The application of hydrogen (H2) energy in Microgrids (MGs) is suppressed by the limited energy conversion efficiency between H2 and electricity, even though the energy density of H2 outperforms the electricity storage. This paper proposed a multi-objective energy management system (EMS) to economically operate a hybrid H2-electricity system under the flexible electricity market, considering the energy loss in not only power-to-gas (P2G) and gas-to-power (G2P) conversion but also the energy consumption of H2 pressurization. Particle swarm optimization (PSO) is deployed to perform a day-ahead schedule for the electricity storage system and H2 storage system separately. The global optimal operation is achieved by the proposed method to allocate renewable energy and shift the cheap electricity. The performance of the proposed EMS is verified through numerical experiments based on real-world data. The influence of prediction methods on the proposed MOEMS is further tested and analyzed to improve the reproducibility of the approach. The results indicate that the proposed MOEMS could effectively schedule the operation of the hybrid storage system.