电力系统低频振荡非线性模态分析方法及分析软件开发
|
摘要
我国电力工业正处于高速发展时期,电网的规模日渐庞大,“重负荷、弱联系、快速励磁、低阻尼”的情况日趋明显。这使得当今的电力系统成了一个极其复杂的强非线性系统,它时常表现出如状态变量和模式交互作用等非线性现象。而这些非线性作用已成为影响系统稳定性的重要因素。
传统的线性化分析方法,即使是全模型数值仿真,都很难计及系统内部非线性动态结构信息,并揭示这些非线性奇异现象的实质。目前,研究这类非线性的理论依据主要有正规形理论和模态级数法。前者的2阶解析解对电力系统2阶非线性特性的分析和应用已经相当成熟,而后者相对较少。基于模态级数法的3阶解析解因其表达式十分复杂而不利于编程实现。可见,对电力系统更高阶非线性特性的分析和应用仍处于亟待研究的阶段。如何方便、可靠地对这些非线性特性进行分析,已日益成为电力领域学者的一项重要工作。
本文从电力系统小信号分析的算法和软件两个方面进行了研究:
1.高阶模态分析方法的研究
本文基于正规形理论推导了形式更加简洁的电力系统3阶解析解表达式,并定义了模态交互作用指标,如3阶非线性参与因子、非线性度指标和非线性相互作用指标等。通过3个系统的仿真,验证了该3阶解析解的有效性。并导出了基于该3阶解析解的振荡分析表达式,进行了振荡机理仿真分析。同时,应用正规形理论的2、3阶解析解分析3机9母线系统的非线性相互作用。结果表明,非线性相互作用极其复杂,模态的高阶交互作用具有多样性,在电力系统模态分析中考虑2阶以上的更高阶项的影响是非常必要的。
2.电力系统高阶模态分析程序的开发
目前国内外广泛使用的电力仿真软件无一例外的只提供了小信号稳定分析(又称小干扰稳定分析)功能,这一功能只能进行线性模态分析,不具备2阶及以上的高阶模态分析功能。针对此空白,本文使用Matlab?编写电力系统高阶分析程序和操作界面,同时集成正规形理论和模态级数法。针对求取系统状态方程的高阶偏导数矩阵的关键和难点,采用了数值微分法来解决。针对高阶偏导数矩阵维数高、稀疏性大的特点,采用现有的稀疏技术保证了计算分析的速度和效率,尽可能减少对内存的占用。同时,该工具采用Java编写了一个接口类来实现了与PSASP的接口,其输入数据由PSASP提供,包括系统数据、潮流和暂态结果数据,并提供m文件格式存储这些输入数据,输出结果采用报表或绘图的形式给出。该分析程序采用界面化操作,可以方便地使用正规形理论和模态级数法进行电力系统振荡等模态分析。
With the growth of interconnected power system, the power system is characterized by the conditions such as‘heavily loaded, weakly connected, fast excitation, low damping’. Nowadays, the power system becomes a complex nonlinear system, in which the nonlinear interaction phenomenon, such as the interaction of the state variables and the modes, generally happens. These nonlinear interactions have become one of the most significant factors affecting the stability of system.
The linear analysis methods, even the full model numerical simulation, are not able to explain the essence of some nonlinear singularity phenomenon, because those methods are difficult to consider the internal nonlinear structural dynamics. Currently, the Normal Form theory and the Modal Series method in vector fields are the basic tools to study the dynamic characteristic of nonlinear power systems. The 2nd-order analytical solutions of the former are widely applied to analyze the 2nd-order nonlinear characteristic of power system, while the later is a little less used. Moreover, the expressions of the 3rd-order analytical solutions based on Modal Series method are so complex that it is not convenient to implement it in programs. It is shown that the further higher order nonlinear analysis of power system and applications should be developed, and methods for convenient and reliable analysis should be found.
The thesis studied the algorithms and software tool of power system small-signal analysis, including two aspects:
1. Study on the higher-order modal analysis methods
Based on the theory of Normal Forms, the much simpler 3rd-order analytical solutions of power systems dynamical equation were deduced, and modal interaction indices, such as the 3rd-order nonlinear participation factors, the nonlinearity index and the nonlinear interaction index, were also defined. The validity of the solutions was proved through the simulations of three power system cases. The expression of oscillation analysis based on the 3rd-order analytical solutions is deduced, and used to analyze the mechanism of oscillation. The nonlinear interaction of 3-generator 9-bus system was analyzed by the 2nd- and 3rd-order analytical solutions based on Normal Form theory. It is shown that the higher-order modal interaction is so complicated that the effect of terms higher than 2nd-order should be considered in power system modal analysis.
2. Development of the power system higher order analysis tool
Nowadays, the widely used power system simulation tools only give the small-signal stability analysis which is linear modal analysis, lacking the function of nonlinear modal analysis. In this thesis, the“power system higher order analysis tool”and its operation interface are developed using Matlab?. Both the Normal Form method and Modal Series method are implemented in this tool set. The numerical differentiation algorithm is used to obtain the higher order partial derivatives of the system state space equation. In order to compress the memory requirements and lower the dimension of the matrices, sparse technology is used in the development of the program. An interface class, which is programmed with Java, is developed to interchange data with PSASP. The input data supplied by PSASP include the system parameters, the power flow and transient stability data. The tool provides a mechanism to store these data, and present the results either in tables or in figures. This tool provides an interface, on which modal analysis, such as oscillation modes of power system, with the Normal Form method and the Modal Series method, can be conviently performed.
传统的线性化分析方法,即使是全模型数值仿真,都很难计及系统内部非线性动态结构信息,并揭示这些非线性奇异现象的实质。目前,研究这类非线性的理论依据主要有正规形理论和模态级数法。前者的2阶解析解对电力系统2阶非线性特性的分析和应用已经相当成熟,而后者相对较少。基于模态级数法的3阶解析解因其表达式十分复杂而不利于编程实现。可见,对电力系统更高阶非线性特性的分析和应用仍处于亟待研究的阶段。如何方便、可靠地对这些非线性特性进行分析,已日益成为电力领域学者的一项重要工作。
本文从电力系统小信号分析的算法和软件两个方面进行了研究:
1.高阶模态分析方法的研究
本文基于正规形理论推导了形式更加简洁的电力系统3阶解析解表达式,并定义了模态交互作用指标,如3阶非线性参与因子、非线性度指标和非线性相互作用指标等。通过3个系统的仿真,验证了该3阶解析解的有效性。并导出了基于该3阶解析解的振荡分析表达式,进行了振荡机理仿真分析。同时,应用正规形理论的2、3阶解析解分析3机9母线系统的非线性相互作用。结果表明,非线性相互作用极其复杂,模态的高阶交互作用具有多样性,在电力系统模态分析中考虑2阶以上的更高阶项的影响是非常必要的。
2.电力系统高阶模态分析程序的开发
目前国内外广泛使用的电力仿真软件无一例外的只提供了小信号稳定分析(又称小干扰稳定分析)功能,这一功能只能进行线性模态分析,不具备2阶及以上的高阶模态分析功能。针对此空白,本文使用Matlab?编写电力系统高阶分析程序和操作界面,同时集成正规形理论和模态级数法。针对求取系统状态方程的高阶偏导数矩阵的关键和难点,采用了数值微分法来解决。针对高阶偏导数矩阵维数高、稀疏性大的特点,采用现有的稀疏技术保证了计算分析的速度和效率,尽可能减少对内存的占用。同时,该工具采用Java编写了一个接口类来实现了与PSASP的接口,其输入数据由PSASP提供,包括系统数据、潮流和暂态结果数据,并提供m文件格式存储这些输入数据,输出结果采用报表或绘图的形式给出。该分析程序采用界面化操作,可以方便地使用正规形理论和模态级数法进行电力系统振荡等模态分析。
With the growth of interconnected power system, the power system is characterized by the conditions such as‘heavily loaded, weakly connected, fast excitation, low damping’. Nowadays, the power system becomes a complex nonlinear system, in which the nonlinear interaction phenomenon, such as the interaction of the state variables and the modes, generally happens. These nonlinear interactions have become one of the most significant factors affecting the stability of system.
The linear analysis methods, even the full model numerical simulation, are not able to explain the essence of some nonlinear singularity phenomenon, because those methods are difficult to consider the internal nonlinear structural dynamics. Currently, the Normal Form theory and the Modal Series method in vector fields are the basic tools to study the dynamic characteristic of nonlinear power systems. The 2nd-order analytical solutions of the former are widely applied to analyze the 2nd-order nonlinear characteristic of power system, while the later is a little less used. Moreover, the expressions of the 3rd-order analytical solutions based on Modal Series method are so complex that it is not convenient to implement it in programs. It is shown that the further higher order nonlinear analysis of power system and applications should be developed, and methods for convenient and reliable analysis should be found.
The thesis studied the algorithms and software tool of power system small-signal analysis, including two aspects:
1. Study on the higher-order modal analysis methods
Based on the theory of Normal Forms, the much simpler 3rd-order analytical solutions of power systems dynamical equation were deduced, and modal interaction indices, such as the 3rd-order nonlinear participation factors, the nonlinearity index and the nonlinear interaction index, were also defined. The validity of the solutions was proved through the simulations of three power system cases. The expression of oscillation analysis based on the 3rd-order analytical solutions is deduced, and used to analyze the mechanism of oscillation. The nonlinear interaction of 3-generator 9-bus system was analyzed by the 2nd- and 3rd-order analytical solutions based on Normal Form theory. It is shown that the higher-order modal interaction is so complicated that the effect of terms higher than 2nd-order should be considered in power system modal analysis.
2. Development of the power system higher order analysis tool
Nowadays, the widely used power system simulation tools only give the small-signal stability analysis which is linear modal analysis, lacking the function of nonlinear modal analysis. In this thesis, the“power system higher order analysis tool”and its operation interface are developed using Matlab?. Both the Normal Form method and Modal Series method are implemented in this tool set. The numerical differentiation algorithm is used to obtain the higher order partial derivatives of the system state space equation. In order to compress the memory requirements and lower the dimension of the matrices, sparse technology is used in the development of the program. An interface class, which is programmed with Java, is developed to interchange data with PSASP. The input data supplied by PSASP include the system parameters, the power flow and transient stability data. The tool provides a mechanism to store these data, and present the results either in tables or in figures. This tool provides an interface, on which modal analysis, such as oscillation modes of power system, with the Normal Form method and the Modal Series method, can be conviently performed.
引文
[1]任元居. 2006年国外重大停电事故解析.广西电业,2007.2(总第83期):20-21
[2]陈向宜,陈允平.构建大电网安全防御体系—欧洲大停电事故的分析及思考.电力系统自动化,2007,31(1):4-8
[3]中华人民共和国电力行业标准《电力系统安全稳定导则》DL755-2001.中华人民共和国国家经济贸易委员会发布.北京:中国电力出版社,2001
[4] Prabha Kundur.电力系统稳定与控制(影印版).北京:中国电力出版社,2001,699-822
[5]刘隽,李兴源,邹全平.互联电网低频振荡的相关问题及研究.继电器,2005,33(16):70-77,84
[6]苗友忠,汤涌,李丹.局部振荡引起区间大功率振荡的机理.中国电机工程学报,2007,27(10):73-77
[7]刘峰,辛焕海,邱家驹.电力系统结构保留模型的回顾、分析和展望.电力系统及其自动化学报,2005,17(2):13-20,57
[8]王锡凡.现代电力系统分析.北京:科学出版社,2003,365-399,326-327
[9]仲悟之.大型电力系统小干扰稳定性分析方法研究和软件开发:[硕士学位论文].北京:中国电力科学研究院,2005
[10] H.M. Shanechi, N. Pariz, E. Vaahedi. General Nonlinear Modal Representation of Large Scale Power Systems. IEEE Trans .Power. Syst, Aug, 2003, vol.18(3): 1103-1109
[11] J. J. Sanchez-Gasca, V. Vittal, M. J. Gibbard, et al. Inclusion of higher order terms for small-signal (modal) Analysis:committee report-task force on assessing the need to include higher order terms for small-signal (modal) analysis. IEEE Trans. Power Syst., Nov, 2005, vol.20(4):1886-1904
[12] Y. Ni, V. Vittal and W. Kliemann. Analysis of structural properties responsible for nonlinear modal behavior of a stressed power system using the normal form technique. IEEE, 39th Midwest symposium., Circuits and Systems ., Aug, 1996, vol.3: 1029-1033
[13] C. M. Lin, V. Vittal, W. Kliemann, and A. A. Fouad. Investigation of modal interaction and its effects on control performance in stressed power systems using normal forms of vector fields. IEEE Trans. Power Syst., May,1996, vol. 11(2): 781–787
[14] Y. X. Ni, V. Vittal, W. Kliemann, and A. A. Fouad. Nonlinear modal interaction in HVDC/AC power systems with D. C. modulation. IEEE Trans. Power Syst., Nov. 1996, vol.11(4): 2011-2017
[15] S. Saha, A. A. Fouad, W. Kliemann, and V. Vittal. Stability boundary approximation of a power system using the real normal form of vector fields. IEEE Trans. Power Syst., May, 1997, vol.12(2): 797–802
[16] V. Vittal,W. Kliemann, Y. X Ni. Determination of Generator Groupings for an Islanding Scheme in the Manitoba Hydro System Using the Method of Normal Forms. IEEE Trans on Power Systems,1998,13(4):1345-1351
[17] J. Thapar, V. Vittal, W. Kliemann, and A. A. Fouad. Application of the normal form of vector fields to predict interarea separation in power systems. IEEE Trans. Power Syst., May,1997,vol. 12(2): 844–850
[18] E. Barocio, and A. R. Messina. Analysis of Nonlinear Modal Interaction in Stressed Power Systems with SVCs. IEEE Power Engineering Society Winter Meeting., Jan.,2002, Vol.2:.1164–1169
[19]李颖晖,张保会.运用非线性系统理论确定电力系统暂态稳定域的一种新方法.中国电机工程学报,2000, Vol. 20(1): 41-44
[20]李颖晖,张保会,李勐.电力系统稳定边界的研究.中国电机工程学报,2002,Vol.22 (3): 72-77
[21] Songzhe Zhu, V. Vittal, W. Kliemann. Analyzing dynamic performance of power systems over parameter space using normal forms of vector fields-part I: identification of vulnerable regions. IEEE Trans. on Power Systems, 2001, Vol. 16 (3): 444-450
[22] Z.Y. Zou, Q.Y. Jiang, Y.J. Cao, H.F. Wang. Investigation of interactions among the multi-control channels of UPFC using normal forms of vector fields. Proc. 39th International Universities Power Engineering Conference, 2004, pp. 343– 347
[23] E. Barocio, A. R. Messina. Normal form analysis of stressed power systems: Incorporation of SVC models. International Journal of Electrical Power and Energy Systems, 2003,Vol.25 (1): 70-79
[24] A. R. Messina, E. Barocio, J. Arroyo. Analysis of Modal Interaction in Power Systems with FACTS Controllers using Normal Forms. IEEE Power Engineering Society General Meeting, July, 2003, Vol. 4: 2111–2117
[25] G. Jang, V. Vittal, W. Kliemann. Effect of nonlinear modal interaction on control performance: use of normal forms technique in control design. I. General theory and procedure. IEEE Trans. on Power Systems, 1998,Vol. 13 (2):401-407
[26] G. Jang, V. Vittal, W. Kliemann. Effect of nonlinear modal interaction on control performance: use of normal forms technique in control design. II. Case studies. IEEE Trans. on Power Systems, 1998,Vol. 13 (2): 408–413
[27] S. Liu; A.R. Messina, V. Vittal. Assessing Placement of Controllers and Nonlinear Behavior Using Normal Form Analysis. IEEE Trans. on Power Systems, 2005,Vol. 20 (3):1486–95
[28] S. Liu, A. R. Messina, V. Vittal. A Normal Form Analysis Approach to Siting Power System Stabilizers (PSSs) and Assessing Power System Nonlinear Behavior. IEEE Trans. Power Systems, 2006,Vol. 21 (4): 1755-1762
[29] N. Pariz, H.M. Shanechi,, and E. Vaahedi. Explaining and Validating Stressed Power Systems Behavior Using Modal Series. IEEE Trans. Power. Syst., May.2003, vol.18, no.2: 778-785
[30]郑云海.模态级数法在交直流电力系统中的应用:[硕士学位论文].成都:四川大学,2005.
[31]刘红超,李兴源,郝巍等.交直流互联电力系统非线性模态分析.电力系统自动化,2006,30(18):8-12,26
[32]李蓓.基于模态级数法的电力系统稳定性分析和控制研究:[硕士学位论文].成都:四川大学,2006
[33]邓集祥,陈武晖.电力系统3阶解析解的推导及验证.中国电机工程学报,2007,27(28):12-18
[34]邓集祥,陈武晖,贺建明.应用模态级数3阶解析解分析大干扰低频振荡机理.电力系统自动化,2007,31(22):27-30.
[35]邓集祥,李佳,刘艳等.大干扰下发电机间三阶非线性相关作用.现代电力,2007,24(6):6-10.
[36]韩祯祥,张琦,徐政.电力系统分析软件的现状与发展.国际电力,电网技术,1999,1:51-54
[37]潘福,韩伟强,张建设.南方电网系统仿真工具应用的分析研究.南方电网技术研究,2005,1(1):14-18
[38] I. Martinez, A.R. Messina, E. Barocio.“Higher-Order Normal Form Analysis of Stressed Power Systems: A Fundamental Study.”Electric Power Components and Systems, 2004, Vol. 32(12): 1301-1317
[39] A.H. Nayfef, B. Balachandran. Modal interactions in dynamical and structural systems. ASEM Appiled Mechanics Reviews.Vol.42: S175-S210
[40] S.A. Nayfeh, A. H. Nayfeh, Nonlinear interactions between two widely spaced modes. Intenrational Jounral of Bifurcationand Chaos, 1993, vol.3: 417-427
[41] I. Martinez, A. R. Messina, E. Barocio. Perturbation analysis of power systems: effects of second-and third-order nonlinear terms on system dynamic behavior. Electric Power Systems Research,2004, 7(1):159-167
[42] V.I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. 2nd ed., New York: Springer-Verlag, 1988, 180-221
[43] P.M.安德逊,A.A.佛阿德.电力系统的控制与稳定(第一卷)(《电力系统的控制与稳定》翻译组译).北京:水利电力出版社,1979,28-29
[44] PSASP电力系统分析综合程序6.22版,基础数据用户手册,中国电力科学研究院.
[45]施妙根,顾厢珍.科学和工程计算基础.北京:清华大学出版社,1995,181-182 .
[2]陈向宜,陈允平.构建大电网安全防御体系—欧洲大停电事故的分析及思考.电力系统自动化,2007,31(1):4-8
[3]中华人民共和国电力行业标准《电力系统安全稳定导则》DL755-2001.中华人民共和国国家经济贸易委员会发布.北京:中国电力出版社,2001
[4] Prabha Kundur.电力系统稳定与控制(影印版).北京:中国电力出版社,2001,699-822
[5]刘隽,李兴源,邹全平.互联电网低频振荡的相关问题及研究.继电器,2005,33(16):70-77,84
[6]苗友忠,汤涌,李丹.局部振荡引起区间大功率振荡的机理.中国电机工程学报,2007,27(10):73-77
[7]刘峰,辛焕海,邱家驹.电力系统结构保留模型的回顾、分析和展望.电力系统及其自动化学报,2005,17(2):13-20,57
[8]王锡凡.现代电力系统分析.北京:科学出版社,2003,365-399,326-327
[9]仲悟之.大型电力系统小干扰稳定性分析方法研究和软件开发:[硕士学位论文].北京:中国电力科学研究院,2005
[10] H.M. Shanechi, N. Pariz, E. Vaahedi. General Nonlinear Modal Representation of Large Scale Power Systems. IEEE Trans .Power. Syst, Aug, 2003, vol.18(3): 1103-1109
[11] J. J. Sanchez-Gasca, V. Vittal, M. J. Gibbard, et al. Inclusion of higher order terms for small-signal (modal) Analysis:committee report-task force on assessing the need to include higher order terms for small-signal (modal) analysis. IEEE Trans. Power Syst., Nov, 2005, vol.20(4):1886-1904
[12] Y. Ni, V. Vittal and W. Kliemann. Analysis of structural properties responsible for nonlinear modal behavior of a stressed power system using the normal form technique. IEEE, 39th Midwest symposium., Circuits and Systems ., Aug, 1996, vol.3: 1029-1033
[13] C. M. Lin, V. Vittal, W. Kliemann, and A. A. Fouad. Investigation of modal interaction and its effects on control performance in stressed power systems using normal forms of vector fields. IEEE Trans. Power Syst., May,1996, vol. 11(2): 781–787
[14] Y. X. Ni, V. Vittal, W. Kliemann, and A. A. Fouad. Nonlinear modal interaction in HVDC/AC power systems with D. C. modulation. IEEE Trans. Power Syst., Nov. 1996, vol.11(4): 2011-2017
[15] S. Saha, A. A. Fouad, W. Kliemann, and V. Vittal. Stability boundary approximation of a power system using the real normal form of vector fields. IEEE Trans. Power Syst., May, 1997, vol.12(2): 797–802
[16] V. Vittal,W. Kliemann, Y. X Ni. Determination of Generator Groupings for an Islanding Scheme in the Manitoba Hydro System Using the Method of Normal Forms. IEEE Trans on Power Systems,1998,13(4):1345-1351
[17] J. Thapar, V. Vittal, W. Kliemann, and A. A. Fouad. Application of the normal form of vector fields to predict interarea separation in power systems. IEEE Trans. Power Syst., May,1997,vol. 12(2): 844–850
[18] E. Barocio, and A. R. Messina. Analysis of Nonlinear Modal Interaction in Stressed Power Systems with SVCs. IEEE Power Engineering Society Winter Meeting., Jan.,2002, Vol.2:.1164–1169
[19]李颖晖,张保会.运用非线性系统理论确定电力系统暂态稳定域的一种新方法.中国电机工程学报,2000, Vol. 20(1): 41-44
[20]李颖晖,张保会,李勐.电力系统稳定边界的研究.中国电机工程学报,2002,Vol.22 (3): 72-77
[21] Songzhe Zhu, V. Vittal, W. Kliemann. Analyzing dynamic performance of power systems over parameter space using normal forms of vector fields-part I: identification of vulnerable regions. IEEE Trans. on Power Systems, 2001, Vol. 16 (3): 444-450
[22] Z.Y. Zou, Q.Y. Jiang, Y.J. Cao, H.F. Wang. Investigation of interactions among the multi-control channels of UPFC using normal forms of vector fields. Proc. 39th International Universities Power Engineering Conference, 2004, pp. 343– 347
[23] E. Barocio, A. R. Messina. Normal form analysis of stressed power systems: Incorporation of SVC models. International Journal of Electrical Power and Energy Systems, 2003,Vol.25 (1): 70-79
[24] A. R. Messina, E. Barocio, J. Arroyo. Analysis of Modal Interaction in Power Systems with FACTS Controllers using Normal Forms. IEEE Power Engineering Society General Meeting, July, 2003, Vol. 4: 2111–2117
[25] G. Jang, V. Vittal, W. Kliemann. Effect of nonlinear modal interaction on control performance: use of normal forms technique in control design. I. General theory and procedure. IEEE Trans. on Power Systems, 1998,Vol. 13 (2):401-407
[26] G. Jang, V. Vittal, W. Kliemann. Effect of nonlinear modal interaction on control performance: use of normal forms technique in control design. II. Case studies. IEEE Trans. on Power Systems, 1998,Vol. 13 (2): 408–413
[27] S. Liu; A.R. Messina, V. Vittal. Assessing Placement of Controllers and Nonlinear Behavior Using Normal Form Analysis. IEEE Trans. on Power Systems, 2005,Vol. 20 (3):1486–95
[28] S. Liu, A. R. Messina, V. Vittal. A Normal Form Analysis Approach to Siting Power System Stabilizers (PSSs) and Assessing Power System Nonlinear Behavior. IEEE Trans. Power Systems, 2006,Vol. 21 (4): 1755-1762
[29] N. Pariz, H.M. Shanechi,, and E. Vaahedi. Explaining and Validating Stressed Power Systems Behavior Using Modal Series. IEEE Trans. Power. Syst., May.2003, vol.18, no.2: 778-785
[30]郑云海.模态级数法在交直流电力系统中的应用:[硕士学位论文].成都:四川大学,2005.
[31]刘红超,李兴源,郝巍等.交直流互联电力系统非线性模态分析.电力系统自动化,2006,30(18):8-12,26
[32]李蓓.基于模态级数法的电力系统稳定性分析和控制研究:[硕士学位论文].成都:四川大学,2006
[33]邓集祥,陈武晖.电力系统3阶解析解的推导及验证.中国电机工程学报,2007,27(28):12-18
[34]邓集祥,陈武晖,贺建明.应用模态级数3阶解析解分析大干扰低频振荡机理.电力系统自动化,2007,31(22):27-30.
[35]邓集祥,李佳,刘艳等.大干扰下发电机间三阶非线性相关作用.现代电力,2007,24(6):6-10.
[36]韩祯祥,张琦,徐政.电力系统分析软件的现状与发展.国际电力,电网技术,1999,1:51-54
[37]潘福,韩伟强,张建设.南方电网系统仿真工具应用的分析研究.南方电网技术研究,2005,1(1):14-18
[38] I. Martinez, A.R. Messina, E. Barocio.“Higher-Order Normal Form Analysis of Stressed Power Systems: A Fundamental Study.”Electric Power Components and Systems, 2004, Vol. 32(12): 1301-1317
[39] A.H. Nayfef, B. Balachandran. Modal interactions in dynamical and structural systems. ASEM Appiled Mechanics Reviews.Vol.42: S175-S210
[40] S.A. Nayfeh, A. H. Nayfeh, Nonlinear interactions between two widely spaced modes. Intenrational Jounral of Bifurcationand Chaos, 1993, vol.3: 417-427
[41] I. Martinez, A. R. Messina, E. Barocio. Perturbation analysis of power systems: effects of second-and third-order nonlinear terms on system dynamic behavior. Electric Power Systems Research,2004, 7(1):159-167
[42] V.I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. 2nd ed., New York: Springer-Verlag, 1988, 180-221
[43] P.M.安德逊,A.A.佛阿德.电力系统的控制与稳定(第一卷)(《电力系统的控制与稳定》翻译组译).北京:水利电力出版社,1979,28-29
[44] PSASP电力系统分析综合程序6.22版,基础数据用户手册,中国电力科学研究院.
[45]施妙根,顾厢珍.科学和工程计算基础.北京:清华大学出版社,1995,181-182 .