## 4 Simulation Results and Discussion
### Online Results  
This section aims at validating the effectiveness of the proposed online controller mentioned above. The objective is to test the achievement of results as close as possible to those obtained offline, considered as reference solutions since in real-time, the driver behavior is not known.  
The resulting initial estimated \(\lambda_{in_{estimated}}(0)\) value is equal to 5.0825 \(\times\) 10\({}^{-5}\) for a required SoE balance \(\Delta\)SoE 70% of around 5338.368 Wh. The self-selected parametric coordinates of the required SoC to be tracked giving a final SoC of about 30% for an electric energy storage system with a full initial charge are (\(\alpha=-0.0059\) and \(\beta=100\)).  
Figure 13 depicts the vehicle velocity, the required vehicle velocity, the state of charge, the required SoC, the multiplier variation, and ICE and EM torques for the proposed online \(\lambda\)-control algorithm for PHEV in the blended mode. Figure 14 presents the resulting ICE  
Figure 12: Validation of the proposed offline L-Control algorithm for PHEV during 10 NEDC cycles.  
operating points within its optimal, maximum, and minimum torque curves. Table 2 outlines the impact of varying \(\lambda\)-control parameters on the engine energy consumption and battery balance for 10 NEDC cycles.  
Figure 14: ICE operating point variation of the proposed L-Control algorithm and RI-based strategy.  
Figure 13: Validation of the proposed online L-Control algorithm for PHEV during 10 NEDC cycles.  
Figure 13 shows both good SoC management and tracking for the desired target as well as for the vehicle speed. The fuel consumption is equal to 1.90 L per 100 km for a SoC balance value of around \(-\)70.20%, and it is the near-optimal solution compared with the offline calculation results.  
Figure 14 reveals that for the PMP method, ICE operating points build a cloud surrounding the engine economic area since it is designed to globally optimize the value of the optimal torque. In counterpart, it is noted that for the rule-based strategies in our previous work [1], the optimization is done instantaneously since its construction is based on a comparison at each given time of the power demand with the optimal thermal torque.  
Table 3 reveals that \(\lambda\)-Control parameters condition the final SoC value and thus the fuel efficiency. The optimal fuel rate is obtained for a periodic update frequency \(N_{\lambda}\) equal to 65 s and \(K_{P}\) equal to 0.65 \(\times\) 10\({}^{-5}\) by balancing the battery energy to the targeted value as closely as possible.  
\(\lambda\)-Control Robustness Assessment  
To test the robustness of the proposed \(\lambda\)-Control, two synthesized cycles are used. The first consists of ten consecutive mixed cycles that combine a hyzem urban cycle (Figure 4) with a EUC NEDC cycle (Figure 3). The second is a succession of six WLTC cycles. Figure 15 shows the state of charge SoC, the required SoC, and the multiplier variation of the proposed online L-Control algorithm for PHEV during ten MIXED cycles, six WLTC, and ten NEDC cycles. Figure 16 presents the vehicle velocity, the required vehicle velocity during ten MIXED cycles, and six WLTC of the proposed algorithm. Although the parameters have been conserved, curves indicate good SoC and vehicle speed tracking. It can be seen that the lambda variation depends on the driving cycle, but its variation is slightly perturbed. The amount of fuel consumed is around 2.33 L per 100 km for a battery balance of around \(-\)70.80% for ten mixed cycles (the best offline result is: 2.1 L for a battery balance of around \(-\)70.2%), which represents a gap of 0.10% in terms of fuel consumption and 0.6% in SoC per 100 km.  
The amount of fuel consumed is around 2.87 L per 100 km for a battery balance of around \(-70.11\%\) for 6 WLTC cycles (the best offline result of 2.7 L is for a battery balance of around \(-69.8\%\)), allowing a small gap from optimal results (0.06% in terms of fuel consumption for a 0.31% in SoC per 100 km).