Wind Turbine Energy Conversion (2D)

Power in the wind and the Betz limit

The kinetic power flowing through the rotor disc of area A = pi R^2 is P_wind = 1/2 rho A v^3. A rotor extracts the fraction Cp of it. Momentum theory (actuator disc) with axial induction a gives Cp = 4a(1-a)^2 and thrust coefficient Ct = 4a(1-a); the maximum is Cp = 16/27 = 0.593 at a = 1/3 (Betz limit). Real rotors stay below it because of wake rotation, blade drag and a finite number of blades.

Cp(lambda, beta)

The tip-speed ratio is lambda = omega R / v and beta is the blade pitch angle. The page uses the widely used empirical fit Cp = c1 (c2/l_i - c3 beta - c4) exp(-c5/l_i) + c6 lambda with 1/l_i = 1/(lambda + 0.08 beta) - 0.035/(beta^3 + 1) and c = (0.5176, 116, 0.4, 5, 21, 0.0068). It peaks near lambda = 8.1, Cp = 0.48 at beta = 0. Pitching a blade out of the wind lowers Cp, which is how power is shed above rated wind.

Drive-train dynamics and control

J d(omega)/dt = T_aero - T_gen - T_loss with T_aero = 1/2 rho pi R^3 v^2 Cp / lambda and rotor inertia J ~ 2.5 R^4 (scaled by the slider). Below rated wind the generator torque follows T_gen = k omega^2 with k = 1/2 rho pi R^5 Cp_max / lambda_opt^3, which holds lambda at its optimum (variable-speed operation). Above rated, torque is limited to P_rated/(eta omega_rated) and a PI controller pitches the blades (rate-limited to 8 deg/s) to hold the rated rotor speed. Rotor speed is also capped by a tip-speed limit of 80 m/s. Electrical power is eta T_gen omega with eta = 0.95. Cut-in is 3 m/s and cut-out 25 m/s, where the blades feather and a brake parks the rotor.

Wind and energy yield

The hub wind is a mean plus an Ornstein-Uhlenbeck turbulence process (time constant 10 s, standard deviation = turbulence intensity x mean), low-pass filtered by the rotor-disc size. The long-term resource is a Weibull distribution f(v) = (k/c)(v/c)^(k-1) exp(-(v/c)^k) with c = v_mean / Gamma(1 + 1/k). Annual energy is 8760 h x integral of P(v) f(v) dv using the steady power curve; the capacity factor is that energy divided by rated power x 8760 h. Air density follows altitude and temperature: rho = p/(287.05 T) with p = 101325 (1 - 2.256e-5 h)^5.256.

Wakes in a row

The Jensen (Park) model gives the speed deficit behind a turbine: delta = (1 - sqrt(1 - Ct)) (R/(R + k x))^2 with wake-decay constant k = 0.075 (onshore). The third turbine sees the root-sum-square of two deficits. Row efficiency is total power divided by three times the power of an isolated turbine.

Honest simplifications

The Cp surface is an empirical fit for a generic large rotor, not a blade-element-momentum solution of this exact blade; no tower shadow, yaw error, wind shear, blade flexibility, gearbox or grid dynamics. Turbines 2 and 3 are quasi-steady (no own rotor dynamics). The Jensen wake is a top-hat engineering model. Annual energy ignores availability, electrical and wake losses beyond the row shown. Values are educational estimates, not a bankable yield assessment.

Model checks: a grid search of Cp over the whole (lambda, pitch) plane against an independent golden-section optimum and the Betz limit; the Weibull pdf integral, mean and cubic moment against Gamma-function closed forms; actuator-disc power from the kinetic-energy flux difference against Cp x wind power; and a headless 900 s steady-wind run of the dynamics.