Constant power cold start mode of proton exchange membrane (PEM) fuel cell is essential in practical applications, however, unlike the constant current and voltage cold start modes, the constant power mode was largely ignored in previous fundamental modeling and experimental studies. In this study, a PEM fuel cell stack cold start model for analyzing the constant power cold start process is developed, and the fundamental differences among the various start-up modes are elucidated. In the constant power cold start mode, the start-up process may fail before the ice fully covers the cathode catalyst layers (CLs), because the stack may not be able to supply the required power output, which is different from the constant current and voltage start-up modes. The initial water content and start-up temperature could limit the power output significantly. The constant power can be controlled at higher current (CPHC) or lower current (CPLC). In the CPHC mode, the current density decreases and the stack voltage increases during the cold start process, while it is reversed in the CPLC mode. Generally, the CPLC mode produces less heat during cold start process than the other start-up modes, because the current density is often kept at low level. (C) 2014 Elsevier Ltd. All rights reserved.