]

550

Frequency [Hz]

6

t [s]

4

10 6

0

7

0

1

1

2

3

t [s]

t [s]

100

250

80

200

Magnitude [V]

Frequency [Hz]

90 70 60 50 40

150 100

30 20

50

10 0

0

1

2

3

4

5

6

0

7

0

1

2

3

t [s]

t [s]

600

14

4

550

2

12

450 400

3

350 300

2

250

10 8

3

6

4

4

200

1

150 100

Magnitude [V]

Frequency [Hz]

500

0

1

2

3

4

1

2

5

6

7

0

0

1

t [s]

2

3

4

t [s]

Fig. 4. Case study-5: estimated spectral components versus time. (a) Frequencies; (b) corresponding amplitudes. Rows 1–2: from the current recording; rows 3–4: from the voltage recording. Table 4 Case study-1: mean and standard deviation (std) of estimated amplitudes, frequencies and initial phases of components over data blocks. (4a) Method-1; (4b) Method-2. Frequency

Magnitude

Initial phase

True

Mean

Std

True

Mean

Std

True

Mean

Std

4a (Method-1) 50.02 58.00 63.00

50.00 60.00 65.00

0.02 2.00 2.00

100.0 0.10 0.10

100.0 0.20 0.13

0.02 0.12 0.06

0.00 0.00 0.00

0.31 −0.38 −0.40

0.36 0.56 0.67

4b (Method-2) 50.02 58.00 63.00

50.02 58.04 62.99

2.72e−05 0.05 0.05

100.0 0.10 0.10

100.0 0.10 0.09

1.69e−03 1.72e−03 5.69e−03

0.00 0.00 0.00

0.00 0.03 −0.04

6.23e−06 0.40 0.34

A. Bracale et al. / Electric Power Systems Research 88 (2012) 112–120

119

Table 5 Case study-2: mean and standard deviation (std) of estimated amplitudes, frequencies and initial phases of components over data blocks. (a) Method-1; (b) Method-2. Frequency

Magnitude

Initial phase

True

Mean

Std

True

Mean

Std

True

5a (Method-1) 50.00 82.00 100.0 150.0 182.0 200.0 250.0 350.0

50.00 80.00 100.0 150.0 180.0 200.0 250.0 350.0

0.00 2.00 0.00 0.00 2.00 0.00 0.00 0.00

100.0 3.00 0.50 3.00 5.00 1.00 4.00 2.50

100.0 2.34 0.52 3.00 3.74 1.06 4.00 2.50

0.19 0.66 0.08 0.15 1.26 0.30 0.08 0.03

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 −0.06 0.01 0.00 −0.06 0.00 0.00 0.00

9.38e−04 0.89 0.29 3.02e−02 0.90 0.35 2.99e−02 2.43e−02

5b (Method-2) 50.0 82.0 100.0 150.0 182.0 200.0 250.0 350.0

50.00 82.00 100.0 150.0 182.0 200.0 250.0 350.0

3.05e−05 1.10e−03 6.81e−03 9.67e−04 6.26e−04 3.13e−03 8.82e−04 1.40e−03

100.0 3.00 0.50 3.00 5.00 1.00 4.00 2.50

100.0 3.00 0.50 3.00 5.00 1.00 4.00 2.50

2.30e−03 2.61e−03 2.32e−03 2.51e−03 2.42e−03 2.23e−03 2.66e−03 2.59e−03

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

2.29e−05 0.81 4.55e−03 8.46e−04 0.81 2.45e−03 5.81e−04 8.32e−04

Table 4 shows the estimated mean values and standard deviations of harmonic and interharmonic amplitudes, frequencies and initial phases in Case study 1 from Method-1 (Table 4a) and Method-2 (Table 4b). Observing the results from the Method-1 in Table 4a, one can see that the estimated interharmonic frequencies are 60 Hz and 65 Hz, both deviate 2 Hz from the true frequencies. It is evident that the 5-Hz frequency resolution of DFT does not allow correct frequency estimation for the interharmonics. Also, spectral leakage results significant errors in the amplitude estimates. Further, observing the results from Method-2 in Table 4b one can see it yielded highly accurate estimation for all spectral components in terms of amplitudes and frequencies. Comparing the results from the proposed scheme in Tables 2 and 4b, one can see that the accuracy of the proposed scheme is similar to that in Method-2. For the estimated initial phase of fundamental component, the proposed method (and Method-2) has a higher accuracy than that of Method-1. Table 5 shows the mean values and standard deviations for the estimated harmonics and interharmonics in Case study 2, from Method-1 (Table 5a) and Method-2 (Table 5b). Table 5a shows that the 5 Hz frequency resolution of the conventional SW DFT results in correct frequency estimates for harmonics, but generates poor frequency estimates for interharmonics. For the estimated amplitudes of interharmonics, errors from the conventional SW DFT are as high as 25.2% for the 182 Hz interharmonic component. The spectral leakage due to non-synchronized interharmonics also affected the estimation of harmonics, resulting in slightly larger standard deviations of amplitudes as compared with the proposed scheme. Comparing the results in Table 3 (the proposed method) and Table 5a (Method-1), it shows that synchronized SW DFT in the proposed scheme performs slightly better than that in Method-1. This is mainly due to the absence of 82 Hz interharmonic in x1 (n) with a reduction of interharmonic-induced spectral leakage in the subsequent SW DFT; consequently, more accurate estimation of harmonics. On the other hand, Table 5a (Method-1) include two interharmonics at 82 and 182 Hz, hence less accurate results are obtained due to the interharmonic-induced spectral leakage. One can see from Table 5b that Method-2 yields an accurate estimation for all spectral components. Once again, results are similar to those obtained from the proposed scheme. Comparing Table 3 (the proposed method) and Table 5b (Method-2) one can see that estimated 3rd and 4th harmonic amplitudes are slightly more accurate in Method-2. This is probably due to the spectral leakage from

Mean

Std

Table 6 Computational time required in the proposed scheme, Method-1 and Method-2 for analyzing waveforms in Case study-1 and 2.

Case study-1 Case study-2

Proposed scheme [second]

Method-1 [second]

Method-2 [second]

0.021 0.469

0.001 0.001

0.734 1.236

the significant 182 Hz component in the signal x1 (n) analyzed in the second step. 5. Computational speed Further, the efficiency of the proposed scheme is evaluated by comparing the computational time. It should be noted that all programs are implemented by Matlab that are not optimized for computational speed, the computational time only gives a rough and relative comparison of efficiency in different methods. Table 6 shows the computational time required for analyzing one window of data by the proposed scheme, Method-1 and Method-2. The computational time in Table 6 is obtained by calculating the time required for estimating all signal components (frequencies, amplitudes and initial phases) of one data block, followed by averaging the times over all data blocks in the data sequence. From the analysis of Table 6 it clearly indicates that the average computational time for analyzing one window of data using the proposed scheme is greater than Method-1 (SW DFT), however is significantly lower than Mathod-2 (conventional SW ESPRIT applied to full band signal). In particular, the computation in Case study-1 is reduced by more than two scale levels while for Case study-2, a reduction of more than 60%. 6. Conclusion The proposed scheme has been tested and evaluated on both waveform recordings measured in power systems as well as synthetically generated data sequences. Results have shown that the proposed joint scheme combining SW ESPRIT and SW DFT is effective and efficient in terms of accurately estimating harmonic and interharmonic components from distorted voltage and/or current waveforms, as well as computational speed. Our tests have also shown that by combining a model-based method (SW ESPRIT)

120

A. Bracale et al. / Electric Power Systems Research 88 (2012) 112–120

and a non-model based method (SW DFT), accurate estimation of time-varying harmonic and interharmonic components is obtained at very high frequency resolution, while the spectral leakage is mitigated from using a synchronized analysis window. The main outcomes of the scheme are: • Yielded an accurate estimation of the fundamental period and harmonics. The proposed scheme significantly reduces the estimation error of harmonics in the SW DFT by using a synchronized analysis window to the fundamental period. • Resulted in highly accurate interharmonic estimates due to the use of the SW ESPRIT. • Resulted in a significant reduction of the computational time as comparing with the conventional SW ESPRIT applied to the full band data sequence, without sacrificing the performance of estimation. Future work includes more systematic tests of the scheme using large numbers of recorded waveforms from measurements; extending the study to include broadband interharmonic distortions; and close examining the spectral splitting that occasionally occurred in the current scheme. References [1] J. Arrillaga, N.R. Watson, Power System Harmonics, 2nd ed., Wiley, Chichester, 2003. [2] P. Caramia, G. Carpinelli, P. Verde, Power Quality Indices in Liberalized Markets, Wiley-IEEE Press, NJ, 2009. [3] M.H.J. Bollen, I.Y.H. Gu, Signal Processing of Power Quality Disturbances, WileyIEEE Press, NJ, 2006. [4] IEC standard 61000-4-30. Testing and measurement techniques—power quality measurement methods, 2003.

[5] IEC standard 61000-4-7. General guide on harmonics and interharmonics measurements, for power supply systems and equipment connected thereto, 2002. [6] A. Testa, D. Gallo, R. Langella, On the processing of harmonics and interharmonics: using Hanning window in standard framework, IEEE Trans. Power Del. 19 (1) (2004) 28–34. [7] D. Gallo, R. Langella, A. Testa, A self tuning harmonics and interharmonics processing technique, Eur. Trans. Electr. Power 12 (1) (2002) 25–31. [8] D. Gallo, R. Langella, A. Testa, Desynchronized processing technique for harmonic and interharmonic analysis, IEEE Trans. Power Del. 19 (3) (2004) 993–1001. [9] G.W.C. Chang, I. Chen, Y.J. Liu, M.C. Wu, Measuring power system harmonics and interharmonics by an improved fast Fourier transform-based algorithm, IET Proc. Gen. Trans. Distribut. 2 (2) (2008) 193–201. [10] G.W. Chang, I.C. Cheng, Measurement techniques for stationary and timevarying harmonics, in: Proceedings of 2010 IEEE PES General Meeting, 2010, pp. 1–5. [11] T. Lobos, Z. Leonowicz, J. Rezmer, P. Schegner, High-resolution spectrumestimation methods for signal analysis in power systems, IEEE Trans. Instr. Meas. 55 (1) (2006) 219–225. [12] I.Y.H. Gu, M.H.J. Bollen, Estimating interharmonics by using sliding-window ESPRIT, IEEE Trans. Power Del. 23 (1) (2008) 13–23. [13] A. Bracale, P. Caramia, G. Carpinelli, Adaptive Prony method for waveform distortion detection in power systems, Int. J. Electr. Power Energy Syst. 29 (5) (2007) 371–379. [14] A. Bracale, G. Carpinelli, Z. Leonowicz, T. Lobos, J. Rezmer, Measurement of IEC groups and subgroups using advanced spectrum estimation methods, IEEE Trans. Instr. Meas. 57 (4) (2008) 672–681. [15] A. Bracale, G. Carpinelli, An ESPRIT DFT-based new method for the waveform distortion assessment in power systems, in: Proceedings of CIRED 2009, 2009, pp. 1–4. [16] R. Roy, T. Kailath, ESPRIT—estimation of signal parameters via rotational invariance techniques, IEEE Trans. Ac. Sp. Sig. Proc. 37 (7) (1989) 984–995. [17] P. Stoica, R. Moses, Introduction to Spectral Analysis, Prentice Hall, NJ, 1997. [18] D.G. Manolakis, V.K. Ingle, S.M. Kogon, Statistical, Adaptive Signal Processing, McGraw-Hill, New York, 2000. [19] A. Bracale, P. Caramia, G. Carpinelli, Optimal evaluation of waveform distortion indices with Prony and RootMusic methods, Int. J. Power Energy Syst. 27 (4) (2007).