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This paper presents a robust and fast control strategy for load-voltage regulation in grid-connected inverters with inductor-capacitor-inductor (LCL) filter. The LCL filter complicates the power circuit dynamics, particularly when the uncertain nature of the grid background distortion and system parameters is considered. Furthermore, grid-voltage control in the presence of LCL filter dynamics yields dynamic coupling between current and voltage control loops. To overcome these difficulties, first, a robust high-bandwidth multiloop current controller is designed to decouple the current and voltage control dynamics and mitigate uncertain interactions between the LCL-filtered current-controlled inverter and grid. Second, a dual-sequence adaptive-gain variable-structure voltage control scheme is proposed for effective mitigation of fast, dynamic, unbalanced, and harmonic voltage disturbances. A theoretical analysis and comparative experimental results are presented to demonstrate the effectiveness of the proposed control scheme in supporting the voltage reliability.