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To decrease fuel-based energy consumption, it is important to investigate the optimal energy management problem for the seaport integrated energy system in a fully distributed manner. A multi-objective energy management model is constructed, considering energy consumption, greenhouse gas emission, and carbon trading, which satisfy the sustainable development of the international shipping industry advocated by the International Maritime Organization (IMO). The carbon exhaust of a seaport is restrained by integrated carbon capture/storage devices. A fully distributed energy management strategy with dynamic-weighted coefficients is proposed to acquire the optimal solutions for the seaport integrated energy system. Moreover, to decrease the communication resources, an event-triggered mechanism between energy bodies was designed against the bandwidth limitation of the seaport. Finally, a rigorous mathematical analysis based on the multi-agent theory and the case studies proves the effectiveness of the proposed method.