Adiabatic Compressed Air Energy Storage (ACAES) is regarded as a promising, grid scale, medium-to-long duration energy storage technology. In ACAES, the air storage may be isochoric (constant volume) or isobaric (constant pressure). Isochoric storage, wherein the internal pressure cycles between an upper and lower limit as the system charges and discharges is mechanically simpler, however, it leads to undesirable thermodynamic consequences which are detrimental to the ACAES overall performance. Isobaric storage can be a valuable alternative: the storage volume varies to offset the pressure and temperature changes that would otherwise occur as air mass enters or leaves the high-pressure storage. In this paper we develop a thermodynamic model based on expected ACAES and existing CAES system features to compare the effects of isochoric and isobaric storage. Importantly, off-design compressor performance due to the sliding storage pressure is included by using a second degree polynomial fit for the isentropic compressor efficiency. For our modelled systems, the isobaric system round-trip efficiency (RTE) reaches 61.5%. The isochoric system achieves 57.8% even when no compressor off-design performance decrease is taken into account. This fact is associated to inherent losses due to throttling and mixing of heat stored at different temperatures. In our base-case scenario where the isentropic compressor efficiency varies between 55% and 85%, the isochoric system RTE is approximately 10% lower than the isobaric. These results indicate that isobaric storage for CAES is worth further development. We suggest that subsequent work investigate the exergy flows as well as the scalability challenges with isobaric storage mechanisms.