fencing under electrical transmission line
Duke Energy has employees whose job is specifically to work on easement issues.  I received a call back from them after a few days.  They talked to me about my fence project and sent a PDF with some general guidance, as well as aerial photos of the property with the easement and lot lines marked.

No written application was required; fence project approved over the phone
16-foot gate required at each end of easement for Duke vehicle access; Duke will supply locks which I could connect to my own locks if desired, so they can access the easement without bothering me or cutting anything
Maximum fence height 8 feet (no problem for me)
No limit on materials, e.g. metal fence would be fine
No special grounding requirements

The Duke rep recommended agricultural gates which are about $200ea.