CHP-MG is important for sustainable energy development and building a green and low-carbon society. The complex coupling relationship between internal energy structure and equipment also brings challenges to its operational optimization. In this paper, the thermoelectric decoupling of cogeneration equipment is realized by using energy storage devices on the supply side, utilizing the mutual complementary relationship of electricity and heat energy on both sides of supply and demand, and the multi-energy supply capacity of the system is enhanced by various energy conversion devices. The load types are classified on the demand side, and the elasticity of electric load and the diversity of heating modes of the system are utilized. A comprehensive energy demand response model including time-shift of electric load, reduction response and conversion response of heating mode is built, and a response compensation mechanism is proposed. On this basis, taking the minimum sum of system operation cost and response compensation cost as the objective, and accounting for the constraints of equipment operation and dispatchable load resources on both sides of supply and demand, a mathematical model of optimal operation of CHP-MG based on multi-energy complementarity is established. Finally, the results and comparisons of an example show that the bilateral collaborative optimization of supply and demand considering multi-energy complementarity can effectively improve the flexibility of energy supply and operational economy of the system. This work is supported by National Natural Science Foundation of China (No. 51607105).