This paper deals with power control strategies for a fixed-pitch direct drive marine current turbine (MCT) when the marine current velocity exceeds the rated value corresponding to the MCT nominal power. At overrated marine current speed, the MCT control strategy is supposed to be changed from maximum power point tracking (MPPT) stage to constant power stage. In this paper, flux-weakening strategy is investigated to realize appropriate power control strategies at high marine current speeds. During flux-weakening operations, the generator can be controlled to produce nominal or over-nominal power for a specific speed range (constant power range). These two power control modes are compared and the constant power range is calculated in this paper. The relationship between the expected constant power range and generator parameters requirement (stator inductance, permanent magnet flux, nominal power coefficient) is analyzed in this paper. A Torque-based control with a robust feedback flux-weakening strategy is then carried out in the simulation. The proposed control strategies are tested in both high tidal speed and swell wave cases; the results validate the analysis and show the feasibility of the proposed control method.