KR20070115523A - A method for accurate estimation of the low frequency and the spectrum angle for real time analysis of dynamic systems - Google Patents

Abstract

A method for estimating a low frequency and a spectrum phase for a real-time analysis of a dynamic system is provided to accurately estimate the low frequency and a phase of a Fourier spectrum by fitting a peak value and a left/right spectrum calculated in discrete Fourier transformation with an exponential attenuation function. A peak frequency and a peak value of a low frequency band for analyzing the dynamic system is calculated from the Fourier spectrum(S130). The frequency and the left/right spectrum of the Fourier transformation are selected by corresponding to the peak value(S140). The selected spectrum is fitted by using the exponential attenuation function(S160). The peak value is selected from the fitted spectrum. A frequency corresponding to the peak value is selected as the accurate frequency and the spectrum phase corresponding to the accurate frequency is selected as the accurate phase(S170). A selection result is stored and outputted(S180).

Description

A method for accurate estimation of the low frequency and the spectrum angle for real time analysis of dynamic systems

1 is a schematic block diagram of hardware for carrying out the present invention;

2 is a flowchart illustrating an operation process of an embodiment of the present invention;

3 is a diagram showing the active power waveform measured during a large power failure in a power system

4 is a conceptual low frequency and phase estimation conceptual diagram,

5 is a Fourier spectrum that is symmetrical,

6 is a Fourier spectrum that is asymmetrical,

7 is a Fourier spectrum and a new spectrum affected by different peaks,

8 is a flowchart of an algorithm for accurate low frequency and phase estimation,

9 shows accurate phase estimation in a continuous spectrum,

10 shows accurate phase estimation in the accumulated spectrum,

<Explanation of symbols for the main parts of the drawings>

10: data acquisition unit 20: input processing unit

30: arithmetic processing unit 40: data output unit

50: data storage unit 60: data transmission unit

The present invention relates to a method for accurate low frequency and phase estimation using exponential decay function fitting in Fourier transform, including system identification, statistical signal processing and estimation of modes or parameters included in discrete time series data. Related.

Until now, the method of estimating a parameter from time series data can be largely divided into a parametric method and a non-parametric method. The parametric method estimates a parameter (mode) included in discrete data by calculating a complex exponential from a linear prediction matrix obtained by converting a signal into an autoregressive moving average (ARMA) model. Way.

The typical parametric method is the Prony method. This algorithm takes a lot of computational time because it needs to solve linear prediction equations and calculate solutions of higher-order equations when fitting discrete signals as complex index functions. You need a computer with a large storage area. In addition, this method is accompanied by an error with respect to noise and has a disadvantage in that the parameter is sensitively changed depending on the interval of the signal data and the size of the data.

Discrete Fourier transforms are widely used in a variety of industries and are mainly used to calculate frequencies in signals. The nonparametric method using the discrete Fourier transform is a method of estimating a parameter included in the discrete data by repeatedly applying the Fourier transform. Nonparametric method repeatedly estimates the vibration mode corresponding to each frequency by applying Discrete Fourier Transform and continuously calculates the magnitude of the spectrum corresponding to each peak, then applies curve fitting to correspond to the peak frequency. Estimate the mode. However, there are disadvantages in that a few Fourier transform must be performed and a large error occurs when spectral leakage occurs.

In estimating the parameters included in the discrete signal as described above, the frequency corresponding to the exact peak value is a factor that determines the accuracy of other parameters. Up to now, the Fourier transform has been mainly applied to estimating the frequencies included in the discrete signal and to accurately estimate the frequencies in the comparative high frequency band.

On the other hand, in dynamic systems, the response to the low frequency region is the main frequency of interest rather than the high frequency region. Low frequencies near the frequency 1.0 Hz can greatly affect the response of a dynamic system. In order to understand the dynamic response characteristics in the dynamic system and to control it, the accurate low frequency must be estimated. The Fourier transform for high frequency analysis can ignore the effects of spectral leakage, while at low frequencies where the response characteristics change slowly, the peak corresponding to the peak due to spectral leakage has a lot of error. And because the spectral phase changes rapidly around the peak spectrum, if the peak frequency is incorrect, the phase contains a very large error. In estimating the exact frequency for the frequency band between 0.1 Hz and 2.0 Hz, which is a relatively low frequency, the inherent nature of the Discrete Fourier Transform is dependent on the resolution, which depends on the number of discrete data given and the data interval interval.

The higher the resolution, the more discrete the Fourier transform, which requires more computational time. Therefore, the decision between the accuracy and the computational speed should be made according to the application purpose. The present invention relates to a method for estimating accurate low frequency and phase by fitting several spectra with exponential attenuation function around the peak of Fourier spectrum at low frequency. Until now, Fourierspectrum is mainly used to calculate relatively high frequencies, and no accurate algorithm has been developed by fitting exponential attenuation functions to low frequencies in the 0.1Hz to 2.0Hz band.

The conventional methods described above include complex algorithms such as linear equations and least squares methods, which require a lot of computation time and require large computer storage. Also, when the signal is small, it is very sensitive to noise and cannot accurately estimate the frequency.

Accordingly, an object of the present invention is to provide a method for estimating the accurate low frequency and Fourier spectrum phase by fitting the peak and left and right spectrums calculated by the Discrete Fourier Transform as an exponential attenuation function in order to solve the problems of the prior art.

[Hardware for Carrying Out the Invention]

1 is a schematic block diagram of hardware for carrying out the present invention. As can be seen from the drawings, the hardware for carrying out the present invention includes a data processor 10, an input processor 20, and an operation processor. 30, the data output unit 40, the data storage unit 50, and the data transmission unit 60 may be configured.

The input processor 20 analog-to-digital converts input time series data in analog form and inputs the same to the calculation processor 30.

The arithmetic processing unit 30 performs a discrete Fourier transform on time series data input from the input processing unit 20 to estimate a peak value and a frequency at a low frequency, from which time series data corresponding to each peak value is derived. Estimate low frequency and spectral phase.

The data output unit 40 outputs the frequency and phase estimation result estimated by the operation processor 30 to the output device.

The data storage unit 50 stores the estimation result of the frequency and phase estimated by the calculation processing unit 30 in the storage device.

The data transmission unit 60 transmits the estimation result of the frequency and phase estimated by the calculation processing unit 30 to the remote device (server) through the network.

Hereinafter, specific details of the present invention performed through the hardware configured as described above will be described.

The present invention relates to a method of obtaining information for interpreting and controlling a system in an output signal of a dynamic system, wherein the dynamic characteristics important for the dynamic system are mainly governed by low frequencies between 0.1 Hz and 2.0 Hz.

For example, in the steady state except the transient state in the power system, the vibration phenomenon according to the dynamic characteristics of the generators is mainly generated at the low frequency as described above. Such a vibration causes the power system to collapse, leading to a large power failure. In FIG. 3, the vibration is normally operated at an initial stage (S210), but is potentially vibrating at 0.25 Hz. It can be seen that the output suddenly decreases in the vicinity of 30 seconds (S220) of FIG. 3, and the vibration gradually increases from the vicinity of 40 seconds (S230) and vibrates at a distinct frequency (0.25 Hz). As a result, the low frequency vibration is not attenuated (S240), the power system is collapsed and the large customer is blackout, which can be seen that the industrial massive losses.

In order to analyze and control this vibration phenomenon, it is necessary to detect the accurate low frequency included in the output signal. Also, in large dynamic systems, many signals exist and the computation time needs to be very fast with minimal computer memory to process them in real time. That is, signal processing is required at the level of digital signal processing (DSP).

The present invention relates to a method for estimating accurate low frequency and spectral phase using a minimum of computer storage. In a commercially available DSP, a fast Fourier transform is performed on 4096 samples within 0.001 second to calculate a frequency. Therefore, this method can be used for fast parameter estimation, but since the storage location is limited, a simple and straightforward algorithm is needed to perform the calculation using the minimum storage location.

In view of the above necessity, the present invention relates to a method for directly obtaining an accurate low frequency and spectral phase for each mode representing a system state in a discrete Fourier transform of time series data, and to estimate an accurate frequency and phase from a Fourier spectrum. It includes an embodiment to.

First, the principle applied to the embodiment of the present invention will be described. The Fourier transform equation for the continuous function is shown in Equation 1.

에 정의된 신호

의 이산 푸리에변환은 수학식 2와 같다.The signal given as actual time series data is most often an aperiodic signal rather than a periodic signal. section

Signal defined in

The discrete Fourier transform of is equal to Equation 2.

이며, 스펙트럼 계수

는 구간

에서 정의 된다. 한편 신호가 비주기적이라면 이산푸리에 변환의 스펙트럼은 연속적으로 표현되는데, 대부분 자연계의 신호는 비주기적이므로 스펙트럼은 연속적으로 표현된다. 따라서 이산푸리에 변환 결과 얻어진 푸리에 스펙트럼에서 첨두 스펙트럼(peak spectrum)을 중심으로 한 좌우 스펙트럼은 도 4에 나타낸 것과 같은 포락선 형태가 된다. 도 4에서 점으로 나타낸 것이 이산푸리에 변환한 스펙트럼인데, 첨두치(S320) S_p에 첨두주파수(S340) ω_p가 대응하고, 동일한 주파수(S340,S380)에서 스펙트럼 위상이 추정된다. 그러나 추정한 첨두치(S320) 및 첨두주파수(S340,S380)가 스펙트럼의 누설로 인하여 부정확한 값임을 알 수 있다. 도 4에 점선으로 나타낸 좌우포락선은 이산푸리에 변환한 스펙트럼으로부터 지수감쇠함수로 적합한 도식을 나타내고 있다. 지수감쇠함수로 적합한 첨두치(S310)와 이에 대응하는 주파수(S330) 및 스펙트럼 위상(S360)은 원래 이산푸리에변환에서 얻은 주파수(S340,S380) 및 위상과 차이가 있음을 알 수 있다.In equation (2)

Spectral coefficient

Is the interval

Is defined in. On the other hand, if the signal is aperiodic, the spectrum of the Discrete Fourier Transform is represented continuously. Most of the natural signals are aperiodic, so the spectrum is represented continuously. Therefore, in the Fourier spectrum obtained as a result of the Discrete Fourier Transform, the left and right spectrums centered on the peak spectrum become an envelope form as shown in FIG. 4. The point shown in FIG. 4 is a discrete Fourier transformed spectrum, and the peak frequency S340 ω _p corresponds to the peak value S320 S _p , and the spectral phases are estimated at the same frequencies S340 and S380. However, it can be seen that the estimated peak value S320 and peak frequencies S340 and S380 are inaccurate values due to the leakage of the spectrum. The left and right envelopes shown by dotted lines in Fig. 4 show a schematic diagram suitable as an exponential decay function from the discrete Fourier transformed spectrum. It can be seen that the peak value S310 suitable for the exponential attenuation function, the frequency S330 and the spectral phase S360 corresponding thereto are different from the frequencies S340 and S380 and phase originally obtained by the Discrete Fourier Transform.

A continuous function x (t) is Assuming that the frequency ω damping factor α and the size A, phase clothing possession attenuation cosine function with φ _1, can be represented as shown in Equation 3, when referred to the complex exponential sine function, equation (4) Can be expressed as:

x (t) = Ae ^-αt cos (ωt + φ ₁ )

x (t) = Ae ^-αt sin (ωt + φ ₁ )

First, in order to consider the exponential decay cosine function, Equation 3 is Fourier transformed.

A phase corresponding to the frequency ω ₁ is called φ ₁ , and when the braking coefficient is small and ω ₁ ≫α is established, it is expressed by Equation (6).

If Equation 5 is shown, it will be the same as the left and right envelopes represented by dotted lines in FIG. 4. Therefore, in the present invention, the left and right spectrum including the peak spectrum in the Fourier spectrum is assumed as a complex exponential decay function (Equation 5). In other words, as a result of discrete Fourier transform of arbitrary time series data, when a peak is formed at a specific frequency, the left and right spectrums around the peak are the same as the Fourier transform of the complex exponential decay function or the complex exponential cosine function.

Therefore, the parameters included in Equation 5 can be estimated from several peaks on the left and right of the peak and peak obtained from the discrete Fourier transform. Using the left and right spectrum including the peak value, it is possible to estimate the parameter required if it is fitted with the exponential attenuation function. To select. The phases corresponding to the new frequencies S330 and S370 are selected as the correct phase S360. Therefore, from the Fourier spectrum of the complex attenuation function, it is possible to estimate the exact low frequency included in the signal and the spectral phase corresponding to each mode.

The Fourier spectrum of the exponential decay cosine function shown in Equation 4 is the same as the exponential decay cosine function, and since only 90 degrees of phase difference occurs, the same applies to the exponential decay cosine function described above.

An embodiment is included to describe the above in more detail.

5 is a diagram illustrating a case where an accurate frequency is estimated as a result of the Discrete Fourier Transform. If the first left and right spectrums S430 and S440 are largely small with respect to the peak value S410, the peak value S410 and the frequency S460 may be determined to be accurate values. More specifically, if the first left and right spectrums (S430, S440) are significantly smaller than the peak (S410), and the sizes of the left and right spectrums (S430, S440) are similar, they can be accurately simulated in the exponential decay function. The frequency S460 may be determined to be an accurate value. For example, when% X ₁ is less than or equal to 10% in Equations 7 and 8, since the peak value S410 has a larger value than the left and right spectrums S430 and S440, the frequency estimated by the Discrete Fourier Transform is an accurate frequency. It can be determined as In addition, when% X ₁ is a relatively large value of 10% or more, if the left and right spectrums (S430, S440) have a similar size, that is, if% X ₂ has a value of 5% or less in Equation 9, it is accurately simulated with an exponential decay function. As a result, the frequency estimated by the discrete Fourier transform can be determined as the correct frequency.

When spectral leakage occurs in the Discrete Fourier Transform due to improper sampling or noise, the exact frequency cannot be estimated. 6 is a diagram illustrating a case where spectral leakage occurs as a result of the Discrete Fourier Transform. It can be seen that a large difference occurs between the first left spectrum S530 and the right spectrum S540 with respect to the peak value S510. At this time, the peak value (S510) and the peak frequency (S560) is a value containing a lot of errors. Therefore, the appropriate spectrum needed to fit with the exponential attenuation function should be selected. First, when% X ₄ shown in Equation 12 is 5% or less, it means that the second left and right spectrums S520 and S550 have similar values. Therefore, in this case, the exact peak and frequency are calculated by using the peak attenuation function (S510), the first left and right spectrums (S530, S540), and the second left and right spectrums (S520, S550) as an exponential decay function (Fig. 4). . If% X ₃ is greater than or equal to 50%, the larger of the two values in the first left and right spectrums (S530, S540) is replaced with S ₀ shown in Equation 13, and then the peak value (S510) and the first left and right spectrums (S530). , S540) and the smallest value of S ₀ and the second left and right spectrums (S520, S550) are used as exponential attenuation functions to calculate the exact peak and frequency.

6 illustrates a case in which the Fourier spectrum is asymmetrical due to the influence of another neighboring spectrum. In this case, the exponential decay function is used by using the spectrum (S620, S630) that is not influenced by other spectra, and using the new S ₀ point and the four points (S610, S620, S630, S660) shown in Equation 12. Estimate the correct frequency and phase.

In real time signal processing, signals are successively inputted. If the discrete Fourier transform is repeatedly performed in the overlapped time intervals, the Fourier spectrum and phase remain similar. 9 is a diagram showing superimposed consecutive time intervals and spectral phases. When discrete Fourier transforms are performed on time intervals t ₁ (S810), t ₂ (S820), and t ₃ (S830) continuously moving with the same sampling, the frequency resolution is the same. Therefore, each frequency section is the same for each time section. On the other hand, when spectral leakage occurs due to noise or lack of samples, the phase changes rapidly around the peak but has a similar slope for similar time intervals. Therefore, in order to obtain a more accurate spectral phase, phases for three time periods S810, S820, and S830 may be used. That is, the exact phase can be selected as the exact phase of the point where the three straight lines intersect the peak value or the point with the smallest deviation between the three straight lines as shown in Equation 14, and also corresponds to the exact phase obtained from the three time intervals. The correct frequency can be selected.

10 is a diagram illustrating overlapping time intervals and spectral phases having the same time interval t ₁ but different sampling intervals Δt ₁ , Δt ₂ , Δt ₃ . If you perform Discrete Fourier Transform on data with different sampling intervals, the frequency resolution will be different, and therefore, each frequency interval will be different. As described above, more accurate spectral phases can be obtained using the respective spectral phases for the three samplings S910, S920, and S930. That is, the exact phase can be selected as the exact phase of the point where the three straight lines intersect the peak value or the point with the smallest deviation between the three straight lines as shown in Equation 14, and also corresponds to the exact phase obtained from the three time intervals. The correct frequency can be selected.

As described above, the present invention is capable of estimating accurate low frequencies and phases representing the characteristics of a dynamic system directly from the Fourier transform of time series data. Unlike conventional methods, the present invention does not rely on simulation, and thus does not repeatedly calculate the Fourier transform. Instead, the low frequency included in the signal and the corresponding spectral phase can be estimated in one discrete Fourier transform.

Since the present invention greatly shortens the low frequency and phase estimation time, it is possible to improve performance in a system requiring fast parameter estimation in real time. In particular, since the present invention is a simple and clear algorithm based on the equation, there is an effect that can be easily applied to all systems applying the Discrete Fourier Transform.

Claims (4)

An accurate low frequency and phase estimation method for real time analysis of dynamic systems, From the Fourier spectrum, the peak and peak values of the low frequency bands required for dynamic system analysis are calculated. Select the frequency and left and right spectrum of the Fourier spectrum corresponding to the peak, Fit the selected spectrum as an exponential attenuation function, Select the peak from the appropriate new spectrum Select the frequency corresponding to the peak to the correct frequency, the spectral phase corresponding to this frequency to the correct phase, And a step of outputting and storing the result of the selection. The method of claim 1, wherein in initially determining the correct frequency, Determining the exact frequency when the magnitudes of the first left and right spectrums are similar at a constant size with respect to the peak value; Determining the exact frequency when the magnitudes of the first and second left and right spectrums are similar at a constant magnitude with respect to the peak, And a step of outputting and storing the result of the determination. An accurate low frequency and phase estimation method using Fourier spectrum attenuation function fitting. The method according to claim 1, wherein the peak and left and right spectra are fitted as Fourier spectra of an exponential decay function. A step of introducing a new spectrum instead of an inaccurate spectrum affected by the other peak from the magnitude of the first left and right spectrum around the peak, A step of fitting the left and right spectrum including the peak value and the new spectrum to the Fourier spectrum of the exponential attenuation function, Selecting a frequency corresponding to a suitable new peak as an accurate frequency, and selecting a spectral phase corresponding to this frequency as an accurate phase, And a step of outputting and storing the result of the selection. An accurate low frequency and phase estimation method for real time analysis of dynamic systems, From each Fourier spectrum over successive time intervals, the peak and peak values of the low frequency bands required for dynamic system analysis are calculated.  Select each spectral phase that corresponds to the spectrum to the left and right of the peak, And selecting a phase having a minimum deviation from the selected spectral phase as an accurate phase, and outputting and storing the result of the selection.

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