## 4 Numerical Results
Instead of a single run using Equation (2) with Equations (5)-(9), two separate runs were performed to optimise parameters of the anode and cathode gas preheaters first (only for Equation (6) and Equation (8)) and, after that, for the temperature of the stack (Equation (9)). In the latter case, we did not use the dynamic information about the preheaters from Equations (5)-(8) but rather the corresponding measurements. The reason for these two separate runs is that the magnitudes of the variables \(y_{1}\), \(y_{3}\) and \(\theta\) are too different (of orders \(10^{-2}\), \(10^{-1}\), and \(10^{3}\), respectively). In this setting, the degree of verification is C3\(-\) (corresponding to (i.b) + (ii.a)). In the S-VERICELL setting, we performed the optimisation for MPC1 in one run, taking into account only the model equations without additional measurements according to  
\[\sum_{t_{k}=T_{b}}^{T_{c}}\left(\left(\frac{y_{1}(t_{k})-y_{m,1}(t_{k})}{ \sigma_{1}}\right)^{2}+\left(\frac{y_{3}(t_{k})-y_{m,3}(t_{k})}{\sigma_{3}} \right)^{2}+\left(\frac{\theta(t_{k})-y_{m,4}(t_{k})}{\sigma_{4}}\right)^{2}\right) \tag{17}\](a weighted sum, \(\sigma_{i}\), \(i=1,3,4\), is the mean value of the corresponding measurement data \(y_{m,i}\) over \([T_{b},T_{e}]\)) as a comparison. Since we could not take into account the models for the preheaters in case of MPL1 (see Section 3.2), we identified only the parameters for the SOFC stack temperature here. For both MPC1 and MPL1, we employed the floating point based tool IPOPT for parameter identification and an interval IVP solver VNODE-LP for simulation, resulting in the overall verification degree of C2 - (situation where (ii.a) is combined with (ii.b)). Besides, we do not only consider (i.b), the solution obtained by the Euler method, but also (i.a), the exact solution. In both settings, \(T_{b}=362\) and \(T_{e}=18000\), which corresponds to the heating stage from the data recorded during an experiment on the SOFC test rig. We identify eight parameters for MPC1: \(\alpha_{i}\) (\(\alpha_{i}=\alpha_{j}=\alpha_{k}\) in our context), \(c_{\text{N},2,0}\), \(c_{\text{CG},2,0}\), \(c\), \(T_{\text{AG}}^{\text{inv}}\), \(T_{\text{CG}}^{\text{inv}}\), \(T_{\text{SL},\text{AG}}^{\text{inv}}\), \(T_{\text{SL},\text{CG}}^{\text{inv}}\) given in Table 2, starting with values from S-MATLAB under S-VERICELL. For MPL1, we exclude the preheater parameters and consider \(c_{\text{CG},1}\), \(c_{\text{N},2,1}\) additionally. As a measure of accuracy, the root mean square error \(e\) is used for the temperature:  
\[e=\sqrt{\frac{\sum\limits_{k=T_{b}}^{T_{e}}(\theta(t_{k})-\theta_{m}(t_{k}))^{ 2}}{T_{e}-T_{b}}}\enspace. \tag{18}\]  
When we speak about the closed-form solution, we assume that \(T_{\text{SL},\text{AG}}^{\text{inv}}\
eq T_{\text{AG}}^{\text{inv}}\). We compare the results to MPQ1, simulated and optimised under S-MATLAB. A comparison of simulated temperatures delivered by MPQ1, MPL1 and MPC1 in Figure 3 shows that the model with linear heat capacities has the best accuracy (cf. also Table 3). The simulated and measured temperatures themselves are shown in Figure 4. Surprisingly, the MPL1 simulation results are even better than those of MPQ1 (which are obtained at higher computing times). Generally, the optimisation in two runs and with measurements for the preheaters delivered better parameter sets for MPC1 than the optimisation with the weighted sum in Equation (17). This is the reason for the large difference between measured and simulated data for the option \"Set 2\" in Figure 4, left. However, the best parameter set for MPC1 is obtained by IPOPT relying on the exact solution (Variant (i.a), Set 3). The computing times are high in this case because we used IPOPT with the exact Hessian matrix obtained by (exact) algorithmic differentiation. None of the computed parameter sets using MPC1 could be proved to be consistent with respect to the condition in Equation (3) (\(c_{i}=15\)). For MPL1, the accuracy is better than that for MPC1, and the computing times are comparable.