= Solution
For a generous illustrative roof area 100 m$^2$, annual rain depth one metre and five-metre usable fall, maximum <potential energy> is $\rho A h_{\mathrm{rain}}gH\simeq4.9$ MJ, only 1.36 kWh per year before conversion losses. Even 25 million roofs give only about 34 GWh annually, around $10^{-4}$ of a 300 TWh electricity system. Tiny intermittent flows and installation embodied <energy> make the practical result worse. Household roof-rain generation is therefore a very small carbon-saving resource under realistic domestic heights; the small gravitational drop and water flux explain why, without needing to assume an inefficient turbine.
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