基于傅里叶变换的精确频率测量算法.pdf

基于傅里叶变换的精确频率测量算法牟龙华,邢锦磊(同济大学电子与信息工程学院,上海市200092)摘要:传统的傅里叶频率测量算法,通过傅里叶算法求出相邻2个周期的相位,采用相位差对采样频率进行修正和迭代,计算量大而精度差。文中根据严格推导得到傅里叶算法计算值的准确数学形式,通过对相位差的三角函数进行分解展开,代入傅里叶算法计算值,即可在不需要计算相位的情况下得到相邻2个周期相位差的准确值,从而得到真实的信号频率。仿真分析结果表明,该算法精度高,计算量小,实现简单,完全适合于微机保护测控类装置的实际应用。关键词:频率测量;傅里叶变换;相位差中图分类号:TM935收稿日期:2008205219;修回日期:2008209208。0 引言电力系统交流采样普遍采用傅里叶算法,要求采样频率和原始信号严格同步,即采样频率是基波频率的整数倍,否则会产生频谱泄漏,不能正确反映被测信号的各种参数,引起傅里叶算法计算误差。为了减少傅里叶变换的频谱泄漏,提高傅里叶算法的计算精度,频率跟踪技术成为现代微机保护装置必不可少的重要组成部分。目前,频率跟踪测量方法主要可分为硬件法和软件法2类123。硬件法通过滤波整形电路和锁相环实现,需要一定的成本,并且实现较复杂,不适应微机保护装置微型化的发展方向。软件测频方案无需额外硬件电路,实现方式灵活,因而得到了广泛的重视。常用的软件跟踪测频算法有傅里叶算法、最小二乘法、卡尔曼滤波、小波分析法等多种算法429。传统的傅里叶频率测量算法,通过计算得到相邻2个周期的相位10212,再用得到的频率偏差值进行迭代修正,需多次迭代运算,计算量大且测量精度得不到保证。本文在对傅里叶算法进行分析的基础上,介绍了一种精确的傅里叶频率测量算法。该算法能在电力系统频率偏离额定值的情况下自动跟踪频率变化测得其精确值。1 基本原理1.1 信号中仅含基波分量假设输入电压或电流信号是理想的正弦波信号,即仅含有基波分量。令基波的理想频率为f0=50 Hz,对应的理想角频率为0,周期为T0。由于电力系统的实际频率通常在50 Hz上下波动,故可以设基波的实际频率为f=f0+f,对应的实际角频率为,周期为T,f表示频差。输入信号可以表示为:x(t)=Umsin(2f0t+2 f t+0)(1)式中:Um和0分别为输入正弦信号的幅值和初相角。由于在0附近波动,所以在实际角频率未知的情况下,通常采用理想角频率0近似计算,得近似傅里叶变换为:a=2TT0 x(t)sintdt2T0T00 x(t)sin0tdtb=2TT0 x(t)costdt2T0T00 x(t)cos0tdt(2)将式(1)代入式(2),并进一步展开,得展开后的正弦项和余弦项分别为:a=2T0T00Umsin(2f0t+2 f t+0)sin 2f0tdt=2Umf0T0f(2f0+f)cos(f T0+0)sin f T0(3)b=2T0T00Umsin(2f0t+2 f t+0)cos 2f0tdt=2Um(f0+f)T0f(2f0+f)sin(f T0+0)sin f T0(4)相位的表达式如下:=arctanba=arctanf0+ff0sin(f T0+0)cos(f T0+0)(5)76第32卷 第23期2008年12月10日Vol.32No.23Dec.10,2008 进一步考虑在第2个采样周期进行的近似傅里叶变换,则有:a=2T2TTx(t)sintdt2T02T0T0 x(t)sin0tdtb=2T2TTx(t)costdt2T02T0T0 x(t)cos0tdt(6)同理,可以得到式(6)展开后的正弦项和余弦项分别为:a=2Umf0T0f(2f0+f)cos(3 f T0+0)sin f T0(7)b=2Um(f0+f)T0f(2f0+f)sin(3 f T0+0)sin f T0(8)根据第2个采样周期得到的相位 为:=arctanba=arctanf0+ff0sin(3 f T0+0)cos(3 f T0+0)(9)分别考察式(3)与式(7)、式(4)与式(8),可以得到:A=aa=cos(f T0+0)cos(3 f T0+0)(10)B=bb=sin(f T0+0)sin(3 f T0+0)(11)对式(10)和式(11)进一步展开分析,得A=1cos 2 f T0-sin 2 f T0sin(f T0+0)cos(f T0+0)(12)B=1cos 2 f T0+sin 2 f T0cos(f T0+0)sin(f T0+0)(13)消去式(12)与式(13)中的sin(f T0+0)/cos(f T0+0)项,化简得:cos 2 f T0=AB+1A+B=ab+abab+ab(14)由此得到傅里叶频率测量算法的基本公式如下:|f|=12T0arccosab+abab+abf=f0|f|(15)在实际应用中,一般采用C语言作为开发工具,可以直接采用C语言库函数中提供的反余弦函数,具有相当高的计算精度。为减少计算量,也可采用如下的近似计算方法13:arccosx=1.998 26X12-0.641 06X32+0.289 96X52-0.076 5X72式中:X=(1-x)/(1+x)。采用该式计算时,其计算误差值约为10-4。1.2 符号选取考虑到余弦函数是偶函数,故式(15)中f取正还是取负将取决于角度2 f T0的符号。分析式(3)得:a=2Umf0T0(2f0+f)sin f T0fcos(f T0+0)令f的变化范围为-12.5 Hz12.5 Hz,该区间已足够满足实际的测频应用要求,则2 f T0的取值范围为(-/2,/2),且满足:2Umf0T0(2f0+f)0sin f T0f0(16)因此,a的符号完全取决于cos(f T0+0);同理,b的符号取决于sin(f T0+0),a 的符号取决 于cos(3 f T0+0),b的 符 号 取 决 于sin(3 f T0+0)。为此,可首先根据a和b的符号,判断 f T0+0所在象限,记为P;然后,根据a 和b 的符号,判断3 f T0+0所在象限,记为Q。显然P和Q将在相同或相邻象限。若 f T0+0和3 f T0+0这2个角度在相同象限,即P=Q,显然可以通过三角函数的增减特性判断角度2 f T0的符号,具体判据如下:1)P和Q为第1或第2象限,余弦函数满足递减特性,若aa,则2 f T0为正;反之为负。2)P和Q为第3或第4象限,余弦函数满足递增特性,若aa,则2 f T0为负;反之为正。若这2个角度在不同象限,具体判据如下:1)若象限P与象限Q相邻,且Q=P+1或Q=P-3,则2 f T0为正。2)若象限P与象限Q相邻,且Q=P-1或Q=P+3,则2 f T0为负。由于式(15)并不需要计算相邻2个周期的相位以得到准确的相位差值,故无需根据频率偏差值对算法进行迭代修正,即可求得系统的真实频率f,显然优于根据相角变化值求频率的方法。1.3 信号中含谐波分量若傅里叶算法以额定频率50 Hz为基础,采样频率固定为N50 Hz(N为每周期采样点数),当频率发生偏移时,谐波频率必然也发生偏移,在此情况下傅里叶算法不能很好地抑制谐波分量,从而可能导致测频误差的明显增大1。862008,32(23)若傅里叶算法基于系统真实频率,采样频率以上一次计算频率为基础进行自适应调整,采样间隔得到修正,则可得到更佳的滤波效果,减少傅里叶变换的误差。实现采样频率自适应调整的测频流程如图1所示,通过循环迭代计算,以软件方法实现频率跟踪、锁相。其中,为迭代计算中设定的频率差门槛值。由式(2)、式(6)可知,迭代1次需2个周期的采样数据。图1 测频流程Fig.1Flowchart of frequency measurement迭代计算的关键是迭代次数与迭代算法的收敛性,考虑到电网频率偏移通常不会很大,为加快迭代计算速度,推荐将迭代频率初值设为额定值50 Hz。由于初始基准频率选为50 Hz,刚开始采样计算时若实际频率偏差较大则存在误差,但后续循环迭代计算可很快减小此误差,一般迭代2次后即可满足精度要求。随着迭代次数的增加,测频的精度也将大大提高。大量仿真计算发现,以上迭代算法不存在发散现象,并且迭代次数在3次以上时,频率精度的提高将十分有限,因此实际应用时可以考虑将迭代次数限制在4次5次。2 算法仿真与分析为验证本文所提出的算法的测频效果,对不同输入信号情况下算法的性能进行了仿真。式(2)和式(6)的积分采用梯形法近似计算,各仿真中迭代频率初值设为50 Hz,每周期采样点数N=24。当测量频率与上次测量频率之差 0.01 Hz,则停止迭代,并将该频率作为真实值;若迭代次数大于迭代上限,则视最后一个测量频率为真实值。2.1 不含谐波时的情况当输入信号为纯正弦波时,设信号为x(t)=sin(2f t+),其频率变化范围为38 Hz62 Hz,初相角随机选择,利用式(15)进行仿真计算。=/6时,频率测量结果如表1所示。表1 不含谐波时频率测量算法仿真结果Table 1Simulation results of frequency measurement for asignal without harmonics实际频率/Hz测量频率/Hz绝对误差/Hz相对误差/(%)计算次数6262.000 10.000 10.000 1615857.999 90.000 10.000 1715555.000 00015151.000 00015050.000 00014949.000 00014545.000 00014242.000 10.000 10.000 2513838.000 10.000 10.000 261表1数据表明,对于纯正弦信号,测频精度可达到0.000 1 Hz,算法的数据窗为2个基波周期,能够满足频率跟踪的要求。2.2 含谐波时的情况为考察含谐波时算法的性能,设输入信号为:x(t)=sin(2f t+1)+0.05sin(22f t+2)+0.2sin(32f t+3)+0.1其频率变化范围为38 Hz62 Hz,初相角随机选择。1=/6,2=/3,3=/2时,频率测量结果如表2所示。表2 含有谐波时频率测量算法仿真结果Table 2Simulation results of frequency measurementfor a signal with harmonics实际频率/Hz1次计算测量频率/Hz2次计算测量频率/Hz3次计算测量频率/Hz3次计算后绝对误差/Hz6261.922 962.000 162.000 005855.244 858.015 058.000 005554.568 655.002 855.000 005151.012 951.000 051.000 005050.000 050.000 050.000 004949.022 449.000 049.000 004545.435 945.004 445.000 004040.429 540.004 840.000 003838.031 338.001 238.000 00可见,在含谐波的情况下,只需3次迭代计算,即可实现对真实频率的精确跟踪,算法本身不存在误差。2.3 含谐波与噪声时的情况为进一步考察含谐波与噪声条件下算法的性能,设输入信号为:x(t)=sin(2f t+1)+0.05sin(22f t+2)+0.2sin(32f t+3)+0.1+其中,为40 dB的零均值高斯白噪声。测量频率变化范围为45 Hz55 Hz,初相角随机选择。1=96 研制与开发 牟龙华,等 基于傅里叶变换的精确频率测量算法/6,2=/3,3=/2时,频率测量结果如表3所示。表3 含有谐波及噪声时频率测量算法仿真结果Table 3Simulation results of frequency measurement for asignal with harmonics and noise实际频率/Hz测量频率/Hz绝对误差/Hz相对误差/(%)计算次数5555.008 40.008 40.015 2745352.999 90.000 10.000 1935150.992 10.007 90.015 4935050.000 00014949.006 50.006 50.013 2724747.007 20.007 20.015 3234544.992 50.007 50.016 675显然,即使在含谐波与噪声的情况下,测频误差的绝对值仍低于0.01 Hz,而迭代次数在5次以内,算法仍具有较高的精度。考虑到实际电力系统中的噪声通常在50 dB70 dB之间12,故此时测频算法受到的影响将更小。3 结语本文提出的基于傅里叶算法的测频新方法,无需计算相邻2个周期相位的准确值,即可得到真实的信号频率。该算法原理简单,不存在理论误差,迭代次数少;传统的傅里叶频率测量算法只能适用于实际频率在50 Hz附近小范围内波动的场合,而本文提出的算法在较大频率波动范围内皆可保证精确的频率测量,且即使存在谐波与噪声成分,仍可达到很高的测量精度。实际应用中,仅一次计算就可以达到相当高的测量精度,并且该算法的微机实现简单,完全适用于电网频率的微机实时测量要求。参 考 文 献1李一泉,何奔腾.一种基于傅氏算法的高精度测频方法.中国电机工程学报,2006,26(2):78281.LIYiquan,HE Benteng.Ahigh2accuracyalgorithm formeasuring frequency of power system based on Fourier filter.Proceedings of the CSEE,2006,26(2):78281.2孙莉.电力系统软件测频的改进方法.继电器,2006,34(19):38241.SUN Li.An improved method of frequency measurement forpower system.Relay,2006,34(19):38241.3张瑛,牟龙华,刘军.电力系统频率测量及跟踪.电力系统及其自动化学报,2003,15(3):35236.ZHANG Ying,MU Longhua,LIU Jun.Frequency measuringand tracking of power systems.Proceedings of the EPSA,2003,15(3):35236.4谢小荣.韩英铎.电力系统频率测量综述.电力系统自动化,1999,23(3):54258.XIE Xiaorong,HAN Yingduo.An overview on power systemfrequencymeasurement.AutomationofElectricPowerSystems,1999,23(3):54258.5赵成勇,胥国毅,何明锋,等.基于改进递归小波的电力系统频率测量.电工技术学报,2005,20(6):62265,76.ZHAO Chengyong,XU Guoyi,HE Mingfeng,et al.Powersystem frequency measurement based on IRWT.Transactions ofChina Electrotechnical Society,2005,20(6):62265,76.6罗谌持,张明.基于Sigma点卡尔曼滤波器的电力频率跟踪新算法.电力系统自动化,2008,32(13):35239.LUO Chenchi,ZHANG Ming.Frequency tracking of distortedpowersignalusingcomplexsigmapointKalmanfilter.Automation of Electric Power Systems,2008,32(13):35239.7江亚群,何怡刚.基于自适应短时傅立叶变换的电频率跟踪测量算法.电子测量与仪器学报,2006,20(2):10215.JIANGYaqun,HEYigang.Newdigitalalgorithmformeasuring electric frequencybased onadaptiveshort2timeFourier transform.Journal ofElectronicMeasurementandInstrument,2006,20(2):10215.8 AGHAZADEH R,LESANI H,SANAYE2PASAND M,et al.New technique for frequency and amplitude estimation of powersystem signals.IEE Proceedings:Generation,Transmission andDistribution,2005,152(3):4352440.9陈平,李庆民,张黎.电网瞬时频率的一种跟踪算法.电力系统自动化,2007,31(1):80284.CHEN Ping,LI Qingmin,ZHANG Li.An algorithm fortracking instantaneous frequency of power grid.Automation ofElectric Power Systems,2007,31(1):80284.10磨少清,李啸骢.一种高精度的改进傅里叶测频方法.电力系统自动化,2003,27(12):48249,54.MO Shaoqing,LI Xiaocong.An improvedhigh accuracyalgorithmforfrequencymeasurementbasedonFouriertransform.Automation ofElectric Power Systems,2003,27(12):48249,54.11张同尊,邵俊松,方勇杰.一种基于离散傅里叶变换的频率测量算法.电力系统自动化,2007,31(22):70272.ZHANG Tongzun,SHAO Junsong,FANG Yongjie.AnalgorithmforfrequencymeasurementbasedonDFT.Automation of Electric Power Systems,2007,31(22):70272.12 SIDHUTS.Accuratemeasurementofpowersystemfrequency using a digital signal processing technique.IEEETrans on Instrumentation and Measurement,1999,48(1):75281.13姜路,李耀华,齐智平,等.伺服系统中复杂函数的处理.电工电能新技术,2000,19(1):125.JIANG Lu,LI Yaohua,QI Zhiping,et al.Processing ofcomplex functions in servo motor control system.AdvancedTechnology ofElectricalEngineeringandEnergy,2000,19(1):125.牟龙华(1963),男,通信作者,教授,博士生导师,主要研究方向:电力系统微机保护、智能电器与电能质量。E2mail:lhmu 邢锦磊(1983),男,硕士研究生,主要研究方向:电力系统微机保护与嵌入式系统开发。(下转第94页 continued on page 94)072008,32(23)An OPC Technology Based SCADA System Design for Wind Power PlantsLU Xiaojun1,DON G Dalong2,SON G Bin1,TA N G Chenghong1(1.Shenzhen NARI Technologies Co.Ltd.,Shenzhen 518057,China;2.Tangshan Power Supply Company,Tangshan 063000,China)Abstract:In view of the coexistence of multiple control systems in wind power plant,this paper introduces an integration of thewind motor control system and electric control system for wind power plants based on the OPC technique and the characteristicsof SCADA system.Thus,the interconnection problem of multiple control systems is solved.At the same time,the issue onnetwork security is discussed and some security strategies and measures are proposed.Finally,the advantages of thissupervisory control scheme in the electric power system are also analyzed.Key words:wind power plant;object linking and embedding for process control(OPC);supervisory control and dataacquisition(SCADA)system;network security(上接第50页 continued from page 50)Left2inversion Soft2sensing Method for the Directly Immeasurable Variables in the Synchronous GeneratorZ HA N G Kaif eng,DA I Xianzhong,MA Chao,Z HA N G Ga(Key Laboratory of Measurement and Control of CSE of Ministry of Education,School of Automation,Southeast University,Nanjing 210096,China)Abstract:The problem of soft2sensing of directly immeasurable variables in the synchronous generator,such as the powerangle,thedandqaxis transient electromotive forces,thedandqaxis stator circuit currents,is investigated.Firstly,themodel of synchronous generator suitable for soft2sensing is proposed,which includes the directly measurable variables ofgenerator,such as the active and reactive powers,the amplitude of current,etc.The proposed model is a type of nonlineardifferential2algebraic sub2systems.Secondly,aiming at the specialty of the proposed model,an expanded left2inversion soft2sensing algorithm is proposed for the soft2sensing of synchronous generator.Then,a left2inversion soft2sensor is designedconsidering the 42order transient practical model of synchronous generator,and all directly immeasurable variables in the modelcan be soft2sensed.Finally,simulations are performed using MATLAB/SimPowerSystems,in which a 62order sub2transientpractical model of generator is used,and the results demonstrate the validity of the proposed method.This work is supported by National Natural Science Foundation of China(No.50507002,No.60574097).Key words:power system;soft2sense;synchronous generator;directly immeasurable variable;practical model(上接第70页 continued from page 70)An Accurate Frequency Measuring Algorithm for Power Systems Based on Fourier TransformMU Longhua,X IN G J inlei(Tongji University,Shanghai 200092,China)Abstract:In the conventional frequency measuring algorithm,first the phases of two adjacent periods are obtained using Fouriertransform,then the phase difference is used to perform modification and iteration for the sampling frequency.This algorithmasks for a lot of calculation but is inaccurate.Based on strict mathematical deduction,this paper presents an accuratemathematical expression using Fourier transform,which is composed of the cosine function for the phase difference.Then aftertrigonometric function decomposition,the cosine function for the phase difference can be expressed with the Fourier transformresults.Hence,the real signal frequency can be obtained without calculating the precise phases of the two periods.Simulationresults show that the new algorithm is suitable for application in microcomputer2based monitoring and protective devices for itshigh accuracy,low calculating workload and easy implementation.Key words:frequency measurement;Fourier transform;phase difference492008,32(23)