## Wind Energy and Power

The **wind power** that can be captured by a wind turbine depends on the air density, the swept area by the blades, and the wind speed.

#### Kinetic Power of the Wind

The kinetic power contained in the wind passing through a given area is:

Where:

: Wind power (W). : Air density ( ), typically at sea level. : Swept area by the turbine blades ( ). : Wind speed ( ).

#### Swept Area by the Blades

The swept area by the blades of a wind turbine (in the case of a horizontal axis turbine) is the area of a circle:

Where:

: Swept area ( ). : Radius of the blades ( ).

## Power Captured by the Turbine

Not all the wind power can be converted into electrical energy. The **captured power** by a wind turbine is limited by the **power coefficient**

#### Power Captured by the Wind Turbine

The actual power that the turbine can generate is given by:

Where:

: Power captured by the turbine (W). : Power coefficient (dimensionless), which depends on the design of the turbine. The theoretical maximum value is (Betz limit).

#### Betz Limit

The **Betz limit** states that no wind turbine can capture more than 59.3% of the kinetic energy of the wind.

#### Betz Limit

## Coefficients and Efficiency of the Turbine

The efficiency of a wind turbine is related to various coefficients that describe the turbine’s ability to convert wind energy into electricity.

#### Power Coefficient ( )

The **power coefficient** is the ratio of the power generated by the turbine to the total power available in the wind.

#### Overall Efficiency of the Wind System

The **overall efficiency of the wind system** takes into account losses in other components, such as the generator, cables, and inverter.

Where:

: Efficiency of the generator. : Efficiency in the transmission through the cables. : Efficiency of the inverter from DC to AC.

## Characteristic Speeds of the Turbine

Wind turbines have certain **speed thresholds** of the wind at which they start generating power and stop functioning for safety reasons.

#### Cut-in Speed ( )

The **cut-in speed** is the minimum wind speed at which the turbine begins to generate electrical power.

#### Rated Speed ( )

The **rated speed** is the wind speed at which the turbine reaches its rated power, i.e., the maximum power it can generate.

#### Cut-out Speed ( )

The **cut-out speed** is the maximum wind speed at which the turbine stops operating to avoid structural damage.

## Forces and Torque in the Turbine

#### Drag Force ( )

The **drag force** is the resistance that the wind exerts on the turbine blades. It is calculated with the following formula:

Where:

: Drag coefficient (depends on the shape of the blades). : Air density ( ). : Projected area ( ). : Wind speed ( ).

#### Lift Force ( )

The **lift force** is the force that acts perpendicular to the wind on the blades, generating rotation in the turbine.

Where:

: Lift coefficient (depends on the aerodynamic profile of the blades).

#### Torque on the Turbine Shaft

The **torque** on the turbine shaft is the result of the lift forces that generate the rotational movement of the blades.

Where:

: Torque ( ). : Lift force (N). : Length of the blades or turbine radius (m).

## Rotational Speed and Tip-Speed Ratio

#### Angular Speed of the Turbine

The angular speed (

Where:

: Angular speed (rad/s). : Wind speed ( ). : Radius of the blades ( ).

#### Tip-Speed Ratio (TSR)

The **tip-speed ratio (TSR)** is the ratio of the tangential speed of the tips of the blades to the wind speed.

Where:

: Tip-speed ratio. : Angular speed of the blades (rad/s). : Radius of the blades (m). : Wind speed ( ).

The **optimal TSR** varies depending on the design of the turbine, and for most modern turbines, it is in the range of

## Electrical Power and Sizing of the Wind System

#### Electrical Power Generated

The electrical power generated by a wind turbine is limited by the efficiency of the system and losses in energy conversion.

Where:

: Electrical power generated (W). : Power captured by the turbine (W). : Overall efficiency of the system (including generator, cables, and inverter).

#### Energy Generated Over a Period of Time

The energy generated by a turbine over a period of time

Where:

: Energy generated (Wh or kWh). : Electrical power generated (W). : Operating time (hours).

## Number of Turbines Needed

The number of turbines needed for a system depends on the energy demand and the power generated by each turbine.

#### Number of Turbines

The number of turbines is calculated by dividing the total required power by the rated power of a single turbine:

Where:

: Number of turbines needed. : Total power required (W). : Rated power of a turbine (W).