The demand for locally emission-free mobility is today higher than ever. Beside the battery electric vehicles, the fuel cell vehicle is a way to realize such a mobility. In particular, the low-temperature fuel cell with polymer electrolyte membrane is suitable for use in the vehicle due to the operating temperatures, the solid electrolyte, the non-toxic reactants and the reaction product water. In this work, a polymer electrolyte membrane fuel cell has been modeled. For this purpose, the known basic equations were extended or modified to account for the effects of air pressure, mass flow and temperature. By a curve fitting, the model of the fuel cell test bench of the Vienna University of Technology could be reproduced with satisfactory accuracy. The model equations were then implemented in a model of a fuel cell system newly developed within the simulation software GT-Suite framework. The system consists of the cell voltage calculation minus the voltage losses, a compressor air supply system that provides the air pressure and the mass flow and a thermal mass of the fuel cell for the temperature control. As part of an existing vehicle model, investigations and simulations have been performed for the extended fuel cell system in terms of maximum fuel cell power, hydrogen consumption, and vehicle range. The maximum fuel cell power has been determined for low as well as for high air pressure (two possible positions of the throttle valve). The calculated power was, according to the pressure dependence of the fuel cell model, also higher for high air pressure. In addition, the temperature dependence of the model was used for the evaluation of an electric heating system for the cold start. An improved energy balance due to the faster higher efficiency of the fuel cell has not been observed. The influence of different battery capacities on the fuel cell performance has also been investigated. Hydrogen consumption decreased with decreasing battery capacity due to shorter charging periods, as a battery with smaller capacity can reach a higher state of charge faster. As a result, the phases with high fuel cell performance, and hence hydrogen consumption, were shorter. A smaller battery, however, is more heavily strained and therefore more limited in its durability.