I just discovered this thread! Lots of useful information in here. Thanks to all who have contributed.
I have an early 2013 S85. Firmware version 2020.24.6.11. I've been having this humming noise for quite a while now. I didn't think to keep a record of it, but I think it has been going on for nearly a year or so. The humming will go on for many hours. And I've been having the same drop in RM as others --- I'll often see a loss of 10 - 13 miles overnight when plugged in. I typically charge my S to 90%. For a while, I was charging to 80%.
In mid 2019, I started noticing errors in the calculation of "time remaining to charge".
Model S: Charge Time Remaining calculation is now incorrect I went to the SC. Before I took the car in to the SC, they did remote diagnostics and told me that my battery might need to be replaced. After taking the car in, I was told the battery simply needed balancing. While I didn't believe that was the root cause, I went through a number of cycles to rebalance the battery. Nothing has changed regarding this time-remaining-to-charge-calculation.
I wonder if this calculation issue is related in any way to the humming sound. They both started at about the same time.