In this paper, we address a novel and comprehensive social welfare maximization (SWM) problem for the optimal energy management in a smart grid. The objective is to maximize the total social welfare of dispatchable devices in the smart grid while satisfying certain constraints. Each device in the smart grid is required to meet its local power constraints, and the system as a whole maintains supply-demand balance, taking into account transmission losses and stochastic output power. To facilitate distributed algorithm design, we initially transform the SWM problem into an equivalent dual problem, which is a distributed composite optimization problem. Subsequently, a novel fully distributed proximal alternating direction method of multipliers (PADMM) is proposed, where each agent can autonomously select non-coordinated step size parameters based solely on local information, independent of other agents and the network structure. Detailed convergence analysis is provided, and a worst-case $\mathcal {O}(1/k)$ convergence rate is established in the non-ergodic sense. Finally, several numerical experiments are conducted to confirm the effectiveness of the proposed algorithm.