<p>The conventional virtual synchronous generator (VSG) is normally designed to meet certain operational and control requirements in the islanded mode. However, once the VSG is switched to grid-connected mode (GCM), the robust operation cannot be guaranteed under different grid conditions. It can lead to, especially in strong grids, poor dynamic performance such as significant oscillation, long settling time, and high overshoot. In order to improve the VSG performance in the GCM, this article first analyzes in depth the inherent coupling between the active and reactive power and its dependence on the grid conditions, such as the short circuit ratio, grid impedance ratio Xg=Rg, and the wide grid impedance variations. Afterwards, an online grid impedance estimation-based adaptive VSG (AVSG) control strategy is proposed to ensure the robust operation of the VSG considering both strong and weak grid conditions. This technique allows the operator to specify the desired settling time of output power and damping ratio.To estimate the grid impedance in real time without extra hardware and reduce the associated impacts on power quality, an online event-based grid impedance estimation algorithm is embedded into the control loop of the AVSG. Both the simulation and experimental results indicate that, compared to the conventional fixed-parameters-based controller design method, the AVSG enables the desired performance such as no oscillation, specified time duration for the settling time, and minimal overshoot regardless of the grid conditions.</p>