TAHA Engineering Toolsby TAHA Knowledge Lab

Energy & Renewables

Wind Turbine Power & Energy Calculator

Wind power in the rotor’s swept area, the estimated electrical output with Cp and efficiency, and the energy from hours or a capacity factor.

Inputs

Wind arrows pass through a rotor disc of diameter D on a tower; formulas for wind, mechanical and electrical power.wind VADP(wind) = ½ ρ A V³P(mech) = P(wind) × CpP(el) = P(mech) × ηCp ≤ 16/27 ≈ 0.593 (Betz)
The wind at speed V flows through the circle swept by the blades (diameter D, area A = πD²/4). The rotor can capture at most the Betz fraction 16/27 of this power; real Cp values are lower. Diagram not to scale.

1.225 kg/m³ is standard air at sea level and 15 °C. It is lower at altitude and in hot weather.

Power rises with the cube of the speed, so an error of 10% in speed gives about 33% in power.

Share of the wind power the rotor captures at this speed, from the turbine data. It cannot exceed the Betz limit 16/27 = 0.593; real rotors reach about 0.35–0.50 at best.

Gearbox, generator and converter together.

Result

Estimated electrical power
8.86683
Swept rotor area A
78.5398
Power in the wind through A
24.6301
Mechanical power at the rotor (Pwind × Cp)
9.85203
Betz-limit maximum at this speed (theoretical)
14.5956
Energy per day
53.201
Energy per month (average)
1,618.2
Energy per year
19,418.4
Capacity factor used
25%
Working
A = π × D² ÷ 4 = π × 10² ÷ 4 = 78.5398 m² P_wind = ½ × ρ × A × V³ = 0.5 × 1.225 × 78.5398 × 8³ = 24.6301 kW P_mech = P_wind × Cp = 24.6301 kW × 0.4 = 9.85203 kW P_el = P_mech × η = 9.85203 kW × 0.9 = 8.86683 kW Energy per day = P_el × 6 h = 53.201 kWh

At this wind speed the turbine gives about 8.86683 kW of electrical power (the wind carries 24.6301 kW through the rotor), or about 19,418.4 kWh per year on this basis.

  • An estimate only, not a prediction of site production. A real turbine follows its power curve: nothing below the cut-in speed, a fixed rated power above the rated speed, and shut-down above the cut-out speed. Turbulence, wind shear, air density changes and availability also matter.
  • The energy assumes the wind blows at exactly this speed for the hours entered. Real wind varies, and because power depends on V³ the average wind speed does not give the average power.

Educational estimate — check the assumptions below.

About this calculator

The power carried by the wind through a rotor grows with the swept area (the square of the diameter) and with the cube of the wind speed. A turbine can only capture part of it: the power coefficient Cp, which in theory can never exceed the Betz limit of 16/27 ≈ 0.593. Gearbox, generator and converter losses reduce it again (efficiency η).

This calculator gives the power at one wind speed (an instantaneous value, in kW) and an estimate of the energy (kWh) from either the hours per day at that speed or a long-term capacity factor — never both. It does not replace the turbine’s power curve or a site wind assessment.

Formula

A = π D² ÷ 4    Pwind = ½ ρ A V³    Pmech = Pwind × Cp    Pel = Pwind × Cp × η

Eday = Pel × t (hours per day)   or   Eday = Pel × 24 h × CF    Eyear = 365 × Eday    Cp ≤ 16/27 ≈ 0.593 (Betz)

ρ = air density (kg/m³), V = wind speed (m/s), D = rotor diameter (m), Cp = power coefficient, η = drive-train and generator efficiency, CF = capacity factor. Power is in W; energy in kWh = W × h ÷ 1000.

Worked example

  • ρ = 1.225 kg/m³, D = 10 m, V = 8 m/s: A = π × 10² ÷ 4 = 78.54 m²; Pwind = 0.5 × 1.225 × 78.54 × 8³ = 24.63 kW.
  • Cp = 0.40: Pmech = 9.852 kW; η = 90%: Pel = 8.867 kW. The Betz maximum at this speed would be 14.59 kW of mechanical power.
  • 6 hours a day at this speed: 53.20 kWh/day, about 19 418 kWh/year. With a 25% capacity factor instead: 8.867 × 24 × 0.25 = 53.20 kWh/day (the same, because 6 h is 25% of 24 h).

Assumptions and limitations

  • Steady, uniform wind at one speed across the rotor. Real wind varies in time and with height.
  • Cp and η are taken as constant at the speed entered. In a real turbine both change with wind speed (the power curve).
  • The Betz limit is a theoretical maximum for an ideal rotor, not a typical operating value.
  • Below cut-in speed a real turbine produces nothing; above rated speed its output is limited to the rated power; above cut-out speed it stops. These are not modelled.
  • A month is taken as 365 ÷ 12 days. An estimate only; it does not predict actual site production.

References

  • Betz, A. (1920) — maximum power coefficient 16/27 of an ideal wind rotor.
  • IEC 61400-12-1 — power performance measurement of wind turbines (power curves).
  • Manwell, J. F., McGowan, J. G. and Rogers, A. L., Wind Energy Explained — power in the wind and capacity factor.