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1、Harmonic PhasorAnalysisBased on ImprovedFFTAlgorithmBo Zeng, Zhaosheng Teng, Yulian Cai, Siyu Guo, and Baiyuan QingI. INTRODUCTIONNONLINEAR loads can introduce large harmonic current into power systems, which may lead to severe problems(e.g., meter malfunctions, equipment overheat, overvoltage, and

2、data loss) 1. In researches on eliminating or at least reducing the impacts of harmonics on power systems, harmonic phasor analysis has been one of the most vital problems that attract most attentions.译文：电力系统中由于非线性负荷的存在产生大量的谐波电流，导致出现例如仪表故障、设备过热、 过压和数据丢失等严重问题 1。在研究消除或至少减少谐波对电力系统的影响时，谐波相量分析已经成为较重要和热门的

3、方法。分析：第一层简述研究领域，确定研究对象电力系统谐波(harmonic of power systems )及研究的主流研究方法一谐波相量分析(harmonic phasor analysis).The existing harmonic phasor analysis methods utilize a variety of techniques, such as the least square algorithms 2, Kalman filter 3, artificial neural network 4, NewtonProny m ethod 6, and state est

4、imation 7. However, when real-time performances are required, these methods do not give satisfactory outcome, and in these sit- uations, the fast Fourier transform (FFT)-based methods are preferable for its availability, understandability, simplicity, and easiness to implement in DSP and advanced RI

5、SCmachineschips. Unfortunately, the fundamental frequency of a power system may vary, and fixed sampling rates are typical for most data acquisition systems 8. Though a number of sampling synchronization methods, such as the adoption of discrete phase-locked loop 9 or adjustable sampling frequency 1

6、0, have been proposed, synchronous sampling is still difficult to achieve.译文:现有的谐波相量分析理论有：最小二乘法2、卡尔曼滤波2、人工神经网络 (ANNs)4、Newton法5、Prony法6及状态估计法7。但是，对于动态信号这些方法难以有满意的效果，在此情况 下，快速傅里叶变换(FFT)因其有效性，易懂，DSP及RISC芯片的轻松实现，得到广泛应用。然而，由于电力系统基波频率实时变化，固定的采样率仅对大多数数据采集系统具有典型性8。尽管已经提出许多同步采样方法，如使用采用离散锁相环技术9,或修正采样频率法10,同步

7、采样仍然很难实现。分析：第一层，综述现有谐波相量分析方法理论，描述解决问题的方法一一快速傅里叶变换(FFT)。The FFT approaches under asynchronous sampling suffer from two serious drawbacks 11, namely, the spectral leakagedue to time limitation and the picket fence effect due to the frequency discretization of the calculated spectrum. As a con- sequenc

8、e,the harmonic phasor of a signal cannot be obtained accurately. The common strategy to cope with these drawbacks is the windowing of the signal sequence for reducing the spectral leakage 12 and spectrum interpolation for reducing the picket fence effect 13, 14.译文：在非同步采样条件下，FFT算法有两个严重的缺点11,即数据截断引起频谱

9、泄漏，频谱离散 化造成栅栏效应，最终导致不能准确获得谐波相量。解决这些缺点的最常见的方法是对信号序列加窗函数 和差值来减少频谱泄露和栅栏效应1314。分析：第一层陈述目前FFT算法存在的不足及研究的关键点一一减少频谱泄露和栅栏效应。Windows with great sidelobe attenuation and high sidelobe decaying rate can sufficiently reduce the spectral leakage and the harmonic interference 15. To achieve these desirable proper

10、ties, different windows have been defined by adjusting coefficients of classical windows 16 -19 or by convoluting parent windows 15, 20, and have been used in replacement of the rectangular window. Although the design of windows is thought to be a quite matured research area, new methods are still e

11、merging, e.g., the new class of adjustable windows based on the cosine hyperbolic function proposed in 21 and the novel method for window parameterization in the frequency domain presented in 22. Nuttall 23 categorized classical windows according to optimal sidelobe behaviors and intro- duced the fa

12、mily of optimal cosine-type windows.To reduce the spectral error caused by the picket fence effect, windowed interpolation algorithms 12 -15 are employed. Among these algorithms, the multipoint interpolation discrete Fourier transform (DFT) algorithms 24 -29 are widely applied to promote the harmoni

13、c phasor analysis accuracy, i.e., the measurementaccuracy of the harmonic frequencies, am- plitudes, and phases, or as called, the phasors. Unfortunately, for complicated windows such as the high-order combined cosine-type windows, the rectification is computationally ex- pensive due to the solution

14、 of high-order equations. Approaches have been proposed to deal with the problem, e.g., Yang et al.30 proposed an accurate phasor estimation algorithm by using an FIR comb filter, and in 31, a method for exact calculation of harmonics using adaptive window width is presented.译文：窗函数的旁瓣衰减和快速衰减特性可有效减少频

15、谱泄露和间谐波15。为了获得满意的特性，不同的适应经典窗系数的窗函数16-19或卷积窗函数1520被提出且已经取代了矩形窗。尽管窗函数已经是一个相当成熟的研究领域，新的理论依旧不断涌现，如在21基于余弦双曲线函数的可调窗函数，在22提出在频域中窗参数新方法，纳托尔在23根据最优窗旁瓣的行为将古典窗分类，并介绍了最优的余弦窗函数族。为了减少栅栏效应所造成的的频谱误差，引入的加窗插值的方法12-15。在这些方法中，多点插值离散傅里叶变换(DFT)算法24 129被广泛应用于提高谐波相量分析的精度，如测量谐波频率的参数，振幅， 相位，即相量。然而，对于复杂的窗函数，例如高阶的余弦型窗，为了计算高阶方

16、程的解，这种改进方法 是计算复杂度很高的。 Yang et al.在文献30中通过使用FIR滤波器提出了一个准确的参数估计算法，在 31中提出变矩形窗宽度的自适应谐波分析算法。分析：第二层针对目前FFT算法出现的两个突出问题，对现有研究进行文献综述。However, variations of the fundamental frequency are usually caused by faults in the power systems, which may leads to the uncertainty of the signal harmonics, and the phasor

17、analysis of arbitrary harmonics under frequency variation is still an openproblem. Besides, interharmonics may exist in the measured signal. Accordingly, a frequency domain approach for power system harmonic phasor analysis under asynchronous sampling is thus proposed and discussed in this paper.译文：

18、基波频率的变化通常是由电力系统故障引起的，这可能会导致该信号的谐波的不确定性，任意 变频谐波的相量分析仍然是一个开放性问题。而且，间谐波也会存在于测量信号中。因此，本文提出并分 析在非同步采样条件下，频域内的电力系统谐波相量分析问题。分析：第三层：提出目前尚未解决的问题，引出本研究动机。In this approach, the four-term fifth derivative Nuttall (FFDN) window, which ex-hibits better sidelobe behaviors compared with classic windows, is chosen t

19、o truncate the harmonic signal. For applications such as weak harmonic analysis of power systems where the spectral leakage appears as the major difficulty, better performance can be achieved by the FFDN. Based on the FFDN, an improved FFT (IFFT) algorithm is presented for calculating the power syst

20、em signal phasors. The proposed method has the major ad- vantages that it has shorter computation time and can get exact solutions, where white noise and interharmonics are present, frequency deviates from nominal frequency, and some harmonic components are extremely weak. The application of the new

21、 algorithm in a type of smart meter demonstrates good perfor-mance, suggesting that it be a preferable choice for real-world applications.译文:在此法中，和经典窗函数相比具有更好的旁瓣效应的4项5阶Nuttall窗(FFDN)被用来获取谐波信号。使用FFDN可以实现弱化电力系统谐波分析中的频率泄露问题。本文提出一种基于FFDN窗的改进FFT (IFFT)算法，此法用于电力系统谐波信号相量分析。这种方法的主要优点是在白噪声和间谐波存在频 率偏离标称频率时，一些

22、谐波组件极度疲弱的情况下缩短计算时间，获得准确结果，这种应用于智能电表 的新算法预示它将在实际运算会是一个理想选择。分析：第四层：介绍本研究，并概括研究的优点或者意义所在。The remainder of the paper is organized as follows. Section II gives a brief review of the spectral leakage of FFT due to asyn-chronous sampling. Spectral characteristics of the Nuttall windows are analyzed in Sect

23、ion III. The FFDN-based IFFT algo-rithm is presented in Section IV, and in Section V are simula-tion results. Section VI shows the application result of the pro-posed method in a smart meter. Finally, conclusions are drawn in Section VII.译文：本文的其余部分组织如下。第二节简要回顾由于非同步采样FFT的频谱泄漏现象。第三节分析了 Nuttall窗的频率特性，第

24、四节提出基于IFFT的FFDN算法，第五节为仿真结果，第六节做出此法在智能电表中的应用效果，第七节得出结论。分析：第四层：简述研究每部分章节划分及研究内容。基于改进FFTM法的谐波相量分析I引言电力系统中由于非线性负荷的存在产生大量的谐波电流，导致出现例如仪表故障、设备 过热、过压和数据丢失等严重问题1。在研究消除或至少减少谐波对电力系统的影响时，谐 波相量分析已经成为较重要和热门的方法。现有的谐波相量分析理论有：最小二乘法2、卡尔曼滤波2、人工神经网络（ANNs）4、Newton法5、Prony法6及状态估计法7。但是， 对于动态信号这些方法难以有满意的效果，在此情况下，快速傅里叶变换（FF

25、T）因其有效性，易懂，DSP及RISC芯片的轻松实现，得到广泛应用。然而，由于电力系统基波频率实时 变化，固定的采样率仅对大多数数据采集系统具有典型性8。尽管已经提出许多同步采样方法，如使用采用离散锁相环技术9,或修正采样频率法10,同步采样仍然很难实现。在非同步采样条件下，FFT算法有两个严重的缺点11,即数据截断引起频谱泄漏，频 谱离散化造成栅栏效应，最终导致不能准确获得谐波相量。解决这些缺点的最常见的方法是 对信号序列加窗函数和差值来减少频谱泄露和栅栏效应1314。窗函数的旁瓣衰减和快速衰减特性可有效减少频谱泄露和问谐波15。为了获得满意的特性，不同的适应经典窗系数的窗函数16-19或卷

26、积窗函数1520被提出且已经取代了矩形 窗。尽管窗函数已经是一个相当成熟的研究领域，新的理论依旧不断涌现，如在21基于余弦双曲线函数的可调窗函数，在22提出在频域中窗参数新方法，纳托尔在23根据最优窗旁 瓣的行为将古典窗分类，并介绍了最优的余弦窗函数族。为了减少栅栏效应所造成的的频谱误差，引入的加窗插值的方法 12-15。在这些方法 中，多点插值离散傅里叶变换（DFT）算法24 J29被广泛应用于提高谐波相量分析的精度，如 测量谐波频率的参数，振幅，相位，即相量。然而，对于复杂的窗函数，例如高阶的余弦型 窗,为了计算高阶方程的解，这种改进方法是计算复杂度很高的。Yang et al.在文献30

27、中通过使用FIR滤波器提出了一个准确的参数估计算法，在31中提出变矩形窗宽度的自适应谐波 分析算法。基波频率的变化通常是由电力系统故障引起的，这可能会导致该信号的谐波的不确定性， 任意变频谐波的相量分析仍然是一个开放性问题。而且，问谐波也会存在于测量信号中。因 此，本文提出并分析在非同步采样条件下，频域内的电力系统谐波相量分析问题。在此法中， 和经典窗函数相比具有更好的旁瓣效应的 4项5阶Nuttall窗（FFDN）被用来获取谐波信号。使用 FFDN可以实现弱化电力系统谐波分析中的频率泄露问题。本文提出一种基于FFDN窗的改进FFT （IFFT）算法，此法用于电力系统谐波信号相量分析。这种方法的主要优点是在白噪声和 问谐波存在频率偏离标称频率时，一些谐波组件极度疲弱的情况下缩短计算时间，获得准确 结果，这种应用于智能电表的新算法预示它将在实际运算会是一个理想选择。本文的其余部分组织如下。第二节简要回顾由于非同步采样 FFT的频谱泄漏现象。第三 节分析了 Nuttall窗的频率特性，第四节提出基于IFFT的FFDN算法，第五节为仿真结果，第 六节做出此法在智能电表中的应用效果，第七节得出结论。