In response to the challenge of achieving simultaneous and rapid quantitative analysis of system reliability improvement needs during the process of energy storage siting and sizing in distribution networks, this paper proposes an optimal configuration model and solution method for distribution network energy storage considering system reliability constraints. First, the system adequacy index is selected to quantify the improvement of system reliability through energy storage configuration. A bi-level optimization model for distribution network energy storage configuration is established, balancing economic and reliability objectives. The upper level aims to minimize the net investment cost of energy storage configuration in the distribution network, while the lower level aims to minimize the load shedding under N-1 contingency conditions in the distribution system. Then, based on the KKT conditions and composite function differentiation rules during the solution of the optimization problem using the interior-point method, analytical expressions for the sensitivity analysis of system reliability indices to the upper limit of energy storage output and capacity are derived. These expressions are incorporated into the linearized system reliability inequality constraints to achieve parameter transmission between the upper and lower models. Finally, the effectiveness and rationality of the designed bi-level optimization model and the derived sensitivity analysis expressions are validated using a modified IEEE RTS-79 test system.