Due to the high penetration of renewable energy resources in microgrids (MGs), the grid inertia becomes low which leads to the grid to be vulnerable to large disturbances. The energy storage devices can play an important role to enhance the inertia of MGs. However, due to the high investment cost of storages or their dp/dt limitation, the installed energy storages cannot cover the challenge of high df/dt. A prominent solution to solve the problem is to use the inertia response of the wind generators. However, relatively high second frequency nadir is the main drawback of using the inertia response of the wind generators which may impose an extensive disturbance to MGs. Accordingly, a coordinated operation strategy for MGs between wind generator and hybrid energy storage (HES) system is proposed in this paper. In addition, to improve the inertia response of the MG; providing high-quality communication infrastructures with low delay and increasing the Ultracapacitor capacity have been paid attention. In this paper, the costs of the installed Ultracapacitor and quality of communication services are defined as the operation cost. Guaranteeing enough frequency damping for the MG with low operation cost are two conflict objectives. Therefore, a multi-objective optimization method is used to set the controllers' values and reduce the operation cost. The results confirmed that the effectiveness of the proposed strategy to control hybrid power storage in coordination with the wind generator and the frequency recovery process is improved. Also, employing the optimum values guaranteed the frequency damping effectively with low operation cost. The Integral Absolute Error (IAE) value and operation cost are reduced by 13.6% and 32%, respectively. Also, the simulation results show that the maximum MG frequency deviation and maximum df/dt is well compatible with different standards in the presence of load perturbations and different wind speeds.INDEX TERMS Microgrid (MG), Frequency control, Wind generator, Hybrid energy storage (HES), Ultra-capacitor, Inertia response, Multi-objective optimization.