基于轨迹模式空间解耦及模式能量序列的振荡分析(二)算法及应用
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摘要
基于状态空间中的模式解耦,及在时域中按其摆次分段,分析振荡能量的时序演化特性。利用互补群惯量中心—相对运动(CCCOI-RM)保稳变换,将非简谐振荡的多机轨迹严格映射为一系列映象上的时变单机系统轨迹,并通过后者在逐次摆动期间振荡能量的演变来刻画原多机系统的振荡行为,在时变单机映象系统的外力—位置平面上分析振荡能量的时空转换,量化其非保守性。文中分别以映象系统轨迹上的动态中心点(DCP)处的动能,及最远点(FEP)处的势能来反映该模式在过去半摆中的振荡总能量;以两者组成的能量序列反映该空间振荡模式的时变性。通过理论分析及数值仿真证实:在描述哈密顿单机系统振荡行为时,轨迹摆次能量序列与特征根分析完全一致,而在分析非哈密顿的单机系统或一般的多机系统时,轨迹摆次能量序列可以克服平衡点特征根的众多缺陷。
This paper analyzes the time evolution of oscillation energy based on the trajectory modes decoupled in state space and the swings divided in time domain. The trajectory of multi-machine system, which is non-harmonic oscillation, can be mapped into a series of trajectories for time-varying image system through the complementary-cluster center-of-inertial relative-motion(CCCOI-RM) transform, so the oscillation behavior of the multi-machine system is described by the oscillation energy variation of the successive swing. Besides, the non-conservative can be quantified through analyzing the time-space variation of the oscillation energy on the force-position plane. In this paper, the kinetic energy at the dynamic center point(DCP) and the potential energy at the far end point(FEP) are both used to reflect the oscillation energy in the past half swing. Therefore, the sequence of them can reflect the time-varying characteristic of the spatial oscillation mode. After that, through theoretical analysis and numerical simulation, the consistency between the trajectory swing energy sequence and the eigenvalue system is proved when they are used to describe the oscillation behavior of Hamilton one machine infinite-bus(H-OMIB), but the trajectory swing energy sequence can also be used to overcome many defects of the eigenvalue in the non H-OMIB or general multi-machine system.
This paper analyzes the time evolution of oscillation energy based on the trajectory modes decoupled in state space and the swings divided in time domain. The trajectory of multi-machine system, which is non-harmonic oscillation, can be mapped into a series of trajectories for time-varying image system through the complementary-cluster center-of-inertial relative-motion(CCCOI-RM) transform, so the oscillation behavior of the multi-machine system is described by the oscillation energy variation of the successive swing. Besides, the non-conservative can be quantified through analyzing the time-space variation of the oscillation energy on the force-position plane. In this paper, the kinetic energy at the dynamic center point(DCP) and the potential energy at the far end point(FEP) are both used to reflect the oscillation energy in the past half swing. Therefore, the sequence of them can reflect the time-varying characteristic of the spatial oscillation mode. After that, through theoretical analysis and numerical simulation, the consistency between the trajectory swing energy sequence and the eigenvalue system is proved when they are used to describe the oscillation behavior of Hamilton one machine infinite-bus(H-OMIB), but the trajectory swing energy sequence can also be used to overcome many defects of the eigenvalue in the non H-OMIB or general multi-machine system.
引文
[1] 薛禹胜,郝思鹏,刘俊勇,等.关于低频振荡分析方法的评述[J].电力系统自动化,2009,33(3):1-8.XUE Yusheng,HAO Sipeng,LIU Junyong,et al.A review of analysis methods for low-frequency oscillations[J].Automation of Electric Power Systems,2009,33(3):1-8.
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[5] 王娜娜,廖清芬,唐飞,等.基于割集能量及灵敏度的强迫功率振荡扰动源识别[J].电力自动化设备,2013,33(1):76-80.WANG Nana,LIAO Qingfen,TANG Fei,et al.Disturbance source identification based on cutset energy and sensitivity for forced power oscillation[J].Electric Power Automation Equipment,2013,33(1):76-80.
[6] 李文锋,郭剑波,李莹,等.基于WAMS的电力系统功率振荡分析与振荡源定位(1)割集能量法[J].中国电机工程学报,2013,33(25):41-46.LI Wenfeng,GUO Jianbo,LI Ying,et al.Power system oscillation analysis and oscillation source location based on WAMS:Part 1 method of cutset energy[J].Proceedings of the CSEE,2013,33(25):41-46.
[7] 商显俊,顾雪平,梁海平,等.计及相量测量单元信息不可观性的强迫功率振荡扰动源定位[J].电力系统自动化,2015,39(10):56-62.SHANG Xianjun,GU Xueping,LIANG Haiping,et al.Location identification method of disturbance source in forced power oscillation considering unobservability of PMU information[J].Automation of Electric Power Systems,2015,39(10):56-62.
[8] 陈月辉,张文朝,徐遐龄,等.基于前K最短路径的电力系统低频振荡源定位方法[J].电力系统保护与控制,2017,45(13):117-123.CHEN Yuehui,ZHANG Wenchao,XU Xialing,et al.Locating method of low frequency oscillation source based on K shortest paths[J].Power System Protection and Control,2017,45(13):117-123.
[9] 薛禹胜.运动稳定性量化理论——非自治非线性多刚体系统的稳定分析[M].南京:江苏科学技术出版社,1999.XUE Yusheng.Quantitative study of general motion stability and an example on power system stability[M].Nanjing:Phoenix Science Press,1999.
[10] 刘庆龙,薛禹胜,陈国平.基于轨迹模式空间解耦及模式能量序列的振荡分析:(一)理论基础[J].电力系统自动化,2019,43(12):1-10.DOI:10.7500/AEPS20190430033.LIU Qinglong,XUE Yusheng,GHEN Guoping.Analysis of power system oscillation based on trajectory mode spatial domain decoupling and swing-energy sequence:Part one theoretical basis[J].Automation of Electric Power Systems,2019,43(12):1-10.DOI:10.7500/AEPS20190430033.
[11] 薛禹胜,潘学萍,ZHANG Guorui,等.计及时变系统完整非线性的振荡模式分析[J].电力系统自动化,2008,32(18):1-7.XUE Yusheng,PAN Xueping,ZHANG Guorui,et al.Oscillation mode analysis considering full nonlinearity of time-varying systems[J].Automation of Electric Power Systems,2008,32(18):1-7.
[12] THAPAR J,VITTAL V,KLIEMANN W,et al.Application of the normal form of vector fields to predict inter-area separation in power systems [J].IEEE Transactions on Power Systems,1977,12(2):844-850.
[13] HIROYUKI A,TERUHISA K,TOSHIO I.Nonlinear stability indexes of power swing oscillation using normal form analysis [J].IEEE Transactions on Power Systems,2006,21(3):18-23.
[14] PARIZ N,SHANECHI H M,VAAHEDI E.Explaining and validating stressed power systems behavior using modal series [J].IEEE Transactions on Power Systems,2003,18(2):778-785.
[15] 刘红朝,李兴源,郝巍,等.交直流互联电力系统非线性模态分析[J].电力系统自动化,2006,30(18):8-12.LIU Hongchao,LI Xingyuan,HAO Wei,et al.Nonlinear modal analysis for HVDC/AC power systems[J].Automation of Electric Power Systems,2006,30(18):8-12.
[16] 鞠平,谢欢,孟远景,等.基于广域测量信息在线辨识低频振荡[J].中国电机工程学报,2005,25(22):56-60.JU Ping,XIE Huan,MENG Yuanjing,et al.Online identification of low-frequency oscillations based on wide-area measurements[J].Proceedings of the CSEE,2005,25(22):56-60.
[17] JIANG T,BAI L,LI G,et al.Estimating inter-area dominant oscillation mode in bulk power grid using multi-channel continuous wavelet transform[J].Journal of Modern Power Systems and Clean Energy,2016,4(3):394-405.
[18] SHEN C,AN Z,DAI X,et al.Measurement-based solution for low frequency oscillation analysis[J].Journal of Modern Power Systems and Clean Energy,2016,4(3):406-413.
[19] YANG D,CAI G,CHAN K.Extracting inter-area oscillation modes using local measurements and data-driven stochastic subspace technique[J].Journal of Modern Power Systems and Clean Energy,2017,5(5):704-712.
[20] 张鹏飞,薛禹胜,张启平.电力系统时变振荡特性的小波脊分析[J].电力系统自动化,2004,28(16):32-35.ZHANG Pengfei,XUE Yusheng,ZHANG Qiping.Power system time-varying oscillation analysis with wavelet ridge algorithm[J].Automation of Electric Power Systems,2004,28(16):32-35.
[21] 郝思鹏,薛禹胜,唐茂林,等.通过轨迹特征根分析时变振荡特性[J].电力系统自动化,2009,33(6):1-5.HAO Sipeng,XUE Yusheng,TANG Maolin,et al.Trajectory eigenvalues analysis time variant oscillation characters[J].Automation of Electric Power Systems,2009,33(6):1-5.
[22] 潘学萍,薛禹胜,张晓明,等.轨迹特征根的解析估算及其误差分析[J].电力系统自动化,2008,32(19):10-14.PAN Xueping,XUE Yusheng,ZHANG Xiaoming,et al.Analytical calculation of power system trajectory eigenvalues and its error analysis[J].Automation of Electric Power Systems,2008,32(19):10-14.
[23] XUE Y,BIN Z.Trajectory section eigenvalue method for nonlinear time-varying power system[J].International Journal of Electrical Power & Energy Systems,2019,107:321-331.
[2] 薛禹胜,郝丽丽,WU Qinghua,等.轨迹断面特征根对受扰轨迹最远点及动态鞍点的诠释[J].电力系统自动化,2010,34(12):1-7.XUE Yusheng,HAO Lili,WU Qinghua,et al.Annotation for FEP and DSP in terms of trajectory section eigenvalues[J].Automation of Electric Power Systems,2010,34(12):1-7.
[3] 余一平,闵勇,陈磊,等.基于能量函数的强迫功率振荡扰动源定位[J].电力系统自动化,2010,34(5):1-6.YU Yiping,MIN Yong,CHEN Lei,et al.Disturbance source location of forced power oscillation using energy functions[J].Automation of Electric Power Systems,2010,34(5):1-6.
[4] 陈磊,闵勇,胡伟.基于振荡能量的低频振荡分析与振荡源定位:(一)理论基础与能量流计算[J].电力系统自动化,2012,36(3):22-27.CHEN Lei,MIN Yong,HU Wei.Low frequency oscillation analysis and oscillation source location based on oscillation energy:Part one mathematical foundation and energy flow computation[J].Automation of Electric Power Systems,2012,36(3):22-27.
[5] 王娜娜,廖清芬,唐飞,等.基于割集能量及灵敏度的强迫功率振荡扰动源识别[J].电力自动化设备,2013,33(1):76-80.WANG Nana,LIAO Qingfen,TANG Fei,et al.Disturbance source identification based on cutset energy and sensitivity for forced power oscillation[J].Electric Power Automation Equipment,2013,33(1):76-80.
[6] 李文锋,郭剑波,李莹,等.基于WAMS的电力系统功率振荡分析与振荡源定位(1)割集能量法[J].中国电机工程学报,2013,33(25):41-46.LI Wenfeng,GUO Jianbo,LI Ying,et al.Power system oscillation analysis and oscillation source location based on WAMS:Part 1 method of cutset energy[J].Proceedings of the CSEE,2013,33(25):41-46.
[7] 商显俊,顾雪平,梁海平,等.计及相量测量单元信息不可观性的强迫功率振荡扰动源定位[J].电力系统自动化,2015,39(10):56-62.SHANG Xianjun,GU Xueping,LIANG Haiping,et al.Location identification method of disturbance source in forced power oscillation considering unobservability of PMU information[J].Automation of Electric Power Systems,2015,39(10):56-62.
[8] 陈月辉,张文朝,徐遐龄,等.基于前K最短路径的电力系统低频振荡源定位方法[J].电力系统保护与控制,2017,45(13):117-123.CHEN Yuehui,ZHANG Wenchao,XU Xialing,et al.Locating method of low frequency oscillation source based on K shortest paths[J].Power System Protection and Control,2017,45(13):117-123.
[9] 薛禹胜.运动稳定性量化理论——非自治非线性多刚体系统的稳定分析[M].南京:江苏科学技术出版社,1999.XUE Yusheng.Quantitative study of general motion stability and an example on power system stability[M].Nanjing:Phoenix Science Press,1999.
[10] 刘庆龙,薛禹胜,陈国平.基于轨迹模式空间解耦及模式能量序列的振荡分析:(一)理论基础[J].电力系统自动化,2019,43(12):1-10.DOI:10.7500/AEPS20190430033.LIU Qinglong,XUE Yusheng,GHEN Guoping.Analysis of power system oscillation based on trajectory mode spatial domain decoupling and swing-energy sequence:Part one theoretical basis[J].Automation of Electric Power Systems,2019,43(12):1-10.DOI:10.7500/AEPS20190430033.
[11] 薛禹胜,潘学萍,ZHANG Guorui,等.计及时变系统完整非线性的振荡模式分析[J].电力系统自动化,2008,32(18):1-7.XUE Yusheng,PAN Xueping,ZHANG Guorui,et al.Oscillation mode analysis considering full nonlinearity of time-varying systems[J].Automation of Electric Power Systems,2008,32(18):1-7.
[12] THAPAR J,VITTAL V,KLIEMANN W,et al.Application of the normal form of vector fields to predict inter-area separation in power systems [J].IEEE Transactions on Power Systems,1977,12(2):844-850.
[13] HIROYUKI A,TERUHISA K,TOSHIO I.Nonlinear stability indexes of power swing oscillation using normal form analysis [J].IEEE Transactions on Power Systems,2006,21(3):18-23.
[14] PARIZ N,SHANECHI H M,VAAHEDI E.Explaining and validating stressed power systems behavior using modal series [J].IEEE Transactions on Power Systems,2003,18(2):778-785.
[15] 刘红朝,李兴源,郝巍,等.交直流互联电力系统非线性模态分析[J].电力系统自动化,2006,30(18):8-12.LIU Hongchao,LI Xingyuan,HAO Wei,et al.Nonlinear modal analysis for HVDC/AC power systems[J].Automation of Electric Power Systems,2006,30(18):8-12.
[16] 鞠平,谢欢,孟远景,等.基于广域测量信息在线辨识低频振荡[J].中国电机工程学报,2005,25(22):56-60.JU Ping,XIE Huan,MENG Yuanjing,et al.Online identification of low-frequency oscillations based on wide-area measurements[J].Proceedings of the CSEE,2005,25(22):56-60.
[17] JIANG T,BAI L,LI G,et al.Estimating inter-area dominant oscillation mode in bulk power grid using multi-channel continuous wavelet transform[J].Journal of Modern Power Systems and Clean Energy,2016,4(3):394-405.
[18] SHEN C,AN Z,DAI X,et al.Measurement-based solution for low frequency oscillation analysis[J].Journal of Modern Power Systems and Clean Energy,2016,4(3):406-413.
[19] YANG D,CAI G,CHAN K.Extracting inter-area oscillation modes using local measurements and data-driven stochastic subspace technique[J].Journal of Modern Power Systems and Clean Energy,2017,5(5):704-712.
[20] 张鹏飞,薛禹胜,张启平.电力系统时变振荡特性的小波脊分析[J].电力系统自动化,2004,28(16):32-35.ZHANG Pengfei,XUE Yusheng,ZHANG Qiping.Power system time-varying oscillation analysis with wavelet ridge algorithm[J].Automation of Electric Power Systems,2004,28(16):32-35.
[21] 郝思鹏,薛禹胜,唐茂林,等.通过轨迹特征根分析时变振荡特性[J].电力系统自动化,2009,33(6):1-5.HAO Sipeng,XUE Yusheng,TANG Maolin,et al.Trajectory eigenvalues analysis time variant oscillation characters[J].Automation of Electric Power Systems,2009,33(6):1-5.
[22] 潘学萍,薛禹胜,张晓明,等.轨迹特征根的解析估算及其误差分析[J].电力系统自动化,2008,32(19):10-14.PAN Xueping,XUE Yusheng,ZHANG Xiaoming,et al.Analytical calculation of power system trajectory eigenvalues and its error analysis[J].Automation of Electric Power Systems,2008,32(19):10-14.
[23] XUE Y,BIN Z.Trajectory section eigenvalue method for nonlinear time-varying power system[J].International Journal of Electrical Power & Energy Systems,2019,107:321-331.