# A Multi-Level Fuzzy Evaluation Method for the Reliability of Integrated Energy Systems
## 4 Results and Discussion
### Results  
During the analysis of the weighting of the first-level indicators, the 1-9 scale method was used to quantify the indicators based on expert experience and an analysis of the cumulative industry data. The system can work stably for a long time and has an emergency and self-healing ability in case of failure, which is most important for an integrated energy system, as reliability is the first priority; a high proportion of clean energy is also one of the goals of an integrated energy system, which can represent environmental protection  
Figure 5: Energy supply and use characteristics of IES. (**a**) Power supply. (**b**) Power consumption. (**c**) Heating. (**d**) Energy consumption of heating water. (**e**) Cooling. to a certain extent. Considering that environmental protection indicators can evaluate the discharge of pollutants, environmental friendliness is the second priority. Network loss will not only affect the economy of the system to some extent but also reflect the advanced technology of the energy network; if the network loss is too large, it will reflect the fault of some parts of the system, so we think it is more important than the demand-side response. Equipment economy is the last consideration. After analysis, we construct the judgment matrix as shown in Equation (20):  
\[\text{C1}= \begin{bmatrix}1&5&7&9&3\\ 1/5&1&2&4&1/2\\ 1/7&1/2&1&2&1/4\\ 1/9&1/4&1/2&1&1/6\\ 1/3&2&4&6&1\end{bmatrix}  \tag{21}\]  
Analytic hierarchy process is used to calculate the initial index weight of the judgment matrix C1, and the maximum eigenvalue \(\lambda_{\text{max}}\), consistency index CI and CR are, respectively,  
\[\lambda\text{ max }=\text{ }5.078\text{ CR }=\text{ }0.0174\text{ CI }=\text{ }0.0195 \tag{22}\]  
where CR is less than 0.1, which meets the consistency requirement. Then, the combined weight w1 is calculated by the entropy formula according to the standardized judgment matrix R1. \[\text{w1}=[0.5323\text{ }\text{ }0.1273\text{ }\text{ }0.0681\text{ } \text{ }0.0401\text{ } \text{ }0.2319] \tag{23}\]  
Taking the secondary indicator of reliability as an example to illustrate the weight calculation method, as shown above, the reliability contains three secondary indicators: average time of disability, reliability of the power supply and energy inefficiencies. Based on the accumulated experience of loss and data analysis, the judgment matrix and standardized judgment matrix of the secondary indicator of reliability are given as follows:  
\[\text{C}_{21}= \begin{bmatrix}1&5&3\\ 1/5&1&1/4\\ 1/3&4&1\end{bmatrix}  \tag{24}\]  
\[\text{R}_{21}= \begin{bmatrix}0.6522&0.5000&0.7059\\ 0.1304&0.1000&0.0588\\ 0.2174&0.4000&0.2353\end{bmatrix}  \tag{25}\]  
The judgment matrix C\({}_{21}\) and initial indicator weights W\({}_{21}\), with large eigenvalues \(\lambda\) max and consistency indicators CI and CR, were calculated using hierarchical analysis as  
\[\text{W}_{21}=[0.6267\text{ }\text{ }\text{ }\text{ }0.0936\text{ }\text{ }\text{ }\text{ }0.2797]^{\text{T}} \tag{26}\]  
Figure 6: Classification of system evaluation indicators. \[\lambda\;\text{max}\;=\;3.0858\;\text{CR}\;=\;0.0739\;\text{CI}\;=\;0.0429 \tag{27}\]  
The consistency requirement is satisfied. The entropy formula is used to calculate the drops E\({}_{21}\), deviation degree d\({}_{21}\), correction coefficient u\({}_{21}\), weight coefficient \(\theta_{22}\) after correction of the weight method and combination weight w\({}_{21}\) for each level of the indicators based on the standardized judgment matrix R\({}_{21}\), respectively. \[\begin{array}{l}\text{E}_{21}=\left[ \begin{array}{ccc}0.7975&0.8587&0.6853\\ 0.2025&0.1413&0.3147\\ \mu_{21}=\left[\begin{array}{ccc}0.3075&0.2146&0.4779\\ \end{array}\right]\\ \theta_{21}=\left[\begin{array}{ccc}0.5562&0.0580&0.3858\\ \end{array}\right]^{\text{T}}\\ \text{w}_{21}=\left[\begin{array}{ccc}0.5915&0.0758&0.3327\\ \end{array}\right]^{\text{T}}\end{array}