An integrated generation, transmission, and energy storage planning model accounting for short-term constraints and long-term uncertainty is proposed. The model allows to accurately quantify the value of flexibility options in renewable power systems by representing short-term operation through the unit commitment constraints. Long-term uncertainty is represented through a scenario tree. The resulting model is a large-scale multi-stage stochastic mixed-integer programming problem. To overcome the computational burden, a distributed computing framework based on the novel Column Generation and Sharing algorithm is proposed. The performance improvement of the proposed approach is demonstrated through study cases applied to the NREL 118-bus power system. The results confirm the added value of modeling short-term constraints and long-term uncertainty simultaneously. The computational case studies show that the proposed solution approach clearly outperforms the state of the art in terms of computational performance and accuracy. The proposed planning framework is used to assess the value of energy storage systems in the transition to a low-carbon power system.