# Battery Sizing for Mild P2 HEVs Considering the Battery Pack Thermal Limitations
## 1 Introduction
The control of the battery temperature can be accomplished by systems of different complexity, from simple passive cooling (heat is conducted through the battery casing and natural convection), to active cooling [17; 18] (resorting to air or liquid as a coolant, or by other means such as using phase change materials [20]). The influence of battery temperature on the performance and fuel consumption of HEV is discussed in [13], considering a 100 Ah capacity Lithium battery. Penalty factors for different ambient temperature and drive cycle combinations are integrated with the cost function of the ECMS supervisor. A battery aging semi-empirical model is taken into account along with its temperature dynamics, showing that cold ambient temperature causes a frequent charge and discharge of the battery due to the battery capacity reduction  
Figure 1: Schematic of the P2 HEV powertrain. in cold ambient temperatures. The limitation of the electric traction due to high battery temperatures is not considered in the work [13]. [PERSON] et al. has studied how battery aging is affected by high or low temperatures, and high SOC [21]. A penalty for using the battery at high temperatures is included in the cost function. Considering a 7 kWh lithium-ion battery, an ambient temperature of 15 \({}^{\circ}\)C running on an Artemis drive cycle, the battery temperature does not exceed 35 \({}^{\circ}\)C. The lower operating temperatures in this work can be attributed to the large battery size. [PERSON] et al. [22] compared different control strategies incorporating the battery ageing model in the cost function. Considering the EPA driving cycle, and a pure electric vehicle with a relatively large 9 kWh lithium-ion battery, high temperatures are not an issue, with the temperature reaching 32 \({}^{\circ}\)C from a 30 \({}^{\circ}\)C ambient temperature. An accurate electro-thermal model of the battery pack is needed to study the influence of the battery temperature in HEV. Battery heat generation induced by the electrical current can be divided into irreversible and reversible components [23; 24]. The former is related to the Joule and polarization effects due to the current flowing through the internal resistance and charge transfer. The latter is due to the entropy change, which is related to the temperature dependence of the open circuit voltage (OCV). [PERSON] and [PERSON] in [25] propose a model that can predict the thermal behaviour of a pouch lithium-ion battery cell based on its current and ambient conditions. The model only takes the effect of the OCV and internal resistance into account. However, the resistive-capacitive (RC) parallel branches are not included in the model of the battery, as the thermal time constant greatly outweighs the electrical one. [PERSON] et al. in [26] review the determination of thermal parameters of a single cell, such as internal resistance, specific heat capacity, entropic heat coefficient, and thermal conductivity. These parameters are then used for the design of a suitable thermal management system. A lumped-parameter thermal model of a cylindrical LiFePO4/graphite lithium-ion battery is developed in [27] to compute the internal temperature. Thermal management systems with active heating and cooling can be adopted for controlling the temperature in the battery pack [28; 29; 30]; however, the additional cost and complexity would not be affordable, especially in 48 V micro or mild hybrid electric vehicles. Conversely, 48 V micro or mild hybrids are potentially characterized by high-temperature fluctuations due to the large C-rate reached in boost and recuperation due to the small battery capacity. The influence of battery temperature limitations on the fuel consumption of mild HEVs and the study of battery capacity, above which temperature does not represent a criticality, seems to be still largely unattended in the vast literature on the topic. Additionally, most of the literature that addresses energy management problems has adopted a simplified analytical model to estimate battery SOC and temperature.