In this paper, we study analytically the transient stability of
grid-connected distributed generation systems with grid-forming (GFM) complex
droop control, also known as dispatchable virtual oscillator control (dVOC). We
prove theoretically that complex droop control, as a state-of-the-art GFM
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 GFM 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