An Efficient Eigenvalue Tracking Method for Time-Delayed Power Systems based on Continuation of Invariant Subspaces

Abstract :  Time delays are inevitable in the power systems with wide-area control signals, which may compromise the control performance and thus jeopardize the stability of the power systems. To analyze the impacts of system parameters on critical oscillation modes of a time-delayed power system (TDPS), a new method for eigenvalue trajectory tracking of TDPS is proposed in this paper based on the continuation theory of invariant subspaces. When small changes are imposed on system parameters, such as parameters of wide-area damping controllers (WADCs) and time delays, the invariant subspaces are continuously predicted and corrected, followed by the eigenvalues and eigenvectors. In the process, the predictor and corrector are computed by solving linear equations sparsely and in blocks, which is highly scalable and computationally efficient for large-scale power system applications. The 16-generator 68-bus test system and two real-world power grids in China are used to test and demonstrate the proposed method. The results show that the proposed method can accurately and efficiently track eigenvalue trajectories for TDPS, especially for large-scale power grids. In addition, the sensitivities of eigenvalues and eigenvectors with respect to WADCs’ parameters and time delays are obtained as by-products of the approach.
 ? The transcription of the Kronecker sum into the matrix pencil form reduces the problem to the existence of the unit circle of generalized eigenvalues of the corresponding matrix pencil. ? The proposed method offered larger delay margin and takes less computation time compared to some existing methods. ? Frequency-based DIS methods are generally built on the verification of the non-existence of purely imaginary system poles for arbitrary delay values. ? Then, the non-existence of any positive real root of D (?) - which is a sufficient DIS condition - was proved by the Déscartes rules of signs.
 ? In continuation methods, the predictor-corrector scheme has been widely applied in power systems, such as continuation power flow. ? Its principle is to predict the next solution point and then correct the prediction to get the next point on the curve. ? The eigenvalue tracking problem addressed in this paper is not an exception. ? This is a challenging problem since the frequencies, damping ratios and their mode shapes may change greatly or jump as WADCs’ parameters change.
 • The proposed direct method provides the central manifold approximation for lossless propagation model without the use of the central manifold theorem and the structure reconstruction. • It was utilized to investigate the local asymptotic stability of the positive equilibrium for the n-dimensional Lotka-Volterra system and to compute the rightmost characteristic roots (poles), respectively. • The applicability of the method was also improved by the introduction of a new class of TDSs and the corresponding transformation into the proposed common canonical form.
 ? Previous weakly damped blackouts and increased power system congestion have prompted new techniques to improve the damping of these oscillation modes. ? By introducing remote signals with global observability and controllability for interarea oscillation modes, wide-area damping controllers (WADCs) can provide additional damping to effectively improve the weakly damped modes and achieve system-level control. ? However, the remote signals can bring inevitable time delays into the widearea closed loop, which may deteriorate WADC performance and even threaten the stability of power systems

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