The exact same thing that would happen today if some hacker ramped up production from a natural gas plant. Either the rest of the grid would respond by ramping down production or voltage/frequency would go out of whack.
Even the fastest gas plant presumably needs a few seconds to power up - enough time for the network to react. Additionaly, the maximum power and current it can deliver is generally known.
In contrast, the plan linked here basically proposes to build a giant distributed capacitor bank - with ever-growing capacity.
From what I understand, purpose-built large capacitor banks are used to generate very short pulses of enormously large currents - e.g. to power experimental lasers, railguns or similar.
Maybe my imagination runs wild here, but I see no reason a hacker with appropriate access couldn't do something similar - e.g. command all batteries to unload with the largest possible churrent at once and setup switches in such a way the current from all batteries is directed to a particular location.
I'm don't know what you could do with the resulting pulse but probably nothing good.