the vent solenoid just sits on top of the alternator on a bracket its not ran from it
normally a low voltage battery or bad earth (battery to body) will cause the PS pump to start slow then build up speed over 30sec till the alternator give it some juice
the alternator can be checked for charging voltage with a mulimeter it should be 14.2v when stone cold and around 13.7v when the engine is up to temperature with nothing on (heaters/lights/radio)
if its lower then 13v then that could indicate a battery or alternator charging problem