Influence of the Weibull Parameters on the Estimation of Wind Turbine ...
Influence of the Weibull Parameters on the Estimation of Wind Turbine Loads Daniel F. A. Lemos1 , Alexandre L. Pereira2, and Everaldo A. N. Feitosa1 1
Brazilian Wind Energy Centre - CBEE, CTG/UFPE, Recife-PE, 50740-530, BRAZIL Phone/Fax: +55 (81) 34532975 e-mail: [email protected] 2 Wind Energy Postgraduate Program – Federal University of Pernambuco Recife – PE, BRAZIL e-mail: [email protected]
ABSTRACT There are many sites around the world characterized by wind distributions with relatively high Weibull shape factors and mean wind speeds but the design codes and Standards working groups only recommend the use of Rayleigh distribution for the wind turbine load calculations. The paper focuses on the effects of high Weibull shape factors on the loads calculation. Extreme and fatigue load cases are used to analyze the influence of different wind speed frequency distributions on the design load of a wind turbine. A real 250kW wind turbine was simulated using an aeroelastic computer model to obtain loads acting in several important components. The fatigue damage is calculated for different wind distributions using the lifetime equivalent load and the extreme load is the steady state calculations for an extreme wind with 50 years recurrence period at hub height. The extreme loads results indicate that wind tubines at higher shape factors sites will face much lower loads due to the reduced expected extreme wind speeds. The fatigue loads have a more complicated pattern of variation with the wind conditions. The influence of the shape factor on dynamic and fatigue loads should be determined for each wind turbine type because it depends on aerolastic and dynamic characterisitcs.
1. INTRODUCTION Although the wind energy community has a consensus of opinion about the use of Weibull distribution to represent all kinds of wind characteristics, the design codes and Standards working groups keep recommending the use of Rayleigh distribution for the wind turbine load calculations. Then one should expect that this approach leads to the worst load conditions, but is that really true? There are many sites around the world characterized by wind distributions with relatively high Weibull shape factors and mean wind speeds. Wind turbines can generate typically 10 to 30% more annually at those sites compared to Rayleigh distributed wind sites. But the frequency distribution of wind speeds also influence the extreme and fatigue loads that have to be considered in the design of a wind turbine. How do the loads compare to the standard (European) sites? This paper tries to answer these questions focusing on the effects of high Weibull shape factors on the loads calculation. Extreme and fatigue load cases are used to analyze the influence of different wind speed frequency distributions on the design load of a wind turbine. The IEC61400-1 [1] and the European Wind Turbine Standards II [2] are used as reference for the load cases, parameters and calculation procedures. A previous work from Møller [3] investigated the extreme and fatigue loads for wind turbines according to the expected Brazilian wind conditions. He showed that the extreme loads calculated for the region of high Weibull shape factors are considerably lower than the loads used in the design because the expected extreme wind speeds for a 50 years recurrence period are reduced. On the other hand, some fatigue loads, caused by gravity effects, are expected to be higher, specially the edwise blade root bending and the main shaft bending moments. The recent wind turbine installations in Brazil, in the region of high Weibull shape factors, motivated the present investigation on the assessment of structural integrity for local wind conditions. In the next sections we show detailed description of wind turbine load calculations for two representative design load cases: a) a design situation of a parked wind turbine with extreme external conditions for analysis of ultimate loads; and b) a normal design situation for analysis of fatigue loads. A real operating wind turbine is modelled and simulated using a validated computational code to provide results that can be checked out against measurements
2. MODELS AND SIMULATION A complete aeroelastic wind turbine model has been configured using the program BLADED [4]. The modeling parameters were taken from a real wind turbine with 29m diameter rotor, 250kW, stall controlled with induction generator, located at the Northeast coast of Brazil near the Brazilian Wind Energy Centre [5].
The three dimensional wind model, based on Veers [6] and Kaimal approaches, can describe the spatial and temporal atmospheric turbulences through the entire rotor disk. For the fatigue calculations, a total of 80 dynamic simulations have been performed using synthetic wind series from 5 to 24 m/s with 20Hz signal and 600s length each. The turbulence intensity is given by Equation 1 using parameter a = 3, which corresponds to the model adopted by the IEC 61400-1 for low level turbulence sites. The I15 is the turbulence intensity at 15 m/s and Ū is the mean wind speed.
I (a 15 / U ) I u 15 ( a 1)
(1)
The results of the simulations are the calculated loads acting upon the wind turbine components. Part of a simulated response for the flapwise bending moment at the blade root is shown in Figure 1 as an illustration.
fl apw ise ben ding (kN. m)
80 70 60 50 40 30 0
10
20
30 tim e (seconds)
40
50
60
Figure 1. Simulated flapwise blade bending moment at 9 m/s.
Fatigue loads The loads cycles were classified and counted using a rainflow algorithm [7], and the cumulative damage caused by the cyclic loads, applying the linear Miner´s rule, is given by n D i N (S i ) i
(2)
where ni is the counted number of cycles at stress level S i, and N(S i) is the number of cycles to failure at stress level Si obtained from the material S-N curve. This work investigates the relative fatigue damage not the components design life, therefore the fatigue loads are expressed by load ranges, R i, and number of cycles. The total fatigue damage, caused by the sum of all load ranges, can also be written as the result of one equivalent load, R eq, acting a number of cycles neq ., as shown in equation 3 below m n i Ri D ni R i m n eq R eq m R eq n eq
1
m
(3)
where m is the Wöhler curve exponent, assumed in the following calculations as 4 for steel components (tower and shaft) and 10 for the fiberglass blades. In addition, a correction for the value of Req has been applied to take into account the influence of the mean load level. The modified Goodman criterion, described in Thomsen [8], is a simple correction procedure that applies a gain factor, calculated from the load cycles, to the equivalent load. The equivalent loads are obtained for each simulated load series and average values are calculated for corresponding mean wind speeds. Although only 8 different wind speeds have been simulated, a complete spectrum of equivalent
loads, from cut-in to cut-out wind speeds, is computed using polynomial interpolation for the mean wind speeds not simulated. In order to compare the fatigue damage for site-specific wind conditions, i.e., different wind distributions, the lifetime equivalent load, Leq , is derived taking into consideration the contribution of the mean equivalent loads for all wind speeds during the design life of a wind turbine: m Req (U ) m n eq P (U ) nT 20 years Leq n eq, L 1
(4)
where U is the mean wind speed relative to a mean equivalent load, P(U) is the probability of occurrence of wind speed U defined by the Weibull distribution function, and nT20years is the occurrence number of ten minutes series in a lifetime of 20 years. Extreme loads There are several load cases described in [1] that must be analyzed for the determination of the ultimate loads. This paper analyzes one of them which is characterized by an extreme wind condition in combination with a normal machine state. The steady state calculations are made considering an extreme wind with 50 years recurrence period at hub height, the wind turbine in a normal state, rotor parked and perpendicular to the wind direction, aerodynamic brakes enabled and “no faults” status for the control system. Several load cases have been considered: flap and lag moments at blade root, torque and thrust at the hub, tower bending moments at the base for two orthogonal directions. The rotor position, or the azimuth angle, that leads to the worst condition for each load case was also included. The extreme wind speed adopted here, which is derived from [2], is the maximum 3s gust with a recurrence period of 50 years, Ve50 . Its relationship with the Weibull distribution parameters is important to compare the ultimate loads for different wind conditions. The basic assumptions are that the wind distribution for the 10min mean values, U, are well modeled by the Weibull function and that the short term (3s) averages, u 3s, are distributed around the 10min mean values with standard deviation 3s, following the Gaussian function
1 u 3 s U 1 f (u 3 s | U ) exp 3s 2 2 3 s
(5)
The conditional density function, f(u3s |U ), expresses the probability of a 3s gust within the 10min interval defined by the mean wind speed U. The total probability of occurrence for a 3s gust, f(u3s), is derived by integrating f(u 3s |U ) and the Weibull density function for all mean wind speeds k 1 k u 3s U 1 k U U f (u 3 s ) f (u3 s | U ) f (U ) exp exp dU 0 2 C C C ( U ) 2 3s 3s
(6)
Integrating the pdf function shown above, we find the cumulative function of the 3 second average wind speed. The cdf function must of the 315 576 th order statistics to take in account the various independent observations [9]. Now, it is possible to determine the Ve50 for any C and k wind condition, which is equivalent to find the wind speed with a 2% probability of being exceeded in a year. The 3s is the part of
the total standard deviation u that remains after using a low pass filter with averaging time of 3 seconds and u varies with the mean wind speed according to Eq. 1. Figure 2 shows the variation of the V e50 for several Weibull shape factors and four different mean wind speeds, 6.0, 7.5, 8.5 and 10.0m/s. The extreme wind speed decreases with the increase of the Weibull parameter k for all wind speeds. The annual mean wind speeds have been chosen to be identical with the ones used by the IEC61400-1 to define the wind turbine classes. The Ve50 used on the IEC calculation procedures are derived from Fig.2 using a shape factor of approximately 1.8. 3. RESULTS The ultimate load analysis has been performed for 12 wind conditions, four mean wind speeds and three shape factors (2, 3 and 4). The extreme wind speed, V e50, for each condition is presented in the Table 1.
The extreme loads for the 12 different wind conditions are presented in Figure 3. The values are normalized against the peak for each load case in order to facilitate the comparison. The edgewise bending moment at the blade root shows a small reduction when changing the shape factor from 2 to 3, and just a slight reduction from 3 to 4. However, the other load cases, flapwise blade and tower bending moments, which are greatly influenced by the value of the extreme wind speed, present a large variation with the factor k. The reduction can be as much as 70% in some cases when changing from a wind condition of k=2 to k=4. The fatigue load analysis is performed comparing the lifetime equivalent load, Leq (Eq. 6), calculated for several wind conditions defined by a combination of four annual mean wind speeds, 6.0, 7.5, 8.5 and 10m/s, and Weibull shape factors from 1.5 to 6.
Table 1. Extreme wind speed, Ve50 , for different wind conditions.
Weibull k
Annual mean wind speed 6 m/s
7,5 m/s
8,5 m/s
10 m/s
2
37,0 m/s
45,9 m/s
51,8 m/s
60,7 m/s
3
24,1 m/s
29,7 m/s
33,4 m/s
38,9 m/s
4
19,8 m/s
24,1 m/s
27,1 m/s
31,4 m/s
90
Extreme Wind Speed Ve 50 [m/s]
80 70 60 50 40 10 m/s 8.5 m/s 7.5 m/s 6 m/s
30 20 10 1
1,5
2
2,5
3
3,5
4
Weibull k
Figure 2. Influence of the shape factor on the V e50 for different annual mean wind speeds.
The calculated fatigue loads are for the blades (flap and lag root moments), main shaft (bending and torsional moments), tower top (tilt and torsional moments) and tower base (bending moments). The edwise blade moment results are presented in Figure 4. For all four mean wind speeds the equivalent load increases almost linearly with the Weibull shape factor. The equivalent load for the main shaft torsional moment, showed in Figure 5, increases with k for sites with mean wind speeds higher than ~7.0m/s and has a tendency to decrease for wind speeds lower than this value. The fatigue damage for the tower base bending moment parallel to the wind direction shows several patterns of variation with k depending of the mean wind speed, see Figure 6. For wind speeds of 7.5 and 8.5m/s the equivalent load has a minimum value around k = 2, while for 10m/s the minimum is at k = 3 and for 6.0m/s at 1.5.
Normalized Extreme Load Blade edge (Mx)
1
Blade flap (My)
Tower (Mx)
Tower (My)
0.9
Extreme load (kNm)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
2
3
2
4
3
4
2
3
4
2
3
4
Weibull k 10 m/s
8,5 m/s
7,5 m/s
6 m/s
Figure 3. Normalized extreme load for different wind conditions.
Turbulence intensity The previous results have considered the turbulence intensity varying with wind speed according to the recommendations in the IEC61400-1 (Eq. 1). In order to investigate the influence of the TI in the variation of the fatigue loads a second model was used assuming TI constant, and equal to 15% for all wind speeds. The Figure 7 presents the variation of the equivalent load for the tower base bending moment parallel to the wind direction with the TI = 15%. The patterns of variation with k have changed significantly for the lowest and highest wind speeds whereas for the 7.5 and 8.5 m/s values the effect was just smoothing down the variation.
Equivalent load [kNm]
65
64
63
62
61
60 1,5
2
2,5
3
3,5
4
4,5
5
Weibull k 6 m/s
7,5 m/s
8,5 m/s
Figure 4. Blade root edgewise bending moment.
10 m/s
Equivalent load [k Nm]
16
15
14
13
12
11 1,5
2
2,5
3
3,5
4
4,5
5
Weibull k 6 m/s
7,5 m/s
8,5 m/s
10 m/s
Figure 5. Main shaft torsional moment. 230
Equivalent load [kNm ]
225 220 215 210 205 200 195 190 185 1,5
2
2,5
3
3,5
4
4,5
5
Weibull k 6 m/s
7,5 m/s
8,5 m/s
10 m/s
Figure 6. Tower base bending moment (wind direction). 220
Equivalent load [kNm]
210 200 190 180 170 160 150 140 1,5
2
2,5
3
3,5
4
4,5
5
Weibull k 6 m/s
7,5 m/s
8,5 m/s
10 m/s
Figure 7. Tower base bending moment (wind direction) with turbulence intensity 15%.
4. CONCLUSIONS The wind distribution, which is defined by the two Weibull parameters, greatly influences both values and variations of the loads in a wind turbine. The paper investigated the effect of high Weibull shape factors on the ultimate and fatigue loads using suitable mean wind speeds for the calculations, facilitating a comparison with the IEC61400-1. The extreme loads calculations presented here indicate that wind tubines at higher k factors sites will face much lower loads due to the reduced expected extreme wind speeds. The difference between two sites with same mean wind speed and shape factors of 2 and 4 can be as much as 70% for some load cases. The results for the extreme static loads can be generalised for any kind of wind turbine because they are driven only by the extreme wind speed. The fatigue loads have a more complicated pattern of variation with the wind conditions. The influence of k on dynamic and fatigue loads should be determined for each wind turbine type because it depends on aerolastic and dynamic characterisitcs. The results of this paper do not lead to general conclusions because some load cases present higher loads with higher k factors, others the opposite variation. The fatigue loads depend also on the mean wind speed and the turbulence intensity.
5. ACKNOWLEDGMENT The authors would like to thank the financial support of the Brazilian Council for Science and Technology Development – CNPq. 6. REFERENCES [1] IEC. IEC 61400-1 Wind Turbine Generator Systems – Part 1: Safety Requirements. International Standard. Genebra, Suíça, 1999. 2 ed. 57 p. [2] DEKKER, J. W. M. et al. European Wind Turbine Standards II. ECN Solar & Wind Energy, Petten, Netherlands, 1998. 59p. [3] MOLLER, T. K. Investigation on Extreme and Fatigue Loads for Wind Turbines in Brazil. Folkecenter for Renewable Energy, Hurup, DK, 1996, 18p. [4] BOSSANYI, E. A. Bladed for Windows: Theory Manual. Garrad Hassan and Partners Limited, Bristol, UK, 1997. 60 p. [5] LEMOS, D. F.A. Análise de Projeto de Turbina Eólica de Grande Porte para as Condições Climáticas da Região Nordeste do Brasil. (in Portuguese) MSc. Thesis, Universidade Federal de Penambuco, Recife, Brazil, 2005. 91p. [6] VEERS, P. S. Three-dimensional wind simulation. SAND88-0152, Sandia National Laboratories, Albuquerque, NM, USA, 1988. [7] IEA. Recommended Practices for Wind Turbine Testing - Part 3: Fatigue Loads. Norma Internacional. Genebra, Suíça, 1990. 2 ed. 32 p. [8] THOMSEN, K. The Statistical Variation of Wind Turbine Fatigue Loads. Risø R-1063 (EN), Risø National Laboratory, Roskilde, Denmark, 1998. 33 p. [9] BËRGSTROM, H. Distribution of Extreme Wind Speed. Wind Energy Report WE 92:2, Upsala University, Departament of Meteorology, Upsala, Sweden. 1992.
Brazilian Wind Energy Centre - CBEE, CTG/UFPE, Recife-PE, 50740-530, BRAZIL Phone/Fax: +55 (81) 34532975 e-mail: [email protected] 2 Wind Energy Postgraduate Program – Federal University of Pernambuco Recife – PE, BRAZIL e-mail: [email protected]
ABSTRACT There are many sites around the world characterized by wind distributions with relatively high Weibull shape factors and mean wind speeds but the design codes and Standards working groups only recommend the use of Rayleigh distribution for the wind turbine load calculations. The paper focuses on the effects of high Weibull shape factors on the loads calculation. Extreme and fatigue load cases are used to analyze the influence of different wind speed frequency distributions on the design load of a wind turbine. A real 250kW wind turbine was simulated using an aeroelastic computer model to obtain loads acting in several important components. The fatigue damage is calculated for different wind distributions using the lifetime equivalent load and the extreme load is the steady state calculations for an extreme wind with 50 years recurrence period at hub height. The extreme loads results indicate that wind tubines at higher shape factors sites will face much lower loads due to the reduced expected extreme wind speeds. The fatigue loads have a more complicated pattern of variation with the wind conditions. The influence of the shape factor on dynamic and fatigue loads should be determined for each wind turbine type because it depends on aerolastic and dynamic characterisitcs.
1. INTRODUCTION Although the wind energy community has a consensus of opinion about the use of Weibull distribution to represent all kinds of wind characteristics, the design codes and Standards working groups keep recommending the use of Rayleigh distribution for the wind turbine load calculations. Then one should expect that this approach leads to the worst load conditions, but is that really true? There are many sites around the world characterized by wind distributions with relatively high Weibull shape factors and mean wind speeds. Wind turbines can generate typically 10 to 30% more annually at those sites compared to Rayleigh distributed wind sites. But the frequency distribution of wind speeds also influence the extreme and fatigue loads that have to be considered in the design of a wind turbine. How do the loads compare to the standard (European) sites? This paper tries to answer these questions focusing on the effects of high Weibull shape factors on the loads calculation. Extreme and fatigue load cases are used to analyze the influence of different wind speed frequency distributions on the design load of a wind turbine. The IEC61400-1 [1] and the European Wind Turbine Standards II [2] are used as reference for the load cases, parameters and calculation procedures. A previous work from Møller [3] investigated the extreme and fatigue loads for wind turbines according to the expected Brazilian wind conditions. He showed that the extreme loads calculated for the region of high Weibull shape factors are considerably lower than the loads used in the design because the expected extreme wind speeds for a 50 years recurrence period are reduced. On the other hand, some fatigue loads, caused by gravity effects, are expected to be higher, specially the edwise blade root bending and the main shaft bending moments. The recent wind turbine installations in Brazil, in the region of high Weibull shape factors, motivated the present investigation on the assessment of structural integrity for local wind conditions. In the next sections we show detailed description of wind turbine load calculations for two representative design load cases: a) a design situation of a parked wind turbine with extreme external conditions for analysis of ultimate loads; and b) a normal design situation for analysis of fatigue loads. A real operating wind turbine is modelled and simulated using a validated computational code to provide results that can be checked out against measurements
2. MODELS AND SIMULATION A complete aeroelastic wind turbine model has been configured using the program BLADED [4]. The modeling parameters were taken from a real wind turbine with 29m diameter rotor, 250kW, stall controlled with induction generator, located at the Northeast coast of Brazil near the Brazilian Wind Energy Centre [5].
The three dimensional wind model, based on Veers [6] and Kaimal approaches, can describe the spatial and temporal atmospheric turbulences through the entire rotor disk. For the fatigue calculations, a total of 80 dynamic simulations have been performed using synthetic wind series from 5 to 24 m/s with 20Hz signal and 600s length each. The turbulence intensity is given by Equation 1 using parameter a = 3, which corresponds to the model adopted by the IEC 61400-1 for low level turbulence sites. The I15 is the turbulence intensity at 15 m/s and Ū is the mean wind speed.
I (a 15 / U ) I u 15 ( a 1)
(1)
The results of the simulations are the calculated loads acting upon the wind turbine components. Part of a simulated response for the flapwise bending moment at the blade root is shown in Figure 1 as an illustration.
fl apw ise ben ding (kN. m)
80 70 60 50 40 30 0
10
20
30 tim e (seconds)
40
50
60
Figure 1. Simulated flapwise blade bending moment at 9 m/s.
Fatigue loads The loads cycles were classified and counted using a rainflow algorithm [7], and the cumulative damage caused by the cyclic loads, applying the linear Miner´s rule, is given by n D i N (S i ) i
(2)
where ni is the counted number of cycles at stress level S i, and N(S i) is the number of cycles to failure at stress level Si obtained from the material S-N curve. This work investigates the relative fatigue damage not the components design life, therefore the fatigue loads are expressed by load ranges, R i, and number of cycles. The total fatigue damage, caused by the sum of all load ranges, can also be written as the result of one equivalent load, R eq, acting a number of cycles neq ., as shown in equation 3 below m n i Ri D ni R i m n eq R eq m R eq n eq
1
m
(3)
where m is the Wöhler curve exponent, assumed in the following calculations as 4 for steel components (tower and shaft) and 10 for the fiberglass blades. In addition, a correction for the value of Req has been applied to take into account the influence of the mean load level. The modified Goodman criterion, described in Thomsen [8], is a simple correction procedure that applies a gain factor, calculated from the load cycles, to the equivalent load. The equivalent loads are obtained for each simulated load series and average values are calculated for corresponding mean wind speeds. Although only 8 different wind speeds have been simulated, a complete spectrum of equivalent
loads, from cut-in to cut-out wind speeds, is computed using polynomial interpolation for the mean wind speeds not simulated. In order to compare the fatigue damage for site-specific wind conditions, i.e., different wind distributions, the lifetime equivalent load, Leq , is derived taking into consideration the contribution of the mean equivalent loads for all wind speeds during the design life of a wind turbine: m Req (U ) m n eq P (U ) nT 20 years Leq n eq, L 1
(4)
where U is the mean wind speed relative to a mean equivalent load, P(U) is the probability of occurrence of wind speed U defined by the Weibull distribution function, and nT20years is the occurrence number of ten minutes series in a lifetime of 20 years. Extreme loads There are several load cases described in [1] that must be analyzed for the determination of the ultimate loads. This paper analyzes one of them which is characterized by an extreme wind condition in combination with a normal machine state. The steady state calculations are made considering an extreme wind with 50 years recurrence period at hub height, the wind turbine in a normal state, rotor parked and perpendicular to the wind direction, aerodynamic brakes enabled and “no faults” status for the control system. Several load cases have been considered: flap and lag moments at blade root, torque and thrust at the hub, tower bending moments at the base for two orthogonal directions. The rotor position, or the azimuth angle, that leads to the worst condition for each load case was also included. The extreme wind speed adopted here, which is derived from [2], is the maximum 3s gust with a recurrence period of 50 years, Ve50 . Its relationship with the Weibull distribution parameters is important to compare the ultimate loads for different wind conditions. The basic assumptions are that the wind distribution for the 10min mean values, U, are well modeled by the Weibull function and that the short term (3s) averages, u 3s, are distributed around the 10min mean values with standard deviation 3s, following the Gaussian function
1 u 3 s U 1 f (u 3 s | U ) exp 3s 2 2 3 s
(5)
The conditional density function, f(u3s |U ), expresses the probability of a 3s gust within the 10min interval defined by the mean wind speed U. The total probability of occurrence for a 3s gust, f(u3s), is derived by integrating f(u 3s |U ) and the Weibull density function for all mean wind speeds k 1 k u 3s U 1 k U U f (u 3 s ) f (u3 s | U ) f (U ) exp exp dU 0 2 C C C ( U ) 2 3s 3s
(6)
Integrating the pdf function shown above, we find the cumulative function of the 3 second average wind speed. The cdf function must of the 315 576 th order statistics to take in account the various independent observations [9]. Now, it is possible to determine the Ve50 for any C and k wind condition, which is equivalent to find the wind speed with a 2% probability of being exceeded in a year. The 3s is the part of
the total standard deviation u that remains after using a low pass filter with averaging time of 3 seconds and u varies with the mean wind speed according to Eq. 1. Figure 2 shows the variation of the V e50 for several Weibull shape factors and four different mean wind speeds, 6.0, 7.5, 8.5 and 10.0m/s. The extreme wind speed decreases with the increase of the Weibull parameter k for all wind speeds. The annual mean wind speeds have been chosen to be identical with the ones used by the IEC61400-1 to define the wind turbine classes. The Ve50 used on the IEC calculation procedures are derived from Fig.2 using a shape factor of approximately 1.8. 3. RESULTS The ultimate load analysis has been performed for 12 wind conditions, four mean wind speeds and three shape factors (2, 3 and 4). The extreme wind speed, V e50, for each condition is presented in the Table 1.
The extreme loads for the 12 different wind conditions are presented in Figure 3. The values are normalized against the peak for each load case in order to facilitate the comparison. The edgewise bending moment at the blade root shows a small reduction when changing the shape factor from 2 to 3, and just a slight reduction from 3 to 4. However, the other load cases, flapwise blade and tower bending moments, which are greatly influenced by the value of the extreme wind speed, present a large variation with the factor k. The reduction can be as much as 70% in some cases when changing from a wind condition of k=2 to k=4. The fatigue load analysis is performed comparing the lifetime equivalent load, Leq (Eq. 6), calculated for several wind conditions defined by a combination of four annual mean wind speeds, 6.0, 7.5, 8.5 and 10m/s, and Weibull shape factors from 1.5 to 6.
Table 1. Extreme wind speed, Ve50 , for different wind conditions.
Weibull k
Annual mean wind speed 6 m/s
7,5 m/s
8,5 m/s
10 m/s
2
37,0 m/s
45,9 m/s
51,8 m/s
60,7 m/s
3
24,1 m/s
29,7 m/s
33,4 m/s
38,9 m/s
4
19,8 m/s
24,1 m/s
27,1 m/s
31,4 m/s
90
Extreme Wind Speed Ve 50 [m/s]
80 70 60 50 40 10 m/s 8.5 m/s 7.5 m/s 6 m/s
30 20 10 1
1,5
2
2,5
3
3,5
4
Weibull k
Figure 2. Influence of the shape factor on the V e50 for different annual mean wind speeds.
The calculated fatigue loads are for the blades (flap and lag root moments), main shaft (bending and torsional moments), tower top (tilt and torsional moments) and tower base (bending moments). The edwise blade moment results are presented in Figure 4. For all four mean wind speeds the equivalent load increases almost linearly with the Weibull shape factor. The equivalent load for the main shaft torsional moment, showed in Figure 5, increases with k for sites with mean wind speeds higher than ~7.0m/s and has a tendency to decrease for wind speeds lower than this value. The fatigue damage for the tower base bending moment parallel to the wind direction shows several patterns of variation with k depending of the mean wind speed, see Figure 6. For wind speeds of 7.5 and 8.5m/s the equivalent load has a minimum value around k = 2, while for 10m/s the minimum is at k = 3 and for 6.0m/s at 1.5.
Normalized Extreme Load Blade edge (Mx)
1
Blade flap (My)
Tower (Mx)
Tower (My)
0.9
Extreme load (kNm)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
2
3
2
4
3
4
2
3
4
2
3
4
Weibull k 10 m/s
8,5 m/s
7,5 m/s
6 m/s
Figure 3. Normalized extreme load for different wind conditions.
Turbulence intensity The previous results have considered the turbulence intensity varying with wind speed according to the recommendations in the IEC61400-1 (Eq. 1). In order to investigate the influence of the TI in the variation of the fatigue loads a second model was used assuming TI constant, and equal to 15% for all wind speeds. The Figure 7 presents the variation of the equivalent load for the tower base bending moment parallel to the wind direction with the TI = 15%. The patterns of variation with k have changed significantly for the lowest and highest wind speeds whereas for the 7.5 and 8.5 m/s values the effect was just smoothing down the variation.
Equivalent load [kNm]
65
64
63
62
61
60 1,5
2
2,5
3
3,5
4
4,5
5
Weibull k 6 m/s
7,5 m/s
8,5 m/s
Figure 4. Blade root edgewise bending moment.
10 m/s
Equivalent load [k Nm]
16
15
14
13
12
11 1,5
2
2,5
3
3,5
4
4,5
5
Weibull k 6 m/s
7,5 m/s
8,5 m/s
10 m/s
Figure 5. Main shaft torsional moment. 230
Equivalent load [kNm ]
225 220 215 210 205 200 195 190 185 1,5
2
2,5
3
3,5
4
4,5
5
Weibull k 6 m/s
7,5 m/s
8,5 m/s
10 m/s
Figure 6. Tower base bending moment (wind direction). 220
Equivalent load [kNm]
210 200 190 180 170 160 150 140 1,5
2
2,5
3
3,5
4
4,5
5
Weibull k 6 m/s
7,5 m/s
8,5 m/s
10 m/s
Figure 7. Tower base bending moment (wind direction) with turbulence intensity 15%.
4. CONCLUSIONS The wind distribution, which is defined by the two Weibull parameters, greatly influences both values and variations of the loads in a wind turbine. The paper investigated the effect of high Weibull shape factors on the ultimate and fatigue loads using suitable mean wind speeds for the calculations, facilitating a comparison with the IEC61400-1. The extreme loads calculations presented here indicate that wind tubines at higher k factors sites will face much lower loads due to the reduced expected extreme wind speeds. The difference between two sites with same mean wind speed and shape factors of 2 and 4 can be as much as 70% for some load cases. The results for the extreme static loads can be generalised for any kind of wind turbine because they are driven only by the extreme wind speed. The fatigue loads have a more complicated pattern of variation with the wind conditions. The influence of k on dynamic and fatigue loads should be determined for each wind turbine type because it depends on aerolastic and dynamic characterisitcs. The results of this paper do not lead to general conclusions because some load cases present higher loads with higher k factors, others the opposite variation. The fatigue loads depend also on the mean wind speed and the turbulence intensity.
5. ACKNOWLEDGMENT The authors would like to thank the financial support of the Brazilian Council for Science and Technology Development – CNPq. 6. REFERENCES [1] IEC. IEC 61400-1 Wind Turbine Generator Systems – Part 1: Safety Requirements. International Standard. Genebra, Suíça, 1999. 2 ed. 57 p. [2] DEKKER, J. W. M. et al. European Wind Turbine Standards II. ECN Solar & Wind Energy, Petten, Netherlands, 1998. 59p. [3] MOLLER, T. K. Investigation on Extreme and Fatigue Loads for Wind Turbines in Brazil. Folkecenter for Renewable Energy, Hurup, DK, 1996, 18p. [4] BOSSANYI, E. A. Bladed for Windows: Theory Manual. Garrad Hassan and Partners Limited, Bristol, UK, 1997. 60 p. [5] LEMOS, D. F.A. Análise de Projeto de Turbina Eólica de Grande Porte para as Condições Climáticas da Região Nordeste do Brasil. (in Portuguese) MSc. Thesis, Universidade Federal de Penambuco, Recife, Brazil, 2005. 91p. [6] VEERS, P. S. Three-dimensional wind simulation. SAND88-0152, Sandia National Laboratories, Albuquerque, NM, USA, 1988. [7] IEA. Recommended Practices for Wind Turbine Testing - Part 3: Fatigue Loads. Norma Internacional. Genebra, Suíça, 1990. 2 ed. 32 p. [8] THOMSEN, K. The Statistical Variation of Wind Turbine Fatigue Loads. Risø R-1063 (EN), Risø National Laboratory, Roskilde, Denmark, 1998. 33 p. [9] BËRGSTROM, H. Distribution of Extreme Wind Speed. Wind Energy Report WE 92:2, Upsala University, Departament of Meteorology, Upsala, Sweden. 1992.
