Evaluation of hysteresis expressions in a lumped voltage prediction model of a NiMH battery system in stationary storage applications

Abstract

As a part of battery system operation, battery models are often used to determine battery characteristics such as the state of charge (SOC) and the state of health (SOH). A phenomenon that has a large impact on battery model accuracy for NiMH batteries is open circuit voltage (OCV) hysteresis, which causes the OCV to differ not only with the SOC of the battery but also with the charge-discharge history. This characteristic is especially influential when running the system in applications with dynamic current patterns. A model including a way to emulate battery hysteresis behavior would improve the battery management system function.
In this study a lumped battery model for cell voltage prediction was expanded to include an OCV hysteresis model. Different expressions to describe the hysteresis behavior were explored. The different models were then evaluated using both synthetic and real-life application experimental data. In all cases the error was reduced by adding a hysteresis component to the model. Using this type of model in the battery management system of stationary energy storage systems based on NiMH batteries could help make the state prediction more accurate. This, in turn, would allow for better optimization of the system operation, something that could help increase system efficiency and lifetime.

Full text

Journal of Energy Storage 48 (2022) 103985
Available online 13 January 2022
2352-152X/© 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Research Papers
Evaluation of hysteresis expressions in a lumped voltage prediction model
of a NiMH battery system in stationary storage applications
Jenny B¨
orjesson Ax´
en
a
,
b
,
*
, Henrik Ekstr¨
om
a
,
c
, Erika Widenkvist Zetterstr¨
om
b
,
G¨
oran Lindbergh
a
a
KTH – School of Chemistry Biotechnology and Health, Department of Chemical Engineering, Division of Applied Electrochemistry - Teknikringen 42, 114 28 Stockholm,
Sweden
b
Nilar AB – B¨
onav¨
agen 55, 806 47 G¨
avle, Sweden
c
COMSOL AB – Tegn´
ergatan 27, 111 40 Stockholm, Sweden
ARTICLE INFO
Keywords:
Battery modelling
OCV hysteresis
NiMH
Voltage Response Model
Physical model
Application verication
ABSTRACT
As a part of battery system operation, battery models are often used to determine battery characteristics such as
the state of charge (SOC) and the state of health (SOH). A phenomenon that has a large impact on battery model
accuracy for NiMH batteries is open circuit voltage (OCV) hysteresis, which causes the OCV to differ not only
with the SOC of the battery but also with the charge-discharge history. This characteristic is especially inuential
when running the system in applications with dynamic current patterns. A model including a way to emulate
battery hysteresis behavior would improve the battery management system function.
In this study a lumped battery model for cell voltage prediction was expanded to include an OCV hysteresis
model. Different expressions to describe the hysteresis behavior were explored. The different models were then
evaluated using both synthetic and real-life application experimental data. In all cases the error was reduced by
adding a hysteresis component to the model. Using this type of model in the battery management system of
stationary energy storage systems based on NiMH batteries could help make the state prediction more accurate.
This, in turn, would allow for better optimization of the system operation, something that could help increase
system efciency and lifetime.
1. Introduction
To ensure that a battery-based energy storage system is efcient and
has as long service life as possible, a battery management system (BMS)
is needed to regulate the interaction of the battery bank with the grid. A
BMS not only regulates the system but also tracks physical parameters
needed to do this, for example voltage and temperature. In addition, it
estimates important characteristics for the regulating function, such as
the state of charge (SOC) or the state of health (SOH). Due to the
complexity of the battery, a battery model is often needed to track these
characteristics.
In this study, a so-called lumped model is used to predict the cell
voltage under operation, a model based on electrochemical principles.
However, unlike more complex physical models, all similar processes
are lumped together and represented by one expression per physical
process. In this case the parameters used in the model are a result of
tting the model to experimental current and cell voltage data and do
not necessarily represent actual physical parameters for the electro-
chemical processes. Unlike the electrical equivalent circuit (EEC) model
type, a lumped model represents the battery processes using electro-
chemical theory, which is more suited to describe the behavior than
electronic components [1]. In addition, the lumped model can use the
same parameters over the whole SOC window, while an EEC model re-
quires parametrization at regular SOC intervals making large look-up
tables necessary. At the same time, due to its simplicity, the lumped
model is more viable to run on-line in a system in comparison to the
more complex physical models. This makes a lumped model a good
contender to use in a BMS. Ekstr¨
om et al. previously described a lumped
model, evaluating three different expressions for the diffusion over-
potentials for a lithium-ion battery NMC cell [2]. Our study aims to
expand this model to reproduce cell voltage data from a stationary en-
ergy storage nickel metal hydride (NiMH) battery system in use. How-
ever, there is a difference in chemistry between the NiMH battery and
the NMC cell in Ekstr¨
om’s study. As a consequence, an additional
* Corresponding Author.
E-mail addresses: jaxen@kth.se (J.B. Ax´
en), heek@kth.se (H. Ekstr¨
om), erika.widenkvistzetterstrom@nilar.com (E.W. Zetterstr¨
om), gnli@kth.se (G. Lindbergh).
Contents lists available at ScienceDirect
Journal of Energy Storage
journal homepage: www.elsevier.com/locate/est
https://doi.org/10.1016/j.est.2022.103985
Received 8 July 2021; Received in revised form 4 January 2022; Accepted 5 January 2022

Journal of Energy Storage 48 (2022) 103985
2
phenomenon needs to be addressed in order for the model to function,
namely Open Circuit Voltage (OCV) hysteresis.
OCV is the potential of a cell when equilibrium has been reached in a
resting state and is a property that varies with battery SOC. In voltage
modeling, the OCV is the base upon which the rest of the model rests,
with transient processes adding overpotentials and ohmic losses to give
the nal cell voltage. OCV hysteresis is when the OCV of the cell at a
certain SOC depends on the charge-discharge path taken to arrive there.
For example, a cell at 50% SOC that was charged from 0% SOC up to that
point will have a higher OCV than if it was discharged to 50% SOC from
a fully charged state. A combination of charge and discharge pulses
made to get to that state will cause the OCV to end up somewhere in
between these two values. While OCV hysteresis has been observed for
Li-ion chemistries, the hysteresis of the NiMH battery is even more
pronounced [3–5]. In the battery tested in this study, any mid-range
OCV value can fall in an SOC range approximately 70% SOC wide
depending on the charge-discharge history, as seen in Figure 1. A dy-
namic real time parametrization method (e.g.[6]) will be subjected to
this behavior, and any attempt to compare the acquired OCV with
tabulated SOC-OCV relationships will result in a gross miss-estimation of
the SOC. Therefore, in order to adequately model the voltage of the
battery and to accurately estimate SOC, a way to estimate the hysteresis
behavior of the battery is essential.
The physical origins of OCV hysteresis are not fully understood, but it
is believed to be caused by processes in the insertion electrode material
[7–9]. One theory is the domain theory, commonly used to explain
hysteresis in magnetism, where the existence of meta-stable states and
their placement in relation to each other affects the potential of avail-
able sites [10,11]. Dreyer et al. further hypothesizes that the existence of
multiple particles in porous electrodes makes the hysteresis phenome-
non persevere, even for particles too small to contain more than one
phase. In the theory they present, the hysteresis is created by the dis-
tribution of particles in the different meta-stable states, rather than a
phase distribution in a single particle [11]. Assat et al. instead discuss
meta-stable paths as an alternative theory, where shifts in the material
structure after the electron transfer reaction forces the reaction to move
along another path once the current is reversed, causing an entropic loss
[12]. This ambiguity surrounding the physical origins of hysteresis
makes it difcult to develop models based on physical principles. As
there are currently no generally accepted models of this type, an
empirical approach to modeling hysteresis is the most feasible.
In order to model hysteresis empirically, Plett et al. developed the
discrete zero state model which ips between the charge and discharge
OCV curve, a method that has also been used in other studies [13,14].
However, due to the size of the hysteresis window for the NiMH battery,
such a model is not likely to give much improvement since the binary
assumption that the OCV will always be located at either the charge or
discharge curve is not accurate. Instead, the dynamic movement of OCV
between the charge and discharge curve has to be modeled, for which
two main approaches have so far been used: the Preisach model, which
is discrete, and models based on different differential equations. The
Preisach model is used to model hysteresis in magnetic systems and has
been used to model hysteresis for both lithium iron phosphate, NiMH
and nickel cadmium batteries [15–18]. Due to its discrete nature, it
combines well with discrete battery models, such as the EEC models.
However, when setting up a continuous model, using a continuous
expression for the hysteresis as well is preferable. As the intended
lumped model in this study is continuous, this would be the preferable
strategy. Examples of the continuous type of hysteresis model have been
described by Verbrugge et al. and Zhang et al. [19,20].
There are both EEC models and physical models that describe the
NiMH battery. The EEC models often include hysteresis but are depen-
dent on extensive parametrization and storing these parameters in look-
up tables; And while the physical models are less dependent on look-up
tables, they are based on a theoretical understanding of the battery
processes and therefore do not include hysteresis [14,21]. Using a
physics derived lumped model in combination with an expression for the
OCV hysteresis behavior could combine the strengths of these two model
types with the following benets: make simulation of dynamic behavior
possible, reduce the number of parameters to be stored compared to an
EEC model, and introduce a model based on electrochemical principles.
In addition, the computational resources needed are less compared to a
conventional physical model. If used in a BMS, a physics derived lumped
model has the potential to improve system function and state estimation
accuracy. To achieve this, this study expands on the lumped model as
described by Ekstr¨
om et al. to better t the NiMH battery by using
various differential expressions to approximate OCV voltage hysteresis
[2]. The models are then evaluated using both experimental synthetic
and real-life application data running both Time Shifting and Phase
Shifting applications.
Figure 1. NiMH cell open circuit voltage as a function of SOC, with the upper line representing E
OCch
and the lower line representing E
OCdch
. The three loops are the
OCV when starting discharge at 35% (blue), 60% (orange) & 80% (gray), discharging 25% of the full capacity before starting charge again to the initial loop
SOC value.
J.B. Ax´
en et al.

Journal of Energy Storage 48 (2022) 103985
3
2. Base Model
The model used in this study describes the battery behavior through
general physical expressions with lumped constants, and is based on the
article by Ekstr¨
om et al. [2]. These constants are then approximated
through tting the model to experimental data. To describe the diffusion
inside the particle, Ekstr¨
om et al. evaluated three different expressions:
one typical Resistor-Capacitor (RC) circuit element, and two expressions
for diffusion inside an idealized particle,
τ
and K
τ
. It should be noted
that the RC circuit element makes the Ekstr¨
om model equivalent to a
traditional EEC, whereas the other two expressions are based on Fick’s
law of diffusion [2]. For this study the
τ
model was chosen, as the three
expressions. had similar accuracies but the
τ
model requires one less
tting parameter [2]. In the model, expressions (1)-(6) are used to
simulate the voltage response.
The total battery voltage is described by the following function
Ebatt =EOCV (SX=1) +
η
ohm +
η
act (1)
where E
OCV
is the open circuit voltage (OCV) as a function of State of
Charge (SOC),
η
ohm
the ohmic resistance overpotential, and
η
act
is the
activation overpotential. These overpotentials are represented by
η
ohm =RohmIbatt (2)
which is the expression for electrical resistance, where R
ohm
is the ohmic
resistance and I
batt
is the cell current. And the expression for the acti-
vation resistance
η
act =2RT
Fsinh−1(Ibatt
2I0)(3)
which is based on an inverted Butler-Volmer expression with a sym-
metry factor of 0.5. R is the ideal gas constant, 8.314 J mol
−1
K
−1
, F is
the Faraday constant, 96485 C mol
−1
, T is the temperature in Kelvin and
I
0
is the exchange current in Ampere.
The reactions are assumed to take place on the surface of a spherical
particle. To represent the SOC distribution in the particle caused by mass
transport limitations the local SOC variable S is used, where S
X=1
is the
SOC at the surface of the particle. The following differential equation is a
generalized expression for diffusion in a particle, where
τ
is a time
constant proportional to the inverse of the diffusion coefcient.
τ
dS
dt + ∇⋅(− ∇S) = 0(4a)
To simplify the model the particle is assumed to be spherical and,
accordingly, spherical coordinates are used. If X is taken to be a space
variable representing the radial position as a fraction of the total radius,
X =r/R, which varies between 0 at the center of the particle to 1 at the
particle surface, the generalized expression above can be written as
follows in spherical coordinates.
τ
dS
dt +1
X2
∂
∂
X(−X2
∂
S
∂
X)=0(4b)
The boundary condition for the particle center is a zero-ux
condition
− ∇S=0X=0(5)
and for the particle surface we have the following Neumann boundary
condition
− ∇S=
τ
Ibatt
3QX=1(6)
which represents the charge carrier ux through the particle surface and
where Q is the battery capacity.
To determine the constants in these expressions, the lumped model is
tted to experimental battery data. The resulting values are then used to
verify the performance of the model when faced with a second set of
data of the same type. Three constants were tted: R
ohm
, I
0
and
τ
. Before
tting, two of them were transformed to avoid the solver changing signs
of the target parameters or dividing by zero. The tted constants and
their tting transformations are found in Table 1.
2.1. OCV model
To account for the hysteresis of the OCV, an OCV model is added to
the lumped model. Since the thermodynamical origin of OCV hysteresis
is not fully understood, the models in this study are not based on
physical theory, but rather mathematical expressions chosen for their
capabilities in capturing the observed hysteresis, similar to the study by
Verbrugge et al. [19]. The hysteresis part of the model is based on a
dimensionless variable,
χ
, that varies between 0 and 1 depending on
charge/discharge history. The open circuit voltage is then given by
EOC =EOCdch +
χ
(EOCch −EOCdch)(7)
where E
OCch
and E
OCdch
is the charge and discharge open circuit voltage
at a given state of charge, respectively. These voltages are extracted
from experimental data. The OCV charge and discharge curves are
shown in Figure 1 as a function of SOC.
The voltage hysteresis behavior of the NiOH positive electrode has
been characterized by Srinivasan et al., who reported a signicant
hysteresis behavior in terms of OCV voltage levels and dynamics [10].
During a charge from 0% to 100% SOC, or a discharge from 100% to 0%
SOC, the OCV follows the upper/lower OCV curve, and
χ
is 1 and
0 respectively. When the battery is subjected to a more dynamic current
pattern, with a mix of charge and discharge,
χ
will vary somewhere in
between 0 and 1, decreasing with discharge and increasing with charge.
As a result, the OCV value ends up somewhere in between the two OCV
curves. When changing current direction at an OCV somewhere in be-
tween the charge and discharge OCV curves, the battery behaves as if the
turning point represents a new boundary, and this affects the change
rate (slope in Figure 1) of the OCV. Thus, when a turn has been made, the
OCV describes the same shape as when turning at the corresponding
boundary, but now in a smaller window. This is called a sub-loop and is
illustrated in Figure 1.
The completed models were solved using COMSOL Multiphysics
version 5.5 and the BDF (Backward Differentiation Formula) time-
dependent solver. The partial differential diffusion equation was
solved using the nite element method, using a computational mesh
consisting of 18 elements, with an even size in the bulk of the particles
and the ve elements closest to the particle surface (X =1) decreasing in
size with the smallest closest to the surface. The models were solved in a
time interval corresponding to the experimental data for the four cases.
Parameter tting to the experimental data was carried out using the
least-squares method and the Levenberg-Marquardt optimization solver.
Parameter tting was carried out separately for each model and case
combination. To validate the models, a second set of data was recorded
for each of the four cases, using the current data and the tted param-
eters to simulate a voltage response. The error between the model-
generated voltage values and the experimental voltage was evaluated
using the RMS method with even time intervals.
Table 1
Fitted parameters used in the base model
Parameter Transformation Meaning Unit
R
ohm
η
ohm, 1C
Ohmic overpotential at 1C V
I
0
log(1/I
0norm
) Logarithm of the invers of the exchange
current, normalized to battery capacity
-
Τ NA Diffusion time constant, inversely
proportional to the diffusion coefcient
s
J.B. Ax´
en et al.

Journal of Energy Storage 48 (2022) 103985
4
3. Experimental
To decide how to best model the hysteresis variable
χ
, experiments
were made to characterize the dynamic hysteresis behavior. In the rst
step, the experimental OCV behavior was studied to give a good starting
point to create a model. The OCV was determined using the GITT
method with a 5% SOC step and charging/discharging with 0.3C from
different SOC levels. In the charge case 20%, 40% and 60% SOC turning
points were tested, and in the discharge case 20%, 35%, 60%, 80% and
95% SOC turning points were tested. All charging measurements were
rst discharged from fully charged to the starting point, and all dis-
charging measurements were rst charged from fully discharged to the
starting point. The relaxation pulses were tted to an EEC model of a
Randles type with double Thevenin circuits [22,23]. The resulting
OCV-values were plotted against SOC. These curves were then used to
develop models for further evaluation.
To evaluate the different models of the OCV under dynamic condi-
tions, four different test cases were selected with two experimental data
sets for each case, one to t the model to and one to evaluate the t with.
Two of the cases were synthetic and tested on a single battery in a lab-
oratory set-up. The other two cases were taken from application data
from systems in use, representing two of the most common applications
for stationary energy storage. Not only did this provide a realistic cur-
rent behavior but also included measurement errors and disturbances
present in a system.
3.1. Synthetic cases
For the two synthetic cases, the experiments were carried out on a
single Nilar EC 10-cell module with a nominal voltage of 12 V and a
rated capacity of 10 Ah. The tests consisted of randomly chosen pulses
with currents that were multiples of 0.1 C up to 0.8 C as well as an
additional possible rate of 0.85 C. The length of the pulses was always
set in even minutes, this is also true for the rest periods. The two cases
were:
•A series of randomized charge pulses followed by a series of ran-
domized discharge pulses
•Randomized charge and discharge pulses, mixed
3.2. Application cases
In the two cases where data was collected from Nilar EC systems
running in application in the eld, the systems chosen were selected to
showcase two different types of run cycles with importance for sta-
tionary energy storage. In both cases, the number of data points used for
tting had to be reduced to be able to run the parameter estimation
optimization algorithm on the available hardware. The downsampling
was made with even distances and without ltering.
3.2.1. Time Shifting
This application is representative for a self-sufcient energy system,
where charging occurs intermittently during the day using solar panels,
and the discharge is primarily in the evening. In an application such as
this, the whole capacity of the bank is frequently used. As a conse-
quence, end of charge and discharge cut off criteria can be used to nd a
starting point for the t and simulation. While the system chosen for
data extraction in this case is connected to the power grid, unlike the
more standard off-grid type, it still follows a representative usage
prole.
3.2.2. Phase Shifting
In Phase Shifting, the energy storage is used to offset current spikes
on the different power phases in a three-phase system. This application
is fully dynamic, and the drive cycle consists of random charge and
discharge currents with an intermittent balancing of the energy storage.
This balancing can be used to calibrate the SOC, and as a known starting
point for the t and simulation. The cycling is primarily carried out at
mid to high range SOC. In the case used for this study, the energy storage
is connected to the power grid and used to smoothen the power con-
sumption prole in an apartment block that has solar panels installed.
4. Results & Discussion
4.1. Hysteresis Model
By analyzing the dynamic OCV behavior, we chose four model can-
didates. The partial charging and discharging OCV behavior is shown in
Figure 2.
Figure 2a depicts the OCV behavior as a function of SOC with
different turning points. Basing the model on replicated data from in-
dividual turning points would require extensive testing and large look
up tables. Instead, we plotted the same data as
χ
versus the distance from
the turning point, Figure 2b.The variable
χ
indicates the position be-
tween the two OCV boundaries, where
χ
=1 falls on the charge
boundary and
χ
=0 falls on the discharge boundary. While the similarity
of the curves shows that the hysteresis behavior is independent of
turning point position, a model based on this type of data would still
have to keep track of the distance from the turning point. To make a
model that is independent of SOC and distance from turning point, the
hysteresis expression can take the form of a differential equation. In this
expression the change rate is dependent on the affected variable,
d
χ
dSOC(
χ
). When studying this behavior seen in Figure 2c, the charging
behavior is found to be relatively consistent. Besides the curve creating a
backwards bend as
χ
approaches 1, the slope is close to linear for all
three turning points. Based on this, expression (8a) was chosen as the
charge expression for all hysteresis models.
d
χ
dSOC =k
χ
(1−
χ
)(8a)
By rewriting it on a time basis, using dSOCdt=Icell/Q, we arrive at
expression (8b), which is the form that will be used in the model.
d
χ
dt =k
χ
Icell
Q(1−
χ
)(8b)
In this expression, and all to follow, Q denotes the capacity of the
cell, and consequently I
cell
/Q represents the normalized current, the so-
called C-rate.
The discharge behavior, however, is less predictable with the shape
of the curve varying with turning point. In addition, the trend is not as
clear as with the linearity of the charge case. Therefore, three different
discharge expressions were evaluated, one linear expression (9), one 2
nd
degree polynomial expression (10), and one polynomial with an un-
known exponent a that was also tted (11). Each of these were combined
with the previously specied charge expression (8) to form the hyster-
esis model.
d
χ
dt =k
χ
Icell
Q
χ
(9)
d
χ
dt =k
χ
Icell
Q
χ
2(10)
d
χ
dt =k
χ
Icell
Q
χ
a(11)
These expressions all contain one or more kinetic parameters, k
χ
, that
inuence the change rate. These kinetic parameters are evaluated
together with all the other tted parameters when tting the compre-
hensive model to the experimental data.
To check the feasibility of these hysteresis models they were tted to
the data of Figure 2a using Matlab as a proof of concept. The tting was
done individually for each hysteresis prole, and the results are found in
J.B. Ax´
en et al.

Journal of Energy Storage 48 (2022) 103985
5
Figure 3. The gure shows that there is a reasonable t for all the
models, but that the
α χ
a
model gives the closest t in the discharge case.
However, this result could prove to be different for the application data,
as the k
χ
constant and the a constant will not be able to vary with the
SOC.
When comparing the slope of the curves in Figure 2c, the slope for
the charge curves and most of the discharge curves is similar with
k
χ
~0.1, except for the turn at 95% discharge curve (green). This would
indicate that using the same k
χ
for both charge and discharge could work
well in some cases, but not necessarily in others. To evaluate this, the
Figure 2. Experimental OCV data from different turning points. Turning points indicated in % SOC. Left: Charge, Right: Discharge. a) OCV voltage as a function of
SOC. b) Hysteresis variable
χ
as a function of distance from turning point in SOC. c) Change in hysteresis variable
χ
as a function of
χ
.
J.B. Ax´
en et al.

Journal of Energy Storage 48 (2022) 103985
6
model where (9) is used as the discharge expression was evaluated for
both a common k
χ
with the same constant for both charge and discharge
and a split k
χ
with separate constants for charge and discharge. The
other two hysteresis models were only evaluated with split constants, as
their shape is not linear. This brings the total up to four different hys-
teresis models. However, only (9) (with both a common and a split k
χ
) is
discussed in detail in this article. This as the performance of the models
was similar. The data for all four hysteresis models can be found in the
supplementary information.
4.2. Model evaluation
The purpose of adding a hysteresis part to the lumped model was to
obtain a better t in dynamic current conditions, especially in real life
application cases. To evaluate the t and compare the models, the
modeled voltage is compared to the experimental cell voltage visually
and using the RMS of the error.
The drive cycles used in this study fall under two main categories.
The rst is represented by the randomized pulses synthetic case and the
Time Shifting case. It describes a behavior where the battery OCV is
following the upper OCV curve through intermittent charging until a
sequence of intermittent discharging starts. In the randomized pulses
synthetic case this is then followed by a charging period. The hysteresis
model is primarily needed to emulate the OCV behavior as it moves from
one OCV curve to the other after the current reversal. The second
category, represented by the mixed randomized pulses synthetic case
and the Phase shifting case, describes a more stochastic OCV behavior.
Here the OCV is moving in cell voltage between the two OCV curves due
to switching between charging and discharging of the battery. Since the
hysteresis model cannot fully capture the sub-loop slope change, these
cases where the current is reversed many times are expected to be
described less well by the models and give larger errors when compared
to the cases with current reversals occurring only from the upper or
lower OCV curve.
4.2.1. Synthetic cases
The evaluation error RMS is listed to the far right of Table 2, with the
lowest marked in bold. Compared to using the model without hysteresis
(
χ
=0.5) the best t cuts the evaluation error RMS to less than half in
both the synthetic cases. However, the mixed randomized pulses case
has RMS values of almost double that of the randomized case. A good
model of the OCV becomes more critical in the mixed case. This since the
drive cycle moves in between the two boundaries without touching them
for a longer period of time as well as engaging the sub loops, which is
reected in the RMS.
Considering the hysteresis rate constant, k
χ
, and rst comparing the
α
χ
model with one and two constants respectively, there is a signicant
difference in evaluation error in the mixed randomized pulses case. Here
the model with two constants is noticeably better, while it’s still better
but not as large as for the randomized pulses case. That there is an
improvement is expected, as more tting parameters usually improves
the possibility for a good t, but the model with a single rate constant
Figure 3. Hysteresis models tted to experimental hysteresis data from different turning points. Each turning point data tted individually to model. Left: Charge,
Right: Discharge.
Table 2
Values of the tted parameters and the corresponding error RMS during tting and validation using two different hysteresis models. Lowest evaluation error RMS for
each case marked in bold.
Symbol R
Ω
I
0norm
τ
k
χ
k
χ
ch
k
χ
dch
RMS
t
RMS
eval
Unit mΩ 10
−3
(dimensionless) ks - - - mV mV
Hysteresis model
Randomized pulses No hysteresis (
χ
=0.5) 5 0.91 36 22 24
α χ
5 12 13 9.0 11 9
α χ
, split k
χ
6 23 16 13.0 4.5 8 7
Randomized pulses, mixed No hysteresis (
χ
=0.5) 5 9.67 15 26 29
α χ
5 85 11 8.2 12 16
α χ
, split k
χ
6 229 12 17.2 6.1 8 12
Time shifting No hysteresis (
χ
=0.5) 15 940 167 27 22
α χ
1 21 160 44.4 17 11
α χ
, split k
χ
2 15 156 4.4 47.1 16 12
Phase shifting No hysteresis (
χ
=0.5) 10 965 507 25 23
α χ
0 362542 62 76.5 17 17
α χ
, split k
χ
4 10034 328 10.5 54.7 10 11
J.B. Ax´
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Journal of Energy Storage 48 (2022) 103985
7
still gives an almost as small error and could be used in those cases
where model simplicity is a key factor. Regarding the value of the rate
constant, an inspection of Figure 2 gives an expected charge value of
k
χ
~10 which is in the same order of magnitude as the tted values in
Table 2.
Comparing the models visually, Figure 5 shows the evaluation cell
voltage data for the randomized pulses case together with the experi-
mental data. It also shows the model errors as a function of time. Both
graphs show a clear improvement when adding hysteresis to the model,
while the difference between the models is less dramatic. The biggest
differences between the models are seen during discharge (3 h <t <7
h), which is expected since the discharge expression is where the models
differ. This is conrmed when looking at the RMS for the different states
in Figure 4a.
In Figure 6 the evaluation data for the randomized mixed pulses case
is plotted together with the experimental data. The error as a function of
time is also shown. As in the randomized pulses case, the improvement
when adding hysteresis to the model is noticeable. However, unlike the
randomized pulses case, the RMS of the different states varies more
randomly between the models, Figure 4b. The models are very similar at
rst and then start to drift apart. This can be explained, since initially all
models have
χ
=1, following the upper OCV curve. When the discharge
pulses start, the different models come into play, creating a spread be-
tween the predicted cell voltages.
4.2.2. Application data cases
The system cases present a bigger challenge for a model than the
synthetic cases; Compared to the synthetic cases, the collection of the
system data is less resolved, resulting in additional modeling difculties.
In addition, due to hardware limitations during parameter estimation,
the data is downsampled in post processing. Using downsampling rather
than shorter datasets was necessary due to the length of the drive cycles,
as both the charge and discharge data is required to evaluate the func-
tion of the hysteresis model. The downsampling of the data amounted to
one data point every 85 s in the Time Shifting case and one data point
every 705 s in the Phase Shifting case, making it more difcult to catch
the fast diffusion processes. Because of this, the diffusion processes are
harder to t in the Phase Shifting case, resulting in a greater variation in
tted parameter regulating diffusion,
τ
, as seen in Table 2.
In addition to the measurement uncertainties, the maximum current
is lower in the application cases than in the synthetic cases. With the
expression for the activation resistance, equation 3, becoming linear at
lower currents, this causes the model to have two elements to t that
vary linearly with the current,
η
ohm
and
η
act
. This gives the optimizer an
Figure 4. RMS
eval
divided into charge, discharge, and rest periods for all models. a) Randomized pulses, b) Randomized pulses mixed, c) Time shifting, d)
Phase shifting.
J.B. Ax´
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Journal of Energy Storage 48 (2022) 103985
8
innite number of local minima to nd, as long as the sum of
η
ohm
and
η
act
is constant. As a result, the R
Ω
and the I
0norm
values vary greatly in
the application cases, as compared to the synthetic cases where they stay
mostly consistent.
All these factors could partly explain the larger tting error RMSs in
the application cases compared to the synthetic cases. However, the
errors are also likely aggravated by the higher complexity of the applied
drive cycles. Compared to using the no hysteresis model,
χ
=0.5, the
best t cuts the evaluation error RMS to half in the Phase shifting case
and to a third in the Time shifting case. Unlike in the synthetic cases,
there is no big difference in error when comparing the two different
applications. The fact that there is such a marked improvement when
applying the hysteresis model, even though the sparsity of the data and
the relatively low currents makes it difcult to t the parameters related
to the transient processes in the battery, is a testament to how important
a functioning hysteresis model is to the overall voltage behavior of the
battery in dynamic operations. If data points close enough to catch the
transient processes could have been used, the errors would likely have
been even lower.
As for the hysteresis rate constant, k
χ
, when comparing the
α χ
model
with one and two constants respectively, there is a marked difference in
error in the Phase Shifting case, while there’s no notable improvement in
the Time Shifting case. Regarding the size of the charge hysteresis rate
constant, it is close to the expected value just like in the synthetic cases.
Moving on to the visual interpretation, the model evaluation data for
the Time Shifting application together with the experimental data for
comparison is shown in Figure 7, as well as the model error. The biggest
difference between the hysteresis models is found during the charge
phase (t <7 h), something that corresponds well to the state RMS values
in Figure 4c. This is in contrast to the randomized pulses synthetic cases,
where the biggest difference is seen during the discharge phase. How-
ever, in this case the shift in error distribution is likely an effect of the
optimization. As the discharge lasts much longer than the charge, the
weight of any error accrued during this period will be larger than errors
accumulated during charge. As a consequence, the model will be tted
to follow the discharge which causes the RMS for the charge period to
exceed that for the discharge period.
When looking at the
χ
=0.5 model, it follows the other models
reasonably well during the charge period, but once the rest and
discharge periods are reached it veers off course, overestimating the
Figure 5. Randomized pulses case - Left: Model validation simulation results. Right: Model Error. Blue:
χ
=0.5 model, orange:
α χ
model, yellow:
α χ
model with split
k
χ
,
and black: corresponding experimental cell voltage
Figure 6. Randomized pulses, mixed case - Left: Model validation simulation results. Right: Model Error. Blue:
χ
=0.5 model, orange:
α χ
model, yellow:
α χ
model
with split k
χ
, and black: corresponding experimental cell voltage
J.B. Ax´
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Journal of Energy Storage 48 (2022) 103985
9
voltage signicantly. In this case the model without hysteresis is
compensating through a high ohmic resistance. The resistance over-
potential is roughly ten times higher than for the models with hysteresis.
During charge this gives a fair result, but during discharge the lack of
handling the hysteresis affects the shape and accuracy of the voltage
curve as the quick initial voltage decrease is caused by the change in
hysteresis state. The
χ
=0.5 model’s failure to capture this not only leads
to differences in the shape of the voltage curve, but it also causes it to
overestimate the discharge voltage for the rest of the discharge period.
In Figure 8 the evaluation data for the Phase Shifting application is
displayed together with the experimental data for comparison. The
model error is also displayed. As in all previous cases the improvement
that stems from adding hysteresis in the model is clearly visible, with the
error of the
χ
=0.5 model increasing over the cycles. The other models
are very similar at the start of the test, but the difference increases as the
test progresses, with the one kinetic parameter model veering off course
the longer the run continues. But the general trend for the hysteresis
models is the same, at the start of the test the models underestimate the
voltage, goes through a phase in the middle with better adherence and
end up overestimating it at the end. There is no clear pattern in the state
RMS for the different models, Figure 4d.
To summarize; In all four cases adding hysteresis improved the ac-
curacy of the model when simulating a second dataset of the same type
that the model was tted to. The error RMS of the best option compared
with a model without hysteresis was halved in three cases and dimin-
ished to a third in one case. This suggests that adding hysteresis to a
model, irrespective of the type, will improve voltage prediction accu-
racy. To further continue this line of investigation, future work will
include comparison with other hysteresis models regarding accuracy,
model complexity and BMS computation requirements. The tting er-
rors are slightly larger for the application cases, most likely due to the
less precise data recording of the system compared to the experimental
set up, as well as the larger complexity of the application drive cycles.
However, while there is a difference, the errors are still in the same order
of magnitude, indicating that the lumped type models adapt well to the
increased uncertainty of the online system.
5. Conclusions
The experimental data show that the NiMH battery has a signicant
OCV hysteresis behavior. This makes any voltage model not including
OCV hysteresis poor under dynamic conditions. So, to improve model
Figure 7. Time Shifting case - Left: Model validation simulation results. Right: Model Error. Blue:
χ
=0.5 model, orange:
α χ
model, yellow:
α χ
model with split k
χ
,
and black: corresponding experimental cell voltage
Figure 8. Phase Shifting case - Right: Model validation simulation results. Left: Model Error. Blue:
χ
=0.5 model, orange:
α χ
model, yellow:
α χ
model with split k
χ
,
and black: corresponding experimental cell voltage
J.B. Ax´
en et al.

Journal of Energy Storage 48 (2022) 103985
10
function for the NiMH battery, different ways of simulating this effect
were studied.
Two different hysteresis models were added to a lumped battery
model for cell voltage prediction and were evaluated for a NiMH system
on experimental data from synthetic laboratory cycles and applications.
Due to the lack of theoretical expressions that accurately explains the
origins of OCV hysteresis, the models were by necessity empirical. In all
four data cases adding hysteresis to the model made it signicantly more
accurate, with no exceptions. This shows that modeling hysteresis is
essential when studying real drive cycles regardless of model type, as
usage is most often dynamic.
As results on application data show, the lumped model format adapts
well to the inherent measurement errors in a system. Consequently,
using this type of model with the inclusion of hysteresis estimation in
stationary applications could improve BMS accuracy and system
function.
Funding
This work was supported by the Swedish Foundation for Strategic
Research, ID16-0111.
CRediT authorship contribution statement
Jenny B¨
orjesson Ax´
en: Conceptualization, Methodology, Valida-
tion, Formal analysis, Investigation, Writing – original draft, Visualiza-
tion. Henrik Ekstr¨
om: Conceptualization, Methodology, Writing –
review & editing, Supervision. Erika Widenkvist Zetterstr¨
om:
Conceptualization, Writing – review & editing, Supervision. G¨
oran
Lindbergh: Conceptualization, Writing – review & editing, Supervision.
Declaration of Competing Interest
The authors declare the following nancial interests/personal re-
lationships which may be considered as potential competing interests:
Jenny B¨
orjesson Ax´
en and Erika Widenkvist Zetterstr¨
om are both
employed at Nilar AB, the manufacturer of the battery and battery
systems used to parametrize and verify the model in the study. Henrik
Ekstr¨
om is employed at COMSOL AB, the company that sells the
modeling software used to make the model in the study.
Supplementary materials
Supplementary material associated with this article can be found, in
the online version, at doi:10.1016/j.est.2022.103985.
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