Age of Information (AoI) has been introduced as a new metric for characterizing data freshness in wireless networks. To optimize the system AoI at wireless-powered networks, we investigate the first work to optimize the weighted sum of average AoI with stochastic arrivals in this paper. Firstly, the average AoI optimization problem with stochastic arrivals is formulated, which tries to minimize the weighted sum of average AoI from the point of directional energy transferring. The lower bound of the achievable AoI by any network operating is derived. Then, to assist in the energy harvesting process of the source nodes, the stationary semi-randomized scheduling policy is introduced and the closed-form expression under the space of such a policy is derived. The optimal stationary semi-randomized scheduling policy is also proposed. Finally, based on the obtained closed-form expression of average AoI, an age-based max-weight charging scheduling policy is proposed with Lyapunov optimization, which is proved to have a bounded approximation ratio.