Optimization formulations with chance constraints have been widely proposed to operate the power system under various uncertainties, such as renewable production and load consumption. Constraints like the system's physical limits are required to be satisfied at high confidence levels. Conventional solving methodologies either make assumptions on the underlying uncertainty distributions or give overly-conservative results. We develop a new distributionally robust (DR) chance constrained optimal power flow formulation in which the chance constraints are satisfied over a family of distributions with known first-order moments, ellipsoidal support, and an assumption that the probability distributions are log-concave. Since most practical uncertainties have log-concave probability distributions, including this assumption in the formulation reduces the objective costs as compared to traditional DR approaches without sacrificing reliability. We derive second-order cone approximations of the DR chance constraints, resulting in a tractable formulation that can be solved with commercial solvers. We evaluate the performance of our approach using a modified IEEE 9-bus system with uncertain wind power production and compare it to standard approaches. We find that our approach produces solutions that are sufficiently reliable and less costly than traditional DR approaches.