基于Hopf分岔理论的电力系统动态电压稳定研究
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摘要
大型互联电力系统电压稳定问题的研究对于我国电力系统的安全稳定运行具有十分重要的意义。电力系统动态电压稳定可行域的边界通常由鞍节、Hopf和奇异诱导三个分岔空间组成的,而Hopf分岔是一种典型的动态分岔,是连接确定解和周期解的桥梁。本文围绕Hopf分岔理论,探讨了Hopf分岔对电力系统动态电压稳定性的影响,主要研究内容有:
(1)提出了一种求解电力系统动态电压稳定Hopf分岔点的简化直接法。该方法对以往的Hopf分岔点直接解法进行了拓展和简化,对于一个n维系统,仅需构造n+2维的拓展系统,该拓展系统由描述电力系统动态特性的微分代数方程组和两个标量方程组成。
(2)提出了一种基于重启动精化Arnoldi算法的动态电压稳定分析方法。该算法基于精化投影思想,对确定的Ritz值λ~i ,在Krylov子空间用达到最小残量的向量(即精化Ritz向量)代替传统的Ritz向量做为待求矩阵的近似特征向量,以达到提高计算效率的目的。将这一算法与追踪电力系统平衡解流形问题的连续法结合,在得到一个平衡点后求出对应的系统雅可比矩阵的关键特征值,再以此来判断是否出现了Hopf分岔点。
(3)提出了一种计及特征值实部关于负荷增长的二阶灵敏度系数的自适应步长调整连续法。该方法利用特征值实部关于负荷增长的一阶和二阶灵敏度系数来自动调整预测步长的大小,在初始点附近利用较大步长来追踪平衡解流形,当运行点靠近Hopf分岔点时,则自动减小预测步长,以避免由于预测步长过大而发生“跳过”Hopf分岔点的情况。
(4)提出了一种通过追踪关键特征值来确定Hopf分岔点的混合方法。该方法从初始状态开始,利用连续性方法追踪平衡解流形,同时计算特征值对负荷增长的灵敏度系数,结合特征值分析选取关键特征值;当关键特征值比较靠近虚轴时,再转入对关键特征值的连续追踪,直至Hopf分岔点。
(5)利用动态电压稳定裕度对控制参数的灵敏度系数,挑选出控制效果较好的控制参数做为主要的控制方式,重点刻画出与此类控制参数所对应的由Hopf分岔所决定的动态电压稳定可行域边界。
全文从Hopf分岔点的求解方法出发,首先研究了求解多机电力系统Hopf分岔点的直接法;然后结合特征值分析方法分别提出了求解Hopf分岔点的连续性方法和对关键特征值的追踪算法;在此基础上,进一步结合动态电压稳定裕度对控制参数的灵敏度分析技术,利用连续性方法刻画出了由Hopf分岔所决定的动态电压稳定可行域边界。
The intensive research on voltage stability of bulk connected power system plays a significant role in the safe, reliable and efficient operation of the power system. Generally, the feasibility boundar of dynamic voltage stability is constructed by three bifurcation space: Saddle, Hopf and singularity induced bifurcation. Hopf bifurcation is a kind of typical dynamic bifurcation which can server as a bridge between the determinate solutions and the periodic ones. The dynamic voltage stability based on Hopf bifurcation theory is studied in this thesis, the main content was shown as below.
The direct method was an effective way to determine the Hopf bifurcation points in power system dynamic voltage stability. However, for an n-dimensional power system a (2n+2)-dimensional augmented system was required in classic direct methods and the computation to solve the augmented system was expensive. A novel approach to calculate the Hopf bifurcation points is presented. In this method, a simple (n+2)-dimensional augmented system is founded which includes the differential-algebraic equations set to describe the dynamic characteristics of the power system and 2 scalar equations. The Hopf bifurcation points can be ascertained by solving the new augmented system.
A restarted-refined Arnoldi method is introduced into the dynamic voltage stability analysis to calculate the critical eigenvalues of bulk power system. The refined Ritz vectors are used as the approximations based on the refined projection theory in order to enrich the information of the eigenvectors in the projective subspace. The method can be combined with the continuous arithmetic which was used to trace the equilibrium manifold. The key eigenvalue can be ascertained by the application of restarted-refined Arnoldi method at each equilibrium point of the power system DAE models. The Hopf bifurcation appears if there is a pair of conjugate eigenvalues with zero real-part.
A step adaptive-adjustion continuous method with consideration of second-order sensitivity of eigenvalue real-part to load increase is put forward. The forecast step is adjusted adaptive by the application of the first and second sensitivity of eigenvalue real-part to load increase. The biggish step is applied to trace the equilibrium manifold at initial point, the step will be decreased automatically when the operation point is close to the Hopf bifurcation in order to avoid jumping over the Hopf bifurcation.
A hybrid method to calculate Hopf bifurcation by the tracing of the key eigenvalue is presented. The equilibrium manifold is traced by the continuous method at beginning. The key eigenvalue is ascertained based on the sensitivity of eigenvalue real-part to load increase. The key eigenvalue is traced when it is closed to the imaginary coordinate-axis until the Hopf bifurcation appeares.
The feasibility boundary based on the selected control parameters is depicted by the application of the continuous method presented. The efficiently control parameters are selected according to the sensitivity of load margin to control parameter.
To sum up the above arguments, the direct method to determine the Hopf bifurcation points of power system is studied at first. Then the step adaptive-adjustion continuous method and the hybrid method to trace the key eigenvalue are presented respectively which are integrated with eigenvalues-analysis. Lastly, the feasibility boundary based on the selected control parameters is depicted by the application of the continuous method presented and the sensitivity of load margin to control parameter.
(1)提出了一种求解电力系统动态电压稳定Hopf分岔点的简化直接法。该方法对以往的Hopf分岔点直接解法进行了拓展和简化,对于一个n维系统,仅需构造n+2维的拓展系统,该拓展系统由描述电力系统动态特性的微分代数方程组和两个标量方程组成。
(2)提出了一种基于重启动精化Arnoldi算法的动态电压稳定分析方法。该算法基于精化投影思想,对确定的Ritz值λ~i ,在Krylov子空间用达到最小残量的向量(即精化Ritz向量)代替传统的Ritz向量做为待求矩阵的近似特征向量,以达到提高计算效率的目的。将这一算法与追踪电力系统平衡解流形问题的连续法结合,在得到一个平衡点后求出对应的系统雅可比矩阵的关键特征值,再以此来判断是否出现了Hopf分岔点。
(3)提出了一种计及特征值实部关于负荷增长的二阶灵敏度系数的自适应步长调整连续法。该方法利用特征值实部关于负荷增长的一阶和二阶灵敏度系数来自动调整预测步长的大小,在初始点附近利用较大步长来追踪平衡解流形,当运行点靠近Hopf分岔点时,则自动减小预测步长,以避免由于预测步长过大而发生“跳过”Hopf分岔点的情况。
(4)提出了一种通过追踪关键特征值来确定Hopf分岔点的混合方法。该方法从初始状态开始,利用连续性方法追踪平衡解流形,同时计算特征值对负荷增长的灵敏度系数,结合特征值分析选取关键特征值;当关键特征值比较靠近虚轴时,再转入对关键特征值的连续追踪,直至Hopf分岔点。
(5)利用动态电压稳定裕度对控制参数的灵敏度系数,挑选出控制效果较好的控制参数做为主要的控制方式,重点刻画出与此类控制参数所对应的由Hopf分岔所决定的动态电压稳定可行域边界。
全文从Hopf分岔点的求解方法出发,首先研究了求解多机电力系统Hopf分岔点的直接法;然后结合特征值分析方法分别提出了求解Hopf分岔点的连续性方法和对关键特征值的追踪算法;在此基础上,进一步结合动态电压稳定裕度对控制参数的灵敏度分析技术,利用连续性方法刻画出了由Hopf分岔所决定的动态电压稳定可行域边界。
The intensive research on voltage stability of bulk connected power system plays a significant role in the safe, reliable and efficient operation of the power system. Generally, the feasibility boundar of dynamic voltage stability is constructed by three bifurcation space: Saddle, Hopf and singularity induced bifurcation. Hopf bifurcation is a kind of typical dynamic bifurcation which can server as a bridge between the determinate solutions and the periodic ones. The dynamic voltage stability based on Hopf bifurcation theory is studied in this thesis, the main content was shown as below.
The direct method was an effective way to determine the Hopf bifurcation points in power system dynamic voltage stability. However, for an n-dimensional power system a (2n+2)-dimensional augmented system was required in classic direct methods and the computation to solve the augmented system was expensive. A novel approach to calculate the Hopf bifurcation points is presented. In this method, a simple (n+2)-dimensional augmented system is founded which includes the differential-algebraic equations set to describe the dynamic characteristics of the power system and 2 scalar equations. The Hopf bifurcation points can be ascertained by solving the new augmented system.
A restarted-refined Arnoldi method is introduced into the dynamic voltage stability analysis to calculate the critical eigenvalues of bulk power system. The refined Ritz vectors are used as the approximations based on the refined projection theory in order to enrich the information of the eigenvectors in the projective subspace. The method can be combined with the continuous arithmetic which was used to trace the equilibrium manifold. The key eigenvalue can be ascertained by the application of restarted-refined Arnoldi method at each equilibrium point of the power system DAE models. The Hopf bifurcation appears if there is a pair of conjugate eigenvalues with zero real-part.
A step adaptive-adjustion continuous method with consideration of second-order sensitivity of eigenvalue real-part to load increase is put forward. The forecast step is adjusted adaptive by the application of the first and second sensitivity of eigenvalue real-part to load increase. The biggish step is applied to trace the equilibrium manifold at initial point, the step will be decreased automatically when the operation point is close to the Hopf bifurcation in order to avoid jumping over the Hopf bifurcation.
A hybrid method to calculate Hopf bifurcation by the tracing of the key eigenvalue is presented. The equilibrium manifold is traced by the continuous method at beginning. The key eigenvalue is ascertained based on the sensitivity of eigenvalue real-part to load increase. The key eigenvalue is traced when it is closed to the imaginary coordinate-axis until the Hopf bifurcation appeares.
The feasibility boundary based on the selected control parameters is depicted by the application of the continuous method presented. The efficiently control parameters are selected according to the sensitivity of load margin to control parameter.
To sum up the above arguments, the direct method to determine the Hopf bifurcation points of power system is studied at first. Then the step adaptive-adjustion continuous method and the hybrid method to trace the key eigenvalue are presented respectively which are integrated with eigenvalues-analysis. Lastly, the feasibility boundary based on the selected control parameters is depicted by the application of the continuous method presented and the sensitivity of load margin to control parameter.
引文
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