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How many KW do Tesla's solar superchargers generate?

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It depends on where you are. I'd recommend using PVWatts Calculator to get a fact-based estimate.

PV Watts is a great tool to get detailed, more precise estimates, but 4.5 hours per average day for coastal California is a pretty good estimate. For the sunny southwest, it's more like 5 hours per average day. Those are averaged over a year. You have to also add a typical 80% efficiency factor going from DC at the panels to AC to the grid.

Cottonwood Grove Annual Comparison shows the logged results of the 9.9 kW DC array at my house in Boulder. It has good southern exposure and is mounted on a south facing 8/12 (34˚) roof. I have had grid tied solar about as long as I have had a Tesla, and since I have owned a Tesla, I have produced far more electricity than my Teslas consume. The grid is a wonderful battery!

Let's look at the average hours of production per day for my Boulder array, averaged over 6 years.

14,182kWh/365.25days/*(9.9kW*80%) = 4.9 hours/day​


I would use 250 Watts for a more typical production PV Panel today.

100*250W*4.5hr*80% = 90 kWh/day​

All those panels at Hawthorne produce about 2 or 3 charges a day... If Tesla is really going to produce enough solar to feed the Supercharger network, they will need some large solar farms in other locations.
 
Here in the Central Valley of California, our modest 2,590-watt system generated 12,722 kWh through our third anniversary date of installation. This averages out to 4,240 kWh/year. this gives an average daily rate of 5.6 kWh from July '11 to July '14.