## 4 Simulation Validation and Results Analysis
### Simulation with Initial SOC of 0.9  
We selected two standard drive cycles, i.e., NEDC and HWFET, to validate the performances of the proposed algorithm, DP based method and rule-based method. Figures 9 and 10 show their speed profile and the driveline power, from which we can find that their maximum speeds and maximum driveline power demand are 120 km/h, 96.40 km/h, 38.3 kW and 33.40 kW, respectively.  
Based on the proposed algorithm, the optimal engine-on power threshold can be calculated by the SA algorithm. After calculation, the optimal engine-on power thresholds are 15.54 kW, 15.15 kW, and 13.52 kW when seven to nine standard NEDC drive cycles are simulated, and 13.25 kW, 13.05 kW, 12.65 kW and 12.50 kW when six to nine standard HWFET drive cycles are simulated.  
Figure 10: Speed and driveline power demand for the Highway Fuel Economy Driving Schedule (HWFET).  
Figure 9: Speed and driveline power demand for the New European Driving Cycle (NEDC).  
Figure 11 shows the battery power commands, the engine output power, the driveline power demand and the engine on/off commands based on the proposed method when eight NEDC cycles are applied. It is clearly observed that the convex optimization-based method can start the engine more frequently, since it considers the global optimization of the battery and engine efficiency. Therefore, the DP based method is optimal and the convex optimization-based method is sub-optimal. For the existence of electrical accessories and energy loss, the battery power is higher than the driveline power demand when the engine is off. Figure 12 compares the SOC trajectory when different control methods are applied. Obviously, the battery SOC decreases more slowly when the proposed algorithm is applied due to its capability of the global optimization. Figure 13 compares the comparison of engine-efficiency points by DP and the proposed method, respectively, from which we can observe that the work efficiency by DP is superior than that by the proposed method and the CD/CS method. In addition, there are also less low efficiency points by the proposed method compared to those by CD/CS method. Actually, the purpose of the proposed algorithm is to increase the opportunity of the engine working in the high efficiency region and thus to improve the fuel economy. As shown in Figure 13, the convex optimization-based method increases the operating chances of engine working near the best operating line.  
In order to compare the fuel consumption, a linearly corrected method is applied to ensure the ending SOC with the same value when applying different strategies [2]. Table 3 list the final fuel consumption \(F\), the terminal SOC and the rate of reducing fuel consumption at different cycles. It can be found that the proposed strategy can reduce the fuel consumption by 9.31%, 8.26%, 8.49% when seven to nine NEDC cycles are simulated, and 8.40%, 7.10%, 6.83%, 6.45% when six to nine HWFET cycles  
Figure 11: (**a**) The result based on dynamic programming (DP) when eight NEDC cycles are simulated; (**b**) The result based on convex optimization and SA when eight NEDC cycles are simulated.  
Figure 12: Battery SOC comparison.  
are simulated. As shown in Table 3, the fuel savings achieved with the convex optimization-based method are approximate to the DP based method for the HWFET and NEDC driving conditions. In addition, the fuel savings based on the proposed method are currently less than that based on the DP method. As shown in Table 4, the computation time of the DP based method is obviously longer than that of convex optimization-based method, based on a laptop computer of an i7 processor and 4 Gigabyte RAM. Thus, it justifies the effectiveness of reducing the fuel consumption based on the proposed method.