There are several ways to supply an excitation system and depending on how the supply is generated it can introduce unexpected system responses in different cases.
The most common supply-type related problem is the response to a bus fault or big load impacts such as direct online motor starting.
A common way to generate power dedicated to the excitation system is the use of a permanent magnet generator (PMG), it is fairly insensitive to bus disturbances and load changes. The PMG output is sensitive to its own load and as long as properly sized and associated with a high initial response AVR it is an acceptable solution. One major issue is to maintain those additional rotating parts.
Unfortunately, very badly supported, it can become extremely expensive to maintain as it gets older. Fortunately there are several solutions available and they do not involve any rotating equipment.
In most cases when using a high initial response AVR the step down transformer connected to the generator output is acceptable and with a proper forcing ratio it can withstand voltage drop below 50%. If a UPS is available why not use it to power your AVR, it will become absolutely insensitive to your generator load and it will cost only a bit of additional load on this supply.
The last solution when your expected disturbances generate short circuit conditions at the generator end with no auxiliary supply available is to use a boosting circuit. This is generally done with a set of current transformers placed on the neutral side of the generator and re-injecting the short circuit energy into the excitation system when necessary through a boosting device connected to the power input or output of the AVR. This is a bit more costly but at least using no rotating parts and well supported