轨迹稳定性与时变因素分析
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摘要
基于Lyapunov稳定性理论的直接法是近几十年来电力系统暂态稳定分析的主流发展方向之一。该类方法依赖于精确的故障后系统模型,通过研究平衡点性质及其稳定域大小来判定分析暂态稳定性。但在实际的工程应用中,很难及时重构扰后系统模型和确定扰后系统初值,且扰前建立的数学模型已无太多参考意义。即使克服了模型重构的困难,直接法算法在处理高维度复杂模型时也很难高效准确。上述问题这意味着直接法在处理这类大扰动影响下的模型信息缺失的复杂大系统的稳定性问题时存在一定局限性。
随着计算机软硬件技术的长足进步,唯一能够准确获取且可靠反映系统动态的便是扰后系统的实测轨迹信息。为此,一类以EEAC(Extended Equal Area Criterion,扩展等面积准则)方法为代表的融合系统状态轨迹和暂态失稳机理的暂态稳定分析混合法被陆续提出。它们通过对系统失稳的机理分析和解释,从系统轨迹中提取出能够体现系统稳定或系统失稳的轨迹几何特征,并以此作为判定分析系统稳定性的依据。具体地,EEAC方法着眼于经典的牛顿运动定律,将单机系统的稳定性机理解释为单刚体动能与势能的完全的相互转换,并从中提取出分别用于失稳和稳定判定的轨迹几何特征,即动态鞍点(Dynamic Saddle Point,DSP)和最远点(Far End Point,FEP)。应该说,EEAC方法的提出主要是迫于直接法在数学模型获取及算法在工程实现上的困难。但其新颖的基于轨迹特征的思想为系统稳定性的分析提供了一条有别于Lyapunov稳定性框架的崭新途径。
EEAC从单机系统轨迹中提取出DSP和FEP轨迹特征点,并可正确应用于定常的单机无穷大(One Machine Infinite Bus,OMIB)系统轨迹中。但是,对于多机系统的暂态稳定分析,EEAC方法的分析对象是投影映射得到的具有时变性的等值OMIB系统,其轨迹因时变性而畸变,并远复杂于定常OMIB系统轨迹。此时,直接把DSP和FEP轨迹特征判据应用其中在理论上和技术上存在诸多问题。因此,有必要深入分析时变性导致的等值OMIB系统轨迹畸变对于EEAC方法的影响,提高其基于轨迹特征判定分析暂态稳定性的有效性和可靠性。围绕这一主题,本文研究的主要内容如下:
1)DSP轨迹特征判据拓展应用至等值OMIB系统的理论基础研究。
A.针对一般系统的运动轨迹,从轨迹分析的角度,提出了轨迹稳定性概念和摆次稳定性概念。并在此基础上,深入分析了DSP、摆次平稳性及轨迹稳定性之间的关系,提出了DSP失稳判据应用于一般非线性非自治系统的充分条件,确定了一类能够使用DSP作为轨迹失稳特征的系统子集。
B.通过仿真计算和机理解释,验证说明了等值OMIB系统满足上述充分条件,从而为EEAC方法应用DSP作为轨迹失稳判据奠定了新的数学理论基础。
2)时变性强弱的量化评估方法以及考虑时变性影响的DSP失稳判据和裕度评估点的改进。
A.考虑到现有时变性强弱指标评估范围的细化程度不足,提出了能够刻画某一个时刻或轨迹点处的时变性强弱程度的量化评估指标和简化的开关评估指标,并给出了相应的结合时域仿真轨迹的指标计算公式。
B.考虑到时变性的影响会降低DSP失稳判据的准确率,需识别出不能给出失稳判定的那些DSP,即病态DSP。为此,提出了DSP在时变性和能量积累两个方面的轨迹特征及相应的量化指标。并籍此给出了两种分别基于上述指标的病态DSP识别方法。两种方法能够可靠识别病态DSP和正常DSP,从而提高了时变性影响下应用DSP轨迹特征进行失稳判定的准确性。
C.考虑到时变性的影响会降低FEP处稳定裕度评估的准确性,进而不利于临界参数的灵敏度搜索。提出了因强时变性导致的病态FEP概念,其不再适合作为裕度评估点。并在此基础上,提出了裕度评估点的改进原则和方法,从而提高了时变性影响下的稳定裕度评估的准确性。
3)考虑映象摆次裕度-参数复杂变化关系的系统临界参数迭代搜索算法的改进。
根据已有文献对映象摆次裕度-参数曲线的复杂特征的分析,发现其中出现的非单调现象和孤立稳定域现象会降低原有迭代搜索算法的准确性和效率。为此,针对原有迭代搜索算法两个主要环节,分别给出了优化改进措施,并在此基础上提出了系统临界参数的改进迭代搜索算法,从而提高了求取多摆失稳临界参数的准确性和搜索效率。
The direct methods which are based on the Lyapunov stability theory are the major branch of transient stability analysis in power systems. These methods depend on the mathematical models of the post-fault power systems and assess the transient stability of power systems by analyzing the stable equilibrium point and its stability domain. However, in the engineering application of direct methods, it is very difficult to reconstruct the model of disturbed systems and calculate its initial value. Even if the difficulty of model reconstruction is overcome, the algorithms of direct methods are not efficient and accurate in complicated models with high dimension. It is implied that there exist some limitations in direct methods when they are used to analyze the stability of large complex systems without enough model information.
With the development of computer technology and simulation algorithm, only the measured trajectories of disturbed systems, which can exhibit system dynamics, are able to be acquired precisely. Therefore, a new type of hybrid methods for transient stability assessment, represented by EEAC (extended equal area criterion) method, was proposed, which combine system trajectories with the mechanism of transient instability. Via the mechanism analysis of instability, they extract the geometric characteristics from system trajectories, which can reflect the stability and instability of power systems. For example, in EEAC, based on the classical Newton's laws of motion, the stability of one machine system is interpreted as the full conversion between the kinetic energy and potential energy of the single rigid body. And, the geometric characteristics of trajectories, i.e. dynamic saddle point (DSP) and far end point (FEP), which can be defined as the instability and stability criterion, are proposed. It should be said that because of the difficulties in obtaining precise mathematical models and the engineering implementation of algorithms in direct methods, EEAC method had to be proposed. Its new idea, which is based on trajectory characteristics and different from the traditional Lyapunov stability framework, is provided for the transient stability analysis of power systems.
From trajectories of one machine system, EEAC method extracts the trajectory characteristic points, i.e. DSP and FEP, which can be correctly applied to the time-invariant OMIB (One Machine Infinite Bus) system. However, for the transient stability analysis of multi-machine systems, the analyzed object of EEAC method is the time-varying equivalent OMIB system, obtained by projection mapping. Its trajectory is distorted by time-variation and much more complicated than the one of time-invariant OMIB system. Hence, there are some theory and technique problems in the application of trajectory characteristic criterions of DSP and FEP to equivalent OMIB systems.
It is necessary to study the trajectory distortion caused by time-varying factor and its effects on EEAC method, in order to improve its effectiveness and reliability for equivalent OMIB systems. The main works are summarized as follows:
1) Theoretical foundation of the instability criterion of DSP for equivalent OMIB systems.
A. Based on the trajectory of a general motion system, the concepts of trajectory stability and swing steadiness etc are proposed. The relationships among the DSP, swing steadiness and trajectory stability are thoroughly studied and the sufficient condition that the instability criterion of DSP can be used in general motion systems is presented. Thus, a system subset, in which trajectory instability can be determined by DSP, is defined.
B. Through the simulation analysis and mechanism interpretation, the above sufficient condition is verified in equivalent OMIB systems. Thus, a theoretical foundation of DSP for instability criterion in EEAC method is developed.
2) The evaluation of the time-varying factor and the improvements of instability criterion of DSP and stability assessment point for reducing the impact of time-variation.
A. To increase the precision of the conventional indicators, a quantification index and a simplified switch index for evaluating the degree of the time-varying factor at one trajectory point are proposed respectively. And, the computation equations of the above indices based on the simulated trajectory are also presented.
B. Because the time-varying factor reduces the accuracy of the instability criterion of DSP, the invalid DSP that can not be instability criterion should be identified. To distinguish invalid DSP, two trajectory characteristics of DSP, i.e. the time-varying factor and the kinetic energy accumulation, are investigated and the corresponding indices of DSP are proposed. Based on the indices, two identification methods for invalid DSP are proposed. Their reliable identification of invalid DSPs contributes to the higher accuracy of instability analysis in time-varying equivalent OMIB systems by DSP.
C. The time-varying factor reduces the accuracy of the stability assessment at FEP and consequently affects the sensitivity-based search for transient stability limits. The concept of invalid FEP is proposed, which is due to strong time-varying factor and not suitable for assessment point any more. With this knowledge, the improvement principle of assessment points of stable swings and the corresponding method are proposed for improving the accuracy of the stability assessment in time-varying equivalent OMIB systems.
3) An improved iterative method for assessment of multi-swing stability limit
It is observed that the complicated variation of stability margin of OMIB-swing versus parameter, such as the non-monotonicity of margin curve and the isolated stability domain, might reduce the efficiency and accuracy of the conventional iterative method in assessing multi-swing stability limits. Therefore, the above effects on the two major stages of the conventional method are thoroughly studied and an improved sensitivity-based iterative method is proposed for searching first- or multi-swing transient stability limit with higher efficiency and accuracy.
随着计算机软硬件技术的长足进步,唯一能够准确获取且可靠反映系统动态的便是扰后系统的实测轨迹信息。为此,一类以EEAC(Extended Equal Area Criterion,扩展等面积准则)方法为代表的融合系统状态轨迹和暂态失稳机理的暂态稳定分析混合法被陆续提出。它们通过对系统失稳的机理分析和解释,从系统轨迹中提取出能够体现系统稳定或系统失稳的轨迹几何特征,并以此作为判定分析系统稳定性的依据。具体地,EEAC方法着眼于经典的牛顿运动定律,将单机系统的稳定性机理解释为单刚体动能与势能的完全的相互转换,并从中提取出分别用于失稳和稳定判定的轨迹几何特征,即动态鞍点(Dynamic Saddle Point,DSP)和最远点(Far End Point,FEP)。应该说,EEAC方法的提出主要是迫于直接法在数学模型获取及算法在工程实现上的困难。但其新颖的基于轨迹特征的思想为系统稳定性的分析提供了一条有别于Lyapunov稳定性框架的崭新途径。
EEAC从单机系统轨迹中提取出DSP和FEP轨迹特征点,并可正确应用于定常的单机无穷大(One Machine Infinite Bus,OMIB)系统轨迹中。但是,对于多机系统的暂态稳定分析,EEAC方法的分析对象是投影映射得到的具有时变性的等值OMIB系统,其轨迹因时变性而畸变,并远复杂于定常OMIB系统轨迹。此时,直接把DSP和FEP轨迹特征判据应用其中在理论上和技术上存在诸多问题。因此,有必要深入分析时变性导致的等值OMIB系统轨迹畸变对于EEAC方法的影响,提高其基于轨迹特征判定分析暂态稳定性的有效性和可靠性。围绕这一主题,本文研究的主要内容如下:
1)DSP轨迹特征判据拓展应用至等值OMIB系统的理论基础研究。
A.针对一般系统的运动轨迹,从轨迹分析的角度,提出了轨迹稳定性概念和摆次稳定性概念。并在此基础上,深入分析了DSP、摆次平稳性及轨迹稳定性之间的关系,提出了DSP失稳判据应用于一般非线性非自治系统的充分条件,确定了一类能够使用DSP作为轨迹失稳特征的系统子集。
B.通过仿真计算和机理解释,验证说明了等值OMIB系统满足上述充分条件,从而为EEAC方法应用DSP作为轨迹失稳判据奠定了新的数学理论基础。
2)时变性强弱的量化评估方法以及考虑时变性影响的DSP失稳判据和裕度评估点的改进。
A.考虑到现有时变性强弱指标评估范围的细化程度不足,提出了能够刻画某一个时刻或轨迹点处的时变性强弱程度的量化评估指标和简化的开关评估指标,并给出了相应的结合时域仿真轨迹的指标计算公式。
B.考虑到时变性的影响会降低DSP失稳判据的准确率,需识别出不能给出失稳判定的那些DSP,即病态DSP。为此,提出了DSP在时变性和能量积累两个方面的轨迹特征及相应的量化指标。并籍此给出了两种分别基于上述指标的病态DSP识别方法。两种方法能够可靠识别病态DSP和正常DSP,从而提高了时变性影响下应用DSP轨迹特征进行失稳判定的准确性。
C.考虑到时变性的影响会降低FEP处稳定裕度评估的准确性,进而不利于临界参数的灵敏度搜索。提出了因强时变性导致的病态FEP概念,其不再适合作为裕度评估点。并在此基础上,提出了裕度评估点的改进原则和方法,从而提高了时变性影响下的稳定裕度评估的准确性。
3)考虑映象摆次裕度-参数复杂变化关系的系统临界参数迭代搜索算法的改进。
根据已有文献对映象摆次裕度-参数曲线的复杂特征的分析,发现其中出现的非单调现象和孤立稳定域现象会降低原有迭代搜索算法的准确性和效率。为此,针对原有迭代搜索算法两个主要环节,分别给出了优化改进措施,并在此基础上提出了系统临界参数的改进迭代搜索算法,从而提高了求取多摆失稳临界参数的准确性和搜索效率。
The direct methods which are based on the Lyapunov stability theory are the major branch of transient stability analysis in power systems. These methods depend on the mathematical models of the post-fault power systems and assess the transient stability of power systems by analyzing the stable equilibrium point and its stability domain. However, in the engineering application of direct methods, it is very difficult to reconstruct the model of disturbed systems and calculate its initial value. Even if the difficulty of model reconstruction is overcome, the algorithms of direct methods are not efficient and accurate in complicated models with high dimension. It is implied that there exist some limitations in direct methods when they are used to analyze the stability of large complex systems without enough model information.
With the development of computer technology and simulation algorithm, only the measured trajectories of disturbed systems, which can exhibit system dynamics, are able to be acquired precisely. Therefore, a new type of hybrid methods for transient stability assessment, represented by EEAC (extended equal area criterion) method, was proposed, which combine system trajectories with the mechanism of transient instability. Via the mechanism analysis of instability, they extract the geometric characteristics from system trajectories, which can reflect the stability and instability of power systems. For example, in EEAC, based on the classical Newton's laws of motion, the stability of one machine system is interpreted as the full conversion between the kinetic energy and potential energy of the single rigid body. And, the geometric characteristics of trajectories, i.e. dynamic saddle point (DSP) and far end point (FEP), which can be defined as the instability and stability criterion, are proposed. It should be said that because of the difficulties in obtaining precise mathematical models and the engineering implementation of algorithms in direct methods, EEAC method had to be proposed. Its new idea, which is based on trajectory characteristics and different from the traditional Lyapunov stability framework, is provided for the transient stability analysis of power systems.
From trajectories of one machine system, EEAC method extracts the trajectory characteristic points, i.e. DSP and FEP, which can be correctly applied to the time-invariant OMIB (One Machine Infinite Bus) system. However, for the transient stability analysis of multi-machine systems, the analyzed object of EEAC method is the time-varying equivalent OMIB system, obtained by projection mapping. Its trajectory is distorted by time-variation and much more complicated than the one of time-invariant OMIB system. Hence, there are some theory and technique problems in the application of trajectory characteristic criterions of DSP and FEP to equivalent OMIB systems.
It is necessary to study the trajectory distortion caused by time-varying factor and its effects on EEAC method, in order to improve its effectiveness and reliability for equivalent OMIB systems. The main works are summarized as follows:
1) Theoretical foundation of the instability criterion of DSP for equivalent OMIB systems.
A. Based on the trajectory of a general motion system, the concepts of trajectory stability and swing steadiness etc are proposed. The relationships among the DSP, swing steadiness and trajectory stability are thoroughly studied and the sufficient condition that the instability criterion of DSP can be used in general motion systems is presented. Thus, a system subset, in which trajectory instability can be determined by DSP, is defined.
B. Through the simulation analysis and mechanism interpretation, the above sufficient condition is verified in equivalent OMIB systems. Thus, a theoretical foundation of DSP for instability criterion in EEAC method is developed.
2) The evaluation of the time-varying factor and the improvements of instability criterion of DSP and stability assessment point for reducing the impact of time-variation.
A. To increase the precision of the conventional indicators, a quantification index and a simplified switch index for evaluating the degree of the time-varying factor at one trajectory point are proposed respectively. And, the computation equations of the above indices based on the simulated trajectory are also presented.
B. Because the time-varying factor reduces the accuracy of the instability criterion of DSP, the invalid DSP that can not be instability criterion should be identified. To distinguish invalid DSP, two trajectory characteristics of DSP, i.e. the time-varying factor and the kinetic energy accumulation, are investigated and the corresponding indices of DSP are proposed. Based on the indices, two identification methods for invalid DSP are proposed. Their reliable identification of invalid DSPs contributes to the higher accuracy of instability analysis in time-varying equivalent OMIB systems by DSP.
C. The time-varying factor reduces the accuracy of the stability assessment at FEP and consequently affects the sensitivity-based search for transient stability limits. The concept of invalid FEP is proposed, which is due to strong time-varying factor and not suitable for assessment point any more. With this knowledge, the improvement principle of assessment points of stable swings and the corresponding method are proposed for improving the accuracy of the stability assessment in time-varying equivalent OMIB systems.
3) An improved iterative method for assessment of multi-swing stability limit
It is observed that the complicated variation of stability margin of OMIB-swing versus parameter, such as the non-monotonicity of margin curve and the isolated stability domain, might reduce the efficiency and accuracy of the conventional iterative method in assessing multi-swing stability limits. Therefore, the above effects on the two major stages of the conventional method are thoroughly studied and an improved sensitivity-based iterative method is proposed for searching first- or multi-swing transient stability limit with higher efficiency and accuracy.
引文
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