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

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.

I. Introduction

Power systems are currently operated close to their stability limits and transient stability analysis is one of the important tools in determining the operating limits. Direct methods of transient stability analysis have been successfully applied in evaluating the stability of power systems [1]–[7]. However, the accuracy of these methods strongly relies on the determination of the controlling UEPs. Despite significant advances in the application of the direct methods in transient stability analysis, the problem of precise determination of the controlling UEPs and the speed of computation are still of concern. The presence of fractal shapes of the convergence regions of the controlling UEPs is the main reason of failure of several numerical methods in finding the correct controlling UEPs [7]–[9]. The Newton method and its variations, for example, converge only if the initial guess lies within the region of convergence of the desired solution; to find the correct controlling UEPs, the initial guesses have to be within a certain neighborhood of the desired solutions. This requirement makes it difficult to find the best guesses and it is computationally expensive.