## 3 The Dynamic Reliability Evaluation and Critical Component Identification Method for Power System
### Polynomial Chaos Expansion  
Polynomial chaos expansion (PCE) is an efficient uncertainty quantification theory. It has a solid mathematical foundation, excellent performance, and is widely used in aerospace, machinery, ships, and other fields [16]. The output response of the power system with random variables is uniquely determined by the system's random variables. If the input variable \(\mathbf{X}=[x_{1},x_{2},\ldots x_{n}]^{\mathrm{T}}\), it can be represented by normal variables \(\mathbf{\xi}=\left[\xi_{1},\xi_{2},\ldots,\xi_{n}\right]^{\mathrm{T}}\). The output response \(\mathbf{Y}\) can then be expressed as follows:  
\[\mathbf{Y}=c_{0}+\sum_{i_{1}=1}^{n}c_{i_{1}}H_{1}\big{(}\xi_{i_{1}}\big{)}+ \sum_{i_{1}=1}^{n}\sum_{i_{1}=1}^{i_{1}}c_{i_{1}i_{2}}H_{2}\big{(}\xi_{i_{1}}, \xi_{i_{2}}\big{)}+\sum_{i_{1}=1}^{n}\sum_{i_{2}=1}^{i_{1}}\sum_{i_{3}=1}^{i_{2 }}c_{i_{1}i_{2}i_{3}}H_{3}\big{(}\xi_{i_{1}},\xi_{i_{2}},\xi_{i_{3}}\big{)}+\cdots \tag{23}\]  
where \(c_{0},c_{i_{1}},c_{i_{1}i_{2}},c_{i_{1}i_{2}i_{3}}\) is the undetermined coefficient, and \(H_{m}(\cdot)\) is the m-order Hermite orthogonal polynomial that can be calculated using Equation (24):  
\[H_{m}\big{(}\xi_{i_{1}},\xi_{i_{2}},\ldots,\xi_{i_{n}}\big{)}=(-1)^{m}e^{\frac {1}{2}\xi^{T}\xi}\frac{\partial^{n}}{\partial\xi_{i_{1}}\partial\xi_{i_{2}} \cdots\partial\xi_{in}}e^{-\frac{1}{2}\xi^{T}\xi} \tag{24}\]  
The first 5-order Hermite orthogonal polynomials can be expanded according to the Equation (25):  
\[ \begin{array}{c}H_{0}(\xi)=1\\ H_{1}(\xi)=\xi\\ H_{2}(\xi)=\xi^{2}-1\\ H_{3}(\xi)=\xi^{3}-3\xi\\ H_{4}(\xi)=\xi^{4}-6\xi^{2}+3\\ H_{5}(\xi)=\xi^{5}-10\xi^{3}+15\xi\end{array}  \tag{25}\]  
Typically, the stochastic response surface method (SRSM) is used to solve non-embedded coefficient problems [17]. The projection method requires the use of Gaussian quadrature to estimate the expectation, which is more difficult and time-consuming. In this paper, the SRSM method is used to minimise the sum of squared errors between the predicted PCE value and the experimental data in order to determine the coefficient. The PCE model can take p-order truncation, and the truncated output response \(\mathbf{Y}\) can be abbreviated as:  
\[\mathbf{Y}\approx\sum_{i=0}^{P}b_{i}\Phi_{i}(\mathbf{\xi}) \tag{26}\]  
The PCE model needs to produce a series of sample points via experimental design and acquire the corresponding response value via simulation in order to construct thecoefficient solution equation. In this paper, the LHS considering the input PDF is used for experimental design. The specific steps are as follows:  
1. Use LHS to obtain samples \(\mathbf{X}=[x_{1},x_{2},\ldots,x_{n}]^{T}\)[18];
2. Use the Nataf method to obtain samples \(\mathbf{\xi}=[\xi_{1},\xi_{2},\ldots,\xi_{n}]^{T}\)[19];
3. Calculate the corresponding response value \(g(\mathbf{x^{S}})=\left[g(x_{1}^{S}),g(x_{2}^{S}),\ldots,g(x_{n}^{S})\right]^{ \mathrm{T}}\);
4. Construct the solving equation of the following coefficient \(\mathbf{b}\). \[ \begin{bmatrix}\[PERSON]_{0}\big{(}\xi_{1}^{s}\big{)}&\[PERSON]_{1}\big{(}\xi_{1}^{s} \big{)}&\cdots&\[PERSON]_{P}\big{(}\xi_{1}^{s}\big{)}\\ \[PERSON]_{0}\big{(}\xi_{2}^{s}\big{)}&\[PERSON]_{1}\big{(}\xi_{2}^{s}\big{)}&\cdots&\[PERSON] _{P}\big{(}\xi_{2}^{s}\big{)}\\ \vdots&\vdots&&\vdots\\ \[PERSON]_{0}\big{(}\xi_{N}^{s}\big{)}&\[PERSON]_{1}\big{(}\xi_{N}^{s}\big{)}&\cdots&\[PERSON] _{P}\big{(}\xi_{N}^{s}\big{)}\end{bmatrix}\begin{bmatrix}b_{0}\\ b_{1}\\ \vdots\\ b_{p}\end{bmatrix} = \begin{bmatrix}g(x_{1}^{S})\\ g(x_{2}^{S})\\ \vdots\\ g(x_{N}^{S})\end{bmatrix} \] (27) where \[\[PERSON]= \begin{bmatrix}\[PERSON]_{0}\big{(}\xi_{1}^{s}\big{)}&\[PERSON]_{1}\big{(}\xi_{1} ^{s}\big{)}&\cdots&\[PERSON]_{P}\big{(}\xi_{1}^{s}\big{)}\\ \[PERSON]_{0}\big{(}\xi_{2}^{s}\big{)}&\[PERSON]_{1}\big{(}\xi_{2}^{s}\big{)}&\cdots& \[PERSON]_{P}\big{(}\xi_{2}^{s}\big{)}\\ \vdots&\vdots&&\vdots\\ \[PERSON]_{0}\big{(}\xi_{N}^{s}\big{)}&\[PERSON]_{1}\big{(}\xi_{N}^{s}\big{)}&\cdots& \Phi_{P}\big{(}\xi_{N}^{s}\big{)}\end{bmatrix} ,\mathbf{b}=