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How does the volume of water going over a hydroelectric dam affect electricity production? Could the flux of the generators not be modulated to different amounts of electricity with the same flow of water? If so, what would the minimum flow be?

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The gravitational potential is transformed in electrical energy. If the height from the generators to the water level of the dam is $h$, the gravitational energy per time is $\rho Qh$, where $\rho$ in the specific weight of the water, and $Q$ the flow of the river that fills the reservoir, in volume per time. Of course there are losses, and that is the maximum available electrical power.

That is a steady state situation. In times of drought, the flow to the generators can be greater than the river flow, and the level decreases. In times of heavy rain and with the dam full, part of the water can bypass the generators.