Wind Speed Dynamics Model

The annual wind speed distribution of West Beacon Farm can be described using the Rayleigh probability function [109]:

𝑓.𝑝𝑑𝑓(𝑣) = _𝛼𝑣2𝑒𝑥𝑝 [− (

𝑣 √2𝛼)

2

] (3-1)

where 𝑣 is the instantaneous wind speed and alpha, 𝛼, is given by:

𝛼 = 𝑣𝑚̅√2/𝜋 (3-2)

which relates to the notably long time scale of the mean wind speed, 𝑣𝑚̅, which is

measured in m/s.

The standard deviation, 𝑆𝐷, and the turbulence intensity, 𝐼, are calculated using Equations (3-3) and (3-4) [18] , respectively

𝑆𝐷 = √_𝑁1 𝑠−1∑ (𝑣 − 𝑣𝑚̅ ) 2 𝑁𝑠 𝑖−1 (3-3) 𝐼 =√𝜎_𝑣 2 𝑚 ̅̅̅ = 𝜎 𝑣𝑚̅̅̅ (3-4)

where 𝑁_𝑠 is the number of samples during each short-time interval and 𝜎 is the standard deviation. Aerodynamic torque Wind speed Model Aerodynamic

Model Drive train _Model

SCIG + Power converter

Model Wind

speed Generator speed

Rotor

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Figure 3-3 shows the annual Rayleigh distribution for West Beacon Farm , Loughborough (solid line), which has a mean wind speed of 4.5 m/s [110, 111] at 24 metres hub height. Considering that the area under the curve is unity, Equation (3-3) provides the standard deviation, 𝑆𝐷 = 0.59 m/s. Thus, the turbulence intensity, over ten-minute period, 𝐼 = 0.13. Also, from this figure, it should be noted that this mean wind speed is not the most commonly occurring wind speed which is somewhat less than this, at around 3.7 m/s. To verify the authentic of the simulated Rayleigh distribution curve, the simulated curve (solid line) is then compared with the Rayleigh distribution that has been measured at Rutherford Appleton Laboratory (RAL), UK (dashed line)[112] . As RAL is located in England and has almost same mean wind speed and standard deviation as estimated in Loughborough, the simulated curve reveals insignificant difference with the measured curve at RAL.

Figure 3-3 Annual wind speed at West Beacon Farm based on the Rayleigh distribution

An equivalent wind speed model is developed to simulate the actual torque experienced by the rotor at West Beacon Farm. The equivalent or actual

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 2 4 6 8 10 12 14 16 18 20 f.p d f( v) Wind speed (m/s)

Rayleigh distribution for Loughborough annual wind speed

Measured Modeling

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instantaneous velocity is a superposition of the turbulence wind and the mean wind speed components and can be written as

𝑣𝑒𝑞(𝑡) = 𝑣𝑚̅(𝑡) + 𝑣𝑡(t) (3-5)

where 𝑣_𝑒𝑞(𝑡) is the equivalent or actual instantaneous wind speed velocity, 𝑣_𝑚̅(𝑡) is the mean wind speed component that determines the turbine’s current operating point, 𝑣_𝑚̅(𝑡) is assumed to be constant after several minutes and 𝑣𝑡(t)

is a turbulence component that displays high frequency oscillations near the operating points. The frequency and amplitude of the wind speed variance is limited by 𝑣𝑡(t).

The equivalent or actual instantaneous wind speed behaviour can be modelled as shown in Figure 3-4. This model makes the following assumptions:

(a) The wind shear effect is ignored. Wind shear is the variation of wind speed with elevation. Wind shear is typically influenced by several factors such as atmospheric stability, surface roughness, changes in surface conditions and terrain shape. For wind power developers, these details of wind shear are important to be obtained to determine the actual potential electricity production and economic feasibility study of a candidate wind turbine site [18, 113, 114], particularly if the turbine blade has a diameter more than 100 metres [113]. This is important because the wind shear plays important role in determining the suitable turbine hub height. However, since these issues are not one of the thesis’s aims in this study, wind shear effect was disregarded. In addition, for small turbine with 10 metres in diameter and turbine hub height of 24 metres, no significant effect may affect the energy yield estimation [113]. Furthermore, ignorance of wind shear in modelling work is acceptable since wind shear extrapolation may have a large ambiguity because the wind shear models do not always describe reality [18].

(b) The wind speed variations in the horizontal direction are not considered. Perfect tracking in the yaw direction is assumed; in practice, this is impossible and results in energy losses at least 1-2% [109] and additional stresses on the components.

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Figure 3-4 Equivalent wind speed simulation model

The mean wind speed, which is also known as the medium- and long-term component, can be denoted as [18, 109]

𝑣_𝑚̅(𝑡) = ∑𝑁𝑖=0𝐴𝑖cos (𝜔𝑖𝑡 + 𝜑𝑖) (3-6)

where 𝑁 is the number of samples during each medium- and long-term interval and 𝜑𝑖 is the phase of the harmonics, which is generated randomly with a

uniform distribution in the range of [𝜋, −𝜋]. If 𝑖 = 0, and 𝜔₀ = 0, 𝜑₀ = 0, then 𝐴₀ = 𝑣_𝑚̅ where 𝑣𝑚̅ is the mean wind speed.

A distributed white noise signal is generated using a random number generator to denote the driving force of the wind. The white noise signal is subsequently smoothed with a signal-shaping filter to transform the white noise signal into a coloured noise signal. The signal-shaping filter can be designed using the Kaimal or Von Karman spectrum model. These models are popular for modelling short-term turbulence characteristics in wind velocity series simulations [4]. In this work, the von Karman spectrum model is used to shape the white noise signal. The transfer function of the Von Karman shaping filter can be expressed as [109]

𝐺𝐹(𝑗𝜔) =_{(1+𝑗𝜔𝑇}𝑉𝐹

𝐹)5/6 (3-7)

where 𝑉𝐹 is the amplification factor, and 𝑇𝐹 is the time constant of the shaping

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The parameter selection for the shaping filter depends on the annual mean wind speed, 𝑣𝑚̅ and the turbulence length scale, 𝐿 , which corresponds to

the site roughness. The turbulence length scale can be expressed as [4]

L = {150 𝑚,_{5𝑧 𝑚, 𝑖𝑓 𝑧 < 30 𝑚} 𝑖𝑓 𝑧 ≥ 30 𝑚 (3-8)

where 𝑧 is the height of the wind turbine hub.

At West Beacon Farm, the wind turbine hub is located 24 m above ground level, and hence the turbulence length is equal to 120 meters. The amplification factor, 𝑉𝐹, and the time constant, 𝑇𝐹, of the shaping-filter parameters can be

obtained as 𝑉_{𝐹 ≈ √} 2𝜋 ℬ(1₂,1₃). 𝑇𝐹 𝑇 (3-9) 𝑇_𝐹 =_𝑣𝐿 𝑚 ̅̅̅ (3-10)

where 𝑇 is the sampling time. Equation (3-9) used 𝑇 = 1 second because this value has been used in several other studies [17]. ℬ denotes the Beta function, which is also known as the Euler integral function. The Beta function can be defined in many forms; in this study, it was defined as

ℬ(x, y) =(x−1)!(y−1)!_{(𝑥+𝑦−1)!}

The wind gust or turbulence, 𝑣𝑡, is subsequently acquired by multiplying

the coloured noise signal by the wind’s standard variance, 𝜎̅𝑣 (a constant), and

can be calculated as 𝑣𝑡(𝑡) = 𝜎̅𝑣. 𝑣𝑐(𝑡) (3-11) where 𝜎̅𝑣 = 𝑘𝜎,𝑣. 𝑣𝑚̅ (3-12) 𝑘_𝜎,𝑣= {0.10 ~ 0.15 𝑜𝑓𝑓 𝑠ℎ𝑜𝑟𝑒 0.15 ~ 0.25 𝑜𝑡ℎ𝑒𝑟𝑠 (3-13)

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The equivalent or actual instantaneous wind speed velocity, 𝑣𝑒𝑞, can be

calculated using Equation 3-5. The white noise, turbulence and the created equivalent velocity signal at West Beacon Farm with 𝑘_𝜎,𝑣= 0.25 and 𝑣_𝑚̅ = 4.5 m/s for 600 seconds with 1 second time step of simulation can be depicted as shown in Figure 3-5(a) to (c).

Figure 3-5 (a) white noise signal, (b) gust/turbulence signal, (c) equivalent or instantaneous velocity signal (d) modified equivalent or

instantaneous velocity signal for all wind speed conditions

0 100 200 300 400 500 600 -1 0 1 time (s) W h it e n o is e

(a) White noise signal

0 100 200 300 400 500 600 3 4 5 6 time (s) V e q ( m s )

(c) Simulated real-time of equivalent wind speed

0 100 200 300 400 500 600

0 10 20

(d) Simulated of equivalent wind speed covering all wind regions

V e q - a ll re g io n s ( m /s ) Time (s) 0 100 200 300 400 500 600 -1 0 1 time (s) G u s t, v t (m /s ) (b) Gust/Turbulence signal

70

However, because this study is focused on the behaviour of a SRVSWT, in the PL, IL and FL regions, the equivalent or instantaneous wind velocity is modified by changing the value of the mean wind speed and the coloured noise signal. Consequently, the designed turbine can be tested at all wind speeds, from approximately 0 m/s to 16 m/s. Figure 3-5(d) depicts the equivalent or actual instantaneous wind speed velocity, 𝑣_𝑒𝑞, which covers all wind speeds and has a mean wind speed of 10.4 m/s for a 600 seconds simulation.