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Accepted to be published in: 2023: IEEE 1st International Conference on
Advanced Innovations in Smart Cities (ICAISC 2023), Jeddah, Saudi Arabia,
January 23-25, 2023.

# Document Provenance and Authentication through Authorship Classification

Muhammad Tayyab Zamir1, Muhammad Asif Ayub2, Jebran Khan, Muhammad Jawad
Ikram,
Nasir Ahmad, Kashif Ahmad

1 Abasyn University Islamabad Campus  2 DCSE, UET, Peshawar, Pakistan,
<EMAIL_ADDRESS><EMAIL_ADDRESS>3 CSIT, Jeddah
International College, Saudi Arabia,
<EMAIL_ADDRESS>4 Department of AI, AJOU University, South Korea,
<EMAIL_ADDRESS>5 Department of Computer Science, Munster Technological
University, Cork, Ireland
<EMAIL_ADDRESS>
###### Abstract

Style analysis, which is relatively a less explored topic, enables several
interesting applications. For instance, it allows authors to adjust their
writing style to produce a more coherent document in collaboration. Similarly,
style analysis can also be used for document provenance and authentication as
a primary step. In this paper, we propose an ensemble-based text-processing
framework for the classification of single and multi-authored documents, which
is one of the key tasks in style analysis. The proposed framework incorporates
several state-of-the-art text classification algorithms including classical
Machine Learning (ML) algorithms, transformers, and deep learning algorithms
both individually and in merit-based late fusion. For the merit-based late
fusion, we employed several weight optimization and selection methods to
assign merit-based weights to the individual text classification algorithms.
We also analyze the impact of the characters on the task that are usually
excluded in NLP applications during pre-processing by conducting experiments
on both clean and un-clean data. The proposed framework is evaluated on a
large-scale benchmark dataset, significantly improving performance over the
existing solutions.

###### Index Terms:

Document Authentication; Provenance; BERT; RoBERTa; Text Classification; Late
Fusion;

##  I Introduction

Recently the reliability of information spread overshared on online social
networks has gained great attention from the community. In this regard, much
emphasis is being put on the provenance and authenticity of the information
shared on social networks. In the case of written documents, an important
aspect of such types of provenance criticism relates to authorship, which
involves assessing the authenticity of information by identifying the
documents’ original author(s) [1]. Thus, one can argue that the development of
computational authorship identification systems, which can assist humans in
various tasks in different application domains, such as journalism, law
enforcement, content moderation, etc., carries great significance. Style
analysis is a primary task in the development of such computational authorship
identification systems. Style analysis mainly involves the identification of
distinct patterns in writing style by utilizing intrinsic features in a
collaboratively authored document. Style change detection (SCD) is one of the
key applications of style analysis, which is relevant to intrinsic plagiarism
detection (IPD). IPD leverages the writing style consistency of the authors
over the length of the document for plagiarism detection independent of any
other reference corpus [2, 3].

The literature already reports several works on SCD in multi-authored
documents. It has also been part of PAN111https://pan.webis.de/index.html,
which is a benchmark shared task on digital text forensics and stylometry, for
several years. The PAN 2021 shared task [4] on SCD involves three sub-tasks,
namely (i) Single vs. Multiple, (ii) Style Change Basic, and (iii) Style
Change Real-World. In this work, we target the first task (i.e., single vs.
multiple) that involves the classification of single and multi-authored
documents. The literature reports several interesting solutions for the task.
However, several aspects of the task are still unexplored. For instance, the
majority of the solutions rely on individual ML and NLP algorithms. However,
based on our previous experience with similar NLP applications [5], we believe
these algorithms can complement each other if jointly utilized in a merit-
based fusion. Similarly, the impact of pre-processing step, which generally
involves cleaning data by removing unnecessary characters (e.g., stop words,
question marks, contractions, etc.,), on the performance of the algorithm in
the task is not been explored yet. We believe that the text generally removed
in the pre-processing step could be useful in SCD. For example, one author may
use stop words, question marks, contractions, etc., more frequently compared
to others, which could be useful in SCD.

To overcome these limitations, we propose a merit-based late fusion-based
framework incorporating several state-of-the-art ML and NLP algorithms both
individually and jointly using several weight optimization and selection
methods for merit-based fusion of the individual algorithms. Moreover, we also
analyze the impact of characters that are generally removed during pre-
processing in NLP applications on SCD by conducting experiments on both clean
and un-clean datasets. The contributions of the work are summarized as
follows.

  * •

We propose a novel document provenance and authentication framework able to
differentiate a document written by single and multiple authors by employing
several merit-based late fusion methods.

  * •

We evaluate several ML and NLP algorithms including classical and state-of-
the-art transformers both individually and jointly in late fusion.

  * •

We conducted several experiments to explore different aspects of the task. The
experiments are conducted on both cleaned and un-clean data to analyze the
impact of characters on style detection that are usually excluded in NLP
applications during pre-processing.

##  II Related Study

Stylometry, which involves authors’ style analysis, assumes writing style as a
quantifiable measure that allows quantifying distinctive features in a given
piece of text. Stylometry involves several tasks aiming to analyze several
aspects of the text, such as authorship attribution, verification, authorship
profiling, stylochronometry, and adversarial stylometry. Authorship
attribution is one of the key and widely explored tasks in stylometry [6, 7].
Author attribution, which aims at the identification of the author of a
document, generally relies on SCD techniques.

The literature reports several interesting solutions, exploring different
aspects of text analysis, for SCD [8]. For instance, in one of the initial
efforts for the task, Glover et al. [9] employed stylometric features for the
identification of author boundaries in documents written by multiple authors.
Akiva et al. [10], on the other hand, proposed a clustering framework for
separating a multi-authored document into threads written by individual
authors. SCD was also included as a shared task in a benchmark initiative
namely PAN style detection for three consecutive years [1, 4]. Most of the
recent works on the topic are based on this shared task. The shared task is
composed of multiple sub-tasks including (i) differentiating between single
and multi-authored documents and identifying the sequences of paragraphs where
the style changes in multi-authored documents considering (i) basic change,
and (ii) style change Real-World. In this paper, we propose a solution for the
first task and will provide an overview of the literature on the first task
only.

In the literature, several interesting solutions for the classification of
single and multi-authored documents are presented. These solutions explore
different aspects of the task. For instance, Zhang et al. [11] proposed a BERT
and a fully connected neural network-based framework to differentiate between
single and multi-authored documents. Strom et al. [12] proposed an ensemble-
based framework to combine the classification scores obtained through
different classifiers trained on two different types of features including
BERT embedding and lexical features. The lexical features include character,
word, sentence, and contracted and functional word features. Singh et al. [13]
also relied on lexical features, where a diversified list of features
including TF-IDF values of n-grams, POS-Tag tri-grams, and special characters,
frequency of function words, and the average number of characters per word,
are used. A logistic regression classifier is then trained on the extracted
features to classify single and multi-authored documents. Deibel et al. [14]
used an MLP algorithm trained on a diversified set of textual features
including corrected type-token ratio, mean sentence and word length, function
word frequency, and linear-write-formula. Nath et al. [15], on the other hand,
proposed a deep-learning Siamese Neural Network architecture with bi-
directional LSTM for the task.

In this work, in contrast to the existing solutions, we employ several
transformers in a merit-based late fusion framework. We also analyze the
impact of data cleaning and balancing techniques on the performance of the
framework.

##  III Methodology

Figure 1 depicts different components of the proposed methodology. The first
component is based on several pre-processing techniques to clean and balance
the dataset. The pre-processing phase is followed by a classification phase
where several algorithms are used. In the final part, we combine the scores of
the individual models using several weight optimization and selection methods.
The details of these components are provided below.

Figure 1: A visual depiction of the proposed methodology.

###  III-A Pre-processing

In the pre-processing phase, two different tasks namely (i) data cleaning, and
(ii) data balancing are carried out. In the data cleaning task, we removed
unnecessary information, such as usernames, URLs, emojis, and stop words.
Moreover, we expanded (i.e., decontracted) the contractions using a Python
library namely Contractions222https://pypi.org/project/contractions/.

In the second part of the pre-processing phase, transpose and Synthetic
Minority Oversampling Technique (SMOTE) methods are used to balance the
dataset. In the transpose method, new samples are generated by swapping two
consecutive/adjacent characters. The SMOTE method, on the other hand,
generates newer samples having a closer resemblance to the samples of the
minority classes using data augmentation techniques.

###  III-B Feature Extraction and Classification

In this phase, we employed both classical ML and DL methods. In classical ML
methods, we choose several state-of-the-art ML algorithms including Random
Forest (RF), Naive Bayes (NB), XGBoost, K-Nearest Neighbour (KNN), Logistic
Regression, Decision Trees (DT), and Support Vector Machines (SVMs). We note
that all these algorithms are trained on TF-IDF (term frequency and inverse
document frequency) features.

On the other hand, several state-of-the-art models are used in DL methods.
These models include BERT (Bidirectional Encoder Representations from
Transformers), Roberta (Robustly Optimized BERT), DistilBert, XML-Roberta, XL-
Net, and ALBERT. These models are selected based on their proven performances
in relevant tasks [16, 5, 17]. A brief overview of these models is provided
below.

  * •

BERT: BERT [18], which is a transformer-based algorithm, is one of the state-
of-the-art NLP algorithms. It is originally pre-trained on a large collection
(Wikipedia) of unlabeled text and can be fine-tuned for any NLP task. BERT has
several key characteristics that make it a suitable choice for several NLP
applications. However, its strength mainly lies in its ability for
bidirectional training. This allows it to read the text in both directions
simultaneously. This bi-direction training helps in the extraction of
contextual information. BERT is available in two different versions having a
different number of layers and parameters. In this work, we considered the
model having 12 layers and attention heads with a total of 110 million
parameters.

  * •

DistilBERT: DistilBERT [19] is a smaller, faster, cheaper, and lighter version
of the BERT model. The model is trained by distilling the BERT base. The model
is able to preserve the 95% performance of the original BERT model on the GLUE
benchmark dataset with a significant (around 40%) reduction in the number of
parameters [19]. The reduction in the number of parameters results in a 60%
faster model than the original BERT base model.

  * •

RoBERTa: RoBERTa [20] is a modified form of BERT, where key hyperparameters of
BERT are modified. For instance, it is trained with a higher learning rate and
larger mini-batch size. More importantly, it removes BERT’s next-sentence
pretraining objective resulting in an improved masked language modeling
objective compared with the original BERT. Moreover, RoBERTa is trained on a
larger collection of data.

  * •

XML-RoBERTa: XLM-RoBERTa is a multilingual version of RoBERTa. The model is
trained on a collection of 2.5TB text documents from 100 different languages.
The selection of the model for this task is based on its proven performance in
similar tasks [5].

  * •

ALBERT: ALBERT [21], which is considered a lighter version of BERT, is built
on top of the BERT base model by targeting three aspects. These include (i)
parameter sharing, (ii) embedding factorization, and (iii) sentence order
prediction. For parameter sharing, instead of using different parameters for
all of the 12 layers, ALBERT uses the same weights/parameter values for all
the layers. This results in a significant reduction in the parameters of
ALBERT compared to the original BERT model. ALBERT also uses a reduced size of
embeddings. Moreover, in contrast to the original model, ALBERT uses sentence
order prediction (SOP) loss based on the coherence of sentences rather than
just predicting the topics.

###  III-C Fusion

In the fusion phase, the scores of the individual models are combined in a
late fusion scheme using Equation 1. In the equation, $S$ is the combined
classification score while $s_{n}$ represents the classification score of the
nth model. $w_{1},w_{2},w_{3},...w_{n}$ represent the weights assigned to the
individual models, which are calculated using several weight optimization and
selection methods.

$S=w_{1}s_{1}+w_{2}s_{2}+w_{3}s_{3}+....+w_{n}s_{n}$ (1)

For weight selection, mainly three merit-based fusion methods namely PSO,
NelderMead, and Powell’s method are used. In the fusion methods, the weights
of the models are optimized/selected on a validation set. In all of the fusion
methods, the fitness function is based on an accumulative classification error
as shown in Equation 2. In the equation, $e$ and $A_{acc}$ represent the
classification error and accumulative accuracy, respectively.

$e=1-A_{acc}$ (2)

The accumulative accuracy is computed using Equation 3, where
$x_{1},x_{2},x_{3},...x_{n}$ are the variables (weight values) to be optimized
while $p_{1},p_{2},p_{3},...p_{n}$ $p_{n}$ represents the probabilities
obtained through the individual models on the validation set.

$A_{acc}=x(1)*p_{1}+x(2)*p_{2}+...+x(n)*p_{n}$ (3)

The selection of the optimization/selection methods is motivated by their
proven performance in similar tasks [5]. The details of these methods are
provided below.

####  III-C1 PSO-based Fusion

Our first merit-based fusion method is based on PSO [22]. PSO is heuristic in
nature, which means that the solution (weights combination in our application)
obtained with PSO is not globally optimal. However, the literature shows that
the solution obtained with PSO is usually near the global optimal. The working
mechanism of PSO is motivated by a group of fish or a flock of birds moving
together, where the individual members of the group help each other by sharing
their discoveries with the rest of the group leading to the best hunt for the
entire group. In PSO, a similar approach is followed in three different steps.
Firstly, an evaluation of each of the candidate solutions is carried out based
on fitness criteria. Secondly, the personal best and global best values are
updated. Finally, the position and velocity of each particle in the swarm are
updated accordingly.

####  III-C2 NelderMead Method-based Fusion

Our second merit-based fusion method is based on the Nelder Mead algorithm,
which can be used for n-dimensional optimization problems where ’n’ can be 1
or greater than 1. It is a numerical method and can find the maximum or
minimum of an objective function, which is based on the accumulative
classification error. Our experiments are based on the implementation of the
method provided by a Python open-source library, namely,
SciPy333https://scipy.org/.

####  III-C3 Powell’s Method-based Fusion

The third method is based on Powell’s method, which is a censored maximum-
gradients technique. The method starts with an initial guess (weights
combination) and starts moving towards a minimum of a function by finding and
calculating a distance in a good direction. One of the main limitations of the
method is the assumption of the step size in the direction. The method is
simple and has been proven very effective in similar tasks [5]. In this work,
our implementation is based on the Python open library
SciPy444https://scipy.org/. Similar to the other methods, our fitness function
for Powell’s method is defined in equation 2.

##  IV Experiments and Results

###  IV-A Dataset

In this work we used the PAN21 authorship analysis dataset [23]. The dataset
is created for SCD task by collecting randomly chosen threads from the popular
StackExchange network of Q&A sites. The dataset consists of 16,000 text
documents drawn from sites related to topics focused on technology. The
topical scope of the text is limited to ensure topical coherence in the
dataset.

The data is cleaned by removing graphical content, URLs, code snippets,
quotations, lists, and very short posts. Moreover, the posts that were edited
after their submission were also removed from the dataset to ensure the
authors’ naturalness. After cleaning the text, it is split into paragraphs,
while dropping the paragraphs with less than 100 characters. The original
dataset for the task of single and multi-authored documents prediction is
imbalanced and it consists of 25% single-authored and 75% multi-authored
documents. The document length ranges from 1,000 to 10,000 words per document.
Moreover, each document has at least 2 and at most 100 paragraphs. The dataset
is split into training, validation, and test data with a ratio of 70%
(11,200), 15% (2400), and 15% (2400), respectively. The training data contains
8,400 multi-authored, and 2800 single-authored documents. Similarly, the test
and validation datasets consists of 18,00 multi-authored and 600 single-
authored documents.

###  IV-B Experimental Setup

The objectives of this work are multi-fold, and we want to explore different
aspects of the task. On one hand, we want to analyze the role of the
characters that are usually eliminated in the pre-processing step, such as
stop words, URLs, separate alphanumeric characters, and contractions. We
believe some of these words especially the use of contraction can help in the
detection of style/author changes in a document. Similarly, we also want to
analyze and evaluate the performance of several NLP algorithms including
classical ML algorithms, such as RF, DT, SVMs, and transformers, such as BERT,
RoBERTa, XLM-RoBERTa, and ALBERT in the task. We also aim to analyze and
evaluate the performance of different fusion methods in combining the results
of the individual models.

To achieve these objectives, we conducted the following experiments.

  * •

We conducted experiments on both clean and raw data.

  * •

We also conducted experiments on the imbalanced and balanced dataset, where
three different methods are used for balancing the dataset.

  * •

We evaluate the performance of several individual models including classical
and DL algorithms.

  * •

We evaluate the performance of several fusion techniques including a naive and
three merit-based late fusion schemes.

###  IV-C Results

Table I provides the results of our first experiment where we evaluate the
performances of the individual models on both clean and un-clean data in terms
of F1-score. Moreover, the models are evaluated on balanced and imbalanced
datasets. The performances of the individual models in different experimental
setups are described below.

  * •

Classical Vs. Transformers-based Models: In total, we used 8 different models
including 7 classical ML algorithms trained on TF-IDF features, and five
transformers. As expected, better results are obtained with transformers
compared to the classical methods on all cleaned, un-clean, balanced, and
imbalanced datasets. Overall, the highest score is obtained with XML-RoBERTa
with an F1-score of 0.84 on un-clean imbalanced data. However, there is no
single winner in all types of datasets. Surprisingly, compared to the
transformers, some of the classical ML algorithms, such as RF, obtained better
results on the clean dataset. This implies that transformers are more suited
for extracting meaningful information from complex data compared to the
classical NLP and ML algorithms.

  * •

Clean vs. Un-clean Dataset: Interesting behavior has been observed for the
models on clean and unclean datasets. The difference in the performance of the
transformers on clean and un-clean datasets is significant while comparable
results are obtained with the classical models on both sets of data. The
highest F1 score obtained on the clean dataset is 0.79 while it is 0.84 on the
un-cleaned dataset. The reduction in the performance of the clean data
indicates that the pre-processing step results in the removal of
words/characters that are useful for the transformers in the author detection
task. We note that during the pre-processing phase, we removed different
special characters, such as usernames, URLs, emojis, and stop words. We also
expanded contractions. In the future, we aim to explore which of the pre-
processing techniques resulted in the removal of useful features by employing
explainable NLP techniques.

  * •

Balanced vs. Un-Balanced Dataset: The performances of the models also vary on
balanced and imbalanced datasets. Overall, better results are observed on the
imbalanced dataset. Moreover, we evaluate the performance of two different
methods, namely (i) SMOTE and (ii) Transpose, for balancing the datasets.
There is no clear winner in the data balancing techniques. In some cases,
SMOTE resulted in a better score while in others Transpose method obtained
better results.

TABLE I: Evaluation of the individual models on the clean, raw, balanced, imbalanced dataset. Model | Un-clean Data | Clean Data
---|---|---
Imbalanced | Balance (SMOTE) | Balance (Transpose) | Imbalanced | Balance (SMOTE) | Balance (Transpose)
Random Forest | 0.62 | 0.6 | 0.61 | 0.63 | 0.49 | 0.56
Naïve bayes | 0.43 | 0.45 | 0.43 | 0.43 | 0.51 | 0.48
XGBoost | 0.62 | 0.62 | 0.62 | 0.62 | 0.48 | 0.36
KNN | 0.49 | 0.48 | 0.49 | 0.49 | 0.51 | 0.48
SVM | 0.57 | 0.46 | 0.21 | 0.57 | 0.45 | 0.47
Decision Tree | 0.51 | 0.53 | 0.51 | 0.52 | 0.49 | 0.46
Logestic Regression | 0.5 | 0.48 | 0.21 | 0.5 | 0.51 | 0.46
BERT | 0.79 | 0.75 | 0.79 | 0.79 | 0.59 | 0.79
Roberta-Base | 0.8 | 0.7 | 0.43 | 0.65 | 0.59 | 0.62
Albert | 0.79 | 0.79 | 0.53 | 0.62 | 0.62 | 0.53
Distillbert | 0.75 | 0.72 | 0.2 | 0.76 | 0.6 | 0.2
XLM-Roberta | 0.8481 | 0.75 | 0.79 | 0.66 | 0.59 | 0.79

Table II provides the results of the fusion experiments, where we evaluate the
performance of several fusion methods. We note that in the fusion methods, we
combine the classification score of the transformers only. These methods
include three merit-based and a simple averaging fusion method. In the
averaging method, all the models are treated equally while in the other
methods weights are optimized/selected for the individual models based on
their performances.

As can be seen in the table, overall, better results are obtained with fusion
compared to the individual models on both cleaned and un-clean datasets. The
improvement over the best individual model on the cleaned dataset is
significant. However, less improvement for the fusion method could be observed
on the un-clean dataset.

Moreover, in fusion methods, better results are obtained by merit-based fusion
methods compared to the naive fusion method. This shows the efficacy of merit-
based fusion methods and insinuates to consider the performance of the
individual models in fusion. As far as the evaluation of the merit-based
fusion methods is concerned, no significant variation has been observed.
Overall, slightly better results are obtained for Powell method-based fusion
compared to the other weight optimization/selection methods.

TABLE II: Evaluation of the fusion methods on the clean, raw, balanced, imbalanced dataset. Fusion Method | Un-clean Data | Clean Data
---|---|---
Imbalanced | Balance (SMOTE) | Balance (Transpose) | Imbalanced | Balance (SMOTE) | Balance (Transpose)
Simple Averaging | 0.8272 | 0.7673 | 0.4546 | 0.6623 | 0.6167 | 0.2074
PSO Method | 0.8486 | 0.8031 | 0.5301 | 0.6511 | 0.6007 | 0.54
Nelder Nead Method | 0.8491 | 0.8031 | 0.52381 | 0.6439 | 0.56575 | 0.5397
Powell Method | 0.8523 | 0.8031 | 0.52381 | 0.6412 | 0.5657 | 0.54

We also provide a comparison of the proposed method against the existing
solutions in Table III. As can be seen in the table, all of the merit-based
fusion methods have significant improvement over the existing literature. Our
best performing fusion method (i.e., Powell Method-based fusion) has an
improvement of 5.7% over the highest-scoring solution in the literature.

TABLE III: Comparison against existing solutions. Method | F1-Score
---|---
Zhang et al. [11] | 0.753
Strom et al. [12] | 0.795
Singh et al. [13] | 0.634
Deibel et al. [14] | 0.621
Nath et al. [15] | 0.704
Nelder Nead Method-based Fusion (This work) | 0.849
PSO-based Fusion (This work) | 0.848
Powell Method-based Fusion (This work) | 0.852

##  V Conclusions

In this paper, we proposed a merit-based fusion framework for the
classification of single and multi-authored documents by employing several
weight selections and optimization methods for the joint use of multiple NLP
algorithms. Moreover, we evaluated the performance of several individual
algorithms including classical ML and deep learning models. We also
experimented with both cleaned and un-clean data to analyze the impact of
characters on style detection that are usually excluded in NLP applications
during pre-processing. We also evaluated the performance of multiple data
balancing techniques. Overall, better results are obtained for the
transformers. A significant improvement in the merit-based fusion techniques
over the individual models is noticed. We also noticed that the characters
that are generally excluded in NLP applications during pre-processing steps
play a key role in style detection. In the future, we aim to extend the work
by exploring the other missing aspects of the task, such as combining the
scores obtained through classifiers trained on lexical features and DL models.

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# Structural Identifiability of a Pseudo-2D Li-ion Battery Electrochemical
Model

Ross Drummond  Stephen R. Duncan  Department of Engineering Science,
University of Oxford, 17 Parks Road, OX1 3PJ, Oxford, United Kingdom.
Email: {ross.drummond, stephen.duncan<EMAIL_ADDRESS>
###### Abstract

Growing demand for fast charging and optimised battery designs is fuelling
significant interest in electrochemical models of Li-ion batteries. However,
estimating parameter values for these models remains a major challenge. In
this paper, a structural identifiability analysis was applied to a pseudo-2D
Li-ion electrochemical battery model that can be considered as a linearised
and decoupled form of the benchmark Doyle-Fuller-Newman model. From an
inspection of the impedance function, it was shown that this model is uniquely
parametrised by 21 parameters, being combinations of the electrochemical
parameters like the conductivities and diffusion coefficients. The well-
posedness of the parameter estimation problem with these parameters was then
established. This result could lead to more realistic predictions about the
internal state of the battery by identifying the parameter set that can be
uniquely identified from the data.

###### keywords:

Li-ion batteries, electrochemical models, structural identifiability.

††thanks: Work supported by the Engineering and Physical Sciences Research
Council through grant EP/P005411/1- “Structured electrodes for improved energy
storage.”

## Introduction

As the importance of electrical energy storage continues to grow and Li-ion
batteries numbers keep on increasing, this technology is becoming ever more
mature. The sheer number of Li-ion batteries being produced is leading to
significant price reductions (Curry (2017)), due primarily to the economics of
scale, meaning that any new battery chemistry hoping to be commercialised will
have to overcome increasingly steep economic barriers before entering the
mass-market. As such, at least in the near future, the next wave of battery
innovation is likely to come from optimising existing technology, rather than
from the introduction of radically new chemistries. Control engineering will
play a pivotal role in this optimisation.

Electrochemical models are important tools for optimising battery use, both
for control purposes within a battery management system (Chaturvedi et al.
(2010)) and for design. These models provide a rich description of the
battery’s response in terms of snapshots of its internal electrochemical
state; however, their accuracy critically relies upon the electrochemical
parameter values. Obtaining accurate estimates of these parameters has thus
emerged as a crucial research topic in recent years. Unfortunately, the
relative complexity of electrochemical models makes estimating their parameter
values difficult. This difficulty was illustrated in Forman et al. (2012)
where it took approximately three weeks’ worth of computation on five quad-
core Intel Q8200 computers to estimate the parameters of the benchmark Doyle-
Fuller-Newman (DFN) model (Doyle et al. (1993)). To bring some clarity to this
problem, Bizeray et al. (2018) recently approached the parameter estimation
problem from a different perspective by performing a structural
identifiability analysis of the simplified single particle model (SPM),
determining the unique six parameters fully describing the SPM’s response. In
such a way, the well-posedness of the parameter estimation problem for this
simplified model could be established. This result built upon existing
literature in this area, including Pozzi et al. (2018) where sensitivity
functions for a polynomial approximations of the single particle model with
electrolyte (SPMe) were obtained, Park et al. (2018) which considered
experiment design for ranking the sensitivity of each electrochemical
parameter, Lai et al. (2019) where analytic sensitivity functions for the
SPM’s parameters were derived from Padè approximations of the impedance and
Alavi et al. (2016) that analysed the structural identifiability of Randles
circuit battery models.

Motivated by Bizeray et al. (2018), this paper develops a framework towards a
structural identifiability analysis of the DFN electrochemical Li-ion battery
model (Doyle et al. (1993)). The DFN model is widely considered as a benchmark
micro-scale model for Li-ion batteries from which simplifications like the SPM
(Ning and Popov (2004); Atlung et al. (1979)) and the SPMe (Moura et al.
(2017); Marquis et al. (2019); Richardson et al. (2019)) are derived. But,
estimating its parameter values in a methodical way is challenging (Forman et
al. (2012)). This paper details preliminary results on this problem by
performing a structural identifiability analysis of a decoupled and linearised
form of the DFN model which can be considered as the SPMe with added double-
layer effects.

To perform the structural identifiability analysis, the DFN model was first
simplified into a linearised form with dynamics decomposed into three
elements: solid-state diffusion, bulk electrolyte mobility and the charge
transfer resistance caused by the relaxation of the overpotential. This
simplification enables a tractable analysis, with the generalised impedance
functions developed in Sikha and White (2007) being too complex, right now, to
determine structural identifiability. It is shown that this decoupled DFN
model is uniquely parametrised from current/voltage data by 21 parameters
formed from combinations of the electrochemical parameters (see Table 2). It
is hoped that the results of this work will provide the theoretical
underpinning behind a generalised parameter estimation method of the DFN model
that will not rely upon substantial a prior knowledge of the cell’s makeup.
The need for accurate, and recursive, estimates of the DFN model’s parameters
is expected to grow still further as the importance of fast charging (Kang and
Ceder (2009)) and battery design become ever clearer. For these applications,
the SPM provides neither sufficient accuracy nor richness.

## Notation

Definitions of the electrochemical model states and parameters are given in
Table 1. Special attention is paid to the definition of the spatial domains,
identified by the subscript $k\in\\{1,2,3\\}$, $k=1$ denoting the anode, $k=2$
the separator and $k=3$ the cathode. For instance, the point $x=L_{12}$
defines the anode/separator boundary and $x=L_{23}$ is the separator/cathode
boundary. $\Omega_{1}~{}:=~{}\\{x:~{}0~{}<~{}x~{}<~{}L_{12}~{}\\}$ denotes the
spatial domain of the anode,
$\Omega_{2}~{}:=~{}\\{x:~{}L_{12}~{}<~{}x~{}<~{}L_{23}~{}\\}$ that of the
separator and $\Omega_{3}~{}=:~{}\\{x:~{}L_{23}~{}<~{}x~{}<~{}L~{}\\}$ that of
the cathode. When a parameter takes different values in each domain then the
subscript $k$ is added, for instance $\varepsilon_{1}$ is the porosity in the
anode. When these parameter subscripts are missing, like in (6), this
indicates that the equation holds in all domains. This notation is adopted for
compactness to avoid duplication of the equations as much as possible.

For spatio-temporal variables (e.g. $c_{e}(x,t)$), tilde will denote
variations (e.g. $\tilde{c}_{e}(x,t))$ around an equilibrium denoted by an ∗
(e.g. $c^{*}_{e}$). In particular, the expansions

$\displaystyle c_{e}(x,t)$ $\displaystyle=c_{e}^{*}+\tilde{c}_{e}(x,t),$ (1a)
$\displaystyle u_{s}(x,r,t)$
$\displaystyle=\tilde{u}_{s}(x,r,t)+\frac{r}{R_{s}}\tilde{u}_{s}^{\text{surf}}(x,t),$
(1b) $\displaystyle u_{s}(x,r,t)$ $\displaystyle=rc_{s}(x,r,t)$ (1c)

are used. The surface concentrations are
$c_{s}^{\text{surf}}(x,t)=c_{s}(x,R_{s},t)$ and the transformed surface
concentrations are $u_{s}^{\text{surf}}(x,t)=u_{s}(x,R_{s},t)$. Overbars
denote Laplace transforms of signals, e.g. $\bar{\tilde{c}}_{e}(x,s)$.

##  1 Problem formulation

Structural identifiability of a decomposed form of the DFN electrochemical
model for Li-ion batteries is considered. In each electrode, the DFN model
electrochemical equations are

$\displaystyle\frac{\partial u_{s}(x,r,t)}{\partial t}$
$\displaystyle=D_{s}\frac{\partial^{2}u_{s}(x,r,t)}{\partial r^{2}},$ (2a)
$\displaystyle\frac{\partial c_{\text{e}}(x,t)}{\partial t}$
$\displaystyle=D_{\text{e}}\frac{\partial^{2}c_{e}(x,t)}{\partial^{2}x}+\frac{a_{\text{s}}\left(1-t_{+}\right)}{\varepsilon}j(x,t),$
(2b) $\displaystyle\eta(x,t)$
$\displaystyle=\phi_{\text{s}}(x,t)-\phi_{\text{e}}(x,t)-U(u_{\text{s}}^{\text{surf}}(x,t)),$
(2c) $\displaystyle i_{0}=k\mathcal{F}(c_{\text{s}}^{\text{max}}$
$\displaystyle-
c_{\text{s}}^{\text{surf}}(x,t))^{\alpha_{\text{a}}}c_{\text{s}}^{\text{surf}}(x,t)^{\alpha_{\text{c}}}c_{\text{e}}(x,t)^{\alpha_{\text{a}}},$
(2d) $\displaystyle j(x,t)$
$\displaystyle=\frac{i_{0}(x,t)}{\mathcal{F}}\sinh\left(\frac{\mathcal{F}}{RT}\eta(x,t)\right),$
(2e) $\displaystyle\frac{i_{\text{s}}(x,t)}{\sigma^{\text{eff}}}$
$\displaystyle=-\frac{\partial\phi_{\text{s}}(x,t)}{\partial x},$ (2f)
$\displaystyle\frac{i_{\text{e}}(x,t)}{\kappa}$
$\displaystyle=-\frac{\partial\phi_{\text{e}}(x,t)}{\partial
x}+K\frac{\partial\tilde{c}_{\text{e}}(x,t)}{\partial x},$ (2g) $\displaystyle
i(t)$ $\displaystyle=i_{\text{s}}(x,t)+i_{\text{e}}(x,t),$ (2h)
$\displaystyle\frac{\partial i_{\text{e}}(x,t)}{\partial x}$
$\displaystyle=a_{\text{s}}\mathcal{F}j(x,t),$ (2i)

and, in the separator, the governing equations are simply

$\displaystyle\frac{\partial c_{\text{e}}(x,t)}{\partial t}$
$\displaystyle=D_{\text{e}}\frac{\partial^{2}c_{e}(x,t)}{\partial x^{2}},$
(3a) $\displaystyle\frac{i(t)}{\kappa}$
$\displaystyle=-\frac{\partial\phi_{\text{e}}(x,t)}{\partial
x}+K\frac{\partial\tilde{c}_{\text{e}}(x,t)}{\partial x}.$ (3b)

In (2g), (3b), it was assumed that $c_{e}^{*}\gg 0$ so that MacInnes’ equation
could be approximated with $\frac{\partial\ln(c_{e})}{\partial
x}\approx\frac{1}{c_{e}^{*}}\frac{\partial\tilde{c}_{e}}{\partial x}$ since a
linearised model is eventually used for the impedance response. This
assumption holds for most concentrated electrolytes. To obtain statements on
structural identifiability of this electrochemical model, three further
assumptions are applied to the DFN model equations.

Assumption 1: All parameters are time independent. Time independence allows
the analysis to be conducted in the frequency domain and the statements on
structural identity to be derived, although, in doing so, thermal and other
state-dependent effects are ignored.

Assumption 2: The applied currents are small enough so that a small signal
analysis can be performed. The states then do not deviate significantly from
their equilibria $c_{e}(x,t)\approx c_{e}^{*}$, $c_{s}^{\text{surf}}\approx
c_{s}^{\text{surf},*}$ and $\partial\eta\approx\partial\phi_{dl}$.

Assumption 3: Simplified relations for the Butler-Volmer inter-facial reaction
rate (2e) are used. As adopted in Ong and Newman (1999) for the analysis of
the fast overpotential dynamics (6c), $j(x,t)$ is simplified through a
linearisation of (2e) with small deviations in concentrations from equilibrium
assumed $i_{0}\approx i_{0}^{*}$ (with $i_{0}^{*}$ denoting the ion exchange
current evaluated at $c_{e}=c_{e}^{*}$ and
$c_{s}^{\text{surf}}=c_{s}^{\text{surf},*}$) so

$\displaystyle j\approx\frac{i_{0}^{*}a_{s}F}{RT}\eta(x,t).$ (4)

For the slower particle (6b) and bulk electrolyte (6a) dynamics, constant
reactivity across the electrode is assumed

$\displaystyle j^{*}(t)$
$\displaystyle=\frac{-I(t)}{a_{s}L\mathcal{F}A},~{}x\in\Omega_{1},~{}j^{*}(t)=\frac{I(t)}{a_{s}L\mathcal{F}A},~{}x\in\Omega_{3}.$
(5)

Constant reactivity across the electrode is used in the control-orientated SPM
and SPMe models (Moura et al. (2017); Marquis et al. (2019)).

Spatial Variables
---
$x$ | Battery spatial variable. | m
$r$ | Particle spatial variable. | m
Variables
$c_{s}(x,r,t)$ | Particle Li concentration. | mol m-3.
$c_{s}^{\text{surf}}(x,t)$ | Li surface concentration. | mol m-3.
$c_{e}(x,t)$ | Effective electrolyte conc. | mol m-3.
$u_{s}(x,r,t)$ | Transformed concentration. | mol m-2.
$u_{s}^{\text{surf}}(x,t)$ | Transformed surface conc. | mol m-2.
$\phi_{s}(x,t)$ | Potential in the solid. | V.
$\phi_{e}(x,t)$ | Potential in the electrolyte. | V.
$\phi_{dl}(x,t)$ |  $\phi_{s}(x,t)-\phi_{e}(x,t)$. | V.
$i_{s}(x,t)$ | Current in the solid. | A m-2
$i_{e}(x,t)$ | Current in the electrolyte. | A m-2.
$j(x,t)$ | Reaction rate. | A m-3.
$\eta(x,t)$ | Overpotential. | V.
$U(u_{s}^{\text{surf}})$ | Open circuit potential. | V.
$i(t)$ | Applied current density. | A m-2.
$V(t)$ | Measured Voltage | V.
Parameters |
$c_{s}^{\text{max}}$ | Max. Li particle concentration. | mol m-3.
$D_{s}$ | Particle diffusion coefficient. | m2 s-1.
$R_{s}$ | Particle radii. | m.
$L_{{k}}$ | Length of domain $k\in\\{1,\,2,\,3\\}$. | m.
$L_{{k}}$ | Separator boundary $k\in\\{12,\,23\\}$. | m.
$L$ | Total length of the battery. | m.
$\varepsilon$ | Porosity coefficient. |
$D_{\text{e}}$ | Effective diffusion coefficient. | m2 s-1.
$a_{s}$ | Specific interfacial area. | m-1.
$t_{+}$ | Transference number. |
$\mathcal{F}$ | Faraday’s constant. | C mol-1.
$\alpha_{c}$/$\alpha_{c}$ | Anode/Cathode transfer coefs. |
$R$ | Universal gas Constant. | mol-1 K-1.
$T$ | Temperature. | K.
$k$ | Exchange rate parameter. | m2.5mol-0.5s-1.
$\sigma$ | Electronic conductivity. | S m-1.
$\kappa$ | Ionic conductivity. | S m-1.
$R_{ctc}$ | Contact resistance. |  $\Omega$.
$K$ | $\frac{2\left(1-t_{+}\right)RT}{c_{e}^{*}\mathcal{F}}$ |
$A_{\text{cc}}$ | Current collector surface areas. |
Sub and super scripts
1 | Denotes anode parameters. |
2 | Denotes separator parameters. |
3 | Denotes cathode parameters. |
12 | Anode/separator boundary. |
23 | Cathode/separator boundary. |
Table 1: Nomenclature of the DFN model. Parameters for solid diffusion
---
$\tau_{c_{s}}^{k}$ | $\frac{D_{s,k}}{R_{s,k}^{2}}$ | Particle time constant.
$\theta_{c_{s}}^{k}$ | $\frac{dU_{k}(\cdot)/dc_{s}^{\text{surf}}}{L_{k}A_{\text{cc}}}$ | OCP slope per unit volume.
Parameters for bulk electrolyte
$\tau_{c_{e}}^{1}$ | $\frac{D_{e,1}}{L_{1}^{2}}$ | Electrolyte time constant.
$\tau_{c_{e}}^{2}$ | $\frac{D_{e,2}}{L_{2}^{2}}$ | Electrolyte time constant.
$\tau_{c_{e}}^{3}$ | $\frac{D_{e,3}}{L_{3}^{2}}$ | Electrolyte time constant.
$\pi_{1}$ | $\frac{a_{\text{s},1}\left(1-t_{+}\right)}{\varepsilon_{1}\mathcal{F}}$ | Reactivity term in anode.
$\pi_{3}$ | $\frac{a_{\text{s},3}\left(1-t_{+}\right)}{\varepsilon_{3}\mathcal{F}}$ | Reactivity term in cathode.
$\theta_{c_{e}}^{1,1}$ | $\sqrt{\frac{D_{e,2}}{D_{e,1}}}$ | Normalised diffusivity.
$\theta_{c_{e}}^{1,2}$ | $\sqrt{\frac{D_{e,2}}{D_{e,3}}}$ | Normalised diffusivity.
$\theta_{c_{e}}^{2,1}$ | $\frac{A_{\text{cc}}K\sigma_{1}}{\sigma_{1}+\kappa_{1}}$ |
$\theta_{c_{e}}^{2,2}$ | $\frac{A_{\text{cc}}K\kappa_{1}}{\sigma_{1}+\kappa_{1}}$ |
$\theta_{c_{e}}^{3,1}$ | $\frac{A_{\text{cc}}K\sigma_{3}}{\sigma_{3}+\kappa_{3}}$ |
$\theta_{c_{e}}^{3,2}$ | $\frac{A_{\text{cc}}K\kappa_{3}}{\sigma_{3}+\kappa_{3}}$ |
Parameters for overpotential
$\tau_{\eta}^{k}$ | $L_{k}\sqrt{\left(\frac{\kappa_{k}+\sigma_{k}}{\sigma_{k}\kappa_{k}}\right)C^{\text{sp}}_{k}}$ | Double layer time constant.
$\theta_{\eta}^{1,k}$ | $\sqrt{\frac{L_{k}^{2}(\sigma_{k}+\kappa_{k})\mathcal{F}a_{s,k}i_{0,k}^{*}}{\sigma_{k}\kappa_{k}RT}}$ | Normalised reactivity term.
$\theta_{\eta}^{2,k}$ | $A_{\text{cc}}L_{k}\left(\frac{\kappa_{k}/\sigma_{k}+\sigma_{k}/\kappa_{k}}{\kappa_{k}+\sigma_{k}}\right)$ | A resistance-like term.
Table 2: Group of twenty one model parameters that make the electrochemical
model structurally identifiable. The overpotential and particle parameters are
defined for both the anode and cathode domains set by the index
$k\in\\{1,\,3\\}$. In this notation, for instance, $\tau_{c_{s}}^{1}$ is the
solid-state diffusion time constant in the anode.

With these assumptions, a state-space form for the DFN model can then be
generated. For this, dynamics for the overpotential are introduced, described
in Ong and Newman (1999), by first assuming that
$\partial\tilde{c}_{e}/\partial x\approx 0$ at the time scales where these
dynamics are important (doing so decouples the dynamics of $\phi_{dl}$ from
$c_{e}$) and by adding a double layer term to the divergence equation (2i).
The obtained linearised state-space form of (2) is then (Drummond et al.
(2019))

$\displaystyle\frac{\partial\tilde{c}_{e}(x,t)}{\partial t}$
$\displaystyle=D_{e}\frac{\partial^{2}\tilde{c}_{e}(x,t)}{\partial
x^{2}}-\frac{a_{\text{s}}\left(1-t_{+}\right)}{\varepsilon F}j^{*}(t),$ (6a)
$\displaystyle\frac{\partial\tilde{u}_{s}(x,r,t)}{\partial t}$
$\displaystyle=D_{s}\frac{\partial^{2}\tilde{u}_{s}(x,r,t)}{\partial r^{2}},$
(6b) $\displaystyle C^{\text{sp}}\frac{\partial\tilde{\eta}(x,t)}{\partial t}$
$\displaystyle=\left(\frac{\kappa\sigma}{\kappa+\sigma}\right)\frac{\partial^{2}\tilde{\eta}(x,t)}{\partial
x^{2}}-\frac{a_{s}i_{0}^{*}\mathcal{F}}{RT}\tilde{\eta}(x,t).$ (6c)

Each of these dynamical equations will be analysed individually to establish
structural identifiability. The electrochemical states of this system are
subject to the boundary conditions

$\displaystyle\frac{1}{R_{s}}\frac{\partial\tilde{u}_{s}(x,r,t)}{\partial
r}\Big{|}_{r=R_{s}}-\frac{\tilde{u}_{s}(x,R_{\text{s}},t)}{R_{s}^{2}}$
$\displaystyle=\frac{-j^{*}(t)}{D_{s}},$ (7a)
$\displaystyle\tilde{u}_{s}(x,0,t)=0,$ (7b)
$\displaystyle\frac{\partial\tilde{\eta}(x,t)}{\partial
x}\Big{|}_{x\in\\{0,L\\}}$ $\displaystyle=-\frac{i(t)}{\sigma},$ (8a)
$\displaystyle\frac{\partial\tilde{\eta}(x,t)}{\partial
x}\Big{|}_{x\in\\{L_{12},L_{23}\\}}$ $\displaystyle=\frac{i(t)}{\kappa},$ (8b)
$\displaystyle\frac{\partial\tilde{c}_{e}(x,t)}{\partial
x}\Big{|}_{x\in\\{0,L\\}}=0,$ (9a) $\displaystyle
D_{e,1}\frac{\partial\tilde{c}_{e}}{\partial
x}\Big{|}_{an}=D_{e,2}\frac{\partial\tilde{c}_{e}}{\partial
x}\Big{|}_{sep},~{}x\ =L_{12},$ (9b) $\displaystyle
D_{e,3}\frac{\partial\tilde{c}_{e}}{\partial
x}\Big{|}_{cat}=D_{e,2}\frac{\partial\tilde{c}_{e}}{\partial
x}\Big{|}_{sep},~{}x\ =L_{23},$ (9c) $\displaystyle\tilde{c}_{e}(x,t)\text{ is
continuous on }x\in\\{L_{12},L_{23}\\}.$ (9d)

Here, the subscript $an$ denotes a derivative normal to the boundary of the
anode, $sep$ is normal to the separator and $cat$ is normal to the cathode.

###  1.1 Voltage expression

The voltage is defined as the difference in solid-phase potentials between
each current collector but requires some reformulation to be expressed in
terms of the electrochemical states. For this, consider

$\displaystyle v(t)$
$\displaystyle=\phi_{s}(L,t)-\phi_{s}(0,t)-R_{\text{ctc}}i(t)$ (10a)
$\displaystyle={\phi}_{dl}(L,t)-{\phi}_{dl}(0,t)+\phi_{e}(L,t)-\phi_{e}(0,t)-R_{\text{ctc}}i(t)$
(10b)
$\displaystyle={\phi}_{dl}(L,t)-{\phi}_{dl}(0,t)+\int_{0}^{L}\frac{\partial\phi_{e}(x,t)}{\partial
x}~{}dx-R_{\text{ctc}}i(t)$ (10c) with $\frac{\partial\phi_{e}}{\partial x}$
evaluated from MacInnes’ equation (2g) $\displaystyle v(t)$
$\displaystyle={\phi}_{dl}(L,t)-{\phi}_{dl}(0,t)$ (10d)
$\displaystyle+\int_{0}^{L}\frac{-i_{e}(x,t)}{\kappa}+K\frac{\partial\tilde{c}_{e}(x,t)}{\partial
x}~{}dx-R_{\text{ctc}}i(t).$

Analysing the integral in (10d) piece-wise across each of the three domains
$\Omega_{1},$ $\Omega_{2}$, $\Omega_{3}$ using the relationship

$\displaystyle i_{e}(x,t)$
$\displaystyle=\frac{\sigma_{k}\kappa_{k}}{\sigma_{k}+\kappa_{k}}\frac{\partial\phi_{dl}}{\partial
x}+\frac{K\sigma_{k}\kappa_{k}}{\sigma_{k}+\kappa_{k}}\frac{\partial\tilde{c}_{e}}{\partial
x}+\frac{\kappa_{k}i(t)}{\sigma_{k}+\kappa_{k}},$
$\displaystyle\quad\quad\quad x\in\Omega_{k},k\in\\{1,3\\},$ (11a)
$\displaystyle i_{e}(x,t)$ $\displaystyle=i(t),~{}x\in\Omega_{2},$ (11b)

allows the voltage to be expressed in terms of the electrochemical states

$\displaystyle v(t)$ $\displaystyle={\phi}_{dl}(L,t)-{\phi}_{dl}(0,t)$ (12)
$\displaystyle+\int_{\Omega_{1}}-\frac{\sigma_{1}}{\sigma_{1}+\kappa_{1}}\frac{\partial\phi_{dl}}{\partial
x}+\frac{K\kappa_{1}}{\sigma_{1}+\kappa_{1}}\frac{\partial\tilde{c}_{e}}{\partial
x}-\frac{i(t)}{\sigma_{1}+\kappa_{1}}~{}dx$
$\displaystyle+\int_{\Omega_{3}}-\frac{\sigma_{3}}{\sigma_{k}+\kappa_{3}}\frac{\partial\phi_{dl}}{\partial
x}+\frac{K\kappa_{3}}{\sigma_{3}+\kappa_{3}}\frac{\partial\tilde{c}_{e}}{\partial
x}-\frac{i(t)}{\sigma_{3}+\kappa_{3}}~{}dx$
$\displaystyle+\int_{\Omega_{2}}K\frac{\partial\tilde{c}_{e}}{\partial
x}-\frac{i(t)}{\kappa_{k}}~{}dx-R_{\text{ctc}}i.$

However, in this paper, the analysis is concerned with local perturbations of
the voltage around the equilibrium of the open circuit voltage (OCV). These
local perturbations are found by substituting the expression for $\phi_{dl}$
from the overpotential definition (2c) into (12) and using the approximation

$\displaystyle U(c_{s}^{\text{surf}}(x,t))\approx
U(c_{s}^{\text{surf},*})+\frac{1}{R_{s}}\frac{dU\left(c_{s}^{\text{surf}}\right)}{dc_{s}^{\text{surf}}}\Big{|}_{c_{s}^{\text{surf}}=c_{s}^{\text{surf},*}}\tilde{u}_{s}^{\text{surf}}.$
(13)

The voltage can then be expressed as

$\displaystyle v(t)=OCV+\tilde{v}(t),$ (14)

with the OCV defined from the $U(c_{s}^{\text{surf},*})$ terms and the
perturbations being

$\displaystyle\tilde{v}(t)=v_{\eta}(t)+v_{u_{s}^{\text{surf}}}(t)+v_{c_{e}}(t)-R_{\text{res}}i(t)$
(15a) where $\displaystyle v_{\eta}(t)$
$\displaystyle=-\frac{\kappa_{1}}{\kappa_{1}+\sigma_{1}}\tilde{\eta}(0,t)-\frac{\sigma_{1}}{\kappa_{1}+\sigma_{1}}\tilde{\eta}(L_{12},t)\
$ (15b)
$\displaystyle+\frac{\sigma_{3}}{\kappa_{3}+\sigma_{3}}\tilde{\eta}(L_{23},t)+\frac{\kappa_{3}}{\kappa_{3}+\sigma_{3}}\tilde{\eta}(L,t),$
$\displaystyle v_{u_{s}^{\text{surf}}}(t)$
$\displaystyle=-\frac{\frac{\kappa_{1}}{R_{s}}\frac{dU_{1}}{dc_{s}^{\text{surf}}}}{\kappa_{1}+\sigma_{1}}\tilde{u}_{s}^{\text{surf}}(0,t)-\frac{\frac{\sigma_{1}}{R_{s}}\frac{dU_{1}}{dc_{s}^{\text{surf}}}}{\kappa_{1}+\sigma_{1}}\tilde{u}_{s}^{\text{surf}}(L_{12},t)$
$\displaystyle+\frac{\frac{\sigma_{3}}{R_{s}}\frac{dU_{3}}{dc_{s}^{\text{surf}}}}{\kappa_{3}+\sigma_{3}}\tilde{u}_{s}^{\text{surf}}(L_{23},t)+\frac{\frac{\kappa_{3}}{R_{s}}\frac{dU_{3}}{dc_{s}^{\text{surf}}}}{\kappa_{3}+\sigma_{3}}\tilde{u}_{s}^{\text{surf}}(L,t),$
(15c) $\displaystyle v_{ce}(t)$
$\displaystyle=-\frac{K\kappa_{1}}{\sigma_{1}+\kappa_{1}}\tilde{c}_{e}(0,t)-\frac{K\sigma_{1}}{\sigma_{1}+\kappa_{1}}\tilde{c}_{e}(L_{12},t)$
$\displaystyle+\frac{K\sigma_{3}}{\sigma_{3}+\kappa_{3}}\tilde{c}_{e}(L_{23},t)+\frac{K\kappa_{3}}{\sigma_{3}+\kappa_{3}}\tilde{c}_{e}(L,t),$
(15d) $\displaystyle R_{\text{res}}$
$\displaystyle=R_{\text{ctc}}+\frac{L_{1}}{\sigma_{3}+\kappa_{3}}+\frac{L_{2}}{\kappa_{2}}+\frac{L_{3}}{\sigma_{3}+\kappa_{3}}.$
(15e)

Under the constant reactivity assumption of the SPM (5), which holds for cells
with sufficiently high electronic conductivities, then the voltage from the
particles’ surfaces can be simplified to

$\displaystyle
v_{u_{s}^{\text{surf}}}\approx\frac{1}{R_{s,3}}\frac{dU_{3}}{dc_{s}^{\text{surf}}}\tilde{u}_{s}^{\text{surf}}(L_{23},t)-\frac{1}{R_{s,1}}\frac{dU_{1}}{dc_{s}^{\text{surf}}}\tilde{u}_{s}^{\text{surf}}(L_{12},t),$
(16)

allowing the particle dynamics to be described by a single particle.

###  1.2 Structural Identifiability

This paper is concerned with structural identifiability of this system
(Bellman and Åström (1970)). Structural identifiability is defined as (Ljung
(2001); Alavi et al. (2016)):

######  Definition 1 (Structural identifiability)

Consider a model structure $\mathcal{M}$ with the transfer function
$H(s,\theta)$ parametrised by $\theta\in D\subset\mathbb{R}^{n}$ where $n$
denotes the number of parameters of the model. The identifiability equation
for $\mathcal{M}$ is given by:

$\displaystyle H(s,\theta)=H(s,\theta^{*})~{}\text{for almost all }s$ (17)

where $\theta,\theta^{*}\in\mathcal{D}$. The model structure $\mathcal{M}$ is
said to be

  * •

globally identifiable if (17) has a unique solution in $\mathcal{D}$,

  * •

locally identifiable if (17) has a finite number of solutions in
$\mathcal{D}$,

  * •

unidentifiable if (17) has a infinite number of solutions in $\mathcal{D}$.

If the model is globally identifiable then it is said to be structurally
identifiable and the parameter estimation problem is well-posed (meaning that
it admits a unique solution). To verify structural identifiability of the
considered model, the minimum number of parameters needed to uniquely
characterise the impedance functions of each electrochemical state is
determined.

##  2 Impedance analysis for structural identifiability

The analysis of the previous section, under the given assumptions, decomposed
the DFN model into three components with three distinct timescales: solid-
state diffusion of $u_{s}$ (slow); bulk electrolyte diffusion (mid) and
relaxation of overpotentials (fast). Structural identifiability of each of the
three decoupled timescales will now be analysed individually.

###  2.1 Low-frequency range: Solid-state diffusion

Solid-state diffusion within the active material particles dominates the low
frequency response (when $\omega\ll 1$ rad s-1), with the structural
identifiability of these dynamics detailed in Bizeray et al. (2018) and
repeated here for completeness. These dynamics are defined by the radial
diffusion of (6b), (7), (16) having Laplace transform

$\displaystyle\frac{\bar{\tilde{c}}^{surf}_{s}(x,s)}{\bar{j}^{*}(s)}=\left(\frac{R_{s}^{2}}{a_{s}\mathcal{F}D_{s}}\right)\left(\frac{\tanh\left(\sqrt{\frac{s}{\tau_{c_{s}}}}\right)}{\tanh\left(\sqrt{\frac{s}{\tau_{c_{s}}}}\right)-\sqrt{\frac{s}{\tau_{c_{s}}}}}\right).$
(18)

Applying the constant reactivity approximation of (5) to (18) gives the
transfer function of the SPM

$\displaystyle\frac{\bar{v}_{u_{s}^{\text{surf}}}}{\bar{I}(s)}$
$\displaystyle=\theta_{c_{s},3}\tau_{c_{s},3}\left(\frac{\tanh\left(\sqrt{\frac{s}{\tau_{c_{s},3}}}\right)}{\tanh\left(\sqrt{\frac{s}{\tau_{c_{s},3}}}\right)-\sqrt{\frac{s}{\tau_{c_{s},3}}}}\right)$
(19)
$\displaystyle-\theta_{c_{s},1}\tau_{c_{s},1}\left(\frac{\tanh\left(\sqrt{\frac{s}{\tau_{c_{s},1}}}\right)}{\tanh\left(\sqrt{\frac{s}{\tau_{c_{s},1}}}\right)-\sqrt{\frac{s}{\tau_{c_{s},1}}}}\right)$
(20)

with
$\theta_{c_{s}^{\text{surf}}}^{k}=\frac{dU_{k}/dc_{s}^{\text{surf}}}{L_{k}A_{cc}}$
and $\tau_{c_{s}}^{k}=R_{s,k}/\sqrt{D_{s,k}}$ where $k\in\\{1,3\\}$. These are
the four parameters that uniquely map the input current to variations in
particle surface concentrations, under the above assumptions.

###  2.2 Mid-frequency range: Movement of bulk electrolyte

Moving up the frequency range to $\omega\approx 1$ rad s-1, polarisation of
the bulk electrolyte becomes the dominant electrochemical effect. Structural
identifiability of the electrolyte requires a transfer function for (6a) and
(15d). Unfortunately, due to the boundary conditions (9a)-(9d) coupling the
electrolyte’s solutions across various domains, the impedance analysis of this
state is more complex to analyse than the other effects considered in the
model. To alleviate some of this complexity, the uniform reactivity
approximation (4) is applied to (6a), converting the dynamics in the
electrodes to

$\displaystyle\frac{\partial\tilde{c}_{e}(x,t)}{\partial t}$
$\displaystyle=D_{e,1}\frac{\partial^{2}\tilde{c}_{e}(x,t)}{\partial
x^{2}}+\pi_{1}i(t),\forall x\in\Omega_{1},$ (21a)
$\displaystyle\frac{\partial\tilde{c}_{e}(x,t)}{\partial t}$
$\displaystyle=D_{e,2}\frac{\partial^{2}\tilde{c}_{e}(x,t)}{\partial
x^{2}},\forall x\in\Omega_{2},$ (21b)
$\displaystyle\frac{\partial\tilde{c}_{e}(x,t)}{\partial t}$
$\displaystyle=D_{e,3}\frac{\partial^{2}\tilde{c}_{e}(x,t)}{\partial
x^{2}}-\pi_{3}i(t),\forall x\in\Omega_{3},$ (21c)

where
$\pi_{k}=\frac{a_{\text{s},k}\left(1-t_{+}\right)}{\varepsilon_{k}\mathcal{F}}$,
$k\in\\{1,3\\}$ is a parameter for the reactivity.

The co-ordinate transformation $\gamma$ is introduced, being defined in each
electrode as

$\displaystyle\gamma(x,t)=\tilde{c}_{e}(x,t)-\pi_{1}\int^{t}_{0}i(\tau)\,d\tau,~{}x\in\Omega_{1},$
(22a)
$\displaystyle\gamma(x,t)=\tilde{c}_{e}(x,t)+\pi_{3}\int^{t}_{0}i(\tau)\,d\tau,~{}x\in\Omega_{3}.$
(22b)

Then, in the electrodes,

$\displaystyle\frac{\partial\gamma(x,t)}{\partial t}$
$\displaystyle=\frac{\partial\tilde{c}_{e}(x,t)}{\partial
t}+\pi_{1}i(t),\forall x\in\Omega_{1},$ (23a)
$\displaystyle\frac{\partial\gamma(x,t)}{\partial t}$
$\displaystyle=\frac{\partial\tilde{c}_{e}(x,t)}{\partial
t}-\pi_{3}i(t),~{}\forall x\in\Omega_{3},$ (23b)
$\displaystyle\frac{\partial^{2}\gamma(x,t)}{\partial x^{2}}$
$\displaystyle=\frac{\partial^{2}\tilde{c}_{e}(x,t)}{\partial x^{2}},$ (23c)

with the dynamics in this co-ordinate system then converting (21) into the
simpler, unforced diffusion equation

$\displaystyle\frac{\partial\gamma(x,t)}{\partial
t}=D_{e,k}\frac{\partial^{2}\gamma(x,t)}{\partial
x^{2}},\,x\in\\{\Omega_{k}\\},\,k\in\\{1,\,3\\},$ (24)

where the forcing reactivity term in (21) has been moved into the boundary
conditions via (9d).

In each domain, the Laplace transform solution for (24) (assuming zero initial
conditions) is

$\displaystyle\bar{\gamma}(x,s)$
$\displaystyle=A_{1}(s)e^{\frac{\sqrt{s}x}{\sqrt{D_{e,1}}}}+B_{1}(s)e^{\frac{-\sqrt{s}x}{\sqrt{D_{e,1}}}},x\in\Omega_{1},$
(25a) $\displaystyle\bar{\tilde{c}}_{e}(x,s)$
$\displaystyle=A_{2}(s)e^{\frac{\sqrt{s}\left(x-L_{12}\right)}{\sqrt{D_{e,2}}}}+B_{2}(s)e^{\frac{-\sqrt{s}\left(x-L_{12}\right)}{\sqrt{D_{e,2}}}},x\in\Omega_{2},$
(25b) $\displaystyle\bar{\gamma}(x,s)$
$\displaystyle=A_{3}(s)e^{\frac{\sqrt{s}\left(x-L_{23}\right)}{\sqrt{D_{e,3}}}}+B_{3}(s)e^{\frac{-\sqrt{s}\left(x-L_{23}\right)}{\sqrt{D_{e,3}}}},x\in\Omega_{3},$
(25c)

with spatial derivatives

$\displaystyle\frac{\partial\bar{\gamma}}{\partial x}$
$\displaystyle=\frac{A_{1}(s)\sqrt{s}}{\sqrt{D_{e,1}}}e^{\frac{\sqrt{s}x}{\sqrt{D_{e,1}}}}-\frac{B_{1}(s)\sqrt{s}}{\sqrt{D_{e,1}}}e^{\frac{-\sqrt{s}x}{\sqrt{D_{e,1}}}},x\in\Omega_{1},$
(26a) $\displaystyle\frac{\partial\bar{\tilde{c}}_{e}}{\partial x}$
$\displaystyle=\frac{A_{2}(s)\sqrt{s}}{\sqrt{D_{e,2}}}e^{\frac{\sqrt{s}\left(x-L_{12}\right)}{\sqrt{D_{e,2}}}}-\frac{B_{2}(s)\sqrt{s}}{\sqrt{D_{e,2}}}e^{\frac{-\sqrt{s}\left(x-L_{12}\right)}{\sqrt{D_{e,2}}}},x\in\Omega_{2},$
(26b) $\displaystyle\frac{\partial\bar{\gamma}}{\partial x}$
$\displaystyle=\frac{A_{3}(s)\sqrt{s}}{\sqrt{D_{e,3}}}e^{\frac{\sqrt{s}\left(x-L_{23}\right)}{\sqrt{D_{e,3}}}}-\frac{B_{3}(s)\sqrt{s}}{\sqrt{D_{e,3}}}e^{\frac{-\sqrt{s}\left(x-L_{23}\right)}{\sqrt{D_{e,3}}}},x\in\Omega_{3}.$
(26c)

The six boundary conditions for the electrolyte (9) allow the six solution
coefficients $(A_{1},\,B_{1},\,A_{2},\,B_{2},\,A_{3},\,B_{3})$ of (25) to be
computed.

Beginning with the no-flux condition at the current collectors (9a), then

$\displaystyle A_{1}(s)$ $\displaystyle=B_{1}(s),$ (27a) $\displaystyle
B_{3}(s)$ $\displaystyle=A_{3}(s)e^{\frac{2\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}.$
(27b)

The constant flux conditions (9b), (9c) at the electrode/separator means

$\displaystyle
A_{1}(s)=\sqrt{\frac{D_{e,2}}{D_{e,1}}}\left(\frac{A_{2}(s)-B_{2}(s)}{e^{\sqrt{\frac{s}{D_{e,1}}}L_{1}}-e^{-\sqrt{\frac{s}{D_{e,1}}}L_{1}}}\right),$
(28a) $\displaystyle
A_{3}(s)=\sqrt{\frac{D_{e,2}}{D_{e,3}}}\left(\frac{A_{2}(s)e^{\sqrt{\frac{s}{D_{e,2}}}L_{2}}-B_{2}(s)e^{-\sqrt{\frac{s}{D_{e,2}}}L_{2}}}{1-e^{\frac{2\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}}\right).$
(28b)

Lastly, solution continuity at the separator interface (9d) ensures

$\displaystyle\pi_{1}\frac{\bar{i}(s)}{s}+$ $\displaystyle
A_{1}(s)\left(e^{\frac{\sqrt{s}L_{1}}{\sqrt{D_{e,1}}}}+e^{-\frac{\sqrt{s}L_{1}}{\sqrt{D_{e,1}}}}\right)=A_{2}(s)+B_{2}(s),$
(29a) $\displaystyle A_{2}(s)e^{\frac{\sqrt{s}L_{2}}{\sqrt{D_{e,2}}}}$
$\displaystyle+B_{2}(s)e^{-\frac{\sqrt{s}L_{2}}{\sqrt{D_{e,2}}}}$ (29b)
$\displaystyle=A_{3}(s)\left(1+e^{\frac{2\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}\right)-\pi_{3}\frac{\bar{i}(s)}{s}.$

We then have four equations to solve for the four remaining unknowns
$A_{1}(s),\,A_{3}(s),\,A_{2}(s)$ and $B_{2}(s)$.

Substituting (28a) into the continuity equation (29a) gives

$\displaystyle\pi_{1}\frac{\bar{i}(s)}{s}=(1-p_{1}(s))A_{2}(s)+(1+p_{1}(s))B_{2}(s)$
(30a) with $\displaystyle
p_{1}(s)=\sqrt{\frac{D_{e,2}}{D_{e,1}}}\frac{\left(e^{\sqrt{\frac{s}{D_{e,1}}}L_{1}}+e^{-\sqrt{\frac{s}{D_{e,1}}}L_{1}}\right)}{\left(e^{\sqrt{\frac{s}{D_{e,1}}}L_{1}}-e^{-\sqrt{\frac{s}{D_{e,1}}}L_{1}}\right)}.$
(30b)

The variables $p_{k}(s),$ $k=1,\,\dots\,,\,7$ are introduced to simplify the
algebra. Similarly, with (28b), the continuity condition (29b) at the right
hand side can be written

$\displaystyle-\pi_{3}\frac{\bar{i}(s)}{s}=p_{2}(s)A_{2}(s)+p_{2}(s)e^{\frac{-2\sqrt{s}L_{2}}{\sqrt{D_{e,2}}}}B_{2}(s)$
(31a) where $\displaystyle p_{2}(s)$
$\displaystyle=e^{\frac{\sqrt{s}L_{2}}{\sqrt{D_{e,2}}}}\left(1-\sqrt{\frac{D_{e,2}}{D_{e,3}}}\left(\frac{1+e^{\frac{2\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}}{1-e^{\frac{2\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}}\right)\right),$
(31b) $\displaystyle p_{3}(s)$
$\displaystyle=e^{-\frac{\sqrt{s}L_{2}}{\sqrt{D_{e,2}}}}\left(1+\sqrt{\frac{D_{e,2}}{D_{e,3}}}\left(\frac{1+e^{\frac{2\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}}{1-e^{\frac{2\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}}\right)\right).$
(31c)

Equating (30a) and (31a) gives an expression for $A_{2}(s)$

$\displaystyle A_{2}(s)$
$\displaystyle=\left(\frac{\pi_{1}p_{3}(s)+\pi_{3}(1+p_{1}(s))}{-\pi_{1}p_{2}(s)-\pi_{3}(1-p_{1}(s))}\right)B_{2}(s).$
(32a) $\displaystyle=p_{4}(s)B_{2}(s).$ (32b)

Subbing back into either (30a) or (31a) gives $B_{2}(s)$ in terms of
$\bar{i}(s)$

$\displaystyle B_{2}(s)$
$\displaystyle=\frac{\pi_{1}}{s(p_{2}(s)(1-p_{1}(s))+1+p_{1}(s))}\bar{i}(s)=p_{5}(s)\bar{i}(s),$
(33a)
$\displaystyle=\frac{-\pi_{3}}{s\left(p_{2}(s)p_{3}(s)+p_{3}(s)\right)}\bar{i}(s),$
(33b)

and so

$\displaystyle A_{2}(s)=p_{4}(s)p_{5}(s)\bar{i}(s).$ (34)

The remaining unknowns $A_{1}(s)$ and $A_{3}(s)$ can then be computed from
(28)

$\displaystyle
A_{1}(s)=\sqrt{\frac{D_{e,2}}{D_{e,1}}}\left(\frac{p_{4}(s)(p_{5}(s)-1)}{e^{\sqrt{\frac{s}{D_{e,1}}}L_{1}}-e^{-\sqrt{\frac{s}{D_{e,1}}}L_{1}}}\right)\bar{i}(s),$
(35a) $\displaystyle
A_{3}(s)=\sqrt{\frac{D_{e,2}}{D_{e,3}}}\left(\frac{p_{4}(s)\left(p_{5}(s)e^{\frac{\sqrt{s}L_{2}}{\sqrt{D_{e,2}}}}-e^{-\frac{\sqrt{s}L_{2}}{\sqrt{D_{e,2}}}}\right)}{1-e^{2\sqrt{\frac{{s}}{{D_{e,3}}}}L_{3}}}\right)\bar{i}(s),$
(35b)

or, more compactly,

$\displaystyle A_{1}(s)=p_{6}(s)\bar{i}(s),$ (36a) $\displaystyle
A_{3}(s)=p_{7}(s)\bar{i}(s).$ (36b)

The impedance function of the bulk electrolyte can then be expressed by

$\displaystyle\frac{\bar{v}_{c_{e}}(s)}{\bar{I}(s)}$
$\displaystyle=-\frac{Kp_{6}(s)}{\sigma_{1}+\kappa_{1}}\left(2\kappa_{1}+\sigma_{1}\left(e^{\frac{\sqrt{s}L_{1}}{\sqrt{D_{e,1}}}}+e^{\frac{-\sqrt{s}L_{1}}{\sqrt{D_{e,1}}}}\right)\right)$
$\displaystyle+\frac{Kp_{7}(s)}{\sigma_{3}+\kappa_{3}}\left(\sigma_{3}+\sigma_{3}e^{\frac{2\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}+2\kappa_{3}e^{\frac{\sqrt{s}L_{3}}{\sqrt{D_{e,3}}}}\right)$
$\displaystyle+K\frac{(\pi_{3}+\pi_{1})}{s},$ (37)

which is uniquely identified by the eleven parameters in Table 2. These
parameters relate to the the ionic time constants, the reactivity terms and
the relative diffusivities of each adjacent domain.

###  2.3 High-frequency response: Overpotential dynamics

The high-frequency response of the cell follows from the localised
overpotential dynamics (6c) coupled to the boundary conditions (8). In
$x\in\Omega_{k}$, $k\in\\{1,3\\}$, the Laplace transform for this system was
given in Ong and Newman (1999) as

$\displaystyle\frac{\bar{\tilde{\eta}}(x,s)}{\bar{i}(s)}=\frac{L_{k}}{\upsilon_{k}(s)\sinh(\upsilon_{k}(s))}$
(38)
$\displaystyle\times\left[\frac{1}{\sigma_{k}}\cosh\left(\frac{\upsilon_{k}(s)x}{L_{k}}\right)+\frac{1}{\kappa_{k}}\cosh\left(\upsilon_{k}(s)\left(1-\frac{x}{L_{k}}\right)\right)\right],$

with $\upsilon_{k}(s)$ given by

$\displaystyle\frac{\upsilon_{k}^{2}}{L_{k}^{2}}=\left(\frac{1}{\kappa_{k}}+\frac{1}{\sigma_{k}}\right)\left(\frac{\mathcal{F}a_{s,k}i_{0,k}^{*}}{RT}+C^{\text{sp}}_{k}s\right).$
(39)

From (38), the transfer function of the overpotential dynamics, corresponding
to the charge transfer resistance, can be computed as

$\displaystyle\frac{v_{\eta}(s)}{\bar{i}(s)}=$
$\displaystyle\left(\frac{\kappa_{3}/\sigma_{3}+\sigma_{3}/\kappa_{3}}{\kappa_{3}+\sigma_{3}}\right)\frac{L_{3}\coth(v_{3}(s))}{v_{3}(s)}$
$\displaystyle-\left(\frac{\kappa_{1}/\sigma_{1}+\sigma_{1}/\kappa_{1}}{\kappa_{1}+\sigma_{1}}\right)\frac{L_{1}\coth(v_{1}(s)}{v_{1}(s)}.$
(40)

This function is parameterised by

$\displaystyle\theta_{\eta}^{1,k}$
$\displaystyle=L_{k}\sqrt{\left(\frac{1}{\kappa_{k}}+\frac{1}{\sigma_{k}}\right)\frac{\mathcal{F}a_{s,k}i_{0,k}^{*}}{RT}},~{}k\in\\{1,3\\},$
(41a) $\displaystyle\theta_{\eta}^{2,k}$
$\displaystyle=A_{\text{cc}}L_{k}\left(\frac{\kappa_{k}/\sigma_{k}+\sigma_{k}/\kappa_{k}}{\kappa_{k}+\sigma_{k}}\right),~{}k\in\\{1,3\\},$
(41b) $\displaystyle\tau_{\eta}^{k}$
$\displaystyle=L_{k}^{2}{\left(\frac{1}{\kappa_{k}}+\frac{1}{\sigma_{k}}\right)C^{\text{sp}}_{k}},~{}k\in\\{1,3\\},$
(41c)

so that

$\displaystyle\upsilon_{k}=\theta^{\eta}_{1,k}+\sqrt{\tau_{\eta,k}}\sqrt{s}$
(42)

and

$\displaystyle\frac{v_{\eta}(s)}{\bar{I}(s)}=$
$\displaystyle\theta^{\eta}_{2,3}\frac{\coth(v_{3}(s))}{v_{3}(s)}-\theta^{\eta}_{2,1}\frac{\coth(v_{1}(s))}{v_{1}(s)}.$
(43)

Six parameters then uniquely determine the response due to heterogeneous
electrode overpotentials.

###  2.4 Combined analysis

When combined, these results show that the current to voltage mapping of the
considered decoupled and linearised battery model is characterised uniquely by
the 21 parameters of Table 2. The model is then structurally identifiable with
respect to these parameters and a unique solution to the parameter estimation
problem for this model then exists when current-voltage data is used. The
obvious next step of this research is then to develop algorithms to estimate
values for these identifiable parameters from data, with the key message of
this paper being that the feasibility of obtaining unique parameter estimates
for these electrochemical models should be considered before any algorithms
are applied.

## Conclusions

A structural identifiability analysis of a decoupled and linearised Doyle-
Fuller-Newman Li-ion battery electrochemical model was applied. It was shown
that the model is structurally identifiable from a group of 21 parameters
(composed of electrochemical quantities like the conductivities and lengths),
with these parameters uniquely characterising the impedance function of this
model. The parameter estimation problem for this model is then well-posed with
respect to these groups of parameters. Future work will aim to exploit this
result to develop an algorithm to recursively estimate the parameter values
for pseudo-2D battery models from generic data.

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