HT Power Transformers

Then load only 5 MVA and 3.5 MVA only.

Check for harmonics.Also check for cooling method, Natural or forced cooling.

Harmonics caused by new inverter loads, such as VFDs are likely a reason that a previously cool transformer becomes hot with a similar ampere load.

Also, the designed primary voltage needs to be held close to original design. It is possible that when installed, it may have been repurposed from some other system voltage, perhaps a bit lower, but now increasing load adding to temperature rise. You can test this with the unit unloaded, check the turns ratio.

Also, if liquid cooled, I have found where some cooling fin banks show no temperature rise, while others on the other side of the tank do. 30 year old transformers, I assumed that sludge blocked the entry of oil into the cooling headers at the bottom of the transformer. These transformers were very lightly loaded for a long time.

Low oil level reduces the heat capacity of the cooling equipment.

The temperature rise is probably the only way to determine the rating of a transformer. But what are their specs for temp. rise? The following link may help determine it:

https://www.copper.org/environment/sustainable-energy/transformers/education/trans_efficiency.html

This may be a long shot, but....

Are you multiplying the line current by 1.732 when calculating MVA? The capacity difference you report is similar to using line to line voltage and line current to calculate MVA. The rated secondary line current of your 10MVA unit is 1750 amperes. However, if you assumed it was 3000 amperes, your transformer would start overloading at 57% load, or 5.77 MVA.

There are standards specifying the methods for various tests to be conducted to test the performance of transformers (IEC standards). You may like to perform open circuit and short circuit heat runs at various currents to get maximum permissible temperature rise ( assuming that you know the class of insulation of the winding and the rated voltage of the windings, cooling parameters i.e. amount of cooling water flow etc. I hope you are not assuming these conditions in absence of rating plate).

Which temperature is limiting the rise of load to your assumed rated load? Is it winding temperature of core temperature? This information is important. A loose core might have resulted in core hot spots. You may like to compare the temperature rise of active parts with older operation logs, if available.

(Usual disclaimer re. ancient knowledge, etc.) To deal with the symptoms, on a very old transformer I would check for sludge limiting oil flow as someone else suggested. You can do a sludge test if I recall correctly - think it is called an acidity test. Not sure what tests one will do to check actual electrical performance ratings.

Slightly off-topic, I would also do a furan (furaldehyde?) analysis - this checks the condition of the paper insulation without the intrusiveness of taking an actual paper sample for a DP test - as acidity degrades the insulation. In fact, on such old transformers I took samples annually and did the full barrage of tests, including DGA (dissolved gas analysis). It allows for trend analysis and early warning of brooding trouble.