# Control Strategies for Piston Trajectory in Ionic Compressors for Hydrogen Storage
## 2 Methodology and Numerical Model
### Numerical Model  
The entire numerical simulation model can be divided into a hydraulic drive system and a hydrogen compression system in which the key dimensions of components such as cylinders, pistons, hydraulic cylinders, valves, hydraulic valves, and pumps are the basis \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline Intake gas pressure (MPa) & 12 \\ Discharge gas Pressure (MPa) & 45 \\ Operating frequency (Hz) & 5 \\ Piston velocity equation & \(v=-\frac{\dot{s}}{2}2\pi fsin(2\pi ft)\) \\ Flow rate (\(\text{Nm}^{3}/\text{h}\)) & 200 \\ Temperature of intake gas (\({}^{\circ}\)C) & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design conditions for the 45 MPa ionic compressor. Figure 3: Different control strategies for the trajectory of the free piston in the compressor. of the numerical simulation. The numerical model of the hydraulically driven hydrogen compressor system utilizing the Dual-PS control method built in AMSEim is provided in Figure 4. The hydrogen compression subsystem was numerically modeled using components in the pneumatic library, pneumatic component design library and signal control library. It predominantly comprises pre-defined components including cylinders, pistons, pressure gauges, valves, gas sources, and signal sources. The van der Waals equation of state for the submodel in the pneumatic library in AMSEim was used in the simulations to account for variations in behavior due to the non-ideal nature of the hydrogen gas. The hydrogen compression subsystem contains a cylinder, piston, valve, and pressure gauge. In the simulation, the pressure gauge measures the cylinder's internal pressure. If the pressure inside the cylinder drops below 12 MPa, the inlet valve receives an open signal, and hydrogen gas is introduced into the cylinder. Conversely, when the pressure inside the cylinder exceeds 45 MPa, the exhaust valve is signaled to open, and the compressed gas is released from the cylinder. The key geometrical dimensions of the hydrogen compression system are the diameter of the compression cavity \(D\), the stroke \(s\), and the valve area for fluid \(A_{e}\), which can be determined by Equations (1)-(3), respectively. \[D=\sqrt[3]{\frac{4V_{st}}{60\pi rf}} \tag{1}\]  
Figure 4: Simulation model of the hydraulically driven hydrogen compressor system with the Dual-PS control method. \[s=\frac{D}{r} \tag{2}\]  
\[A_{e}=\frac{A_{p}}{Nv_{vm}}v_{m} \tag{3}\]  
The stroke volume \(V_{st}\) is calculated using Equation (4), while the average flow rate \(v_{vm}\) can be determined using Equation (5). \[V_{st}=\frac{F_{in}}{\lambda_{d}} \tag{4}\]  
\[v_{vm}=Ma \tag{5}\]  
where \(F_{in}\) is the hydrogen inlet flow rate, determined using Equation (6); \(\lambda_{d}\) is the delivery coefficient, which can be obtained using Equation (7); \(M\) is the average Mach number of the valve gap; and \(a\) is the speed of sound. \[F_{in}=\frac{F_{0}}{60}\times\frac{p_{0}}{p_{in}}\times\frac{T_{in}}{T_{0}} \tag{6}\]  
Figure 3:0  
Figure 3:1  
where \(p\) is the pressure; \(T\) is the temperature; 0 and in denote standard and inlet conditions, respectively; \(\lambda_{V}\) is the volume coefficient; \(\lambda_{p}\), \(\lambda_{T}\), and \(\lambda_{l}\) are the pressure, temperature, and leakage coefficients, respectively; \(r_{CV}\) is the ratio of clearance volume; \(\varepsilon_{c}\) is compression ratio; and \(k_{c}\) is the process coefficient of expansion. According to the design specifications of the ionic compressor, the necessary dimensions for the hydrogen compression model can be calculated using the equations provided above.