## 4 Opposite Normalized Trust-Region Reflective (ONTRR) Algorithm
### Solution Quality and Computational Efficiency  
#### 5.2.1 Six Cases Utilizing the IEF-Based Objective Function  
The optimal results extracted from the IEF-based objective function by the proposed ONTRR are summarized in Table 2. To ensure that the reader can reproduce the RMSE values, the parameter values in Table 2 are correct to 8 decimal places and compared to the results of some state-of-the-art algorithms, such as EJBO [5], GSK [6], GNDO [7], CLRao\(-1\)[8], IGSK [9], IMPA [10], EOTLBO [11], ELBA [12], QRMSCS [13], MIGTO [14], CPMPSO [15], GOFANM [16], WLCSODGM [17], ISCE [18], DOLADE [19], Rcr-IJADE [20], DPDE [21], HDE [22], EJADE [23], ISMA [24], DE/WOA [25], QNMPO [26], LNMHGS [27], EHA-NMS [28] and ABC-TRR [29]. By crosschecking Table 2 and the results in [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29], one can find that there is a less than 0.01 percent differences between them, even though they include different numbers of decimal places. This means that the proposed ONTRR possesses identical accuracy to the compared algorithms. However, considering the consumed number of function evaluations (NFEs), it is evident from Figure 4 that only ONTRR and ABC-TRR always take the least NFEs to achieve the optimal RMSE values among all compared algorithms. This suggests that these two algorithms need the least computational effort to reach the same accuracy, thereby having the highest computational efficiency for parameter extraction of the IEF-based SDM, DDM and TDM, followed by the ISCE [18], EHA-NMS [28] and Rcr-IJADE [20] algorithms.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Cell/Module** & **R.T.C. France Cell** & **PWP201 Module** & \
\begin{tabular}{c} **STM6-40/36** \\ **Module** \\ \end{tabular} &
\begin{tabular}{c} **STP6-120/36** \\ **Module** \\ \end{tabular} \\ \hline
**Model** & **SDM** & **DDM** & **TDM** & **SDM** \\ \hline \(n_{1}\), \(n_{2}\), \(n_{m}\) & 1.48118359 & 1.45101671 & 2.0000000 & 48.64283466 & 1.52030292 & 1.26010347 \\ \(n_{2}\) & & 2.00000000 & 2.0000000 & & & \\ \(n_{3}\) & & 1.45101679 & & & \\ \(I_{\mu\mu}\), \(I_{\mu\mu}\) (A) & 0.76077553 & 0.76078108 & 0.76078108 & 1.03051430 & 1.66390478 & 7.47252992 \\ \(I_{0}\), \(I_{0}\), \(I_{1}\), \(I_{0}\) (u\(\mu\)A) & 0.32302080 & 0.22597412 & 0.57680977 & 3.48226273 & 1.73865682 & 2.33499484 \\ \(I_{0}\) (A) & & 0.74934873 & 0.17253702 & & & \\ \(I_{0}\) (A) & & 0.22597433 & & & \\ \(R_{s}\), \(R_{\mu m}\) (m\(\Omega\)) & 36.37709278 & 36.74043067 & 36.74042998 & 1201.27101346 & 4.27377143 & 4.59463464 \\ \(R_{s}\), \(R_{\mu m}\) (\(\Omega\)) & 53.71852462 & 55.48544266 & 55.48543523 & 981.98216392 & 15.92829404 & 22.21989981 \\ RMSE & \(9.860219\times 10^{-4}\) & \(9.824849\times 10^{-4}\) & \(9.824849\times 10^{-4}\) & \(2.425075\times 10^{-3}\) & \(1.729814\times 10^{-3}\) & \(1.660060\times 10^{-2}\) \\ MaxNFEs & 1000 & 5000 & 10,000 & 1000 & 1000 & 1000 \\ \hline \end{tabular}
\end{table}
Table 2: Optimal results extracted from the IEF-based objective function by ONTRR algorithm.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Cell/Module** & **R.T.C. France Cell** & **PWP201 Module** & \
\begin{tabular}{c} **STM6-40/36** \\ **Module** \\ \end{tabular} &
\begin{tabular}{c} **STP6-120/36** \\ **Module** \\ \end{tabular} \\ \hline
**Model** & **SDM** & **DDM** & **TDM** & **SDM** & **SDM** \\ \hline \(n_{1}\), \(n_{2}\), \(n_{m}\) & 1.48118359 & 1.45101671 & 2.00000000 & 48.64283466 & 1.52030292 & 1.26010347 \\ \(n_{2}\) & & 2.00000000 & 2.0000000 & & & \\ \(n_{3}\) & & 1.4510167