频域谐波分析算法的新解释及其推广

New Explanation of Frequency Domain Methods for Harmonic Analysis and Its Generalization

在电力系统谐波分析的频域方法中,加汉宁窗插值算法应用较为广泛,其某些实现具有相似结构,只是选取的插值点数不同。为深入理解其内在本质,文中利用汉宁窗和矩形窗之间的关系,将已有的两种加汉宁窗算法转化成未加窗的等价形式,找出其中规律,并将其推广到通用的未加窗p点插值算法。在推导过程中,首先建立了多分量信号的未加窗离散傅里叶变换(DFT)值的近似模型,接着利用此近似模型分析出算法系数所需满足的两个条件,最后设计了满足条件的一组系数。当插值点数p为4或5时,所提算法等价于已有的两种加汉宁窗算法;p的最小允许值为3,此时算法具有最高的频域分辨率,并且不存在对应的基于汉宁窗的等价形式。结合改进后的DFT递推公式,所提算法所需计算量仅和插值点数p呈线性关系,因此能够大幅降低计算量,并为电网信号提供实时估计。仿真部分比较了算法采用不同插值点数p时的表现,实验部分将算法应用于三相电机电流信号的实时估计。

Among the frequency domain methods for harmonic analysis,Hann-windowed interpolation algorithms are widely applied.Some of the implementations have similar structures,with merely different interpolation point numbers.By using the relationship between Hann and Rectangular windows,this paper transforms two known Hann-windowed algorithms into their equivalent un-windowed forms,finds out their common features and develops a generalized un-windowed p-point interpolation algorithm.In the process of derivation,an approximate model for the un-windowed discrete Fourier transform(DFT)values of multi-component signals is established,and then two conditions the algorithm coefficients should satisfy are obtained,while a set of coefficients are designed.When the interpolation point number pis 4 or 5,the proposed method is equivalent to two known Hann-windowed methods.The proposed algorithm requires at least 3 interpolation points,when it has the best frequency domain resolution and no corresponding equivalent Hann-windowed form.By combining with the improved DFT recursive formula,the computation amount of the algorithm is linearly related to the interpolation point number p,so the proposed method can greatly reduce the computation amount and provide real-time estimation.Different performances of the proposed algorithms with 3to 6interpolation points are compared in the simulation,and the algorithm is applied to the realtime estimation of the current signal of a three-phase motor in the experiment.