A turbomachine system includes a turbomachine provided with a turbomachine rotor. The turbomachine rotor is comprised of a turbomachine shaft with a first shaft end and a second shaft end. The turbomachine shaft is supported by active magnetic bearings for rotation in a turbomachine casing. The turbomachine system further includes a rotary machine drivingly coupled to the first shaft end, and a first closed cooling circuit adapted to circulate a cooling fluid therein and fluidly coupled to the active magnetic bearings to remove heat therefrom. The closed cooling circuit includes a cooling fluid impeller mounted on the turbomachine shaft for rotation therewith and adapted to circulate the cooling fluid in the closed cooling circuit. The closed cooling circuit further includes a heat exchanger adapted to remove heat from the cooling fluid. A method of operating a turbomachine system is further disclosed.