The integration of wind generation to the grid is growing rapidly across the world. As a result, grid operators have introduced the so-called grid codes (GC), which nowadays include a range of technical conditions and requirements, which wind generators must fulfill. Among these, the low voltage ride through (LVRT) is a requirement for wind turbines to stay connected to the grid and continue to operate during the disturbance. In this study, a control structure, combining inertial kinetic energy storage with a crowbar circuit, is proposed to enhance the ride-through capability of a wind turbine generator (WTG) based on a wound-field synchronous generator (WFSG) under unsymmetrical voltage dips. For the grid-side converter (GSC), a decoupled double synchronous reference frame (DDSRF) d-q current controller is used. Furthermore, a Second-Order Sliding Mode Controller (SOSMC) with Super-twisting (ST) algorithm is proposed for the GSC and the machine-side converter (MSC) to improve the response speed and achieve an accurate regulation of the dq-axis current components simultaneously. The main objectives of the GSC are to achieve a balanced, sinusoidal current and smooth the real and reactive powers to reduce the influence of the negative sequence voltage. A series of simulations are presented to demonstrate the effectiveness of the proposed control scheme in improving the LVRT capability of the WFSG-driven wind turbine and the power quality of the system under unbalanced grid voltage conditions.