## 2 The Comprehensive Vulnerability Rate Model
### Multisource Situational Awareness Analysis
#### 2.3.2 Power Supply Situation  
This indicator reflects the impact of power supply equipment on distribution line operation. Power distribution equipment includes conventional power supply equipment and various types of distributed energy equipment. The external environment has different effects on their operation. (1) Conventional power supply  
As the period increases, the failure rate and the power supply risk that most of the conventional power components such as transformers are also increased. The failure rate of a power component is described by Weibull function, as shown in Figure 1. If the equipment maintenance interval is \(\Delta t\) in the loss period, and the failure rate after maintenance is reduced from \(\lambda_{i}\) to \(\mu_{i}\) times the basic failure rate, the equipment failure rate \(\lambda_{z}\) is:  
\[\lambda_{z}\ =\ \mu_{i}\lambda_{L}\ +\ \frac{\beta_{2i}}{\tau_{2i}^{\beta_{2i}} }(t\ -\ t_{i})^{\beta_{2i}-1} \tag{8}\]  
where \(\tau_{2i}\) is the scale parameter. \(\beta_{2i}\) is the shape parameter, which indicate the influence of the repair on the basic failure rate and the aging of the component. Thus, the power supply index (PSI) of the conventional power supply system is:  
\[\lambda_{k}\ =\ \frac{\sum_{m=1,\cdots,\mu_{\Delta_{k}}}|e_{m,g}|\lambda_{m,z}}{\sum_{m=1,\cdots,b_{1}}\lambda_{m,z}\ +\ \sum_{m=1,\cdots,b_{2}}P_{m,dg}+\ \sum_{m=1,\cdots,b_{3}}P_{m,css}} \tag{9}\]  
(2) Distributed generations (DGs)  
Figure 1: Equivalent curve of actual aging failure rate. Icing weather has a great impact on the output of distributed power, the wind power DG and the photovoltaic power output model is as follows:  
\[P_{f}(v)\ =\ \left\{ \begin{array}{ll}0&v\leq v_{q}\\ \frac{P_{r}}{v_{r}-v_{q}}v-\frac{P_{r}}{v_{r}-v_{q}}v_{q}&v_{q}<v<v_{r}\\ P_{r}&v_{r}<v<v_{r}\\ 0&v\geq v_{t}\end{array} \right. \tag{10}\]  
\[P_{ph}\ =\ A\cdot S\cdot\phi\cdot\eta_{0}[1\ -\ 0.005\cdot(T\ -\ 25)] \tag{11}\]  
where \(P_{r}\) is power rated of the wind turbine. \(v_{r}\) is the rated wind speed. \(v_{q}\) is the cut-in wind speed. \(v_{t}\) is the cut-out wind speed. \(A\) is the number of photovoltaic panels. \(S\) is the area of a single photovoltaic panel. \(\phi\) is the solar radiation intensity. \(\eta_{0}\) is the photoelectric conversion efficiency under standard test conditions. \(T\) is the ambient temperature, the unit is \({}^{\circ}\)C.  
Thus the power supply index (PSI) of DGs is:  
\[\delta_{k}\ =\ \frac{\sum_{m=1,\cdots,\mu_{2,k}}\big{|}e_{m.\text{dg}}\big{|}P_{ m.\text{dg}}}{\sum_{m=1,\cdots,b_{1}}\lambda_{m.2}+\ \sum_{m=1,\cdots,b_{2}}P_{m.dg}+\ \sum_{m=1,\cdots,b_{3}}P_{m.ess}} \tag{12}\]  
(3) Energy storage system (ESS)  
It is assumed that the ESS does not participate in power supply when it is connected to ADN, so only the power supply situation of the ESS in the island is considered in this paper. When the faults occur in the ADN, the area that is powered by the backup power supply (BPS) is called an island. With the development of ESS technology, it plays a more important role in the ADN. Therefore, this paper argues that the BPS consists of the ESS and DGs. Among them, the BPS in the island may include multiple ESSs and DGs. The model of ESS is as follows and constraints of ESS divided into state of charge (SOC) constraints, maximum charge, and discharge constraints. \[\left\{ \begin{array}{ll}E_{j,bat}^{t+1}=E_{j,bat}^{t}+P_{j,ch}^{t}\eta^{ch}T -\frac{P_{j,sh}^{t}T}{\eta^{ls}}\\ E_{j,bat,max}\times 20\%\leq E_{j,bat}^{t+1}\leq E_{j,bat,max}\times 90\%\\ E_{j,bat,max}\times 20\%\leq E_{j,bat}^{t}\leq E_{j,bat,max}\times 90\%\ \ t\in[2,N_{T}]\\ 0\leq P_{j,sh}^{t}\leq D_{j,dis}^{t}P_{j,dis,max}^{t}\\ 0\leq P_{j,ch}^{t}\leq D_{j,ch}^{t}\cap P_{j,ch,max}^{t}\\ D_{j,dis}^{t}+D_{j,ch}^{t}\leq 1\forall t\in[1,N_{T}],\forall j\in[1,N_{ bus}]\end{array} \right. \tag{13}\]  
where, \(E_{j,bat}^{t+1}\) is the remaining capacity of the node \(j\) energy storage device at the end of the period of \(t+1\).