One-Dimensional Simulation of Real-World Battery Degradation Using Battery State Estimation and Vehicle System Models

Abstract

This study aims to develop a method for analyzing real-world battery degradation in electric vehicles in order to identify the optimal battery management system (BMS) during the early digital phase of vehicle development. Battery management of lithium-ion batteries (LiBs) in electric vehicles is important to ensure a stable output and to counteract degradation and thermal runaway. To design the optimal system, it is most effective to use a 1D (one-dimensional) vehicle system simulation model, which connects each unit model inside the vehicle, due to the system’s complexity. In order to create a long-term degradation simulation in a vehicle system model, it is important to reduce computational load. Therefore, in this paper, we studied a suitable battery degradation calculation for the vehicle system model based on an equivalent circuit model (ECM) and degradation approximation formulas. After implementing these models, we analyzed long-term degradation behavior through the real-world operation of an electric vehicle driver. We first implemented a high-accuracy ECM using transient charge–discharge tests and Bayesian Optimization. Next, we formulated approximation formulas for degradation prediction based on calendar and cycle degradation tests. Finally, we simulated real-world degradation behavior using these models. The simulation results revealed that even for users who frequently use electric vehicles, degradation under storage conditions is the dominant factor in overall degradation.

1. Introduction

As environmental and energy concerns grow, such as those regarding the increasing concentration of greenhouse gases and the depletion of fossil fuels, the global movement towards achieving carbon neutrality is accelerating [1]. Consequently, in the transportation sector, electric vehicles such as Battery Electric Vehicles (BEVs), Hybrid Electric Vehicles (HEVs), and other electrified vehicles are being actively introduced [2].

For energy storage batteries in electric vehicles, LiBs are the most viable option. However, it is essential to ensure stable output and implement countermeasures against degradation and thermal runaway [3]. Since these issues are influenced by electrochemical reactions, it is necessary to appropriately manage the battery state, including its temperature and state of charge (SOC). Therefore, the BMS, which coordinates heat and energy flow within the vehicle, is essential.

To design the optimal management system, it is most effective to use a 1D vehicle system simulation model, as shown in Figure 1. This model implements the thermal, electrical, and mechanical functions of each unit within the vehicle in order to study the wide variety of unit layouts and heating/cooling options [4]. However, due to the computational load challenges of the electrochemical reactions within this model, the main objective of such analyses has been to predict the temperature of each component, including the LiB, during short-term driving modes such as the WLTC (Worldwide Harmonized Light Vehicles Test Cycle) [5,6].

While there are research cases of long-term degradation analysis in 1D models, these are limited to models that calculate degradation of the battery and vehicle separately [7]. Furthermore, battery degradation is known to partly depend on the applied current [8], and this current is dynamically controlled by the BMS based on the status of both the battery and vehicle [9]. Therefore, in order to accurately predict the current and simulate degradation correctly, it is necessary to integrate the vehicle and battery models and estimate the battery voltage with high precision.

Consequently, in this paper, we studied a suitable configuration for battery degradation simulation within a vehicle system model, based on a battery state estimation method using ECM [10] and a degradation approximation formula. First, we provide an outline of the proposed battery degradation simulation system. Next, we implement a voltage estimation model based on the ECM using transient charge–discharge test data from a high-capacity ternary LiB. To improve prediction accuracy efficiently, the optimal parameters of the model are studied using Bayesian Optimization [11]. Subsequently, we formulate approximation formulas for degradation prediction based on calendar and cycle degradation tests. Finally, applying these models, we analyze long-term degradation behavior through the real-world operation of a BEV driver.

By implementing this proposed method, it will be possible to reasonably verify the impact of various BMS configurations on battery degradation in the real world within a short period during the early digital phase of vehicle development. This will enable the design of the optimal BMS configuration and is expected to contribute to preventing unnecessary revisions in the later stages of development.

2. Outline of Proposed Battery Degradation Simulation Method

The purpose of this study is to develop a simplified battery degradation simulation method that can be implemented in a vehicle system model to simulate long-term scenarios where the vehicle and battery interact with each other. In this chapter, we first introduce the specifications of the battery used in this research. Next, we provide an outline of the proposed battery degradation simulation method, which is integrated within the vehicle system model.

2.1. Introduction to the Reasearch Battery

To analyze real-world degradation in BEVs, we used a high-capacity lithium-ion battery in this research. The basic battery specifications are shown in

Table 1. Specifications of the research battery.

Cell Format	Prismatic
Cathode material	NMC
Anode material	Graphite
Nominal voltage	V	3.67
Nominal capacity	Ah	50
Dimensions	mm	W148 × H102 × L28
Weight	g	900

. The battery used is a mass-produced prismatic lithium-ion battery with a nominal voltage of 3.67 V and a capacity of 50 Ah. It features an NMC (nickel–manganese–cobalt) cathode and a graphite anode. For EV applications, these cells are typically arranged into modules, with several modules further integrated to form a battery pack. In this paper, we implemented a voltage and degradation estimation model for this battery into our vehicle system model in order to analyze its real-world degradation.

2.2. Overview of Proposed Battery Degradation Simulation Method

To achieve long-term degradation simulation in a vehicle system model, it is important to reduce implicit solver calculations to mitigate the computational load [12]. Therefore, we studied an explicit method to estimate both the electrical and degradation behavior of the battery. The electrical estimation was based on the ECM that calculates the battery voltage, shown below:

$$U=OCV-I{R}_0-\sum _{i=1}^{N_{RC}}\Delta U_{RC,i} (i=\mathrm{1,2},\dots ,N_{RC})$$

(1)

The elements of the above equation are defined as follows:

$U$	Terminal voltage [V];
$OCV$	Open circuit voltage [V];
$I$	Current [A];
$R_0$	Internal resistance [ohm];
$N_{RC}$	Number of RC (resistor–capacitor) circuits [-];
$\Delta U_{RC,i}$	Voltage drop for $i$-th RC circuit [V].

The second term of the equation represents ohmic polarization, and the third term accounts for activation and diffusion polarization. The third RC circuit term calculates the voltage drop using a differential equation, which is an implicit method. To address this, we converted it to an explicit formula as follows.

$$\Delta U_{RC,i}\left[t\right]=\Delta U_{RC,i}\left[t-1\right]+\left(-{\displaystyle \frac{1}{R_i{C}_i}}\Delta U_{RC,i}\left[t-1\right]+{\displaystyle \frac{1}{C_i}}I[t]\right)\Delta t$$

(2)

The elements of the above equation are defined as follows:

$t$	Time [s];
$R_i$	Resistance for $i$-th RC circuit [ohm];
$C_i$	Capacitor for $i$-th RC circuit [F].

Figure 2 shows the ECM configuration employed in this research, which includes a single internal resistance (as shown in Formula (1)) and three RC circuits to model voltage changes occurring over several, tens, and hundreds of seconds. Subsequently, for degradation estimation, we employed approximation formulas that calculate calendar and cycle capacity fade based on the respective accumulated values. The details of these formulas will be explained in a subsequent chapter.

Figure 3 shows an overview of the proposed degradation simulation framework, which integrates the battery state estimation models described above. First, the simulation model inputs a target vehicle speed to the driver model. The driver model then calculates the torque demand via PID feedback based on the difference between the target and actual vehicle speeds. Subsequently, the controller model determines the motor’s target torque, considering the desired torque along with battery condition. It also calculates the auxiliary load based on the ambient temperature. The vehicle model then calculates the vehicle speed in the next step using this motor torque. These procedures form an iterative loop that updates the vehicle speed at each time step.

Within the battery model, the battery power demand is determined from the preceding process, and the current is then calculated as follows:

$$I={\displaystyle \frac{T_{mot}{\omega }_{mot}+P_{loss}+P_{aux}}{U{N}_{cell}}}$$

(3)

The elements of the above equation are defined as follows:

$T_{mot}$	Motor torque [Nm];
${\omega }_{mot}$	Motor angular velocity [rad/s];
$P_{loss}$	Motor loss [W];
$P_{aux}$	Auxiliary load [W];
$N_{cell}$	Number of cells [-].

Notably, this terminal voltage is derived from the aforementioned ECM estimation model. This involves iterating through a process that calculates the ECM’s Formulas (1) and (2), followed by Formula (3). For aging calculations, the degradation formula feeds back an aging factor to the ECM; this factor depends on the elapsed time for calendar degradation and the accumulated current for cycle degradation. The method employed to reflect the aging factor is as follows. First, capacity fade is shown below:

$${SOC\left[t\right]}_{deg}={SOC[t-1]}_{deg}-{\displaystyle \frac{I\left[t\right]\Delta t}{Q_{As}{(1-k}_{cap})}}$$

(4)

The elements of the above equation are defined as follows:

${SOC}_{deg}$	SOC accounting for degradation [-];
$Q_{As}$	Battery capacity [As];
$k_{cap}$	Aging factor representing capacity fade [-].

Subsequently, internal resistance increase is shown below:

$$U_{deg}=OCV-I{R}_0(1+k_{res})-\sum _{i=1}^n\Delta U_{RC,i} (i=\mathrm{1,2},\dots ,N_{RC})$$

(5)

The elements of the above equation are defined as follows:

$U_{deg}$	Terminal voltage accounting for degradation [V];
$k_{res}$	Aging factor representing internal resistance increase [-].

As will be discussed later, we attribute the degradation to the formation of a film on the electrode and structural disorder, which increase the electrode contact resistance. Therefore, in the current equation, we assumed that aging affects only the internal resistance, but in reality, some aging mechanisms can manifest in the RC arrays.

The model sets the ambient and battery temperatures as boundary conditions to clearly compare the differences in degradation progression due to temperature. While using temperature as a boundary condition helps clarify its impact, the absence of a dynamic thermal model may reduce realism under extreme or rapidly changing conditions. In this paper, we prioritized understanding the temperature effects on long-term degradation tendency by adopting boundary conditions. Additionally, we employed GT-SUITE [13] for overall vehicle system simulation and GT-AutoLion1D [14] for detailed battery electrochemical reaction simulation.

3. Implementation of Voltage Estimation Model

In this chapter, we explain battery voltage estimation within a vehicle system model using the ECM. In order to efficiently implement a high-accuracy ECM estimation, we performed the procedures outlined in the following sections. First, we conducted transient charge–discharge tests to observe battery voltage responsiveness. We then implemented the Doyle–Fuller–Newman (DFN) model [15], a battery physics-based model, to efficiently collect battery voltage behavior data for ECM modeling. Finally, we adopted Bayesian Optimization to calibrate the ECM parameters for high-accuracy estimation.

3.1. Introduction of the Transient Charge–Discharge Test

In order to observe and model the battery voltage responsiveness, we conducted transient load tests on a single cell under the conditions presented in

Table 2. Conditions for the transient charge–discharge test.

Test Case	Ambient Temperature	Initial SOC	Sequential Pattern of Current Load
	°C	%
Square wave	25, 45	50
Vehicle driving load (WLTC)		100

. For the first condition, a square current waveform was applied to obtain the voltage change characteristics during both charging and discharging, and upon interruption of the current. Specifically, the procedure involved 1C discharging from 50% SOC for 180 s, followed by an equal rest period. This was then succeeded by 180 s of 1C charging, again followed by an equal rest period. Subsequently, the entire current load application sequence was reversed, beginning with charging. Throughout the test, the ambient temperature was controlled under two conditions (25 °C and 45 °C) using a constant-temperature chamber.

For the second condition, we conducted the test under current loads that mimicked the WLTC mode, which represents actual vehicle driving load fluctuations. During the extra-high driving scenario, the maximum discharge current was approximately 1.5 C to 2 C. The test began from a fully charged SOC, and the ambient temperature conditions were the same as in the first condition. The measurement results, along with the model’s prediction results, will be presented in the next section.

3.2. Implementation of the DFN Model for Data Augmentation

To collect the battery voltage behavior data efficiently, we implemented a battery physics-based model using the test results in Table 2. In this study, we adopted the DFN model, which employs governing equations to describe the electrochemical reactions and ion transport within the battery [15]. The model performs calculations by satisfying the conservation of charge transfer in both the solid and liquid phases, and the conservation of chemical species in the active material and electrolyte. Regarding charge transfer kinetics, the current density of the electrochemical reaction is represented by the Butler–Volmer equation, shown below:

$$j=j_0\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{{\alpha }_{a}F\eta }{RT}}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{{\alpha }_{c}F\eta }{RT}}\right)\right]$$

(6)

The elements of the above equation are defined as follows:

$j$	Current density [A/m2];
$j_0$	Exchange current density [A/m2];
${\alpha }_a$	Charge transfer coefficient for anode [-];
${\alpha }_c$	Charge transfer coefficient for cathode [-];
$F$	Faraday’s constant [C/mol];
$R$	Universal gas constant [J/molK];
$T$	Temperature [K];
$\eta$	Activation overpotential [V].

Additionally, for species conservation, the concentration of neutral lithium in the solid phase is represented by Fick’s second law, shown below:

$${\displaystyle \frac{\partial c_s(r,t)}{\partial t}}={\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left(D_s{r}^2{\displaystyle \frac{\partial c_s(r,t)}{\partial r}}\right)$$

(7)

The elements of the above equation are defined as follows:

$c_s$	Solid-phase concentration [mol/m3];
$r$	Particle radius [m];
$D_s$	Solid-phase diffusion coefficient [m3/s].

Consequently, this model requires the setting of internal battery parameters, such as active material particle size, electrode porosity, and solid- and liquid-phase diffusion coefficients. Therefore, these parameters were fitted to align the simulated voltage behavior with the measured data. For the material property parameters, we utilized baseline values from the simulation software’s [14] database and fitted correction factors for these values.

Figure 4 compares the measured data with the DFN model simulation. Under both the square wave and vehicle driving conditions, the root-mean-square error (RMSE) between the measured and predicted voltages was below 5 mV. This suggests that the model could reproduce the measured voltage behavior with high accuracy. However, the DFN model is not suitable for vehicle layer simulation due to its high computational load. In the next section, we implement the ECM based on simulation data generated by this model.

3.3. Implementation of ECM Voltage Estimation Using Bayesian Optimization

To implement a high-accuracy ECM estimation model, the internal resistance and RC network parameters of the model need to vary with battery temperature and SOC. For this reason, and to prevent noise in the measured data, the prepared DFN model was simulated under the conditions presented in

Table 3. Conditions for the ECM’s parameter fitting.

Test Case	Battery Temperature	Initial SOC	Sequential Pattern of Simulation Current Load
	°C	%
Square wave with vehicle driving load	25, 45	30, 50, 70, 90

; the results were subsequently used for the ECM fitting. The model was simulated at two temperature levels (25 °C and 45 °C) and four SOC levels (30%, 50%, 70%, and 90%), covering normal to high temperatures and low to high SOCs, respectively. Additionally, the simulation current pattern was composed of the aforementioned rectangular wave and vehicle driving load, with rest periods incorporated to achieve an equilibrium state before the last half of the load pattern. This combination was used to reproduce both fast and slow voltage responses by the ECM.

ECM parameter fitting requires a significant number of trials because the parameters for each battery temperature and SOC condition must be optimized separately for both charge and discharge phases. Note that the number of parameters in this case is approximately 70. Therefore, in this study, we adopted Bayesian Optimization [11] instead of general optimization methods such as the Genetic Algorithm [16] to efficiently find the optimized parameters of each condition. Bayesian Optimization is an optimization method based on Bayesian theorem [17] to find the best possible settings of an unknown and complex function. The Bayesian theorem is shown below:

$$P\left(A|B\right)={\displaystyle \frac{P\left(B|A\right)P\left(A\right)}{P\left(B\right)}}$$

(8)

The elements of the above equation are defined as follows:

$P\left(A|B\right)$	Posterior [-];
$P\left(B|A\right)$	Likelihood [-];
$P\left(A\right)$	Prior [-];
$P\left(B\right)$	Marginal [-].

This optimization uses Gaussian Process Regression (GPR) [18] to predict the true function from existing points and an acquisition function to decide the next point to observe. The GPR predictions and the acquisition function’s decision are repeatedly performed to identify the extremum. We defined the observation point as the battery voltage RMSE between the ECM and the DFN model, shown below:

$${RMSE}_k=\sqrt{{\displaystyle \frac{1}{N_k}}\sum _{t=t_{start,k}}^{t_{end,k}}{\left(U_{ECM}\left(t\right)-U_{DFN}\left(t\right)\right)}^2} (k=wave,veh)$$

(9)

$$J_{opt}={\displaystyle \frac{{RMSE}_{wave}+{RMSE}_{veh}}{2}}+w_{bias}\left|{RMSE}_{wave}+{RMSE}_{veh}\right|$$

(10)

The elements of the above equation are defined as follows:

$J_{opt}$	Objective function for optimization [-];
${RMSE}_k$	RMSE for section $k$ [V];
$N_k$	Number of samples for section $k$ [-];
$t_{start,k}$	Start time for section $k$ [s];
$t_{end,k}$	Terminal for section $k$ [s];
$U_{ECM}$	Calculation voltage for ECM [V];
$U_{DFN}$	Calculation voltage for DFN [V];
$w_{bias}$	Weight factor for bias term [-].

In this case, Bayesian Optimization finds the minimum point to minimize the voltage error. In order to prevent accuracy bias between the square wave and vehicle driving load, we adopted an error balance term within the formula. Furthermore, for the optimization procedure, we performed the optimization three times for each condition, with each optimization trial consisting of 200 iterations, to explore a more valid global minimum point. At the first optimization, we explored the ECM parameters within their acceptable range. Then, for the second optimization, we explored the parameters within ±50% of the values obtained from the first optimization. The third optimization further narrowed this exploration to ±25% of the values obtained from the second optimization. Notably, conditions with the same temperature but different SOCs were grouped and optimized together as a single condition to enable continuous parameter variation as a function of SOC.

Figure 5 shows a portion of the comparison results between the ECM and the DFN model simulations. By gradually narrowing the search range, we successfully identified the optimal parameters. Initially, the ECM provided a rough trace of the DFN model’s behavior, with difficulty in reproducing fine details, but by the third optimization, it accurately replicated the minute features. As a result, the voltage RMSE against the DFN model was below 10 mV for all conditions, as shown in

Table 4. Summary of the ECM parameter optimization.

Battery Temperature	Initial SOC	RMSE of Voltage at Each Optimization Iteration (Average of Square Wave and Vehicle Driving Load)
		First	Second	Third	Summary
°C	%	mV
25	30	6.0	4.9	4.2
	50	6.1	5.4	5.0
	70	5.1	4.3	3.9
	90	8.7	7.5	5.4
45	30	5.7	4.8	3.8
	50	6.4	3.8	3.8
	70	3.8	3.8	2.9
	90	5.7	6.4	5.5

. We successfully developed an ECM voltage estimator that achieved comparable performance to the physics-based model.

4. Implementation of Degradation Estimation Model

In this chapter, we explain the battery degradation estimation within a 1D vehicle system model using an approximation formula. In order to predict practical degradation trends, we conducted the procedures outlined in the following sections. We first studied the major factors of battery degradation. We then conducted calendar and cycle degradation tests to observe long-term battery degradation behavior. Lastly, we implemented calendar and cycle approximation formulas based on the studied phenomena.

4.1. Study of Degradation Phenomena

In degradation analysis using a vehicle system model, capturing degradation trends with reduced computational load is prioritized over reproducing detailed degradation phenomena [7]. Therefore, it is necessary to select the major factors for calendar degradation and cycle degradation, respectively, from the overall degradation factors. Figure 6 shows a simplified diagram of battery degradation illustrating the possible major factors based on the analysis software utilized in this study.

The first degradation phenomenon is SEI growth on the anode. On the surface of the anode’s active material, ethylene carbonate from the electrolyte undergoes a side reaction, combining with lithium ions to form an SEI film [19]. This chemical formula is shown below:

$${\mathrm{L}\mathrm{i}}^{+}+{\mathrm{e}}^{-}+\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{e}\to \mathrm{S}\mathrm{E}\mathrm{I}$$

(11)

The next phenomenon is electrode structural disorder. The intercalation and deintercalation of lithium ions within the electrodes lead to the expansion and contraction of the active material. Repeated charging and discharging cycles cause stress from these volumetric changes, which degrades the connections between active material particles, leading to their isolation [20]. Subsequently, the third phenomenon is the transition layer of the cathode. On the surface of the cathode’s active material, repeated charging and discharging have been observed to alter the surface crystal structure [20]. Finally, the last phenomenon is lithium plating on the anode. During cell charging, if the anode potential falls below the lithium plating potential (0.0 V vs. Li/Li+), lithium metal is deposited on the surface of the active material due to the reduction reaction of lithium ions [21]. This chemical formula is shown below:

$${\mathrm{L}\mathrm{i}}^{+}+{\mathrm{e}}^{-}\to \mathrm{L}\mathrm{i}$$

(12)

In each of these degradation phenomena, the capacity decreases due to the consumption or isolation of lithium ions. Additionally, the internal resistance increases as the active material’s reaction area is reduced by film formation or structural changes.

4.2. Introduction of Calendar and Cycle Degradation Tests

To observe and model battery degradation during storage and cycling, we conducted calendar and cycle degradation tests under the conditions presented in

Table 5. Conditions of the calendar and cycle degradation tests.

Test Case	Ambient Temperature	Storage Condition		Number of Cycles	Voltage Range of Cycling	C Rate of Current Load
		Duration	SOC
	°C	Day	%	Cycle	V	C
Calendar	25, 45	Approx. 360	100	-	-	-
Cycle		-		3000	4.2 to 2.8 (Approx. 100% DOD)	±1.0

. For the calendar test condition, the storage duration was approximately 360 days with the battery fully charged, and the ambient temperature was controlled under two conditions (25 °C and 45 °C). The capacity was checked at 25 °C by discharging at a 1/5 C rate (from 4.2 V to 2.8 V) and charging at a 1/5 C rate (from 2.8 V to 4.2 V). Additionally, Direct Current Internal Resistance (DC-IR) was measured at 25 °C and 50% SOC using 10 s charge/discharge pulses at 1/2 C, 1 C, and 2 C rates. Initially, these measurements were taken frequently to observe inflection points in degradation, while in the later stages, they were performed at longer appropriate intervals.

For the cycle test condition, the number of cycles was 3000, with each cycle involving a 1 C rate charging and discharging between 4.2 V and 2.8 V, assuming a DOD (depth of discharge) of 100%. Similarly, the measurement procedures were identical to those of the calendar test, and they were conducted every 200 cycles. The measurement results, along with the model’s prediction results, will be presented in subsequent sections.

4.3. Implementation of Calendar Degradation Estimation

The plots in Figure 7 show the capacity and internal resistance aging progression from the calendar degradation test, specifically displaying the averaged charging and discharging phases. These results indicate that the calendar degradation progresses with a regular dependency on both time and temperature, following a square-root-of-time relationship and increasing under high-temperature conditions.

In previous research, SEI growth has been identified as the primary degradation phenomenon during calendar aging, with its progression reported to follow a square-root-of-time dependency [22]. Therefore, the approximation formula implemented in this study was fundamentally based on the square root of time with the Arrhenius equation. However, it was found that this term alone did not align with the initial degradation rise in the measured data. Based on this finding, we added an additional exponential term to the formula to account for rapid degradation progression in the early phases, with degradation subsequently depending on time. This formula was represented as

$$k_{cal,l}=A_{cal,l}\exp \left({\displaystyle \frac{-E_{cal,l}}{RT}}\right)t^{0.5}+B_{cal,l}\left(1-\mathit{exp}\left(-C_{cal,l}\ast t\right)\right) (l=cap,res)$$

(13)

The elements of the above equation are defined as follows:

$k_{cal,l}$	Calendar aging factor for characteristic $l$ [-];
$A_{cal,l}$	Pre-exponential factor of calendar aging for characteristic $l$ [-];
$E_{cal,l}$	Activation energy of calendar aging for characteristic $l$ [-];
$B_{cal,l}$	Correction factor of the additional exponential term of calendar aging for characteristic $l$ [-].
$C_{cal,l}$	Correction factor in the additional exponential term of calendar aging for characteristic $l$ [-].

To describe this formula physically, after the initial SEI formation, the subsequent growth requires ethylene carbonate to diffuse through the existing SEI layer to reach the active material surface, as shown in Figure 8. Therefore, it was inferred that degradation progresses rapidly in the early phases when the SEI film is thin.

The lines in Figure 7 show the estimation results for calendar capacity and internal resistance degradation. For both capacity and internal resistance, we confirmed that the formula reproduced the measured degradation trends well. Specifically, the coefficient of determination was approximately 1 for capacity degradation and 0.8 for internal resistance degradation. In addition, we used nonlinear least squares fitting to derive the parameters of this approximation formula.

4.4. Implementation of Cycle Degradation Estimation

The plots in Figure 9 show the capacity and internal resistance aging progression from the cycle degradation test, specifically displaying the averaged charging and discharging phases. However, as the calendar degradation factor is also present during cycle tests, the real cycle degradation characteristics were derived by subtracting the calendar degradation incurred during the cycle test from the total degradation observed in the cycle tests.

Consequently, the real cycle degradation (shown in Figure 10) was derived by subtracting the calendar degradation factor using the implemented Formula (13). Since we controlled the charging and discharging times using a specific voltage range, we normalized the x-axis using the accumulated absolute value of the applied current to analyze degradation across different conditions fairly.

In the case of the cycle aging, capacity degradation showed a regular dependency on the integrated current and temperature, while internal resistance degradation exhibited strong nonlinearity. In previous research, structural disorder of the electrode has been identified as the primary degradation phenomenon during cycle aging [19]. Simultaneously, the collapse of SEI due to charging and discharging stress has been reported [19]. This contrasts with the monotonic capacity fade resulting from irreversible lithium-ion consumption, as SEI collapse can also reduce internal resistance, contributing to its more complex degradation behavior. Predicting this complex internal resistance degradation with an approximation formula proves challenging and can lead to erroneous conclusions. Moreover, for the real-world degradation analysis presented in the subsequent chapter, the battery’s accumulated current will not be operated to reach levels that cause significant degradation, such as 1.5 or 2 times the original value. Therefore, we have decided to implement only the capacity degradation formula for cycle aging in this study. The formula of cycle capacity degradation is shown below:

$$k_{cyc,cap}=A_{cyc,cap}\exp \left({\displaystyle \frac{-E_{cyc,cap}}{RT}}\right)\sum _{m=0}^t\left|I\left[m\right]\right|\Delta t$$

(14)

The elements of the above equation are defined as follows:

$k_{cyc,cap}$	Cycle aging factor for capacity [-];
$A_{cyc,cap}$	Pre-exponential factor of cycle aging for capacity [-];
$E_{cyc,cap}$	Activation energy of cycle aging for capacity [-].

This formula includes a linear factor based on the accumulated absolute current and a temperature factor derived from the Arrhenius equation.

The lines in Figure 9 and Figure 10 show the estimation results for cycle capacity degradation. We confirmed that the predicted degradation, incorporating both cycle and calendar effects, reproduced the measured degradation trends well. Specifically, the coefficient of determination was approximately 0.9 for both true cycle degradation and the combination of cycle and calendar degradation.

5. Discussion on Real-World Degradation Analysis

In this chapter, we discuss real-world degradation analysis using a vehicle system model that incorporates the developed battery state estimation model. In order to analyze realistic BEV degradation trends, we conducted the procedures outlined in the following sections. First, we studied a long-term degradation scenario through the real-world operation of a BEV driver. We then compared the calculation performance between the conventional physics-based (DFN) model and the proposed estimation model. Lastly, we conducted a simulation of the studied real-world scenario using the integrated model and analyzed the dominant factors of degradation.

5.1. Study of Real-World Degradation Scenario

The specifications of the BEV for this analysis are shown in

Table 6. Specifications of the BEV used in this research.

Vehicle Segment		C-Segment
Vehicle Weight		kg	1650
Maximum Output of Motor		kW	79.2
Battery System	Number of Cells	Ah	50
	Nominal Voltage	V	355
	Nominal Capacity	kWh	35.5

. The vehicle is a C-segment EV, and its motor has a maximum output of 79.2 kW. Subsequently, the battery system features a nominal voltage of 355 V and a total energy capacity of 35.5 kWh, comprising 192 of the aforementioned cells (Table 1).

Figure 11 shows a sequence of the real-world degradation analysis conducted in this paper. To simulate a user’s typical commute on a working day, we defined a single cycle (totaling 24 h) that includes driving to work, parking at the workplace, driving back home, and subsequent charging and parking at home. In this paper, we employed the FTP-75 (Federal Test Procedure) for the commute driving pattern, as shown in Figure 12a, because it assumes an urban route. We repeated this working day cycle five times, and then simulated weekend cycles. For the first day of the weekend cycles, the commute driving pattern was replaced by a long-trip driving pattern on the first day, with the second day dedicated to rest. The long-trip driving pattern was a combination of the US06 and FTP-75 cycles, assuming travel via highway and urban routes, as shown in Figure 12b. In addition, the load during home charging was set to 3 kW, with the initial SOC of each cycle charged to 95%, and this sequence of procedures was executed for 730 days (two years) in the analysis.

In this paper, we compared the differences in degradation progression due to regional variations for the operation of the same BEV, employing two regions in the United States: as a hot spot (Phoenix) and a cold spot (Seattle).

Table 7. Average ambient temperatures from 1991 to 2020 in Phoenix and Seattle [23,24].

Month	Average Ambient Temperature
	Phoenix	Seattle
	°C
January	13.8	6.1
February	15.5	6.7
March	19.1	8.4
April	22.9	10.8
May	27.8	14.2
June	33.0	16.7
July	35.3	19.5
August	34.6	19.6
September	31.8	17.0
October	25.2	12.0
November	18.4	8.0
December	13.2	5.6
Year	24.2	12.1

shows the mean ambient temperature for each month over the past 20 years for both regions [23,24], and these data were used for the simulation. In the previous chapter, we adopted an approximate formula for the battery degradation model based on a constant temperature throughout the tests. Therefore, we used the yearly average temperatures from this table, augmented by 10 °C, as the battery temperature boundary condition of the model. This offset was chosen because the optimal operating temperature range for LiBs is widely considered to be 15 to 35 °C [25]. We assumed that thermal management control would raise temperatures in colder regions to near the middle of this optimal range, and lower them in hotter regions to below the maximum.

Furthermore, for the auxiliary load, we assumed a 0.25 kW load when the ambient temperature was between 18 °C and 25 °C (non-air-conditioned conditions). As shown in Figure 13, the load increases linearly by up to an additional 3 kW outside of this temperature range (air-conditioned conditions). From the preceding discussion,

Table 8. Simulation conditions of the long-term real-world degradation analysis.

Region	Ambient Temperature	Battery Temperature	Initial SOC (Each Cycle)	Driving Pattern		Charging Load	Auxiliary Load	Simulation Duration
				Working Day	Weekend
	°C	°C	%	km/h		kW	kW	Day
Phoenix	Table 7	34.2	95	Figure 12a	Figure 12b	3.0	Figure 13	730
Seattle		22.1

summarizes the simulation conditions for the real-world degradation analysis.

5.2. Comparison of Calculation Performance Between Physics-Based and Proposed Models

Before conducting the real-world simulation, we implemented both the conventional DFN model and the proposed battery state estimation model in the vehicle system model. We then calculated the first week of the real-world analysis conditions for both models and compared the computation times using the same computer.

Figure 14 shows the comparison of the calculation performance between the conventional and proposed models. The time step of both models was set to 0.5 s by default; however, the DFN model’s time step was increased to 50 s (100 times the basic time step) during periods of no battery load due to the implicit solver. Nevertheless, we successfully reduced the calculation time by 85%, thereby confirming the feasibility of long-term real-world analysis. Since the only change in the model was the battery model, this improvement contributed to the reduction in the calculation load.

5.3. Analysis of Real-World Degradation

Finally, we conducted the real-world simulation, and Figure 15 shows the results of degradation under the conditions in Table 8. Before discussing the results, it should be noted that over the 730-day analysis duration, the total distance traveled reached approximately 28,000 km, and the accumulated absolute current reached 15,000 Ah. As a result, the capacity decreased by 12% in Phoenix and 8.7% in Seattle. The influence of the battery temperature difference, specifically 12 °C, caused a 3% increase in aging progression. Subsequently, for the dominant factor of degradation, it was revealed that calendar degradation was 5 times larger than cycle degradation in both hot and cold conditions. The reason is that when the accumulated current was converted to the number of cycles in the conducted cycle test, it amounted to fewer than 200 cycles, which is a very small number. Additionally, the internal resistance increased by approximately 14% in Phoenix and 5.7% in Seattle due to the calendar degradation, as shown in Figure 16.

In a previous study, simulating real-world degradation of an LiB with a similar material type under hotter regional conditions, specifically Pune, India, it was reported that the capacity degradation was approximately 7% over 40 weeks [7]. This aligns with the degradation amount observed in our results at the same time point. Furthermore, that study indicated calendar degradation as the dominant factor, a view our research confirmed, aligning not only in terms the amount of degradation but also in terms of its contribution to the overall degradation.

Since the contribution of cycle degradation was low, we considered an operation that increases the load of battery charge and discharge. Therefore, we added a power supply from vehicle to home (V2H) to the operation sequence. Figure 17 shows a new sequence of real-world degradation analysis that implements this power supply in each cycle. During working days and the first day of the weekend cycles, we assumed a 3 kW power supply to the house for 3 h before charging after returning home. For the second day of the weekend cycles, we also assumed a 3 kW power supply until the SOC reached 15%. Then, before the rest phase, the battery would be recharged to 95% at 3 kW.

Figure 18 shows the degradation behavior results, including the home power supply operation. Although the total distance traveled remained the same, the accumulated absolute current reached approximately 38,000 Ah, which is over 2.5 times the previous amount. As a result, the capacity decreased by 14% in Phoenix and 11% in Seattle, about 3% more than the previous analysis, due to increased cycle degradation. Therefore, the contribution of degradation shifted, with cycle degradation increasing to 50% of calendar degradation, which is 2.5 times higher than before. However, even in high-load cases, the contribution of calendar degradation remained significant. This simulation revealed that for users who use BEVs for weekday commuting and weekend trips, calendar degradation, specifically the growth of the SEI layer on the anode, is the dominant factor. Consequently, managing battery conditions, such as the temperature during vehicle storage, is considered effective in suppressing overall degradation.

In a previous study on long-term degradation simulation of an NCA (nickel cobalt aluminum) cathode LiB under V2X scenarios, it was reported that the capacity degradation was approximately 15% over 20 months [26]. Even when multiple V2X scenarios—in other words, charge–discharge cycles—were simulated, the variation in degradation was only about 3%. Based on this, it can be considered that our simulation and conclusions are reasonable.

Lastly, regarding this model’s applicability, since we implemented SEI formation and electrode structural degradation, it is also effective for batteries that use graphite in the anode, such as Lithium Iron Phosphate (LFP) batteries and other ternary cathode material batteries. However, due to its simplified formulation, our model cannot predict certain degradation behaviors. For instance, it does not account for the collapse of the SEI due to repeated charging and discharging, nor does it predict lithium plating that occurs during high-rate charging at extremely low temperatures. For such degradation phenomena, a physics-based model would be necessary.

6. Summary

In this paper, we developed a simplified battery degradation simulation method integrated within a vehicle system model to simulate long-term scenarios where the vehicle and battery interact with each other. The study focused on predicting battery voltage and degradation behavior with practical accuracy using an ECM and approximation formulas for calendar and cycle degradation.

First, we introduced the specifications of the high-capacity LiB used in this research, featuring an NMC cathode and a graphite anode. We then outlined the proposed battery degradation simulation method, which integrates the ECM for voltage estimation and degradation prediction using approximation formulas into the vehicle system model.

Subsequently, to implement the voltage estimation model, we conducted transient charge–discharge tests and employed the DFN model for data augmentation. We then adopted Bayesian Optimization to calibrate the ECM parameters, achieving high accuracy in voltage prediction with an RMSE within 10 mV.

For degradation estimation, we conducted calendar and cycle degradation tests to observe long-term battery behavior. The calendar degradation formula was based on the square root of time, with an additional exponential term to account for rapid initial degradation. And the cycle degradation formula incorporated a linear factor based on the accumulated absolute current.

We then analyzed real-world degradation scenarios by simulating a typical BEV operation over two years, considering regional variations in ambient temperature. The results revealed that calendar degradation, specifically the growth of the SEI layer on the anode, was the dominant factor, contributing significantly more than cycle degradation. The capacity decreased by 12% in Phoenix and 8.7% in Seattle, with internal resistance increasing by approximately 14% and 5.7%, respectively.

To study higher-battery-load cases, we added a V2H power supply to the simulation. This increased the accumulated absolute current and shifted the contribution of cycle degradation to 50% of calendar degradation. However, calendar degradation remained significant, emphasizing the importance of managing battery conditions during vehicle storage to suppress overall degradation. Leveraging our model’s fast computational capabilities, we plan to extend this analysis in future work to cover a longer period, such as a full BEV lifecycle.