## 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
The initial, minimum, and maximum capacity values of the HST are represented by \(C_{\text{SOH}}^{\text{init}}\), \(C_{\text{SOH}}^{\text{D}}\), and \(C_{\text{SOH}}^{\text{U}}\), respectively. 5. The operation constraints of the load unit are presented in Formula (9). \(P_{\text{Load}}(t)\) and \(P_{\text{Load}}^{\text{ref}}(t)\) refer to the load's actual power and demand power, respectively. \(\delta_{2}\) is the cut-off rate of the load. \[\left\{ \begin{array}{l}0\leq P_{\text{Load}}(t)\leq P_{\text{Load}}^{\text{ref }}(t)\\ \sum_{t=0}^{T}\left(\left(P_{\text{Load}}^{\text{ref}}(t)-P_{\text{Load}}(t) \right)/P_{\text{Load}}^{\text{ref}}(t)\right)\leq\delta_{2}\end{array} \right.\] (9)(2) The optimization objective function  
This study aimed to analyze the characteristics of a hybrid electrolyzer that combined AWE and PEME. To achieve this, two objective functions for optimization were established, focusing on the economic aspects and the utilization of AWE and PEME. This resulted in a multi-objective optimization problem structured as follows. 1. This study examined the economic aspects of hydrogen production by analyzing the initial investment cost, hydrogen production cost, and sales revenue associated with AWE and PEME. The costs per kilowatt-hour (kWh) of the BESS, PVGs, and WTGs were considered. Subsequently, an objective function \(J_{1}\) was established to represent the economic viability of hydrogen production using a hybrid electrolyzer, as detailed in Formula (10) [25]. The investment cost is directly related to the rated power of the hybrid electrolyzer. In contrast, the hydrogen production cost and sales revenue depend on the actual output power of the electrolyzer, which is constrained by its rated capacity. \[ \begin{array}{l}J_{1}=c_{\text{PV}}\sum\limits_{t=0}^{T}P_{\text{PV}}(t)+c _{\text{WIND}}\sum\limits_{t=0}^{T}P_{\text{WIND}}(t)+c_{\text{BESS}}\sum \limits_{t=0}^{T}(P_{\text{BESS}}^{\text{d}}(t)-P_{\text{BESS}}^{\text{e}}(t))+ \ldots\\ \min\limits_{i\in\{\text{AWE,PEME}\}}\sum\limits_{t=0}^{T}\left(c_{i}^{\text{ mix}}\frac{1000P_{i}^{\text{N}}(t+\gamma)^{N_{i}}}{36S\left((1+\gamma)^{N_{i}}-1 \right)}+c_{i}^{\text{pro}}\frac{10000\bar{t}(t)\eta_{i}}{\lambda_{\text{H}_{ 2}}^{\text{H}}}\right)-\sum\limits_{t=1}^{T}(V_{\text{SELL}}(t)\rho_{\text{H }_{2}\text{SELL}})\end{array} \] (10)
2. This study analyzed the energy loss and idle conditions associated with hydrogen production in a hybrid electrolyzer, focusing on its utilization efficiency. Then, another objective function, \(J_{2}\), was designed, as detailed in Formula (11), where \(P_{i}(t)\left(1-\eta_{i}\right)\) and \((P_{i}^{\text{N}}-P_{i}(t))\) represent the energy loss and the idle power of hydrogen production using AWE or PEME at time \(t\), respectively. \[ \begin{array}{l}\min\limits_{j}=\sum\limits_{i}\sum\limits_{t=0}^{T}\left(P_ {i}(t)(1-\eta_{i})+\left(P_{i}^{\text{N}}-P_{i}(t)\right)\right)\quad i\in\{ \text{AWE,PEME}\}\end{array} \] (11) where \(P_{i}(t)\) and \(V_{\text{SELL}}(t)\) represent the hydrogen production power of AWE or PEME and the volume of hydrogen sold at time \(t\); \(P_{i}^{\text{N}}\), \(N_{i}\), \(\lambda_{\text{H}2}^{i}\), and \(\eta_{i}\) are the rated power, operation life, electric hydrogen ratio, and hydrogen production efficiency of AWE or PEME, respectively; \(c_{\text{PV}}\), \(c_{\text{WIND}}\), and \(c_{\text{BESS}}\) are the cost per kW/h of the PVGs, WTGs, and BESS; \(c_{i}^{\text{inv}}\), \(c_{i}^{\text{Pro}}\), and \(c_{\text{SELL}}\) are the investment cost coefficient, hydrogen production cost of AWE or PEME, and the hydrogen sales revenue, respectively; \(\gamma\) is the discount rate.