电气工程学报, 2015, 10(6): 58-66 doi:

理论研究

基于改进HHT算法的电力系统低频振荡模态辨识研究

金涛1, 褚福亮1, 李威2, 刘奕路3

1.福州大学电气工程与自动化学院电气工程系 福州 350108

2.国网电力科学研究院电网稳定控制技术研究所 南京 210003

3.美国田纳西大学电气工程与计算机科学系 美国诺克斯维尔 37996

Research on Power System Low Frequency Oscillation Modal Identification Based on Improved HHT Method

Jin Tao1, Chu Fuliang1, Li Wei2, Liu Yilu3

1.Fuzhou University Fuzhou 350108 China

2.Research Institute of Power System Stability Control SGEPRI Nanjing 210003 China

3.University of Tennessee Knoxville TN 37996 USA

责任编辑: 郭丽军

收稿日期: 2015-01-19   网络出版日期: 2015-06-25

基金资助: 欧盟FP7国际科技合作基金.  909880
国家自然科学基金.  50907011
福建省杰出青年科学基金.  2012J06012
福建省高校杰出青年人才培育基金.  JA1108

Received: 2015-01-19   Online: 2015-06-25

作者简介 About authors

金 涛 男 1976年生,研究员,博士生导师,主要研究方向为电力系统广域监测和电力系统稳定性分析。

褚福亮 男 1989年生,硕士研究生,主要研究方向为电力系统监测与故障分析。

摘要

针对电力系统低频振荡问题,在运用阻尼转矩对单机无穷大系统分析低频振荡机理与特点的基础上,对低频振荡经验模态分解时存在的端点效应问题进行了理论分析与改进,提出了一种基于端点优化对称延拓法的有效改进EMD分解边界效应的HHT算法对电力系统低频振荡进行辨识。通过对测试信号进行仿真,同时也利用广域FNET监测系统的测试结果进行低频振荡参数辨识及抑制实验,研究了该算法在模式辨识方面的有效性和准确性。仿真和实验表明,基于改进HHT算法的低频振荡辨识方法能快速高精度地辨识出振荡模态信息,并能有效指导电力系统稳定器PSS的配置及参数设计,从而维持电力系统的安全与稳定。

关键词: 低频振荡 ; HHT算法 ; 端点效应 ; 模态辨识 ; 电力系统稳定器

Abstract

Aimed at the problem of power system low frequency oscillation, based on analyzing of oscillation mechanism and oscillation characteristics with single-machine infinite bus system, this paper proposed an improved HHT algorithm using end optimization symmetric extension method to identify modal parameters of power system low frequency oscillation. Through simulations with testing signal and restraining experiments utilizing wide-area FNET measurement system, the paper also analyzed the effectiveness and veracity of the proposed algorithm. Simulations and experiments showed that the low-frequency oscillation pattern information can be effectively identified based on the proposed HHT algorithm. At the same time, the proposed method is also proved to have a good help to guide the disposition and parameters’ design of power system stabilizer, thereby the safety and stabilization of power system are maintained.

Keywords: Low frequency oscillation ; Hilbert-Huang transform algorithm ; boundary effect ; model identification ; power system stabilizer

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本文引用格式

金涛, 褚福亮, 李威, 刘奕路. 基于改进HHT算法的电力系统低频振荡模态辨识研究. 电气工程学报[J], 2015, 10(6): 58-66 doi:

Jin Tao. Research on Power System Low Frequency Oscillation Modal Identification Based on Improved HHT Method. Journal of Electrical Engineering[J], 2015, 10(6): 58-66 doi:

1 引言

随着电力行业技术的不断进步,各区域电网逐步互联,日益形成规模巨大、运行复杂且高维强非线性的现代大型电力系统。高放大倍数快速励磁技术的广泛采用、电网互联的日益扩大也导致电网的运行更加趋近稳定临界点,增大了现代电力系统发生低频振荡从而导致失稳乃至瘫痪的可能性[1]。电力系统中引起低频振荡的因素很多,扰动地点、扰动类型、系统容量和负荷水平等都对低频振荡的受扰轨迹有着明显的影响,反映在扰动轨迹上就有不同的特点,对低频振荡事故的扰动轨迹及其频谱分析可以看出不同扰源(Line Trip、Generator Trip、Load Shedding等)产生的低频振荡在广域测量信号中就具有不同类型的振荡模式、传播规律、波形、阶次、幅值和阻尼等。

目前电力系统低频振荡的分析方法主要归纳为基于系统模型的离线分析方法和基于受扰轨迹的在线分析方法[2]。基于系统模型的离线分析方法有时域仿真法、特征值法和Normal Form方法等。时域仿真法以数值分析为基础,是电力系统暂态稳定分析中广泛采用的分析方法,能充分考虑电力系统非线性因素的影响,对建模几乎没有限制,但由于扰动和时域观测量的选择对时域分析结果影响很大,时域法在大型电力系统小扰动稳定性分析中的实用性较差。特征值分析法是在工作点附近将系统线性化求解系统状态方程矩阵,进而求得特征值,其代表方法是QR法、S矩阵法等,这些算法要么不能应用稀疏矩阵技术存在维数灾问题,要么有可能遗漏其中的弱阻尼模式从而使收敛性得不到保证。Normal Form方法通过非线性向量场的正规形变换将原向量场映射为线性系统,极大地简化了实际问题的复杂性,利用Normal Form理论可以研究系统中的模态谐振现象及模态的非线性相关性,此方面的理论及应用还在进一步的发展之中。

近年来,研究低频振荡的国内外学者逐渐把研究方法集中在受扰轨迹的分析方面,目前针对低频振荡问题所采用的分析方法主要有FFT变换、小波变换法、Prony法、HHT算法和基于类噪声信号的分析方法等[3,4]。FFT变换无法分析出阻尼特性和局部特性,所以不适于非线性、非平稳信号。小波分析的不足是很难提取出信号的衰减特征,且计算复杂度高于实现电力系统快速在线分析。Prony法可以认为是傅里叶算法的扩展并能够提取出低频振荡的模态信息,但有定阶问题不确定及抗噪声能力差等不足。基于类噪声信号的分析法假定电网内持续存在类似噪声信号的小幅波动,该方法的研究与应用还在进一步发展之中[5]。希尔伯特-黄变换(Hilbert-Huang Transform, HHT)算法是近年来广泛研究的新算法,对于处理非线性非平稳电力信号如低频振荡、谐波分析等方面有清晰的物理意义,但其在理论上仍存在一些问题(如端点效应等),还需要进一步研究和完善[6]

本文在分析低频振荡特点的基础上,通过端点优化对称延拓法,提出一种改进的适于电力系统非平稳、非线性信号的HHT低频振荡分析方法,最大程度缩小经验模态分解EMD的误差范围。仿真分析与实验表明,本文提出的方法不仅可以准确提取出电力系统低频振荡的振荡模式和振荡特性,而且能有效指导PSS投放以维持电网安全稳定。

2 低频振荡及其HHT算法分析

低频振荡对电力系统造成了巨大危害,就其机理来说有多种解释,其中目前普遍认可的振荡机理研究方法是运用阻尼转矩的概念对单机无穷大系统SMIB产生低频振荡现象进行分析和解释[7]。如图1所示,设单机无穷大系统发电机采用三阶模型,忽略发电机的励磁绕组和阻尼绕组、线路分布电容等,用X表示电抗大小,且发电机的暂态电动势和机械功率恒定。

图1

图1   单机无穷大系统及发电机向量图

Fig.1   SMIB and generator vectogram


根据图1,利用电磁理论和电力系统理论分析可把该系统相应发电机dq坐标标幺值数学模型表示成

式中,Pe = EqIq + (Xd - Xd)IdIq;Ef为输出励磁电压;Pm为输出机械功率。对方程组工作点附近进行线性化处理,可得到单机无穷大系统的传递函数框图如图2所示。根据图2的传递函数框图可以列出其传递函数方程组如式(2)所示。当发电机运转受到扰动和系统低频振荡时,各个状态量发生偏差,求解式(2)即可得到扰动及低频振荡的模态特征参数。

图2

图2   单机无穷大系统传递函数框图

Fig.2   System transfer function diagram of SMIB


进一步分析可得,系统中电压调节器等所导致的磁链变化所产生的转矩ΔMe2包括与Δδ成比例的同步转矩ΔMSΔδ及与Δω成比例的阻尼转矩ΔMDΔω,即

式中, , 分别称作机械同步转矩系数和机械阻尼转矩系数;ωd为角频率;TEQ为当量转矩,进一步对式(3)进行化简及合并处理,可更直观地分析重负荷及快速励磁装置对励磁系统的影响,表示成

由上式可知,当出现重负荷时,系统可能呈现负阻尼状态(ΔMD<0),在快速高放大倍数励磁装置的励磁作用时,|ΔMD|更大,负阻尼更加严重,当此负阻尼比发电机励磁绕组、阻尼绕组的正阻尼及机械阻尼更大时,系统会出现振荡失稳,即出现增幅型振荡。所以,重负荷输电线容易导致功率振荡,快速高倍数励磁系统更起了加剧恶化的作用。

从以上低频振荡机理分析可以看出,电力系统低频振荡及扰动分析是一个复杂的非线性、非平稳问题,非常适合用具有自适应性的时频分析方法HHT来研究和处理。与傅里叶分解和小波变换须预先设定基函数相比,HHT算法无需此过程,所以在处理非线性、高度复杂电力系统数据上有着明显的优势[8]。它分为经验模态分解(EMD)和希尔伯特转换(HT)两个部分。 EMD是HHT算法的核心部分,其实质是通过筛选方法将非线性、非平稳信号分解为多个并且相互独立的本质模态函数即IMF分量。对于电力系统扰动轨迹信号x(t),可被分解为n个IMF分量ci(t)和一个剩余分量r(t),即

EMD分解时,上、下包络线主要通过信号局部极大值和极小值的三次样条插值算法获取,即通过插值法求信号x(t)的上、下包络线并计算出均值来选取IMF分量,但由于端点处极值的不确定性,很容易导致每次插值时都会产生拟合误差。以常规HHT算法对振荡信号x(t) = 10sin(2πt+π/6)+15sin(4πt+π/6)分解为例,其EMD分解信号包络如图3所示,从图中可以看出由于端点处无法确定是否为极值,导致包络线在边界无法包络信号,而且由于EMD分解是一个不断筛选IMF分量的过程,飞翼误差的不断累积会最终降低整个HHT算法分析的准确性和精度。

图3

图3   改进前HHT 算法的信号包络线

Fig.3   Former HHT decomposed signal envelope curve


端点效应问题是影响HHT算法效果的关键问题之一,研究人员一直在不断改进[9]。基于电力系统低频振荡分析高实时性、高精度等要求,本文提出一种端点优化对称延拓法,该方法通过信号包络线偏差评价函数的最小化计算,获取最合适的数据信号端点值,从而逼近原始信号两端点,最大程度缩小经验模态分解EMD的误差范围。首先对信号x(t1),x(t2),…,x(tn)确定采样步长为Δt,则可得出

xi两端点值用βγ替代,构造数据序列xi

x0xn-1端点为中心对xi分别向两端进行对称延拓得数据序列

式中,i=0,1,…,n-1,利用三次样条插值法得到延拓信号h的包络线S,优化式(8)中两端点x0xn-1的值,使Sihi偏差最小,进而对包络线在端点发散进行抑制,构造hi与对应Si的偏差评价函数

式中,μ为光滑参数,对上式离散化,得

式中,Δx为采样间隔; 为二阶微分算子。由式(9)与式(10)可知,延拓信号h和包络线S取决于端点值x0=β,xn-1=γ,故偏差评价函数同样取决于βγ,式(10)变为

时,ε存在最小值,此时可求出βγ的值。根据式(11)对原信号进行延拓,延拓后的信号包络线端点可以很好地逼近原始信号的端点,从而有效抑制EMD中端点飞翼现象,显著提高IMF分量获取的准确性。EMD分解完成后,对其进行Hilbert变换

构造一个函数z(t),令

式中,a(t)为振幅;ω(t)为相位。解析信号z(t)实部即为Hilbert谱,即式(14)它是一个频率、时间和幅值的三维图像,可以反映出各振荡模式之间的相互作用。而式(15)所表示的Hilbert边际谱是Hilbert频谱在时间上的积分,其反映的是幅值随频率的变化情况,可以直观地分析出振荡信号的主振荡模式。

图4是利用本节提出的改进HHT算法对图3中的EMD分解进行改进后的效果,从图3可以看出,改进后的信号包络线有效地消除了端点效应,极大提高了用于HHT分析的IMF质量,从而最终提高了HHT算法用于电力系统低频振荡分析的准确性和精度。

图4

图4   改进后的HHT 算法信号包络线

Fig.4   Improved HHT decomposed signal envelope curve


3 算法仿真分析

构造如图5所示低频振荡仿真测试信号x(t)

x(t)=5.08e-0.604tsin(1.068πt)+10.14e-0.708tsin(1.626πt)+15.02e-0.812tsin(3.002πt)

图5

图5   低频振荡信号仿真波形

Fig.5   Simulation waveform of testing oscillation signal


该仿真低频振荡测试信号所含振荡信息见表1,从中可以看出该低频振荡信号含有三种振荡模态,振荡频率分别为1.501Hz、0.813Hz、0.534Hz,这三个振荡频率含有区间(0.7~2.5Hz)、局部(0.1~0.7Hz)两种振荡模式,可以作为典型测试信号进行分析。

表1   低频振荡仿真信号特性

Tab. 1  Features of low frequency oscillation simulation signal

模式幅值/V衰减因子振荡频率/Hz
115.02-0.8121.501
210.14-0.7080.813
35.08-0.6040.534

新窗口打开| 下载CSV


仿真设置采样点为1 000,采样时间0~4s,采样频率为250Hz,分别对该信号进行常规HHT算法分析和本文提出的改进HHT算法分析。对改进前后辨识效果进行对比分析,所得分析结果如图6图7表2表3所示。从图6a可以看出,改进前HHT算法的EMD分解共分解出5个IMF分量和一个剩余分量y,没有准确分解出测试信号所包含的低频振荡的三种振荡模态。从图6b和图6c可以看出,采用未改进的HHT算法,测试信号无法被准确辨识出对应振荡模式,其频谱对应频率模式出现混叠现象。对于改进前的测试信号HHT算法分析,通过相关系数法能去除多余的两个IMF 分量,进行谱分析得到表2所示的分析结果,可以看出其与测试仿真信号误差较大、精度较低,无法准确有效辨识出含有多种振荡模式的振荡信号。

图6

图6   仿真信号常规HHT算法分析

Fig.6   Simulation signal analysis using former HHT


图7

图7   仿真信号改进HHT算法分析

Fig.7   Simulation signal analysis using improved HHT


表2   改进前HHT分析结果

Tab.2  Analysis results with former HHT

模式幅值
/V
误差
(%)
衰减
因子
误差
(%)
频率
/Hz
误差
(%)
imf116.8812.4-0.68815.31.28814.2
imf28.6814.4-0.5916.70.72211.2
imf34.2216.9-0.45125.30.46113.7
r3.54-0.019

新窗口打开| 下载CSV


表3   改进后HHT分析结果

Tab. 3  Analysis results with improved HHT

模式幅值
/V
误差
(%)
衰减
因子
误差
(%)
频率
/Hz
误差
(%)
imf 114.890.9-0.7883.01.4880.9
imf 29.892.5-0.6912.00.8021.4
imf 34.883.9-0.5833.50.5212.4
r0.2-0.021

新窗口打开| 下载CSV


图7是采用改进HHT算法的测试仿真信号的低频振荡分析效果。从图7a中可以看出,EMD分解后测试仿真信号所包含的振荡模态被准确分出三个IMF分量和一个剩余分量y。从图7b和图7c中可以明显看出测试仿真信号包含三种振荡模式,其不同模式间没有出现频率混叠等现象。

表3是根据改进后HHT算法得到的分析结果,与表2相比可以看出,其低频振荡模态信息如振幅、衰减因子和振荡频率等的相对误差大幅下降,与测试信号振荡特性基本接近,据此可得出改进型HHT算法可以准确地分解出振荡信号所包含的模式信息,其在辨识准确性和精度方面更具优势,对于后续如何进行电力系统稳定器PSS的投放及配置有相当重要的参考作用。

4 实验研究与讨论

由于目前的电力系统多采用高增益的励磁调节器等,在系统受到小扰动后,发电机的输出功率、转子角速度以及电磁转矩均易发生波动,且波动幅度逐渐增大,致使系统不稳定并发生系统低频振荡,中外学者对此进行了大量研究[10,11]。目前国内外普遍采用广域监测(Wide-Area Measurement System,WAMS)技术如相量测量单元PMU来进行系统低频振荡监测与分析[12]。但PMU的广域测量系统具有设备贵、安装费用高且测试参数多等不足,于是在A.G.Phadke教授提出的PMU理论的基础上,刘奕路教授在弗吉尼亚理工期间于2000年提出了一种广域同步动态频率监测FNET技术,其结构如图8所示,思想是用研发的FDR结合GPS授时信号主要测试电网的同步时间频率等参数从而在线分析电网的扰动特性并准确快速实现系统的在线评估、预警及决策[13]

图8

图8   广域FNET 监测系统结构示意图

Fig.8   Structure of FNET wide-area measurement system


根据Dr.Gardner、金涛等人对2004~2010年在美国EI、ERCOT、WECC三大电网及加拿大电网大量使用的FNET研究结果和数据分析报告看,基于FNET的广域测量系统和基于PMU的广域测量系统几乎有相同的电网扰动测试精度和监控效果,但该系统还具有分析速度快、所需要测试参数少及利于大规模使用的特点,因而在美国、欧洲和中国逐步得到了推广及应用[14]图9为FNET监测系统监测到的美国EI电网某次低频振荡分析处理后的波形图。

图9

图9   美国东部电网某次振荡信号波形

Fig.9   Oscillation signal waveform of Eastern American power grid


利用本文提出的改进HHT算法对图9中的振荡信号波形进行分析,利用谱分析得到的Hilbert谱如图10所示。从图10可以明显看出,系统中发电机受扰动后功率幅值呈振荡且递增趋势,从而使振荡频率随时间发生类似正弦波形的上下变化。进一步应用本文提出的改进HHT低频振荡分析算法对该次振荡进行模态辨识,研究得到表4所示的结果,即本次低频振荡主要由2个振荡模态组成,其主导频率分别是1.31Hz和0.4Hz,衰减因子分别是0.264和0.323。

图10

图10   实验分析信号的Hilbert 频谱

Fig.10   Hilbert spectrum of experiment oscillation signal


表4   改进HHT算法实验分析

Tab.4  Analysis results with HHT of oscillation signal

模式幅值/V衰减因子振荡频率/Hz
imf 11.310.2640.628
imf 20.400.3231.248
r0.120.015

新窗口打开| 下载CSV


在准确辨识出振荡模式后,通过建立相应传递函数,从而能利用相关器件和装置对低频振荡进行快速有效抑制。电力系统稳定器(Power System Stabilizer,PSS)出现于20世纪70年代,是目前抑制电力系统低频振荡最为经济有效的措施之一,其中PSS的参数设计和优化配置问题是涉及到其广泛应用的主要问题,国内外众多学者对此展开了深入研究和探讨[15]

本文中,通过对图9所示低频振荡信号的辨识分析,将其分析结果应用到电力系统稳定器PSS的参数设计中,并基于留数法对PSS进行配置,可以有效消除振荡带来的影响。所用PSS传递函数如图11所示,采用ΔPe和Δω作为PSS的输入信号,以输出ΔUPSS作为励磁系统附加控制信号,PSS经过励磁系统及电机励磁绕组最后对ΔEq起作用,并产生相应的附加电磁力矩。

图11

图11   PSS 传递函数框图

Fig.11   Transfer function blocks of PSS


依据HHT分析后的振荡信息对系统投入电力系统稳定器PSS后,因为励磁系统及发电机励磁绕组传输信号具有滞后作用,所以在实验中PSS设置为超前相位补偿。系统恢复波形如图12所示,结果显示在低频振荡发生后极短时间系统就恢复稳定状态,证明了所提出方法的快速性和有效性。

图12

图12   PSS 投入后振荡信号波形

Fig.12   Oscillation signal waveform with PSS


5 结论

本文针对电力系统低频振荡问题,在分析低频振荡特点及机理的基础上,提出了基于信号端点优化对称延拓法的改进HHT低频振荡分析方法,并对含局部和区间振荡模式的测试信号进行仿真研究,通过与改进前HHT算法辨识结果进行对比,改进后的HHT算法可快速准确地提取出测试信号的振荡模式和特性,验证了其在辨识低频振荡方面的时变性与自适应性优势。同时应用广域频率FNET监测系统进行低频振荡辨识实验研究,利用提出的算法对系统进行PSS设计与配置取得了良好的效果,验证了本文提出的算法及理论在电力系统应用方面的可行性。

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