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Chaos and Control of a Third-Order Power System Model
DOI: 10.12677/DSC.2019.83021, PDF, HTML, XML, 下载: 936  浏览: 1,595  国家自然科学基金支持

Abstract: The dynamical behaviors and control problems are investigated in the third-order model of power system. The bifurcation diagram, Lyapunov exponential diagram and phase diagram of the corre-sponding differential equations are obtained by numerical simulation using the existing third-order mathematical model of power system. Furthermore, the influences of system parameters change on the dynamic behavior of the system are analyzed, and the Hopf bifurcation and chaos phenomena occur with the change of system parameters under certain conditions. By calculating the divergence of a vector field of the system, the dissipation of the system is obtained, and the fractal dimension of the system is calculated. When chaos appears, two control schemes of adaptive control and nonlinear feedback control are presented, respectively, to verify the proposed control schemes by using Lyapunov stability theory. In addition, the controller design scheme is discussed when some parameters are unknown. By constructing Lyapunov function, it is proved theoretically that three schemes can make the system asymptotically stable. Finally, three control schemes are simulated numerically.

1. 引言

2. 电力系统三阶数学模型的建立

Figure 1. Diagram of single machine infinite bus system

$\left\{\begin{array}{l}\stackrel{˙}{\delta }={\omega }_{0}\omega ,\\ \stackrel{˙}{\omega }=\frac{1}{H}\left({P}_{m}-D\omega -\frac{E{V}_{s}}{{{x}^{\prime }}_{d\Sigma }}\mathrm{sin}\delta \right),\\ \stackrel{˙}{E}=\frac{1}{{{T}^{\prime }}_{d0}}\left(-\frac{{x}_{d\Sigma }}{{{x}^{\prime }}_{d\Sigma }}E+\frac{{x}_{d}-{{x}^{\prime }}_{d}}{{{x}^{\prime }}_{d\Sigma }}{V}_{s}\mathrm{cos}\delta +{E}_{f}\right).\end{array}$ (1)

$\left\{\begin{array}{l}\stackrel{˙}{\delta }={\omega }_{0}\omega ,\\ H\stackrel{˙}{\omega }=-\gamma \omega +{P}_{m}-B{V}_{s}E\mathrm{sin}\delta ,\\ \alpha \stackrel{˙}{E}=-\left(1+XB\right)E+XB{V}_{s}\mathrm{cos}\delta +{E}_{f}.\end{array}$ (2)

$\left\{\begin{array}{l}\stackrel{˙}{x}=dy,\\ \stackrel{˙}{y}=-ay+b-cz\mathrm{sin}x,\\ \stackrel{˙}{z}=-z+0.5\mathrm{cos}x+0.5.\end{array}$ (3)

3. 电力系统三阶模型的稳定性分析

3.1. 参数同步角速度和负阻尼系数对系统的稳定性分析

1) 固定参数 $b=0.5$$c=1$$d=0.5$ ，改变参数a。

$a\in \left(-0.1,0\right)$ 时，系统的分岔图和最大Lyapunov指数图如图2所示。由图可知，当 $a\in \left(-0.1,-0.06\right)$ 时，系统的最大Lyapunov指数大于0，系统处于混沌状态；当 $a\in \left(-0.06,0\right)$ 时，系统的最大Lyapunov指数小于0，系统处于稳定状态。

2) 固定参数 $b=0.5$$c=1$$d=1$ ，改变参数a。

$a\in \left(-0.17,0\right)$ 时，系统的分岔图和最大Lyapunov指数图如图3所示。同理得到，当 $a\in \left(-0.17,-0.1\right)$ 时，系统处于混沌状态；当 $a\in \left(-0.1,0\right)$ 时，系统处于稳定状态。

3) 固定参数 $b=0.5$$c=1$$d=2$ ，改变参数a。

$a\in \left(-0.22,0\right)$ 时，系统的分岔图和最大Lyapunov指数图如图4所示。同理得到，当 $a\in \left(-0.22,-0.14\right)$ 时，系统处于混沌状态；当 $a\in \left(-0.14,0\right)$ 时，系统处于稳定状态。

(a) 系统分岔图 (b) 系统最大Lyapunov指数图

Figure 2. The bifurcation diagram and Lyapunov exponents diagram for ${\omega }_{0}=0.5$

(a) 系统分岔图 (b) 系统最大Lyapunov指数图

Figure 3. The bifurcation diagram and Lyapunov exponents diagram for ${\omega }_{0}=1$

(a) 系统分岔图 (b) 系统最大Lyapunov指数图

Figure 4. The bifurcation diagram and Lyapunov exponents diagram for ${\omega }_{0}=2$

(a) a=-0.01 (b) a=-0.0592 (c) a=-0.08 (d) a=-0.1

Figure 5. Different dynamical behaviors with the change of parameter a

${\omega }_{0}=0.5$ 的情形为例，从图2(a)和图5的图像中，这个系统出现了一些丰富的动力学行为。从右到左，随着a的减少，系统出现了稳定的不动点、周期运动、混沌以及系统的崩溃。我们的分析表明不动点由稳定到不稳定经过了Hopf分岔，在本节的后面会进行具体的介绍。从图2(b)，可以确定混沌区域的出现，不同的阻尼系数影响混沌区域的大小和位置。

3.2. 系统的基本性质分析

(a) x-y相图 (b) 时序图

Figure 6. The phase diagram and time series analysis diagram of the system

3.2.1. 耗散性和吸引子的存在性

$\nabla V=\frac{\partial \stackrel{˙}{x}}{\partial x}+\frac{\partial \stackrel{˙}{y}}{\partial y}+\frac{\partial \stackrel{˙}{z}}{\partial z}=0-a-1=-a-1.$

Liouville定理指出了Lyapunov指数和系统Jacobin矩阵(即散度)之间的关系，即

$\underset{i=1}{\overset{3}{\sum }}{\lambda }_{i}=Tr\left(J\right)=divV,$

$divV=\nabla V=\frac{\partial \stackrel{˙}{x}}{\partial x}+\frac{\partial \stackrel{˙}{y}}{\partial y}+\frac{\partial \stackrel{˙}{z}}{\partial z}=Tr\left(J\right)=-a-1=\underset{i=1}{\overset{3}{\sum }}{\lambda }_{i}={\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3}=-0.86,$

$\frac{\text{d}V}{\text{d}t}={\text{e}}^{-\left(a+1\right)}={\text{e}}^{-0.86},$

$V\left(t\right)={V}_{0}{\text{e}}^{-0.86t}.$

$D=j+\frac{1}{|{\lambda }_{j+1}|}\underset{i=1}{\overset{j}{\sum }}{\lambda }_{j}=2+\frac{{\lambda }_{1}+{\lambda }_{2}}{|{\lambda }_{3}|}=2+\frac{0.1599-0.0264}{0.9935}=2.134.$ (4)

3.2.2. 稳定性分析和分岔理论

$J=\left(\begin{array}{ccc}0& {\omega }_{0}& 0\\ -\frac{B{V}_{s}{E}_{0}\mathrm{cos}{\delta }_{0}}{H}& -\frac{\gamma }{H}& -\frac{B{V}_{s}\mathrm{sin}{\delta }_{0}}{H}\\ -\frac{XB{V}_{s}\mathrm{sin}{\delta }_{0}}{\alpha }& 0& -\frac{1+XB}{\alpha }\end{array}\right),$

$|\lambda E-J|=|\begin{array}{ccc}\lambda & -{\omega }_{0}& 0\\ \frac{B{V}_{s}{E}_{0}\mathrm{cos}{\delta }_{0}}{H}& \lambda +\frac{\gamma }{H}& \frac{B{V}_{s}\mathrm{sin}{\delta }_{0}}{H}\\ \frac{XB{V}_{s}\mathrm{sin}{\delta }_{0}}{\alpha }& 0& \lambda +\frac{1+XB}{\alpha }\end{array}|=0,$

${\lambda }^{3}+\left(\frac{H\left(1+XB\right)+\gamma \alpha }{H\alpha }\right){\lambda }^{2}+\frac{\gamma \left(1+XB\right)+\alpha {\omega }_{0}B{V}_{s}{E}_{0}\mathrm{cos}{\delta }_{0}}{H\alpha }\lambda \text{ }+\frac{{\omega }_{0}B{V}_{s}\left(\left(1+XB\right){E}_{0}\mathrm{cos}{\delta }_{0}-XB{V}_{s}{\mathrm{sin}}^{2}{\delta }_{0}\right)}{H\alpha }=0$

$\left(\lambda +p\right)\left({\lambda }^{2}+{q}^{2}\right)=0,$

$\begin{array}{l}{\lambda }^{3}+\left(\frac{H\left(1+XB\right)+\gamma \alpha }{H\alpha }\right){\lambda }^{2}+\frac{\gamma \left(1+XB\right)+\alpha {\omega }_{0}B{V}_{s}{E}_{0}\mathrm{cos}{\delta }_{0}}{H\alpha }\lambda \\ +\frac{{\omega }_{0}B{V}_{s}\left(\left(1+XB\right){E}_{0}\mathrm{cos}{\delta }_{0}-XB{V}_{s}{\mathrm{sin}}^{2}{\delta }_{0}\right)}{H\alpha }={\lambda }^{3}+p{\lambda }^{2}+{q}^{2}\lambda +p{q}^{2}\end{array}$

$p=\frac{H\left(1+XB\right)+\gamma \alpha }{H\alpha },$

${q}^{2}=\frac{\gamma \left(1+XB\right)+\alpha {\omega }_{0}B{V}_{s}{E}_{0}\mathrm{cos}{\delta }_{0}}{H\alpha },$

$p{q}^{2}=\frac{{\omega }_{0}B{V}_{s}\left(\left(1+XB\right){E}_{0}\mathrm{cos}{\delta }_{0}-XB{V}_{s}{\mathrm{sin}}^{2}{\delta }_{0}\right)}{H\alpha },$

$\therefore p{q}^{2}=\frac{H\left(1+XB\right)+\gamma \alpha }{H\alpha }\cdot \frac{\gamma \left(1+XB\right)+\alpha {\omega }_{0}B{V}_{s}{E}_{0}\mathrm{cos}{\delta }_{0}}{H\alpha }=\frac{{\omega }_{0}B{V}_{s}\left(\left(1+XB\right){E}_{0}\mathrm{cos}{\delta }_{0}-XB{V}_{s}{\mathrm{sin}}^{2}{\delta }_{0}\right)}{H\alpha }$

$\left(1+XB\right)H{\gamma }^{2}+\left(\alpha H{\omega }_{0}B{V}_{s}{E}_{0}\mathrm{cos}{\delta }_{0}+\frac{{H}^{2}{\left(1+XB\right)}^{2}}{\alpha }\right)\gamma +{\omega }_{0}X{B}^{2}{V}_{s}{}^{2}{\mathrm{sin}}^{2}{\delta }_{0}=0.$ (5)

$\therefore Hopf$ 分岔曲线的方程由以下方程组成：

$\left\{\begin{array}{l}{P}_{m}=B{V}_{s}{E}_{0}\mathrm{sin}{\delta }_{0},\\ \left(1+XB\right){E}_{0}={E}_{f}+XB{V}_{s}\mathrm{cos}{\delta }_{0},\\ \left(1+XB\right)H{\gamma }^{2}+\left(\alpha H{\omega }_{0}B{V}_{s}{E}_{0}\mathrm{cos}{\delta }_{0}+\frac{{H}^{2}{\left(1+XB\right)}^{2}}{\alpha }\right)\gamma +{\omega }_{0}X{B}^{2}{V}_{s}{}^{2}{\mathrm{sin}}^{2}{\delta }_{0}=0.\end{array}$ (6)

4. 三阶电力系统的控制

4.1. 自适应控制器设计

4.1.1. 控制器设计方案

$U={\left({u}_{1},{u}_{2},{u}_{3}\right)}^{\text{T}}$ ，其中， ${u}_{1}=-{k}_{1}\left(x-\stackrel{¯}{x}\right)$${u}_{2}=-{k}_{2}\left(y-\stackrel{¯}{y}\right)$${u}_{3}=-{k}_{3}\left(z-\stackrel{¯}{z}\right)$$\left(\stackrel{¯}{x},\stackrel{¯}{y},\stackrel{¯}{z}\right)$ 表示系统的平衡点。则该系统的受控系统表示为如下形式：

$\left\{\begin{array}{l}\stackrel{˙}{x}=dy-{k}_{1}\left(x-\stackrel{¯}{x}\right),\\ \stackrel{˙}{y}=-ay+b-cz\mathrm{sin}x-{k}_{2}\left(y-\stackrel{¯}{y}\right),\\ \stackrel{˙}{z}=-z+0.5\mathrm{cos}x+0.5-{k}_{3}\left(z-\stackrel{¯}{z}\right).\end{array}$ (7)

$J=\left(\begin{array}{ccc}-{k}_{1}& d& 0\\ -c\stackrel{¯}{z}\mathrm{cos}\stackrel{¯}{x}& -a-{k}_{2}& -c\mathrm{sin}\stackrel{¯}{x}\\ -0.5\mathrm{sin}\stackrel{¯}{x}& 0& -1-{k}_{3}\end{array}\right),$ (8)

$P\left(\lambda \right)={\lambda }^{3}+{a}_{1}{\lambda }^{2}+{a}_{2}\lambda +{a}_{3},$ (9)

${a}_{1}={k}_{1}+{k}_{2}+{k}_{3}+1+a,$

${a}_{2}={k}_{1}{k}_{2}+{k}_{1}{k}_{3}+{k}_{2}{k}_{3}+a{k}_{1}+a{k}_{3}+{k}_{1}+{k}_{2}+a+cd\stackrel{¯}{z}\mathrm{cos}\stackrel{¯}{x},$

${a}_{3}={k}_{1}{k}_{2}{k}_{3}+a{k}_{1}{k}_{3}+{k}_{1}{k}_{2}+a{k}_{1}+cd\left({k}_{3}+1\right)\stackrel{¯}{z}\mathrm{cos}\stackrel{¯}{x}-0.5c{\mathrm{sin}}^{2}\stackrel{¯}{x}.$

${a}_{1}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{2}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{3}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{1}{a}_{2}-{a}_{3}>0.$ (10)

4.1.2. 数值仿真

(a) 自适应控制下系统相图 (b) 自适应控制下系统时序图

Figure 7. Phase diagram and the time series analysis diagram of the system

4.2. 非线性反馈控制器设计

4.2.1 . 控制器设计方案

$\left\{\begin{array}{l}{q}_{1}=x-\stackrel{¯}{x},\\ {q}_{2}=y-\stackrel{¯}{y},\\ {q}_{3}=z-\stackrel{¯}{z}.\end{array}$ (11)

$\left\{\begin{array}{l}{\stackrel{˙}{q}}_{1}=d{q}_{2},\\ {\stackrel{˙}{q}}_{2}=-a{q}_{2}-c\left({q}_{3}+\stackrel{¯}{z}\right)\mathrm{sin}\left(\stackrel{¯}{x}+{q}_{1}\right)+b,\\ {\stackrel{˙}{q}}_{3}=-{q}_{3}-\stackrel{¯}{z}+0.5\mathrm{cos}\left(\stackrel{¯}{x}+{q}_{1}\right)+0.5.\end{array}$ (12)

$U={\left({u}_{1},{u}_{2},{u}_{3}\right)}^{\text{T}}$

$\left\{\begin{array}{l}{u}_{1}={p}_{1}{q}_{1},\\ {u}_{2}={p}_{2}{q}_{2}-d{q}_{1}+c\stackrel{¯}{z}\mathrm{sin}\left(\stackrel{¯}{x}+{q}_{1}\right)-b,\\ {u}_{3}={p}_{3}{q}_{3}+c{q}_{2}\mathrm{sin}\left(\stackrel{¯}{x}+{q}_{1}\right)-0.5\mathrm{cos}\left(\stackrel{¯}{x}+{q}_{1}\right)+\stackrel{¯}{z}-0.5.\end{array}$ (13)

$\left\{\begin{array}{l}{\stackrel{˙}{q}}_{1}=d{q}_{2}+{u}_{1},\\ {\stackrel{˙}{q}}_{2}=-a{q}_{2}-c\left({q}_{3}+\stackrel{¯}{z}\right)\mathrm{sin}\left(\stackrel{¯}{x}+{q}_{1}\right)+b+{u}_{2},\\ {\stackrel{˙}{q}}_{3}=-{q}_{3}-\stackrel{¯}{z}+0.5\mathrm{cos}\left(\stackrel{¯}{x}+{q}_{1}\right)+0.5+{u}_{3}.\end{array}$ (14)

$\left\{\begin{array}{l}{\stackrel{˙}{q}}_{1}=d{q}_{2}+{p}_{1}{q}_{1},\\ {\stackrel{˙}{q}}_{2}=\left({p}_{2}-a\right){q}_{2}-d{q}_{1}-c{q}_{3}\mathrm{sin}\left(\stackrel{¯}{x}+{q}_{1}\right),\\ {\stackrel{˙}{q}}_{3}=\left({p}_{3}-1\right){q}_{3}+c{q}_{2}\mathrm{sin}\left(\stackrel{¯}{x}+{q}_{1}\right).\end{array}$ (15)

$V=\frac{1}{2}{q}_{1}^{2}+\frac{1}{2}{q}_{2}^{2}+\frac{1}{2}{q}_{3}^{2}$

$\begin{array}{c}\stackrel{˙}{V}={q}_{1}{\stackrel{˙}{q}}_{1}+{q}_{2}{\stackrel{˙}{q}}_{2}+{q}_{3}{\stackrel{˙}{q}}_{3}\\ ={p}_{1}{q}_{1}^{2}+d{q}_{1}{q}_{2}+\left({p}_{2}-a\right){q}_{2}^{2}-d{q}_{1}{q}_{2}-c{q}_{2}{q}_{3}\mathrm{sin}\left(\stackrel{¯}{x}+{q}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({p}_{3}-1\right){q}_{3}^{2}+c{q}_{2}{q}_{3}\mathrm{sin}\left(\stackrel{¯}{x}+{q}_{1}\right)\\ ={p}_{1}{q}_{1}^{2}+\left({p}_{2}-a\right){q}_{2}^{2}+\left({p}_{3}-1\right){q}_{3}^{2}\end{array}$

${p}_{1}<0$ , ${p}_{2} , ${p}_{3}<1$ . (16)

4.2.2 . 数值仿真

(a) 非线性反馈控制下系统相图 (b) 非线性反馈控制下系统时序图

Figure 8. Phase diagram and the time series analysis diagram of the system

4.3. 未知参数控制器设计

4.3.1 . 未知参数控制器设计方案

$\left\{\begin{array}{l}\stackrel{˙}{x}=dy+{u}_{1},\\ \stackrel{˙}{y}=-ay+b-z\mathrm{sin}x+{u}_{2},\\ \stackrel{˙}{z}=-z+0.5\mathrm{cos}x+0.5+{u}_{3}.\end{array}$

$\stackrel{˜}{a}=a-\stackrel{^}{a}$$\stackrel{˜}{b}=b-\stackrel{^}{b}$$\stackrel{˜}{d}=d-\stackrel{^}{d}$ ，式中 $\stackrel{^}{a}$$\stackrel{^}{b}$$\stackrel{^}{d}$ 是未知参数a，b，d的估计值， $\stackrel{˜}{a}$$\stackrel{˜}{b}$$\stackrel{˜}{d}$ 分别表示未知参数a，b，d的误差。

$V=\frac{1}{2}\left({x}^{2}+{y}^{2}+{z}^{2}+{\stackrel{˜}{a}}^{2}+{\stackrel{˜}{b}}^{2}+{\stackrel{˜}{d}}^{2}\right),$

$\begin{array}{c}\stackrel{˙}{V}=x\stackrel{˙}{x}+y\stackrel{˙}{y}+z\stackrel{˙}{z}+\left(a-\stackrel{^}{a}\right)\left(-\stackrel{˙}{\stackrel{^}{a}}\right)+\left(b-\stackrel{^}{b}\right)\left(-\stackrel{˙}{\stackrel{^}{b}}\right)+\left(d-\stackrel{^}{d}\right)\left(-\stackrel{˙}{\stackrel{^}{d}}\right)\\ =dxy-{k}_{1}{x}^{2}-a{y}^{2}-{k}_{2}{y}^{2}+\left(b-\stackrel{^}{b}\right)y-{k}_{3}{z}^{2}-{z}^{2}+\left(a-\stackrel{^}{a}\right){y}^{2}+\left(b-\stackrel{^}{b}\right)\left(-y\right)+\left(d-\stackrel{^}{d}\right)\left(-xy\right)\\ =-{k}_{1}{x}^{2}-{k}_{2}{y}^{2}-{k}_{3}{z}^{2}-{z}^{2}+dxy-dxy+\stackrel{^}{d}xy-a{y}^{2}+a{y}^{2}-\stackrel{^}{a}{y}^{2}\\ =-{k}_{1}{x}^{2}-{k}_{2}{y}^{2}-{k}_{3}{z}^{2}-{z}^{2}+\stackrel{^}{d}xy-\stackrel{^}{a}{y}^{2}\\ =-\left[x,y,z\right]\left(\begin{array}{ccc}{k}_{1}& -\stackrel{^}{d}& 0\\ 0& {k}_{2}+\stackrel{^}{a}& 0\\ 0& 0& {k}_{3}+1\end{array}\right){\left[x,y,z\right]}^{\text{T}}\end{array}$

4.3.2 . 数值仿真

(a) x-y-z时序图 (b) 参数a、b、d的估计值

Figure 9. The time series analysis diagram with unknown parameters and estimates of unknown parameters

5. 结语

 [1] Yu, Y., Jai, H.J., Li, P. and Su, J. (2003) Power System Instability and Chaos. Electric Power Systems Research, 65, 187-195. [2] Ma, J.P., Sun, Y., Yuan, X.M., Kurths, J. and Zhan, M. (2016) Dynamics and Collapse in a Power Sys-tem Model with Voltage Variation: The Damping Effect. PLoS ONE, 11, e0165943. https://doi.org/10.1371/journal.pone.0165943 [3] 王晓东, 陈予恕. 一类电力系统的分岔和奇异性分析[J]. 振动与冲击, 2014, 33(4): 1-6. [4] Wei, D.Q. and Qin, Y.H. (2011) Controlling Chaos in Single-Machine-Infinite Bus Power System by Adaptive Passive Method. 2011 Fourth International Workshop on Chaos-Fractals Theories and Applications, Hangzhou, 19-22 October 2011, 295-297. https://doi.org/10.1109/IWCFTA.2011.8 [5] Wang, X.D., Chen, Y.S., Han, G. and Song, C. (2015) Nonlinear Dynamic Analysis of a Single-Machine Infinite-Bus Power System. Applied Mathematical Modelling, 39, 2951-2961. https://doi.org/10.1016/j.apm.2014.11.018 [6] Ma, M.L. and Min, F.H. (2015) Bifurcation Behavior and Coexisting Motions in a Time-Delayed Power System. Chinese Physics B, 24, Article ID: 030501. [7] 唐梦雪. 电力系统低频振荡混沌机理与混沌控制研究[D]: [硕士学位论文]. 成都: 西南交通大学, 2018. [8] 陈辉. 互联电力系统混沌特性分析与控制研究[D]: [硕士学位论文]. 合肥: 安徽大学, 2018. [9] 闵富红, 马汉媛, 王耀达. 含功率扰动电力系统混沌振荡的动态滑模控制[J]. 通信学报, 2019, 40(1): 119-129. [10] 张振, 刘艳红. 基于特征值的单机无穷大电力系统随机稳定性分析[J]. 郑州大学学报(工学版), 2018, 39(4): 58-63. [11] Abed, E.H. and Varaiya, P.P. (1984) Nonlinear Oscillations in Power Systems. International Journal of Electrical Power & Energy Systems, 6, 37-43. https://doi.org/10.1016/0142-0615(84)90034-6 [12] 王少夫. 电力系统混沌振荡分析及其自适应控制[D]: [硕士学位论文]. 南昌: 南昌大学信息工程学院电气自动化系, 2012. [13] Liu, M.J. and Piao, Z.L. (2009) Study on Chaos Control for Nonlinear Power System. 2009 International Workshop on Intelligent Systems and Applications, Wuhan, 23-24 May 2009, 1-4. https://doi.org/10.1109/iwisa.2009.5073133 [14] 唐梦雪, 王奔, 朱龙. 电力系统混沌振荡的模糊趋近律滑模控制[J]. 电气自动化, 2019, 41(1): 53-55. [15] 胡茗, 杨晓辉, 王毅. 基于鲁棒反演滑模法的电力系统混沌控制[J]. 电测与仪表, 2019, 56(3): 129-132+138. [16] 孙元章, 焦晓红, 申铁龙. 电力系统非线性鲁棒控制[M]. 北京: 清华大学出版社, 2007: 92-140. [17] Schmietendorf, K., Peinke, J., Friedrich, R. and Kamps, O. (2014) Self-Organized Synchronization and Voltage Stability in Networks of Synchronous Machines. The European Physical Journal Special Topics, 223, 2577-2592. https://doi.org/10.1140/epjst/e2014-02209-8 [18] 黄苏海, 田立新. 一个新的四维超混沌系统的动力学分析及混沌反同步[J]. 电路与系统学报, 2011, 16(6): 66-74. [19] 杜文举, 俞建宁, 张建刚, 张莉, 安新磊. 一个新四维混沌系统的Hopf分岔分析[J]. 温州大学学报(自然科学版), 2014, 35(1): 31-38. [20] 刘熙娟, 陈多花. 一类三维自治混沌系统的动力学行为分析[J]. 贵州师范大学学报(自然科学版), 2015, 33(4): 55-61.