  In this paper we are interested in the problem of adaptive synchronization of
a voltage source converter with a possibly weak grid with unknown angle and
frequency, but knowledge of its parameters. To guarantee a suitable
synchronization with the angle of the three-phase grid voltage we design an
adaptive observer for such a signal requiring measurements only at the point of
common coupling. Then we propose two alternative certainty-equivalent, adaptive
phase-locked loops that ensure the angle estimation error goes to zero for
almost all initial conditions. Although well-known, for the sake of
completeness, we also present a PI controller with feedforward action that
ensures the converter currents converge to an arbitrary desired value.
Relevance of the theoretical results and their robustness to variation of the
grid parameters are thoroughly discussed and validated in the challenging
scenario of a converter connected to a grid with low short-circuit-ratio.
