Micro-energy grids have shown superiorities over traditional electricity and heating management systems. This paper presents a hybrid optimization strategy for micro-energy grid dispatch with three salient features. First, to enhance the ability to support new storage equipment, an energy hub model is proposed using the non-supplementary fired compressed air energy storage (NSF-CAES). This provides flexible dispatch for cooling, heating and electricity. Second, considering the unique characteristics of the NSF-CAES, a sliding time window (STW) method is designed for simple but effective energy dispatch. Third, for the optimization of energy dispatch, we blend the differential evolution (DE) with the hyper-spherical search (HSS) to formulate a hybrid DE-HSS algorithm, which enhances the global search ability and accuracy. Comparative case studies are performed using real data of scenarios to demonstrate the superiorities of the proposed scheme.