I want to note that in this system, where the heat is stored to reheat the gas at discharge time, you want to produce heat during compression. Doing adiabatic compression means you are storing more heat, and therefore can produce more energy at discharge time, than you could with isothermal compression (for a given storage volume and pressure.)

This is unlike traditional CAES, where the air is reheated by burning a fuel. There, fuel use is minimized by isothermal compression.