This document describes a hybrid Kalman filtering approach for estimating the state of charge (SOC) of lithium-ion batteries used in low-powered microcontrollers. An electrical equivalent circuit battery model is developed and its parameters are identified through discharge cycle simulations and experiments. Extended Kalman filtering is then used to estimate SOC with errors less than 1% based on simulations and experiments. The algorithm is implemented on an STM microcontroller, where a hybrid Coulomb counting and EKF approach is used to estimate SOC with less than 0.5% error to suit the microcontroller's limited memory.

1. A Hybrid Kalman Filtering for State of Charge
Estimation of Lithium-Ion Battery Used in Low-
Powered Microcontroller: CCM-EKF Approach
Hong Vin Koay
Department of Electrical Engineering
University of Malaya
Kuala Lumpur, Malaysia.
koayhongvin@gmail.com
Joon Huang Chuah
Department of Electrical Engineering
University of Malaya
Kuala Lumpur, Malaysia.
jhchuah@um.edu.my
Kong Yew Tan
AMS IC Design Group, Microsystem
MIMOS Berhad
Kuala Lumpur, Malaysia.
kongyew.tan@mimos.my
Abstract—A new technique that aimed to be flashed into a
low-power microcontroller to estimate the state of charge (SOC)
of lithium-ion battery is introduced. First, an electrical
equivalent model is developed to simulate and model the
behaviour of the battery. The battery used in this work is
Panasonic NCR18650B and 2RC model is chosen to capture the
slow and fast response of the battery. The parameters are
identified through the discharging cycle with the help of
MATLAB Simulink Design Optimization (SDO). Then, the
model is verified through simulations and experiments. Once the
model is verified to be within an acceptable range, it is then set
to run several drive cycles, including constant current discharge
(CCD) and hybrid pulse power characterization (HPPC) test. It
is shown that the SOC estimation error using Extended Kalman
Filter (EKF) is <1% in both simulation and experiment. Finally,
the algorithm is flashed into an STM chip and then external
circuits are used to collect the real-world data. The SOC is then
estimated. A hybrid Columb Counting Method and EKF SOC
estimation are introduced to suit the limited flash memory of the
chip. The on-chip SOC estimation error is <0.5%.
Keywords—Extended Kalman Filter, Lithium-ion Battery,
Battery Management System, State of Charge Estimation
I. INTRODUCTION
The demand for limited fossil fuels has never come to an
end. With the growing concern of the carbon dioxide
emission, an alternative for clean energy and storage systems
were highly researched and developed. These clean energy
applications have grown when Electronic Vehicles (EVs)
slowly taken over the market, which has made the battery
becoming the most prominent energy storage device [1].
Lithium-ion, or commonly known as li-ion, is a promising
power source for EVs due to its high energy density and long
cycle life [2, 3]. Because of its popularity, the price of
lithium-ion battery (LiB) continues to decline, making it a
much competitive choice for EV applications [4]. With these
great benefits, the safety and reliable operation of a battery
pack should be ensured. Thus, it is crucial to make sure that
the Battery Management System (BMS) can manage the state
of charge (SOC) of batteries efficiently. An accurate model
of the battery is needed, especially the SOC of a battery pack
should be accurately estimated [5].
However, it is difficult in determining and estimating the
parameters as they are interdependent and varies among
different batteries [6]. The exact measurement of SOC for
battery plays an important role when charging and
discharging batteries [7]. There exist several methods in
estimating SOC of a battery [8, 9]. Among those methods, the
Coulomb counting method (CCM) is the commonly used
technique. It integrates the measured battery current with
respect to time and it is then used to determine the battery
SOC. Although this method can do the task, the accuracy of
the estimation is highly dependent on the prior knowledge of
initial SOC and may accumulate errors from noise [10]. Other
methods, including the Open Circuit Voltage (OCV) method,
are suffice, but it needs a long relaxation time, making it to
be not suitable for real-time applications [11].
With the advancement in computation technology, several
intelligent approaches have been introduced as a replacement
for CCM and OCV method and offered higher accuracy in
SOC estimation. Among them, the neural network method
and Kalman Filter (KF) method have been reported to have
higher accuracy.
Although LiB provides a lot of favourable characteristics
and advantages, the main drawback is the cost of materials
and manufacturing [12] and limited use cycle due to the
ageing process, which translates into more waste. Thus, a
better implemented BMS would increase the lifetime.
Cost of implementing highly real-time and accurate model
would be not justified when dealing with only several
batteries for minimal project implementation. As for colossal
project implementation such as EV, the manufacturers are
looking for 300,000 cycle range for Hybrid EV and 1,000
cycle range for Battery EV [13]. All of this translates into the
need for accurate yet promising with minimal cost of
estimation and monitoring of battery.
In this work, the focus is on further improving the accuracy
of estimating of SOC for LiB with the use of Extended
Kalman Filter (EKF). EKF is deemed to be computationally
expensive but thanks to its accuracy, it is justifiable especially
for broad implementation, such as EV. In this project, the
balance between accuracy and the computational requirement
is done to make sure it fit for smaller project implementation.
EKF is very dependent on the type of battery and thus this
project also investigates if this technique fits well for LiB.
Thus, many iterations of experiments shall be taken to make
sure the data is consistent. There do exist several uncertainties
and variables, such as ageing of the battery and possible over-
discharging or overcharging of the battery. This has further
increased the difficulty and accuracy of the suggested
algorithm preparing for hardware implementation.
As for hardware implementation, even though there exists
advanced BMS in the market, but they are usually not suitable
for smaller project implementation and expensive. This
project looks into the integration of the algorithm on-chip for
embedded EKF SOC estimation.
2020 IEEE 8th Conference on Systems, Process and Control (ICSPC), 11–12 December 2020, Melaka, Malaysia
978-1-7281-8860-7 ©2020 IEEE

2. II. LITERATURE REVIEW
Extra care should be applied in the operating conditions to
avoid thermal runaways, physical damage and ageing issue,
in order to harvest the maximum potential of LiB. Thus, a
battery management system (BMS) is needed in a precise
measure, regulate and estimate the battery packs. BMS, by
definition, means a system that is capable of managing a
battery [14].
BMS has been used in many fields. The nearest example is
portable electronic devices, such as notebook and mobile
phone, which uses BMS as well. BMS in EV, on the other
hand, is still in its ‘baby stage’ as the number of battery cells
used in EV is a hundred times higher than that of consumer
portable electronic devices.
A. State of Charge
State of Charge (SOC) is defined as the ratio of releasable
or remaining capacity to the actual capacity of the battery
[15], which could be expressed as in Eq. (1).
= × 100% 1
From eq. (1), Cremaining is the maximum available capacity
of battery after a specific period, while Cactual is the maximum
possible limit of charge of battery during the initial full
charge or discharge cycle. Cremaining is influenced by the
degradation of battery due to cycling and corrosion, and Cactual
will not achieve the rated capacity, Crated as provided by the
battery manufacturer. Cactual depends on State of Health
(SOH) of the battery as well as the discharging current rate.
SOC is represented in percentage, where 100% represents
fully charged and 0% represents fully discharged.
B. Battery Modelling
The battery model is the heart of every model-based SOC
estimation. It is used to mimic the battery behaviour. As such,
it is vital to have a reliable yet accurate model. Battery model
can be classified into Empirical Model, Electrochemical
Model (ECM), Electrical Equivalent Circuit Model (EECM),
Electrochemical Impedance Model and Data-Driven Model.
Among all, EECM is the preferable method for online SOC
estimation. EECM is simple, which requires less
computational power and it provides high compatibility for
embedded system applications.
EECM utilizes lumped discrete components, like a
capacitor, voltage source and resistor to picture the battery
dynamic behaviour [16]. In EECM, generally, a resistor
represents the discharging element, capacitor and voltage
source represents the battery OCV. Multiple RC pairs may be
found in the model with different time constants, which the
time constant stands for the diffusion process in the
electrolyte and porous electrodes, the charge transfer and
double-layer effect in the electrode [17].
EECM has a lot of variants, such as Rint [18], Randles
[19], Partnership for a New Generation of Vehicle (PNGV)
[20, 21], 1RC model / Thevenin model [22] and 2RC model
/ Dual Polarization (DP) model [23, 24].
2RC model is more preferred in online SOC estimation. It
contains two RC branches, of which describes slow and fast
transient response caused by charge transfer and diffusion of
lithium-ion battery [24]. 2RC is used to overcome the
inability of 1RC in getting the difference between
concentration polarization and electrochemical polarization,
which causes inaccurate simulation at the end of charging or
discharging [18]. In the 2RC model, battery temperature and
SOC positively influenced the parameter values.
C. Kalman Filter
Kalman filter (KF) theory was introduced in 1960 by
Rudolf Emil Kalman [25]. KF can be classified into linear
and nonlinear. Nonlinear KF includes Extended Kalman
Filter (EKF), Sigma Point Kalman Filter (SPKF), Cubature
Kalman Filter (CKF), etc. Generally, the steps in combining
battery model with KF algorithm can be shown in Fig. 1,
where there are intermediate steps.
From Fig. 1, discretization is needed as EECM contains
nonlinear partial differential equations (PDEs) and needs to
simplify the calculation. Discretization can be done with the
analytical method, integral method approximation, finite
element method, etc. Three discretization methods, which are
finite difference, finite element and differential quadrature
methods, were applied and then compared with the SP model
[26]. It is found that the differential quadrature method is the
best among all given the trade-off between low order and
accuracy. In the EECM, bilinear transformation is the
standard discretization method [27].
Then, the time domain difference equations are obtained
through a discretized battery model. It is needed for battery
parameter identification. By plugging in the identified
parameter into time-domain difference equations, the state
space representation equations are now useful for KF
algorithm.
Past studies on EKF suggested that the average error is
within 1 – 4% [28-33]. Among all KF algorithm, EKF is able
to operate well even in noisy and inaccurate initial conditions.
When utilizing KF algorithms, initial noise covariance
matrix or also known as KF tuning is an essential step in
determining the accuracy of the estimation. The performance
of the filter is set by covariance matrix P, process noise
matrix Q and measurement noise R. It is not possible to tune
the KF parameters correctly in the field of batteries, and
generally, the parameters are determined through past
experiences or experiments. The covariance matrix, P
determines the convergence behaviour, while the process
noise matrix, Q determines the model uncertainty. As for
measurement noise, R, it determines the measurement
uncertainty. The conditions of the parameters and its
consequences are summarized in Table I [27, 32].
III. METHODOLOGY
A. Battery Modelling
This work uses the 2RC model, as shown in Fig. 2.
Battery
Model
Discretization
Representation
in state space
equations
KF
Algorithm
Fig. 1. Steps in combining battery model and KF algorithm
Fig. 2. 2RC model.
R
Uoc VT
+
-
I
R2
C2
V2
R1
C1
V1
2020 IEEE 8th Conference on Systems, Process and Control (ICSPC), 11–12 December 2020, Melaka, Malaysia
978-1-7281-8860-7 ©2020 IEEE

3. By using simple circuit rule, the dynamic behaviour of the
system could be extracted. Applying Kirchhoff’s Voltage
Law (KVL) on the circuit in Fig. 5, the expression of terminal
voltage could be written as,
= − − − 2
KF only works on the discrete-time system and the
continuous model above has to be discretized [34]. By
converting into a discrete-time model, it makes the model
easier to implement on a modern computer which inherently
is limited to finite numerical representations of numbers and
discrete processes.
The final state-space equation is,
,
,
=
1 0 0
0 0
0 0
,
,
+
−
1 −
1 −
3
B. Battery Parameter Identification
The model used in the simulation is a 2RC model, having
six parameters to be collected. The parameters are Em, R0, τ1,
τ2, R1 and R2. These parameters are estimated through a series
of iteration. They are done through a series of curve fitting
and utilizing the Simulink Design Optimization.
C. Extended Kalman Filter
Since EKF utilizes analytic linearization of the model at
every point of time, thus, it is clear that there are some issues,
but it is still popular, thanks to its computational efficiency.
In EKF, the calculations can be split into two steps, which is
prediction and correction steps. Both steps contain three sub-
steps.
In prediction stage, the present value, is calculated
based on the prior information, . Then, the predicted state
estimate error covariance matrix, Σ , is calculated. The
system output is predicted and written as
. In correction
stage, the output prediction error is calculated and then the
estimator gain matrix (also known as Kalman gain), is
determined. The state estimate is calculated based on the
Kalman gain through the state prediction. Here, is the state
vector and is the output from the system.
The central equation of each step is shown below.
D. Software Simulation
Once the battery modelling is done, it is then used to feed
in with variable load and the battery’s output is being
compared with the output given by EKF. The error is being
calculated and being repeated several iterations, excluding
the ageing process.
The battery modelling is done when a new battery is used
as the parameter is varied slightly across batteries.
Discharging data is used for battery modelling and the
discharging cycle is specifically CCD as identification of
pulse will be made more accessible and more accurate
parameter could be obtained.
In order to collect the battery data, the charging and
discharging of the battery should be done accordingly. There
are several ways to do so. In manual data collection, the
charging and discharging is done separately with different
equipment. Another way of data collection is through battery
tester, as the battery tester is able to do the battery charging,
discharging and monitoring all together in one equipment.
E. Data Collection
The charging and discharging of battery are done through
programmable DC power supply and load.
The recommended charging mode is Constant Current
(CC) – Constant Voltage (CV), at a maximum of 0.5C.
Knowing that the typical capacity of a Panasonic NCR18650
Lithium-Ion battery is 3350mAh, thus the maximum current
that should be used to charge the battery is 1675 mA. The CC
and CV region is categorized as such, where initially, when
TABLE I
COVARIANCE MATRIX, PROCESS NOISE MATRIX AND MEASUREMENT
NOISE VALUE AND ITS CONSEQUENCES
Parameter Condition Consequences
Diagonal
elements
of P, Px
Zero 1. Assumption of accurate initial values.
2. Adaptation of incorrect initial value is
slow.
High 1. Fast correction of initial parameters.
2. Unstable behavior.
Diagonal
elements
of Q, Pw
Zero 1. Assumption of perfect model.
2. No correction of states.
High 1. Increase estimation error.
2. Uncertainty of states.
Diagonal
elements
of R, Pv
Small 1. Assumption of high accuracy of
measurement sensors.
2. Calculation based on measurement,
ignoring the model.
High 1. Low Kalman Gain.
2. Correction of measurement is
reduced.
3. Estimation based on model without
corrections.
Step 1a: State-prediction time update
= | = , , |
≈ , ,
Step 1b: Error covariance time update
Σ , =
≈ Σ , + Σ
where =
, ,
,
=
, ,
Step 1c: Predict system output
= | = ℎ , , | ≈ ℎ , , ̅
Step 2a: Estimator gain matrix
= Σ , Σ , = Σ , Σ , + Σ
where =
, ,
, =
, ,
Step 2b: State-estimate measurement update
= + −
Step 2c: Error covariance measurement update
Σ , = Σ , − Σ ,
Battery Data
(Charging /
Discharging)
Battery Modelling
/ Parameter
Identification
SOC
Estimation via
EKF
Fig. 3. Software simulation procedure used in this work.
2020 IEEE 8th Conference on Systems, Process and Control (ICSPC), 11–12 December 2020, Melaka, Malaysia
978-1-7281-8860-7 ©2020 IEEE

4. the battery is drained, the voltage could not reach the
maximum of 4.2 V, as such, it is operated in CC mode at a
maximum of 1675 mA current to charge the battery.
Eventually, when the charging approaches the end, the
battery voltage is now more than 4.2 V and is in the CV
voltage, at which the current would start to fall. When the
current applied to the battery is less than 65 mA, it is said to
be charged and should be removed. The switchover of CC
and CV is determined by the voltage and current limit set to
the power supply.
Battery lifespan is closely related to the level and duration
of stress applied to it, including charging, discharging and
temperature. Usually, remote control-based product, such as
race car, helicopter, etc. will stretch the battery to the
maximum, by applying up to 30C discharging rate. This will
add safety risks and shorten battery life.
One of the indicators used in discharging is the depth of
discharge (DOD). Usually, a lithium-ion will discharge to 3V
per cell. At this time, about 95% of the energy is used, and
voltages will drop sharply if discharging continues. One
could continue to use the battery under the rated value, but in
order to prevent over-discharging, the battery shall cut off
automatically. High load current appliances would usually set
to a lower end-of-discharge voltage to prevent premature
cutoff.
F. Hardware Setup
The simulation is done with sample data and data collected
from the battery used. The parameters of the EECM used in
modelling the battery is collected. The work continues with
real hardware implementation, in which the steps used in the
simulation will be converted to C code and download into
STM32 microcontroller to undergo on-board estimation. The
EKF model is being coded into a Real-Time Operating
System (RTOS) enabled microcontroller to do on-chip SOC
calculation and to make sure that the battery is able to
perform to its optimum performance.
MakerMax SC4P1 is used as it is able to provide real-time
voltage, current, temperature monitoring through LTC2990.
This board is used as an alternative to collect battery data.
This board is able to do on-board charging as well as
discharging. This standalone board should be connected to
either Arduino or STM32 development board in order to
program it. In this work, the STM32 Nucleo-F446RE
development board is used as the driver board as it provides
on-board ADC conversion. The code is done through
Arduino Compiler.
G. Suggested CCM-EKF Method
To overcome the randomness of initial SOC, where the
battery plugged in is not necessarily fully charged or fully
discharge, a small modification is made. The initial SOC is
calculated from the SOC-OCV relationship. The whole
execution process of the estimation is shown in Fig. 8. The
whole idea of simplifying the computationally intensive
algorithm (in the contexts of limited flash memory available
on Nucleo-F446RE), a hybrid of CCM and EKF is used.
Instead of running full EKF at every iteration, the full EKF
will only run every one minute. The result obtained will then
dynamically update the SOC to avoid the error accumulation
by CCM. CCM is used as the primary SOC estimation, as it
is relatively fast and straightforward. It is first tried to run full
EKF at every iteration, but it took around 1.5 seconds to
update every iteration and built up cache, which slows down
the whole processes. To verify the proposed simplified hybrid
CCM-EKF SOC estimation, the data is logged and is tested
against full EKF to visualize the error. The test is run with
full CCM estimation and proposed hybrid CCM-EKF, to
verify the robustness of the proposed algorithm.
IV. RESULTS AND DISCUSSION
The battery used in this test is Panasonic NCR18650B
(without protecting circuit on battery). The battery used in
this test is assumed to be new, with a resting period of around
1 hour after the battery is fully charged. No discharge has
been performed on this battery and it is assumed that the
battery SOH is on tip-top condition.
The testing sequence is set to be ½ hour resting period
followed by 10 minutes of constant current discharge with a
discharge current of 1.6A (0.5C). The test is continued until
the battery is fully discharged as determined by the battery
tester software. Through this test, 20 cycles are completed
and a total of 43266 data points are registered. Only first
42000 data points are used in this simulation. The data is used
in this simulation is shown in Fig. 6. The parameter of the
sample data is obtained through the series of curve fitting and
optimizations, as shown in Fig. 7. It is observed that two time
constants are more than sufficient to capture the response of
the battery. The parameters for the battery model is shown in
Fig. 8 and is consistant among both software simulation and
hardware implementation, since same cell was used.
CC CV
1675
Fig. 4. Charging characteristic of Panasonic NCR18650B
Initialize the SOC with simple parameter look-up table. The
SOC is first obtained through SOC-OCV relationship.
Full EKF is
ran and
adjustment is
made to the
current SOC
estimation
CCM to
estimate
the SOC
60 sec
passed?
Voltage, Current is read from sensor
Update global SOC value
Display the estimated SOC
Y N
Initialization
Main
Loop
Fig. 5. Simplified version of EKF SOC estimation
2020 IEEE 8th Conference on Systems, Process and Control (ICSPC), 11–12 December 2020, Melaka, Malaysia
978-1-7281-8860-7 ©2020 IEEE

5. A. Software Simulation
The collected data is deemed to be experimental data,
where the data will be undergone full EKF as a baseline of
accuracy. The EKF parameters used are shown in Table II.
The results of the simulation are illustrated in Fig. 9 and
10. Overall, the results are acceptable, with the accumulation
of SOC error towards the end. This is due to the nature of the
battery where the last pulse is near to over-discharging of the
battery, which causing a massive drop of voltage when
discharging. The RMS SOC Estimation Error is 0.8507%.
B. Hardware Implementation
The OCV-SOC relationship is determined as shown below
through curve fitting of Em vs SOC from Fig. 11.
= 57.8812 − 252.9521 + 462.0823
−450.4743 + 245.0518 − 70.7306
+8.2241 + 0.7756 + 3.3247 20
The logged voltage and current is shown in Fig. 11. As
shown in both figures, the logged data has many spikes,
indicating that the sensor inaccuracy is high. This is mainly
due to the ADC conversion. Thus, this serves as useful data
to verify the robustness of the EKF, where the sensor error
could be adjusted by the algorithm. Due to the maximum
discharge current allowable by SC4P1, the discharge current
is capped at 0.5A.
The parameter for EKF is set as in Table III. Due to the
highly unknown of the initial state, the initial SOC is set at
0.75, which could demonstrate the ability of the EKF to fix
the error dynamically.
Fig. 6. Pulse identification on the collected data.
Fig. 7. Time constant comparison of the selected pulse.
Fig. 8. All parameters for the model.
TABLE II
PARAMETERS USED IN EKF SOC ESTIMATION
Parameter Value
Diagonal elements of P, Px 0.1
0.1
0.1
Diagonal elements of Q, Pw 1 × 10
Diagonal elements of R, Pv 500
TABLE III
PARAMETERS USED IN CCM-EKF ON-BOARD SOC ESTIMATION
Parameter Value
Diagonal elements of P, Px 0.1
0.9
0.9
Diagonal elements of Q, Pw 1 × 10
Diagonal elements of R, Pv 100000000
Fig. 9. Estimated and measured SOC
Fig. 10. Estimated and measured voltage.
Fig. 11. Logged voltage and current data from Makermax SC4P1
Fig. 12. SOC error between CCM-EKF and EKF.
2020 IEEE 8th Conference on Systems, Process and Control (ICSPC), 11–12 December 2020, Melaka, Malaysia
978-1-7281-8860-7 ©2020 IEEE

6. The SOC estimation done by CCM-SOC is compared with
full EKF estimation. The SOC error is shown in Fig. 12. It is
observed that with such minor changes, it has brought the
possibility to done KF-based estimation on the memory-
limited microcontroller. The RMS SOC estimation error is
shown to be 0.0468% as compared to full EKF estimation.
V. CONCLUSION
This work has illustrated that the ability of low-memory
microcontroller can handle KF algorithm well, when doing
some minor changes to the algorithm, such as combining
them with lookup-based techniques to minimize the
consumption of memory in computing the estimation. This
work suggested the following possible improvements,
• The memory management on-chip. Currently, the code
was coded in Arduino environment, which is not an
optimized version of code, where there is a possibility of
memory overloading.
• Self EKF parameter selection and normalization.
• Multiple cells for on-chip estimation.
• More robust KF family algorithms can be used. This
work has demonstrated that EKF is possible even on
memory-limited chip, which made it possible for more
demanding KF, to deploy on high- memory chip.
ACKNOWLEDGEMENT
This work is partially supported by MIMOS Berhad.
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