Wind farms (WFs) can provide controlled inertia through synthetic inertial control (SIC) to support system frequency recovery after disturbances. This study proposes a model predictive control (MPC)-based SIC for a WF consisting of wind turbines (WTs) and a battery storage energy system (BESS). In the proposed MPC-SIC, the active power output of the WTs and BESS during the SIC are optimally coordinated in order to avoid over-deceleration of the WTs' rotor, and minimise the loss of extracted wind energy during the SIC and degradation cost of the BESS. The IEEE 39-bus system with a WF consisting of 100 WTs and a BESS is used to validate the performance of the proposed MPC-SIC. Case studies show that, compared with the conventional SIC, the minimum rotor speed among all WTs with MPC-SIC can be improved by 0.08-0.11 p.u., the loss of captured wind energy of WF with MPC-SIC can be reduced by 12-64% and the degradation cost of the BESS with MPC-SIC can be reduced by 72-83%. The results proves that with the proposed MPC-SIC, the WF can avoid the over-deceleration of the WTs' rotor and reduce the operation cost of the WF by improving the efficiency of wind energy usage and lifetime of the BESS.