## 2 Models for SOFC and their Verification Degree
### Verification Degree of ODE Based SOFC Models
In most of our recent publications (e.g., [7; 28]), we solved the global optimisation problem in Equation (2) by verified approximation ((i.b) with (ii.b)): The true solution \(y(t)\) of the IVP was approximated by the explicit Euler method in its interval version taking into account numerical errors but not the discretisation error as \([y^{(k)}]:=[y^{(k-1)}]+h\cdot f([y^{(k-1)}],[p])\), where \(f\) denotes the right-hand side of the IVP (componentwise). The interval approximation \([y^{(k)}]\) at \(t_{k}\) was substituted for the exact solution \(y(t_{k})\) in the cost function in Equation (2) and the discretisation error ignored. In our setup, the sampling time \(h=1\) s is by at least two orders of magnitude smaller than the dominant time constant of the thermal process, that is, \([y^{(k)}]\) was an acceptable approximation of the true solution \(y(t_{k})\). Although we could not verify the whole process by applying interval optimisation procedures in this case, the approximated cost function  
\[J_{app}=\sum_{k=T_{b}+1}^{T_{e}}\sum_{i=1}^{N}\left(y_{i}^{(k-1)}-y_{m,i}(t_{ k})+h\cdot f(y_{i}^{(k-1)},p)\right)^{2}\ \, \tag{4}\]  
where \(y^{(T_{b})}\) is the initial condition, was evaluated in a verified way and optimised using an interval algorithm, resulting in the verification degree of C2 for MQ1 and MQ3. Note that the first (and second) derivatives of \(J_{app}\) were computed exactly with the help of algorithmic differentiation. In Figure 2, the results of this study are shown. \begin{table}
\begin{tabular}{c c c|c c|c c} \hline
**Model** & **(i.a)** & & & **(i.b)** & & **(i.c)** \\ \hline  & **(ii.a)** & **(ii.b)** & **(ii.c)** & **(ii.a)** & **(ii.b)** & **(ii.c)** & **(ii.a)** & **(ii.b)** & **(ii.c)** \\ \hline \multicolumn{6}{c}{Models without preheaters, with quadratic \(c(\theta)\), 1, 3 and 9 volume elements, resp.} \\ \hline MQ1 & not possible & [27] & [28] & & [26; 29] & \\ MQ3 & not possible & [27] & [7] & [29] & too slow & too slow \\ MQ9 & not possible & [27] & future work & & too slow \\ \hline \multicolumn{6}{c}{Model without preheaters, with linear \(c(\theta)\), 1 volume element} \\ \hline ML1 & [26] & not interesting & \multicolumn{2}{c}{not interesting} \\ \hline \multicolumn{6}{c}{Models with preheaters, quadratic \(c(\theta)\), 1 and 3 volume elements, resp.} \\ \hline MPQ1 & not possible & [4] & future work & future work \\ MPQ3 & not possible & [4] & future work & future work \\ \hline \multicolumn{6}{c}{Model with piecewise constant preheaters, linear \(c(\theta)\), 1 volume element} \\ \hline MPL1 & this paper & this paper & not interesting \\ \hline \multicolumn{6}{c}{Models with preheaters, constant \(c(\theta)\), 1, 3 and 9 volume elements, resp.} \\ \hline MPC1 & this paper & not interesting \\ MPC3 & future work & not interesting \\ MPC9 & too complex & future work & too slow \\ \hline \end{tabular}
\end{table} 
Table 1: Options for modelling and simulation of SOFCs. As explained above, fully verified versions of the models for the parameter identification can appear if Variant (i.c) with (ii.b) is used. That means that the true solution \(y(t,p)\) needs to be enclosed numerically by a verified IVP solver. For the Variant (i.c), it is more difficult to compute the derivatives \(\partial/\partial p\) of the cost function \(J\) in general, because it requires solving an additional IVP to obtain the sensitivities of the solution. This possibility was explored in [26; 29]. The general outcome here is that although the quality of the obtained parameters improves in this case, the computing times become too high even for MQ1. On the one hand, there is no possibility to use parallelisation on the GPU (and even on a multicore CPU) due to the lack of corresponding implementations (e.g., for verified IVP solvers). On the other hand, the IVP becomes more complex since the derivatives cannot be calculated by using algorithmic differentiation.