In the past fifteen years considerable progress has been made in first swing power system transient stability assessment using the transient energy function (TEF) method;The accuracy of stability assessment provided by the TEF method depends on the determination of the controlling unstable equilibrium point (UEP). The technique that determines the controlling UEP in the current commercial version (Version 3.0) of the TEF method program is based on the so-called 'exit point method' and has also been recently labeled the 'BCU method.';The exit point method consists of two basic steps. They are the detection of the exit point [theta][superscript]e and detection of the minimum gradient point [theta][superscript]o. The controlling UEP is solved for by using [theta][superscript]o as an initial guess;It has been observed that this method lacks robustness in the sense that the following two problems may occur. (Problem 1) There may be no detection of the point [theta][superscript]o. (Problem 2) If [theta][superscript]o is found, it may not be in the domain of convergence of [theta][superscript]u for the particular solving algorithm used. Hence, another equilibrium point, not the controlling UEP will be located;The result of this research has been the development of a new numerical technique for determining the controlling UEP. With the exit point as an initial starting point this technique efficiently produces a sequence of points. A significant part of this dissertation was the formulation of an analytical foundation which shows that under certain assumptions this sequence will converge to the controlling UEP. Hence this new technique exhibits a substantial improvement over the exit point method because of the following reasons: (1) The technique does not attempt to detect the point [theta][superscript]o. (2) The technique can produce a point that is close to [theta][superscript]u thus avoiding a domain of convergence problem;This technique was applied to two realistic, large-scale power systems. In every case an accurate stability assessment was provided.