WO2024040655A1 - 一种电能信号的分数域降噪方法 - Google Patents

一种电能信号的分数域降噪方法 Download PDF

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WO2024040655A1
WO2024040655A1 PCT/CN2022/119608 CN2022119608W WO2024040655A1 WO 2024040655 A1 WO2024040655 A1 WO 2024040655A1 CN 2022119608 W CN2022119608 W CN 2022119608W WO 2024040655 A1 WO2024040655 A1 WO 2024040655A1
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signal
frft
fourier transform
fractional
optimal
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PCT/CN2022/119608
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French (fr)
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陈蓉
杨勇
樊明迪
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苏州大学
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Priority to US18/019,873 priority Critical patent/US20240088657A1/en
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for ac mains or ac distribution networks
    • H02J3/24—Arrangements for preventing or reducing oscillations of power in networks
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/141—Discrete Fourier transforms
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J13/00—Circuit arrangements for providing remote indication of network conditions, e.g. an instantaneous record of the open or closed condition of each circuitbreaker in the network; Circuit arrangements for providing remote control of switching means in a power distribution network, e.g. switching in and out of current consumers by using a pulse code signal carried by the network
    • H02J13/00002—Circuit arrangements for providing remote indication of network conditions, e.g. an instantaneous record of the open or closed condition of each circuitbreaker in the network; Circuit arrangements for providing remote control of switching means in a power distribution network, e.g. switching in and out of current consumers by using a pulse code signal carried by the network characterised by monitoring

Definitions

  • the present application relates to the field of power electronics technology, and in particular to a fractional domain noise reduction method for electric energy signals.
  • Transient power quality disturbance is an important research topic related to power quality in power systems, which has an important impact on both the grid side and the user side.
  • Transient power quality disturbances usually include voltage sags, voltage swells, voltage interruptions, transient pulses and transient oscillation disturbances.
  • power quality monitoring mainly includes the following aspects: noise reduction, feature extraction and classification.
  • noise reduction In practical applications, power quality signals are often contaminated by noise during the transmission, measurement, and reception processes, causing useful signal features to be overwhelmed by noise, which in turn affects the accurate processing and analysis of subsequent signals. Therefore, effective denoising algorithms are crucial for power quality monitoring and analysis.
  • the fractional Fourier transform can characterize signals in the time fractional frequency domain, thereby achieving a high degree of energy accumulation of different signal components.
  • the FRFT kernel is an orthogonal chirp basis, it is very suitable for processing non-stationary signals, especially linear frequency modulation signals.
  • the calculation speed of the discrete FRFT algorithm is comparable to that of the fast Fourier transform algorithm.
  • FRFT can be applied to the noise reduction processing of non-stationary power quality disturbances.
  • Existing technologies have studied the noise reduction and identification methods of common power quality disturbance signals based on the FRFT algorithm. There has been a preliminary discussion on non-stationary linear frequency modulation interference in the existing technology, but the disturbance signal model is not a transient disturbance.
  • this application proposes the characteristics of non-stationary transient disturbance signals in power electronic power systems, and then proposes a new efficient denoising of power quality disturbances based on fractional Fourier transform algorithm.
  • this application proposes a fractional domain noise reduction method for electric energy signals, including:
  • fractional Fourier transform is defined as
  • p is the FRFT transformation order
  • ⁇ is the angle between the FRFT axis and the time axis
  • ⁇ p ⁇ /2
  • K p ( ⁇ ;u;t ) is the kernel function of the fractional Fourier transform, where n is an integer.
  • s ( t ) is the power frequency signal
  • d ( t ) is the transient disturbance signal
  • n ( t ) is Gaussian white noise
  • d ( t ) is the non-stationary transient disturbance signal
  • A is the amplitude of the disturbance signal
  • u ( t ) is the unit step signal
  • t 1 and t 2 are the starting and ending moments of the disturbance signal appearing respectively
  • f 1 is the starting frequency of the linear frequency modulation disturbance signal
  • the window function used in the band-pass filtering includes at least one of the following: rectangular window, Hanning window, Hamming window, and Blackman window.
  • the energy peak value of the power frequency signal is smaller than the energy peak value of the transient disturbance signal.
  • This application proposes an improved noise reduction algorithm based on fractional Fourier transform for the problem of noise reduction of transient power quality signals.
  • This method is not only suitable for transient stationary disturbance signals, such as voltage surges, dips, and interruptions, but also for transient non-stationary disturbance signals, such as linear frequency modulation interference.
  • chirp interference will be filtered out from the original power frequency signal just like noise, but it can be restored through fractional-order inverse Fourier transform, and then the interference signal characteristics can be extracted to analyze the cause of the disturbance.
  • this application also discusses the method of determining the optimal fractional transformation angle. Based on the fourth-order origin moment of the fractional spectrum, the optimal transformation angle can be efficiently determined through one-dimensional peak search. Experimental results show that the improved noise reduction algorithm based on fractional Fourier transform can effectively achieve noise filtering and retain transient disturbance positioning information.
  • Figure 1 is a block diagram of a traditional signal denoising algorithm based on fractional Fourier transform.
  • Figure 2 is a flow chart of the improved electric energy signal denoising method based on fractional Fourier transform of this application.
  • Figure 3 is an energy distribution diagram of the linear frequency modulation signal of the present application on a two-dimensional plane ( ⁇ , u).
  • Figure 4 is a distribution diagram of the fourth-order origin moment of the fractional-order spectrum of the chirp signal of the present application.
  • Figure 5 is a waveform diagram of the voltage swell signal before and after noise reduction and the residual noise of this application.
  • Figure 6 is a waveform diagram of the voltage interruption signal before and after noise reduction and the residual noise in this application.
  • Figure 7 is a waveform diagram of the chirp transient disturbance signal contaminated by noise in this application.
  • Figure 8 is a schematic diagram of the noise reduction processing process of the linear frequency modulation disturbance electric energy signal in this application.
  • Figure 9 is a schematic diagram of the root mean square error of the estimated value of the transient disturbance start and end times under different signal-to-noise ratios in this application.
  • Fractional Fourier transform is a general form of Fourier transform with linear and unitary properties and is defined as
  • p is the FRFT transformation order
  • ⁇ is the angle between the FRFT axis and the time axis
  • ⁇ p ⁇ /2
  • K p ( ⁇ ;u;t ) is the kernel function of the fractional Fourier transform, where n is an integer.
  • the signal x ( t ) can achieve optimal energy aggregation in the fractional domain, that is,
  • the noise-contaminated power quality signal will form different degrees of energy accumulation in different fractional domains.
  • the signal needs to be transformed into the optimal fractional domain to form the best energy aggregation.
  • s ( t ) is the power frequency signal
  • d ( t ) is the transient disturbance signal
  • n ( t ) is Gaussian white noise
  • disturbance signals such as voltage sag, voltage swell, and voltage interruption only change the amplitude of the power frequency signal within a period of time and have no effect on its frequency change rate, such disturbance signals obtain the best energy accumulation in the fractional domain Consistent with the power frequency signal.
  • the optimal fractional Fourier transform of the signal can be obtained by
  • Figure 1 shows the traditional signal denoising algorithm block diagram based on fractional Fourier transform.
  • A is the amplitude of the disturbance signal
  • u ( t ) is the unit step signal
  • t 1 and t 2 are the starting and ending moments of the disturbance signal respectively
  • f 1 is the starting frequency of the linear frequency modulation disturbance signal
  • k is the modulation frequency.
  • the modulation frequency of the linear frequency modulation disturbance signal is not 0, the angle of its optimal fractional Fourier transform will no longer be consistent with that of the power frequency signal, and multiple energy accumulation peaks will appear in the fractional domain. In this case, the traditional FRFT-based noise reduction algorithm is no longer applicable. In order to solve this problem, we can start from the relationship between the energy peak value of the power frequency signal and the disturbance signal and discuss it in two situations.
  • the energy peak value of the power frequency signal is smaller than the energy peak value of the disturbance signal.
  • the optimal FRFT rotation angle of the non-stationary disturbance signal d ( t ) will be estimated first.
  • the estimated value d ⁇ ( t ) of the non-stationary disturbance signal can be obtained by using the band-pass filter and the inverse fractional Fourier transform.
  • the relevant characteristics of transient disturbance can be obtained by analyzing d ⁇ ( t ).
  • Figure 2 shows the flow chart of the improved power signal denoising algorithm based on fractional Fourier transform, which includes the following steps:
  • the noise reduction algorithm based on fractional Fourier transform needs to process the signal in its optimal fractional transform domain. Therefore, how to estimate the optimal fractional transformation angle of the signal is crucial to the performance of the noise reduction algorithm.
  • the optimal transformation angle of FRFT is obtained by searching for peaks on the two-dimensional plane composed of the fractional transformation domain ( u ) and the transformation angle domain ( ⁇ ). The process can be expressed as
  • the fourth-order origin moment of the fractional-order spectrum of the signal x ( t ) is defined as
  • Figures 3 and 4 respectively show the energy distribution
  • the optimal fractional Fourier transform angle can be determined through one-dimensional search according to the fourth-order origin moment of the fractional spectrum of the signal, thereby greatly improving the calculation efficiency.
  • the fourth-order origin moment of the fractional spectrum at different transformation angles during signal-to-noise ratio transformation will increase as a whole as the noise increases. Therefore, the fourth-order origin moment of the fractional-order spectrum has good anti-noise performance, and this application uses it as an estimation algorithm for the optimal transformation angle.
  • bandpass filtering the signal components in the optimal fractional transform domain is an important step to eliminate the influence of noise and non-stationary disturbance signals. Obviously, the performance of the bandpass filter directly affects the noise reduction effect. Therefore, it is necessary to design appropriate bandpass filters based on different window function performances. Common window functions mainly include rectangular windows, Hanning windows, Hamming windows, Blackman windows, etc., and their performance is shown in Table 1.
  • s ( t ) and s ⁇ ( t ) are the signal before noise pollution and the restored signal obtained after noise reduction processing respectively.
  • the noise reduction algorithm based on the improved fractional Fourier transform is used for the voltage swell signal with an input signal-to-noise ratio of 10dB.
  • the waveforms before and after noise reduction and the residual noise are shown in Figure 5.
  • the signal waveform after noise reduction is smoother and the original signal characteristics are well preserved, and the output signal-to-noise ratio is increased to 21.35dB.
  • the residual noise as shown in Figure 5(c), it can be found that the waveform fluctuates greatly near 0.045s and 0.085s, the time points when the voltage surge occurs.
  • the impact of the noise reduction algorithm on the positioning results needs to be further discussed later. Influence.
  • Figure 6 shows the noise reduction results of the voltage interruption signal when the input signal-to-noise ratio is 10dB. Similarly, the noise is effectively filtered and the characteristics of the power quality signal are retained, and the output signal-to-noise ratio is increased to 19.21 dB. The overall residual noise is relatively flat, but the waveform still fluctuates greatly near the voltage interruption time of 0.045s and 0.085s.
  • Figure 7(a) is the original power frequency signal
  • Figure 7(b) is the electric energy signal containing transient linear frequency modulation interference
  • Figure 7(c) is the electric energy signal contaminated by noise, with a signal-to-noise ratio of 10dB.
  • the fourth-order origin moment of the normalized fractional spectrum of the signal is calculated, and the result is shown in Figure 8(a).
  • the energy peak value of the power frequency signal is smaller than the energy peak value of the disturbance signal.
  • the optimal FRFT transformation angle of the linear frequency modulation disturbance signal d ( t ) at this time is 1.099rad.
  • the fractional Fourier transform is performed to obtain the energy distribution in the fractional domain, as shown in Figure 8 ( b) shown.
  • the estimated value d ⁇ ( t ) of the linear frequency modulation disturbance signal can be obtained by using the band-pass filter and the inverse fractional Fourier transform, as shown in Figure 8(c).
  • the modulation frequency of the transient linear frequency modulation disturbance is different from that of the power frequency signal.
  • the noise reduction algorithm can be used to reconstruct and eliminate it first, and then the power frequency signal can be denoised without interference. After noise reduction, the residual noise is relatively flat overall, but still fluctuates slightly at the start and end moments of the interference signal.
  • the non-stationary transient disturbance signal is reconstructed, which is beneficial to further extracting its characteristic parameters in the future.
  • the improved algorithm and discrete wavelet transform (DWT) method proposed in this application are respectively used to denoise voltage swells, voltage interruptions and chirp interference.
  • DWT discrete wavelet transform
  • the input signal-to-noise ratio of the electric energy signal is 10dB
  • t' 1 and t' 2 are detection
  • ⁇ t 1 and ⁇ t 2 are estimation errors.
  • This application proposes an improved noise reduction algorithm based on fractional Fourier transform for the problem of noise reduction of transient power quality signals.
  • This method is not only suitable for transient stationary disturbance signals, such as voltage surges, dips, and interruptions, but also for non-stationary transient disturbance signals, such as transient linear frequency modulation interference.
  • non-stationary transient disturbances can be effectively reconstructed, which is beneficial to extracting the characteristics of the disturbance signal and analyzing the causes of the disturbance.
  • this application also discusses the method of determining the optimal fractional transformation angle. Based on the fourth-order origin moment of the fractional spectrum, the optimal transformation angle can be efficiently determined through one-dimensional peak search. Experimental results show that the improved noise reduction algorithm based on fractional Fourier transform can effectively achieve noise filtering and retain transient disturbance positioning information.

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Abstract

本申请提供一种电能信号的分数域降噪方法，包括：S1、估计原始信号 x( t)的最佳分数阶傅里叶变换角度式（I）；S2、计算原始信号在所述最佳分数阶傅里叶变换角度下的分数阶傅里叶变换，得到式（II）；S3、在最佳分数阶傅里叶变换域做带通滤波，得到式（II）；S4、计算在角度-式（I）下的分数阶傅里叶变换；S5、式（I）判断是否等于 π/2等，如果是则结束降噪，如果否，则消除恢复信号分量式（III），然后再次重复步骤S1-S5，直至估计得到的最佳FRFT旋转角度等于 π/2。本申请不仅适用于暂态平稳扰动信号，如电压暂升、暂降及中断等信号，更可适用于暂态非平稳扰动信号，如线性调频干扰。实验结果表明，改进的基于分数阶傅里叶变换的降噪算法可有效实现噪声滤除和暂态扰动定位信息的保留。

Description

一种电能信号的分数域降噪方法 技术领域
本申请涉及电力电子技术领域，尤其涉及一种电能信号的分数域降噪方法。
 背景技术
新能源大规模并网、特高压交直流输电和智能电网的快速发展，使电网在源、网、荷各环节呈现电力电子化特征。大量电力电子设备和非线性负载的应用，如变频器、变频调速系统、电动汽车充电装置等，导致了严重的信号污染进而引发电能质量（power quality，PQ）问题。高质量的电力系统监控越来越具有挑战性。
 技术问题
暂态电能质量扰动是电力系统中电能质量有关的一个重要研究课题，其对电网侧和用户侧都具有重要的影响。暂态电能质量扰动通常包含电压暂降、电压暂升、电压中断、瞬态脉冲和瞬态振荡扰动。为有效确定扰动原因避免设备损坏，必须实施有效的电能质量监测。通常，电能质量监测主要包含以下几个方面：降噪、特征提取和分类。实际应用中，电能质量信号往往在传输、测量及接收过程中被噪声污染，导致有用信号特征被噪声淹没，进而影响后续信号的准确处理和分析。因此，有效的去噪算法对电能质量的监测和分析至关重要。近年来，有关电能质量去噪涌现了大量研究成果，如基于傅里叶变换、小波变换、S变换、经验模态分解等算法的降噪方法。虽然这些方法具有较好的降噪效果，但它们大部分是在时域和频域对信号进行分析处理。然而，随着大量非线性快速负载的集成应用，暂态电能质量扰动及噪声可能呈现非平稳特性。与常见暂态电能质量扰动不同，非平稳暂态扰动的频率往往随时间变化，能量分布在频域中不再集中，进而导致其无法和工频信号一同与噪声实现有效分离。
作为传统傅里叶变换的推广，分数阶傅里叶变换（fractional Fourier transform，FRFT）可在时间分数阶频域上表征信号，进而实现不同信号分量的能量各自高度聚集。同时由于FRFT内核为正交chirp基，故非常适用于非平稳信号，特别是线性调频信号的处理。此外，离散FRFT算法的计算速度与快速傅里叶变换算法相当。基于上述优点，FRFT可适用于非平稳电能质量扰动的降噪处理。已有现有技术基于FRFT算法对常见电能质量扰动信号的降噪和识别方法进行了研究。已有现有技术对非平稳的线性调频干扰进行了初步讨论，但扰动信号模型不是暂态扰动。
 技术解决方案
有鉴于此，针对这一问题，本申请提出了电力电子化电力系统中非平稳暂态扰动信号的特征，进而提出了一种新的基于分数阶傅里叶变换的电能质量扰动的高效去噪算法。
基于上述目的，本申请提出了一种电能信号的分数域降噪方法，包括：
S1、估计原始信号 x( t)的最佳分数阶傅里叶变换角度 ；
S2、计算原始信号在所述最佳分数阶傅里叶变换角度下的分数阶傅里叶变换，得到 ；
S3、在最佳分数阶傅里叶变换域做带通滤波，得到 ；
S4、计算 在角度- 下的分数阶傅里叶变换；
S5、判断 是否等于 π /2等，如果是则结束降噪，如果否，则消除恢复信号分量 ，然后再次重复步骤S1-S5，直至估计得到的最佳FRFT旋转角度等于 π /2。
进一步地，所述分数阶傅里叶变换，定义为
   
     
其中， p为FRFT变换阶数， α为FRFT轴与时间轴之间的夹角，且 α =p π /2， K p ( α ;u;t) 为分数阶Fourier变换的核函数，式中 n为整数。
进一步地，所述原始信号 x( t)表示为
其中， s( t)为工频信号， d( t)为暂态扰动信号， n( t)为高斯白噪声。
进一步地，原始信号 x( t)的分数阶频谱四阶原点矩定义为
 
则最佳变换角度 可估计得
。
进一步地，
其中 ， ， ， 及分别是信号 x( t)， s( t)， d( t)和 n( t)在最佳变换角度下的FRFT结果。
进一步地， d( t)为非平稳暂态扰动信号，
  [0028] 式中， A为扰动信号幅值， u( t)为单位阶跃信号， t 1和 t 2分别是扰动信号出现的起止时刻， f 1为线性调频扰动信号的起始频率， k为其调频率。
进一步地，所述带通滤波中使用的窗函数包括以下至少一种：矩形窗、汉宁窗、海明窗、和布莱克曼窗。
进一步地，所述工频信号能量峰值小于暂态扰动信号能量峰值。
 有益效果
总的来说，本申请的优势及给用户带来的体验在于：
本申请针对暂态电能质量信号的降噪问题提出了一种改进的基于分数阶傅里叶变换的降噪算法。该方法不仅适用于暂态平稳扰动信号，如电压暂升、暂降及中断等信号，更可适用于暂态非平稳扰动信号，如线性调频干扰。降噪过程中，线性调频干扰会和噪声一样从原始工频信号中滤除，但可通过分数阶傅里叶反变换对其进行恢复，进而提取干扰信号特征以分析扰动产生原因。此外，本申请还对最佳分数阶变换角度的确定方法进行了讨论，基于分数阶频谱四阶原点矩可通过一维峰值搜索高效确定最佳变换角度。实验结果表明，改进的基于分数阶傅里叶变换的降噪算法可有效实现噪声滤除和暂态扰动定位信息的保留。
 附图说明
在附图中，除非另外规定，否则贯穿多个附图相同的附图标记表示相同或相似的部件或元素。这些附图不一定是按照比例绘制的。应该理解，这些附图仅描绘了根据本申请公开的一些实施方式，而不应将其视为是对本申请范围的限制。
图1为传统的基于分数阶傅里叶变换的信号降噪算法框图。
图2为本申请改进后基于分数阶傅里叶变换的电能信号降噪方法流程图。
图3为本申请线性调频信号在二维平面(α，u)上的能量分布图。
图4为本申请线性调频信号的分数阶频谱四阶原点矩分布图。
图5为本申请电压暂升信号降噪前后及残留噪声波形图。
图6为本申请电压中断信号降噪前后及残留噪声波形图。
图7为本申请受噪声污染的chirp暂态扰动信号波形图。
图8为本申请线性调频扰动电能信号的降噪处理过程示意图。
图9为本申请不同信噪比下暂态扰动起止时刻估计值的均方根误差示意图。
 本发明的实施方式
下面结合附图和实施例对本申请作进一步的详细说明。可以理解的是，此处所描述的具体实施例仅用于解释相关发明，而非对该发明的限定。另外还需要说明的是，为了便于描述，附图中仅示出了与有关发明相关的部分。
需要说明的是，在不冲突的情况下，本申请中的实施例及实施例中的特征可以相互组合。下面将参考附图并结合实施例来详细说明本申请。
本申请在分析常见暂态扰动信号的基础上，重点研究了电力电子化电力系统中非平稳 chirp类扰动信号的特征，进而提出了一种新的基于分数域分析的电能质量扰动的高效去噪算法。
1. 分数阶傅里叶变换
分数阶傅里叶变换是傅里叶变换的一般形式，具有线性和单一特性，其定义为
其中， p为FRFT变换阶数， α为FRFT轴与时间轴之间的夹角，且 α =p π /2， K p ( α ;u;t) 为分数阶Fourier变换的核函数，式中 n为整数。
设信号 x( t)的频率为 f( t)，则其频率变化率可定义
根据FRFT定义，可推导得 x( t)在分数域的最佳变换角度可由其频率变化率求得，即
在最佳变换角度下，信号 x( t)可在分数域内实现最佳能量聚集，即| X α ( u)| 2取得最大值。
2. 基于分数阶傅里叶变换的电能质量降噪算法
2.1传统算法
根据变换角度的不同，受噪声污染的电能质量信号会在不同的分数域中形成不同程度的能量聚集。为了最大程度地滤除噪声，需要将信号变换到最佳分数域内形成最佳能量聚集。
设受噪声污染的电能质量信号表示为
其中， s( t)为工频信号， d( t)为暂态扰动信号， n( t)为高斯白噪声。
由于电压暂降、电压暂升、电压中断等扰动信号仅在一段时间内改变工频信号的幅值，而对其频率变化率没有影响，因此，此类扰动信号获得最佳能量聚集的分数域与工频信号一致。对信号进行最佳分数阶傅里叶变换可以得到
其中， 是 s( t)的最佳分数域变换角度， ， ， 及 分别是信号 x( t)， s( t)， d( t)和 n( t)在最佳变换角度下的FRFT结果。鉴于高斯白噪声在分数域无法形成能量聚集，因此可利用带通滤波器 H( u)在信号的最佳分数阶变换域实现信号与噪声的分离，进而得到
随后对 进行- 角度的分数阶傅里叶变换，即分数阶傅里叶反变换，便可获得降噪后的信号。图1展示的即为传统的基于分数阶傅里叶变换的信号降噪算法框图。
2.2改进算法
由2.1节分析可知，当扰动信号与工频信号具有相同的调频率时，两类信号的最佳分数阶变换角度一致，传统的基于分数阶傅里叶变换的算法可有效实现降噪功能。然而，在电力电子化电力系统中，随着非线性、快速负载的大量应用，电能质量扰动可能呈现非平稳行为，即扰动信号的频率在一个观察周期内无法保持恒定，如以暂态线性调频扰动信号为例，暂态非平稳扰动可定义为
式中， A为扰动信号幅值， u( t)为单位阶跃信号， t 1和 t 2分别是扰动信号出现的起止时刻， f 1为线性调频扰动信号的起始频率， k为其调频率。
由于线性调频扰动信号的调频率不为0，因此其最佳分数阶傅里叶变换的角度将不再与工频信号的一致，分数域中将呈现多个能量聚集峰。这种情形下，传统的基于FRFT的降噪算法不再适用。为了解决这一问题，可以从工频信号和扰动信号能量峰值的大小关系出发，分两种情况进行讨论。
（1）工频信号能量峰值大于扰动信号能量峰值。由于工频信号的调频率为0，即其最佳FRFT角度 = π /2。因此，若估计得到的最佳分数阶傅里叶变换角度等于π/2，则此时可直接应用传统降噪算法将调频率不为零的非平稳扰动和高斯白噪声一并滤除。若需进一步分析暂态非平稳扰动信号的特征，则可从原始信号中消除降噪后得到的电能信号，对剩余信号进一步实施基于FRFT的降噪算法，即可得对暂态扰动进行重构。
（2）工频信号能量峰值小于扰动信号能量峰值。此时，非平稳扰动信号 d( t)的最佳FRFT旋转角度将率先被估计得到。利用带通滤波器及分数阶傅里叶反变换可以求得非平稳扰动信号的估计值 d´( t)。而后从原始信号 x( t)中消除 d´( t)，对 x´( t)= x( t)- d´( t)再次应用FRFT降噪算法直至估计得到的最佳FRFT旋转角度等于 π /2。暂态扰动的相关特征可对 d´( t)进行分析得到。
图2给出了改进后的基于分数阶傅里叶变换的电能信号降噪算法流程图，包括如下步骤：
S1、估计原始信号 x( t)的最佳分数阶傅里叶变换角度 ；
S2、计算原始信号在所述最佳分数阶傅里叶变换角度下的分数阶傅里叶变换，得到；
 
其中 ， ， ， 及分别是信号 x( t)， s( t)， d( t)和 n( t)在最佳变换角度下的FRFT结果。
S3、在最佳分数阶傅里叶变换域做带通滤波，得到 ；
S4、计算 在角度- 下的分数阶傅里叶变换；
S5、判断是否等于 π /2等，如果是则结束降噪，如果否，则消除恢复信号分量 ，再次重复步骤S1-S5，直至估计得到的最佳FRFT旋转角度等于 π /2。
2.3最佳分数阶傅里叶变换角度的估计方法
根据第2.2节的分析可知，基于分数阶傅里叶变换的降噪算法需要对信号在其最佳分数阶变换域中进行处理。因此，如何估计信号的最佳分数阶变换角度对降噪算法的性能至关重要。通常，FRFT的最佳变换角度通过在分数阶变换域( u)和变换角度域( α)构成的二维平面上搜索峰值来获得，其过程可表述为
其中 是最佳变换角度的估计值， 是最佳变换域上能量峰值对应的坐标。
显然，二维峰值搜索会导致相当大的计算复杂度。为解决这一问题，基于模糊度函数，二阶FRFT中心矩被证明可用于最佳变换角度的快速获得。然而，二阶FRFT中心矩对噪声十分敏感，其性能不如四阶FRFT中心矩。进一步考虑计算复杂度，分数阶频谱四阶原点矩表现更为优秀。
信号 x( t)的分数阶频谱四阶原点矩定义为
则最佳变换角度可估计得
图3、图4分别展示了线性调频信号在二维平面( α， u)上的能量分布| X α ( u)| 2和其分数阶频谱四阶原点矩 η( α)在一维平面( α)上的分布情况。
与二维峰值搜索相比，根据信号的分数阶频谱四阶原点矩可通过一维搜索确定最佳分数阶傅里叶变换角度，进而使计算效率获得大幅提升。此外，从图4中还可发现，信噪比变换时不同变换角度下的分数阶频谱四阶原点矩会随着噪声的增强而整体抬升。因此，分数阶频谱四阶原点矩具有很好的抗噪性能，本申请采用其作为最佳变换角度的估计算法。
2.4分数域带通滤波器的选择
确定最佳分数阶变换角度后，在最佳分数阶变换域中对信号分量进行带通滤波是消除噪声和非平稳扰动信号影响的重要步骤，显然带通滤波器的性能直接影响降噪效果。因此，需要根据不同的窗函数性能设计合适的带通滤波器。常见的窗函数主要包括矩形窗、汉宁窗、海明窗、布莱克曼窗等，其性能如表1所示。
为了尽可能减少信号能量损失，同时有效滤除噪声干扰，在设计滤波器时希望窗函数的主瓣宽度窄且旁瓣幅值能够尽快衰减。然而，一般窗函数难以同时满足以上两方面的性能要求，工程应用中往往需要根据实际情况进行综合考虑。对比发现，虽然汉宁窗的主瓣宽度为矩形窗的2倍，但其主瓣旁瓣最大峰值和旁瓣峰值衰减性能明显优于矩形窗，其综合性能在常见的窗函数中相对最优，因此本申请采用汉宁窗设计分数域的带通滤波器，进而对降噪算法性能进行验证。
3. 仿真实验
为了验证算法性能，对电压暂升、电压中断和非平稳暂态扰动（以暂态线性调频干扰信号为例）这三种电能质量问题在MATLAB环境下进行仿真实验。信号采样频率为15 kHz，基波频率为50 Hz，带通滤波器选用汉宁窗。定义信噪比（SNR）以评估降噪效果，如式（12）所示。
其中， s( t)和 s´( t)分别是受噪声污染前信号和降噪处理后得到的恢复信号。
3.1降噪处理过程及信噪比分析
3.1.1电压暂降信号降噪实验
对输入信噪比为10dB的电压暂升信号采用基于改进的分数阶傅里叶变换降噪算法，其降噪前后及残留噪声的波形如图5所示。从图5(a)、(b)可以看出，降噪后的信号波形更为光滑且原信号特征得到了较好的保留，输出信噪比提升至21.35dB。进一步缩小纵坐标范围观察残留噪声，如图5(c)所示，可以发现在电压暂升出现的时刻点0.045s和0.085s附近波形波动较大，后续需进一步讨论降噪算法对定位结果的影响。
3.1.2电压中断信号降噪实验
图6给出的是输入信噪比为10dB时电压中断信号的降噪结果。同样的，噪声得到了有效滤除且电能质量信号的特征得到保留，输出信噪比提升至19.21 dB。残留噪声整体较为平坦，但在电压中断时刻0.045s和0.085s附近波形仍波动较大。
从图5、图6的仿真结果可以看出改进算法可有效实现电能信号的降噪。
3.1.3非平稳暂态扰动降噪实验
对于非平稳暂态扰动的仿真结果则如图7和图8所示。首先，图7(a)为原始工频信号，图7(b)为含有暂态线性调频干扰的电能信号，图7(c)是受噪声污染后的电能信号，信噪比为10dB。
为了估计最佳分数阶变换角度，计算信号的归一化分数阶频谱四阶原点矩，结果如图8(a)所示。显然，此时工频信号能量峰值小于扰动信号能量峰值。利用一维峰值搜索求得此时线性调频扰动信号 d( t)的最佳FRFT变换角度为1.099rad，在此角度下做分数阶傅里叶变换得到分数阶域的能量分布，如图8(b)所示。利用带通滤波器及分数阶傅里叶反变换可以求得线性调频扰动信号的估计值 d´( t)，如图8(c)所示。而后从原始信号 x( t)中消除 d´( t)，对剩余信号 x´( t)= x( t)- d´( t)再次应用FRFT降噪算法。图8(d)为 x´( t)的归一化分数阶频谱四阶原点矩，图8(e)为最佳变换角度的分数阶傅里叶变换结果，图8(f)为经降噪处理恢复后的工频信号，图8(g)为残留噪声。
从上述实验可以看到，暂态线性调频扰动的调频率与工频信号不同。当其能量较大时，可以利用降噪算法首先对其进行重构和消除，而后便可在没有干扰的情况下对工频信号进行降噪处理。降噪后，残留噪声整体较为平坦，但在干扰信号的起止时刻仍稍有波动。此外，非平稳暂态扰动信号得到了重构，有利于后续进一步提取其特征参量。
当输入信噪比从10 dB到20 dB变化时，分别采用本申请提出的改进算法和离散小波变换（DWT）方法对电压暂升、电压中断和线性调频干扰进行去噪处理。选择db6小波作为母小波，并执行两级分解和软阈值。表2给出了输出信噪比的比较。
可以看到，两种方法对于电压暂升电能信号的处理结果较为接近，而对于电压中断和线性调频干扰，本申请提出的改进算法的性能优于传统的小波变换算法。
3.2暂态扰动起止时刻的定位性能分析
电能质量信号处理中，暂态扰动起止时刻的定位是一个重要的研究内容。然而，电能信号经降噪处理后，定位信息往往会被滤除或削弱。因此，是否能够有效的保留定位信息是衡量降噪算法性能的一个指标。采用基于FRFT的改进算法分别对含有电压暂升、电压中断和非平稳暂态扰动的电能信号进行降噪处理，而后对降噪后含有电压暂升、电压中断的电能信号以及重构的非平稳暂态信号进行离散小波变换处理，根据细节系数求得暂态扰动的起止时刻，结果如表3所示。其中，电能信号的输入信噪比为10dB，各类暂态扰动的起始时刻理论值为 t 1=0.045s，终止时刻理论值为 t 2=0.085s， t' 1和 t' 2为检测估计值，△ t 1和△ t 2为估计误差。
当信噪比变化时，进一步讨论经降噪算法处理后信号起止时刻的定位效果。令信噪比变化范围为0dB到10dB，每个信噪比下分别进行100次Monte Carlo仿真，图9(a)是不同信噪比下扰动起始时刻测量值的均方根误差值，图9(b)是不同信噪比下扰动终止时刻测量值的均方根误差值。
实验结果表明，经本申请降噪算法处理后，暂态扰动信号的起止时刻信息得到了较好的保留，可以进行准确定位。
4. 总结
本申请针对暂态电能质量信号的降噪问题提出了一种改进的基于分数阶傅里叶变换的降噪算法。该方法不仅适用于暂态平稳扰动信号，如电压暂升、暂降及中断等信号，更可适用于非平稳暂态扰动信号，如暂态线性调频干扰。降噪过程中，可对非平稳暂态扰动进行有效重构，进而有利于提取扰动信号特征，分析扰动产生原因。此外，本申请还对最佳分数阶变换角度的确定方法进行了讨论，基于分数阶频谱四阶原点矩可通过一维峰值搜索高效确定最佳变换角度。实验结果表明，改进的基于分数阶傅里叶变换的降噪算法可有效实现噪声滤除和暂态扰动定位信息的保留。

Claims (8)

  1. 一种电能信号的分数域降噪方法，其特征在于，包括：
    S1、估计原始信号 x(t)的最佳分数阶傅里叶变换角度 ；
    S2、计算原始信号在所述最佳分数阶傅里叶变换角度下的分数阶傅里叶变换，得到 ；
    S3、在最佳分数阶傅里叶变换域做带通滤波，得到 ；
    S4、计算 在角度- 下的分数阶傅里叶变换；
    S5、判断 是否等于 π /2等，如果是则结束降噪，如果否，则消除恢复信号分量 ，然后再次重复步骤S1-S5，直至估计得到的最佳FRFT旋转角度等于 π /2。
  2. 根据权利要求1所述的方法，其特征在于，
    所述分数阶傅里叶变换，定义为
    其中， p为FRFT变换阶数， α为FRFT轴与时间轴之间的夹角，且 α =p π/2， K p ( α;u;t) 为分数阶傅里叶变换的核函数，式中 n为整数。
  3. 根据权利要求2所述的方法，其特征在于，
    所述原始信号 x( t)表示为
    其中， s( t)为工频信号， d( t)为暂态扰动信号， n( t)为高斯白噪声。
  4. 根据权利要求3所述的方法，其特征在于，
    原始信号 x( t)的分数阶频谱四阶原点矩定义为
    则最佳变换角度 可估计得
    。
  5. 根据权利要求4所述的方法，其特征在于，
    其中， ， ， 及 分别是信号 x( t)， s( t)， d( t)和 n( t)在最佳变换角度下的FRFT结果。
  6. 根据权利要求3所述的方法，其特征在于，
    d( t)为非平稳暂态扰动信号，
    式中， A为扰动信号幅值， u( t)为单位阶跃信号， t 1和 t 2分别是扰动信号出现的起止时刻， f 1为线性调频扰动信号的起始频率， k为其调频率。
  7. 根据权利要求1所述的方法，其特征在于，
    所述带通滤波中使用的窗函数包括以下至少一种：矩形窗、汉宁窗、海明窗、和布莱克曼窗。
  8. 根据权利要求3所述的方法，其特征在于，
    所述工频信号能量峰值小于暂态扰动信号能量峰值。
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