IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 1, JANUARY 2008
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Analysis of Harmonics in Power Systems Using the Wavelet-Packet Transform
Julio Barros, Senior Member, IEEE, and Ramón I. Diego
Abstract—This paper proposes a new algorithm based on the wavelet-packet transform for the analysis of harmonics in power systems. The proposed algorithm decomposes the voltage/current waveforms into the uniform frequency bands corresponding to the odd-harmonic components of the signal and uses a method to reduce the spectral leakage due to the imperfect frequency response of the used wavelet ?lter bank. This paper studies the selection of the mother wavelet, the sampling frequency, and the frequency characteristics of the wavelet ?lter bank for the two most common wavelet functions used for harmonic analysis and compares the performance of the proposed method with the results obtained using the discrete Fourier transform (DFT) analysis and the harmonic-group concept introduced by the International Electrotechnical Commission (IEC) under different measurement conditions. Index Terms—Electric power quality, Fourier analysis, harmonic distortion, International Electrotechnical Commission (IEC) standards, wavelets.
the obtained results are compared with the results of the Fourier analysis used in the IEC approach.
II. W AVELETS AND H ARMONIC D ISTORTION Wavelet analysis is a powerful signal-processing tool that is particularly useful for the analysis of nonstationary signals. Wavelets are short-duration oscillating waveforms with zero mean and fast decay to zero amplitude at both ends which are dilated and shifted to vary their time-frequency resolution. In wavelet analysis, the wavelet function is compared to a section of the signal under study, obtaining a set of coef?cients that represent how closely the wavelet function correlates with the signal in that section. Analogously to the Fourier analysis, the discrete wavelet transform (DWT) is the digital representation of the continuous wavelet transform. The DWT decomposes a signal into different frequency components, but unlike the Fourier analysis, this decomposition provides a nonuniform division of the frequency domain (a logarithmic decomposition) instead of the uniform frequency decomposition of the DFT. The DWT can be implemented using a multistage ?lter bank with the wavelet function as the low-pass (LP) ?lter and its dual as the high-pass (HP) ?lter, as shown in Fig. 1, for a threelevel decomposition tree. Downsampling by two at the output of the LP and HP ?lters scales the wavelet by two for the next stage. d(n) and c(n) in Fig. 1 are the outputs of the HP and LP ?lters, respectively, and represent the detailed version of the high-frequency components of the signal and the approximation version of the low-frequency components. The detail and approximation coef?cients of the DWT can be used to compute the root-mean-square (rms) magnitude of the voltage and current waveforms and also of the output frequency bands of the wavelet decomposition tree [2]. However, these coef?cients cannot be used to measure the rms value of the different harmonic components of the signal because the obtained nonuniform output bands do not correspond to the individual harmonic components of the signal. The higher frequency bands of the wavelet decomposition tree cover more harmonic components than the lower frequency bands. Fig. 2 shows the output frequency bands of the three-level decomposition tree in Fig. 1 for a sampling frequency of 1.6 kHz of the input signal. To overcome this limitation of the DWT, the wavelet-packet transform (WPT) can be used to obtain a uniform frequency decomposition of the input signal as in the Fourier analysis. In the WPT, both the detail and the approximation coef?cients
I. I NTRODUCTION HE International Electrotechnical Commission (IEC) recently de?ned a new method for the measurement of harmonics and interharmonics in power supply systems in the IEC Standard 61000-4-7 [1]. The most important contribution of this standard is the de?nition of the harmonic and interharmonic groups and subgroups in order to provide a more accurate representation of these magnitudes in actual power systems. The discrete Fourier transform (DFT) is proposed in the standard as the processing tool for harmonic analysis, using rectangular time windows of ten cycles’ width of the fundamental frequency in a 50-Hz system, providing a resolution of 5 Hz. The standard itself states that the speci?cation of a DFT reference instrument for harmonic and interharmonic measurement does not preclude the application of other analysis principles, such as wavelet analysis. The purposes of this paper are to investigate the use of the wavelet analysis in the study of harmonic distortion in power supply systems and to propose a new method to reduce the spectral leakage due to the imperfect frequency response of the used wavelet ?lter banks. The performance of the proposed new method is studied under different measurement conditions, and
Manuscript received June 19, 2006; revised September 19, 2007. This work, entitled “Estudio y desarrollo de nuevas téchnicas de procesado aplicadas a la detección, medida y evaluación de la calidad de la energía elétrica,” was supported by the Spanish Ministry of Science and Technology under Grant DPI2003-08869-C02. The authors are with the Department of Electronics and Computers, University of Cantabria, 39005 Santander, Spain (e-mail: barrosj@unican.es; diegori@unican.es). Digital Object Identi?er 10.1109/TIM.2007.910101
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0018-9456/$25.00 ? 2008 IEEE
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Fig. 1. Multistage ?lter bank for a three-level decomposition tree.
Fig. 2. Output frequency bands of the three-level decomposition tree in Fig. 1 for a sampling frequency of 1.6 kHz.
are decomposed to produce new coef?cients, this way enabling a uniform frequency decomposition of the input signal to be obtained. By using the WPT and adequately selecting the sampling frequency and the wavelet decomposition tree, the output frequency bands of the multiresolution analysis can be selected to correspond to the frequency bands of the different harmonic components of the input signal, as will be shown in the next section. There is a lot of work in the technical literature dealing with the use and investigating the performance of the waveletbased algorithms for the analysis of time-varying disturbances in power systems, but the use of wavelets for the analysis of harmonic components in voltage and current waveforms has not been thoroughly investigated until now. Pham and Wong [3] propose an approach for the identi?cation of harmonics in power systems using a combination of discrete wavelet analysis and continuous wavelet analysis to quantify harmonic frequency amplitudes and phases. The same authors (in [4]) propose a method to compensate the imperfect frequency response of the ?lters used in the wavelet-transform ?lter banks. Hamid and Kawasaki [5], [6] propose the use of the WPT, with the Vaidyanathan wavelet function with 24 coef?cients, to improve the results obtained using the DWT for the computation of the rms value of the harmonic components in voltage and current waveforms. Parameswariah and Cox [7] investigated the factors in choosing a wavelet function and how the number of coef?cients of the wavelets is important as a factor that affects the energy distribution leakage. The authors propose the use of Daubechies wavelet function with 20 coef?cients as the best solution for harmonic analysis. The frequency characteristics of Daubechies, Coi?et, and Symlet wavelet functions are compared in [8] in order to select the appropriate wavelet ?lter bank for power-quality monitoring.
Fig. 3.
Three-level wavelet-packet decomposition tree.
Finally, Eren and Devaney [9] present an implementation to reduce the computational complexity of the most commonly used ?lter banks in the wavelet-packet decomposition for the application in real-time metering of harmonics. III. P ROPOSED A LGORITHM The proposed algorithm in this paper is based on the WPT to obtain a uniform frequency decomposition of the input signal, which is compatible with the frequency bands of the different harmonic groups. By selecting a sampling frequency of 1.6 kHz and using a three-level decomposition tree, the frequency range of the output is divided into eight bands with a uniform 100-Hz interval, as shown in Fig. 3. The selected sampling window width is ten cycles of the fundamental frequency (200 ms in a 50-Hz system) as in the IEC Standard 61000-4-7. In each output band, the odd-harmonic frequencies are in the center of the band, this way avoiding the edges of the band where the spectral leakage is higher. Using the decomposition
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BARROS AND DIEGO: ANALYSIS OF HARMONICS IN POWER SYSTEMS USING THE WAVELET-PACKET TRANSFORM
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TABLE I RMS VALUES OF THE OUTPUT BANDS OF THE DECOMPOSITION TREE IN FIG. 1 WHEN A 1-p.u. SINGLE TONE AT A FUNDAMENTAL FREQUENCY AND AT ODD-HARMONIC FREQUENCIES IS INTRODUCED AS AN INPUT SIGNAL
tree of Fig. 3, the fundamental component and the oddharmonic components from the third to the 15th order, from coef?cients d1 to d8 , can be investigated in the input signal. However, the even harmonics cannot separately be computed using the proposed method, and their magnitudes are partially included in the magnitudes of the neighboring odd-harmonic components. The rms magnitude of each of the eight output bands is obtained by using the square root of the mean square of the wavelet coef?cients and using the method proposed in [2]. Higher sampling frequencies can be selected to extend the range of harmonics computed in the input signal. The extension of the sampling frequency implies the use of a different wavelet decomposition tree to obtain the same output frequency bands as in Fig. 3. Thus, when the sampling frequency is doubled to 3.2 kHz, one additional level must be added to the wavelet decomposition tree of Fig. 3 and must double the number of harmonics (up to the 31st order) that can be computed in the input signal. The imperfect frequency response of the ?lters used to decompose the input signal into the subbands determines the level of distortion in each output subband. Generally, it is necessary to ensure maximum ?at passband characteristics and good frequency separation. This way, wavelet functions with a large number of coef?cients have less distortion than wavelets with fewer coef?cients. Based on the results in [3] and [7], Vaidyanathan with 24 coef?cients (v24) and Daubechies with 20 coef?cients (db20) present the best frequency characteristics and were selected as the wavelet functions to implement the ?lter bank in Fig. 3. It is important to remark that, for a given wavelet function and a chosen decomposition tree, the type of the ?lter (LP or HP) and the sequence of the ?lters through which the input signal goes determine the frequency response of each of the output bands. Another wavelet function, another decomposition tree, or a different number of points will produce different frequency characteristics. Table I shows the rms value of each of the coef?cients of the output level of the decomposition tree in Fig. 3, from d1 to d8 ,
using the db20 and v24 wavelet functions, when a 1-p.u. single tone at a fundamental frequency (50 Hz) and at odd-harmonic frequencies from the third to the 15th order (150–750 Hz) is introduced as an input signal in Fig. 3. From the results reported in Table I, it can be seen that the decomposition of the signal into the frequency subbands is not ideal, and a different spectral leakage can be observed depending on the frequency of the tone introduced and on the frequency band of the output considered. Both wavelet functions present a similar behavior, with Vaidyanathan v24 showing a slightly better performance in the passband of the ?lters and less spectral leakage. The error in the estimation of the rms value of each harmonic component is very small (less than 1% using the v24 wavelet function), except for the two output bands at the center of the decomposition tree. In order to accurately characterize the frequency response of the selected ?lter bank, the rms value of each of the coef?cients of the output levels, from d1 to d8 , was calculated when a 1-p.u. single tone from 1 to 800 Hz (the frequency range of the output bands in Fig. 3), in steps of 1 Hz, was introduced as the input signal. Fig. 4(a) and (b) shows the frequency bandwidth characteristics of each of the output coef?cients of the decomposition tree of Fig. 3 when using the v24 and db20 wavelet functions, respectively. As can be seen, the frequency response of each band is different and depends on the type and the sequence of the ?lters (LP or HP) through which the input signal goes. The passband characteristics, the cutoff points, and the distortion at the subband edges can be evaluated using the results shown in Fig. 4. Apart from the imperfect frequency response of the ?lter bank used in the analysis, the spectral leakage can also be produced when there is an error in synchronizing the fundamental power-system frequency and the sampling window width used in the measurement system. The spectral leakage produced using the DFT when the power-system frequency ?uctuates around the nominal
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TABLE II HARMONIC DISTORTION IN THE INPUT SIGNAL COMPUTED USING THE IEC AND WAVELET-PACKET METHODS
the harmonic distortion measured in the low-voltage distribution system of our building during the afternoon in a typical weekday. Table II reports the results obtained using the wavelet-packet method with the v24 and db20 wavelet functions and the IEC method. To reduce the spectral leakage caused by the ?ltering characteristics of the method proposed, and shown in Fig. 4, we have used a double-stage process: First, the fundamental component of the input signal is estimated, and then, this component is ?ltered out; second, the proposed algorithm is applied to the resultant signal to compute the rest of the harmonic components without the interference of the spectral leakage due to the fundamental component. The results reported in Table II show that the errors in the estimation of the harmonic components are very small, and they are within the acceptable range for a harmonic measurement instrument. B. Test 2: Nonstationary Signals To evaluate the performance of the proposed method in the case of nonstationary harmonic distortion, we have used different test signals proposed in the IEC Standard 61000-4-7. The precision in the determination of the magnitude of the different harmonic components using both methods is compared with the total rms value of the signal calculated over ten cycles of the fundamental frequency. Fig. 5(a) shows the case of the rms ?fth-harmonic current that is ?uctuating from 3.536 to 0.7071 A, and Fig. 5(b) shows the corresponding spectrum obtained by applying the DFT analysis on a 200-ms rectangular window. The change in the magnitude of the current occurs after 21.25 periods of the ?fth harmonic. The total rms value of the time function calculated over a time interval of 200 ms is 2.367 A. Table III shows the results obtained in the measurement of the rms magnitude of the eight output bands (corresponding to the odd-harmonic groups from the ?rst to the 15th order) with the wavelet method proposed using both the db20 and v24 wavelet functions. The magnitude in the estimation of the ?fth-order harmonic group using the IEC method is 2.332 A, with an error of 1.47% [1]. On the other hand, the estimations obtained using the proposed wavelet method are 2.3486 and 2.3505 A when using the db20 and v24 wavelet functions, respectively (Table III). In this case, the errors are only 0.77% and 0.50%, respectively, which are less than the error obtained with the IEC method.
Fig. 4. Frequency bandwidth characteristics of coef?cients d1 –d8 . (a) Using the v24 wavelet function. (b) Using the db20 wavelet function.
frequency in a range from 49.5 to 50.5 Hz (the frequency-range variation accepted in the European standards) is a function of the magnitude of the synchronization error and of the harmonic order. On the other hand, the error in the determination of the magnitude of the fundamental and odd-harmonic components using the proposed wavelet method is, in all cases, 0%, showing that this method is insensitive to the ?uctuations of the powersystem frequency in the studied range. IV. W AVELET -P ACKET A LGORITHM AND THE IEC A PPROACH In this section, a comparative study of the performance of the proposed and IEC methods is carried out using different test waveforms under two different conditions: in the case of stationary signals with different harmonic components and in the case of nonstationary harmonic distortion. A. Test 1: Stationary Signal To evaluate the performance of the proposed method in the case of stationary harmonic distortion, the test signal used was
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Fig. 5. (a) Large ?fth-harmonic current ?uctuation. (b) Spectral components of the signal.
Fig. 6. (a) Typical waveform of the third-harmonic current produced by a microwave appliance. (b) Spectral components of the signal. TABLE IV RMS VALUES OF THE OUTPUT BANDS FOR THE TEST WAVEFORM OF FIG. 6(a) USING THE db20 AND v24 WAVELET FUNCTIONS
TABLE III RMS VALUES OF THE OUTPUT BANDS FOR THE TEST WAVEFORM OF FIG. 5(a) USING THE db20 AND v24 WAVELET FUNCTIONS
Another test waveform that is used to compare the performance of the proposed method is shown in Fig. 6(a), and its corresponding spectrum, which is obtained by applying the DFT analysis on a 200-ms rectangular window, is shown in Fig. 6(b). This waveform represents the typical waveform of the third-harmonic current produced by a microwave appliance, and it is also used in the IEC Standard 61000-4-7 as a test signal. The average power is controlled by the zero-crossing multicycle method with, in this case, a repetition rate of 5 Hz and a duty cycle of 50%. The total rms current calculated over 200 ms is 0.707 A. Table IV shows the results obtained in the measurement of the rms magnitude of the eight output bands with the wavelet method using both the db20 and v24 wavelet functions. Using the IEC method, the estimation of the rms value of the third-harmonic group is 0.692 A, with an error of 2.12% [1]. The estimations of the rms magnitudes of the third-harmonic group obtained using the proposed wavelet method are 0.6948 and 0.6946 A when using the db20 and v24 wavelet functions, respectively. The errors, in this case, are only 1.72% and 1.75%,
respectively, which are, once again, less than that obtained using the IEC method. Finally, Fig. 7 shows the ?fth-harmonic voltage which ?uctuates around the average rms value of 10 V with a sinusoidal modulation of 20% and 5 Hz, and its spectral components are obtained using the DFT analysis. The total rms value of the time function evaluated over 200 ms is 10.10 V [1]. Table V shows the results obtained in the measurement of the rms magnitude of the eight output bands using the wavelet method. The IEC method obtains the exact value of the voltage signal because the ?fth-harmonic group completely contains the 250-Hz carrier (?fth-harmonic component) and the two sidelines [1]. The estimations of the rms magnitudes of the ?fthharmonic group obtained using the wavelet method are 10.0414 and 10.0386 V when using the db20 and v24 wavelet functions, respectively. The errors, in this case, are 0.58% and 0.60%, respectively. The results reported in the three-case study show the potential of the WPT-based algorithm that is proposed for the analysis of harmonics in power systems.
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Fig. 8. Voltage waveform used in test 1 with an additional 1% white Gaussian noise. Fig. 7. (a) Voltage waveform of the ?fth harmonic with a 20% amplitude ?uctuation. (b) Spectral components of the signal. TABLE V RMS VALUES OF THE OUTPUT BANDS FOR THE TEST WAVEFORM OF FIG. 7(a) USING THE db20 AND v24 WAVELET FUNCTIONS TABLE VI NOISE IMMUNITY OF THE IEC AND WAVELET-PACKET METHODS
C. Test 3: Noise Immunity In the previous sections, the proposed algorithm has been tested under different measurement conditions but using voltage or current waveforms free of noise. The purpose of this section is to investigate the effect of noise on the performance of the proposed method. The white Gaussian noise has been considered in order to study the performance of the waveletpacket algorithm proposed and to compare the results with the IEC approach. Fig. 8 shows the voltage waveform used in test 1 with an additional 1% white Gaussian noise. Table VI shows the results obtained, in the estimation of harmonic distortion of the waveform in Fig. 8, using the proposed and IEC methods. As can be seen, the effect of noise is not the same in the measurement of the different harmonic groups; the algorithm with the v20 wavelet function shows a better performance than using db20 and a similar noise immunity as in the IEC method. V. C ONCLUSION This paper proposes a new wavelet-packet-based algorithm for the analysis of harmonics in power supply systems. The
frequency characteristics and the spectral leakage of the proposed algorithm have been studied using the db20 and v24 wavelet functions. To reduce the spectral leakage caused by the imperfect ?ltering characteristics of the ?lter bank used, a double-stage process is proposed, estimating the fundamental component and then ?ltering it at the input signal to compute the rest of the harmonic groups. The performance of the proposed method has been compared with the results obtained using the harmonic-group concept proposed by the IEC for different measurement conditions, showing the potential of the wavelet analysis as an alternative processing tool for the harmonic estimation in power systems. R EFERENCES
[1] Electromagnetic Compatibility (EMC)—Part 4–7: Testing and Measurement Techniques—General Guide on Harmonics and Interharmonics Measurement and Instrumentation, for Power Supply Systems and Equipment Connected Thereto, 2002. IEC 61000-4-7. [2] W. K. Yoon and M. J. Devaney, “Power measurement using the wavelet transform,” IEEE Trans. Instrum. Meas., vol. 47, no. 5, pp. 1205–1210, Oct. 1998. [3] V. L. Pham and K. P. Wong, “Wavelet-transform-based algorithm for harmonic analysis of power system waveforms,” Proc. Inst. Electr. Eng.—Gener. Transm. Distrib., vol. 146, no. 3, pp. 249–254, May 1999.
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[4] V. L. Pham and K. P. Wong, “Antidistortion method for wavelet transform ?lter banks and nonstationary power system waveform harmonic analysis,” Proc. Inst. Electr. Eng.—Gener. Transm. Distrib., vol. 148, no. 2, pp. 117– 122, Mar. 2001. [5] E. Y. Hamid and Z. Kawasaki, “Wavelet packet transform for rms values and power measurements,” IEEE Power Eng. Rev., vol. 21, no. 9, pp. 49– 51, Sep. 2001. [6] E. Y. Hamid and Z. Kawasaki, “Instrument for the quality analysis of power systems based on the wavelet packet transform,” IEEE Power Eng. Rev., vol. 22, no. 3, pp. 52–54, Mar. 2002. [7] C. Parameswariah and M. Cox, “Frequency characteristics of wavelets,” IEEE Trans. Power Del., vol. 17, no. 3, pp. 800–804, Jul. 2002. [8] A. Domijan, A. Hari, and T. Lin, “On the selection of appropriate ?lter bank for power quality monitoring,” in Proc. IASTED Int. Conf. PowerCon, New York, Dec. 10–12, 2003, pp. 17–21. [9] L. Eren and M. J. Devaney, “Calculation of power system harmonics via wavelet packet decomposition in real time metering,” in Proc. IEEE IMTC, Anchorage, AK, May 21–23, 2002, pp. 1643–1647.
Ramón I. Diego received the M.Sc. and Ph.D. degrees in physics from the University of Cantabria, Santander, Spain, in 2000 and 2006, respectively. Since 2000, he has been with the Department of Electronics and Computers, University of Cantabria, where he is currently a Lecturer. His research area is digital signal processing techniques applied to power quality.
Julio Barros (M’96–SM’02) received the M.Sc. and Ph.D. degrees in physics from the University of Cantabria, Santander, Spain, in 1978 and 1989, respectively. Since 1989, he has been with the Department of Electronics and Computers, University of Cantabria, where he is currently an Associate Professor. His research areas are real-time computer applications in power systems, harmonics, and power quality.
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