A harmonic frequency division amplitude-phase fast detection method suitable for power grid

Abstract

This invention discloses a rapid harmonic frequency-amplitude-phase detection method and system suitable for power grids, belonging to the field of power detection technology. First, the power grid voltage or current signal is acquired, a Hankel matrix is ​​constructed, and singular value decomposition is performed. The effective rank of the signal is adaptively determined by calculating the ratio of adjacent singular values, serving as the model order for the Prony algorithm. Then, the Prony algorithm is executed using this model order to fit the frequency, amplitude, phase, and attenuation factor of each harmonic and oscillation component. A short-time window sliding processing strategy is introduced, dividing the continuous signal into overlapping short-time data windows. Within each window, the improved Prony algorithm is executed to achieve real-time dynamic tracking of each harmonic parameter. This invention maintains high-precision parameter estimation even under low signal-to-noise ratio conditions, effectively suppresses noise interference, and possesses excellent time-varying dynamic tracking capabilities and high-resolution multi-component analysis capabilities. It can be widely applied to power quality monitoring in complex power grid environments such as new energy grid integration and impulsive loads.

Description

Technical Field

[0001] This invention relates to the field of power detection technology, and in particular to a rapid method for detecting harmonic frequency division amplitude and phase in power grids. Background Technology

[0002] With the widespread integration of new energy power generation (such as photovoltaic and wind power) and power electronic nonlinear loads into power systems, the problem of grid harmonic pollution has become increasingly prominent, and the harmonic composition has become increasingly complex. Harmonic sources not only generate integer harmonics, but also frequently induce interharmonics with frequencies that are not integer multiples of the power frequency, and the content of each harmonic changes rapidly and dynamically with the intermittency and fluctuation of the source load. Against this backdrop, achieving independent, rapid, and high-precision detection of the frequency, amplitude, and phase of each harmonic (including interharmonics) has become a key prerequisite for accurate power quality management and the safe and stable operation of the system.

[0003] Existing harmonic detection technologies are mainly divided into two categories. One category is based on instantaneous reactive power theory, which achieves detection by calculating and extracting harmonic currents in real time. This type of method has a clear physical concept and a simple algorithm structure, but its performance is limited by inherent principles: the detection stage relies on low-pass or high-pass filters to separate harmonics from the fundamental component. The detection accuracy and dynamic response speed are significantly limited by the performance of the filter design, and the inherent phase delay of the filter causes a lag in the detection results, making it difficult to meet the requirements of fast real-time tracking. More importantly, most of these methods can only output the total harmonic distortion and cannot independently identify and track coexisting multi-frequency and non-integer harmonic components. The other category is the spectrum analysis method represented by Fast Fourier Transform (FFT), which theoretically can obtain the amplitude and phase information of each frequency component in the signal, realizing a leap from total quantity detection to frequency division detection. However, the frequency resolution of FFT is limited by the sampling time, which makes it insufficient for distinguishing harmonics and interharmonics with similar frequencies. It is also extremely sensitive to asynchronous sampling, which can easily lead to spectral leakage and picket fence effect, resulting in amplitude measurement error and frequency positioning ambiguity. At the same time, the long data window requirement makes its output delay significant, which makes it difficult to meet the real-time tracking requirements of rapid dynamic changes in harmonics.

[0004] To overcome the aforementioned limitations, parameterized spectral estimation methods, represented by the Prony algorithm, have attracted attention. This method directly fits the signal using a complex exponential linear combination, adaptively extracting each component and directly obtaining its complete parameters such as frequency, amplitude, and phase. Theoretically, it is not limited by a fixed resolution and adapts better to short data windows. However, the performance of the Prony algorithm is heavily dependent on the signal-to-noise ratio of the input signal and is extremely sensitive to noise. In actual measurements of noisy power grid signals, it suffers from problems such as difficulty in accurately selecting the model order and susceptibility to noise-induced distortion of parameter estimation, limiting its reliable application in engineering practice. Furthermore, traditional Prony algorithms typically employ offline batch processing, resulting in high computational load and high output latency, making real-time output impossible. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention proposes a rapid harmonic frequency division amplitude and phase detection method suitable for power grids, comprising the following steps: Step 1: Acquire voltage or current signals from the power grid to obtain a discrete sampled signal sequence; Step 2: Construct a Hankel matrix based on the discrete sampled signal sequence, and perform singular value decomposition on the Hankel matrix to obtain the singular values ​​arranged in descending order; Step 3: Calculate the ratio of adjacent singular values, and adaptively determine the effective rank of the signal based on the distribution characteristics of the ratio. Use the effective rank as the model order of the Prony algorithm. Step 4: Execute the Prony algorithm with the model order determined in Step 3 to fit the discrete sampled signal sequence with a complex exponential function, and solve for the frequency, amplitude, phase and attenuation factor of each harmonic and oscillation component, thereby realizing the rapid detection of frequency-division amplitude and phase of power grid harmonics.

[0006] Furthermore, in step three, the effective rank of the signal is adaptively determined based on the distribution characteristics of the ratio of adjacent singular values. Specifically, when the ratio of adjacent singular values ​​gradually changes from a value greater than 1 and begins to approach 1, the singular value index corresponding to the turning point in the ratio change process is determined as the critical point between the signal subspace and the noise subspace, and this critical point is used as the effective rank of the signal.

[0007] Furthermore, step four further includes: Construct and solve the normal equations of the Prony algorithm to obtain the linear prediction coefficients; Construct a characteristic polynomial based on the linear prediction coefficients and solve it to obtain the poles of each component; Based on the poles, the complex amplitude values ​​of each component are solved using the least squares method; Based on the poles and complex amplitudes, the frequency, attenuation factor, amplitude, and initial phase of each harmonic and oscillation component are calculated.

[0008] Furthermore, to address the time-varying characteristics of the signal, a short-window sliding processing strategy is employed when executing the Prony algorithm in step four, including: The discrete sampled signal sequence is divided into multiple short-time data windows that may overlap or not overlap. Within each short-time data window, the Prony algorithm is executed based on the model order determined in step three to obtain the parameters of each component within that window; By sliding the window sequentially, the trajectory of the parameter evolution of each harmonic and oscillation component over time can be obtained, enabling real-time tracking and detection of time-varying harmonic components.

[0009] Furthermore, the window length of each short-time data window corresponds to at least two fundamental frequency periods, and the window sliding step size is smaller than the window length to ensure window overlap.

[0010] Furthermore, it also includes a model order update step: during the sliding window processing, if it is detected that the singular value ratio curve of the signal in the current window has drifted by more than a preset threshold compared with the criterion of the initial effective rank, then steps two and three are re-executed to update the model order of the current window.

[0011] Furthermore, after obtaining the parameters of each component within each window, the estimation results of the frequency, amplitude, phase, and attenuation factor of the same harmonic or oscillation component between adjacent windows are subjected to weighted averaging or Kalman filtering smoothing to eliminate parameter jumps caused by window switching.

[0012] Furthermore, the effective rank determined in step three is the effective rank of the initial full signal; when using the short-time window sliding processing strategy, the first short-time data window directly uses this initial effective rank as the model order of the Prony algorithm, and subsequent windows selectively update the model order according to the change in the singular value ratio of the current window signal.

[0013] Furthermore, the attenuation factor obtained in step four is used to determine the dynamic characteristics of each harmonic or oscillation component: when the attenuation factor is less than zero, it is determined to be an attenuated oscillation component; when the attenuation factor is equal to zero, it is determined to be a constant amplitude oscillation component; and when the attenuation factor is greater than zero, it is determined to be a divergent oscillation component.

[0014] The present invention has the following beneficial technical effects: First, it offers high-precision and robust full-parameter extraction. By using SVD decomposition and ratio method to adaptively lock effective signal components, it significantly suppresses noise interference. Even at low signal-to-noise ratios, it can still stably output high-precision estimates of amplitude, frequency, phase, and attenuation factor. The estimation accuracy is significantly improved compared to traditional methods, providing a reliable data source for advanced applications.

[0015] Second, it has excellent time-varying dynamic tracking capabilities. The algorithm is highly sensitive to dynamic scenarios such as gradual changes in harmonic amplitude, sudden changes, and frequency jumps. Simulations show that it can quickly and accurately capture real-time changes in signal characteristics under various complex operating conditions, meeting the needs of online monitoring of the dynamic behavior of the power grid.

[0016] Third, its high-resolution analytical capability for complex multi-component and non-integer harmonics can effectively separate and accurately identify the coexisting fundamental wave, harmonics, interharmonics, and oscillating components, overcoming the limitations of Fourier methods in frequency resolution and component aliasing, and significantly enhancing its adaptability to complex power grid environments such as new energy grid connection and impulsive loads.

[0017] Fourth, it simplifies the system structure and improves detection speed and reliability. By replacing hardware filtering and phase-locked loops with software algorithms, it fundamentally eliminates the response delay, parameter drift, and aging effects introduced by hardware, resolving the contradiction between detection accuracy and speed, reducing the complexity and cost of the measurement system, and making it highly practical for engineering applications. Attached Figure Description

[0018] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

[0019] Figure 1 This is a distribution chart of singular values; Figure 2 Here is the algorithm flowchart; Figure 3 The waveform is a noisy test waveform; Figure 4 A graph showing the ratio of adjacent singular values; Figure 5 Figures (a), (b), and (c) show the target harmonic detection and reconstruction results under three different operating conditions. Detailed Implementation

[0020] The invention will now be described in further detail with reference to the accompanying drawings. These drawings are simplified schematic diagrams, illustrating only the basic structure of the invention, and therefore only show the components relevant to the invention.

[0021] This embodiment proposes a fast harmonic frequency division amplitude and phase detection method suitable for power grids, which is an improved Prony parameterization identification algorithm based on singular value decomposition (SVD) and adaptive order determination mechanism. This addresses the fundamental problems of traditional Prony methods, such as difficulty in model order selection and parameter estimation distortion in noisy environments. The specific process is as follows: I. Prony Algorithm Modeling and Solution Process The Prony algorithm is a parameter identification method that fits a signal to a linear combination of a series of complex exponential functions. p An exponential term is used to linearly fit the equally spaced sampled data.

[0022] Fitting x (0), x (1), ..., x ( N -1), the fitted data is denoted as Then we have: ; It is the discrete signal value sampled at equal intervals (the first... (Sampling points), with priority given to instantaneous values ​​of grid voltage or current; pThe model order is the number of complex exponential components in the signal, corresponding to the number of oscillation modes or harmonic components. b i For the first i The complex amplitude value of each component is defined as: ; z i For the first i The poles of each component (the basis of the complex exponential function) are defined as follows: ; This is a discrete-time index, corresponding to the time step. ; N This represents the total number of sampled data points. A i For the first i The amplitude of each component (unit: A or V, depending on the measured quantity); θ i For the first i The initial phase of each component (unit: radians or degrees). α i For the first i The attenuation factor of each component (unit: 1 / s). If α i <0 indicates damped oscillation. α i =0 indicates constant amplitude oscillation. α i >0 indicates divergent oscillation; f i For the first i The frequency of each component (unit: Hz); ∆ t Sampling time interval (unit: seconds); j It is the imaginary unit.

[0023] Construct the characteristic polynomial according to equation (1): ; Where l is the component index, and nm represents the time interval m sampling intervals forward relative to the nth sampling point; both sides of equation (5) are multiplied by a m And summing them, we get: ; make Substituting it into equation (5), we get: ; in z 1, z 2, ⋯, z n for The eigenvalues. Whenz = z i When, characteristic polynomial Then we have (assuming) a 0=1) ; To make the analog signal more consistent with the real data, the principle of least square error is adopted, namely: The goal of Prony's algorithm is to find the coefficients of the minimum value in equation (5). a i Based on the least squares method (8) and the simplified final sample function: From this, the normal equation of the Prony algorithm can be derived: in, ,( yes c (the conjugate of the complex number), ε p It is a perturbation variable. This is obtained by solving equation (10). a i Substituting into equation (6) yields the corresponding result. z i .Will z i Substituting into (1) and transforming, we can obtain a result containing The equation of the Vandermonde matrix: Finally, the result was obtained through calculation. b i and z i Further calculations were performed on the characteristic parameters of the mixed signal, including... θ i For the initial phase, A i The signal amplitude, a i As the attenuation factor, f i The signal oscillation frequency is calculated using the following formula: b i The modulus of the complex amplitude; Im( b i ), Re( b i ) represents the imaginary and real parts of the complex amplitude; ln| z i | is the natural logarithm of the modulus of the pole; arg( z i () is the argument of the pole (in radians); II. Real-time prony based on improved SVD a) Automatic order determination preprocessing Traditional Prony algorithms require pre-setting the model order. p However, the number of harmonic components in actual power grid signals is unknown. Insufficient order selection will miss components, while excessively high order will introduce spurious modes. This invention preprocesses the sampled signal by using singular value decomposition (SVD) to determine its rank, and adaptively determines the effective rank of the signal using the singular value ratio criterion. d Then d As the model order of the Prony algorithm, it enables high-precision identification of harmonic and interharmonic components without prior knowledge. The specific steps are as follows: Let the sampled signal sequence be... Construct the Hankel matrix: in , usually take Make the matrix approximately square. For Perform singular value decomposition Obtain the singular values ​​in descending order. .

[0024] In a specific embodiment, this invention estimates the effective order by performing singular value decomposition on the observation matrix and utilizing the difference in the distribution characteristics of singular values ​​between the noise subspace and the signal subspace. Specifically, the amplitude of singular values ​​corresponding to the noise subspace is significantly lower than that of the signal subspace, and their rate of change tends to be stable. The ratio method quantitatively characterizes the relative degree of change of singular values ​​by calculating the ratio change of adjacent singular values. This ratio can effectively reflect the transition characteristics between the signal subspace and the noise subspace: when the ratio gradually approaches 1, it indicates that the singular values ​​begin to stabilize, thus providing a criterion for determining the effective order.

[0025] in i =1,2,⋯ p -2, Ratio criterion, σ i and σ i+2 For the first i The singular value and the first singular value i+2 There are several singular values. Determining the critical points of the signal subspace using this method not only significantly improves the accuracy of parameter estimation, making it closer to the true parameters, but also effectively eliminates noise-dominated invalid data, thereby greatly reducing computational complexity. The specific process is as follows: Figure 2 As shown.

[0026] To verify the effectiveness of the ratio method for order determination of noisy signals, a sampled signal was tested, with specific parameters shown in Table 1. The expression for the sampled signal is as follows:

[0027] In the formula: It is the fundamental frequency signal of the power grid; Harmonics and oscillation signals.

[0028] Table 1 Main signal parameters

[0029] In the power supply systems of industrial and mining enterprises, rectifiers, as a typical representative of nonlinear loads, are the main harmonic sources of the 5th and 7th harmonics in the power grid; while lighting equipment with significant nonlinear characteristics (such as energy-saving lamps) is the main cause of the 3rd harmonic.

[0030] A total of 2000 data points were sampled, but only the first 800 were selected. The algorithm model order is generally chosen to be approximately 0.25 times the total number of sample points. In this embodiment, 0.2 times the total number of sample points was selected as the model order. Furthermore, 20dB of Gaussian white noise was added to the test signal, and the test signal waveform is shown below. Figure 3 As shown.

[0031] To select a reasonable ratio criterion, the relationship curve between the ratio and the singular value sequence is calculated as follows: Figure 4 As shown. When the ratio The ratio then fluctuated between 1 and 0.9. Therefore, when selecting... At that time, d =15, and it is assumed that the singular values ​​after d=15 correspond to the noise subspace.

[0032] b) Real-time prony detection After completing the adaptive order determination based on Singular Value Decomposition (SVD) and the ratio method to obtain the effective rank d, this invention further introduces a short-time Prony analysis strategy to address the dynamic scenario where power grid harmonics and interharmonic components change rapidly over time. The core idea of ​​this strategy is to divide the continuously acquired power grid signal into a series of overlapping or non-overlapping short-time data windows. Within each data window, an improved Prony parameter identification is performed using a predetermined model order d or fine-tuned based on the signal characteristics within the window. This allows the acquisition of the trajectory of the amplitude, frequency, phase, and attenuation factor of each harmonic and oscillation component over time. This method effectively overcomes the limitation of the traditional Prony algorithm, which is only applicable to stationary signals, enabling the tracking and real-time detection of time-varying harmonic components.

[0033] Construction of short-term data windows: Let the continuously sampled voltage or current signal sequence on the power grid side be... The sampling interval is Define the length as A sliding window with a sliding step size of 1. ( < , usually take =2 To ensure overlap rate). In the first Within each window, the signal subsequence is:

[0034] Window length The selection of frequency resolution and dynamic response speed needs to be considered: An excessively large value will lead to a delayed response to parameter changes, while an excessively small value may fail to accurately identify low-frequency components. Based on the actual needs of power grid harmonic analysis, a value is typically chosen... For two fundamental frequency periods, for example at 50Hz, take the window length. Corresponding to two fundamental frequency periods (i.e., 0.04s), if the sampling rate is 2kHz, then =80 points, take a sliding step size S=40 points.

[0035] For each short-time window, the traditional Prony algorithm requires re-determining the model order, which introduces significant computational latency. This invention utilizes the results of improved SVD preprocessing and employs one of the following two strategies to obtain the model order for the current window. .

[0036] For steady-state or slowly varying harmonic scenarios, directly let ,in The effective rank (number of signal components) of the initial full signal is obtained using the ratio method; if a significant drift is detected in the singular value ratio curve of the signal within the window, such as the ratio of adjacent singular values... If the inflection point position changes beyond a preset threshold, then SVD and ratio method ordering are re-executed and updated. This strategy, while ensuring computational efficiency, can adapt to sudden increases or decreases in harmonic components or frequency jumps. In the... Within each window, the determined model order is utilized. Parameter identification is performed following the classic Prony solution steps: Normal equation construction: based on window signal Calculate the sample function:

[0037] And construct the normal equation: in These are the linear prediction coefficients.

[0038] Solve for the characteristic polynomial: Reaching the extreme point Then, the complex amplitude is obtained by using the least squares method. Finally, the frequency, attenuation factor, amplitude, and initial phase of each component are obtained: Repeat the above steps for all sliding windows to obtain the parameter sequences of each harmonic and oscillation component. To avoid parameter jumps caused by window switching, the estimation results of adjacent windows can be smoothed by weighted averaging or Kalman filtering. Finally, the system outputs the complete parameter trajectories of the fundamental frequency, integer harmonics, non-integer interharmonics, and oscillation components in real time, realizing rapid dynamic detection of the frequency-division amplitude and phase of the power grid harmonic components.

[0039] To verify the superiority of the proposed harmonic frequency division amplitude and phase rapid detection method in terms of dynamic performance, resolution accuracy, and anti-interference capability, comparative tests were conducted under the following complex operating conditions. The fundamental signal was a 50Hz sinusoidal voltage, containing various steady-state and dynamic harmonic components (test circuit signal: .

[0040] The first operating condition aims to verify the method's ability to track dynamic changes in harmonic amplitude. Before 0.27s, a 75Hz damped oscillation component is simulated, its amplitude gradually increasing, stabilizing after 0.27s, and beginning to decay after 0.54s. The method's detection performance for harmonic amplitude is as follows: Figure 5 As shown in (a), the algorithm can accurately track the rise, stabilization, and decay of the amplitude, with a rapid response and no significant overshoot. The second operating condition was used to verify the method's adaptability and resolution capability to harmonic frequency abrupt changes. The initial signal contained a 66Hz damped oscillation component, which abruptly changed to 75Hz at 0.27s. The detection results are as follows... Figure 5 As shown in (b), the algorithm can quickly and accurately relock to a new frequency after a frequency abrupt change, and the estimation of amplitude and phase remains continuous and stable throughout the process. The third test scenario examines the method's ability to handle simultaneous abrupt changes in multiple target harmonics. At 0.27s, three interharmonic components—75Hz, 175Hz, and 375Hz—suddenly increase in the signal simultaneously. For example... Figure 5 As shown in (c), this method can simultaneously and accurately detect three interharmonics with similar or significantly different frequencies. The frequency, amplitude and phase of each component are effectively identified, and there is no significant interference between them.

[0041] Experimental results show that the method proposed in this invention can effectively cope with various complex dynamic scenarios such as harmonic amplitude changes, frequency jumps, and multi-component abrupt changes. Compared with traditional methods, it maintains high precision and high resolution while possessing excellent dynamic tracking capabilities and anti-interference performance. It achieves rapid, accurate, and robust frequency-division amplitude and phase detection of power grid harmonics and interharmonic components, providing a reliable technical basis for refined harmonic mitigation.

[0042] The above are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A rapid harmonic frequency division amplitude and phase detection method suitable for power grids, characterized in that, Includes the following steps: Step 1: Acquire voltage or current signals from the power grid to obtain a discrete sampled signal sequence; Step 2: Construct a Hankel matrix based on the discrete sampled signal sequence, and perform singular value decomposition on the Hankel matrix to obtain the singular values ​​arranged in descending order; Step 3: Calculate the ratio of adjacent singular values, and adaptively determine the effective rank of the signal based on the distribution characteristics of the ratio. Use the effective rank as the model order of the Prony algorithm. Step 4: Execute the Prony algorithm with the model order determined in Step 3 to fit the discrete sampled signal sequence with a complex exponential function, and solve for the frequency, amplitude, phase and attenuation factor of each harmonic and oscillation component, so as to realize the rapid detection of frequency-division amplitude and phase of power grid harmonics.

2. The method for rapid detection of harmonic frequency division amplitude and phase in power grids according to claim 1, characterized in that, In step three, the effective rank of the signal is adaptively determined based on the distribution characteristics of the ratio of adjacent singular values. Specifically, when the ratio of adjacent singular values ​​gradually changes from a value greater than 1 and begins to approach 1, the singular value index corresponding to the turning point in the ratio change process is determined as the critical point between the signal subspace and the noise subspace, and the critical point is used as the effective rank of the signal.

3. The method for rapid detection of harmonic frequency division amplitude and phase in power grids according to claim 1, characterized in that, Step four includes: Construct and solve the normal equations of the Prony algorithm to obtain the linear prediction coefficients; Construct a characteristic polynomial based on the linear prediction coefficients and solve it to obtain the poles of each component; Based on the poles, the complex amplitude values ​​of each component are solved using the least squares method; Based on the poles and complex amplitudes, the frequency, attenuation factor, amplitude, and initial phase of each harmonic and oscillation component are calculated.

4. The rapid harmonic frequency division amplitude and phase detection method for power grids according to claim 1, characterized in that, To address the time-varying characteristics of the signal, a short-window sliding processing strategy is employed when executing the Prony algorithm in step four, including: The discrete sampled signal sequence is divided into multiple short-time data windows; Within each short-time data window, the Prony algorithm is executed based on the model order determined in step three to obtain the parameters of each component within the short-time data window; By sequentially sliding through short-time data windows, the trajectory of the parameter evolution of each harmonic and oscillation component over time can be obtained, enabling real-time tracking and detection of time-varying harmonic components.

5. The rapid harmonic frequency division amplitude and phase detection method for power grids according to claim 4, characterized in that, Each short-time data window has a window length corresponding to at least two fundamental frequency periods, and the window sliding step size is smaller than the window length to ensure window overlap.

6. The method for rapid detection of harmonic frequency division amplitude and phase in power grids according to claim 4, characterized in that, It also includes a model order update step: during the sliding window processing, if it is detected that the singular value ratio curve of the signal in the current window has drifted by more than a preset threshold compared with the criterion of the initial effective rank, then steps two and three are re-executed to update the model order of the current window.

7. The method for rapid detection of harmonic frequency division amplitude and phase in power grids according to claim 4, characterized in that, After obtaining the parameters of each component within each window, the estimation results of the frequency, amplitude, phase and attenuation factor of the same harmonic or oscillation component between adjacent windows are subjected to weighted averaging or Kalman filtering smoothing to eliminate parameter jumps caused by window switching.

8. The method for rapid detection of harmonic frequency division amplitude and phase in power grids according to claim 1, characterized in that, The effective rank determined in step three is the effective rank of the initial full signal. When using the short-time window sliding processing strategy, the first short-time data window directly uses this initial effective rank as the model order of the Prony algorithm, and subsequent windows selectively update the model order based on the change in the singular value ratio of the current window signal.

9. The method for rapid detection of harmonic frequency division amplitude and phase in power grids according to claim 1, characterized in that, The attenuation factor obtained in step four is used to determine the dynamic characteristics of each harmonic or oscillation component: when the attenuation factor is less than zero, it is determined to be an attenuated oscillation component; when the attenuation factor is equal to zero, it is determined to be a constant amplitude oscillation component; and when the attenuation factor is greater than zero, it is determined to be a divergent oscillation component.

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