# 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
#### 3.3.3 The Optimization of Energy Management Using DP
To limit the number of calculations, the allowable state and the control values must be quantized as listed in \[J_{f}=\sum_{k=0}^{N-1}\Delta m_{f,k}\left(\omega_{e}\left(k\right),T_{e}\left( k\right)\right)dt \tag{17}\]5  
Step \(N-1\):  
\[J_{f}=\sum_{k=0}^{N-1}\Delta m_{f,k}\left(\omega_{e}\left(k\right),T_{e}\left( k\right)\right)dt \tag{17}\]6  
Step \(k\), for \(0\leq k\leq N-\) 1:  
\[J_{f}=\sum_{k=0}^{N-1}\Delta m_{f,k}\left(\omega_{e}\left(k\right),T_{e}\left( k\right)\right)dt \tag{17}\]7  
where \(\Delta m_{f,k}\left(x_{k},u_{k}\right)\) is the fuel consumption for the engine at time step \(k\). The simulation results for the use of DP for the discretized SHHV model in the Japan 1015 driving cycle are shown from Figure 10 to \[J_{f}=\sum_{k=0}^{N-1}\Delta m_{f,k}\left(\omega_{e}\left(k\right),T_{e}\left( k\right)\right)dt \tag{17}\]8 The lower part of Figure 10 shows the optimal SOC trajectories that correspond to different initial conditions of the accumulator. The constraint on the final value of the SOC ensures that there is no pre-charged energy in the accumulator during the test and that the engine does not provide any unused energy. For each initial condition of the accumulator, the system demonstrates different fuel consumption figures, as shown in \[J_{f}=\sum_{k=0}^{N-1}\Delta m_{f,k}\left(\omega_{e}\left(k\right),T_{e}\left( k\right)\right)dt \tag{17}\]9 When the initial value for the SOC is small, the assistance provided by the accumulator is small. In this case, the engine must operate at high rate of power that is not the optimal region. When the value for the SOC is too high, there is not enough space to store braking energy. A certain amount of braking energy cannot be recovered. The fuel consumption for the system is increased. The figure shows that the lowest fuel consumption for the system is 21.5 km/L if the initial value of the SOC is about 90%. The behavior of the SHHV system during the 480 to 660 s intervals for the Japan 1015 driving cycle is shown in \[P_{w}\left(k\right)=\left(m\frac{v\left(k+1\right)-v\left(k\right)}{T_{s}}+mg \text{sin}\theta+mf_{r}g\text{cos}\theta+0.5 pC_{d}A_{f}v^{2}\left(k\right) \right)v\left(k\right)\]0 It is seen that the vehicle follows the driving cycle with a reasonable deviation in speed. The negative portion of the pump/motor displacement factor indicates that the component is in the motor mode and the positive one indicates that the component is in the pump mode. The negative portion of the accumulator power profile indicates that the accumulator is charged and the positive portion of the accumulator power profile indicates that the accumulator delivers the power to propel the wheels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Description** & **Symbol** & **Operating Range** & **Number of Grid** \\ \hline State & Accumulator SOC & \(x\) & 0:0.025:1 & \(M=40\) \\ Control & Power-Splitting Factor & \(u\) & 0:0.02:1 & \(L=\)50 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: State and Control Input Discretization. Figure 11: Relationship between Fuel Economy and Initial SOC. Figure 12: System Behaviors during 480–660 s Intervals of Japan 1015 Driving Cycle. Figure 10: State-of-Charge (SOC) Trajectories with Different Initial Condition. Interestingly, when the SOC approaches zero, it means that the stored energy is fully used at each vehicle start and the accumulator has enough space for effective regenerative braking. All of the braking energy throughout the driving test schedule is captured and reused. Avoiding transient operation of the engine and taking into account the low energy density characteristic, the accumulator is only used to absorb excess engine power during the tracking phase, although there is no direct charging. Therefore, the power-splitting factor remains between zero and one.