# Investigation of Self-Driven Profiler with Buoyancy Adjusting System towards Ocean Thermal Energy
## 4 Mathematical Model of the Flow Control System
### State Equation Construction  
According to the logical relationship between the variables of the bonding graph, the state space equations can be derived and are shown in Formulas (5)-(11).  
\[\dot{V}_{2}=-\frac{1}{R_{val\_1}C_{acc}}V_{2}+\frac{1}{R_{val\_1}C_{clo}}V_{7} \tag{5}\]  
\[\dot{V}_{7}=\frac{1}{R_{val\_1}C_{acc}}V_{2}-\left(\frac{1}{R_{val\_1}}+\frac{ 1}{R_{val\_2}}\right)\frac{1}{C_{clo}}V_{7}+\frac{1}{R_{val\_2}C_{cli}}V_{21}- \frac{A_{clo}}{I_{clo}}P_{12} \tag{6}\]  
\[\dot{V}_{21}=\frac{1}{R_{val\_2}C_{clo}}V_{7}-\frac{1}{R_{val\_2}C_{cli}}V_{21 }-\frac{A_{cli}}{I_{cli}}P_{28} \tag{7}\]  
\[\dot{P}_{12}=\frac{A_{clo}}{C_{clo}}V_{7}-\frac{1}{C_{clo\_s}}X_{15}-A_{clo}P_ {out}-S_{clo\_s}-F_{clo} \tag{8}\]  
\[\dot{P}_{28}=\frac{A_{cli}}{C_{cli}}V_{21}-\frac{1}{C_{cli\_s}}X_{27}-S_{cli\_ s}-F_{cli} \tag{9}\]  
\[\dot{X}_{15}=\frac{P_{12}}{I_{clo}} \tag{10}\]  
\[\dot{X}_{27}=\frac{P_{28}}{I_{cli}} \tag{11}\]  
In the formulas: \(V_{2}\) is the fluid volume in accumulator, \(V_{7}\) is the increased fluid volume of external cylinder, \(V_{21}\) is the increased fluid volume of internal cylinder, \(P_{12}\) is the momentum of external cylinder piston, \(P_{28}\) is the momentum of internal cylinder piston, \(X_{15}\) is the external cylinder piston area, \(X_{27}\) is the internal cylinder piston area. For integral calculation, the above formula can be expressed in the form of state equations, as indicated by Equation (12).  
\[\dot{X}=AX+b \tag{12}\]  
pressure decreases and the outer cylinder pressure increases, so the flow rate of liquid decreases gradually after the valve has opened. Figure 4c shows that the external cylinder will endure a large pressure shock the moment the solenoid valve is opened. The larger the hydraulic resistance of the valve, the greater the obstruction to the fluid's entry. Thus, the pressure increases more slowly for the outer cylinder if larger hydraulic resistance is used. The details of the pressure shock are shown in Figure 4d. The initial pressure shock frequency is about 2045 Hz, and the oscillation amplitude gradually decreased over time. According to the formula of the natural frequency of the hydraulic cylinder \(f=\frac{1}{2\pi}\sqrt{\frac{K\Lambda_{\text{dij}}}{\text{T-L}_{\text{lo}}}}\), \(f=2046.01\) Hz when the length of the initial liquid column \(l=19\) mm, and it is consistent with the simulation results.  
When the profiler is diving, solenoid valve 1 closes and solenoid valve 2 opens. The oil flows from the external cylinder to solenoid valve 2 and then the internal cylinder. The integral calculation is carried out by the Runge-Kutta method, and the dynamic characteristic curve of this process is shown in Figure 5. As shown in Figure 5a-c, reducing the valve-port fluid resistance leads to increasing the speed of the external cylinder piston and the internal cylinder piston, and leads to the external cylinder pressure changes speed faster, which means that the flow speed of the liquid increases. As shown in Figure 5a,d, pressure impact is produced at the moment of opening the solenoid valve, and the pressure impact is relatively small and will not cause damage to the equipment.  
Figure 5: Hydraulic system status curve during submersible control.