## 3 Simplifying the Models  
The previous section demonstrated that a trade-off between a high verification degree and an acceptable computing time is necessary. That is, if we would like to arrive at high verification degrees in a real-time simulation or control, we might need to simplify the models, the possibilities for which are described in this section. Besides, we study the quality of such models and try to find their areas of validity in an empirical way in the next section.  
In this paper, we consider the nitrogen as the only anode gas for the heating phase without any chemical reactions of gases. Besides, we work with a single volume element to describe the temperature of the stack as a whole. In [26], we pointed out ways to simplify the model for the SOFC stack with one volume element and no preheaters. In this paper, we consider two preheaters, since this improves the overall quality of SOFC models from the point of view of their control. Parameters denoted by \(m\) describe mass flows of, for example, nitrogen and are assumed to be piecewise constant (applies to all _control variables_ from Table 2). They are spatially invariant in the complete test rig; there are no relevant storage effects due to compressibility. Note that although the notation \(\dot{x}\) denotes the derivative of \(x\) wrt. time and \(\dot{m}\) could have been modeled using appropriate IVPs, we use the information from measurements to describe these values if they are shown in Table 2. All other parameters are constant. Here, we consider the models MPC1 and MPL1 in detail, cf. Table 1.  
The simplified model MPL1 relies on linear polynomial approximations for the heat capacities. However, this results in a Riccati differential equation if we assume that preheater states are not constant inside the considered time interval (of the width 1s in our case). This differential equation can be transformed into a system of two linear differential equations with non-constant coefficients which is not easy to solve analytically in our case. For piecewise constant preheater states, it is possible to calculate the analytical solution similarly to [26]. In this paper, we study the properties of MPL1 in addition to MPC1.  
\begin{table}
\begin{tabular}{c l} \hline \hline  & **Parameters (to Identify)** \\ \hline \(\alpha_{i}\), \(\alpha_{j}\), \(\alpha_{k}\) & coefficients of heat convection in \(\frac{\text{W}}{\text{m}^{2}\cdot\text{K}}\) \\ \(c_{\text{N2,0}}\), \(c_{\text{N2,1}}\) & heat capacity of nitrogen as \(c_{\text{N2}}(\theta)=c_{\text{N2,0}}+c_{\text{N2,1}}\cdot\theta\) in \(\frac{\text{J}}{\text{kg}\cdot\text{K}}\) \\ \(c_{\text{CG,0}}\), \(c_{\text{CG,1}}\) & heat capacity of the cathode gas as \(c_{\text{CG}}(\theta)=c_{\text{CG,0}}+c_{\text{CG,1}}\cdot\theta\) in \(\frac{\text{J}}{\text{kg}\cdot\text{K}}\) \\ \(T_{\text{TAV}}^{\text{inv}}\) & inverse time constant of the anode gas preheater in s\({}^{-1}\) \\ \(T_{\text{SL,AG}}^{\text{micro}}\) & inverse time constant of the cathode gas preheater in s\({}^{-1}\) \\ \(T_{\text{SL,CG}}^{\text{micro}}\) & inverse time constant of the anode gas supply line in s\({}^{-1}\) \\ \(c\) & specific heat capacity of the stack module in \(\frac{\text{J}}{\text{kg}\cdot\text{K}}\) \\ \(m\) & mass of the stack module in kg \\ \(c_{m}^{\text{inv}}\) & \(=1/(c\cdot m)\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Model parameters, control inputs and state variables.