Power in the wind and Betz. The kinetic power flowing through the rotor disc is P_w = 1/2 rho A v^3 (rho = 1.225 kg/m3, A = pi R^2 with R = 63 m). Actuator-disc momentum theory with axial induction a gives Cp = 4a(1-a)^2, maximal at a = 1/3 where Cp = 16/27 = 0.593, the Betz limit. The first experiment finds this by brute-force scan.
Cp(lambda, pitch). Real rotors lose more to wake rotation, tip losses and drag. The lab uses a published empirical fit (Slootweg et al. form) Cp = 0.73 (151/l_i - 0.58 b - 0.002 b^2.14 - 13.2) e^(-18.4/l_i) with tip-speed ratio lambda = w R / v, giving Cp,max = 0.44 at lambda = 6.9, about 74 % of Betz. This is a stand-in for blade-element/momentum results, not a design of a real blade; thrust is taken from momentum theory, Ct = 4a(1-a) with a solved from the same Cp.
Region 2 (below rated). Keep pitch at 0 and let the generator torque follow T = k w^2 with k = 1/2 rho pi R^5 Cp,max / lambda_opt^3. Then the rotor settles at lambda = lambda_opt and P = eta k w^3, so power rises as v^3. A short region 2.5 ramps torque to its rated value so the power does not jump.
Region 3 (above rated). Torque is held at rated (constant power or constant torque, your choice) and a PI loop pitches the blades toward feather to hold w = 12.1 rpm: beta_cmd = Kp (w - w_r) + Ki * integral(w - w_r) dt. Shaft dynamics: J dw/dt = T_aero - T_gen with J = 4.3e7 kg m2 on the low-speed shaft. The pitch drive is rate-limited to 8 deg/s, which is what makes a high-gain controller saturate. Gain scheduling divides the gains by 1 + beta/beta_k because the aerodynamic torque sensitivity dT/dbeta changes with the operating point.
Tower. One fore-aft mode, m x'' + c x' + k x = Thrust, f = 0.32 Hz, modal mass 4.5e5 kg. Thrust uses the relative wind (v - x'), which gives natural aerodynamic damping; the optional tower-velocity feedback adds pitch proportional to x'. Fast pitch loops near the tower frequency fight this damping and raise load. Tower deflection is exaggerated 8x on screen.
Wind. Mean + Ornstein-Uhlenbeck turbulence (time constant 7 s, intensity set by you) + the IEC 61400-1 style extreme operating gust v = -0.37 A sin(3 pi t/T)(1 - cos(2 pi t/T)), T = 10.5 s. The same seed gives the same wind, so the challenge is repeatable.
Honest simplifications. Rigid rotor: no blade flap/edge flexibility, no drivetrain torsion, no yaw error, wind shear or tower shadow, no wake or rotor-averaging of turbulence, uniform wind over the disc, no stall-delayed lift model, and the electrical generator is an ideal torque source with constant efficiency 94.4 %. Numbers resemble a 5 MW reference turbine but are not a certified model; do not use them for design or for tuning a real machine. Overspeed above 120 % of rated trips the brake and feathers the blades.