基于四谱线插值FFT的谐波分析快速算法
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摘要
由于非同步采样和非整周期截断导致快速傅里叶变换(FFT)不能准确地分析出谐波参数,加窗和插值算法经常被用来改善FFT的计算精度。在信号加窗条件下,基于四谱线插值的FFT算法基础上进行快速算法研究。该算法通过分析加窗后信号的频域表达式,利用真实谐波点附近的4根最大谱线值确定实际谱线的位置,对该次谐波进行频率、幅值和相位等参数估计。并且通过多项式拟合的方式推导出了4种典型窗函数的修正公式。根据窗函数主瓣内任意相邻谱线相位相差π的规律,提出一种快速算法,计算某次谐波开方计算量仅需要1次,大大节约了计算复杂度和计算时间。仿真实验表明,四谱线插值算法在拟合阶次较低的情况下,不仅可以获得比常用双谱线和三谱线更高的精度,还具有对偶次谐波检测精度远胜于双、三谱线插值算法的优点。
In the case of non-synchronous sampling and in non-integral period truncation, the fast Fourier transform(FFT) can't analyze harmonic parameters correctly. So the window functions and interpolation algorithms are made to improve the accuracy of harmonic parameter computation by FFT. This paper derives an algorithm based on four-spectrum-line interpolation. This algorithm uses the four most spectrum lines around true location to computer the accuracy position of the harmonic spectrum line through analyzing the discrete-time Fourier transform of the windowed signal. Then the true values of frequency, amplitude and phase can be calculated. Based on four-spectrum-line interpolation algorithm, this paper gives four practical rectification formulas by using the polynomial approximation method. It is proved that the phase difference of any two adjacent spectrum is π. According to this rule, this paper induces a rapid algorithm so that square root is only calculated one time when analyzing some harmonic parameters. The algorithm can save computing complexity and computing time greatly. The simulation results show that the four-spectrum-line interpolation algorithm has higher precision than double/triple- spectrum-line interpolation algorithm especially when estimating the even harmonics and the order of polynomial is lower than two others if obtaining same precision.
In the case of non-synchronous sampling and in non-integral period truncation, the fast Fourier transform(FFT) can't analyze harmonic parameters correctly. So the window functions and interpolation algorithms are made to improve the accuracy of harmonic parameter computation by FFT. This paper derives an algorithm based on four-spectrum-line interpolation. This algorithm uses the four most spectrum lines around true location to computer the accuracy position of the harmonic spectrum line through analyzing the discrete-time Fourier transform of the windowed signal. Then the true values of frequency, amplitude and phase can be calculated. Based on four-spectrum-line interpolation algorithm, this paper gives four practical rectification formulas by using the polynomial approximation method. It is proved that the phase difference of any two adjacent spectrum is π. According to this rule, this paper induces a rapid algorithm so that square root is only calculated one time when analyzing some harmonic parameters. The algorithm can save computing complexity and computing time greatly. The simulation results show that the four-spectrum-line interpolation algorithm has higher precision than double/triple- spectrum-line interpolation algorithm especially when estimating the even harmonics and the order of polynomial is lower than two others if obtaining same precision.
引文
[1]黄银龙,乐健,毛涛,等.一种新型的高压直流输电系统直流侧谐波电压测量方法[J].电工技术学报,2015,30(22):144-152.HUANG Yinlong,LE Jian,MAO Tao,et al.A novel measurement method of the harmonic voltage on the DC-side of HVDC transmission system[J].Transactions of China Electrotechnical Society,2015,30(22):144-152.
[2]赵鲁,李耀华,葛琼璇,等.特定谐波消除及优化脉宽调制单相整流器的研究[J].电工技术学报,2014,29(10):57-64.ZHAO Lu,LI Yaohua,GE Qiongxuan,et al.Study of selective harmonic elimination and optimization pulse width modulation single-phase rectifier[J].Transactions of China Electrotechnical Society,2014,29(10):57-64.
[3]王丰元,邓朴,王攀,等.牵引供电系统的无功补偿与谐波治理研究[J].陕西电力,2015,43(1):86-89.WANG Fengyuan,DENG Pu,WANG Pan,et al.Study on reactive power compensation and harmonic control in traction power supply system[J].Shaanxi Electric Power,2015,43(1):86-89.
[4]杨新华,雷洋洋,吴丽珍,等.交流性微网指定次电压谐波主动补偿策略[J].电网与清洁能源,2015,31(5):21-26.YANG Xinhua,LEI Yangyang,WU Lizhen,et al.Autonomous selective voltage harmonics compensation control method in an AC microgird[J].Power System and Clean Energy,2015,31(5):21-26.
[5]CHANG G W,CHEN C I,LIU Y J,et al.Measuring power system harmonics and interharmonics by an improved fast Fourier transform-based algorithm[J].IET Gener,Transm,and Distrib,2008,2(2):193-201.
[6]庞浩,李东霞,俎云霄,等.应用FFT进行电力系统谐波分析的改进算法[J].中国电机工程学报,2003,23(6):49-54.PANG Hao,LI Dongxia,ZU Yunxiao,et al.An improved algorithm for harmonic analysis of power system using FFT technique[J].Proceeding of the CSEE,2003,23(6):49-54.
[7]GRANDKE T.Interpolation algorithms for discrete Fourier transform of weighed signals[J].IEEE Transactions on Instrumentation and Measurement,1983,32(2):350-355.
[8]周俊,王小海,祈才君.基于Blackman窗函数的插值FFT在电网谐波信号分析中的应用[J].浙江大学学报(理学版),2006,33(6):650-653.ZHOU Jun,WANG Xiaohai,QI Caijun.Estimation of electrical harmonic parameters by using the interpolated FFT algorithm based on Blackman window[J].Journal of Zhejiang University(Science Edition),2006,33(6):650-653.
[9]许珉,张鸿博.基于Blackman-harris窗的加窗FFT插值修正算法[J].郑州大学学报(工学版),2005,26(4):99-101.XU Min,ZHANG Hongbo.The correction algorithm based on the Blackman-harris windows and interpolated FFT[J].Journal of Zhengzhou University(Engineering Science),2005,26(4):99-101.
[10]曾博,唐求,卿柏元,等.基于Nuttall自卷积窗的改进FFT谱分析方法[J].电工技术学报,2014,29(7):59-65.ZENG Bo,TANG Qiu,QING Baiyuan,et al.Spectral analysis method based on improved FFT by Nuttall self-convolution window[J].Transactions of China Electrotechnical Society,2014,29(7):59-65.
[11]翟瑞淼,英超,任国臣,等.基于Nuttall窗的三峰插值谐波算法分析[J].电力系统保护与控制,2015,43(10):38-43.ZHAI Ruimiao,YING Chao,REN Guochen,et al.An approach for harmonic analysis based on Nuttall window and triple-spectral-line interpolation[J].Power System Protection and Control,2015,43(10):38-43.
[12]曾博,滕召胜,温和,等.莱夫-文森特窗插值FFT谐波分析方法[J].中国电机工程学报,2009,29(10):115-120.ZENG Bo,TENG Zhaosheng,WEN He,et al.Harmonic analysis based on Rife-Vincent window interpolated FFT[J].Proceeding of the CSEE,2009,29(10):115-120.
[13]WEN He,TENG Zhaosheng,GUO Siyu.Triangular self-convolution window with desirable side lobe behaviors for harmonic analysis of power system[J].IEEE Transactions on Instrumentation&Measurement,2010,59(3):543-552.
[14]ZENG Bo.Parameter estimation of power system signals based on cosine self-convolution window with desirable side-lobe behaviors[J].IEEE Transactions on Power Delivery,2011,26(1):250-257.
[15]WEN He,TENG Zhaosheng,GUO Siyu,et al.Hanning self-convolution window and its application to harmonic analysis[J].Sci China Ser E,2009,52(2):467-476.
[16]汪立,刘翠琳.一种高精度加窗插值FFT谐波分析方法[J].陕西电力,2013,41(5):1-4.WANG Li,LIU Cuilin.An accurate method for interpolated FFT harmonic analysis based on a new window[J].Shaanxi Electric Power,2013,41(5):1-4.
[17]王玲,徐柏榆,盛超,等.一种新的余弦组合窗插值FFT谐波分析算法[J].武汉大学学报(工学版),2014,47(2):250-254.WANG Ling,XU Baiyu,SHENG Chao,et al.An approach for harmonic analysis based on a new type of cosine combination window interpolation FFT[J].Engineering Journal of Wuhan University,2014,47(2):250-254.
[18]许珉,刘玮.加8项余弦窗插值FFT算法[J].电力系统保护与控制,2015,43(11):27-32.XU Min,LIU Wei.An interpolation FFT algorithm based on 8-term cosine window[J].Power System Protection and Control,2015,43(11):27-32.
[19]徐柏榆,王玲,汪立,等.多项Rife-Vincent窗插值谐波分析方法[J].武汉大学学报(工学版),2014,47(5):636-641.XU Baiyu,WANG Ling,WANG Li,et al.An approach for harmonic analysis based on multi-term Rife-Vincent window interpolation FFT[J].Engineering Journal of Wuhan University,2014,47(5):636-641.
[20]叶景清,邓辉,年俊华,等.基于矩形卷积窗算法的电子式电流互感器在线校验系统[J].电网与清洁能源,2015,31(11):26-30,35.YE Jingqing,DENG Hui,NIAN Junhua,et al.The on-site calibration system for electronic current transformers based on rectangular convolution window algorithm[J].Power System and Clean Energy,2015,31(11):26-30,35.
[21]AGREZ D.Weighted multipoint interpolated DFT to improve amplitude estimation of multi-frequency signal[J].IEEE Transactions on Instrumentation and Measurement,2002,51(2):287-292.
[22]WU J,ZHAO W.A simple interpolation algorithm for measuring multi-frequency signal based on DFT[J].Measurement,2009,42(2):332-327.
[23]牛胜锁,梁志瑞,张建华,等.基于三谱线插值FFT的电力谐波分析算法[J].中国电机工程学报,2012,32(16):30-136.NIU Shengsuo,LIANG Zhirui,ZHANG Jianhua,et al.An algorithm for electrical harmonic analysis based on triple-spectrum-line interpolation FFT[J].Proceedings of the CSEE,2012,32(16):130-136.
[24]牛胜锁,梁志瑞,张建华,等.基于四项余弦窗三谱线插值FFT的谐波检测方法[J].仪器仪表学报,2012,33(9):2002-2008.NIU Shengsuo,LIANG Zhirui,ZHANG Jianhua,et al.Harmonic detection approach based on 4-term cosine window triple-spectrum-line interpolation FFT[J].Chinese Journal of Scientific Instrument,2012,33(9):2002-2008.
[25]黄冬梅,龚仁喜,焦凤昌,等.莱夫-文森特窗三谱线插值的电力谐波分析[J].电力系统保护与控制,2014,42(2):28-34.HUANG Dongmei,GONG Renxi,JIAO Fengchang,et al.Power harmonic analysis based on Rife-Vincent window and triple-spectral-line interpolation[J].Power System Protection and Control,2014,42(2):28-34.
[26]郝柱,顾伟,褚建新,等.基于四谱线插值FFT的电网谐波检测方法[J].电力系统保护与控制,2014,42(19):107-113.HAO Zhu,GU Wei,CHU Jianxin,et al.A power network harmonic detection method based on the four-spectrumline interpolation FFT[J].Power System Protection and Control,2014,42(19):107-113.
[27]蔡晓峰,张鸿博,鲁改凤.应用三谱线插值FFT分析电力谐波的改进算法[J].电力系统保护与控制,2015,43(2):33-39.CAI Xiaofeng,ZHANG Hongbo,LU Gaifeng.Improvement algorithm for harmonic analysis of power system using triple-spectrum-line interpolation algorithm based on window FFT[J].Power System Protection and Control,2015,43(2):33-39.
[2]赵鲁,李耀华,葛琼璇,等.特定谐波消除及优化脉宽调制单相整流器的研究[J].电工技术学报,2014,29(10):57-64.ZHAO Lu,LI Yaohua,GE Qiongxuan,et al.Study of selective harmonic elimination and optimization pulse width modulation single-phase rectifier[J].Transactions of China Electrotechnical Society,2014,29(10):57-64.
[3]王丰元,邓朴,王攀,等.牵引供电系统的无功补偿与谐波治理研究[J].陕西电力,2015,43(1):86-89.WANG Fengyuan,DENG Pu,WANG Pan,et al.Study on reactive power compensation and harmonic control in traction power supply system[J].Shaanxi Electric Power,2015,43(1):86-89.
[4]杨新华,雷洋洋,吴丽珍,等.交流性微网指定次电压谐波主动补偿策略[J].电网与清洁能源,2015,31(5):21-26.YANG Xinhua,LEI Yangyang,WU Lizhen,et al.Autonomous selective voltage harmonics compensation control method in an AC microgird[J].Power System and Clean Energy,2015,31(5):21-26.
[5]CHANG G W,CHEN C I,LIU Y J,et al.Measuring power system harmonics and interharmonics by an improved fast Fourier transform-based algorithm[J].IET Gener,Transm,and Distrib,2008,2(2):193-201.
[6]庞浩,李东霞,俎云霄,等.应用FFT进行电力系统谐波分析的改进算法[J].中国电机工程学报,2003,23(6):49-54.PANG Hao,LI Dongxia,ZU Yunxiao,et al.An improved algorithm for harmonic analysis of power system using FFT technique[J].Proceeding of the CSEE,2003,23(6):49-54.
[7]GRANDKE T.Interpolation algorithms for discrete Fourier transform of weighed signals[J].IEEE Transactions on Instrumentation and Measurement,1983,32(2):350-355.
[8]周俊,王小海,祈才君.基于Blackman窗函数的插值FFT在电网谐波信号分析中的应用[J].浙江大学学报(理学版),2006,33(6):650-653.ZHOU Jun,WANG Xiaohai,QI Caijun.Estimation of electrical harmonic parameters by using the interpolated FFT algorithm based on Blackman window[J].Journal of Zhejiang University(Science Edition),2006,33(6):650-653.
[9]许珉,张鸿博.基于Blackman-harris窗的加窗FFT插值修正算法[J].郑州大学学报(工学版),2005,26(4):99-101.XU Min,ZHANG Hongbo.The correction algorithm based on the Blackman-harris windows and interpolated FFT[J].Journal of Zhengzhou University(Engineering Science),2005,26(4):99-101.
[10]曾博,唐求,卿柏元,等.基于Nuttall自卷积窗的改进FFT谱分析方法[J].电工技术学报,2014,29(7):59-65.ZENG Bo,TANG Qiu,QING Baiyuan,et al.Spectral analysis method based on improved FFT by Nuttall self-convolution window[J].Transactions of China Electrotechnical Society,2014,29(7):59-65.
[11]翟瑞淼,英超,任国臣,等.基于Nuttall窗的三峰插值谐波算法分析[J].电力系统保护与控制,2015,43(10):38-43.ZHAI Ruimiao,YING Chao,REN Guochen,et al.An approach for harmonic analysis based on Nuttall window and triple-spectral-line interpolation[J].Power System Protection and Control,2015,43(10):38-43.
[12]曾博,滕召胜,温和,等.莱夫-文森特窗插值FFT谐波分析方法[J].中国电机工程学报,2009,29(10):115-120.ZENG Bo,TENG Zhaosheng,WEN He,et al.Harmonic analysis based on Rife-Vincent window interpolated FFT[J].Proceeding of the CSEE,2009,29(10):115-120.
[13]WEN He,TENG Zhaosheng,GUO Siyu.Triangular self-convolution window with desirable side lobe behaviors for harmonic analysis of power system[J].IEEE Transactions on Instrumentation&Measurement,2010,59(3):543-552.
[14]ZENG Bo.Parameter estimation of power system signals based on cosine self-convolution window with desirable side-lobe behaviors[J].IEEE Transactions on Power Delivery,2011,26(1):250-257.
[15]WEN He,TENG Zhaosheng,GUO Siyu,et al.Hanning self-convolution window and its application to harmonic analysis[J].Sci China Ser E,2009,52(2):467-476.
[16]汪立,刘翠琳.一种高精度加窗插值FFT谐波分析方法[J].陕西电力,2013,41(5):1-4.WANG Li,LIU Cuilin.An accurate method for interpolated FFT harmonic analysis based on a new window[J].Shaanxi Electric Power,2013,41(5):1-4.
[17]王玲,徐柏榆,盛超,等.一种新的余弦组合窗插值FFT谐波分析算法[J].武汉大学学报(工学版),2014,47(2):250-254.WANG Ling,XU Baiyu,SHENG Chao,et al.An approach for harmonic analysis based on a new type of cosine combination window interpolation FFT[J].Engineering Journal of Wuhan University,2014,47(2):250-254.
[18]许珉,刘玮.加8项余弦窗插值FFT算法[J].电力系统保护与控制,2015,43(11):27-32.XU Min,LIU Wei.An interpolation FFT algorithm based on 8-term cosine window[J].Power System Protection and Control,2015,43(11):27-32.
[19]徐柏榆,王玲,汪立,等.多项Rife-Vincent窗插值谐波分析方法[J].武汉大学学报(工学版),2014,47(5):636-641.XU Baiyu,WANG Ling,WANG Li,et al.An approach for harmonic analysis based on multi-term Rife-Vincent window interpolation FFT[J].Engineering Journal of Wuhan University,2014,47(5):636-641.
[20]叶景清,邓辉,年俊华,等.基于矩形卷积窗算法的电子式电流互感器在线校验系统[J].电网与清洁能源,2015,31(11):26-30,35.YE Jingqing,DENG Hui,NIAN Junhua,et al.The on-site calibration system for electronic current transformers based on rectangular convolution window algorithm[J].Power System and Clean Energy,2015,31(11):26-30,35.
[21]AGREZ D.Weighted multipoint interpolated DFT to improve amplitude estimation of multi-frequency signal[J].IEEE Transactions on Instrumentation and Measurement,2002,51(2):287-292.
[22]WU J,ZHAO W.A simple interpolation algorithm for measuring multi-frequency signal based on DFT[J].Measurement,2009,42(2):332-327.
[23]牛胜锁,梁志瑞,张建华,等.基于三谱线插值FFT的电力谐波分析算法[J].中国电机工程学报,2012,32(16):30-136.NIU Shengsuo,LIANG Zhirui,ZHANG Jianhua,et al.An algorithm for electrical harmonic analysis based on triple-spectrum-line interpolation FFT[J].Proceedings of the CSEE,2012,32(16):130-136.
[24]牛胜锁,梁志瑞,张建华,等.基于四项余弦窗三谱线插值FFT的谐波检测方法[J].仪器仪表学报,2012,33(9):2002-2008.NIU Shengsuo,LIANG Zhirui,ZHANG Jianhua,et al.Harmonic detection approach based on 4-term cosine window triple-spectrum-line interpolation FFT[J].Chinese Journal of Scientific Instrument,2012,33(9):2002-2008.
[25]黄冬梅,龚仁喜,焦凤昌,等.莱夫-文森特窗三谱线插值的电力谐波分析[J].电力系统保护与控制,2014,42(2):28-34.HUANG Dongmei,GONG Renxi,JIAO Fengchang,et al.Power harmonic analysis based on Rife-Vincent window and triple-spectral-line interpolation[J].Power System Protection and Control,2014,42(2):28-34.
[26]郝柱,顾伟,褚建新,等.基于四谱线插值FFT的电网谐波检测方法[J].电力系统保护与控制,2014,42(19):107-113.HAO Zhu,GU Wei,CHU Jianxin,et al.A power network harmonic detection method based on the four-spectrumline interpolation FFT[J].Power System Protection and Control,2014,42(19):107-113.
[27]蔡晓峰,张鸿博,鲁改凤.应用三谱线插值FFT分析电力谐波的改进算法[J].电力系统保护与控制,2015,43(2):33-39.CAI Xiaofeng,ZHANG Hongbo,LU Gaifeng.Improvement algorithm for harmonic analysis of power system using triple-spectrum-line interpolation algorithm based on window FFT[J].Power System Protection and Control,2015,43(2):33-39.