What does the transformer test report indicate? (% excitation current).

I'm guessing the utility charges a power factor penalty. Consider power factor correction.

So is the high bill from the KWH, or from the demand?
The transformer should not use much energy for the load it serves.
Maybe the demand is from the equipment you have in service.

We had this issue after annual maintenance when we energised the transformer. There was nearly no load on the plant. We are aware of this issue of high MVA demand. At start up, we atart each big motor after 15 min interval until we are synchronised and start to export MW to the utility grid.

The on load tap changer has 14 positions. If after maintenance, the tap changer was at an extreme position and at the same time the transformer wa energised, can this be the cause of the high MVA import?

38.3A current when transformer is on no-load is high, considering rated current to be ~480A at 66kV. Could it be that the grid voltage is > 66kV or the transformer tap is below nominal (i.e. towards extreme tap-14) when this measurement was done?
In either case, the transformer is overfluxed causing it to draw higher magnetising current, i.e. drawl of high MVArs from the 66kV grid.
High MVArs (for a given MW) translates in to higher MVA demand.

You do also know that a high demand will stay on a bill for some time after the peak is set? I think normally a year.

Yes we are aware of this.

A utility I'm familiar with calculates average pf using the kWh and kVarh during the month. The percentage points below 95 are applied on top of the real 15 min. demand. To ensure the highest possible "billing demand" charge, load your service heavily for 15 minutes then let all transformers idle for the rest of the month.

What can be done to prevent this from happening again? Can we add overfluxing protection in transformer relay?