Sufficient and convenient fast-charging facilities are crucial for the effective integration of electric vehicles. To construct enough fast electric vehicle-charging stations, station owners need to earn a reasonable profit. This paper proposed an optimization framework for profit maximization, which determined the combined planning and operation of the charging station considering the vehicle arrival pattern, intermittent solar photovoltaic generation, and energy storage system management. In a planning horizon, the proposed optimization framework finds an optimal configuration of a grid-connected charging station. Besides, during the operation horizon, it determines an optimal power scheduling in the charging station. We formulated an optimization framework to maximize the expected profit of the station. Four types of costs were considered during the planning period: the investment cost, operational cost, maintenance cost, and penalties. The penalties arose from vehicle customers’ dissatisfaction associated with waiting time in queues and rejection by the station. The simulation results showed the optimal investment configuration and daily power scheduling in the charging station in various environments such as the downtown, highway, and public stations. Furthermore, it was shown that the optimal configuration was different according to the environments. In addition, the effectiveness of solar photovoltaic, energy storage system, and queue management was demonstrated in terms of the optimal solution through a sensitivity analysis.