Methods, devices and systems capable of performing a hot restart of a compressor driven by a turbine are provided. The turbine has a high speed shaft and a low speed shaft connected to the compressor. A base slow roll speed at which a speed of the low speed shaft is in a slow roll range is maintained at the high speed shaft for a predetermined time. Then, when a vibration amplitude of the compressor is larger than a vibration limit, the maintaining is repeated. Otherwise, a set speed of the high speed shaft is increased, until a current speed of the low speed shaft reaches a minimum operating speed. If, while increasing the set speed, the vibration amplitude becomes larger than the vibration limit, the maintaining is repeated.