## 5 Conclusions  
In this paper, we showed how modelling possibilities for SOFC systems can be analysed using the process-oriented V&V procedure. Additionally, we pointed out two possibilities to simplify complicated SOFC models in such a way as for them to have closed-form solutions. We studied the models with constant and linear heat capacities and one volume element for the temperature extensively. The overall verification degree of C2\(-\) could be attained. The simulations in UniVerMeC were carried out in a verified way, preventing numerical errors. A good parameter set was identified using IPOPT and the exact solution for MPC1. However, this set was still not as good as the parameter set identified with the help of the original, more complicated model with quadratically approximated heat capacities. Overall, the simplified model described the measured data well enough, even though none of the computed parameter sets for MPC1 could be proven to be consistent with respect to the given conditions. The difference between closed-form (i.a) and Euler approximation-based (i.b) versions for MPC1 was dramatic, the former allowing us to identify a much better parameter set. Especially in the context of nonlinear model predictive control, such models can be used in temperature areas with small error ranges in a real-time setting, which we plan to study in the future.  
An interesting observation was that the best parameter set sofar (even compared to the usual, non-simplified case) was identified using the model with linearly approximated heat capacities \(\mathrm{MPL1}\). In this paper, we confirmed for the one-dimensional temperature model that that kind of simplification was very promising. We simulated that model under 1% uncertainty in parameters and showed that a complete coverage of measured data in case of \(\mathrm{MPL1}\) could be achieved. However, the parameter identification using the derived closed-form solution remains a topic for the further research. We  
Figure 5: Simulations with uncertainty in the temperature parameters for \(\mathrm{MPL1}\). For Set 5, the uncertainty of 1% in all parameters can be handled very well (green curves). For Set 6 (here with parameters identified for Set 4), only the uncertainty of \(5\times 10^{-6}\%\) leads to meaningful results (red curves). “Cf” means that the closed-form solution was used, otherwise VNODE-LP.  
also plan to use the \"verified approximation\" (i.b) version of this model for higher dimensions in our future work.  
Models relying on ODEs using more than one volume element to describe the temperature of the SOFC stack are necessary for better control of the temperature inside the stack. However, they cannot be solved analytically in general for non-constant heat capacities. Even in the \"constant\" case, the exact solutions might get too complicated to bring any measurable quality improvement. However, simulations with linearized or constant approximations to heat capacities might reduce overall computing times for higher-dimensional models, which is also a topic for our future work.  
Partially funded by the German Research Council (DFG).  
[PERSON] and [PERSON] developed the general form of the considered SOFC models and supplied the data. [PERSON] analyzed the procedure from the point of view of V&V analysis, identified possible simplifications and developed analytical solutions. [PERSON] ran the simulations using his framework UniVerMeC. [PERSON] and [PERSON] wrote the paper.  
The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.