Journal of Power Sources 513 (2021) 230519

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Journal of Power Sources

journal homepage: www.elsevier.com/locate/jpowsour

Data driven analysis of lithium-ion battery internal resistance towards

reliable state of health prediction
Mohammad A. Hoque a ,∗, Petteri Nurmi a , Arjun Kumar b , Samu Varjonen a , Junehwa Song b ,
Michael G. Pecht c , Sasu Tarkoma a
a
Nodes Lab, Deaprtment of Computer Science, Exactum, P.O. Box 68 (Pietari Kalmin katu 5) 00014, University of Helsinki, Finland
b School of Computing Korea Advanced Institute of Science and Technology (KAIST) 335 Gwahangno, Yuseong-gu, Daejeon 34141, Republic of Korea
c CALCE Center for Advanced Life Cycle Engineering, 1103 Engineering Lab Building, University of Maryland, College Park, MD 20742, USA

HIGHLIGHTS

• Internal resistance offers accurate early-stage health prediction for Li-Ion batteries.
• Prediction accuracy is over 95% within the first 100 cycles at room temperature.
• Demonstrated that internal resistance dynamics characterize battery homogeneity.
• Homogeneous batteries can share the same early-stage prediction models.
• Internal resistance dynamics reliably capture usage pattern and ambient temperature.

ARTICLE INFO ABSTRACT

Keywords: Accurately predicting the lifetime of lithium-ion batteries in the early stage is critical for faster battery
Lithium-ion battery production, tuning the production line, and predictive maintenance of energy storage systems and battery-
State of health powered devices. Diverse usage patterns, variability in the devices housing the batteries, and diversity in their
Battery capacity
operating conditions pose significant challenges for this task. The contributions of this paper are three-fold.
Internal resistance
First, a public dataset is used to characterize the behavior of battery internal resistance. Internal resistance has
health prediction
non-linear dynamics as the battery ages, making it an excellent candidate for reliable battery health prediction
during early cycles. Second, using these findings, battery health prediction models for different operating
conditions are developed. The best models are more than 95% accurate in predicting battery health using the
internal resistance dynamics of 100 cycles at room temperature. Thirdly, instantaneous voltage drops due to
multiple pulse discharge loads are shown to be capable of characterizing battery heterogeneity in as few as five
cycles. The results pave the way toward improved battery models and better efficiency within the production
and use of lithium-ion batteries.

1. Introduction degrading accurately. In addition, due to the long life cycle of lithium-
ion batteries, there is limited performance data available for building
Fast and accurate prediction of the lifetime of lithium-ion batteries and validating prediction models.
is vital for many stakeholders. Users of battery-powered devices can Battery lifetime is traditionally estimated using physical models
understand the effect their device usage patterns have on the life
that estimate capacity loss using factors, such as the growth of the
expectancy of lithium-ion batteries and improve both device usage
solid-electrolyte interface on battery anode [8,9], the loss of active
and battery maintenance [1–3]. Battery manufacturers can enhance
their battery production and verify their production methods with the materials [10,11], lithium plating [12,13], or impedance increase [14].
help of faster prediction [4]. Enabling accurate prediction, however, is These approaches are successful in prediction, however, the chemical
highly challenging as heterogeneity in battery production processes [5], factors are subject to variations due to production heterogeneity, oper-
hardware components of complex devices [6], diversity in usage pat- ating conditions, and usage patterns. Thus, translating the models into
terns [6], variations in device operating conditions [7], and other devices that are actively used is challenging. Another possibility is to
factors make it difficult to model how and when the batteries are

∗ Corresponding author.Nodes Lab, Deaprtment of Computer Science, Exactum, P.O. Box 68 (Pietari Kalmin katu 5) 00014.,
E-mail address: mohammad.a.hoque@helsinki.fi (M.A. Hoque).

https://doi.org/10.1016/j.jpowsour.2021.230519
Received 30 June 2021; Received in revised form 21 August 2021; Accepted 11 September 2021
Available online 1 October 2021
0378-7753/© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
M.A. Hoque et al. Journal of Power Sources 513 (2021) 230519

create models using battery voltage curves captured during charging

or discharging. For example, discharge voltage curves correlate with
battery health degradation [15–17]. Differential voltage curves also
have been used for understanding battery health degradation [18,19]
and for predicting remaining capacity [20]. These approaches depend
on suitable battery instrumentation and specific cycling configurations.
Similarly, voltage curves constructed during charging can be used to as-
sess battery health [21–23]. Alternative feature representations are also
possible, e.g., Lu et al. [24] modeled battery degradation as a function
of four geometrical features of the charging voltage curve. Generally,
these approaches have limited generality, and their performance is
heavily reliant on the charging algorithm [25]. Fig. 1. (a) Thevenin battery model, (b) voltage drops for a discharge load, and (c)
Data-driven approaches for battery lifetime modeling have recently V-edge, i.e., voltage drop across the internal resistance 𝑅𝑏 .
gained traction. Since rechargeable lithium-ion batteries have a long us-
age time, gathering adequate data on battery health is time-consuming.
The example approaches rely on various machine learning techniques, 2. Background
such as support vector machine [26], particle filtering [27], Bayesian
predictive modeling [28], deep learning [29] and Gaussian process
This section first describes how to estimate the internal resistance
regression [30]. These solutions require sufficient amounts of data,
of lithium-ion batteries from the voltage patterns due to pulsed charge
and in most cases, the collected data covers over 25% of battery life
and discharge currents. Next, the behavior of battery internal resistance
degradation. These models are not for early-stage prediction as they
for discharging currents, different operating temperatures, and state of
rely on features that are difficult to estimate in practical use and require
a large dataset to reach high accuracy. Indeed, the lack of sufficient charge (SoC) are discussed.
data is a significant challenge for early-stage battery life prediction.
Severson et al. [31] analyzed battery discharge voltage curves from the 2.1. Measuring battery internal resistance
cycling information of 124 lithium-ion batteries. The authors extracted
numerous statistical features from the voltage curves and developed
early-stage battery life prediction models for fast charging. The features Fig. 1 illustrates battery voltage across the battery’s internal resis-
extracted during the first five cycles were also used to classify batteries tance for a pulsed discharge/charging current of 3 A for an equivalent
as low and high-lifetime batteries. This approach relies on specialized battery model (Thévenin model). For a discharge current 𝐼, there is
measurements and statistical features that are difficult to estimate a sharp drop in the battery voltage as soon as the load begins. The
during the operation of a device. reason for this behavior is the battery’s internal resistance 𝑅𝑏 . This
This paper contributes by presenting a data-driven analysis of sharp change in the voltage is referred to as the V-edge value 𝑉𝑒𝑑𝑔𝑒 .
battery internal resistance using a comprehensive publicly available Formally,
dataset of lithium cobalt oxide (LCO) batteries [32,33]. This analysis
𝑉𝑒𝑑𝑔𝑒 = 𝑅𝑏 × ▵ 𝐼 = 𝑅𝑏 × 𝐼 − 𝑅𝑏 × 𝐼0 , (1)
is applied to create improved early-stage battery lifetime prediction
and battery characterization methods that are general and able to where 𝐼0 is the baseline current load, e.g., when the device is in sleep
operate without complex instrumentation of the battery. First, it is mode but draws some current from the battery. After the sharp drop,
demonstrated that battery internal resistance reliably captures various the battery voltage decreases almost linearly as long as the load contin-
aspects of battery cycling, such as discharge current, operating condi-
ues. This linearity is due to the battery’s double-layer capacitance (𝐶𝑜 ),
tion (temperature), and the battery usage pattern in cycling. Second,
and the polarization resistance (𝑅𝑜 ) [34]. The length of the linearity
based on these findings, early-stage battery health prediction models
depends on the duration of the discharge load. The internal resistance
are constructed. The resistance behavior at room temperature enables
due to a charging current in Fig. 1(c) can be expressed using the same
predicting battery capacity with more than 95% accuracy in 100 cycles.
equation.
The models for higher cycles can be used to predict the capacity of
other batteries with similar accuracy, given that their internal resis-
tance characteristics and operating conditions are identical. Finally, 2.2. Charging/discharging current
such features of batteries can be identified using voltage drop due to
internal resistance, which also determines the heterogeneity among
The internal resistance also depends on the amount of charging
the cells of the similar model during the very early stage of cycling,
e.g., five cycles. This paper overcomes the challenge of limited data or discharging current applied to a battery in a pulse. Fig. 2 (Left)
points by capturing aging as a function of many pulse discharges shows that voltage drop across battery internal resistance increases
in a cycle. Together, these contributions pave the way toward more linearly with the pulse discharging loads for a battery. However, the
accurate battery health predictions that can operate robustly across resistance is inversely proportional to the applied current. Therefore,
different lithium-ion battery chemistries, usage patterns, and working the resistance decreases exponentially as the pulse current increases
conditions. (Right). These plots are constructed using a dataset [32], which is
The rest of the article is organized as follows. Section 2 provides described in the next section.
an overview of lithium-ion battery internal resistance. The dataset is
explained in Section 3. The internal resistance behavior for different
2.3. Battery internal resistance and SoC
cycling conditions is analyzed in Section 4 to understand the dynamics
as battery health degrades with cycling. Section 5 constructs the predic-
tion models and evaluates their performance. It also demonstrates how Battery internal resistance also changes as SoC changes. For exam-
to characterize lithium-ion batteries according to the internal resistance ple, Chen et al. [35] showed that the internal resistance is higher when
dynamics. Finally, we discuss the generality of the approach and future a battery is fully charged or discharged. Such a pattern is consistent for
work in Section 6. The paper concludes in Section 7. different pulse discharge loads.

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M.A. Hoque et al. Journal of Power Sources 513 (2021) 230519

Fig. 2. (Left) V-edge Vs. discharge pulse current, (Right) battery internal resistance vs. discharge pulse current.

Table 1 Table 2
Four discharge profiles and corresponding 16 batteries in the dataset. Distribution of discharge events for different discharge load currents.
Discharge Profile Room temperature 40 ◦ C Low Skew (LS) High Skew (HS)
Low Current Skew LoCus-RT LoCus-40C 0.5A 7.2% 0.5A 2.0%
1.0A 14.8% 1.0A 2.4%
battery-id 13,14,15,16 21,22,23,24
1.5A 19.3% 1.5A 3.6%
RW Cycles 1110,1119,1124,897 511,531,505,523
2.0A 21.6% 2.0A 6.0%
Reference Cycles 22,22,22,18 11,12,11,11
2.5A 14.6% 2.5A 9.2%
RW discharge events 21510,22426,26606,20049 20170,19468,19102,20629
3.0A 10.0% 3.0A 11.8%
High Current Skew HiCus-RT HiCus-40C 3.5A 6.5% 3.5A 17.2%
battery-id 17,18,19,20 25,26,27,28 4.0A 4.0% 4.0A 23.4%
RW Cycles 1307,1384,1343,1284 664,605,611,613 4.5A 1.5% 4.5A 19.4%
Reference Cycles 26,27,27,26 13,12,12,12 5.0A 0.5% 5.0A 5.0%
RW discharge events 15245,12739,13705,12739 11022,9755,7447,9452

current skew distribution (LoCus); another eight were cycled according

2.4. Battery internal resistance and temperature to a high current skew distribution (HiCus) as presented in Table 2.
The experiments were conducted at two operating temperatures: room
The internal resistance value is the same for the same charging and temperature and 40 ◦ C. Consequently, there are four cycling profiles,
discharging current and for a given temperature. However, the internal as shown in Table 1.
resistance behaves differently at different temperatures. It was shown
that as the temperature increases to room temperature, the resistance of 3.2. Ground truth for battery health
26665 (LiFePO4) lithium-ion battery exponentially decreases and then
increases again [35]. The relation is expressed in Eq. (2). After every fifty RW cycles, a reference discharge was performed on
2 the batteries in the dataset. The reference discharge differs from RW
𝑅𝑏 = 𝑎 × 𝑇 + 𝑏 × 𝑇 + 𝑐 (2)
cycles, repeatedly applying a sequence of operations consisting of a 1A
discharge load for 10 min and a 20 min rest period until the battery
3. Dataset was fully discharged. The capacity in Ah is computed by integrating the
discharge current with time for a reference discharge sequence. These
To analyze battery internal resistance and to construct prediction values serve as ground truth on battery health for evaluating prediction
models for battery lifetime prediction, a publicly available lithium- models. Since there are only a few reference cycles (Table 1), the
ion battery dataset [32,33] is used. The dataset contains the cycling capacity loss for individual RW cycles is estimated from the reference
information of 24 lithium cobalt oxide (LCO) 18650 batteries of 2.2 Ah cycles using linear regression models. The fits of the resulting models
initial/design capacity. This paper considers a subset of 16 batteries consistently have 𝑅2 values higher than 95%. The model coefficients
cycled with a 1 min discharge pulse of different currents. The reason for are later used to estimate the State-of-Charge (SoC) and to construct
excluding the remaining 8 batteries is that they had five minutes of dis- new battery health prediction models in the following sections. Note
charge pulses. Hence, the models constructed from these measurements that a non-linear model provides a similar fit for the data points [30],
would not be directly comparable to the other batteries. and we use the simpler model following the parsimony principle.

3.1. Battery cycling 3.3. Extracting V-edges

Table 1 shows the cycling configuration of 16 batteries. A Random A discharge current from the distribution (Table 2) is applied for
Walk (RW) cycle on a battery is a sequence of random discharge current one minute at a time. A sampling frequency of 1 Hz results in a
loads followed by resting or idle events. The duration of each discharge voltage sequence 𝑉 𝑑𝑖 : {𝑉 𝑑1 , 𝑉 𝑑2 , ..., 𝑉 𝑑60 }, in chronological order. The
event is one minute. In an idle event, the battery is neither charged nor discharge events are separated by a few milliseconds of a rest event of
discharged. The duration of an idle event is only a few milliseconds. The two samples 𝑉 𝑟𝑖 : {𝑉 𝑟1 , 𝑉 𝑟2 }. Fig. 1(c) demonstrates how to compute
discharge loads in a cycle follow the distributions presented in Table 2. V-edge for a discharge load, and a V-edge is computed as (𝑉 𝑟2 − 𝑉 𝑑1 ),
Battery voltage, temperature, and discharge currents were sampled at where 𝑉 𝑟2 is the rest voltage before the discharge load is introduced.
1 Hz during the discharge events. The battery internal resistance, 𝑅𝑏 , for the events can be computed
Table 1 also shows the number of such discharge events and the using Eq. (1) given that the idle load, 𝐼0 , is zero.
corresponding RW cycles per battery, which vary according to the us- After computing the discharge current specific 𝑅𝑏 , we separate 𝑅𝑏 s
age or discharge profiles. Eight batteries were cycled according to low according to SoC. As the dataset does not contain actual measurement

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M.A. Hoque et al. Journal of Power Sources 513 (2021) 230519

Table 3 two operating temperatures (see Table 1). Nevertheless, the correlation
Correlation between battery internal resistance (𝑅𝑏 ) and capacity at 100th cycle for a
trends presented in Fig. 3 hint about the operating conditions during
representative battery in each profile for different SoC levels.
the early phase of cycling.
Discharge profile 1.0A 1.5A 2.0A 2.5A
Both linear and non-linear models are investigated in this section
LoCus-RT(20%) −0.91 −0.94 −0.88 −0.94
to analyze the internal resistance behavior as the battery capacity de-
LoCus-RT(50%) −0.93 −0.92 −0.91 −0.90
LoCus-RT(80%) −0.91 −0.94 −0.92 −0.90 grades through usage cycles. The intuition is that battery characteristics
and operating conditions determine the best fitting model type, and
LoCus-40C(20%) 0.55 0.6 0.62 0.63
LoCus-40C(50%) 0.54 0.60 0.63 0.64 thus different kinds of models are needed. To this end, battery-specific
LoCus-40C(80%) 0.53 0.61 0.63 0.65 models are developed according to the profile and discharge loads
Discharge profile 3.0A 3.5A 4.0A 4.5A rather than combining data from multiple batteries. Although some
HiCus-RT(20%) −0.85 −0.85 −0.92 −0.86
previous studies have combined data from various batteries [4,30],
HiCus-RT(50%) −0.88 −0.86 −0.89 −0.85 later in Section 5.3 it is demonstrated that the internal resistance of
HiCus-RT(80%) −0.87 −0.83 −0.93 −0.87 similar new batteries can be different, which can affect the prediction
HiCus-40C(20%) 0.65 0.65 0.60 0.70 accuracy.
HiCus-40C(50%) 0.63 0.63 0.62 0.69 Linear Models: Linear prediction models are investigated as the sim-
HiCus-40C(80%) 0.61 0.65 0.63 0.72 plest potential models. Such models are easy to integrate even with
low-end battery-powered devices as the required computations are
efficient and straightforward. The linear model fits a function 𝑦 = 𝑥𝛽+𝑏,
of SoC, the number of measurements, duration, discharging current, where 𝑦 is the internal resistance value, 𝛽 is the feature vector rep-
and the estimated capacity values are used to determine 20%, 50%, resenting the relationship between variables, and 𝑥 holds the training
and 80% SoC. data points for battery capacity. However, in the early stage of cycling,
a small fraction of observation data points are available. Therefore the
4. Internal resistance dynamics and predicting battery capacity model may struggle to learn the correct relationship between internal
resistance and battery capacity.
In this section, the internal resistance dynamics of batteries are The relationship between variables 𝛽 and internal resistances 𝑦 can
analyzed. It is shown that internal resistance dynamics capture battery be estimated using standard least-squares fitting of the regression lines
health degradation due to cycling and are resilient to operating con- by taking minimum from the sum of squares of the vertical deviation
ditions. Motivated by these results, novel battery capacity prediction of the regression line from each data point. Therefore, a 𝛽 is required
∑
models are developed and evaluated. that minimizes squared loss, i.e., 𝑛𝑖=1 𝜖𝑖2 . That is
∑
𝑛 ∑
𝑛
4.1. Validity of internal resistance dynamics 𝛽̂ = arg min 𝜖𝑖2 = arg min (𝑥𝑖 𝛽 + 𝑏 − 𝑦𝑖 )2 , (4)
𝛽 𝑖=1 𝛽 𝑖=1

This section demonstrates that internal resistance indeed captures where the arg min function aims to find the coefficient values that
battery aging in the early stage due to cycling. This is accomplished minimizes the argument.
by computing the Pearson correlation 𝜌 (Eq. (3)) between internal Non-Linear Models: As for the second class, non-linear power mod-
resistance (𝑋) and battery capacity (𝑌 ) for different SoC levels. Table 3 els are considered. This is because battery capacity degradation is
shows statistically significant, negative correlations between internal non-linear [7,39]. The power model is formulated as
resistance and battery capacity when the batteries are discharged at
𝑦 = 𝑎𝑥𝑏 + 𝑐. (5)
room temperature. Such correlation implies that the internal resistance
of batteries increases as the capacity degrades. The strong negative The parameters 𝑎, 𝑏, and 𝑐 can be solved by applying a logarithmic
correlations suggest that internal resistance is an excellent candidate transformation on the first-order component, which results in a linear
feature for battery health estimation. model where the parameters 𝑎 and 𝑏 can be estimated. This model can
∑( ) (∑( ) ∑( ))
be substituted for the original equation to obtain a linear equation.
𝑁 𝑋𝑌 − 𝑋 𝑌
𝜌= √ (3)
( ∑ (∑ )2 )( ∑ (∑ )2 ) 𝑦 = 𝑎𝑧 + 𝑐, (6)
𝑁 𝑋2 − 𝑋 𝑁 𝑌2 − 𝑌
At 40 ◦ C, the relationship in Table 3 is less obvious. This is due where 𝑧𝑖 = 𝑥𝑏𝑖 . Therefore, if 𝑏 is given, the values of 𝑎 and 𝑐 can be
to both temperature and discharge affecting the internal resistance. computed. Similarly to the sum of square in Eq. (4), the value of 𝑏 can
To better understand the effects of temperature, the correlations 𝜌 for be obtained by minimizing the sum of squared loss, i.e.,
all the discharge profiles are further investigated. Fig. 3 shows that ∑
𝑛
𝜌 is strongly positive at the beginning but steadily decreases through 𝑏̂ = arg min (𝑎𝑧𝑖 + 𝑐 − 𝑦𝑖 )2 , (7)
𝑏 𝑖=1
active usage and time. After 150 − 250 cycles, the magnitude of the
correlation becomes zero, and from this point onward, the correlation Next, 𝑦 can be predicted by selecting the values of 𝑎 and 𝑐 for the 𝑏
𝜌 becomes increasingly negative and approaches −1. Previous studies with the smallest loss and using Eq. (5).
have identified that such decreasing internal resistance originates from Model Goodness: While the least square method finds the best fitting
anode due to cycling [36]. In other words, the resistance of anode coefficients, it does not tell how good the model is in explaining the
decreases. For this reason, initially the effect of temperature subsumes data. Therefore, the models are complemented using 𝑅2 to measure the
the effect of discharge on cycling [37,38]. These correlation trends and goodness of model fit. The 𝑅2 values are between 0 and 1. The closer
operating temperature can be the indicators for constructing accurate the value to 1, the better the model is predicting battery health. 𝑅2 is
prediction models, as demonstrated in the next section. computed as
∑𝑛
𝑆𝑆𝑟𝑒𝑠𝑖𝑑 (𝑦𝑖 − 𝑦̂𝑖 )2
4.2. Prediction model development 𝑅2 = 1 − = 1 − ∑𝑖=1
𝑛 , (8)
𝑆𝑆𝑡𝑜𝑡𝑎𝑙 2
𝑖=1 (𝑦𝑖 − 𝑦̄𝑖 )
Internal resistance behavior is modeled as a function of battery where 𝑆𝑆𝑟𝑒𝑠𝑖𝑑 sum of square of the residuals from the regression, and
capacity degradation for a particular discharge load and operating tem- 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 is the sum of the squared differences of the response variable
perature. In the analysis, the models are constructed separately for the from the mean.

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M.A. Hoque et al. Journal of Power Sources 513 (2021) 230519

Fig. 3. Pearson correlation trend between battery internal resistance (𝑅𝑏 ) and battery capacity as the batteries are discharged within 80% SoC level. At room temperature, the
correlation becomes strongly negative within the first 100 cycle. At 40 ◦ C, the correlation gradually proceeds towards negative.

Evaluation Metrics: The evaluation assesses the prediction perfor- Table 4

Goodness of model fit (𝑅2 ) for four representative batteries according to usage profile
mance of our models using standard error measures for regression.
and discharge load at 80% SoC.
Specifically, the Root Mean Square Error (RMSE) and the median
Model ax+b ax𝑏 +c ax+b ax𝑏 +c
percentage error (MPAE) are considered to evaluate the models’ per-
LoCus-RT 0.95(1.5A) 0.98(1.5A) 0.95(2A) 0.98(2A)
formance. RMSE is defined as
√ HiCus-RT 0.95(3.5A) 0.96(3.5A) 0.96(4.0A) 0.96(4.0A)
∑𝑛 2
𝑖=1 (𝑦𝑖 − 𝑦̂𝑖 ) Model ax+b ax+b
𝑅𝑀𝑆𝐸 = (9)
𝑛 0.94(1.5A) 0.91(2A)
HiCus-40C 0.81(3.5A) 0.84(4A)
and (MAPE):
∑𝑛
|𝑦𝑖 − 𝑦̂𝑖 |
%𝑒𝑟𝑟𝑜𝑟 = × 100. (10)
𝑖=1
𝑦𝑖
model is also investigated for the LoCus-RT and HiCus-RT profiles.
Table 4 shows slightly poorer fits compared to the power degree model.
4.3. Fit of the prediction models
Hence, simple linear models might be sufficient to predict battery
capacity changes within a smaller cycle range. In general, the 𝑅2 values
This section examines the prediction models to explain how inter-
for all profiles in Figs. 4 and 5 are approximately 0.95, which indicates
nal resistance changes as battery capacity reduces using the metrics
that variance in internal resistance values is almost around its mean
presented in the earlier section. Fig. 4 illustrates the fitted regression
irrespective of the ambient temperature. However, the 𝑅2 values for
models of internal resistance values against the battery capacity at
the HiCus profiles at 40 ◦ C are slightly lower at 0.85. Thus, the results
room temperature. The presented coefficients are shown with 95% con- imply that the model’s performance is sensitive to temperature and
fidence bounds. The model fits in all cases are above 0.95, irrespective higher discharge rates, with a slightly decreasing model fit.
of the discharge loads and the discharge profiles. The figure also shows Figs. 4 and 5 demonstrate internal resistance dynamics when the
that the internal resistance behavior follows the second-order power SoC is 80%. Fig. 6 shows that the pattern overlaps for 20% and 50% SoC
function as the capacity reduces due to cycling. However, the behavior as well for a battery. These results highlight that the battery’s internal
of internal resistance differs significantly according to the discharge resistance dynamics are not affected by SoC. As demonstrated in Figs. 4
load. Consequently, the coefficients also vary among similar profiles and 5, the dynamics of internal resistance significantly differ in room
as shown in Fig. 4(c) & (d). We observed similar patterns with all the temperature and in 40 ◦ C temperature. However, it was shown that
batteries of LoCus-RT and HiCus-RT profiles presented in Table 1. battery internal resistance has similar values at 17 ◦ C and 40 ◦ C [35],
In contrast, Fig. 5 depicts the behavior of internal resistance when though the trend is decreasing and increasing respectively as the tem-
the batteries are cycled at 40 ◦ C. Similar to the correlation patterns perature increases from 17 ◦ C. However, we need further investigation
shown in Section 4.1, the figure shows that internal resistance initially to understand the resistance dynamics at such lower temperature.
decreases sharply from a higher value and then gradually increases.
Meanwhile, the batteries lose a small fraction of the initial capacity. 5. Performance evaluation
The pattern follows the second-order exponential function, further
explaining the variations in correlation patterns shown in Fig. 3. Such This section constructs the prediction models using data from early
a pattern is consistent across multiple discharge loads, profile variation cycles and evaluates their performance. Early-stage prediction models
presented in Fig. 5, and across other batteries with similar profiles are essential to understand the performance of the batteries before
presented in Table 1. The only exception was battery 26. We believe significant capacity degradation occurs. Among others, they enable
that there were some measurement anomalies as the dynamics do not re-calibration of device load and selecting the appropriate battery tech-
follow the models discussed earlier. nology to use. The early prediction of battery health involves predicting
Fig. 5 also shows that a linear model can have a very good fit battery health using internal resistance (80% SoC) at the early life
by skipping early-stage usage data where the effect of temperature stage, i.e., with as few cycles as possible.
dominates the effect of discharge. The batteries with LoCus-40C profile
can have fitness as good as 0.95 once the early-stage data is skipped. 5.1. Early prediction of battery health
For HiCus-40C batteries, model goodness slightly deteriorates due to
more sparse measurements than room temperature. However, even in Although the models presented earlier may have very good fits,
this case the overall fit is excellent, consistently being above 0.85 their performance can vary significantly when only small amounts of
(Fig. 5(c) & (d)). data are available from cycling. This section demonstrates that the
To examine whether other types of models better explain the rela- internal resistance-based models have good performance even when
tionship between internal resistance and capacity, a linear regression small amounts of data are available. Data from the first 100 −200 cycles

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M.A. Hoque et al. Journal of Power Sources 513 (2021) 230519

Fig. 4. The changes in internal resistance, 𝑅𝑏 , as the capacity degrades. The internal resistance values are collected when the SoC is above 80%. Figure (a,b), Second order Power
regression model captures internal resistance dynamics for the batteries with LoCus-RT profile. Figure (c,d), Similar Power function fits for the batteries with HiCus-RT profile.

Fig. 5. The internal resistance dynamics at 80% SoC, 𝑅𝑏 , as the battery capacity reduces due to RW cycling at 40 ◦ C. Figure (a,b), Linear first degree polynomial fits for the
battery with LoCus-40C profile after 100 cycles. Figure (c,d), First degree polynomial fit for another battery with HiCus-40C profile after 100 cycles.

Fig. 6. Battery internal resistance dynamics when the battery state of charges are 50% and 20% respectively.

are used to construct the models. The data from successive 50–100 5.2. Transfer learning
cycles are used to evaluate model performance.
The models are constructed by progressively increasing the cycle This section assesses the transfer capability of the models and
count until the model reaches a reasonable mean absolute percentage their generality. Specifically, the models are constructed from the first
error (MAPE). Table 5 shows that the linear models for the batteries battery (i.e., the one with the lowest identifier in the dataset) in each
with LoCus-RT and HiCus-RT profiles suffer from 1%–5% error in category shown in Table 5 and the model for that battery is used to
predicting battery capacity. In other words, these room temperature predict the capacity of the other three batteries. The internal resistance
models achieve more than 90% accuracy within 100 cycles. of the remaining batteries is not considered in constructing the models
The prediction models are excellent during the early cycles for 12 in this evaluation. Similar prediction models, demonstrated in Figs. 4
of the 16 batteries, corresponding to 3 of the 4 usage profiles. The and 5, are derived from longer cycling data.
HiCus-40C and LoCus-40C models are constructed with 100 cycles Fig. 7 demonstrates that the resulting predictions are closely aligned
data and have 92%–99% accuracy. However, the resistance values from with the actual values. In line with the other results, the performance
the first 50–100 cycles are ignored due to the exponential pattern is best for the non-linear models constructed for batteries operating
presented in Fig. 5. The batteries of LoCus-40C profiles have better at room temperature, for which the MAPE is within 5%. In fact, the
accuracy compared to the HiCus-40C models. Fig. 5 also shows that remaining six batteries of LoCus-RT and HiCus-RT profiles had MAPE
the data points are more scattered with this profile HiCus-40C. Thus, it less than 8%.
is possible to construct good prediction models with internal resistance In contrast, only two batteries from the LoCus-40C & HiCus-
from a smaller amount of battery pulse discharge information. This may 40C profiles can share the linear models with other batteries with
vary from battery to battery, even though they are cycled with similar good performance. The remaining six batteries of these profiles have
usage profiles. MAPE within 10%–20%. Such performance is expected, as the internal

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M.A. Hoque et al. Journal of Power Sources 513 (2021) 230519

Fig. 7. Transfer learning and the performance of the regression models in Figs. 4 and 5 for the remaining batteries.

Table 5 6. Discussions and future work

The mean absolute percentage error (MAPE) of the linear regression models constructed
from the partial active usage data of the 16 batteries (80% SoC). The room temperature
models are constructed from the internal resistance of the first 100 RW cycles data, To summarize, internal resistance-based models can accurately ex-
whereas the higher temperature (40 ◦ C) models are constructed by skipping first 50–200 plain the impact of different discharge loads and temperatures on
cycles. battery capacity. Best performance results under normal operating tem-
LoCus-RT (2A) Bat-13 Bat-14 Bat-15 Bat-16 peratures, and as the temperature or battery discharge profile changes,
Training RW Cycles 0–100 0–100 0–100 0–100 the performance degrades slightly. Even in this case, our models’
Testing RW Cycles 101–200 101–200 101–200 101–200 accuracy remains consistently high, reaching over 90% from the early
MAPE 0.02 0.02 0.02 0.01 stage usage data. Utilizing a battery model for another battery depends
LoCus-40C (2A) Bat-21 Bat-22 Bat-23 Bat-24 on how similar the batteries are in their internal resistance. Besides,
Training RW Cycles 5–150 50–150 50–150 50–150 the resistance values from multiple homogeneous batteries should be
Testing RW Cycles 151–250 151–250 151–250 151–250 used to construct more accurate prediction models. Overall, these
MAPE 0.02 0.02 0.01 0.08
are very encouraging results, suggesting that internal resistance is an
HiCus-RT (4A) Bat-17 Bat-18 Bat-19 Bat-20 outstanding candidate feature to predict battery health.
Training RW Cycles 0–100 0–100 0–100 0–100 Table 6 compares recent data-driven health prediction approaches
Testing RW Cycles 101–200 101–200 101–200 101–200 with this work. A recent work considered a large dataset containing the
MAPE 0.03 0.03 0.01 0.05
cycling information of 124 lithium-ion batteries [4]. The batteries were
HiCus-40C (4A) Bat-25 Bat-26 Bat-27 Bat-28
discharged with continuous discharge currents and charged with 72
Training RW Cycles 50–150 – 50–150 100–200 different fast charging conditions. The authors developed the prediction
Testing RW Cycles 151–200 – 151–250 201–250
models from the resulting voltage curves from many batteries during
MAPE 0.01 – 0.01 0.01
the early stage of cycling (100 cycles). Those models could predict bat-
tery health in an early stage with more than 95% accuracy but require
extensive amounts of data, and the resulting features are sensitive to
resistance across the batteries of having the same capacity can vary, as the way the battery is used.
demonstrated in Section 5.3. Richardson et al. [30] investigated the Random Walk dataset to
However, the performance demonstrates that a model from a used predict battery health. They relied on Gaussian process regression to
battery can be applied to another new battery as long as the operating predict battery health based on usage patterns. They considered the
conditions are sufficiently similar—the possibility to share models in- distributions of current, the discharge rate of stored charge, voltage,
creases when the batteries are used at room temperature. The variation and temperature for the discharge load patterns as the input features.
of the internal resistance among new batteries also contributes to Their training includes data from 50% percent cells in the dataset and
different prediction models, as demonstrated in the next section. derived 96% accurate models. These models rely on features that are
difficult to estimate in practical use, and require a large dataset to reach
high accuracy.
5.3. Characterizing batteries of same model In contrast, this paper demonstrates that the internal cell resistance
is an excellent candidate feature for battery health prediction using
This section characterizes batteries using the V-edge values but for the same dataset. Its dynamics correlate very well with battery health
at the early stage of cycling, and simple models are sufficient to
multiple discharge loads, as the relation is linear compared to the
capture the dynamics. Internal cell resistance can be estimated from
internal resistance (see Fig. 2). Fig. 8 shows the linear models for the
information available to the battery interface, in contrast to the models
16 batteries cycled at the four different operating conditions described
described above, which require external details on operating conditions
in Table 1. The models are constructed from the first 5 RW cycles of
or device usage. The best early-stage prediction models are achieved
each battery, and the fits for all models exceed 95%.
at room temperature and are more than 95% accurate. Furthermore,
The regression plots in Fig. 8 show that most of the batteries in it is demonstrated how to separate batteries based on their internal
the dataset have similar internal resistance models in every operating resistance characteristics. This can be used, e.g., to select the optimal
condition. Only one or two batteries in each group have coefficients prediction model to use.
that differ significantly. This indicates that the differences between the This paper has investigated LCO batteries; however, the resistance
linear models can be used to separate batteries according to their inter- dynamics and prediction models should apply to lithium-ion batter-
nal resistance characteristics. Fig. 8 also illustrates that the resistance is ies with different chemistries. For example, Kiel [40] presented that
particularly similar when the discharge current is low, which suggests lithium nickel cobalt aluminum oxide (NCA) batteries exhibit non-
that a single resistance value might not be sufficient to construct the linear patterns at 40 ◦ C, similar to those shown in Fig. 5. Nevertheless,
necessary characterization. the coefficients of the prediction models should vary according to the

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M.A. Hoque et al. Journal of Power Sources 513 (2021) 230519

Fig. 8. The regression models for internal resistance and discharge loads for 16 batteries. The resistance values are taken from the first 5 RW cycle and the Linear Regression
fits have very good fits are computed for four discharge profiles. Figure (a, b), The cells have similar regression coefficients. Figure (c,d), At least one of the cells has different
regression coefficients.

Table 6
Comparison among the data-driven approaches for predicting battery life.
Model Accuracy Dataset (Cells) Features Early prediction Production heterogeneity
Linear Models [4] 95% 128 Discharge Voltage Curve Yes Yes
Gaussian Process [30] 96% 24 Distributions of discharge Voltage, current, temperature, discharge No No
rate of stored charge
Linear/ non-Linear Models 95% 16 Internal resistance, V-edge Yes Yes

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