IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 12, DECEMBER 2012
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Hybrid Wavelet and Hilbert Transform With Frequency-Shifting Decomposition for Power Quality Analysis Norman C. F. Tse, Member, IEEE, John Y. C. Chan, Student Member, IEEE, Wing-Hong Lau, Senior Member, IEEE, and Loi Lei Lai, Fellow, IEEE
Abstract—The wavelet transform, the S-transform, the Gabor transform, and the Wigner distribution function are popular techniques for power quality (PQ) analysis in electrical power systems. They are mainly used to identify power harmonics and power disturbances and to estimate power quantities in the presence of nonstationary power components such as root-mean-square values and total harmonic distortions. Recently, the Hilbert–Huang transform has been also used in PQ analysis. These techniques have proven to be useful in PQ analysis; however, their performances depend on the types of PQ events. In this paper, a novel frequency-shifting wavelet decomposition via the Hilbert transform is introduced for PQ analysis. The proposed algorithm overcomes the spectra leakage problem in the discrete wavelet packet transform and can be used for estimating power quantities accurately and for detecting flickers. The effectiveness of the proposed algorithm was verified by computer simulations and experimental tests. Index Terms—Discrete wavelet transforms, frequency shift, harmonic analysis, Hilbert transform (HT), power quality (PQ), wavelet packets.
I. I NTRODUCTION
V
ARIOUS techniques have been used for analyzing power quality (PQ) in electrical power systems. The wavelet transform (WT), the S-transform, the Gabor transform (GT), and the Wigner distribution function (WDF) have received much attention. While these techniques are useful in identifying some PQ issues, they have some limitations as discussed below. The WT is a multiresolution analysis technique that decomposes the signal into nonuniform frequency bands. The discrete version of the WT is called the discrete WT (DWT). Both the WT and the DWT are commonly used for detecting power Manuscript received February 20, 2012; revised May 9, 2012; accepted June 12, 2012. Date of publication September 12, 2012; date of current version November 15, 2012. This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Project RGC CityU 121409. The Associate Editor coordinating the review process for this paper was Dr. E. Fiorucci. N. C. F. Tse and W. H. Lau are with the City University of Hong Kong, Kowloon, Hong Kong (e-mail:
[email protected]). J. Y. C. Chan is with the City University of Hong Kong, Kowloon, Hong Kong, and also with the School of Engineering and Mathematical Sciences, City University London, London EC1V 0HB, U.K. L. L. Lai is with the Director of Energy Strategy, Planning, Policy Support, and Research and Development Center, State Grid Energy Research Institute, Beijing 100052, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2012.2211474
Fig. 1.
Relationship between sampling frequency and integer harmonics [31].
disturbance [1]–[6]. The discrete wavelet packet transform (DWPT) is an extended version of the DWT that decomposes signals into uniform frequency bands, thus better for calculating power quantities such as root-mean-square (RMS) values and total harmonic distortion (THD) as in [7]–[17]. It has been used to analyze nonstationary waveforms as well. As illustrated in [7], the DWPT exhibits varying spectra leakages in frequency bands due to differences in transition length. When integer harmonics are to be analyzed, the leakage errors in the DWPT can be minimized by carefully selecting the sampling frequency so that the integer harmonics in question are located at the center of each frequency bands, as suggested in [13]. This approach is illustrated in Fig. 1, in which a sampling frequency of 400 Hz should be selected for the identification of 50, 100, and 150 Hz. This approach is difficult to implement when both integer and interharmonics exist in a power waveform. The ST [18] is closely related to the short-time Fourier transform (STFT) and the continuous WT (CWT). It uses Gaussian windows and is regarded as a generalized form of the STFT. This technique enables good time resolution [19]–[21] for localizing power disturbances but is not suitable for harmonic analysis, particularly when high-order harmonics are present. In addition, both the GT [22] and the WDF [23] are well known for PQ analysis due to their capability in the time–frequency domain analysis. However, the GT has a low clarity, and the WDF introduces cross terms. The Gabor–Wigner transform (GWT) [24] combines the advantages of the GT and the WDF that has been applied to detect PQ disturbances as in [25]. The GWT produces a new correlated spectrum through locating the intersection of the time–frequency spectrums generated from the GT and the WDF. The GWT is useful for identifying and locating harmonics and disturbances but is not suitable for estimating power quantities such as RMS values. Recently, the Hilbert–Huang transform (HHT) [26] was used for PQ analysis as in [27] and [28]. The HHT makes use of the empiricalmode decomposition (EMD) to decompose the power signal
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into intrinsic-mode functions (IMFs), which are then analyzed by the Hilbert transform (HT) for estimating instantaneous frequency, amplitude, and phase. A prerequisite is that IMFs must be monotone signals, and this is accomplished by using an empirical technique. As the optimum stopping criteria of the EMD are to be guessed [29], special assistances are needed in the iteration process as in [27] and [30]. In this paper, a novel frequency-shifting wavelet decomposition via the HT algorithm is introduced. The proposed algorithm is able to estimate power quantities and harmonics accurately, and can identify common PQ problems including flickering. The proposed algorithm also overcomes the leakage problem in the DWPT. The structure of this paper is as follows: The HT and the DWT are described in Sections II and III, respectively. Section IV describes the proposed novel algorithm. Section V presents the performance of the proposed algorithm on analyzing common PQ events, including sag, harmonics, transient, and flicker, and Section VI contains conclusions and future works. II. HT The HT [26] is time invariant and linear, defined as ∞ 1 x(τ ) y(t) = H [x(t)] = p.v. dτ π t−τ
(1)
−∞
where p.v. represents the Cauchy principal value. It convolutes a monotone signal x(t) with 1/πt and shifts each frequency components of x(t) by 90◦ . The HT is related to the Fourier transform, as shown in (2), and can be implemented by the fast Fourier transform (FFT), i.e., Y (ω) = F [H [x(t)]] = −j · sgn(ω) · X(ω).
(2)
The HT can convert x(t) into an analytic function by adding y(t) as the imaginary part, i.e., z(t) = x(t) + j · y(t) = a(t) · ejθ(t) (3) y(t) where a(t) = x(t)2 + y(t)2 , and θ(t) = tan−1 x(t) . The instantaneous amplitude a(t) and phase θ(t) in (3) can be estimated using Euler’s formula. These instantaneous amplitude and phase are useful for harmonic analysis only when x(t) is a monotone signal, as shown below. Let x(t) is defined as x(t) = cos (ωt + α).
If x(t) contains more than one frequency component, e.g., x(t) = cos (ωt) + cos (2ωt)
(8)
from (3), its instantaneous amplitude is a(t) = (cos (ωt)+cos (2ωt))2 +(sin (ωt)+sin (2ωt))2 . (9) Hence, the instantaneous amplitude a(t) and phase θ(t) of an individual harmonic component cannot be estimated. A. Frequency Shifting HT can be used for shifting frequency components in a signal, commonly known as single side-band modulation [31]. Frequency shifting is accomplished by multiplying ejω1 t to the analytic function, in which ω1 is the frequency shift in the spectrum. The shifted signal is readily obtained from the real part of the signal as (10) sω1 (t) = Re (x(t) + j · H [x(t)]) · ejω1 t . As ejω1 t shifts the whole spectrum by ω1 , including both the negative and positive frequencies as in (11), it creates a new redundant frequency component in the time domain, i.e., ∞ j(ω0+ω1 )t j(ω0 −ω1 )t
+e e F cos (ω0 t)·ejω1 t (ω) = 2 −∞
1 e−jωt dt = [δ(ω−ω0 −ω1 )+δ(ω+ω0 −ω1 )] . 2 (11) The analytic function helps to remove the negative-frequency component, and thus, the redundant frequency will not be produced as in F (cos (ω0 t) + j · sin (ω0 t)) · e−jω1 t (ω) ∞ =
ej(ω0 +ω1 ) e−jωt dt = −δ(ω) − ω0 − ω1 ). t
(12)
−∞
Hence sω1 (t) = Re (cos ω0 t + j · sin ω0 t) · e−jω1 t = cos ((ω0 + ω1 )t) .
(13)
(4)
The HT gives
III. WT ◦
y(t) = cos (ωt + α + 92 ) = sin (ωt + α).
(5)
From (3), the analytic function of (4) is z(t) = cos (ωt + α) + j · sin (ωt + α) = ej(ωt+α)
(6)
A. DWT
where a(t) = 1, and θ(t) = ωt + α. The instantaneous (angular) frequency ω(t) is ω(t) =
dθ(t) . dt
The WT provides a time–frequency analysis for the analyzing signal and is particularly useful for analyzing nonstationary signals. The CWT and the DWT are commonly used. Only the DWT will be discussed here.
(7)
The DWT is an orthonormal transform that is conveniently implemented by conjugate mirror filters and downsampling of the output sequence, and is thus called fast (orthogonal) WT
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Fig. 3. Frequency bands of the DWPT. Fig. 2.
Frequency response of the “db20” filter N . Levels (a) 1 and 2.
[32]. It decomposes a sequence with N coefficients into two sequences with N /2 coefficients by downsampling the output of the high-pass filter (wavelet filter, g[n]) and the low-pass filter (scaling filter, h[n]). Its output coefficients are called detail coefficients dj+1 and approximation coefficients aj+1 as in dj+1 [n] = aj+1 [n] =
+∞ m=−∞ +∞
aj [n]g[2n − m]
(14)
aj [n]gh[2n − m].
(15)
m=−∞
Only the approximation coefficients are decomposed in the next stage. B. DWPT The DWPT is an extended version of the DWT that decomposes both the detail coefficients and the approximation coefficients in each stage. The frequency bands produced have the same bandwidth, which is a desirable feature for harmonics analysis as in [7]–[17]. However, a spectra leakage problem exists due to unequal transition lengths in different frequency bands. This is particularly evident for frequency bands at the center of the frequency spectrum, as illustrated in [7]. This leakage problem is caused by the hierarchy structure of the DWPT, filters g[n] and h[n], and the downsampling of coefficients. As the same filters are applied in each level in the DWPT process but only the frequency spectrum and the length of the signal are halved, the transition lengths of the filters are also halved in each level. Fig. 2 shows the transition lengths of the “db20” mother wavelet in the first and second levels. When a signal with a frequency spectrum of 0–800 Hz is passed through filter h[n] at level 1, the output contains the spectra from 0 to 550 Hz (including a transition length of 150 Hz). At level 2, the filters separate the signal into two bands: the first band (0–200 Hz) contains the spectra from 0 to 275 Hz (including a transition length of 75 Hz), and the second band (200–400 Hz) contains the spectra from 125 to 550 Hz (including a transition length of 75 + 150 Hz). The frequency band at the center of the frequency spectrum suffers most, whereas the bands at the sides suffer the least. Fig. 3 shows the frequency spectrum of the decomposed bands by the DWPT as in [7]. A total of 16 frequency bands (50-Hz bandwidth) are decomposed in which the leakage problem is obvious. As suggested in [13], the leakage problem can be minimized by locating the integer harmonics at the center of each fre-
Fig. 4. Basic frequency-shifting concept of the proposed algorithm.
quency band. This technique relies on the selection of a proper sampling frequency in the signal capturing stage and is only applicable for detecting integer harmonics. Another approach suggested in [10] is to compensate the distortion caused by the filters. The technique will become very complicated when a frequency band contains more than one frequency component. IV. H YBRID D ISCRETE WAVELET D ECOMPOSITION —HT The proposed algorithm makes use of the DWT and the HT to decompose the electrical waveform into uniform frequency bands. The electrical waveform is iteratively decomposed by shifting the spectra of the waveform to the frequency band at the lowest side of the frequency spectrum, thus avoiding the leakage problem. The basic approach of the proposed algorithm is illustrated in Fig. 4. First, the DWT is applied to obtain the approximation coefficients aj of the waveform. The number of levels of decomposition is dependent on the desired bandwidth of the approximation coefficients. With a sampling frequency of 1600 Hz, a bandwidth of 200 Hz would require two levels of decomposition. In any case, the desired bandwidth must be larger than the spectra offset ω1 , or otherwise, the spectrum cannot be fully covered and some frequency components will be lost. Second, the approximation coefficients are converted into time-domain data by inverse DWT (iDWT) and are
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Fig. 6.
Frequency bands of the proposed algorithm.
Fig. 7. Combined frequency bands of the proposed algorithm adapted for integer harmonics.
Fig. 5. Flowchart of the proposed algorithm.
subtracted from the input waveform. This is to make sure that the positive-frequency components in the spectrum will not be shifted to the negative-frequency region, which will create unwanted frequency components in the next step. Third, the spectra of the waveform are shifted by a fixed offset ω1 (e.g., 25 Hz) to the lower frequency spectrum using the HT. The extent of spectra shifting is dependent on the frequency contents of the waveform. While the FFT would be used to provide a rough idea of the frequency spectrum of the waveform, it is found that a 25/30-Hz fixed offset is sufficient for common PQ analysis. The aforementioned steps are repeated until the entire spectrum is shifted to 0 Hz. All approximation coefficients thus obtained can be used for estimating harmonics (e.g., RMS and THD). The instantaneous amplitude, frequency, and phase are estimated by the HT. Fig. 5 shows the flowchart of the proposed algorithm. Detailed steps are described below. Initial: Step 1: s−1 [n] = Re((x[n] + j · H[x[n]]) · ejω1 n ). Step 2: a0 [n] = DWT(s−1 [n]). Step 3: r0 [n] = iDWT(a0 [n]). Step 4: s0 [n] = x[n] − Re((r0 [n] + j · H[r0 [n]]) · e−jω1 n . Loop: Steps 5–7, where qm = 1, 2, 3 . . . (fs/(2 · ω1 ) − 1) (until the spectra are shifted to 0 Hz) Step 5: am [n] = DWT(sm−1 [n]). Step 6: xm [n] = sm−1 [n] − iDWT(am [n]). Step 7: sm [n] = Re((xm [n] + j · H[xm [n]]) · e−jω1 n )
where x[n] is the input waveform data sequence; is the approximation coefficient; an is the shifted waveform; sn is the spectra shifting offset (25/30 Hz); ω1 Fs is the sampling frequency of x[n]; is the transition variable for steps 3 and 4. r0 It should be noted that the waveform is first shifted by ω1 in step 1 instead of −ω1 as in step 7. This would ensure that the first output of the algorithm will have the same bandwidth as subsequent outputs. In order to avoid aliasing problem in the upper part of the frequency spectrum due to the increase in frequency (ω1 ), in step 4, r0 is shifted back to its original frequency spectrum and is subtracted from x[n], as the original x[n] is not affected by the aliasing problem. The iterative processes in steps 5–7 are the same as described: In step 5, the shifted waveform is decomposed by THE DWT. In step 6, the approximation coefficients are inversely transformed and subtracted from xm [n]. In step 7, xm [n] is shifted again for the next iteration. Fig. 6 shows the frequency property of the proposed algorithm. For illustration purposes, the sampling frequency of the waveform (x[n]) and the desired bandwidth are set as 1600 and 50 Hz, respectively. The decomposition levels in the DWT and the shifting offset (ω1 ) are 4 and 25 Hz, respectively. Thereafter, monotone signals from 0 to 800 Hz in a step of 0.1 Hz are fed into the algorithm individually to show the frequency property of the algorithm. In order to show the frequency property clearly, the output approximation coefficients are inversely transformed into the time domain and shifted back to their original spectrum. The amplitude is calculated by averaging the instantaneous amplitude of the output of the HT. Thirty-two frequency bands with a uniform bandwidth of 25 Hz are decomposed. Except for the frequency bands at the two ends of the frequency spectrum, any two 25-Hz frequency bands can
TSE et al.: HYBRID WAVELET AND HT WITH FREQUENCY-SHIFTING DECOMPOSITION
Fig. 8.
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Synthesized waveform with integer and noninteger harmonics. TABLE I H ARMONIC E STIMATION R ESULTS
Fig. 9. Time–frequency analysis result of the synthesized waveform using the proposed algorithm.
be combined into a 50-Hz frequency band, as shown in Fig. 7, which is then adapted for analyzing integer harmonics. The 25-Hz frequency bands can be also combined in any adaptive manner for the analysis of noninteger harmonics. V. T ESTS Five computer simulations and an experimental test were conducted to evaluate the performance of the proposed algorithm for PQ analysis. The settings of the tests were as follows: Sampling frequency 6400 Hz; DWT decomposition level six levels; 25 Hz; Shifting offset ω1 Wavelet–Daubechies “db20” (length of 40); Sampling window 0.2 s (10/12 cycles, IEC610004-7). The Daubechies “db20” wavelet is one of the suitable wavelets for PQ analysis. Some researchers have made use of it for various PQ applications [7], [11]. A. Simulation Test—Harmonics and Interharmonics The synthesized waveform defined by (16) is shown in Fig. 8. There are nine harmonic components including seven integer harmonics and two interharmonics (475 and 775 Hz), as detailed in Table I, i.e., x[n] = am cos (2πnfm + θm ). (16) m
In order to estimate the instantaneous frequencies, amplitudes, and phases, all outputs (approximation coefficients) of the algorithm are inversely transformed into the time domain
Fig. 10. DWPT.
Time–frequency analysis result of the harmonics waveform by the
and then analyzed by the HT. The time–frequency plot of the output is shown in Fig. 9. The output frequency bands are combined according to the frequency contents in each band. In this case, frequency bands 25–50 and 50–75 Hz are combined into 25–75 Hz for integerharmonic identification, and frequency bands 450–475 and 475–500 Hz are combined into 450–500 Hz for nonintegerharmonic identification. The frequencies, amplitudes, and phases of the harmonic components are calculated by taking the means of the instantaneous values produced by the HT. The results are presented in Table I. It is shown that the proposed algorithm performs well compared with the DWPT. Fig. 10 shows the time–frequency analysis results of the DWPT [10] using the same wavelet (db20). For comparison sake, the DWPT results were inversed into the time domain and analyzed by the HT. It can be seen that the spectrum is blurred in the middle due to the leakage problem of the DWPT, as discussed in Section III. Thus, the monotone signal cannot be provided for the HT.
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Fig. 11. Synthesized waveform with 20% voltage sag at 0.1 s.
Fig. 15. Time–frequency analysis result of the waveform with an oscillating transient by the proposed algorithm. Fig. 12. Time–frequency analysis result of the voltage sag by the proposed algorithm.
Fig. 16.
Synthesized waveform with voltage fluctuation.
Fig. 13. Instantaneous amplitudes in the frequency band 25–75 Hz.
Fig. 14. Synthesized waveform with an oscillating transient.
B. Simulation Test—Voltage Sag The synthesized voltage waveform is shown in Fig. 11. It is a pure sinusoidal waveform at 50 Hz, and a sag to 80% happened at 0.1 s. A voltage sag to 80% can be commonly caused by motor starting. The RMS value calculated from the sampled data is 0.6403. Fig. 12 shows the time–frequency plot of the output of the proposed algorithm. The frequency of the voltage waveform is identified as 50 Hz, and the voltage sag appears as small spectra leakage found at around 0.1 s, with a frequency of approximately 100 Hz. Fig. 13 shows the instantaneous amplitudes of the HT in the frequency band 25–75 Hz, in which the amplitude change is evident. The transitional amplitude variation near 0.1 s is due to the finite wavelet length, which is an inherent characteristic of the HT. The RMS value estimated from the proposed algorithm is 0.6402. The RMS value calculated from the DWPT [17] and
Fig. 17. Time–frequency analysis result of the waveform with voltage fluctuation by the proposed algorithm.
the IEC61000-4-70 spectra grouping method are 0.6402 and 0.6401, respectively. C. Simulation Test—Oscillating Transient A 600-Hz oscillating transient is superimposed onto a pure sinusoidal voltage waveform at 50 Hz, as shown in Fig. 14. The time–frequency plot by the proposed algorithm is shown in Fig. 15. As shown, the fundamental component has a frequency of 50 Hz and exhibits constant amplitude; the oscillating transient has a frequency of 600 Hz and is found near 0.09 s, as indicated by the abrupt amplitude change. D. Simulation Test—Voltage Fluctuations (Flicker) The synthesized waveform defined by (17) is shown in Fig. 16. The amplitude of the waveform fluctuates between 0.8 and 1.2 times of the rated value. Fig. 17 shows the
TSE et al.: HYBRID WAVELET AND HT WITH FREQUENCY-SHIFTING DECOMPOSITION
Fig. 18. Instantaneous amplitudes in the frequency band 25–75 Hz.
Fig. 21. Instantaneous frequencies in the frequency band 25–75 Hz.
Fig. 19. 0.1 s.
Fig. 22. Equipment setup for the experimental test.
Synthesized waveform with frequency change from 50 to 52 Hz at
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Fig. 23. Voltage sag generated by the power supply unit. Fig. 20. Time–frequency analysis result of the waveform with frequency variation by the proposed algorithm.
time–frequency plot generated by the proposed algorithm; the voltage fluctuation is identified by the amplitude variation at 50 Hz, i.e., x[n] = cos (2πnf0 ) · (1 − vf · cos (2πnff ))
(17)
where f0 is 50 Hz, vf is 0.2, and ff is 10 Hz. Fig. 18 shows the instantaneous amplitudes estimated by the HT in the frequency band 25–75 Hz. The amplitude variation is clearly shown. As a result, the proposed algorithm is able to estimate the fundamental frequency of 50 Hz and the amplitude fluctuation of the synthesized waveform accurately. E. Simulation Test—Frequency Variation In the synthesized waveform, the fundamental frequency of the waveform changes from 50 to 52 Hz at 0.1 s, as shown in Fig. 19. Fig. 20 shows the time–frequency plot generated by the proposed algorithm, and the frequency variation is identified at 0.1 s. Fig. 21 shows the instantaneous frequency estimated by the HT in the frequency band 25–75 Hz; the frequency variation is clearly shown. The transitional frequency variation, as shown in Fig. 21, is due to the finite wavelet length. Overall speaking, the proposed algorithm is able to estimate the frequency variation accurately.
Fig. 24. Current drawn by the lamp box.
F. Experimental Test In the experimental set, as shown in Fig. 22, a “Kikusui model PCR-2000LA” multifunctional alternatingcurrent power supply unit was used to provide the voltage supply, and a lamp box containing electronic ballast-controlled fluorescent lamps and incandescent lamps served as the load. The power supply unit is programmed to generate an approximately 10% voltage sag, at a fundamental frequency of 50 Hz. Fig. 23 shows the voltage waveform. The actual voltage sag as measured by the waveform recorder is 11%, and it appears from 0 to 0.08 s. Fig. 24 shows the current waveform drawn by the lamp box. The abrupt change in supply voltage causes a current transient at approximately 0.077 s. The time–frequency plot is shown in Fig. 25. The voltage frequency is identified as 50 Hz, and the high-frequency component at around 0.07–0.08 s denotes the amplitude change. Fig. 26 shows the instantaneous amplitudes of the voltage waveform generated by the proposed algorithm. It is shown
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TABLE II H ARMONIC C URRENT E STIMATED BY THE P ROPOSED A LGORITHM
Fig. 25. Time–frequency analysis result of the captured voltage sag by the proposed algorithm.
VI. C ONCLUSION AND F UTURE W ORKS
Fig. 26. Instantaneous amplitudes in the frequency band 25–75 Hz of the voltage sag estimated by the proposed algorithm.
A novel electrical power waveform decomposition and analysis algorithm has been introduced in this paper. The proposed algorithm utilizes the DWT and the HT to decompose electrical power waveforms iteratively into narrow frequency bands, such that the leakage problem in frequency bands is reduced, as compared with the DWPT, as shown in Figs. 3 and 7. The output approximation coefficients of the proposed algorithm can be used to estimate power quantities readily. As revealed by the simulation results in Section V.A, the parameters of the frequency components estimated by the proposed algorithm are more accurate than those estimated by the DWPT method. The approximation coefficients can be also inversely transformed into the time domain; the HT can be then applied to analyze the time–frequency characteristics of electrical power waveforms. It is useful in detecting PQ problems such as flickers. Future works include the identification of multiple PQ events by the proposed algorithm, together with intelligence approaches such as neural network, and the extension of the proposed algorithm for analyzing a new type of voltage flickers produced by interharmonics, which are mostly generated by power electronics [33]. R EFERENCES
Fig. 27. Time–frequency analysis result of the current by the proposed algorithm.
that the amplitude variations of the supply voltage are clearly identified and the voltage sag is 11%. Fig. 27 shows the time–frequency plot of the current waveform estimated by the proposed algorithm. The transient in the current waveform is shown as widespread abrupt amplitude changes around 0.06–0.08 s. The current transient is caused by the amplitude change in the voltage that happened around 0.07–0.08 s. Table II presents the current harmonics estimated by the proposed algorithm, obtained by taking the means of their respective instantaneous values estimated by the HT.
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[7] [8] [9] [10]
[11] [12] [13] [14] [15] [16]
[17] [18] [19] [20]
[21] [22] [23] [24] [25] [26] [27]
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Norman C. F. Tse (M’09) received the B.S. degree in electrical engineering from Hong Kong Polytechnic University, Kowloon, Hong Kong, in 1985, the M.Sc. degree from the University of Warwick, Coventry, U.K., in 1994, and the Ph.D. degree from the City University London, London, U.K. in 2007. He is currently working with the Centre for Power Electronics, City University of Hong Kong, Kowloon. His research interests include power quality measurement, Web-based power quality monitoring, and harmonics mitigation for building low-voltage electrical power distribution system and energy efficiency.
John Y. C. Chan (S’10) received the B.Eng. degree in computer engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 2009. He is currently working toward the Ph.D. degree in electrical engineering from the City University London, London, U.K. His research interest includes electrical power quality, embedded system design, and signal processing.
Wing-Hong Lau (M’88–SM’06) received the B.Sc. and Ph.D. degrees in electrical and electronic engineering from the University of Portsmouth, Portsmouth, U.K., in 1985 and 1989, respectively. Since 1990, he has been with the City University of Hong Kong, Kowloon, Hong Kong, where he is currently an Associate Professor with the Centre for Power Electronics. His current research interests include the area of digital signal processing, digital audio engineering, and pulsewidth-modulation spectrum analysis. Dr. Lau was a recipient of an IEEE Third Millennium Medal.
Loi Lei Lai (SM’92–F’07) received the B.Sc. (first class honors) and Ph.D. degrees from the University of Aston, Birmingham, U.K., and the D.Sc. degree the from City University London (CUL), London, U.K. Currently, he is the Director of Energy Strategy, Planning, Policy Support, and Research and Development Center, State Grid Energy Research Institute, Beijing, China. He is also an Honorary Visiting Chair Professor with CUL; a Visiting Professor with the Southeast University, Nanjing, China; a Guest Professor with Fudan University, Shanghai, China; and the Pao Yu Kong Chair Professor with Zhejiang University, Hangzhou, China. Prof. Lai was a recipient of an IEEE Third Millennium Medal, the 2000 IEEE Power Engineering Society United Kingdom and Republic of Ireland Chapter Outstanding Engineer Award, the 2003 IEEE Power Engineering Society Outstanding Large Chapter Award, a Prize Paper by the IEEE Power and Energy Society Power Generation and Energy Development Committee in June 2006 and July 2009, and a high-quality paper prize from the International Association of Desalination, USA, in 1995. He is a Fellow of The Institution of Engineering and Technology.
