Wind turbines are increasingly being expected to provide oscillation damping to the power system to which they are connected. In this study, power oscillation damping control of variable speed wind turbines is studied. An energy storage device with a bidirectional DC/DC converter connected to the DC link of a fully rated converter-based wind turbine is proposed. As system oscillation is often induced by an AC fault, it is desirable for wind turbines to ride through the fault first and then provide a damping effect. During the fault period, the energy storage system (ESS) is controlled to assist the fault ride through process, and the line side converter (LSC) is controlled to provide AC voltage support in accordance with the grid code. Methods based on regulating the active power output of the ESS and modulation of reactive power output of the LSC are proposed so as to damp the oscillations of the power system. Matlab/Simulink simulations based on a simplified Irish power system demonstrate the performance of the ESS and LSC during fault periods and validate the damping effect of the proposed system.