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

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.


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]. e 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. is 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][4][5][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][8][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.
ere 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. is behavior is related to the trailing vorticity. e swirl speed represents the blades' angular momentum, so it is directly related to the torque and power extracted by the rotor. e ideal free vortex assumption for the wake is equivalent to the Joukowsky model of the blades in which the bound circulation is constant. is 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. e 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. ey concluded that when the platform experiences surge motions periodically, the Betz limit can be exceeded by the instantaneous power coefficient.
is 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. e 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. e conclusions are discussed in Section 4.

Mathematical Proposal
We consider Figure 1, which represents airflow through a turbine, where V 1 is defined as upstream wind speed, V wind speed in the turbine, V 2 downstream wind speed, and S 1 and S 2 cross-sectional area upstream and downstream, respectively. e turbine extracts part of the wind energy, so downstream wind speed is necessarily smaller than upstream wind speed in practical terms. Besides, S 2 is greater than S 1 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: (1) 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 P 1 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 by where 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) (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 P 1 and downstream P 2 can be expressed as follows: e 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) e final result in (12) represents the optimization function; since ρ and S are constants, it is enough to derive f (V 1 , V 2 ) with respect to V 2 as it is shown in (13) and then equal to zero to obtain the critical numbers: e 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, e second derivative is negative, so the function is concave downwards, and consequently, the maximum wind power extracted by the turbine is obtained when V 2 � V 1 /3. Substituting (10) and (12) in (3)

results in
where speed ratio V q � V 2 /V 1 . en, making V 2 � V 1 /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 � S 1 ; nevertheless, according to the fundamentals of the physical laws established S 1 is the cross-sectional area upstream. From (14) and (15), it is demonstrated that maximum theoretical wind power extraction is obtained when V 2 � V 1 / 3, so mixing (1) and (2) and substituting V 2 � V 1 /3 results, If this area quotient is substituted in (17), then the value of C p is is result is especially important, so it represents the real value of C p corresponding to the value of V 2 which maximizes the wind power extraction, that is, maximizes the function f (V 1 , V 2 ) 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 (V 1 , V 2 ) 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 V 2 /V 1 but also on quotient of areas S/ S 1 (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 (1) and (2) and substituting As it is shown in (20), the speed ratio V q affects the quotient of areas S/S 1 . Now, substituting this result in (16), the ideal C p model can be expressed: In the traditional Betz analysis, it is assumed that S/S 1 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/S 1 , and finally, (21) represents a novel and ideal model for power coefficient where quotient S/S 1 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. 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. is 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. e issue in this model is that it simplifies C p equation considering S/ S 1 as one. Figure 3 is generated from (23), and it represents the C p curve considering S/S 1 � 3/2, the value that was obtained using Froude's formula, and the substitution of V q derived from the optimization process for f (V 1 , V 2 ). 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. e issue in (22) and (23) is that these expressions were obtained by using critical numbers obtained when maximizing f (V 1 , V 2 ) equation instead of the power coefficient equation where area quotient S/S 1 is not a constant.

Results and Discussion
Finally, Figure 4 is the graphical representation of (21), and it represents C p curve considering S/S 1 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, S 2 can increase to infinity when V 2 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 � V 1 � V 2 ; therefore, S � S 1 � S 2 , and consequently, any energy is extracted from the wind and P � 0. 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. e 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.
e 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. e 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).

Conclusions
In this research, the analysis of the power coefficient resulted in two equations based on the relationship among ideal wind    Mathematical Problems in Engineering tunnel cross-sectional areas with no dimensional limits. e 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. is result considers a more appropriated value of quotient S/S 1 than the used by traditional analysis where the model is simplified.
On the contrary, (21) was obtained considering S/S 1 as a V 2 /V 1 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
e authors declare that they have no conflicts of interest.