# The Development of an Optimal Control Strategy for a Series Hydraulic Hybrid Vehicle
## 3 Control System Development
### Dynamic Programming (DP) Optimal Control Strategy with DP Application
#### 3.3.2 The Backward Model for the SHHV  
For a given driving cycle, the power demand is a known priority. A DP algorithm is to find the optimal power-splitting factor that distributes the demand for power to the engine and the accumulator in order to minimize the total fuel consumption for the engine. To evaluate the fuel consumption of the engine for a given driving cycle using DP, a backward model of the SHHV is developed in this section. The diagram for the backward model of the proposed SHHV is shown in Figure 8.  
The demand for power to propel the vehicle is estimated using the following equation:  
\[P_{w}=\left(m\frac{dv}{dt}+mg\text{sin}\theta+mf_{r}g\text{cos}\theta+0.5\rho C _{d}A_{f}v^{2}\right)v \tag{6}\]  
The pump/motor power is determined by:  
\[P_{p/m}=\left\{ \begin{array}{ll}P_{w}/\eta_{p/m}&\text{if}\,P_{w}>0\,( \text{motor mode})\\ P_{w}\eta_{p/m}&\text{if}\,P_{w}<0\,(\text{pump mode})\end{array} \right. \tag{7}\]  
where \(\eta_{p/m}\) is the total efficiency of the pump/motor which is based on the Pourmovahed model [19]. During the propulsion phase, the requested power, \(Pp/m\), can be provided only by either the engine or the accumulator. For convenience, a power-splitting factor, \(u\in[-\infty,1]\), is used to illustrate. It is obvious that for a given power demand, the power-splitting factor is arbitrary within a range that depends on the engine power rate. The power flows and the relationship for the power-splitting factor is briefly described in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \(u\) & \(P_{acc}\) & \(P_{p}\) & **Description** \\ \hline
0 & 0 & \(P_{p/m}\) & Engine Propelling \\
1 & \(P_{p/m}\) & 0 & Accumulator Propelling \\
0\(<u<\)1 & Propelling & Propelling & Engine and Accumulator Propelling \\ \(u<0\) & Charging & Propelling & Engine Propelling and Accumulator Charging \\ \hline \end{tabular}
\end{table} 
Table 2: Power-splitting factor and power flows coordinating. Figure 8: Backward Model of the SHHV. Using power-splitting factor as a control input, the engine power is  
\[ \begin{array}{c}P_{p}=\left(1-u\right)P_{p/m}\\ P_{e}=P_{p}/\eta_{p}\end{array}  \tag{8}\]  
Using the experimental data for the engine, as shown in Figure 9, the engine speed and the engine torque are interpolated for a given power demand as follows:  
\[ \begin{array}{c}\omega_{e}=l_{2}\left(P_{e}-l_{1}\right)^{3}+l_{3}\left(P_{e }-l_{1}\right)^{2}+l_{4}P_{e}+l_{5}\\ T_{e}=\frac{\pi}{30}\frac{1000P_{e}}{\omega_{e}}\end{array}  \tag{9}\]  
where \(l_{1}=37.75\), \(l_{2}=0.0311\), \(l_{3}=0.044\), \(l_{4}=52.83\), and \(l_{5}=387.5\) are the fitting coefficients. At each particular operating point, the fuel consumption for the engine is approximately estimated by:  
\[\dot{m}_{f}=k_{00}+k_{10}\omega_{e}+k_{20}\omega_{e}^{2}+k_{11}\omega_{e}T_{e} +k_{01}T_{e}+k_{02}T_{e}^{2} \tag{10}\]  
Since DP is used to minimize the total fuel consumption over the duty cycle, the objective function is represented by:  
\[J_{f}=\int\limits_{0}^{T}\dot{m}_{f}\left(\omega_{e},T_{e}\right)dt \tag{11}\]  
where \(T\) is the length of the duty cycle's time. The power provided/absorbed by the accumulator is given by:  
\[P_{a}=uP_{p/m} \tag{12}\]  
In terms of the accumulator power, the dynamic for pressure in the accumulator is  
\[\dot{p}=\frac{1.4}{V_{a}}p_{pr}^{-1/1.4}p^{1/1.4}P_{a} \tag{13}\]  
The state-of-charge (SOC) of the accumulator and not the pressure represents the current available energy in a hydraulic accumulator. The SOC for an accumulator is defined by:  
\[SOC=\frac{p-p_{\min}}{p_{\max}-p_{\min}} \tag{14}\]  
Figure 9: Engine Torque and Engine Speed Estimation. where \(p\) is the current pressure, and \(p_{\min}\) and \(p_{\max}\) are the minimum and maximum working pressures for the accumulator. When the SOC = 0, there is no available energy in the accumulator and when the SOC = 1, the accumulator is fully charged.