Abstract: The fault recovery in distribution networks is an important guarantee for the safe and economic operation of the power grid. To eliminate the risk of distributed power generation uncertainty on grid operation during fault recovery process, this paper constructs the mixed integer second-order cone programming (MISOCP) stochastic optimization model for distribution network fault recovery based on the stochastic response surface method (SRSM), taking into account the influence of the uncertainty in the entire periods. To accurately evaluate the operational risk of the energy storage system (ESS), a probability equivalence scheme is applied to the randomness of the state of charge of ESS at each time period. Based on the characteristic coefficients of Hermite chaotic polynomials, the convex function of opportunity constraints for distribution network fault recovery is constructed to ensure the global uniqueness of the solution. To reduce the complexity of the optimization model while maintaining the accuracy of the stochastic process fitting, a simplified probabilistic load flow model is proposed to reduce the number of items in the second-order cone relaxation constraint. Finally, the Monte Carlo simulation is employed to compare and validate the proposed stochastic optimization model in this paper. The results demonstrate that the proposed method accurately describes the probability characteristics of random response and maximizes the allocation of sources to minimize outage losses, while adhering to the opportunity constraints of safe operation.