Large-scale fast-varying sinusoidal disturbances (FVSDs) strongly impact the operation of Grid-connected Converters (GcCs). A conventional extended state observer (ESO) is not sufficiently capable to deal with these rapid disturbances. This critical weakness has limited ESO's applications to GcCs. Therefore, a generalized integrator-ESO (GI-ESO) is developed from ESO to address such issues. Knowing that grid common disturbances can be expressed as a phase, frequency, or magnitude disturbed sinusoidal signal, multi-GIs of selective resonant frequencies are, hence, applied into the disturbance estimation loop, which enables FVSD to be observed with relatively low bandwidth. Besides, a frequency-adaptive mechanism is introduced in order to mitigate the effects of frequency deviations. The proposed GI-ESO resolves the tradeoff between the observer's bandwidth and noise filtering. A case of the three-phase phase-locked loop control system of disturbed regimes is, in particular, studied to assess the performance of GI-ESO and the stability of the resulting controlled system. Theoretical analysis and experimental results have proven its effectiveness in applications to GcCs in the presence of various disturbances.