10. Aerodynamic subsystem modeling

The wind turbine blades and its interaction with the wind make up the aerodynamic subsystem to be modeled. The aerodynamic modeling block diagram is shown in Figure 5.

The blades of a wind turbine rotate due to kinetic energy from the wind which is defined by the wind speed. An object with mass (m) which moves at velocity (v) has kinetic energy in the air given by [17] as:

$$E = \frac{1}{2} \times m \times v^2 \tag{2}$$

The power contained in the moving blades assuming constant velocity is equal to the differential of this kinetic energy with respect to time as given in (Eq. (3)).

$$P\_w = \frac{dE}{dt} = \frac{1}{2} \times m \times v^2 \tag{3}$$

where m represents the mass flow rate per second.

When the air crosses the area "A" brushed by blades of the rotor, the power in air can be calculated with (Eq. (4)).

$$P\_w = \frac{1}{2} \times \boldsymbol{\nu}^3 \times \boldsymbol{A} \times \boldsymbol{\rho} \tag{4}$$

Figure 5. Block diagram of the aerodynamic model.

8. Variable speed and fixed speed wind turbines

Wind Solar Hybrid Renewable Energy System

the wind, the turbine efficiency is reduced.

9. Methodology

is shown in Figure 4.

Block diagram of the conversion system model.

Figure 3.

52

The difference between the variable speed and fixed speed wind turbines is whether the rotor is designed to run at different speed or constrained to move at a particular speed. Early wind turbine designs generally operated at constant speed. In this type, the rotor speed does not change regardless of wind speed changes. A converter of power electronic frequency is needed in order to link the variablefrequency output of the wind turbine to the constant electrical system frequency. Power electronics required for different speed wind turbines may be more costly, but they make up for the higher costs by spending more time than fixed turbines working at optimum aerodynamic efficiency [33]. A graph of the performance coefficient versus the tip speed ratio shows this difference clearly. Tip speed ratio is known as the ratio between the angular velocity of the blade tips of a turbine and the wind velocity as shown in Eq. (7). In wind turbine with fixed speed, ω is constant, corresponding to a specific wind speed. Hence for any other speed from

The aim of the wind turbine with variable speed is to always run at optimal efficiency, with tip speed ratio consistency, corresponding to the maximum performance coefficient, by adapting the velocity of the blades to variations of wind speed. Therefore, wind turbines with variable speed designs are ideal for efficient power generation, regardless of the wind speed. Then again, as a result of the fixed speed operation for constant speed turbines, any variations in the speed of the wind are communicated as instabilities in the mechanical torque and then as instabilities in the electrical power grid [17]. This as well as an increased energy capture capability of the variable speed turbine makes the power electronics cost effective [33].

This section introduces the model of the wind turbine while analyzing all three blocks. The equations used in the design model for each block are derived and analyzed as well. The wind energy conversion system modeling is reduced into three subsystems which are the aerodynamics block, mechanical block, and the

The aerodynamics block is responsible for the extraction of power from the wind in the form of kinetic energy necessary to propel the blades. The mechanical block then converts this kinetic energy into mechanical energy used to drive the generator which in turn is turned into electrical energy by the electrical block.

The modeled wind turbine system is designed and simulated with the MATLAB SIMULINK software. The simulation model diagram showing all three subsystems

Therefore, wind turbines with variable-speed are more preferable.

electrical block as shown in the block diagram in Figure 3.

ρ = the density of air.

Density of air can be conveyed as a component of the turbine's rise above sea level (H) as shown in Equation (5).

$$
\rho = \rho\_0 - \mathbf{1.194} \times \mathbf{10^{-4}} \times H \tag{5}
$$

where ρ<sup>0</sup> = 1.225 kg/m3 which is the air density at sea level at temperature T = 298 K. The power extracted from wind is defined as [32],

$$P\_{\text{BLADE}} = \text{Cp}(\lambda, \beta) \times P\_w = \text{Cp}(\lambda, \beta) \times \frac{1}{2} \times v^3 \times A \times \rho \tag{6}$$

Cp = 0.593 (Betz law). The rotor power coefficient is a function of both the tip speed ratio "λ" and the blade pitch angle "β" (in degrees). The blade pitch angle is characterized as the angle between the blade cross-area and the plane of rotation. It alludes to changing the attack angle to best suited angles to adjust the rotation speed of the blades hence adjusting generated power.

Tip speed ratio is defined in Eq. (7),

$$
\lambda = \frac{w\_m \times R}{v} \tag{7}
$$

11. Mechanical subsystem modeling

Simulation schematic diagram of the aerodynamic block.

DOI: http://dx.doi.org/10.5772/intechopen.85064

Figure 7.

55

Figure 6.

Mechanical subsystem of a wind turbine.

The wind turbine mechanical subsystem is known as the drive train. It comprises of a blade pitching component, a hub with blades, a rotor shaft and a gearbox (Figure 7). The mechanical model of the wind turbine will be modeled based on two lumped

Modeling and Simulation of a 10 kW Wind Energy in the Coastal Area of Southern Nigeria…

dt <sup>¼</sup> Te <sup>þ</sup>

where Te is Electromagnetic torque,Tm is Mechanical torque,Tw is wind torque. Since the wind turbine shaft and generator are linked utilizing a gearbox, the shaft of the turbine is not viewed as stiff. Hence there will be movement in the

Tm n

dt <sup>¼</sup> Tw � Tm (14)

(13)

masses assumptions: the gear box mass and the wind wheel mass.

Hg �

shaft. The equation of motion of the drive train shaft is computed as

Hm �

The induction generator equation of motion as given [35] is defined as:

dWg

dWm

wm is angular velocity of the rotor, R blade radius, and "wm\*R" is the blade tip speed. The rotor torque is therefore defined as,

$$\mathbf{T\_w} = \frac{P\_{BLADE}}{w\_m} = \frac{\mathbf{C}p(\lambda, \boldsymbol{\beta}) \times \frac{1}{2} \times \boldsymbol{\nu}^3 \times \mathbf{A} \times \boldsymbol{\rho}}{w\_m} \tag{8}$$

And, A, the area covered by the blade

$$A = \pi \times \mathbb{R}^2\tag{9}$$

Substitute Eq. (9) into Eq. (8), giving Eq. (10).

$$\mathbf{T\_w} = \frac{P\_{BLADE}}{w\_m} = \frac{\mathbf{C}p(\lambda, \beta) \times \frac{1}{2} \times \boldsymbol{\nu}^3 \times \boldsymbol{\pi} \times \mathbf{R}^2 \times \boldsymbol{\rho}}{w\_m} \tag{10}$$

The power coefficient Cp can be expressed as shown in Eq. (11).

$$\mathbf{C}p(\hbar,\beta) = c\_1 \times \left(c\_2 \times \frac{1}{\mathcal{Y}} - c\_3 \times \beta - c\_4 \times \beta^x - c\_5\right) \times e^{\frac{-c\_6}{\mathcal{Y}}} \tag{11}$$

where gamma "y" is given as [32],

$$\frac{1}{y} = \frac{1}{\lambda + 0.08\beta} - \frac{0.035}{1 + \beta^3} \tag{12}$$

where c1–c6 are the aerodynamic coefficients given as c1 is 0.5176; c2 is 116; c3 is 0.4; c4 is 5; c5 is 21; and c6 is 0.0068.

The simulation schematic of the aerodynamic modeling is shown in Figure 6.

The equations used are (Eq. (7)) for the Lambda, the beta is the pitch angle for the wind turbine blades. The gamma function block uses (Eq. (11)), the power coefficient is modeled using (Eq. (11)) and the wind torque is modeled using (Eq. (10)).

Modeling and Simulation of a 10 kW Wind Energy in the Coastal Area of Southern Nigeria… DOI: http://dx.doi.org/10.5772/intechopen.85064

Figure 6. Simulation schematic diagram of the aerodynamic block.
