Toward a physics-based model of power coefficient in horizontal-axis wind turbine

Abstract

Global energy is in a major shift from energy using fossil sources to renewable energy using more sustainable energy sources. The wind is the clean and inexhaustible one of renewable energy resources that is available in most parts of the world. In order for wind power generation to meet more ambitious targets around the world, it is necessary to understand all the physics behind process of wind generation. The most interesting of this process is the power coefficient which involves the real characteristics of the wind turbine as a function of the generated power. Up to now, the power coefficient does not yet have a general form but are fitted as mathematical functions from manufacturer data measured from popular wind turbines in the world. In this paper, we develop an analytical physics-based method to estimate the power coefficient general for various wind generation systems of horizontal-axis wind turbines (HAWT). Instead of 12 parameters in semi-experimental exponential formula , our power coefficient uses only 5 parameters with their physical meanings. By varying these five parameters we also obtain the power coefficients fitted from experimental data.

Keywords

Wind power generation horizontal-axis wind turbine power coefficient

Introduction

Increasing global electricity demand leads to environmental degradation, manifested by higher carbon dioxide (CO₂) emissions and increased greenhouse effect (Liu et al., 2023; Qadir et al., 2021). Human’s dependence on fossil fuels, resulting “climate change” and increasing energy costs have spurred interest in alternative sources (Al-Shetwi, 2022). These alternative sources are renewable energies and they have now contributed to the diversification of energy production (Androniceanu and Sabie, 2022; Chudy-Laskowska and Pisula, 2022) in which wind energy and solar energy are popular today. Accompanying the process of increasing wind power projects in the world, many projects have been implemented in Vietnam, aiming to comply with the Kyoto Protocol and reduce CO₂ emissions to zero by 2050, even there are many other motivations for developing these alternative energy sources.

Although wind is a clean and inexhaustible energy everywhere in the world, there are regions where resource is very abundant. Therefore, understanding an energy conversion process from wind to turbine is essential. Various mathematical statistical methods have been proposed to calculate wind energy density, average wind speed, turbine capacity, generation capacity, and its load factor for many particular geographical areas (Li et al., 2023). Although this allows to identify suitable areas for installing wind farms with the desired capacity, it is not universal for all types of wind turbines in the world. This paper hopes to find a general rule about useful capacity for all types of wind turbines, thereby proposing designs for the construction of desired wind turbines, or contributing to surveys and assessments of the efficiency of current wind turbines.

The process of converting wind energy into electricity begins with the wind colliding with turbine blades, turning a generator. In fact, wind has a random nature, so turbine blades absorb energy cumulatively, so wind power is also a cumulative function. This accumulation gives us a useful capacity suitable for generating electricity. The ratio between the useful power received by the turbine and the power of the wind colliding with the blades is the power coefficient ( $C_{p}$ ). This coefficient has been evaluated by the International Engineering Commission (IEC) as one of the most important coefficients expressing the relationship between actual generation capacity and the energy from the surrounding wind transmitted to the turbine blades (IEC 61400-12-1, 2005; Xia et al., 2013). In addition, due to the random nature of wind, the voltage amplitude and frequency generated by the generator change continuously. To regulate the voltage of a wind turbine’s generator to meet the loads with amplitude and frequency values specified to the regulations of a country, an electronic system is used to perform this conversion (Mayilsamy et al., 2023; Rekioua, 2014). Therefrom, the torque coefficient ( $C_{t}$ ) is added to evaluate the power generation ability of the wind turbine. These two coefficients are necessary to determine the production efficiency of a wind turbine or wind farm, along with other parameters such as air turbulence, density, and ambient temperature. They are also coefficients that distinguish one wind turbine from another. Therefore, understanding the characteristics of these parameters helps design and operate wind turbines more effectively. Many statistical data from many parts of the world have been used to analyze all the properties of these parameters (Hwangbo et al., 2017; Niu et al., 2018). Most of these statistics provide mathematical functions used to represent the behavior of the power coefficient as a function of the useful wind speed and the angle of wind impact on the blade, thereby establishing the general expressions for better analysis and comparison.

The power coefficient is related to characterizing the wind speed at a specific location or area. This behavior is reflected, to a certain extent, in the power delivered by a wind turbine generator (WTG; Carrillo et al., 2004, 2009; Pidre et al., 2003). Due to the random nature of the wind, the power coefficient does not exactly follow any known statistical distribution (Rocha et al., 2012; Tuller and Brett, 1984). Therefore, the normal way to characterize the wind speed at a specific location is to perform “in situ” measurements, which should last several years. The manufacturer data is used to fit approximately to expected mathematical functions almost in three forms: polynomial, sinusoidal, and exponential (Ahmed et al., 2014; Bustos et al., 2012; Gao et al., 2008; Khajuria and Kaur, 2012; Kotti et al., 2014; Llano et al., 2014; Ovando et al., 2007; Shi et al., 2013). However, this is not a common method for all types of turbines, because it is too complicated and taken much time. So, this leads to the application of the probability distribution functions (PDFs). The use of these functions (Burton et al., 2011; Carta et al., 2009; Celik et al., 2010; Usta and Kantar, 2012) is useful or unavoidable in certain studies, for example, specialization of wind energy resources (WAsP, 2024), simulations of WTG behaviors (Feijoo et al., 1999), development of site-matching approaches (Hu and Cheng, 2007; Huang and Wan, 2009), etc. However, difficulty arises in choosing the best PDF that fits the wind speed distribution (Carta et al., 2009; Celik et al., 2010). The PDFs most commonly used by researchers that study wind characteristic at wind sites are the Weibull function and its interpolations (Rayleigh and exponential), which appear to be related to the nature of the wind in certain conditions (Edwards and Hurst, 2001; Tuller and Brett, 1984).

Hundreds of papers have been written on the Weibull distribution and its application to TWG, which has been attracting the attention of engineers for about half a century now. Almost the Weibull distribution is comprehensively used for delineating the wind power potential at a destined site. The two-parameter Weibull distribution is the most widely used and accepted owing to its wide array of applicability, flexibility, and usefulness for describing the occurrence of high wind speeds. A three-parameter Weibull distribution has also been utilized in some studies and was improved flexibility than the two-parameter Weibull distribution. However, some authors have indicated that the Weibull distribution should not be used in a generalized way, as it is unable to represent some wind regimes. The question is, how are the parameters of the Weibull distribution related to power growth, power peak, power cutoff, and pitch angle?

A large number of studies have been published presenting the use of a variety of probability density functions to describe wind speed distributions. However, there are no studies that can answer the above question. To be able to overcome this question, we are restricted ourselves to starting with the simplest interpolation of the Weibull distribution—the exponential distribution.

This article focuses on the analysis of horizontal-axis and three-blades turbines since they are the most common types. The article is organized as follows. We start by considering the characteristics of wind turbines to identify the most important coefficients that determine its operation and provide each turbine in the wind energy conversion process. Then we build a physics-based analytical model established based on wind cumulative distribution. After the analytical expression of the power coefficient was obtained, we determine the physical parameters in this power coefficient expression and compare them with functions obtained from manufacturer data of eight wind turbines that have been evaluated in some popular places in the world. Finally, evaluation and conclusion of the scientific results of the paper are given.

The experiment-based models

Wind is air movement in the Earth’s atmosphere, and it produces the power

P_{w} = \frac{1}{2} ρ A v^{3}

(1)

where, $ρ$ is the air density, $v$ is the wind speed, $A$ is the circular cross-section of the swept area covered by the blades of the turbine $A = π R^{2}$ , and $R$ is radius of rotor (Wiser et al., 2016).

It is worth noting that wind power increases with the third power of wind speed and therefore wind speed is one of the deciding factors when wanting to use wind energy.

Wind power can be used, for example, through a wind turbine to generate electricity, but the energy produced is much less than the energy of the wind flow because of the speed of the wind behind a turbine cannot be reduced to zero. In theory, a maximum of 59.3% of the energy existing in the wind can be extracted (called Betz’s Law, discovered by Albert Betz in 1919 (Betz, 2013). The value of the ratio between the power extracted from the wind and the wind power is called the power coefficient ( $C_{p}$ ). Hence, the power in the turbine ( $P_{t}$ ) is

P_{t} = \frac{1}{2} ρ A v^{3} C_{p}

(2)

The power coefficient ( $C_{p}$ ) depends on several parameters, such as the type of turbine (horizontal or vertical axis), the number of blades (in the case of a horizontal axis), the specific speed or tip speed ratio ( $λ = R ω / v$ , where $ω$ is the rotational speed of the turbine), and the pitch angle of the blades ( $θ$ ; Hau and Von Renouard, 2006).

In fact, $C_{p} (λ, ω)$ are different depending on the specific case of commercial wind turbines. Normally, manufacturers measure and provide data where the $C_{p}$ behaviors are plotted as graphs. These graphs, then, are used to obtain mathematical approximations (i.e. by fitting to expected functions). These are carried out by means of optimal numerical approximations (Nouira et al., 2012), almost are expressed through three mathematical forms:

• Polynomial functions

• Sinusoidal functions

• Exponential functions

Castillo et al. (2023) have some comments on the power coefficients of HAWTs fitted to the above three forms. Fitting to polynomial forms depends only on $λ$ , since considering $θ$ as a constant, and they are often used in low-power wind turbines where θ is not controlled. In addition, in some cases it is necessary to limit $λ$ so that the power coefficient is feasible and valid. For values $λ$ larger than the limit, this coefficient does not occur in real turbines (Arifujjaman et al., 2006).

Fitting based on sinusoidal functions also has the disadvantage of needing to limit the value of $λ$ for each power coefficients. In addition, for high values of both $λ$ and $θ$ some power coefficients have problems such as decreasing and increasing again or vice versa, which are not suitable for real wind turbines. In some other cases, power coefficients seem more appropriate, but they tend to infinity when changing both $λ$ and $θ$ (Merahi et al., 2014).

Regarding exponential-based coefficients, there is considerable variety, but they all have the same rule of starting at 0 for $λ = 0$ and gradually increasing and reaching a maximum point and then descending to zero at large $λ$ . This does not happen in most cases data fitted to polynomial or sinusoidal forms. Furthermore, for non-zero values of $θ$ , the maximum of $C_{p}$ decreases more strongly as $θ$ is larger, thus evaluating the true performance of the turbine. Moreover, the observed behaviors of the torque coefficients also agree well with reality (Ahmed et al., 2014; Bustos et al., 2012; Gao et al., 2008; Khajuria and Kaur, 2012; Kotti et al., 2014; Llano et al., 2014; Ovando et al., 2007; Shi et al., 2013) .

Thus, through evaluating the above three types of analytical functions, all power coefficients expressed as exponential functions are most suitable for actual wind turbines and are grouped into a general equation expressed as summarized in Castillo et al. (2023):

C_{p} (λ, θ) = c_{0} (\frac{c_{1}}{\tilde{λ}} - c_{2} θ - c_{3} θ \tilde{λ} - c_{4} {\tilde{λ}}^{c_{5}} - c_{6}) e^{- \frac{c_{7}}{\tilde{λ}}} + c_{8} λ

(3)

where

\frac{1}{\tilde{λ}} = \frac{1}{λ + c_{9} θ + c_{10}} - \frac{c_{11}}{1 + θ^{3}}

(4)

and the parameters $c_{i}$ (i = 0, 1, 2, … 11) of this expression are identified by the papers (Ahmed et al., 2014; Bustos et al., 2012; Gao et al., 2008; Khajuria and Kaur, 2012; Kotti et al., 2014; Llano et al., 2014; Ovando et al., 2007; Shi et al., 2013) and they are shown in Table 1.

Table 1.

Parameters of the power coefficients in the exponential functions (Castillo et al., 2023).

Refs.	$c_{0}$	$c_{1}$	$c_{2}$	$c_{3}$	$c_{4}$	$c_{5}$	$c_{6}$	$c_{7}$	$c_{8}$	$c_{9}$	$c_{10}$	$c_{11}$
Kotti	0.5	116	0	0.4	0	0	5	21	0	0.08	0	0.035
Khajuria	0.5	116	0.4	0	0	0	5	21	0	0	0.088	0.035
Ovando	0.518	116	0.4	0	0	0	5	21	0.007	0.08	0	0.035
Gao	0.22	116	0.4	0	0	0	5	12.5	0	0.08	0	0.035
Llano	0.5	72.5	0.4	0	0	0	5	13.13	0	0.08	0	0.035
Shi	0.73	151	0.58	0	0.002	2.14	13.2	18.4	0	0.02	0	0.003
Bustos	0.44	125	0.4	0	0	0	6.94	17.05	0	0.08	0	0.001
Ahmed	1	110	0.4	0	0.002	2.2	9.6	18.4	0	0.02	0	0.03

These eight power coefficients as the functions depending on $λ$ for $θ = 0$ are shown in Figure 1. From the figure we realize that no coefficient exceeds the Betz limit. All power coefficients have a general rule of starting from zero and increasing until reaching a maximum point and then decreasing to zero at large λ.

Figure 1.

The power coefficients based on an exponential functions (Castillo et al., 2023).

All above functions describing the power coefficient of HAWTs are fitted from experimental data. Although these results are good for specific HAWTs typical of a certain region, they do not describe all HAWTs installed around the world. For this reason, the problem is how to find a general expression for all types of HAWTs installed and will be installed anywhere on earth. Below we will try to find this function based on physics and the meaning of the parameters.

The physics-based model

Based on physics, the received power in the turbine ( $P_{t}$ ) can be expressed by using an exponential cumulative distribution to the useful wind speed $v$ as follows

P_{t} (v) = \int_{0}^{v} P_{w} (u) f (u) du

(5)

where

P_{w} (u) = \frac{1}{2} ρ A u^{3}

(6)

is the wind power and

f (u) = \frac{1}{ζ} e^{- \frac{u - μ}{ζ}}

(7)

is the exponential probability density function. Here, $ζ$ is scale factor, and $μ$ is location parameter.

Equation (5) has the following physical meaning: with a given wind speed $v$ , and therefore wind power $P_{w}$ , the received turbine power is the accumulated energy corresponding to a characteristic distribution. The most general distribution suitable for this case is the Weibull distribution (Edwards and Hurst, 2001). Here we use the simplest approximation of this distribution—the exponential distribution.

Taking the integral (5) analytically, we obtain:

P_{t} (v) = \frac{1}{2} ρ A v^{3} C_{p}

(8)

where

C_{p} = e^{\frac{μ}{ζ}} [6 \frac{ζ^{3}}{v^{3}} - e^{- \frac{v}{ζ}} (1 + 3 \frac{ζ}{v} + 6 \frac{ζ^{2}}{v^{2}} + 6 \frac{ζ^{3}}{v^{3}})]

(9)

Putting $ζ = v_{t} / χ$ , $v_{t} = ω R$ , $χ$ is a new parameter, and $μ = \bar{v}$ , then

\frac{μ}{ζ} = \frac{χ}{\bar{λ}}

(10a)

\frac{v}{ζ} = \frac{χ}{λ}

(10b)

where $λ = v_{t} / v$ , $\bar{λ} = v_{t} / \bar{v}$ .

Putting $η = μ / ζ$ and $y = v / ζ$ , then instead of parameters $μ$ , $ζ$ (or $χ$ ), and variable $v$ we have a new parameter $η$ and a new variable $y$ . Thus, equation (9) becomes

C_{p} = e^{η} [\frac{6}{y^{3}} - e^{- y} (1 + \frac{3}{y} + \frac{6}{y^{2}} + \frac{6}{y^{3}})]

(11)

Comparing with (3) and (4), we arrive at

y = α (\frac{1}{λ + γ θ} - \frac{β}{1 + θ^{3}}),

(12)

Thus, starting from the idea that the turbine power is received from the process of accumulating wind power, we obtain (11) in a general form for all types of HAWTs. To compare with the approximate expression (3) that is fitted with some manufacturer data, we will first find out the meaning of the parameters in (11). It’s convenient that instead of 12 parameters in (3) and (4), our model only needs 4 parameters: $η$ , $α$ , $β$ , and $γ$ as shown in equations (11) and (12). Below we will examine the meaning of these four parameters. First, we consider the case $θ = 0$ and choose a set of three parameters $η$ , $α$ , and $β$ suitable in the region of the above eight semi-empirical models. The results are shown in Figure 2.

Figure 2.

The power coefficients for $θ = 0$ based on our physical model (the red solid line). The cyan lines and shading area are exponential functions of Refs. (Ahmed et al., 2014; Bustos et al., 2012; Gao et al., 2008; Khajuria and Kaur, 2012; Kotti et al., 2014; Llano et al., 2014; Ovando et al., 2007; Shi et al., 2013) .

Keeping the three parameters unchanged and changing the remaining parameter, we obtain Figures 3 to 5 (for $θ = 0$ ).

Figure 3.

The power coefficients for $θ = 0$ based on our physical model as $η$ changes. As $η$ increases, the maximum value of $C_{p}$ also increases.

Figure 4.

The power coefficients for $θ = 0$ based on our physical model as $α$ changes. As $α$ increases, the slope in the small $λ$ region of $C_{p}$ decreases.

Figure 5.

The power coefficients for $θ = 0$ based on our physical model as $β$ changes. As $β$ increases, the breakpoint of $C_{p}$ decreases.

Three parameters $η$ , $α$ , $β$ clearly show the physical properties of $C_{p}$ as well as the wind turbine power $P_{t}$ . $η$ is related to the magnitude of the maximum value of $C_{p}$ , that is, related to the maximum power value of the wind turbine. $α$ is related to the increase in $C_{p}$ as well as the wind turbine power. And $β$ is related to the cutoff point of turbine power.

Comparing with (3) and (4), we see that

α = c_{7}, β = c_{11}, γ = c_{9}, e^{η} = c_{0} c_{6}

(13)

Hence, all the semi-empirical curves obtained by Kotti et al. (2014), Khajuria and Kaur (2012), Ovando et al. (2007), Gao et al. (2008), Llano et al. (2014), Shi et al. (2013), Bustos et al. (2012), and Ahmed et al. (2014) can also be obtained from (11) and (12) corresponding to suitable parameter sets of $η$ , $α$ , $β$ , and $γ$ .

Now we consider the change of $C_{p}$ when $θ$ is non-zero. Realizing that when $θ$ increases, the maximum value of $C_{p}$ decreases as well as the breakpoint of $C_{p}$ also decreases, so $η$ and $β$ need to be adjusted as follows

η \to η g (θ)

(14a)

β \to β g^{- 1} (θ),

(14b)

where $g (θ)$ is a correction function that satisfies the condition $g (θ) = 1$ when $θ = 0$ and $g (θ) < 1$ when $θ \neq 0$ ( $g$ is an even function of $θ$ ).

Compare with the results of the dependence of $C_{p}$ on $θ$ in Castillo et al. (2023), we derive the correction function $g (θ)$ in simplest ansatz is

g (θ) = e^{- δ θ^{2}}

(15)

where $δ$ is a new parameter characterizing the influence of pitch angle on wind turbine power. The influence of the parameter $δ$ on $C_{p}$ is shown in Figure 6.

Figure 6.

The power coefficients for $θ = 20^{°}$ based on our physical model as $δ$ changes. As $δ$ increases, the maximum value and the breakpoint of $C_{p}$ decreases.

Thus, by requiring the maximum value and the breakpoint of $C_{p}$ decreases, the fifth parameter $δ$ is added. Note that here the fourth parameter $γ$ does not have any significant contribution, so in fact there are only four physically meaningful parameters: $η$ , $α$ , $β$ , and $δ$ .

Hence, the general function of the power coefficient reads

C_{p} = e^{η g (θ)} [\frac{6}{y^{3}} - e^{- y} (1 + \frac{3}{y} + \frac{6}{y^{2}} + \frac{6}{y^{3}})],

(16)

where

y = α (\frac{1}{λ + γ θ} - \frac{β g^{- 1} (θ)}{1 + θ^{3}}),

(17)

with ansatz

g (θ) = e^{- δ θ^{2}} .

(18)

Here, the four physics parameters $α$ , $η$ , $β$ , and $δ$ characterize the changes of power growth, power peak, power cutoff, and pitch angle, respectively.

The variation of $C_{p}$ with respect to $λ$ and $θ$ with new $η$ and $β$ in (14) using $g (θ)$ in (15) with choosing $δ = 1$ is shown in Figures 7 and 8.

Figure 7.

The power coefficients for $θ \neq 0$ based on our physical model as $θ$ changes with $δ = 1$ . As $θ$ increases, the maximum value and the breakpoint of $C_{p}$ decreases.

Figure 8.

The power coefficient $C_{p}$ versus $λ$ and $θ$ based on our physical model with $δ = 1$ .

Finally, by adjusting these five parameters $η$ , $α$ , $β$ , $γ$ , and $δ$ , we can obtain any functional form of $C_{p}$ . For example, the exponential functions of Gao et al. (2008) and Bustos et al. (2012) can be obtained from our analytical function corresponding to the parameter sets shown in Figures 9 and 10.

Figure 9.

The power coefficient $C_{p}$ versus $λ$ at $θ = 20^{°}$ based on our physical model fitted with Gao et al. (2008).

Figure 10.

The power coefficient $C_{p}$ versus $λ$ at $θ = 20^{°}$ based on our physical model fitted with Bustos et al. (2012).

Table 2 summaries the four parameters $η$ , $α$ , $β$ , and $γ$ obtained by our analytical functions fitted from Kotti et al. (2014), Khajuria and Kaur (2012), Ovando et al. (2007), Gao et al. (2008), Llano et al. (2014), Shi et al. (2013), Bustos et al. (2012), and Ahmed et al. (2014).

Table 2.

Parameters of the power coefficients in our functions at $δ = 1$ .

Refs.	$η$	$α$	$β$	$γ$
Kotti	1.26	28.5	0.078	0.08
Khajuria	1.25	26.5	0.078	0.08
Ovando	1.40	27.5	0.075	0.08
Gao	1.32	16.5	0.078	0.08
Llano	1.25	18.2	0.104	0.08
Shi	1.32	25.2	0.090	0.08
Bustos	1.36	23.5	0.057	0.08
Ahmed	1.33	25.0	0.117	0.08

Conclusion

The computation of WTG is the top of a portfolio of wind farm projects that will help the industry understand how to dependably integrate large quantities of wind energy into system operations, as well as to advance capabilities that will enable these new wind installation systems to actively improve the quality of electric grids. To decide if an investment is to be made for a planned wind plant, it takes the assistance of the wind speed data of that region. Usually, the wind distribution of that region is measured for many years. Since wind speed is a variable factor, it is expected to fit a probability distribution. This will reduce measurement time and make project evaluation faster.

In this study, by using the exponential cumulative distribution function a physical model is derived for the power of the wind turbine, in particular the power coefficient function is found by analytical calculation results. Instead of 12 parameters that have no physical meaning in the power coefficient fitted as exponential function by manufacturer data, this analytic function has only 5 parameters that characterize the properties of real wind turbine power, such as power growth, power peak, breakpoint, and pitch angle.

In summary, our research obtained the following main results:

An analytical expression for the power coefficient common to all horizontal axis wind turbines is derived not by fitting manufacturer data but by using the exponential distribution, the simplest variation of the Weibull distribution.

The number of main parameters in the power coefficient is only four, and they have physical meaning corresponding to the power growth, power peak, power cutoff, and pitch angle.

In the article we used the exponential cumulative distribution, to expand it we can use a more general distribution function, such as the Weibull distribution.

f (u) = \frac{κ}{ζ} {(\frac{u - μ}{ζ})}^{κ - 1} e^{- {(\frac{u - μ}{ζ})}^{κ}} .

(18)

With the Weibull distribution function, there is an additional configuration parameter $κ$ . The significance of this parameter is possible for fine adjustments of wind turbine power. The power coefficient function with additional configuration parameters may be a prospective study for us in the future.

Footnotes

Acknowledgements

We thank our colleagues in the Faculty of Energy Technology, Electric Power University for their valuable contributions and discussions regarding our studies.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Tuan-Anh Nguyen

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