## 4 Results and Discussion  
To evaluate the fuel economy of the proposed control strategy along with the dual-mode engine, six standard driving cycles were selected for simulation validation, as shown in Figure 9. Among those driving cycles, the UDDS cycle represents urban driving conditions and contains a period of medium-high-speed driving; the HWFET cycle represents highway driving without any stops; the MANHATTAN cycle and NYCC cycle represent low-speed urban driving with traffic and frequent start/stops; the US06 cycle represents aggressive driving behavior with high speed and high acceleration; and the WLTC cycle is a comprehensive driving cycle containing multiple driving conditions and covering a large range of vehicle speeds. The simulation was performed on a personal computer with Intel Core) i7-10710U [EMAIL_ADDRESS] GHz, in a MATLAB/Simulink (R2024a) environment with a well-calibrated model of the target SHEV powertrain system. The simulation was divided into two steps:  
Step 1: Offline calibration--the first five driving cycles were conducted in the simulation with the dual-loop PSO algorithm to find the optimal EFs for different driving conditions; the average iteration time for each driving cycle is around 112 min. Step 2: Online validation--the WLTC cycle was used to evaluate the control performance when the EF was fixed after optimization and a single-loop PSO algorithm was applied to solve the ECMS problem; the calculation was finished within 70 s, which is less than 4% of the cycle duration of 1800 s, which shows that the PSO algorithm fulfills the requirement of real-time implementation. In order to investigate the impact of driving conditions on the optimal EF, and to evaluate the benefit of dual-mode engine operation, three strategies were applied in the simulation for comparison. The results are shown in Table 2.  
Figure 9: Standard driving cycles for simulations in this study. * Strategy 1: SI combustion without NVH constraints. Under this strategy, the internal combustion engine will only operate in the SI mode. The instantaneous engine power is determined by the optimal ECMS solution without NVH constraints. This is the ideal case, which could achieve the optimal fuel economy for SI mode operation. * Strategy 2: SI combustion with NVH constraints. NVH constraints will be considered in this case, including the maximum allowed engine power due to NVH and the minimum engine-on time to avoid frequent engine on/off switching. Under this strategy, the engine could operate away from the efficient range under some of the driving conditions; therefore, the fuel economy will be worse than in Strategy 1. This strategy will also be considered as a baseline for comparison with Strategy 3. * Strategy 3: SI/HCCI dual-mode combustion with NVH constraints. This is the proposed strategy in this study. The internal combustion engine can operate in either SI or HCCI modes depending on the optimization result, and multiple NVH constraints are applied. This allows certain assessments, e.g., considering the maximum allowed power at low speed or low load, and avoiding frequent engine start/stop and frequent mode switches. ECMS combined with PSO is applied in each strategy to determine the engine on/off status and engine-generator output power. For the purpose of investigating the benefit of the dual-mode combustion engine in an SHEV in the charge-sustaining mode with NVH constraints, the other energy management strategies, such as thermostat and power follower, are not considered for comparison in this study. By choosing a proper weighting in Equation (24), all the cases have achieved a minimum SOC deviation by the end of the cycle; thus, it is fair to evaluate the fuel economy by MPG (miles per gallon). It can be seen that Strategy 3--which applies dual-mode combustion--can achieve a higher MPG for all the simulated driving cycles than Strategy 2; this is roughly the same as that achieved in Strategy 1.