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Stiff dynamics continue to pose challenges for power system dynamic state estimation. In particular, models of inverters with control schemes designed to support grid voltage and frequency, namely, grid-forming inverters (GFMs), are highly prone to numerical instability. This paper develops a novel analytical modeling technique derived from two cascading subsystems, namely synchronization and dq-frame voltage control. This allows us to obtain a closed-form discrete-time state-space model based on the matrix exponential function. The resulting model enables a numerically stable and decentralized dynamic state estimator that can track the dynamics of GFMs at standard synchrophasor reporting rates. In contrast, existing dynamic state estimators are subject to numerical issues. The proposed algorithm is tested on a 14-bus power system with a GFM and compared with the standard algorithm whose process model is discretized using well-known Runge-Kutta methods. Numerical results demonstrate the superiority of the proposed method under various conditions. 

We devise a novel formulation and propose the concept of modal participation factors to nonlinear dynamical systems. The original definition of modal participation factors (or simply participation factors) provides a simple yet effective metric. It finds use in theory and practice, quantifying the interplay between states and modes of oscillation in a linear time-invariant (LTI) system. In this paper, with the Koopman operator framework, we present the results of participation factors for nonlinear dynamical systems with an asymptotically stable equilibrium point or limit cycle. We show that participation factors are defined for the entire domain of attraction, beyond the vicinity of an attractor, where the original definition of participation factors for LTI systems is a special case. Finally, we develop a numerical method to estimate participation factors using time series data from the underlying nonlinear dynamical system. The numerical method can be implemented by leveraging a well-established numerical scheme in the Koopman operator framework called dynamic mode decomposition. 

We propose a reformulation of the streaming dynamic mode decomposition method that requires maintaining a single orthonormal basis, thereby reducing computational redundancy. The proposed efficient streaming dynamic mode decomposition method results in a constant-factor reduction in computational complexity and memory storage requirements. Numerical experiments on representative canonical dynamical systems show that the enhanced computational efficiency does not compromise the accuracy of the proposed method. 

The stiff dynamic model of grid-forming inverters (GFMs) presents numerical challenges in time discretization for real-time simulation, estimation, and control. To address this challenge, this paper proposes modeling GFMs as two cascading subsystems: a reference-frame-synchronization system and an in-reference-frame system. This paper demonstrates that these two subsystems are approximately linear, allowing their time discretization to be derived in closed form using the matrix exponential. The proposed discretization method is tested on the IEEE 14-bus power system with a GFM connected and compared with second- and fourth-order Runge-Kutta methods. The numerical results validate the proposed method and demonstrate its potential for discretization with longer step sizes.