This paper investigates model predictive control (MPC) of a single sea wave energy converter (WEC). By using control schemes which constrain certain quantities, such as the maximum size of the feedback force, the energy storage for actuators and relative heave motion, it is possible for control to not only improve performance but to directly impact strongly on design and cost. Motivated by this fact, a novel objective function is adopted in the MPC design, which brings obvious benefits: First, the quadratic program (QP) derived from this objective function can be easily convexified, which facilitates the employment of existing efficient optimization algorithms. Second, this novel design can trade off the energy extraction, the energy consumed by the actuator and safe operation. Moreover, an alternative QP is also formulated with the input slew rate as optimization variable, so that the slew rate limit of an actuator can be explicitly incorporated into optimization. All these benefits promote the real-time application of MPC on a WEC and reduced cost of hardware.