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Each lighting section would have a backup generator that can last the rest of the race. I see your point regarding safety. Since the FIA is the sanctioning body for F1, their ultimate concern is safety during any race. They have quite a few issues to work out before night racing commences, and they still need to secure a track to hold the night race.

Here's an idea:

LED headlighting can provide sufficient forward lighting for night racing and can be integrated into the front wing assembly without much modification and without aerodynamic penalty (huge + if they are allowed to be enclosed and integrated). They would have to be setup in an array with either a magnifying glass or projector, or both. As is typical in F1, think small and light.