This paper presents a novel successive flow direction enforcing (SFDE) algorithm for solving power system operation models with flexible AC transmission system (FACTS) devices. The power flow control capabilities of FACTS devices can improve the utilization of the existing transmission network. However, the added computational complexity of modeling FACTS devices in power system operation models is one of the primary challenges for FACTS deployment. A prominent component is the nonlinearity introduced by variable-impedance FACTS devices to the linear DC power flow equations. According to previous studies, such nonlinearity can be tackled by preassigning the power flow directions on lines equipped with FACTS to restore the linearity of DC power flow equations. However, this method has the issue of converging to suboptimal solutions in some cases, due to suboptimal flow direction assignment. The algorithm presented in this paper can address the issue of suboptimality, while still achieving computational efficiency improvements. Power system operation models with FACTS are solved iteratively with flow directions enforced, and the predetermined flow directions are adjusted successively based on the results of the previous iteration. Simulation studies confirm the effectiveness of the method in converging to the globally optimal solution within a few iterations in almost all practical cases.