# Energy Regeneration Hydraulic System via a Relief Valve with Energy Regeneration Unit
## 3 Control System
### Working Pressure of the HA  
The HA is the main component of the HERU and must be properly designed to achieve the two goals of the control system. The key parameters of the HA include the rated volume, the pre-charge pressure, as well as the minimum and maximum working pressure. The appropriate parameters of the HA are beneficial in controlling how much energy can be regenerated. For example, if the working pressure of the HA is high, which means that the outlet pressure of the PRV is high, the pressure drop of PRV is consequently small and the energy lost in the orifice is low. However, the amount of oil that the HA can store is less. In comparison, if the working pressure of the HA is low, which means that the outlet pressure of the PRV is low, the energy lost in the orifice of the PRV is subsequently high. However, the amount of oil that the HA can store is more. The HA is ideal for those confronted with frequent and short start-stop cycles in adequate space. The key advantage of using HA as energy regeneration components in HERU is its seamless interface. They can be easily integrated into a hydraulic circuit of a HE. However, the major disadvantage of a HA is that the energy storage density is severely limited compared to other competing technologies, such as the battery. Therefore, the major objective of working pressure optimization for the HA is to improve the energy density of the HA. According to the [PERSON]'s law, the gas in the accumulator follows the ideal gas law as  
\[p_{a0}V_{a0}{}^{n}=p_{a1}V_{a1}{}^{n}=p_{a2}V_{a2}{}^{n}=\mathrm{C} \tag{6}\]where \(p_{a0}\) is the pre-charge pressure of HA; \(V_{a0}\) is the rated volume of HA; \(V_{a1}\) and \(p_{a1}\) are the gas volume and pressure when the HA is at the minimum working pressure, respectively; \(V_{a2}\) and \(p_{a2}\) are the gas volume and pressure when the HA is at the maximum working pressure, respectively; \(n\) is the polytrophic exponent; and C is a constant. The HA used in the HE mainly functions during the starting and braking process, which lasts only a short period of time, while the working cycle of the HE is about 20 s. Following this, the compression process when the HA is charged can be considered as the adiabatic process. The gas index \(n\) is set to 1.4. The volume of the HA is calculated by the equation  
\[V_{a0}=\Delta V_{a1}\frac{\left(\frac{p_{a1}}{p_{a0}}\right)^{\frac{1}{n}}}{1- \left(\frac{p_{a1}}{p_{a2}}\right)^{\frac{1}{n}}} \tag{7}\]  
where \(\Delta V_{a1}\) is the maximum oil volume that is stored in HA when the pressure in HA increases from \(p_{a1}\) to \(p_{a2}\). The energy density \(E_{\rho}\) of the HA can be calculated as  
\[E_{\rho}=p_{2}\frac{r^{\frac{1}{n}}-r}{1-n} \tag{8}\]  
where  
\[r=\frac{p_{a1}}{p_{a2}} \tag{9}\]  
As seen from the above equations, increasing the volume of the HA, improving the pre-charge pressure and the maximum working pressure or reducing the minimum working pressure can improve the energy storage. The volume of the HA is constrained by the installation space of the HE. Furthermore, the pre-charge pressure, minimum and maximum pressure all are related to one another In order to obtain the relationships, the derivative of Equation (8) is used as  
\[\frac{d(E_{\rho})}{dr}=0 \tag{10}\]  
Accordingly, the maximum energy density based on the optimum pressure ratio can be represented in the form  
\[r=n^{\frac{n}{1-n}} \tag{11}\]  
On the other hand, the HA works frequently in the process of charging and releasing. To prolong the service life of the HA, the peak volume change of the air bladder of the HA cannot be too large. Hence, the pressures should satisfy  
\[0.25p_{a2}<p_{a0}<0.9p_{a1} \tag{12}\]  
For example, if the rated pressure of the PRV is 21.5 MPa, the pressure drop of the PRV should be at least 1 MPa, which is the same as the pressure drop in the throttle control valve. This is needed to guarantee the fact that the PRV can release the redundant flow of the hydraulic system.