Renewable energy sources such as wind and solar power exhibit intermittency and volatility due to weather conditions, which can compromise the reliable operation of multi-energy complementary systems. Hydrogen energy, as a high-quality secondary energy source, offers advantages of being green, pollution-free, and possessing high energy density. To address the uncertainty in new energy output, this paper constructs a multi-energy complementary cogeneration system model. This system integrates a thermal power unit, wind turbines, photovoltaic generators, an electric boiler, and a hydrogen storage system, incorporating waste heat recovery to enhance system flexibility and energy utilization efficiency. Based on this, an optimization scheduling model is established with the dual objectives of minimizing total operating costs and reducing carbon emissions. For this model, an improved multi-objective simulated annealing particle swarm optimization algorithm is proposed, effectively accelerating convergence and preventing local optima. Simulation analysis using a case study from a region in Shandong province demonstrates that the proposed method reduces the system’s total operating costs by an average of 12.51% and carbon emissions by 5.53%, validating the feasibility and superiority of the developed model and algorithm.