## 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  
Section 2.1 presents the structure of the wind-photovoltaic-battery system combined with a hybrid electrolyzer, specifically under stand-alone conditions. A mathematical model of a stand-alone wind-photovoltaic-battery system with the hybrid electrolyzer was established. Many operation constraints and two optimization objective functions focused on enhancing both the efficiency and the economic feasibility of the system. (1) The operation constraints  
In the operation of a stand-alone wind-photovoltaic-battery system with a hybrid electrolyzer, several equations and inequalities govern the system's performance. The operational framework incorporates several crucial elements: real-time power and energy equilibrium, the PVG and WTG outputs, and BESS dynamics (including power output and state of energy) along with the hydrogen production, storage, and distribution constraints. This includes the power from AWE and PEME, the state of hydrogen (SOH) in the hydrogen storage tank (HST), and the rate of hydrogen sales. 1. The constraints of the real-time power balance of the system are expressed in Formula (1) and indicates the current balance of the system's operating power in real-time. The system variables are defined as follows [31]: \(P_{\text{PV}}(t)\)--photovoltaic generation output; \(P_{\text{WIND}}(t)\)--wind turbine generation output; \(P_{\text{BESS}}^{\text{d}}(t)\)--battery energy storage system discharge power; \(P_{\text{BESS}}^{\text{c}}(t)\)--battery energy storage system charge power; \(P_{\text{Load}}(t)\)--active load demand; \(P_{\text{AWE}}(t)\)--alkaline water electrolyzer hydrogen production power; \(P_{\text{PEME}}(t)\)--proton exchange membrane electrolyzer power. \(P_{\text{BESS}}^{\text{c}}(t)\) is negative, and the others are positive. \[P_{\text{PV}}(t)+P_{\text{WIND}}(t)+P_{\text{BESS}}^{\text{d}}(t)+P_{\text{ BESS}}^{\text{c}}(t)=P_{\text{Load}}(t)+P_{\text{PEME}}(t)+P_{\text{AWE}}(t)\] (1)
2. Formula (2) outlines the operation constraints of the PVGs and WTGs [25]. The power generation capacity of the renewable energy system is subject to operational constraints, where \(P_{\text{PV}}(t)\) and \(P_{\text{PV}}^{\text{max}}(t)\) indicate the instantaneous output power and maximum available power from the PVGs, respectively, and \(P_{\text{WIND}}(t)\) and \(P_{\text{WIND}}^{\text{max}}(t)\) correspondingly represent these parameters for the WTGs. \(P_{\text{PV}}^{\text{N}}\) and \(P_{\text{WIND}}^{\text{N}}\) refer to the rated power of the PVGs and WTGs, respectively. Additionally, \(\delta_{1}\) represents the abandonment rate of the new energy source. \[\left\{ \begin{array}{l}0\leq P_{\text{PV}}(t)\leq P_{\text{PV}}^{\text{max }}(t)\leq P_{\text{PV}}^{\text{N}}\\ 0\leq P_{\text{WIND}}(t)\leq P_{\text{WIND}}^{\text{max}}(t)\leq P_{\text{WIND }}^{\text{N}}\\ \sum_{t=0}^{T}\big{(}(P_{\text{PV}}^{\text{max}}(t)-P_{\text{PV}}(t))/P_{\text {PV}}^{\text{max}}(t)+\big{(}P_{\text{WIND}}^{\text{max}}(t)-P_{\text{WIND}}(t )\big{)}/P_{\text{WIND}}^{\text{max}}(t)\big{)}\leq\delta_{1}\end{array} \right.\] (2)
3. In the operation constraints of the BESS, there are specific relationships between the charging/discharging power and SOE of the BESS [31; 32]. It outlines potential limitations regarding the BESS's operating power and stored energy capacity. Formula (3) shows the initial, minimum, and maximum values of the SOE. Then, Formula (4) describes the BESS's charging and discharging power range. \[\left\{ \begin{array}{l}C_{\text{SOE}}(t+\Delta t)=C_{\text{SOE}}(t)-\big{(} P_{\text{BESS}}^{\text{d}}(t)/\eta_{\text{d}}+P_{\text{BESS}}^{\text{c}}(t) \times\eta_{\text{c}}\big{)}\times\Delta t\times 100/S_{\text{BESS}}^{\text{N}}\\ C_{\text{SOE}}(0)=C_{\text{SOE}}(T)=C_{\text{SOE}}^{\text{init}}\\ C_{\text{SOE}}^{\text{D}}\leq C_{\text{SOE}}(t)\leq C_{\text{SOE}}^{\text{U}} \end{array} \right.\] (3)\[\left\{