基于FFT和神经网络的非整数次谐波分析改进算法_英文_王公宝

第28卷第4期中国电机工程学报V ol.28 No.4 Feb.5, 2008
102 2008年2月5日 Proceedings of the CSEE ©2008 Chin.Soc.for Elec.Eng. 文章编号：0258-8013 (2008) 04-0102-07 中图分类号：TM 935 文献标识码：A 学科分类号：470⋅40
基于FFT和神经网络的非整数次谐波分析改进算法
王公宝，向东阳，马伟明
（海军工程大学，湖北省武汉市430033）
Improved Algorithm for Non-integer Harmonics Analysis
Based on FFT Algorithm and Neural Network
WANG Gong-bao, XIANG Dong-yang, MA Wei-ming
(Naval University of Engineering, Wuhan 430033, Hubei Province , China )
ABSTRACT: By using an artificial neural network (ANN) model, high measurement accuracy of integer harmonics can be obtained. Combining the windowed fast Fourier transform (FFT) algorithm with the improved ANN model, the paper provides an improved algorithm for analysis of non-integer harmonics in electric power systems. Firstly, the Hanning- windowed FFT algorithm processes the sampled signal. By this time, the number of harmonics and the orders of harmonics are obtained. Secondly, choose the number of neural nodes according to the number of harmonics. Thirdly, choose the initial values of orders of harmonics according to the result obtained from the Hanning-windowed FFT algorithm. Moreover, an adaptive algorithm for the adjusting step of the order of harmonic is presented. Finally, by using the improved linear ANN model obtained in the paper, non-integer harmonics can be detected precisely. Through such processing, the time of iterations is shortened and the convergence rate of neural network is raised thereby. The simulation results show that close non-integer harmonics can be separated from a signal with higher accuracy and better real-time by using the improved algorithm presented.
KEY WORDS: power systems; fast Fourier transform; artificial neural network; Hanning-window; harmonics analysis
摘要：运用神经网络模型进行整数次谐波检测可达到较高的检测精度，但该种线性神经元模型不适合非整数次谐波的检测。

为精确检测非整数次谐波，该文提出一种改进的线性人工神经元模型，并将加汉宁窗的FFT算法和改进的线性人工神经元模型结合起来，提出一种改进的非整数次谐波分析算法。

首先，对采样信号用加汉宁窗的FFT算法进行预处理，得到谐波个数和精度不高的谐波次数；其次，根据谐波个数设定神经元的个数，根据预处理后得到的谐波次数设
基金项目：高等学校优秀青年教师教学科研奖励计划基金项目(教
育部人事司2001-182)。

定神经网络谐波次数迭代的初始值；为了提高迭代速度，提
出了谐波次数迭代步长自适应调整的算法。

最后对改进后的
人工神经网络进行训练，实现了非整数次谐波的精确检测。

仿真实例表明，该方法能将频率相近的非整数次谐波分离，可有效提高谐波参数的检测精度和速度。

关键词：电力系统；快速傅里叶变换；人工神经网络；汉宁窗；谐波分析
0 INTRODUCTION
With the extensive applications of non-linear loads in power systems, harmonics in electric network are becoming more complicated. There are not only integer harmonics but also many non-integer harmonics[1-2]. Harmonics are harmful to the normal operation of power equipment. So we must manage to reduce the effects of harmonics. The accurate detection and analysis of harmonics is a prior step for realizing the compensation of harmonics.
The precise measurement for integer harmonics can be realized by using the Fast Fourier Transform (FFT) algorithm provided that the periodic signal is synchronous sampling. But the FFT algorithm is not fit for non-integer harmonics analysis due to picket fence effects and spectrum leakage. And the FFT algorithm is not also suitable to be used for integer harmonics analysis when the periodic signal is not synchronous sampling because of the same reasons mentioned above. The spectrum leakage error of the FFT algorithm is analyzed in Ref. [3-4] and the interpolated FFT algorithm with windows is presented. The algorithm is used for integer harmonics analysis
第4期 王公宝等： 基于FFT 和神经网络的非整数次谐波分析改进算法 103
and the measurement accuracy is improved. In Ref.
[5-6], the spectrum leakage is analyzed and an improved algorithm is provided by means of simple transform from the results of the FFT algorithm. This algorithm can diminish the spectrum interference of non-integer harmonics and high measurement accuracy is obtained. In recent years, with the development of the artificial intelligence technology, linear artificial neural network (Adaline ANN) has been applied to integer harmonics analysis in power systems [7-9]. In Ref. [7], the high measurement accuracy of integer harmonics parameters is obtained by use of the FFT–Adaline algorithm. In Ref. [10-11], an improved FFT–Adaline algorithm is presented and applied to non-integer harmonics analysis. In Ref. [12-14], wavelet transform is used to detect non-integer harmonics or integer harmonics. In Ref. [15-16], the authors apply interpolation FFT algorithm to the detection of inter- harmonics.
For non-integer harmonics analysis, the true parameters of harmonics can’t be obtained by the FFT algorithm due to picket fence effects and spectrum leakage. But a maximum will appear near the spectral lines of each frequency component of the original signal in the corresponding spectrum chart.
For example, the spectrum distribution of 1k k r =+th harmonic (1k is an integer,01r <<) doesn’t concentrate on a single spectral line, but distribute in the whole frequency domain. The amplitude of 1n k =or 11n k =+th spectral line is maximal. With the increase of 1n k r −−, the corresponding amplitude of n th spectral line is descending [5].
The spectrum leakage will result in the spectrum interference of non-integer harmonics. As a result, it will reduce the measurement accuracy of harmonics. Here we take the following signal for example. Suppose that a signal comprises 1.4th, 2.6th and 3.8th harmonics. The length of sampled data is 64 within 4 fundamental periods. We process the sampled signal by using the Hanning-windowed 64-point FFT algorithm [3]. The result is shown in Fig.1.
As shown in Fig.1, the spectral lines have three maxima, which indicates that the signal includes three
kinds of harmonics, i.e. 1.5th, 2.5th and 3.75th harmonics. It can be seen that the result is imprecise, but the number of harmonics is detected.
In this paper, we combine the Hanning- windowed FFT algorithm with the improved linear ANN model and present an improved algorithm for non-integer harmonics analysis. The simulation results show that the improved algorithm presented in this paper can separate close non-integer harmonics from a signal with higher accuracy and better real-time.
0 2 4 6 8
The order of harmonics 0
0.51.01.52.0A m p l i t u d e /p u
图1 信号加汉宁窗FFT 变换频谱特性
Fig. 1 Amplitude spectrum obtained by the
Hanning-windowed FFT algorithm
1 IMPROVED LINEAR ANN MODEL
Suppose that a periodic signal comprising harmonics is expressed as:
1
()sin()m
i i i x t A i t ωϕ==+∑ (1)
where 2f ω=π. Then we have
1()[cos sin()sin cos()]m
i i i i i x t A i t A i t ϕωϕω==+∑ (2)
where f is the frequency of the fundamental component, i is the order of harmonic, A i is the amplitude of the i th harmonic, ϕi is the phase of the i th harmonic, and m is the maximal order of harmonics.
The conventional Adaline ANN model for integer harmonics analysis is shown in Fig. 2 (real line segments diagram) [7].
The neuron functions can be expressed as follows:
()[sin(),cos(),sin(2),t t t t φωωω=
cos(2),...,sin(),cos()]t m t m t ωωω (3)
Let ˆ()()(),i i e i x t x t =−where ()i x t is the
sampling value of signal at i t , and ˆ()i x
t is the
104 中 国 电 机 工 程 学 报 第28卷
output of Adaline ANN. Take ()e i as the control condition of measurement accuracy. After the training process is accomplished, the output of the network is the weight vector: 1111{cos ,sin ,...,cos ,m m A A A ϕϕϕ
sin }.m m A ϕ
图2 自适应线性人工神经元模型 Fig. 2 Adaline ANN model
Since the neuron functions of the conventional
Adaline-ANN model are definite, the algorithm is fit only for integer harmonics analysis. The algorithm isn’t fit for non-integer harmonics analysis because the orders of harmonics are unknown. If a signal comprising non-integer harmonics is analyzed by this algorithm, the result will not be reliable.
In order to detect non-integer harmonics, we provide the following improved Adaline ANN model. In the new model, the neuron functions aren’t definite. Speaking specifically, we will take the orders of harmonics as the training object like network weights. Thus we get a new network model for non-integer harmonics analysis shown in Fig. 2 (adding broken line segments).
The operation of the new network model is as follows:
The inputs are ,
1,2,...,i t i k =, where k is the length of sampled data.
The number of neurons is 2h . The neuron functions are
112()[sin(),cos(),sin(),t a t a t a t φωωω= 2cos(),...,sin(),cos()]h h a t a t a t ωωω
where l a is the order of harmonic ).,...,2,1(h l =
The outputs are
21
1
ˆ()()[cos sin()h
h
i j j i l l l i j l x
t w t A a t φϕω====+∑∑ sin cos()]l l l i A a t ϕω (4)
where .,...,2,1k i =
Let ˆ()()()i i e i x t x
t =− (5) where )(i t x is the sampling value of signal at i t (i =1,2,…,k ).
The error index is
21
()2
()e i V i =, 1,2,...,i k =
2
1
1
1()2()k
k i i e i V V i ====
∑∑ For the learning algorithm of the network, we adopt the basic inertia algorithm [17].
Let 001
1()()(1)()()(1)j j j l l l V i w i w i w V i a i a i a αηαη∂⎧
Δ=−+Δ−⎪∂⎪
⎨∂⎪Δ=−+Δ−⎪∂⎩ (6)
where i =1,2,…,k ; j =1,…,2h ; l =1,…,h ; Δw j (0)=0,
(0)0l a Δ=; 01010, 0; 10, 10.ααηη>>>>>>0α is the adjusting step of weights, and 1α is the adjusting step of harmonic order. 0η and 1η are the inertia coefficients. In the formula (6), ()/j V i w ∂∂,
()/l V i a ∂∂represent the partial derivatives of ()V i with respect to w j and a l respectively. Then w j (i +1) = w j (i )+Δw j (i ), (
1)()()l l l a i a i a i +=+Δ. After the training process is accomplished, the outputs of the network are the weight vector and the order vector of harmonics:
1111{cos ,sin ,...,cos ,sin }h h h h w A A A A ϕϕϕϕ=
12[,,...,]h a a a a =.
The phases are
angle(cos j sin ), 1,...,l l l l l A A l h ϕϕϕ=+= where j is the imaginary unit.
In the following discussion, we take 00.1α=，00.2η=, 10.2η=. Set the global approximation
error 410J −<. And 1α
is determined by the adaptive algorithm for the adjusting step of harmonic order explained in Section 3. The simulation results in Section 4 show that higher accuracy for analysis of non-integer harmonics can be obtained.
2 FFT-ANN ALGORITHM
Because of the inherent characteristic of ANN,
the error function may have many local minimal points and platforms when there are several parameters to be adjusted. The training process of the network perhaps needs plenty of time. The network
第4期王公宝等： 基于FFT和神经网络的非整数次谐波分析改进算法105
doesn’t convergent sometimes. Therefore, the choice of initial values of parameters is very important for the network.
In order to detect non-integer harmonics precisely, we will adopt the following approach. For convenience sake, we call the approach the FFT-ANN algorithm. Firstly, the Hanning-windowed FFT algorithm processes the sampled signal. By this time, the number of harmonics and the orders of harmonics are obtained. Secondly, choose the number of neural nodes according to the number of harmonics. Thirdly, choose the initial values of orders of harmonics according to the result obtained from the Hanning- windowed FFT algorithm.Moreover, an adaptive algorithm for the adjusting step of the order of harmonic is presented(See Section 3). Finally, by using the improved linear ANN model stated in Section 1, non-integer harmonics can be detected precisely. The simulation examples in Section 4 show that the improved FFT-ANN algorithm can guarantee the higher accuracy as well as better real-time for analysis of non-integer harmonics parameters.
3 ADAPTIVE ALGORITHM FOR THE AD- JUSTING STEP OF HARMONIC ORDER In this paper, the Hanning-windowed FFT algorithm processes the sampled signal first, and then the initial values of orders of harmonics are chosen close to their real values. In order to ensure the convergence of the network, the adjusting step of harmonics orders α1 should be sufficient small. We have proved that this is true through a lot of simulations. On the other side, the weights w j can’t be obtained in the frequency spectrum chart. In this paper, we may choose the initial values of all weights to be 1. On this assumption, from the formulas (4), (5) and (6), it is not difficult to find that α1is the main factor with regard to the convergence of the network. Now, the remaining question is how to set α1 so that the network is convergent. In the following discussion, we present an adaptive algorithm for the determination of α1, the adjusting step of harmonic order.
For a certain sampled signal, in the beginning of the training, ()
e i is usually large. From the formula
(6), We can choose the initial value of α1 so that
|()|
l
a i
Δis small sufficiently. In the process of the subsequent training, along with )(i
w
j
closing to its real value and the decrease of ()
e i, |()|
l
a i
Δremains sufficient small. For 1
i=, |(1)|0.
l
a
Δ= Now, we turn to consider the choice of α1 for 2
i=. Through lots of simulation samples and analysis, we
find that if |(2)|0.000 1
l
a
Δ<, then the network is convergent rapidly. Moreover, this method is of adaptability for signals comprising non-integer harmonics.
The adaptive algorithm for the adjusting step of the order of harmonic is as follows: ① According to the method stated in Section 2, choose the initial values of the orders of harmonics , and set the initial values of all weights to be 1; ②For2
i=, calculate
(2)
e from the formulas (4) and (5); ③Choose1
l=，1
0.002
α=; ④Calculate |)2(
|
l
a
Δ from the formula
(6); ⑤If|(2)|0.000 1
l
a
Δ>, let
11
:0.95
αα
=, and turns
to ④; if |(2)|0.000 1
l
aΔ≤, then turns to ⑥; ⑥Let :1
l l
=+; if l h
≤, then turns to ④; if l h
>, then turns to ⑦; ⑦Give the output of α1 and choose this value as the adjusting step of harmonic order.
The simulation results in Section 4 validate that the algorithm is effective for detection of non-integer harmonics.
4 SIMULATION EXAMPLES
4.1 A Signal Comprising Three Non-integer Harmonics Analysis
Suppose the signal is
() 2.6sin(2.232/9) 1.75sin(2.43
2/6) 2.3sin(3.052/10)
x t ft
ft ft
=×π+π+×
π+π+×π+π
The sampling time is 0.12 seconds, and the length of sampled data is 64. We process the sampled signal by using the Hanning-windowed 64-point FFT algorithm. The result is shown in Fig.3.
As shown in Fig.3, the spectral lines have three maxima and the corresponding orders of harmonics are 2.166 7, 2.5 and 3. According to the algorithm stated in Section 2, we choose 6 as the number of neural nodes and 2.166 7, 2.5 and 3 as the initial
106 中 国 电 机 工 程 学 报 第28卷
values of orders of harmonics respectively. All the initial values of the weights are chosen to be 1. By use of the adaptive algorithm introduced in Section 3, we
get 51 4.493 410α−=×. The results obtained after 31
iterations of training are shown in Tab.1. And it takes 0.047 s to obtain the results by using the FFT-ANN algorithm. In Tab.1, we provide the comparison on the results of the FFT-ANN algorithm, the three points
algorithm and the five points algorithm [5].
2 4 6
The order of harmonics
1
2
A m p l i t u d e /p u
图3 信号加汉宁窗FFT 变换频谱特性 Fig. 3 Amplitude spectrum obtained by the
Hanning-windowed FFT algorithm
表1 不同算法的检测结果比较
Tab. 1 Comparison of the results of three different algorithms
Algorithm Order Amplitude/pu Phase/(°)Order Amplitude/pu Phase/(°)Order Amplitude/pu Phase/(°)Theoretical values 2.23 2.6 20 2.43 1.75 30 3.05 2.3 18 Three points algorithm 2.220 4 2.364 3 32.706 4 2.409 8 1.672 0 141.961 7 3.048 6 2.308 3 19.889 8Five points algorithm 2.205 3 2.125 3 44.700 3 2.358 2 1.616 7 198.216 9 3.050 7 2.296 6 17.104 2FFT-ANN algorithm
2.229 9
2.601 2
20.093 2
2.430 1
1.751 0
29.878 6
3.050 0
2.299 5
17.993 3
4.2 A Signal Comprising Four Harmonics Analysis Suppose the signal is
()10sin(1.52/6)sin(22/4) 6sin(2.52/3)3sin(3.12/10)
x t ft ft ft ft =×π+π+×π+π+
×π+π+×π+π
The sampling time is 0.12 seconds, and the
length of sampled data is 64. We process the sampled signal by using the Hanning-windowed 64-point FFT algorithm. The result is shown in Fig. 4. As shown in Fig.4, the spectral lines have four maxima and the corresponding orders of harmonics are 1.5, 2, 2.5 and 3.166 7 respectively. Similarly, we choose 8 as the number of neural nodes and 1.5, 2, 2.5 and 3.166 7 as the initial values of orders of harmonics respectively. All the initial values of the weights are chosen to be 1. By use of the adaptive algorithm introduced in Section 3, we get 51 1.068 710α−=×. The results obtained
after 39 iterations of training are shown in Tab.2. And it takes 0.062 s to obtain the results by using the FFT-ANN algorithm. In Tab.2, we provide also the comparison on the results of the FFT-ANN algorithm, the three points algorithm and the five points algorithm [5].
2 4 6
The order of harmonics
48
A m p l i t u d e /p u
12
图4 信号加汉宁窗FFT 变换频谱特性 Fig. 4 Amplitude spectrum obtained by the
Hanning-windowed FFT algorithm
表2 不同算法的检测结果比较
Tab. 2 Comparison of the results of three different algorithms
Algorithm Order Amplitude/pu Phase/(°) Order Amplitude/pu Phase/(°)Order Amplitude/pu Phase/(°) Order Amplitude/pu Phase/(°)Theoretical values
1.5
10 30
2
1 45
2.5
6 60
3.1
3 18 Three points algorithm 1.500 010.000 1 29.998 3 2.000 20.999 5 44.548 52.501 1 5.991 6 58.640 8 3.100 0 3.000 0 18.000 4Five points algorithm 1.495 110.003
4 35.262 0 1.903 2 1.144 0 149.610 22.498 1 6.004 1 62.113 0 3.100 0 3.000 0 18.000 0FFT-ANN algorithm 1.500 0
10.000 5
29.997 0 1.999 8
1.000 2
45.288 12.500 0
5.999 9
59.996 1 3.100 0
3.000 5
17.981 8
4.3 A Signal Comprising Higher Harmonic Orders Analysis
Suppose the signal is ()0.88sin(22ππ/3)0.9sin(19.21 2ππ/6)0.75sin(20.332ππ/9)
x t ft ft ft =×++×
++×+
The sampling time is 0.1 s, and the length of
sampled data is 256. We process the sampled signal by using the Hanning-windowed 256-point FFT algorithm. The result is shown in Fig.5. As shown in Fig.5, the spectral lines have three maxima and the corresponding orders of harmonics are 2, 19.2 and 20.4 respectively. Similarly, we choose 6 as the
第4期 王公宝等： 基于FFT 和神经网络的非整数次谐波分析改进算法 107
number of neural nodes and 2, 19.2 and 20. 4 as the
initial values of orders of harmonics respectively. All the initial values of the weights are chosen to be 1. By use of the adaptive algorithm introduced in Section 3, we get 31109000.1−×=α. The results obtained after 46 iterations of training are shown in Tab. 3. And it takes 0.14 s to obtain the results by using the FFT-ANN algorithm. In Tab. 3, we provide also the comparison on the results of the FFT-ANN algorithm, the three points algorithm and the five points algorithm [5].
A lot of simulation examples show that the FFT-ANN algorithm is applicable to the detection of non-integer harmonics. From Tab.1, Tab.2 and Tab.3, it follows that close non-integer harmonics including the case of higher harmonic orders can be separated from signals with higher accuracy and better real- time by using the improved FFT-ANN algorithm
presented in the paper. Moreover, the FFT-ANN algorithm has higher accuracy than three points algorithm as well as five points algorithm for a signal with close harmonic orders or distant harmonic amplitudes (See Tab.1 and Tab.2). And the three different algorithms have about the same effect for a signal with distant harmonic orders and close harmonic amplitudes (See Tab.3).
0 5 10 15 20 25 30
The order of harmonics 0
0.3A m p l i t u d e /p u
0.60.9
图5 信号加汉宁窗FFT 变换频谱特性
Fig. 5 Amplitude spectrum obtained by the
Hanning-windowed FFT algorithm
表3 不同算法的检测结果比较
Tab. 3 Comparison of the results of three different algorithms
Algorithm Order Amplitude/pu Phase/(°)Order Amplitude/pu Phase/(°)Order Amplitude/pu Phase/(°)Theoretical values 2 0.88 60 19.210.9 30 20.33 0.75 20 Three points algorithm 2.000 0 0.880 0 59.999 919.209 90.900 1 30.024 020.330 0 0.750 0 19.942 8Five points algorithm 2.000 0 0.880 0 60.000 019.210 00.900 0 29.980 620.330 0 0.750 0 20.011 7FFT-ANN algorithm
1.999 8
0.880 0
60.093 8
19.210 1
0.899 9
29.877 2
20.328 9
0.750 1
20.167 5
5 CONCLUSIONS
In this paper, an improved algorithm with higher accuracy for non-integer harmonics analysis based on the FFT algorithm and the improved linear ANN model is presented. Moreover, an adaptive algorithm for the adjusting step of harmonic order is also provided. According to the algorithm, at first we process the sampled signal by use of the Hanning-windowed FFT algorithm. Then we choose the number of neural nodes and the initial values of orders of harmonics according to the result obtained from the Hanning-windowed FFT algorithm. Through such processing, the convergence rate of the network can be improved and non-integer harmonics can be detected effectively. Simulation examples are provided to demonstrate the validity of the improved algorithm. The algorithm developed in the paper provides a new approach for the analysis of non-integer harmonics in electric power systems.
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收稿日期：2007-08-14。

作者简介：
王公宝(1962—)，男，博士，教授，从事小波分析、神经网络及其在电力系统中的应用研究工作，wanggongbao8@；
向东阳(1978—)，男，博士研究生，研究方向为神经网络理论与应用、电力系统谐波检测；
马伟明(1960—)，男，博士，教授，博士生导师,中国工程院院士，主要从事特种电源和电力系统电磁兼容性等领域的研究工作。

(编辑王剑乔)。