## 3 Optimization-Based Strategies
### Global Information-Driven Optimization
#### 3.2.2 Deterministic Dynamic Programming (DDP)  
Applying deterministic dynamic programming (DDP) [78; 79] in the EMS of HEVs could greatly improve the fuel economy over the whole driving cycle, which provides a benchmark for assessing the optimality of other energy management strategies despite the off-line optimization. Assuming that the state equation of a nonlinear system is expressed as:  
\[\dot{x}(k+1)=f(x(k),u(k),k) \tag{13}\]  
where \(x\) is the state variables, \(u\) is the control variables, \(k=0,1,2,\ldots,N-1\), and \(N\) is the stage number of the whole driving cycle. Under the action of the control variables \(u\), the cost function can be expressed as:  
\[J=\sum_{k=0}^{N-1}U(x_{k},u_{k}) \tag{14}\]  
where \(U\) is the instantaneous cost at every moment. The core of dynamic programming is [PERSON]'s optimal principle [80], which can be solved by dealing with the optimization of a sequence of sub-problems, and is implemented backward from the terminal state to the initial state of the driving cycle by searching for the optimal trajectory among each state. Unlike to [PERSON]'s minimum principle, the principle of optimality provides sufficient conditions for optimality, which need to solve the nonlinear Hamilton-Jacobi-Bellman (HJB) equation. According to the principle of optimality, the basic recursive equation of dynamic programming can be obtained:  
\[J_{k}^{*}(x_{k})=\min_{u_{k}}\left[U(x_{k},u_{k})+J_{k+1}^{*}(x_{k+1})\right] \tag{15}\]  
where, \(J_{k}^{*}\) is the optimal cost-to-go function at state \(x_{k}\) in the \(kth(0\leq k\leq N-1)\) step, \(x_{k+1}\) is the next state, which is obtained under \(u_{k}\) being applied to the current state \(x_{k}\) at the \(kth\) step. The objectives of DP in HEVs are to find optimal control sequences to obtain the optimal SOC trajectory and minimize fuel consumption over a given driving schedule. That is, the optimal cost function of each step can be regarded as the minimum fuel consumption for that stage. Consequently, how to determine the minimum fuel consumption of each stage and then search for the optimal trajectory to obtain global optimal fuel economy are crucial issues for the DDP-based strategy. In addition, because that dynamic programming is a numerical algorithm, the state variables, control variables and continuous time should be discretized before formulating the DP, and the boundary issue of their feasible regions should be considered when implementing DP to solve the optimal control problem of HEVs. In addition, the physical constraints on the states and the inputs should be considered to ensure safe operation of the components (such as the engine, motors and battery). Generally, the state variables usually include the vehicle speed \(v\) and SOC, and the control variables could be selected from among engine torque, engine speed, motor torque, engine speed, and power-split ratio. Based on the Rint model, the battery SOC of the battery can be formulated as [20]:  
\[SOC_{k+1}=SOC_{k}-\frac{U_{OC}-\sqrt{U_{OC}^{2}-4R_{int}\cdot P_{ batt}}}{2R_{int}\cdot C} \tag{16}\]where \(SOC_{k+1}\) is the battery SOC at the \((k+1)\)_th_ step. \(U_{OC}\) is the open-circuit voltage, which is related to the SoC and the battery temperature \(T_{\text{{htt}}}\) and can be obtained from the interpolation function \(U_{OC}=f(SOC,T_{\text{{htt}}})\). \(R_{\text{{htt}}}\) is the internal resistance of the battery, which is related to SOC, \(T_{\text{{htt}}},I\) and can be obtained from the interpolation function \(R_{\text{{int}}}=f(\text{SOC},T_{\text{{htt}}},I)\), \(P_{\text{{htt}}}\) is the electric power, and \(C\) is the nominal capacity of the battery, which is a fixed value. Therefore, the main steps of DDP-based strategy can be concluded as follows, and the main technical routes are shown in Figure 7:  
1. [label=Step 0]
2. Discretize the relevant variables and determine the boundaries of the feasible regions.