## 2 Plug-In Hybrid Architecture Modelling  
The plug-in hybrid electric vehicle is a complex system whose energy management must be well-designed and evaluated to achieve the expected performance. The implementation of the global system optimal control is then necessary to ensure global control for effective management between the two energy sources. A systemic approach based on the modeling of the vehicle is considered where this multipart system is seen as a coupling of different subsystems interacting with each other and with the vehicle environment. Tools for modeling and representation are thus considered. Depending on the performance requirements and the objectives of the overall PHEV modeling, the direct model, also known as the \"forward\" model, makes it possible to determine the evolution of the dynamic system and the control strategy over time, deduced from the speed and acceleration as a function of the required torque. Several representation methods have been introduced in the literature to facilitate modeling studies to design and control complex multi-physics systems, such as:  
* Bond graphs [23; 24]. * Energetic Macroscopic representation [25; 26; 27]. * The mixed representation block diagram and graph bond is based on the representation of the subsystems in the form of block diagrams connected between them by energy variables. The implementation of an energy management strategy needs to be evaluated, and its performance assessed through the use of the driving cycle, which represents speed versus time and highly influences consumption and pollutant emissions. The most known in Europe are the standard cycles, such as the New European Driving Cycle (NEDC) (Figure 2, more recently replaced by WLTC (World Light vehicle Test Cycle) (Figure 3), and the real-life cycles based on experimental studies, such as Hyzem cycles (Figure 4). There are many tools for simulating PHEVs' behavior and validating their energy management. They depend on the type of approach and representations used for their modeling. Figure 4: Urban Hyzem cycle, Road Hyzem cycle, and Highway Hyzem cycle. Figure 3: European Standard cycle WLTC (CLASSES3/BV). Figure 2: European standard cycle NEDC. MATLAB/Simulink(r)seems to be one of the proper platforms to implement a mixed representation that allows various components models. VEHLIB, an open-source software developed by Gustave Eiffel University, uses this approach. Excepting the control, most of the models used for our PHEV come from this software [28; 29; 30]. In the context of technological developments, choosing the appropriate PHEV architecture is an important and cost-effective solution to achieve sustainable mobility and reduce the ecological footprint of the transport system. In this framework, the layout adopted is a parallel PHEV powertrain where a mechanical coupling is inserted between the ICE and the e-motor fitted in between the clutch and the gearbox. This link-up allows the thermal torque to be used in conjunction with the electric torque after the electrochemical energy conversion flow via the battery or separately. Thus, an all-electric self-driven mode can power the system, which meets the ecological requirements in urban areas. The global EMS-PHEV model consists of four main sub-blocks, as presented in Figure 5: the driver request cycle block, the EMS controller block, the PHEV transmission components block, and the vehicle Dynamics block. For the modeling of the powertrain, both the ICE engine and the electric drive parts that combines the electric motor with the converters are based initially on the efficiency operational maps (Figures 6 and 7). In order to obtain mathematical models that could be easily used in our proposed EMS methods, these curves are approximated by polynomial expression. Equations (1) and (2) describe the fuel flow \(S_{fuel}\)\((T_{ice}(k).W_{ice})\) and the required electric battery power \(P_{BATT}(T_{em}.W_{em})\) after operating the proposed interpolation. \[