基于自适应陷波滤波器的电力信号时频分析
|
摘要
电力系统电压和电流可归结为多个正弦分量叠加的有限带宽概周期信号,其时频分析主要包括把输入信号分解为不同频率正弦成分之和,并估计每个正弦分量的频率、幅值、相角,以提供电力系统状态评价和保护的基本依据。本文提出频率自适应梳状滤波器的时频分析,属于非线性微分动力系统,不同于常用积分变换方法。
首先,提出了二维线性正弦跟踪器以及幅值和相角估计算法。该跟踪器是依据最小方差原则和梯度下降方法经旋转变换而得到的二维线性常微分方程。本文给出了暂态和稳态响应算式,说明算法的频率参数和带宽参数的物理意义,分析了参数大小对暂态和稳态性能的影响,通过仿真比较,验证算法较离散傅立叶变换和非线性正弦跟踪器的优点。
其次,提出了线性梳状滤波器。采用多个二维线性正弦跟踪器并联,形成带宽可调的多频点线性梳状滤波器。证明了一致渐近稳定性,给出频率响应算式,分析了参数值对动态和稳态响应性能影响。说明不论是谐波还是间谐波,只要输入信号所有正弦分量的频率都落在梳状滤波器频率分割点上时,滤波器就能够同时准确跟踪所有分量,并准确估计每个分量的幅值、相角。通过仿真比较,验证了算法性能优于离散傅立叶变换和非线性梳状滤波器。
第三,用两级线性梳状滤波器分析基波电压闪变。当电压信号中包含谐波和间谐波成分时,国际电工委员会(IEC)推荐的闪变检测方法存在理论误差。本文先用一个梳状滤波器实现电压信号分解,除去谐波和间谐波成分,获得包含闪变的基波电压幅值跟随,再用另一个梳状滤波器分析基波幅值跟随信号,提取基波电压的恒定值和波动分量,然后对电压波动进行加权滤波、平方、平滑滤波后得到瞬时视感度。仿真说明算法有效抑制了间谐波造成的平方解调法检测闪变度的误差,具有较好的抗干扰性能。
第四,基于二维线性正弦跟踪器提出了非归一化和归一化两种频率估计算法。两种估计器都是三维自适应陷波滤波器,具有渐近稳定的缓慢积分流形,能够准确估计频率未知的单个正弦信号的频率、幅值和相角。非归一化频率估计器的动态响应速度随着输入信号幅值的减小而变慢,而归一化频率估计器的动态响应速度基本不受输入信号幅值大小的影响,对幅值变化具有更好的鲁棒性。文中分析了带宽参数和频率自适应增益的大小对动态响应速度与噪声抑制性能的影响。
第五,基于线性梳状滤波器提出非归一化基波频率估计算法。用李雅普诺夫定理和平均方法证明缓慢自适应积分流形的存在性和稳定性。对于基波频率未知的多谐波信号,如果已知信号中的谐波和间谐波次数,并据此设定线性梳状滤波器的频率参数,则算法是一致渐近稳定的,能够准确跟随基波频率与各个正弦成分及其幅值。仿真结果验证了算法性能,说明了减小滤波器带宽参数和自适应增益能够获得更好的噪声抑制性能。
第六,提出了频率自适应梳状滤波器。本文运用慢积分流形实现状态估计和频率估计两个多维非线性微分方程之间的解耦,获得关于多个估计频率的概周期非线性动力系统,再应用平均方法导出估计频率的非线性自治方程。在给定的估计频率空间中,估计频率的平均方程的平衡点有三种情况:孤立平衡点、平衡点连续体和没有严格的平衡点。分析了孤立平衡点的指数稳定性与平衡点连续体的半稳定性以及没有平衡点时系统的鲁棒性。频率估计的暂态响应速度主要决定于频率自适应增益,幅值估计的暂态响应速度主要决定于带宽参数,频率和幅值的稳态精度决定于频率自适应增益与带宽参数的乘积。若把频率轴划分为多个区间,只要每个正弦分量仅仅位于一个频率区间内,本算法就能准确估计每个正弦分量的频率、幅值和相角。通过仿真验证了算法性能。
最后,研究了算法在非仿真环境或嵌入式系统中的实现。针对离散时间信号,用四阶龙格-库塔方法实现本文所提出的新算法,证明离散算法的收敛性和稳定性。推导了离散算法的计算公式,通过了仿真验证公式的正确性。
The voltage and the current in a power system is commonly regarded as an almost periodic signal of which the time-frequency analysis mainly concerns the different sinusoidal components that are tracked and estimated for exact amplitudes, initial phases as well as the frequencies, and thus forms part of the basis of the evaluation and protection of power system. In this dissertation, a new method for time-frequency analysis that is in nature a nonlinear differential dynamic system in contrary to the traditional integral methods is put forward and may be called as frequency adaptive comb filter.
A linear sinusoid tracer that is a two-dimensional linear ordinary differential equation deduced by using least square error principle, gradient descent method and rotation transformation can yield the estimates of amplitude and initial phase. The transient and steady state responses are studied with descriptions of the physical significances of the parameters such as frequency and bandwidth as well as their effects on the transient and steady state performance. The comparison of the simulation results validated the superior performance of this algorithm to DFT and nonlinear sinusoid tracers.
The concept of an adjustable-bandwidth multi-frequency linear comb filter composed of a number of two-dimensional linear sinusoid tracers connected in parallel is then put forward and with its asymptotic stability proved. The expression of the frequency response and the effect of the bandwidth on transient and steady state performance are provided. It shows that the instantaneous value as well as the amplitude and the phase angle of each sinusoidal component can be accurately estimated while the frequencies of all the components whether harmonic or inter-harmonic of input signal are located on the division of the filter. The better performance than FFT and nonlinear comb filter is validated by simulation results comparison.
The method of voltage flicker measurement recommended by IEC brings theoretical error while the voltage signal consists of harmonics and inter-harmonics, so a two-stage linear comb filter is tried to detect the fundamental voltage flicker. The first stage is used to decompose the input voltage to obtain the fluctuating fundamental signal with the deletion of the harmonics and inter-harmonics. The second stage is used to analyze such signal to get the fundamental amplitude and its fluctuation. Then the instantaneous flicker level can be obtained by weighted filtering, squaring and smoothing. Simulation verified that the new method can effectively restrain the flicker error caused by inter-harmonics and provide good ability of anti-jamming.
Based on two-dimensional linear sinusoid tracer, two algorithms for frequency estimation, i.e. the normalized and the non-normalized, are presented. Being a three-dimensional adaptive notch filter, these two estimators form slow integral manifold of which the existence and asymptotic stability are proved. The influence of the bandwidth and the frequency adaptive gain on transient and steady state performance are discussed. The frequency, amplitude and phase angle of an unknown sinusoid signal can be exactly estimated. The decreased amplitude results the slow transient response of the non-normalized estimator while the speed of the normalized algorithm is hardly influenced by input amplitude due to the better robustness.
A linear comb filter based non-normalized fundamental frequency estimator is also proposed. The method forms slow adaptive integral manifold of which the existence and stability are proved by Lyapunov stability theorem and averaging method. With the frequency parameters of the filter adjusted to the same orders of the harmonics and inter-harmonics in the input signal, then the integral manifold is uniformly asymptotically stable and the fundamental frequency as well as the amplitudes of all the harmonics and inter-harmonics can be precisely traced in exponential convergence. The validity of the proposed algorithm is verified by simulation and it is pointed out that better noise characteristic can be achieved by decreasing bandwidth and adaptive gain.
It is put forward at last an algorithm of frequency adaptive comb filter. The algorithm includes two coupled nonlinear differential equations respectively for state estimate and frequency updating, and the decoupling is carried out by slow integral manifolds to reach an almost periodic nonlinear dynamic system for multi frequencies estimation, then the nonlinear autonomous equation for the frequency estimation deduced by averaging. With the frequency estimate space given, the equilibrium of the average system can be an isolated equilibrium, a continuum of equilibriums, or non-strict equilibrium. The exponential stability of the isolated equilibrium, the semi-stability of the continuum of equilibriums, and the robustness of system with no equilibrium are investigated respectively. The convergence speed of the frequencies and the amplitudes is governed by the adaptive gain and the bandwidth parameter respectively. The frequency axis is divided into a number of zones corresponding to the high and low limits of the frequency estimates setup beforehand. The frequency, amplitude and phase angle of each sinusoidal component can be exactly estimated if it is in one of the zones. The algorithm is validated by simulations.
Finally, the implementation of the proposed algorithms without simulation tools or in embedded systems is discussed. The new algorithm can be implemented by forth order Runge-Kutta method for discrete time signal with the convergence and stability verified.
首先,提出了二维线性正弦跟踪器以及幅值和相角估计算法。该跟踪器是依据最小方差原则和梯度下降方法经旋转变换而得到的二维线性常微分方程。本文给出了暂态和稳态响应算式,说明算法的频率参数和带宽参数的物理意义,分析了参数大小对暂态和稳态性能的影响,通过仿真比较,验证算法较离散傅立叶变换和非线性正弦跟踪器的优点。
其次,提出了线性梳状滤波器。采用多个二维线性正弦跟踪器并联,形成带宽可调的多频点线性梳状滤波器。证明了一致渐近稳定性,给出频率响应算式,分析了参数值对动态和稳态响应性能影响。说明不论是谐波还是间谐波,只要输入信号所有正弦分量的频率都落在梳状滤波器频率分割点上时,滤波器就能够同时准确跟踪所有分量,并准确估计每个分量的幅值、相角。通过仿真比较,验证了算法性能优于离散傅立叶变换和非线性梳状滤波器。
第三,用两级线性梳状滤波器分析基波电压闪变。当电压信号中包含谐波和间谐波成分时,国际电工委员会(IEC)推荐的闪变检测方法存在理论误差。本文先用一个梳状滤波器实现电压信号分解,除去谐波和间谐波成分,获得包含闪变的基波电压幅值跟随,再用另一个梳状滤波器分析基波幅值跟随信号,提取基波电压的恒定值和波动分量,然后对电压波动进行加权滤波、平方、平滑滤波后得到瞬时视感度。仿真说明算法有效抑制了间谐波造成的平方解调法检测闪变度的误差,具有较好的抗干扰性能。
第四,基于二维线性正弦跟踪器提出了非归一化和归一化两种频率估计算法。两种估计器都是三维自适应陷波滤波器,具有渐近稳定的缓慢积分流形,能够准确估计频率未知的单个正弦信号的频率、幅值和相角。非归一化频率估计器的动态响应速度随着输入信号幅值的减小而变慢,而归一化频率估计器的动态响应速度基本不受输入信号幅值大小的影响,对幅值变化具有更好的鲁棒性。文中分析了带宽参数和频率自适应增益的大小对动态响应速度与噪声抑制性能的影响。
第五,基于线性梳状滤波器提出非归一化基波频率估计算法。用李雅普诺夫定理和平均方法证明缓慢自适应积分流形的存在性和稳定性。对于基波频率未知的多谐波信号,如果已知信号中的谐波和间谐波次数,并据此设定线性梳状滤波器的频率参数,则算法是一致渐近稳定的,能够准确跟随基波频率与各个正弦成分及其幅值。仿真结果验证了算法性能,说明了减小滤波器带宽参数和自适应增益能够获得更好的噪声抑制性能。
第六,提出了频率自适应梳状滤波器。本文运用慢积分流形实现状态估计和频率估计两个多维非线性微分方程之间的解耦,获得关于多个估计频率的概周期非线性动力系统,再应用平均方法导出估计频率的非线性自治方程。在给定的估计频率空间中,估计频率的平均方程的平衡点有三种情况:孤立平衡点、平衡点连续体和没有严格的平衡点。分析了孤立平衡点的指数稳定性与平衡点连续体的半稳定性以及没有平衡点时系统的鲁棒性。频率估计的暂态响应速度主要决定于频率自适应增益,幅值估计的暂态响应速度主要决定于带宽参数,频率和幅值的稳态精度决定于频率自适应增益与带宽参数的乘积。若把频率轴划分为多个区间,只要每个正弦分量仅仅位于一个频率区间内,本算法就能准确估计每个正弦分量的频率、幅值和相角。通过仿真验证了算法性能。
最后,研究了算法在非仿真环境或嵌入式系统中的实现。针对离散时间信号,用四阶龙格-库塔方法实现本文所提出的新算法,证明离散算法的收敛性和稳定性。推导了离散算法的计算公式,通过了仿真验证公式的正确性。
The voltage and the current in a power system is commonly regarded as an almost periodic signal of which the time-frequency analysis mainly concerns the different sinusoidal components that are tracked and estimated for exact amplitudes, initial phases as well as the frequencies, and thus forms part of the basis of the evaluation and protection of power system. In this dissertation, a new method for time-frequency analysis that is in nature a nonlinear differential dynamic system in contrary to the traditional integral methods is put forward and may be called as frequency adaptive comb filter.
A linear sinusoid tracer that is a two-dimensional linear ordinary differential equation deduced by using least square error principle, gradient descent method and rotation transformation can yield the estimates of amplitude and initial phase. The transient and steady state responses are studied with descriptions of the physical significances of the parameters such as frequency and bandwidth as well as their effects on the transient and steady state performance. The comparison of the simulation results validated the superior performance of this algorithm to DFT and nonlinear sinusoid tracers.
The concept of an adjustable-bandwidth multi-frequency linear comb filter composed of a number of two-dimensional linear sinusoid tracers connected in parallel is then put forward and with its asymptotic stability proved. The expression of the frequency response and the effect of the bandwidth on transient and steady state performance are provided. It shows that the instantaneous value as well as the amplitude and the phase angle of each sinusoidal component can be accurately estimated while the frequencies of all the components whether harmonic or inter-harmonic of input signal are located on the division of the filter. The better performance than FFT and nonlinear comb filter is validated by simulation results comparison.
The method of voltage flicker measurement recommended by IEC brings theoretical error while the voltage signal consists of harmonics and inter-harmonics, so a two-stage linear comb filter is tried to detect the fundamental voltage flicker. The first stage is used to decompose the input voltage to obtain the fluctuating fundamental signal with the deletion of the harmonics and inter-harmonics. The second stage is used to analyze such signal to get the fundamental amplitude and its fluctuation. Then the instantaneous flicker level can be obtained by weighted filtering, squaring and smoothing. Simulation verified that the new method can effectively restrain the flicker error caused by inter-harmonics and provide good ability of anti-jamming.
Based on two-dimensional linear sinusoid tracer, two algorithms for frequency estimation, i.e. the normalized and the non-normalized, are presented. Being a three-dimensional adaptive notch filter, these two estimators form slow integral manifold of which the existence and asymptotic stability are proved. The influence of the bandwidth and the frequency adaptive gain on transient and steady state performance are discussed. The frequency, amplitude and phase angle of an unknown sinusoid signal can be exactly estimated. The decreased amplitude results the slow transient response of the non-normalized estimator while the speed of the normalized algorithm is hardly influenced by input amplitude due to the better robustness.
A linear comb filter based non-normalized fundamental frequency estimator is also proposed. The method forms slow adaptive integral manifold of which the existence and stability are proved by Lyapunov stability theorem and averaging method. With the frequency parameters of the filter adjusted to the same orders of the harmonics and inter-harmonics in the input signal, then the integral manifold is uniformly asymptotically stable and the fundamental frequency as well as the amplitudes of all the harmonics and inter-harmonics can be precisely traced in exponential convergence. The validity of the proposed algorithm is verified by simulation and it is pointed out that better noise characteristic can be achieved by decreasing bandwidth and adaptive gain.
It is put forward at last an algorithm of frequency adaptive comb filter. The algorithm includes two coupled nonlinear differential equations respectively for state estimate and frequency updating, and the decoupling is carried out by slow integral manifolds to reach an almost periodic nonlinear dynamic system for multi frequencies estimation, then the nonlinear autonomous equation for the frequency estimation deduced by averaging. With the frequency estimate space given, the equilibrium of the average system can be an isolated equilibrium, a continuum of equilibriums, or non-strict equilibrium. The exponential stability of the isolated equilibrium, the semi-stability of the continuum of equilibriums, and the robustness of system with no equilibrium are investigated respectively. The convergence speed of the frequencies and the amplitudes is governed by the adaptive gain and the bandwidth parameter respectively. The frequency axis is divided into a number of zones corresponding to the high and low limits of the frequency estimates setup beforehand. The frequency, amplitude and phase angle of each sinusoidal component can be exactly estimated if it is in one of the zones. The algorithm is validated by simulations.
Finally, the implementation of the proposed algorithms without simulation tools or in embedded systems is discussed. The new algorithm can be implemented by forth order Runge-Kutta method for discrete time signal with the convergence and stability verified.
引文
[1]佟为明,宋雪雷,李凤阁,等.电网信号四象限功率测量原理及实现方法.电力系统自动化,2006,30(21):49-53.
[2]李生虎.线路距离III段保护动作裕度的节点功率控制算法.电力系统自动化,2006,30(16):27-32.
[3]郭创新,朱传柏,曹一家,等.电力系统故障诊断的研究现状与发展趋势.电力系统自动化,2006,30(8):98-103.
[4]程浩忠,艾芊,张志刚,等.电能质量.北京:清华大学出版社,2006,128-250.
[5] E L Owen. A History of Harmonics in Power Systems. IEEE Industry Applications Magazine, 1998, 4(1):6-12.
[6]王兆安,杨君,刘进军.谐波抑制和无功功率补偿.北京:机械工业出版社,1998.
[7] Shuter T C ,Vollkommer H T. Survey of harmonic levels on the American Electric power distribution system. IEEE Transactions on Power Ddlvery, 1989, (4): 2204- 2213.
[8] Hu Chung-Hsing,Wu Chi-Jui,et al. Survey of harmonic voltage and current at distribution substation in Northern Taiwan. IEEE Transactions on Power Delivery,1997,12(3):1275-1284.
[9] Jiao Lianwei,Diao Qinhua,et al. Harmonic survey and solutions of Xining power grid in western China. Internationa1 Conference on Power System Technology,Perth,2000,Vol.3:1161-1165.
[10]罗安.电网谐波治理和无工补偿技术及装备.北京:中国电力出版社,2006年.
[11]Srinivasan K. Digital measurement of the voltage flicker. IEEE Transactions on Power Delivery, 1991, 6(4):1593-1998.
[12] Hung S, Hsie. Application of continues wavelet transform for study of voltage flicker-generated signals . IEEE Transactions on Aerospace and Electric Systems, 2000, 36(3):925-932.
[13] AL-Hasawi W M, ELNaggar K M. A genetic based algorithm for voltage flicker measurement. IEE/ Melecon 2002, Cairo, Egypt, 2002: 600-604.
[14]Andria G, Savino M, Trotta A. Windows and interpolation algorithms to improve electrical measurement accuracy, IEEE Trans. Instrumentation and Measurement, 1989, 38(4):856-863.
[15]Hidalgo R M, Fernandez G, Rivera R R, et al. A Simple Adjustable Window Algorithm to Improve FFT Measurements. IEEE Trans. instrumentation and measurement, 2002, 51(1):31-36.
[16]张伏生,耿中行,葛耀中.电力系统谐波分析的高精度FFT算法.中国电机工程学报,1999,19(3):63-66.
[17]Zhang Fu-sheng, Geng Zhon-gxing, Yuan Wei. The Algorithm of InterpolatingWindowed FFT for Harmonic Analysis of Electric Power System. IEEE Trans. power delivery, 2001, 16(2):160–164.
[18]Testa A, Gallo D, Langella R. On the Processing of harmonics and interharmonics: Using Hanning window in standard framework. IEEE Trans. power delivery, 2004, 19(1):28–34.
[19]黄纯,江亚群.谐波分析的加窗插值改进算法.中国电机工程学报,2005,25(15):26-32.
[20]钱昊,赵荣祥.基于插值FFT算法的间谐波分析.中国电机工程学报,2005,25(21):87-91.
[21]陈明凯,郑翔骥,汪晓强.减少频谱泄漏的一种等角度间隔采样递推算法.电工技术学报,2005,20(8):94-98.
[22]赵凤展,杨仁刚.基于短时傅立叶变换的电压暂降扰动检测.中国电机工程学报,2007,27(10):28-34.
[23]薛雪东,程旭德,徐兵,李友利.基于STFT的高压电气设备局放信号时频分析.高电压技术,2008, 34(1):70-72.
[24]李弼程,罗建书.小波分析及其应用.北京:电子工业出版社,2003.
[25]杨桦,任震.基于小波变换检测谐波的新方法.电力系统自动化,1997,21(10):39-42.
[26]杜天军.基于抗混叠小波理论的电力系统谐波检测与抑制研究.电子科技大学博士学位论文,2006.
[27]赵成勇,何明锋.基于复小波变换相位信息的谐波检测算法.中国电机工程学报,2005,25(1):35-42.
[28]Tarasiuk T. Hybrid Wavelet–Fourier Method for Harmonics and Harmonic Subgroups Measurement—Case Study. IEEE Trans. power delivery, 2007, 22(1):4–17.
[29]Hamid E Y, Kawasaki Z I. Instrument for the quality analysis of power systems based on the wavelet packet transform. IEEE Power Engineering Review, 2002, 22(3): 52–54.
[30]王成山,王继东.基于小波包分解的电能质量扰动分类方法.电网技术,2004,28(15):78-82.
[31]陈祥训,刘兵,赵波.分析电能质量扰动的实小波域修正相子法.中国电机工程学报,2006,26(24):37-42.
[32]牛发亮,黄进.基于电磁转矩复解析小波变换的感应电机转子故障检测.电工技术学报,2005,20(7):100-105.
[33]陈丽安,张培铭.基于小波变换的短路故障早期检测门限值的研究.电工技术学报,2005,20(3):64-69.
[34]谭杨红,何怡刚.模拟电路故障诊断的小波方法.电工技术学报,2005,20(8):89-93.
[35]周林,夏雪,万蕴杰,等.基于小波变换的谐波测量方法综述.电工技术学报,2006,21(9):67-74.
[36]李天云,赵妍,李楠.基于EMD的Hilbert变换应用于暂态信号分析.电力系统自动化,2005,29(4):49-52.
[37]庞浩,王赞基,陈建业,赵伟.基于2对Hilbert移相滤波器的无功功率测量方法.电力系统自动化,2006,30(18):45-48.
[38]舒泓,王毅.基于数学形态滤波和Hilbert变换的电压闪变检测.中国电机工程学报, 2008, 28(1):111-114.
[39]丁仁杰,刘健,张隽,赵玉伟.一种基于瞬时无功功率理论的SVC控制方法.电工技术学报,2006,21(5):47-51.
[40]李鑫,曾光,苏彦民,张静钢.基于瞬时无功功率理论的SVC无功电流检测方法.电力电子技术,2006,40(5):121-123.
[41]李圣清,朱英浩,周有庆,彭玉楼.基于瞬时无功功率理论的四相输电谐波电流监测方法.中国电机工程学报, 2004, 24(3):12-17.
[42]张萍,余建明,魏磊.用瞬时无功功率理论求电压闪变参数.高电压技术,2007,33(4):134-137.
[43] P A Regalia. An improved lattice-based IIR notch filter. IEEE Trans. Signal Process, 1991, 39(9):2124–2128.
[44] M Bodson, S C Douglas. Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency. Automatica. 1997, 33(12): 2213–2221.
[45] L Hsu, R Ortega, G Damm. A globally convergent frequency estimator. IEEE Trans. Automatic Control, 1999, 44(4): 698–713.
[46] Mojiri M, Bakhshai A R. An adaptive notch filter for frequency estimation of a periodic signal, IEEE Trans. Autom. Control, 2004, 49(2):314–318.
[47] Karimi-Ghartemani M, Ziarani A K. A nonlinear adaptive filter for online signal analysis in power systems: Applications. IEEE Trans. on power delivery, 2002, 17(2): 617–622.
[48] Karimi-Ghartemani M,Ziarani A K.Periodic orbit analysis of two dynamical systems for electrical engineering applications.Journal of Engineering Mathematical,2003,45(2):135-154.
[49] Karimi-Ghartemani M, Ziarani A K. Performance characterization of a nonlinear system as both an adaptive notch filter and a phase-locked loop. International Journal of Adaptive Control Signal Process, 2004, 18(1):23–53.
[50] Ziarani A K,Konrad A.A method of extraction of nonstationary sinusoids.Signal Process,2004,84(8):1323-1346.
[51] Karimi-Ghartemani M,Ziarani A K.A nonlinear time-frequency analysis method.IEEE Trans.Signal Process,2004,52(6):1585-1595.
[52] Karimi-Ghartemani M, Iravani M R. Measurement of Harmonics/Inter-harmonics of Time-varying Frequencies. IEEE Trans. Power Delivery, 2005, 20(1): 23–31.
[53] McNamara D M, Ziarani A K, Ortmeyer T H. A New Technique of Measurement of Nonstationary Harmonics. IEEE Trans. power delivery, 2007, 22(1): 387–395.
[54] Mojiri M, Karimi-Ghartemani M, Bakhshai A. Time-Domain Signal Analysis Using Adaptive Notch Filter , IEEE Trans. Signal Process, 2007, 55(1): 85–93.
[55] Tarek M, Melchilef S, Rahim N A. Application of adaptive notch filter for harmonics currents estimation. In: Proceedings of IPEC 2007 Meeting Singapore, SINGAPORE: IEEE, 2007, 1236-1240.
[56] Mvuma A, Nishimura S, Hinamoto T. Tracking analysis of an adaptive IIR notch filter using gradient-based algorithm. In: Proceedings of IEEE International Symposium on Circuits and Systems. Seattle, WA: IEEE, 2008, 1148-1151.
[57] Torbjo rn Johansson A, White P R. Instantaneous frequency estimation at low signal-to-noise ratios using time-varying notch filters. Signal Processing, 2008, 88(5): 1271-1288.
[58] M Ta, DeBrunner V. Adaptive notch filter with time-frequency tracking of continuously changing frequencies. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing. Las Vegas, NV: IEEE, 2008. 3557-3560.
[1] Zhang Fu-sheng, Geng Zhon-gxing, Yuan Wei. The Algorithm of Interpolating Windowed FFT for Harmonic Analysis of Electric Power System. IEEE Trans. power delivery, 2001, 16(2):160–164.
[2] Karimi-Ghartemani M, Ziarani A K. A nonlinear adaptive filter for online signal analysis in power systems: Applications. IEEE Trans. on power delivery, 2002, 17(2): 617–622.
[3] Karimi-Ghartemani M,Ziarani A K.Periodic orbit analysis of two dynamical systems for electrical engineering applications.Journal of Engineering Mathematical,2003,45(2):135-154.
[4] Karimi-Ghartemani M, Ziarani A K. Performance characterization of a nonlinear system as both an adaptive notch filter and a phase-locked loop. International Journal of Adaptive Control Signal Process, 2004, 18(1):23–53.
[5] Ziarani A K,Konrad A.A method of extraction of nonstationary sinusoids.Signal Process,2004,84(8):1323-1346.
[6] Girgis A A, Chang W B, Makram E B. A digital recursive measurement scheme for on-line tracking of power system harmonics. IEEE Trans. power delivery, 1991, 6(3):1153–1160.
[7]储昭碧,张崇巍,冯小英.一种可调带宽的线性电力信号实时分析新算法.中国电机工程学报,2009,29(1):94-99.
[1] Owen E L. A history of harmonics in power systems. IEEE Industry Application Magazine, 1998, 4(1):6-12.
[2] Hidalgo R M, Fernandez J G, Rivera R R, et al. A Simple Adjustable Window Algorithm to Improve FFT Measurements. IEEE Trans. instrumentation and measurement, 2002, 51(1):31-36.
[3] Hidalgo R M, Fernandez J G, Rivera R R, et al. A Simple Adjustable Window Algorithm to Improve FFT Measurements. IEEE Trans. instrumentation and measurement, 2002, 51(1):31-36.
[4] Zhang Fu-sheng, Geng Zhon-gxing, Yuan Wei. The Algorithm of Interpolating Windowed FFT for Harmonic Analysis of Electric Power System. IEEE Trans. power delivery, 2001, 16(2):160-164.
[5] Testa A, Gallo D, Langella R. On the Processing of harmonics and interharmonics: Using Hanning window in standard framework. IEEE Trans. power delivery, 2004, 19(1):28-34.
[6] Sozanski K P. The shunt active power filter with better dynamic performance. In: proceedings of IEEE Lausanne Powertech. Lausanne, SWITZERLAND, IEEE, 2007. 1504-1508.
[7] Hamid E Y, Kawasaki Z I. Instrument for the quality analysis of power systems based on the wavelet packet transform. IEEE Power Engineering Review, 2002, 22(3): 52-54.
[8] Tarasiuk T. Hybrid Wavelet–Fourier Method for Harmonics and Harmonic Subgroups Measurement—Case Study. IEEE Trans. power delivery, 2007, 22(1):4-17.
[9]周林,夏雪,万蕴杰,等.基于小波变换的谐波测量方法综述.电工技术学报,2006,21(9):67-74.
[10]Karimi-Ghartemani M,Ziarani A K.A nonlinear time-frequency analysis method.IEEE Trans.Signal Process,2004,52(6):1585-1595.
[11]吴麒,王诗宓.自动控制原理(下册).北京:清华大学出版社,2006:448-450.
[12]张贤达.矩阵分析与应用.北京:清华大学出版社,2004:56-71.
[13]贾秀芳,赵成勇,胥国毅,等. IEC闪变仪误差分析及改进设计.电工技术学报, 2006, 21(11):121-126.
[14]李庆,任普春,赵海翔,等.针对风电引起闪变的数字化测试系统仿真.电力自动化设备,2008,28(1):69-72.
[15]徐福华,周国梁,王建增,等.基于DSP的数字化闪变仪研制.电测与仪表,2008,45(5): 12-15.
[16]孙佐,王念春,许卫兵.双DSP结构电网谐波监测及保护一体化装置研究.电子测量与仪器学报,2007,21(5):84-89.
[1] Y. Xiao, Y. Takeshita, K. Shida, Steady-state analysis of a plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 2001, 48(7): 733–740.
[2] M Karimi-Ghartemani, A K Ziarani. Performance characterization of a nonlinear system as both an adaptive notch filter and a phase-locked loop. International Journal of Adaptive Control Signal Process, 2004, 18(1):23–53.
[3] David M. McNamara, Alireza K. Ziarani, Thomas H. Ortmeyer. A New Technique of Measurement of Nonstationary Harmonics. IEEE Trans. on power delivery, 2007, 22(1): 387–395.
[4] P A Regalia. An improved lattice-based IIR notch filter. IEEE Trans. Signal Process, 1991, 39(9):2124–2128.
[5] M Bodson, S C Douglas. Adaptive algorithms for the rejection of sinusoidaldisturbances with unknown frequency. Automatica. 1997, 33(12): 2213–2221.
[6] L Hsu, R Ortega, G Damm. A globally convergent frequency estimator. IEEE Trans. Automatic Control, 1999, 44(4): 698–713.
[7] M. Mojiri, A. R. Bakhshai. An adaptive notch filter for frequency estimation of a periodic signal, IEEE Trans. Autom. Control, 2004, 49(2):314–318
[8] Mohsen Mojiri, Masoud Karimi-Ghartemani, Alireza Bakhshai. Time-Domain signal analysis Using Adaptive Notch Filter. IEEE Trans. Signal Process, 2007, 55(1): 85–93.
[9] B Riedle, P Kokotovic. Integral manifolds of slow adaptation. IEEE Trans. Automatic Control, 1986, AC-31(4):316–324.
[10]林振声.概周期微分方程与积分流形.上海:上海科学技术出版社,1982:166–219.
[1] Karimi-Ghartemani M, Iravani R M. Measurement of Harmonics/Inter-harmonics of Time-Varying Frequencies. IEEE Trans. Power Delivery, 2005, 20(1): 23-31.
[2] Khanshan A H, Amindavar H. Spectral estimation through parallel filtering. In: Proceedings of Australasian Telecommunication Networks and Applications Conference, Christchurch, New Zealand: IEEE, 2007.420-424.
[3] Riedle B, Kokotovic P. Integral manifolds of slow adaptation. IEEE Trans. Automatic Control, 1986, AC-31(4): 316-324.
[4] Khalil H K著,朱义胜,董辉,李作洲,等译.非线性系统.北京:电子工业出版社, 2005: 78-311.
[5] Bhat S P, Bernstein D S. Arc-length-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria. In: Proceedings of the American Control Conference, Denver, USA: IEEE, 2003. 2961-2966. [6 ]Hui Q, Haddad W M, Bhat S P. Lyapunov and Converse Lyapunov Theorems for Semistability. In: Proceedings of IEEE conference on Decision and Control, New Orleans, USA: IEEE, 2007. 5870-5874.
[1] Steven C. Chapra, Raymond P. Canale,著.唐玲艳,田尊华,刘齐军译.工程数值方法.北京:清华大学出版社,2007.
[2]李瑞遐,何至庆.微分方程数值方法.上海:华东理工大学出版社,2005.
[2]李生虎.线路距离III段保护动作裕度的节点功率控制算法.电力系统自动化,2006,30(16):27-32.
[3]郭创新,朱传柏,曹一家,等.电力系统故障诊断的研究现状与发展趋势.电力系统自动化,2006,30(8):98-103.
[4]程浩忠,艾芊,张志刚,等.电能质量.北京:清华大学出版社,2006,128-250.
[5] E L Owen. A History of Harmonics in Power Systems. IEEE Industry Applications Magazine, 1998, 4(1):6-12.
[6]王兆安,杨君,刘进军.谐波抑制和无功功率补偿.北京:机械工业出版社,1998.
[7] Shuter T C ,Vollkommer H T. Survey of harmonic levels on the American Electric power distribution system. IEEE Transactions on Power Ddlvery, 1989, (4): 2204- 2213.
[8] Hu Chung-Hsing,Wu Chi-Jui,et al. Survey of harmonic voltage and current at distribution substation in Northern Taiwan. IEEE Transactions on Power Delivery,1997,12(3):1275-1284.
[9] Jiao Lianwei,Diao Qinhua,et al. Harmonic survey and solutions of Xining power grid in western China. Internationa1 Conference on Power System Technology,Perth,2000,Vol.3:1161-1165.
[10]罗安.电网谐波治理和无工补偿技术及装备.北京:中国电力出版社,2006年.
[11]Srinivasan K. Digital measurement of the voltage flicker. IEEE Transactions on Power Delivery, 1991, 6(4):1593-1998.
[12] Hung S, Hsie. Application of continues wavelet transform for study of voltage flicker-generated signals . IEEE Transactions on Aerospace and Electric Systems, 2000, 36(3):925-932.
[13] AL-Hasawi W M, ELNaggar K M. A genetic based algorithm for voltage flicker measurement. IEE/ Melecon 2002, Cairo, Egypt, 2002: 600-604.
[14]Andria G, Savino M, Trotta A. Windows and interpolation algorithms to improve electrical measurement accuracy, IEEE Trans. Instrumentation and Measurement, 1989, 38(4):856-863.
[15]Hidalgo R M, Fernandez G, Rivera R R, et al. A Simple Adjustable Window Algorithm to Improve FFT Measurements. IEEE Trans. instrumentation and measurement, 2002, 51(1):31-36.
[16]张伏生,耿中行,葛耀中.电力系统谐波分析的高精度FFT算法.中国电机工程学报,1999,19(3):63-66.
[17]Zhang Fu-sheng, Geng Zhon-gxing, Yuan Wei. The Algorithm of InterpolatingWindowed FFT for Harmonic Analysis of Electric Power System. IEEE Trans. power delivery, 2001, 16(2):160–164.
[18]Testa A, Gallo D, Langella R. On the Processing of harmonics and interharmonics: Using Hanning window in standard framework. IEEE Trans. power delivery, 2004, 19(1):28–34.
[19]黄纯,江亚群.谐波分析的加窗插值改进算法.中国电机工程学报,2005,25(15):26-32.
[20]钱昊,赵荣祥.基于插值FFT算法的间谐波分析.中国电机工程学报,2005,25(21):87-91.
[21]陈明凯,郑翔骥,汪晓强.减少频谱泄漏的一种等角度间隔采样递推算法.电工技术学报,2005,20(8):94-98.
[22]赵凤展,杨仁刚.基于短时傅立叶变换的电压暂降扰动检测.中国电机工程学报,2007,27(10):28-34.
[23]薛雪东,程旭德,徐兵,李友利.基于STFT的高压电气设备局放信号时频分析.高电压技术,2008, 34(1):70-72.
[24]李弼程,罗建书.小波分析及其应用.北京:电子工业出版社,2003.
[25]杨桦,任震.基于小波变换检测谐波的新方法.电力系统自动化,1997,21(10):39-42.
[26]杜天军.基于抗混叠小波理论的电力系统谐波检测与抑制研究.电子科技大学博士学位论文,2006.
[27]赵成勇,何明锋.基于复小波变换相位信息的谐波检测算法.中国电机工程学报,2005,25(1):35-42.
[28]Tarasiuk T. Hybrid Wavelet–Fourier Method for Harmonics and Harmonic Subgroups Measurement—Case Study. IEEE Trans. power delivery, 2007, 22(1):4–17.
[29]Hamid E Y, Kawasaki Z I. Instrument for the quality analysis of power systems based on the wavelet packet transform. IEEE Power Engineering Review, 2002, 22(3): 52–54.
[30]王成山,王继东.基于小波包分解的电能质量扰动分类方法.电网技术,2004,28(15):78-82.
[31]陈祥训,刘兵,赵波.分析电能质量扰动的实小波域修正相子法.中国电机工程学报,2006,26(24):37-42.
[32]牛发亮,黄进.基于电磁转矩复解析小波变换的感应电机转子故障检测.电工技术学报,2005,20(7):100-105.
[33]陈丽安,张培铭.基于小波变换的短路故障早期检测门限值的研究.电工技术学报,2005,20(3):64-69.
[34]谭杨红,何怡刚.模拟电路故障诊断的小波方法.电工技术学报,2005,20(8):89-93.
[35]周林,夏雪,万蕴杰,等.基于小波变换的谐波测量方法综述.电工技术学报,2006,21(9):67-74.
[36]李天云,赵妍,李楠.基于EMD的Hilbert变换应用于暂态信号分析.电力系统自动化,2005,29(4):49-52.
[37]庞浩,王赞基,陈建业,赵伟.基于2对Hilbert移相滤波器的无功功率测量方法.电力系统自动化,2006,30(18):45-48.
[38]舒泓,王毅.基于数学形态滤波和Hilbert变换的电压闪变检测.中国电机工程学报, 2008, 28(1):111-114.
[39]丁仁杰,刘健,张隽,赵玉伟.一种基于瞬时无功功率理论的SVC控制方法.电工技术学报,2006,21(5):47-51.
[40]李鑫,曾光,苏彦民,张静钢.基于瞬时无功功率理论的SVC无功电流检测方法.电力电子技术,2006,40(5):121-123.
[41]李圣清,朱英浩,周有庆,彭玉楼.基于瞬时无功功率理论的四相输电谐波电流监测方法.中国电机工程学报, 2004, 24(3):12-17.
[42]张萍,余建明,魏磊.用瞬时无功功率理论求电压闪变参数.高电压技术,2007,33(4):134-137.
[43] P A Regalia. An improved lattice-based IIR notch filter. IEEE Trans. Signal Process, 1991, 39(9):2124–2128.
[44] M Bodson, S C Douglas. Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency. Automatica. 1997, 33(12): 2213–2221.
[45] L Hsu, R Ortega, G Damm. A globally convergent frequency estimator. IEEE Trans. Automatic Control, 1999, 44(4): 698–713.
[46] Mojiri M, Bakhshai A R. An adaptive notch filter for frequency estimation of a periodic signal, IEEE Trans. Autom. Control, 2004, 49(2):314–318.
[47] Karimi-Ghartemani M, Ziarani A K. A nonlinear adaptive filter for online signal analysis in power systems: Applications. IEEE Trans. on power delivery, 2002, 17(2): 617–622.
[48] Karimi-Ghartemani M,Ziarani A K.Periodic orbit analysis of two dynamical systems for electrical engineering applications.Journal of Engineering Mathematical,2003,45(2):135-154.
[49] Karimi-Ghartemani M, Ziarani A K. Performance characterization of a nonlinear system as both an adaptive notch filter and a phase-locked loop. International Journal of Adaptive Control Signal Process, 2004, 18(1):23–53.
[50] Ziarani A K,Konrad A.A method of extraction of nonstationary sinusoids.Signal Process,2004,84(8):1323-1346.
[51] Karimi-Ghartemani M,Ziarani A K.A nonlinear time-frequency analysis method.IEEE Trans.Signal Process,2004,52(6):1585-1595.
[52] Karimi-Ghartemani M, Iravani M R. Measurement of Harmonics/Inter-harmonics of Time-varying Frequencies. IEEE Trans. Power Delivery, 2005, 20(1): 23–31.
[53] McNamara D M, Ziarani A K, Ortmeyer T H. A New Technique of Measurement of Nonstationary Harmonics. IEEE Trans. power delivery, 2007, 22(1): 387–395.
[54] Mojiri M, Karimi-Ghartemani M, Bakhshai A. Time-Domain Signal Analysis Using Adaptive Notch Filter , IEEE Trans. Signal Process, 2007, 55(1): 85–93.
[55] Tarek M, Melchilef S, Rahim N A. Application of adaptive notch filter for harmonics currents estimation. In: Proceedings of IPEC 2007 Meeting Singapore, SINGAPORE: IEEE, 2007, 1236-1240.
[56] Mvuma A, Nishimura S, Hinamoto T. Tracking analysis of an adaptive IIR notch filter using gradient-based algorithm. In: Proceedings of IEEE International Symposium on Circuits and Systems. Seattle, WA: IEEE, 2008, 1148-1151.
[57] Torbjo rn Johansson A, White P R. Instantaneous frequency estimation at low signal-to-noise ratios using time-varying notch filters. Signal Processing, 2008, 88(5): 1271-1288.
[58] M Ta, DeBrunner V. Adaptive notch filter with time-frequency tracking of continuously changing frequencies. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing. Las Vegas, NV: IEEE, 2008. 3557-3560.
[1] Zhang Fu-sheng, Geng Zhon-gxing, Yuan Wei. The Algorithm of Interpolating Windowed FFT for Harmonic Analysis of Electric Power System. IEEE Trans. power delivery, 2001, 16(2):160–164.
[2] Karimi-Ghartemani M, Ziarani A K. A nonlinear adaptive filter for online signal analysis in power systems: Applications. IEEE Trans. on power delivery, 2002, 17(2): 617–622.
[3] Karimi-Ghartemani M,Ziarani A K.Periodic orbit analysis of two dynamical systems for electrical engineering applications.Journal of Engineering Mathematical,2003,45(2):135-154.
[4] Karimi-Ghartemani M, Ziarani A K. Performance characterization of a nonlinear system as both an adaptive notch filter and a phase-locked loop. International Journal of Adaptive Control Signal Process, 2004, 18(1):23–53.
[5] Ziarani A K,Konrad A.A method of extraction of nonstationary sinusoids.Signal Process,2004,84(8):1323-1346.
[6] Girgis A A, Chang W B, Makram E B. A digital recursive measurement scheme for on-line tracking of power system harmonics. IEEE Trans. power delivery, 1991, 6(3):1153–1160.
[7]储昭碧,张崇巍,冯小英.一种可调带宽的线性电力信号实时分析新算法.中国电机工程学报,2009,29(1):94-99.
[1] Owen E L. A history of harmonics in power systems. IEEE Industry Application Magazine, 1998, 4(1):6-12.
[2] Hidalgo R M, Fernandez J G, Rivera R R, et al. A Simple Adjustable Window Algorithm to Improve FFT Measurements. IEEE Trans. instrumentation and measurement, 2002, 51(1):31-36.
[3] Hidalgo R M, Fernandez J G, Rivera R R, et al. A Simple Adjustable Window Algorithm to Improve FFT Measurements. IEEE Trans. instrumentation and measurement, 2002, 51(1):31-36.
[4] Zhang Fu-sheng, Geng Zhon-gxing, Yuan Wei. The Algorithm of Interpolating Windowed FFT for Harmonic Analysis of Electric Power System. IEEE Trans. power delivery, 2001, 16(2):160-164.
[5] Testa A, Gallo D, Langella R. On the Processing of harmonics and interharmonics: Using Hanning window in standard framework. IEEE Trans. power delivery, 2004, 19(1):28-34.
[6] Sozanski K P. The shunt active power filter with better dynamic performance. In: proceedings of IEEE Lausanne Powertech. Lausanne, SWITZERLAND, IEEE, 2007. 1504-1508.
[7] Hamid E Y, Kawasaki Z I. Instrument for the quality analysis of power systems based on the wavelet packet transform. IEEE Power Engineering Review, 2002, 22(3): 52-54.
[8] Tarasiuk T. Hybrid Wavelet–Fourier Method for Harmonics and Harmonic Subgroups Measurement—Case Study. IEEE Trans. power delivery, 2007, 22(1):4-17.
[9]周林,夏雪,万蕴杰,等.基于小波变换的谐波测量方法综述.电工技术学报,2006,21(9):67-74.
[10]Karimi-Ghartemani M,Ziarani A K.A nonlinear time-frequency analysis method.IEEE Trans.Signal Process,2004,52(6):1585-1595.
[11]吴麒,王诗宓.自动控制原理(下册).北京:清华大学出版社,2006:448-450.
[12]张贤达.矩阵分析与应用.北京:清华大学出版社,2004:56-71.
[13]贾秀芳,赵成勇,胥国毅,等. IEC闪变仪误差分析及改进设计.电工技术学报, 2006, 21(11):121-126.
[14]李庆,任普春,赵海翔,等.针对风电引起闪变的数字化测试系统仿真.电力自动化设备,2008,28(1):69-72.
[15]徐福华,周国梁,王建增,等.基于DSP的数字化闪变仪研制.电测与仪表,2008,45(5): 12-15.
[16]孙佐,王念春,许卫兵.双DSP结构电网谐波监测及保护一体化装置研究.电子测量与仪器学报,2007,21(5):84-89.
[1] Y. Xiao, Y. Takeshita, K. Shida, Steady-state analysis of a plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 2001, 48(7): 733–740.
[2] M Karimi-Ghartemani, A K Ziarani. Performance characterization of a nonlinear system as both an adaptive notch filter and a phase-locked loop. International Journal of Adaptive Control Signal Process, 2004, 18(1):23–53.
[3] David M. McNamara, Alireza K. Ziarani, Thomas H. Ortmeyer. A New Technique of Measurement of Nonstationary Harmonics. IEEE Trans. on power delivery, 2007, 22(1): 387–395.
[4] P A Regalia. An improved lattice-based IIR notch filter. IEEE Trans. Signal Process, 1991, 39(9):2124–2128.
[5] M Bodson, S C Douglas. Adaptive algorithms for the rejection of sinusoidaldisturbances with unknown frequency. Automatica. 1997, 33(12): 2213–2221.
[6] L Hsu, R Ortega, G Damm. A globally convergent frequency estimator. IEEE Trans. Automatic Control, 1999, 44(4): 698–713.
[7] M. Mojiri, A. R. Bakhshai. An adaptive notch filter for frequency estimation of a periodic signal, IEEE Trans. Autom. Control, 2004, 49(2):314–318
[8] Mohsen Mojiri, Masoud Karimi-Ghartemani, Alireza Bakhshai. Time-Domain signal analysis Using Adaptive Notch Filter. IEEE Trans. Signal Process, 2007, 55(1): 85–93.
[9] B Riedle, P Kokotovic. Integral manifolds of slow adaptation. IEEE Trans. Automatic Control, 1986, AC-31(4):316–324.
[10]林振声.概周期微分方程与积分流形.上海:上海科学技术出版社,1982:166–219.
[1] Karimi-Ghartemani M, Iravani R M. Measurement of Harmonics/Inter-harmonics of Time-Varying Frequencies. IEEE Trans. Power Delivery, 2005, 20(1): 23-31.
[2] Khanshan A H, Amindavar H. Spectral estimation through parallel filtering. In: Proceedings of Australasian Telecommunication Networks and Applications Conference, Christchurch, New Zealand: IEEE, 2007.420-424.
[3] Riedle B, Kokotovic P. Integral manifolds of slow adaptation. IEEE Trans. Automatic Control, 1986, AC-31(4): 316-324.
[4] Khalil H K著,朱义胜,董辉,李作洲,等译.非线性系统.北京:电子工业出版社, 2005: 78-311.
[5] Bhat S P, Bernstein D S. Arc-length-based Lyapunov tests for convergence and stability in systems having a continuum of equilibria. In: Proceedings of the American Control Conference, Denver, USA: IEEE, 2003. 2961-2966. [6 ]Hui Q, Haddad W M, Bhat S P. Lyapunov and Converse Lyapunov Theorems for Semistability. In: Proceedings of IEEE conference on Decision and Control, New Orleans, USA: IEEE, 2007. 5870-5874.
[1] Steven C. Chapra, Raymond P. Canale,著.唐玲艳,田尊华,刘齐军译.工程数值方法.北京:清华大学出版社,2007.
[2]李瑞遐,何至庆.微分方程数值方法.上海:华东理工大学出版社,2005.