## 3 Simplifying the Models
### Simplified Model MPC1: Constant Heat Capacities

Table 2: _Cont._\[y_{1}(t)=u_{1}-\left(u_{1}-y_{1}^{\rm ic}-T_{\rm AG}^{\rm inv}(u_{1}-y_{0}^{\rm ic })(t-t_{0})\right)e^{-T_{\rm AG}^{\rm inv}(t-t_{0})}\enspace. \tag{12}\]  
The solution for the cathode gas preheater looks the same, only the parameters are different: all labels \"N2\" (or AG) should be changed to \"CG\". The solution for the temperature can be obtained easily after inserting the corresponding expressions for \(y_{3}(t)\) and \(y_{1}(t)\) into Equation (9) by variable separation and variation of the constant. It has the general form  
\[\theta(t)=\mathcal{I}_{\rm N2}(t)+\mathcal{I}_{\rm CG}(t)+k_{\theta}e^{-c_{m} ^{\rm inv}\cdot k_{\rm lin}(t-t_{0})}-\frac{k_{\rm const}}{k_{\rm lin}}\enspace, \tag{13}\]  
where  
\[\mathcal{I}_{\rm N2}(t) :=\frac{u_{1}\cdot c_{\rm N2,0}}{k_{\rm lin}}+c_{m}^{\rm inv} \cdot c_{\rm N2,0}\Bigg{(}-\frac{T_{\rm SL,AG}^{\rm inv}}{(T_{\rm SL,AG}^{\rm inv }-T_{\rm AG}^{\rm inv})(c_{m}^{\rm inv}k_{\rm lin}-T_{\rm AG}^{\rm inv})}(u_{1 }-y_{0}^{\rm ic})e^{-T_{\rm AG}^{\rm inv}(t-t_{0})}\] \[\quad+\frac{k_{\rm N2}}{c_{m}^{\rm inv}k_{\rm lin}-T_{\rm SL,AG}^ {\rm inv}}e^{-T_{\rm SL,AG}^{\rm inv}(t-t_{0})}\Bigg{)}\enspace,\]  
\(\mathcal{I}_{\rm CG}(t)\) analogously, and \(k_{\theta}=\theta^{\rm ic}-\mathcal{I}_{\rm CG}(t_{0})-\mathcal{I}_{\rm N2}(t _{0})+\frac{k_{\rm const}}{k_{\rm lin}}\). Note that only the situation in which \(T_{\rm SL,AG}^{\rm inv}\
eq T_{\rm AG}^{\rm inv}\) and \(T_{\rm SL,CG}^{\rm inv}\
eq T_{\rm CG}^{\rm inv}\) is relevant for the considered SOFC test rig. (Note that an analytical formula for \(\theta(t)\) can be obtained also for the case of identical time constants using the same technique as above.) If we assume that the preheater outlet temperatures are piecewise constant in time and employ the recorded sensor data instead of the dynamic model given by Equations (5)-(8), the corresponding solution for the temperature becomes much simpler:  
\[\theta(t)=-\frac{\tilde{k}_{\rm const}}{k_{\rm lin}}+\left(\theta^{\rm ic}+ \frac{\tilde{k}_{\rm const}}{k_{\rm lin}}\right)\cdot e^{-c_{m}^{\rm inv}\cdot k _{\rm lin}\cdot(t-t_{0})}\enspace, \tag{14}\]  
where \(\tilde{k}_{\rm const}=-\theta_{A}\cdot k_{a}-(c_{\rm CG,0}\cdot y_{3}^{\rm ic }+c_{\rm N2,0}\cdot y_{1}^{\rm ic})\). Because the control variables are piecewise constant, the IVP is solved separately in each time interval \([t_{n-1},t_{n}]\), \(n=T_{b}+1,\ldots,T_{e}\). The initial values in the next step are the results from the previous step, \(t_{0}=t_{n}\). If the possibility (ii.b) is used, we work with the interval hull of the values of these control variables, because we do not know the exact value inside \([t_{n-1},t_{n}]\), only in \(t_{n-1}\) and \(t_{n}\). The same approach is possible if larger numbers of volume elements are used for modelling the temperature of the SOFC stack. However, the results in Section 4 suggest that the expressions for the exact solutions are so complex that there is almost no gain in using them from the point of view of the computing time. That is, option (i.a) might work too slow for MPC9, for example. Note that using the numerical computing environment MATLAB did not work very well for obtaining closed-form expressions (which had to be derived \"by hand\"). Although it was possible to compute the closed-form solution using MATLAB for \(y_{0}\), \(y_{1}\), \(y_{2}\) and \(y_{3}\), the expressions were numerically unstable, leading to overflows with increasing times \(t\). The expression for the temperature \(\theta\) could not be computed in MATLAB for dynamic preheater representations. A MATLAB solution under the assumption that the preheater states are piecewise constant (the case similar to Equation (14)) was also numerically unstable.