HARMONIC and interharmonic measurements are one of

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1 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 2, APRIL A Method to Improve the Interharmonic Grouping Scheme Adopted by IEC Standard Jin Hui, Honggeng Yang, Member, IEEE, Wilsun Xu, Fellow, IEEE, and Yamei Liu Abstract The interharmonic group concept adopted by the IEC is very useful for addressing the need to monitor interharmonics in power systems. This paper shows that grouping spectral bins into interharmonics and harmonic components of 5 Hz resolution could yield misleading information for cases where an interharmonic component s frequency is very close to a harmonic frequency. Since interharmonics in such cases can cause more severe waveform modulation, the interharmonic grouping scheme needs to be improved. A signal-processing method that can separate interharmonic and harmonic components in close proximity is, therefore, proposed in this paper to address this need. The method is based on the estimation of leakage values caused by interharmonics, at harmonic frequencies. Simulation studies and field data analysis showed that the proposed method was able to extract interharmonic and harmonic components correctly. Moreover, this paper also discusses how to apply the method to enhance the spectral grouping scheme of the IEC Index Terms Discrete Fourier transform (DFT), harmonic analysis, interharmonics, spectral leakage. I. INTRODUCTION HARMONIC and interharmonic measurements are one of the common tasks for power-quality (PQ) monitoring and troubleshooting [1], [2]. The discrete Fourier transform (DFT) and its fast algorithm fast Fourier transform (FFT) are the most commonly used techniques to process the measured data. However, these techniques have accuracy problems when a waveform contains interharmonics. These problems are due to the spectral leakage and picket fence effects that cannot be avoided when interharmonics are present. To address this problem and standardize the measurement approach, the International Electrotechnical Commission established a harmonic and interharmonic measurement protocol in IEC [3]. DFT is still the processing tool, and the recommended window width is 12 fundamental frequency cycles (for a 60 Hz system). This width yields a frequency resolution of 5 Hz. The DFT results are grouped into harmonics and interharmonics components. This approach is easy to implement and can deal with spectral leakage issues in most cases. However, Manuscript received August 12, 2011; revised November 17, 2011; accepted December 27, Date of publication February 20, 2012; date of current version March 28, Paper no. TPWRD J. Hui, H. Yang, and Y. Liu are with the College of Electrical Engineering and Information Technology, Sichuan University, Chengdu , China ( hj4655@163.com; yangsi@mail.sc.cninfo.net; huijean@live.cn). W. Xu is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4 Canada ( wxu@ualberta.ca). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TPWRD if the frequency of an interharmonic component is very close to a harmonic frequency, the spectral leakage problem caused by 5 Hz frequency resolution can corrupt both the harmonic and interharmonic group results. The harmonic results are often overestimated, and the interharmonic results underestimated. As a result, the interharmonic component cannot be characterized properly [1]. Researchers have found that interharmonics in close proximity of a harmonic frequency can cause much more significant waveform fluctuations (modulations) [4], [5] and, therefore, have the highest need for their proper measurement and characterization. Some studies have been published on reducing the impact of spectral leakage [6] [15]. However, if an interharmonic component has a distance less than 5 Hz (a frequency resolution) from a harmonic frequency, strong interharmonic-harmonic spectral interference will occur and will be beyond the processing capability of the methods proposed in these studies. In this paper, the authors propose a method to solve the aforementioned problem and to improve the accuracy of the IEC interharmonic group results. The idea is to group harmonics based on the harmonic spectrum (the total spectrum of all harmonic components) and to group interharmonics based on the interharmonic spectrum (the total spectrum of all interharmonic components). In this way, the spectral interference between the harmonics and interharmonics is eliminated. The determination of the harmonic spectrum and interharmonic spectrum is achieved by calculating the interharmonic leakage contributions at harmonic frequencies. This paper is structured as follows. Section II introduces the interharmonic group concept defined in IEC In Section III, the proposed method is described. In Section IV, the simulations and field experiment results are provided to show the effectiveness of the proposed method. Section V concludes this paper. II. IEC INTERHARMONIC GROUP CONCEPT The IEC defined harmonic subgroup includes one harmonic bin and two bins adjacent to the central harmonic. For example, 115, 120, 125 Hz form the 2nd harmonic subgroup for a 60 Hz system. The magnitude of this group is defined as the rms value of the three bins where and represent the magnitude of the DFT results and the harmonic order, respectively, refers to the frequency (1) /$ IEEE

2 972 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 2, APRIL 2012 Fig. 1. Harmonic and interharmonic subgroups based on IEC resolution (about 5 Hz), and refers to the fundamental frequency (about 60 Hz). The other spectral bins between the two adjacent harmonic subgroups form the interharmonic subgroup. For example, 70, 75, 80, 110 Hz form the 1.5th interharmonic subgroup. The magnitude of this group is defined as the rms value of all these nine bins Fig. 1 illustrates the group concept defined before, where HSG and IHSG are short for harmonic subgroup and interharmonic subgroup, respectively. The spectrum is obtained by performing DFT on a 12-cycle waveform snapshot of a signal containing five frequency components (Harmonics: 60, 120, 180 Hz. Interharmonics: 117.5, Hz). If interharmonics are in close proximity with a harmonic, the aforementioned grouping approach can yield misleading results. Fig. 1 is an example of this phenomenon. According to the definition of a harmonic subgroup, the DFT bins located at 115 Hz (bin ), 120 Hz (bin ), and 125 Hz (bin ) are all included as the 2nd harmonic component (subgroup). A large interharmonic component (bin ) is thus misclassified as a harmonic component. On the other hand, the magnitude of the 1.5th interharmonic subgroup will be underestimated. Based on these results, a PQ meter user could mistakenly conclude that the measured waveform has a large harmonic component at 120 Hz and an insignificant interharmonic component between 60 Hz and 120 Hz. Such a conclusion can easily mislead troubleshooting efforts. Interharmonics, which produce worst flicker levels, can be seen as the most harmful interharmonics. The literature [4] shows an experimental result from a study of the acceptable magnitude of a single interharmonic component which causes perceptible light flicker by various lamps. The result reveals that the frequencies of the lowest acceptable magnitudes are all close to the harmonic frequencies. In summary, the accuracy of the IEC interharmonic and harmonic group results needs to be improved. One of the direct benefits will be the availability of accurate measurement results for PQ troubleshooting. (2) Fig. 2. Separated harmonic and interharmonic spectra. (a) Harmonic spectrum. (b) Interharmonic spectrum. III. PROPOSED SPECTRAL-PROCESSING TECHNIQUE The proposed method uses only the 12-cycle data available to an IEC-compliant measurement device. Its basic idea is to separate the DFT spectrum produced from a 12-cycle snapshot into a harmonic spectrum and an interharmonic spectrum. The RMS values of the harmonic and interharmonic groups are calculated based on the two separated spectra. Fig. 2 shows the separated spectra from the spectrum shown in Fig. 1. Note that all of these calculations are based on the IEC frequency resolution (5 Hz). A. Harmonic and Interharmonic Spectra Separation The IEC standard requires the DFT window length to be synchronized with the real system frequency by using a phase-locked loop (PLL) or other frequency synchronization techniques. Therefore, the harmonic energies will be all concentrated only at the harmonic frequencies in the original DFT spectrum, and no leakage will take place. In contrast, the interharmonic energies will still leak at all frequencies. Thus, the spectral bins at the harmonic frequencies will be determined by the harmonics and the interharmonics while the bins at the interharmonic frequencies will be determined only by the interharmonics. For instance, bin in Fig. 1 is determined by both the 2nd harmonic and the two interharmonics while bin and bin are determined only by the two interharmonics. Therefore, the whole spectrum separation problem can be transformed into the problem of separating the harmonic and interharmonic contributions at the harmonic frequencies (e.g., by separating bin in Fig. 1 into bin and bin in Fig. 2). From a mathematical perspective, the interharmonic leakage values (contributions) at the harmonic frequencies (e.g., ) can be calculated by knowing the spectral lines at the interharmonic frequencies (e.g., bins and ), as long as the number of interharmonic components in the signal is known. Thus, the DFT results at the interharmonic frequencies and the calculated leakage

3 HUI et al.: A METHOD TO IMPROVE THE INTERHARMONIC GROUPING SCHEME ADOPTED BY IEC STANDARD values at harmonic frequencies form the final interharmonic spectrum, and the remaining spectrum forms the harmonic spectrum. In the following text, the leakage calculation processes for considering one, two, and three interharmonics in the signal are introduced. Since the DFT result is nonlinear with respect to the frequency, the calculation process will be complex and timeconsuming. To speed up and simplify the calculation, this paper simplifies the DFT results to transform the nonlinear problem into a simple liner problem. 1) Simplification of the Interharmonic DFT Result: Consider an interharmonic component (ignore the negative frequency part) where,, and are the frequency, amplitude, and phase angle of the component, respectively. Digitize the signal with the sample interval of where, is the sample number per cycle, and is the cycle number in the time window. Its DFT is (3) (4) Based on (8), the leakage contribution of this interharmonic at a frequency of times the frequency resolution away from the th harmonic can be expressed as where (9) (10) (11) For the readability, is represented by in the following text. 2) Interharmonic Leakage Calculation by Considering one Interharmonic Component: If only one interharmonic is considered (with frequency ) around the th harmonic, the spectral values around Hz will be contributed by the considered interharmonic only (12) where. Deform the above equation (5) where means the total spectral value contributed by all interharmonics, and means the spectral value only contributed by the interharmonic with frequency. As stated before, the interharmonic spectral values adjacent to Hz directly correspond to the DFT results and have the expressions (6) (13) (14) where. As long as the frequency of the item to be summed in (6) (i.e., ) is very low in comparison with the sampling frequency, the summation in (6) can be replaced by integrating The concerned spectral bin, in reality, is usually located close enough to make the above assumption true. For instance, when 253 Hz,, 37 Hz 7.4, 12, 64, the result of by using (7) is, while the exact value by using (5) is. The difference is very small compared to the absolute value. Based on the simulation studies by the authors, the sample frequency of more than 100 points per cycle is acceptable to make (7) hold. (7) (8) After simple mathematical transformation, two linear equations can be formed as where (15) (16) By solving the above linear equations, the interharmonic leakage value at Hz can be obtained by (17) A simple numerical example will be used here to illustrate the calculation process. In this example, the interharmonic leakage value at 120 Hz in Fig. 1 is calculated by considering only the Hz interharmonic component, as the impact of the 177.5

4 974 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 2, APRIL 2012 Hz component is small. The spectral values located at 115 Hz and 125 Hz (bins and ) are (18) (19) The interharmonic leakage value at 120 Hz (2) (bin in Fig. 2) can be obtained by using (15) and (17) with the two known values (2) and (2). The result is, which is a good estimation compared to the real value of. The difference is due to the influence of the Hz component. 3) Interharmonic Leakage Calculation by Considering two Interharmonic Components: If two interharmonics are considered around the th harmonic, the total leakage values contributed by the two interharmonics around Hz are (20) where and refer to the frequencies of the two interharmonics. Through a series of mathematical transformations, the four spectral lines surrounding Hz will meet the following linear equations: where As a result, the interharmonic leakage value at solved by (21) (22) Hz can be (23) The interharmonic leakage value at 120 Hz in Fig. 2 (bin ) was again calculated by considering two interharmonic components. In this case, the four DFT results centered with 120 Hz were used. The calculated leakage result is, which is better than the result from considering only one interharmonic since the new result includes the impact of the Hz interharmonic. The small error is due to the negative frequency components. 4) Calculate Interharmonic Leakage Values by Considering Three Interharmonic Components: Similar to the two aforementioned cases, if three interharmonic components are considered, the six DFT results adjacent to Hz will satisfy six linear equations (the mathematical expression is omitted here to save space), and the interharmonic leakage values can be obtained by solving the corresponding equations. Here, the interharmonic leakage value at 120 Hz in Fig. 2 (bin ) was estimated once again by considering three interharmonic components. The result is, which is exactly the same as the true value. It is better than both the previous results since more interharmonic components (three instead of only one or two) are included. The aforementioned results show that as long as the real interharmonic number is less than or equal to the considered number, accurate results can be obtained. However, the number of interharmonic components is usually unknown before measuring a raw signal. How to determine the number of the considered interharmonics will be discussed further in Section IV. B. Improved Grouping Scheme After calculating the interharmonic leakage values at harmonic frequencies one-by-one, the pure harmonic and interharmonic spectra can thus be established. Then, the group results can be calculated based on the separated spectra: 1) For harmonics, the spectral lines at the harmonic frequencies of the harmonic spectrum can be obtained by subtracting the calculated interharmonic leakage values at the harmonic frequencies from the original DFT and the results directly correspond to the real harmonic results. 2) However, for interharmonics, their energy is not concentrated at specific frequencies in the interharmonic spectrum, but is usually spread out to all frequency points [shown in Fig. 2(b)] due to asynchronous truncation. If an interharmonic constituent is near a harmonic, the main energy of the interharmonics will distribute around the harmonic frequency. Therefore, the three spectral bins centered with the harmonic frequency [as shown in Fig. 2(b) in the interharmonic spectrum] cannot be ignored and should be grouped into the right interharmonic group. Based on the attenuation trend of a component s spectral bins, the three bins should be included into the interharmonic group to which the larger one of the two side bins belongs. For instance, in Fig. 2(b), bins,, and should be grouped into the 1.5th interharmonic group, since bin is larger than bin. Similarly, the spectral lines at , and 185 Hz should be included in the 2.5th interharmonic group. C. Complete Method The complete method is summarized as follows. 1) Capture a 12-cycle snapshot of the waveform and perform DFT on the snapshot. 2) Calculate the interharmonic leakage values at the harmonic frequencies one-by-one. 3) Obtain separated harmonic and interharmonic spectra. 4) Calculate harmonic and interharmonic group results based on their own spectra. If one needs to know the frequencies, magnitudes, and phase angles of the main interharmonics, interpolation algorithms [8] [15] can be employed to meet this requirement since the interference from the harmonics has already been eliminated by this spectrum separation technique discussed before. IV. VERIFICATION STUDIES In this section, we use two groups of data to assess the performance of the IEC recommended method and the proposed

5 HUI et al.: A METHOD TO IMPROVE THE INTERHARMONIC GROUPING SCHEME ADOPTED BY IEC STANDARD TABLE I PARAMETERS OF THE TEST SIGNAL TABLE II CALCULATED INTERHARMONIC LEAKAGE VALUES AT HARMONIC FREQUENCIES. Fig. 3. DFT spectrum of the 0.2-s waveform snapshot. improved method. The first group is from the computer simulation, which represents the ideal cases. The second group is a real-world case study of an industrial VFD. A. Simulation Study Assume that the test signal consists of six tones, the parameters of which are listed in Table I. For the sake of simplicity and without loss of generality, all phase angles of the frequency components are set to be zeros. Fig. 3 shows the 12-cycle spectrum based on the recommendation of the IEC standard, which reveals that the DFT results at the harmonic frequencies no longer correspond to the actual harmonic parameters even though the sampling is synchronized with the fundamental frequency. This finding occurred because the spectral lines at harmonic frequencies were interfered with by interharmonics. The severity of the interference is determined by the closeness of the harmonic to the nearby interharmonic and its relative amplitude. For the fundamental component, since it has the strongest energy and the interharmonics are relatively far away, the interharmonic interference has little impact and can be ignored. However, for the second and third harmonic tones, since they are located in close proximity to the interharmonics, whose energies are comparable to theirs, the interference is strong. On the other hand, for the interharmonics with frequencies Hz and Hz, the spectral lines in their main lobes are also disturbed by the two harmonics. To eliminate the interference discussed before, the proposed spectrum separation method was used to obtain the pure harmonic and interharmonic spectra. The first step of the separation method is to calculate the interharmonic leakage values at harmonic frequencies. Table II lists the calculated results (IH is short for the interharmonic, and H is short for the harmonic in the following). The test results show that for each harmonic, the three results obtained by considering different interharmonic numbers are comparable, and all are around the actual values. When three interharmonics are considered, the calculated IH leakage values are almost the same as the true values since the real number of interharmonic components is also three. After obtaining the interharmonic leakage values at the harmonic frequencies, the pure interharmonic spectrum can be established. Then, the harmonic spectrum can also be formed by subtracting the calculated interharmonic leakage values from TABLE III HARMONIC AND INTERHARMONIC GROUP RESULTS the original DFT harmonic spectrum. Based on the improved grouping scheme, the group results for both harmonics and interharmonics were calculated, and the results are shown in Table III. The results show that for the fundamental 0.5th and 2.5th groups, the results from the IEC method are acceptable and close to the real values. However, for the 2nd, 3rd, 1.5th, and 3.5th groups, the results are far from the true values because the interharmonic constituents in the two interharmonic groups are located close to the two nearby harmonics. The IEC measurement method irrationally grouped the interharmonic bins of 115 Hz and 125 Hz into the second harmonic group and grouped the interharmonic bins of 175 Hz and 185 Hz into the third harmonic group. Thus, the harmonic results were overestimated, and the interharmonic results were underestimated. In contrast, the results from the improved method are all acceptable and accurate since it groups the main bins of the interharmonics into the correct interharmonic groups. Furthermore, to verify the antinoise performance, the calculation was also performed for a noise environment in which the signal-to-noise ratios (SNRs) varied in the range of db at an increment of 20 db. The results are shown in Fig. 4. (Improved methods 1 3 refer to the proposed methods by considering 1 3 IHs, respectively, which suggests that the proposed method was still effective in dealing with noise-disturbed signals.) If one wants to know the magnitudes, frequencies, and phase angles of the main interharmonics, the interpolation algorithm developed in other studies can meet this requirement. In this paper, the transformed discrete Fourier transform (TDFT) developed by [10] was used to obtain the detailed information

6 976 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 2, APRIL 2012 if no real interharmonics are present. Too small interharmonic bins may produce large errors in estimating the true interharmonic leakage impact on certain harmonics. Consequently, for each harmonic, a criterion has to be specified to filter out the noise and the ignorable interharmonic interference (24) Fig. 4. Results of interharmonic groups with the presence of white Gaussian noise. (a) The 0.5th interharmonic group. (b) The 1.5th interharmonic group. (c) The 2.5th interharmonic group. (d) The 3.5th interharmonic group. TABLE IV RESULTS OF INTERHARMONIC PARAMETERS of the two main interharmonics. The results are tabulated in Table IV, which shows that the TDFT gave accurate interharmonic parameters. Note that the spectrum separation makes the interpolation algorithms, such as TDFT, usable. If the interharmonic and harmonic spectra are interfered with, as is shown in Fig. 3, the details of the interharmonic tones cannot be located by using any interpolation methods. B. Practical Issues and the Solutions To put the proposed method into practice, some practical issues must be discussed and addressed as follows. 1) In practice, due to the limit acquiring resolution of the instrument and the measurement noise, the spectral bins around the harmonic frequencies are always nonzero even Practical investigations by the authors show that the impact of the measurement noise is usually much larger than that of the instrument acquiring resolution for real power signals, so the coefficient in (24) should be user-defined and determined by the noise level. 2) The test signal in the simulation section contains exactly three interharmonic constituents. Strictly, three interharmonics should be considered in calculating the interharmonic leakage values at the three harmonic frequencies. However, the result implies that considering one interharmonic tone is enough since the numbers of the interharmonics close to the harmonics are all less than two in this case. For practical signals, the number of dominant interharmonics interfered with by harmonics is also usually less than two. Therefore, considering one interharmonic can meet the measurement requirement for practical uses. 3) On the other hand, the results obtained by considering two and three interharmonics are also needed since they can be used to confirm the number of interharmonics near certain harmonics and to crosscheck the results by considering one interharmonic component. If only one interharmonic is near a harmonic, the results by considering one, two, and three interharmonics will be almost the same since the other interharmonics effects are ignorable. If the three results are all far from each other, the interharmonic bins adjacent to the harmonic frequencies are likely to be caused by the irregular fluctuation of the harmonic component. In these cases, the IEC subgroup method can be used directly, and the adjacent interharmonic bins should be grouped into the harmonic groups to evaluate the final rms values. C. Real Case Study The real case involves a 25-kV distribution feeder commonly seen in rural North America. The VFD is one of the largest loads supplied by the feeder. Since the VFD had been installed, lighter flicker problems have been reported due to significant interharmonics. A measurement was taken at the secondary side of a stepdown transformer connecting the VFD. The sampling frequency is Hz. Fig. 5 shows the waveform snapshots and the 12-cycle DFT results. The DFT results show that the 120 Hz and 240 Hz harmonics are strongly interfered by the nearby interharmonics. Thus, it can be concluded that the IEC based group scheme will give misleading answers in calculating the 2nd and 4th harmonic groups and the interharmonic groups nearby. Ten continuous 12-cycle waveform snapshots were studied. By filtering out the negligible interharmonic leakage effect and noise using (24) ( was chosen as 0.3% in this case), only the

7 HUI et al.: A METHOD TO IMPROVE THE INTERHARMONIC GROUPING SCHEME ADOPTED BY IEC STANDARD Fig. 5. Waveforms and spectrum of the measured signal. (a) The 2-s waveform snapshot. (b) The 0.2-s waveform snapshot. (c) DFT spectrum of the 0.2-s waveform snapshot. Fig. 7. Calculated interharmonic leakage values at the 4th harmonic frequency. (a) Magnitudes of the leakage values. (b) Phase angles of the leakage values. Fig. 6. Calculated interharmonic leakage values at the 2nd harmonic frequency. (a) Magnitudes of the leakage values. (b) Phase angles of the leakage values. interharmonic interferences at 120 Hz and 240 Hz were considered and extracted for most of the snapshots. The calculated interharmonic DFT results at the two harmonic frequencies are shown in Figs. 6 and 7. These figures show that the three leakage results by considering three different interharmonic numbers are consistent and concentrated. This finding not only implies that the results are correct, but also indicates only one dominant interharmonic was in close proximity to the nearby harmonic. Thus, for this case, considering one interharmonic is enough and credible. After separating the original DFT spectrum, the harmonic and interharmonic group results were calculated for each snapshot. In order to verify the accuracy of the calculated results, the group results of the 2-s waveform snapshot were used as the references. The mean group magnitudes obtained by using the three different methods are shown in Fig. 8. For the harmonics, Fig. 8. Comparison among group results from different methods. (a) Results of harmonic groups. (b) Results of interharmonic groups. the IEC standard procedure gives always overestimated results especially when interharmonics are nearby. The results reach an overestimation of about 70% for the 2nd harmonic group and of 320% for the 4th harmonic group. In contrast, for the interharmonic groups, the IEC standard procedure gives underestimated results for the 1.5th and the 3.5th interharmonic group results. By comparison, the proposed method gives accurate results for the harmonic and interharmonic groups. It was discovered that the 1.5th and 3.5th interharmonics are significant. Furthermore, the magnitudes, frequencies, and phase angles of the two interharmonic tones near 120 Hz and 240 Hz were calculated based on the separated interharmonic spectrum and the TDFT algorithm [10]. The results are shown in Fig. 9. The

8 978 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 2, APRIL 2012 can remain as recommended by IEC (i.e., 5 Hz), and provide more accurate information on the harmonics and interharmonics present in the signal. The specific characteristics of the proposed method are: 1) harmonics and interharmonics can be processed separately in the frequency domain with the proposed spectrum separation technique, even if the frequency difference between a harmonic and an interharmonic is less than one frequency resolution (5 Hz); 2) the new interharmonic grouping scheme can group the main energy of the interharmonic tones into the correct groups when the interharmonic is in close proximity with a harmonic; 3) more accurate signal-processing techniques, such as spectral interpolation, can be performed to obtain the detailed interharmonic information since the interference from harmonics is strongly restrained. Fig. 9. Parameters of the two studied interharmonics. (a) Results of the magnitudes. (b) Results of the frequencies. (c) Results of the phase angles. most obvious finding is that the frequency and phase angle distances between the 120 Hz tone and the nearby interharmonic are almost the same as the ones between the 240 Hz tone and the nearby interharmonic. This phenomenon exactly fits the index [16]. (The frequency distance between the two main interharmonics produced by VFD is equal to twice the fundamental frequency.) From this perspective, the interharmonic answers are correct. Moreover, along with this verification, other evidence also shows the correctness of the results. In Fig. 5(a), the fluctuation frequency of the waveform presents a down and up trend. As is known, for most electrical signals, the waveform fluctuation frequency is determined mainly by the frequency distance between the main interharmonics and the nearby harmonics [17]. Therefore, the interharmonic frequency should also have the same trend, and the results shown in Fig. 9 are consistent with the waveform fluctuation trend. V. CONCLUSION A comprehensive solution to the problems caused by the leakage and picket fence effects is to select a frequency resolution that is a common divider of the frequencies of all components in the signal. This would not be practical (among other reasons) because the window width would become too large. If the window width being examined becomes too large, the risk of dealing with nonstationary signals increases. This paper proposed a method to improve the grouping scheme adopted by IEC. By using the method, the frequency resolution REFERENCES [1] C. Li, W. Xu, and T. Tayjasanant, Interharmonics: Basic concepts and techniques for their detection and measurement, Elect. Power Syst. Res., vol. 66, pp , [2] T. X. Zhu, Exact harmonics/interharmonics calculation using adaptive window width, IEEE Trans. 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9 HUI et al.: A METHOD TO IMPROVE THE INTERHARMONIC GROUPING SCHEME ADOPTED BY IEC STANDARD Jin Hui was born in Wuxi, China, in He received the B.S. degree in electrical engineering from the College of Electrical Engineering and Information Technology, Sichuan University, Chengdu, China, in 2007 and is currently pursuing the Ph.D. degree in electrical engineering at Sichuan University, Chengdu, China. His main research interests are power system harmonic analysis and evaluation. Wilsun Xu (M 90 SM 95 F 05) received the Ph.D. degree in power engineering from the University of British Columbia, Vancouver, BC, Canada, in He was an Engineer with BC Hydro from 1900 to Currently, he is a Professor and an ESERC/iCORE Industrial Research Chair at the University of Alberta, Edmonton, AB, Canada. His research interests are power quality and harmonics. Honggeng Yang (M 07) was bon in Chengdu, China, in He received the M.S. degree in electrical engineering from Harbin Institute of Technology, Harbin, China, in 1985, and the Ph.D. degree in electrical engineering from Liege University, Leige, Belgium, in Since 2002, he has been a Professor at Sichuan University, Chengdu. His research interests include power quality and reactive power/voltage control. Yamei Liu received the M.S. degree from Chengdu University of Science and Technology, Chengdu, China, in 1999, and is currently pursuing the Ph.D. degree in electrical engineering at Sichuan University, Chengdu. Her main research interests are power system harmonics and interharmonic analysis.