NEW METHOD FOR ESTIMATING DIRECTIONAL EXTREME WIND SPEED BY CONSIDERING THE CORRELATION AMONG EXTREME WIND SPEED IN DIFFERENT DIRECTIONS

Transcription

1 The Eighth Asia-Pacific Conference on Wind Engineering, December 0 4, 203, Chennai, India NEW ETHOD FO ESTIATING DIECTIONAL EXTEE WIND SPEED BY CONSIDEING THE COELATION AONG EXTEE WIND SPEED IN DIFFEENT DIECTIONS ZHANG Bingchao, QUAN Yong 2, GU ing 3 asteral Student, State Key Laboratory of Disaster eduction in Civil Engineering, Tongji University, Shanghai,China, zhangbc052@hotmail.com 2 Associate Professor, State Key Laboratory of Disaster eduction in Civil Engineering, Tongji University, Shanghai,China, quanyong@tongji.edu.cn Professor, State Key Laboratory of Disaster eduction in Civil Engineering, Tongji University, Shanghai, China, minggu@tongji.edu.cn ABSTACT Current research on ectional extreme wind speeds is incomplete, especially in the area of correlation among wind speeds in different wind ections. An improved method of independent storm is proposed in this paper. The proposed method is appropriate for ectional extreme wind speed sampling. The asymptotic independence of wind speeds in different ections for normal wind climates is affirmed by calculating the tail dependence coefficient. A new estimation technique for design wind speeds is proposed based on this information by considering wind ection. esults are presented and discussed with Nanjing City as the example. Keywords: Normal wind, Wind ection, Extreme wind speed, Correlation Introduction Davenport (964; 977; 96, 983) pointed out that researchers must consider wind ection to accurately estimate wind load on a structure. However, only annual maximum wind speeds are used in the traditional wind extremum estimation method. Thus, information that can help estimate ectional extreme wind speeds is lacking. Cook (982) developed a stable estimation method based on independent storm records to improve data utilization rate, thus making it possible to analyze wind speed in each wind ection. Cook (983; 999) proposed a practical method for estimating ectional design wind speeds based on a previous technique, which was a significant development in the field at that time. However, this method has several problems, such as disregarding a number of strong wind speed samples in certain ections and the correlation among extreme wind speeds in different wind ections. esearch on the correlation among extreme wind speeds in different wind ections is minimal. Haraguchi and Kanda (999) and Kanda and Itoi (200) proposed their two correlated Gumbel probability models for ectional wind speeds separately. However, these two models are not applied in ectional wind extremum estimation. Itoi and Kanda (2002) compared these two models and pointed out that this kind of method requires optimization. An improved method of independent storm (IS) is proposed in this paper. The asymptotic independence of wind speeds in different wind ections for normal wind climates is affirmed by calculating the tail dependence coefficient. A new estimation technique for design wind speeds is proposed based on this information by considering wind ection. esults are presented and discussed with Nanjing City as the example. Proc. of the 8th Asia-Pacific Conference on Wind Engineering Nagesh. Iyer, Prem Krishna, S. Selvi ajan and P. Harikrishna (eds) Copyright c 203 APCWE-VIII. All rights reserved. Published by esearch Publishing, Singapore. ISBN: doi:0.3850/

2 Directional IS Sampling ethod The basic idea behind IS as proposed by Cook (982) is to consider the maximum wind speed of each independent storm as statistical samples. An independent storm refers to the continuous record of wind speed beyond the selected threshold value. The IS adopted in the present study is different from Cook s IS. As shown in Fig., maximum wind speed from each wind ection in the independent storm should be selected. Samples from the same wind ection should be entirely independent of one another because they were obtained from different independent storms. Observed wind speed (m/s) NE Time series Present IS points Cook's IS points NE ESE NE NNE A very large value in ection NE threshold 2 Time Fig.. Comparison of the proposed IS and Cook s IS. Compared with Cook s IS, the proposed IS has the following advantages. () The current method improves the quantity for each wind ection to ensure the independence of the samples. (2) As shown in Fig., strong wind speed samples from several ections are disregarded in Cook s IS because only one value is selected from an independent storm. This problem can be avoided in the present method. (3) The samples selected through the present method contain information on the correlation of wind speeds in different wind ections because such samples were selected from a single independent storm. Independence Test The independence of the samples should be ensured by selecting a good threshold value. The run test (Bendat & Piersol, 200) and Kendall τ test (Prokhorov, 200) are useful for confirming the independence of samples. Only Kendall τ test was described because the former was described by Cook (982). If a set of samples is made up of v, v 2,..., v n, we can rearrange this sample set in the form of ( xi, y i), where xi = vi, yi = v i +, and i =,2,..., n. Any pair of ( xi, y i) and ( xj, y j ) is said to be concordant if ( xi xj)( yi yj) > 0 ; the pair is said to be discordant if ( xi xj)( yi yj) < 0. The Kendall τ coefficient is defined as 2( Nc Nd) τ =, () ( n )( n 2) where N c and N d are the number of concordant pairs and the number of discordant pairs. This coefficient can be used as a test statistic in a statistical hypothesis test to establish whether the variables are statistically dependent. The sampling distribution of τ is expected to have a value of zero under the null hypothesis of independence. For larger samples, it is common to use approximation to the normal distribution with mean zero and variance. 326

3 2(2n + 3) Var( τ ) = 9 nn ( 3) The rejection region can be ascertained with a given confidence level. (2) Fitting Distribution Function The cumulative distribution function (CDF) of annual maximum wind speed in wind ection was fitted by XIIS (Harris, 2009). CDF is presented as follows: F () exp exp( w v = av b ), (3) where identifies wind ections and =,2,...,. is the number of wind ections. The wind ection in China is often divided into 6 sectors. We let identify the sector in the north, 2 identify the sector in the north by east, and so on. This manner of dividing and numbering was adopted in this study. Correlation among Extreme Wind Speeds in Different Wind Directions Asymptotic Independence Wind speeds in different wind ections are correlative. However, one phenomenon in extreme statistics is that the correlation between the extreme values of two random variables can be weaker than the correlation between the two variables themselves. For example, when the wind speed in ection A reaches its maximum once in 50 years, the wind speed in ection B may be strong but seldom reaches the same level. Hence, the correlation between the maxima once in 50 years in two ections is weak. In such event, the two variables are said to be asymptotically independent (Coles, 200). Tail dependence coefficients χ and χ between two variables were calculated to determine whether the variables are asymptotically independent. For two random variables X and Y with the distribution functions FX ( X ) and FY ( Y ), we introduced χ ( u) and χ ( u) presented by ln Pr { FX( X) < u, FY( Y) < u} χ( u) = 2, (4) ln u 2ln( u) χ( u) =. (5) ln Pr F ( X) > u, F ( Y) > u { } X χ and χ are defined as χ = lim χ( u) and χ = lim χ( u). The correspondence between the u u tail dependence coefficients and the correlation is shown in Table. Table. Correspondence between tail the dependence coefficients and correlation. tail dependence coefficients Y correlation χ χ parent extremum χ = 0 0< χ < χ = χ = entirely negatively correlated < χ < 0 negatively correlated χ = 0 entirely independent 0< χ < positively correlated χ = positively correlated entirely positively correlated asymptotic independent asymptotic dependent 327

4 xi y i, i=,2,..., n is available, the empirical estimator of χ ( u) and χ ( u) can be presented as ln a ( ) 2, ( ( )) a χ u = Var χ u =, (6) ln u na ln u When a set of samples (, ) ( u) ( ) 2 ( b) ( u) 4 ( ln ) 2ln 4 ln χ( u) =, Var( χ( u) ) =, (7) ln b nb b n a= I F x < u F y < u { Xn( i ), Yn ( i ) }, (8) n i = n b= I F x > u F y > u { Xn( i ), Yn ( i ) }, (9) n i = where FXn () and FYn () verify whether X and Y are asymptotically independent by plotting ( u) are the empirical distribution of X and Y, respectively. We can χ and χ ( u). Application to Wind Speed Data The wind speeds in two different ections were regarded as the aforementioned X and Y. Fig. 2 shows the tail dependence coefficients of wind speeds in two adjacent ections in Nanjing. Although the confidence interval is too large to ascertain the accurate value of the tail dependence coefficients owing to the lack of wind speed data, the upper limit of the confidence interval of χ ( u) remains smaller than when u. Hence, the wind speeds in these two ections exhibit asymptotic independence. The same procedure was performed for wind speeds in dozens of Chinese inland cities, and results similar to those in Fig. 2 were obtained. This finding affirms the asymptotic independence of wind speeds in different wind ections value of χ 95% confidence interval of χ value of χ(u) 95% confidence interval of χ(u) χ u χ u Fig.. χ ( u) and χ ( u) of wind speeds in NNW and N ections in Nanjing. However, this conclusion is only applicable to normal wind climates. The probability that a strong typhoon will generate maximum wind speeds from several wind ections in one year is significantly high. In this case, extreme wind speeds in different wind ections are correlated. Estimating Design Wind Speeds Theory of Directional Extreme Wind Speed The definition of ectional design wind speed v, for =,2,..., at return period is as follows. () The probability that annual maximum wind speed V is,max 328

5 smaller than design wind speed v, in all ections is. That is, the probability of exceedance for any one year is as shown in the following equation: Pr V < v, =,2,..., =. (0) {,max, } (2) The probability of exceedance in each wind ection is constant in one year and can be expressed as follows: {,max, } Pr V < v = const, =, 2,...,. () We considered these two extreme cases. Case : Wind speeds in different wind ections are entirely positively correlated, which means that wind speeds in all ections always reaches their maximum once in years at the same moment. Hence, Pr { V,max < v,, =,2,..., } = Pr{ V,max < v, } =. (2) Case 2: Wind speeds in different wind ections are entirely independent. In this case, Pr { V,max < v,, =,2,..., } = Pr{ V,max < v, }. (3) Combining Eqs. (0) and () provides Pr{ V,max < v, } =. (4) In reality, wind speeds in different wind ections are positively correlated although this correlation might be weak. Thus, the true value of Pr{ V,max < v, } should be in the interval { V,max v, } According to the proposed IS, Pr{ V,max v} < Pr < <. (5) < can be replaced by F ( v, ). Thus, < F ( v, ) <. (6) Estimating Directional Design Wind Speeds The accurate value of Pr{ V,max < v, } should be derived based on the tail dependence coefficients. However, the confidence interval is too large because of the limit of wind speed data. The theory of asymptotic independence is useful in this case. When return period is large, the weak correlation can be disregarded. Therefore, the independence among extreme wind speeds with a small overall probability of exceedance in different wind ections can be assumed. eplacing v, with v, to indicate that v, is under the aforementioned assumption, we obtain F ( v, ) =. (7) By incorporating Eq. (3) into Eq. (7), we derive w b v, = + ln ln ln a a a. (8) This explicit v, can be utilized to estimate v,. 329

6 Estimating Design Wind Speed without Considering Wind Direction If the ectional design wind speeds v, have the same value in Eq. (), i.e., v, = v for =,2,...,, v is the design wind speed without considering wind ection. We can neglect the weak correlation in the same manner. v, the estimator of v, can then be ascertained by F ( v ) =. (9) Eq. (9) can be easily solved by numerical calculation because the left side of the equation contains a monotone increasing function. Calculation Example The daily wind data used in this study were obtained from the meteorological station in Nanjing. These data include 2,95 daily maximum wind speeds with an average of 0 min from January, 95 to December 3, 200. By using the proposed IS with a threshold of 9.2 m/s, 335 independent storms and 735 wind speed samples in different wind ections were obtained. un test and Kendallτ test were utilized to determine the independence of the aforementioned data set. The average number of samples per year and the CDF of wind speed samples from each ection were obtained easily. Design wind speeds were derived with Eqs. (8) and (9). Wind speed with a return period of 50 years (m/s) WNW W WSW NW SW NNW SSW N S SSE NNE NE SE ENE E ESE design wind speed (without considering wind ections) ectional design wind speed Fig. 3. Design wind speeds in Najing at a return period of 50 years. After calculation, the design wind speeds at a return period of 50 years were determined to be m/s (disregarding wind ection), m/s in ection W (which is the maximum), and 6.0 m/s in ection SSW (which is the minimum). The design wind speeds are shown in Fig. 3. Discussion Coefficient γ is introduced to obtain 330

7 F ( v, ) = γ. (20) γ can be considered the number of wind ections in which wind speeds are entirely independent. According to Eq. (7), < γ <. However, obtaining an accurate value for γ is difficult because tail dependence information is unavailable because of the lack of wind speed data. Considering the asymptotic independence among wind speeds in different wind ections, the value of γ should be very close to. By assuming γ =, we obtained v, in Eq. (6) as the estimator of v,. The error caused by assuming that γ = was estimated. The accurate value of ectional design wind speeds v, for =,2,..., can be represented by w b v, = + ln ( γ ) ln ln a a a. (2) v, is larger than v, because γ < ; thus, using v, as the estimator is secure. However, given that Δv, dv, =, (22) w Δγ dγ aγ w ( v, ) the relative error can be derived as follows: Δ v, γ γ γ Δ Δ = η Δ. (23) w w v, aw ( v, ) γ aw ( v, ) γ γ The value of η is between 6% and 9% with Nanjing as an example. Considering that γ and Δγ 0, the relative error should be significantly smaller than the value of η. Using v, as the estimator is accurate enough for engineering applications. Conclusion An improved IS is proposed in this study. The asymptotic independence of wind speeds in different wind ections for normal wind climates was affirmed by calculating the tail dependence coefficient. A new estimation technique for design wind speeds is proposed based on this information by considering wind ection. esults are presented and discussed with Nanjing City as the example. The following conclusions were obtained. () Cook s IS was improved to make it more suitable for ectional extreme wind speed estimation. (2) The asymptotic independence of normal wind speeds in different wind ections was confirmed, which means that the correlation among extreme wind speeds in different ections can be neglected. (3) A new definition and estimator of ectional design wind speeds are proposed according to asymptotic independence. The estimator is secure and suitable for engineering applications. Acknowledgements This work is supported by the National Natural Science Foundation of China (Project No ) and the Fundamental esearch Funds for Central Universities. The financial support is gratefully acknowledged. We also thank the China eteorological Data Sharing Service System ( for providing meteorological data. 33

8 eferences Bendat, J.S., & Piersol, A.G. (200). andom Data: Analysis and easurement Procedures: Wiley. Coles, Stuart. (200). An Introduction to Statistical odeling of Extreme Values: Springer. Cook, N. J. (982). "Towards better estimation of extreme winds". Journal of Wind Engineering and Industrial Aerodynamics, 9(3), Cook, N. J. (983). "Note on ectional and seasonal assessment of extreme winds for design". Journal of Wind Engineering and Industrial Aerodynamics, 2(3), Cook, Nicholas J., & iller, Craig A. (999). "Further note on ectional assessment of extreme winds for design". Journal of Wind Engineering and Industrial Aerodynamics, 79(3), Davenport, A. G. (964). "Note on the distribution of the largest value of arandom function with application to gust loading". Proceedings of the Institution of Civil Engineers, 28, Davenport, A. G. and Surry D. et al. (977). "Wind loads on low rise buildings: Final report of phases I and II BLWT-SS8-977.". London,Ontario, Univ.of Western Ontario. Davenport, A.G. (96). "The application of statistical concepts to the wind loading of structures". Proc. Inst. Civ. Eng., 9, Davenport, A.G. (983). "The relationship of reliability to wind load". Journal of Wind Engineering and Industrial Aerodynamics, 3, Haraguchi, K., & Kanda, J. (999). "Probability model for annual maximum wind speeds in multi-ection". Paper presented at the Wind Engineering into 2st Century, ICWE. Harris,. I. (2009). "XIIS, a penultimate extreme value method suitable for all types of wind climate". Journal Of Wind Engineering And Industrial Aerodynamics, 97(5-6), Itoi, Tatsuya, & Kanda, Jun. (2002). "Comparison of correlated Gumbel probability models for ectional maximum wind speeds". Journal of Wind Engineering and Industrial Aerodynamics, 90(2-5), Kanda, J., & Itoi, T. (200). "Correlated Gumbel probability model for ectional wind speeds,". Conference Name, Conference Location. Type retrieved rom UL Prokhorov, A.V.. (200). Kendall coefficient of rank correlation. In. Hazewinkel (Ed.): Encyclopedia of athematics, Springer. 332