# Power System Transient Stability Assessment

Power System Transient Stability Assessment Based on Quadratic Approximation of Stability Region

This paper presents an approach to estimate the Critical Clearing Time (CCT) of the multi-machine power systems based on the quadratic surface which approximates the boundary of stability region relating to the controlling unstable equilibrium point. A decomposition method is developed to obtain the coefficients of the quadratic approximation surface. The CCT is determined by the crossing point of the quadratic surface and the continuous faulted trajectory. Simulations in research 9-bus and New England system show the effectiveness of the proposed approach.

IN the power system transient stability assessment (TSA), the concept of stability region (attraction region) and Controlling Unstable Equilibrium Point (CUEP) has been well recognized . The boundary of the stability region for the power systems are composed of the stable manifolds of the unstable equilibrium point on the boundary , the CUEP is the unstable equilibrium point whose stable manifold is crossed by the continuous faulted trajectory. The CCT is determined by the crossing point of the stable manifold of the CUEP and the continuous faulted trajectory. Thus the description of the stable manifold of the CUEP plays an important role in TSA, various approximations to the stable manifold have been proposed. One way is to use the equal energy surface, which is determined by certain transient energy function, these approximations may always give out good results but have limitation for the non-existence of transient energy function for general power system models. Another way is to obtain truncated approximations by applying the Taylor series expansion or the normal form method to the partial differential equation describing the stable manifold of the CUEP. With the idea of Taylor series expansion, ref. obtained the hyper-plane approximation using the first order term and ref. derived a hyper-surface approximation using the second order term. Recent work also presented a quadratic approximation, but it lacks theoretical justification and need the energy correction procedure. Based on the extension of Poincare’s classical result on normal formal theory to approximate the stable manifold, the early work first proposed an algorithm to compute relevant coefficient of stable manifold in power series presentation. The work originally used the normal form to compute the boundary of stability region, it got the exact series representation of the stable manifolds characterizing the stability boundary. This series can be computed recursively. However, no numerical tests have been conducted. With the real normal form, ref. obtained the second order approximation for the stability region boundary, but this method requires the computation for all the eigenvectors and eigenvalues of the Jacobian matrix, which results in burdensome computation. To improve the normal form computation, ref. presented another method to calculate the quadratic approximation based on similarity transformation which avoiding the computation for all the eigenvectors, unfortunately, this approximation is not correct . The above normal form approximations are based on the implicit equation which describing the stable manifold in a new coordinate. Recently, discovered the explicit equation which describing the stable manifold of type-i equilibrium point in the original coordinate. Furthermore, they presented a quadratic approximation for the stable manifold of CUEP, which is the same as the hypersurface proposed in . This paper applies the proposed quadratic approximation to determine the CCT of the multimachine power systems with the network-reduced structure. While obtaining the coefficients of quadratic approximation, we explore the special structure of the model and develop the decomposition method to reduce the computation. The estimated CCT is determined by the crossing point of the quadratic surface and the continuous faulted trajectory. The simulations in research 9-bus