Reduced-order electrochemical model for lithium-ion battery with

Energy 221 (2021) 119662
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Energy
journal homepage: www.elsevier.com/locate/energy
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
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 27 October 2019
Received in revised form
23 November 2020
Accepted 17 December 2020
Available online 19 January 2021
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.
Keywords:
Lithium-ion battery
Electrolyte diffusion
Domain decomposition technique
Reduced-order electrochemical model
Battery management system
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 batteries, 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 currentvoltage characteristics) of the batteries since the model parameters
have no immediate physical meaning. The electrochemical models
* Corresponding author.
E-mail addresses: [email protected] (C. Li), [email protected] (N. Cui).
https://doi.org/10.1016/j.energy.2020.119662
0360-5442/© 2020 Elsevier Ltd. All rights reserved.
(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 electrode thickness [13], active material type [14], etc. Although the
EM-based battery management technologies have also been conducted in some literature [15e18], it is hard to realize onboard
application because solving the full-order EM with strong nonlinearity is rather challenging for today’s BMSs. The P2D model consists 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 equations. Bizeray et al. [21] and Cai et al. [22] discretized the PDEs of
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
SOC0
0
tþ
T
Vcell
Nomenclature
a
A
brugg
ce
c0e
diff
ce
csteady
e
cs
c0s
specific surface area [m1 ]
electrode plate area [m2 ]
Bruggeman number
electrolyte concentration [mol m3]
initial electrolyte concentration [mol m3]
aa
ac
b
εe
εs
difference between ce and c0e [mol m3]
electrolyte concentration at steady state [mol m
solid phase concentration [mol m3]
initial solid phase concentration [mol m3]
3
cavg
s
cmax
s
average concentration of solid particle [mol m3]
maximum solid phase concentration [mol m3]
cs
surf
concentration at solid particle surface [mol m3]
cdiff
s
De
difference between csurf
and cavg
[mol m3]
s
s
2 1
electrolyte ionic diffusivity [m s ]
Deff
e
Ds
F
i0
Iapp
j
javg
k
l
OCP
OCPsurf
OCVcell
Qcell
R
Rctct
Rs
RSEI
effective electrolyte ionic diffusivity [m2 s1]
solid phase ionic diffusivity [m2 s1]
Faraday constant [96487 C mol1 ]
exchange current density [A m2]
applied current [A]
pore wall flux [mol m2 s1]
average pore wall flux [mol m2 s1]
kinetics reaction rate constant [A m2.5 mol1.5]
thickness of porous regions [m]
open circuit potential of solid phase [V]
open circuit potential at particle surface [V]
battery open circuit voltage [V]
battery capacity [Ah]
ideal gas constant [8.314 J mol1 K1]
contact resistance [U]
radius of the spherical solid particle [m]
SEI film resistance [U m2]
]
hctct
he
he;con
he;ohm
hkin
hsum
kin
hSEI
hsum
SEI
q
ke
keff
e
keff
D
ss
seff
s
initial state of charge for battery
electrolyte transference number
battery temperature [K]
battery terminal voltage [V]
anodic transfer coefficient
cathodic transfer coefficient
electrolyte activity coefficient
volume fraction of electrolyte
volume fraction of solid phase
contact resistance overpotential [V]
electrolyte overpotential [V]
electrolyte diffusion overpotential [V]
electrolyte ohmic overpotential [V]
kinetics reaction overpotential [V]
sum reaction overpotential of two electrodes [V]
SEI film ohmic overpotential [V]
sum SEI film overpotential of two electrodes [V]
lithium-ion stoichiometric number in solid phase
electrolyte ionic conductivity [S m1]
effective electrolyte ionic conductivity [S m1]
effective ionic diffusion conductivity [S m1]
solid phase conductivity [S m1]
4e
4s
effective solid phase conductivity [S m1]
electrolyte potential [V]
solid phase potential [V]
Subscripts
i
n
n=c
p
p=c
sep
substitution of n, sep or p
negative electrode
negative-electrode/current-collector interface
positive electrode
positive-electrode/current-collector interface
separator
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 thickness 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 diffusion, Rahimian et al. [31] approximated the electrolyte concentration 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 Pade
approximation. For further achieving a generalized analytical solution, Yuan et al. [33] isolated the electrolyte diffusion PDEs of
both electrodes by introducing modified boundary conditions and
P2D model using orthogonal collocation methods. These techniques 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 techniques, 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 researches, among which the single particle (SP) model can be
considered as the most representative one [23]. The SP model describes 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 concentrations of the solid particle using several first-order inertia
links. Forman et al. [26] obtained the linearized transfer function of
approximation.
solid-phase diffusion system based on Pade
2
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
the conclusion.
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 implementation 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 approximation 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 combination with other in-cell dynamics, which is a five-state diagonal
system with high computational efficiency. (3) By performing galvanostatic 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 3 presents the newly proposed simplification procedure of
electrolyte diffusion. Section 4 focuses on the formation of the
ROEM, and its performances are validated in Section 5. Section 6 is
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 configurated 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 particles 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 electrode, 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 : Lix C6 /6C þ xLiþ þ xe
Positive electrode : Li1x CoO2 þ xLiþ þ xe /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 particles at different electrode positions, and the electrolyte diffusion
process takes place along the x-axis. The charge conservation effects 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, separator 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 transformed into Eq. (10) at the inner nodes.
Fig. 1. Schematic of the lithium-ion cell sandwich during discharge.
3
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
Table 1
Governing equations of the P2D model for lithium-ion battery.
Governing Equations
Boundary Conditions
Lithium-ion diffusion in electrolyte phase, ce ðx; tÞ:
vce
v
vce
0
¼
εe;i
j
Deff
þ ai 1 tþ
e;i
vx
vt
vx
where; ai ¼ 3εs;i =Rs;i ;
Charge conservation in electrolyte phase, 4e ðx;tÞ:
0 ð1 þ bÞkeff =F(3a)
2RT 1 tþ
e;i
Deff
e;i
¼
8 vc
vce
e
>
>
> vx x¼0 ¼ 0; vx x¼l ¼ 0
>
cell
>
>
>
>
>
>
ce x¼ln ¼ ce x¼lþn ; ce x¼lnþsep ¼ ce x¼lþnþsep
>
>
<
eff vce
eff vce
>
>
De;n
¼ De;sep
>
>
vx x¼ln
vx x¼lþn
>
>
>
>
>
>
vc
vce
>
e
eff
eff
>
: De;sep
¼ De;p
vx x¼lnþsep
vx x¼lþnþsep
(2a)
De εbrugg
e;i
v
v4
v
vlnce
keff e þ
keff
¼ ke εbrugg
;
¼ Fai jwhere; keff
e;i
e;i
vx e;i vx
vx D;i vx
(2b)
keff
¼ 8
D;i
v4e
v4e
>
¼ 0;
¼0
>
>
vx x¼0
vx x¼lcell
>
>
>
>
>
4
¼ 4ex¼lþ ;
4e x¼lnþsep ¼ 4e x¼lþnþsep
>
>
n
> e x¼ln
<
v4e
v4e
>
keff
¼ keff
>
e;n
e;sep
>
>
vx x¼ln
vx x¼lþn
>
>
>
>
>
>
v4
> eff
eff v4
:
ke;sep e ¼ ke;p e þ
vx x¼lnþsep
vx x¼lnþsep
vcs
1 v
vcs
¼ 2
Ds;i r2
(4a)
Lithium-ion diffusion in solid phase, cs ðx; r; tÞ:
vt
vr
r vr
Charge conservation in solid phase, 4s ðx; tÞ:
v
v
v
v
eff
eff
4 ¼ Fai jwhere; ss;i ¼ ss;i εs;i
4 ¼ Fai jwhere; ss;i ¼ ss;i εs;i (5a)
seff
seff
s;i vx s
s;i vx s
vx
vx
(3b)
8
vcs
>
>
¼0
>
<
vr r¼0
>
vc
s
>
>
¼ j
: Ds;i
vr r¼Rs;i
(4b)
8 v4
v4s
s
>
>
> vx x¼ln ¼ 0; vx x¼lnþsep ¼ 0
>
>
>
>
>
<
Iapp
v4s
¼
seff
s;n
>
vx x¼0
A
>
>
>
>
>
>
Iapp
>
: seff v4s
¼
s;p
vx x¼lcell
A
(5b)
aa;i F
a F
hkin exp c;i hkin
exp
RT
RT
(6)
Butler-Volmer kinetics reaction, jðx; tÞ hkin ðx; tÞ:
ac;i
a
surf aa;i
c
where; i0 ¼ ki ce a;i cmax
csurf
s
s
s;i
j¼
i0
F
Solid/Electrolyte interface film ohmic effect:hSEI ¼ RSEI Fj (7)
OCPsurf ¼ OCP csurf
=cmax
(8)
s
s;i
Solid/Electrolyte interface potential balance: 4s 4e ¼ OCPsurf þ hkin þ hSEI where;
Battery terminal voltage, Vcell ðtÞ:Vcell ¼ 4sx¼lcell 4sx¼0 Rctct Iapp (9)
ðiÞ
dcs
ðiþ1Þ
cs
ðiÞ
2cs
h2
ði1Þ
cs
ðiþ1Þ
cs
ði1Þ
cs
þ
2
¼ Ds
þ
ði 1Þh
2h
Ds i 2 ði1Þ
i ðiþ1Þ
ðiÞ
cs
c
¼ 2
i ¼ 2; 3; …; 19
2cs þ
i1 s
h i1
dt
totally 20 (30 þ 30) ¼ 1200 DAEs for solid-phase diffusion,
including 18 (30 þ 30) ¼ 1080 ODEs and 2 (30 þ 30) ¼ 120 AEs.
In the same way, the electrolyte diffusion equation can be transformed into 30 þ 20þ30 ¼ 80 DAEs (74 ODEs and 6 AEs) in x-axis.
Besides, there are 30 þ 20þ30 ¼ 80 AEs for the charge conservation
!
(10)
where h ¼ Rs =19 is the node spacing and the superscript of cs denotes 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 ð1Þ
ð2Þ
ð3Þ
3cs þ 4cs cs
¼0
2h
(11)
Ds ð18Þ
ð19Þ
ð20Þ
¼ j
cs 4cs þ 3cs
2h
(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
Fig. 2. Schematic illustration of the nodes.
4
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
in electrolyte, 30 þ 30 ¼ 60 AEs for charge conservation in electrodes, and 30 þ 30 ¼ 60 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.
3.1. Computation domain decomposition
The red solid line in Fig. 3 denotes a typical electrolyte concentration 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 c0e . 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,
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 iteratively 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 approximate polynomials.
Three fundamental assumptions are given as follows:
Assumption 1: the effect of solid and electrolyte concentrations
on the dynamics parameters (De , Ds and ke ) 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,
where De;i ði ¼ n; sep; pÞ is the effective electrolyte ionic diffusivity
at the porous regions and follows,
8
Iapp
avg
>
>
¼
j
>
< n
FAan ln
De;i ¼ De εe;i
>
Iapp
>
>
: javg
p ¼
FAap lp
8
2 steady
eff v ce
0 avg
>
>
1
t
jn
þ
a
0 ¼ De;n
n
>
þ
>
>
vx2
>
>
>
>
<
steady
v2 ce
x ¼ ln ; lnþsep
0 ¼ Deff
e;sep
>
>
vx2
>
>
>
>
>
steady
>
>
v2 c e
:
0 avg
0 ¼ Deff
jp
þ ap 1 tþ
e;p
2
vx
eff
(13)
i ¼ n; p
(15)
x ¼ lnþsep ; lcell
i ¼ n; sep; p
(16)
By doubly integrating Eq. (15), the electrolyte concentration
distribution at steady state yields,
8
>
>
M x2 þ M2 xn þ M3 þ c0e
>
< 1 n
steady
ce
¼ M4 xsep þ M5 þ c0e
>
>
>
: M x2 þ M x þ M þ c0
where ai ði ¼ n; pÞ is the specific surface area of electrode and
follows,
3εs;i
ai ¼
Rs;i
brugg
x ¼ ½0; ln (14)
6 p
Assumption 3: the electrolyte concentration distribution profiles are approximated as polynomials within the porous regions,
i.e., negative electrode, separator and positive electrode.
7 p
8
e
xn 2½0; ln xsep 2 0; lsep
(17)
xp 2 0; lp
where Mk ðk ¼ 1; 2; 8Þ is the coefficient to be determined. xi
ði ¼ n; sep; pÞ denotes the transformed coordinate for the porous
regions and follows,
8
< xn ¼ x
xsep ¼ x ln
:
xp ¼ lcell x
x2½0; ln x2 ln ; lnþsep
x2 lnþsep ; lcell
According to Eq. (17), the boundary conditions in Eq.(2b) at
steady state follows,
8
>
>
>
> M2 ¼ 0
>
>
>
>
> M7 ¼ 0
>
>
>
< M l2 þ M l þ M ¼ M
5
1 n
2 n
3
> M4 lsep þ M5 ¼ M8
>
>
>
>
>
> 2M1 ln þ M2 ¼ M4
>
>
>
>
> M4 ¼ 2M6 lp 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.
5
C. Li, N. Cui, C. Wang et al.
0
1 tþ
lð
cell
I
app
FA
Energy 221 (2021) 119662
¼ Deff
e;sep M4
εe;i csteady
ðxÞdx ¼
e
0
where xequ and zequ are the distances of the equilibrium point away
from the negative and positive electrodes respectively, as marked in
Fig. 3.
As shown in Fig. 3, the equilibrium point decomposes the cell
sandwich into two computation domains, i.e., domain 1 and
domain 2. The x-axis is the coordinate axis for domain 1 that starts
at the interface between the negative electrode and current collector (named neg/cc interface). The z-axis is the transformed coordinate axis for domain 2, which is opposite to the x-axis and
starts at the interface between the positive electrode and current
collector (named pos/cc interface). Besides the original boundary
conditions of Eq.(2b), the invariable electrolyte concentration at
equilibrium point provides an additional boundary condition,
(19)
lcell
ð
εe;i c0e dx
(20)
0
Solving the equations of Eqs. (18), (19) and (20), the analytic
solution of Mk ðk ¼ 1; 2; 8Þ in Eq. (17) is obtained as,
0
8
1 tþ
1
Iapp
¼
M
>
1
>
brugg
>
2εe;n ln De FA
>
>
>
>
>
>
>
M2 ¼ 0
>
>
>
>
>
9
8
>
>
εe;sep ln lsep þ εe;p ln lp εe;p lsep lp
>
>
>
>
>
>
>
þ
þ
>
>
>
>
>
brugg
brugg
>
>
>
>
2ε
ε
>
>
>
e;n
e;sep
=
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
1brugg 2
1brugg 2
1brugg 2 >
>
>
>
>
>
>
ε
l
þ
3ε
l
þ
2ε
l
>
n
e;sep
e;p
>
n
sep
p>
;
:
>
0
>
>
1 tþ
>
6
>
Iapp
¼
M
>
3
>
l
þ
ε
l
þ
ε
l
D
FA
ε
>
e;n n
e;sep sep
e;p p
e
>
>
>
<
0
1 1 tþ
>
Iapp
M4 ¼ brugg
>
>
>
εe;sep De FA
>
>
>
>
>
!
>
>
0
>
1 tþ
ln
>
>
>
Iapp
M
¼
þ
m
5
3
>
brugg
>
De FA
>
2εe;n
>
>
>
>
>
>
>
0
>
1 tþ
1
>
>
Iapp
M6 ¼ brugg
>
>
>
D
FA
>
2εe;p lp e
>
>
>
>
>
>
>
M7 ¼ 0
>
>
>
>
!
>
>
0
>
>
1 tþ
lsep
lp
ln
:
Iapp
M8 ¼ brugg brugg brugg þ m3
De FA
2εe;n
εe;sep
2εe;p
ce
After decomposing the computation domain, the polynomial
approximation method can be applied to each domain individually.
Taking the computation domain 1 as an example, quadratic polynomial and linear polynomial are applied for describing the concentration distributions in the negative electrode and the partial
separator, respectively,
8
< a ðtÞx2 þ b ðtÞx þ c ðtÞ þ c0 x2½0; l n
1
1
1
e
ce ðx; tÞ ¼
: d1 ðtÞðx ln Þ þ e1 ðtÞ þ c0 x2 ln ; ln þ xequ
where a; b; c; d; e are the time-varying polynomial coefficients and
the subscript 1 represents the domain 1.
There are five unknown coefficients in Eq. (25), and thus to
obtain the electrolyte concentration distribution in domain 1, five
equations need to be constructed. Three original boundary conditions for domain 1 in Eq.(2b) and the newly introduced boundary
condition Eq. (24) can provide four algebraic equations,
8
>
>
b1 ðtÞ ¼ 0
>
>
>
>
< 2Deff l a ðtÞ ¼ Deff d ðtÞ
e;n n 1
e;sep 1
2
>
> ln a1 ðtÞ þ c1 ðtÞ ¼ e1 ðtÞ
>
>
>
>
: x d ðtÞ þ e ðtÞ ¼ 0
(22)
Thus, the position of the concentration equilibrium point is
determined as,
ln þx
ð equ
diff
Qe;1 ðtÞ ¼
¼ εe;n
(25)
e
According to Assumption 4 and Eq. (17), the concentration
equilibrium point should meet the following equation in the
separator region,
8
>
< xequ ¼ M5
M4
>
:
zequ ¼ lsep xequ
(24)
3.2. Polynomial approximation solution for electrolyte
concentration
(21)
M4 xequ þ M5 þ c0e ¼ c0e
0
x¼ln þxequ ¼ce jz¼lp þzequ ¼ce
equ 1
(26)
1
The amount variation of lithium-ion in domain 1 for per unit
diff
electrode area, Qe;1
, can be calculated by the x-dimensional inte-
(23)
gration as follows,
εe;i ce ðx; tÞ c0e dx
0
1
1
1
a1 ðtÞl3n þ b1 ðtÞl2n þ c1 ðtÞln þ εe;sep d1 ðtÞx2equ þ e1 ðtÞxequ
3
2
2
6
(27)
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
8
< a ðtÞz2 þ b ðtÞz þ c ðtÞ þ c0 z2 0; lp
2
2
2
e
ce ðz; tÞ ¼
: d2 ðtÞ z lp þ e2 ðtÞ þ c0 z2 lp ; lp þ zequ
diff
Besides, the Qe;1 can also be calculated by the time-dimensional
integration as follows,
(31)
e
diff
Qe;1 ðtÞ ¼
ðt eff vce
0 Iapp
þ De;sep
1 tþ
dt
FA
vt x¼ln þxequ
where the subscript 2 represents the domain 2.
Using the same analysis process applied to domain 1, the polynomial coefficients for domain 2 in Eq. (31) can be expressed as,
(28)
0
8
eff
De;sep Q2
>
>
ðtÞ
¼
c2 ðtÞ
a
>
> 2
eff
>
>
2De;p lp
>
>
>
>
>
>
>
b ðtÞ ¼ 0
>
>
< 2
where vce =vtx¼ln þxequ ¼ d1 ðtÞ from Eq. (25). Eq. (28) means that the
amount variation of lithium-ion in domain 1 is caused by two
physical processes: the lithium-ion insertion or extraction process
at the solid/electrolyte interface, and the lithium-ion diffusion
process at the interface between domain 1 and domain 2.
The material balance of lithium-ion in domain 1 contributes as
one physical constraint for Eq. (25) by combining Eqs. (27) and (28),
and its differential form is given as,
εe;n
eff
0
De;sep
1 tþ
>
>
>
c2 ðtÞ Iapp ðtÞ
c_2 ðtÞ ¼
>
>
P2
FAP2 Q2
>
>
>
>
>
>
> d2 ðtÞ ¼ Q2 c2 ðtÞ
>
>
>
:
e2 ðtÞ ¼ zequ Q2 c2 ðtÞ
!
!
x2equ _
l3n
l2
a_ 1 ðtÞ þ n b_ 1 ðtÞ þ ln c_1 ðtÞ þ εe;sep
d1 ðtÞ þ xequ e_1 ðtÞ
3
2
2
I ðtÞ
app
0
¼ 1 tþ
þ Deff
e;sep d1 ðtÞ
FA
(29)
where
!
8
eff
>
De;sep 2
1
>
2
>
¼
ε
l
þ
l
z
P
> 2
e;p
p equ εe;sep zequ
p
>
>
2
<
3Deff
e;p
>
>
>
>
>
>
: Q2 ¼ Combing Eqs. (26) and (29), the polynomial coefficients for
domain 1 can be deduced as follows,
8
eff
De;sep Q1
>
>
ðtÞ
¼
c1 ðtÞ
a
>
> 1
eff
>
>
2De;n ln
>
>
>
>
>
>
>
b1 ðtÞ ¼ 0
>
>
<
eff
0
De;sep
1 tþ
>
>
>
c1 ðtÞ þ
Iapp ðtÞ
c_1 ðtÞ ¼
>
>
P1
FAP1 Q1
>
>
>
>
>
>
> d1 ðtÞ ¼ Q1 c1 ðtÞ
>
>
>
:
e1 ðtÞ ¼ xequ Q1 c1 ðtÞ
(32)
2Deff
e;p
Deff
e;sep lp
þ 2Deff
e;p zequ
3.3. State-space representation for electrolyte diffusion
and that at
The concentration variation at neg/cc interface cdiff
e;n=c
(30)
diff
pos/cc interface ce;p=c can be derived from Eqs. (25) and (31),
respectively.
8
>
< cdiff ðtÞ ¼ ce ðx ¼ 0; tÞ c0e ¼ c1 ðtÞ
e;n=c
>
: cdiff
ðtÞ ¼ ce ðz ¼ 0; tÞ c0e ¼ c2 ðtÞ
e;p=c
where
diff
(33)
diff
Taking ce;n=c and ce;p=c as state variables, the state-space representation describing the electrolyte diffusion process is obtained by
assembling Eqs. (30), (32) and (33).
!
8
>
Deff
1
e;sep 2
>
>
l þ ln xequ εe;sep x2equ
P1 ¼ εe;n
>
>
eff n
>
2
<
3De;n
x_ e ðtÞ ¼ Ae xe ðtÞ þ Be ue ðtÞ
Coef e ðtÞ ¼ Ce xe ðtÞ þ De ue ðtÞ
(34)
>
eff
>
>
2De;n
>
>
>
: Q1 ¼ eff
De;sep ln þ 2Deff
e;n xequ
iT
h
diff
where xe ¼ cdiff
is state vector and ue ¼ Iapp is input
e;n=c ; ce;p=c
Similarly, the approximate polynomials for concentration distribution in domain 2 have the form as,
variable. Coef e ¼ ½ a1 ; c1 ; d1 ; e1 ; a2 ; c2 ; d2 ; e2 T is output vector. The
parameter matrices, Ae , Be , Ce and De , are,
7
C. Li, N. Cui, C. Wang et al.
2
6
1
6 P1
6
Ae ¼ Deff
e;sep 6
4
0
Energy 221 (2021) 119662
2 Deff Q
e;sep 1
6 2Deff l
6
e;n n
6
6
61
6
6
6
6Q
6 1
2
3
3
6
1
6
6
0 7
6
7
6 xequ Q1
0 6 P1 Q1 7
7
1
t
þ
6
7; B e ¼
7; Ce ¼ 6
6
6
7
7
6
FA 4
1 5
1 5
6
60
P2
P2 Q2
6
6
6
6
6
60
6
6
6
60
4
0
3
0
7
7
7
7
7
0
7
7
7
7
0
7
7
7
7
0
7
7
7; D e ¼ 0
7
eff
De;sep Q2 7
7
7
7
2Deff
l
e;p p 7
7
7
7
1
7
7
7
7
Q2
5
zequ Q2
diff
dcs;i ðtÞ
It is worth noting that the electrolyte diffusion process is
approximated as a diagonal system with only two independent
state variables eventually. Compared with the FDM in Section 2, the
number of ODEs for electrolyte diffusion process is reduced from 74
to 2.
dt
¼ 30Ds;i
R2s;i
cdiff
ðtÞ s;i
6 avg
j ðtÞ
Rs;i i
(38)
During battery operation, the lithium-ions are extracted from
one electrode and inserted into the other electrode with the same
amount. The amount variation of lithium-ion in electrode determines the state of charge (SOC) for battery, and their relationship
follows,
8
0
0
< Aln εs;n cavg
s;n cs;n ¼ 3600 SOC SOC Qcell =F
avg
0
: Al ε
0
p s;p cs;p cs;p ¼ 3600 SOC SOC Qcell =F
4. Reduced-order electrochemical model
In this section, other physical processes inside lithium-ion batteries, including the solid-phase diffusion, kinetics reaction and
ohmic effect, are combined with the simplified electrolyte process
to establish the complete ROEM.
(39)
where c0s;i ði ¼ n; pÞ is the initial lithium-ion concentration in electrode. SOC0 denotes the initial state of charge and the battery SOC
follows,
4.1. Open circuit voltage
_
SOCðtÞ
¼ Based on Assumption 2, the solid-phase diffusion is no longer an
x-dependent process and each electrode can be represented by a
single spherical particle. To simplify the solid-phase diffusion, the
three-parameter parabolic approximation method proposed in
Ref. [24] is adopted here. The average lithium-ion concentration
avg
surf
cs;i and surface lithium-ion concentration cs;i
dt
.
.
surf
surf
cmax
OCPn cs;n
cmax
OCVcell ¼ OCPp cs;p
s;p
s;n
of the particle are
diff
dt
þ
30Ds;i
surf
(35)
R2s;i
number at the surface of the electrode particle. OCPi ði ¼ n; pÞ is the
open circuit potential of the electrode as a function of the lithiumion stoichiometric number.
Substituting Eqs. (36) and (39) into Eq. (41), the OCV is written
as,
(36)
avg
cdiff
ðtÞ
s;i
(41)
where cs;i =cmax
s;i ði ¼ n; pÞ represents the lithium-ion stoichiometric
javg ðtÞ
þ3 i
¼0
Rs;i
ðtÞ ¼ csurf
ðtÞ cavg
ðtÞ
cdiff
s;i
s;i
s;i
dcs;i ðtÞ
(40)
The open circuit voltage is defined as the potential difference
between the positive and negative electrodes. Thus the OCV is
obtained as,
derived as,
avg
dcs;i ðtÞ
Iapp ðtÞ
3600Qcell
Rs;i dji ðtÞ
6 avg
¼0
þ
j þ
Rs;i i
35Ds;i
dt
(37)
OCVcell ¼ OCP
p
The response caused by the variation of cdiff
to javg
is relatively
s;i
i
slow, thus the differential term of
And Eq. (37) is reduced as,
javg
i
c0s;p þ cdiff
s;p
OCP
in Eq. (37) can be neglected.
n
8
cmax
s;p
c0s;n þ cdiff
s;n
cmax
s;n
þ
!
3600 SOC0 SOC Qcell
FAlp εs;p cmax
s;p
!
3600 SOC0 SOC Qcell
FAln εs;n cmax
s;n
(42)
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
diff
diff
The SOC, cs;n and cs;p in Eq. (42) are time-varying variables.
According to Eqs. (13), (38) and (40), the state equation for the OCV
system is given as,
x_ OCV ðtÞ ¼ AOCV xOCV ðtÞ þ BOCV uOCV ðtÞ
(43)
iT
h
diff
where xOCV ¼ SOC; cdiff
is state vector, uOCV ¼ Iapp is input
s;n ; cs;p
variable, and the parameter matrices of the three-state OCV system
are,
20
6
6
60
6
AOCV ¼ 6
6
6
4
0
0
0
30Ds;n
2
Rs;n
0
0
3
7
7
7
7
7; BOCV
7
7
30Ds;p 5
R2s;p
2
3
1
6 3600Q 7
6
cell 7
6
7
6
7
2
6
7
¼6
7
6 FAln εs;n 7
6
7
6
7
2
4
5
FAlp εs;p
Fig. 4. Block diagram of the proposed ROEM.
4.2. Electrolyte overpotential
0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
8
!
u
>
avg
u Fjavg 2
>
>
Fj
2RT
Bt
C
>
i
>
þ1 þ i A
>
< hkin;i ¼ F ln@
2i0;i
2i0;i
>
>
>
>
0:5 >
>
surf 0:5
surf 0:5
: i ¼ k c0
cmax
cs;i
0;i
i
e
s;i cs;i
With Assumption 2, the electrolyte potential governing equation Eq.(3a) is rewritten as,
The sum kinetics reaction overpotential of the two electrodes is
given as,
v
v4e
v 2RT eff vlnce
avg
0
1
t
ð1
þ
¼ Fai ji
keff
b
Þ
k
þ
e;i vx
vx e;i vx
vx
F
hsum
kin ¼ hkin;p hkin;n
where keff
ði ¼ n; sep; pÞ is the effective electrolyte ionic conduce;i
hsum
SEI ¼ hSEI;p hSEI;n ¼ tivity and follows,
(49)
4.4. Terminal voltage
Based on the above calculating results of OCV and different
overpotentials, the battery terminal voltage is given as,
sum
Vcell ¼ OCVcell þ he þ hsum
kin þ hSEI þ hctct
he ¼ 4e jx¼lcell 4e jx¼0
!
ce;p=c
2RT 0
1 tþ
ð1 þ bÞln
F
ce;n=c
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
(50)
where hctct is the overpotential caused by contact resistance and
follows,
hctct ¼ Rctct Iapp
he;con
!
Iapp
lsep
lp
ln
eff eff eff
þ
A
2ke;n ke;sep 2ke;p
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Rs;p RSEI;p Rs;n RSEI;n Iapp
þ
εs;p lp
εs;n ln
3A
(45)
The electrolyte potential 4e as a function of x-coordinate can be
derived by analytically solving Eq. (44) [37]. Afterward, the electrolyte overpotential he is calculated as the electrolyte potential
difference between pos/cc and neg/cc interfaces,
¼
(48)
Substituting Eq. (13) into Eq.(7), the sum SEI film overpotential
of the two electrodes is written as,
(44)
brugg
keff
¼ ke εe;i
i ¼ n; sep; p
e;i
(47)
(46)
he;ohm
Eq. (46) means that the electrolyte overpotential consists of two
parts: he;con caused by electrolyte diffusion process, and he;ohm
caused by the ohmic effect of electrolyte.
4.3. Kinetics reaction overpotential and SEI film overpotential
The kinetics reaction is governed by the Butler-Volmer equation.
The transfer coefficients of aa and ac in Eq.(6) are usually assumed
to equal 0.5, thus the kinetics reaction overpotential hkin;i is derived
as,
Fig. 5. Experimental setup for battery testing.
9
(51)
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
Fig. 6. (a) Terminal voltages under different galvanostatic discharges, (b) electrolyte concentration distributions under 1 C discharge at different times, and (c) electrolyte concentration distributions at t ¼ 80 s under different discharge currents.
Table 2
Errors of the termianl voltage under various conditions.
cTerminal voltage error (mV)
0.5C Disch.
1 C Disch.
1.5C Disch.
2 C Disch.
2 C FUDS
2 C UDDS
ROEM vs.
Measured
P2D vs.
Measured
ROEM vs.
P2D
MAE
RMSE
MAE
RMSE
MAE
RMSE
10.1
14.0
31.1
27.4
13.7
11.5
17.1
15.8
36.2
32.6
28.4
24.2
11.6
8.0
24.6
30.9
14.6
13.7
17.3
10.7
30.2
38.2
26.3
22.8
5.66
10.6
12.1
18.5
6.6
6.7
6.18
12.2
16.6
24.6
9.9
10.6
From the discussion above, the ROEM for lithium-ion batteries is
represented by Fig. 4 in the block diagram form. The resulting
ROEM is described by five state variables, and its state equation is
given as Eq. (52) by assembling Eqs. (34) and (43).
x_ ROEM ðtÞ ¼ AROEM xROEM ðtÞ þ BROEM uROEM ðtÞ
(52)
where xROEM ¼ ½xOCV ; xe is state vector and uROEM ¼ Iapp is input
variable. All parameter matrices are reformulated from the original
parameters of the P2D model and given as,
AROEM ¼
AOCV
0
0
BOCV
; BROEM ¼
Be
Ae
Fig. 7. (a) Electrolyte concentration variations at neg/cc and pos/cc interfaces, (b)
electrolyte diffusion overpotential, and (c) overpotential error during 1 C discharge.
The state equation of the ROEM, Eq. (52), is solved iteratively with
T
5. Validation and discussions
the initial condition xROEM ð0Þ ¼ SOC0 ; 0; 0; 0; 0
after time
discretization.
Overall, the P2D model is reformulated to a five-state electrochemical model after order-reduction. It is worth mentioning that
the state matrix of the reduced system is in diagonal canonical
form. The number of ODEs to be solved in the ROEM is only five,
which is far less than 1154 in the P2D model as illustrated in Section
2.
This section investigates the accuracy of the proposed ROEM in
predicting the battery external and internal behaviors by performing the galvanostatic discharge and dynamic tests. Besides, the
computation time of the ROEM is compared with that of the P2D
model to demonstrate its potential for real-time application.
10
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
summarized in Table 2. As can be seen from Table 2, with the increase of discharge C-rate, the deviation between ROEM and P2D
results becomes larger but acceptable.
Fig. 6(b) shows the electrolyte concentration distributions
through the thickness of the cell sandwich at different time points
under 1 C galvanostatic discharge. It indicates that the concentration distributions of the ROEM are in good agreement with that of
the P2D model. The maximum concentration errors are 5.96 mol/
m3, 6.33 mol/m3, 5.10 mol/m3 and 5.44 mol/m3 at the time points of
5 s, 10 s, 20 s and 80 s, respectively. Fig. 6(c) shows the electrolyte
concentration distributions at t¼80 s (steady state) under different
discharge currents. The RMSE shows an increasing trend with
applied discharge C-rate, saying 2.04 mol/m3 for 0.5C, 4.02 mol/m3
for 1 C, 5.92 mol/m3 for 1.5C and 7.86 mol/m3 for 2 C.
Fig. 7(a) shows the electrolyte concentration variations at neg/cc
and pos/cc interfaces during 1 C galvanostatic discharge. The RMSEs
are only 12.96 mol/m3 and 13.14 mol/m3 for neg/cc and pos/cc interfaces respectively. According to Eq. (46), the electrolyte concentrations at neg/cc and pos/cc interfaces determine the
electrolyte diffusion overpotential he;con directly. Accordingly,
Fig. 7(b) gives the electrolyte diffusion overpotential responses of
the ROEM and P2D model during 1 C discharge, and Fig. 7(c) shows
the maximum error of the overpotential is less than 6.5 mV. The
electrolyte diffusion overpotential of the ROEM reaches a constant
after a period of time, while that of the P2D model still keeps
changing slightly. The main reason for this distinction is that the
P2D model can capture the nonuniform distribution of pore wall
flux within electrodes [29].
To verify the internal performance under different galvanostatic
discharges, the MAEs and RMSEs of electrolyte concentration and
electrolyte diffusion overpotential between the ROEM and P2D
solutions are summarized in Table 3. The results show clearly that
all errors become larger with the increasing discharge C-rate
Table 3
Errors of the electrolyte concentration and electrolyte diffusion overpotential under
various conditions.
Condition
0.5C Disch.
1 C Disch.
1.5C Disch.
2 C Disch.
2 C FUDS
2 C UDDS
ce;n=c (mol/m3)
ce;p=c (mol/m3)
he;con (mV)
MAE
RMSE
MAE
RMSE
MAE
RMSE
6.25
11.55
15.43
17.86
3.86
5.83
6.89
12.96
17.06
19.18
5.10
7.13
6.54
12.46
17.35
20.71
4.90
7.27
6.75
13.14
18.29
21.96
6.17
8.90
1.6
3.1
4.4
5.4
1.0
1.5
1.7
3.3
4.7
5.7
1.3
1.8
5.1. Experimental setup
A LiCoO2/graphite lithium-ion battery is tested to verify the
ROEM, whose parameters are given in Appendix A. To maintain the
tested battery always at a constant temperature (25+C) during
operation, it is placed inside an isothermal calorimeter whose
working mechanism can be found in Refs. [38,39]. As shown in
Fig. 5, the experimental setup consists of a battery tester (ARBIN BT5HC), an isothermal calorimeter (HEL iso-BTC) and their host
computer. The sampling frequency is 1 Hz for all tests.
5.2. Galvanostatic discharges
The battery is discharged at the current of 0.5C, 1 C, 1.5C and 2 C,
respectively. The lower cutoff voltage is set as 2.75 V for all tests.
Fig. 6(a) shows the terminal voltages of the experiment, ROEM and
P2D model. It can be seen that the simulated results match well
with the measured results, which demonstrates the reliability of
the ROEM and P2D model. The mean absolute errors (MAEs) and
root mean squared errors (RMSEs) between the three results are
Fig. 8. (a) Terminal voltages of experiment, ROEM and P2D model under 2 C FUDS cycles, (b) local detail for the terminal voltages, and (c) cumulative percentiles of absolute error
between any two results.
11
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
Fig. 11. (a) Electrolyte concentration variations at neg/cc and pos/cc interfaces, (b)
electrolyte diffusion overpotential, and (c) overpotential error during one 2 C UDDS
cycle.
Fig. 9. (a) Electrolyte concentration varaitions at neg/cc and pos/cc interfaces, (b)
electrolyte diffusion overpotential, and (c) overpotential error during one 2 C FUDS
cycle.
Table 4
Comparison of the computation times between the ROEM and P2D model.
Conditions
Duration-time
ROEM
P2D
Ratio (P2D/ROEM)
0.5C Disch.
1 C Disch.
1.5C Disch.
2 C Disch.
2 C FUDS
2C UDDS
7092 s
3407 s
2101 s
1417 s
14547 s
7769 s
18.5
17.5
16.4
15.7
22.3
19.1
39.15 s
19.93 s
13.51 s
9.73 s
80.18 s
44.01 s
2116
1139
824
620
3596
2304
ms
ms
ms
ms
ms
ms
desired maximum discharge current of 2 C in this study.
The simulated and measured terminal voltages under 2 C FUDS
cycles are shown in Fig. 8(a). Overall, the simulated results of P2D
and ROEM keep similar dynamics with the measured results.
Relatively larger errors only occur at the late of the test where the
SOC is less than 10%, which is caused by the asymmetricButlerVolmer kinetics [41,42] and the large OCV gradient. The MAEs
and RMSEs between the terminal voltages of the experiment, ROEM
and P2D model are also included in Table 2. The MAEs and RMSEs
are less than 14.6 mV and 28.4 mV, respectively. Fig. 8(c) shows the
cumulative percentiles of absolute error between the measured
and simulated terminal voltages. It can be seen that, during 2 C
FUDS cycles, over 80% of the absolute errors are less than 20 mV for
any comparisons.
Fig. 9(a) shows the electrolyte concentration variations at neg/cc
and pos/cc interfaces under one 2 C FUDS cycle. Results indicate
that the approximate electrolyte concentration of the ROEM can
match the P2D solution accurately, with the maximum errors of
14.06 mol/m3 and 17.59 mol/m3 at the neg/cc and pos/cc interfaces
respectively. Fig. 9(b) shows the electrolyte diffusion overpotential
response under the 2 C FUDS cycle. The corresponding error is less
than 3.9 mV, as shown in Fig. 9(c). It can be seen that the maximum
error occurs as the maximum current is applied, which is mainly
because that the non-uniformity of the pore wall flux is enhanced
by the increasing current [29].
Fig. 10. (a) Terminal voltages of experiment, ROEM and P2D model under 2 C UDDS
cycles, (b) local detail for the terminal voltages, and (c) cumulative percentiles of absolute error between any two results.
because of the non-uniformity of pore wall flux.
5.3. Dynamic conditions
To validate the ROEM under daily driving situations, the fullycharged battery is tested under the current profiles of two standard dynamic tests, the Federal Urban Driving Schedule (FUDS) and
the Urban Dynamometer Driving Schedule (UDDS), which are
commonly adopted in many studies to evaluate the performance of
batteries used in EVs. The EV model in Ref. [40] is used to generate
the current profile based on the velocity profile of the dynamic
driving cycles, in which the current profile is scaled to meet the
12
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
dynamics, the ROEM for lithium-ion batteries is derived as a fivestate system with diagonal structure.
According to the experimental and simulated results, the ability
of the ROEM in predicting the battery internal and external behaviors is thoroughly verified. The electrolyte concentrations of the
ROEM are in good agreement with the P2D solutions, and the
RMSEs of the resulting electrolyte diffusion overpotential are less
than 5.7 mV under both galvanostatic discharge and dynamic tests.
Under the same conditions, the RMSEs between the simulated and
measured terminal voltage are less than 36.2 mV. Besides, the
computation time of the ROEM is reduced by over 600 times
compared with the P2D model.
As the battery performance is affected by the electrode material,
working condition and other factors, the parameters of the ROEM
need to be updated to ensure accurate and efficient battery management. In the future, we will study the parameter identification
method for the model and analyze the dependence of the parameters on battery temperature and aging state.
UDDS test is also performed to verify the model performance
under dynamic conditions. Fig. 10(a) shows the terminal voltage
responses of the experiment, ROEM and P2D model under 2 C
UDDS cycles. Both the ROEM and P2D model can predict the terminal voltage accurately. The RMSE between the ROEM and
measured results is 24.2 mV, and 22.8 mV between the P2D and
measured results. Moreover, Fig. 10(c) indicates that over 80% of the
absolute errors are less than 20 mV for any comparisons. Fig. 11(a)
shows the electrolyte concentration variations at neg/cc and pos/cc
interfaces under one 2 C UDDS cycle. The maximum errors are only
18.67 mol/m3 and 23.83 mol/m3 at neg/cc and pos/cc interfaces
respectively. As shown in Fig. 11(b) and (c), the simulated electrolyte diffusion overpotential of the ROEM is in good agreement with
that of the P2D model, with the maximum error less than 5.3 mV.
Like the FUDS test, the UDDS test also proves the excellent prediction ability for the ROEM under dynamic conditions.
5.4. The computational efficiency
To compare the computational efficiencies of the ROEM and P2D
model quantitatively, their computation times of the aboveinvolved test conditions are listed in Table 4. Both the ROEM and
P2D model are constructed on the MATLAB platform, and the
programs run on a computer with Intel Core i5 CPU @ 1.6 GHz, 8 GB
RAM and Windows operating system. It can be seen that the ROEM
can be simulated within 22 ms for all conditions. The computation
time of the ROEM is less than one six-hundredth of the P2D model.
Therefore, the ROEM can acquire a much better computational efficiency than the P2D model, which makes it an ideal candidate for
practical application in BMSs.
Declaration of competing interest
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Natural Science
Foundation of China (NO. 61633015, U1864205, U1864202), the Key
Technology Research and Development Program of Shandong
Province (No. 2019JZZY020814), and the National Natural Science
Foundation of China (NO. U1764258).
6. Conclusion
The high computational complexity is the major obstacle for the
real-time application of the full-order P2D electrochemical model,
thus a reduced-order electrochemical model for lithium-ion batteries is proposed in this paper. We first focus on the simplification
of the electrolyte diffusion process in batteries. The domain
decomposition technique is introduced to decouple the polynomials which describe the electrolyte concentration distribution
in the thickness direction of cell sandwich. As a result, the electrolyte diffusion is reduced to a two-state diagonal system. Incorporating the solid-phase diffusion, kinetics reaction and other
Appendix A. Model parameters for tested battery
Table A1 provides the parameters of the tested battery for model
validation in Section 5, and Fig. A1 shows the electrode OCP profiles
as functions of stoichiometric number.
Table A.1
Model parameters for the tested LiCoO 2/graphite lithium-ion battery at 25+C.
Parameters
Units
L
m
76:5 Rs
m
εs
εe
1
1
mol/m3
12:6 106
0.67
0.3
24895
mol/m3
mol/m3
m2/s
c0s
cmax
s
c0e
Ds
ss
De
ke
RSEI
brugg
K
0
tþ
aa ; ac
В
S/m
m2/s
S/m
U m2
1
A m2.5/mol1.5
Neg.
Sep.
106
20 Pos.
106
69:5 Sources
106
[meas.]
0.4
6:2 106
0.62
0.3
22220
[adj.]
[est.]
[est.]
31080
1200
1200
51555
1200
[43,44]
[est.]
1:4 1014
100
1 1014
100
3:01 1010
3.46
0.02
1.5
1.5
0
1.5
[adj.]
[43,44]
[44]
[adj.]
[43]
[est.]
[est.]
[adj.]
1
7:76 105
0.363
0.363
3:32 105
0.363
[44]
1
1
0.5, 0.5
3.57
3.57
0.5, 0.5
3.57
[43]
[adj.]
(continued on next page)
13
C. Li, N. Cui, C. Wang et al.
Energy 221 (2021) 119662
Table A.1 (continued )
1
2
3
Parameters
Units
Neg.
Parameters
Qcell
A
Rctct
Units
Ah
m2
Value
2.32
0.072
Sep.
U
OCPn
OCPp
V
V
77:5 103
cf. red line in Fig. A.1
cf. blue line in Fig. A.1
Pos.
Sources
Sources
[meas.]
[meas.]
[adj.]
[43]
[45]
[meas.]: measured.
[adj.]: adjusted.
[est.]: estimated.
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