By considering the specific characteristics of random variables in active distribution grids, such as their statistical dependencies and often irregularly-shaped probability distributions, we propose a non-parametric chance-constrained optimization approach to operate and plan energy storage units in power distribution girds. In particular, we develop new closed-form stochastic models for the key operational parameters in the system. Our approach is analytical and allows formulating tractable optimization problems. Yet, it does not involve any restricting assumption on the distribution of random parameters, hence, it results in accurate modeling of uncertainties. Different case studies are presented to compare the proposed approach with the conventional deterministic and parametric stochastic approaches, where the latter is based on approximating random variables with Gaussian probability distributions.