Flexible multi-state switches (FMSSs), also called soft open point (SOP), are regarded as the core devices in flexible interconnected distribution network. However, due to their more complex control structure and nonlinear characteristics, there is a tight and complex coupling relationship between the transmission power limit and stability of two (or more) converter sides, which brings difficulties to the planning and power flow dispatching control of flexible interconnected distribution networks. In this paper, taking the back-to-back converter based FMSS (BTBC-FMSS) as the research object, the dynamic equivalent circuit connected to the distribution network is constructed, and the transmission active and reactive power expression of the converter on both sides considering PLL dynamics is deduced. The variation rules of transmission power limits on both sides of the device with various structures and control parameters are summarized. Furthermore, the stable operation domain of the BTBC-FMSS is constructed, and the effective measures to improve the transmission power stability of the device are given according to the analysis of the stable operation domain. Finally, a simulation model is built in Matlab/Simulink to verify the effectiveness of the analysis results.