S type volts too low?

My 05 S Type idling before run volts are 15.5v, after good run volts at battery are 13.5v, alternator problem?

Are you seeing any electrical warnings? On most cars that’s natural. The alternator varies output to match demand. After startup, the battery has been drained significantly to run the starter so the alternator increases output to recharge it. Once the demand is filled, it backs off and only supplies what is needed.

Unless the voltage overall falls into the 12v range I would not worry about it, especially if there are no warnings and no signs that the battery is not being charged properly.

Dave