## 2 The Stand-Alone Wind-Photovoltaic-Battery System with the Hybrid Electrolyzer
### Mathematical Model of the Stand-Alone Wind-Photovoltaic-Battery System with the Hybrid Electrolyzer
\begin{array}{l}0\leq P_{\text{BESS}}^{\text{d}}(t)\leq u_{\text{BESS}}^{ \text{d}}(t)\times P_{\text{BESS}}^{\text{N}}\\ -u_{\text{BESS}}^{\text{e}}(t)\times P_{\text{BESS}}^{\text{N}}\leq P_{\text{ BESS}}^{\text{c}}(t)\leq 0\\ u_{\text{BESS}}^{\text{c}}(t)+u_{\text{BESS}}^{\text{d}}(t)\leq 1\end{array} \right. \tag{4}\]  
where \(C_{\text{SOE}}(t)\) represents the SOE of the BESS at time \(t\). \(S_{\text{BESS}}^{\text{N}}\) denotes the rated capacity of the BESS, while \(\eta_{\text{d}}\) and \(\eta_{\text{c}}\) are the discharge and charge efficiency of the BESS, respectively. The variables \(u_{\text{BESS}}^{\text{d}}(t)\) and \(u_{\text{BESS}}^{\text{c}}(t)\) are the binary optimization variables. If \(u_{\text{BESS}}^{\text{d}}(t)=1\), it indicates that the BESS is discharging. Conversely, if \(u_{\text{BESS}}^{\text{e}}(t)=1\), it indicates that the BESS is charging. The initial state of energy is specified as \(C_{\text{SOE}}^{\text{init}}\), with the minimum and maximum capacity limits defined as the CD SOE and \(C_{\text{SOE}}^{\text{U}}\), respectively. Additionally, \(\Delta t\) represents the sampling time, and \(T\) indicates the ending time. 4. The operation constraints of the production, storage, and sale of hydrogen units are defined as Formulas (5)-(8). The equation relationship between hydrogen production power and volume is Formula (5) [29]. Then, Formula (6) outlines the production, storage, and sale relationships among AWE, PEME, and HST [31; 32]. The operation power range and climbing rate for AWE and PEME are illustrated in Formula (7) [28]. Finally, Formula (8) represents the range of hydrogen sales volume [32]. \[\left\{ \begin{array}{l}V_{\text{AWE}}(t)=\dfrac{1000P_{\text{AWE}}(t)\eta_{ \text{AWE}}}{\rho_{\text{H2}}\lambda_{\text{H2}}^{\text{AWE}}}\\ \\ V_{\text{PEME}}(t)=\dfrac{1000P_{\text{PEME}}(t)\eta_{\text{PEME}}}{\rho_{\text{ H2}}\lambda_{\text{H2}}^{\text{PEME}}}\end{array} \right.\] (5)  
\[\left\{ \begin{array}{l}C_{\text{SOH}}(t+\Delta t)=C_{\text{SOH}}(t)+(V_{ \text{AWE}}(t)+V_{\text{PEME}}(t)-V_{\text{SELL}}(t))\times\Delta t\times 100/V_{ \text{HESS}}^{\text{N}}\\ C_{\text{SOH}}(0)=C_{\text{SOH}}(T)=C_{\text{SOH}}^{\text{init}}\\ C_{\text{SOH}}^{\text{D}}\leq C_{\text{SOH}}(t)\leq C_{\text{SOH}}^{\text{U}} \end{array} \right. \tag{6}\]  
\[\left\{ \begin{array}{l}P_{\text{PEME}}^{\text{min}}\leq P_{\text{PEME}}(t) \leq P_{\text{PEME}}^{\text{max}}\leq P_{\text{PEME}}^{\text{N}}\\ P_{\text{AWE}}^{\text{min}}\leq P_{\text{AWE}}(t)\leq P_{\text{AWE}}^{\text{max }}\leq P_{\text{AWE}}^{\text{N}}\\ \left|P_{\text{PEME}}(t+\Delta t)-P_{\text{PEME}}(t)\right|\leq\alpha_{\text{ PEME}}\times P_{\text{PEME}}^{\text{N}}\\ \left|P_{\text{AWE}}(t+\Delta t)-P_{\text{AWE}}(t)\right|\leq\alpha_{\text{AWE}} \times P_{\text{AWE}}^{\text{N}}\end{array} \right. \tag{7}\]  
\[0\leq V_{\text{SELL}}(t)\leq V_{\text{SELL}}^{\text{max}} \tag{8}\]  
where \(C_{\text{SOH}}(t)\), \(V_{\text{SELL}}(t)\), \(V_{\text{AWE}}(t)\), and \(V_{\text{PEME}}(t)\) represent the SOH of the HST, the volume of hydrogen sold, and the volume of hydrogen produced by AWE and PEME at time \(t\), respectively. The hydrogen density (\(\rho_{\text{H2}}\)) is 0.9 kg/m\({}^{3}\); \(V_{\text{HESS}}^{\text{N}}\) is the rated capacity of the HST, and \(V_{\text{SELL}}^{\text{max}}\) is the maximum volume of hydrogen sold. The parameters \(\eta_{\text{AWE}}\), \(\lambda_{\text{H2}}^{\text{AWE}}\), \(\alpha_{\text{AWE}}\), \(P_{\text{AWE}}^{\text{min}}\), \(P_{\text{AWE}}^{\text{max}}\), and \(P_{\text{AWE}}^{\text{N}}\) represent the hydrogen production efficiency, electric hydrogen ratio, climbing rate, minimum power, maximum power, and rated power of AWE, respectively. Similarly, \(\eta_{\text{PEME}}\), \(\lambda_{\text{H2}}^{\text{PEME}}\), \(\alpha_{\text{PEME}}\), \(P_{\text{PEME}}^{\text{min}}\), \(P_{\text{PEME}}^{\text{max}}\), and \(P_{\text{PEME}}^{\text{N}}\) reflect the corresponding properties for PEME.