## 2 Kalman Filter Design for Real-Time Estimation of Electric Power Characteristic Of Fuel Cells  
### Fundamental Current-Voltage and Current-Power Characteristics of SOFCs  
According to the measured data shown in Figure 5, which were determined for a constant hydrogen mass flow \(\dot{m}_{\mathrm{H_{2},in}}\approx 5.1\cdot 10^{-6}\ \frac{\mathrm{kg}}{\mathrm{s}}\) at the available test rig, the typical dependency between the terminal voltage \(U\) of the fuel cell stack and its electric current \(I\) (measured at a variable external electric resistance as a load device) becomes clearly visible. For the presented experiments, an SOFC stack consisting of 60 planar fuel cells in electric series connection was employed. This stack has a maximum electric power output of 1.3 kW. To have sufficient potential for identifying the optimal power point, the stack was operated in part load conditions. In Figure 5d,e, three different regions can be identified:  
* Region A of activation polarization (characterized by small currents \(I\) close to the theoretical Nernst voltage characterizing the SOFC's open-circuit behavior);
* Region B of Ohmic polarization corresponding to an almost linear reduction of the terminal voltage for increasing currents \(I\); and
* Region C of concentration polarization, in which the terminal voltage \(U\) rapidly gets close to zero. After computing the instantaneous electric power according to  
\[P_{\mathrm{EL,}k}=U_{k}\cdot I_{k} \tag{1}\]  
at each sampling instant \(k\) with the sampling interval \(t_{k+1}-t_{k}=1\,\mathrm{s}\), one can clearly see the existing power maximum in Figure 5e. For the sake of maximizing the electric power production in the following section, it is desirable to identify the maximum of the current-power characteristic in real time, while preventing excessively large currents \(I\) coinciding with a negative slope of the current-power characteristic. Such operating points should be avoided in practice by the use of maximum power point tracking approaches because they lead to fuel starvation that can go along with an accelerated degradation of the SOFC module. Note that maximum power point tracking approaches are well-known also from several other fields of applications. For example, they are extensively used for maximizing the power of photovoltaic systems. However, in recent publications from this area (cf. [18; 19]), preventing overshooting the maximum power point is not as crucial as in the SOFC application considered in this paper. Therefore, the approaches applied to photovoltaic systems perform incremental control adaptation schemes without directly accounting for estimation uncertainty in the electric power characteristic. However, for preventing overshooting, this information about uncertainty is essential and can be provided in real time by applying the Kalman filter-based identification scheme derived in this paper. Figure 5: Measured electric current \(I\), terminal voltage \(U\), and electric power \(P_{\mathrm{EL}}\) of the SOFC for a relatively small, constant hydrogen mass flow \(m_{\mathrm{H_{2},in}}\approx 5.1\cdot 10^{-6}\,\frac{\mathrm{kg}}{\mathrm{s}}\). In Figure 5d,e, analytic approximations, including logarithmic terms inspired by [PERSON]'s equation [20, 21], for the terminal voltage  
\[U_{k}=a_{0}+a_{1}\cdot I_{k}+a_{2}\cdot I_{k}^{2}+a_{3}\cdot\ln\left(I_{k}\right) +a_{4}\cdot\dot{m}_{\mathrm{H_{2},in,}k}+a_{5}\cdot I_{k}\cdot\dot{m}_{\mathrm{ H_{2},in,}k}+a_{6}\cdot I_{k}\cdot\dot{m}_{\mathrm{H_{2},in,}k}^{2} \tag{2}\]  
as well as for the electric power  
\[P_{\mathrm{EL},k}=\left(a_{0}+a_{1}\cdot I_{k}+a_{2}\cdot I_{k}^{2}+a_{3}\cdot \ln\left(I_{k}\right)+a_{4}\cdot\dot{m}_{\mathrm{H_{2},in}}+a_{5}\cdot I_{k} \cdot\dot{m}_{\mathrm{H_{2},in,}k}+a_{6}\cdot I_{k}\cdot\dot{m}_{\mathrm{H_{2}, in,}k}^{2}\right)\cdot I_{k} \tag{3}\]  
are compared with the measured data under the assumption of a constant hydrogen mass flow.