Simple Interpolated FFT Algorithm Based on Minimize ... - IEEE Xplore

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 9, SEPTEMBER 2011

Simple Interpolated FFT Algorithm Based on Minimize Sidelobe Windows for Power-Harmonic Analysis He Wen, Zhaosheng Teng, Yong Wang, Bo Zeng, and Xiaoguang Hu

Abstract—This paper focuses on the low-computation harmonicanalysis procedure with sufficient suppression of spectral leakage and picket-fence effect. The interpolated FFT algorithm based on the minimized sidelobe window is considered, and its calculate procedure and formulas are given, which is free of solving highorder equations. The implementation of the proposed algorithm in the digital-signal-processor (DSP) based three-phase harmonic ammeter is also introduced. The proposed algorithm has the major advantages that the calculate formulas for harmonic parameters can be easily implemented by hardware multipliers, making the method a good choice for real-time applications. The simulation and application results validate the accuracy and efficiency of the proposed algorithm. Index Terms—Digital signal processor (DSP), harmonic analysis, interpolated FFT, minimize sidelobe windows (MSWs), picketfence effect, spectral leakage.

I. INTRODUCTION ARMONIC analysis is an important issue in powersystem signal-processing tasks because the existence of harmonics greatly affects the operation of electric power system in safety and economy [1]–[3]. Large classes of harmonicanalysis algorithms have been adopted, which can be classified as either time-domain (parametric) or frequency-domain (nonparametric) methods [4]. Time-domain methods, such as autoregressive models [5], Prony’s method [6], MUSIC algorithm [7], estimate the harmonic parameters with high accuracy, but require intensive computational algorithms to determine the model order [4]. Frequency-domain approaches based on the well known discrete Fourier transform (DFT) are of appreciable features in real-time processing because they allow the use of nonparametric and low-computational effort procedures [8]–[11]. To obtain accurate harmonic parameters, the direct application of the DFT has the requirement on the periodicity of signal

H

Manuscript received November 22, 2010; revised January 20, 2011; accepted January 25, 2011. Date of current version September 16, 2011. This work was supported in part by the National Natural Science Foundation of China under Grant 61002035, in part by the Research Fund for the Doctoral Program of Higher Education of China under Grant 20100161120004, and in part by the China Postdoctoral Science Foundation under Grant 20100471213. Recommended for publication by Associate Editor V. Staudt. H. Wen, Z. Teng, Y. Wang, and B. Zeng are with the College of Electrical and Information Engineering, Hunan University, Changsha City 410082, China. X. Hu is with the School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China. Digital Object Identifier 10.1109/TPEL.2011.2111388

and the synchronization of sampling process [12]. However, it is difficult to implement strict synchronous sampling even when using the discrete phase-locked-loop technique because of the fundamental frequency instability, fundamental variations, timevarying harmonics, and harmonic interference in electric power systems. Without necessary synchronization, the DFT has inherent performance limitations, namely, spectral leakage and picket-fence effect [13], [14]. It is well known that a considerable reduction of spectral leakage error can be achieved by weighting the time samples by a suitable window [9], [15], such as the rectangular window, the triangular window, the combined cosine windows [16], the maximum-sidelobe-decay windows (also known as the class I Rife-Vincent windows) [9], [17], [18], the polynomial windows [19], the rectangular convolution window [20], the extremely flat-top convoluted windows [21], the Hanning self-convolution window [22], and the triangular selfconvolution window (TSCW) [23], and the error caused by the picket-fence effect can be eliminated by adopting interpolated fast Fourier transform (FFT) algorithms with dual-spectrumline or multispectrum-line [4], [13], [18], [24]–[27]. Recently, a high-accuracy multipoint-interpolated FFT algorithms with the maximum-sidelobe-decay windows has been proposed without polynomial approximations generally required in the interpolation FFT method [4], [9], which also proved that the systematic errors caused by the nonsynchronous sampling decrease as the number of interpolation points and/or the window order increases [17], [18]. It is also worth to point out that the windowed interpolation FFT algorithms have the advantages of robustness toward signal model inaccuracies and low computational effort [9], [18], [28]. It has been shown that the performances of the windowed interpolated FFT (WIFFT) algorithms are mainly affected by the data windows [29], [30]. In particular, the WIFFT can be implemented with a simple calculation formula when the Rectangular window and Hanning window are adopted [29], [31]. However, when the window function is complex, for instance, the highorder combined cosine windows, the WIFFT algorithms involve solving high-order equations on the harmonic parameter estimation [27]. Different solving methods, including Chebyshev best approximation theory [27], iterative method [32] and neural network algorithm [33]–[35], etc., have been presented to accurately determine harmonic parameters. However, presented methods are not convenient to be implemented through hardware and the computational complexity is very high [36], [37]. Moreover, for the multifrequency signals of a power system, the spectral leakage and harmonic interference may not be

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WEN et al.: SIMPLE INTERPOLATED FFT ALGORITHM BASED ON MINIMIZE SIDELOBE WINDOWS

effectively suppressed and weak harmonic components can easily be obscured by nearby strong harmonics due to the spectral leakage [15], [23]. There is thus a necessity to develop new harmonic-analysis algorithms to satisfy the requirements for accurate dynamic complicated signal-parameter estimation and its high-speed implementation in embedded terminals [23], [38], [39]. The using of a window with good sidelobe behavior is of great importance for the high-accuracy harmonic analysis because the width of the major lobe in a window’s spectrum relates to the frequency resolution, while the amplitude and decay of the sidelobe determine leakage effect. The minimize sidelobe windows (MSWs), which give low-peak sidelobe levels (a maximum value as –93 dB) along with high sidelobe-decaying rates (a maximum value as 41 dB/oct), can achieve desirable performance for the suppression of spectral leakage. Hence, this paper proposed a simple interpolated DFT algorithm based on the MSW, in which the sampled signals are weighted with MSW, and the harmonic parameters are derived by the polynomial approximation theory without solving high-order equations. The proposed algorithm has the major advantages that the calculate formulas for harmonic parameters can be easily implemented by hardware multipliers, making the method a good choice for real-time applications. The effectiveness and practicability of the proposed algorithm have been verified by its application to the three-phase multifunctional harmonics ammeter, which can achieve a precision of class 0.2 S for electrical power measurement. The organization of this paper is as follows. The basic theories of harmonic analysis are introduced in Section II. The MSW and its characteristics are given in Section III. The MSW-based interpolated DFT algorithm is presented in Section IV. Simulation and implementation results are given in Section V. Finally, conclusions are drawn in Section VI.

II. HARMONIC ANALYSIS BASED ON DFT A. Processing With Synchronous Sampling In the following, we consider the sampled multifrequency signal of length N (n = 0, 1, . . ., N − 1):

x (nTs ) = x (t) |t=n T s =

H 

Ah sin (2πfh nTs + ϕh )

(1)

h=1

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samples can be written as ¯ (λΔf ) = X

N −1 

x (nTs ) e−j 2π k λ/N

(2)

k =1

where Δf = 1/(N Ts ) is the frequency resolution and λ = 0, 1, . . ., N − 1. From (2), the harmonic components of the signal x (nTs ) fall at frequencies, which are integer multiples of Δf in the case of a synchronous sampling. It is thus possible to determine amplitude and phase of the phasor X˙ h = Ah exp (jϕh ) from (2), which can be written as ¯ (λΔf ) 2j X X˙ λ = . N

(3)

B. Processing With Nonsynchronous Sampling It is well known that a synchronized sampling is purely theoretical because of either the practical requirement of processing finite-duration records or unknown fluctuations of the input signal frequency. As a consequence, the spectral leakage is inevitable, and the samplings are weighted by the time window to suppress the leakage. The DFT of the windowed samples x(n) w(n) on N sampled points can be written as follows: H   Ah  j ϕ h e Wf (k − kh ) − e−j ϕ h Wf (k + kh ) 2j h=1 (4) where k = 0, 1, . . ., N – 1, Wf is the DFT of the adopted window, kh is the signal frequency divided by the frequency resolution Δf of the window w(n), and can be written in two parts

X (k) =

kh =

fh N = lh + ζh fs

(5)

where lh is an integer value and the fractional part ζh (0 ≤ ζh ≤ 1) is caused by the nonsynchronous sampling. From (5), the spectral line exactly corresponding to the hth harmonic may lie between the two greatest spectral lines, namely, the lh th and the (lh +1)th, or the lh th and the (lh –1)th. The lh can be easy found by the peak-search procedure, and then, the ζh can be determined through the interpolated algorithms; finally, the harmonic parameters, i.e., frequency, amplitude, and phase, can be calculated accordingly. III. WINDOWS AND CHARACTERISTICS A. Choice of Window Characteristics

where H is the number of fundamental and harmonic components, fh , Ah , and ϕh are, respectively, the frequency, amplitude, and phase of the hth harmonic, for h = 1, f1 , A1 , and ϕh are the parameters of fundamental component, Ts = 1/fs is the sampling interval, and fs is the sampling rate. By assuming that the Nyquist frequency fm ax = 1/(2Ts ) is larger than all of the fh , the aliasing effects that arise from finite sampling rate can be neglected. If the synchronous sampling condition is satisfied, i.e., Nf1 /fs is an integer, the complex coefficients of the DFT relevant to the sampled signal (1) on N

Seeing the DFT of the windowed samples, the second term in (4) is caused by the negative-frequency component in a real signal, and the harmonic interference between adjacent components should also be eliminated to achieve high accuracy. Hence, the following two factors should be considered. 1) The major lobe width of the adopted window: In practical harmonic analysis, the kh th spectral line, which represents the hth harmonic component, should be frequently well separated. Therefore, the minimum allowed difference between two adjacent frequency components

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TABLE I COEFFICIENTS AND CHARACTERISTICS OF MSW

|fh –fh +1 | should satisfy the following inequality: ψM LW
40 dB, the accuracies achieved by the proposed algorithm with MSW 4(II) are greater than those of the other algorithms. It should be noted that with high noise level, the harmonic components, including the weak component, can be accurately detected by using the proposed algorithm with MSW 4(II). Finally, from the simulation results summarized in Tables IV, VI, and VII, and Figs. 3–5, it is reasonable to say that the in-

Fig. 6.

Block diagram of the DSP-based three-phase harmonics ammeter.

terference among harmonics or white-noise variation can be restrained effectively. The signal parameters, including evenorder harmonics and weak components, can be accurately determined by using the proposed algorithm, which can completely meet the practical requirements of harmonic analysis in power systems. D. Results of the Applications The signal in a real power system is affected not only by noise or harmonic distortions but also by other types of disturbances. The acquired data in these situations can be used to analyze the algorithm capabilities to perform under such unfavorable conditions. The proposed algorithm is used in the high-accuracy three-phase harmonics ammeter. As shown in Fig. 6, the ammeter is developed based on the structure of ADC+ digital-signal processor (DSP), where the ADC is ADS8364 with a sampling frequency of 2.5 kHz, and

WEN et al.: SIMPLE INTERPOLATED FFT ALGORITHM BASED ON MINIMIZE SIDELOBE WINDOWS

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TABLE VIII VOLTAGES OF HARMONIC COMPONENTS IN THE EXPERIMENTAL TEST

TABLE IX TEST RESULTS OF ACTIVE POWER WITH VOLTAGE FLUCTUATION

Fig. 7. Setup of the experimental. (a) Clou CL303B multifunction electrical power standard. (b) DSP-based three-phase harmonics ammeter.

Fig. 8. Percentage errors of amplitude and phase difference of the voltage and current of phase-C for each frequency component with different load characteristics by the three-phase harmonic ammeter.

the DSP is TMS320VC5502, which is a fixed point processor. The ammeter weights the voltage signals from a three-phase 220 V/50 Hz power system by using the proposed algorithm with the MSW 4(II) with a length of N = 512 to estimate parameters of the fundamental and harmonics. To comply with the real-time requirement on the harmonic measurement, the main core clock rate of the DSP is set to 300 MHz. Moreover, the preprocess circuit including a low-pass filter is used in the measurement process to avoid the aliasing phenomenon. As shown in Fig. 7, the three-phase harmonics ammeter is used to measure the multifrequency waveform generated by the Clou CL303B multifunction electrical power standard. The output resolution of voltage and current of CL303B is 10 ppm and the frequency accuracy is 0.001 Hz. The testing results, which include the amplitude and phase difference of the voltage and current of phase-C for each frequency component with different load characteristics, are shown in Fig. 8. And the measurement results of the 21st harmonic voltage of three phase in the experimental test are shown in Table VIII. In Table IX, the “true

values” are given by the CL303B, and the “measured values” are given by the DSP-based three-phase harmonics ammeter. Table IX gives the test results of the fundamental active power T_error with the fluctuation of the fundamental voltage, where In = 1.5 A, Im ax = 6 A, GB_e0 is the standard error given by GB/T 17215.322-2008 [40]. The results show that the ammeter achieves a precision of class 0.2 S for electrical power measurement, and completely fulfills the requirements for class A power harmonics measurement stated in GB/T14549-1993 Power Quality, Harmonics in Public Power Grid. The results also demonstrate the high accuracy and efficiency in real-world application of the proposed method. VI. CONCLUSION The spectral leakage and picket-fence effect are always the main causes of the harmonic-analysis errors in power system. This paper chooses the MSWs as the weighting window in time domain, and proposed a simple interpolation FFT algorithm based on the MSWs for accurate harmonic analysis. The proposed algorithm calculates harmonic parameters by simple formulas which are deduced through the polynomial approximation theory without solving high-order equations. The utilization of a simple calculate procedure is significantly important because a simple calculate procedure means less computation and low cost in implementation, which could lead to more effective and less costly embedded system designs for harmonicanalysis applications. The application of the proposed algorithm in the three-phase harmonic ammeter without adopting phaselocked-looped mechanisms has verified the effectiveness of the proposed algorithm. By observing the estimation results for the

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simulated and actual measured signals, it is seen that the proposed method is more accurate and needs less computation than the other WIFFT algorithms under comparison, while the computational efficiency is maintained. However, it is usually difficult for the FFT-based methods to obtain accurate harmonic analysis results when the measured signal is highly time varying or contains interharmonics. Because for the highly time-varying signal, it is difficult to determine a suitable frequency resolution with desirable suppressions of spectral leakage. Therefore, an important future work is the dynamical adjustment of the parameters of the adopted window to effectively fulfill the requirements on the frequency resolution and sidelobe behaviors for the minimum harmonic estimation errors in power system. REFERENCES [1] T. Bhavsar and G. Narayanan, “Harmonic analysis of advanced busclamping PWM techniques,” IEEE Trans. Power Electr., vol. 24, no. 10, pp. 2347–2352, Oct. 2009. [2] A. Z. Albanna and C. J. Hatziadoniu, “Harmonic modeling of hysteresis inverters in frequency domain,” IEEE Trans. Power Electr., vol. 25, no. 5, pp. 1110–1114, May 2010. [3] H. C. Lin, “Inter-harmonic identification using group-harmonic weighting approach based on the FFT,” IEEE Trans. Power Electr., vol. 23, no. 3, pp. 1309–1319, May, 2008. [4] D. Belega, D. Dallet, and D. Petri, “Accuracy of sine wave frequency estimation by multipoint interpolated DFT approach,” IEEE Trans. Instrum. Meas., vol. 59, no. 11, pp. 1–8, Nov. 2010. [5] R. P. Kane, “Some methods of spectral analysis: comparison with MEMMRA combination,” Appl. Math. Comput., vol. 181, pp. 949–957, Oct. 2006. [6] J. Zygarlicki, M. Zygarlicka, J. Mroczka, and K. J. Latawiec, “A reduced prony’s method in power-quality analysis—Parameters selection,” IEEE Trans. Power Delivery, vol. 25, no. 2, pp. 979–986, Apr. 2010. [7] T. Lobos, Z. Leonowicz, J. Rezmer, and P. Schegner, “High-resolution spectrum-estimation methods for signal analysis in power systems,” IEEE Trans. Instrum. Meas., vol. 55, no. 1, pp. 219–225, Feb. 2006. [8] G. W. Chang, C. I. Chen, Y. J. Liu, and M. C. Wu, “Measuring power system harmonics and interharmonics by an improved fast Fourier transformbased algorithm,” IET Gener. Transm. Dis., vol. 2, pp. 193–201, Mar. 2008. [9] D. Belega and D. Dallet, “Frequency estimation via weighted multipoint interpolated DFT,” IET Sci., Meas. Technol., vol. 2, pp. 1–8, Jan. 2008. [10] M. , T. Hagh, H. Taghizadeh, and K. Razi;, “Harmonic minimization in multilevel inverters using modified species-based particle swarm optimization,” IEEE Trans. Power Electr., vol. 24, no. 10, pp. 2259–2267, Oct. 2009. [11] M. Odavic, M. Sumner, P. Zanchetta, and J. C Clare, “A theoretical analysis of the harmonic content of PWM waveforms for multiple-frequency modulators,” IEEE Trans. Power Electr., vol. 25, no. 1, pp. 131–141, Jan. 2010. [12] D. Agrez, “Dynamics of frequency estimation in the frequency domain,” IEEE Trans. Instrum. Meas., vol. 56, no. 6, pp. 2111–2118, Dec. 2007. [13] Z. Fusheng, G. Zhongxing, and Y. Wei, “The algorithm of interpolating windowed FFT for harmonic analysis of electric power system,” IEEE Trans. Power Delivery, vol. 16, no. 2, pp. 160–164, Apr. 2001. [14] D. Belega, D. Dallet, and D. Petri, “A high-performance procedure for effective number of bits estimation in analog-to-digital converters,” IEEE Trans. Instrum. Meas., vol. PP, no. 99, pp. 1–11, Oct. 2010. [15] J. Schoukens, Y. Rolain, and R. Pintelon, “Analysis of windowing/leakage effects in frequency response function measurements,” Automatica, vol. 42, pp. 27–38, Jan. 2006. [16] F. C. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” in Proc. IEEE, Jan, 1978, vol. 66, pp. 51–83. [17] D. Belega, D. Dallet, and D. Slepicka, “Accurate amplitude estimation of harmonic components of incoherently sampled signals in the frequency domain,” IEEE Trans. Instrum. Meas., vol. 59, no. 5, pp. 1158–1166, May 2010.

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WEN et al.: SIMPLE INTERPOLATED FFT ALGORITHM BASED ON MINIMIZE SIDELOBE WINDOWS

He Wen was born in Hunan, China, in 1982. He received the B.Sc., M.Sc, and Ph.D. degrees in electrical engineering from Hunan University, Hunan, China, in 2004, 2007, and 2009, respectively. He is currently a Lecturer with the College of Electrical and Information Engineering, Hunan University. His current research interests include powersystem-harmonic measurement and analysis, power quality, and digital signal processing.

Zhaosheng Teng was born in Hunan, China, in 1963. He received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering, all from Hunan University, Changsha, China, in 1984, 1995, and 1998, respectively. From 1999 to 2000, he was a Postdoctoral Research Fellow with the National University of Defense Technology. Since 2000, he has been a Professor at Hunan University. His current research interests include power-quality monitoring, harmonic analysis, information fusion, and electrical measurement.

Yong Wang received the B.Sc. degree from Shenyang Institute of Aeronautics Engineering, Shenyang, China, in 2002, and the M.Sc. degree in 2009 from the College of Computer Science and Engineering, Hunan University, Changsha, China, where he is currently working toward the Ph.D. degree. His current research interests include electric power-system-harmonic measurement and information fusion.

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Bo Zeng received the B.Sc. and M.Sc. degrees, in 2005 and 2008, respectively, from the College of Electrical and Information Engineering, Hunan University, Changsha, China, where he is currently working toward the Ph.D. degree. His current research interests include electric power-system-harmonic measurement and analysis, and intelligent information processing.

Xiaoguang Hu received the B.Sc. degree from Dongbei Electric Power University, Dongbei, China, in 1983, the M.Sc. degree from the Wuhan University of Hydraulic and Electrical Engineering, Wuhan, China, in 1987, and the Ph.D degree from the Harbin Institute of Technology, Harbin, China, in 2003. Since 2004, she has been a Professor with Beijing University of Aeronautics and Astronautics, Beijing, China. Her current research interests include power electronics, power-system analysis, and signal processing.