The advent of synchronized phasor measurements allows the problem of real time prediction of instability and control to be considered. The use of direct methods for these on-line applications is assessed.

The classical representation of a power system allows the use of two reference frames: Center of angle and one machine as reference. Formulae allowing transition between the two reference frames are derived. It is shown that the transient energy in both formulations is the same, and that line resistances do not dampen system oscillations.

Examples illustrating the mathematical characterization of the region of attraction, exit point, closest u.e.p. and controlling u.e.p. methods are presented.

Half-dimensional systems (reduced-order systems) are discussed. The general expression for the gradient system which accounts for transfer conductances is derived without making use of the infinite bus assumption. Examples illustrating the following items are presented: a) Effect of the linear ray approximation on the potential energy (inability to accurately locate the u.e.p.’s); b) Comparison of Kakimoto’s and Athay’s approach for PEBS crossing detection; c) BCU method and; d) One·parameter transversality condition.

It is illustrated that if the assumption of the one-parameter transversality condition is not satisfied, the PEBS and BCU methods may give incorrect results for multi-swing stability. A procedure to determine if the u.e.p. found by the BCU method lies on the stability boundary of the original system is given. This procedure improves the BCU method for off~line applications when there is time for a hybrid approach (direct and conventional), but it does not improve it for on-line applications due to the following: a) It is time consuming and b) If it finds that the u.e.p. does not belong to the stability boundary it provides no information concerning the stability/instability of the system.