电力系统本质上是一个高维、非线性的复杂动态系统，微分方程是其描述的基本形式之一。其稳定边界构成异常复杂，求解十分困难，目前大多采用某种形式的近似等效稳定域的部分边界，从而实现电力系统暂态稳定分析。边界近似精度将直接影响暂态稳定分析结果。如何评价近似边界的精度，以及在获取近似精度后如何改进近似边界正是本论文研究的主要问题。研究了在一定假设条件下的非线性微分动力系统的不变流形求解方法。该方法利用Taylor级数展开计算位于稳定边界上n-1维稳定流形、和实数特征根对应的n维空间中一维特征不变流形、以及和复数特征根对应的n维空间中二维特征不变流形。在求解其Taylor级数系数的过程中，根据所满足偏微分方程的特点，利用符号运算工具包将方程组两端Taylor级数展开各阶系数化成具有递推关系的线性方程组以简化计算，并从理论上证明了该简化方法的正确性。为评价近似边界有效范围和在有效范围之外进行边界拓展奠定了理论基础。改进和扩展了非线性动力系统稳定域近似边界可信域的定义和计算方法。该方法将最初仅适用于低维系统的可信域推广到n维。可信域是近似边界的有效范围，是近似边界与真实边界在给定误差阈值下的最大重合域。真实边界解析表达式甚至高阶级数近似计算十分困难。特征不变流形维数最高为二维，其高阶Taylor级数展开和求解容易实现。将之替代真实边界，从而使得近似边界可信域数值实现成为可能。理论上可信域的计算不受边界近似方法和电力系统发电机模型的影响。提出了一种联合可信域和暂态能量函数的边界拓展方法。该方法首先计算近似边界可信域，在可信域范围之外利用能量函数常值能量面拓展边界，从而分段近似电力系系统暂态稳定域边界。为进一步提高边界近似精度，在等势能面选取时考虑了故障轨迹与可信域位置信息。将该拓展边界用于电力系统暂态稳定临界清除时间后，与多项式近似的稳定边界和纯能量函数常值能量面比较，该方法在一定程度上克服了能量函数的保守性，又不会出现多项式近似边界可能的偏冒进现象，更加接近数值仿真结果。

Power systems are large-scale dynamic systems with very complex characteristics. Generally they are described by ordinary nonlinear differential equations with high dimension. It is a tough task to obtain an analytical expression of the stability boundary of such a nonlinear system. In practice, the stability boundary is approximated by Taylor’s series form or others. The accuracy of the approximation boundary has a direct influence to the precise of critical clearing time calculation in power system transient analysis. How to assess the accuracy of the approximated boundaries and extend the boundary out of the valid range are the main projects of the dissertation.Based on the differential geometry theory of nonlinear system with given assumptions, a numerical computation method by resolving a series of linear equations recursively is developed to calculate the coefficients of the Taylor series of the invariant manifolds lying on the stability boundary. The multivariate Taylor series expansion is adopted to compute the partial differential equations associated with the 1-dimensional and 2-dimensional characteristics invariant manifolds corresponding to the real eigen value and complex eigen value respectively. The proposed method is the theoretical basis for evaluating the valid range of the approximated boundary and extending the approximated boundary out of its valid range.Secondly, with the original definition of credible region which can only be used in 3-dimensional nonlinear dynamical system or less, a modified algorithm is presented to find the credible region of the approximation boundaries for a general nonlinear dynamic system. The credible region is the valid range of the approximated boundary under giving threshold compared with the actual boundary. Due to its complication and the dimension disaster, it is almost impossible to obtain the actual boundary or its high-order Taylor series expansion in practice. Instead, low dimension invariant manifolds such as the 1-dimensional or 2-dimensional characteristics invariant manifolds which are the subset of the actual boundary are used to estimate the credible region. Theoretically, the approach to calculate the credible region can be used in a power system with detailed generator model.Finally, a novel method of boundary extension using the aforementioned credible region and transient energy function originated from the Lyapunov function is proposed. The stability boundary is the union of the credible region and the constant potential energy surface at the point on the boundary of the credible region which is the minimum distance to its controlling unstable equilibrium point. Furthermore, the fault-on trajectory can be used to choose the position of the boundary of the credible region where the constant energy surface passes through. The proposed method not only eliminates the limitation of local validation of the boundary approximated by the quadratic surface but also overcomes the conservation of transient energy function, and therefore enhances the accuracy of the direct methods.