Power Coefficient of a Wind Turbine

The power coefficient of a wind turbine, known as Cp, is a key performance indicator for understanding how efficiently a wind turbine converts wind energy into mechanical energy. The power coefficient reveals the maximum percentage of wind energy that can be harnessed. With the increasing focus on wind energy, improving the power coefficient is essential to maximise power generation and reduce environmental impact. This article will explore what the power coefficient is, its mathematical model and how wind turbine design affects the Cp. Therefore, understanding the concept of the power coefficient is critical to optimise wind power and maximise wind energy production.

What is the power coefficient?

The efficiency of a wind turbine is defined as the power coefficient (C_p), which is specific to the design of each turbine. The efficiency of a wind turbine is calculated as the ratio between the mechanical power generated by the blades (P_mec) and the wind power (P_w). The wind power is defined as the kinetic energy per mass of wind moving (V_w) across a blade-swept surface (A), which is defined by the air density (). The air density is approximately equal to 1,225 kg/m³ at sea level and at a temperature of 15°C.

The maximum theoretical value of the power coefficient is obtained through the application of Betz’s law, which indicates that the maximum value is approximately equal to 59%. This implies that the wind turbine is capable of converting a maximum of 59% of the kinetic energy of the wind into mechanical energy.

Mathematical model of the Power Coefficient

The power coefficient surface of an individual wind turbine model can be described as a function of its aerodynamic characteristics, the tip-speed ratio (λ), and the pitch angle (β). Commonly used models for defining the power coefficient (C_p) include the exponential model, which is outlined as follows:

The tip speed ratio is defined as the ratio between the magnitude of the tangential velocity and the magnitude of the wind speed.

Where is the angular velocity of the wind turbine and R_t is the radius of the wind turbine.

The figure below shows the power coefficient surface as a function of λ (rad) from 0 to 15 rad and β (sexagesimal degrees) varying from 0° to 45°. The maximum value of the power coefficient is approximately equal to 0.425 and is given for a pitch angle (β) equal to 0° and a tip-speed ratio (λ) equal to 6.9.

It can be observed that an increase in pitch angle (β) results in a reduction in power coefficient. This establishes the fundamental concept of mechanical power control by pitch angle in zone 4 of the wind turbine power curve.

Each power coefficient curve generated by different values of beta exhibits a maximum value. When the wind turbine is operating at partial load, the angle β is equal to zero.

Fig. 2. Power coefficient variation

Relationship between the number of blades and wind turbine efficiency

Figure 3 depicts the distinct power coefficient curves for various categories of wind turbines. In a series of wind tunnel experiments, Poul La Cour reached the following conclusion: “More blades does not mean more power”. This shows that an increase in the number of blades on a wind turbine results in a corresponding decrease in efficiency. American multi-bladed wind turbines are the least efficient, whereas the most efficient are three-bladed and two-bladed wind turbines. The rationale for selecting a three-bladed wind turbine, despite its slightly lower efficiency compared to the two-bladed, is based on technical and economic considerations. For example, to generate the same power, a three-bladed wind turbine requires a lower rotational speed than a two-bladed one.

Fig. 3. Different power coefficient curves for each type of wind turbine. Source: Público

For different rotor types, the optimum tip-speed ratio varies. For multi-bladed wind turbines, λ has a value of approximately one, and as the number of blades decreases, λ increases. For example, the power coefficient curves in Figure 2 correspond to a three-bladed wind turbine. This type of wind turbines, which are the most widely used at the industrial level, have an optimum tip speed between 6 and 8 rad.

Conclusion

The power coefficient (C_p) is a fundamental parameter for evaluating the efficiency of a wind turbine, as it quantifies the fraction of wind energy that is converted into mechanical energy. According to Betz’s Law, the theoretical maximum value of C_p is 59%. Various factors, including the tip-speed ratio (λ) and pitch angle (β), exert a direct influence on C_p. Three-bladed wind turbines exhibit the highest efficiency at the industrial scale. By optimizing C_p through rigorous design and control, it is possible to maximize wind power generation.