## 2 Solid Oxide Fuel Cell Control Method
### SOFC Working Principle  
Solid oxide fuel cells are one of a variety of fuel cells that use hydrogen to generate electricity, water, and heat. Due to their high operating temperature requirements, they have been applied for waste heat recovery in many industrial processes [24; 25]. In SOFC chemistry, the anode provides hydrogen as fuel, oxygen molecules in the cathode decompose into oxygen ions, and the directional movement of oxygen ions form a current that drives the load and passes to the anode and hydrogen in the form of oxygen vacancies for electrochemical reactions that release heat, and the anode and cathode are electrolyzed to achieve heat exchange [26]. The model structure of the SOFC system considers the principles of electrochemistry and thermodynamics. The specific structure and detailed parameters are shown in Figure 2 and Table 1, respectively. In the figure, \(q_{f}\) is the hydrogen flow rate, and the \(I_{fc}\) and \(V_{fc}\) are the current and output voltage of the SOFC, respectively. \(T_{out}\) is the output temperature. Figure 1: Block diagram of offshore wind SOFC system structure. During the actual electrochemical reaction, the fuel cell output voltage is reduced because the current flow makes the equilibrium between the electrode and the cell disrupted, a phenomenon called electrode polarization. The degree of polarization can be measured by the overpotential, and the polarization of the SOFC can be divided into activation polarization, concentration polarization, and ohmic polarization, i.e., three types of voltage loss. So, the actual output voltage \(V_{fc}\) is calculated by  
\[V_{fc}=V_{0}-V_{act}-V_{olm}-V_{conc} \tag{1}\]  
where \(V_{act}\) is the activation pressure loss. \(V_{olm}\) is the ohmic pressure loss. \(V_{conc}\) is the concentration pressure loss. \(V_{0}\) is the voltage generated by the electron transport in the SOFC, and \(V_{0}\) which can be calculated by the Nerst formula with the following equation:  
\[V_{0}=\frac{\Delta g_{f}}{2F}+\frac{RT}{2F}\ln\frac{P_{H_{2}}P_{O_{2}}^{0.5}}{P _{H_{2}O}} \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Parameter** \\ \hline \(K_{H_{2}}\) & Valve molar constant for hydrogen \\ \(K_{H_{2}O}\) & Valve molar constant for steam \\ \(K_{O_{2}}\) & Valve molar constant for oxygen \\ \(K_{r}\) & Reaction constant \\ \(\tau_{H_{2}O}\) & Response time for steam flow \\ \(\tau_{O_{2}}\) & Response time for oxygen flow \\ \(\tau_{H_{2}}\) & Response time for hydrogen flow \\ \(\tau_{f}\) & Fuel processor response time \\ \(T_{int}\) & Initial temperature \\ \(T_{0}\) & Constant temperature \\ \(\lambda_{s}\) & Thermal conductivity \\ \(h_{eff}\) & Thickness \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Meaning of involved SOFC system parameters. Figure 2: Partial model block diagram of SOFC system. where \(\Delta g_{f}\) is the molar Gibbs free energy change. \(F\) is the Faraday constant with a value of 96,487 C/mol. \(R\) is the gas constant. \(T\) is the operating temperature of the SOFC. \(P_{H_{2}}\), \(P_{H_{2}O}\), and \(P_{O2}\) are the hydrogen, water vapor, and oxygen partial pressures, respectively, and are related to the input hydrogen and oxygen flow rates [27]. The calculation equation is as follows:  
\[P_{H_{2}}=\frac{1/K_{H_{2}}}{1+\tau_{H_{2}}s}\Bigg{(}\frac{1}{1+\tau_{f}s}q_{f }-2K_{r}I_{fc}\Bigg{)} \tag{3}\]  
\[P_{O_{2}}=\frac{1/K_{O_{2}}}{1+\tau_{O_{2}}s}\Bigg{(}\frac{1/r_{H_{2}O}}{1+ \tau_{f}s}q_{f}-K_{r}I_{fc}\Bigg{)} \tag{4}\]  
\[P_{H_{2}O}=\frac{1/K_{H_{2}O}}{1+\tau_{H_{2}O}s}2K_{r}I_{fc} \tag{5}\]  
Activation loss \(V_{act}\) is the loss due to electrochemical reactions, which is calculated by  
\[V_{act}=\frac{RT}{F}(z+\sqrt{1+z^{2}}) \tag{6}\]  
\[z=I_{fc}/2I_{0} \tag{7}\]  
\[I_{0}=A\cdot\exp(-E_{act}/RT) \tag{8}\]  
where \(I_{fc}\) is the external current. \(A\) is the exponential prefactor. \(E_{act}\) is the activation energy related to the electrode material [28].