Torque Rotor-speed diagrams

For a wind turbine, the power harvested is given by the following formula:

${\displaystyle P={\frac {1}{2}}\rho \pi R^{2}v^{3}C_{p}(\lambda )}$

where ${\displaystyle P}$ is the aerodynamic power and ${\displaystyle \rho }$ 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, ${\displaystyle Q}$, on the rotor shaft is given by the ratio of the power extracted to the rotor speed:

${\displaystyle Q={\frac {P}{\omega }}}$

Thus we can get the following expressions for torque and power:

${\displaystyle P={\frac {1}{2\lambda ^{3}}}\rho \pi R^{5}\omega ^{3}C_{p}(\lambda )}$

and

${\displaystyle Q={\frac {1}{2\lambda ^{3}}}\rho \pi R^{5}\omega ^{2}C_{p}(\lambda )={\frac {1}{2\lambda }}\rho \pi R^{3}v^{2}C_{p}(\lambda )}$

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, ${\displaystyle C_{p}}$.^[3] This diagram is presented below:

Stall regulated

As discussed earlier, a wind turbine would ideally operate at its maximum efficiency for below rated power. Once rated power has been hit, the power is limited. This is for two reasons: ratings on the drivetrain equipment, such as the generator; and second to reduce the loads on the blades. An operating strategy for a wind turbine can thus be divided into a sub-rated-power component, and a rated-power component.

Below rated power

Below rated power, the wind turbine will ideally operate in such a way that ${\displaystyle C_{p}=C_{p~max}}$. On a Torque-rotor speed diagram, this looks as follows:

where the black line represents the initial section of the operating strategy for a variable speed stall-regulated wind turbine. Ideally, we would want to stay on the maximum efficiency curve until rated power is hit. However, as the rotor speed increases, the noise levels increase. To counter this, the rotor speed is not allowed to increase above a certain value. This is illustrated in the figure below:

Rated power and above

Once the wind speed has reached a certain level, called rated wind speed, the turbine should not be able to produce any greater levels of power for higher wind speeds. A stall-regulated variable speed wind turbine has no pitching mechanism. However, the rotor speed is variable. The rotor speed can either be increased or decreased by an appropriately designed controller. In reference to the figure illustrated in the blade forces section, it is evident that the angle between the apparent wind speed and the plane of rotation is dependent upon the rotor speed. This angle is termed the angle of attack.

The lift and drag co-efficients for an airfoil are related to the angle of attack. Specifically, for high angles of attack, an airfoil stalls. That is, the drag substantially increases. The lift and drag forces influence the power production of a wind turbine. This can be seen from an analysis of the forces acting on a blade as air interacts with the blade (see the following link). Thus, forcing the airfoil to stall can result in power limiting.

So it can be established that if the angle of attack needs to be increased to limit the power production of the wind turbine, the rotor speed must be reduced. Again, this can be seen from the figure in the blade forces section. It can also be seen from considering the torque-rotor speed diagram. In reference to the above torque-rotor speed diagram, by reducing the rotor speed at high wind speeds, the turbine enters the stall region, thus bringing some limiting to the power output.

Pitch regulated

Pitch regulation thus allows the wind turbine to actively change the angle of attack of the air on the blades. This is preferred over a stall-regulated wind turbine as it enables far greater control of the power output.

Below rated power

Identical to the stall-regulated variable-speed wind turbine, the initial operating strategy is to operate on the ${\displaystyle C_{p~max}}$ curve. However, due to constraints such as noise levels, this is not possible for the full range of sub-rated wind speeds. Below the rated wind speed, the following operating strategy is employed:

Above rated power

Above the rated wind speed, the pitching mechanism is employed. This allows a good level of control over the angle of attack, thus control over the torque. The previous torque rotor-speed diagrams are all plots when the pitch angle, ${\displaystyle \beta }$, is zero. A three dimensional plot can be produced which includes variations in pitch angle.

Ultimately, in the 2D plot, above rated wind speed, the turbine will operate at the point marked 'x' on the diagram below.

Grid Connections

Consider a variable speed wind turbine with a permanent magnet synchronous generator. The generator produces AC electricity. The frequency of the AC voltage generated by the wind turbine is a function of the speed of the rotor within the generator:

${\displaystyle N={\frac {120f}{P}}}$

where ${\displaystyle N}$ is the rotor speed, ${\displaystyle P}$ is the number of poles in the generator, and ${\displaystyle f}$ is the frequency of the output Voltage. That is, as the wind speed varies, the rotor speed varies, and so the frequency of the Voltage varies. This form of electricity cannot be directly connected to a transmission system. Instead, it must be corrected such that its frequency is constant. For this, power converters are employed, which results in the de-coupling of the wind turbine from the transmission system. As more wind turbines are included in a national power system, the inertia is decreased. This means that the frequency of the transmission system is more strongly affected by the loss of a single generating unit.