The large-scale integration of renewable energy into power grids has led to a significant reduction in power system rotational inertia. To ensure system frequency security, a frequency-constrained unit commitment model based on linear AC power flow is proposed. First, a unit commitment model based on mixed integer second-order cone relaxation is developed. The AC power flow model is combined with the unit commitment model to establish a unit commitment model that can accurately characterize power flow distribution within the network. Second, to ensure the security and stability of the system frequency, frequency-related constraints are incorporated into the model, and a frequency-constrained unit commitment model based on mixed-integer second-order cone relaxation is constructed. Finally, a two-stage progressive optimization strategy is proposed to improve the accuracy of the solution by collaboratively solving the frequency-constrained unit commitment model and the multi-period optimal power flow problem. The effectiveness of the proposed model is verified using a modified IEEE 118-bus system.