# A Multi-Level Fuzzy Evaluation Method for the Reliability of Integrated Energy Systems
## 3 Methods
### Multi-Level Fuzzy Evaluation  
The basic idea of fuzzy evaluation is to use the principle of fuzzy linear transformation and the principle of maximum subordination to consider all evaluation factors and quantify them uniformly, assign weights according to the degree of influence of different factors on the judged object and then make a reasonable evaluation [43; 44]. The evaluation process is shown in Figure 3.  
For the IES discussed in this paper, all possible rubrics used to represent the outcome are assumed to be m, denoted V = (v1, v2,... vm) and called the set of rubrics. Through analysis and experimentation, the affiliation of each first-level rubric to each rubric \(\mathrm{v_{i}}\) (I = 1, 2,..., m) is determined, and the single-factor rubric results are obtained as follows.  
\[\mathrm{r_{n_{j}}^{i}}=\left\{\mathrm{r_{n_{j}}^{i},r_{n_{2}}^{i},\cdots r_{n_ {m}}^{i}}\right\} \tag{16}\]  
Figure 3: Process of AHP multi-level fuzzy evaluation.  
The single-factor evaluation is actually a fuzzy mapping of the set of factors to the set of rubrics, and the fuzzy evaluation matrix \(\mathrm{R_{i}}\) for \(\mathrm{U_{i}}\) is  
\[\mathrm{R_{i}}=\begin{pmatrix}\mathrm{r_{1}^{i}}\\ \mathrm{r_{2}^{i}}\\ \vdots\\ \mathrm{r_{3}^{i}}\end{pmatrix}=\begin{pmatrix}\mathrm{r_{11}^{i}}&\mathrm{r_ {12}^{i}}&\cdots&\mathrm{r_{1m}^{i}}\\ \mathrm{r_{21}^{i}}&\mathrm{r_{22}^{i}}&\cdots&\mathrm{r_{21}^{i}}\\ \vdots&\vdots&\ddots&\vdots\\ \mathrm{r_{n_{1}}^{i}}&\mathrm{r_{n_{1}}^{i}}&\cdots&\mathrm{r_{n_{m}}^{i}} \end{pmatrix} \tag{17}\]  
Among them, the affiliation \(\mathrm{r_{i}^{i}}\)\(\in\)[0,1].  
For multi-level fuzzy synthesis evaluation (taking level 2 as an example), it is necessary to perform level 1 fuzzy evaluation first and then use the result value as the factor value for level 2 synthesis evaluation [45; 46]. The result of the first-level fuzzy evaluation is the synthesis result of the fuzzy evaluation matrix and the weight vector, as shown in Equation (17).  
\[\mathrm{B_{i}}=\mathrm{W}*\mathrm{R_{i}}=(\mathrm{b_{i1}},\mathrm{b_{i1}}, \cdots\mathrm{b_{im}}),(\mathrm{i}=1,\ 2,\cdots,\mathrm{N}) \tag{18}\]  
Using the above results as the factor values for the second-level fuzzy comprehensive evaluation, the second-level fuzzy relationship matrix \(\mathrm{R_{T}}\) is  
\[\mathrm{R^{T}}=\begin{pmatrix}\mathrm{B_{1}}\\ \mathrm{B_{2}}\\ \vdots\\ \mathrm{B_{N}}\end{pmatrix}=\begin{pmatrix}\mathrm{W_{1}}*\mathrm{R_{1}}\\ \mathrm{W_{2}}*\mathrm{R_{2}}\\ \vdots\\ \mathrm{W_{N}}*\mathrm{R_{N}}\end{pmatrix}=\begin{pmatrix}\mathrm{b_{11}}& \mathrm{b_{12}}&\cdots&\mathrm{b_{1m}}\\ \mathrm{b_{21}}&\mathrm{b_{22}}&\cdots&\mathrm{b_{2m}}\\ \vdots&\vdots&\ddots&\vdots\\ \mathrm{b_{N1}}&\mathrm{b_{N1}}&\cdots&\mathrm{b_{Nm}}\end{pmatrix} \tag{19}\]  
Normalize the fuzzy composite judging results.  
\[\mathrm{B^{\prime}}=\left(\mathrm{b_{1}^{\prime}},\mathrm{b_{2}^{\prime}}, \cdots,\mathrm{b_{m}^{\prime}}\right)=\frac{1}{\sum_{j=1}^{m}\mathrm{b_{j}}}( \mathrm{b_{1}},\mathrm{b_{2}},\cdots,\mathrm{b_{m}}) \tag{20}\]  
Taking \(\mathrm{B^{\prime}}\) as the final judgment result, \(\mathrm{b_{i}^{\prime}}\) is the probability that the evaluation object belongs to rubric i. According to the principle of maximum affiliation, \(\mathrm{max(b_{i}^{\prime})}\) is the final judgment result of the evaluation object.