Modified algorithm to trace critical eigenvalues of power system with sensitivities via continuation of invariant subspaces

The critical eigenvalue tracing in reference [1] is further modified to extract further useful information. This includes the direction and the speed of movement of eigenvalues. The algorithm is both robust and efficient. The calculation of invariant subspaces is basically solving Riccati equation, which is equivalent to solving bordered matrix equations of Sylvester type. The bordered Bartels-Stewart algorithm is used to solve it effectively. The subspace continuation technique allows us to uniquely identify the image of the movement of the set of the critical eigenvalues w.r.t. the change of the continuation parameter (such as system load level etc.). Furthermore, the eigenvalue and eigenvector sensitivities can also be obtained as by-products. An eigenvalue index is proposed to determine the critical eigenvalue that might affect the stability change of the system. It can be used to estimate the oscillatory stability margin boundary of the system during the continuation by linear estimation. Finally, the numerical techniques are applied to study the New England 39 - bus system.