Maximum Power Coefficient Analysis in Wind Energy Conversion Systems: Questioning, Findings, and New Perspective

González-Hernández, José Genaro; Salas-Cabrera, Rubén

doi:https://doi.org/10.1155/2021/9932841

Mathematical Problems in Engineering

On this page

Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9932841 | https://doi.org/10.1155/2021/9932841

Maximum Power Coefficient Analysis in Wind Energy Conversion Systems: Questioning, Findings, and New Perspective

José Genaro González-Hernández^1,2and Rubén Salas-Cabrera¹

Academic Editor: Juan C. Jauregui-Correa

Received03 Apr 2021

Accepted12 Jul 2021

Published20 Jul 2021

Abstract

For decades, maximum power coefficient limit, known as the Betz limit, has been accepted as a theoretical optimum value for wind turbine power extraction; nevertheless, some reports, exceeding this limit, have already been published. To explain this phenomenon and show a different point of view, a novel theoretical and ideal analysis based on flow conservation law and areas’ quotient is presented, supported by a review of works related to surpassing the power coefficient limit approached from different perspectives.

1. Introduction

Wind power has become an important source of energy for humanity; in recent years, the installed capacity has grown considerably to reach hundreds of Giga Watts as it was reported in [1]. The importance of wind energy nowadays is crucial to face the energetic problems around the world; as established by [2], wind energy has evolved from an emerging technology to a near-competitive technology. This fact, combined with an increasing global focus on political desire in the energy supply diversification and environmental concern, promotes wind energy has an important role in the future electricity market.

For example, in recent times, specialized methods in aerostatic stability and wind fragility analysis have been made as well as sophisticated methods based on aeroelastic models to improve structural reliability have also been proposed, for example [3–6]. Among all wind studies, energy efficiency is a highlight; because of this, new and sophisticated control equipment is currently developed to extract the most energy from the wind, and several works about overviews of emerging technologies of wind energy conversion systems (WECS) have already been published in recent years, for example [7–9], but it is necessary to move back 100 years to the past to identify and understand the origin of wind turbine power extraction limit, or at least, as it has been known for decades.

A century has passed since the German scientist Albert Betz published the results of the power coefficient (C_p) limit as approximately 0.593, a result whose authorship was the subject of debate for years. According to [10], Lanchester was very close to determining the famous wind turbine limit efficiency in a paper that he published in 1915; unfortunately, he did not include Froude’s result, which establishes that the velocity in the disk is the average of velocities upstream and downstream. Joukowsky and Betz used vortex theory to support Froude’s result and obtained theoretical maximum power coefficient simultaneously, just as Newton and Leibnitz developed calculus theory simultaneously.

Betz deduction has been presented in several ways; for example, Ochieng and Ochieng [11] present a mathematical series power expansion method to obtain the Betz equation functional form to determine optimum wind power coefficient. It is important to note that Betz analysis supposes an ideal turbine, which, according to [12], is the most efficient wind turbine and has so many blades so that there will not be blade tip loss as well as root losses to diminish efficiency; the lift-drag ratio of any blade element is infinite. In another study, Okulov [13] presented analytical solutions for optimizing wind turbines with specialized rotor models in the case of an infinite number of blades.

For decades, Betz limit has been accepted for any device that harvests power torque or drag in a horizontal-axis rotational motion in a right angle referred to the wind field, which also includes airborne wind energy systems according to [14]; however, it does not include the losses provoked by the rotation of the wake; therefore, it represents a conservative upper maximum as it has been described by [15].

In contrast, there have been reports of wind turbines’ designs as well as tidal turbines whose performance has exceeded 0.593, which seems to question the Betz limit, as it is reported by [16], whose investigation demonstrates that, despite flow reduction, it is still possible for wind turbines in a farm to exceed the Betz limit of a single isolated turbine, emphasizing the importance to note that there are experimental results which exceed the Betz limit for an isolated turbine.

There exist some interesting issues related to these reported results; for example, Cueva and San-Andres [17] state that the maximum energetic behavior of an ideal system extracting energy from an ideal wind exists only when the system slows down the wind speed to 2/3 of the wind speed of the undisturbed flow upstream; besides, Betz optimization is a pure energetic concept, and it does not establish any repercussion neither on the structural bearings of the system nor on the lifetime depletion associated to this energetically optimum condition.

Vaz[18] establish that there have been many analyses of horizontal-axis wind turbines operating at low tip speed ratio (λ), and at least two have suggested that power coefficient may exceed Betz limit as λ tends to zero. This behavior is related to the trailing vorticity. The swirl speed represents the blades’ angular momentum, so it is directly related to the torque and power extracted by the rotor. The ideal free vortex assumption for the wake is equivalent to the Joukowsky model of the blades in which the bound circulation is constant. This causes an infinite velocity along the axis of rotation theoretically.

In [19], an interesting wind power derivation research starts from the physical work concept, which derived a similar formulation to the Betz limit procedure. The model yielded close results within less than 10% relative difference compared to Betz limit; however, the results were always better than 0.593.

In [20], a remarkable study about the power coefficient overshoot of an offshore floating wind turbine in surge oscillations is developed. They concluded that when the platform experiences surge motions periodically, the Betz limit can be exceeded by the instantaneous power coefficient. This phenomenon only happens when λ is close to its optimal value to ensure a mean power coefficient approaching the Betz limit and when the platform surge is severe enough to produce a high variation in the apparent wind speed and the resultant power output.

Another study, published by Khamlaj and Rumpfkeil [21], states that some wind turbine models have been developed based on experimental data to highlight their potential to surpass the Betz limit. It is also mentioned that shrouded turbines equipped with a flange can overcome the Betz limit.

As it can be seen, several studies existed and experimental results were reported which establish that the Betz limit has been surpassed; therefore, in this paper, an explanation about this phenomenon is presented. First, the traditional Betz limit is calculated based on the flow conservation law; then, a relationship between the transverse surfaces of a wind tunnel is obtained to derive a better result. Finally, completely ideal conditions are assumed to obtain the theoretical maximum wind power extraction based on nonlimitations of physical conditions.

The rest of this manuscript is organized as follows. Section 2 presents a mathematical model based on flow conservation law that derives in a higher power coefficient limit than the traditional value. Section 3 shows the obtained results. The conclusions are discussed in Section 4.

2. Mathematical Proposal

We consider Figure 1, which represents airflow through a turbine, where V₁ is defined as upstream wind speed, V wind speed in the turbine, V₂ downstream wind speed, and S₁ and S₂ cross-sectional area upstream and downstream, respectively. The turbine extracts part of the wind energy, so downstream wind speed is necessarily smaller than upstream wind speed in practical terms. Besides, S₂ is greater than S₁ because, by definition, the flow rate must be the same everywhere within the tube. It is important to emphasize the importance of equation (1) since it represents the analysis basis:

Besides, the wind speed in the turbine is represented by (2) as the average of wind speeds’ upstream and downstream. Froude established this result in 1889 according to [10]

As it can be found in [22], by definition, the power coefficient C_p in a wind turbine is given by (3), where P represents the wind power extracted by turbine and P₁ is the complete wind power, that is, wind power upstream:

Now, in general, the energy E of an air mass in a wind tube is given bywhere V is the wind speed and m is the air mass which can be obtained by equation (5) as follows:where ρ represents the air density and x is the air mass thickness in the tube. Substituting (5) in (4) results in

(6) can also be written in its differential form as

Now, power P is defined by

Combining (7) and (8) results in

Accordingly, power upstream P₁ and downstream P₂ can be expressed as follows:

The power that the wind turbine extracts from the wind is the difference between the wind power upstream and downstream as it is expressed in (11):

Using (1) and (2) in (11) results in

The final result in (12) represents the optimization function; since ρ and S are constants, it is enough to derive f (V₁, V₂) with respect to V₂ as it is shown in (13) and then equal to zero to obtain the critical numbers:

The critical numbers are

Critical number V_2a is absurd in practice, since speed cannot be negative; then, applying second derivative test and evaluating in V_2b results,

The second derivative is negative, so the function is concave downwards, and consequently, the maximum wind power extracted by the turbine is obtained when V₂ = V₁/3. Substituting (10) and (12) in (3) results inwhere speed ratio V_q = V₂/V₁. Then, making V₂ = V₁/3, maximum theoretical wind power extraction is obtained:

Here is the issue, in order to simplify results, traditional maximum wind power extraction analysis assumes S = S₁; nevertheless, according to the fundamentals of the physical laws established S₁ is the cross-sectional area upstream. From (14) and (15), it is demonstrated that maximum theoretical wind power extraction is obtained when V₂ = V₁/3, so mixing (1) and (2) and substituting V₂ = V₁/3 results,

If this area quotient is substituted in (17), then the value of C_p is

This result is especially important, so it represents the real value of C_p corresponding to the value of V₂ which maximizes the wind power extraction, that is, maximizes the function f (V₁, V₂) established in (12), and which can explain results surpassing Betz limit in research reported. It is important to remark that this equation corresponds to extracted power by the turbine; however, it does not represent power coefficient; C_p is represented by (3), so there are two phenomena to be studied in the analysis; one is the function f (V₁, V₂) which is directly proportional to wind power extracted by turbine, assuming ρ and S constants, and the other one is C_p which depends not only on speed ratio V₂/V₁ but also on quotient of areas S/S₁ (quotient that strictly speaking is not constant but can also be represented as a function of speed ratio V_q in a more daring analysis). In (16), V_q was defined as V_q = V₂/V₁, so V₂ = V₁V_q. Combining (1) and (2) and substituting V₂ = V₁V_q results,

As it is shown in (20), the speed ratio V_q affects the quotient of areas S/S₁. Now, substituting this result in (16), the ideal C_p model can be expressed:

In the traditional Betz analysis, it is assumed that S/S₁ equals one, so using (16), C_p can be expressed by

On the contrary, using the result obtained in (18) and substituting in (16), C_p would be determined as follows:

In summary, (22) represents traditional wind power coefficient, (23) presents C_p expression considering Froude's result to obtain the quotient S/S₁, and finally, (21) represents a novel and ideal model for power coefficient where quotient S/S₁ was substituted as a function of V_q according to (20). Of course physical limitations related to areas are not considered, this is an ideal mathematical model.

3. Results and Discussion

Figure 2 is generated from (22), and it represents Betz traditional power coefficient curve. It can be appreciated that the limit value of 0.593 is reached when V_q = 1/3. When V_q tends to one, C_p diminishes as expected, finally reaching the zero value. This limit is perfectly logical in practical terms, since it means that upstream and downstream winds’ speed are equal and power extraction is zero. On the contrary, when V_q tends to zero, in the graphic, C_p equals 0.5, but in practical terms, this result does not make sense, since it does not match real measurements for these conditions. The issue in this model is that it simplifies C_p equation considering S/S₁ as one.

Figure 3 is generated from (23), and it represents the C_p curve considering S/S₁ = 3/2, the value that was obtained using Froude's formula, and the substitution of V_q derived from the optimization process for f (V₁, V₂). C_p limit equals 8/9, and it is reached when V_q = 1/3, and this is a remarkable result, since it provides an explanation for results reported in papers where Betz limit is surpassed and presents one of the contributions of this work. In the right section of the curve, when V_q tends to one, the decreasing behavior of C_p perfectly matches with practical results reaching zero at the end of horizontal axis. The issue in (22) and (23) is that these expressions were obtained by using critical numbers obtained when maximizing f (V₁, V₂) equation instead of the power coefficient equation where area quotient S/S₁ is not a constant.

Finally, Figure 4 is the graphical representation of (21), and it represents C_p curve considering S/S₁ as a function of speed quotient V_q. It is important to mention that (21) represents the ideal C_p equation to be maximized. No limitations of cross-sectional dimensions are considered, so theoretically, S₂ can increase to infinity when V₂ tends to zero in order to maintain flow conservation; as a result, maximum C_p is the unit, as it is shown in the left section of Figure 4, which means all wind power has been extracted. Naturally, in practice, this phenomenon does not occur, since continuous flow is needed to ensure wind power extraction and cross-sectional areas are obviously limited by physical restrictions; nevertheless, there is no reason to assume a decreasing behavior of C_p for values of V_q lower than 1/3 as it has been described in some studies, for example, [22].

In practical terms, all representations of C_p match perfectly the zero value of power coefficient when V_q = 1, which implies that V = V₁ = V₂; therefore, S = S₁ = S₂, and consequently, any energy is extracted from the wind and . Figure 5 presents a comparison among the different C_p models presented in this work, while Table 1 shows some values of C_p obtained from (21)–(23).

It is important to mention that recent works on power coefficient curve modeling of innovative wind turbines were reported to exceed the Betz limit; in some cases, the power coefficient was higher than the maximum C_p obtained in (23), but all are within the limits of (21).

In [23], a mathematical model of a horizontal-axis shrouded wind turbine was proposed. The investigation highlights that a nonoptimized ducted wind turbine cannot exceed C_p = 0.8, while a well-made ducted wind turbine can reach C_p = 0.93, but the Betz limit is surpassed in both cases.

In [24], a design of experiment approach applied to the analysis of diffuser-augmented wind turbines was proposed. The study concluded that the appropriate combination of factors such as the duct thickness, camber, chord and stagger positively contribute to C_p with a maximum value of 0.953.

In another study, the wind-lens efficiency is increased by designing the shroud and turbine shape and flange height through optimization; thus, all optimized wind-lens turbines exceed the Betz limit for power production, in a power coefficient span from 0.817 to 0.954 [21].

In [25], an investigation about a ducted wind turbine optimization and sensitivity to rotor position is developed; the design of a ducted wind turbine modeled using an actuator disc maximized the power coefficient exceeding the Betz limit with a total C_p of 0.67.

Other research presents an experimental validation of a ducted wind turbine design strategy. In the study, a synergistic design strategy for ducted horizontal-axis wind turbines (DWTs), utilizing the numerical solution of a ducted actuator disk system as the input condition for a modified blade element momentum method, is presented. The ducted configuration generated values are about C_p = 0.85–0.90 [26].

As it can be noticed, evidence supports that the Betz limit can be surpassed when the wind turbine design is appropriated. Table 2 presents a comparison among maximum C_p in different investigations; some values surpassed even 8/9, but those results can be explained by the ideal and novel proposed equation (21).

4. Conclusions

In this research, the analysis of the power coefficient resulted in two equations based on the relationship among ideal wind tunnel cross-sectional areas with no dimensional limits. The maximum value of curves generated by the models derived from the analysis in this work overcame the Betz limit, providing an alternative to explain published papers dealing with surpassing power coefficient classical limit, a reality that needed an explanation and cannot be denied.

(23) presents a maximum C_p = 8/9, representing a more accurate equation to explain some turbine behaviors, especially those reported in tidal turbines models performance. This result considers a more appropriated value of quotient S/S₁ than the used by traditional analysis where the model is simplified.

On the contrary, (21) was obtained considering S/S₁ as a V₂/V₁ function generating an ideal curve, where maximum C_p is the unit. It is important to remark that the analysis was based on the principle of flow conservation, so if the output speed tends to zero, then the correspondent area grows disproportionally to infinity, so theoretically it would be possible to extract all the power from the wind, but, in practice, is evident that this is not possible, since a wind turbine cannot be considered as a part of a strictly closed system such as an ideal wind tunnel; besides, there are physical restrictions related to the increasing of the areas in order to maintain the flow. Finally, the purpose of the authors is not to criticize traditional calculations but offer the interested readers a general overview to contrast maximum power coefficient value from a mathematical point of view which does not consider physical limitations.

Data Availability

All the datasets are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the support of the Tecnológico Nacional de México-Instituto Tecnológico de Ciudad Madero and Consejo Nacional de Ciencia y Tecnología during every stage of this research.

References

GWEC, Global Wind Report, Global Wind Energy Council, Brussels, Belgium, 2019, https://gwec.net/global-wind-report-2019/.
G. C. Larsen and A. T. Crespo, “Wind turbine wakes for wind energy,” Wind Energy, vol. 14, no. 7, pp. 797-798, Oct. 2011.
View at: Publisher Site | Google Scholar
G. Data, A. Shahoo, and S. Bhattacharjya, “Wind fragility analysis of RC chimneywith temperature effects by dual response surface method,” Wind and Structures, vol. 31, no. 1, pp. 59–73, 2020.
View at: Google Scholar
G. Zamiria and S. R. Sabbagh-Yazdi, “Nonlinear aerostatic analysis of long-span suspension bridge by element free Galerkin method,” Wind and Structures, vol. 31, no. 1, pp. 71–84, 2020.
View at: Google Scholar
I. Ibrahim, H. Aboshosha, and A. El-Damatty, “Numerical characterization of downburst wind field at Wind EEE dome,” Wind and Structures, vol. 30, no. 3, pp. 231–243, 2020.
View at: Google Scholar
C. Demartino and F. Ricciardelli, “Probabilistic versus deterministic assessment of the minimum structural damping required to prevent galloping of dry bridge hangers,” Journal of Structural Engineering, vol. 145, no. 8, pp. 1–19, 2019.
View at: Publisher Site | Google Scholar
V. Yaramasu, B. Wu, P. C. Sen, S. Kouro, and M. Narimani, “High-power wind energy conversion systems: state-of-the-art and emerging technologies,” Proceedings of the IEEE, vol. 103, no. 5, pp. 740–788, 2015.
View at: Publisher Site | Google Scholar
S. Y. A. De Falani, M. O. A. González, F. M. Barreto, J. C. De Toledo, and A. L. V. Torkomian, “Trends in the technological development of wind energy generation,” International Journal of Technology Management & Sustainable Development, vol. 19, no. 1, pp. 43–68, 2020.
View at: Publisher Site | Google Scholar
M. Cheng and Y. Zhu, “The state of the art of wind energy conversion systems and technologies: a review,” Energy Conversion and Management, vol. 88, pp. 332–347, 2014.
View at: Publisher Site | Google Scholar
V. L. Okulov and G. A. M. Van Kuik, “The Betz-Joukowsky limit: on the contribution to rotor aerodynamics by the British, German and Russian scientific schools,” Wind Energy, vol. 15, no. 2, pp. 335–344, 2012.
View at: Publisher Site | Google Scholar
R. M. Ochieng and R. O. Ochieng, “A power series formulation of the Betz criterion and equation in a wind turbine,” International Journal of Energy, Environment and Economics, vol. 23, no. 2, pp. 291–299, 2015.
View at: Google Scholar
H. Jiang, Y. Li, and Z. Cheng, “Performances of ideal wind turbine,” Renewable Energy, vol. 83, pp. 658–662, 2015.
View at: Publisher Site | Google Scholar
V. L. Okulov, “Limit cases for rotor theories with Betz optimization,” Journal of Physics Conference Series, vol. 524, pp. 1–8, 2014.
View at: Publisher Site | Google Scholar
M. De Lellis, R. Reginatto, R. Saraiva, and A. Trofino, “The Betz limit applied to airborne wind energy,” Renewable Energy, vol. 127, pp. 32–40, 2018.
View at: Publisher Site | Google Scholar
V. L. Okulov and J. N. Sørensen, “Refined Betz limit for rotors with a finite number of blades,” Wind Energy, vol. 11, no. 4, pp. 415–426, 2008.
View at: Publisher Site | Google Scholar
R. Venell, “Exceeding the Betz limit with tidal turbines,” Renewable Energy, vol. 55, pp. 277–285, 2013.
View at: Google Scholar
A. Cueva and A. San-Andres, “The extended Betz-Lanchester limit,” Renewable Energy, vol. 30, no. 5, pp. 783–794, 2005.
View at: Google Scholar
J. R. P. Vaz and D. H. Wood, “Performance analysis of wind turbines at low tip-speed ratio using the Betz-Goldstein model,” Energy Conversion and Management, vol. 126, pp. 662–672, 2016.
View at: Publisher Site | Google Scholar
S. Zekai, “Modified wind power formulation and its comparisson with Betz limits,” Journal of Energy Research, vol. 13, no. 8, pp. 959–963, 2013.
View at: Google Scholar
W. Binrong, T. Xinliang, D. Xingjiang, P. Zhike, and Z. Wenming, “On the power coefficient overshoot of an offshore floating wind turbine in surge oscillations,” Wind Energy, vol. 21, no. 11, pp. 1076–1091, 2018.
View at: Google Scholar
T. A. Khamlaj and M. P. Rumpfkeil, “Analysis and optimization of ducted wind turbines,” Energy, vol. 162, pp. 1234–1252, 2018.
View at: Publisher Site | Google Scholar
F. D. Bianchi, H. De-Batista, and R. J. Mantz, Wind Turbine Control Systems: Principles, Modeling and Gain Scheduling Design, Springer, London, UK, 2007.
N. Keramat Siavash, G. Najafi, T. Tavakkoli Hashjin, B. Ghobadian, and E. Mahmoodi, “Mathematical modeling of a horizontal axis shrouded wind turbine,” Renewable Energy, vol. 146, pp. 856–866, 2020.
View at: Publisher Site | Google Scholar
R. Bontempo, R. Carandente, and M. Manna, “A design of experiment approach as applied to the analysis of diffuser-augmented wind turbines,” Energy Conversion and Management, vol. 235, Article ID 113824, 2021.
View at: Publisher Site | Google Scholar
N. Bagheri-Sadeghi, B. T. Helenbrook, and K. D. Visser, “Ducted wind turbine optimization and sensitivity to rotor position,” Wind Energy Science, vol. 3, no. 1, pp. 221–229, 2018.
View at: Publisher Site | Google Scholar
B. Kanya and K. D. Visser, “Experimental validation of a ducted wind turbine design strategy,” Wind Energy Science, vol. 3, no. 2, pp. 919–928, 2018.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 José Genaro González-Hernández and Rubén Salas-Cabrera. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies