A turbomachine system (1; 101) comprising: at least one turbomachine (3; 103) comprising a turbomachine rotor (7; 113) having a turbomachine shaft (9; 115) with a first shaft end (9.1; 115.1) and a second shaft end (9.2; 115.2), the turbomachine shaft (9; 115) being supported by active magnetic bearings (13, 15, 17; 117, 119, 121) for rotation in a turbomachine casing (11; 112), and the first shaft end (9.1; 115.1) extending outside the turbomachine casing (11; 112); characterized by at least one rotary machine (5; 105) drivingly coupled with the first shaft end (9.1; 115.1) of the turbomachine (3; 103); wherein the rotary machine (5; 105) comprises a second casing (6; 106), separated from the turbomachine casing (11; 112); and wherein a shaft line (40; 104) extending between the turbomachine casing (11; 112) and the second casing (6; 106) drivingly connects the turbomachine (3; 103) to the rotary machine (5; 105); a first closed cooling circuit (56; 133) adapted to circulate a cooling fluid therein and fluidly coupled to at least one of the active magnetic bearings (13, 15, 17; 117, 119, 121), which drivingly support the turbomachine shaft (9; 115) in the turbomachine casing (11; 112), to remove heat therefrom; wherein the first closed cooling circuit (56; 133) comprises a cooling fluid impeller (57; 131) mounted on the turbomachine shaft (9; 115) for rotation therewith and adapted to circulate the cooling fluid in the first closed cooling circuit (56; 133); and wherein the first closed cooling circuit (56; 133) further comprises at least one heat exchanger (59; 135) adapted to remove heat from the cooling fluid. at least one turbomachine (3; 103) comprising a turbomachine rotor (7; 113) having a turbomachine shaft (9; 115) with a first shaft end (9.1; 115.1) and a second shaft end (9.2; 115.2), the turbomachine shaft (9; 115) being supported by active magnetic bearings (13, 15, 17; 117, 119, 121) for rotation in a turbomachine casing (11; 112), and the first shaft end (9.1; 115.1) extending outside the turbomachine casing (11; 112); characterized by at least one rotary machine (5; 105) drivingly coupled with the first shaft end (9.1; 115.1) of the turbomachine (3; 103); wherein the rotary machine (5; 105) comprises a second casing (6; 106), separated from the turbomachine casing (11; 112); and wherein a shaft line (40; 104) extending between the turbomachine casing (11; 112) and the second casing (6; 106) drivingly connects the turbomachine (3; 103) to the rotary machine (5; 105); a first closed cooling circuit (56; 133) adapted to circulate a cooling fluid therein and fluidly coupled to at least one of the active magnetic bearings (13, 15, 17; 117, 119, 121), which drivingly support the turbomachine shaft (9; 115) in the turbomachine casing (11; 112), to remove heat therefrom; wherein the first closed cooling circuit (56; 133) comprises a cooling fluid impeller (57; 131) mounted on the turbomachine shaft (9; 115) for rotation therewith and adapted to circulate the cooling fluid in the first closed cooling circuit (56; 133); and wherein the first closed cooling circuit (56; 133) further comprises at least one heat exchanger (59; 135) adapted to remove heat from the cooling fluid.