Abstract: As the proportion of distributed renewable energy sources continues to increase in the distribution grid, distributed energy storage systems emerge as an effective means to address the uncertainty of renewable energy output, highlighting their growing importance. A two-stage robust optimization model for distributed energy storage planning with a min-max-min structure is established, aimed at optimizing the operation cost of the distribution grid while considering network constraints. The model incorporates operational constraints for distributed energy storage, demand response loads, small gas turbines, distributed photovoltaic systems, and distributed wind power. By employing second-order cone relaxation techniques, the flow constraints are transformed into convex relaxations based on a branch flow model. Utilizing the Karush-Kuhn-Tucker(KKT) conditions and the column- and-constraint generation algorithm, the original problem is decomposed into a master problem and subproblems characterized by mixed-integer linear features. Through iterative optimization, the optimal solution is obtained. Finally, simulation validation in the IEEE 33-node system demonstrates the effectiveness of the proposed energy storage planning method in managing the uncertainty of renewable energy output, providing a reference for distribution grid investors in energy storage planning and investment.