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Transient stability analysis of a power system is concerned with the system's ability to remain in synchronism following a disturbance. In utility planning, transient stability is studied by numerical simulation. The long CPU run times for simulation preclude their use for on-line security analysis. Interest has therefore shifted toward the Lyapunov direct method of stability analysis. This paper provides a critical review of research on direct methods since 1970. Considerable progress has been made on both theoretical properties of energy functions and on criteria suitable for on-line implementation. Current theory provides a satisfactory treatment of voltage-dependent reactive power demand, transfer conductances, and flux decay. However, it cannot incorporate the exciter control Proposed on-line criteria appear to work very well on sample examples; but, they still lack rigorous justification. Finally, recent work has shown that power systems can exhibit chaotic behavior. This surprising fact demonstrates that our understanding of the dynamics of power systems remains incomplete.