1、 Abstract With the continuous increase of wind power grid-connected capacity,the random fluctuation of wind power makes subsynchronous oscillations(SSO)exhibit time-varying amplitudes and frequencies characteristics.This paper proposes an on-line monitoring method to identify subsynchronous oscillat
2、ion modals(including frequencies,amplitudes and damping factors,etc.)with time-varying amplitudes and frequencies based on sliding window FFT.First,windows and interpolation are used to reduce spectral leakage and fence effect,and decrease the FFT identification errors.Then,through the sliding of th
3、e time window,the signal intercepted by each time window is transformed to obtain a dynamic sequence of oscillation frequency and oscillation amplitude,that is,varying frequency and amplitude with time.The dynamic sequence of the damping factor is obtained by analyzing and calculating the time-varyi
4、ng oscillation amplitude.Finally,using the ideal non-stationary signal and the measured signal of the power grid as test case,the comparison results with the Prony and HHT algorithms show that this method not only can eliminate the mode mixing,but also have anti-noise ability.The method can effectiv
5、ely identify the random time-varying oscillation modals and realize on-line modal onitoring and analysis of subsynchronous oscillation modals.Index Terms Subsynchronous oscillation,Time-varying amplitude frequency,Modal identification,Sliding window,FFT I.INTRODUCTION S environmental pollution and e
6、nergy crisis continue to become more serious,new energy power generation is growing1.New energy units are connected with power grids through power electronic converters.The converters interact with the power grid,which may cause subsynchronous Manuscript received July 15,2018.This work was supported
7、 by the National Nature Science Foundation of China(51407071，51637005)and Fundamental Research Funds for the Central Universities(2018MS006)oscillations(SSO)problems and endanger the safe and stable operation of the power grid2.In recent years,the grid-connected capacity of wind power generation has
8、 been increasing.Due to the random fluctuation of wind speed,the subsynchronous oscillations are characterized by random time-varying amplitude and frequency3.In order to ensure the stability of the operation of the power system and real-time monitor of the oscillation mode,it is necessary to identi
9、fy the oscillation mode quickly and accurately.The existing mode identification methods based on signal analysis mainly include Prony4,Hilbert-Huang transform5(HHT),stochastic subspace identification6(SSI)and so on.The Prony method can analyze the deterministic oscillation signal and can obtain the
10、accurate amplitude,frequency,phase and damping factor of the signal7.The Prony algorithm has poor accuracy for the analysis of random signals.The Prony algorithm is very sensitive to noise,and its anti-noise ability is not ideal8.HHT can process the random oscillation signal to obtain a curve of sig
11、nal frequency and amplitude as a function of time9.However,if there are two oscillation components with a frequency ratio less than 1.5 in the signal10,mode mixing occurs in HHT11.It is not suitable for power systems with multiple SSO modes,and HHT has an end effect12,the frequency and amplitude ide
12、ntification errors at the boundary are large13.The stochastic subspace identification6 obtains the identification result through a series of matrix operations,and the calculation process is complicated and the calculation time is long.In order to monitor SSO online accurately and quickly,this paper
13、proposes sliding window FFT method to analyze random time-varying amplitude frequency subsynchronous oscillation signals.First,add a time window to the signal,and then perform FFT on the signal truncated by the time window to The On-line Monitoring of Time-varying Amplitude and Frequency Characteris
14、tic of Sub-synchronous Oscillation Based on Sliding Window FFT SHUTAO PENG1,JING YANG2,JUNCHEN LI1,JUN DENG1,XIAOTENG LI1,JILIANG JIN1,TONG WANG2 1、State Grid Shaanxi Electric Power Research Institute,Xian 710000 China 2、State Key Laboratory for Alternate Electrical Power System with Renewable Energ
15、y Sources,North China Electric Power University Beijing 102206 China A 2018 China International Conference on Electricity Distribution Tianjin,17-19 Sep.2018CICED2018 Paper No.201805270000010 Page1/6 1625 obtain the amplitude and frequency.The amplitude and frequency is considered to be the amplitud
16、e and frequency of the whole signal in this time window midpoint.Moving the time window,and the same processing is performed on the signal to obtain the time-varying amplitude-frequency identification result of the signal.Finally,the sliding window FFT is compared with the Prony and HHT algorithms b
17、y using an example to verify the effectiveness and superiority of the sliding window FFT.II.SLIDING WINDOW FFT A.Fast Fourier Transform Fourier transform can transform the time domain signal into the frequency domain signal,which is an important tool for signal analysis processing.Discrete Fourier t
18、ransform(DFT)can be used to analyze discrete signals.However,the calculation of DFT is very large.Fast Fourier transform is a method to reduce the amount of DFT calculation.Frequency extraction is a commonly used algorithm.The main steps are as follows.x(t)is a sequence of length N,and its discrete
19、Fourier transform formula is as in(1)10()()0 1NtkNtX k x t A k N(1)In the formula 2jNNAe.Let the sequence point number N=2M,and M be a positive integer.The N-point DFT is divided into two N/2-point DFTs according to the parity of f.N/2 is still an even number,so each N/2 point DFT can be decomposed
20、into even and odd arrays.Such decomposition can be performed M times,and the result obtained after M operations is the result X(k)of the N point DFT of x(t).B.Windows and Interpolated FFT There are three main sources of error in the fast Fourier transform:aliasing,leakage and fence14.If the sampling
21、 signal frequency does not satisfy the sampling theorem,aliasing will occur.Because the signal is truncated by adding the time window,the spectrum obtained after Fourier transform has some false spectral lines with smaller amplitudes at other frequency points on both sides of the real spectral line,
22、causing leakage phenomenon and bringing errors to the transformation result.The fence phenomenon is due to the fact that the discrete Fourier transform obtains the spectral values at N sampling points,and the spectral values between these points cannot be obtained.By using a time window with a narro
23、w main lobe width and a fast side lobe attenuation speed,spectrum leakage can be improved.The interpolation algorithm can be used to correct the error caused by the fence phenomenon.The Hanning window15 has a fast sidelobe attenuation rate,a simple interpolation formula,a small amount of calculation
24、,and a relatively easy programming.Therefore,the method of adding a Hanning window is used,and the window function is as shown in(2).2()0.5 0.5cos()twtN(2)The polynomial approximation and the bimodal spectral line correction method can be used to obtain the correction formula for the Hanning window
25、frequency0f and amplitude A as shown in(3)-(5).2121=1.5yyyy(3)100.5SkfFN(4)22146()(2.26557103 1.22719978 0.37607775 0.09767389)/A y yN(5)In the formula,the number of spectral line on the left and right sides of the true signal frequency are k 1 and k 2,respectively,k 2=k 1+1,y 1 and y 2 are the ampl
26、itudes of the corresponding spectral lines k 1 and k 2,F S is the sampling frequency,and N is the number of sampling points.C.Sliding Window FFT For a continuous signal,it is infinitely long,and it needs to be truncated when analyzing the signal,which can be seen as adding a rectangular time window
27、to the signal.For a time-varying amplitude-frequency signal,when the length of the time window is short,it can be considered that the waveform characteristics of the signal are unchanged during the time window,and the segment signal can be regarded as a stationary signal.By sliding the time window o
28、n the time axis and performing FFT on the signals in each time window,the spectrum at different times can be obtained.From this,we can obtain a curve of signal frequency and amplitude as a function of time For a discrete signal x(t)with a sampling frequency of Fs,n c time windows are taken,the numbe
29、r of sampling points is N c,and the interval between each window is c j points.Firstly,a segment of the signal in the first time window of the signal leftmost end is analyzed,and the segment signal is multiplied by(2)to add a Hanning window,and the windowed signal is subjected to fast Fourier transf
30、orm to obtain amplitude-frequency characteristics.The frequency is then interpolated and corrected to obtain the frequency,amplitude and phase information of the signal.Then,the time window is moved once every c j points,and the FFT of the signal intercepted by each window is performed,and the corre
31、sponding amplitude-frequency characteristics can be obtained.The time coordinate of the frequency and amplitude 2018 China International Conference on Electricity Distribution Tianjin,17-19 Sep.2018CICED2018 Paper No.201805270000010 Page2/6 1626 of each time window is the time of the midpoint of eac
32、h time window,and finally the relationship between the amplitude A(t)and the frequency f(t)as a function of time is obtained.Suppose a mode component can be expressed as:00()cos(2)tx t A e ft(6)The relationship between the sliding window FFT identification result A(t)and(6)is 0()tA t A e(7)Take the
33、logarithm of(7)0ln()ln A t A t(8)The slope of the straight line in(8)is the damping factor,and the vertical ordinate of the intersection with the vertical axis is the logarithm of the amplitude A 0.The oscillation parameter amplitude A 0 and the damping factor can be obtained by linear least squares
34、 fitting of lnA(t).For the non-stationary subsynchronous oscillation component,the damping factor()t can be obtained by the following(9):ln(/)/()()s j sjF A t c F A ttc(9)III.SIMULATION AND VERIFICATION A.Subsynchronous Oscillating Signal with Modes Having Similar Frequency A SSO signal with similar
35、 frequencies is constructed as in(10).0.10.2()cos(2 15)2 cos(2 20)3ttx t e tet(10)The signal is identified by a sliding window FFT.The signal sampling frequency is 100 Hz,the length of the time window is 100 points,and the number of time windows is 200.The relationship between the signal frequency a
36、nd the amplitude as a function of time is shown in Fig.1.It can be seen from Fig.1 that the signal has two modes,and the blue curve and the green curve respectively represent one mode,the frequencies of the two modes are constant,and the amplitude is attenuated.According to the time-varying amplitud
37、e-frequency data identified in Fig.1,the attenuation factor is calculated by linear least-squares fitting of the logarithm of the amplitude by(8),and the sliding window FFT identification results are listed in Table I.TABLE I IDENTIFICATION RESULTS OF SLIDING WINDOW FFT Frequency/Hz Amplitude Dampin
38、g Factor Mode 1 15.0007 1.0002-0.1 Mode 2 19.9999 2.0013-0.2 Fig.1.The frequency and amplitude vary with time Comparing the data in Table I with(10),the sliding window FFT can accurately identify the signal frequency,amplitude and damping factor.The signal is fitted according to the data in Table I.
39、The fitting result is shown in Fig.2.The solid blue line is the signal fitted according to the identification result,and the red dotted line is the original signal.The two are very close and almost coincide.Fig.2.Fitting results of signals with similar frequencies The signal is analyzed by Hilbert-H
40、uang transform.The first two components of the empirical mode decomposition result and the spectrum of the SSO signal are shown in Fig.3.The upper part of Fig.3 is the first intrinsic mode function IMF1 and its spectrum obtained by empirical mode decomposition of the original signal,and the lower pa
41、rt is the second intrinsic mode function IMF2 and its spectrum.It can be seen from Fig.3 that the IMF1 component contains two frequency components of 15 Hz and 20 Hz,and the empirical mode decomposition does not correctly separate the two modes in the signal.Both modes exist in the IMF1 component,an
42、d HHT has a mode mixing phenomenon on the signal processing,which cannot accurately identify the amplitude and frequency of the 0.5 1 1.5 2 2.50.511.52Time(s)Amplitude Mode1Mode20.5 1 1.5 2 2.51416182022Time(s)Frequency(Hz)Mode1Mode20.5 1 1.5 2 2.5-3-2-10123Time(s)x Fitting signalOriginal signal2018
43、 China International Conference on Electricity Distribution Tianjin,17-19 Sep.2018CICED2018 Paper No.201805270000010 Page3/6 1627 signal.The IMF2 component has no practical meaning and is a false component.Compared with HHT,sliding window FFT has no mode mixing,and can accurately separate and identi
44、fy signal,showing obvious superiority.Fig.3.Empirical mode decomposition results B.Noisy Synchronous Oscillation Signal Noise of 20 dB is added to the SSO signal shown in the equation(10),and the noise-containing synchronous oscillation signal is processed by the sliding window FFT.The signal sampli
45、ng frequency is 100 Hz,the length of the time window is 100 points,and the number of time windows is 200.The relationship between the signal frequency and the amplitude as a function of time is shown in Fig.4.It can be seen from Fig.4 that the original signal contains two oscillation modes with freq
46、uencies of about 15 and 20 Hz,respectively,and the amplitude is continuously attenuated with time.In comparison with Fig.1,it can be seen that the frequency and amplitude sequences of the two modes do not change much compared to the signal processing results before the noise is added.Since the fast
47、Fourier transform itself has strong anti-noise ability,the signal noise has little effect on the sliding window FFT identification,and the two oscillation modes of the sub-synchronous oscillation signal can still be identified.Further,the damping factor is calculated,and the identification result of
48、 the noisy signal by the sliding window FFT is shown in Table II.The errors between the data in Table II and the true value is small.After adding the noise to the subsynchronous oscillating signal,the sliding window FFT can accurately identify the frequency,amplitude and damping factor of the signal
49、,and has certain anti-noise ability.The Prony algorithm is used to analyze the noisy synchronization signal.The Prony and sliding window FFT fitting results are shown in Fig.5.The blue curve in the figure is the original signal,the green curve is the Prony fitting signal,and the red curve is the sli
50、ding window FFT fitting signal.From the fitting results,the Prony fitting signal and the sliding window FFT fitting signal are close to the original signal.In order to compare the fitting results of the two methods,the absolute values of the Prony and sliding window FFT fitting errors are shown in F
