A turbocharging system comprising: a first compressor and a first turbine; a second compressor and a second turbine which are coaxially connected to each other; an intake line through which intake air pressurized by the first compressor is supplied to the second compressor; an exhaust line through which exhaust gas which has driven the second turbine is supplied to the first turbine; an electric motor which is connected to a shaft end of the first compressor; a generator which is connected to a shaft end of the first turbine; a bypass exhaust line through which the exhaust gas bypasses the first turbine; wherein an exhaust on-off valve is provided in the bypass exhaust line, characterized in a cooler (51; 71; 81; 91; 101; 111; 121; 131; 141) which cools the generator; and a control device which is configured to operate the cooler (51; 71; 81; 91; 101; 111; 121; 131; 141) when the generator is driven by the exhaust gas which has driven the first turbine, the cooler (111; 121; 131; 141) includes a cooling line (L11; L13; L15; L18) through which a cooling medium is introduced to the generator so as to cool the generator and a cooling on-off valve (112; 122; 132; 142) which is provided in the cooling line (L11; L13; L15; L18), the control device is configured to close the cooling on-off valve (112; 122; 132; 142) to stop an operation of the cooler (51;71;81;91;101;111;121;131;141) when opening the exhaust on-off valve, and the control device is configured to open the cooling on-off valve (112; 122; 132; 142) to operate the cooler (51;71;81;91;101;111;121;131;141) when closing the exhaust on-off valve to drive the generator. a first compressor and a first turbine; a second compressor and a second turbine which are coaxially connected to each other; an intake line through which intake air pressurized by the first compressor is supplied to the second compressor; an exhaust line through which exhaust gas which has driven the second turbine is supplied to the first turbine; an electric motor which is connected to a shaft end of the first compressor; a generator which is connected to a shaft end of the first turbine; a bypass exhaust line through which the exhaust gas bypasses the first turbine; wherein an exhaust on-off valve is provided in the bypass exhaust line, characterized in a cooler (51; 71; 81; 91; 101; 111; 121; 131; 141) which cools the generator; and a control device which is configured to operate the cooler (51; 71; 81; 91; 101; 111; 121; 131; 141) when the generator is driven by the exhaust gas which has driven the first turbine, the cooler (111; 121; 131; 141) includes a cooling line (L11; L13; L15; L18) through which a cooling medium is introduced to the generator so as to cool the generator and a cooling on-off valve (112; 122; 132; 142) which is provided in the cooling line (L11; L13; L15; L18), the control device is configured to close the cooling on-off valve (112; 122; 132; 142) to stop an operation of the cooler (51;71;81;91;101;111;121;131;141) when opening the exhaust on-off valve, and the control device is configured to open the cooling on-off valve (112; 122; 132; 142) to operate the cooler (51;71;81;91;101;111;121;131;141) when closing the exhaust on-off valve to drive the generator.