In extended equal area criterion（EEAC）， dynamic saddle points（DSP） encountered in disturbed trajectories can be regarded as not only the criterion of power system instability， but also the observation point for the quantitative analysis of stability degree. However， the strong time-varying of image systems might result in some special trajectory points， which although satisfy the definition of DSP， are not suitable for accurately assessing the stability degree. If the above ill-conditioned DSPs（IDSPs） can be distinguished， the computational burdens can be further reduced in the engineering application via early termination of the integration process. The studies show that around IDSPs， image systems not only exhibit the strong time-varying characteristic， but also exist the potential energy. With this knowledge， an ill-condition index of DSP and the corresponding identification method for IDSP are proposed. They are thoroughly verified in the New England system. This research also contributes to the study on stability mechanisms of general nonlinear dynamic systems.