电力系统中的间谐波频谱分析
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摘要
电能是现代社会最重要的二次能源,无论是工农业生产,还是人们的日常生活都离不开电能。因此,电网的稳定安全和电能质量水平都直接影响国民经济。随着我国经济的快速发展,各种用电负荷大量涌入电网,其中很多是非线性负载和冲击性负载,产生了很多新的电能质量问题。这些问题不仅威胁电网的安全运行也影响用户的正常使用。谐波问题是电网中常见的电能质量问题,供电部门和用户对谐波都已有充分的认识,但电网中除了整数次谐波外还存在非整数次谐波——间谐波。间谐波除了造成与谐波相同的危害外还会引起照明设备闪变、干扰甚至损坏无功补偿或滤波装置等特有的危害。因此,间谐波问题也必须引起足够重视。
治理电力系统中的间谐波,前提是要准确的分析间谐波的频谱。目前IEC标准中关于间谐波分析的主要有IEC 61000-4-15中的闪变仪和IEC 61000-4-7与IEC 61000-4-30中的基于离散傅立叶变换的谐波、间谐波分族的方法。除此之外,AR模型、神经网络、支持向量机和小波分析等也都应用于间谐波分析。但这些方法存在分析范围窄、时间窗长、易受噪声干扰或频谱泄漏混叠等问题,为此,本文提出了一种基于空间谱分析的超分辨率方法。本文主要内容如下:
(1)本文综述了对间谐波的认识,其中包括定义、危害和限值等,并结合实际经验,详细介绍了每一种间谐波源:变频装置、感应电机、电弧炉、轧钢机和音频控制信号。针对间谐波与闪变的关系进行了深入研究,建立IEC闪变仪的仿真模型测试其对间谐波的检测能力及测量范围,建模分析了间谐波引起电压有效值波动和峰值波动的机理及对不同照明设备的影响。
(2)结合实例指出了IEC间谐波的分族方法在间谐波频率确定、相邻间谐波识别、同步采样、矩形窗和处理时变信号等方面存在的不足。即使通过先将采样数据补零获得更光滑的频谱曲线估计谱峰,再应用加窗插值进一步校正间谐波频率的改进方法,仍存在所需数据长度过长、临近谱峰难以区分等问题。AR模型虽然可以突破频率分辨率的限制,但其受阶数和噪声的严重影响;自适应神经网络在训练权值前,神经元的初始设定是以插值傅立叶变换结果为基础;支持向量机分析间谐波时需要做高维的线性回归,并且频谱泄漏严重;小波分析在时间轴上滑动不同通带的滤波器获得间谐波谱峰,但小波分析在各频段混叠严重,频率分辨率不高。解决上述方法的困难,需要事先估计间谐波参数模型正弦成分数,因此本文将空间谱分析引入到间谐波的分析中。
(3)介绍了空间谱分析的数学模型,证明了间谐波信号经欧拉公式或希尔伯特变换后可以转变为空间线性阵列接收的空域信号,正弦成分数对应空域信号源数,正弦成分频率对应空域信号频率。由于间谐波信号等效的空域阵列接收数据也满足阵列模型的统计特性,并且其变换后的空域信号之间互不相干,可以应用空间谱理论分析间谐波。
(4)应用空间谱分析中的信号源数估计方法来确定间谐波参数模型中的正弦成分数。这些算法有信息论方法、盖氏圆盘方法和典型相关方法。信息论方法提出了基于最大似然的判别信号源数的AIC、MDL和HQ准则;盖氏圆盘方法通过酉阵变换将协方差矩阵中对应信号和噪声的盖氏圆盘互相分开,并且噪声盖氏圆盘半径远小于信号的盖氏圆盘半径;典型相关方法是通过判断两个空间不同线阵接收数据的典型相关系数来获得信号源数。经仿真测试说明信息论中的MDL准则、盖氏圆盘方法和典型相关方法都适用于间谐波参数模型的正弦成分数估计。
(5)信号源数的准确估计可以使子空间类算法准确地划分信号子空间和噪声子空间,利用信号子空间和噪声子空间互相正交或利用信号子空间旋转不变性质估计正弦成分(间谐波)的频率。分析时主要应用子空间类算法中的Root-MUSIC算法和TLS-ESPRIT算法,其中Root-MUSIC算法不像MUSIC算法那样需要进行谱峰搜索,可以直接求解正弦成分的频率。通过计算多级维纳滤波器可以直接获得信号子空间的信息,无需对协方差矩阵做特征值分解,使ESPRIT算法计算量显著减少,然后通过总体最小二乘法考虑总体扰动就可以求出正弦成分的频率。仿真试验说明了Root-MUSIC算法和TLS-ESPRIT算法分析正弦成分频率的性能。
(6)最后,在幅值和相位信息估计后,本文提出了完整的间谐波分析算法。间谐波参数模型的正弦成分数和频率由空间谱分析获得,在低信噪比时使用支持向量机或遗传算法估计幅值和相位信息。IEC改进方法未能给出准确频谱的间谐波信号时,基于空间谱估计的方法均可以给出准确的分析结果。对实际间谐波数据进行分析,结果表明基于空间谱估计的间谐波方法优于IEC方法、支持向量机和Prony算法。
Nowadays electric energy is most important secondary energy. Electric power plays an important role in industry production and human lives. The stability of power system and the quality of electric power directly affect national economy. Along with the rapid development of economy, all kinds of loads are rushing into electric power system, some of which are nonlinear or impact loads. So new problems about power quality appear which threaten not only the safe operation of power system but also normal production of users. Harmonic is a common power quality event, and the utilities and users have paid much attention to it. Besides harmonic whose frequency is integer multiples of basic frequency, there is also interharmonic whose frequency is not integer multiples. Interharmonic can cause flicker and disturb reactive power compensation devices or filters besides the same hazards of harmonic, so we should also pay attention to interharmonic.
The precondition of repressing interharmonic is accurately analyzing the spectrum of interharmonic. IEC has provided two methods to evaluate interharmonic, IEC flickermeter proposed in IEC 61000-4-15 and grouping method based on Fourier transform in IEC 61000-4-7 and IEC 61000-4-30. Besides the aforementioned methods AR model, neural network, SVMs(Support Vector Machines) and wavelet analysis are also applied to analyze interharmonic. However these methods have shortcomings of narrow analyzing range, long time window, easily being influenced by the noise, spectrum leakage and aliasing. So a stepwise parametric method based on spatial spectrum analysis is presented in this paper. The main contents of this paper are shown as follows.
(1) Firstly, this paper introduced the basic concepts about interharmonic, including definitions, hazards and limited values. Then main interharmonic sources were shown, such as converter devices, inductive motors, arc furnaces, rolling mills and ripple control signals. In order to investigate the relationship between interharmonic and flicker, a simulation model of IEC flickermeter was built up, and numerous simulations tested the performance that IEC flickermeter evaluated interharmonic. Furthermore the mechanism that interharmonic incured R.M.S. and peak fluctuation was discussed and the influence of interharmonic to lighting devices were also shown.
(2) Actual instances showed the measuring method of interharmonic in IEC had flaws on estimating interharmonic frequency, estimating the number of interharmonics, synchronized sampling, the use of rectangular window, handling time-varying interharmonic and so on. Though the modified method of IEC padded zeroes at the end of sampling data to smooth the Fourier spectrum and Hanning window and interpolation were performed to correct the interharmonic information from DFT, it still cannot solve the problems at long sampling data and close interharmonic recognition. The methods based on AR model can break through the restriction from frequency resolution, but its performance was easily influenced by the order of AR model and the noise. It's not reliable that the number of neurons and its original value was estimated from the results of interpolation DFT of interharmonic signals before a self-adapting neural network trained weight parameters. When analyzing interharmonic, SVMs need carry out a higher dimensional linear regression, and had severe spectrum leakage. The wavelet analysis obtains interharmonic frequency by sliding the filters with different pass band on time axis, but easily influenced by wavelet aliasing. In order to solve the mention above problems, we need a method which can accurately estimate the number of sinusoidal components in interharmonic parametric model at first.
(3) After the mathematic model of spatial spectrum analysis was introduced, interharmonic signals can be transformed to spatial signals received by linear sensor array through Euler's formula or Hilbert transform. The number of sinusoidal components of the interharmonic parametric model was corresponding to the source number of transformed spatial signals, and the frequencies of sinusoidal components were corresponding to that of transformed spatial signals. The transformed received data also had the same statistic character with sensor array model and the transformed spatial signals were not coherent each other, so the methods of spatial spectrum analysis can be applied to evaluate the spectrum of interharmonic signals.
(4) The source number estimating methods were applied to estimate the number of sinusoidal components of the interharmonic parametric model, which were information theory, Gerschgorin's Disk and Canonical Correlation method. Based on maximum likelihood estimation AIC, MDL and HQ criterions from information theory were present to estimate the source number. The unitary transform of covariance matrix made the Gerschgorin's disks of signals and that of noise separated on complex plane, and Gerschgorin's radiis of signals are obviously greater than that of noise. Canonical Correlation method estimated source number by surveying the canonical correlation coefficients of the received data matrices form two space-separated linear sensor array. Numerous simulations proved that MDL of information theory, Gerschgorin's Disk and Canonical Correlation method had well performance on estimating the number of sinusoidal components.
(5) The estimated source number made it possible to accurately separate signal subspace from noise subspace. Using the orthogonality between signal subspace and noise subspace or the invariance of rotating signal subspace the frequencies of every sinusoidal component can be figured out. Root-MUSIC and TLS-ESPRIT algorithm belonging to subspace methods were used to analyze the frequency information of interharmonic signal. Root-MUSIC can directly obtain the frequencies through calculation roots of a polynomial, while MUSIC need searching the spectrum peaks. ESPRIT avoided eigenvalve decomposition of covariavance matrix after multi-stage Wiener filters were obtained to form signal subspace through forward recursive algorithm, so the calculation load significantly cut off. Based on the signal subspace the total least square method can get frequencies. Numerous simulations tested the performance of Root-MUSIC and TLS-ESPRIT.
(6) Finally, after the information of amplitudes and angle phases were obtained, we proposed an integrated method to analyze interharmonic spectrum. The number of sinusoidal components and their frequencies were obtained by the spatial spectrum analysis, and SVMs or GA can calculate amplitudes and angle phases when SNR was low. The proposed method succeeded in analyzing the simulated interharmonic signal while IEC modified method failed. Sampling data from actual interharmonic sources, inductive motor and rolling mill, were applied to test the proposed method, and the results showed that it performed better than IEC modified method, SVMs and Prony algorithm.
治理电力系统中的间谐波,前提是要准确的分析间谐波的频谱。目前IEC标准中关于间谐波分析的主要有IEC 61000-4-15中的闪变仪和IEC 61000-4-7与IEC 61000-4-30中的基于离散傅立叶变换的谐波、间谐波分族的方法。除此之外,AR模型、神经网络、支持向量机和小波分析等也都应用于间谐波分析。但这些方法存在分析范围窄、时间窗长、易受噪声干扰或频谱泄漏混叠等问题,为此,本文提出了一种基于空间谱分析的超分辨率方法。本文主要内容如下:
(1)本文综述了对间谐波的认识,其中包括定义、危害和限值等,并结合实际经验,详细介绍了每一种间谐波源:变频装置、感应电机、电弧炉、轧钢机和音频控制信号。针对间谐波与闪变的关系进行了深入研究,建立IEC闪变仪的仿真模型测试其对间谐波的检测能力及测量范围,建模分析了间谐波引起电压有效值波动和峰值波动的机理及对不同照明设备的影响。
(2)结合实例指出了IEC间谐波的分族方法在间谐波频率确定、相邻间谐波识别、同步采样、矩形窗和处理时变信号等方面存在的不足。即使通过先将采样数据补零获得更光滑的频谱曲线估计谱峰,再应用加窗插值进一步校正间谐波频率的改进方法,仍存在所需数据长度过长、临近谱峰难以区分等问题。AR模型虽然可以突破频率分辨率的限制,但其受阶数和噪声的严重影响;自适应神经网络在训练权值前,神经元的初始设定是以插值傅立叶变换结果为基础;支持向量机分析间谐波时需要做高维的线性回归,并且频谱泄漏严重;小波分析在时间轴上滑动不同通带的滤波器获得间谐波谱峰,但小波分析在各频段混叠严重,频率分辨率不高。解决上述方法的困难,需要事先估计间谐波参数模型正弦成分数,因此本文将空间谱分析引入到间谐波的分析中。
(3)介绍了空间谱分析的数学模型,证明了间谐波信号经欧拉公式或希尔伯特变换后可以转变为空间线性阵列接收的空域信号,正弦成分数对应空域信号源数,正弦成分频率对应空域信号频率。由于间谐波信号等效的空域阵列接收数据也满足阵列模型的统计特性,并且其变换后的空域信号之间互不相干,可以应用空间谱理论分析间谐波。
(4)应用空间谱分析中的信号源数估计方法来确定间谐波参数模型中的正弦成分数。这些算法有信息论方法、盖氏圆盘方法和典型相关方法。信息论方法提出了基于最大似然的判别信号源数的AIC、MDL和HQ准则;盖氏圆盘方法通过酉阵变换将协方差矩阵中对应信号和噪声的盖氏圆盘互相分开,并且噪声盖氏圆盘半径远小于信号的盖氏圆盘半径;典型相关方法是通过判断两个空间不同线阵接收数据的典型相关系数来获得信号源数。经仿真测试说明信息论中的MDL准则、盖氏圆盘方法和典型相关方法都适用于间谐波参数模型的正弦成分数估计。
(5)信号源数的准确估计可以使子空间类算法准确地划分信号子空间和噪声子空间,利用信号子空间和噪声子空间互相正交或利用信号子空间旋转不变性质估计正弦成分(间谐波)的频率。分析时主要应用子空间类算法中的Root-MUSIC算法和TLS-ESPRIT算法,其中Root-MUSIC算法不像MUSIC算法那样需要进行谱峰搜索,可以直接求解正弦成分的频率。通过计算多级维纳滤波器可以直接获得信号子空间的信息,无需对协方差矩阵做特征值分解,使ESPRIT算法计算量显著减少,然后通过总体最小二乘法考虑总体扰动就可以求出正弦成分的频率。仿真试验说明了Root-MUSIC算法和TLS-ESPRIT算法分析正弦成分频率的性能。
(6)最后,在幅值和相位信息估计后,本文提出了完整的间谐波分析算法。间谐波参数模型的正弦成分数和频率由空间谱分析获得,在低信噪比时使用支持向量机或遗传算法估计幅值和相位信息。IEC改进方法未能给出准确频谱的间谐波信号时,基于空间谱估计的方法均可以给出准确的分析结果。对实际间谐波数据进行分析,结果表明基于空间谱估计的间谐波方法优于IEC方法、支持向量机和Prony算法。
Nowadays electric energy is most important secondary energy. Electric power plays an important role in industry production and human lives. The stability of power system and the quality of electric power directly affect national economy. Along with the rapid development of economy, all kinds of loads are rushing into electric power system, some of which are nonlinear or impact loads. So new problems about power quality appear which threaten not only the safe operation of power system but also normal production of users. Harmonic is a common power quality event, and the utilities and users have paid much attention to it. Besides harmonic whose frequency is integer multiples of basic frequency, there is also interharmonic whose frequency is not integer multiples. Interharmonic can cause flicker and disturb reactive power compensation devices or filters besides the same hazards of harmonic, so we should also pay attention to interharmonic.
The precondition of repressing interharmonic is accurately analyzing the spectrum of interharmonic. IEC has provided two methods to evaluate interharmonic, IEC flickermeter proposed in IEC 61000-4-15 and grouping method based on Fourier transform in IEC 61000-4-7 and IEC 61000-4-30. Besides the aforementioned methods AR model, neural network, SVMs(Support Vector Machines) and wavelet analysis are also applied to analyze interharmonic. However these methods have shortcomings of narrow analyzing range, long time window, easily being influenced by the noise, spectrum leakage and aliasing. So a stepwise parametric method based on spatial spectrum analysis is presented in this paper. The main contents of this paper are shown as follows.
(1) Firstly, this paper introduced the basic concepts about interharmonic, including definitions, hazards and limited values. Then main interharmonic sources were shown, such as converter devices, inductive motors, arc furnaces, rolling mills and ripple control signals. In order to investigate the relationship between interharmonic and flicker, a simulation model of IEC flickermeter was built up, and numerous simulations tested the performance that IEC flickermeter evaluated interharmonic. Furthermore the mechanism that interharmonic incured R.M.S. and peak fluctuation was discussed and the influence of interharmonic to lighting devices were also shown.
(2) Actual instances showed the measuring method of interharmonic in IEC had flaws on estimating interharmonic frequency, estimating the number of interharmonics, synchronized sampling, the use of rectangular window, handling time-varying interharmonic and so on. Though the modified method of IEC padded zeroes at the end of sampling data to smooth the Fourier spectrum and Hanning window and interpolation were performed to correct the interharmonic information from DFT, it still cannot solve the problems at long sampling data and close interharmonic recognition. The methods based on AR model can break through the restriction from frequency resolution, but its performance was easily influenced by the order of AR model and the noise. It's not reliable that the number of neurons and its original value was estimated from the results of interpolation DFT of interharmonic signals before a self-adapting neural network trained weight parameters. When analyzing interharmonic, SVMs need carry out a higher dimensional linear regression, and had severe spectrum leakage. The wavelet analysis obtains interharmonic frequency by sliding the filters with different pass band on time axis, but easily influenced by wavelet aliasing. In order to solve the mention above problems, we need a method which can accurately estimate the number of sinusoidal components in interharmonic parametric model at first.
(3) After the mathematic model of spatial spectrum analysis was introduced, interharmonic signals can be transformed to spatial signals received by linear sensor array through Euler's formula or Hilbert transform. The number of sinusoidal components of the interharmonic parametric model was corresponding to the source number of transformed spatial signals, and the frequencies of sinusoidal components were corresponding to that of transformed spatial signals. The transformed received data also had the same statistic character with sensor array model and the transformed spatial signals were not coherent each other, so the methods of spatial spectrum analysis can be applied to evaluate the spectrum of interharmonic signals.
(4) The source number estimating methods were applied to estimate the number of sinusoidal components of the interharmonic parametric model, which were information theory, Gerschgorin's Disk and Canonical Correlation method. Based on maximum likelihood estimation AIC, MDL and HQ criterions from information theory were present to estimate the source number. The unitary transform of covariance matrix made the Gerschgorin's disks of signals and that of noise separated on complex plane, and Gerschgorin's radiis of signals are obviously greater than that of noise. Canonical Correlation method estimated source number by surveying the canonical correlation coefficients of the received data matrices form two space-separated linear sensor array. Numerous simulations proved that MDL of information theory, Gerschgorin's Disk and Canonical Correlation method had well performance on estimating the number of sinusoidal components.
(5) The estimated source number made it possible to accurately separate signal subspace from noise subspace. Using the orthogonality between signal subspace and noise subspace or the invariance of rotating signal subspace the frequencies of every sinusoidal component can be figured out. Root-MUSIC and TLS-ESPRIT algorithm belonging to subspace methods were used to analyze the frequency information of interharmonic signal. Root-MUSIC can directly obtain the frequencies through calculation roots of a polynomial, while MUSIC need searching the spectrum peaks. ESPRIT avoided eigenvalve decomposition of covariavance matrix after multi-stage Wiener filters were obtained to form signal subspace through forward recursive algorithm, so the calculation load significantly cut off. Based on the signal subspace the total least square method can get frequencies. Numerous simulations tested the performance of Root-MUSIC and TLS-ESPRIT.
(6) Finally, after the information of amplitudes and angle phases were obtained, we proposed an integrated method to analyze interharmonic spectrum. The number of sinusoidal components and their frequencies were obtained by the spatial spectrum analysis, and SVMs or GA can calculate amplitudes and angle phases when SNR was low. The proposed method succeeded in analyzing the simulated interharmonic signal while IEC modified method failed. Sampling data from actual interharmonic sources, inductive motor and rolling mill, were applied to test the proposed method, and the results showed that it performed better than IEC modified method, SVMs and Prony algorithm.
引文
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