Motor vehicles with at least one electric motor as a drive motor must be charged at charging stations in order that they can store the energy necessary for driving the motor vehicle.  For this purpose, the motor vehicle has an electrical energy store and a connector, which can be connected directly or indirectly to a power supply station by means of a charging cable assembly.  In order to speed up the charging operation, fast charging stations are used as power supply stations, whereby strong electric currents are transmitted.  These electric currents sometimes cause the current conductor cables provided in the charging cable assembly to heat up.  In order to achieve small cable cross sections in this case, it is necessary for there to be a cooling device in the charging cable assembly. 
The prior art, such as for example EP 2 652 754 B1, which is incorporated by reference herein, DE 10 2011 100 389 A1, which is incorporated by reference herein, or U.S. Pat.  No.  9,321,362 B2, which is incorporated by reference herein, discloses in this respect charging cable assemblies in which at least one current conductor cable with a coolant line having a heat transfer surface to cool the current conductor cable is provided.  However, it has been found that with such assemblies a high cooling effort is required to limit excessive heating of the current conductor cables.