# Multiparadigm Modeling Framework to Evaluate the Impacts of Travel Patterns on Electric Vehicle Battery Lifespan

Authors: Shubham Agrawal; Srinivas Peeta; Mohammad Miralinaghi
Journal: Journal of Advanced Transportation
 (2023)

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## Abstract

The widespread adoption of electric vehicles (EVs) can help attain economic and environmental sustainability by reducing oil dependency and greenhouse gas emissions. However, several issues need to be addressed before EVs can become a popular vehicle choice among the general public. A key issue is the perpetual reduction in EV battery capacity caused by battery degradation over time with usage. This can lead to a reduced driving range and cause “range anxiety” for EV drivers. This becomes even more critical in developing countries where consumers are highly sensitive to battery replacement costs. Thus, to promote EVs in developing economies, policymakers and vehicle manufacturers need to develop attractive incentive schemes and warranty strategies preceded by a thorough assessment of the useable EV battery lifespan for a wide range of users. This paper develops a multiparadigm modeling framework to compute battery degradation for a large population of EVs by capturing the effects of travel patterns, traffic conditions, and ambient temperature. The proposed framework consists of four different building blocks: (i) a microscopic traffic simulation model to generate speed profiles, (ii) an EV power consumption model, (iii) a battery equivalent circuit model, and (iv) a semiempirical battery degradation model. The proposed framework can also be used to assess the battery life-cycle of electric-powered automated vehicles by adjusting their travel patterns accordingly. A case study is presented using travel diary data of around 700 households from the U.S. National Household Travel Survey of 2009 to simulate household travel patterns and corresponding battery lifespan distribution.


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## Body

## 1. Introduction

The transportation sector’s share of energy consumption has been steadily increasing in developing economies like India[1]. This makes the transportation sector in developing countries one of the major contributors to urban air pollution [2]. Electric vehicles (EVs) have zero tailpipe emissions, making them a promising alternative to conventional fossil fuel-based vehicles for curbing greenhouse gas emissions [3]. EVs can also potentially reduce national oil dependency as the renewable and nuclear energy sectors grow. Several automobile companies have already introduced EVs to the market. For example, in 2021, Volvo announced a plan to cease ICEV production from 2030 onwards and to produce only BEVs and PHEVs [4]. Policymakers are also currently offering incentives to encourage the purchase of EVs [5]. For example, under the clean vehicle rebate project [6], the California Environmental Protection Agency (CalEPA) provides a rebate of up to $7,000 for EV purchases.Despite international efforts, the greater adoption of EVs still faces several challenges. For example, the reduced maximum driving range due to battery degradation can accentuate range anxiety (i.e., the fear of running out of battery charge before completing the trip) among EV drivers over time [7, 8]. The costs associated with battery replacement from degradation can further deter EV purchases. This impending inconvenience and economic burden make affordable EVs unattractive for new automobile buyers, especially in developing countries where consumers are more sensitive towards costs. The scarcity of information on battery degradation and contributing factors also plays a critical role in increasing consumer skepticism. A thorough understanding of the effects of travel patterns, driving characteristics, and environmental factors on EV battery lifespan would assist vehicle manufacturers and policymakers in designing appealing incentive schemes and warranty strategies to increase EV adoption [9]. It can also help consumers make informed decisions based on their travel patterns and driving behaviors.In this context, this study proposes a multiparadigm modeling framework to estimate the useable battery lifespan of a large population of EVs. The useable battery lifespan is defined as the duration within which the maximum EV battery capacity degrades below a certain threshold of its original capacity and needs to be replaced for regular use. It is an important factor in deriving the energy consumption estimation model for EVs [10]. The generally accepted battery degradation threshold before it needs replacement is considered to be 20 to 30% of its original capacity [11, 12]. The proposed modeling framework consists of four different building blocks: (i) a network-level microscopic traffic simulation model to obtain realistic drive cycles or speed profiles (i.e., a series of vehicle speeds versus time) of the vehicles; (ii) an EV power consumption model to compute the power profile from the speed profile; (iii) a battery circuit model that converts the power profile to a battery current flow profile; and (iv) a semiempirical battery degradation model that simulates battery lifespan based on current flows and ambient temperature. The multiparadigm approach provides the flexibility to use the most suitable modeling methods at each step. Some studies have used multiparadigm or multistage approaches in EV-related research, including digital battery lifespan management [13]; however, to the best of our knowledge, they have not been used to understand the impacts of travel patterns on the battery lifespan.Battery degradation or battery aging can be classified into two different mechanisms: calendar aging (during storage) and cycle aging (during use). Aging happens as a result of structural disordering, variation in electrolyte composition, or loss of active material caused by thermodynamic instability [14, 15]. Calendar aging is mainly caused by the growth of a protective layer at the anode called solid electrolyte interphase (SEI), which results in the loss of active material (e.g., lithium) and increased electrode impedance [14, 16]. Cycle aging is mainly caused by structural changes and chemical decomposition of active material at the cathode and changes in SEI at the anode from electrolyte reactions during charging and discharging [17]. Although total degradation is considered as the summation of both calendar aging and cycle aging, their degradation mechanisms are not independent, and interactions occur [17]. This study focuses on modeling Li-ion battery degradation since most commercially available EVs use lithium-ion (Li-ion) battery packs [8]. Due to the inherent complexity of aging mechanisms and their interactions, several semiempirical battery degradation models [11, 18–21] and statistical models [22] based on experimental data have been proposed in the past literature.Several efforts have been made in the past to quantify the useable battery lifespan of EVs. For example, Marano et al. [23] and Onori et al. [24] proposed a model to compute battery degradation for plug-in hybrid electric vehicles (PHEVs) using depth-of-discharge (DoD) (i.e., the percentage of battery capacity used before recharging) and battery temperature using linear combinations of standard driving schedules like the Urban Dynamometer Driving Schedule (UDDS) [25], which limits its applicability to a large population of EVs with diverse travel patterns and driving behavior. Guenther et al. [26] investigated vehicle-to-grid applications for synthetic drive cycles using a simplistic energy-based battery model, which ignores the effects of internal resistance and cell voltage at different state-of-charge (SOC) (i.e., the percentage of battery capacity available). Peterson et al. [27] analyzed vehicle-to-grid applications using UDDS and concluded that using a realistic drive cycle is important to quantify battery degradation as DoD may provide misleading results. Some studies have evaluated the impacts of battery recharging strategies on battery degradation using a combination of DoD, SOC, and/or temperature [28, 29]. However, these models assume a very simplistic drive cycle with little or no variation in speed and, hence, the current flow through the battery. Remmlinger et al. [30] presented a method to compute the degradation index relative to the degradation of new batteries by using the measurement values of cell voltage and current flow. Pelletier et al. [31] provide a brief overview of several Li-ion battery degradation models that can be used for EV applications. The proposed multiparadigm framework extends the existing battery lifespan computation frameworks by introducing a microscopic simulation layer to assess heterogeneity in travel patterns and driving behavior. Yang et al. [32] developed a novel analytical framework to determine battery degradation based on travel demand and the ambient high temperature of the battery. Their results show that battery life ranges from approximately 5 years in Florida to 13 years in Alaska, the United States. They also showed that if an EV continues to operate after the 30% battery degradation limit, the greenhouse gas emissions and energy consumption can be significantly increased. Xu et al. [33] proposed a Q-learning-based strategy to minimize Li-ion battery degradation and energy consumption. The Q-learning method is an adaptive optimal control algorithm that uses the Bellman equation of dynamic programming. It is shown that Q-learning decreases the battery capacity loss and increases the lifespan by 13-20%. A summary of the literature on battery degradation models is presented in Table 1.Table 1
Summary of the literature on battery degradation models.
StudyDegradation modelCycle lifeCalendar lifeInputMarano et al. [23]Damage accumulation modelYNCurrent severity relative to battery size (i.e., C-rate), temperature, DoD, and SOCPeterson et al. [27]Integrated driving and energy use profile modeling frameworkYNC-rate, discharge power rate, DoD, and driving profileRemmlinger et al. [30]Internal resistance dependent degradation modelYNTemperature, SOC, and power demandOnori et al. [24]Weighted Ah-throughput modelYNTemperature and DoDGuenther et al. [26]Energy-based battery modelYYTemperature and power demandOuyang et al. [34]Prognostic and mechanistic modelYNC-rate and temperatureYang et al. [32]Pseudo two-dimensional battery capacity fading modelYYTemperature, travel demand, and driving profileCalearo et al. [35]Integrated thermal and SOC dynamics modelYYTemperature and SOCMotapon et al. [36]Physical degradation modelYNC-rate, temperature, and DoDOlmos et al. [37]Empirical degradation modelNYC-rate, temperature, DoD, and SOCXu et al. [33]Control-oriented cycle-life modelYYC-rate, temperature, and SOCOur studySemiempirical traffic simulation-based degradation modelYYC-rate, temperature, travel demand, and driving profileThe key contributions of this research are threefold. First, this study proposes a multiparadigm modeling methodology to derive the useable lifespan of battery of a large population of EVs. Second, this study integrates a microscopic traffic simulation model to account for the driving behavior heterogeneity for battery lifespan estimation. Third, each component of the multiparadigm method offers the flexibility to incorporate even newer models or real-world data without changing the framework’s overall structure. For instance, real-world vehicle speed data, if available, can replace the traffic simulation model.The following section introduces the multiparadigm modeling framework to quantify battery lifespan for a large population of EVs and discusses each building block in detail. Then, a case study for the city of Indianapolis, U.S., is presented, and the results are discussed to illustrate the impacts of travel patterns, driving behavior, and temperature on battery lifespan. The paper concludes with a discussion on potential applications of the proposed framework and insights for policymakers, vehicle manufacturers, and other stakeholders to aid greater EV adoption.


## 2. Methodology: Multiparadigm Modeling Framework

The proposed framework is composed of four modeling stages. This framework requires household vehicle travel patterns as input, which can be obtained using resources such as the U.S. National Household Travel Survey [38]. At a minimum, household vehicle travel patterns should include details about departure time, travel time, and distance traveled for all trips made by a particular vehicle on a given day. The microscopic traffic simulation model based on a real-world road network is calibrated using available traffic demand data to generate realistic drive cycles. Each trip in household travel patterns is then matched to a suitable drive cycle generated by the traffic simulation model. The speed profiles are then fed into an EV power consumption model to compute the power profiles. A battery equivalent circuit model is then employed to get a time series of current flow and SOC of the EV battery. These data are subsequently inputted to a semiempirical battery degradation model to calculate the battery state-of-health (SOH) (i.e., the ratio of current battery capacity to its original capacity).


### 2.1. Microscopic Simulation Model

Traffic simulation models are widely used to capture nonlinear interactions between vehicles and infrastructure at a microscopic level. Such models can simulate vehicle dynamics and output dynamic variables like position, speed, and acceleration for a large number of vehicles. In this framework, a time series of vehicle speeds is needed to compute the power requirement from the EV battery for propulsion. The simulation model needs to be calibrated using travel demand data between origin-destination pairs for the given road network. The aforementioned information can be gathered from multiple sources, like regional traffic management websites and open-source online resources (e.g., OpenStreetMap). The realism of the simulation model can be further enhanced by including more information such as traffic management infrastructure (e.g., traffic signals) and driver behavior (e.g., car-following model parameters). The simulated drive cycles are then used as input for the EV power consumption model discussed below.


### 2.2. Electric Vehicle Power Consumption Model

EV power consumption depends directly on the vehicle speed profile. Since the kinetic energy of a vehicle depends on its speed, it is necessary to capture changes in the speed at a microscopic level to accurately compute the vehicle’s power requirements. Some simulation packages such as ADVISOR [39] and Autonomie [40] can compute energy consumption and MPGe (miles per gasoline equivalent) using drive cycle data. These tools simulate a detailed EV drivetrain system and hence are computationally expensive. Since a key application of the proposed framework is to quantify battery degradation for a large population of EVs, it is important to select a computationally efficient model at each step. This study adopts a physical model proposed by Van Haaren [41] to compute power consumption. This approach is used for the following reasons: (i) the model parameters are fitted using real-world Tesla Roadster data [42], and (ii) the computational runtime is significantly lower than the simulation packages mentioned before. This framework provides flexibility to use other similar EV power consumption models that take speed data as a primary input [43, 44].The physical model used in this study computes the net vehicle power loss/gain (Pnet) as the sum of two components. First, the power required to maintain a constant speed (Pcons). Second, vehicle power requirements at variable speed during loss/gain in kinetic energy while accelerating or braking (Pkin). The power loss/gain due to road grade is assumed to be zero but can be added to this model by considering the power gain/loss due to the change in potential energy of the vehicle (Ppot). The total power loss (in Watts) at constant speed is the sum of power losses due to aerodynamics (Par), drivetrain (Pdr), rolling friction (Prr), and ancillary systems (Pan) as expressed in equations (1)-(6). The parameter definitions and values are presented in Table 2. Some of the parameter values are adapted to match the [43] model, a plug-in electric compact car that accounted for more than 23% of plug-in EV sales in the U.S. in 2013 [45].(1)Par=12ρAvehCdV3,(2)Pdr=αdrVmph3+βdrVmph2+γdrVmph+cdr,(3)Prr=crrmgV,(4)Panc=180,(5)Ppot=mgVsinθ,(6)Pcons=Par+Pdr+Prr+Pan+Ppot.Table 2
EV power consumption model parameters.
ParameterDefinitionValueVVehicle speed in meters per secondVmphVehicle speed in miles per hourρAir density (kg/m3)1.225AvehVehicle front area (m2)2.27CdDrag coefficient0.28αdrDrivetrain coefficient 10.004βdrDrivetrain coefficient 20.5γdrDrivetrain coefficient 329.3cdrDrivetrain coefficient 4375crrRolling resistance coefficient0.0075mVehicle mass (kg)1520gGravity (m/s2)9.81θRoad grade0ΔtDiscrete time step (s)1βeffBattery to motor efficiency0.85βrbsRegeneration efficiency0.4PmaxMaximum power output (W)100,000PminMinimum power loss or maximum regeneration/recharging power gain (W)−7,000The total kinetic energy (in Joules) of the vehicle (Ekin) consists of linear kinetic energy (Elin) and rotational kinetic energy (Erot). For simplicity, the model assumes that the rotational kinetic energy is approximately 5% of the linear kinetic energy. The power loss/gain at variable speed is the change in kinetic energy (ΔEkin) for the given time interval (Δt) as expressed in equations (7)-(9).(7)Elin=12mV2,(8)Ekin=Elin+Erot≅1.05Elin,(9)Pkin=ΔEkinΔt.The net power loss is multiplied by the battery-to-motor efficiency factor to account for the inefficiencies in electrical to kinetic energy conversion. Similarly, the regeneration efficiency factor is multiplied to account for the energy recuperation from the regenerative braking system in case of net power gain. The net power consumption of EVs is bounded by their battery limits as expressed in equation (10). This framework assumes these limits as −7 kW (Pmin) and 100 kW (Pmax).(10)Pnet=Pmax,Pcons+Pkinβeff≥Pmax,Pcons+Pkinβeff,Pcons+Pkin≥0,βrbsPcons+Pkin,Pcons+Pkin<0,Pmin,βrbsPcons+Pkin≤Pmin.


### 2.3. Battery Equivalent Circuit Model

Cycle aging primarily depends on the current flow through the battery during charging or discharging (or C-rate). A 1 C rate is defined as the theoretical discharging current drawn from the battery that will discharge the entire battery in an hour at its rated nominal voltage. A 2 C rate implies double the amount of discharging current corresponding to the 1 C rate; that is, it will discharge the battery in half an hour. This framework implements a battery equivalent circuit as illustrated in equations (11)-(16). The parameter values for Reference [46] are obtained from its owner’s manual [46] and advanced vehicle testing activity data [47]. Model parameter definitions and values are presented in Table 3. The model uses 1-D lookup tables to get the cell internal resistance Rintt during charging/discharging and open-circuit voltage VOCt at different SOC, as illustrated in Figures 1 and 2, respectively. Due to the lack of data, the internal resistance and open-circuit values are assumed to be constant until the battery’s end-of-life (EOL). This assumption can be relaxed by using 2-D lookup tables containing internal resistance and open-circuit voltage values with respect to both battery SOC and SOH, if such data are available.(11)Vcellt=VOCt−Rintt⋅Icellt−1,(12)Pchgmaxt=−VoutmaxVoutmax−VOCt⋅NsRintt⋅Nt,(13)Pdismaxt=VoutminVOCt⋅Ns−VoutminRintt⋅Nt,(14)P˜outt=PdismaxtPoutt≥PdismaxtPouttPdismaxt≥Poutt≥Pchgmaxt,PchgmaxtPoutt≤Pchgmaxt.(15)Icellt=P˜outtVoutct⋅Ns⋅Np,Iratet=IcelltKcell⋅δsoh,(16)δsoct=δsoct−1−Iratet⋅Δt.Table 3
Battery equivalent circuit model parameters.
ParameterDefinitionValueNsNumber of cells in series in each module96NpNumber of parallel modules2NtTotal number of cells in battery pack192VOCtCell open-circuit voltage at timet (V)Using lookup-tableRinttCell internal resistance at timet (ΩUsing lookup-tableIcelltCell current at timet (A)Icell0=0VcelltCell terminal voltage at timet (V)VoutminMinimum battery terminal voltage (V)336VoutmaxMaximum battery terminal voltage (V)403.2PouttBattery power requirement at timet (W)P˜outtActual battery power output at timet (W)PdismaxtMaximum battery power output while discharging at timet (W)PchgmaxtMaximum battery power input while charging at timet (W)IratetC-rate at timetKcellRated cell capacity (Ah)33.1δsoctBattery state-of-charge at timetδsohBattery state-of-healthΔtDiscrete time step (seconds)1Figure 1
Cell internal resistance values at different SOC.Figure 2
Cell open-circuit voltage values at different SOC.


### 2.4. Battery Degradation Model

Battery degradation is affected by several factors such as battery temperature, SOC, C-rate, and total current throughput (Ah-throughput) [48]. Some models approximate the total Ah-throughput as a product of constant depth-of-discharge and the number of cycles [23]; thereby, ignoring the effects of SOC and C-rate. This framework adopts a semiempirical battery degradation model proposed by Wang et al. [21] that includes three important parameters: time (or battery age), temperature, and C-rate. It computes calendar aging (Qcal) as a function of time (τ) and temperature (T) and cycle aging (Qcyc) as a function of temperature, C-rate (Irate), and lifetime Ah-throughput (Iah). The model parameters are fitted using experimental aging data for high-power density 1.5 Ah, 18650 cylindrical cells with LiMn1/3Ni1/3Co1/3 + LiMn2O4 (NCM + LMO) cathode and a graphite anode. Their results indicate that the predicted values are within ±5% of the measured battery capacity loss. The model can be expressed as equations (17)-(19).(17)QT=Qcal+Qcyc,(18)Qcal=fτ0.5⋅exp−EaRT,(19)Qcyc=aT2+bT+c⋅expdT+e⋅Irate⋅Iah.To account for the variable C-rate and temperature, the model is modified by taking the differentiating total capacity loss function with respect to a discretized time step (Δτ). For each time step, the current flow is assumed to be constant, and instantaneous capacity loss (DT) is computed as the sum of instantaneous capacity loss Dcap and instantaneous cycle loss Dcyc by taking the differential of their respective functions. The sum of instantaneous capacity loss is updated to get the total capacity loss until the battery replacement threshold limit δrep is reached. Hence, the battery lifespan is equal to the number of days elapsed before the total capacity loss reaches the specified threshold limit. Because Reference [49] has a rated nominal cell capacity of 33.1 Ah and the data are calibrated for 1.5 Ah cells, a correction factor βcorr equal to the ratio of rated nominal capacities is multiplied by the cycle loss. Equations (20)-(22) describe the modified model, and Table 4 shows the parameter values.(20)DT=Dcal+Dcyc,(21)Dcal=0.5fτ−0.5⋅exp−EaRT,(22)Dcyc=aT2+bT+c⋅expdT+e⋅Irateτ⋅Icellτ⋅βcorr.Table 4
Battery degradation parameter values and units.
ParameterValue and unitParameterValue and unita8.61e −  (1/Ah-K2)IahLifetime current throughput (Ah)b−5.125e −  (1/Ah-K)IrateτC-rate at timeτc0.7629 (1/Ah)IcellτCell current at timeτ (A)d−6.7e − 3 (1/K-(C-rate))βcorr1.5/33.1e2.35 (1/(C-rate))Ea24500 (J/mole)f14876 (1/day0.5)R8.314 (J/K-mole))τTime (in days)TTemperature (K)ΔτDiscretized time step (in days)δrep30%This model uses ambient temperature as a proxy for battery temperature, thereby assuming that the thermal effects on internal resistance and cell current are negligible. This assumption can be addressed by including a suitable electrochemical-thermal model in the framework (see Reference [50] for a review) or by using a 1D lookup table to link battery temperature with the environment temperature and battery cell characteristics (e.g., [51, 52]. The model parameters are calibrated for discharging current only. Thus, it is assumed that charging and discharging will have a similar impact on cycle aging based on the absolute value of Irate.


## 2.1. Microscopic Simulation Model

Traffic simulation models are widely used to capture nonlinear interactions between vehicles and infrastructure at a microscopic level. Such models can simulate vehicle dynamics and output dynamic variables like position, speed, and acceleration for a large number of vehicles. In this framework, a time series of vehicle speeds is needed to compute the power requirement from the EV battery for propulsion. The simulation model needs to be calibrated using travel demand data between origin-destination pairs for the given road network. The aforementioned information can be gathered from multiple sources, like regional traffic management websites and open-source online resources (e.g., OpenStreetMap). The realism of the simulation model can be further enhanced by including more information such as traffic management infrastructure (e.g., traffic signals) and driver behavior (e.g., car-following model parameters). The simulated drive cycles are then used as input for the EV power consumption model discussed below.


## 2.2. Electric Vehicle Power Consumption Model

EV power consumption depends directly on the vehicle speed profile. Since the kinetic energy of a vehicle depends on its speed, it is necessary to capture changes in the speed at a microscopic level to accurately compute the vehicle’s power requirements. Some simulation packages such as ADVISOR [39] and Autonomie [40] can compute energy consumption and MPGe (miles per gasoline equivalent) using drive cycle data. These tools simulate a detailed EV drivetrain system and hence are computationally expensive. Since a key application of the proposed framework is to quantify battery degradation for a large population of EVs, it is important to select a computationally efficient model at each step. This study adopts a physical model proposed by Van Haaren [41] to compute power consumption. This approach is used for the following reasons: (i) the model parameters are fitted using real-world Tesla Roadster data [42], and (ii) the computational runtime is significantly lower than the simulation packages mentioned before. This framework provides flexibility to use other similar EV power consumption models that take speed data as a primary input [43, 44].The physical model used in this study computes the net vehicle power loss/gain (Pnet) as the sum of two components. First, the power required to maintain a constant speed (Pcons). Second, vehicle power requirements at variable speed during loss/gain in kinetic energy while accelerating or braking (Pkin). The power loss/gain due to road grade is assumed to be zero but can be added to this model by considering the power gain/loss due to the change in potential energy of the vehicle (Ppot). The total power loss (in Watts) at constant speed is the sum of power losses due to aerodynamics (Par), drivetrain (Pdr), rolling friction (Prr), and ancillary systems (Pan) as expressed in equations (1)-(6). The parameter definitions and values are presented in Table 2. Some of the parameter values are adapted to match the [43] model, a plug-in electric compact car that accounted for more than 23% of plug-in EV sales in the U.S. in 2013 [45].(1)Par=12ρAvehCdV3,(2)Pdr=αdrVmph3+βdrVmph2+γdrVmph+cdr,(3)Prr=crrmgV,(4)Panc=180,(5)Ppot=mgVsinθ,(6)Pcons=Par+Pdr+Prr+Pan+Ppot.Table 2
EV power consumption model parameters.
ParameterDefinitionValueVVehicle speed in meters per secondVmphVehicle speed in miles per hourρAir density (kg/m3)1.225AvehVehicle front area (m2)2.27CdDrag coefficient0.28αdrDrivetrain coefficient 10.004βdrDrivetrain coefficient 20.5γdrDrivetrain coefficient 329.3cdrDrivetrain coefficient 4375crrRolling resistance coefficient0.0075mVehicle mass (kg)1520gGravity (m/s2)9.81θRoad grade0ΔtDiscrete time step (s)1βeffBattery to motor efficiency0.85βrbsRegeneration efficiency0.4PmaxMaximum power output (W)100,000PminMinimum power loss or maximum regeneration/recharging power gain (W)−7,000The total kinetic energy (in Joules) of the vehicle (Ekin) consists of linear kinetic energy (Elin) and rotational kinetic energy (Erot). For simplicity, the model assumes that the rotational kinetic energy is approximately 5% of the linear kinetic energy. The power loss/gain at variable speed is the change in kinetic energy (ΔEkin) for the given time interval (Δt) as expressed in equations (7)-(9).(7)Elin=12mV2,(8)Ekin=Elin+Erot≅1.05Elin,(9)Pkin=ΔEkinΔt.The net power loss is multiplied by the battery-to-motor efficiency factor to account for the inefficiencies in electrical to kinetic energy conversion. Similarly, the regeneration efficiency factor is multiplied to account for the energy recuperation from the regenerative braking system in case of net power gain. The net power consumption of EVs is bounded by their battery limits as expressed in equation (10). This framework assumes these limits as −7 kW (Pmin) and 100 kW (Pmax).(10)Pnet=Pmax,Pcons+Pkinβeff≥Pmax,Pcons+Pkinβeff,Pcons+Pkin≥0,βrbsPcons+Pkin,Pcons+Pkin<0,Pmin,βrbsPcons+Pkin≤Pmin.


## 2.3. Battery Equivalent Circuit Model

Cycle aging primarily depends on the current flow through the battery during charging or discharging (or C-rate). A 1 C rate is defined as the theoretical discharging current drawn from the battery that will discharge the entire battery in an hour at its rated nominal voltage. A 2 C rate implies double the amount of discharging current corresponding to the 1 C rate; that is, it will discharge the battery in half an hour. This framework implements a battery equivalent circuit as illustrated in equations (11)-(16). The parameter values for Reference [46] are obtained from its owner’s manual [46] and advanced vehicle testing activity data [47]. Model parameter definitions and values are presented in Table 3. The model uses 1-D lookup tables to get the cell internal resistance Rintt during charging/discharging and open-circuit voltage VOCt at different SOC, as illustrated in Figures 1 and 2, respectively. Due to the lack of data, the internal resistance and open-circuit values are assumed to be constant until the battery’s end-of-life (EOL). This assumption can be relaxed by using 2-D lookup tables containing internal resistance and open-circuit voltage values with respect to both battery SOC and SOH, if such data are available.(11)Vcellt=VOCt−Rintt⋅Icellt−1,(12)Pchgmaxt=−VoutmaxVoutmax−VOCt⋅NsRintt⋅Nt,(13)Pdismaxt=VoutminVOCt⋅Ns−VoutminRintt⋅Nt,(14)P˜outt=PdismaxtPoutt≥PdismaxtPouttPdismaxt≥Poutt≥Pchgmaxt,PchgmaxtPoutt≤Pchgmaxt.(15)Icellt=P˜outtVoutct⋅Ns⋅Np,Iratet=IcelltKcell⋅δsoh,(16)δsoct=δsoct−1−Iratet⋅Δt.Table 3
Battery equivalent circuit model parameters.
ParameterDefinitionValueNsNumber of cells in series in each module96NpNumber of parallel modules2NtTotal number of cells in battery pack192VOCtCell open-circuit voltage at timet (V)Using lookup-tableRinttCell internal resistance at timet (ΩUsing lookup-tableIcelltCell current at timet (A)Icell0=0VcelltCell terminal voltage at timet (V)VoutminMinimum battery terminal voltage (V)336VoutmaxMaximum battery terminal voltage (V)403.2PouttBattery power requirement at timet (W)P˜outtActual battery power output at timet (W)PdismaxtMaximum battery power output while discharging at timet (W)PchgmaxtMaximum battery power input while charging at timet (W)IratetC-rate at timetKcellRated cell capacity (Ah)33.1δsoctBattery state-of-charge at timetδsohBattery state-of-healthΔtDiscrete time step (seconds)1Figure 1
Cell internal resistance values at different SOC.Figure 2
Cell open-circuit voltage values at different SOC.


## 2.4. Battery Degradation Model

Battery degradation is affected by several factors such as battery temperature, SOC, C-rate, and total current throughput (Ah-throughput) [48]. Some models approximate the total Ah-throughput as a product of constant depth-of-discharge and the number of cycles [23]; thereby, ignoring the effects of SOC and C-rate. This framework adopts a semiempirical battery degradation model proposed by Wang et al. [21] that includes three important parameters: time (or battery age), temperature, and C-rate. It computes calendar aging (Qcal) as a function of time (τ) and temperature (T) and cycle aging (Qcyc) as a function of temperature, C-rate (Irate), and lifetime Ah-throughput (Iah). The model parameters are fitted using experimental aging data for high-power density 1.5 Ah, 18650 cylindrical cells with LiMn1/3Ni1/3Co1/3 + LiMn2O4 (NCM + LMO) cathode and a graphite anode. Their results indicate that the predicted values are within ±5% of the measured battery capacity loss. The model can be expressed as equations (17)-(19).(17)QT=Qcal+Qcyc,(18)Qcal=fτ0.5⋅exp−EaRT,(19)Qcyc=aT2+bT+c⋅expdT+e⋅Irate⋅Iah.To account for the variable C-rate and temperature, the model is modified by taking the differentiating total capacity loss function with respect to a discretized time step (Δτ). For each time step, the current flow is assumed to be constant, and instantaneous capacity loss (DT) is computed as the sum of instantaneous capacity loss Dcap and instantaneous cycle loss Dcyc by taking the differential of their respective functions. The sum of instantaneous capacity loss is updated to get the total capacity loss until the battery replacement threshold limit δrep is reached. Hence, the battery lifespan is equal to the number of days elapsed before the total capacity loss reaches the specified threshold limit. Because Reference [49] has a rated nominal cell capacity of 33.1 Ah and the data are calibrated for 1.5 Ah cells, a correction factor βcorr equal to the ratio of rated nominal capacities is multiplied by the cycle loss. Equations (20)-(22) describe the modified model, and Table 4 shows the parameter values.(20)DT=Dcal+Dcyc,(21)Dcal=0.5fτ−0.5⋅exp−EaRT,(22)Dcyc=aT2+bT+c⋅expdT+e⋅Irateτ⋅Icellτ⋅βcorr.Table 4
Battery degradation parameter values and units.
ParameterValue and unitParameterValue and unita8.61e −  (1/Ah-K2)IahLifetime current throughput (Ah)b−5.125e −  (1/Ah-K)IrateτC-rate at timeτc0.7629 (1/Ah)IcellτCell current at timeτ (A)d−6.7e − 3 (1/K-(C-rate))βcorr1.5/33.1e2.35 (1/(C-rate))Ea24500 (J/mole)f14876 (1/day0.5)R8.314 (J/K-mole))τTime (in days)TTemperature (K)ΔτDiscretized time step (in days)δrep30%This model uses ambient temperature as a proxy for battery temperature, thereby assuming that the thermal effects on internal resistance and cell current are negligible. This assumption can be addressed by including a suitable electrochemical-thermal model in the framework (see Reference [50] for a review) or by using a 1D lookup table to link battery temperature with the environment temperature and battery cell characteristics (e.g., [51, 52]. The model parameters are calibrated for discharging current only. Thus, it is assumed that charging and discharging will have a similar impact on cycle aging based on the absolute value of Irate.


## 3. Data: Case Study

A case study is presented using the proposed framework for the city of Indianapolis, Indiana, U.S. Real-world household vehicle travel pattern data were extracted from U.S. National Household Travel Survey (NHTS) of 2009 [38]. The NHTS dataset contains 1-day travel diary data of 821 vehicles (with vehicle type as “car”) in the state of Indiana. The key variables of the NHTS dataset include household ID, vehicle ID, vehicle type, trip departure time, trip arrival time, and trip travel time for all trips made by the household in a single day. Since most affordable EVs have a driving range of about 80-100 miles, vehicles with trip distances exceeding 80 miles for any single trip are excluded from the analysis to avoid situations with EVs running out of battery in the middle of a trip. In the preliminary data analysis, we observed that several vehicles having longer total daily distance did not have sufficient time between trips to recharge the battery. Thus, based on our preliminary data inspection, we also excluded vehicles with total daily distance exceeding 120 miles from the analysis. These distance-based exclusion criteria reduced our dataset to 760 vehicles. A microscopic traffic simulation model is created to generate realistic drive cycles for the vehicles using the simulation software Aimsun [53]. A detailed road network of Indianapolis containing all major freeways, most urban roads, and some minor roads is built in AIMSUN (see Figure 3). A dynamic 15-minute time period origin-destination traffic demand matrix is simulated for a 24-hour time horizon with an additional 1-hour warm-up period. The traffic demand is calibrated using NHTS trip data for Indiana. The departure time period, distance traveled, travel time, and speed profile of each simulated vehicle are recorded. We generated 41,736 unique simulated drive cycles. Each NHTS trip is matched to a simulated drive cycle which has the least Euclidean distance in terms of both trip distance and average speed. NHTS households with missing simulated drive cycle data for any number of trips are excluded from the analysis. Since most affordable EVs have a driving range of about 80-100 miles, vehicles with a trip distance exceeding 80 miles for any single trip or total daily distance exceeding 120 miles are also excluded. In the end, a total of 3,225 trips made by 697 vehicles were analyzed. Most parameter values in the battery equivalent circuit model and degradation model are taken for Reference [43] with a 24 kWh Li-ion battery. The battery degradation threshold is taken as 30%, that is, the battery is considered to be unusable once its SOH is 70%. Daily average temperature values for Indianapolis in the year 2018 (see Figure 4) are used in a loop to compute both calendar and cycle aging [54]. Since degradation computation is performed for each time step, it allows the use of higher resolution temperature data (e.g., hourly temperature) for more accuracy. Due to data limitations, it is assumed that household vehicle travel patterns remain unchanged until the end of useable battery life. This assumption can be relaxed with the availability of a richer travel pattern dataset. The vehicle is assumed to be fully charged to its SOH level at the start of every day. Opportunistic charging behavior is assumed during the day with a constant charging power of 7 kWh.Figure 3
Road network of Indianapolis, U.S., used in the simulation.Figure 4
Daily average temperature values for Indianapolis, U.S., in 2018.


## 4. Results and Discussion

The impacts of driving behavior and travel patterns on battery lifespan are analyzed using the proposed multiparadigm modeling framework. Figure5 shows the daily distance traveled distribution for the case study data. More than 80% of households travel less than 50 miles per day. Using simulated drive cycles allows the analysis of heterogeneity in driving behaviors such as average speed and speed deviation. The average speed and speed deviation for all households’ combined daily drive cycles of all trips are illustrated in Figure 6. The population means of average speed and speed deviation are 37.9 mph and 16.4 mph, respectively.Figure 5
Daily distance traveled distribution.Figure 6
Speed deviation vs. average speed for combined daily drive cycles of households.Most automobile manufacturers offer battery warranties based on either battery age, total distance traveled, or their combination as a threshold. For example, Reference [46] covers the necessary repairs needed to return battery capacity to about 75% of the original capacity for a period of 5 years or 60,000 miles, whichever comes first [49]. The case study analysis results indicate that almost 50% of the batteries reach their SOH threshold within 5 years and 90% within 8 years. The useable battery lifespan distribution is illustrated in Figure 7. In terms of total distance traveled before EOL, the 50% and 90% quantiles for battery lifespan are approximately 44,500 miles and 72,500 miles, respectively.Figure 7
Useable battery lifespan distribution.To analyze the impact of driving behavior, the average speed and speed deviation for each household are classified as “higher” or “lower” groups based on their value compared to their respective population means. The results indicate that there is little or no difference between the higher and lower average speed groups for the same total distance traveled before EOL (see Figure8). On the contrary, an apparent decrease in battery lifespan can be observed among households with higher speed deviation than those of lower speed deviation (see Figure 9). A higher speed deviation entails more fluctuations in kinetic energy due to acceleration and deceleration (see Section 2.2), which increases the vehicle’s overall power consumption. Hence, the difference in lifespan is caused by this additional amount of power required, and thereby, the current flows through the battery to balance the fluctuating kinetic energy at variable speed. These findings illustrate the importance of using realistic and diverse drive cycles over combinations of standard driving schedules to account for differences in driving behaviors.Figure 8
Battery lifespan vs. total distance traveled for average speed-based classification.Figure 9
Battery lifespan vs. total distance traveled for speed deviation-based classification.


## 5. Concluding Comments

This study presents a multiparadigm modeling framework to quantify the useable battery lifespan of a large population of EVs. The inclusion of a microscopic traffic simulation model in the framework enables the analysis of driving behavior heterogeneity in battery lifespan estimation. Each building block of the multiparadigm approach provides the flexibility to include newer models or real-world cases without affecting the framework structure. For example, the traffic simulation model can be replaced by real-world vehicle speed data if such data are available. This study uses the vehicle and battery specifications of Reference [49] in the case study, but the proposed framework can be used to assess other EVs, such as e-rickshaws, by changing the parameter values accordingly. The impacts of vehicle travel patterns and driving behavior (e.g., speed deviation) on EV battery lifespan under regional temperature trends can provide useful information for vehicle owners, policymakers, and vehicle manufacturers. The framework can be used by vehicle owners to assess the lifetime cost of EV ownership, including battery maintenance cost, insurance cost, and battery resale value, based on their travel needs, driving behavior, and geographic location. In addition, policymakers can use battery lifespan distribution for regional temperature and traffic conditions to design incentive strategies, such as tax credits and extended battery warranties, to regulate EV adoption trajectories. It allows vehicle manufacturers to factor in regional conditions and consumer driving behaviors while evaluating the performance and economics of different batteries by modifying the battery equivalent circuit model and degradation model accordingly. The proposed framework can be utilized to enhance EV research applications such as battery management strategies (e.g., maximum/minimum SOC range) to enhance battery lifespan, battery recharging strategies, vehicle-to-grid applications, etc., by including realistic vehicle travel patterns and driving behavior. Furthermore, there is an ongoing argument among researchers to promote electric-powered autonomous vehicles (AVs) over gasoline-powered AVs to limit greenhouse gas emissions and mitigate negative environmental impacts. The proposed framework can be used to assess the battery life and, thereby, life cycle environmental impacts of a large population of electric-powered AVs. This research can be extended in different directions. The first potential future research direction is to integrate the electrochemical-thermal effects into the battery degradation model. This integration enables for more accurate battery health state monitoring as it accounts for the external measurements of terminal voltage, applied current, and surface temperature of the battery [13, 55, 56]. The second research direction is to compare the effects of travel patterns on fuel consumption between EVs and traditional internal combustion engine vehicles. The third research direction is to consider the battery degradation model for different types of batteries used in EVs. The battery degradation of different EV batteries varies based on the charging/discharging rate, number of cycles, and temperature. The detailed information for some recent battery technology is presented by Zhao et al. [57] and Mathieu et al. [58]. The disaggregate battery degradation model should be calibrated for each type of EV to increase the accuracy of the results.



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*Source: 1689075-2023-02-16.xml*