Electro-Thermal Modeling and Aging Evaluation of Lithium Battery Packs for Electric Vehicles

Batteries are critical components of battery electric vehicles (BEVs) and yet their expected lifetime is hard to estimate. Various determinants play a significant role, among which cell temperatures and usage profile are notably the most predominant, but their impact is complex to estimate. The variability of the vehicle usage patterns makes it difficult to estimate typical stresses on the battery, and cell temperatures are often unknown conversely to the ambient temperature, that is usually used in degradation models. For these reasons, in this study we present a methodology for estimating the expected battery lifetime in electric vehicles (EVs) through a comprehensive powertrain, electrical and thermal model of the vehicle and its battery, including a battery degradation model. Supported by experimental measures, the whole model, including the battery cooling system, is calibrated with data obtained from the onboard Controller Area Network (CAN) of a BEV with average error below 1°C. The case study to represent typical traffic conditions on Italian routes highlights that the approach is robust, expected battery lifetimes are suitable for typical vehicle usage, and that appropriate battery thermal modelling can reduce errors in the lifetime assessment of beyond 40-50%. The study emphasizes the importance of active thermal management in preserving battery health, and the methodology provides a foundation for further multi-disciplinary modeling of EV battery aging.

SECTION I. Introduction

Battery Electric Vehicles (BEVs) are necessary technologies for sustainable transportation. However, as their onboard battery is the primary source of motion, its lifetime and performances are a critical factor for widespread adoption [1]. Estimating the expected lifetime is however extremely challenging [2]. Accurately modelling battery degradation is indeed an open research question, as it depends on various complex chemistry reactions and the uncertain weather and usage conditions they operate [3]. In particular, the specific cell temperature and battery charging/discharging pattern have direct impact [4]. However, within the realm of BEVs, accurately estimating these factors has traditionally proven challenging, especially when integrating them with suitable models for predicting the battery’s lifetime.

Literature is rich in powertrain simulators for BEVs. They are traditionally focused on estimating the economics of transitioning from Internal Combustion Engines to BEVs, estimating the best battery management strategies, or its control system, as in [5] and [6]. In the literature, commercial tools like Dymola, MATLAB-Simulink, Amesim, SimulationX, are notable standard software that enable to smoothly simulate BEVs operation and easily integrate thermal battery modelling, alongside overall powertrain, and electric dynamics.

Modern BEVs rely on large battery packs smartly arranged in the vehicle to minimize volumes while ensuring battery cooling and performances. The thermal management of the battery is indeed critical to preserve the battery lifetime [7]. Manufactures are increasingly including active cooling technologies and pre-heating before charging, to avoid battery damages. Therefore, the battery aging-temperature behaviour depends not only on the ambient temperature, but also on the cooling strategy both during operation and parking conditions [8]. However, rarely this has been considered. Indeed, for simplicity, battery degradation studies assume the onboard battery of BEVs to operate at the ambient temperature or selected average values across the year, despite the large inertia battery have, the active cooling system and the intrinsic heating that occurs during operation [8], [9], [10]. In this case, the use of detailed electro-thermal modelling can overcome the challenge.

Various modelling approaches exist [11], [12], [13] to determine electro-thermal models of cells and modules. The most advanced techniques leverage Computational Fluid Dynamics (CFD) methodologies to conduct comprehensive simulations encompassing electrical, thermal and chemical aspects [14], [15], [16]. In contrast, Equivalent Circuit Models (ECM) represents a good compromise between simplicity and reliability, providing a reasonable approximation of the primary electrical and thermal behaviors, as demonstrated both in the literature [17], [18] and in the work carried out by some of the authors of this paper [19], [20]. The accuracy is further regarded adequate for degradation analyses, as well as their complexity is compatible with powertrain and electric simulations of BEVs, which makes them appropriate for this study.

Battery degradation models are the counter stone for estimating BEVs lifetime. Several empirical and theoretical approaches are proposed from basic total energy throughput models, eventually with weighted formulations [21], to rainflow-based techniques [4], [22], and even chemical ones [23]. Recent models are semi-empiric formulations where fitting functions are selected to resemble known equations from electrochemistry [24], the Arrhenius one being the most used [25]. Degradation models aim to capture degradation due to charging and discharging patterns, also refer to as “cycling degradation”, as well the so-called “calendar” aging, related to degradation when battery is in stand-by [26], that are both needed for vehicle applications. Basic energy throughput models cannot capture calendar ageing [21], while calendar-only models are not suitable for considering the variable operation of BEVs’ batteries [24]. Recent battery rainflow-based models have demonstrated the capabilities of conjointly capturing calendar and cycling aging, yet rarely they have been fed with detailed estimation of battery temperatures, especially for vehicle applications. Similar activities involve estimating the battery temperature with average weather data [2], [4], [26], which can lead to estimation errors. Therefore, in this paper, we used a literature-based rainflow battery degradation model, calibrated using experimental data, and improved to account for detailed cell temperatures, rather than average ambient values.

The main contribution of this paper is therefore to show a holistic approach in electro-thermal modelling and aging evaluation of battery packs, using a comprehensive powertrain, thermal and electrical modelling of the battery tied with an improved degradation model. The approach we propose proves to be relevant for modeling various real-world battery packs found in market-ready BEVs. Supported by experimental data, three main models have been developed and synergically interfaced: (a) the electro-thermal model of the battery to estimate cell temperatures, (b) the overall vehicle model combining powertrain and the battery and (c) the battery degradation model, which is fed by the estimated cell temperature, current and power profile. The procedure estimates the battery stresses starting from the usage pattern of the vehicles, considering the mechanical, thermal, and electrical characteristics of the vehicle, the battery and its cooling system. Therefore, it can flexibly adapt to various vehicle conditions, hence successfully overcome the challenges in the estimation of EV battery lifetime.

The remainder of the paper is as follows. Section II presents the proposed comprehensive methodology. Section III, IV and V detail the electro-thermal model of the battery, the powertrain representation, and the battery degradation model. Section VI describes the case studies whose results are detailed in Section VII. Finally, conclusions are drawn.

SECTION II. Comprehensive Vehicle and Battery Modelling

To successfully estimate the longevity of EV batteries given the characteristics of the vehicle, the battery and the behavioral usage of the vehicle, in this paper we propose the comprehensive methodology depicted in Fig. 1 and mathematically detailed in the subsequent sections. Provided the model with the behavioral usage of the vehicle and the characteristics of the system, including the cooling of the battery, the proposed methodology estimates the comprehensive stresses on the battery: current, power and cell-specific temperature profiles.

FIGURE 1.

Comprehensive methodology to estimate the battery degradation.

To meet the target, an iterative procedure is proposed to simulate the battery operation for every day of utilization, subject to progressive degradation due to each daily operation. For each daily vehicle usage behavior, a powertrain simulation is executed to estimate the expected power the EV battery must deliver. The powertrain model enables to simulate the electro-mechanical dynamics of the system and identify the power the battery needs to deliver to meet the mission profile, as discussed in Section IV. Then, the iterative simulation in takes place with the goal of estimating the daily battery degradation, due to weather conditions, the cooling system of the battery, the behavioral usage of the vehicle and the recharging strategy. As the battery degradation is affected by the temperature and current, the advanced thermo-electric model of the EV battery, depicted in Fig. 1 and detailed in Section III, is developed. The formulation captures the relation of the battery temperature with the weather conditions, the battery cooling system, and the joule-induced losses, which represent a novelty of this study. The current and temperature estimates are then fed to the battery degradation model to estimate the daily battery degradation, as presented in Section V.

The proposed comprehensive procedure stands out as a groundbreaking contribution, as, to the best of the authors’ knowledge, no other paper has conjointly proposed a powertrain-electro-thermal model of the vehicle, to estimate the battery lifetime, subject to various EV usage patterns with detailed daily simulations and advanced rainflow-based battery degradation model. As further explained in Section V, thanks to the proposed integrated technique, current-induced degradation phenomena are also modelled through the induced thermal losses, phenomenon that was neglected in previous studies [2], [4], [22], and represents a further contribution of this work.

In the following sections, the detailed representation of the model is presented and demonstrated with practical case studies.

SECTION III. Modeling Lithium Battery Packs

A. Battery Electro-Thermal Model

In [27] we already described how the electro-thermal model of a whole battery pack can be extrapolated from the electro-thermal model of a single cell, which is the basic component of the typical battery cell-module-pack structure. Here below, some remarks are presented, as the basic assumptions from the electrical and thermal point of view. As mentioned, greater attention must be devoted to thermal hypothesis in order to accurately simulate thermal interactions among cells, as well as among modules.

The electrical connection between the cells in a series/parallel configuration, combined with the series/parallel configuration of the modules, defines the architecture of the battery pack to be modelled. In order to emulate the main electrical and thermal characteristics of a lithium cell, and therefore of the battery pack, it is useful to represent their equivalent behavior using the circuits shown in Fig. 2. The adopted electrical model is displayed in Fig. 2 (a) and its respective thermal model is shown in Fig. 2 (b): both provide useful information about the voltage, current and temperature response as a function of different power demands and heat exchange conditions.

FIGURE 2.

Comprehensive methodology to estimate the battery degradation.

The electrical cell model used is a 1-block R-C circuit where the existing parameters ($R_{0}$ , $R_{1}$ , $C_{1}$ and $V_{OC}$ ) are a function of the State of Charge (SOC) and the internal temperature $T_{core}$ of the cell. The output variables are the cell voltage (V) and current (I). The SOC of the cell is defined by equation (1), where $SOC_{0}$ is the initial value and $C_{n}$ is the nominal capacity of the cell.

$$\begin{equation*} SOC\left ({{ t }}\right)=SOC_{0}-\frac {\int {I\left ({{ t }}\right)dt}}{C_{n}} \tag {1}\end{equation*}$$

The thermal cell model is a two-temperature model that includes the core temperature ($T_{core}$ ) and the surface temperature ($T_{surf}$ ). It is possible to adopt this type of thermal modelling because the geometric and heat exchange conditions ensure that the ‘single cell’ object can be represented by a first order lumped parameter thermal model. This assumption is verified in the literature, as demonstrated by the same authors in [19] and [20]. The other parameters of the thermal model are the equivalent thermal capacity, $C_{th}$ , the conductive thermal resistance between the centre and the surface of the cell, $R_{cond}$ , and the convective thermal resistance between the surface of the cell and the external environment, $R_{conv}$ . The coupling factor between the electrical and thermal models is represented by $Q_{gen}$ , which is the thermal power generated by the cell. Its value is related to the sum of two contributions, as can be seen in equation (2), which contains two terms: the irreversible heat ($R_{0}I^{2}+R_{1}I_{1}^{2}$ ) due to the Joule effect associated with the resistances $R_{0}$ and $R_{1}$ , and the reversible heat (${\partial V}_{OC}/ \partial T_{core} I T_{core}$ ), caused by internal entropy variations, depending on the cell current I, the internal temperature $T_{core}$ and the “entropy coefficient” $dV_{OC}/dT_{core}$ .

$$\begin{equation*} Q_{gen}=R_{0}I^{2}+R_{1}I_{1}^{2}+\frac {{\partial V}_{OC}}{\partial T_{core}} I T_{core} \tag {2}\end{equation*}$$

All parameters not depending on external operating conditions, such as $R_{0}$ , $R_{1}$ , $C_{1}$ , $V_{OC}$ , ${\partial V}_{OC}/ \partial T_{core}$ , $C_{th}$ and $R_{cond}$ , can be obtained through specific experimental tests, as can be seen in [19], or through a literature research. Numerical examples of the parameterisation of the circuits in Fig. 2 for lithium cells with different chemical compositions, different geometries and different sizes are given in the Appendix (Fig. 11–​12 and Table 6).

TABLE 1 Parameters of Electro-Thermal Pack Model

TABLE 2 Simulated Mission Profiles Information

TABLE 3 Simulated Mission Profiles Information

TABLE 4 Simulated Mission Profiles Information

TABLE 5 Simulated Mission Profiles Information

TABLE 6 Parametrization of Equivalent Thermal Circuit of Fig. 2, for Different Lithium Chemistry, Size, and Geometry

FIGURE 3.

Schematization of the proposed battery pack thermal model.

FIGURE 4.

Powetrain and battery pack model in Modelica environment.

FIGURE 5.

Examples of vehicle usage profiles (one-way trip visualization): (a) “Urban” profile; (b) “Mixed” profile; (c) “Highway” profile, with vehicle speed profiles, distance traveled, and required electric power.

FIGURE 6.

Simulation results in the case of “Mixed” mission profile, 25 kWh battery and 50 kW charging power.

FIGURE 7.

Results of all simulated scenarios as a function of mission profile, charging power, and battery size. (a) battery electrical stress in term of maximum C-rate; (b) battery thermal stress in term of maximum temperature increase, (c) maximum temperature difference between the hottest and coldest modules of the battery.

FIGURE 8.

Results of the proposed methodology for the Real thermal battery management, subject to the sensitivity over the battery capacity, mission profile, and recharge strategy; error-bars show the maximum and minimum variation generally within 6% due to the recharge strategy.

FIGURE 9.

Effect of the recharge power into the expected lifetime (a) and temperature (b); values of expected lifetime are normalized with respect to the lowest recharge rate.

FIGURE 10.

Effect of the thermal modelling on the expected lifetime: the solid lines represent the normalized lifetime and the dashed line represent the average temperature by scenario. Error bars denote maximum and minimum values for all the tested recharging strategies.

FIGURE 11.

Parametrization of the electrical equivalent circuit for lithium cells as a function of the cell SOC and temperature: (a) lithium LFP cell 60 Ah [19]; (b) lithium NMC cell 20 Ah [23], [24].

FIGURE 12.

Parametrization of the entropic coefficients for lithium cells as a function of the cell SOC, for different chemistries and sizes [6], [7], [16] and [25].

The electro-thermal cell model can be scaled up and taken to higher levels to create the model of a module and, consequently, of the whole battery pack. Analytically, this means changing the value of the parameters in Fig. 2 in a way that depends on the series/parallel connections for the electrical model and in a way that depends on the geometrical arrangement of the elements for the thermal model.

The electrical model of the battery pack is obtained in two steps: the first step consists in moving from the cell model to the module model, and the second in connecting several modules together to obtain the model of the entire battery pack. Under this light, each module will have a number of cells in series and in parallel equal to $N_{s}^{c}$ and $N_{p}^{c}$ respectively. Hence, the total number of cells in the battery pack connected in series and in parallel will be equal to $N_{s}=N_{s}^{c}\mathrm {\cdot }N_{s}^{m}$ and $N_{p}=N_{p}^{c}\mathrm {\cdot }N_{p}^{m}$ respectively, resulting in a total number of cells equal to $N_{cell}=N_{s}\mathrm {\cdot }N_{p}$ . Values of the scaling coefficients are in [27].

The battery thermal model follows a scaled approach as well to define its circuital parameters. However, some considerations must be made, leading to the definition of the thermal model adopted in this work. The first observation concerns the number of basic elements present in the system, $N_{cell}$ , which can reach very high values, corresponding to hundreds or thousands of equivalent circuits as the one of Fig. 2 (b). This introduces complexity into the system and relative difficulty in analysing the results. The second concerns the possibility of maintaining a good degree of reliability of the model while simplifying it, taking advantage of the modularity of the physical structure of the battery pack combined with the characteristics of the materials which it is typically made with. From this perspective, we choose to adopt the simplified assumptions listed below, to obtain the simplified battery pack thermal model, shown in Fig. 3:

• the cells are placed in close contact with each other to ensure their integrity against mechanical stress. In addition, the structure containing the cells is made of metallic material (e.g. aluminium), which therefore has a high thermal conductivity;

• given the high compactness of the structure making up the module, it is assumed that each module has a thermal capacity equal to the sum of the thermal capacities of the individual cells, and that each module is characterized by an internal and a surface temperature;

• each module exchanges heat with the external environment by convection ($Q_{ex}$ in Fig. 3) through an equivalent surface area, $A_{module}$ (highlighted in orange in Fig. 3) and a suitably calibrated convective coefficient, h;

• the temperature of the fluid with which each module exchanges heat is not considered as a variable in the model, but only as its input. This means that it is set as a constant within the whole fluid flow path (fluid inlet to fluid outlet), although the authors are aware that this assumption is not entirely realistic. However, this condition allows for maintaining the status of a simplified model, pointing out that the main objective in this section is to correctly estimate the temperatures of the modules ($T_{surf,1},\ldots,T_{surf,n}$ ) and not the cooling fluid temperature, which is here always set equal to $T_{amb}$ .

With these assumptions, it is possible to implement a simplified thermal model, which facilitates the model construction, the simulation time, and the analysis of results. Like the electrical model, the thermal model can be scaled by appropriately varying its parameters. Thanks to its simplicity, it can also be parameterized to represent different cooling modes, depending on the desired simulation scenario. By defining the value of the convection coefficient h, it is possible to analyze the cases of heat exchange by natural convection with air, where $h =5$ W/m² K, forced convection with air where h varies from 15 to 25 W/m² K, and cases of forced convection with a refrigerant, where h can vary from 60 to 200 W/m² K, depending on the configuration of the cooling circuit [28]. The thermal model is characterised by two temperatures for each module (the internal temperature and the surface temperature), with the single cells characterized by their respective module temperatures.

B. Model of A Real Ev Battery Pack

The vehicle selected to calibrate the electro-thermal battery model presented in the previous sections is the Renault Zoe ZE50, which shares the same cell-module-pack battery structure as described in Fig. 3, as reported in [29], whose battery pack elements’ characteristics are given in Table 1. The missing information for the calibration of the model is the temperature of the cooling fluid with which the modules exchange heat and the values of the equivalent convection coefficients ($T_{amb}$ and $h_{1}\mathrm {,\ldots,}h_{n}$ respectively in Fig. 3). The latter variables are obtained by reading onboard CAN signals, considering them sufficiently accurate. To obtain the missing parameters, a full recharging test was carried out at a power of 50 kW, as reported in [27].

The procedure for calibrating the convection coefficients, described in [27], makes it possible to use the hottest and coldest experimental temperature profiles to determine the convection coefficients, by means of an optimization process aimed at reducing the error between simulated and measured temperatures. This leads to a value of $h_{cold} =25$ W/(m2 K) and a value of $h_{hot} =13$ W/(m2 K), which are in line with the values suggested by studies using a similar thermal modelling approach, such as in [28]. The values of the other coefficients were scaled linearly within the range of the values obtained.

Finally, with this set of electrical and thermal parameters, as show in [27] and reported in Appendix (Fig. 13), the simulated battery voltage deviates from the experimentally measured voltage with an average percentage error of 0.6% and a maximum percentage error of 2.2%. For the battery temperatures, the model reproduces the experimental curves with an average error of 0.8°C and a maximum error of 1.4°C for the hottest module and an average error of 0.5°C and a maximum error of 1.2°C for the coldest one.

FIGURE 13.

Results of the electro-thermal battery pack model as in [18]: comparison between model and experimental profiles during the full charging test of the BEV under study. Fist plot: battery voltage; second plot: temperature of the coldest and the hottest modules of the battery.

SECTION IV. Powertrain Simulation Model

A. Model Description

In [27] we already described how the electro-thermal model of the battery was incorporated into the powertrain model of a Battery Electric Vehicle (BEV) to simulate the battery’s behaviour during typical vehicle usage scenarios. The powertrain model used for this purpose is an existing model, briefly presented in [27] and developed within the Modelica environment [30]. It appears as depicted in Fig. 4.

This model takes a set-point speed profile as input and generates the corresponding electric power profile as output. The electric traction drive is accurately calibrated based on real-world consumption patterns. Considering a BEV resembling the Renault Zoe or similar vehicles, the model is fine-tuned to achieve a consumption rate of 148 Wh/km for the standard WLTC certification cycle, aligning with the standards set by major electric vehicle manufacturers [31]. In addition to the battery, vehicle, and driver models, Fig. 4 also encompasses the following components:

1. “Mission profile”: it is the block through which it is possible to select the reference speed profile to be executed by the vehicle. The block consists of a feedback control that directly regulates the traction drive by eliminating the error between the measured and desired speed. Further details about the implementation of this block are described in [2]. In order to simulate different vehicle usage scenarios, this block allow to select different vehicle speed profiles, such as urban or highway paths, as will be detailed in Section III-B;

2. “BTMS” (Battery Thermal Management System) block: it defines the inputs of the battery pack thermal port ($T_{amb}$ , $h_{cold}$ and $h_{hot}$ ). These values are modified based on the conditions in which the vehicle operates. If the vehicle is not operating or the battery cooling system is not active, $T_{amb}$ is equal to the outside air temperature ($T_{outside}$ ) and both the convective coefficients are set to 5 W/m² K. Instead, when the cooling system is on, $T_{amb}$ is set to 10°C and the convective coefficients vary between the modules: $h_{cold} =25$ W/m² K and $h_{hot} =13$ W/m² K. As an additional condition for turning the battery cooling system on or off, we rely on general rules that tend to keep the battery temperature between 25 and 30°C [29], choosing to activate the cooling system if the hottest battery temperature rises above 35°C and to deactivate it when it drops below 30°C;

3. “Charging Station”: it is the block that defines the adopted charging profile by setting an input electric power profile for the battery. The chosen charging levels that can be simulated are: 3 kW for domestic use, 22 and 50 kW for urban charging stations, and 120 kW for fast highway charging. The recharge phase ends when a desired final SOC is reached. It should also be noted that charging profiles typically do not take place entirely at constant power. Moreover, to maintain such levels for an extended SOC window range, peak power levels should be even higher.

B. Definition of Mission Profile

Each simulated mission profile has a duration of 24 hours, starting ideally at 8am in the morning. This allows us to simulate different typical days of vehicle usage. Each mission profile is characterized by a speed-time profile that determines the daily kilometers traveled, as detailed in Table 2 and shown in Fig. 5. It should be noted that the consumption results are obtained taking into account a vehicle mass of 1500 kg and a driving powertrain efficiency ranging from 74% to 88% (in order to meet the specifications of the Renault Zoe under analysis, as referenced in [29]). As a general pattern, each daily profile follows the same structure, which means it is assumed that the vehicle covers half of the total kilometers on an outbound route and then retraces the same path symmetrically on the return route, with an 8-hour intermediate break. At the end of the day, a recharge is implemented to restore the battery SOC. The only exception is the “Highway” profile, which is instead composed by one single outbound route without the backward route. Fig. 6 shows the one-way section of the Urban, Mixed, and Highway profiles, with vehicle speed, kilometers traveled, and required electric power profiles.

C. Simulation Results

Table 3 reports the three tested usage profiles, the considered charging levels (3, 22, 50, and 120 kW), and the selected battery sizes 25, 50, 75, and 100-kWh, for a total of 48 possible scenarios. To modify the battery size, we need to adjust the number of cells in parallel per module $N_{p}^{c}$ (see Table 2). Taking as a reference the battery model developed in Section III, which represents a 50-kWh battery with $N_{p}^{c} =2$ , battery models of 25, 75, and 100-kWh are obtained by setting $N_{p}^{c}$ values to 1, 3, and 4, respectively. As additional conditions, we set the ambient temperature at 20°C for each simulation, as well as the starting SOC at 95%. From an electrical perspective, we evaluate the variations in the maximum current passing through the battery (in terms of C-rate, which is calculated by the ratio between the battery current expressed in ampere and the battery capacity expressed in ampere hours) as well as the maximum Depth of Discharge (DOD) reached for the different system combinations. From a thermal perspective, we analyze the simulation results by reporting, for each case, the highest temperatures reached by the hottest and coldest modules within the battery. In this way it is possible to evaluate which of the simulated variables has the greatest impact on the battery’s thermal increase and, at the same time, estimate the maximum thermal gradients among the internal modules.

As an initial outcome, Table 3 presents the DOD as a function of various mission profiles and the variation of the battery sizes. Notably, for the highway profile, the 25-kWh size is not adequate to achieve a 150 km range, resulting in a table value indicating DOD >100%.

As an example, we choose to display in Fig. 6 the results of one simulated scenario, reporting the evolution of the most significant output variables during the 24 h time horizon considered. In this way, we can imagine that the vehicle starts its first outbound route at simulation time equal to zero, meaning around 8 am in the morning. The second displacement is time located at around 4:30 pm (around 8.5 h simulation time), with a recharging phase at 6 pm (10 h simulation time) reasonably when the vehicle is back home.

To demonstrate a notably challenging thermal scenario, we present the case involving a “Mixed” mission profile, 25 kWh battery size and 50 kW charging power. In fact, this case represents a scenario in which the BTMS needs to activate the battery cooling system because the threshold of 35°C is reached. As can be seen in Fig. 6, the battery temperature, starting from a value of 20°C in equilibrium with the outer environment, exceeds the value of 35°C in correspondence of the high power 50 kW charge phase. This is the trigger condition that switches on the battery cooling system, increasing the $h_{cold}$ (for the coldest module) and $h_{hot}$ (for the hottest module) respectively to 25 and 13 W/(m2K), and setting the battery convective exchanging temperature $T_{amb}$ to 10°C (in Fig. 6, this condition is highlighted as “cooling mode: ON”). As this occurs, a temperature gradient of around 2°C appears between the hottest and the coldest modules ($T_{hot}$ and $T_{cold}$ in Fig. 6) which returns to zero during the subsequent thermal relaxation phase following the completion of the charging phase.

Fig. 7 presents the results for all the 48 simulated scenarios. In particular, Fig. 7 (a) shows the battery electrical stress in terms of the maximum C-rate achieved, while the battery thermal stress in terms of the maximum temperature increase reached by the hottest module (maintaining an external temperature of 20°C for all cases) is depicted in Fig. 7 (b). Finally, Fig. 7 (c) reports the maximum temperature difference between the hottest and coldest module.

Typically, lithium-ion batteries for automotive use can withstand peak currents on the order of a maximum of 2C (meaning numerically double the nominal capacity). Based on this, Fig. 7 (a) indicates which cases exceed this limit: the 120 kW charging is prohibitive for the 25 kWh battery while push at its limit the size of 50 kWh, precisely in the case of “Highway” mission profile.

These latter results can be explained as follows: focusing on the 50 kWh results for 120 kW charging power (yellow bars in chart Fig. 7 (a)) we notice a similar C-rate around 1.9 for the “Urban” and “Mixed” mission profile, and a C-rate of 2C for the “Highway” case. This is due to the battery voltage level reached before the 120 kW charging starts. For the “Urban” and “Mixed” profiles the maximum DOD levels for the 50 kWh size battery are reasonably narrow: respectively 2 and 15%, which means reaching high and very similar battery voltage levels. Instead, the DOD reached during the “Highway” case is 58%, and this leads the battery having a much lower voltage when the 120 kW charge starts, resulting in a higher current value. For all other cases, the electrical stress in terms of battery remains below 2C. Also, from Fig. 7 (a), notice that when varying the charging power within the range from 3 to 50 kW, with the same battery size and usage profile, there are no significant variations in the maximum C-rate. This implies that, for these cases, the maximum current intensity is reached during the driving phase and not during the charging phase.

From Fig. 7 (b), one can immediately observe which cases are the most critical in terms of the maximum temperature reached by the battery. By maximum temperature, we mean the highest temperature reached by the internal modules (specifically, the hottest module). The critical cases are always those associated to high charging power intensity and smaller battery size: specifically, 120 kW charging power with 25 kWh and 50 kWh sizes. It is notable that the greatest impact occurs for the “Mixed” profile for the 25 kWh size and the “Highway” profile for the 50-kWh size. This is because these are the cases in which the charging lasts more. In fact, from Table 3, we know the respective maximum DODs and, consequently, the time required for charging in both cases. Restoring 32% for the 25 kWh size takes about 4 minutes at a constant power of 120 kW, corresponding to an average cell current of approximately 300 A; restoring 58% for the 50 kWh size takes about 15 minutes at a constant power of 120 kW, with average cell current of approximately 150 A. The thermal impact in both cases is similar: an increase of +37°C compared to the 20°C ambient temperature for the 25 kWh size and +34°C for the 50 kWh size. Therefore, both cases would be prohibitive from a thermal perspective (or at the limit of battery performance) because the typical operating temperature range for lithium cells is between 50°C and 55°C.

For all other cases, the maximum temperatures reached by the battery are below these limits, making the simulation estimates plausible and reasonable. It is worth noting that, at the same charging power, the trend of the results is similar: a decrease in the maximum temperature reached with increasing battery size and decreasing intensity of the working cycle. In essence, from a thermal perspective, what impacts the most is the average working power and the duration for which it is sustained.

Finally, for lithium battery packs, a maximum temperature difference of 5°C between the cells is allowed, due to thermal and electrical homogeneity constraints [32]. As observed in Fig. 7 (c), this limit is guaranteed for all simulated scenarios. The worst-case scenario closely resembles the experimental test described in Section III: a long charge (35 minutes) at 50 kW for the 50 kWh size and “Highway” profile. In this case, the maximum temperature difference recorded is 3.5°C, in line with the approximately 5°C measured experimentally for the similar case study, with the only differences being the charging duration and ambient temperature: approximately one hour for the experimental measurement instead of 35 minutes and approximately 42°C instead of 20°C.

In conclusion, one can note that the cases that are prohibitive in terms of electrical stress (expressed in C-rates) are the same ones approaching thermal limits. Charging at 120 kW is proven to be viable only for battery sizes of 75 kWh and 100 kWh. Furthermore, the results show that for urban and peri-urban usage, the 25 kWh battery size is sufficient, ensuring robustness from both an electrical and thermal perspective, even when considering charging up to 50 kW.

SECTION V. Battery Degradation Model

The proposed battery degradation model accounts for calendar and cycling degradation using a rainflow-based technique, in agreement to the literature leveraging on state-of-the-art rainflow-based models described in [2], [4], and [22]. Contrary to previous studies that assumed the battery to be kept at constant temperature or at ambient temperature, in this paper, we account for an accurate battery degradation model that calculates the battery pack temperature using the modelling described in Sections III and IV, accounting also for current-induced effects.

In particular, the proposed method involves the execution of the following steps:

1. load SOC and battery temperature profiles, calculated using the model in Section III, in agreement to the procedure in Fig. 1;

2. identify the characteristic parameters to model the two reported degradation components [2]:

   1. Cycling degradation: to model cycling life, the rainflow algorithm is used to break down a SOC profile into a series of elementary cycles, each one equivalent to the application of a charge and discharge profile of the same current amplitude. Each elementary cycle is identified by an average state of charge $\sigma _{r}$ and a cycle depth of $\delta _{r}$ . Moreover, given the duration of the cycle, the average temperature value $T_{r}^{B}$ to which the battery is exposed during each elementary cycle is also calculated.

   2. Calendar degradation: calendar life depends on the battery’s storage temperature, the average SOC to which it is maintained, and the temporal duration. To simplify the calculation, the acceptable range of temperature and SOC is discretized into intervals. For each pair of intervals ($T_{s,\theta }^{B}$ temperature and $\sigma _{s,\theta }$ SOC), it is calculated the time $\Delta _{s,\theta }$ during which the battery was exposed to this combination of stress factors.

3. Use of equations (3)–​(11) to calculate the SOH.

Equation (3) illustrates the representation of life as a function of the incremental degradation parameter $\Delta ft^{d}$ compared to the value obtained at the end of the previous iteration, $f_{t-1}^{d}$ . This coefficient, in turn, is derived by combining calendar degradation $\Delta f_{t}^{d,cal}$ and degradation due to cycling battery use ${\Delta f}_{t}^{d,cyc}$ , shown in (4). Their mathematical representation is described in equations (5) and (6), quantifying the stresses due to the average SOC ($S^{\sigma }$ ), cycle depth ($S^{\delta }$ ), battery cell temperature ($S^{\theta,cal\mathrm {/}cyc}$ ), and the direct effect of elapsed time ($S^{t}$ ).

$$\begin{align*} SoH& =e^{-\left ({{ f_{t-1}^{d}+{\Delta f}_{t}^{d} }}\right)} \tag {3}\\ {\Delta f}_{t}^{d}& =\sum \limits _{s,\theta }^{N^{S},{N^{\vartheta }}} {\mathrm {\Delta }f}_{s,\theta }^{d,cal} +\sum \limits _{r}^{N^{R}} {\Delta f}_{r}^{d,cyc} \tag {4}\\ {\Delta f}_{t}^{d,cal}& =S^{\sigma }\left ({{ \sigma _{s,\theta } }}\right){\cdot S}^{\theta,cal}\left ({{ T_{s,\theta }^{B} }}\right){\cdot S}^{t}\left ({{ \Delta _{s,\theta }^{B} }}\right) \tag {5}\\ {\Delta f}_{t}^{d,cyc}& =S^{\delta }\left ({{ \delta _{r} }}\right)\cdot S^{\sigma }\left ({{ \sigma _{r} }}\right)\cdot S^{\theta,cyc}\left ({{ T_{r}^{B} }}\right) \tag {6}\end{align*}$$

The mathematical equations representing the calendar degradation model are shown in (7)–(9), while those of the cycling degradation model are highlighted in (10) and (11). The quantities $k^{\sigma }$ , $\sigma _{ref}$ , $T_{ref}^{B}$ , $k^{\theta }$ , $k^{t}$ , and $k^{\delta \mathrm {\ast }}$ are coefficients obtained through interpolation of experimental data, as defined in [4].

$$\begin{align*} S^{\sigma }\left ({{ \sigma _{s, \theta } }}\right)& =e^{k^{\sigma }\left ({{ \sigma _{s,\tau }-\sigma _{ref} }}\right)} \tag {7}\\ { S}^{\theta,cal}\left ({{ T_{s, \theta }^{B} }}\right)& =e^{k^{\theta }\left ({{ T_{s, \theta }^{B}-T_{ref}^{B} }}\right)\frac {T_{ref}^{B}}{T_{s, \theta }^{B}}} \tag {8}\\ S^{t}\left ({{ \Delta _{s, \theta } }}\right)& =k^{t}\Delta _{s,\theta } \tag {9}\end{align*}$$

Equations (10) and (11) demonstrate the calculation of degradation functions specific to modelling the cycling degradation phenomenon of the battery. The function $S^{\sigma }$ is analogous to the calendar life counterpart function.

$$\begin{align*} S^{\delta }\left ({{ \delta _{r} }}\right)& =k^{\delta 1}\delta _{r}^{k^{\delta 2}}+k^{\delta 3}\delta _{r} \tag {10}\\ {S}^{\theta,cyc}\left ({{ T_{r}^{B} }}\right)& =e^{k^{\theta }\left |{{ T_{s, \theta }^{B}-T_{ref}^{B} }}\right |\frac {T_{ref}^{B}}{T_{s, \theta }^{B}}} \tag {11}\end{align*}$$

SECTION VI. Case Study

In this section, we describe a case studie to support the value of the proposed methodology. In particular, we consider the vehicle described in Section IV for the environmental conditions of Northern Italy, see [2], and we perform a sensitivity analysis on (a) the battery capacity, (b) the mission profile, (c) the charging strategy and (d) the modelling of the cooling strategy of the EV battery. Three battery sizes are considered (25, 50 and 100 kWh), the three mission profiles and four recharging strategies (3, 22, 50 and 120 kW), as described in Section IV. Finally, three cooling strategies: Real (R), Ideal (I) and Static (S). In the “Ideal” model we assume a continuous forced convection is applied to the battery that makes the battery to operate approximately at the ambient temperature. In the Static configuration, we assume the battery is kept at the average yearly temperature, as similarly to other studies [22]. In the “Real” formulation, the realistic cooling system is considered as described in Section IV, where the total heat transfer coefficient varies in the range 12.5-2 W/m²K, depending on the cooling system status. Table 4 reports the scenarios selected for the application of the degradation algorithm, matching the notation in Sections III–​V.

The simulations estimate the entire battery life with daily time-horizon at high temporal resolution (seconds), subject to the proposed sensitivity for a total of 42 simulations. The simulations are terminated if the battery degrades to a capacity level below 80% of its initial value or up to 40 years. When the simulation terminates because of the temporal limit (40 years), the expected battery lifetime is assessed using a linear interpolation of the average degradation. The degradation model is characterised by the characteristic parameters described in the Appendix (Table 7), calibrated based on experimental data, according to the approach in [2].

TABLE 7 Degradation Algorithm Parameters

SECTION VII. Results

A. Expected Lifetime

The results of the proposed methodology on the “Real” thermal management of the battery are shown in Fig. 8 and Table 5. Results highlight a strong variability in the expected lifetime related to the mission profile and the battery size, discussed in the following. Negligible variability is identified for the charge pattern, as highlighted by the error bars in Fig. 8. Results confirm the battery lifetime to exceed 100,000-200,000 km of range, which is range of typical vehicle usage [2].

The mission profile has a dramatic effect in the estimation. In the Highway scenarios, the battery lifetime is estimated around 2 to 20 years, for the 50-kWh and 100-kWh batteries respectively, due to the extreme stresses the battery is subject to. On the other hand, results for Urban cases experienced expected lifetime exceeding 40 years, as batteries are subject to limited stress with cycling below 2-5%/day, which means that batteries mainly degrade because of calendar aging, a domain where NMC cells demonstrated notable resilience [4]. However, experimental datasets for calendar aging are generally limited to few years of experimentation. Therefore, results suggest that for Urban applications the expected lifetime is long and the current battery size of EVs are well suited for the application. Moreover, realistic vehicle usage patterns experience a combination of the proposed mission profiles, which means that expected lifetimes are within the extremes shown in Fig. 8. For these scenarios, having longer experimental campaigns for calendar studies appear mandatory in order to facilitate drawing conclusive insights in next studies.

Battery size also plays a significant role, especially in the heavier mission profiles of Mixed and Highway cases, as shown in Table 5. In the Urban case, the predominant degradation phenomenon is calendar aging, thus the effect of the capacity size is within 30% the expected lifetime, whereas in the Highway scenario doubling the battery capacity extends the lifetime by 9 times. Indeed, for the 50 kWh battery, a Highway scenario discharges the battery twice a day at 58% DOD each time, whereas in the highway case with 100-kWh battery they correspond to about 27% DOD, due to the higher capacity and lower losses. This suggests that vehicles expected to experience significant Highway usage are recommended to mount larger battery capacities. For Urban-only applications, even 25 kWh batteries appear to be oversized with respect to the target application. More numerical details are further detailed in Table 5.

B. Effect of the Charging Power

The effect of the charging current on the expected lifetime and temperature is depicted in Fig. 9, where to emphasize the differences, the expected lifetime is normalized with respect to values at 3 kW or 50 kW, depending on the scenario. Results highlight that in Mixed and Highway scenarios higher recharging rates can increase the expected lifetime of batteries, especially in high recharging ratios. Indeed, the increased heating of the batteries, in the range of 10°C for heavier cycles for the proposed case study, induces a lower specific degradation rate during the cycling operation as the average temperature is warmer and closer to the reference 25°C. On the other side, in Urban scenarios where calendar aging is the predominant aging phenomenon the expected lifetime is nearly constant (marginal decrease below 1-2%). The results agree with the literature for NMC batteries that highlight negligible battery degradation due to current phenomena, conversely to the thermal ones effectively simulated in this study.

C. Effect of the Thermal Modelling

To clarify the benefit of the proposed thermal model with respect to typical approximations, Fig. 10 shows the normalized expected lifetime (solid lines) and average battery temperature (dashed lines) subject to a sensitivity on the thermal modelling. In particular, the proposed advanced thermal model discussed in Sections III and IV is compared to the Ideal case, where batteries are isothermal to the ambient temperature (Ideal case), and the Static case, where batteries are operated at the average yearly temperature (Static case).

Results show that the proposed thermal model is the preferred as the Static and Ideal ones induce errors in the estimation. In particular, the Static one leads to overestimated lifetime (at least 10-20%), whereas, using the Ideal one, underestimated lifetime (even beyond 20-40%) is experienced. Indeed, the Static formulation neglects the temporal fluctuations that induce non-linear degradation on the battery, whereas the Ideal one may overestimate the fluctuations as well as the current-induced heating of the battery. While differences for the Urban case are limited, for the Mixed and Highway scenarios using the Ideal case can lead to underestimating the battery lifetime beyond 20-40%. That arises from the underestimation of the battery’s operating temperature, which leads to accelerated degradation during cycling. However, that doesn’t apply for cycles with limited stress on the battery such as the Urban cases or Mixed ones with larger batteries. Indeed, in these scenarios, the battery operates at temperature closer to the average yearly temperature as the thermal inertia of the battery partially smoothens the variation of the ambient temperature. However, for heavy cycles such as Mixed or Highway case where cycling is the main aging mechanism, errors can be as great as 40%. For these scenarios detailed representations using the proposed method is indeed recommended.

Results highlight that the thermal modelling of the battery can play a significant role in the appropriate estimation of the battery lifetime and more advanced modelling are recommended for high-intensive scenarios. In particular, results highlight that the general assumption that the battery is isothermal with the ambient can lead to significant errors in the estimation. Considering the yearly average of the ambient temperature can be a safer approach when the detailed thermo-electric modelling proposed in this study is not possible. However, results confirm that detailed thermo-electric modelling are advisable, especially when heavy battery cycling is expected.

SECTION VIII. Conclusion

In this paper, we successfully presented a comprehensive methodology for estimating the battery lifetime of Battery Electric Vehicles (BEVs) supported by a comprehensive powertrain, electrical and thermal model of the vehicle, the battery packs and its cooling system, taking into account different use profiles, battery sizing and charging scenarios. The model, calibrated using experimental data for a BEV and NMC batteries, is employed in over 40 aging simulations to provide guidance for practitioners and modelers and derive recommendations for battery cooling strategies.

Results highlight that the NMC battery technology has a long-life exceeding the typical requirements for BEVs. Even in the extrema scenarios of daily vehicle usage across tested usage patterns (Urban, Mixed and Highway), lifetimes span across several years, corresponding to over 100,000-200,000 km, which is beyond typical vehicle lifetime. While in Mixed and Highway scenarios the cycling aging is the predominant degradation phenomenon, in Urban scenarios the calendar aging appears to be the primary one. This suggests that especially in Urban application, the battery can support other uses, e.g. vehicle-to-grid, so as to reduce the overall lifetime cost.

The proposed thermo-electric and powertrain modelling has proven robust and well-representative of the experimental data. Maximum battery voltage errors are approximately 2% and battery most critical temperature maximum errors below 3°C, which confirm robustness of the model. The powertrain simulations have also successfully accommodated various driving profiles, battery sizes and charging power levels.

The proposed thermo-electric model has further highlighted that typical approximations, where the battery operates at ambient temperature, can lead to significant errors in the expected battery lifetime, even exceeding 40%, for the proposed Italian case study. Therefore, modelers shall better estimate battery lifetime using appropriate thermal modelling or simplified techniques that consider expected thermal averages, rather than the hourly ambient temperature.

This paper lays the foundations and guidelines for further conjoint mechanical, electrical and thermal modelling of BEVs for ageing evaluations and provides practical insights for practitioners and modelers.

The paper has some limitations; therefore, further studies can investigate how to maximize the battery lifetime, subject to different usage vehicles, second-life phenomena, different weather conditions and battery chemistries, as well to potentially investigate improved cooling strategies of the battery of BEV. With larger data availability, the methodology can also be easily extended to other battery chemistries and vehicle systems.

NOTE

Open Access provided by 'Università di Pisa' within the CRUI CARE Agreement

Appendix

See Figures 11–​13 and Tables 6 and 7.