#### 2.1. Wind Turbine Model

The extracted power from the wind at any wind speed and turbine rotational speed is given by [

18]

Equation (1) indicates that the power coefficient is influenced by the TSR and the blade pitch angle, which is always below the rated wind speed. In wind energy systems, when the pitch angle is kept constant, the power captured will be a function only in the turbine rotational speed, maximum at a certain speed. Hence, to maximize the captured power, $\lambda $ should be kept at its optimum value ${\lambda}_{opt}$, and then the captured power is technically computed from the wind speed and the blade radius. When the wind speed is above the rated speed, the pitch angle adjusts to reduce the impact of excessive wind speed on the generated power. Here, β is kept constant for simplicity.

Figure 2 presents the turbine blade’s power variation with wind speed and rotational speed. The maximum power output occurs at a particular rotational speed.

#### 2.2. Wind Shear and Tower Shadow Model

Different wind speeds at different altitudes and the passing of the blade in front of the tower result in periodic torque pulsations in wind turbines.

Based on the height, the wind speed changes as follows [

19,

20]:

Table 1 summarizes the geographical features and represents a general wind shear index [

1].

As a result of height-dependent wind speeds, the turbine blades oscillate in response to the wind shear. Additionally, the amplitude of the wind speed increases with the height for different rotating angles from 0° to 360°. The wind speed in Equation (2) can be rewritten with respect to the speed at the turbine hub as follows [

21]:

Considering the blade’s rotation and the unique wind shear at each angle, instead of expressing the wind shear in only the

$z$ direction, the wind shear in Equation (6) can be expressed as a function of the blade rotational angle

${\psi}_{b}$ and radial distance

$r$ from the rotor axis. The wind shear torque can be modeled as [

22]:

which can be written as

Equation (5) can be represented as a function of the wind shear disturbance

$F\left(r,{\psi}_{b}\right)$:

Due to wind shear, the wind speed can be expanded using Fourier transform as shown in the following Equation [

22]:

In low altitude, wind shear is influenced by the large amount of change in wind speed due to the friction of the surface. In addition, the longer the blade length of the wind turbine, the wider the turning radius of the rotor, and the larger the wind shear effect is.

Figure 3 shows the wind shear effect of the turbine blade length,

r hub height, and

H empirical wind shear index as in Equation (7).

In the case of a large-diameter wind turbine, the blades experience a wide range of wind speeds in each revolution. With three-blade wind turbines, the frequent rotation through this wide range of speeds results in torque and power oscillations at three times the rotor rotational speed, called 3p frequency.

The tower radius (

a), the distance of the blade origin from the tower midline (

x), and the lateral distance of the blade from the tower midline (

y) affect the tower shadow amplitude, as shown in

Figure 4.

In front of the tower, the disturbance is nonnegligible and affects the wind speed and turbine output power. The wind speed reduces by a value of

$\Delta $ in front of the tower and within the region

W of 3–6 times of the tower diameter, as shown in

Figure 5.

The tower shadow as a function of

x and

y is given by [

23]

Expressing the tower shadow as a function of

$r$ and

${\psi}_{b}$,

where

$m=1+\frac{\alpha \left(\alpha -1\right){R}^{2}}{8{H}^{2}}$.

Figure 6 shows wind shear and tower shadow effects at the tip of each blade at an average wind speed of 8 m/s. The tower shadow shape was calculated for different radial circulating points, from 8 m to 20 m in diameter, measured from the hub center, with an incremental distance of 8 m, as shown in

Figure 6. When the diameter increases, the points on this circle perimeter experience short tower shadow intervals. The inner curve in

Figure 6 indicates the circle with a diameter of 20 m, which has the lowest tower shadow interval. In contrast, the outer and widest curve in

Figure 7 illustrates the shadow effect at a radius of 4 m. The parameters used in this test are

$R=10\mathrm{m},H=25\mathrm{m},\alpha =0.15,a=0.75\mathrm{m},$ and

$x=3\mathrm{m}.$ If the radial distance

r is kept constant and the distance between the blade origin to the tower midline

x varies from 3 m to 5 m, as

Figure 8 illustrates, the tower shadow effect increases when the distance

x is small. Therefore, the blades should be installed far from the tower to reduce the tower shadow effect. The tower shadow is given by

For any wind speed value, the turbine torque

T_{t} can be expressed as:

The wind shear module implements the disturbance in torque due to wind shear given as

and the tower shadow module implements the disturbance in torque due to the tower shadow expressed as

Due to turbine torque, wind shear, and tower shadow, the resultant aerodynamic torque is defined as Equation (15a).

However, the wind speed at the hub level is not accurate when modeling the turbine torques. Hence, a wind speed estimation method can be used to estimate the average wind speed to resemble real characteristics. The resultant torque in Equation (15a) can be expressed as Equation (15b):