How this calculator works
A hydroelectric plant converts the potential energy of water held above the turbine into electricity. Every cubic metre of water that falls through a head of H metres releases 1,000 x 9.80665 x H joules, and the flow in cubic metres per second turns that into power. A turbine and generator capture most of it: large Francis and Pelton machines reach 90 to 95 percent, while small or old installations are lower. A tidal or in-stream turbine works differently, using the kinetic energy of moving water, which rises with the cube of the current speed: doubling the current gives eight times the available power. The efficiency there is lower because the turbine cannot slow the whole current, and the theoretical Betz limit of 0.593 applies as it does to wind. Annual energy is the power multiplied by the operating hours; most plants run more than 350 days because water flows even when the demand is low.
Formula
dam: power (W) = density x gravity x flow (m3/s) x head (m) x efficiency. tidal: power = 0.5 x density x swept area x current speed^3 x efficiency. channel flow (m3/s) = cross-sectional area x velocity. annual energy (kWh) = power (W) / 1000 x 24 x operating days. revenue = annual energy x tariff.
Worked example
A small dam with a 20 m head, 5 m3/s of flow and 90 percent efficiency produces 1,000 x 9.80665 x 5 x 20 x 0.9 = 882,599 W, or about 883 kW. Running 350 days a year it generates 883 x 24 x 350 = 7,417,000 kWh, or 7.4 GWh, worth $593,000 at $0.08 per kWh. A tidal turbine with a 100 m2 swept area in a 2.5 m/s current produces 0.5 x 1,000 x 100 x 15.625 x 0.9 = 703 kW.
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Frequently asked questions
What is a realistic efficiency?
Large hydro turbines typically convert 90 to 95 percent of the water power into electricity. Small micro-hydro units are often 70 to 85 percent, and a tidal turbine in an open current is lower still because much of the water flows around the rotor. Enter the figure from your equipment, or use 90 percent as a large-plant default.
Why does the tidal formula use the cube of speed?
Kinetic energy rises with the square of velocity and the mass of water passing per second also rises with velocity, so the power in the current scales with velocity cubed. A site with a 3 m/s current has seven times the power of one with 1.5 m/s.
Does the calculator include transmission losses?
No. The efficiency you enter covers the turbine and generator. Transformer and transmission losses, typically a few percent, and the electricity the plant itself uses are on top. Include them by lowering the efficiency slightly.
Can this size a run-of-river project?
Yes, if you know the available head and the diverted flow. Run-of-river schemes use the natural drop in the river rather than a storage dam, so their output follows the seasonal flow, and the annual energy estimate should use an average flow rather than the peak.