## 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 \(\text{BO}_{\text{M}}\) and \(\text{OW}_{\text{M}}\) vectors represent the central BO and OW vectors, respectively, and they can be determined using Equations (13) and (14):  
\[\text{BO}_{\text{M}}=\left[m_{B1}\quad m_{B2}\quad\cdots\quad m_{Bn}\right] \tag{13}\]  
\[\text{OW}_{\text{M}}=\left[m_{1W}\quad m_{2W}\quad\cdots\quad m_{nW}\right] \tag{14}\]  
where \(m_{Bn}\) represents the most likely value in \(\left[l_{Bn}\quad m_{Bn}\quad u_{Bn}\right]\) and \(m_{nW}\) represents the most likely value in \(\left[l_{nW}\quad m_{nW}\quad u_{nW}\right]\). Step 4. Determine the central weight of each index. Based on the \(\text{BO}_{\text{M}}\) and \(\text{OW}_{\text{M}}\) vectors, the central weight of each index can be determined using Equations (15) and (16):  
\[\frac{\omega_{B}^{M}}{\omega_{n}^{M}}=m_{Bn}(n=1,2,\cdots,6) \tag{15}\]  
\[\frac{\omega_{n}^{M}}{\omega_{W}^{M}}=m_{nW}(n=1,2,\cdots,6) \tag{16}\]  
where \(\omega_{n}^{M}\) represents the central weight of the \(n\)th index and \(\omega_{B}^{M}\) and \(\omega_{W}^{M}\) represent the central weights of \(l_{B}\) and \(l_{W}\), respectively. However, in general, the central weights of the six indices determined from the BO and OW vectors cannot fully satisfy Equations (15) and (16) simultaneously. Therefore, the optimization problem in  
Figure 4: Membership functions of weight ratio. Equation (17) can be adopted to minimize the maximum absolute difference \(\left|\frac{\omega_{B}^{M}}{\omega_{n}^{M}}-m_{Bn}\right|\) and \(\left|\frac{\omega_{n}^{M}}{\omega_{W}^{M}}-m_{n}w\right|\), and the central weight of every index can be determined [30; 34]. \[\left\{ \begin{array}{l}\underset{j}{\mathrm{minmax}}\left\{\left|\frac{\omega _{B}^{M}}{\omega_{n}^{M}}-m_{Bn}\right|,\left|\frac{\omega_{n}^{M}}{\omega_{W} ^{M}}-m_{n}w\right|\right\}\\ s.t.\\ \sum\limits_{n=1}^{6}\omega_{n}^{M}=1\\ \omega_{n}^{M}\geq 0,n=1,2,\cdots,6\end{array} \right. \tag{17}\]  
The optimization problem in Equation (17) is equivalent to the optimization problem in Equation (18):  
\[\left\{ \begin{array}{l}\underset{s.t.}{\mathrm{min}}\xi\\ \left|\frac{\omega_{n}^{M}}{\omega_{n}^{M}}-m_{Bn}\right|\leq\xi,\\ \sum\limits_{n=1}^{6}\omega_{n}^{M}=1\\ \omega_{n}^{M}\geq 0\end{array} \right. \tag{18}\]  
Step 5. Check the consistency. When \(m_{Bn}\) and \(m_{nW}\) satisfy Equation (19), the comparison is fully consistent. However, it is usually impossible to satisfy due to the subjectivity and ambiguity of the experts in their judgment. \[m_{Bn}m_{nW}=m_{BW} \tag{19}\]  
Therefore, we need to calculate the consistency ratio by Equation (20) to check the consistency. \[\mathrm{CR}=\frac{\xi}{\mathrm{CI}} \tag{20}\]  
where CI represents the consistency index, \(\xi\) is the value determined by programming Equation (18) and CR is the consistency ratio. The value of CI can be determined using Equation (21). It can also be obtained by Table 3 directly if \(m_{BW}\) is an integer. \[\mathrm{CI}=\frac{2m_{BW}+1-\sqrt{8m_{BW}+1}}{2} \tag{21}\]  
The value of CR is distributed in the interval \(\left[0\quad 1\right]\). The closer the value of CR is to 0, the better the consistency is; on the contrary, the closer the value of CR is to 1, the worse the consistency is. In fact, we can take the consistency relation shown in Equation (19) to check the consistency intuitively, as presented in Table 4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(m_{BW}\) & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline CI & 0.00 & 0.44 & 1.00 & 1.63 & 2.30 & 3.00 & 3.73 & 4.47 & 5.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The value of the consistency index (CI). If \(m_{Bn}\times m_{nW}\) is too different from \(m_{BW}\), then we can identify that the \(n\)th index is the cause of the possible low consistency. Therefore, we can revise \(m_{Bn}\) and \(m_{nW}\) to improve the consistency [30]. Step 6. Determine the interval weight of each index.