A power supply and cooling system for a floating structure having a tank, includes a supply circuit having at least one compression device, the supply circuit being configured to supply gas to a gas-consuming device, and a cooling circuit having a heat exchanger configured to participate in managing the internal pressure of the tank, the cooling circuit being connected to the supply circuit on either side of the compression device. The compression device includes two compression stages, and the power supply and cooling system includes a control device configured to connect the compression stages in series or in parallel.