## Coeffecient of Performance

Recall that the power available in the wind can be expressed as

P=½AρV3

where ρ is the density of the air, A is the capture area, and V is the wind speed.

The power actually captured by the wind turbine rotor, PR, is some fraction of the available power, defined by the coefficient of performance, Cp, which is essentially a type of power conversion efficiency:

Cp = PR/P

The maximum theoretical value of the coefficient of performance is 0.593, a value determined by a fluid mechanics constraint known as the Betz limit. Actual coefficients of performance are less than this limit due to various aerodynamic and mechanical losses. For a given turbine design, Cp is a function of tip speed ratio (TSR). As shown in the curves in Figure 1, there is a tip speed ratio for which the power capture is a maximum. Comparisons of the various wind turbine types in Figure 1 shows how inefficient the drag-based Savonius turbine is compared to the lift-based turbines. The Darrieus turbines and the HAWT have similar values of the maximum coefficient of performance, but the HAWT can operate at much higher tips speed ratios (faster rotation speeds or lower wind speeds).

## Power Curve

The electrical power output from the generator is less than the power captured by the rotor, due to losses in both the gear train and generator:

PT=Cpηgηb(½AρV3)

where ηg and ηb are efficiencies (power output over power input) for the generator and the gearbox. Gearbox efficiencies are typically 90- 95%, while generator efficiencies range from 50% (for a car alternator) to better than 80% for a high quality, grid-connected model.

The power curve for a wind turbine shows this net power output as a function of wind speed. As shown in Figure 2 (for an 82 m diameter wind turbine), these curves feature three key wind speeds:

1. Cut in wind speed: This is the wind speed at which the wind turbine will start generating power— typical cut-in wind speeds are 3 to 5 m/s.
2. Nominal wind speed: This is the lowest speed at which the wind turbine reaches its nominal power output. Above this speed, higher power outputs are possible, but the rotor is controlled to maintain a constant power to limit loads and stresses on the blades.
3. Cut-out wind speed: This is the highest wind speed which the turbine will operate at. Above this speed, the turbine is stopped to prevent damage to the blades.

## Annual Energy Generation and Capacity Factor

The power curve combined with the annual wind speed distribution can be used to estimate how much energy a wind turbine could generate in typical year. Specifically, the power at each wind speed is multiplied by the number of hours per year that the wind blows at that speed to estimate how much energy is generated at each wind speed (red curve in Figure 3). This is then summed to get the annual energy generated. For the Vestas V82 example shown, a 1.7 MW turbine operating in Boston, 3,800 MWh are generated each year. This is enough energy to power approximately 350 homes.

The capacity factor of a wind turbine is the total annual energy generated divided by the energy that could be generated if it were running continuously at rated capacity 24 hrs a day for 365 days a year. For the V82 example, the capacity factor is found from:

3,800 MWh/(1.7 MW*8760 hr) = 25%.

This value is on the low end of the typical range of 25 to 40%, mainly because Boston is not as windy as the locations typically chosen for wind farms.