Prediction of Battery Degradation Using Hidden Markov Model

A Scalable Online System
Predicting Battery Degradation
A SpaceTime Insight White Paper

© 2017 Space Time Insight, Inc. 2
Table of Contents
Executive Summary .................................................................................................................3
Introduction ..............................................................................................................................4
What Do We Predict?...............................................................................................................5
Battery Basics ..........................................................................................................................6
Algorithm ..................................................................................................................................7
The Importance of Modeling Consumption and Temperature in Particular.......................8
Example Results.....................................................................................................................12
Conclusions............................................................................................................................16
A Scalable Online System for
Predicting Battery Degradation
Zaid Tashman, Jason Lenderman, and Paul Hofmann

Executive Summary
Today’s modern, scalable technologies, affordable computing power, and advanced
analytics technologies – especially when combined with real-time visualization – enable
enterprise operations systems to become more predictive and prescriptive. Our
approach merges our artificial intelligence system with IoT data and broad data from
the enterprise, operations, and external sources to predict the time when a part or
machine will fail in any point of an event spectrum; past, present, or future. In practice,
our model has correctly predicted the capacity degradation in ninety-eight out of one
hundred batteries compared to actual field observations with hold out samples, saving
the customer millions of dollars by avoiding excess warranty accruals.
This unique, comprehensive approach can also prescriptively optimize a company’s operations by
efficiently and effectively addressing the predicted event and visualizing the insights for actionable
decision making.
This white paper’s use case demonstrates how to apply machine learning to predict battery capacity
for any type of battery, predict battery degradation, and relay insight into the degradation process
and remaining useful life at any point in the event spectrum for the life of the battery. This system of
predictive analytics, prescriptive analytics for optimal operations planning and scheduling
automation, and promotion of necessary human feedback, is extensible to other asset types in a
variety of industries. It provides bottom line improvements in terms of cost, labor, and effective use
of an organization’s resources.

A prevalent part of our modern life, batteries
power a wide range of devices: from cell
phones and laptops, to our electric vehicles,
to our homes. In fact, batteries are essential
to enabling the wide range of renewable
distributed energy resources. For instance,
wind energy generation is extremely variable,
often producing the highest energy output
when demand is at a low point. Therefore,
energy storage coupled with wind farms can
increase the effectiveness of these solutions.
Energy storage comprised of batteries
requires a robust health monitoring and
prognostics system to ensure reliable,
secure, and efficient operations.
Our solution comprises an online scalable
prognostics battery system that estimates
the Time of Failure or Remaining Useful Life
(the “RUL”) and the failure time of equipment,
merging this insight with operations
intelligence. This will improve the reliability,
availability, and security of the storage
systems while reducing replacement and
maintenance costs. Appropriate
maintenance and planning decisions are
automatically optimized, scheduling the
necessary materials, parts, and human
resources before failure occurs. This is crucial
in such mission critical systems, where the cost
of failure is expensive and may lead to
cascading problems.
Our approach is prescriptive. It predicts the RUL,
then optimizes operations by providing the best
action to take that minimizes cost, maximizes
safety and reliability, and ensures that spare
parts are readily available with the right crew to
perform the required tasks, given the current
and future predictions. Having an accurate
prediction of the battery’s failure time is critical,
as the failure may lead to dangerous explosions
or chemical poisoning. Our goal is to provide an
insight to battery asset reliability while
maximizing storage fleet utilization and
minimizing warranty risk.
Introduction

By predicting the capacity degradation of a battery over time, we answer:
1. What is the current capacity of the battery?
2. What kind of degradation process is the battery experiencing?
3. When will the battery reach X% capacity limit? (Typically, a vendor
or battery operator guarantees Y years of a minimum X% battery
capacity, committing to replace the battery at no cost if the unit
fails to maintain capacity).
There are two types of factors that affect the performance of a lithium-
ion (Li-ion) battery. The first set of factors are external, related to the
treatment of the battery and the environment around it. When the
batteries are used by residential customers, the battery degradation
process is largely affected by how the customer treats the battery (the
consumption profile) and environment around it, such as the
temperature. Our experience has shown (described in figure 7) that it is
important to profile each user's consumption and their ambient
temperature to accurately model how the battery degrades in future
cycles.
The second set of factors describes the variability in battery
performance due to variations from the typical aging processes evolving
over time within a battery. These can include the irreversibility of
What Do We Predict?

losses inside the battery as impedances in a
lumped parameter model:
The Li-ion battery degradation process is
nonlinear in nature. Typically, the degradation
rate speeds up rapidly at a later stage in the
battery life. This phenomenon, often referred
chemical processes, aging of the polymer
separator, etc. Variations due to aging are
typically caused in the manufacturing process.
For example, excessive porosity of the polymer
separator hinders the ability of the pores to
close, which is vital to allow the separator to
shut down an overheated battery. Our model
can learn the variation in performance if there
are enough data from a large population of
batteries.
In principle, these two types of factors are
statistically independent. The external factors,
captured by the user profile, and the variation in
performance pertinent to the battery
manufacturing process can be modeled
independently.
Battery Basics
A typical battery consists of a pair of electrodes
immersed in an electrolyte. The chemical driving
force across the cell is due to the difference in
the chemical potentials of its two electrodes. To
simplify this process, we model the various
Figure 1: Lumped Battery
Parameter Model
to as "rapid-fade," makes it challenging to
predict the capacity of the battery in the long
term, especially if the batteries of interest are
only a few years old, and none of them has yet
experienced this rapid increase in the
degradation rate. Our algorithms predict future
states, in this case capacity degradation,
including detection of rapid-fade even if it has
been observed only in a controlled lab
experiment but not in the field. In this case, our
approach combines data from the laboratory
experiment with the field data recorded from all
batteries to predict the battery capacity in the
long term. This approach of learning from the
lab data to predict the rapid fade in the field has
been promising, and we have been able to verify
it with customers. In addition, the proposed
approach continuously learns new behavior
from the live system, and once the rapid fade
starts to appear on some of the older batteries
in the field, the system will be able to detect it
and update the model accordingly.
The IR drop due to the electrolyte resistance
is denoted by RE, the activation polarization is
modeled as a resistive RCT and capacitive CDL,
and the concentration polarization effect is
modeled as RW. The value of these internal
parameters change with different aging and
fault processes, such as plate sulfation, pas-
sivation, and corrosion. Climate and ambient
temperatures also have a large effect on the
battery’s degradation process. In our ap-
proach of modeling the RUL we are interested
in the RE and RCT parameters, since both val-
ues correlate with the battery capacity C and
the degradation process over time.

Algorithm
We apply a hidden Markov model (HMM) of
topic mixtures. This sophisticated machine
learning tool can capture a rich family of
probability distributions that more accurately
model real-world phenomena. Most other
mainstream approaches rely on overly
simplifying assumptions to make problem
solving tractable. These approaches typically
result in a single, stationary distribution for
analysis, albeit a well-formed one. Our approach
allows a sequence of possibly nonstationary
mixture models that blend many simple
distributions together to create a more complex
and highly realistic probability distribution
function. As presented in Figure 2, the result is a
powerful representation of time-series data,
whether for battery internal measurements or
other phenomena.
Machine learning occurs within a space of
hierarchically structured probability
distributions. In other words, our model reasons
about probability distributions of distributions, it
learns how to learn. This means that our
HMM approach is more advanced than most
contemporary methods in that it does not
require strict assumptions and categorizing
the events a priori. Rather, the hierarchical
HMM of topic mixtures can infer the relevant
events and categorical values while
observing and learning the time-series data.
This ability is a salient feature of our model with
distributions of distributions (probabilities of
probabilities) that discretizes the measurable
space of interest. Therein the natural hierarchies
and groupings (both logical and statistical) that
exist in most real-world data are fully leveraged.
Figure 2: Hidden Markov model with
topic mixtures

Figure 3: Capacity degradation and HMM of topic mixtures schematically
This means faster training and retraining that
can scale up very well to big datasets.
In the case of battery health, the important
questions boil down to whether the battery will
provide enough power during the current
discharge cycle, and what kind of degradation
process the battery is experiencing. This allows
insight to how many more discharge cycles the
battery can produce. This is captured by the
State of Charge (SOC) and the Remaining Useful
Life (RUL) (see Figure 3). Even though our
approach has the capability to model and
predict both SOC and RUL metrics, our model
focuses on the more long-term prediction, RUL.
This improves planning and optimization of the
entire battery fleet, maximizing asset utilization.
The Importance of Modeling Consumption and
Temperature in Particular
As mentioned in the introduction, there are two types of factors that affect the performance of
batteries: external factors that we call consumption profiles like temperature, SOC, and power,
and internal manufacturing-related factors.

The temperature is one of the most important
factors that affect the degradation of the
battery. Temperature is known to have a
significant impact on the performance and cycle
lifetime of Li-ion batteries. The formation and
modification of the surface films on the
electrodes as well as structural changes in the
electrodes are found to be the main contributors
to the degradation rate increasing with
temperature. Typically, battery systems are
installed across multiple geographic regions
that experience different temperature
conditions. The effect of temperature on the
degradation can also be seen clearly in lab
experiments. Figure 4 shows the capacity
degradation of four batteries, cycled at the
same conditions except that two of them
were kept at 45°C and the other two were
kept at room temperature (25°C).
Let’s look at the power output of the battery
in watts. Power output depends, for example,
on the type of appliances used by the
Figure 5: Pair-wise distance measure between the consumption
patterns of 400 random Li-ion example batteries
customer. The consumption is a continuous
variable, and varies between cycles. To show
that there is a significant difference in
consumption patterns between different
batteries we calculated a pair-wise distance
measure between the batteries’ power-out
distributions. Figure 5 shows the pair-wise
distance between all and each consumption of
an example population in an Asian country. A
value of 1 indicates completely different usage
Figure 4: Capacity curves from lab samples
capacity
cycle

patterns, and a value of 0 indicates an identical
usage. Notice that the values along the diagonal
are exactly 0, which is a result of comparing a
battery usage with itself. This plot shows that
there are indeed differences in how customers
use their battery. Therefore, it is reasonable and
well-advised to model the consumption
including the different temperature distribution
over the geography of a country. We reasonably
assume that consumption is independent from
battery degradation, therefore we model it
separately.
Typically, we train a hierarchical mixture model
to learn the consumption profile (temperature
observations, customer usage, threshold
settings, etc.) and observe the battery capacity.
This model will essentially describe the
dependencies between the observations'
sequence from the battery and the degradation
rate.
Hierarchical mixture models are sophisticated
models in machine learning that can capture a
rich family of probability distributions that
accurately model large, complex, real-world
data. The model blends many simple
distributions together to create a more
complex and highly-realistic probability
distribution function. As a result, we capture
a rich family of probability distributions that
accurately model the dependence between
the battery’s daily observations and
degradation. In addition, we are able to
profile each battery based on observations of
its consumption and achieve a more
accurate capacity prediction.
With profiling, we can learn archetypes or
topics that are combined with so-called
hierarchical mixtures – i.e. distributions of
distributions. These topic mixtures describe the
battery consumption pattern of customers and
help make a more accurate prediction of battery
capacity degradation. Each battery/customer
will have their unique profile, which changes as
we gather more data from the battery over time.
Generally, such a profile converges to a stable
mixture of archetypes. The stable mixture
allows estimation of the battery behavior, and
thus, its degradation. In Figure 6 we show an
example of a battery where the profile and its
entropy converge in less than a year of
observations and remain stable for the
Figure 6: Example of convergence of a battery profile and its entropy in less than a year

Figure 7: Profiling influence on the prediction. Each line shade represents the predicted capacity curve computed using different amounts of historical data.
Thicker lines use more historical data, whereas lightly shaded lines use fewer historical data.
remaining life of the battery. In this example, we
clearly see the typical reduction in entropy – the
measure of uncertainty – after about 600 cycles.
To further demonstrate the power of profiling, we
show in Figure 7 how the prediction of the future
capacity changes as the profile of the battery is
learned with higher accuracy due to more observed
data. Notice how the prediction changes at
October 2015. From that point on, the model has
enough information - using data from day 1 until
October 2015 - to accurately predict the future
capacity. Online learning is a unique feature of our
approach, which enables the model to improve the
prediction in a continuous fashion.
In addition to the capacity predictions, the
generative model allows us to explore the
dependencies between battery observations and
degradation. This is another advantage of using a
generative model, as opposed to a black-box
approach. For example, we found surprising
anomalies between the customer minimum
capacity threshold setting and the expected
degradation.
Battery Capacity Prediction

Example Results
As mentioned, battery degradation is highly
nonlinear. One often observes rapid-fade since
the degradation process changes at a later
stage of battery life. Lithium deposits form on
electrodes during each cycle, but some
deposits do not completely dissolve. The more
a battery is used, the more permanent deposits
build and the less capacity is left. This causes
rapid-fade degradation. It is important for the
model to capture the two degradation patterns,
normal and rapid-fade.
Hidden Markov models provide a very powerful
tool that allows us to learn the complex
sequential dependencies between all the
observations mentioned before. We will not go
into the technical details of our modified and
constrained hidden Markov model
implementation as that is outside the scope of
this report. However, we can distinguish the
exponential degradation of batteries from the
rapid-fade in the field, even if we observed
degradation below the fifty-percent mark only
from lab data.
Figure 8 shows a typical example of
randomly chosen batteries. As more data is
fed into the model, the prediction updates,
taking into account all the information
available up to the time of the prediction.
Also, the belief about the current state of the
battery and the future behavior of the battery
will become clearer. For example, young
batteries will most likely have high uncertainty in
state belief in the first few cycles.
As more evidence becomes available, the
uncertainty lessens, giving a more accurate
prediction of the capacity. This is a unique
feature of our Bayesian reasoning approach,
which not only provides an estimate of the
mean predicted capacity in the future, but that
Figure 8: Example of results from our learned models for some randomly chosen batteries.

Figure 9: Architecture and
data flow for dynamic hier-
archical Bayesian networks
to predict the degradation
of battery capacity.
Figure 10: Change in predicted
capacity as more and more
data is revealed.
mean capacity is accompanied with an
uncertainty value (i.e. variance) that is very
important in making decisions about these
batteries. Two batteries can have the same
predicted capacity, but the difference in variance
could provide a second level of insight that
makes the decision making process more
insightful and data-driven. Note that this does
not hold for the typical deep learning
approaches. Figure 9 shows the architecture
with two tiers of the Bayesian model. It learns
the model from historical data and updates it or,
in other words, it reasons by calculating the
Bayesian posterior in real time from the
streaming online data.
In Figure 10 we demonstrate the improvement
of the prediction over time when more data is
revealed. For the selected battery, the capacity
over the next eight years or so is calculated
every cycle, each time with one extra day of
information. For instance, on day 1 the capacity
is predicted over the next eight years without
any historical data. This is shown in the 3D plot

Figure 11: Predicted capacity and pertinent 95% confidence interval (above) and probability of failure f(t) (below)
in Figure 10 by the degradation curve aligned
with "1" on the axis labeled "Day of Prediction".
One can see that during the initial iterations, the
predicted capacity curves change slightly from
one cycle to another as the uncertainty in the
belief of the state is higher. However, moving
along the "Day of Prediction" axis, more data is
fed into the model, adding more confidence in
the belief of the battery state given the evidence
so far, and the predicted capacity curve
becomes more stable. That is, the information
or the data fed into the model is consistent with
the belief about the battery state, therefore the
uncertainty goes down, achieving a more
accurate prediction. It is important to note that
this example is related to this specific battery,
and that each battery can have very different
uncertainty in the predicted capacity values.
We conclude with an example of a typical failure
analysis, answering the question, “How many
batteries will fail?” To answer this question, we
will define the failure event to be reaching a
capacity of fifty percent. We are interested in the
probability distribution of the time of this event,
for each battery. Fifty percent is an arbitrary
value that could be changed to a different
value, such as to define when a warranty
policy can be invoked. The terms “event” and
“failure” are used interchangeably in this
section.
Suppose that the failure times of the battery,
Time, is a random variable, and the function
that describes the likelihood of observing
Time at time t relative to all other failure
times is known as the probability density
function (pdf), or f(t). The cumulative
distribution function (cdf), F(t), describes the
probability of observing Time less than or equal
to some time t, or p(t ≥ Time):
For each battery, we can estimate the probability
of failure f(t) using the predicted mean capacity
at each time step; the lower curve in Figure 11.

Figure 12: The probability of failure f(t) and the aggregated failure count for the sample batteries
The upper curve of figure 11 shows the variance associated with this prediction at the 95%
confidence interval.
We can now calculate the expected failure count, i.e., number of batteries that are expected to fail
on each day, by aggregating the probability of failure over all the batteries. The expected failure
count for the sample batteries is shown in figure 12.
“As more evidence becomes
available, the uncertainty lessens,
giving a more accurate prediction of
the capacity. This is a unique feature
of our Bayesian reasoning approach,
which not only provides an estimate
of the mean predicted capacity in the
future, but that mean capacity is
accompanied with an uncertainty
value (i.e. variance) that is very
important in making decisions about
these batteries. Two batteries can
have the same predicted capacity,
but the difference in variance could
provide a second level of insight that
makes the decision making process
more insightful and data-driven.
Note that this does not hold for the
typical deep learning approaches.”

Conclusions
Modeling Li-ion battery capacity is complex and
requires advanced modeling techniques. We use
our machine learning approach of hidden Mar-
kov models of topic mixtures to learn the com-
plex dependencies between customer consump-
tion and battery degradation patterns, resulting
in an accurate prediction of the battery capacity.
In typical examples we achieve about 2% predic-
tion accuracy (that is, 98% of our predictions are
correct) compared to actual observations in the
field with hold out samples. This has saved one
customer many millions of dollars. Our software
ameliorates the customer’s existing practice of
accruing money for warranty claims long before
the batteries are likely to cross below the war-
ranted capacity threshold.
We believe our solution to predicting rapid-fade
is unique, and it lays a solid foundation for fur-
ther industrial-grade battery prediction. In this
paper we only scratched the surface of what
can be done in modeling this system of predic-
tion, combined with prescriptive analytics for
optimal operations planning, scheduling au-
tomation, and financial risk mitigation based
on the battery failure predictions.
To summarize, our model offers significant
benefits:
Learns Continuously: As demonstrated, our
model learns continuously from new obser-
vations, updating the model with the newest
evidence, then recomputing the predictions,
achieving more accurate prediction results.
Predicts Assets Individually: Our model
treats each battery in the population distinct-
ly. Each battery encounters unique condi-
tions, producing a distinct and unique profile
which is then used in predicting the future
behavior of the battery.
Eliminates Uncertainty: Our model gives us a
confidence interval of the prediction in a
mathematically proper way. Understanding
the uncertainty in the prediction is extremely
important for making better data-driven deci-
sions. In addition, our ability to estimate the
expected failure count is a unique feature in
our model. This is particularly important for fi-
nancial calculations, such as warranty and ac-
cruals.
Enables Transfer Learning: Our approach does
not make any specific assumptions about the
make and model of these batteries. Therefore,
the same analytical model can be retrained on a
different set of batteries from different manu-
facturers or of different sizes, assuming similar
data is available. This holds true for similar pre-
dictive models of other assets types in a multi-
vendor, industrial asset fleet.

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650.513.8550
www.spacetimeinsight.com
@spacetimeinsght
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About SpaceTime Insight
SpaceTime Insight enables organizations in asset-intensive industries to
generate more value from their people, processes, and assets. Our award-
winning analytics and industrial internet of things applications optimize
operations in motion, in context and in real time. Teams at some of the
largest organizations in the world, including transportation and energy firms
and some of the world’s largest utilities, use SpaceTime Insight software to
power mission-critical systems. SpaceTime is headquartered in San Mateo,
CA with offices in Canada, UK, India, and Japan.

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