shemoneh esrei - Shalom Rav / Sim Shalom - by Mincha according to Nusach Ashkenaz

Nusach Ashkenaz says Shalom Rav at Mincha most days. However, when there is Birchas Kohanim (such as a fast day), they say Sim Shalom. Is there a connection between Birchas Kohanim and Sim Shalom? If there is a connection, what is it?