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Citation: Martí-Florences, M.; Cecilia,
A.; Costa-Castelló, R. Modelling and
Estimation in Lithium-Ion Batteries:
A Literature Review. Energies 2023,
16, 6846. https://doi.org/10.3390/
en16196846
Received: 31 July 2023
Revised: 20 September 2023
Accepted: 25 September 2023
Published: 27 September 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
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distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
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4.0/).
energies
Review
Modelling and Estimation in Lithium-Ion Batteries:
A Literature Review
Miquel Martí-Florences 1,2 , Andreu Cecilia 1and Ramon Costa-Castelló 1,*
1ETSEIB, ESAII, Universitat Politécnica de Catalunya, Avinguda Diagonal, 647, 08028 Barcelona, Spain;
miquel.marti.florences@upc.edu (M.M.-F.); andreu.cecilia@upc.edu (A.C.)
2Insitut de Robòtica i Informàtica Industrial CSIC-UPC, C/Pau Gargallo, 14, 08028 Barcelona, Spain
*Correspondence: ramon.costa@upc.edu
Abstract: Lithium-ion batteries are widely recognised as the leading technology for electrochemical
energy storage. Their applications in the automotive industry and integration with renewable energy
grids highlight their current significance and anticipate their substantial future impact. However,
battery management systems, which are in charge of the monitoring and control of batteries, need
to consider several states, like the state of charge and the state of health, which cannot be directly
measured. To estimate these indicators, algorithms utilising mathematical models of the battery
and basic measurements like voltage, current or temperature are employed. This review focuses on
a comprehensive examination of various models, from complex but close to the physicochemical
phenomena to computationally simpler but ignorant of the physics; the estimation problem and a
formal basis for the development of algorithms; and algorithms used in Li-ion battery monitoring.
The objective is to provide a practical guide that elucidates the different models and helps to navigate
the different existing estimation techniques, simplifying the process for the development of new
Li-ion battery applications.
Keywords:
Li-ion batteries; battery modelling; single-particle model; observers; state of charge
estimation; state-of-health estimation
1. Introduction
Lithium-Ion Batteries (LIBs) are currently the most popular Energy Storage System
(ESS). Their high demand in electric vehicles (EVs) and renewable-energy-powered grids
make them a device of great importance on the road towards decarbonisation of industrial
societies. There are several reasons behind this popularity. First, their flexibility makes them
suitable for aggressive, rapidly-changing applications such as electric vehicles (EVs), as well
as for static and slow-changing cases like grid-connected storage or small applications
such as laptops or phones. Moreover, their rotund-trip efficiency is typically rated between
85 and 95%. Additionally, LIBs offer a good balance between power density and energy
density [1,2], and their cost has been decreasing as production increases.
There exist some notable alternatives to LIBs. Zinc-ion batteries are potentially safer
and low-cost but are still in development [
3
,
4
]. Lithium-selenium batteries are another
emerging technology with potential for high energy density [
5
]. Aqueous ammonium-ion
batteries are environmentally friendly, with potential for static applications [
6
]. Superca-
pacitors are characterised by a low energy density and can therefore be applied to fewer
applications if not combined with other ESS [
7
]. Finally, vanadium redox-flow batteries are
suitable for static applications, as power can be decoupled from energy [
8
,
9
]. Compared to
the alternatives, LIBs emerge as a mature and versatile technology in terms of energy and
power density, as well as the number of applications.
Lithium-ion batteries do not refer to a single type of device. Instead, there is a wide
variety of chemistries and mechanical formats, each with different characteristics [
10
].
While the working principles of these batteries are the same, commercial cells use different
Energies 2023,16, 6846. https://doi.org/10.3390/en16196846 https://www.mdpi.com/journal/energies
Energies 2023,16, 6846 2 of 36
materials for the cathode and anode. Moreover, LIBs can come in multiple formats, the
most typical of which are cylindrical, prismatic and pouch cells.
To successfully implement a battery system in a real application, the use of a battery
monitoring system (BMS) is needed. BMSs are used to monitor and manage several states
of the cell to keep the battery in safe and efficient conditions and actuate in case of abuse
or malfunctioning [
11
,
12
]. To do so, estimation algorithms must compute the cell status
using available measurements, such as temperature, voltage and current, as well as a
mathematical description of the system dynamics. Hence, battery modelling is a key
part of the estimation design process. The development of control-oriented models in
recent years has led to a myriad of modelling approaches, which can usually be classified
into electrochemical-based models, equivalent circuit models (ECMs) and data-driven
models [
13
,
14
]. The differences between diverse models are discussed in this paper. A
general idea is that the closer the model is to the real physics of the cell, the more detailed
the description of the battery behaviour but at the expense of increased computational cost.
In an attempt to maintain the benefits of an accurate model while sufficiently reducing
the computational cost for real-time applications, many simplification methods have been
developed in the literature.
With knowledge of the behaviour of the battery, the next step in its management is
the estimation of unmeasured internal variables. Typically, state of charge (SoC) and state
of health (SoH) are key indicators of the cell that cannot be directly measured. Battery
parameters also play an important role in battery management. While they vary slowly
(typically due to aging processes), obtaining information about them is highly desirable.
While some offline experimentation or open-loop calculation (i.e., Coulomb counting for
SoC calculation [
15
]) can provide an approximation, the online closed-loop estimation of
these indicators is crucial for robust and accurate real-time monitoring. However, the large
variety of estimation algorithms can make the choice of a concrete algorithm difficult.
Some approaches are based on data-driven models that operate by means of machine
learning algorithms. Other procedures rely on the use of observers. While variations in
the extended Kalman filter (EKF) [
16
–
18
] or other linear observers are widely used, the
use of a linear observer to estimate the states of a nonlinear system can result in failed
estimation if the initial state of the algorithm is far away from the current state of the
cell [
19
]. To address this, nonlinear observers can be a better solution if their inherent
difficulties are successfully overcome.
The contributions of this article can be summarised as follows:
•
Insight is provided on the fundamentals of Li-ion batteries regarding the main and
side reactions, with the aim of explaining the basics in a comprehensive way;
•
The modelling of LIBs is summarised and explained in a straightforward way from a
state-estimation perspective;
•
A complete single-particle model (SPM) is provided, as well as a review of the simpli-
fication methods for electrochemical models. Finally, the proposed SPM is simplified
to extract a reduced order state-space model;
•
A picture of the landscape of observers is provided, with explanations of the strengths
and weaknesses of each mentioned observer, again with the intention of comprehen-
sively elucidating the fundamentals;
• A classification of estimation methods is provided for SoC and SoH.
The article structure can be found in Figure 1. In Section 2, the fundamentals of
LIBs are discussed, with mentions of potential side reactions and the provision of some
indicators. Additionally, a comparison between different Li-ion chemistries is included.
Section 3covers the modelling of LIBs, and models are classified as electrochemically
based ECM and data-driven models. Furthermore, we provide a brief discussion about
the simplification of electrochemical models from a control perspective. Finally, Section 4
provides an overview of the estimable information for LIBs and a definition and justification
for observers. Moreover, we offer an account of observers used in LIBs.
Energies 2023,16, 6846 3 of 36
Figure 1. Schematic representation of the article layout.
2. Fundamentals of Li-Ion Cells
In this section, we provide a general description of the working mechanisms of a
Li-ion battery, followed by a summary of the indicators to be taken into account to monitor
the battery.
2.1. Working Principles
Lithium-ion batteries are composed of three essential components: a positive electrode
known as the cathode, a negative electrode known as the anode, an electrolyte and two
current collectors. The cathode is typically made from lithium metal oxide, while the anode
is commonly constructed using graphite or other carbon-based materials. The electrolyte is
a solution that contains lithium salts. The current collectors are responsible for conducting
the electrical current outside the battery, serving as an interface between the cell and the
external circuit.
Simply put, a battery is an electrochemical device that is capable of bidirectionally
transforming chemical energy into electrical energy. This conversion is achieved through a
bidirectional process involving the electrolyte stored within the battery electrodes. The inter-
face connecting the electrode and electrolyte facilitates reduction–oxidation reactions [
20
],
as shown in Figure 2.
When a current flows through the battery, the oxidation reaction releases electrons,
which are then directed through the external circuit by the reduction reaction. Simultane-
ously, ions released from the electroactive species travel between the positive and negative
regions of the battery, facilitated by the movement of the electrolyte.
The battery voltage is measured as the difference in potential between the current
collectors and is related to the concentration of
Li+
intercalated in the electrode particles
(see Section 3.1.4).
The main reactions, described in a general form for a carbon anode, are
(1)
for the
charge process and (2) for the discharge process.
Charge reactions:
At the anode: LinC6←−− Li0C6+nLi++ne−;
At the cathode: Lim−nMO2+nLi++ne−←−− LimMO2;
Global reaction: LinC6+Lim−nMO2←−− Li0C6+LimMO2.
(1)
where
MO2
is the general form of the cathode chemistry, as an oxide of a metal (M), and
n
and
m
are the stoichiometric coefficients of lithium involved in the reaction. During the
charge, lithium ions move from cathode to anode through the electrolyte, while the electrons
move through the external circuit. During discharge, the process is reversed.
Energies 2023,16, 6846 4 of 36
Discharge reactions:
At the anode: LinC6−−→ Li0C6+nLi++ne−;
At the Cathode: Lim−nMO2+nLi++ne−−−→ LimMO2;
Global reaction: LinC6+Lim−nMO2−−→ Li0C6+LimMO2.
(2)
Typically these reactions are exothermic, as the transport of lithium is associated with
heat generation. A comparison of different materials for LIBs is provided in Section 2.4. As
an example, for a cathode of
LiMn2O4
and a carbon anode (see Section 2.4), the equations
would be:
At the anode: LinC6−−*
)−− Li0C6+nLi++ne−;
At the cathode: Lim−nMn2O4+nLi++ne−−−*
)−− LimMn2O4;
Global reaction: LinC6+Lim−nMn2O4−−*
)−− Li0C6+LimMn2O4.
(3)
Figure 2facilitates the visualisation of the structure of the cell, as well as detailing
where and when reactions happen. The external circuit is also depicted.
Figure 2.
Operation of a Li-ion battery. While the electrons cross through the external circuit, the
Li+
ions travel diluted in the liquid phase through the separator. During the charge process, the electrons
move from cathode to the anode, and the
Li+
moves from anode to cathode, while during discharge,
the movement direction of electrons and ions reverses. The equations depicted over the diagram are
reversible depending on the process. For instance, the anode releases electrons during discharge, and
the cathode absorbs them. The situation is reversed during a charge.
MO2
is the general form of the
cathode chemistry, as an oxide of a metal (M),
n
and
m
are the stoichiometric coefficients of lithium
involved in the reaction.
2.2. Side Reactions
Side reactions occur within LIBs and differ from the main reactions described in
Section 2.1. These side reactions have the potential to occur under specific circumstances,
Energies 2023,16, 6846 5 of 36
and their effects can be potentially harmful. Many of these reactions appear when the
battery is excessively charged (overcharged) or almost
discharged [14,21]
. The former
usually generates a high output voltage, while the latter generates a low output voltage.
While only an overview is provided here, more information on degrading side reactions can
be found in [
22
,
23
]. Finally, a detailed mathematical model is found in [
24
]. A summary of
side reactions can be found in Table 1.
Table 1.
Side reactions by region and cause. The effects caused by overvoltage, undervoltage and
high C rates in a concrete region are listed [14,21].
Region Overvoltage Undervoltage High Currents
Negative electrode Lithium plating Solid electrolyte interphase Particle fracture
Positive electrode Oxidation of electrolyte Solid electrolyte interphase -
2.2.1. Overvoltage
During the charging process of LIBs, lithium ions (
Li+
) migrate from the positive
electrode to the negative electrode and intercalate within it. This process increases the cell’s
potential. However, if the battery undergoes a strong overcharge (a situation that occurs
when the stored charge is close to or in excess of the battery capacity), the cell potential can
rise to a point where the following phenomena occur.
•
Lithium plating: The graphite electrodes become saturated with lithium, leading to an
effect known as lithium deposition or lithium plating. This results in the formation of
dendrites within the anode, which obstruct the flow of ions. Over time, dendrites can
extend into the separator, causing short circuits [25,26].
•
Oxidation of the electrolyte: In the cathode, high voltages can induce the oxida-
tion of electrolyte salts as they react with the cathode material, leading to capacity
fading [24,27].
2.2.2. Undervoltage
On the other hand, a sudden voltage drop appears when the battery is overdischarged.
In this situation, the properties of the electrolyte change, creating an effect known as solid
electrolyte interphase (SEI).
•
SEI: An undervoltage situation leads to the precipitation of insoluble products onto the
electrode surface in the anode, forming a passive film known as the SEI. While the SEI
is necessary to prevent unwanted reactions between the electrolyte and the electrode,
it is convenient to avoid the excessive growth of the SEI, as it can also contribute to
capacity fading [28,29].
2.2.3. High Currents
A current much greater than the nominal current of the cell can also have a negative effect:
•
Particle fracture: The particles may change volume and thus stress the electrode
materials. The effect of particle fracture can indirectly cause the growth of the SEI
or loss of active material [30].
2.2.4. High Temperatures
Finally, high temperatures can influence the battery behaviour in two ways: first, as an
accelerator for undesired reactions; and second, by melting the separator, creating internal
short circuits or thermal runaway [23].
2.3. Indicators
The use of indicators that enable the evaluation and diagnosis of battery behaviour
through monitoring is highly desirable. Consequently, the following indicators are usu-
ally defined.
Energies 2023,16, 6846 6 of 36
•
Capacity (
Qmax
) [Ah]: The maximum amount of charge that can be stored inside the
battery and delivered during a full discharge cycle. The capacity (usually expressed in
Ah) varies according to the quantity of material in the electrodes;
•
Nominal capacity (
QN,max
) [Ah]: The original capacity of the battery when it is new
and no degradation has occurred;
•
State of Charge (SoC): The primary indicator in LIBs, providing information about the
remaining energy inside the battery. SoC can be defined as in Equation
(4)
, where
Q
is
the actual capacity and
Qmax
is the maximum capacity of the cell, both measured in
Ah
;
SoC =Q
Qmax (4)
•
SoH: The ratio between the current maximum available capacity and the rated avail-
able capacity, which indicates the battery’s aging condition. It can be expressed
mathematically as follows:
SoH =Qmax
QN,max
(5)
where QN,max represents the capacity before any degradation occurs;
•
C rate: The charging or discharging current relative to capacity. A 1C rate corresponds
to full charging or discharging the battery in one hour, while 0.5C corresponds to two
hours of charging. The C-rate for a 2 Ah cell being charged or discharged at 2 A is
1C, and if the current were 6 A, it would be 3C. In general, a value above 3–4 C is
considered a high C-rate, although this varies notably depending on the chemistry.
2.4. Li-Ion Materials
According to [
10
], LIBs can be categorised into several categories depending on the
electrode material. In the case of negative electrode materials [
10
], carbon-based and
lithium titanate electrodes (LTO, Li4Ti15O12) are the most common materials for LIBs.
The most common positive electrode materials [
10
] are lithium cobalt oxide (LCO,
LiCoO2
), lithium nickel oxide (LNO,
LiNiO2
), lithium manganese oxide (LMO,
LiMn2O4
),
lithium iron phosphate (LFP,
LiFePO4
), lithium nickel manganese cobalt oxide (NMC,
Li(NixMnyCo1−x−y)O2
) and lithium nickel cobalt aluminium oxide (NCA,
Li(NixCoyAl1−x−y)O2
). xand yrefer to the particular alloy used by manufacturers, which
may vary. The material in the cathode is mainly responsible for the battery cost and perfor-
mance in terms of energy, power, lifespan and safety, as shown in Table 2, where different
families are compared with on a scale of 1–4. In [
10
], a comparison was made in terms
of cost, energy (Ah), power (W), safety, lifespan and performance. While cost, capacity
and power are measurable quantities, the rest of the variables require further explanation.
Safety can be related to the temperature at which the cell degrades beyond recovery in
what is known as the thermal runaway; the highest temperature of the thermal runaway
corresponds to the highest cell safety level. Lifespan is directly computed based on the
number of cycles that a cell can withstand. Finally, performance refers to the behaviour of
the cell at hot and cold temperatures, where typically capacities are reduced [31,32].
Table 2.
Different LIB families compared on a scale of 1–4 in terms of cost, energy (Ah), power (W),
safety, lifespan and performance [10]. The highest score is 4.
Chemistry Family Capacity Specific Power Safety Performance Lifespan Cost
LCO 4 2 2 3 2 3
LMO 3 3 3 2 2 3
LFP 2 4 4 3 4 3
NMC 4 3 3 3 3 3
NCA 4 3 2 3 3 2
LTO 2 3 4 4 4 1
Energies 2023,16, 6846 7 of 36
3. Modelling
In order to achieve the estimation of key indicators, it is essential to have a mathe-
matical model that can accurately represent the behaviour of the system, which, in turn,
provides a comprehensive understanding of the interconnected working principles. Finally,
mathematical models serve as a basis for the development of real-time control, estimation
and monitoring algorithms. There exists a wide range of models for LIBs, varying in
complexity and level of detail. Naturally, the computational time and feasibility are directly
related to the complexity of the model.
Such models can be classified as follows [13].
•Mechanistic models
are based on the physical and chemical phenomena occurring
within the battery, providing a detailed, accurate and interpretable representation of
its internal behaviour.
•ECMs
simulate the causality between the battery current and the voltage by construct-
ing an electric circuit using resistors and capacitors, although they do not inherently
represent the physical effects.
•Data-driven models
are developed solely based on measured data and with minimum
to zero use of the battery first principles, which makes them less effective in describing
internal physicochemical phenomena. However, they have the potential to capture
complex behaviour that is not yet fully understood from a physical perspective.
3.1. Mechanistic Models
When discussing mechanistic models, one can delve deep enough to the point of
treating each particle individually. Although these models can be valuable in material
design for batteries, they may not be practical for operational purposes, especially when
conducting whole-cell simulations over time. In such cases, simpler models that treat
the battery as a continuous medium rather than accounting for individual particles are
necessary. Typically, mechanistic models incorporate distributed behaviour, which refers to
the spatial distribution of system variables and parameters. However, when it comes to
control and estimation applications that require dynamic models, further simplification is
required (see Section 3.1.5). In this section, we present a concise overview of microscale
and homogenised models that serve to illustrate the simplification of physics prior to
reaching the Doyle-Fuller-Newman (DFN), also known as the Pseudo2-dimensional (P2D)
model, which is probably the most popular and foundational model for simplified rep-
resentations. A single-particle model (SPM) is a simplified version of DFN, but as it
still employs distributed parameters, we present certain simplifications in Section 3.1.5.
A simple comparison of the mentioned models is provided in Figure 3.
Before continuing, a necessary clarification must be made. It has been stated that
mechanistic models incorporate the spatial distribution of system states, which necessitates
consideration of the dimensions of the cell models. A battery is a device with three spatial
dimensions, but it does not always need to modelled by considering all three dimensions.
Of the models presented in this section, the microscale and homogenised models can
be used to describe the battery in two or three spatial dimensions; the DFN model is,
as mentioned, pseudo-two-dimensional, and the SPM is essentially one-dimensional.
Energies 2023,16, 6846 8 of 36
(a)
(b)
(c)
Figure 3.
Comparative illustration of mechanistic models. Note that the electrolyte is present in the
three regions of the cell as a liquid. Solid particles can be seen in red and blue in their respective
areas. The separator prevents the solid particles from spreading across the whole cell. (
a
) Microscale
and homogenised model. The difference between the models is that fluxes and parameters are
homogenised in the latter, but the geometry of the particle still needs to be solved. (
b
) DFN model.
With homogenised parameters, the geometry of particles is assumed to be spherical. (
c
) Single-particle
model. If the properties of all particles are assumed to be similar, they can be approximated by a
single particle.
Energies 2023,16, 6846 9 of 36
3.1.1. Microscale Model
The first model that must be mentioned is the microscale model, which offers a contin-
uous description of charge transport at the level of individual electrode particles [
33
]. This
model serves as the basis for other battery models and requires an accurate microstruc-
tural representation. It encompasses the transport of lithium ions and electrons within the
electrodes, as well as the role of the electrolyte in facilitating ion and charge movement.
However, the computational requirement of determining the geometry of each electrode
particle presents a significant hurdle to overcome.
The concentration of lithium intercalated in the electrode molecules in the cathode
and anode is known as the solid phase. Despite the difficulty of determining the geometry
of the particles, solid transport is commonly modelled using a diffusion equation, which
allows for the quantification of the concentration at each point of the electrode (see Figure 3).
Concerning the electrolyte, as it is responsible of the transport of the lithium ions (
Li+
),
it can be directly coupled with the charge flow. Finally, the Butler–Volmer equation is
utilised to capture the exchange of lithium ions between the electrode particles and the
electrolyte [14].
Diffusion demonstrates the multidimensional characteristic of this model, as well as
how other models simplify this phenomenon. Diffusion is modelled using Fick’s second
law of diffusion [34], as expressed in a general three-dimensional form in Equation (6):
∂c
∂t=D(x,y,z,c)∂2c
∂x2+∂2c
∂y2+∂2c
∂z2, (6)
where cis the concentration; x, y and z are the spatial coordinates; and Dis the diffusion
coefficient, which is dependent on x, y and z, as well as the concentration. We return to this
example for explanation of the subsequent models.
3.1.2. Homogenised Model
A simplification of the microscale model can be obtained by assuming that the porous
material and its spatial structure form a uniform continuum. This simplified approach [
35
],
known as a homogenised model, can be achieved based on the solutions of the microscale
model, which indicate that several variables do not significantly vary over the length scale
of the microstructure. To carry out the simplification, averaged variables and effective
parameters are employed instead of relying on the spatial region. Following the example
introduced in
(6)
, here, as we assume that the solid phase is homogeneous and that the diffu-
sion coefficient no longer depends on the spatial coordinates but only on the concentration.
∂c
∂t=D(c)∂2c
∂x2+∂2c
∂y2+∂2c
∂z2, (7)
More information about the microscale and the homogenised models can be found
in [14]. With some additional simplification, the DFN model can be achieved.
3.1.3. Doyle–Fuller–Newman Model
The DFN model [
36
] is arguably one of the most popular mechanistic models for
lithium-ion batteries. Evidence of its popularity can be found in various derivations (SPMs)
and its applications in the field [37–40].
The DFN model is derived from the aforementioned homogenised model but with the
assumption that all particles are perfectly homogeneous spheres, thereby eliminating the
need to compute the geometry of each particle. Additionally, the electrodes and separator
are considered to have a one-dimensional geometry, simplifying the equations to be solved
on a one-dimensional axis (typically the horizontal axis). As the morphology of the particles
is typically unknown, these assumptions significantly facilitate the characterisation of
the cell.
Energies 2023,16, 6846 10 of 36
Solid Phase
Returning to the example of
(6)
and
(7)
, Fick’s second law can be applied to perfect
spheres using spherical coordinates, as shown in
(8)
, setting the origin at the centre of
the particle:
∂c
∂t=D(c)1
r2
∂
∂rr2∂c
∂r+1
r2sinθ
∂
∂θ sinθ∂c
∂θ +1
r2sin2θ
∂2c
∂φ2. (8)
The angle variations (the terms relative to the angle (
θ
and
φ
)) can be neglected, as the
particles are assumed to be homogeneous and the variation inside a particle only depends
on the radius. This is reflected in the following equation.
∂c
∂t=D(c)1
r2
∂
∂rr2∂c
∂r, (9)
where
c
is the concentration of the solid phase,
D
is the diffusion of the solid phase and
r
is
the radius of the particle. A complete summary of the nomenclature used can be found in
Table 3.
Note that here, the concentration in each particle only depends on the radial distance
to the centre of the particle, but there are several particles distributed along the xaxis (as in
Figure 3). Thus, the model can be called pseudo-two-dimensional, as it depends on two
axes that are not independent of one another.
In Equation
(9)
, the boundary conditions must be evaluated at the radial coordinate
(r=0 or r=Rs, where Rsis the radius of the particle; see Figure 4) [41].
When
r=
0, the constraint results in solid particles that cannot diffuse from the centre
of the particle, as the mass would disappear as in a black hole, which would break the
conservation-of-mass assumption inside each particle.
∂cs,i
∂r
r=0=0 (10)
When the radius is that of the particle (
r=Rs
, see Figure 4), diffusion occurs in the
form of flux of lithium ions, which are linked to the current that charges or discharges the
battery. This boundary is held by the assumption that the particles (de)lithiate uniformly,
which means that lithium ions (de)intercalate in the solid at the same rate.
∂cs,i
∂r
r=Rs
=±j(x,t)
F(11)
where jis the molar flux and Fis the Faraday constant.
Figure 4. A particle in the solid phase.
With Equations (10) and (11), the boundary conditions for Equation (15) are met.
Energies 2023,16, 6846 11 of 36
Table 3. Parameters for DFN and SPM.
Symbol Definition Unit
xsParameter/variable related to solid -
xeParameter/variable related to electrolyte -
cxConcentration of xmol
m3
DxDiffusion of xm2
s
rRadial coordinate m
RxRadius of particle xm
FFaraday constant C
s
axElectroactive surface area of xm−1
ACell cross-sectional area m2
LThickness m
ePorosity -
esVolume fraction of the solid electrode material in the porous electrode -
jMolar flux A
m3
t+Transference number -
idensity Current density A
m2
ICurrent A
αTransport coefficient -
i0Exchange current density A
m2
ke f f Effective electrolyte conductivity
Electrolyte
In order to define the DFN model with respect to transport in the solid and electrolyte
phases, further explanation is required.
The electrolyte is in charge of the transport of diluted lithium ions across the whole
cell. As shown in Figure 3, it permeates the three areas of the cell. Figure 5represents
the cell, focusing only on the electrolyte. It is important to understand the limits of the
spacial coordinates.
Figure 5.
A depiction of the area permeated by the electrolyte and that to which the electrolyte
equations are applied.
The governing equations are introduces as follows. Mass transport in the electrolyte
can be described using (12) [14,42], which is based on mass balance for the whole cell.
e∂ce,i
∂t=∂
∂xDe∂ce
∂x+1−t+±as·j
F(12)
where
t+
is the transference number (
t+∈R
, the fraction of the current carried by positive
ions [
14
]);
e
is the porosity; and
as
is the electroactive surface area, which can be described
as
as=3es
Rs
, where
e
is the active volume fraction of solid phase. A slight variation of
Equation (12) can be found in [41], where Dedepends on ce.
The first term on the right-hand side of
(12)
can be thought of as the change in
concentration due to diffusion [
37
]. Notice that the first term comes from Fick’s second
Energies 2023,16, 6846 12 of 36
law of diffusion
(6)
, ignoring the diffusion on the
y
and
z
axes. The second term reflects
the change in concentration due to variations in current. As the electrolyte permeates the
whole cell,
(12)
must be applied to the three regions, but as in the separator, there is no
charge transfer, so the last term of (12) disappears.
The boundaries of Equation
(12)
can be evaluated at the extremes of the cell and at the
extremes of the three different areas. In the first case, Equation
(13)
prevents the diffusion
of electrolyte outside the cell, which would result in its destruction.
∂ce
∂x
x=0−
=∂ce
∂x
x=0+
=0 (13)
To reflect the fact that the electrolyte permeates the entire cell and that, therefore, the
flux and concentration are continuous between electrodes and separators [
37
], the evalua-
tion of Equation
(12)
in the region where the electrodes meet the separator is expressed as
Equation (14)
, indicating that force is continuous, with uniform flux between the anode,
separator and cathode.
∂ce,−
∂x
x=L−
=∂ce,sep
∂x
x=0sep
∂ce,+
∂x
x=Lsep
=∂ce,+
∂x
x=L+
(14)
where xand Lare the same as in Figure 5.
With this set of equations, the DFN model is complete. In [
14
], a complete derivation
from the homogenised model was provided. The missing parts required to determine the
described behaviour of the whole cell are the cell voltage and the thermal model. However,
for the sake of simplicity, they are only described for the SPM, as there is no simple way of
describing them without the assumptions of the SPM.
3.1.4. SPM
Diverse simplified versions of the DFN model fall under the SPMs, still written in
the form of PDE but significantly simpler computationally. The main assumption of these
models is that a single particle can be representative of the whole electrode, assuming that
the behaviour is similar across multiple particles.
Therefore, according to [
43
], the assumptions under which the SPM is derived from
the DFN model can be summarized as follows:
•
The concentration of solid particles is homogeneous in a radial sense so that the
concentration only varies on the radial coordinate (r);
• The current density in each electrode is uniformly distributed;
•
The number of moles in the electrolyte and that in the solid phase are both conserved.
This can establish a proportional relation between current and flux;
• The transport coefficients (α) of the anode and cathode are equal.
The diverse ways in which the DFN model has been simplified has led to a wide
variety of SPMs that can be classified, according to [
14
], as those take into account the
electrolyte dynamics (SPMe) [44] and those that do not [45].
However, all of these methods the same working principles and the same governing
equations. Figure 6shows the simplest representation of the cell, with only one particle in
the anode and cathode, respectively, and three separated regions.
Energies 2023,16, 6846 13 of 36
Figure 6.
SPM considering variations in the spacial coordinate (x) and assuming that the behaviour
of the electrodes can be described by a single solid particle.
In summary, the SPM can be established as a 1D geometric model that only depends
on the spatial coordinate (
x
) in the electrolyte or the radial coordinate (r) in the solid phase.
The model has three separate areas and considers flux as homogeneous. Mass transport in
lithium ions and the electrolyte are needed to successfully describe the model.
Due to the aforementioned assumptions, the subsequent equations describe an SPM,
describing first the diffusion in the solid phase, as well as taking into account electrolyte
dynamics. After describing the diffusion phenomena in the three areas, we explain the
cell potential. We also provide a brief discussion about incorporation of the effects of
temperature in the SPM.
Solid Phase
As mentioned, the dynamics of the concentration in the solid phase can be described
using Fick’s second law of diffusion, which serves as an example to illustrate the simplifica-
tion from the microscale (6) model to here. Equation (9) can be rewritten in a simpler way
using the chain rule as:
∂cs,i
∂t=Ds2
r
∂cs,i
∂r+∂2cs,i
∂r2(15)
Note the suffix i, which refers to the negative electrode or the positive electrode,
and the suffix s, which reflects the fact that the parameter or variable is related to a
solid. This notation is used later in this paper.
Ds
is assumed to be constant under the
assumptions according to which the SPM model is derived. Therefore,
Ds∈R
; however, if
this assumption were not considered, it would be a function that depends on cs.
The boundary conditions in Equation
(15)
are the same as those in
(9)
; however,
as current is proportional to flux (according to SPM assumptions), when the radius is that
of the particle (r=Rs, see Figure 4), the boundary condition can be rewritten as:
∂cs,i
∂r
r=Rs
=±Ibat
FasDsA Li
(16)
where
F
is the Faraday constant,
as
is the electroactive surface area,
A
is the cell area and
Li
is the thickness of the region i(see Figure 4).
With Equations (10) and (16), the boundary conditions for Equation (15) are met.
Energies 2023,16, 6846 14 of 36
Electrolyte
In the SPM, where electrolyte dynamics are considered, the general equations are
similar to those of DFN. As mentioned, SPM assumes that the flux is proportional to the
current, which enables (12) to be rewritten as:
e∂ce,i
∂t=∂
∂xDe∂ce
∂x+1−t+±Ibat
F·A·Li
(17)
The boundaries are the same as in DFN, as described in
(13)
and
(14)
. We note that
(17)
must be evaluated in the three areas, but as there is no charge transfer in the separator,
the last term would disappear.
Cell Potential
The remaining issue concerning the governing equations for the cell is the potential.
The potential can be measured between the two current collectors and is the effective
voltage that affects any device connected to the cell. A typical curve of the cell potential is
shown in Figure 7, for an LFP cell.
(18) is a general equation based on [14,41].
V=φp−φn=U+η−φe−RfIbat (18)
where
U
is the Open Circuit Voltage (OCV),
η
is the overpotential,
φe
is the drop in potential
in the electrolyte and Rfis the film resistance related to SEI.
Overpotential (
η
) is calculated using different approaches in the literature. In the case
of [43], the solution was computed as:
η=RT
αFsinh−1Ibat
2asLi0(19)
Figure 7.
Discharge curve at different C rates for the ANR26650M1-B cell. The discharge curve shows
a typical flat area characteristic of LFP cells. The image was extracted from the data sheet provided
in [46].
Different approaches are available to solve the Butler-Volmer (B-V) relation:
j=1
Fi0eαF
RT η−e−αF
RT η(20)
Energies 2023,16, 6846 15 of 36
In
(19)
and
(20)
,
R
is the gas constant,
α
is the transport coefficient (previously assumed
to be equal for both the anode and cathode) and i0is the exchange current density.
The potential drop in the electrolyte (
φe
) is also expressed in several ways in the
literature. Again, according to [43]:
φe=ReIbat +2RT(1−t+)kf
Flnce(0+)
ce(0−)(21)
where
kf
is the diffusional conductivity, as reported in [
37
,
43
] as a function dependent on
the electrolyte concentration that can be approximated as a uniform parameter if assumed
to be constant in the space. On the other hand, the authors of [
41
] defined
kf
as an empirical
equation.
Re
is defined in
(22)
. On the other hand, the authors of [
14
,
47
] completely
neglected the effect of the concentration on the drop in potential of the electrolyte.
Re=1
2A
L++2Lsep +L−
ke f f (22)
where
ke f f
is the effective solid conductivity, typically defined in the literature according
to Bruggeman’s relation [
14
,
41
,
43
,
47
] as
ke f f =ke1,5
. Bruggeman’s relation refers to the
effective conductivity inside the microstructure of the cell.
Thermal Modelling
The SPM presented earlier is treated as isothermal, but the temperature cannot be
neglected. Thermal management is also a popular topic; therefore, thermal modelling has
been widely discussed in the literature [
48
–
51
]. However, predicting the temperature is
highly complicated, as heat generation is nonlinear and thermal properties are anisotropic.
Moreover, a commercial cell is large enough to exhibit a temperature gradient throughout
the entire cell. Hence, the temperature variation becomes a three-dimensional problem,
given the significant temperature variation across the structure. This results in a five-
dimensional model in the case of a DFN thermal model or a four-dimensional model if the
SPM is used.
Nevertheless, it has been found that lithium diffusivity, reaction rates, and electrolyte
diffusivity and conductivity can be described by the Arrhenius relationship, as expressed
in (23).
f(T) = fre f e
Ea
R(1
Tref −1
T)(23)
This relation was used in [
41
,
48
] to extract a temperature-dependent SPM. While
the authors of [
48
] neglected the electrolyte voltage drop, the authors of [
41
] considered
it. In both of the abovementioned studies, OCV was considered temperature-dependent
as follows:
U(T) = U(Tre f ) + ∂U
∂T(T−Tref )(24)
In other works, such as [47], a purely empirical equation is provided.
Heat generation is not explicitly calculated and takes the following form:
ρcp∂T
∂t=q+˙
Qirr +˙
Qrev (25)
where
q
is cooling according to Newton’s Law,
˙
Qirr
is the irreversible generated heat and
˙
Qrev
is the generated reversible heat. The calculations of these terms vary depending on
the source, as shown in [14,48,51].
3.1.5. Finite-Order Models
Several methods are available to transition from a physics-based model, such as
the DFN, to a simpler and more manageable model for control-oriented purposes. One
apparent approach is to employ a simplified physics approach, resulting in the SPM,
Energies 2023,16, 6846 16 of 36
as discussed in Section 3.1.4. However, it should be noted that the SPM remains an infinite-
order model, as it is described using PDEs.
To successfully represent a system, such as in a state-space representation, a finite-
order model is necessary. In response to this requirement, several simplification techniques
have been developed. The authors of [
42
] reviewed the techniques used in the literature to
achieve such reductions and classified them in the following families.
•
Spatial discretization: Well-known techniques such as Finite-difference Method (FDM)
or Finite-volume Method (FVM) are used;
•
Function approximation: Spatiotemporal variables are approximated by a finite
weighted sum of assumed trial functions that are fitted using optimisation techniques;
•
Frequency domain approximation: The frequency response is obtained means of Padé
approximation or the residue grouping method, among other techniques;
•
Physics simplification: Typically achieved using diverse SPM approaches with sev-
eral variations.
Spatial Discretization
The spatial coordinate is discretized to obtain a continuous-time description, making
it suitable for control applications. This strategy is also known as the method of lines.
The most common and direct methods of spatial discretization are the finite-difference
method [
52
] and the finite-volume method [
53
]. These methods can preserve most model
properties within a wide range of conditions. The main difference between them is that
FVM is mass-conservative. However, their complexity lies in the number of control volumes
or mesh points used. In the appendix of [
12
], the general equations for discretization using
FVM and FDM are provided. Here, the equations for FVM are presented in
(26)
. The differ-
ence between FVM and FDM can be seen in Figure 8. FVMs are mass-conservative, since
they enforce the conservation principle and balance the net flow of conserved quantities,
while FDMs lack explicit conservation enforcement within their discretization.
First-order derivatives: ∂ci(x,r)
∂r=ci+1(t)−ci(t)
h0
r
;
Second-order derivatives: ∂2ci(x,r)
∂r2=ci+1(t)−2ci+1(t) + ci(t)
h02
r
;
(26)
Figure 8.
Comparison between FVM and FDM. The FDM places elements at each point where the
calculation is performed, while the FVM assumes a constant volume.
Function Approximation
The behaviour of the model is approximated by a function that has time-varying
coefficients, approximating a trial function dependent only on the spatial variables. These
functions can be polynomials, sinusoidal, logarithmic or a combination of several thereof.
These methods are also known as projection-based methods, and the fitting of the param-
eters or trial functions is achieved using optimisation techniques. An example can be
found in [
54
], where a decomposition of the data from the original DFN model is presented.
Energies 2023,16, 6846 17 of 36
If instead of minimising the error, it is integrated and weighted over the evaluated domain
such that the result of the integral is zero, these methods can be referred to as spectral
methods. The renowned Galerkin method also falls into this category [55].
Frequency Domain Approximation
Various approaches are available to obtain the frequency response of a system. The
eigenfunction technique [
56
] calculates all periodic roots of the transcendental function, trun-
cating the infinite series to obtain a finite-order model. The residue grouping method [
57
]
also calculates and truncates periodic poles and zeros, grouping poles and approximating
them through frequency response cost function optimisation. While residue grouping offers
improved accuracy across a broad frequency range, it is less suitable for real-time systems
due to computational inefficiency, sensitivity to initial guesses, and lack of guaranteed
convergence and global optimality.
In contrast, Padé approximation [
58
] linearises transcendental transfer functions into
rational ones in the s domain, allowing for direct system order reduction through moment
matching. The rational polynomial coefficients incorporate physical cell parameters and
can be easily updated for operational changes. Higher-order Padé approximations provide
increased accuracy at the expense of additional computational requirements, making them
suitable for EVs, while low-order approximations suffice for stationary battery applications.
Physics Simplification
After discussing the SPM in Section 3.1.4, there is little left to discuss regarding physics
simplification. However, Ref. [
59
] presents a different simplification of the DFN model
in which the electrolyte concentration remains constant and the diffusion equation of the
solid is approximated.
Before continuing, let it be noted that considering the equations, it becomes evident
that the SPM is not a finite-order model. One potential solution is provided in [
41
], where
the FDM is applied to the SPM.
Another option is the use of the FVM instead of the FDM, which offers the advantage
of conserving mass. This document provides the discretized equations and the state space
representation (see Section 4.4) using the mentioned method for a set of two control volumes
per particle and three control volumes for the electrolyte, as seen in Figure 9.
Figure 9. Description of the volumes used for discretization.
Energies 2023,16, 6846 18 of 36
Concentration in the Solid
With boundaries
(10)
and
(16)
and applying the FVM with Equation
(26)
, the concen-
trations for the solid in the four control volumes shown in Figure 9can be obtained using
Equation (15).
˙
c1=˙
cn,1 =Dn8
R2(cn,2 −cn,1)(27)
˙
c2=˙
cn,2 =Dn8
R2(cn,1 −cn,2)±4
R
Ibat
FaDnALn(28)
˙
c3=˙
cp,1 =Dp8
R2(cp,2 −cp,1)(29)
˙
c4=˙
cp,2 =Dp8
R2(cp,1 −cp,2)±4
R
Ibat
FaDpAL p(30)
The concentration in every control volume is dependent on itself, as well as the
concentrations of its neighbours.
Concentrations in the Electrolyte
With boundaries
(14)
and applying the FVM with Equation
(26)
, the concentrations
for the electrolyte in the three control volumes shown Figure 9can be obtained using
Equation
(17)
. Note that when
(17)
is applied to the separator, the second term disappears,
as there is no mass transfer in that area. This implies that
(32)
is not dependent on the
battery current.
˙
c5=˙
ce,1 =De
e
4
(L−+Lsep)2(ce,1 −ce,2) + (1−t+)
e
±Ibat
0.5FA(L−+Lse p)(31)
˙
c6=˙
ce,2 =De
e
4
(L−+Lsep)2(ce,2 −ce,1) + De
e
4
(L++Lsep)2(ce,2 −ce,3)(32)
˙
c7=˙
ce,3 =De
e
4
(L++Lsep)2(ce,3 −ce,2) + (1−t+)
e
±Ibat
0.5FA(L++Lse p)(33)
3.2. ECMs
ECMs are, as mentioned, electric circuits designed to mimic the behaviour of a battery.
Although they do not inherently represent the internal states of the battery, their simple
form makes them easily understandable for non-experts on the topic.
The authors of [
13
] established two subcategories of ECM: those that are based on the
electrochemical process and those that are not.
3.2.1. Phenomenological ECMs
Phenomenological ECMs represent a simple and popular way to reproduce battery
dynamics without considering internal phenomena. Easily scalable, they usually takes
the shape of an ideal voltage source corresponding to an OCV, series resistance and a
variable number of RC nets. This can be seen in Figure 10, where a first and second order
are presented. The models are scaled-up by adding more RC nets. The order of the model
and, thus, its accuracy can be increased at the expense of computational effort [60].
A comparative study of the ECM family applied to several cell chemistries was
reported in [
60
]. Experiments showed that the first-order model achieved superior accuracy
on the performed tests.
Energies 2023,16, 6846 19 of 36
(a) First-Order ECM. (b) Second-order ECM.
Figure 10. Different equivalent circuit models.
3.2.2. Electrochemical ECM
The equivalent circuit aims to replicate the cell’s dynamics by incorporating electrical
elements that recreate the battery’s structure. Since certain dynamics cannot be adequately
reproduced by a simple RC network, the equivalent circuit also utilises ZARC elements
(which account for the phase influence in the impedance and are represented by a net
composed of a resistor and a constant phase element [
61
]) and Warburg impedances. ZARC
elements are commonly employed to represent the SEI, while Warburg impedances are used
to characterise diffusion. The use of these elements and their phase shape can are described
in [
13
,
62
]. The use of these frequency-dependent elements requires precise characterisation.
Electrochemical impedance spectra (EIS) is a method consisting of the application of a pulse
of current or voltage to the cell of a determined frequency and amplitude and, with the
phase shift and variation of amplitudes recorded. This yields the response of the cell in the
form of a Nyquist diagram.
3.3. Data-Driven Models
Data-driven models utilise data to predict the behaviour of the battery. Their primary
advantage is that there is no need to comprehend the complex physicochemical phenomena
that determine the behaviour of a cell. However, the result of the model highly depends on
the quality of data used for calibration.
The objective of data-driven models is to establish a relationship between measurable
information about the battery and key indicators, such as SoC and SoH, commonly using
machine learning algorithms to extract a model from data. Typical families of algorithms
include artificial neural networks (ANNs), Gaussian process regression (GPR), linear
regression (LR) and support vector machines (SVMs). To successfully develop an estimator
using these techniques, measurements are needed in advance. Datasets are composed of
inputs such as voltage, current or temperature and typically include the SoC as output.
Models can be trained to extract an accurate output with respect to the past and present
inputs. Depending on whether these models rely solely on data or employ a model (such
as those described in the preceding sections, for instance) and data to calibrate it, the
authors of [
13
] categorise them as black-box or grey-box models, respectively. Of the
later, the combination of Kalman filter with one of the mentioned algorithms is a popular
choice [18,63–65].
Clearly, the quality of the data used to train the models has a considerable impact
on the accuracy of the estimation. The collection of these datasets can be a long process,
as it may require long-term experimentation. It is also important that the accuracy of
the data be compared to that required for the application in which the estimation will be
used. In [
66
], some of the reviewed models were reported to use data that are not fully
representative of the case under study. The process of training the models can also be
a source of inaccuracy, as randomness is intrinsic to many machine learning algorithms.
To overcome this situation, the authors of [
66
] recommend training the same model several
times with the same dataset.
Energies 2023,16, 6846 20 of 36
Without detailing of the myriad of algorithms that fall into the machine learning
family (which are described in reviews such as [
67
,
68
]), here, we focus on the type of
data used to train them. In the case of SoC estimation, while the authors of [
69
] relied
solely on voltage and current, most methods [
70
–
72
] also use temperature. It should be
mentioned that all these techniques require the “true” SoC for the training dataset. For SoH
estimation, a greater number of combinations exists. Refs. [
73
–
75
] used only current, while
the method described in [
76
] operates only using voltage. The authors of [
77
] employed
both, whereas the authors of [
78
,
79
] considered temperature in addition to current and
voltage. Finally, the authors of [80,81] relied on current, voltage and capacity.
Machine learning algorithms can also be used to forecast how a battery will behave
in the future. The authors of [
82
–
84
] used different methods to predict a battery’s future
behaviour using only information from a single cycle. The authors of [
82
] used voltage,
current and SoC to estimate the trajectories of the remaining capacity and internal resistance.
With the same goal, the authors of [
84
] employed the capacity and internal resistance of one
cycle. On the other hand, the authors of [
83
] used voltage, current, capacity and temperature
to estimate the remaining useful life (RUL), as computed in charge/discharge cycles.
4. Estimation in Li-Ion Batteries
4.1. Estimable Information
Estimation in Li-ion batteries is essential for proper battery management. Typically,
the target is estimation of internal (and immeasurable) states, but estimation techniques also
allow for estimation of information about system parameters, increasing the efficiency with
which a model is adjusted. It is also possible to estimate parameters in online applications
using a model that evolves with time and is therefore always accurate.
4.1.1. SoC
There are several ways to compute the SoC without using estimation techniques. Some
observers use estimation techniques as part of their algorithms. However, each method has
distinct disadvantages that, if used in an open-loop configuration, can cause errors.
•Coulomb counting method
: Coulomb counting is the most straightforward approach
to compute SoC in a cell. It involves integrating the current over time, thereby cal-
culating the extracted capacity of the cell. However, this technique has two main
drawbacks: the initial SoC is usually unknown, and the capacity may change depend-
ing on the C rate or the temperature, as well as the cell’s aging. Additionally, Coulomb
counting is susceptible to various sources of error, as listed in [
15
], including current
measurement error, current integration error, timing error, and measurement and
process noise.
•OCV method
: The OCV is closely related to SoC, meaning if one is known, the other
can be determined. However, OCV can only be measured in the absence of current and
after the battery has been at rest, as current causes voltage to deviate from the OCV
curve, and due to hysteresis, it takes a long time to recover. Although this method
is precise and straightforward, it is unsuitable for online applications. Nevertheless,
it holds value in providing an OCV–SoC curve, as required in many models. The
authors of [85] provided a detailed guide for OCV characterisation.
•Internal resistance method
: By applying a fast current pulse and measuring the
resulting voltage variation, the internal resistance can be determined and linked to
SoC [
86
]. This method performs quite well at lower levels of SoC, where voltage
tends to decrease rapidly. However, in other ranges, especially with a typical voltage
plateau, SoC estimation becomes much less precise.
•Model and Look-Up table
: Using a simple model properly experimentally calibrated
under static conditions, OCV can be extracted. Then, for online applications, SoC can
be computed using a look-up table that relates OCV to SoC.
Energies 2023,16, 6846 21 of 36
More information about SoC computation can be found in [
11
]. A different path to
SoC estimation is the use of observers. A discussion of the most suitable observers for LIBs
is provided later in this section.
4.1.2. SoH
Several methods are available to estimate SoH using measurements.
•Direct measurement
: Periodical experiments to measure the capacity can directly
compute Equation
(5)
. A typical experiment used to achieve this involves the use
of charge/discharge cycle at a determined C rate (usually 1C to extract nominal
capacity) [60].
•Internal resistance measurement
: As a battery loses capacity and degrades, the inter-
nal resistance increases [
87
]. Periodically measuring it with current pulses, always
at the same level of SoC, can provide information about the SoH. Internal resistance
characterisation is an easier experimental method than capacity measurement and can
be performed in a shorter time.
•Impedance measurement
: Measurement of the impedance can also be related to
SoH [
88
], as the battery impedance is affected by degradation. EIS is needed to
characterise this phenomenon, providing valuable information, as electrochemically
based models (Section 3.2.2) also use EIS to extract model parameters.
Other approaches involve the use of estimation algorithms, which lead to observation
problems, as discussed in Section 4.2.
4.1.3. Parameters
Parameter identification for LIBs is a crucial process in understanding and characteris-
ing their behaviour.
Before continuing, we note the difference between parameters and states. While states
are variables that evolve over time, are influenced by system inputs and are an indicator of
the condition of the system, parameters are constant values that characterise the system.
Parameters can change over time in response to changes in the system itself. In the case of
LIBs, parameters may change due to aging.
When developing a model, the accuracy of its results depends on the selected param-
eters. Thus, parameter identification can be achieved through experimental testing and
by later fitting experimental results to the model. Experimental tests involve the appli-
cation of known inputs to the battery and measurement of its responses. These data are
then fitted to appropriate mathematical models to extract the relevant parameters [
89
,
90
].
However, since temperature and aging can impact the cell’s behaviour, some approaches
adapt the measured data to these two factors. This can be achieved both offline and online.
In offline methods, the procedure involves conducting tests at different aging stages or
temperatures ([
91
] to obtain an OCV–SoC–temperature relationship). Online parameter
identification usually involves the use of estimation algorithms, which are discussed in the
following sections.
Another important aspect related to parameter identification is the dynamics of the
processes. For instance, charge transfer is significantly faster than diffusion processes. Esti-
mation algorithms may take this into consideration and attempt to solve the identification
task according to the appropriate dynamics [92].
4.2. The Observation Problem
Mathematical models of Li-ion batteries describe the dynamic behaviour of internal
variables of the system, e.g., species concentrations or temperature, when a specific in-
put profile is introduced to the system. This mathematical depiction allows for virtual
simulation of the system which, in turn, can be utilised to further improve the design of
Li-ion batteries. Another ambitious objective is the deployment of mathematical models
that run in real time and in parallel with the true Li-ion battery. Real-time models can
be used to retrieve information that, on the one hand, is not directly available from sen-
Energies 2023,16, 6846 22 of 36
sor measurements but, on the other hand, is necessary for control and monitoring of the
Li-ion battery.
Up to this point, we have focused on presenting computationally efficient models
that can be operated in real time. In this section, we focus on how to exploit these models
and the measurements of available Li-ion battery sensors to retrieve internal information
about the system. The process of estimating unknown dynamic internal variables from the
measurement signals is commonly referred to as the observation problem.
In most cases, the observation problem is formulated using the state-space formal-
ism; that is, the dynamics of the system are depicted through a multi-input, multioutput
(potentially nonlinear) system in the following form:
˙x =f(x,u)
y=h(x),(34)
where
x∈Rn
is defined as the state vector, that is, the vector of internal variables of the
system. The states are not directly measured and are assumed to be unknown. The term
u∈Rq
depicts the vector of controlled inputs, that is, a set of signals that are measured
and can be modified. The term
y∈Rm
is the set of measured signals. As a mathematical
technicality and in order to simplify the analysis, it is common to assume that the vector
functions
f
,
h
are sufficiently smooth and that the solutions of the system in
(34)
are
bounded and unique.
We define
ψt(x0
,
u)
as the values of the states (
x
) of
(34)
at time
t
with the input
profile (
u
) and initial condition (
x0
). Moreover, using this notation,
h(ψt(x0
,
u))
depicts
the value of the measured output (
y
) of
(34)
at time
t
with the input profile (
u
) and initial
condition (x0).
In the so-called observation problem, the objective is to generate a time-varying signal
(
ˆx
), which is defined as a state estimation, based on the known values (
u
and
h(ψt(x0
,
u))
)
such that
ˆx
eventually converges to the true
ψt(x0
,
u)
. One may wonder if this problem
can be immediately satisfied by inverting the output equation, that is, by generating the
estimation as
ˆx=h−1(y),
where
h−1
is an inverse function of the output map (
h
) such that
h−1(h(x)) = x
. However,
Li-ion battery systems have fewer sensors than the dimensions of the state to be estimated
(
m<n
). Therefore, the inverse function (
h−1
) does not exist, and this solution is not
feasible. This fact implies that a single sample of the output (
y
) is not sufficient to infer
the states’ values. For this reason, the observation problem has to be solved using model
in Equation
(34)
and the full trajectory of the measured signals (
u
and
h(ψt(x0
,
u))
) in the
time range (t∈[0, t]).
The observation problem can be solved using different theoretical methodologies.
As we assumed the solutions of the system
(34)
, a solution to the observation problem
is obtained by estimating the initial condition of the system
(34)
and forward simulating
the model to generate the full state trajectory. A straightforward method to implement
this idea is to forward simulate the system
(34)
for a set of different initial conditions and
gradually eliminate all the initial conditions that do not agree with the measured signal
(
y
). This method can be implemented by means of a stochastic framework [
93
–
98
] (see [
99
]
for more technical details about the approach) or a deterministic
framework [100–102]
(see
[103,104]
for more technical details about the approach). Moreover, under some
observability assumptions, it can be proven that this method eventually retrieves the true
initial condition. Although this is an interesting solution to the observation problem, it
requires an adequate initialisation of the algorithm in a region around the true initial state.
Moreover, the model of the system needs to be very accurate in order to achieve a precise
forward simulation of the states.
Energies 2023,16, 6846 23 of 36
Alternatively, the set of adequate initial conditions for the system can be computed by
minimising a cost function online in the following form form [105–108]:
ˆx0∈arg min
x0∈XZt
t−T|y(s)−ψt(x0,u(s))|ds, (35)
for some positive constant (
T
), time (
t≥T
) and set (
X∈Rn
). Examples of this strategy in
Li-ion batteries can be found in [109–114].
The limitation of this approach is that if the dynamics of the system are described by
nonlinear differential equations (as is the case for Li-ion batteries), the optimisation in
(35)
becomes a nonlinear non-convex problem, which presents the obstacles of computational
complexity, non-convexity and multiple local minima.
Up to this point, two possible routes for solving the observation problem have been
presented. However, these routes are limited in terms of implementation in Li-ion batteries.
Therefore, we need an alternative method to solve the problem.
Notice that the challenges involved in resolving the observation problem stem from
three key factors: the unknown initial conditions of the system, uncertainties present in the
analytical model and the nonlinearities within the model.
To begin with, the issue of unknown initial conditions can be addressed by ensuring
that the algorithm’s trajectories ’forget’ their initial conditions over time. This characteristic
can be seen as a stability property, where the error between the true states and the estimated
states remains controlled.
In order to handling model uncertainty, the algorithm is required to exhibit robustness
in the face of these uncertainties. Stability and robustness are fundamental concepts in con-
trol theory, which inspired the development of a state estimation algorithm incorporating
the feedback concept. This specific algorithm is referred to as an observer.
Lastly, the matter of model nonlinearities serves as the motivation for designing an
algorithm from a nonlinear control theory perspective and the development of nonlin-
ear observers.
The main message of this section is that an observer corresponds to an adequate
algorithm to solve estimation problems in Li-ion batteries. The remainder of this section
is devoted to properly defining the concept of observer and proceeding with a review of
observer-based techniques implemented in Li-ion batteries.
4.3. Observer Definition
Let us recall that the objective is to estimate the unknown states (
x
) of a system
depicted by a model
(34)
. The observer approach revolves around creating and employing
a device known as an “observer” to generate real-time estimations, denoted as
ˆx
. Figure 11
illustrates the structure of observers.
Figure 11. Diagram of the general structure of an observer.
In essence, an observer can be likened to a dynamic system that utilises the mathe-
matical model of the actual system to replicate the state trajectories of the genuine plant.
To compensate for model uncertainty, unmodelled disturbances and uncertain initial con-
Energies 2023,16, 6846 24 of 36
ditions, the observer incorporates a feedback component that adjusts the estimation by
comparing the measured outputs with the observer’s estimations of these outputs. A more
rigorous mathematical definition of an observer provided below (see also [
115
,
116
], [
117
]
(Chapter 1) and [
118
] for more technical details on observer design for nonlinear systems).
Consider the following set of dynamics:
˙
ˆ
ξ=ϕ(ˆ
ξ,u) + κ(ˆ
ξ,y−h(ˆx)),ˆx=Γ(ˆ
ξ)(36)
where
ˆ
ξ⊆Rnξ
, with
nξ≥n
, is the observer state. The function
κ
is the output feedback
term that satisfies
κ(ˆ
ξ
, 0
) =
0 for all
ˆ
ξ∈Rnξ
. Finally, there exists a right inverse
Γ−R
of the
function ψsuch that x=Γ(Γ−R(x)) for all x∈Rn.
Such a structure is an observer if it satisfies the following properties:
•ˆx(0) = x(0)⇒ˆx(t) = x(t),∀t≥0
•|ˆx(t)−x(t)| → 0as t →∞.
Notice that according to this definition of observer, we allow the following possibilities:
•
The observer dynamics
(36)
are depicted in a different set of coordinates (
ˆ
ξ
) than the
original system coordinates (
x
). Consequently, the observer includes a map (
Γ
) that
relates the observer coordinates to the coordinates of the original system.
• The dimensions of the observer may be larger than the original system dimensions.
•
Not only can the dynamics of the system be nonlinear, but the observer feedback term
(κ) may also be a nonlinear function.
4.4. State-Space Model of Li-Ion Batteries
In Section 4.2, observer development is commonly formulated by means of state-space
representation. To successfully represent the battery model in state space, the system must
be described in the following form:
˙x=f(x,u)
y=h(x),(37)
In this case, the input (u) and the output (y) of the system are:
y=V
u=Ibat
(38)
where Vis the equation for the potential of the cell as expressed in
(18)
, and
Ibat
is the
battery current.
Therefore, xis be a vector containing all the states (Equations (27)–(33)):
x=
cn,1
cn,2
cp,1
cp,2
ce,1
ce,2
ce,3
=
c1
c2
c3
c4
c5
c6
c7
(39)
The equation for voltage
(18)
should be rewritten in a form that clearly indicates the
dependencies on the current and on the states:
V=U(cn,1,cn,2 ,cp,1,cp,2) + η(Ibat)−ReIbat −2RT(1−t+)kf
Flnce,3
ce,1 −RfIbat
V=U(cn,1,cn,2 ,cp,1,cp,2)−2RT(1−t+)kf
Flnce,1
ce,3 +η(Ibat ) + Re
RfIbat
(40)
Energies 2023,16, 6846 25 of 36
The expression for
η(Ibat )
can be found in Equation
(19)
. Finally,
˙x
can be represented
in the form of ˙x=Ax+Bu, where A and B correspond to the following matrices:
A=
−Dn8
R2Dn8
R20 0 0 0 0
Dn8
R2−Dn8
R20 0 0 0 0
0 0 −Dp8
R2Dp8
R2000
0 0 Dp8
R2−Dp8
R2000
0 0 0 0 De
e4
(L−+Lsep )2−De
e4
(L−+Lsep )20
0 0 0 0 −De
e4
(L−+Lsep )2
De
e4
(L−+Lsep )2+De
e4
(L++Lsep )2−De
e4
(L++Lsep )2
0 0 0 0 0 −De
e4
(L++Lsep )2
De
e4
(L++Lsep )2
(41)
b=
0
4
R1
FaALn
0
4
R1
FaALp
(1−t+)
e1
0.5FA(L−+Lse p)
0
(1−t+)
e1
0.5FA(L++Lse p)
(42)
4.5. Observers in Li-Ion Batteries
In this section, we provide a review of observer techniques implemented in Li-ion
batteries. Given the definition of observer provided in the previous sections, Figure 12
shows how the mentioned structural links to the battery model.
Figure 12. A depiction of the observation scheme for Li-ion cells.
Energies 2023,16, 6846 26 of 36
4.5.1. Linear State Observers
Although Li-ion batteries models are depicted by nonlinear differential equations,
the model can be locally approximated by a linearly, which largely simplifies the observer
design process at the cost of having to initialise the observer states “close enough” to the
true battery states.
More precisely, the idea is to consider a pair
(x∗(t)
,
u∗(t))
that depicts a nominal
trajectory of the nonlinear system to be observed. Then, first-order Taylor expansion of
system (34) around (x∗(t),u∗(t)) leads to a linear time-variant system:
˙xδ=A(t)xδ+b(t)uδ,yδ=c(t)xδ(43)
A(t) = ∂f(x,u)
∂x
x∗(t),u∗(t)
;b(t) = ∂f(x,u)
∂u
x∗(t),u∗(t)
;c(t) = ∂h(x)
∂x
x∗(t)
.
Then, since
(43)
is a good linear approximation of the battery model, a linear observer
can be implemented to estimate the states.
The major limitation of this approach is that, since the true states of the system (
x
), are
unknown, the nominal trajectory
(x∗(t)
,
u∗(t))
is also unknown, and the system has to be
linearised around the estimated trajectory
(ˆx(t)
,
u∗(t))
. It is for this reason that the linear
approximation
(43)
is no longer adequate if the observer is not initialised “close enough” to
the true trajectory, i.e., |ˆx(0)−x(0)|<e, where eis a sufficiently small constant.
Although this is a significant limitation, the simplicity of the implementation of
observers based on linearisation motivated their adoption in Li-ion batteries. In this context,
there are three major observer techniques that can be implemented in a linearised system.
•Extended Kalman filter
[
16
,
17
,
119
–
125
]: This represents a classical approach to state
observation for dynamic systems that effectively converts (locally) a nonlinear system
into a linear one. This transformation is achieved by computing the first-order Taylor
series expansion, specifically the Jacobian matrix, around the estimated operating
point at each time step. Consequently, the nonlinear system is approximated as
a continuum of linearised points. Additionally, assumptions are made regarding
the measurement noise and process perturbations, assuming them to be zero-mean,
Gaussian and independent of each other. However, it may prove inaccurate when
applied to highly nonlinear systems. Moreover, there is no guarantee of convergence if
the initial values of the observer estimation deviate significantly from the actual values.
•H∞Observer
[
126
]: This method endeavours to identify corresponding states that
satisfy a mathematical optimisation problem formulated using the
H∞
norm of the
observer. Its primary goal is to achieve an optimal solution for a range of diverse plants
representing varying levels of uncertainty or noise. Consequently, it offers certain
advantages over an extended Kalman filter, including heightened robustness against
model uncertainties and the ability to handle unknown noise statistics. However,
the implementation of this approach demands a significant level of mathematical
comprehension and relies heavily on the specific plants employed during its design.
Furthermore, if the actual operating conditions differ from those used in the observer’s
design, convergence is not guaranteed.
4.5.2. Nonlinear State Observers
The observers based on a linearised model presented in the last section present the
significant benefit of a simple design process. Nonetheless, they present two major draw-
backs. First, the linear observer has to be initialised “close enough” to the true state.
Second, the linearisation process can destroy the structural properties of the battery dy-
namics. For this reason, the current trend is to directly design nonlinear observers using
the nonlinear model of the battery.
The process of designing nonlinear observers is more convoluted than the process of
designing linear observers but, in general, results in observers with better performance in
Energies 2023,16, 6846 27 of 36
terms of robustness and transient behaviour. In this section, we provide a review of the
major nonlinear observer techniques that have been implemented in Li-ion battery systems.
•Unscented Kalman filter
[
127
–
132
]: An unscented Kalman filter (UKF) is a nonlinear
variant of a Kalman filter and typically demonstrates superior performance com-
pared to the extended Kalman filters when confronted with highly nonlinear systems.
The key to its effectiveness