A turbo uses exhaust energy to drive it. The f1 turbos also have a motor generator which, when the turbo is being driven hard, harvests electricity for their DRS. When the turbo slows, and "turbo lag" becomes a factor, then electricity is used to drive the turbo to keep the speed up. This is a method of developing power but at the expense of fuel. Yes, fuel economy is a factor in f1, but power is first and foremost the aim (as Honda will vouch). Electric superchargers, on the other hand, will give you power (or economy) without burning petrol, however they are hugely expensive and massively energy consuming. A plant I worked on used them, but they were 3 phase, 415 volt beasts with phenomenal start up current.
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