How To Calculate Jpdf

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1 The Seventh Asia-Pacific Conference on Wind Engineering, November -, 9, Taipei, Taiwan AN EMPIRICAL JOINT PROBABILITY DENSITY FUNCTION OF WIND SPEED AND DIRECTION Jun Chen and Xiaoqin Zhang Associate Professor, Department of Building Engineering & State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University,Shanghai, China Postgraduate, Department of Building Engineering, Tongji University Shanghai, China, ABSTRACT An empirical joint probability density function (JPDF) of mean wind speed and direction is presented in this paper. The proposed JPDF model is built up by marginal distributions of wind speed and wind direction that are assumed as an Extreme-Value equation. Details of the JPDF model are first discussed with focus on application procedure for both unimodal and bimodal wind data and approaches for determining the modal parameters. It is then applied to field measured joint probability density of wind sample from parent population and extreme value data as well. Design wind prediction for given return period using the proposed JPDF is also briefly discussed. The performance of the proposed JPDF model is assessed by goodness-of-fit criteria. It is concluded from the results that the proposed JPDF model can represent quite well the joint distribution properties of wind speed and direction from different data source. KEYWORDS: JOINT PROBABILITY DENSITY FUNCTION, WIND SPEED, WIND DIRECTION Introduction Field measurement of wind turbulent properties and analysis is a long-term and fundamental task for structural wind engineering since uncertainties of wind load is in fact the key factor that affects the analysis accuracy of wind-resistant structures. Among many turbulent wind parameters as wind spectrum, wind profile, turbulent intensity and so on, the joint probability density function of mean wind speed and direction (short as JPDF hereafter) is an very important but less addressed property. The importance of wind directionality for design wind prediction, wind-induced fatigue analysis and wind-energy assessment has been underlined and emphasized by many researchers. Without considering wind directionality, as pointed out by Moriarty [Moriarty 93], the design extreme wind speed might be overestimated. In the current non-directional wind-induced fatigue analysis procedure, it assumes that the wind blows with constant direction during the whole life of the structure concerned, which in turn leads to in general conservative prediction [Repetto and Solari,, Xu and Chen ]. In the evaluation of wind-power resources available at a given site, the knowledge of joint probability density function is crucial for positioning the wind turbines to maximize the capturable energy [Carta ]. Several JPDF models have been proposed by researchers in the past decades that can be broadly classified into continuous function and discrete function. McWilliams [McWilliams 979] suggested an isotropic Gaussian model based on the assumptions that the wind speed component along the prevailing wind direction follows normal distribution with non-zero mean and a given variance, while the wind speed component along the direction orthogonal to the prevailing direction is independent and normally distributed with zero mean

2 The Seventh Asia-Pacific Conference on Wind Engineering, November -, 9, Taipei, Taiwan and same variance. Weber [Weber 99] extended McWilliams model to the anisotropic Gaussian model in which the same variance limitation was released. Angular-linear distribution function is often adopted to model the JPDF, and normally the wind speed is modeled as Gaussian distribution and wind direction is modeled as von-mises distribution [Marida ]. More recently, Carta [Carta ] proposed a new JPDF model based on the angular-linear distribution originally suggested by Jonhson [Jonhson and Wehrly 97 ]. This model is derived using marginal PDF of wind speed and wind direction, which are assumed as normal Weibull mixture distribution and von Mises distribution respectively. As for discrete JPDF model, the basic idea is divide the whole circle into several sub-sections, in which the probability density of wind speed is assumed as Weibull, Gumbel or other functions and some measures are introduced to account for the correlation between each section [Gu 999, Ge and Xiang, Matsui, Xu and Chen ]. None of the above JPDF models, however, enjoy universal acceptance nowadays. More attempts are necessary to archive a better JPDF model. In this connection, an empirical JPDF model is presented in this paper based on the classical directional statistical theory on angular-linear distribution. The proposed JPDF model is actually built up by marginal distributions of wind speed and wind direction whose distribution is assumed as an Extreme- Value equation. The framework of the proposed JPDF model is introduced in the following section. The performance of the model is then assessed by applying to field measured JPDFs from different sources. Proposed Empirical JPDF model Basic functions There has been extensive research carried out over the last forty more years to address separately the probability distribution of mean wind speed or wind direction. Therefore, the most distinct way to form the joint probability density function of wind speed and direction is using the information of the marginal distributions. In wind-induced structural fatigue analysis, for instance, the general approach is to divide the wind direction into several subsections and apply identical probability function say Weibull distribution for wind speed in each subsection and assume statistical independence among subsections [Ge ]. For angular-linear distribution, Johnson and Wehrly [Johnson 97] suggested to construct the joint probability density using marginal distribution as ( ) ( ) ( ) ( ) f v, θ = πg ζ f v f θ ; θ π; v () VΘ V where fv ( v), fθ ( θ ) ζ is a circular variable and g ( ζ ) is its probability density function, and ( ) Θ is probability density function for mean wind and direction respectively; g ζ can be derived from cumulative probability distribution of mean wind speed and direction. Inspired by Eq. the following empirical JPDF model is suggested based on the following two assumptions: () the existence of one or multi- prevailing wind directions, and the whole circle [~π ] section can be accordingly divided into several sub-section (usually two) each possessing only one prevailing wind direction; () in each sub-section the JPDF model is identical as: ( ) ( ) fω U, θ = a* f,, * (,, ) * (,, )* (,, ) i U v b c + d fθ θ e f + g fu v b c fθ θ e f () where Ω i is the ith sub-section, a,b,c,d,e,f, g are seven unknown model parameters to be determined; function fu ( vbc,, ), fθ ( θ, e, f) are the extreme-value equation as

3 The Seventh Asia-Pacific Conference on Wind Engineering, November -, 9, Taipei, Taiwan v b v b fu ( v, b, c) = exp exp + c c (3) θ e θ e f ( θ, e, f ) = exp exp + < θ π () f f Given the above JPDF, the marginal PDF for wind direction/wind speed can then be calculated by integrating over θ /U. Estimation of model parameters Two technical issues associated with the proposed JPDF model are definition of each sub-section and determination of modal parameters. Taking the most common situation of two prevailing wind directions as example, suppose θ is the wind direction having the lowest occurrence frequency of wind data and θ is another wind direction having the lowest occurrence frequency of wind data in the range of [ θ+ π π /, θ+ π + π /]. The two subsections can then be defined as { θ ( θ, θ ) modπ} ( ), Ω= { θ θ, θ modπ} Ω = () The prevailing wind direction in each sub-section can then be calculated as ( ) β β ( ) f f + f f f f θ = = β + β β ( ) k k + Ω k k k f f f+ f f+ f (7) where ( β β ] k is the upper and lower bound of the wind direction interval having the maximum wind occurrence; f is the wind occurrence in this interval, f and f + are the wind occurrence value in the adjacent interval [Mardia ]. Finally, the unknown modal parameters can be identified by fitting the model to field measured data using non-linear least square algorithm., k Prediction of design wind speed A simplified procedure is used in this paper to predict the design wind speed at a certain wind direction zone θ j Δ β /, θ +Δβ / using the proposed JPDF. By this way, the value obtained can be directly compared with that determined by other discrete JPDF model. In particular, the design wind speed at certain wind direction zone θ j Δ β /, θ +Δβ / in return period R (year) can be determined by the following equation Δβ θ j + Umax Δβ P( u, θ) dudθ θ j = () R where θ j Δ β and U max is, respectively, wind direction, interval and design wind speed. Application and Discussion Various types of field measured wind data are employed in this section to assess the applicability of the proposed model. In specific, unimodal and bimodal wind data and wind

4 The Seventh Asia-Pacific Conference on Wind Engineering, November -, 9, Taipei, Taiwan sample from parent population and extreme value data are adopted. The verification procedure is as follows: () compute the discrete joint probability density function using field measured mean wind speed and direction; () fit the proposed model to the measured one to determine the modal parameters. The performance of the proposed model is judged by goodness-of-fit criteria. Data Source The proposed JPDF is applied to three field measured discrete JPDFs, denoted as Case to 3. In Case, the measured JPDF was computed from hourly mean wind speed and wind direction of a city for year, which is considered as bimodal sample from parent population [Chen 9]. In Case, the measured JPDF was computed from weekly extreme value of wind speed and direction of a city for more than thirty year, which is considered as bimodal sample from extreme value [Yang ].. In Case 3, the measured JPDF was calculated from 5-year records of hourly mean wind speed and direction from a 5 m high mast [Chen 7], which is considered as unimodal sample from parent population. For each case, the whole circle is divided into sections each of.5 degree interval, the number of wind speed falling in each section are counted and the occurrence rate is calculated accordingly through Eq.9, where N j, N is the number of wind data in the section and total wind data. The measured JPDF (wind occurrence rate) for Case, Case and Case 3 are given in Table, Table and Table 3 respectively. P ij = N ij N* ΔU* Δθ (9) Table : Measured JPDF of Case (%) m/s N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Total Comparison of Measured and Calculated JPDF The field measured JPDFs for case to 3 are depicted on left side of Fig., and the proposed JPDFs are shown on right side of Fig. accordingly. The identified modal parameters and the Chi-square error for the three cases are given in Table. It is observed

5 The Seventh Asia-Pacific Conference on Wind Engineering, November -, 9, Taipei, Taiwan from Fig. that the proposed JPDF model can represent the field measurement results quite well. The Chi-square error for Case, Case and Case 3 is larger than.9, the minimum value is.9 for case 3 which indicates a very good fit for all cases. Furthermore, introducing the identified seven unknown parameters into Eq.5, the calculated value is unity for each cases which also indicates a proper PDF model. Table : Measured JPDF of Case m/s N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW Table 3: Measured JPDF of Case 3 m/s N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW Case Sect ion Table : Modal Parameters and Chi-square Error Modal parameters a b c d e f g Chi-square error γ Ω Ω Ω Ω Ω Determination of Design Wind Speed Taking Case as an example, the design wind speed for y return period is calculated and given in Table 5. In [Ge, Yang ] the joint distribution probability model is suggested by assuming identical probability model with different model parameters for each wind direction section. The prediction of design wind speed for y for Gumbel, Frechet,

6 The Seventh Asia-Pacific Conference on Wind Engineering, November -, 9, Taipei, Taiwan Weibull distribution are also given in Table 5 for comparison. Figure plots the design wind speed at each wind direction. It is seen from Fig. that some kind of smoothing effect can be observed for values obtained by the proposed JPDF model, since the value for relatively small occurrence wind direction become large while those for relatively large occurrence wind direction become small. Generally, the value obtained by the proposed JPDF mode is small than that from separated Gumbel/Frechet/Weibull distribution. Further work is necessary to explore the meaning of directional design wind speed. Frequency Probability Density Wind Speed (m/s) Wind Speed (m/s) (a) Bimodal wind sample from parent population Probability Density(%) Probability Density (%) Wind Speed (m/s) Wind Speed (m/s) (b) Bimodal wind sample of extreme value 7 7 Probability Density(%) 5 5 Probability Density Wind Direction(Degree) Wind Speed (m/s ) (c) Unimodal wind sample from parent population Figure : Filed measured JPDF (left) and Computed JPDF (right) Wind Speed (m/s )

7 The Seventh Asia-Pacific Conference on Wind Engineering, November -, 9, Taipei, Taiwan Gumbel Frechet Weibull JPDF Figure : Comparison of design wind speed for Case Table 5: Design wind speed prediction at returen period Y Direction Gumbel distribution Frechet distribution Weibull distribution JPDF N 5.* NNE NE ENE E ESE SE SSE S SSW.... SW WSW W WNW NW NNW W *Unit: m/sec Concluding Remarks An empirical joint probability density function (JPDF) of mean wind speed and direction is suggested in the paper. The proposed JPDF model is built up by marginal distributions of wind speed and wind direction that are assumed as an Extreme-Value equation. Different marginal distribution can be adopted for wind speed and wind direction thereby forming different JPDF model. The proposed JPDF model is applied to various field measured wind sample of extreme value or from parent population with unimodal or bimodal distribution. It is concluded from the results that the proposed JPDF model can represent quite well the joint distribution properties of wind speed and direction from different data source. There are still some problems remains for the JPDF model and needs further work. For

8 The Seventh Asia-Pacific Conference on Wind Engineering, November -, 9, Taipei, Taiwan instance, the value of JPDF may less than zero according to the definition, and the meaning of the design wind speed and the joint distribution function in the angular-liner expression is not very clear. References Carta J A, Ramirez P, Bueno C. (), A joint probability density function of wind speed, and direction for wind energy analysis. Energy Conversion and Management, 9(), Chen J., Michael C. H. Hui and Y. L. Xu, (7), A comparative study of stationary and non-stationary wind models using field measurements, Boundary-Layer Meteorology,:, 5- Chen J. Zhao Xu-dong, (9), Analytical method of joint probability density function of wind speed and direction from parent population, J. of Dsaster Prevention and Mitigation Engineering, 9:, 3-7 (in Chinese) Ge Y J, Xiang H F. (), Statistical study for mean wind velocity in Shanghai area. J. of Wind Engineering and Industrial Aerodynamics, 9(-5), Gu, M., et al., Fatigue life estimation of steel girder of Yangpu cable-stayed bridge due to buffeting. J. of Wind Engineering and Industrial Aerodynamics, 999. (3): p Johnson RA, Wehrly TE. (97), Some angular linear distributions and related regression models. J Am Statist Associat,73:. Mardia, KV. and Jupp P., Directional Statistics (nd edition), John Wiley and Sons Ltd.,. McWilliams B, Newmann MM, Sprevak D. (979), The probability distribution of wind velocity and direction. Wind Eng, 3:9 73. Moriarty, W.W. and Templeton, J.I. (93), On the estimation of extreme wind gusts by direction section, J. of Wind Engienrring and Industrial Aerodynamics, 3, 7-3 Matsui, M., T. Ishihara, and K. Hibi, Directional characteristics of probability distribution of extreme wind speeds by typhoon simulation. J. of Wind Engineering and Industrial Aerodynamics,. 9(-5): p Repetto M P, Solari G. (), Directional wind-induced fatigue of slender vertical structures. J. of Structural Engineering, ASCE, 3(7), 3-. Weber R. (99), Estimator for the standard deviation of wind direction based on moments of the Cartesian components. J Appl Meteorol;3:3 53. Xu Y L, Chen J, Ng C L, (), Occurrence probability of wind-rain-induced stay cable vibration, Advances in Structural Engineering, (), Yang Y X., Ge Y J. and Xiang H F., Statistic analysis of wind speed based on the joint distribution of wind speed and wind direction, Structural Engineers,, (3):9-3 (in Chinese) Acknowledgement Financial support from The th Young Teacher Research Fund by Fok Ying Tung Education Foundation through project No 77 Field measurement and modeling of joint probability density function of wind speed and direction to the first author is highly appreciated. The wind data in case are collected from open source for academic research purpose. Any opinion and conclusions presented in this paper are entirely those of the authors.