# Optimal Experimental Design for Large-Scale Inverse Problems via Multi-PDE-
constrained Optimization

Andrea Petrocchi<EMAIL_ADDRESS>Matthias K. Scharrer
<EMAIL_ADDRESS>Franz Pichler<EMAIL_ADDRESS>Stefan Volkwein
<EMAIL_ADDRESS>
###### Abstract

Accurate parameter dependent electro-chemical numerical models for lithium-ion
batteries are essential in industrial application. The exact parameters of
each battery cell are unknown and a process of estimation is necessary to
infer them. The parameter estimation generates an accurate model able to
reproduce real cell data. The field of optimal input/experimental design deals
with creating the experimental settings facilitating the estimation problem.
Here we apply two different input design algorithms that aim at maximizing the
observability of the true, unknown parameters: in the first algorithm, we
design the applied current and the starting voltage. This lets the algorithm
collect information on different states of charge, but requires long
experimental times ($60\,000\text{\,}\mathrm{s}$). In the second algorithm, we
generate a continuous current, composed of concatenated optimal intervals. In
this case, the experimental time is shorter ($7000\text{\,}\mathrm{s}$) and
numerical experiments with virtual data give an even better accuracy results,
but experiments with real battery data reveal that the accuracy could decrease
hundredfold. As the design algorithms are built independent of the model, the
same results and motivation are applicable to more complex battery cell models
and, moreover, to other applications.

###### keywords:

lithium-ion batteries , parameter estimation , optimal input design

[1] organization=University of Konstanz, addressline=Universitaetsstrasse 10,
postcode=78464, city=Konstanz, country=Germany,

[2] organization=Virtual Vehicle Research GmbH, addressline=Inffeldgasse 21a,
postcode=8010, city=Graz, country=Austria,

##  1 Introduction

Mobile appliances and vehicles containing lithium-ion batteries have become
ubiquitous over the recent decade. It is thus paramount to model and simulate
lithium-ion cells efficiently and accurately. High accuracy in simulation
requires the internal states of a cell to be known. These internal states may
include a continuum of physical quantities, e.g., potentials, concentrations
and temperature, i.e., a virtually infinite number of individual values. Very
few quantities may actually be measured non-intrusively, e.g., cell potential
and temperature on the outside.

Typical high-fidelity simulation models are based on the equations described
by Doyle-Fuller-Newman (DFN) model in Newman and Tiedemann (1993); Newman and
Thomas-Alyea (2004). The DFN model is a homogenized system of PDEs, that
typically comprise of more than 50 parameters, a vast majority of which cannot
be determined experimentally.

This model may be simplified further to a Single Particle Model (SPM) by
neglecting spatially resolved effects, e.g., concentration differences among
particles of the same electrode. Although it is known to result in less
accuracy in general, it is still possible to achieve very high model accuracy
by taking the SPM’s limitations into account, e.g., currents below
$1\text{\,}\mathrm{C}$, i.e., the equivalent of a full discharge in more than
1 hour, as described in Santhanagopalan et al. (2006); Ning and Popov (2004).

The parameter estimation for lithium-ion battery models is usually extremely
time consuming. One needs very long input currents and optimization time; for
example, in Reddy et al. (2019), in order to approximate 44 battery parameters
an input current longer than $200\,000\text{\,}\mathrm{s}$ (approximately 55
hours) is used, and even the optimization method lasts for several hours. The
estimation process is based on a fitting argument: chosen a certain input
current, we observe the output voltage of the battery cell. The parameter
estimation is then an optimization problem based on minimizing the discrepancy
between the mathematical model and the real experimental response of the
battery.

Treating non-convex optimization problems, it is possible to converge to a
local minimum, possibly far from the true hidden parameter. For many
optimization problems this can depend on the user-defined input, whose choice
change drastically the results of the parameter estimation.

In the parameter estimation framework the input current is arbitrarily chosen
and, as observed in many works (e.g., López et al. (2016)), the current
impacts strongly on the identification of some of the parameters. For example,
in Liu et al. (2016) an optimized battery cycling is linked to an improved
parameter identifiability and therefore to robust battery health and
experiment safety conditions.

The field of optimal design of experiment can help define, modify and improve
the battery modeling framework. In the last decades, the literature concerning
design of experiment has been extensive; we recommend some work about classic
design of experiment in Goodwin and Payne (1977); Pázman (1986); Rafajłowicz
(1986); Pukelsheim (2006); Atkinson et al. (2007); Pronzato and Pázman (2013).
Different techniques of design of experiment have been used for building new
battery models, or manufacturing optimized layouts in order to improve
conditions such as ageing, thermal design and charging properties; see, e.g.,
Franceschini and Macchietto (2008); Román-Ramírez and Marco (2022).

The focus of this work focuses on the design of optimized inputs in the
context of parameter estimation; see, e.g., Mehra (1974) for a survey on
optimal input design for parameter estimation in dynamic systems. The optimal
input design is strictly related to the optimal design of experiment, and the
main purpose of such techniques in the framework of estimation is to improve
the observability of the system.

It is well known that the many parameters in electrochemical lithium-ion
battery models have different magnitude of observability – and hence of
identifiability. Some parameters are for example more sensitive to extreme
_states of charge_ , and some are more sensitive to fast changes of applied
current – the so-called _high frequency dynamics_. Standard methods of
experimental design analyze the sensitivities of the parameters, namely
measuring how the single parameters impact the observable quantities. In
Schmidt et al. (2010) the parameters of a modified single-particle model are
divided using a grouping algorithm and it is suggested that using experimental
design methods it would be sensible to generate targeted data for the
different groups.

In Park et al. (2018) an optimized excitation input for a parameter estimation
problem is generated from a collection of predefined inputs. On the other
hand, works that do not define a-priori a collection of inputs include Pozzi
et al. (2018) and Pozzi et al. (2020), where the input design of experiment
for parameter identification problem is analyzed for both an isothermal and
not single particle models. In particular, in Pozzi et al. (2020) global
sensitivities are analyzed in a more rigorous approach, and a more precise
parameter estimation is achieved.

In this work we start with an adaptive algorithm that, starting from some
initial guesses for the parameter ${\bm{\mu}}^{0}$ and sequentially generate
new input arrays ${\mathrm{u}}^{[n]}$ at each iteration $n>0$. Each of these
inputs is designed in order to maximize the observability of the current
parameter, and with it we generate data and optimize the parameter
${\bm{\mu}}^{n}$; the inputs are chosen in order to improve or facilitate the
parameter optimization and the convergence to the unknown underlying parameter
${\bm{\mu}}^{*}$, by collecting different pieces of information. Hence, the
result of the input design algorithm is not intended to be a single “optimal”
input, but rather a collection of inputs that collects the most information.
In a later variation of the input design algorithm, we also try to optimize
the experiment resources, such as minimizing the length of the excitation
input. In this framework we defined optimized current sub-intervals, with
ideas similar to the settings in Pozzi et al. (2018).

To quantify the information of a parameter ${\bm{\mu}}$ carried by the current
experiment, an information matrix is used, defined by using the so-called
sensitivities, which measures how the output is sensitive to change in the
model parameters. This information matrix resembles the Fisher information
matrix, a common used statistical object and known lower bound to the
covariance matrix generated by the parameter estimation. Another
interpretation of the Fisher information matrix is mentioned, among others, in
Uciński (2005), where it is seen as an approximation of the Hessian of the
least-squares problem given by the parameter estimation problem. Hence,
improving the well-conditioning of the approximated information matrix can
also help in making the cost function more convex around the unknown parameter
– especially useful given the general non-convexity of the estimation problem.

In the standard battery cell parameter estimation applied currents are many
hours long, in order to observe the battery as long as possible as collect
enough information to infer the tens of parameters. Our goal is to create a
collection of input current profiles much shorter than these, lasting just a
few minutes, to infer 9 of the most observable parameters. One reason to treat
the result of the design as a collection of inputs rather than a single
“optimal” input is that in such a complex setting, we cannot expect to
maximize the observability of all the hidden parameters with just one such
short inputs, and thus we hope to capture the different information with
different settings – or inputs.

##  2 Methods

This section has the following structure: we begin in Section 2.1 with the
description of the numerical model. Then, we proceed to discuss the input
design algorithm in Section 2.2, followed by the discretization of the design
optimization and parameter estimation problems in Sections 2.3 and 2.4. In
Section 2.5 we propose a modification of the design in order to improve a
weakness of the previous framework and cut the experimental time. We consider
the analysis of real battery data in Section 2.6 in order to study the quality
of the generated optimal design.

###  2.1 The numerical model

The numerical model used in this work is based on the assumption that an
electrode may be fully represented by one spherical symmetric particle – hence
it is referred to as SPM. In each particle per electrode $i\in\\{A,C\\}$, with
$A$ and $C$ marking cathode and anode, respectively, the scaled lithium
concentration $\xi_{i}\in[0,1]$ along the respective radius ($r\in[0,R_{i}]$)
for any time $t\in[0,t_{\mathsf{f}}]$ ($t_{\mathsf{f}}>0$) is governed by the
system of equations

$\displaystyle{\dot{\xi}_{i}}{(t,r)}-\frac{1}{r^{2}}\partial_{r}\big{(}r^{2}{D_{i}}\partial_{r}{\xi_{i}}{(t,r)}\big{)}$
$\displaystyle=0,\quad{(t,r)\in(0,t_{\mathsf{f}})\times}(0,R_{i}),$ (1)

where $D_{i}>0$ is each particle’s scaled diffusion constant. The spheres’
symmetry is exploited by setting the natural center boundary condition at
$r_{i}=0$ to $\partial_{r}{\xi_{i}}(t,0)=0$. The boundary condition at the
particles’ outer border $R_{i}$ is modeled by

$\displaystyle-{D_{i}}\partial_{r}{\xi_{i}}(t,R_{i})$
$\displaystyle=\dfrac{1}{{\rho_{i}}{c_{m,i}}}{j_{i}}(\xi_{i}(t,R_{i}),u_{i}(t)),\quad{t\in(0,t_{\mathsf{f}})},$
(2)

where $\rho_{i}c_{m,i}$ is the scaling constant of ${\xi_{i}}{(t,r)}$. The
connection between mass transport and the electric domain is modeled in (2) by
the Butler-Volmer expression

$\displaystyle{j_{i}}({\xi_{i}}{(t,R_{i}),u_{i}(t)})$
$\displaystyle={j^{0}_{i}}({\xi_{i}}{(t,R_{i})})\sinh\\!\left(\tfrac{F}{R\vartheta}(u_{i}(t)-{O_{i}}({\xi_{i}}{(t,R_{i})})\right),$
(3)

where $R$ is the universal gas constant and $\vartheta$ is the temperature. As
the SPM formulation is expressed mainly by the open-circuit potentials
$O_{i}({\xi_{i}}{(t,R_{i})})$ of the two electrodes, these are represented by
a Redlich-Kister sum as derived in Karthikeyan et al. (2008)

$\displaystyle\begin{aligned}
&{O_{i}}({\xi_{i}(t,R_{i})})={U_{0,i}}+\tfrac{R\vartheta}{F}\ln\dfrac{1-{\xi_{i}(t,R_{i})}}{{\xi_{i}(t,R_{i})}}\\\
&\qquad+\tfrac{R\vartheta}{F}\sum\limits_{k=0}^{n_{i}}{A_{i,k}}\bigg{(}(2{\xi_{i}(t,R_{i})}-1)^{k+1}-\dfrac{2{\xi_{i}(t,R_{i})}k(1-{\xi_{i}(t,R_{i})})}{(2{\xi_{i}(t,R_{i})}-1)^{k-1}}\bigg{)},\end{aligned}$
(4)

where $U_{0,i}$ is the open-circuit potential offset for each electrode.
Throughout the rest of the paper, we fix $U_{0,A}=0$ and use $U_{0}$ as a
simplified substitute for $U_{0,C}$. The basic exchange current is modeled by
where $k_{i}$ is the proportional reaction rate factor. For implementation,
$\tilde{k}_{i}\colonequals\log k_{i}$ is used. The electric boundary condition
is applied by the current $i_{\mathsf{cell}}(t)$

$\displaystyle\dfrac{3{m_{i}}}{{\rho_{i}}{R_{i}}}F{j_{i}}(\xi_{i}(t,R_{i}),u_{i}(t))$
$\displaystyle=-{i_{\mathsf{cell}}(t)},$ (5)

where $F$ is Faraday’s constant. The resulting cell voltage
$v_{\mathsf{cell}}$ is then defined as

$\displaystyle{u_{C}}(t)+{u_{A}}(t)+{i}_{\mathsf{cell}}(t){R_{I}}$
$\displaystyle={v_{\mathsf{cell}}(t)},$ (6)

where $R_{I}$ is the total inner resistance.

Our work concerns the estimation of nine parameters of this model, selected on
the basis of their high impact on the model behavior – since, as mentioned for
the cases of Schmidt et al. (2010); Park et al. (2018), different parameters
are more or less identifiable.

Parameter | Meaning | Lower bound | Upper bound
---|---|---|---
$D_{C}$ | Cathode diffusion coefficient | 0.0001 | 0.01
$D_{A}$ | Anode diffusion coefficient | 0.0001 | 0.01
$\xi_{A}$ | Anode initial state of charge | 0.007 | 0.2
$R_{I}$ | Inner resistance | 0.003 | 0.07
$m_{C}$ | Cathode active mass | 0.01 | 0.052
$m_{A}$ | Anode active mass | 0.006 | 0.034
$\tilde{k}_{C}$ | Cathode reaction rate | -20 | -1
$\tilde{k}_{A}$ | Anode reaction rate | -23 | 10
$U_{0}$ | Cathode 0th RK parameter | 3 | 4
Table 1: Unscaled parameter names and bounds.

The collection of considered parameters is depicted in Table 1, together with
lower and upper bounds. In particular, the diffusion coefficients
$D_{C},D_{A}$ appear in (1) as well as the initial state of charge of the
anode $\xi_{A}$ – since $\xi_{C}$ can be recovered algebraically from a given
cell voltage $v|_{t=0}$. The total inner resistance $R_{I}$ appears in the
voltage equation (6), while the masses $m_{C},m_{A}$ are involved in the
current equation (5). The reaction rate logarithms
$\tilde{k}_{C},\tilde{k}_{A}$ are relevant in the exchange current formula
(LABEL:eq:models:spm:final_exchange_current) and, finally, the first Redlich-
Kister term $U_{0}$ is considered, while all the other model parameters are
set to experimental acceptable standards.

As these values have different cardinalities, we apply the following scalings:

$\displaystyle\mu_{1}:=\log_{10}\left(\frac{D_{C}}{D_{C}^{L}}\right),$
$\displaystyle\mu_{2}:=\log_{10}\left(\frac{D_{A}}{D_{A}^{L}}\right),$
$\displaystyle\mu_{3}:=\frac{\xi_{A}}{(\xi_{A}^{L}+\xi_{A}^{U})/2},$
$\displaystyle\mu_{4}:=\frac{R_{I}}{(R_{I}^{L}+R_{I}^{U})/2},$
$\displaystyle\mu_{5}:=\frac{m_{C}}{(m_{C}^{L}+m_{C}^{U})/2},$
$\displaystyle\mu_{6}:=\frac{m_{A}}{(m_{A}^{L}+m_{A}^{U})/2},$
$\displaystyle\mu_{7}:=\tilde{k}_{C}-(\tilde{k}_{C}^{L}+\tilde{k}_{A}^{U})/2,$
$\displaystyle\mu_{8}:=\tilde{k}_{A}-\tilde{k}_{C},$
$\displaystyle\mu_{9}:=\frac{U_{0}}{(U_{0}^{L}+U_{0}^{U})/2}.$

Finally, we report the scaled parameter bounds in Table 2.

${\bm{\mu}}^{L}$ | ${\bm{\mu}}^{U}$
---|---
0 | 2
0 | 2
0.0676328502415459 | 1.93236714975845
0.0821917808219178 | 1.91780821917808
0.32258064516129 | 1.67741935483871
0.3 | 1.7
-9.5 | 9.5
-3 | 11
0.857142857142857 | 1.14285714285714
Table 2: Bounds for the scaled parameters

###  2.2 The input design algorithm

From now on, we use the battery model as a black box system that takes a model
parameter vector ${\bm{\mu}}\in{\mathbb{R}}^{d}$ and a current density
function $i:[0,t_{\mathsf{f}}]\to{\mathbb{R}}$ input variables, whereas the
voltage $v_{{\bm{\mu}},i}:[0,t_{\mathsf{f}}]\to{\mathbb{R}}$ is the output
variable.

We use an adaptive algorithm that starts from an arbitrary parameter
${\bm{\mu}}^{0}$ and, for $n\geq 1$ (called _design iteration_), the algorithm
iterates on three phases:

  * 1.

In the first phase we generate a new current function $i^{[n]}$.

  * 2.

In the second phase we read and store some data $w^{[n]}$ derived by using the
current $i^{[n]}$.

  * 3.

The last phase is the parameter optimization/estimation using previous data
$\\{w^{[1]},\dots,w^{[n]}\\}$, with which we obtain the parameter
${\bm{\mu}}^{n}$, approximation of the true hidden parameter ${\bm{\mu}}^{*}$.

There are different ways of ending this algorithm. One possible way is to stop
the algorithm as soon as the last generated current function is sufficiently
close to one of the already computed currents $i^{[k]}$, $k=1,\ldots,n-1$, in
our collection. Therefore, we use the stopping criterion

$\min_{i\in\\{i^{[1]},\ldots,i^{[n-1]}\\}}\|i^{[n]}-i\|_{L^{2}(0,t_{\mathsf{f}})}<\varepsilon$
(17)

with a properly chosen tolerance $\varepsilon>0$. However, a known limitation
is that we have no guarantee of convergence. This fact can potentially
generate an infinite number of current inputs. In the cases, where we have
used this criterion, we have also set a maximum number of iterations as a
safeguard. An alternative termination is to fix _a priori_ a determinate
number of inputs and forcing the algorithm to generate inputs sufficiently far
from each other, by penalizing the closeness between them. We will later
describe in detail this forced generation of inputs.

Next, we introduce a so-called symmetric information matrix in order to
quantify the uncertainty produced by a certain input $i$. As in Atkinson et
al. (2007), we define the information matrix
${\mathrm{M}}\in{\mathbb{R}}^{d\times d}$ by using the sensitivity variables,
described in details in the next subsection. Then, the uncertainty given by an
information matrix is defined as the inverse by applying the D-optimal design,
or D-criterion (see, e.g., Goodwin and Payne (1977); Atkinson et al. (2007)),
i.e.,

$\phi({\mathrm{M}})\colonequals-\log\left(\det\left({\mathrm{M}}\right)\right),$

which is equivalent to minimizing the generalized variance of the parameters –
as discussed, e.g., in Atkinson et al. (2007). We can also define the
uncertainty function by using the eigenvalues of ${\mathrm{M}}$ – especially
convenient when the dimension $d$ is high and evaluating the determinant gets
computationally expensive. Indeed, if
$\Lambda:=\\{\lambda_{1},\dots,\lambda_{d}\\}$ is the spectrum of
${\mathrm{M}}$, then

$\phi({\mathrm{M}})\colonequals\log\left(\prod_{\lambda_{j}\in\Lambda}\frac{1}{\lambda_{j}}\right).$

Observe that we can also interpret this criterion as minimization of the
volume of the confidence ellipsoid for the parameters.

###  2.3 Approximation of the uncertainty measure

We fix the time step to $\delta=0.1$ seconds, so that we have $K=600$ time
steps in an interval of $60\text{\,}\mathrm{s}$. Time steps $t_{k}:=k\delta$
for $k=0,\dots,K$ are discretized uniformly, and all current functions will be
discretized as step function evaluated on such time steps, so that the current
information information is compressed into the _input array_ ${\mathrm{u}}$ –
or, simply, _input_. Given $n_{\mathrm{u}}\in\mathbb{N}^{+}$ and
$\tau_{\mathrm{u}}\colonequals t_{\mathsf{f}}/n_{\mathrm{u}}$, the array

${\mathrm{u}}\colonequals[u_{1},\dots,u_{n_{\mathrm{u}}},v_{0}]\in{\mathbb{R}}^{n_{\mathrm{u}}+1}$
(18)

corresponds to the current function

$i_{\mathrm{u}}(t)\colonequals\left\\{\begin{aligned} &u_{1}&&\text{for }0\leq
t<\tau_{\mathrm{u}},\\\ &u_{2}&&\text{for }\tau_{\mathrm{u}}\leq
t<2\tau_{\mathrm{u}},\\\ &&&\vdots\\\ &u_{n_{\mathrm{u}}}&&\text{for
}(n_{\mathrm{u}}-1)\tau_{\mathrm{u}}\leq t\leq
t_{\mathsf{f}},\end{aligned}\right.$

and the last component, $v_{0}$, is the initial voltage of the battery, which
is required by the numerical model. With
$v_{{\bm{\mu}},{\mathrm{u}}}:[0,t_{\mathsf{f}}]\to\mathbb{R}$ we indicate the
output voltage given the parameter ${\bm{\mu}}$ and input array ${\mathrm{u}}$
(hence given current $i_{\mathrm{u}}$ and initial voltage $v_{0}$). We seldom
use its discretization ${\mathrm{v}}_{{\bm{\mu}},{\mathrm{u}}}$ evaluated on
the time steps $\\{t_{k}\\}_{k=0}^{K}$.

The sensitivity variables (or, simply, _sensitivities_) are usually evaluated
by solving the so-called _sensitivity equations_. However, in a black-box
model they can only be approximated by using finite differences. For $1\leq
j\leq d$, the $j$-th discretized sensitivity
${\mathrm{s}}^{j}={\mathrm{s}}^{j}_{{\bm{\mu}},{\mathrm{u}}}\in{\mathbb{R}}^{K+1}$
is defined as

${\mathrm{s}}^{j}_{{\bm{\mu}},{\mathrm{u}}}:=\frac{{\mathrm{v}}_{{\bm{\mu}}+\nu{\mathrm{e}}_{j},{\mathrm{u}}}-{\mathrm{v}}_{{\bm{\mu}},{\mathrm{u}}}}{\nu}\approx\frac{{\mathrm{d}}}{{\mathrm{d}}\mu_{j}}{\mathrm{v}}_{{\bm{\mu}},{\mathrm{u}}},$

where ${\mathrm{e}}_{j}$ is the $j$-th Euclidean basis element, and
$\nu:=10^{-3}$. Let us mention that the approximation of the sensitivities can
be improved by using other (and more accurate) methods to evaluate the partial
derivatives, such as automatic differentiation (see, e.g., Rall (1981)).

Then, the information matrix
${\mathrm{M}}={\mathrm{M}}_{\bm{\mu}}^{\mathrm{u}}$, evaluated at some
parameter ${\bm{\mu}}$ and input array ${\mathrm{u}}$, is defined as

$\left({\mathrm{M}}\right)_{j,l}\colonequals\sum_{k=0}^{K-1}\frac{{\mathrm{s}}^{j}_{k}{\mathrm{s}}^{l}_{k}+{\mathrm{s}}^{j}_{k+1}{\mathrm{s}}^{l}_{k+1}}{2}(t_{k+1}-t_{k}),\qquad
1\leq j,l\leq d,$

namely, as the approximation of the $L^{2}$ inner product $\langle
s^{j},s^{l}\rangle_{L^{2}(0,t_{\mathsf{f}})}$ via the trapezoidal rule, where
the sensitivities are evaluated at the parameter ${\bm{\mu}}$ and the current
$i_{\mathrm{u}}$.

As we want to maximize the total information coming from a collection of
inputs, we modify the information matrix of each design iteration. In
particular, we simply sum the information matrices, thus at design iteration
$n$ we define the _optimal design function_

$\Phi({\mathrm{u}})\colonequals-\log_{10}\left(\det\bigg{(}\sum_{m=1}^{n-1}{\mathrm{M}}_{{\bm{\mu}}^{n-1}}^{[m]}+{\mathrm{M}}_{{\bm{\mu}}^{n-1}}^{\mathrm{u}}\bigg{)}\right)+\gamma\,{\|{\mathrm{u}}\|}^{2}_{2},$

where ${\mathrm{M}}^{[m]}$, $m=1,\dots,n-1$, denotes the information matrix
corresponding to input array ${\mathrm{u}}^{[m]}$, $\gamma\colonequals
10^{-4}$ accompanies a regularization term and $\|\cdot\|_{p}$ indicates the
$\ell^{p}$-norm for $1\leq p\leq\infty$.

Hence, to find a new input array we solve the optimization problem

$\min_{{\mathrm{u}}\in{\mathbb{R}}^{n_{\mathrm{u}}+1}}\Phi({\mathrm{u}})\quad\text{subject
to}\quad{\mathrm{u}}^{\mathsf{min}}\leq{\mathrm{u}}\leq{\mathrm{u}}^{\mathsf{max}},$
(19)

where ${\mathrm{u}}^{\mathsf{min}}$ and ${\mathrm{u}}^{\mathsf{max}}$ are the
bounds in ${\mathbb{R}}^{n_{\mathrm{u}}+1}$ chosen as

$\displaystyle-8.8\equalscolon u^{\mathsf{min}}_{j}\leq u_{j}\leq
u^{\mathsf{max}}_{j}\colonequals 8.8,\quad j=1,\dots,n_{\mathrm{u}},$
$\displaystyle 3.3\equalscolon u^{\mathsf{min}}_{n_{\mathrm{u}}+1}\leq
v_{0}\leq u^{\mathsf{max}}_{n_{\mathrm{u}}+1}\colonequals 4.1.$

in our experiments, where the unit of current is  and the unit of the initial
voltage is .

The optimization problem (19) is solved numerically using the routine
fmin_l_bfgs_b from the scipy Python library; see, e.g., Virtanen et al.
(2020).

As mentioned earlier, when we want the algorithm to generate a fixed number of
inputs, we need to make them distinct, since there is no information gain when
the input generated is equal to one we already have. We therefore define an
alternative (nonsmooth) optimization function that enforces the new inputs to
be sufficiently far from each other

$\hat{\Phi}({\mathrm{u}})\colonequals\Phi({\mathrm{u}})+{\textstyle\sum\limits_{\begin{subarray}{c}{\mathrm{u}}^{[n]}\text{
previous}\\\
\text{controls}\end{subarray}}}\frac{1}{1+100\,{\|{\mathrm{u}}-{\mathrm{u}}^{[n]}\|}_{\infty}}.$
(20)

The penalizing function, as well as the scalar 100 have been chosen in an
intuitive way in order to guarantee that we stop our algorithm provided our
current control is sufficiently far from the already computed controls, but
also not to modify the initial optimization problem too strongly.

###  2.4 Parameter estimation

Assuming now that the input vector ${\mathrm{u}}$ (hence the current $i$) is
fixed, we want to find the parameter vector ${\bm{\mu}}$ minimizing the
discrepancy between our parametrized model and data coming from the real
application. We assume, for the moment, that the data is given by the voltage
corresponding to some unknown parameter ${\bm{\mu}}^{*}$, evaluated on the
time array ${\mathrm{t}}$, i.e.,

$w_{\mathrm{u}}=v_{{\bm{\mu}}^{*},{\mathrm{u}}}:[0,t_{\mathsf{f}}]\to\mathbb{R},$

and its discretization is indicated as
${\mathrm{w}}_{\mathrm{u}}\colonequals{\mathrm{v}}_{{\bm{\mu}}^{*},{\mathrm{u}}}\in{\mathbb{R}}^{K+1}$.
This means that we consider noiseless virtual data.

For the numerical optimization we use a least squares method (from
scipy.optimize, see Branch et al. (1999)) to fit
${\mathrm{v}}_{{\bm{\mu}},{\mathrm{u}}}$ to ${\mathrm{w}}_{\mathrm{u}}$. In
particular, we minimize the relative error
${\mathrm{e}}_{{\mathrm{u}}}({\bm{\mu}})\colonequals({\mathrm{v}}_{{\bm{\mu}},{\mathrm{u}}}-{\mathrm{w}}_{\mathrm{u}})/{\mathrm{w}}_{\mathrm{u}}$
(where the division is intended component-wise) by in particular solving

$\min_{{\bm{\mu}}\in\mathcal{P}_{\mathsf{ad}}}J({\bm{\mu}})\colonequals\frac{1}{2}{\left\|\frac{{\mathrm{v}}_{{\bm{\mu}},{\mathrm{u}}}-{\mathrm{w}}_{\mathrm{u}}}{{\mathrm{w}}_{\mathrm{u}}}\right\|}^{2}_{2},$
(21)

where $\mathcal{P}_{\mathsf{ad}}\subset\mathbb{R}^{d}$ is the closed, convex
set of admissible parameters – in our case the box-constrained set defined by
${\bm{\mu}}^{L}\leq{\bm{\mu}}\leq{\bm{\mu}}^{U}$ in Table 2.

In the framework of multiple data, we can simply merge information by stacking
the data vectors together. In particular, given
${\mathrm{u}}^{[1]},\ldots,{\mathrm{u}}^{[n]}$ input arrays generated by the
algorithm, and ${\mathrm{w}}^{[1]},\ldots,{\mathrm{w}}^{[n]}$ associated data
vectors (with
${\mathrm{w}}^{[n]}\colonequals{\mathrm{w}}_{{\mathrm{u}}^{[n]}}={\mathrm{v}}_{{\bm{\mu}}^{*},{\mathrm{u}}^{[n]}}$),
then we want to fit ${\mathrm{v}}_{{\bm{\mu}},{\mathrm{u}}^{[1]}}$ to
${\mathrm{w}}^{[1]}$ and so on. We concatenate all vectors of relative error
into one vector
$[{\mathrm{e}}_{{\mathrm{u}}^{[1]}}({\bm{\mu}}),\ldots,{\mathrm{e}}_{{\mathrm{u}}^{[n]}}({\bm{\mu}})]\in{\mathbb{R}}^{n(K+1)}$
and we minimize its $\ell^{2}$-norm. The parameter generated at OID iteration
$n$ is then the optimizer ${\bm{\mu}}^{n}$. We mention that in order to
optimize computing time, we evaluate voltage vectors in a parallel way for all
different current inputs.

We later explore the effect of the input design on the parameter optimization
problem by analyzing the Hessian matrix $H$ of the least-squares function $J$
defined in (21), evaluated in the optimum ${\bm{\mu}}^{*}$. In particular, we
analyze the value

$\beta\colonequals\lambda_{\mathsf{max}}(H)/\lambda_{\mathsf{min}}(H),$ (22)

i.e., the conditioning number of the Hessian, where at iteration $n$ this
value will be indicated as $\beta_{n}$ – as the Hessian matrix changes at each
OID iteration $n$. When this value decreases, the smallest eigenvalue is
growing because in this construction the biggest eigenvalue cannot decrease,
hence indicating positivity of the Hessian and convexity of the optimization
problem. Let us comment on the fact that we cannot use the term $\beta$ in the
definition of the optimal input design optimization problem, since this
quantity depends on knowing the hidden parameter ${\bm{\mu}}^{*}$.
Nonetheless, we hope to minimize this value indirectly.

###  2.5 Another design framework

Generating ten different short inputs sounds like a good idea if we want to
look at the behavior of a battery cell in all different states of charge:
starting from different initial voltages serves exactly this purpose. But
there might be a problem in the actual use of these different current
profiles, since we cannot evaluate all inputs at the same time in a parallel
way – as we do in the numerical model. On the contrary, before and after every
input we need a phase of preparation to bring the battery cell at the right
initial voltage, and a long phase of rest to let the internal electro-chemical
dynamics stabilize. For this reason, even if ten different inputs of
$60\text{\,}\mathrm{s}$ each together sum up to $600\text{\,}\mathrm{s}$,
actually setting up and running the experiment can take more than
$60\,000\text{\,}\mathrm{s}$.

Hence, we have modified the design framework and structure in order to avoid
this problem. The idea is to generate a sequence of new input intervals
successively, rather than to use them as entirely different current profiles.
As a consequence, we may not choose the initial voltage of each interval,
because this will be given by the ending voltage of the previous interval.
Rather, we may only choose the initial voltage, namely the starting voltage of
the first interval; this is chosen as $3.9\text{\,}\mathrm{V}$, equivalent to
appr. $75\text{\,}\mathrm{\char 37\relax}$ SOC.

We also realize that we need a resting interval in order to capture the long-
term dynamics. Hence, our input is translated into six “jumps” of a
$120\text{\,}\mathrm{s}$ high-frequency time interval followed by a
$600\text{\,}\mathrm{s}$ resting phase. An example of the first input with
arbitrary jumps is plotted in Figure 1. This time we are not arbitrarily
choosing the first input ${\mathrm{u}}^{[1]}=(u^{[1]}_{1},\dots,u^{[1]}_{6})$,
but it is found in the first step of the OID algorithm. In the following
iterations, we find input array ${\mathrm{u}}^{[n]}$ by optimizing the
uncertainty of the full-concatenated current input, i.e., we add the new input
to the old one and hence consider only one longer input. See Figure 1 for a
visual explanation.

Figure 1: Visual arbitrary example of the concatenated-inputs setting. Above:
initial current function, where the input variables define the 6 jumps and a
successive rest phase is imposed. Below: once the first current function is
fixed, the next input array ${\mathrm{u}}^{[2]}$ is optimized by optimizing
the concatenated current function $i^{[2]}$.

###  2.6 Analysis on real battery data

While we evaluate these optimal inputs using virtual data
$w_{\mathrm{u}}=v_{{\bm{\mu}}^{*},{\mathrm{u}}}$ in the algorithm, we want to
check how beneficial they are in a real-world parameter estimation problem. To
this end, we have run experiments on a real battery cell and collected the
measurement data. The measured voltage data obtained through cycling two
Molicel INR21700-P45B commercial cells is used in this work. For a given
starting voltage, the cell is first fully charged at a constant current of
$4.5\text{\,}\mathrm{A}$ and then, at reaching the top voltage of
$4.2\text{\,}\mathrm{V}$, switch to constant voltage mode until the magnitude
of the current tapered down below $0.05\text{\,}\mathrm{A}$. After a resting
period of $600\text{\,}\mathrm{s}$, the cell is discharged at
$4.5\text{\,}\mathrm{A}$ to the given voltage followed again by a constant
voltage step until $0.05\text{\,}\mathrm{A}$ is underrun. Then, the designed
current profile is applied in a “current simulation” step. All tests were
carried out using an Arbin LBT21084HC battery testing system and a Memmert
incubator with Peltier cooling (model IPP600) for maintaining the temperature
at $25\pm 2.\text{\,}\mathrm{\SIUnitSymbolCelsius}$ by forced air cooling.

We want to use these real-world data to measure the effect of the optimal
input design on the parameter estimation. This data also includes preparation
steps at the beginning and the resting phase at the beginning, and depending
on the test we will use information coming from the extra phases or not.
Furthermore, the time stepping was chosen depending on the situation, varying
between a minimum of $0.0002\text{\,}\mathrm{s}$ (used in high-frequency
dynamics) and a maximum of $1.0\text{\,}\mathrm{s}$ (used in the resting
phases).

##  3 Results and Discussion

###  3.1 First input design: collection of inputs

We have applied the input design algorithm with piecewise constant inputs
consisting of 24 jumps, therefore each continuous sub-interval is
$2.5\text{\,}\mathrm{s}$ long and the input dimension is 25. We furthermore
set the maximum number of inputs to 10 and arbitrarily fix the first input as

${\mathrm{u}}^{[1]}\colonequals[-1,+1,-1,+1,\dots,-1,+1,3.7]\in{\mathbb{R}}^{25}.$
(23)

This corresponds to a quickly alternating input applied to a roughly mid-
charge battery. The generated inputs are plotted in Figure 2.

Figure 2: Input functions $i^{[n]}$, $n=1,\dots,10$ generated by the
algorithm. In the lower left corner the initial voltages $v_{0}$ are
indicated.

Notice how some of the inputs look very similar, but the penalization function
introduced in (20) guarantees that we do not use the exact same input array
more than once.

Figure 3: Top, blue: information corresponding to each design iteration $n$.
First line: uncertainty function and cost evaluated in the optimal parameter.
Second line: Euclidean norm of the error between the optimal parameter and the
true parameter and value $\beta_{n}$ given in (22). Bottom: qualitative
statistical study for the input design algorithm. Dark bars (blue) show the
results of 100 parameter estimations using only the input given by
${\mathrm{u}}^{[1]}$. Light bars (orange) show the results using the optimal
collection of inputs. The red crosses mark the hidden parameters.

In order to analyze the effect of the input design algorithm, in Figure 3 we
plot the change in different quantities, such as the design algorithm the
uncertainty function $\hat{\Phi}$, the parameter optimization cost $J$, the
relative error
$e^{\mathsf{rel}}({\bm{\mu}}^{n}):=\|{\bm{\mu}}^{*}-{\bm{\mu}}^{n}\|_{2}/\|{\bm{\mu}}^{*}\|_{2}$
and the sequence $\beta_{n}$ defined in (22). In the same plot, in the lower
area we plot a qualitative analysis of the input design, where we compare the
results of 100 parameter optimizations with randomized initial parameter at
the beginning and at the end of the algorithm, i.e., comparing the information
before (in dark colored bars) and after (in light colored bars). Firstly, we
can comment on the properties of the initial input ${\mathrm{u}}^{[1]}$
defined in (23): while it looks like the error $J$ is already low ($1.77\cdot
10^{-10}$) at the initial parameter, the relative error of the parameter reads
in fact 2.22. Namely, the initial input holds little information and it does
not provide data such that the parameter optimization approximates
${\bm{\mu}}^{*}$. We can observe this in the bar plot, too: the blue bars do
not approximate the parameter indicated with the red cross most of the time.

During the design algorithm the uncertainty function, the parameter
optimization cost and the relative error do not decrease monotonously. The OID
algorithm converges to the true parameter after some preliminary steps, in
particular reaching the neighborhood of ${\bm{\mu}}^{*}$ in the fourth
iteration. This could be explained by the behavior of the sequence
$\beta_{n}$, which decreases monotonously, reaching a plateau after four
iterations, when the algorithm finds the hidden parameter with a relative
error of approximately $3.73\cdot 10^{-10}$.

In the bar plots of Figure 3 we can observe that the input design improves the
parameter estimation significantly: with a non-informative input (dark bars),
the parameter estimation fails almost every time, while using the collection
of selected inputs (light bars) the convergence to the hidden parameter (the
red cross) improves greatly.

Although the converge to the optimal parameter does not happen always, the
optimization results cluster around the optimal parameters. Overall, we can
see a degree of bias in the bar plots of Figure 3 – namely, we can guess the
hidden parameter in an easier way. However, this framework might find in its
applicability some unexpected counter-effects, which we discuss in the next
section.

###  3.2 Second input design: concatenated input

In this setting we generate 9 inputs, each lasting $120\text{\,}\mathrm{s}$:
therefore, concatenating the respective jumps and rest phases together we
reach an “optimal” input lasting $6480\text{\,}\mathrm{s}$, i.e. 108 minutes.
This optimal current $i^{[9]}$ generated by the algorithm is plotted in Figure
4 together with the corresponding output voltage
$v_{{\bm{\mu}}^{*},{\mathrm{u}}^{[9]}}$ produced from the battery cell model
using such input. Further information for each OID iteration are also plotted
in Figure 4. In this case we can observe a non monotone decrease of the
quantity $\beta_{n}$, but still a good convergence of the parameter
optimization in later iterations, reaching a relative parameter error of
$9.74\cdot 10^{-12}$.

Figure 4: Top left, in black: optimal input generated by the algorithm and
corresponding output voltage generated by the battery cell model. Top right,
in blue: plots of different quantities at OID iteration $n$. Bottom bar plots:
qualitative statistical results of the optimal parameter of 100 parameter
estimations. The red crosses mark the true parameters ${\bm{\mu}}^{*}$.

Similar to what we did previously, in the bottom part of Figure 4 we plot the
result of 100 separate executions of the parameter optimization in order to
show the estimation error for randomized initial parameters. We observe an
overall better consistence and accuracy in the convergence to the true
parameter; considering the fact that we the overall experiment time will be
reduced at least 4 times, as shown in the next section, we regard this as a
great improvement of the input design presented in the previous section.

###  3.3 Real battery data

We consider four different situations, depending on two different factors:
whether we use the collection of inputs derived in Section 3.1 or the
concatenated input derived in Section 3.2, and whether we consider the
additional data provided by the preparation and resting phases, or not. These
choices are summarized in Table 3, and some of the results given in each test
are summarized in Table 4.

| Cut data: | Full data:
---|---|---
| only impulses | impulses + rest phases
Collection of inputs | Test 1 | Test 2
Concatenated input | Test 3 | Test 4
Table 3: Considered tests for the analysis of real data. | TESTS
---|---
| 1 | 2 | 3 | 4
Experiment time ($s$) | 600 | 63,362 | 6,480 | 14,199
Data points | 20,350 | 83,160 | 216,002 | 223,727
Optimization time ($s$) | 800 | 154 | 550 | 450
Table 4: First line: total input time $t_{\mathsf{f}}$; second line: number of
data points; third line: average time for the parameter optimization.

In each of these situations we run 100 parameter optimizations in order to
infer the unknown battery cell parameters. We analyze the quality of the
estimation using a bar plot, and we try to infer the true parameter by
proposing an optimal parameter. Since this is unknown, we finally perform a
qualitative error analysis on the proposed parameters given in each test.

In particular, we will propose three parameters ${\bm{\mu}}^{\mathsf{opt},2}$,
${\bm{\mu}}^{\mathsf{opt},3}$ and ${\bm{\mu}}^{\mathsf{opt},4}$ inferred from
test 2, 3 and 4, respectively. For each of these parameters we plot the output
voltage produced by the numerical model $v_{{\bm{\mu}},{\mathrm{u}}}(t)$, as
well as the relative error given by the discrepancy with the real battery data
$w(t)$, namely

$e^{\mathsf{rel},[n]}_{\bm{\mu}^{\mathsf{opt},k}}(t):=(w^{[n]}(t)-v_{{\bm{\mu}}^{\mathsf{opt},k},{\mathrm{u}}^{[n]}}(t))/w^{[n]}(t)\qquad\text{for
}n=2,3,4\ \text{and }k=2,3,4.$

The results are shown in the upcoming figures.

#### Test 1

By counting only the 10 one-minute intervals, we get a relatively small
experiment time and number of data points. Although the optimization is longer
than in other cases, we cannot identify the unknown parameters with
confidence. Even if we are using the same input collections as the results
shown in Figure 3, the bar plots shown in Figure 5 look much worse. Therefore,
in this case we conclude that this input design is not successful, as we
derive no hint about where the hidden parameters might be.

Figure 5: Results for Test 1. Bar plots of qualitative analysis of 100
parameter estimations starting from randomized initial guess.

#### Test 2

Since the production of the data necessary for Test 1 produces, by default,
the data used in Test 2, this test is necessarily more informative and hence
the estimation of the true parameter is more successful. However, the
experiment lasts for over 17 hours, with an average optimization time of about
two and a half minutes. As shown in Figure 6, we can identify a candidate
${\bm{\mu}}^{\mathsf{opt},2}$, indicated with the blue mark.

The analysis of the model error given by the proposed parameter confirms the
quality of such parameter: for the inputs ${\mathrm{u}}^{[2]}$ and
${\mathrm{u}}^{[3]}$ the relative error oscillates in the order of $10^{-3}$,
while for the input ${\mathrm{u}}^{[4]}$ the order of the relative error is
$10^{-2}$. The numerical model tuned with such parameter is therefore an
accurate simulator of the battery cell taken into consideration.

Figure 6: Results for Test 2. Top: bar plots of qualitative analysis of 100
parameter estimations starting from randomized initial guess; the blue cross
finds the proposed parameter ${\bm{\mu}}^{\mathsf{opt},2}$; bottom: data,
output and error generated by the proposed parameter.

#### Test 3

By construction, the time of the generated experiment is 108 minutes, and the
optimization over 216,002 points takes in average several minutes. However, we
can see that in this case the estimations gravitate around different local
minima. With ${\bm{\mu}}^{\mathsf{opt},3}$ we have indicated the most common
optimizer, indicated in Figure 7 with the blue mark.

This parameter produces accurate results for the data $w^{[3]}$, but
accumulates a too high error in the higher voltage changes occurring in data
observations $w^{[2]}$ and $w^{[4]}$, hence not capturing very well the full
range of battery cell dynamics.

Figure 7: Results for Test 3. Top: bar plots of qualitative analysis of 100
parameter estimations starting from randomized initial guess; the blue cross
finds the proposed parameter ${\bm{\mu}}^{\mathsf{opt},3}$; bottom: data,
output and error generated by the proposed parameter.

#### Test 4

As mentioned already, Test 4 is obtained by using the data of Test 3 together
with the measurements available in the preparation and rest phases. The
candidate parameter is definitely more concentrated around a single parameter
${\bm{\mu}}^{\mathsf{opt},4}$, plotted in Figure 8 with the blue cross.
Counter-intuitively, the optimization over a larger set of data produces in
this case a faster convergence in the parameter optimization. We interpret
this effect as a result of the well-posedness of the Hessian matrix and the
convexity of the parameter optimization.

Figure 8: Results for Test 4. Top: bar plots of qualitative analysis of 100
parameter estimations starting from randomized initial guess; the blue cross
finds the proposed parameter ${\bm{\mu}}^{\mathsf{opt},4}$; bottom: data,
output and error generated by the proposed parameter.

We can see the results in Figure 8. The proposed parameter
${\bm{\mu}}^{\mathsf{opt},4}$ holds good approximation of the 3 sets of data,
comparable to the one given by parameter ${\bm{\mu}}^{\mathsf{opt},2}$. The
construction introduced in Section 2.5 leads to major time cuts with the only
impact being the optimization time, which averages to about
$154\text{\,}\mathrm{s}$ for Test 2 and $450\text{\,}\mathrm{s}$ for Test 4,
since the number of data points has almost tripled.

### Comments on the accuracy of the optimizers

At a first glance, it seems that we cannot find a parameter to perfectly
replicate the data. This might depend on a number of factors. The first
justification comes from the fact that we are working with a simplified
numerical model with a reduced number of active, unknown parameters, and
therefore the approximation of a real battery cell data is indeed impossible.
We can also imagine that adjusting a number of parameters which now are fixed
– for example, the RK parameters – we can fit better to the data.

On the bright side, we can make considerations about the merit of the input
design. It has in fact improved robustness the parameter estimation, because
it has made it harder, i.e., it happens more rarely, to fall in other local
optima. While it is possible that a better fitting parameter is hiding in
another local optimum, it is highly unlikely.

It looks like the parameter inferred from Test 2, i.e., the longest
experiment, is the one better approximating the data. Test 4, which is over 4
times shorter also produces a parameter that approximates well the real data
dynamics. As usual, using more expensive experiments produces data of better
quality, while using shorter experiments might save time by sacrificing some
of the accuracy. We believe the main advantage of using the input design
algorithm is the enforcement of the well-posedness of the Hessian matrix
corresponding to the parameter optimization problem. The short optimization
times might be the best experimental benefit, as many optimization methods
work very well with convex problems.

The input design might be even more effective when applied to more complex
battery models, as it could be able to accurately represent the battery cell
in all conditions: it can indeed happen that our single particle model fails
at reproducing real battery cells in the settings chosen by the input design
where valuable information lies, for example, for very high or very low states
of charge.

We can, of course, also argue that the parameter estimation has failed because
the input design was badly set up. We have selected inputs with very
restricted shapes and lengths in an arbitrary and intuitive way. For example,
with longer current jumps, or longer pauses, or in general longer experiment,
the input design could have found the optimal parameter reproducing the data.
Of course, the possibilities of generalization are endless, and we encourage
future works to obtain better results.

##  4 Conclusions

In this work we have applied an algorithm for the optimal input design for the
parameter estimation of a lithium-ion battery model. After introducing the
numerical model, we have described a general iterative input design algorithm,
where inputs are chosen in order to maximize the observability of the
parameters, quantified using an approximated Fisher information matrix. By
discretizing the information matrix we then define the input design problem as
a finite dimensional optimization problem.

At first, we generate a collection of different input functions in order to
maximize the overall observation, i.e., the collective information. The data
given by this collection of input is indeed effective for the parameter
estimation, and we show the effect of the input design on the convexity of the
optimization problem. However, in the lab experiments, the use of the optimal
inputs results in extremely long experiment times, since before and after each
input some extra time needs to be taken into account, either for a resting
phase or for taking the battery to a certain state of charge.

For this reason, a second input design algorithm is used, where a continuous,
longer input is obtained by concatenation of optimal sub-intervals. In this
way, the problem of resting and state of charge preparation is avoided – or at
least done only once – and the overall experiment is shorter.

We derive four tests to interpret the effect of the input design and the
accuracy of the parameter estimation using real battery data. In three out of
four tests the input design can indeed make the parameter estimation easier,
in the way that it makes us aware of a common local optimum, which we propose
as the approximator of the unknown parameter.

Although we infer different parameters with different discrepancy between the
numerical output and the real battery voltage, we observe how the input design
was able to iteratively generate a current profile facilitating the parameter
estimation, both in accuracy and in optimization time.

As mentioned, we believe that the advantage of the input design algorithms
would be more evident in more complex battery cell models, where all the
electro-chemical dynamics are reproduced and the data can be completely
reproduced by the numerical model. In a wider scope, the techniques of optimal
input/experimental design propose a goal-oriented way of defining experiment
inputs and settings. Such designs might be built to guarantee several
mathematical properties and rigor. The management of resources, such as time
or energy, could be imposed a-priori in the design of the experiments in order
to prevent waste.

## Acknowledgments

The authors would like to acknowledge the financial support within the COMET
K2 Competence Centers for Excellent Technologies from the Austrian Federal
Ministry for Climate Action (BMK), the Austrian Federal Ministry for Labour
and Economy (BMAW), the Province of Styria (Dept. 12) and the Styrian Business
Promotion Agency (SFG). The Austrian Research Promotion Agency (FFG) has been
authorised for the programme management. The authors further acknowledge
support by the Deutsche Forschungsgemeinschaft for the project Localized
Reduced Basis Methods for PDE-constrained Parameter Optimization under
contract VO 1658/6-1.

## Author contributions

Andrea Petrocchi: Conceptualization, Methodology, Software, Validation, Formal
analysis, Writing - Original Draft, Visualization. Matthias K. Scharrer:
Software, Resources, Writing - Original Draft. Franz Pichler: Methodology,
Software. Stefan Volkwein: Methodology, Writing - Review & Editing,
Supervision, Project administration, Funding acquisition.

## References

  * Atkinson et al. (2007) Atkinson, A., Donev, A., Tobias, R., 2007.  Optimum experimental designs, with SAS. volume 34 of Oxford Statistical Science Series.  Oxford University Press, Oxford.  doi:10.1093/oso/9780199296590.001.0001.
  * Branch et al. (1999) Branch, M., Coleman, T., Li, Y., 1999.  A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems.  SIAM Journal on Scientific Computing 21, 1–23.  doi:10.1137/S1064827595289108.
  * Franceschini and Macchietto (2008) Franceschini, G., Macchietto, S., 2008\.  Model-based design of experiments for parameter precision: State of the art.  Chemical Engineering Science 63, 4846–4872.
  * Goodwin and Payne (1977) Goodwin, G., Payne, R., 1977\.  Dynamic system identification. volume 136 of Mathematics in Science and Engineering.  Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London.
  * Karthikeyan et al. (2008) Karthikeyan, D.K., Sikha, G., White, R.E., 2008.  Thermodynamic model development for lithium intercalation electrodes.  Journal of Power Sources 185, 1398 – 1407.  URL: http://www.sciencedirect.com/science/article/B6TH1-4T4XR9X-2/2/c0b51ae8a4bdea0067dbefc7ed39d7a2, doi:DOI:10.1016/j.jpowsour.2008.07.077.
  * Liu et al. (2016) Liu, J., Rothenberger, M., Mendoza, S., Mishra, P., Jung, Y.s., Fathy, H.K., 2016\.  Can an identifiability-optimizing test protocol improve the robustness of subsequent health-conscious lithium-ion battery control? an illustrative case study, in: 2016 American Control Conference (ACC), IEEE. pp. 6320–6325.
  * López et al. (2016) López, C., Diana, C., Wozny, G., Flores-Tlacuahuac, A., Vasquez-Medrano, R., Zavala, V., 2016\.  A computational framework for identifiability and ill-conditioning analysis of lithium-ion battery models.  Industrial & Engineering Chemistry Research 55, 3026–3042.  doi:10.1021/acs.iecr.5b03910.
  * Mehra (1974) Mehra, R., 1974.  Optimal input signals for parameter estimation in dynamic systems–survey and new results.  IEEE transactions on automatic control 19, 753–768.
  * Newman and Thomas-Alyea (2004) Newman, J., Thomas-Alyea, K.E., 2004\.  Electrochemical Systems.  3rd ed., John Wiley & Sons, Inc., Hoboken, New Jersey.  doi:ISBN:978-0471477563.
  * Newman and Tiedemann (1993) Newman, J., Tiedemann, W., 1993\.  Potential and current distribution in electrochemical cells.  Journal of The Electrochemical Society 140, 1961–1968.  URL: http://link.aip.org/link/?JES/140/1961/1, doi:10.1149/1.2220746.
  * Ning and Popov (2004) Ning, G., Popov, B.N., 2004\.  Cycle life modeling of lithium-ion batteries.  Journal of The Electrochemical Society 151, A1584–A1591.  URL: http://link.aip.org/link/?JES/151/A1584/1, doi:10.1149/1.1787631.
  * Park et al. (2018) Park, S., Kato, D., Gima, Z., Klein, R., Moura, S., 2018\.  Optimal experimental design for parameterization of an electrochemical lithium-ion battery model.  Journal of The Electrochemical Society 165, A1309.
  * Pozzi et al. (2018) Pozzi, A., Ciaramella, G., Volkwein, S., Raimondo, D., 2018\.  Optimal design of experiments for a lithium-ion cell: parameters identification of an isothermal single particle model with electrolyte dynamics.  Industrial & Engineering Chemistry Research 58, 1286–1299.  doi:10.1021/acs.iecr.8b04580.
  * Pozzi et al. (2020) Pozzi, A., Xie, X., Raimondo, D.M., Schenkendorf, R., 2020\.  Global sensitivity methods for design of experiments in lithium-ion battery context.  IFAC-PapersOnLine 53, 7248–7255.
  * Pronzato and Pázman (2013) Pronzato, L., Pázman, A., 2013\.  Design of experiments in nonlinear models.  Lecture notes in statistics 212.
  * Pukelsheim (2006) Pukelsheim, F., 2006.  Optimal design of experiments.  SIAM.
  * Pázman (1986) Pázman, A., 1986.  Foundations of optimum experimental design.  Mathematics and its applications, D. Reidel.  URL: https://cir.nii.ac.jp/crid/1130282269915354752.
  * Rafajłowicz (1986) Rafajłowicz, E., 1986.  Optimum choice of moving sensor trajectories for distributed-parameter system identification.  International Journal of Control 43, 1441–1451.  doi:10.1080/00207178608933550.
  * Rall (1981) Rall, L., 1981.  Automatic Differentiation: Techniques and Applications. volume 120 of Lecture Notes in Computer Science.  Springer, Berlin.  doi:10.1007/3-540-10861-0.
  * Reddy et al. (2019) Reddy, S., Scharrer, M., Pichler, F., Watzenig, D., Dulikravich, G., 2019.  Accelerating parameter estimation in doyle–fuller–newman model for lithium-ion batteries.  COMPEL-The international journal for computation and mathematics in electrical and electronic engineering 38, 1533–1544.
  * Román-Ramírez and Marco (2022) Román-Ramírez, L.A., Marco, J., 2022\.  Design of experiments applied to lithium-ion batteries: A literature review.  Applied Energy 320, 119305\.
  * Santhanagopalan et al. (2006) Santhanagopalan, S., Guo, Q., Ramadass, P., White, R.E., 2006\.  Review of models for predicting the cycling performance of lithium ion batteries.  Journal of Power Sources 156, 620 – 628.  URL: http://www.sciencedirect.com/science/article/B6TH1-4GM4620-7/2/022854c2137ca57cdc1f45fb0d5542e2, doi:DOI:10.1016/j.jpowsour.2005.05.070.
  * Schmidt et al. (2010) Schmidt, A.P., Bitzer, M., Imre, Á.W., Guzzella, L., 2010\.  Experiment-driven electrochemical modeling and systematic parameterization for a lithium-ion battery cell.  Journal of Power Sources 195, 5071–5080.
  * Uciński (2005) Uciński, D., 2005.  Optimal measurement methods for distributed parameter system identification.  Systems and Control Series, CRC Press, Boca Raton, FL.
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S., Brett, M., Wilson, J., Millman, K., Mayorov, N., Nelson, A., Jones, E., Kern, R., Larson, E., Carey, C., Polat, İ., Feng, Y., Moore, E., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E., Harris, C., Archibald, A., Ribeiro, A., Pedregosa, F., van Mulbregt, P., SciPy 1.0 Contributors, 2020.  SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python.  Nature Methods 17, 261–272.