## 3 Mathematical Model
### Electro-Hydraulic Actuator  
The electro-hydraulic actuator is a key part of the SCRS, which consists of a mechanical unloader, a hydraulic cylinder, and a high-speed solenoid valve. When the regulating system is working, the high-pressure hydraulic oil flows into and out of the hydraulic cylinder through the high-speed solenoid valve and controls the extension and retraction of the mechanical unloader. The dynamic response performance of the solenoid valve has a great influence on the system's performance. The dynamic performance of the solenoid valve mainly includes four parts: opening delay, opening moving time, closing delay, and closing moving time. If the excitation voltage is fixed, the supply hydraulic pressure has little effect on the opening delay and opening moving time, but it has a great influence on the closing delay and closing moving time [22]. Hence, the dynamic performance of the solenoid valve is determined and can be obtained by model calculation or experimental test, as mentioned in the literature [22]. In order to simplify the mathematical model of the electro-hydraulic actuator, the opening and closing processes of the solenoid valve are simplified. The opening delay, opening moving time, closing delay, and closing moving time are \(t_{1}\), \(t_{2}\), \(t_{3}\), and \(t_{4}\), respectively. The solenoid valve control port flow area can be expressed on the time axis of a single operation cycle T of the compressor as follows. \[A(t,t_{c})=\left\{ \begin{array}{ll}A_{h}&t_{on}\leq t\leq t_{c}+t_{off}\\ 0&\text{otherwise}\end{array} \right., \tag{8}\]  
Figure 4: Simplified physical model of gas pipeline. where \(t_{on}=t_{1}+t_{2}/2\) is the equivalent opening delay of the solenoid valve. Similarly, \(t_{off}=t_{3}+t_{4}/2\) represents the equivalent closing delay of the solenoid valve. \(t_{c}\) is the hold time of switching on the control signal output from the solenoid valve driver. The dynamic characteristic model of the electro-hydraulic actuator can be expressed as follows. \[\left\{ \begin{array}{l}Q_{h}(u)=C_{h}A_{h}(t,t_{c})\sqrt{\frac{2(P_{all}-P_{ A})}{P_{all}}+C_{h}\widetilde{A_{h}}(t,t_{c})\sqrt{\frac{2}{P_{all}}P_{A}}}\\ Q_{h}(u)=A_{A}\dot{x}+C_{l}P_{A}\\ F_{hu}=P_{A}A_{A}-K_{f}(x+x_{0})=m_{A}\ddot{x}+B\dot{x}\end{array} \right., \tag{9}\]  
where \(Q_{h}\) represents the hydraulic fluid flow of the control port of the high-speed solenoid valve, \(C_{h}\) represents the flow coefficient, \(P_{all}\) represents the supply pressure, \(P_{A}\) represents the pressure in the hydraulic cylinder, \(\rho_{all}\) represents the hydraulic oil density, \(A_{A}\) represents the piston area of the oil cylinder, \(x\) is the load displacement, \(C_{l}\) is the oil cylinder leakage coefficient, B represents the cylinder damping coefficient, \(x_{0}\) represents the pre-compression amount of the loaded spring, \(m_{A}\) represents the equivalent mass of the cylinder piston and load, and \(K_{f}\) represents the reset spring stiffness of the unloader, which is an extremely important parameter that has a great impact on the regulation effect, but it is also a component that easily degrades. Therefore, the influence of the reset spring stiffness is thoroughly analyzed in the following Section 4. At a certain time, the switch-off control signal output is sent from the solenoid valve driver. The hydraulic force is released, and the mechanical unloader is lifted up by the spring force. Then all the suction valves will close together with the mechanical unloader. The hold time \(t_{c}\) is determined by the compressor load ratio \(\eta\), as indicated below [10]. \[\eta V_{s}+V_{exp}+V_{de}=V_{de}+V_{str}\Big{(}1-cos\theta_{r}+\frac{\lambda}{ 2}sin^{2}\theta_{r}\Big{)}, \tag{10}\]  
\[t_{c}=30\theta_{r}/n\pi-t_{off}-t_{0}, \tag{11}\]  
where \(n\) represents compressor speed, \(\theta_{r}\) is the inlet valve plate closing angle, and \(t_{0}\) is a constant dependent on where the actuator is extended.