## 2 Materials and Methods
### AL-Copula Three-Dimensional Wind Speed Probability Distribution Model  
[PERSON] and [PERSON] [26] derived the angular-linear distribution model from the maximum entropy principle, and used it to describe the joint probability distribution of wind speed and direction. The joint probability density function is  
\[f(v,\theta)=2\pi g(\zeta)f(v)f(\theta) \tag{2}\]  
Among them, \(f(v)\) is the wind speed probability density function of the full wind direction, which can be described by commonly used wind speed probability distribution models. \(f(\theta)\) is the wind direction angle probability density function, which can be described by the mixed von Mises distribution (mvM) [10]. Its probability density function is  
\[f(\theta)=\sum_{i}^{N}\omega_{i}\frac{\exp[k_{i}\cos(\theta-\mu_{i})]}{2\pi I _{0}(k_{i})} \tag{3}\]  
In Equations:  
\[\theta:wind\ \text{angle},0\leq\theta<2\pi\] \[\mu_{i}:average\ \text{wind direction},0\leq\mu_{i}<2\pi\] \[\omega_{i}:weight,0\leq\omega_{i}<1,\sum\omega_{i}=1\] \[k_{i}:\text{scale parameters},k_{i}\geq 0\] \[I_{0}(k_{i}):\text{zero-order modified type I }[PERSON]\ \text{equation}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function** & **Expression** & **Parameter** & **Clan** \\ \hline Gumbel- & \(\exp\left(-\left[\sum\limits_{i=1}^{N}\left(-\ln u_{i}\right)^{\theta}\right]^{ 1/\theta}\right)\) & \(\theta\in[1,+\infty)\) & [PERSON]/Extremum \\ Frank- & \(\frac{1}{\theta}\ln\left(1+(\frac{1}{i=1}(\exp(-\theta u_{i})-1))/(\exp(- \theta)-1)^{N-1}\right)\) & \(\theta\in(-\infty,+\infty)\) & Archimedes \\ Clayton- & \(\left(\sum\limits_{i=1}^{N}u_{i}^{-\theta}-N+1\right)^{-1/\theta}\) & \(\theta\in(0,+\infty)\) & Archimedes \\ Gaussian- & \(\Phi_{\theta}(\Phi^{-1}(u_{1}),\Phi^{-1}(u_{2}),\cdots\Phi^{-1}(u_{N}))\) & \(\theta\in(-1,1)\) & Ellipse \\ Student’s t- & \(T_{\theta,\lambda}(T_{\lambda}^{-1}(u_{1}),T_{\lambda}^{-1}(u_{2}),\cdots T_{ \lambda}^{-1}(u_{N}))\) & \(\theta\in(-1,1),\lambda\in(0,+\infty)\) & Ellipse \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Commonly used copula functions. \(g(\zeta)\) is the probability density function of parameter \(\zeta\), which is obtained by fitting the mvM. Among them  
\[\zeta=\left\{ \begin{array}{c}2\pi[F(v_{i})-F(\theta_{i})]\,F(v_{i})\geq F( \theta_{i})\\ 2\pi+2\pi[F(v_{i})-F(\theta_{i})]\,F(v_{i})<F(\theta_{i})\end{array} \right. \tag{4}\]  
When fitting the high order mixed von Mises distribution, giving a better initial value of parameters can significantly improve the accuracy and efficiency of the fitting. [PERSON] [10] proposed a calculation method to determine the initial value of parameters in the mvM. Firstly, the wind direction data are divided into the following \(N\) groups: \(0^{\circ}-\theta_{1},\theta_{1}-\theta_{2},\cdots\theta_{i-1}-\theta_{i}, \theta_{N-1}-\theta_{N}(\theta_{N}=360^{\circ})\). Among them, the number of samples in the \(i\)-th group \([\theta_{i-1}-\theta_{i})\) is \(n_{i}\). The initial value can be calculated using the following formula:  
\[\omega_{i}=\frac{n_{i}}{\sum\limits_{i=1}^{N}n_{i}} \tag{5}\]  
\[\mu_{i}= \begin{cases}\arctan(\frac{s_{i}}{c_{i}})&,s_{i}\geq 0,c_{i}>0\\ \frac{\pi}{2}&,s_{i}>0,c_{i}=0\\ \pi+\arctan(\frac{s_{i}}{c_{i}})&,c_{i}<0\\ 2\pi+\arctan(\frac{s_{i}}{c_{i}})&,s_{i}<0,c_{i}>0\\ \frac{3\pi}{2}&,s_{i}<0,c_{i}=0\end{cases}  \tag{6}\]  
\(k_{i}\) can be obtained by the following formula:  
\[\frac{I_{1}(k_{i})}{I_{0}(k_{i})}=(s_{i}^{2}+c_{i}^{2})^{0.5} \tag{7}\]  
\[s_{i}=\frac{\sum\limits_{i=1}^{n_{i}}\sin\theta_{j}}{n_{i}},c_{i}=\frac{\sum \limits_{j=1}^{n_{i}}\cos\theta_{j}}{n_{i}} \tag{8}\]  
The previous copula function is used to connect the distribution of one-dimensional random variables. This paper attempts to connect the joint distribution of two-dimensional random variables with the copula function. Firstly, the AL model is used to describe the joint distribution of wind speed and direction at each height \(F_{V_{1}\theta_{1}}(v_{1},\theta_{1}),F_{V_{2}\theta_{2}}(v_{2},\theta_{2}),\cdots\)\(F_{V_{N}\theta_{N}}(v_{N},\theta_{N})\).