## 3 Two-Stage Stochastic Scheduling Model for the Hybrid AC/DC Distribution Network
### Scenario Generation and Reduction  
The wind power prediction error is regarded as a random variable, which obeys a normal distribution, then the wind power output in real time can be expressed by [24]:  
\[\overline{P}_{t}^{W}=\hat{P}_{t}^{W}+\Delta\overline{P}_{t}^{W},\Delta \overline{P}_{t}^{W}\sim G(0,\delta_{t}^{2}), \tag{22}\]  
where \(\Delta\overline{P}_{t}^{W}\) is the wind power prediction error at time \(t\); \(\hat{P}_{t}^{W}\) is the day-ahead wind power prediction value; \(\delta_{t}\) is the standard deviation of the prediction error; and \(G\) represents the normal distribution function. The scenario generation and reduction techniques are implemented to handle wind power uncertainty. Since the Monte Carlo simulation method based on simple random sampling suffers from a higher computational burden, the Latin hypercube sampling (LHS) is adopted to obtain the original wind power scenario set, which can achieve good coverage from the entire distribution of the random variable [25]. The procedures of LHS can be described as follows:  
Step 1: Assuming the number of samples for the random variable, \(\Delta\overline{P}_{t}^{W}\), is \(N\), the interval \([0,1]\) can be equally divided into \(N\) intervals, and a uniformly distributed random number within the range of \([(k\!-\!1)/N,k/N]\) is generated for \(k=1\) to \(N\);  
Step 2: Disorder the order of the above \(n\) random numbers obtained by Step 1;Step 3: The \(n\) random numbers by Step 2 are regarded as the probability values of samples, and the value of the random variable, \(\Delta\overline{P}_{t}^{W}\), can be obtained according to the inverse function of the cumulative distribution function (CDF) for \(\Delta\overline{P}_{t}^{W}\):  
\[\Delta P_{t}^{(k)}=F_{t}^{-1}(\frac{k-0.5}{N}),\ k=1,\cdots,N, \tag{23}\]  
where \(F_{t}(\cdot)\) represents the CDF of random variable \(\Delta\overline{P}_{t}^{W}\), and \(\Delta P_{t}^{(k)}\) is the \(i\)th sample value of \(\Delta\overline{P}_{t}^{W}\). Cholesky decomposition (CD) is introduced to alleviate the undesired correlations between random variables at different time points [26]. The sample matrix for \(\Delta\overline{P}_{t}^{W}\) by LHS can be expressed as:  
\[\Delta\mathbf{P}=\left[ \begin{array}{cccc}\Delta P_{1}^{(1)}&\Delta P_{1}^{ (2)}&\cdots&\Delta P_{t}^{(N)}\\ \Delta P_{2}^{(1)}&\Delta P_{2}^{(2)}&\cdots&\Delta P_{t}^{(N)}\\ \vdots&\vdots&\ddots&\vdots\\ \Delta P_{T}^{(1)}&\Delta P_{T}^{(2)}&\cdots&\Delta P_{t}^{(N)}\end{array}  \right]_{\mathbf{T}\times N}. \tag{24}\]  
The procedures of CD are as follows:  
Step 1: A \(T\times N\) ordering matrix, \(\mathbf{L}\), is generated, and the element at the \(t\)th row and \(i\)th column represents the position of the sample, \(\Delta P_{t}^{(k)}\), in the matrix, \(\Delta\mathbf{P}\), to be arranged;  
Step 2: The elements of each row in \(\Delta\mathbf{P}\) are arranged according to the ordering matrix, \(\mathbf{L}\);  
Step 3: The corresponding correlation matrix of \(\mathbf{L}\) noted as \(\boldsymbol{\rho}_{L}\) can be calculated and decomposed by the Cholesky decomposition:  
\[\boldsymbol{\rho}_{L}=\mathbf{D}\mathbf{D}^{\mathrm{T}}. \tag{25}\]  
Step 4: A \(T\times N\) matrix, \(\mathbf{G}\), can be obtained by:  
\[\mathbf{G}=\mathbf{D}^{-1}\mathbf{L}. \tag{26}\]  
Step 5: The elements of each row in the matrix, \(\mathbf{L}\), are arranged according to the order of elements in the corresponding row in \(\mathbf{G}\);  
Step 6: The elements of each row in the matrix, \(\Delta\mathbf{P}\), are arranged according to the updated ordering matrix, \(\mathbf{L}\). Then, the simultaneous backward reduction method is utilized to decrease the number of wind power scenarios [27]. For the matrix, \(\Delta\mathbf{P}\), obtained by the LHS and CD techniques, the original scenario \(k\) noted as \(\Delta\mathbf{P}^{(k)}\) is a vector composed of the elements of the kth row in \(\Delta\mathbf{P}\), for which the occurrence probability is \(\pi^{(k)}\).