OK, now that we have established a portion of the story, here are some more issues to deal with.
Using a 3600RPM motor and driving it at 6600RPM with a VFD poses an entire new set of problems. That is 83% overspeed for that motor design (110Hz), which means that you will have significantly less torque available at 6600RPM than you did at 3600RPM. If you don't understand that concept, learn about over speeding AC induction motors with a VFD. Briefly, once you go over base speed you are in constant HP mode because the voltage can no longer increase along with frequency since you will be at max voltage when you get to 60Hz. The torque will then drop rapidly as you overspeed, and in fact, you will have slightly less than 1/2 of the motor's torque at that speed (taken from a chart, not calculated). There are tricks to overcome this, but it is all in the motor design, so hopefully someone making this proposal to you has that under consideration.
That brings up another question. HP is a function of speed and torque. You cannot possibly need the same HP motor at 2 different speeds like that, so somewhere that issue must be dealt with. If the compressor needs 6000HP @ 6600RPM, you must take that as your primary requirement. Therefore, considering the above VFD issue, to be sure to get that torque at that speed you will need a 12,500HP motor and VFD!
One trick might be to use a 6000HP 2300V motor and feed it with the 4160V drive so that you can continue the voltage increase without losing torque over base speed, but that would be something to be discussed at length with the motor supplier first. The motor would be bigger and might cost more than a 12,500HP 4000V, but either way it will be a custom motor. The VFD however would be smaller and less expensive at 6000HP 4160V.
The gearbox and lower speed issue has some considerations as well. 6000HP @ 6600RPM = 4772 ft. lbs. of torque at that speed. At 1200RPM, a 6000HP motor provides 26,250 ft.-lbs. of torque, and when geared up at 5.5:1 to get to 6600RPM, you will have the required 4772 ft.-lbs. in theory, except for the losses. Gearbox losses are difficult to pin down, but a "rule of thumb" is to estimate 2% losses through each mesh. So a simple input/idler/output gear set would represent 4% losses. At 6000HP that is 240HP, which leaves you with only 5760HP or 4581 ft.-lbs. of torque at the compressor shaft, significant enough to be considered in the prime mover. You may need to go to 6500HP to be sure.
Using a VFD of course essentially eliminates any discussion of power factor, because the power factor presented by the VFD to the line will be corrected to about .95 anyway. In also introduces harmonics as an issue, and at 12,000HP that can be a BIG issue.
The use of a synchronous motor can also mean running at unity or even over unity PF to correct for other induction loads (if any), so that is a good option from that standpoint.
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