## 4 DRCE-IS Combined with D-Vine Copula and SNMM to Assess the Power System Reliability
### Overview of CE-IS  
Denote all the RVs in reliability assessment by a vector \(\mathbf{x_{ss}}\), i.e.,  
\[x_{ss}=[x^{L},x^{CU},x^{WT},x^{LW}], \tag{15}\]  
where \(\mathbf{x^{L}}\), \(\mathbf{x^{CU}}\) and \(\mathbf{x^{WT}}\) are the RV vectors representing the states of transmission lines, conventional units and WTGs. \(\mathbf{x^{LW}}\) is the RV vector representing the values of the load and wind speed at different sites.  
Then the reliability index \(\gamma\) can be determined by:  
\[\gamma=\int I(x_{ss})f(x_{ss};\theta)dx_{ss}, \tag{16}\]  
where \(I(\mathbf{x_{ss}})\) is a function to indicate the system state, e.g., in the normal or failure states. \(f(\mathbf{x_{ss}};\theta)\) is the original joint PDF of \(\mathbf{x_{ss}}\) with \(\theta\) as the parameter vector.  
Equation (16) can be further transformed by the optimal joint PDF \(g(\mathbf{x_{ss}})\) used in importance sampling (IS), which is called IS PDF in this paper. The transformed (16) is shown as:  
\[\gamma=\int I(x_{ss})\frac{f(x_{ss};\theta)}{g(x_{ss})}\cdot g(x_{ss})dx_{ss}= E_{g}\bigg{[}I(x_{ss})\frac{f(x_{ss};\theta)}{g(x_{ss})}\bigg{]}, \tag{17}\]  
where the rightmost term is an expectation in which \(g(\mathbf{x_{ss}})\) and \(I(\mathbf{x_{ss}})Q(\mathbf{x_{ss}})\) act as the PDF and a function of the RVs.  
The theoretical expression of the optimal IS PDF is [28]:  
\[g(x_{ss})=\frac{I(x_{ss})f(x_{ss};\theta)}{\int I(x_{ss})f(x_{ss};\theta)dx_{ss} }=\frac{I(x_{ss})f(x_{ss};\theta)}{\gamma}, \tag{18}\]  
where \(g(\mathbf{x_{ss}})\) is obviously entangled with \(\gamma\), which means that the explicit expression of \(g(\mathbf{x_{ss}})\) is unavailable.  
To solve the problem, the concept of cross entropy is introduced into the process of IS to form the traditional CE-IS method. The basic idea is to select a function form in advance, regardless of the actual form of the optimal PDF. Denote the selected function form as \(f(\cdot;\mathbf{\omega})\), in which \(\mathbf{\omega}\) is the parameter vector unknown. By minimizing the Kullback-Leibler (KL) divergence between \(f(\cdot;\mathbf{\omega})\) and \(g(\cdot)\), we can derive a satisfactory suboptimal PDF as the IS PDF.  
The KL divergence \(K(g,f)\) is expressed as:  
\[K(g,f)=\int g(x_{ss})\ln g(x_{ss})dx_{ss}-\int g(x_{ss})\ln f(x_{ss};\omega)dx_ {ss}=C-\int g(x_{ss})\ln f(x_{ss};\omega)dx_{ss}, \tag{19}\]  
where the first integral term is independent of \(\mathbf{\omega}\), so it can be regarded as a constant \(C\) when we search for the optimal \(\mathbf{\omega}\).  
Then, we plug (18) into (19) and get the objective function used for optimizing \(\mathbf{\omega}\), shown as:  
\[\omega=\underset{\omega}{\operatorname{argmin}}K(g,f)=\underset{\omega}{ \operatorname{argmin}}\gamma K(g,f)=\underset{\omega}{\operatorname{argmin}} \gamma\int I(x_{ss})\ln f(x_{ss};\omega)f(x_{ss};\theta)dx_{ss}. \tag{20}\]The integral term in (20) can be further converted as:  
\[\begin{array}{ll}\omega&=\underset{\omega}{\operatorname{argmax}}\int I(x_{ss}) \frac{f(x_{ss};\omega)}{f(x_{ss};\omega)}f(x_{ss};\omega)\ln f(x_{ss};\omega)dx_{ ss}=\underset{\omega}{\operatorname{argmax}}E_{\omega}\Big{[}I(x_{ss})\frac{f(x_{ss}; \omega)}{f(x_{ss};\omega)}\ln f(x_{ss};\omega)\Big{]}\\ &=\underset{\omega}{\operatorname{argmax}}E_{\omega}[I(x_{ss})Q(x_{ss})\ln f( x_{ss};\omega)]\approx\underset{\omega}{\operatorname{argmax}}\frac{1}{n_{s}} \sum\limits_{s=1}^{n_{s}}I(x_{ss,s})Q(x_{ss,s})\ln f(x_{ss,s};\omega),\end{array} \tag{21}\]  
where \(E_{\omega}(\cdot)\) stands for the expectation of a function with \(f(\cdot;\omega)\) as the PDF. \(n_{s}\) is the sample size for estimating the expectation. \(Q(x_{ss})\) is called the likelihood ratio function (LRF), which can be divided into several parts, shown as:  
\[Q(x_{ss})=Q(x^{L})Q(x^{CU})Q(x^{WT})Q(x^{LW}). \tag{22}\]  
There is no constraint imposed on (21), so the best \(\omega\) can be achieved when the derivative of the last term in (21) is equal to zero.