## 3 Triangular Fuzzy BW Method to Determine the Interval Weight of Steady State Power Quality Index
### Determine the Weight of Steady State Power Quality Index by TFBW Method  
The comprehensive evaluation of power quality first requires determining the weight of each index. According to the TFBW method proposed in this paper, an interval weight of each indicator can be obtained, and the specific steps are as follows:  
Step 1. Identify the most important index and the least important index. These two indices are used as reference indices. After they are determined, they can be compared with the remaining indices. The most important index of the six power quality indices in this paper, according to the consulted experts, is the frequency deviation, which is represented by \(I_{B}\); the least important one is the three-phase voltage imbalance, which is represented by \(I_{W}\)[32]. Step 2. Determine the best-to-others (BO) vector and the others-to-worst (OW) vector. Suppose there are \(n\) indices to be evaluated. Then, we compare \(I_{B}\) with all indices and compare all indices with \(I_{W}\) to determine the BO vector and \(\mathrm{OW}\) vector via Equations (9) and (10):  
\[\mathrm{BO}=\left[ \begin{bmatrix}I_{B1}&m_{B1}&u_{B1}\end{bmatrix} \right]\ \ \left[ \begin{bmatrix}I_{B2}&m_{B2}&u_{B2}\end{bmatrix} \ \ \cdots\ \ \left[ \begin{bmatrix}I_{Bn}&m_{Bn}&u_{Bn}\end{bmatrix} \right] \tag{9}\]  
\[\mathrm{OW}=\left[ \begin{bmatrix}I_{1W}&m_{1W}&u_{1W}\end{bmatrix} \right]\ \ \left[ \begin{bmatrix}I_{2W}&m_{2W}&u_{2W}\end{bmatrix} \ \ \cdots\ \ \left[ \begin{bmatrix}I_{nW}&m_{nW}&u_{nW}\end{bmatrix} \right]\right] \tag{10}\]  
where \(\left[ \begin{bmatrix}I_{Bn}&m_{Bn}&u_{Bn}\end{bmatrix} \right]\) and \(\left[ \begin{bmatrix}I_{nW}&m_{nW}&u_{nW}\end{bmatrix} \right]\) are triangular fuzzy numbers. \(m_{Bn}\) is the midpoint of \(I_{Bn}\) and \(u_{Bn}\), and it is also the most likely value in \(\left[ \begin{bmatrix}I_{Bn}&m_{Bn}&u_{Bn}\end{bmatrix} \right]\); \(m_{nW}\) is the midpoint of \(l_{nW}\) and \(u_{nW}\), and it is also the most likely value in \(\left[ \begin{bmatrix}I_{nW}&m_{nW}&u_{nW}\end{bmatrix} \right]\)[33]. The acquisition of the BO and \(\mathrm{OW}\) vectors requires consultation with experts to judge the importance of each index. Experts can make a judgment on the importance between the \(i\)th and \(j\)th indices by a triangular fuzzy number with the membership function shown in Figure 4[28]. Figure 3: Comparison importance of index by best-worst (BW) method. Figure 2: Comparison of importance of index by analytic hierarchy process (AHP) method. The boundary value of the triangular fuzzy number could be a non-integer, which can reflect the experts' understanding of the importance of each index more accurately. As shown in Figure 4, \(\left[l_{Bn}\quad m_{Bn}\quad u_{Bn}\right]=\left[1\quad 1\quad 1\right]\) when \(n=B\) and \(\left[l_{nW}\quad m_{nW}\quad u_{nW}\right]=\left[1\quad 1\quad 1\right]\) when \(n=W\). If there are a total of \(k\) opinions by \(k\) experts, we can take their average using Equations (11) and (12) [33]:  
\[\left[l_{Bn}\quad m_{Bn}\quad u_{Bn}\right]=\left[\frac{l_{Bn1}+l_{Bn2}+\cdots +l_{Bnk}}{k}\quad\frac{m_{Bn1}+m_{Bn2}+\cdots+m_{Bnk}}{k}\quad\frac{u_{Bn1}+u_ {Bn2}+\cdots+u_{Bnk}}{k}\right] \tag{11}\]  
\[\left[l_{nW}\quad m_{nW}\quad u_{nW}\right]=\left[\frac{l_{nW1}+l_{nW2}+\cdots +l_{nWk}}{k}\quad\frac{m_{nW1}+m_{nW2}+\cdots+m_{nWk}}{k}\quad\frac{u_{nW1}+u_ {nW2}+\cdots+u_{nWk}}{k}\right] \tag{12}\]  
where \(\left[l_{Bnk}\quad m_{Bnk}\quad u_{Bnk}\right]\) and \(\left[l_{nWk}\quad m_{nWk}\quad u_{nWk}\right]\) represent the opinion of the \(k\)th expert. Step 3. Determine the \(\text{BO}_{\text{M}}\) and \(\text{OW}_{\text{M}}\) vectors.