## 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 most likely value in a triangular fuzzy number is actually the midpoint of the upper and lower bounds; we assume that the central weights obtained by Equations (13) and (14) are different from the interval weights obtained by Equations (9) and (10) with a radius [34]. Therefore, we can present the interval weight of the \(n\)th index by Equation (22):  
\[\omega_{n}=\left[\omega_{n}^{M}-d_{n}\quad\omega_{n}^{M}+d_{n}\right] \tag{22}\]  
where \(\omega_{n}\) represents the interval weight of \(n\)th index and \(d_{n}\) represents the radius of \(\omega_{n}\). The radius is obtained based on interval regression analysis, which is to find the estimated intervals to include the original data. In this problem, \(m_{Bn}\) and \(m_{nW}\) are approximated as \(\left[l_{Bn}\quad u_{Bn}\right]\) and \(\left[l_{nW}\quad u_{nW}\right]\). Thus, the interval weights of the indices should satisfy the constraints of Equations (23) and (24) [34; 35]:  
\[\left[l_{Bn}\quad u_{Bn}\right]\subseteq\left[ \begin{matrix}\omega_{B}^{M}-d_ {B}&\omega_{B}^{M}+d_{B}\\ \omega_{n}^{M}-d_{n}&\omega_{n}^{M}+d_{n}\\ \end{matrix} \right] \tag{23}\]  
\[\left[l_{nW}\quad u_{nW}\right]\subseteq\left[ \begin{matrix}\omega_{n}^{M}-d_ {n}&\omega_{n}^{M}+d_{n}\\ \omega_{n}^{M}-d_{W}&\omega_{n}^{M}+d_{W}\\ \end{matrix} \right] \tag{24}\]  
\[=\left[ \begin{matrix}\omega_{n}^{M}-d_{n}&\omega_{n}^{M}+d_{n}\\ \omega_{n}^{M}+d_{W}&\omega_{n}^{M}-d_{W}\\ \end{matrix} \right]\]  
where \(d_{B}\) represents the radius of \(IB\)'s interval weight and \(d_{W}\) represents the radius of \(I_{W}\)'s interval weight. Thus, the interval weight can also be described by the middle point and radius using Equation (25) [36]:  
\[\omega_{n}=\left[ \begin{matrix}\omega_{c}&\omega_{r}\\ \end{matrix} \right] \tag{25}\]  
where \(\omega_{c}\) represents the middle point of interval weight and \(\omega_{r}\) represents the radius of interval weight. Under the constraints of Equations (23) and (24), the radius of each index's interval weight should be minimized. To find the minimum value, we can use the optimization constraint to solve the problem, as shown in optimization problem in Equation (26) [34; 35]. \[\left\{ \begin{array}{ll}\min\lambda\\ s.t.\\ \frac{\omega_{n}^{M}-d_{B}}{\omega_{n}^{M}+d_{B}}\leq l_{Bn},&\frac{\omega_{n}^{ M}+d_{B}}{\omega_{n}^{M}-d_{B}}\geq u_{Bn}\\ \frac{\omega_{n}^{M}-d_{n}}{\omega_{n}^{M}+d_{W}}\leq l_{nW},&\frac{\omega_{n}^ {M}+d_{B}}{\omega_{W}^{M}-d_{W}}\geq u_{nW}\\ d_{n}\leq\lambda\\ \omega_{n}^{M}-d_{n}\geq 0\end{array} \right. \tag{26}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Indices** & \(\mathbf{I}_{1}\) & \(\mathbf{I}_{2}\) & \(\ldots\) & \(\mathbf{I}_{B}\) & \(\ldots\) & \(\mathbf{I}_{W}\) & \(\ldots\) & \(\mathbf{I}_{n}\) \\ \hline \(I_{B}\to I_{n}\) & \(m_{B1}\) & \(m_{B2}\) & \(\ldots\) & \(1\) & \(\ldots\) & \(m_{BW}\) & \(\ldots\) & \(m_{Bn}\) \\ \hline \(I_{n}\to I_{W}\) & \(m_{1W}\) & \(m_{2W}\) & \(\ldots\) & \(m_{BW}\) & \(1\) & \(\ldots\) & \(m_{nW}\) \\ \hline \(m_{Bn}\times m_{nW}\) & \(m_{B1}\times m_{1W}\) & \(m_{B2}\times m_{2W}\) & \(1\times m_{BW}\) & \(m_{BW}\times 1\) & \(\ldots\) & \(m_{Bn}\times m_{nW}\) \\ \(m_{BW}\) & \(m_{BW}\) & \(m_{BW}\) & \(m_{BW}\) & \(m_{BW}\) & \(m_{BW}\) & \(m_{BW}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Consistency check table. The interval weight of each power quality index shows the acceptable range for experts.