15V

15V is too high, 99% of the time this is NOT the VR at fault. Easy to test. EASY

1....You say battery is 13.1. Either the battery has a "surface charge" or your meter is wrong. Check your meter. Go to another vehicle known charged and check resting battery. Should be 12.6

2...There are two ways to determine what is wrong, sometimes both test methods help determine. Also, you must check GROUND circuit

2A......With key in "run" but engine stopped, stick one probe of your meter directly onto battery + post. Stick other probe onto the VR power (IGN) terminal. You should read a VERY low voltage the lower the better. Do this with all wiring normally connected. If you have a breaker points ignition, measure this with points CLOSED. To determine, measure coil+ voltage with key "in run." With points open, that reading will be "same as battery." With points closed, the reading will be much lower, 10-11V or a little less.

Anyhow the test reading, should be LOW less than say, .3V (3/10) of a volt

2A1........Second way to test this same situation is run engine at high idle with battery as closed to "normalized" as you can and WARMED UP. Check voltage from VR power/ IGN terminal to battery ground. If it shows about 14 (13.8--14.2) not below 13.5 and not above 14.4, VR is OK. Double check battery direct to ground and confirm that it is still HIGHER This shows VR is OK, voltage drop in the harness is cause

3.....CHECK GROUND Run engine at high idle, run this check first with all loads off, and again with lights, heater, etc running. Stab one probe of your meter into the battery NEG post. Stab other into the VR mounting flange. You should read almost nothing, again, the less the better, zero is perfect. ANY voltage here, .2V (2/10 or more) you need to improve grounding.