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WTF? Unless you're building "mine is taller than yours" luxury spires for the uber-rich, the structural and economic inefficiencies of ultra-tall buildings are ridiculous. Use a bigger weight and far-shorter building. Or maybe better - a vertical shift dug in hard rock, some place where the low water table makes that cheap.
Adding to this battery fires are glorious. A skyscraper battery fire would end up in the history books. Such a massive storage system must not be anywhere near people and modules must be engineered as compartments to burn themselves out by design so that firefighters need not risk their lives. I like your idea of a vertical shaft. Fire suppression chemicals can at least move downward fast and could be engineered so that gravity takes it where it needs to go in the correct proportions and flows.
Um...I don't think this design includes chemical batteries, for such fires to be an issue:

> ...design for a skyscraper that would use a motor powered by electricity from the grid to elevate giant blocks when energy demand is low. These blocks would store the electricity as “potential” energy. When there is demand, the blocks would be lowered, releasing the energy, which would be converted into electricity.

Though it could still be very bad if there was a cable, pulley, clutch, or brake failure.

I assumed I guess incorrectly they were storing the energy from the blocks in batteries otherwise once cables stop everyone loses the added power or power offset. Doing this without some type of capacitive storage suggests in my mind a Rube Goldberg machine of sorts. I guess I will wait to see what this complex beast looks like.