Stability boundary approximation of a power system using the real normal form of vector fields

Presently interconnected power Systems are much more stressed than ever due to heavier loading of the transmission network. This stress in a power network exhibits several interesting nonlinear phenomena. This nonlinear complex behavior is not adequately analyzed with existing tools. The work reported upon in this paper aims at explaining one aspect of this behavior, namely the size of the region of stability and the shape of its boundary. The paper proposes the use of real normal form of vector fields, a comparatively new tool in the domain of power system analysis, to study the stability boundary of a stressed power system. The approach consists of Taylor series expansion, linear analysis, normal form method, energy method, and time simulation. The key idea of this paper is to approximate the stable manifold of an unstable equilibrium point (UEP) using the normal form method. The original nonlinear system is transformed to a linear system using the nonlinear coordinate transform around the UEP of interest. Then the stable eigenspace of the normal form system is transformed back to the original system to approximate the stable manifold up to some degree. This approximate stable manifold is used to find the stability boundary of a stable equilibrium point (SEP). Thus, the normal form method enables us to approximate the stable manifold of this UEP which is otherwise very difficult to compute numerically for a practical size power system. The proposed methodology has been tested on an 11-generator test system. The shape of the stability boundary and the region of attraction of the postfault SEP have been studied for different system loading conditions. Certain attributes of the stability boundary (e.g., curvature) are analyzed. The behavior of the stable and unstable system trajectories have been observed as they approach the stability boundary. The following are among the findings reported upon in the paper. • It has been found that the stability boundary is more curved in the direction of the manifolds corresponding to the machines advanced in the UEP. • The size of the region of stability is reduced as the system becomes more stressed with loading. • The curvature of the boundary near the UEP, as displayed by the approximated manifolds, increases with the increase in system stress. • As the system becomes more stressed, the UEPs and SEPs tend to move closer; in extreme situations they may combine. • As the unstable faulted trajectory leaves the boundary, it follows the unstable manifold near the UEP.