﻿ 电力系统稳定域确定及算法特性研究
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 浙江大学学报(工学版)  2019, Vol. 53 Issue (1): 200-206  DOI:10.3785/j.issn.1008-973X.2019.01.023 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2019.01.023
[复制中文]
ZHOU Chi, LI Ying-hui, ZHENG Wu-ji, WU Peng-wei, DONG Ze-hong. Study on stability region determination and algorithm characteristics of power system[J]. Journal of Zhejiang University(Engineering Science), 2019, 53(1): 200-206.
dx.doi.org/10.3785/j.issn.1008-973X.2019.01.023
[复制英文]

### 作者简介

orcid.org/0000-0002-5088-3919.
E-mail：1148342949@qq.com.

### 通信联系人

orcid.org/0000-0002-6024-4547.
E-mail：liyinghui66@163.com
.

### 文章历史

Study on stability region determination and algorithm characteristics of power system
ZHOU Chi , LI Ying-hui , ZHENG Wu-ji , WU Peng-wei , DONG Ze-hong
Aeronautics Engineering College, Air Force Engineering University, Xi'an 710038, China
Abstract: A new method based on manifold theory method was proposed in order to solve the shortcomings of strong conservatism in calculating the stability region of power system by conventional method. The stability boundary of dynamic system consists of the union of the stable manifolds of all I type unstable equilibrium points on the stability boundary. All the equilibrium points were obtained by solving a nonlinear power system. Then the stable manifolds of each I type unstable equilibrium point were computed by trajectory arc length method. The union of the stable manifold was taken as the stable boundary of the system. The single machine and infinite bus system was taken as the research object. The comparison of the stability region determined by Monte Carlo method and reachable set theory was conducted. Results show that the manifold theory can be used to solve the stability region of high-dimensional power system with high accuracy.
Key words: stability region    manifold theory    trajectory arc length method    Monte Carlo    reachable set

1 流形理论构造稳定域

1.1 流形的基本理论

 ${{\dot x}} = f({ x}).$ (1)

 \left.\begin{aligned} {W^{\rm s}}({{ x}_0}) = \left\{ {{ x}\left| {\mathop {\lim }\limits_{t \to \infty } \phi (t,{ x}) = {{ x}_0}} \right.} \right\}\, , \\ {W^{\rm u}}({{ x}_0}) = \left\{ {{ x}\left| {\mathop {\lim }\limits_{t \to - \infty } \phi (t,{ x}) = {{ x}_0}} \right.} \right\}\, . \end{aligned} \right\} (2)

Chiang等[20]指出，当满足以下条件(A1~A3)时，非线性动力学系统的稳定边界是由稳定平衡点的稳定边界上的I型不稳定平衡点（特征值中只有其中一维的特征值具有正实部)的稳定流形的并集构成.

1）所有在稳定边界上的平衡点都是双曲平衡点.

2）在稳定边界上平衡点的稳定和不稳定流 形都必须满足横截性条件.

3）当 $t \to \infty$ 时，稳定边界上的每一条解轨线都收敛于某一个稳定平衡点.

1.2 筛选出位于稳定边界上的不稳定平衡点

1) 以I型不稳定平衡点 ${{ x}_i}$ 为中心，取半径为 $\varepsilon$ n维超球体. 在超球体内部每个球面上均匀取点，第 $m$ 个球面上的点 ${{ y}_m} = [ - \varepsilon ,{d_1},{d_2},\cdots,{d_{n - 1}},$ $\varepsilon ] + {{ x}_i}$ ，其中 ${d_j}(j = 1,2,\cdots,n - 1)$ ${\rm{( - }}\varepsilon ,\varepsilon {\rm{)}}$ 内均取的点， $m = 1,2,\cdots,n$ .

2) 以步骤1)中取的点为初始点，对非线性系统进行反时间积分. 若所得轨线都能够保持在超球体内部，则进入3）；反之，将球体半径取为 $\alpha \varepsilon {\rm{(0}} < \alpha < {\rm{1)}}$ ，返回1）.

3） 以满足步骤2）中所取的 ${{ x}_i}$ 上不稳定流形上的各点为初始点，对系统进行正向数值积分. 判定当 $t \to \infty$ 时，积分轨线是否会收敛于稳定平衡点xs.

4） 若步骤3）中的积分轨线会收敛于稳定平衡点xs，则不稳定平衡点 ${{ x}_i}$ 为稳定平衡点xs稳定边界上的点.

1.3 非线性系统稳定域边界构造

 图 1 轨道弧长法画稳定域流程图 Fig. 1 Flow chart of stable region of trajectory arc length method

1）确定初始圆. 首先利用不稳定平衡点xi的Jacobian矩阵求特征向量，将稳定的特征向量张成稳定流形的特征子空间（即切平面）；然后在确定的切平面上，以 ${{ x}_i}$ 为中心，r为半径作圆，其中r的取值略大于计算机浮点计算精度. 在圆上均匀取N个点 $\left\{ {{p_{1,1}},{p_{1,2}},\cdots,{p_{1{\rm{,}}N}}} \right\}$ .

2）求解轨线. 以点集 $\left\{ {{p_{1,1}},{p_{1,2}},\cdots,{p_{1,N}}} \right\}$ 为初始点，分别对系统进行反时间积分，当轨线长度达到设定值L时停止，得到第一代轨线 $\left\{ {{T_{1,1}}},{T_{1,2}},\cdots,\right.$ $\left.{{T_{1,N}}} \right\}$ ，将各条轨线的终点记为 $\left\{ {{p_{2,1}},{p_{2,2}},\cdots,{p_{2{\rm{,}}N}}} \right\}$ .

3）轨线间疏密程度判定. 检验各条轨线间的距离，若2条轨线间的距离大于Dmax，则轨线太疏，在这两条轨线的初始点间插入一个新初始点. 反之，若2条轨线间的距离小于Dmin，则2条轨线过密，删除其中一条轨线的初始点. 最后重新调整初始点集，返回2），直至满足要求，进入4）.

4） 以步骤3）中确定的轨线终点为二代初始点，重复步骤2）和3），迭代次数达到Zmax时停止.

5）连接相邻代的轨线，并将始于相邻初始点的轨线连接，形成边界面.

6）将稳定平衡点稳定边界上各个不稳定平衡点利用轨道弧长法所构造的边界面全部拼接在一起，所得界面为系统稳定域的边界.

2 可达集计算稳定域

2.1 可达集的基本理论

 $\dot{ x}{\rm{ = }}f{\rm{(}}{x}{\rm{,}}t{\rm{,}}{u}{\rm{)}}.$ (3)

 图 2 目标集与可达集的关系 Fig. 2 Relation between target set and reachable set

 $\frac{{\partial \phi {\rm{(}}{x}{\rm{,}}t{\rm{)}}}}{{\partial t}}{\rm{ + }}f \cdot \Delta \phi {\rm{ = }}0.$ (4)

 ${G_0}{\rm{ = \{ }}{x} \in {{\bf{R}}^n}{\rm{|}}\phi {\rm{(}}{x}{\rm{,}}0{\rm{)}} \leqslant {\rm{0\} }}{\rm{.}}$ (5)

 \left.\begin{aligned} \displaystyle\frac{{\partial \phi \left( {{ x},t} \right)}}{{\partial t}} + \min \left[ {0,\;H\left( {{ x},\displaystyle\frac{{\partial \phi \left( {{ x},t} \right)}}{{\partial { x}}}} \right)} \right] = 0,\;{x} \in {{\bf{R}}^n},\;{{t < 0}};\\ \phi {\rm{(}}{x}{\rm{,}}t{\rm{) = }}\phi {\rm{(}}{x}{\rm{),}}\;{x} \in {{\bf{R}}^n},\;{t}{\rm{ = }}0 . \end{aligned}\right\} (6)

Hamilton函数 $H{\rm{(}}{x}{\rm{,}}{p}{\rm{)}}$

 $H{\rm{(}}{x}{\rm{,}}{p}{\rm{) = }}\mathop {\max }\limits_{{u} \in U} {{p}^{\rm{T}}}f{\rm{(}}{x}{\rm{,}}t{\rm{,}}{u}{\rm{)}}{\rm{.}}$ (7)

 ${{u}^{\rm{*}}}{\rm{(}}{x}{\rm{,}}{p}{\rm{) = arg max }}\;{{p}^{\rm{T}}}f{\rm{(}}{x}{\rm{,}}t{\rm{,}}{u}{\rm{)}}.$ (8)

 ${P_\tau }{\rm{(}}{G_{\rm{0}}}{\rm{) = \{ }}{x} \in {{\bf{R}}^n}{\rm{|}}\phi {\rm{(}}{x}{\rm{,}}\tau {\rm{)}}\leqslant {\rm{0\} }}.$ (9)
2.2 可达集计算

 $H\left({x}{\rm{,}}{{ p}^{\rm{ + }}}{\rm{,}}{{ p}^{\rm{ - }}}\right): = H\left({x}{\rm{,}}\frac{{{ p}{\rm^{ + }}{\rm{ + }}{{ p}^{\rm{ - }}}}}{2}\right) - \frac{{\rm{1}}}{{\rm{2}}}{{ k}^{\rm T}}\left({{ p}^{\rm{ + }}}{\rm{ - }}{{ p}^{\rm{ - }}}\right){\rm{.}}$ (11)

$p_i^{\rm{ - }}{\rm{ = }}\displaystyle\frac{{\phi \left({x_i}{\rm{,}}t\right) - \phi {\rm{(}}{x_{i{\rm{ - }}1}}{\rm{,}}t{\rm{)}}}}{{{x_i}{\rm{ - }}{x_{i{\rm{ - }}1}}}}$ $p_i^{\rm{ + }}{\rm{= }}\displaystyle\frac{{\phi {\rm{(}}{x_{i{\rm{ + }}1}}{\rm{,}}t{\rm{) - }}\phi {\rm{(}}{x_i}{\rm{,}}t{\rm{)}}}}{{{x_{i{\rm{ + }}1}}{\rm{ - }}{x_i}}}$

 ${k_i}{\rm{ = ma}}{{\rm{x}}_{{p_i} \in {\rm{[}}{p}_{i{{\rm{min},}}}{p}_{i{{\rm{max]}}}}}}\left| {\frac{{\partial {{H}}}}{{\partial {p_i}}}} \right|{\text{，}}$ (12)

3 应用实例 3.1 三阶单机无穷大系统

 \left. {\begin{aligned}{} {{{T'}_{{\rm do}}}{{\dot E'}_q} = {E_{\rm f}} - {{E'}_q} - \left( {{X_d} - {{X'}_d}} \right){I_d}}, \\ {{T_{\rm j}}\dot \omega = {P_{\rm m}} - {P_{\rm e}} - D\left( {\omega - 1} \right)}, \\ {\dot \delta = {\omega _{\rm b}}\left( {\omega - 1} \right)}. \end{aligned}} \right\} (13)

 \left. {\begin{aligned}{} {{I_d} = {\left.{\left( {{{E'}_q} - U\cos \delta } \right)}\right/{\left( {X + {{X'}_d}} \right)}}}, \\ {{I_q} = {{U\sin \delta }\left/{\left( {X + {X_q}} \right)}\right.}}, \\ {{P_{\rm e}} = {{E'}_q}{I_q} - \left( {{{X'}_d} - {X_q}} \right){I_d}{I_q}}, \\ {X = {X_{\rm l}} + {X_{\rm t}}} . \end{aligned}} \right\} (14)

 $\left.\begin{split}{} {X_q} = 0.72 ,\quad{T_{\rm j}} = 8.75 ;\\ {X_d} = 1.035\;4, \quad {{T'}_{{\rm do}}} = 8.0; \\ {{X'}_d} = 0.36,\quad D = 3.0 ;\\ {X_{\rm l}} = 0.413,\quad U = 1.0 ; \\ {X_{\rm t}} = 0.15 ,\quad{P_{\rm m}} = 0.7;\\ {\omega _{\rm b}} = 2\pi \times 50 .\\ \end{split}\right\}$ (15)
3.2 利用流形理论计算三阶单机无穷大系统稳定域

AB两点的特征值分别为

A \begin{aligned} {\lambda _{1{\rm{,}}2}} = - 0.05 \pm 0.241\;1{\rm i} , {\lambda _3} = - 0.117 ;\end{aligned}

B \begin{aligned} {\lambda _{1{\rm{,}}2}} = - 0.164 \pm 0.176\;5{\rm i} , {\lambda _3} = 0.109\;6 .\end{aligned}

 图 3 利用流形理论计算三阶单机无穷大系统稳定域 Fig. 3 Calculation of stability region of SMIB by manifold theory
3.3 利用蒙特卡洛方法计算三阶单机无穷大系统稳定域

 图 4 利用蒙特卡洛方法计算三阶单机无穷大系统稳定域 Fig. 4 Calculation of stability region of SMIB by Monte Carlo

 图 5 蒙特卡洛方法与流形理论法对比图 Fig. 5 Comparison diagram of Monte Carlo and manifold theory method

3.4 利用可达集理论计算三阶无穷大系统稳定域

2章对如何求非线性系统的可达集进行了较详细的介绍，主要过程如下. 首先在系统（13）的状态空间中，找到一个较小的稳定初始区域作为目标集；然后对目标集进行逆时间求解，得到可达集；最后将求得的可达集，作为电力系统的稳定域. 利用流形理论与可达集方法确定稳定域的对比图如图6所示.

 图 6 流形理论与可达集方法对比图 Fig. 6 Comparison diagram of manifold theory and reachable set method

4 仿真分析

5 结　论

（1）利用流形方法能够精确地构造电力系统的稳定域，改善了传统方法构造稳定域保守性较强的缺点；该方法与蒙特卡洛方法及可达集方法相比，具有耗时短的优点，为实现电力系统安全稳定分析打下基础.

（2）基于可达集方法在一定程度上虽然能够精确地确定稳定域，但该方法主要受限于初始网格点数，网格点数取得越密，精度越高，但是会导致较长的耗时，因此在一定程度上限制了可达集方法在控制领域的应用前景，对该方法进行改良是实验室下一步的研究目标.

（3）流形方法相较于传统方法和可达集方法，更加适用于电力系统的稳定域确定，在电力系统的安全性分析及控制领域具有更广的应用前景.

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