A variable speed wind turbine is one which is specifically designed to operate over a wide range of rotor speeds. It is in direct contrast to fixed speed wind turbine where the rotor speed is approximately constant. The reason to vary the rotor speed is to capture the maximum aerodynamic power in the wind, as the wind speed varies. The aerodynamic efficiency, or coefficient of power, C p {\displaystyle C_{p}} for a fixed blade pitch angle is obtained by operating the wind turbine at the optimal tip-speed ratio as shown in the following graph.
Tip-speed ratio is given by the following expression,
λ = ω R v {\displaystyle \lambda ={\frac {\omega R}{v}}}
where ω {\displaystyle \omega } is the rotor speed (in radians per second), R {\displaystyle R} is the radius of the rotor, and v {\displaystyle v} is the wind speed. As the wind speed varies, the rotor speed must be varied to maintain peak efficiency.
Background Before the need to connect wind turbines to the grid, few turbines were fixed-speed. This was not a problem because turbines did not have to be synchronized with the frequency of the grid. All grid-connected wind turbines, from the first one in 1939 until the development of variable-speed grid-connected wind turbines in the 1970s, were fixed-speed wind turbines. As of 2003, nearly all grid-connected wind turbines operate at an exactly constant speed (synchronous generators) or within a few percents of constant speed (induction generators).
History The Gamma 60 wind turbine - a 1.5 MW two-bladed yaw control turbine, which is ongoing further development by Seawind Ocean Technology B.V., was the world's first variable speed wind turbine with a teeter hinge.
Torque Rotor-speed diagrams For a wind turbine, the power harvested is given by the following formula:
P = 1 2 ρ π R 2 v 3 C p ( λ ) {\displaystyle P={\frac {1}{2}}\rho \pi R^{2}v^{3}C_{p}(\lambda )}
where P {\displaystyle P} is the aerodynamic power and ρ {\displaystyle \rho } is the density of the air. The power coefficient is a representation of how much of the available power in the wind is captured by the wind turbine and can be looked up in the graph above. The torque, Q {\displaystyle Q} , on the rotor shaft is given by the ratio of the power extracted to the rotor speed:
Q = P ω {\displaystyle Q={\frac {P}{\omega }}}
Thus we can get the following expressions for torque and power:
P = 1 2 λ 3 ρ π R 5 ω 3 C p ( λ ) {\displaystyle P={\frac {1}{2\lambda ^{3}}}\rho \pi R^{5}\omega ^{3}C_{p}(\lambda )}
and
Q = 1 2 λ 3 ρ π R 5 ω 2 C p ( λ ) = 1 2 λ ρ π R 3 v 2 C p ( λ ) {\displaystyle Q={\frac {1}{2\lambda ^{3}}}\rho \pi R^{5}\omega ^{2}C_{p}(\lambda )={\frac {1}{2\lambda }}\rho \pi R^{3}v^{2}C_{p}(\lambda )}
From the above equation, we can construct a torque-speed diagram for a wind turbine. This consists of multiple curves: a constant power curve which plots the relationship between torque and rotor speed for constant power (green curve); constant wind speed curves, which plot the relationship between torque and rotor speed for constant wind speeds (dashed grey curves); and constant efficiency curves, which plot the relationship between torque and rotor speed for constant efficiencies, C p {\displaystyle C_{p}} . This diagram is presented below:
Notes Green curve: Plot of power = rated power so that P = Q ω {\displaystyle P=Q\omega }
Grey curve: Wind speed is assumed constant so that Q ∝ ω 2 C p ( λ ) {\displaystyle Q\propto \omega ^{2}C_{p}(\lambda )}
Blue curve: Constant C p ( λ ) {\displaystyle C_{p}(\lambda )} so that Q ∝ ω 2 {\displaystyle Q\propto \omega ^{2}}
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