## 2 Problem Formulation For SOFC  
According to the previous research [7; 8; 9; 10], the working principles of fuel cells are essentially similar. In short, the SOFC system contains the fuel process part and the fuel cell stack. Functions of the SOFC are the same as those of the inverse device for water electrolysis. The general structure of the SOFC is depicted in Figure 1. The single cell for the SOFC includes several parts, such as the anode, the cathode and the solid oxide electrolyte. Fuel oxidation and oxidant reduction occur in the anode and cathode, respectively. At the same time, the catalyst, which can accelerate the electrochemical reaction of the electrode, is widely available in the anode and cathode. Based on the above introduction of the SOFC, it can be speculated that the actual effectiveness of the SOFC is equivalent to a DC voltage source in that the SOFC system can provide electricity continuously via the electrochemical reactions. Comparing the structures of the SOFC and DC source, it is indicated that the anode of the SOFC is equivalent to the negative power supply of the DC source, while the cathode of the SOFC is equivalent to the positive power supply of the DC source. The basic process of the chemical reaction in an SOFC is introduced in Figure 1. The fuel gas, similar to hydrogen (H\({}_{2}\)), methane (CH\({}_{4}\)) and city gas (CO), must be supplied on the anode side continuously. Then, the surface of the anode, which has a porous structure, adsorbs the fuel gas and sends it to the insideof the anode and electrolyte. O\({}_{2}\) can be transformed into O\({}^{2-}\) when the catalytic action inside the cathode gives O\({}_{2}\) an electron. Due to the impact of the chemical potential, O\({}^{2-}\) combines with the solid oxygen ion conductor, which can be realized as an electrolyte. Under the influence of the concentration gradient, O\({}^{2-}\) can react with the fuel gas when O\({}^{2-}\) diffuses into the inside of the electrolyte and anode. Then, all lost electrons return to the cathode to combine with O\({}_{2}\). Overall, this is a cyclic process. The main electrochemical processes of the SOFC can be described as follows:  
\[\text{Anode}:\text{H}_{2}+\text{O}^{2-}\rightarrow\text{H}_{2} \text{O}+2\text{e}^{-}\] \[\text{Cathode}:\text{O}_{2}+4\text{e}^{-}\to 2\text{O}^{2-}\]  
Figure 1: The general structure of the solid oxide fuel cell (SOFC). In this paper, we adopt the widely-accepted nonlinear dynamical model of the SOFC, which was presented in [27; 30]. This SOFC model is depicted in Figure 2. Taking advantage of [PERSON]'s equation with the consideration of ohmic, concentration and activation losses (i.e., \(\eta_{ohmic}\), \(\eta_{conc}\) and \(\eta_{act}\)), the function of the stack output voltage \(V_{dc}\) associated with the natural gas flow rate \(q_{f}\) can be defined as:  
\[V_{dc}=V_{0}-\eta_{ohmic}-\eta_{conc}-\eta_{act} \tag{1}\]  
therein  
\[V_{0}=N_{0}\left[E_{0}+\frac{R_{0}T_{0}}{2F_{0}}\ln\frac{p_{\mathrm{H_{2}}} \sqrt{p_{\mathrm{O_{2}}}/101,325}}{p_{\mathrm{H_{2}O}}}\right] \tag{2}\]  
\[p_{\mathrm{H_{2}}}=\frac{1}{K_{\mathrm{H_{2}}}(1+\tau_{\mathrm{H_{2}}}s)} \left(\frac{1}{1+\tau_{f}s}q_{f}-2K_{r}I\right) \tag{3}\]  
\[p_{\mathrm{O_{2}}}=\frac{1}{K_{\mathrm{O_{2}}}(1+\tau_{\mathrm{O_{2}}}s)} \left(\frac{1/\tau_{\mathrm{H-O}}}{(1+\tau_{f}s)}q_{f}-K_{r}I\right) \tag{4}\]  
Figure 1.0  
Figure 1.1  
Figure 1.2  
For the nonlinear dynamic system of the SOFC, the parameters and corresponding values are shown in Table 1[24]. Figure 1.3 SOFC system dynamic model. In the design process of this dynamic control system for the SOFC, we mainly consider three aspects, which are described as follows:  
1. It is found that the current load \(I\) is related to the stack output voltage \(V_{dc}\) based on the above dynamic model (see Figure 2), and changes of the current load \(I\) can influence the output voltage \(V_{dc}\) directly.