Facing the inability of cut-plane to accurately describe the neighborhood of the critical point on the voltage stability limit surface

a bi-level programming model based on second-order standard approximation of limit surface considering uncertainties of the load increase direction is proposed to solve the the critical point on the voltage stability limit surface. First

the critical point along the original load increase direction is obtained

and the Weingarten mapping is performed to attain the principal curvature of the limit surface at the point. Then the second-order standard approximation near the critical point is obtained. Finally

an iterative process is developed to search for the closest critical point of limit surface based on the trust region algorithm and quadratic programming. Saddle node bifurcation (SNB) and limit induced bifurcation (LIB) are all included in the proposed model. The simulations on IEEE systems and a real system in Eastern China show the validity of the proposed method. And the analysis not only proves the efficiency of the proposed method

but also suggests that the proposed method is fast and robust and have good global search ability

which is able to find the weak area of power system and be utilized in the real large system. This work is supported by the Key Program of National Natural Science Foundation of China (No. 51337005) and National Basic Research Program of China (973 Program) (No. 2014CB23903).