## 4 Results Obtained with the Modified Nelder-Mead Method  
We investigated the behavior of our modified Nelder-Mead parameter extraction method with two kinds of benchmarks. The first benchmark compares the overall error of the parameter fitting performed with the new method with the overall error of the prior brute-force approach used in the Delphi4 LED project. This comparison was chosen because the brute-force-based method--though very slow and computationally intensive--provides very good fits (less than a 1% error for \(V_{F}\) and less than a 5% error for \(\Phi_{e}\)) for all the LED samples characterized in the Delphi4 LED project [41]. Therefore, this comparison offers a valuable insight into the fit errors of the new parameter extraction procedure. However, caution is needed for this comparison, as there are several significant differences between the two methods. * The old method requires predefined junction temperatures, so the measurement points that do not meet this requirement are not taken into account. * Slight differences in junction temperatures (\(\pm\)0.5 \({}^{\circ}\)C) are neglected in the old method. * The old and the new parameter extraction methods also differ in the objective functions that they minimize during the fitting process. The old method uses the sum of square errors as the objective function to be minimized, while in the new method, the sum of the relative square error is used. Therefore, for the used figure of merit for the comparison of the two methods, the maximal relative error is much higher at low forward currents when applied to evaluate the fit error of the old method. The new model uses the sum of the relative square error, because many end-users of the SPICE-like multi-domain LED model may require accurate simulations across a wide range of forward currents, spanning two decades (for our sample LEDs: from 20 mA up to 1000 mA). As a result, the LED model should be equipped with a parameter set that ensures accurate results, even below the greatest decade in the forward current (\(I_{F}<100\) mA). To address this need, we compared the performance of the two parameter extraction methods, both with and without including low-forward-current operating points. In the comparisons for the high current domain, we completely discarded the low-current operating points of the LEDs during the fitting procedures. The parameter extraction procedures were applied to 52 high-power LED samples measured in the Delphi4 LED project: 11 phosphor-converted white LED samples of the same type, 5 red LED samples of the same type, 5 under LED samples of the same type, and 31 blue LED samples with 11 samples from the same LED type and 20 other samples of other types. The maximum relative error values obtained with the old and new parameter extraction procedures are shown in Figure 5 for the electrical branch, Figure 6 for the optical branch, and Figure 7 for luminous flux, applied to the high-forward-current operating points only (\(I_{F}>100\) mA) and for the full range of operating points (involving the low-forward-current ones as well). In the high-forward-current region (above 100 mA), where the \"linearization effect\" of the series resistance starts to appear in the LED characteristics, the strong dependence of the accuracy of the parameters of the pn-junction-related parts of the multi-domain LED model is increasingly suppressed as the forward current increases. Figure 5: Maximum relative errors for the electrical branch of the multi-domain LED model. The errors are shown for parameter sets estimated using the prior brute-force method from the Delphi4 LED project and the novel modified Nelder–Mead method. Results are presented for different LED types, for forward currents above 100 mA, and for the full range of forward currents, including the low current regime. Figure 6: Maximum relative errors for the optical branch of the multi-domain LED model with parameter sets estimated using the prior brute-force method, compared to the modified Nelder–Mead-based parameter extraction process.