I know that APUs are power plants for non-propulsion related purposes and that they are turned on before takeoff.
What powers an airplane prior to the APU being turned on?
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Other than the APU, there are multiple ways to provide electrical power to an aircraft:
Ground Power: Most (maybe all) airliners can accept a connection to a Ground Power Unit (GPU), which will supply AC electrical power to run all aircraft systems on the ground. These can be either mobile units (typically powered by Diesel):
or direct cable connections to the airport power grid:
(taken from this question)
Engine: This would be the method of last resort, but you can run an engine to provide electrical power via its generator. Note however that starting a jet engine would typically require bleed air from the APU or an Air Start Unit (ASU). Some turboprops can run an engine with a stopped propeller to provide power (Hotel mode). The engines also provide all electrical power during the flight, which is also used to re-charge the battery.
Running an APU on the apron is typically discouraged or restricted1 by the airport because of the noise and pollution. It does however provide one advantage over ground power: the APU also supplies bleed air to the air conditioning systems. Without it, air conditioning is only possible via a Pre-Conditioned Air (PCA) hose. See What is this tube connected to a 757 for? for details.
The APU is usually switched off after starting the main engines. It can however be used in the air on some aircraft. See In what conditions is the APU used in midair? for details.
1 Some airports have restrictions on when an aircraft is allowed to run the APU, e.g. Amsterdam:
The use of aircraft APU is forbidden in these stands in the period between 2 minutes after blocks for the arrivals and 5 minutes before off blocks for departure.
The APU will still be used to start the main engines during push-back.
Onboard batteries for DC and a ground power supply for AC.
The ground supply can come either from an airport vehicle or from the stand itself.
Since the standard AC in aviation is 115V and 400Hz, the usual ground power supply needs to be converted to a higher frequency. This used to be done with a rotating converter, i.e. a motor (at 110V and 60Hz or 220V and 50Hz) driving a generator (at 115V and 400Hz), but can probably be done with solid state electronics these days. As for why the 400Hz, there is a good answer here but it boils down to: higher frequencies can work with lighter transformers, but also radiate more, so 400Hz was chosen as a compromise.
Of course, on more advanced aircraft with weight to spare, one can find ways to convert DC into AC and viceversa. DC can be converted to AC using an inverter, while the oppoosite is achieved through a rectifier (typically the rectifier is only one component in something more elaborate like a switched-mode power supply, since by itself it would give a pretty choppy "DC")
Worked at an airport for 3 years. Aircraft were either powered by battery, GPU, jet bridge, or engines. But being powered by engines presented a problem, in that all the blast zones would be deadly and severely limit what work can be done on the aircraft, if any.
Generally the procedure was to get the ground power connected as soon as the aircraft was parked. Alternatively, the pilots could use their APU, but that costs them fuel, and it's less expensive for them to have us use our fuel (though I'm sure that cost is passed into the parking fee the airline pays for each gate.) Also, working on an aircraft that had an APU running was extra loud, even with hearing protection.