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Without specs or details on implementation, I assume Relay provides for slow adjustments with tap change and lifetime limit on current load switches is unknown.
This is common for some high power (MVA) distribution or feeder transformers to have active load switch taps with relay control.
Servo implies stepper motor controlled variac which also has lifetime limit on wiper motion under load current but possibly has smoother voltage regulation. This would likely be more expensive due to variac $/VA
efficiency depends on load loss and regulator loss, so no idea without specs.
the common use of stabilizers at end-user is due to infrastructure cannot support load variation. So in North America and EU, these are rarely used.
SunnySkyguy ,I am retired now (14 years!), but its a fact that all analogue VHF and UHF TV transmitters have to use what we call AVRs ( Automatic Voltage Regulators) or it seems others call "stabilizers. This is because of the requirement to keep the HV +- .25% at a load of up to 250 kW. This is because the thermionic device has to be used over an extremely non linear part of its transfer characteristics so the video has to be pre-corrected for this non linearity.
Now the UK has gone digital, at least 400 AVRs have been dumped.
Do you still have analogue TV in the States?, if you do you will still find AVRs.
So an air con unit should handle +-10% on its nominal mains, BUT requires the ability to handle high inrush currents when the compressor starts. So the easiest way to go is a relay powered tap changer with a buck/boost transformer and a bit of logic and voltage detection to get it working. You don't want a servo type far too sophisticated for an air con unit.