## 2 Analysis of the Operating Mechanism of a Hybrid Plant Station Primary Frequency Regulation
### Mathematical Model and Dynamic Characteristics Analysis of the Electric Hydrogen Production Device  
There are three main types of electrolysis cells, including alkaline electrolytic cells, which are suitable for large-scale engineering applications. Solid oxide electrolytic cells are still in the laboratory research stage, while proton exchange membrane electrolytic cells offer good technical indications but high pricing. Alkaline electrolytic cells have strong economics and a high level of technological maturity, making them appropriate for hybrid plant stations. The following is the mathematical model used to create an alkaline electrolytic cell [32]. The single alkaline electrolytic cell voltage expression is shown in Equation (8):  
\[U_{\text{cell}}=U_{\text{rev}}+\frac{r_{1}+r_{2}T_{\text{el}}}{A_{ \text{el}}}I_{\text{el}}+\left(s_{1}+s_{2}T_{\text{el}}+s_{3}T_{\text{el}}^{2} \right)\times\] \[\text{lg}\left(\frac{t_{1}+\frac{t_{2}}{A_{\text{el}}}-\frac{t_{ 3}}{A_{\text{el}}}}{A_{\text{el}}}I_{\text{el}}+1\right) \tag{8}\]  
In the equation, \(U_{\text{cell}}\) is the electrolytic cell voltage; \(I_{\text{el}}\) is the electrolytic cell current; \(U_{\text{rev}}\) is the reversible voltage; \(r_{1}\), \(r_{2}\) are the ohmic parameters of the electrolyte; \(s_{1}\), \(s_{2}\), \(s_{3}\), \(t_{1}\), \(t_{2}\), and \(t_{3}\) are the electrode overvoltage parameters; \(A_{\text{el}}\) is the electrode area; and \(T_{\text{el}}\) is the electrolyte temperature. The voltage of the electric hydrogen production device composed of the alkaline electrolyzer in series is shown in Equation (9):  
\[U_{\text{el}}=N_{\text{el}}U_{\text{cell}} \tag{9}\]  
In the equation, \(N_{\text{el}}\) is the electrolytic cell number in series, and \(U_{\text{el}}\) is the voltage value of the electric hydrogen production device. The operating state and dynamic properties of the electric hydrogen generation device govern how it responds to the frequency regulation power. The dynamic properties of the alkaline electrolytic cell are analyzed using appropriate parameters [33], and the analysis's findings are displayed in Table 1:  
Table 1 shows that although the alkaline electrolyzer can be swiftly started, it cannot be repeatedly started and stopped in a single day. The alkaline electrolyzer has a restricted operating range. The alkaline electrolyzer will shut off if the input power drops below 1/4 of the rated power, and its working characteristics will limit its capacity to climb power. Equation (10) illustrates the frequency regulation power command that the alkaline electrolyzer can reply to, taking into account its operational characteristics:  
\[\frac{1}{4}N_{\text{el}}P_{\text{el}\_\text{max}}\leq P_{\text{P2H}}\leq N_{ \text{el}}P_{\text{el}\_\text{max}} \tag{10}\]  
In the equation \(P_{\text{el}\_\text{max}}\) is the rated power of the electrolyzer. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Main Technical Indicators**} & **Indicators Value** \\ \hline \multirow{2}{*}{Start-stop} & Start-stop delay \(\alpha\)/h & 0 \\  & Start-stop capability \(\gamma_{\text{max}}/Z_{\text{max}}\) & \(1/1\) \\ \hline \multirow{2}{*}{Power} & Work scope [\(P_{\text{min}}\), \(P_{\text{max}}\)] & [25\%\(P_{\text{el}\_\text{max}}\), 100\%\(P_{\text{el}\_\text{max}}\)] \\  & Gradeability \(\Delta P_{\text{max}}\) & \(<\)(1/7200)\(P_{\text{el}\_\text{max}}\)/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main technical indexes of the alkaline electrolytic cell. Furthermore, within the restrictions of its dynamic properties, the alkaline electrolyzer's frequency regulation power must meet the following power climbing capacity requirements:  
\[|P_{\text{el,t}}-P_{\text{el,t-1}}|\leq\Delta P_{\text{max}} \tag{11}\]  
In Equation (11), \(P_{\text{el,t}}\) is the power command the electrolyzer needs to undertake at moment t; \(\Delta P_{\text{max}}\) is the power change threshold per unit time.