# Battery Sizing for Mild P2 HEVs Considering the Battery Pack Thermal Limitations
## 1 Introduction  
Powertrain hybridization seems to be a viable mid-term solution for the reduction of vehicle fuel consumption and tailpipe emissions [1; 2; 3; 4]. Hybrid Electric Vehicles (HEVs) can be of series, parallel or combined (series-parallel, power-split) configurations [1; 2]. Parallel configurations offer a cost-effective solution due to: easier integration of the HEV modules in an existing powertrain; the presence of a single electric motor (EM); and a small electric battery (especially in mild HEVs) [2; 3]. Among parallel HEV architectures, the so-called P2 configuration is gaining the utmost attention of vehicle manufacturers as well as researchers due to its scalability and an increased number of operating modes [2; 3; 4]. As shown in Figure 1, in a P2 configuration, the electric machine (EM) is installed on the input shaft of the gearbox after the main clutch C0. Opening the C0 allows for driving in pure electric as well as for an efficient regeneration of the braking energy. By adding the clutch C1, the EM can be used as a starter to crank the ICE as well as for gear shifting. Depending on the location of the EM, the P2 configuration can be on-axis or off-axis. The off-axis P2 HEVs have a stage of parallel axis gear, chain or belt drives (Figure 1) with advantages in terms of the axial size of the powertrain and the potential to have a smaller electric machine running at higher speed than the ICE. Owing to the possibility of scaling the EM and the battery module, P2 is compatible with both mild and plug-in HEV, the mild P2 being characterized by a relatively smaller EM power and battery capacity [5; 6]. An analysis of commercially available MHEVs shows that mostly they have a battery capacity in the range of 0.5-2 kWh [6; 7]. Fuel consumption reduction is the main design goal of Mild HEVs, and this requires an integrated design of the powertrain components as well as the control strategy to avoid jeopardizing the vehicle's dynamic performance [8; 9]. A wide range of controllers has been proposed to this end in the literature. Rule-based supervisory controllers are the simplest to implement, however, they have limited performance when compared to optimization-based counterparts [10]. Optimization-based controllers like dynamic programming (DP) [9; 10; 11], equivalent consumption minimization strategy (ECMS) [11; 12; 13], and model predictive control (MPC) [14] are more performant in finding a global minimum of fuel consumption. However, they need a higher computational effort, require prior knowledge about the driving cycle, and hence can mainly be used in offline control strategies [10; 13]. Comparable results to the ECMS can be obtained by implementing fuzzy logic rules, which can be very complex in the case of a large number of control variables [15]. The ECMS was first proposed by [12] as a real-time controller offering a near-optimal solution without prior knowledge of the drive cycle [13; 16]. Furthermore, it allows for the constraint of various system variables by integrating them in a cost function by means of penalty functions. The main principle of the ECMS is to minimize the total equivalent fuel consumption at each time instant, including the contribution coming from the electrical energy to or from the battery. The optimization is constrained by the battery state of charge (SOC) and the maximum/minimum torque of the EM and ICE [12; 15]. Additionally, it is well known that the battery is a safety-critical system that might cause a fire at high temperatures. Hence, most battery management systems (BMS) monitor the battery temperature in real-time to operate at less than 55 \({}^{\circ}\)C [16; 17; 18]. A review of fire incidence associated with electric and HEVs can be found in [19].