Use of homotopy-based approaches in finding Controlling Unstable Equilibrium Points in transient stability analysis

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This paper introduces the use of homotopy-based approaches in computing the Controlling Unstable Equilibrium Points (controlling UEPs) in transient stability analysis using direct methods. It is well known that the regions of convergence of the controlling UEPs are very sensitive to the initial guesses, and traditional iterative methods fail to find the correct controlling UEPs if the initial guesses lie outside their regions of convergence. On the other hand, homotopy-based approaches are very reliable in finding solutions because they are globally convergent. However, homotopy-based approaches are intrinsically slow if the initial point is far from the desired solution because these methods map the trajectory of the solution from an easy and known solution to the desired solution. This paper proposes an algorithm that uses a homotopy-based approach with the exit point as an initial point to reliably find the correct controlling UEP. To reduce computational effort, the proposed method uses an approximate exit point rather than computing an accurate exit point as is common practice in finding controlling UEPs. Further, this method eliminates the necessity of computing the Minimum Gradient Point (MGP), which makes the homotopy-based approaches comparable with the other iterative methods in terms of the speed of computation. An explicit characterization of the region of convergence of a controlling UEP and its boundary starting from an exit point for a typical power system is derived. The method is applied on the WECC and the NE 39 test systems to demonstrate its effectiveness in finding the controlling UEPs.