There is an eccentric inventor on YouTube named Robert Murray-Smith, who has been working towards the megafarad capacitor using graphene oxide, and has recently made progress (without much funding).
Certainly, the best use of coal is not to burn it - but to turn it into graphene (and CNT, Bucky-balls, filaments, etc.). Mass-produced graphene oxide megacapacitors should be cheaper than any other imaginable energy storage medium like lithium batteries, since the main raw material is coal. Murray-Smith figures that the average house, and the power needs of two adults, could be handled by 2-3 megafarad caps, about the size of a single regular car battery. They could be recharged off-peak in a few minutes, or slowly with solar. The current Tesla automobile would need about 10 of them and could be charged in less time than the home, since the limitation on the home is the puny grid connection (assuming a charging station has a high power charger) I agree with Robin that the ADGEX and possibly the STEORN are using an advanced capacitor system, which could be made from graphene oxide or not. Their capacitors are probably in the kilo-farad range but megafarad will happen soon. There is probably an antenna in these units for capturing Wi-Fi as well. RCA developed this tech 5 years ago. https://www.youtube.com/watch?v=1wOJEIMf12I&html5=1 It is very possible that an additional self-charging anomaly has been discovered as one get capacitance levels up to the kilofarad range. This does not necessarily violate CoE, so long as the device cools as it recharges. This magneto-cooling phenomenon was seen with the Arthur Manelas device. Jones

