Optimal Input Design for Parameter Identification in an Electrochemical Li-ion Battery Model

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Optimal Input Design for Parameter Identification
in an Electrochemical Li-ion Battery Model
Saehong Park, Dylan Kato, Zach Gima, Reinhardt Klein, Scott Moura
Abstract—We consider the problem of optimally designing
an excitation input for parameter identification of an electro-
chemical Li-ion battery model. The optimized input is obtained
by solving a relaxed, convex knapsack problem. In contrast to
performing parameter identification with standard test cycles,
we consider the problem as designing an optimal input trajec-
tory that maximizes parameter identifiability. Specifically, we
analytically derive sensitivity equations for the electrochemical
model. This approach enables parameter sensitivity analysis
and optimal parameter fitting via a gradient-based algorithm.
The simulation results show that the optimized inputs achieve
faster parameter identification compared to standard test cycles
and tighten the parameter estimation confidence intervals.
Keywords—Electrochemical Model, Sensitivity Analysis, Sys-
tem Identification, Input Design, Levenberg-Marquardt
I. INTRODUCTION
Batteries are a key enabling technology behind electrified
transportation, portable consumer electronics, and more. To
enhance the safety and performance of these devices, one
must understand their electrochemical behavior, particularly
in extreme operating conditions. To this end, battery systems
researchers are deeply interested in mathematical electro-
chemical models. An experimentally validated model can
be used for design purposes or online battery management
systems (BMS) [1], [2]. Identifying the unknown model
parameters, however, is challenging for multiple reasons.
First, battery cell manufacturers do not disclose this informa-
tion on data sheets. Second, one can only measure voltage,
current, and temperature - at best. Third, characterizing cer-
tain properties, e.g. diffusivities, requires destructive testing.
Finally, the model parameters are nonlinear with respect to
the measured signals, and the dynamics are governed by
coupled nonlinear partial differential-algebraic equations.
In BMS research, battery models can be categorized into
two groups: equivalent circuit models (ECMs), and electro-
chemical models. ECMs represent the input-output behavior
using circuit elements, such as resistors and capacitors. How-
ever, ECMs do not directly capture the physical phenomena
inside the battery, such as lithium transport, solid-electrolyte
interphase dynamics, and degradation mechanisms. Electro-
chemical models directly incorporate diffusion, intercala-
tion, and electrochemical kinetics. Although these models
can accurately explain the internal behavior of the battery,
their mathematical structure is relatively complicated, which
motivates the active sub-field of electrochemical model
Saehong Park, Dylan Kato, Zach Gima, and Scott Moura are with
the Energy, Controls and Applications Lab (eCAL) at the University of
California, Berkeley, CA 94720, USA (E-mail: {sspark, dkkato, ztakeo,
smoura}@berkeley.edu)
Reinhardt Klein is with Robert Bosch LLC, Research and Technology
Center, Palo Alto, CA 94304 USA (E-mail: reinhardt.klein@us.bosch.com)
reduction. Several important innovations in electrochemical
model reduction include the single particle model [3]–[5],
quasilinearization with Pad´
e approximation [6], spectral
methods [7], residue grouping [8] and more.
Non-invasive parameter identification of electrochemical
models has become an emergent research topic, due to
interests in electrochemical models for simulation, analysis,
and control purpose. The authors in [9], propose parameter
identification via Fisher information analysis and a group-
ing strategy, applied to an extended single particle model.
Parameter optimization is performed using nonlinear least
squares. In contrast, [10] identifies the parameters via an
all-in-one approach from driving cycle data using a genetic
algorithm. They validate the identified parameter values with
experiments and perform Fisher information analysis ex post
factor. Similarly, [11] uses a multi-objective genetic algo-
rithm for parameter identification. They use terminal voltage
and surface temperature curves as identification objectives.
Rather than identify all the parameters, some researchers
focus on identifying specific subsets of parameters, such as
battery health-related or kinetic parameters. For example,
[11] mainly target the physical parameters, such as diffusion
coefficients, and activation energies. [12] focuses on the the
electrochemical parameters that govern power and capacity
prediction, as well as their temperature dependence under a
variety of charge sustaining and depleting experiments. In
[13], the parameter estimation for kinetic parameters and
sensitivity analysis via analysis of variance for nonlinear
models is proposed.
Most existing literature on battery parameter identifica-
tion focuses on parameter fitting, namely, matching model
output to experimental data. However, it is unclear if the
experimentally data is “sufficiently rich” to identify the
parameters. A small set of publications in the battery pa-
rameter identification literature deal with this problem by
formulating an input trajectory optimization problem [14],
[15]. This work optimizes the amplitude and frequency of a
sinusoidal input signal to maximize the Fisher information
matrix, in some sense, for an ECM and single particle model.
One could exploit a series of inputs that excite specific
parameter sensitivity, however, collecting the required data
from experiments can be cost intensive. The motivating
question is: which inputs should be considered to maximize
parameter identifiability in a systematic way? In addition, the
estimated parameters should be characterized by confidence
intervals. These question motivate optimal experimental
design (OED), which provides an important link between
experimental design and modeling [16]. In this paper, we
propose an electrochemical model-based optimal experiment
design framework that yields parameter estimates and confi-
dence intervals. Instead of formulating a nonlinear trajectory
2018 Annual American Control Conference (ACC)
June 27–29, 2018. Wisconsin Center, Milwaukee, USA
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Figure 1: Schematic of the first-principle electrochemical
model, called pseudo two-dimensional model (P2D model).
optimization problem, we propose a convex input selection
problem. We summarize our key contributions as follows.
First, a sensitivity analysis for the full-order electro-
chemical model is executed, by analytically deriving the
sensitivity differential algebraic equations. To the best of
authors’ knowledge, an analytic sensitivity analysis for all
the electrochemical parameters has never been executed
before. We place emphasis on this sensitivity analysis, as
it plays a crucial role in computing the Fisher information
matrix, and the parameter estimation algorithm. Secondly,
we formulate an optimal experimental design via convex
optimization. Rather than solving a large-scale nonlinear
optimal control problem, we propose a (relaxed) “knapsack
problem” where the optimal inputs are selected from a
discrete set. Lastly, we estimate the parameters and their
confidence intervals via nonlinear least squares with the
Levenberg-Marquardt algorithm [17].
The paper is organized as follows. Section II briefly
presents the electrochemical Li-ion battery model. Sec-
tion III details the parameter identification framework. Sec-
tion IV presents simulation results with visual examples. In
Section V, we summarize our work and provide perspectives
on future work.
II. ELECTROCHEMICAL BATTE RY MOD EL
In this section we describe the electrochemical model con-
sidered in the paper. It consists of two electrically separated
porous electrodes and a separator, as shown in Fig. 1. The
lithium ions are transported by a diffusion process inside
the active particles along the r-axis in the solid phase. They
traverse the particle-electrolyte interface via Butler-Volmer
kinetics. The ions dissolved in the electrolyte pass through
the separator to the opposite electrode along the x-axis.
The diffusion, intercalation, and electrochemical kinetics
account for the internal battery dynamics expressed by a
combination of partial differential equations (PDEs) and
ordinary differential equations (ODEs). The state variables
are lithium concentration in the solid c±
s(x, r, t), lithium
concentration in the electrolyte ce(x, t), solid electric po-
tential φ±
s(x, t), electrolyte electric potential φe(x, t), ionic
current in the electrolyte i±
e(x, t), and molar ion fluxes
between electrodes and electrolyte j±
n(x, t). We summarize
the governing equations for j∈ {−,sep,+},
∂c±
s
∂t (x, r, t) = 1
r2
∂
∂r D±
sr2∂c±
s
∂r (x, r, t),(1)
εj
e
∂cj
e
∂t (x, t) = ∂
∂x Deff
e(cj
e)∂cj
e
∂x (x, t) + 1−t0
c
Fij
e(x, t),
(2)
σeff,±·∂φ±
s
∂x (x, t) = i±
e(x, t)−I(t),(3)
κeff(ce)·∂φe
∂x (x, t) = −i±
e(x, t) + κeff(ce)2RT
F(1 −t0
c)
×1 + dln fc/a
dln ce
(x, t)∂ln ce
∂x (x, t),
(4)
∂i±
e
∂x (x, t) = a±
sF j±
n(x, t),(5)
j±
n(x, t) = 1
Fi±
0(x, t)heαaF
RT η±(x,t)−e−αcF
RT η±(x,t)i,
(6)
where t∈R+represents time. Some parameters, such
as De, κ, fc/a are functions of the states, ce(x, t).Deff
e=
De(ce)·(εj
e)brug,σeff =σ·(εj
s+εj
f)brug,κeff =κ(ce)·(εj
e)brug
are the effective electrolyte diffusivity, effective solid con-
ductivity, and effective electrolyte conductivity given by the
Bruggeman relationship. In (6), the exchange current density
i±
0(x, t)and over-potential η±(x, t)are expressed:
i±
0(x, t) = k±c±
ss(x, t)αcce(x, t)c±
s,max −c±
ss(x, t)αa,
(7)
η±(x, t) = φ±
s(x, t)−φe(x, t)−U±(c±
ss(x, t))
−F R±
fj±
n(x, t),(8)
where css is the solid phase surface concentration
c±
ss(x, t) = c±
s(x, R±
s, t),U±is the open-circuit potential,
and c±
s,max is the maximum possible concentration in the
solid phase.
A complete exposition on the model equations and bound-
ary conditions can be found in [1]. The input to the model
is the applied current density I(t)[A/m2], and the output is
the voltage measured across the current collectors:
V(t) = φ+
s(0+, t)−φ−
s(0−, t) + RcI(t).(9)
In contrast to an ECM, which has a limited ability to in-
terpret internal battery dynamics, the electrochemical model
has parameters that directly explain physical phenomenon
inside the battery. Note that some of the parameters are
measurable by disassembling the cell, such as geometric
parameters, while other parameters are not directly measur-
able and change over time. These fixed and internal model
parameters are listed separately in Table I-II.
Symbol Description [SI units]
L−Thickness of negative electrode [ m ]
Lsep Thickness of separator [ m ]
L+Thickness of positive electrode [ m ]
AElectrode current collector area [m2]
Table I: Fixed geometric parameters.
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Symbol Description [SI units]
D−
sSolid-phase diffusion coefficients [ m2/sec ]
D+
sSolid-phase diffusion coefficients [ m2/sec ]
R−
sSolid-phase particle radii [m]
R+
sSolid-phase particle radii [m]
σ−Solid-phase conductivity [Ω−1m−1]
σ+Solid-phase conductivity [Ω−1m−1]
De(·)Electrolyte diffusion coefficient [ m2/sec]
−
eElectrolyte volume fraction [-]
sep
eElectrolyte volume fraction [-]
+
eElectrolyte volume fraction [-]
κ(·)Electrolyte conductivity [Ωm]
t0
cTransference number [-]
dln fc/a
dln ce(·)Activity coefficient [-]
k−Kinetic rate constants [(A/m2)(m3/mol)(1+α)]
k+Kinetic rate constants [(A/m2)(m3/mol)(1+α)]
R−
fFilm resistance [Ωm2]
R+
fFilm resistance [Ωm2]
ce0Initial Li-ion concentration in electrolyte [mol/m3]
Table II: Electrochemical model parameters.
DAE Variables P2D Variables
xc−
s, c+
s, ce
zφ−
s, φ+
s, i−
e, i+
e, φe, j−
n, j+
p
uI
θParameters in Table II
Table III: DAE notations for electrochemical model.
For the subsequent sections, we use the notation in
Table III to describe the general dynamic system model
and parameter identification framework. In this context, we
can formulate the system model as a system of differential
algebraic equations (DAEs):
˙
x=f(x,z, u, θ),x(t0) = x0,(10)
0=g(x,z, u, θ),z(t0) = z0,(11)
y=h(x,z, u, θ),(12)
III. PARAMETER IDENTIFICATI ON FR AM EWORK
A. Sensitivity Analysis
Sensitivity analysis is used to understand how a model’s
output depends on variations in the parameter values [18].
Based on nominal parameter values, local sensitivity analysis
measures the effects that small changes in the parameters
have on the output. For continuous dynamic systems, the
local sensitivities are defined as the first-order partial deriva-
tives of the system output with respect to the parameters.
In this section, we briefly introduce how to derive local
sensitivities in dynamical system described in (10)-(12) and
develop this approach toward parameter estimation frame-
work via Fisher information.
Let us define sensitivity variables as follows:
Sx=∂x
∂θ, Sz=∂z
∂θ, Sy=∂y
∂θ,(13)
Sx, Sz, Syare sensitivity matrices where the i, j matrix
element is defined as the partial derivative of the i-th state
to the j-th parameter:
si,j (t) = ∂xi(t)
∂θj
.(14)
Then, we can derive the sensitivity differential algebraic
equations (SDAEs):
˙
Sx=∂f
∂xSx+∂f
∂zSz+∂f
∂θ, Sx(0) = Sx0(15)
0 = ∂g
∂xSx+∂g
∂zSz+∂g
∂θ, Sz(0) = Sz0(16)
Sy=∂h
∂xSx+∂h
∂zSz+∂h
∂θ.(17)
The advantage of SDAEs is that they provide a fundamen-
tal mathematical computation of the sensitivities, compared
to a perturbation method where sensitivity is obtained by
perturbing each parameter slightly and calculating the output
difference with respect to nominal parameters. Note that
SDAEs are linear time-varying DAEs, where the Jacobians
are computed at each time step. In particular, we use CasADi
[19], which efficiently computes the first and second-order
derivatives. In this work, the battery model DAEs and its
SDAEs are simulated by using the IDAS integrator provided
by SUNDIALS via the CasADi interface.
B. Parameter Grouping
In this section, we introduce the methodology of param-
eter grouping for parameter identification. It is well known
that the entire electrochemical parameter vector θis weakly
identifiable from the measured output, since the system is
nonlinear in the parameters. This is mainly due to linear
dependence between the parameter sensitivity vectors [9].
Therefore, it is necessary to analyze the linear dependency
among electrochemical parameters, and rank/organize them
into groups to avoid linear dependence during the parame-
ter identification process. For parameter grouping, we first
perform sensitivity analysis across a library of input values.
The input library includes a variety of profiles, including (i)
pulses, (ii) sinusoids, and (iii) driving cycles. A normaliza-
tion is applied to each category to fairly evaluate candidate
profiles. In this work, a total of 738 profiles across these
three input categories were analyzed. Across all categories,
profiles have been normalized to 1 Ah of charge processed
in both charge and discharge cases.
After calculating the sensitivities via the process in Sec-
tion III-A, we apply the Gram-Schmidt process on ST
ySy
to reveal the sensitivity ranking and linear dependence.
Figure 2 visualizes the average sensitivity magnitudes via
Graham-Schmidt orthonormalization over 738 profiles.
Based on this sensitivity analysis, we group the param-
eters based on their orthogonalized sensitivity magnitude.
The resultant groups are shown in Table IV. The criteria for
classification is determined by the frequency of times each
parameter is ranked in the top or bottom in each subgroup
of the input library. For instance, Group 1 parameters
positioned the highest rank for all subgroups, while Group
4 parameters are the lowest rank in this analysis.
C. Optimal Experiment Design
In statistical experiment design, the amount of the infor-
mation about parameters θcontained in the observation y
from an experiment is calculated by the Fisher information
matrix, F[20]. The Fisher information matrix is defined as:
F=Ztf
0
ST
y(t)Q(t)−1Sy(t)dt, (18)
2302

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
Sensitivity Magnitude [log scale]
Figure 2: The mean of orthogonalized sensitivity magnitudes
in the input library.
Group 1 Group 2 Group 3 Group 4
R−
sD−
sR−
fσ−
R+
sD+
sk−σ+
-ε−
eε+
eεsep
e
-κ(·)ce0k+
-De(·)R+
ft0
c
-dln fc/a
dln ce(·)
Table IV: The result of electrochemical parameters grouping.
where t∈[0, tf], and Q(t)is the covariance matrix of
the measurement error. Since the true parameters θ∗are
unknown, the sensitivity is calculated at nominal parameter
values θ0. The deviation of the parameter estimates from
their true values can be expressed as the covariance matrix
Σ. According to the Cramer-Rao bound [21], the inverse of
the Fisher information matrix provides a lower bound on
Σ≥F−1,
Our objective is to find inputs that minimize the lower
bound of the parameter estimation error, thus improving the
parameter estimation quality. To optimize the amount of in-
formation, a proper scalarization of Fshould be considered,
namely,
D-optimality : min det(F−1)
A-optimality : min tr(F−1)
E-optimality : min λmax(F)
D. Optimal Experiment Design via Convex Programming
We formulate a procedure to optimize experiment design
to produce inputs that are maximally informative for pa-
rameter estimation. To bypass the challenge of solving a
large-scale nonlinear optimal control problem, we pursue a
different approach. Specifically, we seek the combination of
inputs from an input library which maximizes the Fisher in-
formation matrix. This process yields a convex optimization
program, which can be rapidly solved via computationally
efficient open-source solvers, such as cvx [22].
Suppose we have a set of Lexperimental inputs ui(t), i =
1,2,·· · , L. Given any input profile ui(t), we obtain a
corresponding sensitivity vector Sy,i(t)by solving (10)-
(12) and (15)-(17) simultaneously. The basic idea is that
we can select Minputs from a large fixed set of Linputs.
Amongst these Linputs ui(t), i = 1,2, . . . , l, we select M
inputs that are maximally informative as measured by the
Fisher information matrix, F. Let mjdenote the number of
experiments with index number jthat are executed from the
input library. Then, the total number of experiments is
m1+m2+. . . +ml=M(19)
We can then rewrite the Fisher information matrix as:
F=
L
X
i=1
miST
y,iQ−1
iSy,i.(20)
We now formulate a combinatorial optimization problem
to maximize F:
minimize
mi
det l
X
i=1
miST
y,iQ−1
iSy,i!−1
,(21)
subject to mi≥0, m1+. . . +ml=M, (22)
mi∈Z.(23)
This problem is an integer program, where the optimal
number of experiments miis the solution. In addition, large-
scale combinatorial problems are NP-hard. So, in this work,
we relax the last integer constraint (23), yielding a relaxed
optimization problem that is convex. Let ηi=mi/M be the
fraction of experiment type ito execute. Then the Fisher
information (20) can be re-written as
F=M
L
X
i=1
ηiST
y,iSy,i ,(24)
where η∈RL,1Tη= 1. Thus, our ultimate convex optimal
experiment design problem is:
minimize
ηlog det L
X
i=1
ηiST
y,iSy,i !−1
(25)
subject to η0,1Tη= 1 (26)
E. Optimal Parameter Fitting via Nonlinear Least Squares
After obtaining experiment data, we now seek to optimally
fit the parameters. The optimization problem for parameter
identification can be formulated as a nonlinear least squares
problem:
minimize
ˆ
θ
M
X
i=1
tf
X
t=1
yi(t)−ˆ
yi(t;ˆ
θ)
σ2
yi
,(27)
where Mis the number of optimized input profiles from
OED, and σ2
yiis the measurement variance for input profile
i. The Levenberg-Marquardt algorithm is used to update the
parameters ˆ
θiteratively to solve the nonlinear optimization
problem (27). This algorithm adaptively updates the param-
eter estimates via a hybridization of the gradient descent
update and the Gauss-Newton update [23]:
hJTWJ +γdiag(JTWJ)i∆θ=JTW(y−ˆy),(28)
where J=∂ˆ
y/∂ ˆ
θ, the local sensitivity of the output ˆ
y,
and Wis the inverse of the measurement error covariance
2303

matrix, W=Q−1. The value of γweighs gradient de-
scent update against Gauss-Newton update. Conveniently,
the Levenberg-Marquardt algorithm utilizes the Jacobians
already computed in the sensitivity analysis in Section III-B.
After optimally fitting the parameters, estimation error
statistics are calculated according to
ρθ=JTWJ,(29)
σθ=qdiag[JTWJ]−1,(30)
where ρis the parameter covariance matrix, and σis the
standard parameter error. Lastly, confidence intervals for the
parameter estimates are calculated as follows:
ˆ
θ−t(1−0.05,n)
σθ
√n≤θ∗≤ˆ
θ+t(1−0.05,n)
σθ
√n,(31)
where nis the number of observations, and tis the upper
critical value for the t-distribution with n−1freedom.
IV. SIMULATION RESULTS
A. Optimal Input Validation
First we compute the sensitivity trajectories for each input
in the library described in Section III-B by exploiting paral-
lel computing. After running optimal experiment design via
convex programming, we obtain a series of optimal inputs.
In order to examine the identifiability of the inputs, we plot
the L2norm of the model error with respect to parameter
estimates, for instance θ1= (R−
s−R−
s)/(R−
s−R−
s)and
θ2= (R+
s−R+
s)/(R+
s−R+
s)plotted in Fig. 3a, 3b for the
optimized and standard input cycles, respectively.
Note the optimal input profile has a more pronounced
bowl shape, while the pulse input has flatter shape. By ap-
propriately optimizing the input, we can improve the conver-
gence rate of the parameter estimation process. For example,
if we start our initial guess around ˆ
θ= (0.25,0.25), then the
optimal input yields estimates that converge to the optimal
points in just five iterations, while the non-optimized input
does not converge clearly. Although this example is specific
to R−
s, R+
s, pulse inputs and a selected optimized input, the
observed trend is consistent when comparing optimized and
standard input cycles across the electrochemical parameters.
B. Parameter Identification Validation
Next, we compare the parameter estimation performance
for optimized versus standard inputs. In this scenario, we
assume the true parameter values are known. We perturb four
parameters, R±
s,R±
f, from their true values and attempt to es-
timate them. Note that parameters are selected from different
groups, namely, R±
sare in Group 1, and R±
fare located in
Group 3. The convex OED program generates optimal input
sequences that maximize parameter identifiability. To test if
our optimal input profiles outperform other simple inputs, a
1C discharge/charge profile is chosen for comparison. The
simulation results are summarized in Table V-VI.
The results indicate that the optimal input sequences
provide more accurate parameter estimates and tighter confi-
dence intervals than the 1C charge/discharge cycle. Note the
estimated value for R−
fstill exhibits non-trivial error. This
is expected, since R−
fis weakly identifiable from voltage,
as demonstrated by the sensitivity analysis in Fig. 2. A
consequence of achieving lower parameter estimation error
(a) Optimal input profile
(b) Standard input profile
Figure 3: L2 norm of model error ky∗−y(θ1, θ2)kfor a
pulse input. The points indicate parameter estimates every 5
iterations during the parameter estimation process. Parame-
ter values are normalized by Min/Max scaling.
Parameter Estimated Value True Value 95% C.I.
R−
s[m] 11.4561E-06 10.9E-06 ±3.18864E-07
R+
s[m] 9.69823E-06 10.9E-06 ±2.49251E-07
R−
f[Ω] 8.35184E-04 5.0E-04 ±4.70283E-04
R+
f[Ω] 1.00000E-03 1.0E-03 ±9.01101E-04
Table V: Estimated values of the parameters and the corre-
sponding confidence intervals using optimal input profiles.
Parameter Estimated Value True Value 95% C.I.
R−
s[m] 12.4151E-06 10.9E-06 ±2.05626E-06
R+
s[m] 8.97366E-06 10.9E-06 ±1.15361E-06
R−
f[Ω] 1.00000E-03 5.0E-04 ±6.02711E-04
R+
f[Ω] 1.00000E-03 1.0E-03 ±2.73964E-03
Table VI: Estimated values of the parameters and the cor-
responding confidence intervals using 1C Charge-Discharge
input.
increases voltage prediction accuracy for input cycles that
are different from those used for identification. In contrast,
incorrect parameter estimates from non-optimized inputs
may yield poor prediction accuracy for different inputs
from those used for identification. So, it is necessary to
design optimal inputs that improves parameter estimation
performance.
2304

V. CONCLUSION
This paper addresses optimal experiment design for pa-
rameter identification of an electrochemical Li-ion battery
model. Sensitivity analysis is explored to understand each
electrochemical parameter’s identifiability. We observe that
each parameter has different levels of identifiablity, which
motivate a parameter grouping strategy. Optimal inputs for
the parameter groups are designed to maximize (a scalar-
ization of) the Fisher information matrix. We propose an
input selection problem via convex programming. Once
the optimal input profiles are designed, the Levenberg-
Marquardt algorithm is used to solve a nonlinear least square
problem for optimal parameter fitting. In order to test our
proposed framework, we simulate the overall process using
a model-to-model comparison. These results demonstrate
how optimized inputs enhance identifiability and ultimately
yield data that improves parameter estimation performance.
On-going work involves validating our proposed parameter
estimation framework with experimental data, thermal dy-
namics, and various types of Li-ion batteries.
REFERENCES
[1] N. A. Chaturvedi, R. Klein, J. Christensen, J.
Ahmed, and A. Kojic, “Algorithms for advanced
battery-management systems”, IEEE Control Systems,
vol. 30, no. 3, pp. 49–68, Jun. 2010.
[2] R. Klein, N. A. Chaturvedi, J. Christensen, J. Ahmed,
R. Findeisen, and A. Kojic, “Electrochemical model
based observer design for a lithium-ion battery”, IEEE
Transactions on Control Systems Technology, vol. 21,
no. 2, pp. 289–301, 2013.
[3] S. Santhanagopalan and R. E. White, “Online es-
timation of the state of charge of a lithium ion
cell”, Journal of Power Sources, vol. 161, no. 2,
pp. 1346–1355, 2006.
[4] D. Di Domenico, A. Stefanopoulou, and G. Fiengo,
“Lithium-Ion Battery State of Charge and Critical
Surface Charge Estimation Using an Electrochem-
ical Model-Based Extended Kalman Filter”, Jour-
nal of Dynamic Systems, Measurement, and Control,
vol. 132, no. 6, p. 061 302, 2010.
[5] S. Park, D. Zhang, and S. Moura, “Hybrid electro-
chemical modeling with recurrent neural networks
for li-ion batteries”, in American Control Conference
(ACC), 2017, IEEE, 2017, pp. 3777–3782.
[6] J. C. Forman, S. Bashash, J. L. Stein, and H. K.
Fathy, “Reduction of an electrochemistry-based li-ion
battery model via quasi-linearization and pade ap-
proximation”, Journal of the Electrochemical Society,
vol. 158, no. 2, A93–A101, 2011.
[7] A. M. Bizeray, S. Zhao, S. R. Duncan, and D. A.
Howey, “Lithium-ion battery thermal-electrochemical
model-based state estimation using orthogonal col-
location and a modified extended Kalman filter”,
Journal of Power Sources, vol. 296, pp. 400–412,
2015.
[8] K. A. Smith, C. D. Rahn, and C.-Y. Wang, “Model Or-
der Reduction of 1D Diffusion Systems Via Residue
Grouping”, Journal of Dynamic Systems, Measure-
ment, and Control, vol. 130, no. 1, p. 011012, 2008.
[9] A. P. Schmidt, M. Bitzer, ´
A. W. Imre, and L.
Guzzella, “Experiment-driven electrochemical mod-
eling and systematic parameterization for a lithium-
ion battery cell”, Journal of Power Sources, vol. 195,
no. 15, pp. 5071–5080, 2010.
[10] J. C. Forman, S. J. Moura, J. L. Stein, and H. K.
Fathy, “Genetic identification and fisher identifiability
analysis of the Doyle-Fuller-Newman model from
experimental cycling of a LiFePO 4 cell”, Journal
of Power Sources, pp. 263–275, 2012.
[11] L. Zhang, L. Wang, G. Hinds, C. Lyu, J. Zheng,
and J. Li, “Multi-objective optimization of lithium-
ion battery model using genetic algorithm approach”,
Journal of Power Sources, pp. 367–378, 2014.
[12] J. Marcicki, M. Canova, A. T. Conlisk, and
G. Rizzoni, “Design and parametrization analy-
sis of a reduced-order electrochemical model of
graphite/LiFePO4 cells for SOC/SOH estimation”,
Journal of Power Sources, vol. 237, pp. 310–324,
2013.
[13] J. Vazquez-Arenas, L. E. Gimenez, M. Fowler, T.
Han, and S. K. Chen, “A rapid estimation and sensi-
tivity analysis of parameters describing the behavior
of commercial Li-ion batteries including thermal anal-
ysis”, Energy Conversion and Management, pp. 472–
482, 2014.
[14] M. J. Rothenberger, J. Anstrom, S. Brennan, and
H. K. Fathy, “Maximizing Parameter Identifiability of
an Equivalent-Circuit Battery Model Using Optimal
Periodic Input Shaping”, in 2014 ASME Dynamic Sys-
tems and Control Conference, 2014, V001T19A004.
[15] J. Liu, M. Rothenberger, S. Mendoza, P. Mishra,
Y.-s. Jung, and H. K. Fathy, “Can an Identifiability-
Optimizing Test Protocol Improve the Robustness
of Subsequent Health-Conscious Lithium-ion Battery
Control? An Illustrative Case Study”, 2016 American
Control Conference, no. ii, pp. 6320–6325, 2016.
[16] G. Franceschini and S. Macchietto, “Model-based
design of experiments for parameter precision: State
of the art”, Chemical Engineering Science, vol. 63,
no. 19, pp. 4846–4872, 2008.
[17] K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods
for non-linear least squares problems”, 2004.
[18] H. Yue, M. Brown, F. He, J. Jia, and D. B. Kell,
“Sensitivity analysis and robust experimental design
of a signal transduction pathway system”, Interna-
tional Journal of Chemical Kinetics, vol. 40, no. 11,
pp. 730–741, 2008.
[19] J. Andersson, “A General-Purpose Software Frame-
work for Dynamic Optimization”, PhD thesis, Aren-
berg Doctoral School, KU Leuven, Oct. 2013.
[20] E. Walter and L. Pronzato, Identification of paramet-
ric models from experimental data. Springer Verlag,
1997.
[21] T. M. Cover and J. A. Thomas, Elements of informa-
tion theory. John Wiley & Sons, 2012.
[22] M. Grant and S. Boyd, CVX: Matlab software for
disciplined convex programming, version 2.1, http://
cvxr.com/cvx, Mar. 2014.
[23] K. Levenberg, “A method for the solution of certain
non-linear problems in least squares”, Quarterly of
applied mathematics, vol. 2, no. 2, pp. 164–168, 1944.
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