## 3 Optimization-Based Strategies
### Global Information-Driven Optimization
#### 3.2.2 Deterministic Dynamic Programming (DDP)
The shape of the SOC feasible domain is determined by the highest SOC, the lowest SOC, initial SOC, terminal SOC and the maximum charge/discharge current limitation. 3. Determine the SOC value between each adjacent grid point. According to the start time of SOC rising/falling and the terminal time of SOC rising/falling, the feasible region of SOC would be divided into several districts, and then the highest and lowest SOC at each moment would be determined. Limited by the maximum SOC discrete interval, the number of grid points (i.e., discrete points) at each moment can be calculated in order to obtain the SOC value of each grid points. 4. Determine the possible operation modes according to the variation of vehicle speed, vehicle acceleration and battery SOC. In general, the operation modes roughly include motor-only mode, engine-only mode, hybrid mode (motor-assist), regenerative braking mode, mechanical braking mode, recharging mode, and stop mode. 5. Choose the operation mode with the minimum fuel consumption, and the fuel consumption will be stored in the form of a matrix. Moreover, using a three-dimensional matrix to record fuel consumption between each moment contributes to improving the computational efficiency. 6. Search for all the optimal SOC trajectories among the discretized grid points using the improved Dijkstra algorithm based on the fuel matrix. Figure 7: Depiction of the main technical areas of the DDP-based strategy. * [100]
* Obtain the optimal control sequences of the engine and motors and the optimal SOC trajectory domain. In general, DP is employed to locate the optimal actions for the engine at each stage by minimizing the fuel consumption cost function over a certain driving cycle in HEVs [81]. In [20], the torque \(T_{e}\) and \(T_{TM}\) were chosen as the independent control variables and the vehicle speed \(v\) and \(SOC\) reflecting the state of the PHEB were chosen as state variables. As a result, the results of the DDP-based strategy were an approximately 20% improvement in fuel economy compared with the traditional control strategy. In implementing the DDP-based strategy, researchers usually find all possible operation modes of the HEV and all possible control solutions in each mode in order to obtain all control solutions at every grid point of the battery SOC, and then the optimal controls of each stage are obtained by backward solution based on the specified initial SOC or interpolation; however, this results in a heavy computational burden. When using the DP algorithm, it is necessary to find a compromise between the optimization accuracy and the computational burden. One of the reasons for the tremendous computational burden in the calculation process is the storage of the fuel matrix. There are three different manners in which to store the fuel matrix. An intuitive one is point to point, which uses the loop inside the loop at each time. The second is an intuitive matrix method based on an \(M\times M\) matrix, which usually leads to a highly sparse matrix. The third is a method using a dense matrix with column by column (rather than line by line), which contributes to having a smaller loop to construct it, thereby reducing the computational time. To effectively solve the offline issue of DP, the different possibilities for reducing the computational time were investigated in [82; 83], and were introduced from two perspectives: restraining the area of the feasible domain, and changing the solutions from an intuitive manner to a complex matrix manner. As for the feasible area of SOC, the drawback of the non-regular grid is that the maximum discharge limits have virtually no chance of corresponding to an effective edge, while the regular grid creates errors on the SOC variations, and these are accumulated throughout the whole cycle.