This paper applies model predictive control (MPC) for the power processing of an oscillating water column (OWC) wave energy conversion (WEC) system to achieve smooth power delivery to the grid. The particular air turbine design adopted in this study produces large power pulses ranging from 0 to 1 MW in magnitude, and thus, direct connection to the grid is practically impossible, especially in weak grid conditions. Therefore, energy storage is an essential element that should be integrated into this particular WEC system in order to absorb power pulses and thereby ensure smooth delivery of power to the grid. Taking into account the repetitive nature, duration, and magnitude of the power pulses, this study has chosen “supercapacitor” as the suitable energy storage technology. The supercapacitor energy storage (SCES) is integrated into the dc-link of the back-to-back power converter of the WEC system through a bidirectional dc-dc converter. In order to achieve the desired operation of this complex power converter arrangement, a finite control set MPC strategy is proposed in this paper. Performance of the proposed energy storage system (ESS) and control strategy are evaluated through computer simulations. Simulation results show that the proposed SCES system and the control strategy are able to achieve smooth power delivery to the grid amidst power pulses coming from the generator.