## 3 Simplifying the Models
### Simplified Model MPC1: Constant Heat Capacities  
The system of ODEs we can derive if we assume all heat capacities to be constant in temperature (MPC1) is shown below:  
\[\dot{y}_{0} = T^{\text{inv}}_{\text{AG}}\cdot(u_{1}-y_{0}) \tag{5}\] \[\dot{y}_{1} = T^{\text{inv}}_{\text{SL,AG}}\cdot(y_{0}-y_{1})\] (6) \[\dot{y}_{2} = T^{\text{inv}}_{\text{CC}}\cdot(u_{2}-y_{2})\] (7) \[\dot{y}_{3} = T^{\text{inv}}_{\text{SL,CG}}\cdot(y_{2}-y_{3})\] (8) \[\dot{\theta} = -c^{\text{inv}}_{m}\cdot(k_{\text{const}}-(c_{\text{CG,0}}\cdot y _{3}+c_{\text{N2,0}}\cdot y_{1})+k_{\text{lin}}\theta)\enspace, \tag{9}\]  
with  
\[k_{a}=234000\alpha_{i}+448500\alpha_{j}+345000\alpha_{k}\enspace,\] \[k_{\text{const}}=-\theta_{A}k_{a}\enspace,\] \[k_{\text{lin}}=k_{a}+\dot{m}^{\text{in}}_{\text{CG}}\cdot c_{ \text{CG,0}}+\dot{m}^{\text{in}}_{\text{N2}}\cdot c_{\text{N2,0}}\enspace,\]  
and initial conditions \(y_{i}(t_{0})=y^{\text{ic}}_{i}\), \(i=0,1,2,3\), \(\theta(t_{0})=\theta^{\text{ic}}\). That is, we obtain a linear set of ODEs for the temperature including the preheater states from the energy balance equations that were explained in Section 2.1. Physical units of the numerical values included in the defintion of \(k_{a}\) are not listed explicitly, because all parameters are assumed to be given in SI derived units. Equations (5)-(6) describe the first preheater (for the anode gas), Equations (7)-(8) the second (for the cathode gas), and Equation (9) the stack temperature.The meaning of all parameters, variables, and control inputs is given in Table 2. For \(T^{\text{inv}}_{\text{SL,AG}}\
eq T^{\text{inv}}_{\text{AG}}\), there is a simple analytical solution to the model according to  
\[y_{0}(t) = u_{1}-(u_{1}-y^{\text{ic}}_{0})e^{-T^{\text{inv}}_{\text{AG}}(t-t _{0})} \tag{10}\] \[\dot{y}_{1} = T^{\text{inv}}_{\text{SL,AG}}\cdot(y_{0}-y_{1})\]0  
for the anode gas preheater, where \(k_{\text{N2}}=y^{\text{ic}}_{1}-u_{1}+\frac{T^{\text{inv}}_{\text{SL,AG}}}{T^{ \text{inv}}_{\text{SL,AG}}-T^{\text{inv}}_{\text{CAG}}}(u_{1}-y^{\text{ic}}_{0})\). If \(T^{\text{inv}}_{\text{SL,AG}}=T^{\text{inv}}_{\text{AG}}\), we obtain a different formula for \(y_{1}(t)\): \begin{table}
\begin{tabular}{c l} \hline \multicolumn{2}{c}{**Variables**} \\ \hline \(y_{0}=v_{\text{N2}}\) & product of the mass flow and the temperature \(\dot{m}^{\text{in}}_{\text{N2}}\cdot\theta_{\text{N2}}\) of the anode \\  & gas (nitrogen) at the preheater outlet in \(\frac{\text{kg}\cdot\text{K}}{\text{S}}\) \\ \(y_{1}=v^{\text{in}}_{\text{N2}}\) & product \(\dot{m}^{\text{in}}_{\text{N2}}\cdot\dot{\theta}^{\text{in}}_{\text{N2}}\) for the nitrogen at the stack inlet in \(\frac{\text{kg}\cdot\text{K}}{\text{S}}\) \\ \(y_{2}=v_{\text{CG}}\) & product \(\dot{m}^{\text{in}}_{\text{CG}}\cdot\theta_{\text{CG}}\) for the cathode gas at the preheater outlet in \(\frac{\text{kg}\cdot\text{K}}{\text{S}}\) \\ \(y_{3}=v^{\text{in}}_{\text{CG}}\) & product \(\dot{m}^{\text{in}}_{\text{CG}}\cdot\theta^{\text{in}}_{\text{CG}}\) for the cathode gas at the stack inlet in \(\frac{\text{kg}\cdot\text{K}}{\text{S}}\) \\ \(\theta\) & temperature of the stack in K \\ \hline \multicolumn{2}{c}{**Control Inputs**} \\ \hline \(\dot{m}^{\text{in}}_{\text{N2}}\) & mass flow of anode gas in \(\frac{\text{kg}}{\text{S}}\) (recorded data) \\ \(\dot{m}^{\text{in}}_{\text{CG}}\) & mass flow of cathode gas in \(\frac{\text{kg}}{\text{S}}\) (recorded data) \\ \(\dot{\theta}_{A}\) & ambient temperature in K \\ \(\theta^{\text{d}}_{\text{AG}}\) & desired temperature of the anode gas in K (recorded data) \\ \(\dot{\theta}^{\text{d}}_{\text{CG}}\) & desired temperature of the cathode gas in K(recorded data) \\ \(u_{1}=v^{\text{d}}_{\text{N2}}\) & desired \(v_{\text{N2}}=\theta^{\text{d}}_{\text{AG}}\cdot\dot{m}^{\text{in}}_{\text{N2}}\) in \(\frac{\text{kg}\cdot\text{K}}{\text{S}}\) \\ \(u_{2}=v^{\text{d}}_{\text{CG}}\) & desired \(v_{\text{CG}}=\theta^{\text{d}}_{\text{CG}}\cdot\dot{m}^{\text{in}}_{\text{CG}}\) in \(\frac{\text{kg}\cdot\text{K}}{\text{S}}\) \\ \hline \end{tabular}
\end{table}