Adiabatic compressed air energy storage (A-CAES) has shown great application potentials in integrated hybrid energy systems (HES) in recent years. The integration requires A-CAES to store intermittent renewables power to meet the fluctuating load demand. Therefore, the operation of compressors and turbines must be adjusted by renewable power output and load demand, respectively. This paper presents an improved energy management operation strategy (I-EMOS) to enhance the A-CAES system's utilization for decentralized applications. In doing so, besides integrating thermal energy storage (TES) unit into CAES, several limitations of an A-CAES unit, such as its conversion process mode, dynamic characteristics, power input/output constraints of compressor/turbine train, air pressure constraint, and thermal storage capacity limitations are considered. An HES consisting of PV, A-CAES, and grid to answer the building's electric demand is simulated. The results are validated against data from an existing A-CAES pilot. It is concluded that adopting features and capacity limitations of the A-CAES system's power conversion unit in operational strategies contributes to improved management of energy flow in HES. It is observed that for certain air tank volume and maximum/minimum pressures under I-EMOS, an enhanced performance can be achieved. This analysis presents a feasibility assessment approach for decentralized A-CAES integrated with renewables implemented and tested in a real-world case study.