Transient Synchronous Stability Analysis and Assessment of PLL-Based VSC Systems by Bifurcation Theory

For the transient synchronous stability of phase-locked loop based voltage source converter grid-tied systems, the generalized swing equation (GSE) is believed as very important. In this paper, the GSE is studied deeply by bifurcation theory. For instance, the co-dimension-2 Bogdanov-Takens (BT) bifurcation is rigorously analyzed, and the formulations for three co-dimension-1 bifurcations near the BT bifurcation point including the generalized saddle-node, Hopf, and homoclinic bifurcations are explicitly obtained. Moreover, all Hopf bifurcations in the GSE have been theoretically proven to be subcritical by the normal form method. Therefore, after the Hopf and homoclinic bifurcations, two different types of basin of attraction occur widely including a closed loop surrounded by an unstable limit cycle and a fish-like pattern surrounded by the stable manifold of unstable equilibrium point (UEP), respectively. In addition, for the transient synchronous stability assessment after the homoclinic bifurcation, based on the linear approximation of the stable manifold of the UEP, an index for the distance between the stable equilibrium point and the approximated stability boundary is formulated. All these theoretical results are well verified and supported by numerical calculations and real-time simulations, making all previous phenomenal observations of the GSE stand on the solid foundation of rigorous mathematics.