## 2 Modeling and Optimization Methodology
### System Modeling
#### 2.2.1 SOFC Subsystem  
The SOFC is of the planar type and its model is presented in detail in this section. The values of the constant parameters in the SOFC subsystem are given in Table 1. The fuel cell effective cross-sectional area is set at 100 cm\({}^{2}\)[16]. The fuel utilization factor and the fuel cell operating temperature are fixed at 0.9 and 800 \({}^{\circ}\)C, respectively [17; 18; 19]. The temperature at the fuel preheater exit is fixed at 450 \({}^{\circ}\)C [20], while the temperature of the flue gas at the HEx4 exit is fixed at 55 \({}^{\circ}\)C [21]. The temperature at the inlet of water pump 1 is fixed at 40 \({}^{\circ}\)C [15]. The return and supply temperature values in the hot water storage tank are fixed at 40 and 65 \({}^{\circ}\)C, respectively [20]. Figure 1: Configuration diagram of the hybrid photovoltaic (PV)-solid oxide fuel cell (SOFC) system under study. The SOFC stack model includes the direct internal reforming (DIR) and fuel cell reaction processes. The DIR process takes place at the catalyst surface [22] and it is equivalent to the steam reforming process [15]. A 0-dimensional model is considered, to allow coupling to the other system components. Therefore, changes in thermodynamic and electrochemical properties are considered negligible along the fuel cell. The SOFC temperature and pressure are fixed at a constant value throughout the fuel cell, because the outlet radial temperature gradients are negligible. Inlet air provides oxygen for the fuel cell reaction and cooling for the fuel cell stack (N\({}_{2}\) is considered inert both at cathode and anode). The Gibbs free energy, the reversible voltage, and the open circuit voltage, respectively, are defined as [23]:  
\[\Delta g^{o}{}_{f}=1\cdot g_{\text{H2O}}-0.5\cdot g_{\text{O2}}-1\cdot g_{ \text{H2}} \tag{1}\]  
\[E_{\alpha v,0}=\frac{-\Delta g^{o}{}_{f}}{2\cdot F} \tag{2}\]  
\[E_{\alpha v}=E_{\alpha v,0}+\frac{R\cdot T_{fc}}{2\cdot F}\cdot\text{ln}\left( \frac{\frac{g_{\alpha v,\text{H2P}}\gamma_{fc}}{p_{amb}}\cdot\sqrt{\frac{g_{ \alpha v,\text{H2O}}\gamma_{fc}}{P_{amb}}}}{\frac{g_{\alpha v,\text{H2O}} \gamma_{fc}}{P_{amb}}}\right) \tag{3}\]  
The activation losses in the anode and cathode can be determined as follows [17]:  
\[V_{act,ano}=\frac{2\cdot R\cdot T_{fc}}{n_{e}\cdot F}\cdot arcsinh\left(\frac {i}{2\cdot i_{o,ano}}\right) \tag{4}\]  
\[V_{act,cat}=\frac{2\cdot R\cdot T_{fc}}{n_{e}\cdot F}\cdot arcsinh\left(\frac {i}{2\cdot i_{o,cat}}\right) \tag{5}\]  
\[V_{act}=V_{act,ano}+V_{act,att} \tag{6}\]  
The Ohmic losses are determined as follows [17]:  
\[V_{oltm}=i\cdot\left(\frac{L_{ele}}{\sigma_{ele}}+\frac{L_{ano}}{\sigma_{ano}}+ \frac{L_{cat}}{\sigma_{cat}}+\frac{L_{int}}{\sigma_{int}}\right) \tag{7}\]  
where \(\sigma_{ele}\), \(\sigma_{ano}\), \(\sigma_{cat}\) and \(\sigma_{int}\) are the ionic conductivity of the electrolyte, the electronic conductivity of the anode, the electronic conductivity of the cathode, and the electronic conductivity of the interconnect, respectively. These are determined as follows:  
\[\sigma_{ele}=(C_{1 ele})\cdot exp\left(\frac{C_{2 ele}}{T_{fc}}\right) \tag{8}\]  
Figure 1:0 \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Parameter Description** & **Value** \\ \hline \(A_{fc}\) & Fuel cell effective cross-sectional area & 100 cm\({}^{2}\) \\ \(U_{f}\) & Fuel utilization factor & 0.9 \\ \(T_{fc}\) & Fuel cell operating temperature & 800 \({}^{\circ}\)C \\ \(T_{6}\) & Temperature at fuel preheater exit & 450 \({}^{\circ}\)C \\ \(T_{22}\) & Temperature of flue gas at HEx4 exit & 55 \({}^{\circ}\)C \\ \(T_{25}\) & Temperature at water pump 1 inlet & 40 \({}^{\circ}\)C \\ \(T_{28}\) & Return temperature in hot water storage tank & 40 \({}^{\circ}\)C \\ \(T_{29}\) & Supply temperature in hot water storage tank & 65 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fixed parameter values of the SOFC subsystem.