## 2 Modeling the Electrochemical Behavior  
In an SOFC, both electric and thermal energy are produced simultaneously by the cold combustion. However, the overall SOFC process can be split into its thermal and electrochemical behavior for the mathematical description. Since the aim in the presented paper is to design an electric power controller, we restrict ourselves to the electrochemical behavior. All other phenomena, such as the inevitable variation of the stack temperature caused by exothermal reaction enthalpies which are not fully suppressed by adjustments of the cathode gas inlet temperature [10; 12; 13; 14; 15], are treated as non-modeled disturbances from the point of view of the electrochemical subsystem.  
In general, the electric current of a fuel cell is a result of the redox-reaction [16]  
\[\begin{array}{ccc}2\mathrm{H}_{2}+2\mathrm{O}^{2-}\ \rightarrow\ 2\mathrm{H}_{2} \mathrm{O}+4e^{-}&\qquad\text{(oxidation at the anode)}\\ \mathrm{O}_{2}+4e^{-}\ \rightarrow\ 2\mathrm{O}^{2-}&\qquad\text{(reduction at the cathode)}\\ 2\mathrm{H}_{2}(g)+\mathrm{O}_{2}(g)\ \rightarrow\ 2\mathrm{H}_{2}\mathrm{O}(g)& \qquad\text{(redox-reaction).}\end{array} \tag{1}\]  
As shown in Figure 1, the supplied hydrogen gas reacts with oxygen ions at the anode. Free electrons flow through an external circuit, which is implemented at the available test rig as a variable Ohmic resistor. In such a way, the fuel cell, respectively, the series concatenation of multiple cells in a corresponding stack module, serves as a directed current source for the consumer. Finally, the produced flow of electrons reaches the cathode to react with oxygen that is supplied in terms of preheated ambient air.  
The resulting instantaneous power, which can be calculated from the measured current and voltage at the SOFC's terminals should be controlled, which is the aim of this paper under the assumption that the electric current is kept constant by the controllable load resistor. In previous work (cf. [10; 11; 12]), some experiments were performed to solve modeling problems especially related to the control-oriented characterization of the thermal fuel cell behavior in various operating points. Based on these measured data, the linear time-invariant transfer function of the SOFC system  
\[\begin{array}{ll}G_{\mathrm{S}}(s)&=\dfrac{P_{\mathrm{el}}(s)}{\dot{m}_{ \mathrm{H}_{2}}(s)}\\ &=\dfrac{2.903\times 10^{5}s^{5}+5.189\times 10^{4}s^{4}+7.504\times 10^{3}s^{3}+8. 643\times 10^{2}s^{2}+2.908\times 10^{1}s+3.262\times 10^{-1}}{s^{6}+8.229\times 10^{-2 }s^{5}+2.018\times 10^{-2}s^{4}+8.309\times 10^{-4}s^{3}+9.233\times 10^{-5}s^{2}+1. 558\times 10^{-6}s+1.033\times 10^{-8}}\end{array} \tag{2}\]  
was determined with the help of the Matlab transfer function estimator [9], to describe the correlation between the electric power \(P_{\mathrm{el}}\) as the measured output and the hydrogen mass flow \(\dot{m}_{\mathrm{H}_{2}}\) as the input under the aforementioned assumption of a constant electric current and a reliable temperature control keeping the average stack temperature in the close vicinity of the operating point that is equally used for identification and control purposes.  
**Remark 1**.: _Oscillations in the depicted step response in Figure 2 are mostly related to long gas supply tubes and the fluidic inertia of the anode gas mass flow between the control value and the SOFC stack. For visualization purposes, \(\dot{m}_{\mathrm{H}_{2}}=4\times 10^{-6}\frac{\mathrm{kg}}{\mathrm{s}}\) has been used._  
Figure 1: Schematic representation of a SOFC.