  It is shown that proportional-integral (PI) control in the synchronously
rotating (d, q)-reference frame and proportional-resonant (PR) control in the
stationary ({\alpha}, \b{eta})-reference frame, both with anti-windup, are
equivalent if and only if their implementation is done correctly in state space
and the controller parameters and the initial values are identical. It is shown
that an equivalence in the frequency domain does only hold if simplifying
assumptions are satisfied. As consequence of the equivalence, both closed- loop
control performances are identical with respect to closed-loop dynamics and
steady-state accuracy. The control performance will only differ if their
implementation is not done correctly or the time delay induced by the voltage
source inverter becomes significant. To the best knowledge of the author,
equivalence of PR and PI controllers with anti-windup has not been shown before
(in particular not in state space).
