Aggregations of electric loads, like heating and cooling systems, can be controlled to help the power grid balance supply and demand, but the amount of balancing reserves available from these resources is uncertain. In this paper, we investigate data-driven optimization methods that are suited to dispatching power systems with uncertain balancing reserves provided by load control. Specifically, we consider a chance-constrained optimal power flow problem in which we aim to satisfy constraints that include random variables either jointly with a specified probability or individually with different risk tolerance levels. We focus on the realistic case in which we do not have full knowledge of the uncertainty distributions and compare distribution-free approaches with several stochastic optimization methods. We conduct experimental studies on the IEEE 9-bus test system assuming uncertainty in load, load-control reserve capacities, and renewable energy generation. The results show the computational efficacy of the distributionally robust approach and its flexibility in trading off between cost and robustness of solutions driven by data.