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Wind energy

Abdul R. Beig

S.M. Muyeen Department of Electrical Engineering, The Petroleum Institute, Abu Dhabi, UAE

Abstract

Wind energy is one of the promising renewable energy sources. This chapter presents the wind energy basics, the technology used in harvesting wind energy, and the physics related to wind energy. Brief discussions about the different types of wind energy turbines and generator configurations are presented. The chapter concludes with the advantages and disadvantages of wind energy systems.

Keywords

gearbox

horizontal axis

lift to drag ratio

power coefficient

pitch angle control

rotor blades

vertical axis

wind energy

wind farm

wind turbine

wind generator

4.1. Introduction

Conventional energy sources such as natural gas, oil, coal, or nuclear are finite but still hold the majority of the energy market. However, renewable energy sources like wind, fuel cells, solar, biogas/biomass, tidal, geothermal, etc. are clean and abundantly available in nature and hence are competing with conventional energy sources. Among the renewable energy sources wind energy has a huge potential of becoming a major source of renewable energy for this modern world. Wind power is a clean, emissions-free power generation technology. As per the Global Wind Energy Council (GWEC) 2013 statistics, cumulative global capacity has reached to a total of 318 GW, which shows an increase of nearly 200 GW in the past 5 years. GWEC predicts that wind power could reach nearly 2000 GW by 2030, supply between 16.7% and 18.8% of global electricity and help save over 3 billion tons of CO₂ emissions annually. From this scenario, it is clear that wind power is going to dominate the renewable as well as the conventional energy market in the not too distant future. Wind energy is the only power generation technology that can deliver the necessary cuts in CO₂ emissions from the power sector in the critical period up to 2020, when greenhouse gas emissions must peak and begin to decline if we are to have any hope of avoiding the worst impacts of climate change. However, grid integration, voltage, and power fluctuation issues should adequately be addressed due to the huge penetration of wind power to the grid.

4.2. Wind turbine

The wind turbine is the most essential part of the wind energy system. It converts the kinetic energy associated with wind (known as wind energy) into mechanical energy and then to electrical energy. Historically windmills are used for lifting water, where wind energy is converted to mechanical energy [1]. The application of windmills for water lifting purpose dates back to as early as 644AD [1]. Several different types of windmills were in use until the twentieth century in different parts of the world for different types of application, such as lifting water, pumping water, lifting heavy materials such as logs, milling grains, etc. In 1891, Poul La Cour of Denmark first produced electricity in direct current (DC) form from a wind turbine [1]. His wind turbine was mainly based on the traditional windmill technology and was capable of producing small amounts of electricity. Since then there have been major improvements in wind turbine technology and currently there are several wind farms successfully installed in different parts of world generating large amounts of power of the order of a few thousand megawatts. The following section will briefly explain different parts of the wind turbine: rotor blades, gearbox, generator, tower, yawing, brakes, cables, anemometer, and pitch angle. These parts for the horizontal axis type turbine are shown in Figure 4.1.

Figure 4.1 Parts of the horizontal axis wind turbine.

4.2.1. Rotor blades

Rotor blades are the most important parts of a wind turbine in terms of performance and cost of the wind power system. The shape of the rotor blades has a direct impact on performance as this decides the conversion of kinetic energy associated with the wind to mechanical energy (torque). In these types of wind turbines, the blades are designed to have a high lift to drag ratio, based on aerodynamic principles. The number of blades is selected for aerodynamic efficiency, component costs, and system reliability. Theoretically, an infinite number of blades of zero width are the most efficient, operating at a high value of tip speed ratio. But other considerations such as manufacturing, reliability, performance, and cost restrict wind turbines to only a few blades. The majority of wind turbines are the horizontal axis type with three blades. The turbine blades must have low inertial and good mechanical strength for durable and reliable operation. The blades are made up of aluminum or fiberglass reinforced polyester, carbon fiber reinforced plastics, or wood or epoxy laminates [2,3]. A schematic diagram of a rotor blade is given in Figure 4.2. The exterior shape of the blades is based on aerodynamics but the interior is determined by attention to strength. In low power turbines, the blades are directly bolted to the hub and hence are static. In high power turbines, the blades are bolted to the pitch mechanism, which adjusts their angle of attack according to the wind speed to control their rotational speed. The pitch mechanism is bolted to the hub. The blade consists of a spar, which is a continuously tapered longitudinal beam that provides necessary stiffness and strength to withstand the wind load and carry blade weight. The spar is integrated to the hub. Around the spar two aerodynamically shaped shells are mounted and two edges of the shells are sealed. The hub is fixed to the rotor shaft, which drives the generator directly or through a gearbox. Dirt will deposit on the rotor blade surfaces, which will affect the performance of the turbine. Therefore, the blade surfaces will need frequent cleaning and, if required, polish to keep the blade performance uniform.

Figure 4.2 Schematic diagram of the blade.

4.2.2. Nacelle

A nacelle is a box type structure that sits at the top of the tower and is attached to the rotor. The nacelle houses all of the generating components in a wind turbine, including the generator, gearbox, drive train, and brake assembly. The nacelle is made up of fiberglass and protects the wind turbine components from the environment. Modern large farms have a helicopter-hoisting platform built on top of the nacelle, capable of supporting service personnel.

4.2.3. Gearboxes

Mechanical power from the rotation of the wind turbine rotor is transferred to the generator rotor through the main shaft, the gearbox, and high-speed shaft. The wind turbine rotates at very slow speed. This requires a large number of poles to the generator. For economic and optimal design it is necessary to have the gearbox between the wind turbine shaft and generator shaft to increase the speed. The gearbox is of a fixed speed ratio and mainly increases speed. The gear ratio is in the range 20–300 [3]. Lubricating oil is used in the gearbox to reduce friction.

4.2.4. Generators

The generators convert the mechanical energy into electrical energy. The generator has widely varying mechanical input. It is usually connected to the grid in high power systems. In low power ratings the generator may be working in isolation supplying power to the local grid. The generator generally produces variable frequency, variable voltage, three-phase alternating current (AC). This voltage is usually converted to DC, then to regulated and fixed frequency AC using an AC-to-DC/DC-to-AC converter.

4.2.5. Tower

Towers are used to mount the wind turbine. Wind energy yield increases with the height. But optimal design limits the height of the tower, as the cost of the tower will be very high if it is too tall. Towers are usually made of tubular steel or concrete. Tubular towers are conical in shape with their diameter decreasing toward the tip. Steel towers are expensive. An alternate solution is concrete towers.

4.2.6. Yaw mechanism

Horizontal axis wind turbines use forced yawing where generators and gearboxes keep the rotor blades perpendicular to the direction of the wind. The upwind machines use brakes on the yaw mechanism. The yaw mechanism is activated by automatic control, which monitors the rotor. Cable carries the current from the wind turbine down through the tower. The yaw mechanism also should be equipped to protect the cable should it become twisted. Besides the role of tracking wind direction, the yaw mechanism also places an important role on connecting the tower with the nacelle.

4.2.7. Brakes

There are three main types of braking mechanism, namely aerodynamic brakes, electro brakes, and mechanical brakes. In the case of aerodynamic brakes, the blades are tuned such that the lift effect disappears. In electro blades, the electrical energy is dumped into a resistor bank. In mechanical type brakes, the disc or drum brakes are used to lock the blades.

4.2.8. Protection of turbines

Wind turbines need to be protected against overheating, overspeed, and overloading. Vibration is one of the main sources of turbine failure. So a vibration monitoring and protection system is to be installed with the turbine. Wind direction and wind speed are also important for the satisfactory operation of the turbine. The cup anemometer is used for this purpose. Since the wind turbine has rotating parts, lubricating systems are required. This is either a forced circulation or pressurized lubricating system. The other measuring or sensors are temperature of the gearbox, temperature of the generator, voltage–frequency measurement, speed measurement, etc.

4.3. Kinetic energy of wind

Kinetic energy in a parcel of air of mass m flowing at speed v_w in the x direction is:

$E_{w} = \frac{1}{2} m v_{w}^{2} = \frac{1}{2} (ρ A x) v_{w}^{2}$ $E_{w} = \frac{1}{2} m v_{w}^{2} = \frac{1}{2} (ρ A x) v_{w}^{2}$

(4.1)

where, E_w is the kinetic energy in joules, A is the cross-sectional area in m², ρ is the air density in kg/m³, and x is the thickness of the parcel in m.

If we visualize the parcel as in Figure 4.3 with side x moving with speed v_w (m/s), and the opposite side fixed at the origin, we see the kinetic energy increasing uniformly with x, because the mass is increasing uniformly. The power in the wind P_w is the time derivative of the kinetic energy, given by (4.2):

$P_{w} = \frac{d E_{w}}{d t} = \frac{1}{2} ρ A v_{w}^{2} \frac{d x}{d t} = \frac{1}{2} ρ A v_{w}^{3}$ $P_{w} = \frac{d E_{w}}{d t} = \frac{1}{2} ρ A v_{w}^{2} \frac{d x}{d t} = \frac{1}{2} ρ A v_{w}^{3}$

(4.2)

Figure 4.3 Packet of air moving in the x direction with speed v_w.

Thus, the wind power is directly proportional to the cross-sectional area and the cube of the wind velocity.

4.4. Aerodynamic force

4.4.1. Ideal wind turbine output

Ideal wind turbine output can be viewed as the power being supplied at the origin to cause the energy of the parcel to increase according to Equation (4.1). A wind turbine will extract power from side x with Equation (4.2) representing the total power available at this surface for possible extraction.

The physical presence of a wind turbine in a large moving air mass modifies the local air speed and pressure as shown in Figure 4.4. The picture is drawn for a conventional horizontal axis propeller type turbine.

Figure 4.4 Circular tube of air flowing through ideal wind turbine.

Consider a tube of moving air with initial or undisturbed diameter d₁, speed v_w1, and pressure p₁, as it approaches the turbine. The speed of the air decreases as the turbine is approached, causing the tube of air to enlarge to the turbine diameter d₂. The air pressure will rise to the maximum just in front of the turbine and will drop below atmospheric pressure behind the turbine. Part of the kinetic energy in the air is converted to potential energy in order to produce this increase in pressure. Still more kinetic energy will be converted to potential energy after the turbine, in order to raise the air pressure back to atmospheric. This causes the wind speed to continue to decrease until the pressure is in equilibrium. Once the low point of wind speed is reached, the speed of the tube of air will increase back to

v_{w 4} = v_{w 1}

$v_{w 4} = v_{w 1}$

as it receives kinetic energy from the surrounding air [2,4,5].

It can be shown [2,4,5] that under optimum conditions, when maximum power is being transferred from the tube of air to the turbine, the following relationships hold:

$v_{w 2} = v_{w 3} = \frac{2}{3} v_{w 1}; v_{w 4} = \frac{1}{3} v_{w 1}$ $v_{w 2} = v_{w 3} = \frac{2}{3} v_{w 1}; v_{w 4} = \frac{1}{3} v_{w 1}$

(4.3a)

$A_{2} = A_{3} = \frac{3}{2} A_{1}; A_{4} = 3 A_{1}$ $A_{2} = A_{3} = \frac{3}{2} A_{1}; A_{4} = 3 A_{1}$

(4.3b)

The mechanical power extracted is then the difference between the input and output power in the wind:

$P_{m, ideal} = P_{1} - P_{4} = \frac{1}{2} ρ (A_{1} v_{w 1}^{3} - A_{4} v_{w 4}^{3}) = \frac{1}{2} ρ (\frac{8}{9} A_{1} v_{w 1}^{3})$ $P_{m, ideal} = P_{1} - P_{4} = \frac{1}{2} ρ (A_{1} v_{w 1}^{3} - A_{4} v_{w 4}^{3}) = \frac{1}{2} ρ (\frac{8}{9} A_{1} v_{w 1}^{3})$

(4.4)

This states that eight-ninths of the power in the original tube of air is extracted by an ideal turbine. This tube is smaller than the turbine, however, and this can lead to confusing results. The normal method of expressing this extracted power is in terms of the undisturbed wind speed v_w1 and the turbine area, A₂. This method yields:

$P_{m, ideal} = \frac{1}{2} ρ [\frac{8}{9} (\frac{2}{3} A_{2}) v_{1}^{3}] = \frac{1}{2} ρ (\frac{16}{27} A_{2} v_{1}^{3})$ $P_{m, ideal} = \frac{1}{2} ρ [\frac{8}{9} (\frac{2}{3} A_{2}) v_{1}^{3}] = \frac{1}{2} ρ (\frac{16}{27} A_{2} v_{1}^{3})$

(4.5)

The factor 16/27 = 0.593 is called the Betz coefficient. It shows that an actual turbine cannot extract more than 59.3% of the power in an undisturbed tube of air of the same area. In practice, the fraction of power extracted will always be less because of mechanical imperfections. A good fraction is 35–45% of the power in the wind under optimum conditions. A turbine, which extracts 40% of the power in the wind, is extracting about two-thirds of the amount that would be extracted by an ideal turbine. This is rather good, considering the aerodynamic problems of constantly changing wind speed and direction as well as the frictional loss due to blade surface roughness [5].

4.5. Power output from practical turbines

The fraction of power extracted from the power in the wind by a practical wind turbine is usually given by the symbol C_p, standing for the coefficient of performance or power coefficient. Using this notation and dropping the subscripts of Equation (4.3), the actual mechanical power output can be written as:

$P_{m} = C_{p} (\frac{1}{2} ρ A v_{w}^{3}) = \frac{1}{2} ρ π R^{2} v_{w}^{3} C_{p} (λ, β)$ $P_{m} = C_{p} (\frac{1}{2} ρ A v_{w}^{3}) = \frac{1}{2} ρ π R^{2} v_{w}^{3} C_{p} (λ, β)$

(4.6)

where, R is the blade radius of wind turbine (m), v_w is the wind speed (m/s), and ρ is the air density (kg/m³). The coefficient of performance is not constant, but varies with the wind speed, the rotational speed of the turbine, and turbine blade parameters like angle of attack and pitch angle. Generally it is said that power coefficient C_p is a function of tip speed ratio λ and blade pitch angle β (°).

4.6. Tip speed ratio

Tip speed ratio is the ratio of the circumferential velocity of the rotor at the end of the blade, that is, the maximum velocity v_m and the wind velocity v_w in front of the rotor blade. Originally it was defined as:

$λ = \frac{v_{m}}{v_{w}}$ $λ = \frac{v_{m}}{v_{w}}$

(4.7)

A more popular form of tip speed ratio in the wind industry is as follows:

$λ = \frac{ω_{R} R}{v_{w}}$ $λ = \frac{ω_{R} R}{v_{w}}$

(4.8)

where, ω_R is the mechanical angular velocity of the turbine rotor in rad/s and v_w is the wind speed in m/s.

4.7. Coefficient of performance and turbine efficiency

There will be energy loss in the mechanical components of the rotor, gear system, and generator. So the overall efficiency can be obtained as:

$η = C_{p} η_{m} η_{g}$ $η = C_{p} η_{m} η_{g}$

where η_m is the mechanical efficiency and η_g is the generator efficiency.

$η = \frac{P_{o}}{(1 / 2) ρ A v_{w}^{3}}$ $η = \frac{P_{o}}{(1 / 2) ρ A v_{w}^{3}}$

where P_o is the electrical output power.

Modeling of a wind turbine rotor is somewhat complicated. According to the blade element theory, modeling of blade and shaft needs complicated and lengthy computations. Moreover, it also needs detailed and accurate information about rotor geometry. For that reason, considering only the electrical behavior of the system, a simplified method of modeling of the wind turbine blade and shaft is normally used.

Typical C_p–λ curves for MOD-2 wind turbine is shown in Figure 4.5 for different values of β [6,7].

Figure 4.5 C_p–λ curves for different pitch angles.

4.8. Operating range of wind turbine

Wind turbines are allowed to run only in a well-defined range of wind speed. A minimum wind speed is required for the blades to overcome inertia and friction. This minimum speed is called cut-in wind speed (v_cut-in). The typical value of cut-in wind speed is 3–5 m/s. At very high wind speed, say 25 m/s, in order to avoid damage to wind turbines, the wind turbines are stopped from rotating. This is called cut-out speed (v_cut-out). The operating range of a wind turbine can be best explained by a wind power curve as shown in Figure 4.6.

Figure 4.6 Output power versus wind speed.

In the normal wind speed range the turbine will be able to produce rated power. Rated wind speed (v_r) is the wind speed at which the turbine generated rated power. A typical value is about 12–16 m/s. This corresponds to the point at which the conversion efficiency is maximum.

4.9. Classifications of wind turbines

Wind turbines can be classified into two categories, namely (1) horizontal axis and (2) vertical axis wind turbines based on their constructional design.

4.9.1. Horizontal axis wind turbine

Almost all of the commercially established wind energy systems use horizontal type wind turbines. The axis of rotation is horizontal. The major advantage of the horizontal type wind turbine is that by using blade pitch control, the rotor speed and power output can be controlled. Also blade pitch control protects the wind turbine against overspeed when the wind speed becomes dangerously high. The basic principle of a horizontal axis wind turbine is based on propeller-like concepts, so the technological advances of the propeller design are readily incorporated to develop modern highly efficient wind turbines [1]. Figure 4.1 shows the schematic arrangement of a horizontal axis wind turbine. Details of the different parts of the horizontal wind turbine are given in Section 4.1.

4.9.2. Vertical axis wind turbine

The vertical turbine proposed in 1925 by Darrius had some promising features for modern wind energy farms. The blades are curved and the rotor has vertical axis rotation. Figure 4.7 shows the schematic diagram of the Darrius type vertical axis turbine. Compared to the horizontal axis the shape of vertical axis blades is complicated, making them difficult to manufacture. The H-rotor vertical axis wind turbine uses straight blades instead of curved blades as shown in Figure 4.8. The blades are fixed to a rotor though struts. There are other types of vertical axis wind turbines, namely the Savonius type and V-shaped vertical axis turbines [1,2]. These have very low tip speed ratio and low power coefficient, hence they are used only in very low power wind energy systems.

Figure 4.7 Darrius type vertical axis turbine.

Figure 4.8 H type vertical axis turbine.

The vertical axis type generator has a simple design. The shaft is vertical, so the generator is mounted on the ground and the tower is required only to mount the blades. The disadvantages are the tip speed ratio and power output are very low compared to horizontal axis generators. The turbine needs an initial push to start; it is not self-starting. Also it is not possible to control the power output by pitching the rotor blades. Support wires or guy wires are required in addition to the tower. Due to these reasons not much attention is given to vertical axis wind turbines.

4.10. Types of wind turbine generator systems

A wind turbine generator system (WTGS) transforms the energy present in the wind into electrical energy. As wind is a highly variable resource that cannot be stored, operation of the WTGS must be done according to this feature. Based on the rotational speed of the wind turbine, WTGSs can be broadly classified under two major categories, namely fixed speed and variable speed.

4.10.1. Fixed speed wind turbine generator system

A fixed speed WTGS consists of a conventional, directly grid coupled squirrel cage induction generator, which has some superior characteristics such as brushless and rugged construction, low cost, maintenance free, and operational simplicity. The slip, and hence the rotor speed of a squirrel cage induction generator, varies with the amount of power generated. These rotor speed variations are, however, very small, approximately 1–2% from the rated speed. Therefore, this type of wind energy conversion system is normally referred to as a constant speed or fixed speed WTGS. The advantage of a constant speed system is that it is relatively simple. Therefore, the list price of constant speed turbines tends to be lower than that of variable speed turbines. However, constant speed turbines must be more mechanically robust than variable speed turbines [3]. Since the rotor speed cannot be varied, fluctuations in wind speed translate directly into drive train torque fluctuations, causing higher structural loads than with variable speed operation. This partly cancels the cost reduction achieved by using a relatively cheap generating system (Figure 4.9).

Figure 4.9 Schematic diagram of a fixed speed wind turbine generator system.

4.10.2. Variable speed wind turbine generator system

The currently available variable speed wind turbine (VSWT) generator system topologies are shown in Figures 4.10 and 4.11. To allow variable speed operation, the mechanical rotor speed and the electrical frequency of the grid must be decoupled. Therefore, a power electronic converter is used in the variable speed wind generator system. In the doubly fed induction generator, a back-to-back voltage source converter feeds the three-phase rotor winding. In this way, the mechanical and electrical rotor frequencies are decoupled and the electrical stator and rotor frequencies can be matched independently of the mechanical rotor speed. In the direct drive synchronous generator system (PMSG or WFSG), the generator is completely decoupled from the grid by a frequency converter. The grid side of this converter is a voltage source converter, that is, an insulated gate bipolar transistor (IGBT) bridge. The generator side can either be a voltage source converter or a diode rectifier.

Figure 4.10 Schematic diagram of a variable speed wind turbine generator system using a wound rotor induction motor.

Figure 4.11 Schematic diagram of a variable speed wind turbine generator system using a synchronous generator.

The generator is excited using either an excitation winding (in the case of a WFSG) or permanent magnets (in the case of a PMSG). In addition to these three mainstream generating systems, there are some other varieties as explained in Refs [6,7]. One that must be mentioned here is the semivariable speed system. In a semivariable speed turbine, a wound rotor induction generator is used. The output power is regulated by means of rotor resistance control, which is achieved by means of power electronics converters. By changing the rotor resistance, the torque/speed characteristic of the generator is shifted and about a 10% rotor speed decrease from the nominal rotor speed is possible. In this generating system, a limited variable speed capability is achieved at relatively low cost. Other variations are a squirrel cage induction generator or a conventional synchronous generator connected to the wind turbine through a gearbox and to the grid by a power electronics converter of the full rating of the generator.

For each instantaneous wind speed of a VSWT, there is a specific turbine rotational speed, which corresponds to the maximum active power from the wind generator. In this way, the maximum power point tracking (MPPT) for each wind speed increases the energy generation in the VSWT [8,9]. This is illustrated in Figure 4.12.

Figure 4.12 Turbine characteristic with maximum power point tracking.

When the wind speed changes, the rotational speed is controlled to follow the maximum power point trajectory. It should be mentioned here that the measurement of the precise wind speed is difficult. Therefore, it is better to calculate the maximum power, P_max, without the measurement of wind speed as shown in the succeeding section.

4.11. Wind farm performance

Output of a wind farm mainly depends on wind speed pattern. In the case of a VSWT, maximum power is extracted from the wind, which maximizes the wind generator output power. The voltage fluctuation issue is handled by the power electronics converters equipped with variable speed wind generator [9].

However, in the case of a fixed speed wind generator, the power and terminal voltage of a wind generator or wind farm varies randomly. This is because wind is stochastic and intermittent, which causes power variation at the wind generator or wind farm terminal. Figure 4.13 shows wind speeds that are used to generate electric power in a wind farm where there exist five wind generators. Figure 4.14 shows the output power of a wind farm. A capacitor bank is usually connected to the terminal of a fixed speed wind generator, which is designed in such a way to maintain unity power factor at rated wind speed. Therefore, at lower wind speed we have excess or surplus reactive power at the terminal of the wind generator, which will cause an overvoltage issue in the wind farm. This is depicted in Figure 4.15.

4.12. Advantages and disadvantages

4.12.1. Advantages

• Wind energy is environment friendly as no fossil fuels are burnt to generate electricity from wind energy.

• Wind turbines take up less space than the average power station.

• Modern technologies are making the extraction of wind energy much more efficient. Wind is free, so only installation cost is involved and running costs are low.

• Wind energy is the most convenient resource to generate electrical energy in remote locations, where conventional power lines cannot be extended due to environmental and economic considerations.

4.12.2. Disadvantages

• The main disadvantage of wind energy is varying and unreliable wind speed. When the strength of the wind is too low to support a wind turbine, little electricity is generated.

• Large wind farms are required to generate large amounts of electricity, so this cannot replace the conventional fossil fueled power stations. Wind energy can only substitute low energy demands or isolated low power loads.

• Larger wind turbine installations can be very expensive and costly to surrounding wildlife during the initial commissioning process.

• Noise pollution may be problem if wind turbines are installed in the densely populated areas.

4.13. Summary

This chapter explained the various parts of a wind turbine. Different types wind turbines were presented. Wind dynamics and basic principles of converting wind energy to electric energy were discussed and the different types of wind turbine configurations and generation systems were described.

Problems

1. Find the diameter of a wind turbine rotor that will generate 100 kW of electrical power in a steady wind speed of 8 m/s. Assume that the air density is 1.225 kg/m³, C_p = 16/27, and η = 1.

2. A 50-m diameter, three-bladed wind turbine produces 800 kW at a wind speed of 15 m/s. The air density is 1.225 kg/m³. Find:

a. The rotational speed of the rotor at a tip speed ratio of 5.0.

b. The tip speed.

c. The gear ratio if the generator speed is 1600 rpm.

d. The efficiency of the wind turbine system.

3. A wind turbine is operating in steady winds at a power of 1800 kW and a speed of 30 rpm. Suddenly the connection to the electrical network is lost and the brakes fail to apply. Assuming that there are no changes in the aerodynamic forces, how long does it take the operating speed to double? Take J = 4 × 10² kg/m².

4. A standalone single-phase wind turbine generator generates 220 V, 50 Hz AC. The output of the generator is connected to a diode bridge full-wave rectifier, which produces a fluctuating DC voltage.

a. Find the average DC voltage if the output of the generator is rectified using a full-wave diode rectifier.

b. Find the average DC voltage if the output of the generator is rectified using a full-wave fully controlled SCR rectifier. The firing angle is 45°.

5. A four-pole induction generator is rated at 500 kVA and 400 V. It has the following parameters X_LS = X_LR = 0.15 Ω, R_S = 0.014 Ω, R_R = 0.013 Ω, X_M = 5 Ω.

a. How much power does it produce at a slip of −0.025 (take synchronous speed as 1500 rpm)?

b. Find its speed.

c. Find the torque and power factor.

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