Wind power curve and generator topologies
P = ½·ρ·A·v³·C_p, Betz limit, tip-speed ratio, pitch control, and Types 1–4 (SCIG → WRIG+R → DFIG → full-converter PMSG).
Step 1 — Wind power equation: P_wind = ½·ρ·A·v³ · Cp
Reference notes
Use Next → on the narrator above to step through wind turbine physics and generator topologies — the cubic-in-wind-speed power equation, Betz limit, tip-speed ratio, four power-curve regions, pitch control, and four generator types (Type 1 through Type 4).
The wind power equation
where:
- ρ = air density (1.225 kg/m³ at sea level, 25 °C)
- A = swept rotor area = π R² (m²)
- v = wind speed (m/s)
The cubic dependence on v is wind engineering's single most important parameter. Double the wind speed → 8× the available power. A site averaging 8 m/s produces roughly twice the energy of one averaging 6.3 m/s.
The Betz limit
A rotor cannot extract 100 % of the wind energy passing through it — that would require the air to stop, which conservation of mass + momentum forbids. Albert Betz (1919) computed the maximum extractable fraction for an idealized rotor:
This is a fundamental physical bound, not an engineering limitation. Real horizontal-axis turbines achieve C_p ≈ 0.45–0.50 at optimum — about 80 % of Betz. Vertical-axis turbines: C_p ≈ 0.35–0.40.
Tip-speed ratio λ and C_p(λ, β)
Ratio of blade-tip speed (ωR) to wind speed (v). Performance coefficient C_p is a function of both λ and the pitch angle β:
- For fixed β, C_p peaks at a specific λ called λ_opt — typically ≈ 7 for a 3-blade horizontal-axis wind turbine (HAWT).
- Below λ_opt: blades stall (angle of attack too high).
- Above λ_opt: blades over-speed (angle of attack too low).
- Increasing β at any λ reduces C_p — the blades "spill" wind energy.
The four-region power curve
| Region | Wind speed | Control | Output |
|---|---|---|---|
| I (below cut-in) | v < v_cut-in (≈ 3 m/s) | Rotor parked | 0 W |
| II (MPPT) | v_cut-in to v_rated | Track λ = λ_opt by varying ω | Rises as v³ |
| III (pitch) | v_rated to v_cut-out (12 → 25 m/s) | Increase β to hold P = P_rated | Constant at rated |
| IV (cut-out) | v > v_cut-out (≈ 25 m/s) | Feather blades, park rotor | 0 W |
Pitch vs stall control
- Pitch control: actively rotate blades around their long axis (β increases) above rated wind speed to spill aerodynamic energy and hold output at rated. Standard on modern variable-speed turbines.
- Stall control: fixed-blade-pitch design where blades enter aerodynamic stall on their own at high wind, naturally limiting output. Simpler, no actuators, but less precise output regulation. Used on older fixed-speed turbines (Type 1).
- Active stall: small pitch adjustments to assist stall onset. Hybrid approach used on some 1990s-era machines.
Generator topologies (Type 1–4)
Type 1 — SCIG (squirrel-cage induction generator)
Stator and rotor of a standard SCIG, directly grid-connected (no power converter). Fixed speed (synchronous + 1–2 % slip). Cheapest topology. Used on early 1980s–90s wind turbines (< 1 MW). Major drawback: cannot ride through grid faults, draws large reactive current during dips.
Type 2 — WRIG with variable rotor resistance
Wound-rotor induction generator with external resistors in the rotor circuit controlled by power electronics. Allows limited variable speed (~±10 % of synchronous) by varying torque-speed curve via R_2. Used in 1990s 600 kW–1.5 MW turbines.
Type 3 — DFIG (doubly-fed induction generator)
Stator directly grid-connected; rotor fed by a back-to-back IGBT converter sized for ~30 % of rated. The converter handles only the slip power. Variable speed across ±30 % of synchronous. Dominant utility-scale topology installed during the 2000s and 2010s (e.g., GE 1.5/1.6/1.7 MW machines). Trade-off: 30 % converter is cheap, but DFIG has limited low-voltage ride-through capability and is increasingly disadvantaged on grids with strict grid-code requirements.
Type 4 — Full-converter
Generator (usually a permanent-magnet synchronous machine, sometimes a synchronous wound-rotor or induction machine) connected to the grid via a full-rated back-to-back converter. Every watt of generated power passes through power electronics. The rotor is decoupled from grid frequency and can spin at any speed. Provides:
- Low-voltage ride-through (LVRT) without losing synchronization.
- Controllable reactive power across the full range (acts like a STATCOM).
- No magnetizing inrush on connection.
- Direct-drive topologies (no gearbox) become practical because the converter handles low rotor frequency.
More expensive than Type 3 but the standard for offshore and large onshore turbines built since ~2015.
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- On the power-curve view, click + / − to sweep wind speed and watch the operating point traverse the four regions.