## 3 Control Strategy
### Control Strategy  
In the PERS, there is a certain proportional relationship between the control signal of the hydraulic motor and the descending velocity of the lifting cylinder. Therefore, the generator is controlled by the speed mode to improve the control performance when the lifting cylinder descends. The potential energy regeneration control strategy consists of stop, single HMG and double HMGs, respectively, shown in Figure 6. Point A is the minimum signal of handle corresponding to the switch from the stop mode to the single HMG mode and the minimum recovery speed of 1#HMG. Point B is the handle signal corresponding to the switch from the single HMG mode to the double HMG mode and the speed of 1#HMG. Point C is the handle signal corresponding to the switch from the single HMG mode to the double HMG mode and the minimum recovery speed of 2 #HMG. Point D is the handle signal corresponding to the double HMG mode and the maximum recovery speed of 1#HMG and 2 #HMG. Suppose the coordinates of the four points of A, B, C and D are A = (\(Y_{\rm{pA}}\), \(n_{\rm{min}}\)), B = (\(Y_{\rm{pB}}\), \(n_{\rm{max}}\)), C = (\(Y_{\rm{pC}}\), \(0.5n_{\rm{max}}\)) and D = (\(Y_{\rm{pD}}\), \(n_{\rm{max}}\)), respectively. Where, \(Y_{\rm{pA}}\), \(Y_{\rm{pB}}\), \(Y_{\rm{pC}}\) and \(Y_{\rm{pD}}\) are the handle signals of points of A, B, C and D, respectively. \(n_{\rm{min}}\) is the minimum speed when using HMG unit for single HMG mode. \(n_{\rm{max}}\) is the maximum efficient speed of 1#HMG or 2 #HMG for the double HMGs mode. To ensure the linear relationship between the descending velocity of the lifting cylinder and the handle signal and no abrupt changing of the boom velocity, the increment of the sum of the target speed is positively correlated with the increment of the handle signal no matter whether it is in the single HMG mode or in the double HMGs mode. That is to say, the curves are proportional. \[k_{\rm{t}}=k_{1}+k_{2} \tag{1}\]  
where, \(k_{\rm{t}}\) is the proportional relationship between the total target speed and the handle signal. \(k_{1}\) is the proportional relationship between the target speed controlled by 1#HMG and the handle signal. \(k_{2}\) is the proportional relationship between the target speed controlled by 2#HMG and the handle signal. Figure 6: Relation diagram between the motor speed and handle opening. In the stage of \"Stop\", namely, \(Y_{\rm P}\in(0,Y_{\rm pA})\), both 1#HMG and 2#HMG do not work, namely  
\[n_{1}=n_{2}=0 \tag{2}\]  
In the stage of \"single HMG\", namely, \(Y_{\rm P}\in(Y_{\rm pA},\,Y_{\rm pB})\), the relationship between speed of 1#HMG and 2#HMG and the handle signal can be given by Equation (3) or Equation (4) according to the temperature of generators. Additionally, 1#HMG takes precedence over 2#HMG. Hence, if the temperature of the generator 1 is below the maximum allowable operating temperature, 1#HMG works and 2#HMG does not work. Otherwise, 2#HMG works and 1#HMG does not work. \[\left\{ \begin{array}{c}n_{1}=k_{\rm AB}(Y_{\rm P}-Y_{\rm pA})+n_{\rm min}\\ n_{2}=0\end{array} \right. \tag{3}\]  
\[\left\{ \begin{array}{c}n_{1}=0\\ n_{2}=k_{\rm AB}(Y_{\rm P}-Y_{\rm pA})+n_{\rm min}\end{array} \right. \tag{4}\]  
where, \(k_{\rm AB}\) is the line slope from point A to point B. \(n_{\rm min}\) is the minimum working speed of 1#HMG and 2#HMG. \(n_{\rm min}\) can be deduced by the efficiency map of the hydraulic motor and generator. In the stage of \"double HMGs\", namely, \(Y_{\rm P}\in(Y_{\rm pB},\,Y_{\rm max})\), 1#HMG and 2#HMG are exactly the same. Hence, the speed of the 1#HMG and 2#HMG is the same. Additionally, the relationship between the speed of 1#HMG and 2#HMG and the handle signal can be calculated as:  
\[n_{1}=n_{2}=k_{\rm CD}(Y_{\rm P}-Y_{\rm pC})+0.5n_{\rm max} \tag{5}\]  
where, \(k_{\rm CD}\) is the line slope from point C to point D. \(n_{\rm max}\) is the maximum high efficient speed of 1#HMG and 2#HMG. \(n_{\rm max}\) can also be deduced by the efficiency map of the hydraulic motor and generator.