This paper proposes a novel user-preference-aware power scheduling scheme for application at an electric vehicle (EV) charging facility. Here, the preference of the EV user is characterized as a utility function by considering two different factors: 1) a satisfaction factor according to the charged energy, and 2) payment for the received charging service. As a key component of the power scheduling method, this paper proposes a two-stage power charging method and analyzes the advantage of the proposed method from a monetary perspective. In addition, this paper analyzes the economic benefits of a charging facility and EVs using the single-leader, multi-follower Stackelkberg game. By showing the existence of a unique best response for each participant, this paper presents the improvement of financial profit of EV users and charging facility through comparison with the case where the charging facility does not apply a proper power management scheme. Based on the actual world datasets, this paper shows that the proposed power scheduling scheme is feasible for actual environment. With satisfying the constraints, it is possible to reduce overall electricity cost up to 8.59% compared to the case without considering the peak power in EV charging facility.