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Yes: IIRC some large generators work exactly like this, as the energized rotor gives a lot more flexibility in managing frequency and power output.
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Not just some, approximately all of them. It greatly complicates the logistics of a black start. † Of course that situation has additional complexity due to the need for substantial additional power in order for the various fuel supply systems to operate but I digress.

https://en.wikipedia.org/wiki/Black_start

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Generator excitation is not the hard part of a black start. You have to run coal feeders, blowers, and water pumps for an hour before you can spin the generator. Then you get power instantly upon applying power to the field windings.
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And even that's not the hard part of a black start. The plant control is relatively easy. What's hard is grid coordination.

All generation and consumption have to be almost perfectly balanced every second of every day. And the power company doesn't have good addressability of load. Worse when you restore power to an area all their stuff turns on in parallel giving an inrush that could be 3x or more the steady state.

A black start is a very drawn out process of bringing generation and load online in a balanced way and with wait times between load increases for stabilization.

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> You have to run coal feeders, blowers, and water pumps for an hour before you can spin the generator.

That's probably the reason most grid black start facilities in my country (Brazil) are hydroelectric dams, which need none of these.

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After watching a Munro video about it, I see your point. In the motor shown, the rotor gets its magnetic field simply by inducing a current and a field in it in reaction to the stator's field. There are no electromagnets in the rotor like I expected. In that case, I'm not sure either... I'd say more likely than not but it's complicated since the stator basically needs to induce a field and at the same time recover energy from the field that comes back from the rotor. I would further guess that the phase shift between the two components makes it possible to treat them separately.

Previous comment: Don't see why not - the "field" coils (the ones that replace the permanent magnets) need to be energized, which can initially come from the batteries if necessary.

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There are electromagnets in the rotor, it is directly energized.
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