## 4 Numerical Results
Note that the simulation times (wall time 2) under S-VERICELL in Table 3 are given for the verified, interval-based solver VNODE-LP and not just for the normal floating point algorithm (as, for example, in Line 7). Generally, we use wall clock times as computing times (if available) in Table 3 since they give a general idea about the actual amount of elapsed time. The times are provided only as a reference since we compare between MATLAB and C++ on different computers. Besides, values of exact derivatives are computed in parallel in UniVerMeC, which explains short wall times for optimisation for Sets 2 and 5 in Table 3.  
In [10], we showed that introducing uncertainty into MPC1 did not allow us to cover measured data well. For example, the option \"Set 2\" lead to large overestimation of the true solution set. In Figure 5, we show the results of simulations for MPL1 with 1% uncertainty in all parameters for parameter sets computed by Variant 5 from Table 3. The closed-form-based Variant 6 cannot handle uncertainty very well. Only the uncertainty of \(5\times 10^{-6}\%\) leads to meaningful results for Set 6 with nominal parameters as in Set 4. However, we see in both cases that the solution sets cover the measured data very well. Sets in Variant 5 are not prone to much overestimation. Figure 4: Simulated temperatures for different parameter sets (cf. Table 3) in comparison to measurements in case of MPC1 (left) and MPL1 (right). Since the curves are very similar for MPL1, we show only a plot close-up in the Figure. “Cf” means that the closed-form solution was used, otherwise VNODE-LP. A MATLAB solver was employed to identify the parameters from Sets 1 and 4. For parameters in Sets 2, 3 and 5, the solver IPOPT was used. The parameters for Set 6 are the same as for Set 5. \begin{table}
\begin{tabular}{c l l l l l} \hline
**Set** & **Model/Setting** & **Accuracy \(e\)** & **WT 1** & **WT 2** & **Degree** \\ \hline
1 & MPC1/S-MATLAB & 4.8255 K & – & – & C3— \\
2 & MPC1/S-VERICELL(i.b) & 35.595 K & 39 s & 348 s & C2— \\
3 & MPC1/S-VERICELL(i.a) & 3.5259 K & 636 s & \(<\)1 s & C2— \\
4 & MPL1/S-MATLAB & 2.1260 K & – & – & C3— \\
5 & MPL1/S-VERICELL(i.b) & 0.5310 K & 95 s & 232 s & C2— \\
6 & MPL1/S-VERICELL(i.a) & n.d. & n.d. & \(<\)1 s & C2— \\
7 & MPQ1/S-MATLAB & 2.1641 K & \(\approx\)18 h & \(\approx\)1800 s & C3— \\ \hline \end{tabular}
\end{table} 
Table 3: Results for MPC1/MPL1 under different conditions. Results for MPQ1 as a comparison. WT1 is the wall time for the optimisation, WT2 for the simulation. The parameter identification using the closed form solution and IPOPT have not been carried out yet for MPL1 due to reasons explained in Section 3.2, therefore “n.d” for Set 6 means “not done”. Times in MATLAB are given only once as a reference. Figure 3: Differences in the quality between MPC1 (in red), MPL1 (in green), and MPQ1 (in blue) under S-MATLAB, “c” stands for heat capacities. Overall we can say that the availability of the exact solution improved the quality of the identified parameters. However, the enclosures obtained with uncertainties in parameters were better if VNODE-LP was employed than if we used the closed-form expressions given by Formulas from Section 3. The reason is that the dependency problem in the closed-form might lead to high overestimation for interval arithmetic. This is the case, for example, if \(a_{i}\) is identified as negative for the model MPC1. However, the CPU times are better. To improve the quality of enclosures here, we plan to use Taylor models in the future (also for the evaluation of the goal function during parameter identification).