International Journal of Mechanical and Electrical Engineering 2014, Volume 1, Issue 1, Pages 1-9 Published Online 30-12-2014 in IJMEE http://www.ijmee.org

www.atlantisrepublic.com

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License Copyright © 2014 by author(s) and IJMEE Page 1

An Approximate Method for Calculating Stochastic Loads on Wind Turbines

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos* and Antonios Stathis School of Mechanical Engineering, National Technical University of Athens, Athens, Greece

*e-mail: cost@central.ntua.gr, web page: http://www.mech.ntua.gr Received 10-12-2014

Abstract

The actual wind loads which are applied on the various wind turbine parts and components are varying in amplitude and direction. Due to the stochastic nature of the wind force, this variation is stochastic too. Calculating these stochastic loads is not easy and many techniques have been proposed by researchers. In areas where wind turbines are installed, statistical data regarding the wind behavior are obtained using proper equipment and data recorders. These data usually are further processed using modern statistical tools in order to find the characteristics of the wind forces and their trend. The two parameter Weibull distribution and the Rayleigh distribution seem to fit with the actual data. Using time series analysis, the aim is to identify the distribution parameters and its ‘shape factor’. In this work, we perform an analysis of these stochastic loads on wind turbines and we propose a method for assigning an equivalent dynamic load on the main parts of the wind turbine in order to calculate and estimate their useful operational life. Knowing their estimated life can be helpful during the initial selection of these parts and when establishing a maintenance plan for the wind turbine.

Keywords

Static, dynamic, stochastic loads, Weibull distribution, Rayleigh distribution, wind turbines

1. Introduction

Every wind turbine has a rotor with attached airfoil shaped blades which harvest the energy from the wind. The rotor is located on a hub and it is connected to the generator which produces the electricity through the main drive shaft and the gear box.

Wind turbines are divided into two categories regarding the direction of the main rotating shaft which can be vertical or horizontal. Market have prevailed turbines horizontal axis, with two or three blades in various sizes. This is the most common wind turbine design where the axis of blade rotation is parallel to the wind flow and to the ground. Wind turbines operate in different environmental conditions and this variability complicates the comparison between different turbine types.

Generally, a typical horizontal axis wind turbine consists from the following parts:

The blades and the rotor, which converts the energy in the wind to rotational shaft energy.

The main shaft bearings The main shaft is usually made of steel and used to carry the torsional and bending moments of rotor gearbox [1].

The gearbox which adjusts the rotational speed of the rotor to the synchronous speed of the generator while providing support for the weight of the rotor [2].

The electric generator which is connected to the output of the multiplier and converts mechanical energy into electrical

A tower which supports all the electromechanical installation

the guidance, which aligns the rotational axis of the rotor in parallel to the wind direction

The control system (electronic table and control panel) that coordinates and controls all the functions of the wind- generator.

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos and Antonios Stathis

www.ijmee.com Page 2

Figure 1. A typical horizontal wind turbine and its major parts.

Dimensioning of a wind generator depends on the

wind potential, the expected wind speeds and the chosen type. These factors are critical in calculating the applied loads on each part of the wind turbine and the proper dimensioning of them. The design loads can be divided into two major categories:

a. Gravitational and inertial loads [3].

Gravitational and inertial loads are static and dynamic loads that result from gravity, vibration, rotation and seismic activity.

b. Aerodynamic loads. Aerodynamic loads are

static and dynamic loads that are caused by the airflow and its interaction with the stationary and moving parts of wind turbines. The airflow is dependent upon the average wind speed and turbulence across the rotor plane, the rotational speed of the rotor, the density of the air, and the aerodynamic shapes of the wind turbine components and their interactive effects, including aero elastic effects.

2. Theory Overview

The aerodynamic forces are generated by the wind interacting with the blade. The primary focus of wind turbine aerodynamics is the magnitude and distribution of this force. The major aerodynamic forces on the rotor blades are Drag and Lift. The direction of the drag force D is parallel to the relative

wind, while the lift force L is perpendicular. According to the Buckingham’s theorem, these forces can be calculated from the following equations:

𝐶𝐿 =𝐿

12𝜌𝐴𝑤2

→ 𝐿 = 𝐶𝐿

1

2𝜌𝐴𝑤2 (1)

𝐶𝐷 =𝐷

12𝜌𝐴𝑤2

→ 𝐷 = 𝐶𝐷

1

2𝜌𝐴𝑤2 (2)

where: CL = the lift coefficient

CD = the drag coefficient w = the relative wind as experienced by the

wind turbine blade A = the affected area

The relative speed w is a vector subtraction

between the wind speed u and the speed v at each point of a blade.

�⃗⃗⃗� = �⃗⃗� − �⃗� (3)

The speed at the tip of the blade is usually used for

this purpose, and is written as the product of the blade radius and the rotational speed of the blade (v=ω*r, where ω = rotational velocity in radians/second and r = blade radius).

Equations (1) and (2) can be written as:

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos and Antonios Stathis

www.ijmee.com Page 3

The actual loads on the wind turbine parts occur

due to static loads (gravitational) and aerodynamic loads. Calculating the actual loads demands the calculation of the stochastic aerodynamic loads which in turn depend on the actual wind speed. As wind speed is time-varying, a method for approaching the wind speed by statistical methods is necessary.

2.1. Calculations on the wind speed

As it is shown in equations (4) and (5) the aerodynamic forces on the wind turbine are relevant to the square of the wind speed u. Wind speed is not constant and it is changing in amplitude during time. There are many distributions describing the frequency distribution the speed of the wind, but the most widespread and widely used are the Weibull and the Rayleigh distributions. The Weibull distribution describes satisfactorily the wind data in temperate regions, up to the height of 100 meters and has been used to estimate the Wind energy in many countries [4]. This distribution is determined by two parameters and identifies the probability of wind speed being located in an area of speed u [5]. The Weibull curve is given from the following equation:

𝑓(𝑢) =𝑘

𝑐(𝑢

𝑐)𝑘−1

𝑒−(𝑢𝑐)𝑘

(6)

where k is (shape parameter) and c is (scale

parameter). The parameter c is the average speed of the wind under the Instrument [6].

�̅� = 𝑐𝛤 (1 +1

𝑘) (7)

where with Γ denotes an arithmetic function ΄΄

gamma ΄΄. According to the analysis of [7], replacing the

distribution function of wind speed by Weibull in this relationship gives:

�̅� = ∫ 𝑢𝑘

𝑐(𝑢

𝑐)𝑘−1

𝑒−(𝑢𝑐)𝑘

∞

0

𝑑𝑢 (8)

Which can be written as:

�̅� = 𝑘 ∫ (𝑢

𝑐)𝑘

𝑒−(𝑢𝑐)𝑘

∞

0

𝑑𝑢 (9)

Taking

𝑥 = (𝑢

𝑐)𝑘

, 𝑑𝑢 =𝑐

𝑘𝑥(

1𝑘−1)𝑑𝑥 (10)

we arrive at:

�̅� = 𝑐 ∫ 𝑒−𝑥𝑥1

𝑘⁄

∞

0

𝑑𝑥 (11)

Which is in the form of the function ΄΄Γ΄΄ and can

also be written as:

𝛤𝑛 = ∫ 𝑒−𝑥𝑥𝑛−1

∞

0

𝑑𝑥 (12)

Therefore, by equation (11) we can express the

mean wind speed as given in the relation (8). The shape parameter k is inversely proportional to the square of dispersion of speeds in average speed [1]:

𝜎2 = 𝑐2 {𝛤 (1 +2

𝑘)− [𝛤 (1 +

1

𝑘)]

2

} (13)

From the above relationship conclude that larger

values of k give less deviation and therefore a greater concentration of speeds wind around the mean. Parameters k and c vary depending on location because depend on weather conditions and terrain. Generally, the parameter c shows the intensity of the winds prevailing in an area and the parameter k shows how sharp is the Weibull curve [8].

As it can be seen from figures 2 and 3, lower values of c result in higher value of maximum probability distribution, while smaller values of k result in lower value of the maximum likelihood distribution.

The parameter k ranges from 1.5 to 3 for most cases [9]. Depending on the geographical location of the region, the parameter k will have price 1 in Arctic and tropical areas (where the wind speeds are relatively low), 2 in areas dominated by westerly winds as areas in NW Europe and 3 in areas near the equator characterized by relatively steady wind [5]. Depending on the terrain parameter k takes values close to 1.6 for mountain areas and rates close to 2 in coastal areas [10].

𝐿 = 𝐶𝐿

1

2𝜌𝐴(�⃗⃗� − �⃗� )

2→

𝐿 = 𝐶𝐿

1

2𝜌𝐴(�⃗⃗� 2 − 2�⃗⃗� �⃗� + �⃗� 2)

(4)

𝐷 = 𝐶𝐷

1

2𝜌𝐴(�⃗⃗� − �⃗� )

2→

𝐷 = 𝐶𝐷

1

2𝜌𝐴(�⃗⃗� 2 − 2�⃗⃗� �⃗� + �⃗� 2)

(5)

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos and Antonios Stathis

www.ijmee.com Page 4

Figure 2. Effect of parameter c in the Weibull equation

for k = 1.7

Figure 3. Effect parameter ΄΄k΄΄ in equation Weibull for

c=6m/s

Generally small values of k mean small wind speeds [11], but large wind power available due to the greater speed range shown in region , Besides, remember that the wind power potential depends on the cube of the wind speed [10]. Large values of k means that some days are windy and an equal number days described by low wind speeds.

Areas with k = 2 characterized by several days having smaller wind speeds than average and only a few days have strong winds. Large values of the parameter c characterized by high intensity winds. For a value of k = 2, which is quite realistic value for the Aegean region [1], the Weibull distribution is similar to the Rayleigh distribution [12] which is described by the following equation:

𝑓(𝑢) =𝜋𝑢

2�̅�2𝑒

−(𝜋𝑢2

4𝑢2) (14)

For this reason, the Weibull distribution with two

parameters is more general [13] and provides a more accurate calculation of average speed and power of

the wind over the Rayleigh distribution [14]. Another useful curve for the determination of

Wind energy at a certain place is the total density probability function plot which gives the probability of the velocity u to be less than a specific value u0 and it is given in equation (18):

𝐹(𝑢 ≤ 𝑢0) = 1 − 𝑒−(𝑢0𝑐

)𝑘

(15)

As it can be seen from equation (15), the total

probability density depends upon the parameters k and c.

2.2. Methods for determining the parameters of Weibull

There are many methods for determining the parameters k and c. The most important are: the graphical method (minimum squares), the maximum likelihood method, the modified method maximum likelihood method and the standard deviation.

Graphical Method This method involves the construction of which

derives directly from the logarithm of equation (15).

𝐹(𝑢 ≤ 𝑢0) = 1 − 𝑒−(𝑢0𝑐

)𝑘

⇒ 𝑒−(𝑢0𝑐

)𝑘

= 1 − 𝐹(𝑢 ≤ 𝑢0)𝑙𝑛⇒

(16)

𝑙𝑛 [𝑒−(𝑢0𝑐

)𝑘

] = ln[1 − 𝐹(𝑢 ≤ 𝑢0)] ⇒

−(𝑢0

𝑐)𝑘

= ln[1 − 𝐹(𝑢 ≤ 𝑢0)] ⇒

(17)

𝑘𝑙𝑛 (𝑢0

𝑐) = ln{−𝑙𝑛[1 − 𝐹(𝑢 ≤ 𝑢0)]} (18)

Equation (18) has the form of:

𝑌 = 𝐵𝜒 + 𝐴 (19) Where

𝑌 = 𝑙𝑛{−𝑙𝑛[1 − 𝐹(𝑢 ≤ 𝑢0)]} 𝐵 = 𝑘 𝜒 = 𝑙𝑛 (𝑢0) 𝐴 = −𝑘𝑙𝑛(𝑐)

The parameter k is given by (19) while the

parameter c is given by the relation

𝑐 = 𝑒−𝐴𝐵 (20)

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos and Antonios Stathis

www.ijmee.com Page 5

Subdividing the data of wind speed by incidence, we can find the various points of equation (17) and together with the points that result from the equation (18) (where F(u≤u0) is the probability of the wind speed u to be less than u0) we plot the least squares line describing relation (16), the slope of which gives us the constant k.

Method of maximum likelihood This method has been proposed by Stevens and

Smulders [15] and is the most used among others. It's pretty accurate, efficient and easy to use, but it needs many data. This method ignores null values of wind speed [16]. The shape parameter k and the scale parameter c are given by [17]:

𝑘 = [∑ 𝑢𝑖

𝑘𝑙𝑛(𝑢𝑖)𝑛

𝑖=1

∑ 𝑢𝑖𝑘𝑛

𝑖=1

−∑ ln (𝑢𝑖)

𝑛

𝑖=1

𝑛]

−1

(21)

𝑐 = (1

𝑛∑𝑢𝑖

𝑘

𝑛

𝑖=1

)

1𝑘

(22)

Where ui is the wind speed at time i and n is the

non-zero values of wind speed. In order to calculate the parameters, we start from

a random value of k (usually k = 2) and apply it to the formula (18) finding another value of k. Repeating this step leads to a new value of k. This process is repeated until we reach a steady value. Then, with equation (20), the scale parameter c is calculated.

Modified maximum likelihood This method is used when the wind speeds are

given in the form of frequency allocation. It is based on the previous method described above but in this case the shape parameter k and the scale parameter c in this case are given by [17, 18]:

𝑘 = [∑ 𝑢𝑖

𝑘𝑙𝑛(𝑢𝑖)𝑃(𝑢𝑖)𝑛

𝑖=1

∑ 𝑢𝑖𝑘𝑛

𝑖=1𝑃(𝑢𝑖)

−∑ 𝑙𝑛(𝑢𝑖)𝑃(𝑢𝑖)

𝑛

𝑖=1

𝑃(𝑢 ≥ 0)]

−1

(23)

𝑐 = (1

𝑃(𝑢 ≥ 0)∑𝑢𝑖

𝑘𝑃(𝑢𝑖)

𝑛

𝑖=1

)

1𝑘

(24)

where here ui are the values in the chart of wind speeds , n is the number of different frequencies (columns of the bar graph therefore ), P(ui) is the frequency with which each speed value ui is within a certain range i and P(u ≥0) is probability that the wind speed is greater than or equal to zero.

Method standard deviation Combining [9] the relations (10) and (16) we have:

(𝜎

�̅�)2

= 𝛤 (1 +

2𝑘)

𝛤2 (1 +1𝑘)− 1 (25)

Calculating σ and ū in a data series and replacing

the results in equation (28) gives the value of k. Then the parameter c is given by equation:

𝑐 =�̅�

𝛤 (1 +1𝑘)

(26)

Approximately parameter k and c can be given by

the relations [19]:

𝑘 = (𝜎

�̅�)−1.086

(27)

𝑐 =2�̅�

√𝜋 (28)

Also, the parameter c can be more precisely

calculated [20] with the equation below:

𝑐 =�̅�𝑘2.6674

0.184 + 0.816𝑘2.73855 (29)

3. WIND SPEED DISTRIBUTION CASE STUDY ON NAXOS ISLAND

Wind data collected by the National Meteorological Service (EMY) from stations that are located in the island of Naxos. The wind data contain measurements on the speed of wind, wind direction, pressure and temperature at a height of 10 meters above the ground.

Graphical Method Following the procedure described construct the

straight least squares according to the data from the table and calculate the parameter values k and c.

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos and Antonios Stathis

www.ijmee.com Page 6

Table 1. Statistical Data of Wind Speed on Naxos Island

Speed Number of

occurrences Frequency

Cumulative Frequency

U0 x y

0-1 8978 0,113769 0,113769 1 0 -2,1138

1-2 4029 0,051056 0,164825 2 0,693147 -1,71417

2-3 7590 0,096181 0,261006 3 1,098612 -1,19579

3-4 6426 0,081430 0,342436 4 1,386294 -0,86938

4-5 5339 0,067656 0,410092 5 1,609438 -0,63906

5-6 5808 0,073599 0,483691 6 1,791759 -0,41393

6-7 5427 0,068771 0,552462 7 1,945910 -0,21816

7-8 5456 0,069139 0,621601 8 2,079442 -0,02860

8-9 4654 0,058976 0,680576 9 2,197225 0,132113

9-10 3913 0,049586 0,730162 10 2,302585 0,269976

10-11 4430 0,056137 0,786299 11 2,397895 0,433844

11-12 3521 0,044618 0,830917 12 2,484907 0,575133

12-13 3434 0,043516 0,874433 13 2,564949 0,729920

13-14 2684 0,034012 0,908445 14 2,639057 0,871633

14-15 1649 0,020896 0,929341 15 2,708050 0,974517

15-16 1657 0,020998 0,950338 16 2,772589 1,099453

16-17 1141 0,014459 0,964797 17 2,833213 1,207953

17-18 1101 0,013952 0,978749 18 2,890372 1,348424

18-19 634 0,008034 0,986783 19 2,944439 1,464703

19-20 321 0,004068 0,990851 20 2,995732 1,546304

20-21 313 0,003966 0,994817 21 3,044522 1,660587

21-22 146 0,001850 0,996667 22 3,091042 1,741161

22-23 129 0,001635 0,998302 23 3,135494 1,852898

23-24 63 0,000798 0,999100 24 3,178054 1,947827

24-25 20 0,000253 0,999354 25 3,218876 1,993923

25-26 29 0,000367 0,999721 26 3,258097 2,102312

26-27 9 0,000114 0,999835 27 3,295837 2,164605

27-28 5 6,34E-05 0,999899 28 3,332205 2,218842

28-29 4 5,07E-05 0,999949 29 3,367296 2,291506

29-30 2 2,53E-05 0,999975 30 3,401197 2,359246

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos and Antonios Stathis

www.ijmee.com Page 7

Figure 4. Graphical method of determining Weibull parameters

By applying the method of least squares, the

equation of the line is:

𝑦 = 1.45𝑥 − 2.8 (30)

Thus based on the equations (19) and (20) and Weibull parameters can be calculated as k=1.45 and c=6.99.

But if you ignore the zero points (shown with the red dot in figure 4), the line equation is transformed into:

𝑦 = 1.57𝑥 − 3.14 (31) Yielding in k =1.57 and c=7.42 Method of maximum likelihood Applying relations (23) and (24) and applying the

methodology described parameters, Weibull derived as follows:

Figure 5. Weibull distribution to Naxos for k=1.67 και c=8.44

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos and Antonios Stathis

www.ijmee.com Page 8

Figure 6. Duration curve for Naxos for k=1.67 και c=8.44

4. Conclusions

Wind turbines are subjected to constant loads, like gravitational forces, and variable loads, mainly due to aerodynamic loads. After approaching the wind loads on a turbine, the loads on each part of the wind turbine can be calculated and proper dimensioning can be made during the design stage. The aerodynamic loads depend on the wind speed and can be well described by the two-parameter Weibull distribution. Regarding the main shaft bearings, it seems that the loads occurring by implementing the mean wind speed from the Weibull distribution can be satisfactory.

Several methods of approaching the values of these parameters have been proposed. The results from the implementation of the graphical method and the method of maximum probability on an Aegean island wind speed data showed that:

a. The values of Weibull distribution are quite

different when calculated by the maximum likelihood method with respect to graphical method because of the many zero wind speed value records.

b. The maximum likelihood method gives more

accurate results from the graphical method. A substantial difference between these two

methods is that the maximum likelihood method, in contrast to the graphical method, does not account for zero values of wind speed. Consequently, in accordance with the aforementioned case study, this is the reason why the values of the Weibull parameters are considerably different in the relevant calculations.

References

[1] Kaldellis I.K., 2005, Management of Wind Energy, 2nd Edition, Stamoulis, Athens. [2] Manwell J.F., Mc Gowan J.G and Rogers A.L., 2009, Wind Energy Explained - Theory, Design and Application, 2nd Edition, Wiley and sons publishers, Sussex. [3] International Standard IEC 61400-1. [4] Voyiatzis N., Kotti K., Spanomitsos S. and Stoukides M., 2004, “Analysis of Wind Potential and Characteristics in North Aegean, Greece”, Renewable Energy, 29, pp. 1193-1208. doi:10.1016/j.renene.2003.11.017 [5] Redlinger R.Y., Andersen P.D. and Morthorst P., 2002, Wind Energy in 21

st Century, Palgrave publishers, New York.

[6] Burton T., Sharpe D., Jenkins N. and Ervin B., 2001, Wind Energy Handbook, John Wiley and Sons publishers, Sussex. [7] Sathyajith M., 2006, Wind Energy- Fundamentals, Resource Analysis and Economics, Springer publishers, New York. [8] Bivona S., Burlon R. and Leone C., 2003, “Hourly Wind Speed Analysis in Sicily”, Renewable Energy, 28(9), pp. 1371-1385. doi:10.1016/S0960-1481(02)00230-6 [9] Akpinar E.K. and Akpinar S., 2004, “Statistical Analysis of Wind Energy Potential on the Basis of the Weibull and Rayleigh Distributions for Agin-Elazig, Turkey”, Journal of Power and Energy, 218(8), pp. 557-565. doi:10.1243/0957650042584357 [10] Di Piazza A., Di Piazza M.C., Ragusa A. and Vitale G., 2010, “Statistical Processing of Wind Speed Data for Energy Forecast and Planning”, International Conference on Renewable Energies and Power Quality, Grenada, Spain, pp. 23-25. [11] Fung J.C.H. , Yim S.H.L., Lau A.K.H. and Kot S.C., 2008, “Wind Energy Potential in Guangdong Province”, presented at the 9th WRF users’ workshop, June 2008 pp. 23-27. [12] Stiebler M., 2008, Wind Energy Systems for Electric Power Generation, Springer publishers, Berlin.

Georgios Gyparakis, Dimitrios Koulocheris, Theodoros Costopoulos and Antonios Stathis

www.ijmee.com Page 9

[13] Pavia E.G. and O’Brien J., 1986, “Weibull Statistics of Wind Speed Over the Ocean”, Journal of Applied Meteorology, 25, pp. 1324-1332. doi:10.1175/1520-0450(1986)025<1324:WSOWSO>2.0.CO;2 [14] Jowder F., 2006, “Weibull and Rayleigh Distribution Functions of Wind Speeds in Kingdom of Bahrain”, Wind Engineering, 30(5) pp. 439-445. doi:10.1260/030952406779502650

[15] Stevens M.J.M. and Smulders P.T., 2006, “The Estimation of Parameters of the Weibull Wind Speed Distribution for Wind Energy Utilization Purposes”, Journal of Wind Engineering, 3(2), pp. 132-145. [16] Seguro J.V. and Lambert T.W., 2000, “Modern Estimation of Parameters of the Two Weibull Wind Speed Distribution for Wind Energy Analysis”, Journal of Wind Engineering and Industrial Aerodynamics, 85(1), pp. 75-84. doi:10.1016/S0167-6105(99)00122-1

[17] Young L. and Xie M., 2003, “Efficient Estimation of the Weibull Shape Parameter Based on a Modified Profile Likelihood”, Journal of Statistical Computation and Simulation, 73(2), pp 115-123. doi:10.1080/00949650215729 [18] Ahmad S., Hussin M.A., Bawadi M.A. and Sanusi S.A., 2003, “Analysis of Wind Speed Variations and Estimation of Weibull Parameters for Wind Power Generation in Malaysia”, The 2nd Dubrovnik Conference on Sustainable Development of Energy, Water and Environment Systems, Dubrovnik, Croatia. [19] Bowden G.J., Barker P.R., Shestopal V. and Twidell J.W., 1983, “The Weibull Distribution Function and Wind Statistics”, Journal of Wind Engineering, 7(2), pp. 85-98. [20] Balouktsis A., Chassapis D. and Karapantsios T.D., 2002, “A Nomogram Method for Estimating the Energy Produced by Wind Turbine Generators”, Solar Energy, 72(3), pp. 251-259. doi:10.1016/S0038-092X(01)00099-8