Lightning-induced breaking accidents of medium-voltage insulated conductors pose a serious threat to the safety of distribution networks, and the key cause lies in the establishment and sustained combustion of the power-frequency follow-current arc after lightning overvoltage breakdown. This paper systematically investigates the formation mechanism and critical conditions of power-frequency follow-current arcs using combined simulation and experimental approaches. Based on the streamer discharge theory, a lightning breakdown model was established and combined with the arc energy balance equation, revealing that the establishment of power-frequency follow-current arcs is essentially determined by the post-breakdown energy competition process. The simulation results show that the required anode electric field strength for lightning breakdown is not less than 3 kV/mm. When the power-frequency voltage reaches 10 kV, Joule heating of the arc continuously exceeds heat dissipation loss, enabling restrike after zero-crossing and sustaining stable burning. Experiments verified this voltage threshold and further revealed that the arc establishment rate exhibits nonlinear growth with increasing power-frequency voltage, exceeding 90% at power-frequency voltages ≥ 10 kV. The study also reveals that increased gap distance reduces the arc establishment rate, while the introduction of insulators can enhance it by approximately 20%. This study clarifies the energy criterion for power-frequency follow-current arc establishment and the influence patterns of key parameters, providing theoretical basis and engineering reference for lightning protection design and arc suppression in medium-voltage insulated lines.