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This paper proposes a distribution network fault emergency power supply recovery strategy based on 5G base station energy storage. This strategy introduces Theil's entropy and modified Gini coefficient to quantify the impact of power supply reliability in different regions on base station backup time, thereby establishing a more accurate base station's backup energy storage capacity model that can fully utilize the base station's energy storage resources to participate in the emergency power supply of distribution network faults. First, the optimal Copula function in each cycle is determined through the Akaike information criterion and squared Euclidean distance to establish a wind power-photovoltaic output scenario. Secondly, the traditional Gini coefficient is modified using the weighted average node degree and influence rate indicators. A comprehensive vulnerability model is established through the modified Gini coefficient and Theil's entropy indicators. Based on the comprehensive vulnerability model, a backup energy storage time model and a modified backup energy storage capacity model of the base station affected by power supply reliability are established. Thus, the callable energy storage capacity of base stations in different areas is obtained. Finally, a two-stage robust optimization model is introduced to minimize system operating costs to solve the volatility of 5G base station communications and wind-solar output, thereby establishing an emergency power supply recovery model for base station energy storage and wind-solar output. Simulated with the improved IEEE-33 node model, the results show that the proposed base station's energy storage model improves the utilization of the base station energy storage resources and, at the same time, effectively reduces the loss of load during the fault phase of the distribution network and improves the absorption of the PV output. • Modeling of 5G base station backup energy storage. Aiming at the shortcomings of existing studies that ignore the time-varying characteristics of base station's energy storage backup, based on the traditional base station energy storage capacity model in the paper [ 18 ], this paper establishes a distribution network vulnerability index to quantify the power supply reliability of the distribution network nodes by adopting the voltage Theil's entropy and the modified Gini coefficient. Based on the power supply reliability of power grid nodes and combined with load level weights, a model for the backup energy storage time of base stations affected by power supply reliability is established. • The model of 5G base station energy storage participating in power supply recovery In view of the impact of changes in communication volume on the emergency power supply output of base station energy storage in distribution network fault areas, this paper introduces a two-stage robust optimization model to deal with the uncertainty of communication volume. Based on the base station energy storage capacity model established in contribution (1), an objective function is established to minimize the system operating cost in the fault area, and the base station energy storage owned by mobile operators is used as an emergency power source to participate in power supply restoration. Establish sub-objective functions of the loss cost of base station energy storage charging and discharging, the subsidy cost of base station energy storage charging and discharging, and the base station energy storage operating cost to supplement the shortcomings of existing research. [ABSTRACT FROM AUTHOR]