## 4 DRCE-IS Combined with D-Vine Copula and SNMM to Assess the Power System Reliability
### Pre-Simulation Stage of DRCE-IS
#### 4.2.2 IS PMFs of Homogeneous 2- and 3-State Units  
The wind farms in this paper are all centralized and integrated with the transmission network, which means that the WTGs on the same wind farm are definitely connected to the same bus in the power system. Hence, the WTGs of the same type on the same wind farm make equal contributions to the power system. These WTGs should have the same IS PMF. Similarly, the 2-state conventional units, which are of the same type and connected to the same bus, should have the same IS PMF. These units/WTGs are called \"homogeneous units\" in this paper. If we search for the IS PMFs of the homogeneous units individually, the IS PMFs may significantly differ from each other due to the randomness of the samples, which are also far away from the theoretical optimal PMF. Besides, the state of each unit is regarded as a RV, which results in a high-dimension problem and a bulky solution space, increasing the difficulty and time in searching for optimal PMFs. To solve the problem, we aggregate the homogeneous units into a single group and represent the group by one RV. For the group of 2-state conventional units, the RV follows binomial distribution, while for the group of 3-state WTGs, the RV obeys trinomial distribution. By this means, the dimension of the solution space is reduced, which improves the quality of the IS PMFs. Denote the number of 2-state conventional units in the \(c\)-th group as \(n_{c}\). The failure probability is marked as \(\omega_{fc}\). Denote \(x_{fc}\) as the number of units in the failure state in the \(c\)-th group. The PMF of the \(c\)-th group \(p(x_{c})\) is shown as:  
\[p(x_{c})=\frac{n_{c}!(\omega_{fc})^{x_{fc}}(1-\omega_{fc})^{x_{fc}}}{x_{fc}!(n _{c}-x_{fc})!}, \tag{28}\]  
where \(x_{c}=[x_{fc},n_{c}-x_{fc}]\). When we update \(\omega_{fc}\), we only retain the terms containing \(\omega_{fc}\) in the objective function, i.e.,  
\[\max_{\omega_{fc}}\sum_{s=1}^{n_{s}}I(x_{ss,s})Q(x_{ss,s})\ln p(x_{c,s})+C. \tag{29}\]  
Similarly with the processing of transmission lines, we can obtain the solution of (29), which is the updating formulae for \(\omega_{fc}\), shown as:  
\[\omega_{fc,l}=\sum_{s=1}^{n_{s}}I(x_{ss,s})Q(x_{ss,s})x_{fc,s}/n_{c}\sum_{s=1 }^{n_{s}}I(x_{ss,s})Q(x_{ss,s})\,. \tag{30}\]  
Equation (30) is valid for any \(c=1,\ldots,n_{cg}\). \(n_{cg}\) is the number of conventional unit groups. The set \(\{\mathbf{x}_{c}\,|\,c=1,\ldots,n_{cg}\}\) forms the vector \(\mathbf{x}^{\mathbf{CU}}\) in (22). Then the LRF of conventional units \(Q(\mathbf{x}^{\mathbf{CU}})\) can be expressed as:  
\[Q(x^{CU})=\prod_{c=1}^{n_{cg}}\left(\theta_{fc}\right)^{x_{fc}}\left(1-\theta _{fc}\right)^{n_{c}-x_{fc}}/\prod_{c=1}^{n_{cg}}\left(\omega_{fc}\right)^{x_{fc }}\left(1-\omega_{fc}\right)^{n_{c}-x_{fc}}, \tag{31}\]  
where \(\theta_{fc}\) is the original failure probability of the \(c\)-th group of conventional units. Denote the number of 3-state WTGs in the \(w\)-th group as \(n_{w}\). The failure and derated probabilities are marked as \(\omega_{f,w}\) and \(\omega_{d,w}\), respectively. The PMF of the \(w\)-th group \(p(x_{w})\) is shown as:  
\[p(x_{w})=\frac{n_{w}!(\omega_{f,w})^{x_{f,w}}(\omega_{d,w})^{x_{d,w}}(\omega_{ 0,w})^{x_{w,w}}}{x_{f,w}!x_{d,w}!x_{0,w}!}, \tag{32}\]  
where \(x_{w}=[x_{f,w},x_{d,w},x_{o,w}]\).