This circuit consists of a pad mount transformer stepping down 34.5kV to 12kV, a 12kV switch-gear, and a secondary transformer stepping down 12kV to 400V and 690V. One fuse keeps blowing on a single phase of the 690V side every time a moderate load is put on it. ...
Is the 12kV to 690/400V transformer a wye-wye with a 3 leg core? If so, maybe the scenario below could be happening. If not, you can ignore the rest of this post.
With a Yg-Yg three-leg transformer, if there's a blown fuse on one of the line inputs then the missing phase will be regenerated on the secondary of the transformer, and the voltage on the missing phase will be close to the proper value if it's lightly loaded. But the voltage on that regenerated leg will start to drop as the transformer becomes more heavily loaded because of the higher impedance from that leg of the secondary winding. This is described in the paper at the link below.
The front end of the power converter using IGBTs that was described in the OP's post #11 will synchronously rectify the line inputs, keeping the DC bus at a voltage proportional to the L-L input voltages. If there is no output load connected to the DC bus, then there should be a relatively small amount of current drawn by the line inputs to the converter. But if there is some other significant AC load current on the regenerated output leg, then I'd expect the IGBTs connected to that leg to start supplying current to that load to maintain its proper voltage. This is because the conversion between AC and DC using PWM of IGBTs is bidirectional.
The two legs of the transformer that have primary fuses intact would supply the necessary rectified current to the DC bus for powering the DC-to-AC conversion on the regenerated phase to maintain its proper voltage while under load.
https://selinc.com/api/download/124320
And so you might want to check for a blown fuse on one of the 12kV lines to the transformer. Just measuring the voltage may not be adequate without further measures, because in the scenario above the voltage on the open leg of both the primary and secondary would be regenerated. This problem is mentioned in the paper linked above. If the power converter is disconnected or shut down from operating, and a relatively heavy load on the transformer is applied, then a voltage drop on regenerated phase should be detectable. Alternatively, if there's a significant load on the transformer, then you can see if having the converter operating actually increases the voltage on one of the phases vs. it not operating (this would be indicative of a regenerated phase).
An open phase on one of the transformer inputs might also lead to ferro-resonance happening, which might create transient voltages that could damage more sensitive equipment, as the OP has mentioned is occurring.