A novel composite calculation model for power

Energy 126 (2017) 821e829

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Energy journal homepage: www.elsevier.com/locate/energy

A novel composite calculation model for power coefficient and flapping moment coefficient of wind turbine Chao Peng*, Jianxiao Zou, Yan Li, Hongbing Xu, Liying Li School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, 610054, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 February 2016 Received in revised form 15 March 2017 Accepted 17 March 2017

In recent years, with continuous increasing capacity of wind turbine, mechanical fatigue problems caused by the vibration of load on wind turbine have become more and more serious. The load is composed mainly by oscillating load and flapping load, which can be measured by power coefficient and flapping moment coefficient. Both of them are important for wind turbine load analysis and operational control to reduce these load vibration. The existing calculation models for wind turbine load only focus on the power coefficient and always neglect flapping moment coefficient. In this paper, a novel composite power coefficient and flapping moment coefficient calculation model based on Blade Element Momentum Theory is proposed. A modified Blade Element Momentum model is built to calculate axial induction factor, tangent induction factor and torque force coefficient of wind turbine. By using them, a new calculation model for power coefficient and flapping moment coefficient is presented. Then, a composite calculation model based on iteration procedure and nonlinear fitting is built for calculating power coefficient and flapping moment coefficient simultaneously. Finally, the proposed calculation model is implemented to calculate the power coefficient and flapping moment coefficient of NREL 5 MW wind turbine and the results demonstrate its effectiveness. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Wind turbine Composite calculation model Power coefficient Flapping moment coefficient Induction factor

1. Introduction Currently, wind power generation has become one of the most widely used renewable energy power technology. With the development of wind turbine technology, the capacity of wind power turbine, as well as the size and weight of mechanical components of wind turbines increase gradually. These would cause that the flexibility of mechanical components of wind turbines increases accordingly and the natural frequency of wind turbine mechanical system decreases significantly. Thus, under wind load with randomness, fluctuation and mutation, it is more likely that the mechanical oscillation and vibration of mechanical components of wind turbine would occur. This would lead to a large mechanical fatigue load, which brings severe fatigue damage to the wind turbine mechanical components, even the fracture of wind turbine blades and other mechanical components. The problem of mechanical fatigue and damage of wind turbine has become one of the key problems, which hinders and limits the development and

* Corresponding author. E-mail address: [email protected] (C. Peng). http://dx.doi.org/10.1016/j.energy.2017.03.086 0360-5442/© 2017 Elsevier Ltd. All rights reserved.

construction of wind power. Thus, in recent years, the analysis and calculation of mechanical load of large wind turbine, especially for the MW class large wind turbine, has attracted increasing attentions in the field of wind turbine technology [1,2]. The kinetic wind energy is converted into the wind turbine mechanical energy by wind turbine blade. Thus, wind turbine blade is the main component of wind turbine under wind force actions and endures mechanical load, which is the source of other mechanical components in wind turbine. The existing research results have classified the load on the wind turbine into aerodynamic load, gravity load and inertial load [3]. The main load, i.e., aerodynamic load is composed by the oscillating load and flapping load according to Blade Element Momentum Theory [4], the oscillating load which represents the vibration load of rotating surface of wind wheel, can be normally measured by power coefficient, and the flapping load, which represents the vibration load of the vertical rotating surface of wind wheel, can be measured by flapping moment coefficient. The power coefficient is closely related to the structure of the blades of wind turbines. Thus, different wind turbines have different power coefficients [5,6]. Most researchers studied many power coefficient calculation models based on statistical data or

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C. Peng et al. / Energy 126 (2017) 821e829

Nomenclature

Symbols and Meaning of the symbols F The thrust force acting on the circular ring of the wind wheel r Density of air flow v1 Wind speed in front of the wind turbine r Radius of the circular ring a Axial induction factor l Tip speed ratio b Pitch angle T The aerodynamic torque acting on the circular ring of the wind wheel U Rotational speed of the wind turbines b Tangent induction factor B Number of wind turbine blades R Radius of wind turbines Ф Inflow angle of blade element r0 Radius of the hub c Chord length of the blade element Ct Thrust force coefficient Cn Normal force coefficient

experimental data, which have been built to calculate the power coefficient for different wind turbines [7e16]. A calculation model with sinuous function of tip speed ratio and blade pitch angle is proposed to calculate power coefficient in Ref. [7]. In Refs. [8] and [9], a power coefficient calculation model is approximated by using a nonlinear function based on the turbine characteristics. A calculation model with power function of tip speed ratio and four constant variables is used in Refs. [10] and [11]. A calculation model with function of pitch angle tip speed ratio and six constant variables is used in Ref. [12]. A complicate calculation model with third power of pitch angle and its power function is applied to compute the power coefficient in Ref. [13]. In Ref. [14], a calculation model with integral of tip speed ratio is built by using epigenetic linear genetic programming. A nonlinear function of wind speed, shaft speed and pitch angle based calculation model for power coefficient is built in Ref. [15]. In Ref. [16], a power coefficient calculation model with cylinder speed ratio, tip-speed ratio and cylinder aspect ratio, is built by using numerical solutions of the Blade Element Momentum Theory and symbolic regression. Most above power coefficient calculation models are used to analyze aerodynamic load or provide information to control system to maximize power captured by wind turbine, they always neglect the flapping moment coefficient, i.e., flapping load. On the other side, the flapping moment coefficient of the wind turbine blade which is a critical factor to produce the load vibration on the wind turbine blade, drive chain and tower structure etc., will cause the load fatigue of wind turbines accordingly. However, few researchers have devoted to the investigation of flapping moment coefficient in details presently [17,18]. As seen above, it is obvious that most of existing power coefficient calculation models neglect the calculation of flapping moment coefficient. The calculation of power coefficient and flapping moment coefficient are separated. Analyzing and calculating the power coefficient and flapping moment coefficient comprehensively, could be more practical and reflect the actual operation load status of wind turbine more accurately than the above mentioned calculation models. Developing a composite calculation model for power coefficient and flapping moment coefficient is of practical importance for wind

turbine load status analysis [19], wind turbine numerical simulation [20], blade design [21] and proving load status to wind turbine control system, which could be used to reduce its mechanical load fatigue [22], optimize the maximum power point tracking control [23], constant power control [24] and so on. Therefore, to combine the calculation of power coefficient and flapping moment coefficient, a novel composite calculation model is proposed in this paper. The organization of rest of this paper is given as follows, a modified Blade Element Momentum model is presented in Section 2, calculation models for power coefficient and flapping moment coefficient are built in Section 3, based on the above works, a novel composite calculation model for power coefficient and flapping moment coefficient is built in Section 4. In Section 5, the proposed calculation model is implemented in NREL 5 MW wind turbine and its results are discussed. The conclusion is given in Section 6. 2. Modified Blade Element Momentum model In this section, a modified Blade Element Momentum model is built to calculate the aerodynamic torque coefficient of wind turbine. Blade Element Momentum theory is a classical model to evaluate the aerodynamic performance of wind turbines [25,26], which can be categorized as the element theory and momentum theory. Specifically, the momentum theory takes the air as a research object, and the wind wheel is roughly regarded as an approximation of infinite rotating blades. Therefore, the wind speed acting on the wind wheel assumes to be stable and consistent. Based on the momentum theory, the thrust force acting on the circular ring of the wind wheel can be written as follows,

dF ¼ 4prv21 að1  aÞrdr

(1)

where dr is the length of the blade element. By using the equation of momentum moment, the aerodynamic torque acting on the circular ring of the wind wheel can be written as follows,

dT ¼ 4prv1 Ubð1  aÞr 3 dr

(2)

Since there is an assumption that the circular ring of the wind wheel has infinite blade numbers implied in the momentum theory, the momentum theory needs to be modified and the correction factor K is introduced. It could be written as,

K ¼ K1 K2

(3)

where K1 is the tip loss correction, which could be defined as follows [27],


 2 BðR  rÞ K1 ¼ cos1 exp  2r sin 4 p

(4)

K2 is the hub loss correction, which could be expressed by following equation [28],


 2 Bðr  r0 Þ K2 ¼ cos1 exp  2r0 sin 4 p

(5)

By using the obtained correction factor K, the thrust force F and aerodynamic torque T also could be written as follows,

dF ¼ 4prv21 að1  aÞrKdr

(6)

dT ¼ 4prv1 Ubð1  aÞr 3 Kdr

(7)

C. Peng et al. / Energy 126 (2017) 821e829

The blade element theory divides the wind turbine blade into several small segments, which are called blade elements. In this theory, it is assumed that there is no radial interaction between the flows on each blade element. Thus, the thrust force and aerodynamic torque on each blade element can be calculated by the aerodynamic characteristics of wind turbine airfoils. The thrust force acting on the circular ring of the wind wheel could be written as follows,

823

factor is large by using Equ. (15). Therefore, the axial induction factor should be corrected, and based on the empirical formula proposed by Glauert [29], when axial induction factor is higher than 0.4, it must be calculated as follows [30],

a¼

18K  20  3

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CF ð50  36KÞ þ 12Kð3K  4Þ 36K  50

(17)

where CF is the thrust force coefficient, which is defined as

1 dF ¼ Brcv20 Cn dr 2

(8)

where v0 is the air flow velocity acting on the blade element, it is given as follows,

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v0 ¼ ð1  aÞ2 v21 þ ð1 þ bÞ2 U2 r 2

(9)

According to the aerodynamic characteristics of wind turbine airfoils, the aerodynamic torque acting on the circular ring of wind wheel could be obtained by the following equation,

dT ¼

1 Brcv20 Ct rdr 2

(10)

where Ct is the tangential force coefficient. Combining the momentum theory with the blade element theory for the thrust force, makes the Equ. (6) equal to Equ. (8), and the axial induction factor can be written as:

að1  aÞ ¼

s 4

v20 v21

1 Cn K

(11)

Similarly, the aerodynamic torque should be equal to the corresponding aerodynamic torque from the blade element theory. The Equ. (7) equals to Equ. (10) and the tangent induction factor b can be written as

bð1  aÞ ¼

sv20 1 Ct 4v1 Ur K

(12)

From the triangle of velocities on each blade element, it is observed that there is a relationship between inflow angle and wind speed, which is deduced as follows,

sin 4 ¼

ð1  aÞv1 v0

ð1 þ bÞUr cos 4 ¼ v0

(14)

1 2

4K sin 4 sCn

b ¼ 4K

(15)

þ1

1

(16)

It is noted that the above Equs. (15) and (16) are only suitable for the load condition when the axial induction factor is smaller than 0.4. When the axial induction factor is larger than 0.4, the tradition momentum theory is inaccurate for the turbulent wake flow of wind turbine blade, i.e., the calculation error of axial induction

(18)

By substituting Equ. (8) into Equ. (18), the thrust force coefficient can ben given as follows, 1 Brcv2 C dr ð1  aÞ2 s n ¼ CF ¼ 21 2 0 ðCl cos 4 þ Cd sin 4Þ sin2 4 2 rv1 2prdr

(19)

In similar, the aerodynamic torque coefficient CT can be written as follows,

CT ¼ 1

dT

(20)

2 2 rv1 rdA

By using Equs. (11) and (20), the aerodynamic torque coefficient can be written as follows, 1 Brcv2 C rdr 0 t 2 2 rv1 r2prdr

CT ¼ 21

¼

ð1  aÞ2 s sin2 4

ðCl sin 4  Cd cos 4Þ

(21)

3. Power coefficient and flapping moment coefficient calculation model In this section, a power coefficient and flapping moment coefficient calculation model is built. According to Equs. (8) and (10), the thrust force and aerodynamic torque for the wind turbines can be obtained as follows,

F¼

1 Br 2

T¼

1 Br 2

Z

cv20 Cn dr ¼

N 1 X Br ci v20i Cni dri 2 i¼1

Z cv20 Ct rdr ¼

N 1 X Br ci v20i Cti ri dri 2 i¼1

(22)

(23)

where F is the thrust force of the wind turbine blades and T is the corresponding aerodynamic torque. It is obvious that the aerodynamic torque will cause the vibration of wind turbine and the thrust force will cause the wind turbine to flap. Now, the flapping moment could be defined as follows,

M¼

1 sin 4 cos 4 sCt

dF

2 2 rv1 dA

(13)

where v1 ¼ UR=l. According to Equs. (11) and (12), the axial induction factor and tangent induction factor can be simplified as follows,

a¼

CF ¼ 1

1 Br 2

Z cv20 Cn rdr ¼

N 1 X Br ci v20i Cni ri dri 2 i¼1

(24)

According to the aerodynamic of wind turbine, the aerodynamic torque determines the mechanical energy generation of wind turbine. The flapping moment would cause the vibration of wind turbine blades, drive chain and structural tower system, which will cause their mechanical fatigues and operation instability. Similar to the definition of power coefficient, the flapping moment coefficient could be defined as follows,

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C. Peng et al. / Energy 126 (2017) 821e829

M 2 2 rv1 AR

CM ¼ 1

(25)

Initializes calculation parameters, ( tip speed ratio, pitch angle and so on)

According to Equ (24), the flapping moment coefficient can be further rewritten as follows,

CM ¼

1 Br 2

Z

cv20 Cn rdr

1 rv2 AR 2 1

B ¼ pR3

Z

ð1  aÞ2 sin2 4

cCn rdr

(26)

Generally, the mechanical power could be calculated by the product of aerodynamic torque and the rotational angular velocity of wind turbines, the mechanical power of the swept area of the turbine blades usually can be written as follows,

P ¼ TU ¼

1 UBr 2

ZR cv20 Ct rdr

(27)

1 3 1 rv A ¼ rv31 pR2 2 1 2

P CP ¼ P0

Calculates the force coefficients of each blade element

ak = a 'k bk = b 'k

(29)

By substituting the Equs. (27) and (28) into Equ.(29), the power coefficient of wind turbines can be written as following equation.

UB pR2

Calculate the angle of attack of each blade element

(28)

The power coefficient of wind turbines can be written as follows,

CP ¼

Calculate the inflow angle of each blade element

ro

The total wind power acting on the wind wheel can be written as follows,

P0 ¼

Initializes axial and tangent induction factors of each blade element.

Z

v30 cCt rdr B ¼ pR2 v31 v0

Z

ð1  aÞ3 cos 4 cCt dr ð1 þ bÞ sin3 4

Updates axial induction and tangent induction factors of each blade element

(30)

a 'k -ak ≤ ε a and b 'k -bk ≤ ε b N

4. Composite calculation model for power coefficient and flapping moment coefficient Combining above modified Blade Element Momentum model and calculation models, a novel composite calculation model for power coefficient and flapping moment coefficient is built in this section. 4.1. Iteration calculation for axial induction factor and tangent induction factor The iteration based calculation procedure for axial induction factor and tangent induction factor is descript in detail as follows and shown in Fig. 1. Step 1: Initializes calculation parameters. According to the actual condition of wind turbine, sets the range of tip speed ratio l as ½lmin ; lmax  and divide it into N points fli g; ði ¼ 1; 2; /NÞ with interval ðlmax  lmin Þ=N. Sets the range of pitch angle b as ½0; bmax  and divide it into M points fbj g; ðj ¼ 1; 2; /MÞ with interval bmax =M. Divides wind turbine blades into blade elements with a certain interval, the number of wind elements is set as D. Sets the i ¼ 1; j ¼ 1. Step 2: Initializes axial and tangent induction factors of each blade element. Set the axial induction factors ak and tangent induction factors bk as 0, i.e., ak ¼ 0; bk ¼ 0; k ¼ 1; 2; /; D. Step 3: Calculates the inflow angle of each blade element 4k by using following equation.

Y Iteration calculation for axial induction and tangent induction factors ends Fig. 1. Iteration calculation procedure for axial induction factor and tangent induction factors.

4k ¼ arctan

ð1  ak Þv1 ð1 þ bk ÞUrk

(31)

where v1 is the wind speed in front of the wind turbine, v1 ¼ UR=li , rk is the radius of kth blade element. Each blade element has a certain radius range, the minimum value in this range is always considered as radius of blade element. Step 4: Calculates the angle of attack of each blade element ak by using following equation.

ak ¼ 4k  bj

(32)

Step 5: Calculates the coefficients of each blade element, including the lift coefficient Cl;k , drag coefficient Cd;k , tangential

C. Peng et al. / Energy 126 (2017) 821e829

force coefficient Ct;k , and normal force coefficient Cn;k . Their calculation equations are given as follows [31,32],

Cl;k ¼ 2 sin ak cosak ; Cd;k ¼ 2 sin2 ak ;

(33)

Ct;k ¼ Cl;k sin 4k  Cd;k cos 4k ; Cn;k ¼ Cl;k cos 4k þ Cd;k sin 4k ; (34) Step 6: Updates axial induction factors and tangent induction factors of each blade element. By using Equs. (4)e(5) and (15), calculate axial induction factor of kth blade element a0k . If a0k > 0:4, corrects a0k by Equ. (17). Calculates the tangent induction factors b0k by Equ. (16). Step 7: Judges whether the error of axial induction factors and tangent inductionfactors of each blade element satisfy the re  quirements, i.e., if a0k  ak   εa and b0k  bk   εb , where εa and εb are the preset threshold value of error, and always are selected as 1e-4. If the conditions are satisfied, continue. Otherwise, ak ¼ a0k , bk ¼ b0k and goto Step 3. Step 8: Iteration calculation for axial induction factors and tangent induction factors ends.

Table 1 Distributed aerodynamic parameters of NREL 5 MW wind turbine blade. Blade Element

Radius/m

Chord/m

Twist Angle/degree

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

2.8667 5.6000 8.3333 11.7500 15.8500 19.9500 24.0500 28.1500 32.2500 36.3500 40.4500 44.5500 48.6500 52.7500 56.1667 58.9000 61.6333

3.542 3.854 4.167 4.557 4.652 4.458 4.249 4.007 3.748 3.502 3.256 3.010 2.764 2.518 2.313 2.086 1.419

13.308 13.308 13.308 13.308 11.480 10.162 9.011 7.795 6.544 5.361 4.188 3.125 2.319 1.526 0.863 0.370 0.106

4.2. Nonlinear fitting based composite calculation model Now, by using above obtained axial induction factors and tangent induction factors and Equs. (26) and (30), calculate the power coefficient and flapping moment coefficient at ith tip speed ratio li and jth pitch angle bi , i.e., CP;ij and CM;ij . According to the N points of tip speed ratios, M points of group pitch angles, the corresponding power coefficient CP;ij and flapping moment coefficient CM;ij , fits the calculation model of power coefficient CP ¼ f1 ðl; bÞ and calculation model of flapping moment coefficient CM ¼ f2 ðl; bÞ by using following composite calculation model, which is derived the sinusoidal function based calculation model [33].


  p l  hx;3 Cx ¼ hx;1  hx;2 b sin hx;4  hx;5 b   hx;6 l  hx;3 b ðx ¼ P; MÞ

5. Illustrative example 5.1. Aerodynamic parameters of wind turbine The aerodynamic characteristics parameters of wind turbine airfoils are the main parameters used to calculate axial induction factor and tangent induction factor. The determination of wind turbine airfoil parameters can be generally divided into two stages [34]. Before the 1990s, the design of wind turbine airfoils has continued to use the existing traditional aviation airfoils. For example, National Advisory Committee for Aeronantics (NACA) have developed a series of wind turbine airfoils and provided the airfoil code which is composed of the letter “NACA” and a column of digits. Then, based on the geometric parameters related to the digits, a specific equation was used to calculate the exact shape of the airfoil. As the number of the digits increasing, currently, the NACA airfoil family mainly consists of 4 digits, 5 digits, 6 digits, 7 digits, 8 digits and 16 digits. However, the performance of airfoil will be greatly reduced if the tradition air wing for the aircrafts is used for the wind turbine, because there is a large difference in the working condition between the wind turbines and aircrafts. Therefore, the special airfoil for the wind turbines has been carried out gradually by the United States and Europe since the last century

0.10

0.20

0.05

0.15

0.00

0.10

-0.05

TS R: 5 . 0 TS R: 7 . 0 TS R: 9 . 0

0.00

0

10

20

30

40

50

(35)

where hx;1 , hx;2 , hx;3 , hx;4 , hx;5 , hx;6 are the parameters needs fitting.

0.25

0.05

825

TS R: 5 . 0 TS R: 7 . 0 TS R: 9 . 0

-0.10

65

-0.15

0

10

20

Fig. 2. Axial induction factor and tangent induction factor of blade element.

30

40

50

65

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C. Peng et al. / Energy 126 (2017) 821e829

0.5

0.8 TSR: 5.0 TSR: 7.0 TSR: 9.0

0.7 0.6

0.3

0.5

0.2

0.4

0.1

0.3

0

0.2

-0.1

0.1 0

10

20

30

40

50

TSR: 5.0 TSR: 7.0 TSR: 9.0

0.4

65

-0.2 0

10

20

30

40

50

65

Fig. 3. Thrust coefficient and torque coefficient of blade element.

of 90's. After decades of development, the airfoil family specific to the wind turbine has mainly consists of (1) S series in United States, (2) DU series in Holland, (3) RISO series in Denmark, (4) FFA series in Sweden. Among them, since the NREL 5 MW wind turbine airfoil [35] has been used by numerous research projects, such as the Integrated European Union Upwind Research Program and the International Energy Agency Wind Annex XXIII Subtask 2 Offshore Code Comparison Collaboration. The NREL offshore 5-MW baseline wind turbine has also been used to establish the reference specifications for a number of research projects. Therefore, the NREL 5 MW wind turbine airfoil is selected in this paper as a reference blade. This wind turbine airfoil has three blades. The radius of each blade is designed as 63 m and the radius of the hub is 1.5 m. As indicated by Jonkman et al. [36], the main aerodynamic parameters is shown in Table 1, including the node location of each blade element, the aerodynamic twist angle and the chord length at the center of the blade element. 5.2. Parameters analysis By using the proposed calculation model and the aerodynamic parameters of wind turbine blade airfoil in Table 1, the axial

induction factor, tangent induction factor, thrust force coefficient, aerodynamic torque coefficient, power coefficient and flapping moment coefficient are calculated by MATLAB. Without loss of generality, the flow velocity of the wind in front of the blade hub location is assumed as [VX, VY, Vz] ¼ [12.0 m/s, 0, 0] and the corresponding yaw angle is set as zero. 1) Influence of Tip Speed Ratio To investigate the influence of the tip speed ratio to the axial induction factor, tangent induction factor, thrust force coefficient and aerodynamic torque coefficient, the influence could be analyzed with varying the tip speed ratio from 5.0 to 9.0 with the interval of 2.0, and fixing the pitch angle at 5 . The distributions of the axial induction factor and tangent induction factor on each blade element are obtained, which is shown in Fig. 2. As seen in Fig. 2, the axial induction factors vary from almost 0.00 to 0.25 with wind turbine blade elements. The maximum value of the axial induction factor locates around the tip location, which indicates that the axial velocity at this location is the largest one. Meanwhile, the minimum value of the axial induction factor locates at 8 m and 40 m position. It is obvious that the influence of the tip speed ratio on the axial induction factor is remarkable. The value of the tangent

0.25

0.10

0.20

0.05

0.15

0.00

0.10

-0.05 Pitch : 3 o

Pitch : 3 o 0.05

0.00

0

10

20

30

40

Pitch : 5

o

Pitch : 7

o

50

Pitch : 5 o

-0.10

Pitch : 7 o 65

-0.15

0

10

20

Fig. 4. Axial induction factor and tangent induction factor of blade element.

30

40

50

65

C. Peng et al. / Energy 126 (2017) 821e829

827

0.6

0.6

Pitch : 3 o

0.5

0.5

Pitch : 5 o Pitch : 7 o

0.4 0.4

0.3 0.2

0.3

0.1

Pitch : 3 o

0.2

Pitch : 5

o

0.0

Pitch : 7 o

0.1

0

10

20

30

40

50

65

-0.1

0

10

20

30

40

50

65

Fig. 5. Thrust coefficient and torque coefficient of blade element.

Fig. 6. Power coefficient and flapping moment coefficient of wind turbine.

induction factor varies from 0.15 to 0.10. As seen in Fig. 2, the tangent induction factors are negative around the hub location, which indicates the rotational speed of the wind at those locations is opposite to the rotational speed of the wind turbine blade. Besides, it is shown that the influence of the tip speed ratio on the tangent induction factor is relatively small, and only change at the range of the 10 me40 m position of the wind turbine blade element. The thrust force coefficient and the aerodynamic torque coefficient are calculated by using Equs. (19) and (21) respectively. The calculation results are shown in Fig. 3. As seen in Fig. 3, the values of the thrust force coefficient range from 0.1 to 0.8, and they vary with the wind turbine blade element from hub to tip. The maximum value of the thrust force coefficient locates around the 10 m position and the minimum value locates around 8 m position. With the increasing of the tip speed ratio, the fluctuation of the thrust force coefficient on the blade becomes more obvious. The value of the aerodynamic torque coefficient is approximately between 0.1 and 0.4. The aerodynamic torque coefficient is negative around the hub location, which indicates that this position may be the weakest part of the wind turbine blade, and the load at this position is most unfavorable to the wind turbine blade and easy to be broken. The influence of the tip speed ratio on the torque coefficient is insignificant.

2) Influence of Pitch Angle To investigate the influence of pitch angles on the axial induction factor, tangent induction factor, thrust force coefficient and aerodynamic torque coefficient, the influence could be analyzed by changing the pitch angle from 3 to 7 with the spacing of 2 , and the tip speed ratio is fixed at 5. The distribution of axial induction factor and tangent induction factor along with the wind turbine blade is shown in Fig. 4. As seen in Fig. 4, it is obvious that the pitch angle has considerable influence to the axial induction factor from 20 m location to the tip location of the blade. With the increasing of the pitch angle, the value of the axial induction factor gradually decreases. The influence of the pitch angle on the tangent induction factor is quite small, which means the adjusting of pitch angle contribute little influence to the distribution of tangent induction factors along with the wind turbine blade.

Table 2 Parameter value for power coefficient and flapping moment coefficient calculation model. Parameter

hx;1

hx;2

hx;3

hx;4

hx;5

hx;6

SSE

CP(x ¼ P) CM(x ¼ M)

0.48 0.66

0.30 0.03

1.12 0.51

18.45 25.64

0.96 1.26

0.036 0.005

0.82 0.20

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C. Peng et al. / Energy 126 (2017) 821e829

0.5

0.5

0

0

-0.5

-0.5 10

10

10

10 5

5

5

5

0 0

0 0 Fig. 7. Surface of power coefficient of wind turbine.

The thrust force coefficient and the aerodynamic torque coefficient are calculated by using Equs. (19) and (21) respectively, the results are shown in Fig. 5. As seen in Fig. 5, the influence of the pitch angle on the thrust force coefficient mainly falls into the ranges from 20 m location to the tip of the wind turbine blade. Specifically, the values of the thrust force coefficient gradually decrease with the increasing of the pitch angle at these influential locations. However, the influence of the pitch angle on the aerodynamic torque coefficient is relatively small. The aerodynamic torque coefficient is negative at the locations close to the turbine hub. Those positions would be the weakest part of the wind turbine blade since the direction of torque force at those locations will be opposite to the direction of rotational speed. 5.3. Calculation results and analysis The power coefficient and flapping moment coefficient can be calculated by using Equs. (26) and (30) respectively, it is obvious that these two coefficients are mainly dependent on the tip speed ratio and pitch angle. The power coefficient and flapping moment coefficient of the NREL 5 MW wind turbine are calculated with

changing the tip speed ratio from 1.0 to 12.0 with an interval of 0.1 and varying the pitch angle from 0 to 8 with an interval of 2 . The calculation results are shown in Fig. 6. As seen in Fig. 6, the maximum power coefficient of the NREL 5 MW wind turbine is 0.4827 when the tip speed ratio reaches 7.7 and the pitch angle is at 0 , and relative flapping moment coefficient is 0.5214. The power coefficient and flapping moment coefficient decrease with the increasing of pitch angle. By using the proposed calculation algorithm in Section 4, the power coefficient and flapping moment coefficient of NREL 5 MW wind turbine calculation formula are developed as Equ. (35), The value of fitting parameters hx;1 ; hx;2 ; hx;3 ; hx;4 ; hx;5 ; hx;6 is calculated and are given in Table 2, where SSE is the sum of squared fitting error. The comparison of the fitting surfaces and computational surfaces are shown in Figs. 7 and 8 respectively. As seen in Table 2, Figs. 7 and 8, the sum of squared fitting error of power coefficients between TRS [0, 15] and pitch angle [0, 10 ] is 0.82, the sum of squared fitting error of flapping moment coefficients between TRS [0, 15] and pitch angle [0, 8 ] is 0.20. It is obvious that the fitting accuracy of power coefficients and flapping moment coefficients is high.

0.5

0.5

0

0

-0.5 15

8 6

10

4 5

2

-0.5 15

8 6

10

4

5

2 0

0 Fig. 8. Surface of flapping moment coefficient of wind turbine.

C. Peng et al. / Energy 126 (2017) 821e829

6. Conclusion In this paper, a novel composite calculation model is proposed to calculate the power coefficient and flapping moment coefficient comprehensively, which can be used for wind turbine aerodynamic load analysis and provide oscillating load and flapping load status information for wind turbine control. A modified Blade Element Momentum model is built to obtain the axial induction factor, tangent induction factor and aerodynamic torque coefficient of wind turbine. Then, the calculation model for power coefficient and flapping moment coefficient is presented. Based on above works, a composite calculation model based on iteration procedure and nonlinear fitting is designed to calculate power coefficient and flapping moment coefficient simultaneously. The proposed calculation model is implemented to evaluate the power coefficient and flapping moment coefficient of NREL 5 MW wind turbine. Meanwhile, the influence of pitch angle and tip speed ratio on the axial induction factor, tangent induction factor, thrust force coefficient and aerodynamic torque coefficient is analyzed. Compared with the existing calculation models, the proposed composite calculation model not only has a higher calculation accuracy of power coefficient, but also can obtain the flapping moment coefficient in the same time. Acknowledgments This work was supported by National Natural Science Foundation of China under Grant No. 61201010 and Fundamental Research Funds for the Central Universities under Grant no. ZYGX2015J073.

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