1. Introduction
As a common natural phenomenon, wind profoundly affects our lives. On the one hand, with the development of society and the global economy, humankind’s demand for energy is increasing, and wind energy has been widely used as a clean energy in recent years [
1,
2]. On the other hand, the volume of human buildings is gradually increasing, and more and more skyscrapers and large-span bridges have appeared, which are particularly sensitive to wind [
3]. Whether it is for wind energy evaluation or structural wind resistance design, the study of wind characteristics is of fundamental importance. The most direct and effective way of studying wind characteristics is to measure them. For huge measured data, using the appropriate distribution model of wind speed and direction can simply and effectively describe its law. Many scholars have carried out extensive research on this.
Christopher Jung et al. [
4] evaluated 115 different wind speed distribution models, as proposed in 46 articles from 2010 to 2018, according to the quantity and quality of analysis data; the results showed that five-parameter Wakeby distribution and four-parameter Kappa distribution scored the highest. Ilhan Usta et al. [
5] proposed an innovative method, PWMBP (probability weighted moments based on the power density method), which was developed and proposed for estimating the Weibull parameters in wind energy applications. Jianzhou Wang et al. [
6] took four locations in central China as examples to compare commonly used wind speed probability distribution models and estimation methods of corresponding parameters. The results showed that the nonparametric model had better fitting accuracy and operation simplicity, and that the random heuristic algorithm was better than the widely used estimation method. Christopher Jung et al. [
7] compared the goodness of fit between 24 single-component probability density functions, 21 mixed probability density functions, and empirical wind speed probability density functions worldwide. They found that the five-parameter Wakeby probability density function was suitable for land wind speed and the four-parameter Kappa probability density function was suitable for sea wind speed. The two-parameter Weibull probability density function is only optimal for a few wind speeds. Fatma Gul Akgul et al. [
8] used the inverse Weibull (IW) distribution to model wind speed, and the results showed that, in most cases, IW distribution based on ML and MML parameter estimation had better results than Weibull distribution, based on corresponding estimation. Talha Arslan et al. [
9] used generalized Lindley (GL) distribution and power Lindley (PL) distribution to model wind speed data. The results show that both GL distribution and PL distribution can provide the best fit according to different evaluation criteria.
Many scholars have carried out research on the distribution of wind speed, but in many cases, wind direction also needs to be focused on; therefore, some scholars have conducted research on the distribution of wind direction and the joint distribution of wind direction and speed, through which the influence of wind direction can be considered. Jose A. Carta et al. [
10] used von Mises (vM-pdf) finite mixed distribution to represent the distribution of directional wind speed. Analysis of wind direction data from several weather stations in the Canary Islands shows that the model is suitable for wind conditions in areas with multiple patterns of prevailing wind directions. Wang Hao et al. [
11] predicted the basic wind speed of Sutong Bridge based on the joint distribution of wind speed and direction. X.W. Ye et al. [
12] proposed extended parameter estimation algorithms for multivariate and multi-peak cyclic distribution to build a joint distribution model of wind speed and direction. It was found that the model had good representativeness, and that the algorithm could save significant amounts of time in parameter estimation. Qinkai Han et al. [
13] proposed the use of non-parametric kernel density (NP-KD) and non-parametric JW (NP-JW) models. It was shown that the non-parametric models (NP-KD, NP-JW) generally outperformed the parametric models (AG, Weibull, Rayleigh, JW-TNW, JW-FMN) and showed a more robust performance in fitting the joint speed and direction distributions. Zheng Xiaowei et al. [
14] used the multiplication theorem and the AL model to model the joint probability distribution of wind speed and direction, respectively. The results show that the joint probability density function of wind speed and direction derived from the AL model is better than that based on the multiplication theorem, and that ignoring the effects of wind direction significantly improves estimates of extreme wind speeds. Dong Sheng et al. [
15,
16] proposed a new method for establishing a joint distribution model, based on a wind rose diagram using a continuous AL joint distribution model, drawing a new wind speed and direction distribution diagram. The results show that the statistical model has high reliability and a strong correlation with the original data distribution.
Previous researchers have mainly focused on the modeling of wind speed distribution or joint distribution of wind speed and direction, i.e., one or two-dimensional joint distribution models; however, for wind power in small geographic areas, in areas with complex tall buildings [
17,
18] and so on, it is necessary to take three-dimensional joint distribution into consideration. Specifically, the spatial correlation of wind power is concerned when clusters of wind generators are spread over small geographic areas, and a suitable three-dimensional joint distribution model can describe the correlation suitably. For flexible tall buildings, wind-induced dynamic responses are three-dimensional, so the three-dimensional wind load should be clearly described. For the third dimension, the height direction, the change in wind speed is usually considered by exponential law and logarithmic law; however, some studies [
19,
20] show that these laws do not describe the variation law of wind speed in some areas. Therefore, the three-dimensional modeling of wind speed probability distribution is necessary. Based on the AL (angular–linear) model and the copula function, this paper proposes a modeling method to describe the three-dimensional distribution of wind speed.
4. Conclusions
In this paper, the formula for connecting two-dimensional distribution function with the copula function is derived and a modeling method is given for the three-dimensional joint distribution of wind speed, wind direction, and height. The method started with a one-dimensional wind speed distribution model and used mvM to describe a wind direction distribution. A two-dimensional joint distribution model of wind speed and wind direction at each height was established through the AL model. Then, the two-dimensional joint distributions at each height were connected by the copula function, from which a three-dimensional joint distribution model of wind speed, wind direction, and height was formed. Our main conclusions are as follows:
(1) The copula function can connect not only one-dimensional distribution but also two-dimensional or even multi-dimensional distribution. In practice, the joint distribution function of variables that have a clear relationship between each other can be obtained first, and then the copula function can be used to connect these joint distribution functions to form the overall joint distribution function.
(2) This kind of joint distribution model has a good fitting effect, can make full use of the original data, and is not affected by the characteristics of wind speed data. The example in this paper is based on the measured data in a flat area. Because of the compatibility of its function, it can be applied to complex terrains, such as mountain canyons, in subsequent research.
(3) The two-dimensional joint distribution model can describe the relationship between wind speed and wind direction; a suitable two-dimensional distribution model can be adopted to fit the data. The distribution model proposed in this paper can be used for three-dimensional distribution fitting. The number of different heights will not affect the establishment of a low-dimensional model. This will only affect the parameter value of the copula function. In practical applications, the influence of bad data points on the final results can be avoided.
(4) The wind speed in the area chosen for this study is in good compliance with the Weibull distribution parameters, and the north and south wind prevails in this area. In addition, the probability of prevailing wind direction decreases as height increases.
(5) The three-dimensional distribution model was used to obtain the wind field characteristics and to further calculate the wind load of the structure. It provides a basis for wind resistance design of high-rise buildings or long-span bridges. The model can also be applied to the field of wind power to evaluate the potential wind energy in a certain area.