Vertical axis wind turbines (VAWTs) provide promising solutions for wind energy harvesting in complex flow environment. However, it is challenging to guarantee satisfactory self-starting capability and high power efficiency simultaneously in a VAWT design. To address this challenge, a new hybrid Darrieus-Modifed-Savonius (HDMS) VAWT is designed and numerically tested using a fluid-structure interaction approach based on high fidelity computational fluid dynamics. A systematic study is conducted to analyze the effects of the moment of inertia, turbine structure, and external load on the self-starting capability and power efficiency. It is found that compared with the Darrieus VAWT, the HDMS design has better self-starting capability due to the torque provided by the MS rotor at small tip speed ratios (TSRs). The larger the MS rotor is, the better the self-starting capability is. However, there is penalty on power efficiency when the size of the MS rotor increases. With an appropriately sized MS rotor, the HDMS design can maintain high power efficiency comparable with the Darrieus VAWT at large TSRs. The key flow physics is that the HDMS design can keep accelerating at small TSRs due to the inner MS rotor, and can suppress dynamic stall on the Darrieus rotor at large TSRs.