In order to study the interaction between modular multilevel converter (MMC) and the complex power grid, an universal fast simulation model of MMC is proposed, which is generally applicable to various topologies. Firstly, by using controllable voltage source, ideal switch and diode to simulate the circuit structure of different topologies in various operating states, the universal simplified circuit for the arm is obtained, which can accurately simulate the dynamics characteristics and control bevavior of MMC. Secondly, by analytically calculating the dynamics averages of the arm after averaging the switching and modulation process of sub-modules, the dynamic solution of large-scale node admittance matrix can be avoided and the simulation efficiency can be significantly increased. Based on PSCAD/EMTDC, the accuracy and speed of the fast simulation model are verified. In the end, the key control strategies and DC protection of multi-terminal system are studied by using the MMC fast simulation model.