Aircraft air conditioning system and method of operating an aircraft air conditioning system

An aircraft air conditioning system including an ambient air supply line having a first end connected to an ambient air inlet and a second end connected to a mixer of the aircraft air conditioning system so as to supply ambient air to the mixer. A first compressor is arranged in the ambient air supply line and is configured to compress the ambient air flowing through the ambient air supply line. A bleed air supply line allows a flow of bleed air bled off from an engine or an auxiliary power unit therethrough. A bleed air turbine is driven by the bleed air flowing through the bleed air supply line and is coupled to the first compressor so as to drive the first compressor.