With the growth of interconnected power system, and especially the deregulation of the power market, the problems related to small-signal stability have become a critical issue for the power system security. Better methods of analyzing the oscillations would lead to more accurate determination of these limits and the ability to operate the power system closer to the stability margin. An analytical tool to trace the movement of critical eigenvalues with respect to the changing system conditions will help analyze and investigate the cause of the problem. If the oscillatory stability margin and the damping margin can be pre-determined for a specified scenario which might happen in real time, it could provide operators the user guide in operating the power systems when dealing with the potential oscillation or damping problems. In the dissertation, a novel comprehensive framework of invariant subspace-based methods to deal with the above challenging problems for power system computation and analysis is proposed.;We first propose an improved continuation of invariant subspace (ICIS) for the eigenvalue analysis. The ICIS provides us an efficient tool to trace any set of critical eigenvalues of interest. With proper re-initialization, the eigenvalue sensitivities can be successively extracted as by-products of the algorithm during the tracing process. The extracted eigenvalue sensitivity from ICIS is proved mathematically and verified numerically. At each iteration, we not only know the location of each traced eigenvalue, but also the direction and speed of the eigenvalue movement. The extracted eigenvalue sensitivities can be used to automatically adjust the step size in the continuation iteration to improve the efficiency of calculation. From this information, a step size control strategy is proposed to speed up the oscillatory stability margin and damping margin identification. We also propose an improved initialization and update of invariant subspaces, especially for the least damping ratio eigenvalues. The simulation results and computation performance on New England 39-bus system and IEEE 145-bus system are demonstrated in details to show the effectiveness of the algorithm. Results have shown that the ICIS method is an accurate, fast, and robust method in eigenvalue calculation and margin identification.;The ICIS provides us an efficient and accurate way to trace a specified subset of eigenvalues of interest for power system small-signal stability analysis, such as rightmost eigenvalues and least damping ratio eigenvalues. It is also a robust method in tracking close or multiple eigenvalues where the conventional methods usually fail to converge. In addition, the ICIS is integrated with the equilibrium point tracing for overall bifurcation analysis in power systems for the identification of voltage stability margin and other eigenvalue-related margins.
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