# Practical Battery Health Monitoring using Uncertainty-Aware Bayesian Neural
Network
††thanks: This work was supported in part by A*STAR under its MTC Programmatic
(Award M23L9b0052), MTC Individual Research Grants (IRG) (Award M23M6c0113),
the Ministry of Education, Singapore, under the Academic Research Tier 1 Grant
(Grant ID: GMS 693), SIT’s Ignition Grant (STEM) (Grant ID: IG (S) 2/2023 –
792), and Future Communications Research & Development Programme (FCP) under
Grant FCP-SIT-TG-2022-007.

Yunyi Zhao, Wei Zhang∗ ∗ Corresponding author. Email:
<EMAIL_ADDRESS>Singapore Institute of Technology
{yunyi.zhao<EMAIL_ADDRESS>Qingyu Yan  Man-Fai Ng  Nanyang
Technological University, Singapore
<EMAIL_ADDRESS>Agency for Science, Technology and Research
<EMAIL_ADDRESS>B. Sivaneasan  Singapore Institute of Technology
<EMAIL_ADDRESS>Cheng Xiang  National University of Singapore
<EMAIL_ADDRESS>
###### Abstract

Battery health monitoring and prediction are critically important in the era
of electric mobility with a huge impact on safety, sustainability, and
economic aspects. Existing research often focuses on prediction accuracy but
tends to neglect practical factors that may hinder the technology’s deployment
in real-world applications. In this paper, we address these practical
considerations and develop models based on the Bayesian neural network for
predicting battery end-of-life. Our models use sensor data related to battery
health and apply distributions, rather than single-point, for each parameter
of the models. This allows the models to capture the inherent randomness and
uncertainty of battery health, which leads to not only accurate predictions
but also quantifiable uncertainty. We conducted an experimental study and
demonstrated the effectiveness of our proposed models, with a prediction error
rate averaging 13.9%, and as low as 2.9% for certain tested batteries.
Additionally, all predictions include quantifiable certainty, which improved
by 66% from the initial to the mid-life stage of the battery. This research
has practical values for battery technologies and contributes to accelerating
the technology adoption in the industry.

###### Index Terms:

battery health monitoring, end-of-life prediction, machine learning,
industrial artificial intelligence

##  I Introduction

Electric mobility is increasingly recognized as the future of transportation,
largely driven by the need for sustainability and reduced greenhouse gas
emissions. Electric vehicles (EVs) currently make up about 15% of new car
sales globally, and the market share is anticipated to rise to 35% by 2030,
according to the International Energy Agency. In countries like China, the
percentage could be even higher, potentially reaching 40%. Central to the EV
ecosystem is battery, a critical component that determines not only EV’s
mobility performance such as range, but also its impact on sustainability and
emission. Battery therefore has become a research topic for researchers across
various fields, e.g., material science and computer science, and attracted
significant investment worldwide.

Monitoring battery health is a critical aspect of battery technologies. A
degraded battery poses significant safety risks and potentially can lead to
hazardous situations such as fires and explosions. Accurate health monitoring
can provide early warnings to ensure the safety of EV owners and passengers.
Besides, the monitoring helps access the battery’s operational condition and
estimate its end-of-life (EoL). As the battery approaches its EoL, it can be
recycled or repurposed [1] for less demanding applications like power grid
secondary energy storage. This is especially beneficial given the high cost of
batteries, which can exceed ten thousand dollars for a typical capacity of
100kWh.

Existing works for battery health monitoring can be broadly categorized into
several groups. One group utilizes traditional methods such as electrical
engineering and simulations. This includes practices like monitoring voltage
and current or employing multi-physics simulations to infer battery health.
Another group follows data-driven and machine learning (ML) strategies. Its
initial effort is time-series-based forecasting where various ML forecasting
algorithms have been explored and tested. [2] proposed a fusion neural network
(NN) model combining a broad learning system and long short-term memory
(LSTM), and [3] developed an LSTM model integrated with incremental capacity
analysis for predicting battery capacity in the next cycle. Additionally, some
studies argue that the above methods do not fully utilize the data available
and extract features from the raw data to better correlate with battery
health. For instance, [4] demonstrated the effectiveness of the features
derived from degradation voltage curves, while [5, 6] examined feature
selection schemes. These features are further integrated with complex ML
models to enhance prediction accuracy. For example, [7] introduced an auto-
encoder approach, and [8] designed a relevance vector machine combined with
convolutional NN achieving merely 12% of EoL prediction error. However, the
primary emphasis of these works remains on prediction accuracy, often at the
expense of broader and practical insights into other critical aspects like
uncertainty. Some works like [9] took uncertainty in battery health
prediction, but the focus is the next cycle prediction where uncertainty is
not an urgent concern.

In this paper, we present practical solutions for predicting the EoL of
batteries by integrating uncertainty into our models and dynamically updating
the predictions at various stages of battery usage. We provide battery owners
detailed information with not only the expected EoL but also a probability
distribution that indicates potential earlier or later EoL occurrences in
practical scenarios. This approach brings various benefits, e.g., enabling
preparations for EoL earlier than expected when batteries begin to pose
significant risks. Given more and more battery health information over the
course of the battery’s usage, our predictions will be refined and we expect
more accurate EoL estimations and increased certainty.

Specifically, we adopt a Bayesian neural network (Bnn), an ML algorithm within
the NN family. Bnn is known for its capability of quantifying prediction
uncertainty. Our approach involves a sensor network that measures and monitors
various aspects of battery health, e.g., discharge capacity and temperature.
We process the raw sensor data to extract features for Bnn training, with
customized configurations tailored to our specific application requirements.
The trained Bnn models are adopted to predict the EoL of new batteries and
provide detailed certainty quantification based on their sensor readings.
Additionally, the Bnn models are designed to be adaptive to update the
predictions as new sensor data becomes available, improving both the accuracy
and certainty of the predictions. In summary, we have the following main
contributions in this paper.

  * •

We propose a system architecture of battery health monitoring with
uncertainty-aware EoL prediction based on real-world battery usage settings;

  * •

We design and develop the Bnn models customized for battery EoL prediction
with both expected EoL and quantifiable prediction certainty;

  * •

We conduct an experimental study to show the effectiveness of the proposed Bnn
models, achieving on average 13.9%, and as low as 2.9% prediction error rate
with 66% certainty improvement from cycle 100 to 400.

The rest of the paper is organized as follows. We present our system
architecture of Bnn-based battery EoL prediction in Section II. We describe
our methodology of data and technical details of Bnn in Section III. In
Section IV, we present the experimental study and results analysis. Finally,
we conclude this paper in Section V.

##  II System Architecture and Description

In this section, we present the system architecture of battery EoL prediction
using the uncertainty-aware Bnn.

###  II-A Overview of A Battery Health Monitoring System

Battery health monitoring is typically part of a battery management system
(BMS), which uses various sensors to track the battery’s dynamics during
charge and discharge throughout its lifetime. Some sensor readings are used
directly for health monitoring while some others require data transformation
and feature extraction to show a stronger correlation with battery health.
Identifying a set of features that optimally correlates with battery health is
a significant area of research, though not the focus of this paper. Once
suitable features are identified, various methods, ML- and non-ML-based, can
be utilized to access or predict the battery’s health and its remaining useful
life. The prediction results are then communicated to the BMS, for subsequent
battery usage and related services. An illustration of the system is presented
in Fig. 1.

Figure 1: An illustration of the system architecture of ML-based battery EoL
prediction. Using the battery’s health data about charging and discharging, an
ML model predicts the battery’s expected EoL together with a quantifiable
certainty as a 95% confidence interval (CI). The prediction results are used
by the BMS to offer early EoL alerts and other health-related services.

###  II-B Battery Health Indicators

Battery health can be accessed through various measures and many focus on the
battery’s maximum capacity. During a battery’s usage, the state-of-charge
(SoC) indicates the current charge level, as a percentage calculated as
$(Q^{\text{current}}/Q^{\max})\times 100\%$, where $Q^{\text{current}}$ and
$Q^{\max}$ represent the battery’s current and maximum capacity, respectively.
Typically, $Q^{\max}$ and 100% SoC are achieved when a new battery is fully
charged. Due to battery aging, the maximum achievable SoC declines with
increased charging/discharging cycles. In EVs, a battery is considered to have
reached its EoL when its SoC fails to exceed 80% following a full charge,
marked as cycle $c^{\text{EoL}}$. This paper aims to predict such
$c^{\text{EoL}}$, interchangeable with EoL in this paper for simplicity.

###  II-C Uncertainty-Aware EoL Prediction

Most ML models for EoL prediction typically generate a single predicted value
for each tested battery. This value indicates the expected EoL, i.e.,
$\mathbb{E}(c^{\text{EoL}})$, at which the EoL is the most likely to happen.
However, relying solely on expected EoL can be insufficient in practical
applications and sometimes even misleading. In addition to
$\mathbb{E}(c^{\text{EoL}})$, it would be useful for the BMS to understand
that there is high confidence that the battery will not reach EoL before
$\mathbb{E}(c^{\text{EoL}})-\Delta c$, where $0<\Delta
c<\mathbb{E}(c^{\text{EoL}})$ is an integer. Beyond
$\mathbb{E}(c^{\text{EoL}})-\Delta c$, battery health should become a concern,
as the probability of reaching EoL increases significantly, even if it is yet
$\mathbb{E}(c^{\text{EoL}})$. Otherwise, there is a risk of unexpected battery
failures, which can be particularly hazardous during operations like driving.

Besides, data scarcity underscores the need for certainty-aware EoL
prediction. The limited availability of battery health data presents a
daunting challenge for many battery-related research efforts. The scarcity is
often due to the expensive data collection for both cost and time, as well as
the limited diversity of battery data across different usage settings and
environments. For example, the well-known Stanford-MIT dataset [4], which is
used in this study, consists of only about a hundred battery aging curves.
Such a limited dataset is far from enough for training an ML model that can
deliver predictions with high certainty and confidence. Such limitation should
be reflected in battery monitoring, as $\Delta c$ could be too significant to
overlook. Thus in this paper, we aim to develop ML models for EoL predictions
that are uncertainty-aware and risk-minimized. Specifically, we propose to use
Bnn to address the challenges.

###  II-D Bnn for EoL Prediction

In this paper, we choose Bnn for ML-based EoL prediction due to Bnn’s key
advantage of capturing prediction uncertainties. Like most ML models, a Bnn
model consists of a set of parameters. However, unlike traditional ML models
that treat parameters as fixed values, Bnn models each parameter as a
distribution. This approach allows the parameter values to vary according to
the probability distributions with varying predictions even for the same
input. This variability is the main feature of Bnn and brings it the
capability of modelling uncertainty effectively by extracting statistical
features from a list of predictions. We defer the detailed technical
description of Bnn to Section III. Our main point here is that Bnn not only
predicts the expected EoL but also provides quantifiable confidence, e.g., 95%
CI, that is essential for practical battery health monitoring. Overall, the
description of the system architecture paves the way for our detailed
technical description in the following methodology section.

##  III Methodologies

Our methodology begins with an overview of the data used in this research,
followed by the technical details of the Bnn.

###  III-A Battery Health Monitoring Data and Features

Here, we present the dataset and the features for Bnn.

####  III-A1 Dataset

We focus on Li-ion batteries and use the popular Stanford-MIT dataset [4].
This dataset comprises 124 commercial cells, each with a 1.1 Ah nominal
capacity, tested in a temperature-controlled environmental chamber. Though the
dataset is not considered as large by ML standards, it provides several key
advantages for battery health monitoring. These include a wide range of EoL
from 150 to 2,300 cycles, different charging policies, and comprehensive
recordings of internal impedance, voltage, current, cell temperature, etc.
Such diversity in data is crucial for our ML analysis.

####  III-A2 Features

Besides the dataset, [4] also identified some features that significantly
correlate with battery health and EoL. The paper evaluated three ML models,
each utilizing a different set of features, and found that the full model
incorporating all the features performs the best. In our research, we follow
this finding and use all features in Bnn. We group them into three categories
detailed in Table I.

TABLE I: The features used in Bnn for EoL estimation at cyle $c$. Three feature categories are for $\Delta Q$, discharge, and others. Category | Cycles | Features
---|---|---
$\Delta Q_{c-10}(V)$ curve | $10,c$ | minimum $\Delta Q$ in the curve
$10,c$ | variance $\Delta Q$ in the curve
discharge capacity fade curve | $2,\ldots,c$ | slope of the curve’s linear regression
$2,\ldots,c$ | intercept of the curve’s linear regression
$2$ | discharge capacity
others | $1,\ldots,5$ | average charge time
$2,\ldots,c$ | integral of temperature over time
$2,\ldots,c$ | minimum internal resistance
$2,c$ | difference of internal resistance

The first category is based on the empirical knowledge that a battery’s
discharge voltage curve $Q(V)$ tends to flatten with increased battery
charge/discharge cycles. So, the difference between the two curves from
different cycles becomes a critical indicator for estimating the battery’s
degradation. We define this difference as $\Delta Q_{c-c^{\prime}}(V)$ for
cycles $c$ and $c^{\prime}$ where $c>c^{\prime}$. In our study, we use Bnn to
monitor EoL at specific cycles, e.g., $c=200$, with $c^{\prime}$ fixed to 10
for simplicity. This approach allows us to represent the curve as $\Delta
Q_{c-10}$ and we extract the minimum and variance (subject to interpolation)
of the curve as the first two features.

The features in the remaining two categories are relatively easier to obtain.
One category involves the discharge capacity fade curve, derived from the
maximum capacity after a full charge. Intuitively the curve declines
eventually when a battery ages. To simplify, we use a linear regression to
approximate this curve. From the regression line, we extract the slope and
intercept as features, and the capacity of the second cycle is used as a
reference point. The last category includes five features that are relatively
independent of each other. One is the average charge time over the first five
cycles. The rest features consider all the cycles from the second onward,
including the integral of temperatures and both the minimum and difference in
internal resistance.

###  III-B Bnn for Uncertainty-Aware EoL Prediction

The features presented above are used as the input of our Bnn model. Bnn
quantifies uncertainty along with its prediction and this feature serves our
research objective well for incorporating uncertainty into ML-based EoL
prediction. An ML model consists of parameters, which commonly remain
unchanged once the model is trained. Bnn has a different concept that a
parameter should not be a constant, instead it should vary based on its
probability distribution. The technical details of Bnn are presented as
follows.

####  III-B1 Training

During model training, a Bnn model is first initialized with an assumed prior
distribution $p(w)$ for each parameter $w$. This assumption expresses the
model’s initial belief about the parameter, and such belief is often just a
randomized state without observing any data. After initialization, dataset
$\mathcal{D}=(X,C^{\text{EoL}})$ with features $X$ and ground-truth
$C^{\text{EoL}}$ is used to optimize the distribution for each parameter. This
essentially is a process of searching for posterior distribution
$p(w|\mathcal{D})$ of each parameter $w$ by considering $\mathcal{D}$. Such
distribution can be computed through Bayes’ rule as,

$p(w|\mathcal{D})=p(w|X,C^{\text{EoL}})=\frac{p(C^{\text{EoL}}|X,w)\times
p(w)}{p(C^{\text{EoL}}|X)},$ (1)

where the two components in the numerator are easy to calculate and the
denominator is computing intensive. Specifically, $p(C^{\text{EoL}}|X,w)$ is
the likelihood derived by applying the latest Bnn parameters to the given
input $X$, a typical computing-lite ML inference process. $p(w)$ reflects the
model’s latest belief and is readily available. $p(C^{\text{EoL}}|X)$ is the
marginal likelihood where all the possible parameters need to be considered,
i.e.,

$p(C^{\text{EoL}}|X)=\int_{w^{\prime}}p(C^{\text{EoL}}|X,w^{\prime})\times
p(w^{\prime})dw^{\prime},$ (2)

where the computing potentially involves high-dimension space. In practice,
such computationally demanding task is replaced with approximation methods
such as variational inference or Monte-Carlo for efficient training. This
however is not our focus and please refer to [10] for technical details.

####  III-B2 Inference - Prediction

With a trained Bnn model, we make a prediction of a new battery based on the
model’s posterior distributions with sampled parameter values. Different from
typical ML models, a Bnn model has different sets of parameter values and each
set allows the model to make one prediction. Overall, we have a list of
predictions for the same input and aggregate the predictions to generate the
final output, e.g., statistical information about the predictions. In this
paper, we present the aggregated results as a distribution, from which we can
quantify the prediction uncertainty with expected EoL. Often, such results
follow a Gaussian distribution based on the central limit theorem. Given a
list of $n$ predictions
$\hat{\mathbf{c}}^{\text{EoL}}=\hat{c}^{\text{EoL}}_{1},\ldots,\hat{c}^{\text{EoL}}_{n}$,
both mean $\mu$ and standard deviation (SD) $\sigma$ can be calculated to fit
a Gaussian distribution
$\mathcal{N}(\hat{\mathbf{c}}^{\text{EoL}};\mu,\sigma^{2})$, with
$\mu=\frac{1}{n}\sum_{i=1}^{n}\hat{c}^{\text{EoL}}_{i}$ for expected EoL and
$\sigma=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(\hat{c}^{\text{EoL}}_{i}-\mu)^{2}}$
for uncertainty.

##  IV Experimental Study

We evaluate and discuss Bnn’s EoL prediction performance in this section and
let us present our experimental setup first.

###  IV-A Experimental Setup

This part includes Bnn’s model configuration, training strategy, and other
information like computing resources.

####  IV-A1 Model Configuration

Bnn’s model configuration is an important part of our experimental study. The
model begins with an input layer, which takes in the features we extracted
from battery data. The layer is connected with multiple DenseFlipout layers, a
fully connected (FC) layer with 2 neurons, and an IndependentNormal sampling
layer. In each DenseFlipout layer, there is a Flipout estimator [11] for
performing a Monte-Carlo approximation of Bnn’s posterior distribution for
variational inference. The estimator can achieve lower variance for kernel and
bias compared to other estimators. The IndependentNormal is configured by
assuming Gaussian distributions of the final prediction and naturally the
layer samples the predicted EoL following the distribution information of both
mean and SD. Note that there is an FC layer right before the IndependentNormal
layer. This is mainly for mapping the output features from the last
DenseFlipout layer to the IndependentNormal layer as specific values
corresponding to the mean and SD of EoL.

####  IV-A2 Training Strategy

Our Bnn searches for its optimal posterior parameter distributions with a
multi-stage training strategy. We set a relatively large initial learning rate
of 0.05, for coarse-grained fast search. When the training performance,
measured by mean absolute error (MAE), has not improved for 10 epochs, we
reduce the learning rate by half and trigger a new training stage with a
finer-granularity search. We perform multiple stages with a decreased learning
rate until it reaches 0.001. Furthermore, the training can stop earlier given
the model performance remains non-improved for 30 epochs.

####  IV-A3 Others

For results analysis, we sample the Bnn output 100 times and estimate the EoL
distribution of the tested battery with mean and SD. All experiments are run
on a workstation with an AMD Ryzen 9 5950X processor and NVIDIA GTX 3080 GPU.
The reported experiment results are based on 10 independent runs by default
for cross-validation, each run with randomly selected 80% batteries for
training and 20% for testing. Some batteries from the dataset reached EoL in
the early stages ($c^{\text{EoL}}<500$) and are excluded.

Figure 2: A case study of Bnn’s performance on a specific battery cell showing
prediction distribution at cycles 100, 200, 300, and 400. In each sub-figure,
the battery’s actual EoL is shown as a dashed vertical line at cycle 788. Both
the lower limit and upper limit of the 95% CI in each sub-figure are shown in
dotted vertical lines. Bnn’s EoL prediction achieves improved accuracy and
certainty with more cycles of information.

###  IV-B Bnn Case Study for EoL Prediction

A case study is the most direct and intuitive way to show Bnn’s performance
for EoL prediction. Here, we show the results of one tested battery in Fig. 2.
There are four sub-figures that show the EoL prediction results at cycles 100,
200, 300, and 400, respectively. The Bnn model’s prediction in each sub-figure
is reflected as a distribution statistically derived from the model’s
prediction samples. The mean is the cycle that corresponds to the peak of the
distribution and SD measures the spread. A well-performed model should produce
a mean EoL that is close to the actual EoL and a small SD which indicates high
certainty of the prediction. Besides, we quantify the certainty with 95% CI,
which is calculated as $[\mu-z\times\sigma,\mu+z\times\sigma]$, where $z=1.96$
as the $z$-score.

####  IV-B1 Expected EoL

Seen from Fig. 2, Bnn estimates the EoL well where the peak of the
distribution is not far from the actual EoL. Bnn is optimistic about the
tested battery, with the distribution peak occurring tens of cycles after the
actual EoL. As the battery approaches its EoL, the prediction accuracy
improves. For prediction at cycle 100, the gap between the expected EoL and
ground truth is 60.4 cycles. With more updated battery data at cycle 200, the
gap reduces to 53.3 with 11.8% improvement. The gap continues to shrink to
40.5 for cycle 300 and eventually to 23.9 for cycle 400. Such a minimal gap
implies 60.4% of the prediction error reduction compared to cycle 100 and
shows Bnn’s competitive performance for this battery with merely 3% prediction
error.

####  IV-B2 Prediction Uncertainty

Bnn also quantifies the prediction certainty. For the prediction at cycle 100,
Bnn predicts that battery health would not be an issue before cycle 725 and
the EoL can be up to 972 with significant likelihood based on the 95% CI.
Indeed, the actual EoL lies within the interval. Such certainty-aware
prediction is important to trigger a non-delayed warning indicating that the
battery may reach its EoL earlier by up to 123 cycles with non-negligible
probability. This is especially important when the model is optimistic about
the battery health. Because it is a huge risk to assume a normal working
condition of the battery until its expected EoL of 848, while the battery
reaches EoL 60 cycles before at cycle 788. ML’s EoL prediction accuracy is
likely to be better in the future, but a full certainty of its prediction may
never be achieved. Quantifying and understanding the limits of an ML model is
among the prerequisites of deploying it for battery health monitoring in
practice with confidence.

####  IV-B3 Uncertainty Trend

A trend of the Bnn’s uncertainty can be observed where Bnn shows increased
confidence about its prediction when it is made in the later stage of battery
usage. The CI as shown in the sub-figures continues to narrow from $\sim$250
for cycle 100 prediction to below 200 with 400 cycles information, with a
sharper distribution. The peak probability of the distribution increases from
about 0.6% to about 0.8% which is over 30% higher. Such a trend is encouraging
as it matches well with real-world battery usage. Typically, the battery users
are not too concerned in the beginning stage and become more sensitive to the
prediction certainty when the battery is close to its EoL. Our Bnn models can
produce more and more certain predictions over time.

TABLE II: Bnn’s overall performance at different prediction cycles. We report MAE and MAPE of $\mu$ and $\sigma$ for certainty. Prediction Cycle | MAE | MAPE (%) | Uncertainty $\sigma$
---|---|---|---
Train | Test | Train | Test | Train | Test
100 | 160.8 | 183.5 | 17.3 | 18.3 | 60.4 | 66.5
200 | 154.7 | 160.0 | 15.4 | 16.4 | 52.1 | 61.1
300 | 113.1 | 131.2 | 11.4 | 14.6 | 49.2 | 58.3
400 | 84.2 | 105.6 | 10.4 | 13.9 | 34.6 | 40.1

###  IV-C Comprehensive Performance Analysis

Besides the case study of one tested battery, we report the prediction
performance for all batteries and show the results in Table II. We report the
statistical performance in terms of MAE and mean absolute percentage error
(MAPE) for both training and testing. A unique advantage of Bnn is certainty
quantification and we include $\sigma$ as well as a certainty indicator.

####  IV-C1 Trends based on Statistical Information

The results reveal similar trends as we discussed in the case study. The MAE
measures the distance between the expected EoL and the actual EoL. It is over
180 for the prediction at cycle 100. Then it becomes smaller with more cycling
information available, e.g., 160 for cycle 200 and eventually to a bit over
100 with 400 cycles. MAPE describes errors as a percentage of the actual EoL
and its results show the same trend. The percentage is relatively high at
18.3% at cycle 100 and improves to 13.9% at cycle 400 with a 24% error
reduction. Bnn not only is more accurate but also reduces uncertainty with
more cycling data, with $\sigma$ reducing from 66.5 for cycle 100 to 40.1 for
cycle 400 where the initial uncertainty is 66% higher. Overall, Bnn offers
both accurate and certainty-aware EoL predictions which improve throughout a
battery’s lifetime.

####  IV-C2 Over-fitting

Another observation is that Bnn models can be over-fitted. In most runs of
experiments, the prediction performance is more competitive during training
compared to testing. At cycle 400, Bnn’s prediction error is 84.2 for the
batteries used for training and 105.6 when new batteries are tested, with a
25% higher error. The prediction uncertainty $\sigma$ also increases from 34.6
to 40.1 by 16%. Various Bnn model configurations have been tested and such a
performance gap between training and testing remains true in different
settings. One issue is the dataset size, just over a hundred. The scale of the
data fails to offer sufficient diversity for an ML model to learn to be
generic. As a result, a model’s optimized parameters for training data cannot
fully generalize to the testing data. We expect that the over-fitting issue
will be alleviated with more data available in the future.

Figure 3: A comparison study of Bnn and its counterparts with the same data
but different ML models for MAE in Fig. 3 and MAPE in Fig. 3. Bnn achieves
competitive prediction accuracy besides its quantifiable uncertainty.

###  IV-D Comparison Study

Bnn offers uncertainty quantification besides accuracy-based prediction common
in most ML models. It is a natural question if such additional benefit is at
the cost of other aspects. Here, we compare the MAE and MAPE between Bnn and
other ML models, with some of them presented in Fig. 3.

####  IV-D1 Models

Bnn is one of the NN families, and we consider a standard NN as the first ML
algorithm for comparison. Two other algorithms are $k$-nearest neighbors
($k$NN) and support vector regression (SVR), widely used in other studies. The
last one is elastic net (EN) [12], demonstrated to be effective in [4]. The
optimal model configurations are searched for all comparison models and the
best configurations are chosen for this study. Note that none of the
comparison models is capable of incorporating certainty in predictions.

####  IV-D2 Results

Seen from Fig. 3, Bnn, despite its additional uncertainty awareness, achieves
similar prediction accuracy to its counterparts. For MAE, Bnn shares the same
level of performance with NN with both achieving an average error of about 100
cycles. $k$NN and SVR are traditional ML models proposed decades ago, with an
average mis-prediction of 127 and 183 cycles, respectively. Bnn outperforms
both. EN has been shown to be suitable for battery health analysis and is
about ten cycles more accurate than Bnn. Worth mentioning that EN however does
not capture uncertainty in its predictions. This can be one factor that EN is
not adopted in certain applications where informing users of prediction
certainty is of vital importance.

The MAPE results in Fig. 3 show similar observations, e.g., Bnn performs
similarly to NN and $k$NN and outperforms SVR. However, the performance gap
between different models varies compared to Fig. 3, e.g., $k$NN is not worse
than Bnn in terms of MAPE. This is mainly because of the inherent randomness
of our experiments. A big absolute error does not always imply a big
percentage error, as the actual EoL can be large for the tested battery. On
the contrary, when the actual EoL is small, even a minor mis-prediction may
change the percentage significantly. This is one of the practical
considerations of real-world deployment of battery health monitoring
solutions. A more suitable performance evaluation metric that better serves
the application shall be selected.

##  V Conclusion

In this paper, we presented a practical battery health monitoring system with
uncertainty-aware EoL prediction using Bnn. We recognized the importance of
quantifying uncertainty for battery health monitoring with practical benefits
such as safety and sustainability. We designed Bnn-based solutions for
analyzing the battery’s sensor readings and extracted features to predict the
battery’s expected EoL with uncertainty quantification. We conducted an
experimental study and demonstrated Bnn’s important roles of integrating
certainty with EoL prediction with a case study and statistical analysis. Our
Bnn models achieved an average prediction error rate of 13.9%, and the rate
can be as low as 2.9% for certain tested batteries. We also observed the
trends that Bnn becomes more accurate with improved certainty when more
cycling information is available, and the certainty improvement is 66% for the
predictions from cycle 100 to 400. Moreover, Bnn’s prediction accuracy is
competitive compared to several popular ML algorithms, despite Bnn’s
additional benefit of certainty awareness. Overall, Bnn can be an important
health monitoring module in the BMS and support various battery health-related
services.

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