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Carbon Super Super Capacitor

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According to this: Graphene supercapacitor breaks storage record - physicsworld.com the energy density should be 85.6 Wh/kg at room temperature and 136 Wh/kg at 80 °C. Pretty amazing for a device you can make with a common dvd-burner!

That's what they already achieved, I think. They do have bigger plans:

"Our goal is to make a supercapacitor that stores as much energy as the best lithium-ion batteries (for the same weight) but which can still be recharged in less than two minutes," said Jang.

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A post about related research, combining graphene with carbon nanotubes ("quantum wires"):

quantum wires - Page 3
 
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Wouldn't the solution be to load up a supercapacitor at the supercharger station with 85kWh of energy and then when a Tesla Model S plugs in then the power is dumped from the charging station's supercapacitor into the car's supercapacitor?

So almost like a "battery swap" where you don't do a physical swap you just swap the power in one supercapacitor into another.

The charging station's supercapacitor has to be large enough to act as a buffer or cache so that if there are a lot of cars in a short amount of time they can all get a super super super fast charge.

This ofcourse. But you would need one hell of a contactor/plug for the energy transfer. Perhaps something that attaches to a large surface revealed underneath the car as it parks above the charging station, which is in turn buried under ground. That would be cool, esthetic and probably a lot safer than the station above ground and the contact on either side of the car. The ultracapacitor at the station could overdimensioned and charging at a constant rate all day long, say 200kW which would give it 4,8MWh per 24 hours. It would be a lot cheaper too to run the station since there would be an even draw, not high draw on-demand. Also it could charge itself more at night when power is cheaper, effectively acting as a nice buffer for the power-grid, maybe also selling power back during the day hence breaking even? Though, as ultracapacitors do become available at reasonable prices I guess the demand problems of today's power infrastructure will no longer be an issue.
 
Never. Superconduction wires can reduce power losses. But at a certain current density, they lose superconductivity all at once. See Quenching.
You have to keep the wires very cool and manage the current density with large safety margin to prevent from possible destructive quenching events. IOW big expense for little gains.