Optimal power flow (OPF) has been used for energy dispatching in active distribution networks. To satisfy constraints fully and achieve strict operational bounds under the uncertainties from loads and sources, this paper derives an interval optimal power flow (I-OPF) method employing affine arithmetic and interval Taylor expansion. An enhanced I-OPF method based on successive linear approximation and second-order cone programming is developed to improve solution accuracy. The proposed methods are benchmarked against Monte Carlo simulation (MCS) and stochastic OPF. Tests on a modified IEEE 33-bus system and a real 113-bus distribution network validate the effectiveness and applicability of the proposed methods.