for a 20kw generator one most likely could get by with a #4 copper or a #2 aluminum wire, longer distances up a size, for the line from the generator to the transfer switch, they only can produce about 80 amps per leg,
and I would guess most of the time one would only be running 30 or less amps per leg,
mount the transfer switch on the meter pole or near, and the large wire from the meter to the transfer switch would be very short,
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and the smaller the generator the smaller the wire need, on a 10 kw generator if the wire length was short a #10 would carry the 25-30 amps the generator can produce,
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the wire chart in the quote IMO is wrong when it comes to 12 volt capacity, I am not saying that auto 12 volt wiring is under sized in most instances as it carries in such short bursts of power, thus able to mataine the load for the duration, but for continuous loads the chart IMO is not accurate, for example you feed a 100 amps in a #10 wire for any length of time your insulation is going to melt, try it some time if you do not believe me, amps is amps, the voltage of the line has little to do with how many amps the wire can carry, (the amount of power it can carry is a combination of volts and amps), the higher the voltage the more power is carried for a given amp load, that is why transmission lines have high voltage, they can move a lot of power on a fairly small wire, and then a transformer drops the voltage down, and the total amount of power would now need a very large wire to carry the same power,
the KW rating is the sum of that power, where voltage and amps is the mix that makes up the KW rating, (kW Times 1000) divided by Volts =amps