  In this paper, we analytically study the transient stability of
grid-connected converters with grid-forming complex droop control, also known
as dispatchable virtual oscillator control. We prove theoretically that complex
droop control, as a state-of-the-art grid-forming control, always possesses
steady-state equilibria whereas classical droop control does not. We provide
quantitative conditions for complex droop control maintaining transient
stability (global asymptotic stability) under grid disturbances, which is
beyond the well-established local (non-global) stability for classical droop
control. For the transient instability of complex droop control, we reveal that
the unstable trajectories are bounded, manifesting as limit cycle oscillations.
Moreover, we extend our stability results from second-order grid-forming
control dynamics to full-order system dynamics that additionally encompass both
circuit electromagnetic transients and inner-loop dynamics. Our theoretical
results contribute an insightful understanding of the transient stability and
instability of complex droop control and offer practical guidelines for
parameter tuning and stability guarantees.
