The increase in renewable energy sources (RESs), like wind or solar power, results in growing uncertainty also in transmission grids. This affects grid stability through fluctuating energy supply and an increased probability of overloaded lines. One key strategy to cope with this uncertainty is the use of distributed energy storage systems (ESSs). In order to securely operate power systems containing renewables and use storage, optimization models are needed that both handle uncertainty and apply ESSs. This paper introduces a compact dynamic stochastic chance-constrained DC optimal power flow~(CC-OPF) model, that minimizes generation costs and includes distributed ESSs. Assuming Gaussian uncertainty, we use affine policies to obtain a tractable, analytically exact reformulation as a second-order cone problem (SOCP). We test the new model on five different IEEE networks with varying sizes of 5, 39, 57, 118 and 300 nodes and include complexity analysis. The results show that the model is computationally efficient and robust with respect to constraint violation risk. The distributed energy storage system leads to more stable operation with flattened generation profiles. Storage absorbed RES uncertainty, and reduced generation cost.