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When a hexagonal converter (Hexverter) is applied to high-power renewable energy generation such as offshore low-frequency transmission, using traditional PID control to ensure the stable global operation and strong robustness of the system is difficult due to the random fluctuation of renewable energy, system parameter uptake, its non-linear characteristics, and the possible unbalanced grid situation. This paper proposes a non-linear passivity-based control (PBC) strategy for Hexverter under an unbalanced grid. First, the passivity and stability of Hexverter objects are analyzed based on PBC theory. The unbalanced PBC strategy of Hexverter is then designed and derived. Three control objectives are designed according to different control requirements. Finally, the simulation system of the Hexverter under an unbalanced grid is built on Matlab/Simulink to verify the effectiveness and superiority of the proposed unbalanced PBC method. The simulation results show that the proposed method has fewer control parameters, faster response, lower harmonic content, and better overall control effect than PID control. [ABSTRACT FROM AUTHOR]