LAUSR

Several models were developed in the literatures for studying the system impedance characteristics during sub-synchronous oscillations (SSO). However, the doubly-fed induction generator (DFIG) impedance presents a $2\times 2$ matrix with the frequency coupling component at the off-diagonal position, which increases the difficulty for analyzing the impedance characteristics of DFIG. The coupling effect in this case is caused by the asymmetric impedance characteristics of controllers, which are composed of voltage source converters (VSCs). Accordingly, simplifications are often conducted to decouple the impedance matrix into single sequences. Outer loop controller and phase-locked loop (PLL), which are often neglected in simplifications, are the main causes of the asymmetry of impedance matrix. Such simplifications could also hamper proper system stability analyses. In this paper, a comprehensive impedance model is proposed in which the outer loop controller, PLL, and grid-side controller (GSC) are all considered. The proposed comprehensive impedance model can provide exact stability data, in which stable boundaries for compensation level and transmission distances are presented. Furthermore, the proposed model is converted to simplified models in which the impacts of outer loop controller, PLL, GSC and coupling components on boundary stability of SSO are analyzed. The results are validated by time domain simulations and eigenvalue analyses, and deviations of the simplified models are analyzed for different operating parameters and conditions leading to simplifications.