# Overflow Energy Loss Recovery System Based on Hydraulic Motor-Electric Generator
## 3 Performance Analysis
### Mathematical Model of Traditional Proportional Relief Valve  
For the convenience of analysis, the structure schematic diagram of traditional relief valve can be simplified as shown in Figure 4.  
When the relief valve stops working, the main valve core and the pilot valve core are closed. The hydraulic oil from the pump reaches the front cavity 1 of the main valve core, enters the spring chamber 5 and the front chamber 6 of the pilot valve core through the pilot oil circuit and fixed damping hole R. when the hydraulic pressure acting on the pilot valve core is not enough to overcome the electromagnet output force, the pilot valve core 7 is closed, and the main valve core 2 is also closed. With the continuous oil supply of the hydraulic pump, the oil pressure rises. When the pilot valve core 7 is opened, the pilot oil will drain back to the oil tank separately. The oil in the pilot oil circuit flows, and the pressure drop in the spring chamber is less than the inlet pressure due to the pressure drop through the fixed damping hole R. At this time, the resultant force on the main valve core is shown in Equation (3). The resultant force is greater than zero, and the main valve core is opened and the main valve port overflows.  
\[\sum F=p_{1}A_{1}-p_{x}A_{1}-F_{0} \tag{3}\]  
Figure 4: Simplified structure diagram of traditional proportional relief valve. 1—front chamber of main valve core, 2—main valve core, 3—fixed damping, 4—spring cavity, 5—return spring, 6—front chamber of pilot valve core, 7—pilot valve core, 8—proportional electromagnet.  
Dynamic force balance equation of pilot valve core can be described as  
\[A_{0}p_{x}-F_{EM}-F_{fy}-F_{fay}=m\frac{dy^{2}}{dt}+D_{y}\frac{dy}{dt}+K_{sy}(y_{ 0}+y) \tag{4}\]  
\[F_{fy}=\pi{\alpha_{D}}^{2}yp_{x}d\sin 2\beta_{1} \tag{5}\]  
Flow continuity equation of pilot valve port can be given as  
\[Q_{y}-Q_{R}=A_{1}\frac{dx}{dt}+\frac{V_{x}}{E}\frac{dp_{x}}{dt} \tag{6}\]  
\[Q_{y}=\alpha_{D}\pi dy\sin\beta\sqrt{\frac{2}{\rho}p_{x}} \tag{7}\]  
\[Q_{R}=\alpha_{D}\frac{\pi{d_{R}}^{2}}{4}\sqrt{\frac{2}{\rho}(p_{1}-p_{x})} \tag{8}\]  
Dynamic force balance equation of main valve core can be expressed as  
\[A_{1}P_{1}-A_{1}P_{x}=M\frac{dx^{2}}{dt}+D_{x}\frac{dx}{dt}+K_{sx}(x_{0}+x)+F_ {fx}+F_{fax} \tag{9}\]  
\[F_{fx}=\pi{\alpha_{D}}^{2}Dxp_{1}\sin 2\beta_{2} \tag{10}\]  
Flow continuity equation of main valve port can be given as  
\[Q_{1}-Q_{R}-Q_{2}=A_{1}\frac{dx}{dt}+\frac{V_{1}}{E}\frac{dp_{1}}{dt} \tag{11}\]  
\[Q_{2}=\alpha_{D}A(x)\sqrt{\frac{2}{\rho}(p_{1}-p_{2})} \tag{12}\]  
The Laplace transformation is taken for Equations (4)-(12). The transfer function can be deduced as  
\[(A_{0}-K_{fy})P_{x}(s)-F_{EM}(s)=(ms^{2}+D_{y}s+K_{sy}+K_{fy})Y(s) \tag{13}\]  
\[K_{R}P_{1}(s)-K_{qy}Y(s)+A_{1}sX(s)=\biggl{(}-\frac{V_{x}}{E}s+K_{R}K_{qp} \biggr{)}P_{x}(s) \tag{14}\]  
\[\bigl{(}A_{1}-K_{Fy}\bigr{)}P_{1}(s)-A_{1}P_{x}(s)=\Bigl{(}M{\rm s}^{2}+D_{x} s+K_{sx}+K_{Fx}\Bigr{)}X(s) \tag{15}\]  
\[Q_{1}(s)+K_{R}P_{x}(s)-\bigl{(}K_{Qx}+A_{1}s\bigr{)}X(s)=\biggl{(}K_{R}+K_{Qp }+\frac{V_{1}}{E}s\biggr{)}P_{1}(s) \tag{16}\]  
For the convenience of simplification, the sign definition can be given as  
\[G_{1}(s)=\frac{1}{ms^{2}+D_{y}s+K_{sy}+K_{fy}} \tag{17}\]  
\[G_{2}(s)=\frac{1}{-\frac{V_{x}}{E}s+K_{R}K_{qp}} \tag{18}\]  
\[G_{3}(s)=\frac{1}{M{\rm s}^{2}+D_{x}s+K_{sx}+K_{Fx}} \tag{19}\]  
\[G_{4}(s)=K_{Qx}+A_{1}s \tag{20}\]\[G_{5}(s)=\frac{1}{K_{R}+K_{Qp}+\frac{V_{1}}{L}s} \tag{21}\]  
\[H_{1}(s)=A_{1}-K_{fp} \tag{22}\]  
\[H_{2}(s)=A_{0}-K_{fp} \tag{23}\]  
In the steady-state working process of traditional proportional relief valve, the output force of electromagnet \(F_{EM}\)(s) is the input signal, and the inlet pressure \(P_{1}\)(s) is the output signal. The system is a constant flow system. The inlet flow \(Q_{1}\)(s) is basically unchanged. The influence on the system is small. The interference term \(Q_{1}\)(s) can be ignored. As shown in Figure 5, the block diagram of traditional proportional relief valve system can be obtained by combining each link of Equations (13)-(16).