Is it possible for a transformer which is rated 12kV/480V to have the secondary voltage less than 480 (say 460) even if the primary is at 12kV?

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Absolutely.
Partially Shorted windings
So how do you know the primary matches the data plate?
Example. How do you know there isn’t 11,485 volts on the primary?
 
Is nameplate 12kV or 12.47 kV?

If it actually is 12.47 then inputting 12 is very near same ratio as 480:460 and sounds like what you should get.
 
Absolutely.
Partially Shorted windings
So how do you know the primary matches the data plate?
Example. How do you know there isn’t 11,485 volts on the primary?
I do not have all the information yet, but I was told the Powerlogic meter shows primary voltage between 12kV- 12.1 kV. And the secondary reading is around 460V.
 
I do not have all the information yet, but I was told the Powerlogic meter shows primary voltage between 12kV- 12.1 kV. And the secondary reading is around 460V.
Go into the Powerlogic meter setup and see what the PT information is.
 
First, 460 is well within a normal range of +/-5% and gives you still 5% wiggle room. You shouldn’t be having a problem.

Taps are typically -5/+5%, -2.5/+2.5%, 0%. So unless you are at the minimum there is plenty of room.

When you first power up with no load typically it’s a little high or gets tapped to match 480. Later on it should be checked again. But by then the power distribution contractor has long gone.

Under load that 4% drop is not unusual at all. Just be careful. If you tap up under load so you are now at +1% if you lose power and come back with no load that you don’t exceed +10%!! Often tap settings are a compromise.

Also don’t forget your local utility often taps down purposely in summer to reduce their kVA load at peak then taps back up in fall. But they won’t ever admit it. So if you tap up now you may find that’s a mistake in a couple months.

So if I were you I would wait. Check where you are at in a couple months and during a weekend, nights, and an outage. If it’s all good then by all means raise it. Just be smart about the decision. More than once I’ve had an engineer think they found gold then had to change it back.
 
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