The community integrated energy system (CIES) can provide users with reliable and economical energy supply through centralized utilization of various energy equipment. However, due to the significant difference in inertia between different energy sources, the response speed of equipment using different energy sources varies greatly, making it difficult to cooperate well and limiting their ability to cope with source-load uncertainty. Based on this consideration, a two-stage optimal operation strategy for CIES is proposed to utilize dynamic and static energy separately. First, the dynamic and static energy are defined and divided, and their different functions are analyzed. Based on this, a two-stage optimal operation architecture is proposed. Then, a day-ahead interval optimal operation model considering dynamic and static energy matching is established. In which, the uncertainty of source and load is characterized by nonlinear interval numbers, and the interval possibility degree is used for deterministic transformation of nonlinear interval numbers. Thirdly, a dynamic and static energy replacement mechanism based on model predictive control is given, and an intra-day rolling optimal operation model considering dynamic and static energy replacement is established. Finally, a CIES is used as an example to demonstrate the rationality and effectiveness of the proposed strategy. Simulation results show that the proposed strategy can effectively utilize different types of energy equipment and significantly reduce the adverse effects of source-load uncertainty.