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中文摘要：
针对电力系统中静态稳定和避免奇异诱导分岔、鞍结分岔、Hopf分岔几类典型的系统稳定问题，通过分析Jacobian矩阵特征值的性质，并结合数学上谱函数的半光滑特性，分别建立相应的几类系统稳定平衡解的数学模型。新模型为具有半光滑不等式约束的非线性方程组。模型的特点是不仅具有较好的数学性质，且可满足平衡解处的稳定性要求。针对数学模型的半光滑特性，利用光滑化函数将模型进行转换，进而建立模型求解的一类光滑化牛顿型算法。该算法理论上享有良好的与传统牛顿法相同的全局与局部收敛性能。通过电力系统的一个避免鞍结分岔的实例，测试模型及算法的可行性。
英文摘要：
According to steady state stability and avoiding the singularity induced bifurcation, saddle-node bifurcation, and Hopf bifurcation in power systems, this paper presents corresponding mathematical models for stability equilibrium solutions respectively based on analyzing the property of eigenvalues of Jacobian matrix and the semismooth property of spectral functions in mathematics. The new models are composed of a nonlinear system of equations with semismooth inequality constraints. The characteristic of the models is that they have not only good mathematical property, but also they can satisfy the stability requirement at equilibrium points. As to the semismooth property of mathematical models, the models are transformed using smoothing function. Furthermore, a class of smoothing Newton-type algorithm is set for solving the stability models. The proposed approach possesses the nice global and local convergence as the same as the classical Newton method. A simple system for avoiding-saddle-node bifurcation is tested for the effect of models and the algorithm.
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