## 2 Models for SOFC and their Verification Degree
### Verification Degree of ODE Based SOFC Models  
Experimental identification or improvement of model parameters is an important task in the context of control-oriented SOFC modelling. As already mentioned, we are interested in control-oriented SOFC models of the general form  
\[y^{\prime}=f(t,y,p),\quad y(t_{0})=y_{0}\enspace, \tag{1}\]  
with the independent variable \(t\) usually assassinated with time in seconds and constant (but unknown) parameters \(p\). To solve the identification problem, the upper bound \(\overline{J}\) (if interval techniques are used) of the cost function  
\[J=\sum_{k=T_{k}}^{T_{e}}\sum_{i=1}^{N}\left(y_{i}(t_{k},p)-y_{m,i}(t_{k}) \right)^{2} \tag{2}\]  
is minimized with respect to the parameters \(p\). Here, \(y(t_{k},p)\) is the solution to the model equations (e.g., as given in Equations (5)-(9)) at the time \(t_{k}\), \(k=T_{b},\ldots,T_{e}\). The symbols \(T_{b}\), \(T_{b}\) denote the beginning and the end times in seconds, respectively. In our context, \(t_{k}=t_{T_{b}},t_{T_{b}}+1,\ldots,t_{T_{e}}\). Without loss of generality, we assume that the \(N\) states that can be measured using the SOFC test rig are the first \(N\) ones in the vector \(y(t_{k},p)\). The solution is obtained from the initial value problem (IVP) such as in (1) corresponding to the used SOFC model and optimised with respect to the measured values \(y_{m,i}(t_{k})\), \(i=1\ldots N\). That is, the function \(J\) quantifies deviations between the simulated results and the measured output vector acquired with \(T=T_{e}-T_{b}\) samples and a constant sampling time \(h=1\) s.  
According to the number of finite volume elements that are used to discretize the differential equations for the stack temperature, we might have different numbers of equations for this quantity. Until now, we considered one (MQ1), three (MQ3), and nine volume elements (MQ9) without preheaters, cf. Table 1 and [27]. Finer discretisations are possible in principle but are expensive computationally. Notations such as MQ1 are labels for the corresponding models (e.g., the one with a single volume element) and allow us to refer to them in a compact form according to the following system. The letter \(M\) just means _model. Q,L,C_ mean using the approximations to the heat capacities by a \(q\)quadratic, a linear and a \(c\)constant polynomial. Letter \(P\) indicates the use of the model for the \(p\)reheaters, and the number denotes the number of volume elements. An empirically motivated additional condition is induced by the accuracy of the measurements:  
\[y_{i}(t_{k})\subseteq\left[y_{m,i}(t_{k})-c_{i},y_{m,i}(t_{k})+c_{i}\right]= :\left[\Delta y_{m}(t_{k})\right]\quad\text{for}\quad k=T_{b},\ldots,T_{e},\ i=1 \ldots N \tag{3}\]  
with constants \(c_{i}>0\) chosen in accordance with the sensor data sheets of the actual SOFC test rig. The models described above might have the following basic forms depending on:  
(i): the way the simulated solution \(y(t_{k},p)\) of the IVP is obtained in Equation (2): (i.a): \(y(t_{k},p)\) is computed analytically, (i.b): \(y(t_{k},p)\) is approximated by an analytic expression (e.g., using the Euler method) and the approximation error is neglected, which is justified in cases in which the integration step size is smaller by several orders of magnitude than the smallest time constant of an asymptotically stable ODE system (cf. Equation (4)), and (i.c) \(y(t_{k},p)\) is computed using a \"black box\" numerical solver (no explicit expression for the solution); and
2. the kind of used arithmetic: (ii.a) traditional floating point arithmetic, (ii.b) interval arithmetic, and (ii.c) affine arithmetic, Taylor models, etc. Generally, forms (i.a) and (i.c) combined with (ii.b) or (ii.c) correspond to the complete verification of the model (C1), if optimisation is carried out in a completely verified way. If a certain best suitable interval vector is chosen from the list of candidates produced by global optimisation according to a heuristic technique, then the corresponding degree is C2.