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The difference is way less than you are making out.
Be the water storage a built lagoon or river basin the concept is how you make out.
So you would let the Thames fill during the rising tide, then just before falling tide shut the barrier, then once the tide has receded far enough open the flow through the generators. Open the rest of the barriers to equalise then rinse and repeat for the opposing tide.
It’s the same as the lagoon, just instead of man making the entire thing, you block the mouth of the estuary.
And yes, the Thames is not as drastic of tide height, but a large proportion of it is prebuilt… perfect for a proof of concept.
I would call it a tidal buffer.