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This paper investigates the privacy preservation issue of distributed dc optimal power flow (OPF) methods while considering the line capacity constraints in smart grids with radial topology. We show that private information, such as the local energy consumption, local power generation, and parameters in the local cost function can be possibly exposed in traditional OPF methods without proper privacy preserving mechanism. Then we design a privacy-preserving alternating direction method of multipliers based distributed OPF algorithm by adding stochastic noise into the communication messages between neighbor buses, and the analysis on the convergence and optimality of the proposed algorithm is provided. Extensive simulations on a nine-bus power system validate the theoretical results and demonstrate the feasibility of the proposed algorithm.