Global climate change places greater demands on the process of decarbonizing power systems. Battery energy storage can effectively cope with the uncertainty of renewable energy sources and reduce wind power curtailment. Because of the high cost, reasonable economic optimization is required for energy storage planning considering the uncertainty of wind. The traditional stochastic optimization is optimistic while the robust optimization is conservative. To overcome this problem, a two-stage distributionally robust energy storage planning model is proposed in this paper based on the historical information of wind power output. The economics of the energy storage investment scheme is considered in the first stage objective, and the second stage performs the system day-ahead dispatching. This paper used the norm-1 and norm-inf comprehensive constraints to establish the probability distribution uncertainty set of wind power, then tried to find the optimal solution of the model under the worst probability distribution. The column and constraint generation algorithm is used to improve the efficiency of the solution. The numerical test results demonstrate the effectiveness of the model and the method in this paper.