The traditional empirical probability distribution is used to describe the uncertainty of distributed PV output but it is difficult to accurately obtain the probability distribution parameters. Therefore, this paper proposes a distributionally robust optimization method based on Kullback-Leibler divergence to evaluate the maximum grid-connected capacity of distributed PV in a distribution network. First, the distance between the actual probability distribution and the empirical probability distribution is quantified by Kullback-Leibler divergence, and the fuzzy set of distributed PV output uncertainty is established. Considering the nodal voltage and transmission capacity over limits chance constraints, a distributionally robust optimization model of distributed PV grid-connected capacity is established. This model is transformed into a mixed integer programming problem using value at risk and a sample average approximation method. The effectiveness and robustness of the method are demonstrated by simulation results on a modified IEEE-33 bus system. This work is supported by the Science and Technology Plan Projects of Hebei Province (No. 19212102D) and the Science and Technology Project of State Grid Hebei Electric Power Co., Ltd. (No. kj2020-049).