Reduced-order electrochemical model for lithium-ion battery with

Reduced-order electrochemical model for lithium-ion battery with
domain decomposition and polynomial approximation methods
Changlong Li, Naxin Cui
*
, Chunyu Wang, Chenghui Zhang
School of Control Science and Engineering, Shandong University, Jingshi Road 17923, Jinan, 250061, China
article info
Article history:
Received 27 October 2019
Received in revised form
23 November 2020
Accepted 17 December 2020
Available online 19 January 2021
Keywords:
Lithium-ion battery
Electrolyte diffusion
Domain decomposition technique
Reduced-order electrochemical model
Battery management system
abstract
The pseudo-two-dimensional (P2D) electrochemical model can give insight into the internal behavior of
lithium-ion batteries, which is of great significance for intelligent battery management. However, the
computational complexity of the P2D model greatly limits its onboard application. This paper devotes to
develop a reduced-order electrochemical model (ROEM) with high fidelity yet low computational cost.
First, for the simplification of the electrolyte diffusion process in batteries, domain decomposition
technique is applied to divide the whole cell sandwich into two computation domains. The polynomial
approximation method is adopted to describe the electrolyte concentration distribution in each domain,
so that the electrolyte diffusion process is represented by two independent state variables. Next,
combining the simplified electrolyte diffusion process and other dynamics in lithium-ion batteries, the
complete ROEM is obtained as a five-state diagonal system. Finally, the prediction accuracy of the ROEM
on electrolyte concentration, electrolyte diffusion overpotential and terminal voltage is verified by
comparing with the P2D model and experimental results. Moreover, the proposed ROEM has an ideal
computing speed for real-time application, which is at least 600 times faster than the rigorous P2D
model.
©2020 Elsevier Ltd. All rights reserved.
1. Introduction
Under the low-carbon economy background, the development
of electric vehicles (EVs) becomes an important way of energy
saving and emission reduction [1,2]. Compared with the traditional
lead-acid and nickel-cadmium batteries, lithium-ion batteries have
become the most favorable choice for automobile manufacturers
due to their high energy density, high power density and long cycle
life [3,4]. To enhance the safety and efficiency of lithium-ion bat-
teries, a battery management system (BMS) is indispensable [5e7].
An accurate yet simple battery model is often embedded in BMSs
for the realization of state estimation, fault diagnosis and so on
[8,9].
Currently, the equivalent circuit models (ECMs) are widely used
in BMSs due to the simplicity and acceptable accuracy. However,
the ECMs can only capture the external behavior (i.e., the current-
voltage characteristics) of the batteries since the model parameters
have no immediate physical meaning. The electrochemical models
(EMs), represented by the pseudo-two-dimensional model, can
give insight into the battery internal behaviors such as the mass
transportation and kinetics reaction [10]. These behaviors reflect
the battery degradation mechanisms and pave foundations for the
battery optimal control, which is the greatest advantage of EMs
over the ECMs [11,12].
Traditionally, the EMs are usually used to study the effect of
design parameters on the battery performance, such as the elec-
trode thickness [13], active material type [14], etc. Although the
EM-based battery management technologies have also been con-
ducted in some literature [15e18], it is hard to realize onboard
application because solving the full-order EM with strong nonlin-
earity is rather challenging for today’s BMSs. The P2D model con-
sists of several coupled nonlinear partial differential equations
(PDEs). In order to reduce the computational complexity, some
researchers employed advanced numerical methods to solve the
P2D model instead of the traditional finite difference method
(FDM), finite element method (FEM), etc. Dao et al. [19] applied
Galerkin projection to the PDEs governing the concentration and
potential of electrolyte. Xu et al. [20] employed the large scale
systems theory on FDM to solve the solid-phase diffusion equa-
tions. Bizeray et al. [21] and Cai et al. [22] discretized the PDEs of
*Corresponding author.
E-mail addresses: [e-mail] (C. Li), [email protected] (N. Cui).
Contents lists available at ScienceDirect
Energy
journal homepage: www.elsevier.com/locate/energy
https://doi.org/10.1016/j.energy.2020.119662
0360-5442/©2020 Elsevier Ltd. All rights reserved.
Energy 221 (2021) 119662

P2D model using orthogonal collocation methods. These tech-
niques can improve the computational efficiency of P2D model to
some extent with acceptable accuracy. However, there are still a
considerable number of equations to be solved to maintain the
integrity of the original model, causing certain calculation and
memory burdens for the vehicular microprocessor.
Analyzing the numerical-method-based simplification tech-
niques, it is clear that the computational burden is ultimately a
matter caused by the model itself. Simplification methods directly
targeting the model structure have been conducted in some re-
searches, among which the single particle (SP) model can be
considered as the most representative one [23]. The SP model de-
scribes each electrode as one spherical particle and neglects the
potential and concentration differences within the electrolyte. Thus
the solid-phase diffusion process in two representative particles
occupies the main computation resource when solving the SP
model. To simplify the solid-phase diffusion, Subramanian et al.
[24] approximated the lithium-ion concentration distribution
along particle radius with a three-parameter polynomial. Han et al.
[25] described the difference between surface and average con-
centrations of the solid particle using several first-order inertia
links. Forman et al. [26] obtained the linearized transfer function of
solid-phase diffusion system based on Pad e approximation.
However, the linearized transfer function is applicable only to low
frequency range. Further, Sabatier et al. [27] approximated the
original transfer function using the fractional order theory, which
can work well in the full frequency range.
While the SP model can meet the demand for fast computing, it
manifests a poor prediction quality under high C-rates [28]. On the
one hand, this is because the SP model neglects the nonuniform
reaction effect within electrodes. To address this issue, Deng et al.
[29] and Li et al. [30] approximated the reaction flux in the thick-
ness direction of the electrode as a parabolic or cubic polynomial,
and the polynomial coefficients can be effectively obtained without
too much increase in computational complexity. On the other hand,
the electrolyte dynamics is not included in the SP model. Thus,
extended SP models with a simplified electrolyte diffusion process
also have drawn much attention. To simplify the electrolyte diffu-
sion, Rahimian et al. [31] approximated the electrolyte concentra-
tion distribution with one quadratic polynomial in the separator
and two cubic polynomials in the positive and negative electrodes.
Marcicki et al. [32] converted the electrolyte diffusion PDE to a
linearized transfer function using Laplace transformation and Pad e
approximation. For further achieving a generalized analytical so-
lution, Yuan et al. [33] isolated the electrolyte diffusion PDEs of
both electrodes by introducing modified boundary conditions and
Nomenclature
aspecific surface area [m
1
]
Aelectrode plate area [m
2
]
brugg Bruggeman number
c
e
electrolyte concentration [mol m
3
]
c
0
e
initial electrolyte concentration [mol m
3
]
c
diff
e
difference between c
e
and c
0
e
[mol m
3
]
c
steady
e
electrolyte concentration at steady state [mol m
3
]
c
s
solid phase concentration [mol m
3
]
c
0
s
initial solid phase concentration [mol m
3
]
c
avg
s
average concentration of solid particle [mol m
3
]
c
max
s
maximum solid phase concentration [mol m
3
]
c
surf
s
concentration at solid particle surface [mol m
3
]
c
diff
s
difference between c
surf
s
and c
avg
s
[mol m
3
]
D
e
electrolyte ionic diffusivity [m
2
s
1
]
D
eff
e
effective electrolyte ionic diffusivity [m
2
s
1
]
D
s
solid phase ionic diffusivity [m
2
s
1
]
FFaraday constant [96487 C mol
1
]
i
0
exchange current density [A m
2
]
I
app
applied current [A]
jpore wall flux [mol m
2
s
1
]
j
avg
average pore wall flux [mol m
2
s
1
]
kkinetics reaction rate constant [A m
2.5
mol
1.5
]
lthickness of porous regions [m]
OCP open circuit potential of solid phase [V]
OCP
surf
open circuit potential at particle surface [V]
OCV
cell
battery open circuit voltage [V]
Q
cell
battery capacity [Ah]
Rideal gas constant [8.314 J mol
1
K
1
]
R
ctct
contact resistance [ U ]
R
s
radius of the spherical solid particle [m]
R
SEI
SEI film resistance [ U m
2
]
SOC
0
initial state of charge for battery
t
0
þ
electrolyte transference number
Tbattery temperature [K]
V
cell
battery terminal voltage [V]
a
a
anodic transfer coefficient
a
c
cathodic transfer coefficient
b electrolyte activity coefficient
ε
e
volume fraction of electrolyte
ε
s
volume fraction of solid phase
h
ctct
contact resistance overpotential [V]
h
e
electrolyte overpotential [V]
h
e;con
electrolyte diffusion overpotential [V]
h
e;ohm
electrolyte ohmic overpotential [V]
h
kin
kinetics reaction overpotential [V]
h
sum
kin
sum reaction overpotential of two electrodes [V]
h
SEI
SEI film ohmic overpotential [V]
h
sum
SEI
sum SEI film overpotential of two electrodes [V]
q lithium-ion stoichiometric number in solid phase
k
e
electrolyte ionic conductivity [S m
1
]
k
eff
e
effective electrolyte ionic conductivity [S m
1
]
k
eff
D
effective ionic diffusion conductivity [S m
1
]
s
s
solid phase conductivity [S m
1
]
s
eff
s
effective solid phase conductivity [S m
1
]
4
e
electrolyte potential [V]
4
s
solid phase potential [V]
Subscripts
isubstitution of n,sep or p
nnegative electrode
n=cnegative-electrode/current-collector interface
ppositive electrode
p=cpositive-electrode/current-collector interface
sep separator
C. Li, N. Cui, C. Wang et al. Energy 221 (2021) 119662
2

the same procedure in Ref. [32] was used to solve the uncoupled
PDEs.
Among the existing simplification techniques for electrolyte
diffusion, the polynomial approximation method can present an
intuitive description of the lithium-ion concentration distribution
through the thickness of the cell sandwich [31]. However, the
polynomials are often coupled together and solved in one
computation domain, i.e., the whole cell sandwich, causing certain
system complexity. Theoretically, a diagonal linear system with
independent state variables is much favorable for control purposes.
Hence to draw a more concise mathematical description for the
electrochemical battery model and facilitate its easy implementa-
tion on BMSs, domain decomposition technique is introduced to
simplify the electrolyte diffusion process for the first time in this
paper. Three main contributions of this study include: (1) The
domain decomposition method divides the whole cell sandwich
into two computation domains by introducing the electrolyte
concentration equilibrium point. Applying polynomial approxi-
mation in each domain, a two-state electrolyte diffusion system is
derived. (2) Based on the simplified electrolyte diffusion, an
isothermal ROEM for lithium-ion batteries is established in com-
bination with other in-cell dynamics, which is a five-state diagonal
system with high computational efficiency. (3) By performing gal-
vanostatic discharge and dynamic tests, the ROEM is proved with
high accuracy in predicting the battery internal and external
behaviors.
The next section desribes the P2D electrochemical model briefly.
Section 3presents the newly proposed simplification procedure of
electrolyte diffusion. Section 4focuses on the formation of the
ROEM, and its performances are validated in Section 5. Section 6is
the conclusion.
2. The rigorous electrochemical model
2.1. Description of P2D model
As illustrated in Fig. 1, the cell sandwich consists of three main
regions, i.e., negative electrode, separator and positive electrode.
Each region is a porous structure whose pores are filled with
polymer electrolyte. Besides, a metal current collector is con-
figurated at the end of each electrode to conduct electrons from
internal or external circuits. For the discharge process shown in
Fig. 1, lithium-ions diffuse from inner to surface of the solid parti-
cles within negative electrode. Then the oxidation kinetics reaction
occurs at the particle surface, and the generated lithium-ions travel
across the separator towards the positive electrode via the ionic
diffusion and conduction in electrolyte. Within the positive elec-
trode, the reduction kinetics reaction occurs at the solid particle
surface and the lithium-ions consecutively diffuse into the particles
until the discharge process ends. Taking the LiCoO2 battery for
example, the kinetics reaction equations at both electrodes during
discharge are shown in Eq. (1), and the similar reverse reaction and
diffusion processes occur when the battery is charged.
Negative electrode :LixC6/6C þxLiþþxe
Positive electrode :Li1xCoO2þxLiþþxe/LiCoO2
(1)
The P2D model treats each electrode as numerous spherical
particles with the same size. It describes the in-cell dynamics with
two dimensions, i.e., the x-dimension and r-dimension in Fig. 1. The
x-dimension denotes the thickness direction of the cell sandwich,
and the r-dimension refers to the radial direction of the spherical
solid particles. Table 1 summaries the governing equations of the
P2D model. Eqs.(2e5) govern the lithium-ion diffusion and charge
conservation in electrolyte and solid phases. Based on Fick’s second
law, the lithium-ions diffuse along the r-axis within the solid par-
ticles at different electrode positions, and the electrolyte diffusion
process takes place along the x-axis. The charge conservation ef-
fects within solid and electrolyte phases occur along the x-axis
according to Ohm’s law. Kinetics reaction and ohmic effect caused
by SEI film obey Eqs.(6) and (7), respectively. The potential balance
equation at solid/electrolyte interface, Eq.(8), acts as a bridge
among the above processes. After solving these internal processes,
the battery terminal voltage is determined by Eq.(9).
2.2. The finite difference method
In this paper, we solve the P2D model with the finite difference
method [34], and take the P2D solutions as reference to verify the
proposed ROEM. As shown in Fig. 2, the negative electrode, sepa-
rator and positive electrode are evenly discretized with 30, 20 and
30 nodes in x-axis, and the solid particles are evenly discretized
with 20 nodes in r-axis.
Taking the lithium-ion diffusion in one single solid particle as an
example, based on the central finite difference, Eq.(4a) is trans-
formed into Eq. (10) at the inner nodes.
Fig. 1. Schematic of the lithium-ion cell sandwich during discharge.
C. Li, N. Cui, C. Wang et al. Energy 221 (2021) 119662
3

dcðiÞ
s
dt ¼Ds cðiþ1Þ
s2cðiÞ sþcði1Þ s
h2þ2
ði1Þh
cðiþ1Þ
scði1Þ s
2h!
¼Ds
h2i2
i1cði1Þ
s2cðiÞ sþi
i1cðiþ1Þ
si¼2;3;…;19 (10)
where h¼R
s
=19 is the node spacing and the superscript of c
s
de-
notes the node number.
Particularly, at the center and surface of the solid particle (i.e.,
the 1st and 20th nodes), the lithium-ion concentrations are
described as Eqs. (11) and (12) according to the boundary condition
of Eq.(4b).
1
2h3cð1Þ
sþ4cð2Þ scð3Þ s¼0 (11)
Ds
2hcð18Þ
s4cð19Þ sþ3cð20Þ s¼j(12)
As a result, one solid-phase diffusion equation is reformulated as
20 differential algebraic equations (DAEs), including 18 ordinary
differential equations (ODEs) for inner nodes and 2 algebraic
equations (AEs) for boundaries. Because the solid-phase diffusion
in r-axis exists everywhere along the x-axis of electrodes, there are
totally 20 (30 þ30) ¼1200 DAEs for solid-phase diffusion,
including 18 (30 þ30) ¼1080 ODEs and 2 (30 þ30) ¼120 AEs.
In the same way, the electrolyte diffusion equation can be trans-
formed into 30 þ20þ30 ¼80 DAEs (74 ODEs and 6 AEs) in x-axis.
Besides, there are 30 þ20þ30 ¼80 AEs for the charge conservation
Table 1
Governing equations of the P2D model for lithium-ion battery.
Governing Equations Boundary Conditions
Lithium-ion diffusion in electrolyte phase, c
e
ðx;tÞ:
ε
e;i
vc
e
vt¼v
vxD
eff
e;i
vc
e
vxþa
i
1t
0
þ
j
where;a
i
¼3ε
s;i
=R
s;i
;D
eff
e;i
¼D
e
ε
brugg
e;i
(2a) 8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
vc
e
vx
x¼0
¼0;vc
e
vx
x¼l
cell
¼0
c
ex¼l
n
¼c
ex¼l
þ
n
;c
ex¼l
nþsep
¼c
ex¼l
þ
nþsep
D
eff
e;n
vc
e
vx
x¼l
n
¼D
eff
e;sep
vc
e
vx
x¼l
þ
n
D
eff
e;sep
vc
e
vx
x¼l
nþsep
¼D
eff
e;p
vc
e
vx
x¼l
þ
nþsep
(2b)
Charge conservation in electrolyte phase, 4
e
ðx;tÞ:v
vx
k
eff
e;i
v4
e
vxþv
vx
k
eff
D;i
vlnc
e
vx¼Fa
i
jwhere; k
eff
e;i
¼ k
e
ε
brugg
e;i
; k
eff
D;i
¼
2RT1t
0
þ
ð1þ
b Þ k
eff
e;i
=F(3a) 8
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
:
v4
e
vx
x¼0
¼0;v4
e
vx
x¼l
cell
¼0
4
ex¼l
n
¼4
ex¼l
þ
n
;4
ex¼l
nþsep
¼4
ex¼l
þ
nþsep
k
eff
e;n
v4
e
vx
x¼l
n
¼ k
eff
e;sep
v4
e
vx
x¼l
þ
n
k
eff
e;sep
v4
e
vx
x¼l
nþsep
¼ k
eff
e;p
v4
e
vx
x¼l
þ
nþsep
(3b)
Lithium-ion diffusion in solid phase, c
s
ðx;r;tÞ:vc
s
vt¼1
r
2
v
vrD
s;i
r
2
vc
s
vr(4a) 8
>
>
>
<
>
>
>
:
vc
s
vr
r¼0
¼0
D
s;i
vc
s
vr
r¼R
s;i
¼j
(4b)
Charge conservation in solid phase, 4
s
ðx;tÞ:
v
vx
s
eff
s;i
v
vx4
s
¼Fa
i
jwhere; s
eff
s;i
¼ s
s;i
ε
s;i
v
vx
s
eff
s;i
v
vx4
s
¼Fa
i
jwhere; s
eff
s;i
¼ s
s;i
ε
s;i
(5a) 8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
v4
s
vx
x¼l
n
¼0;v4
s
vx
x¼l
nþsep
¼0
s
eff
s;n
v4
s
vx
x¼0
¼I
app
A
s
eff
s;p
v4
s
vx
x¼l
cell
¼I
app
A
(5b)
Butler-Volmer kinetics reaction, jðx;tÞ
h
kin
ðx;tÞ:
j¼i
0
Fexp
a
a;i
F
RT
h
kin
exp
a
c;i
F
RT
h
kin
where;i
0
¼k
i
c
a
a;i
e
c
max
s;i
c
surf
s
a
a;i
c
surf
s
a
c;i
(6)
Solid/Electrolyte interface film ohmic effect:
h
SEI
¼R
SEI
Fj (7)
Solid/Electrolyte interface potential balance: 4
s
4
e
¼OCP
surf
þ h
kin
þ h
SEI
where;OCP
surf
¼OCPc
surf
s
=c
max
s;i
(8)
Battery terminal voltage, V
cell
ðtÞ:V
cell
¼4
sx¼l
cell
4
sx¼0
R
ctct
I
app
(9)
Fig. 2. Schematic illustration of the nodes.
C. Li, N. Cui, C. Wang et al. Energy 221 (2021) 119662
4

in electrolyte, 30 þ30 ¼60 AEs for charge conservation in elec-
trodes, and 30 þ30 ¼60 AEs for kinetics reactions. Eventually, for
the main physical processes in the P2D model, 1154 ODEs and 326
AEs need to be calculated at each time step.
3. Simplified electrolyte diffusion process
This section presents the order-reduction of the electrolyte
diffusion process via domain decomposition and polynomial
approximation methods, which is the main contribution of this
paper. The polynomial approximation method deploys polynomials
to describe the electrolyte concentration distributions in the
thickness direction of negative electrode, separator and positive
electrode. The polynomial coefficients can be obtained by itera-
tively solving the equations established through the boundary
conditions and other physical constraints in P2D model. In previous
studies, however, the coefficients of different polynomials are
coupled together and solved in the computation domain of the
whole cell sandwich. To describe the electrolyte diffusion process in
a more concise way, the domain decomposition technique is
adopted here to decouple the relationships between the approxi-
mate polynomials.
Three fundamental assumptions are given as follows:
Assumption 1: the effect of solid and electrolyte concentrations
on the dynamics parameters (D
e
,D
s
and k
e
) is negligible, and their
values keep constant during operation.
Assumption 2: the pore wall flux of lithium-ion at the solid
particle surface is uniform within each electrode, thus the average
pore wall flux is given as,
8
>
>
>
<
>
>
>
:
javg
n¼Iapp
FAanln
javg
p¼ Iapp
FAaplp
(13)
where a
i
ði¼n;pÞis the specific surface area of electrode and
follows,
ai¼3εs;i
Rs;i
i¼n;p(14)
Assumption 3: the electrolyte concentration distribution pro-
files are approximated as polynomials within the porous regions,
i.e., negative electrode, separator and positive electrode.
3.1. Computation domain decomposition
The red solid line in Fig. 3 denotes a typical electrolyte con-
centration distribution profile in the thickness direction of cell
sandwich during discharge. At any given time, there always exists a
point, named electrolyte concentration equilibrium point, whose
concentration equals the initial value c
0
e
. Theoretically, the location
of equilibrium point changes over time but not much, thus another
assumption is introduced here:
Assumption 4: the position of the electrolyte concentration
equilibrium point remains invariable over time, and is assumed to
be the same as that under steady state.
Based on Assumption 2, the electrolyte concentration would
reach steady state after a period of galvanostatic discharge [35,36].
At that time, the effects of kinetics reaction and electrolyte diffusion
are in dynamic balance. All transient terms in Eq.(2a) equal zero
under steady state, and the Eq.(2a) becomes,
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
0¼Deff
e;n
v2csteady
e
vx2þan1t0
þjavg
nx¼½0;ln
0¼Deff
e;sep
v2csteady
e
vx2x¼ln;lnþsep
0¼Deff
e;p
v2csteady
e
vx2þap1t0
þjavg
px¼lnþsep;lcell
(15)
where D
e;i
ði¼n;sep;pÞis the effective electrolyte ionic diffusivity
at the porous regions and follows,
Deff
e;i¼Deεbrugg e;ii¼n;sep;p(16)
By doubly integrating Eq. (15), the electrolyte concentration
distribution at steady state yields,
csteady
e¼8
>
>
>
<
>
>
>
:
M1x2
nþM2xnþM3þc0 exn2½0;ln
M4xsep þM5þc0
exsep20;lsep
M6x2
pþM7xpþM8þc0 exp20;lp
(17)
where M
k
ðk¼1;2;8Þis the coefficient to be determined. x
i
ði¼n;sep;pÞdenotes the transformed coordinate for the porous
regions and follows,
8
<
:
xn¼xx2½0;ln
xsep ¼xlnx2ln;lnþsep
xp¼lcell xx2lnþsep;lcell
According to Eq. (17), the boundary conditions in Eq.(2b) at
steady state follows,
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:
M2¼0
M7¼0
M1l2
nþM2lnþM3¼M5
M4lsep þM5¼M8
2M1lnþM2¼M4
M4¼2M6lpM7
(18)
Besides, at the steady state, the material balance of lithium-ion
in the electrode and the whole cell sandwich follows Eqs. (19) and
(20) respectively.
Fig. 3. Electrolyte concentration equilibrium point and decomposed computation
domains.
C. Li, N. Cui, C. Wang et al. Energy 221 (2021) 119662
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