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Successive eigenvalue sensitivities extracted from continuation of invariant subspaces (CIS) are proposed for efficient identification of oscillatory stability margin and damping margin in power systems. Spectrum transformation-based methods are introduced to calculate the critical eigenvalues of interest for CIS initialization. The predictor-corrector method is applied to trace the movements of eigenvalues as power system parameter changes. The eigenvalue sensitivities are by-products of the algorithm. From this information, a step size control strategy is proposed to speed up the oscillatory stability margin and damping margin identification. The proposed method is numerically stable, robust, and converges rapidly. The simulation results and computation performance on New England 39-bus system and IEEE 145-bus system are demonstrated in details.