# A Multi-Level Fuzzy Evaluation Method for the Reliability of Integrated Energy Systems
## 2 Problem Formulation
### Device Models and Constraints  
1. Photovoltaic (PV) model  
The output power \(\mathrm{P_{PV}}\) of PV array is strongly linear [30], and its mathematical model is as follows:  
\[\mathrm{P_{PV}}=\mathrm{P_{STC}}\frac{\mathrm{G_{T}}}{\mathrm{G_{STC}}}[1+ \mathrm{K_{C}}(\mathrm{T_{C}}-\mathrm{T_{STC}})] \tag{1}\]  
where \(\mathrm{P_{STC}}\) is the maximum power under standard conditions (sunlight incident intensity \(\mathrm{G_{STC}}\) is 1000 \(\mathrm{W/m^{2}}\) and ambient temperature \(\mathrm{T_{STC}}\) is 25 \({}^{\circ}\)C). \(\mathrm{G_{T}}\) is the amount of radiation the panel is exposed to; \(\mathrm{K_{C}}\) is the work coefficient; and \(\mathrm{K_{C}}\) is the operating temperature of the panel.  
2. Wind Turbine (WT) Model  
The output power of a WT \(\mathrm{P_{wt}}\) is closely related to the wind speed \(\mathrm{v}\)[31], and the following expression is given to describe the relationship between them.  
\[\begin{cases}\mathrm{P_{WT}}=0,&v<\mathrm{v_{ci}}\\ \mathrm{P_{WT}}=\frac{\mathrm{v^{3}-v_{ci}^{3}}}{\mathrm{v_{i}^{2}-v_{ci}^{3}}},&\mathrm{v_{ci}}\leqslant v\leqslant\mathrm{v_{r}}\\ \mathrm{P_{WT}}=\mathrm{P_{WTr}},&\mathrm{v_{r}}\leqslant v\leqslant\mathrm{v _{co}}\\ \mathrm{P_{WT}}=0,&v>\mathrm{v_{co}}\end{cases} \tag{2}\]
3. Electricity to Gas (EG) Model  
Figure 1: Basic structure of IES.  
Electricity to gas mainly refers to the use of electricity to react water and carbon dioxide within natural gas equipment. The specific chemical reactions are as follows:  
Step 1: Water splits into oxygen and hydrogen;  
Step 2: Hydrogen reacts further with carbon dioxide to form methane (CH4, the main component of natural gas).  
The relational expression is shown in Equation (3).  
\[\left\{\begin{array}{l}\mathrm{G_{EG,t}=\mathrm{r_{IEG}\times P_{EG}/q_{hg}} }\\ 0\leq\mathrm{P_{EG,t}\leq P_{EG}^{max}}\end{array}\right. \tag{3}\]  
where \(\mathrm{G_{EG,t}}\) is the gas production, \(\mathrm{n_{IEG}}\) is the efficiency of transformation, \(\mathrm{P_{EG}}\) is the EG power consumption, and \(\mathrm{q_{hg}}\) is the high calorific value of natural gas.  
4. Energy Storage (ES) Model  
An integrated energy system usually contains a variety of energy storage technologies, as an energy storage system has the advantages of a fast response and high conversion efficiency. State of charge (SOC) is usually used to describe the state of the energy storage equipment.  
\[\mathrm{SOC}=\mathrm{E_{SB}/E_{SB,r}} \tag{4}\]  
When the device is charged, its state changes meet the following requirements:  
\[\mathrm{SOC_{SB,t}=(1-\sigma_{SB})\times SOC_{SB,t-1}+\mathrm{n_{SB,in}}\ \times P_{SB,in,t}\times\mathrm{\Delta t/E_{SB,r}}} \tag{5}\]  
When the device discharges, its status changes as follows:  
\[\mathrm{SOC_{SB,t}=(1-\sigma_{SB})\times SOC_{SB,t-1}-\mathrm{n_{SB,out}}\ \times P_{SB,out,t}\times\mathrm{\Delta t/E_{SB,r}}} \tag{6}\]  
where \(\mathrm{E_{SB}}\) is the amount of energy stored by the current energy storage device, \(\mathrm{E_{SB,r}}\) is the maximum amount of energy stored by the energy storage device, \(\mathrm{P_{SB,in,t}}\) is the amount of energy absorbed by the energy storage device, \(\mathrm{P_{SB,out,t}}\) is the amount of electricity provided by the energy storage device, and \(\mathrm{n_{SB,in}}\) is the conversion efficiency of the amount of energy absorbed by the energy storage device. \(\mathrm{n_{SB,out}}\) is the conversion efficiency of the power supplied by the energy storage device, and \(\sigma_{SB}\) is the power consumption rate of the energy storage device.