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The transient energy function method is usually used to assess the first swing stability of a power system. This paper uses the rate of dissipation of transient energy as a measure of system damping. The technique is then applied to determine the additional damping provided by a SSSC. The control strategy of the SSSC is selected very carefully to satisfy the Lyapunov's stability criterion and increase the rate of dissipation of transient energy. An analytical expression for additional damping provided by the SSSC is also derived. The proposed method is then tested on a single machine infinite bus system with a SSSC and the results found are observed to be in complete agreement with the theory described in the paper.