Apart from Humming sound, we can found the following ways:-

1. OTI (Oil temperature indicator) shows the winding
temperature value
2.WTI  (Winding temperature indicator)shows the winding 
temperature value
 
but its not correct way because if we disconnect the TxR (after 
some hours) charging both may show current values 

1)by sensetion (vibration ) of transformer .
2)by sensing the temperature of transformer.

1.BY SENSEING THE VIBRATION IN TRANSFORMER.
2.BY SEEING SOME INDICATION LAMP R,Y,B IS IN CHARGE R NOT 
IN TRANSFORMER.
3.BY SEEING THE RTC PANEL BOARD.
4.BY SEEING OIL TEMP INDICATOR(ITS NOT CORRECT BUT IT 
POSSIBLE)

During Normal operation of tranformers,you should check 
both winding as well as oil temperature which should be 
below the maximum rated and check the breather condition it 
should be blue than pink. And there should not be any 
leakage of transformer oil on its reflectors.

If the transformer is working and if you go near the 
transformer you can sense the heat generated by the 
transformer even if no WTI or OTI installed on the 
transformer.