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.   

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.

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.

Reference

[1] Burton, T., etc., Sharpe, D., Jenkins, N., & Bossanyi, E. (2001). Wind Energy Handbook. John Wiley & Sons.

[2] Ackermann, T. (Ed.). (2012b). Wind Power in Power Systems: Ackermann/Wind Power in Power Systems. Wiley-Blackwell.

The parts that make up a wind turbine are as follows:

1. Blades

The blades of a wind turbine are the components that directly interact with the wind, which is why they are designed with a profile that maximizes their aerodynamic efficiency. Most blades are manufactured using polyester or epoxy reinforced with fiberglass. This material consists of a matrix of synthetic resin and fiberglass fibers, providing the blades with lightness, durability, and resistance to corrosion and moisture. Carbon fiber can also be used as a reinforcement material to enhance the strength and rigidity of the blades. However, carbon fiber comes at a higher cost and poses challenges in terms of recycling.

Transporting the blades can be a major challenge. Larger wind turbines require longer blades, which can complicate their transport to the wind farm.

2. Hub

The hub of a wind turbine is the component responsible for connecting the blades to the shaft that transmits motion to the gearbox in the case of a Doubly Fed Induction Generator (DFIG) or to the generator shaft in the case of a Direct-Drive Permanent Magnet Synchronous Generator (PMSG). The hub contains mechanisms for changing the pitch angle of the blades. The hub is typically made of steel or cast iron and is aerodynamically designed to prevent the formation of turbulence and allow for maximum free airflow.

3. Blade orientation system

It consists of a series of mechanical, electrical, or hydraulic mechanisms that allow for adjusting the pitch angle of the blades. Their purpose is to maintain constant mechanical power when the wind speed exceeds the nominal value (as can be seen in the wind turbine’s power curve). This system can also be considered a protective measure to prevent the wind turbine from exceeding its nominal speed.

The blades move relative to their axis to control the mechanical power. This type of control is called “pitch angle control”.

4. Wind turbine orientation system

The nacelle rotates along the vertical axis of the tower using an active and rotating orientation control system, which consists of electric actuators. The wind direction and speed are continuously monitored by sensors located on the nacelle’s cover, called anemometers and wind vanes. The aim is to align the wind turbines in the direction of the prevailing wind to maximize wind energy capture.

5. Gearbox

The function of the gearbox is to connect the shaft that joins the blades at the hub with the generator shaft. Its purpose is to multiply the turbine’s rotational speed to an efficient speed for the electrical generator. Without a gearbox, the electrical generator would need to rotate optimally between 10 and 25 rpm, meaning a generator with many poles would be required.

Therefore, the gearbox can be thought of as a mechanical transformer. It can multiply the wind turbine’s rotation speed, for example, 1:50, 1:70, 1:80, 1:90, 1:100, 1:110. The higher the multiplication factor, the more complex the gear system and gearbox stages will be. The gearbox provides a higher rotation speed for the electrical generator but lower mechanical torque.

6. Electric generator

6.1. Squirrel Cage Induction Generator (SCIG):

This configuration corresponds to the so-called “Danish concept”, which was very popular in the 1980s. This type of electric generator consumes reactive power, which it can absorb from the grid or a bank of capacitor batteries. To connect it to the grid, a soft-start system is used, based on thyristors, in order to limit the starting current. This generator can cause power oscillations that are directly transferred to the grid. Its main advantages are its smaller size, lower cost, and the simplicity of the control system.

6.2. Doubly Fed Induction Generator (DFIG):

It has been used since the year 2000. An electronic power converter is connected in a back-to-back configuration. Through the use of the electronic power converter, the frequency and current in the rotor can be flexibly controlled, thus extending the range of variable speeds to a satisfactory level. Control is achieved by injecting variable current into the rotor winding, both in magnitude and frequency. The speed variation range is about ± 30% of the rated speed. The stator is directly connected to the grid, while the rotor is connected through power converters with a fraction of power between 20% or 30% of the rated power. Disadvantages include having slip rings, requiring a multi-stage gearbox to connect to the wind turbine, incurring higher maintenance costs, and experiencing increased losses mainly due to the mechanical transmission train.

6.3. Permanent Magnet Synchronous Generator (PMSG):

This type of generator is the most commonly used alongside the DFIG generator. Unlike the DFIG, it integrates full-scale power electronic converters to connect to the electrical grid. This generator type has a greater capacity to support the electrical grid and requires less maintenance. It utilizes rare earth magnets in the rotor, which have a high magnetic flux density. Its primary disadvantage is the cost of the magnet, as it is subject to global uncertainty and can potentially demagnetize due to electrical faults in the generator.

7. Power electronics interface

The power electronics interface plays a crucial role in wind turbines. It is responsible for the transformation, control, and optimization of energy. Power converters are used to control the flow of active and reactive power in both steady and dynamic states, from the electric generator to the power grid. The rotational speed of variable-speed wind turbines is decoupled from the electrical frequency due to the electronic interface; in other words, they do not rotate in sync with the power grid, as a conventional synchronous generator would. The Back-to-back connection is the most commonly used and consists of two voltage-source converters (VSC).

These converters are composed of semiconductor devices such as the Insulated Gate Bipolar Transistor (IGBT), which functions as a controlled switch. IGBTs are controlled using pulse-width modulation techniques. The power converters, along with their control system, act as the “brain” of the wind turbine. It is vital that their operation is appropriate and optimal to ensure the proper functioning of the entire wind energy conversion system.

8. Transformer (LV-MV)

The electrical output power of the wind turbine is generally low voltage and is converted to medium voltage through a transformer to reduce transmission losses by connecting to the medium voltage grid. The transformer is installed either in the nacelle or at the base of the tower. It is a dry-type transformer with two windings.

Conclusion

In conclusion, wind turbines are complex systems as they encompass electrical, electronic, mechanical, aerodynamic, and structural subsystems. Wind turbines are the pillars of renewable energy generation due to their increased capacity in recent years. Each component, from the blades to the electrical generators, plays a vital role in capturing and transforming wind energy into electricity. The blades are aerodynamically designed and constructed with advanced materials to maximize efficiency and durability. The orientation systems ensure optimal operation by continuously adjusting the blades and nacelle based on wind conditions. The gearbox and various types of electrical generators, such as SCIG, DFIG, and PMSG, enable effective conversion of mechanical energy to electrical energy, each with its specific advantages.

The power electronics interface is the “brain” of the wind turbine, managing the power flow and decoupling the rotational speed of the wind turbine from the electrical grid frequency. This control and optimization capability is crucial for the overall system performance. Finally, the transformer raises the generated low voltage to a medium voltage level, minimizing transmission losses and ensuring efficient integration with the electrical grid.

Overall, the engineering behind each part of the wind turbine is fundamental to its efficiency and reliability, making these machines a key element in the transition to cleaner and more sustainable energy sources.

What is the wake effect in wind energy?

The wake effect in wind energy occurs when the wind flow is influenced by the presence of a wind turbine, which impacts nearby turbines. When the wind encounters a wind turbine, it creates a zone of low pressure behind it, resulting in slower and less energetic wind for the subsequent turbines. This situation reduces the efficiency and performance of downstream turbines, known as the wake effect.

Optimizing the arrangement of wind turbines is crucial to maximize the utilization of available space in a wind farm. On one hand, it is necessary to maintain an appropriate spacing between turbines to avoid the influence of wake shadows and generated turbulence, which can reduce the overall production of the wind farm. On the other hand, the turbines should be positioned close enough to optimize the available area and minimize the costs of the internal medium-voltage grid of the wind farm, as well as energy losses in the grid.

The design of wind farms poses a challenge due to the phenomenon of multiple wakes. Calculating the wake effect is a significant issue in wind farms and requires accurate modeling to minimize power losses caused by the wake effect both at close and far distances.

In this video, we will explore how Siemens Gamesa Renewable Energy is maximizing wind energy production through wake optimization. We will delve into their innovative approach to strategically placing wind turbines and utilizing cutting-edge technology, enabling them to generate more energy efficiently and profitably. With the use of NVIDIA Modulus and Omniverse, designers in the wind energy industry have the ability to merge traditional methods with physics-based super-resolution artificial intelligence models. This allows for the generation of high-resolution simulation data at significantly faster rates, resulting in the creation of more precise and detailed engineering wake models.

Impact of the wake effect on the performance of wind farms

The impact of the wake effect on a wind farm is significant and can have several consequences on the efficiency and overall performance of the system. Below are some of the main impacts:

a) Decrease in Energy Production:

The wake effect causes a reduction in the speed and energy of the wind reaching the downstream wind turbines due to the upstream wind turbines. This diminishes the energy production of the downstream wind turbines as they receive a less powerful wind flow. Moreover, the effect can be cumulative and detrimental to the wind turbines in the last row of the distribution within the wind farm.

b) Loss of Efficiency:

Due to the decrease in wind speed, wind turbines affected by the wake effect operate at suboptimal speeds. This results in a reduction in energy conversion efficiency, as they fail to fully harness the available wind potential. Efficiency (ɳ) can be defined as:

Where is the measured power output of turbine , is la power output for free stream conditions, and is the number of turbines.

c) Uneven Wear of Wind Turbines:

The wake effect can also lead to uneven wear on the turbines within the wind farm. The downstream wind turbines experience fluctuating wind loads and turbulence, meaning that the wind speed is not evenly distributed across the entire blade sweep area. This can result in increased mechanical wear and shorten the lifespan of the wind turbines.

d) Limitations in Wind Turbine Placement:

The wake effect imposes restrictions on the optimal placement of wind turbines within a wind farm. It is necessary to strike a balance between maximizing energy production and minimizing the impact of wake effect. This involves considering the distance and configuration of the turbines to avoid interference between the generated wakes. Moreover, it heavily depends on the geography of the installation site, as a particular location on the terrain may be optimal for wind resource, but not suitable for tower installation.

Criteria for Wind Turbine Distribution

Wind turbines are arranged in one or multiple alignments perpendicular to the most energetic component of the wind, as indicated by the energy wind rose. The following suggestions are commonly used as criteria for spacing between turbines:

• For the spacing between turbines within the same alignment, a distance ranging from 2 to 3 times the rotor diameter (D) is typically established.

• For the spacing between alignments, a distance of 6 to 8 times the rotor diameter (D) is applied.

The spacing between turbines is determined based on the “wind shadow” they may create, which depends on the prevailing wind direction.

In Figure 3, the distribution of 48 wind turbines in a 120 MW wind farm is shown with a row orientation of 25 degrees relative to the reference direction.

Conclusion

Wake effect in wind energy is a crucial factor that influences the performance of wind farms. Its impact results in decreased energy production, efficiency loss, and uneven wear on turbines. To maximize production and minimize negative effects, optimizing the strategic placement of wind turbines is essential. By combining traditional methods with high-resolution artificial intelligence models, such as those offered by NVIDIA Modulus and Omniverse, faster and more accurate simulation data can be generated, enhancing the ability to mitigate wake effect and maximize efficiency in wind energy generation. Proper turbine distribution, considering appropriate spacing and configuration, is crucial for optimizing wind farm performance and minimizing interference among wake patterns.

Extractable Wind Power

The following assumptions are used to analyze the power that can be extracted from the rotor of a wind turbine:

• Homogeneous, incompressible, steady state fluid (constant density).
• No frictional drag.
• Infinite number of blades.
• Uniform thrust in the disc or rotor area.
• Non-rotating wake.
• The static pressure before and after the rotor is equal to the undisturbed ambient static pressure.

The calculation is based on momentum theory. The Rankine-Froude model is used. This model considers the volume swept by the wind turbine blades as an infinitely thin disk. An air flow through the area swept by the blades of the analyzed wind turbine is established in a control volume that can be divided into 4 sections.

Where:

V₁: Upstream wind speed (m/s).
V₂: Wind speed just before interacting with the disk.
V₃: Wind speed immediately after disk interaction.
V₄: Downstream wind speed (m/s).
p: Air pressure.
F: Force acting on the disk.
A_w: Area of actuating disk.

The disk divides into two control volumes. Therefore, Bernoulli’s principle applies to the downstream volume:

Bernoulli’s principle is also applied to the upstream volume:

It is assumed that the upstream and downstream pressures are equal (p₁ = p₄) and that the wind speeds immediately upstream and downstream of the disc are equal (V₂ = V₃). By solving the above two equations, a solution can be found for (p₂ – p₃):

The force on the actuator disk is:

Substituting the value of (p₂ – p₃) into the previous equation, we obtain:

The fluid exerts a force F on the disk. This force can be calculated from the change in the amount of motion:

Equating the two expressions for the force F on the disk, we obtain:

Therefore, the wind speed near the actuator disc is approximately equal to half the sum of the upstream (V₁) and downstream (V₄) wind speeds.

Axial Induction Factor (a) is defined as the fractional decrease in wind speed between free stream and rotor plane:

Then, as a function of the input speed V₁ and the axial induction factor (a), the speeds V₂ and V₄ are obtained:

The product of force (F) and speed (V₂) is the power (P) that the airflow transfers to the wind turbine rotor.

By substituting the values of V₂ and V₃ as a function of V₁ and the axial induction factor (a), the mechanical power of the wind turbine can be obtained:

Where:

P_m: Mechanical power.
A_r: Area swept by wind turbine blades (m²).
V_w: Upstream wind speed (m/s).
ρ: Air density (kg/m³).

The power transfer limiting factor is 4a(1-a)². Therefore, the wind turbine has an energy extraction limit that comes from the wind speed.

Calculation of the Betz’s Limit

The factor that limits the mechanical power of the wind turbine is called the power coefficient (Cp). The maximum value of Cp is obtained by derivation with respect to the axial induction factor (a).

The maximum value is obtained when a=1/3:

Figure 2 shows a plot of the power coefficient as a function of the axial induction factor. For values greater than 0.5, the power coefficient is physically meaningless.

Albert Betz stated that the maximum energy that can theoretically be extracted from the wind is only 59.3% of the kinetic energy carried by the wind. This limit is not due to a faulty design of the wind turbine, but to the control volume in which the analysis was performed. Furthermore, in his book “Wind Energie”, he describes much of the knowledge about wind energy at that time.

The theory of wind energy: The legacy of Albert Betz

The book “Wind energie” by Albert Betz is a classic text in the field of wind energy. Originally published in German in 1926, it is considered one of the first academic works in this field.

In the book, Betz presents a detailed theoretical analysis of wind energy and wind turbines. He develops a mathematical theory to describe the behavior of wind and how wind turbines can capture and convert wind energy into electrical energy. The book also discusses the design of wind turbines and how they can be optimized to maximize energy production. Betz provides detailed calculations and graphs to illustrate his concepts.

Although the book was written almost a century ago, it is still relevant today and remains a valuable resource for those working in the field of wind energy. Figure 3 shows the cover of the book.

In summary, Betz’s Law imposes a fundamental limit on the maximum amount of energy that can be extracted from the wind by a wind turbine, which has important implications for wind farm design and efficiency. Although Betz’s Law is a theoretical limit, technological advances in wind turbines and energy control systems are enabling wind farms to achieve ever higher efficiencies and make the most of wind energy.

“Betz’s Law states that only 59% of the kinetic energy of the wind can be converted into mechanical energy to move the wind turbine”.

Reference

[1] Burton, T., Jenkins, N., Sharpe, D., & Bossanyi, E. (2014). Wind Energy Handbook. Wiley.

[2] Manwell, J. F., McGowan, J. G., & Rogers, A. L. (2010). Wind energy explained: Theory, design and application. John Wiley & Sons.

Power surface

The power surface contains all possible points where the wind turbine can operate. Figure 1 shows this surface depending on the wind speed (4 – 20 m/s) and the speed of the wind turbine (8 – 20 rpm). By changing the power coefficient (C_p), different power curves can be obtained, where the black highlighted curve is called the optimal power curve. This curve is where the wind turbine will operate throughout its lifetime.

The optimal power curve can be projected on the Mechanical Power (Pm) vs. RPM (Ω) or Mechanical Power (Pm) vs. Wind Speed (Vw) axes. The graph of the projected power curve on the above axes is shown below.

Power Curve (P_m vs. V_w)

This curve is obtained by projecting on the axes of mechanical power and wind speed (P-V curve). Figure 2 shows the P-V curve of a 2 MW wind turbine. This curve is provided by the wind turbine manufacturer.

The following important points can be distinguished in the P-V curve:

Cut-in wind speed (V_a): This is the wind speed at which the turbine begins to deliver useful power. It is usually in the range of 3 to 5 m/s.

Rate wind speed (V_n): This is the wind speed at which the nominal power of the wind turbine is reached. It is usually in the range of 10 to 15 m/s and depends on the manufacturer.

Cut-out Wind Speed (V_c): It is the wind speed at which the rotor is stopped by the regulation and control systems in order to avoid the risk of damage that a high wind speed can cause. It is in the range of 20 to 30 m/s.

Wind Turbine Operating Zones

The power-speed curve (Figure 2) shows the five operating zones of a wind turbine. The following is a description of the operating zones:

Zone 1: The wind speed is not strong enough to overcome the internal friction (inertia) of the wind turbine. In this zone, the wind turbine cannot produce useful power because the wind is not able to rotate the turbine blades to the minimum rotational speed. Therefore, it is not possible to extract maximum power from the wind. This is generally found at wind speeds below 3 m/s or 5 m/s (depending on the manufacturer).

Zone 2: This is the operating zone where the Maximum Power Point (MPPT) is tracked. It is also the zone where the wind turbine is most likely to operate. It is generally found between wind speeds of 5 m/s to 12 m/s.

Zone 3 (optional): A zone that occurs in some wind turbine designs when maximum speed is reached but not maximum power is generated. It is a small band of operation before Zone 4, which in some wind turbines is part of Zone 2. The speed must be maintained at the maximum value, even if it is not possible to capture the maximum power from the wind, so it does not operate at the point of maximum power.

Zone 4: This zone begins when the captured power is equal to the rated power. The mechanical power generated and the speed of the wind turbine remain constant at their rated values. To achieve this, the pitch angle is modified when the wind speed varies above the rated wind speed.

Zone 5: This is the zone where the wind speed becomes dangerous for the wind turbine, causing it to rotate at higher than nominal speeds, causing mechanical stress and possible destruction of the wind turbine. The angle is set between 45° and 90° so that the mechanical power extracted from the wind is approximately zero. In addition, the mechanical braking system of the entire drive shaft prevents it from rotating until a wind speed suitable for its operation is detected.

Power Curve (P_m vs. Ω)

This curve is obtained by projecting on the mechanical power and speed axes of the wind turbine (P-Ω curve). Figure 3 shows the P-Ω curve of a 2 MW wind turbine.

Ω_min: Minimum wind turbine speed.
Ω₀: Initial speed of the wind turbine where it starts to produce optimal electrical energy.
Ω₁: Final speed of the wind turbine, which is very close to the maximum speed.
Ω_max: Maximum speed of the wind turbine.

The operation of a wind turbine depends on the wind speed and the rotational speed. On the power surface is the power curve of the wind turbine at which it operates optimally, limited by the blade angle control. The P-V curve shows what the mechanical power of a wind turbine will be at different average wind speeds. This curve can be used to calculate the energy generated by a wind turbine in a given period of time, usually a year.

Reference

[1] Munteanu, I., Bratcu, A. I., Cutululis, N.-A., & Ceanga, E. (2010). Optimal control of wind energy systems: Towards a global approach. Springer.

[2] Anaya-Lara, O., Jenkins, N., Ekanayake, J. B., Cartwright, P., & Hughes, M. (2011). Wind energy generation: Modelling and control: Modelling and control (1a ed.). John Wiley & Sons.