# LMEC: Learnable Multiplicative Absolute Position Embedding Based Conformer
for Speech Recognition

Yuguang Yang  Ximalaya Inc., ShangHai, China  Yu Pan  University of Alberta,
Edmonton, Canada  Jingjing Yin  Ximalaya Inc., ShangHai, China  Heng Lu
Ximalaya Inc., ShangHai, China

###### Abstract

This paper proposes a Learnable Multiplicative absolute position Embedding
based Conformer (LMEC). It contains a kernelized linear attention (LA) module
called LMLA to solve the time-consuming problem for long sequence speech
recognition as well as an alternative to the FFN structure. First, the ELU
function is adopted as the kernel function of our proposed LA module. Second,
we propose a novel Learnable Multiplicative Absolute Position Embedding (LM-
APE) based re-weighting mechanism that can reduce the well-known quadratic
temporal-space complexity of softmax self-attention. Third, we use Gated
Linear Units (GLU) to substitute the Feed Forward Network (FFN) for better
performance. Extensive experiments have been conducted on the public
LibriSpeech datasets. Compared to the Conformer model with cosFormer style
linear attention, our proposed method can achieve up to 0.63% word-error-rate
improvement on test-other and improve the inference speed by up to 13% (left
product) and 33% (right product) on the LA module.

Index Terms: Conformer, cosFormer, Linear Attention, Position Embedding, Gated
Linear Units

##  1 Introduction

Over the last decade, end-to-end models have been applied as mainstream
approaches for state-of-the-art automatic speech recognition (ASR) systems.
One major architecture is attention based encoder-decoder (AED)[1, 4, 5, 6]. A
recent trend of building an encoder-decoder ASR model is using stacks of
Transformer block or its variations with joint cross entropy and Connectionist
Temporal Classification (CTC) loss[3].

Compared with the network of vanilla transformer structure, the Conformer[2]
block combines the advantages of convolution’s ability to capture local
information and the transformer’s ability to observe global features.
Furthermore, since the transformer structure lacks the capture of time series
information, position embedding is used, like relative position
encoding[7](XL-RPE), absolute position embedding (APE)[4], and so forth.
Generally, XL-RPE performs better than APE in the Conformer, especially on ASR
tasks.

However, the Conformer block still has some critical points that can be
improved. The main drawback is the quadratic temporal-space complexity due to
the core softmax self-attention operation, which is especially important for
long input sequences. In addition, the Feed Forward Network (FFN) layer can
also be optimized, as it occupies most of the computation in the Conformer
block and plays a decisive role in the effect of the model performance.

Many excellent works[8] have been proposed to tackle the high computational
and memory cost in Transformer-style blocks. The bottleneck mainly comes from
the calculation of $O(N^{2})$ in self-attention, where $N$ refers to the
length of the input sequence. To be specific, in order to reduce the amount of
calculation while minimizing quality loss, some works focus on sparse
attention maps or through low-rank decomposition such as[9, 10, 11].
Meanwhile, some other works’ direction is to turn the calculation of the
original left product into a right product[13, 14, 15]. Besides, [12] proposes
a synthetic self-attention module to approximate attention weights. Efficient
Conformer[21] try to use grouped multi-head attention to reduce its
complexity. Recently, kernel based self-attention[14, 15, 16, 17, 18] has
demonstrated its advantages over vanilla transformers, especially in long
sequence modeling and inference speed.

In this paper, inspired by cosFormer[14], Performer[15] and RFA[18], which
decouple the $O(N^{2}d)$ operation to $O(Nd^{2})$ by replacing self-attention
calculation from $(QK^{T})V$ to $Q(K^{T}V)$, we hope to realize this kind of
self-attention decoupling in ASR Conformer block. In the actual inference
phase for the ASR task, the audio input range may vary from a few seconds to
tens of seconds, which will cause the model to fluctuate in the amount of
calculation and memory usage. Due to the complexity of $O(N^{2})$ operation,
it will make the inferential computing of long audio more difficult.

Moreover, such decoupling for self-attention somewhat conflicts with the
implementation of XL-RPE in the Conformer block, because the scale product of
$Q$ and $K$ as well as relative shift operation cannot be applied to the
linear complexity output $Q(K^{T}V)$. Several attempts have been made to
introduce other RPE to the LA module[9, 14]. Nevertheless, these complex
position embedding approaches are not that cost-effective in ASR tasks.

Another concern in the Conformer block is the Macaron-style FFN module, which
usually consists of two fully connected layers and an activation function.
Although its infrastructure is simple, there are still some works to optimize
this part through sharing parameters like grouped feed forward module[19] or
changing the computing paradigm like[20].

The main contributions of our work are as below:

  * •

We propose a new linear attention kernel, which is much lighter than conformer
self-attention and cosFormer attention. We apply this kernel to the Conformer
and test it on the ASR task.

  * •

We propose to apply the Gated Linear Units layer instead of the Feed Forward
Network in Conformer block.

  * •

We recommend different computational strategies for training and inference.
Left product is used during training to ensure model quality. Dynamically
choosing the product mechanism ensures the inference speed during inference.

To sum up, our goal is to design a model architecture with better performance
in ASR tasks, lower latency in actual use, and better overall cost performance
than Conformer.

##  2 Related Work

Since transformer is widely used, many workers in classic tasks hope to have a
better self-attention paradigm. For example, [22] and [23] both tried to
design a new linear attention kernel and applied it on the ASR task. Their
aims on linear attention (LA) part are the same as ours, which is decoupling
the self-attention operation from attention score calculation and changing
computational complexity from $O(N^{2}d)$ to $O(Nd^{2})$.

###  2.1 Transformer Block

Let’s review the structure of the standard vanilla transformer block in Fig.1
(a), which is a stack of self-attention module with feed forward layer.

The key structure of transformer is undoubtedly self-attention module. Suppose
hidden features from the previous layer is $x$, it will be projected into
$Q=xW_{q}$, $K=xW_{k}$ and $V=xW_{v}$ matrix. Then, self-attention output can
be represented as below:

$O_{attn}=SA(Q,K,V)=\textit{softmax}(\frac{QK^{\intercal}}{\sqrt{d_{k}}})V$
(1)

where $SA(\cdot)$ stands for self-attention operation, Q, K and V stands for
query, key and value matrix, $d_{k}$ is the hidden dimension.

###  2.2 Conformer Block

Combining the transformer’s advantages in capturing global information with
CNN’s advantages in capturing local information, the Conformer block has made
great achievements and has already become a benchmark in speech recognition
tasks.

As shown in Fig.1 (b), Conformer block[2] is a stack of feed forward module,
multi-head attention module, convolution module and feed forward module again.
Suppose the input of $i$-th layer is $x_{i}$, the specific calculation method
is as follows:

$\begin{split}y_{i}=LN^{\tiny{+⃝}}(\frac{1}{2}FFN^{\tiny{+⃝}}(\text{Conv}^{\tiny{+⃝}}(SA^{\tiny{+⃝}}(\frac{1}{2}FFN^{\tiny{+⃝}}(x_{i}))))\end{split}$
(2)

Here the residual function is expressed as $F^{\tiny{+⃝}}(x)=x+F(x)$.

It is easy to see from the figure that self-attention and feed forward modules
are the most computationally intensive net-work structures, which we will
focus on later in this paper.

###  2.3 cosFormer Kernel

A self-attention that can disassemble left and right multiplication has been
redesigned in cosFormer[14]. In cosFormer, ReLU linear attention activation
and cos-based re-weighting mechanism is used as the substitute for softmax
operator. And the row-wise similarity function of $Q$ and $K$ is expressed as:

$s(Q_{i}^{\prime},K_{j}^{\prime})=\psi(Q_{i})\psi(K_{j})cos(\frac{\pi}{2}\times\frac{i-j}{M})$
(3)

where $\psi(\cdot)$ stands for linear attention activation operation. And the
formulation can be decomposed as:

$\begin{split}s(Q_{i}^{\prime},K_{j}^{\prime})=&(\psi(Q_{i})cos(\frac{\pi
i}{2M}))(\psi(K_{j})cos(\frac{\pi j}{2M}))^{\intercal}\\\
+&(\psi(Q_{i})sin(\frac{\pi i}{2M}))(\psi(K_{j})sin(\frac{\pi
j}{2M}))^{\intercal}\end{split}$ (4)

Then the new version of self-attention operation could be derivatived from
Eq.1 to

$\begin{split}O_{attn}=&S(Q,K)V\\\ =&(Q^{cos}K^{cos}+Q^{sin}K^{sin})V\\\
=&Q^{cos}(K^{cos}V)+Q^{sin}(K^{sin}V)\end{split}$ (5)

This approach eliminates the computation bottleneck of long sequence
$O(N^{2})$ calculation compared to the vanilla transformer through clever
formula disassembly and proves its effectiveness in NLP tasks.

From another perspective, this cos-based multiplicative item in cosFormer
realizes a new form of RPE. In order to facilitate the subsequent comparation,
we named the cosFormer kernel as a multiplicative relative position embedding
(M-RPE) LA.

Fig. 1:  The whole structure of Our Mentioned Blocks

##  3 Proposed Methodology

In this section, we propose a Conformer-style block called LMEC based on
kernelized LA and GLU module. First, we focus on designing our linear
attention kernel by choosing an appropriate activate function and improving
the re-weighting mechanism. The primary purpose is to acquire the
concentration ability for the attention weights’ distribution, just retaining
the property of softmax attention, but with a much more straightforward
calculation. Then, we introduce variations of Additive RPE and No PE LA to
fully illustrate the effect of different positional embedding. Next, we adopt
GLU module and use it to substitute FFN layer of Conformer block. Finally, we
explore the differences between left product and right product training. The
overall architecture of our LMLA module is shown Fig.1 (c).

###  3.1 MLA: M-APE Based Linear Attention Kernel

Here we start with two most important points in MLA, the selection of Linear
Attention Activation and the Re-weighting mechanism.

####  3.1.1 Linear Attention Activation Function

Following the decompositon approach in cosFormer[14], we adopt the
decomposable kernel function to rewrite the attention matirx as:

$s(Q_{i}^{\prime},K_{j}^{\prime})=\psi(Q_{i})\psi(K_{j})$ (6)

where $\psi(\cdot)$ maps each row of $Q$ and $K$ to their hidden
representations $Q_{i}^{\prime}$ and $K_{j}^{\prime}$.

However, what kind of mapping method can achieve a better effect in ASR task
is worth exploring. The previous experience of scholars in the NLP task is
that it is more beneficial to the performance and convergence of the model
after mapping the input hidden representation matrix from the upper layer to
the non-negative range[14, 18, 24, 25, 15]. We have selected four common
activation functions as our kernel function for experimental comparison:

$\begin{split}\psi_{relu}(x)&=Relu(x)\\\ \psi_{sigmoid}(x)&=Sigmoid(x)\\\
\psi_{tanh}(x)&=0.5+Tanh(x)+0.5\\\ \psi_{elu}(x)&=ELU(x)+1\\\ \end{split}$ (7)

####  3.1.2 M-APE Based Re-weighting Mechanism

Expected to improve the re-weight mechanism, we make some efforts in
introducing diverse RPE forms to the decomposed LA kernel. But it seems not
cost-effective for ASR tasks. Results of different positional embedding we
will illustrate in later experiments. Here, we concentrate on how we optimize
cosFormer RPE, as the twice $QK^{\intercal}$ style calculation in which still
has redundant parts.

We degenerate original Eq.3 from relative position coding into absolute
position coding and is expressed as follows:

$s(Q_{i}^{\prime},K_{j}^{\prime})=\psi(Q_{i})\psi(K_{j})cos(\frac{\pi}{2}\times\frac{j}{M})$
(8)

which full name is called multiplicative absolute positional embedding linear
attention (M-APE) LA.

###  3.2 LMLA: LM-APE Based Linear Attention Kernel

Furthermore, some works[26, 27] show that random learnable position vectors
could achieve better results on specific tasks. Therefore, we simplify the
Eq.8 as:

$s(Q_{i}^{\prime},K_{j}^{\prime})=\psi(Q_{i})\psi(K_{j})cos(R_{j})$ (9)

where $R$ stands for a random vector. $R_{j}$ is a learnable value for each
position $j$ only applied to the key matrix, so that there is no need to
calculate the scale product operation of QK twice as in Eq.5. Uniformly, we
call this linear attention with learnable multiplicative absolute positional
embedding (LM-APE) as LMLA.

Fig. 2:  Explaination of Re-weighting Mechanism in LMLA

As mentioned above, our proposed APE-based re-weighting approaches are more
straightforward and faster than the M-RPE of cosFormer in Eq.4. To improve the
model’s expressiveness, we extend the embedding of a specific position $j$
from a value to a vector as compensation, slightly increasing the parameters
and retaining the calculation cost. Fig2 illustrates how LM-APE in Eq.9 is
applied to each element $K_{j}^{\prime}\in\mathbb{R}^{1\times d_{k}}$. If
$R_{j}\in\mathbb{R}^{1\times 1}$, it will be automatically broadcasted to
$\mathbb{R}^{1\times d_{k}}$ before element-wise production of
$\psi(K_{j})cos(R_{j})$. In our design,
$R_{j}=[r^{0},r^{1},\dots,r^{d_{k}-1}]^{\intercal}$, where each $r$ is
individually learnable. For M-APE, we use a simple affine transform
$W_{ext}\in\mathbb{R}^{1\times d_{k}}$ to extend
$\cos(\frac{\pi}{2}\times\frac{j}{M})$ mentioned in Eq.8 to a $d_{k}$
dimension vector for making the comparison fair. Our subsequent experimental
conclusions are also adopted in this way by default.

###  3.3 Variations for Comparison

####  3.3.1 Additive Relative Position Embedding Linear Attention

In order to involve positional information by product operation, we also tried
to add information through bias. Besides, the basic calculation of self-
attention with relative position embedding (XL-RPE), which is slightly
different from Eq.1 is as below:

$SA(Q,K,V)=\textit{softmax}(\frac{QK^{\intercal}+S_{bd}}{\sqrt{d_{k}}})V$ (10)

where $S_{bd}$ stands for the $b$ and $d$ option from transformer-xl, which is
actually calculated by $Q$ and linear mapped position encoding with relative
shift operation. Due to space limitations, those interested in specific
details can refer to [7]. Here, we derive the value of $S_{bd}$ to
$\cos(\frac{\pi}{2}\times\frac{i-j}{M})$ in order to decouple $QK$
calculation. Here, we call it additive relative positional embedding (A-RPE)
LA module.

####  3.3.2 No Position Embedding Linear Attention

We hypothesize there is no performance degradation after changing the position
embedding method from RPE into APE in the linear attention paradigm, In order
to prove the effectiveness of APE, we also compare the variation of linear
attention with no positional embedding (NPE). On the basis of our proposed
formula, if the positional embedding part is removed, its formula can be
expressed as Eq.6, where other settings are totally the same as LMLA block
except for the position embedding.

###  3.4 Gated Linear Units

The Conformer block usually applies Convolutional Neuron Network (CNN) layer
followed by Feed Forward Network (FFN) layer, which is represented as below
(here we omit bias representation):

$O_{ffn}=\sigma(xW_{1})W_{2}$ (11)

where $\sigma(\cdot)$ stands for activation functions.

Besides, $W_{1}\in\mathbb{R}^{h_{o}\times h_{ffn}}$ and
$W_{2}\in\mathbb{R}^{h_{ffn}\times h_{o}}$, where $h_{o}$ is the hidden
dimension of encoder output, $h_{ffn}$ is the hidden dimension of FFN layer.
We propose to replace the FFN layer with the Gated Linear Units layer[20],
which represents as below:

$O_{glu}=(\sigma(xW_{1})\otimes xW_{2})W_{3}$ (12)

where $\otimes$ stands for component-wise product operation,
$W_{1},W_{2}\in\mathbb{R}^{h_{o}\times h_{glu}}$,
$W_{3}\in\mathbb{R}^{h_{glu}\times h_{o}}$. For keeping the number of
parameters unchanged, we set $h_{glu}=\frac{2}{3}h_{ffn}$.

###  3.5 Training and Inference Method

In linear attention designed by us, the results of left product (L-Prod)
calculations and right product (R-Prod) calculations are consistent. But in
the actual training process, the gradients update differently. Suppose the
simplified LMLA formula of left product and right product training is as
below:

$O_{attn}=\underbrace{({\color[rgb]{1,0,0}Q^{\prime}}({\color[rgb]{1,0,0}K^{\prime}\otimes
R^{cos}}))V}_{{\color[rgb]{1,0,0}L-Prod}}=\underbrace{Q^{\prime}(({\color[rgb]{0,0,1}K^{\prime}\otimes
R^{cos}}){\color[rgb]{0,0,1}V})}_{{\color[rgb]{0,0,1}R-Prod}}$ (13)

where $Q^{\prime}=\psi(Q)$, $K^{\prime}=\psi(K)$, $R^{cos}=cos(R)$.

Based on our experiments, we suggest an L-Prod approach for training and a
dynamic chosen inference method. In other words, in the training stage, we try
to use L-Prod calculation to ensure training effect and convergence stability.
When $N\leq d$, we use L-Prod inference, and when $N\geq d$, we use R-Prod
inference to maximize the inference speed.

##  4 Experiments

Table 1: WER(%) results on LibriSpeech for different models

|  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
Model | Activation | LA Style |  with GLU |  Heads | Test Clean | Test Other
|  |  ctc greedy |  attention rescore |  ctc greedy |  attention rescore
Conformer0[2] | - | XL-RPE |  $\times$ |  4 |  3.64 |  3.26 |  9.28 |  8.51
Conformer1[2] | - | XL-RPE |  $\times$ |  8 |  3.51 |  3.31 |  9.12 |  8.51
cosFormer0[14] | ReLU | M-RPE-LA |  $\times$ |  4 |  3.79 |  3.38 |  9.80 |  8.95
cosFormer1[14] | ReLU | M-RPE-LA |  $\times$ |  8 |  3.85 |  3.44 |  9.80 |  9.00
LBLA0[23] | Sigmoid | M-RPE-LA |  $\times$ |  4 |  3.63 |  3.19 |  9.35 |  8.51
LBLA1[23] | Sigmoid | M-RPE-LA |  $\times$ |  8 |  3.60 |  3.23 |  9.38 |  8.68
LMEC0 (Ours) | ELU | LM-APE-LA |  ✓ |  4 |  3.54 |  3.20 |  9.13 |  8.32
LMEC1 (Ours) | ELU | LM-APE-LA |  ✓ |  8 |  3.57 |  3.17 |  9.18 |  8.38

  * •

Model{0/1}: Model with 4 or 8 attention head on attention module;
{cosFormer/LBLA}{0/1}: Conformer with cosFormer/LBLA attention module;

###  4.1 Experimental Dataset & Settings

In this work, we evaluate our proposed model and other state-of-the-art models
on the open source dataset LibriSpeech[28]. LibriSpeech consists of 1000 hours
of 16kHZ labeled English audio, which is divided into train, dev and test. We
train all models on the LibriSpeech training dataset which contains
approximately 960 hours of 16kHZ English speech with its corresponding text,
and evaluate them on the LibriSpeech test-clean and test-other datasets.

We first use SentencePiece[29] to build a byte pair encoding tokenizer, which
generates 5000 subword pieces from the transcripts of LibriSpeech. And the
audios are converted to 80-dimensional filter-bank feature sequences, and
Spec-Augment[30] is used during training. All models are trained with 120
epochs, and optimized by Adamw[31], where the original learning rate is 0.001
and warm up step is 10000 with Cosine Annealing[32] scheduler. In terms of
model structure for encoder, 12 layers of the Conformer style block is used,
in which convolutional kernel size is 15, model dimension is 256, hidden
parameter of FFN layer is 2048. The decoder has 6 layers single directional
transformer with 4 attention heads and FFN dimension is 2048. All our
experiments are conducted on WeNet[1] Toolkit and uniformly trained under the
condition of 4 A100 GPUs.

###  4.2 Main Results

In this section, we compare the overall performance of our proposed LMEC model
with some state-of-the-art Conformer-style models, including baseline
Conformer[2], cosFormer[14], LBLA[23]. For sufficient comparison, we compare 4
results between our model and the references on LibriSpeech test-clean/test-
other. Two most common attention heads (4 / 8) and two evaluation methods (ctc
greedy search / attention rescoring) are adopted in our experiments.

Compared with Conformer, our proposed LMEC has 3 out of 4 results that perform
better, achieving 0.2$\sim$4.2% relative WER reduction on test-clean. Only one
performs slightly worse. And similar conclusions can be achieved on test-
other. More, the linearized attention kernel (LMLA) can be computed more
easily and quickly than the softmax attention.

In contrast to cosFormer and LBLA, which are also linear kernels, our
performance gains are more significant. Compared with cosFormer, LMLA can
achieve up to 5.3$\sim$7.8% relative improvement on test-clean and
6.3$\sim$7.0% on test-other. LMLA also shows 0.1$\sim$2.5% (test-clean) and
2.13$\sim$3.46% (test-other) relative promotion than LBLA. All the results
validate the effectiveness of the proposed LMEC model.

Fig. 3:  Inference Time Cost of cosFormer and our LMLA

We use the input signal from 100 to 2000 frames (each frame stands for 40ms
audio) with the batch size of 100 on a single A100 GPU to test the inference.
All tests adopted GPU warm-up inference 1000 times and averaged. One of the
most significant features of Fig.3 is that as the input sequence gets longer,
the speed advantage of right-product computing becomes more evident than left-
product computing. Compared with cosFormer, LMLA reduces the averaged left-
product time cost from 66s to 57s (rel -13%) and the right-product time cost
from 15s to 10s (rel -33%) when the input sequence length is 2000 (80s).

In conclusion, our proposed model LMLA obtains better performance and fewer
inference costs than Conformer, cosFormer and LBLA.

###  4.3 Ablation Studies

####  4.3.1 Effect of Our Proposed LA Module

In this section, we evaluate the effectiveness of our proposed

Table 2:  Effect of Our Proposed LA Module

|  |  |  |  |  |
---|---|---|---|---|---|---
Model |  LA Acti-vation |  LA Style | Test Clean | Test Other
|  |  |  ctc greedy |  attention rescore |  ctc greedy |  attention rescore
LMLA-R |  ReLU |  LM-APE |  3.74 |  3.29 |  9.37 |  9.05
LMLA-T |  Tanh |  LM-APE |  3.78 |  3.33 |  9.73 |  8.91
LMLA-S |  Sigmoid |  LM-APE |  3.67 |  3.28 |  9.37 |  8.51
LMLA-E |  ELU |  LM-APE |  3.70 |  3.32 |  9.37 |  8.47
MLA-R |  ReLU |  M-APE |  3.97 |  3.53 |  9.87 |  9.04
MLA-T |  Tanh |  M-APE |  3.91 |  3.48 |  9.63 |  8.79
MLA-S |  Sigmoid |  M-APE |  3.73 |  3.29 |  9.43 |  8.60
MLA-E |  ELU |  M-APE |  3.80 |  3.39 |  9.36 |  8.52
V0-R |  ReLU |  A-RPE |  3.79 |  3.35 |  9.62 |  8.82
V0-T |  Tanh |  A-RPE |  3.85 |  3.39 |  9.73 |  8.82
V0-S |  Sigmoid |  A-RPE |  3.90 |  3.44 |  9.97 |  9.06
V0-E |  ELU |  A-RPE |  3.68 |  3.33 |  9.65 |  8.73
V1-R |  ReLU |  NPE |  3.95 |  3.54 |  9.85 |  9.07
V1-T |  Tanh |  NPE |  3.87 |  3.47 |  9.77 |  8.87
V1-S |  Sigmoid |  NPE |  3.89 |  3.44 |  9.59 |  8.75
V1-E |  ELU |  NPE |  3.75 |  3.36 |  9.35 |  8.64

  * •

[L]MLA: [Learnable] Multiplicative Linear Attention;
V{0/1}: Linear Attention Variations with {Additive RPE / No PE}
*-{R/T/S/E}: LA block with $\psi_{\\{relu/tanh/sigmoid/elu\\}}(\cdot)$

activation function $\psi_{elu}(\cdot)$ and LM-APE in LMLA block.

The results are illustrated in Table2. Here, all the number of head in encoder
is set to 8 and GLU is not used for fair. Next, we will take the decoding
results of attention rescoring as an example to analyze the key points we put
forward.

Under four different linear attention paradigm, $\psi_{elu}(\cdot)$ obtained
an average of 3.35/8.59% on test-clean/test-other. About the other three,
$\psi_{relu}(\cdot)$ obtained average 3.43/9.00%, $\psi_{tanh}(\cdot)$
achieves 3.42/8.85%, $\psi_{sigmoid}(\cdot)$ achieves 3.36/8.73%. Obviously,
$\psi_{elu}(\cdot)$ is the best choice in our design.

In comparison, our proposed LMLA performs better than other variations with
different styles of positional embedding, which achieves 3.30/8.74% on
average. Otherwise, MLA performs an average of 3.42/8.74%, which shows the
effectiveness of learnable embeddings. V0 with A-RPE gets the results of
3.38/8.86%. Compared with APE, we can see that the advantage of RPE in linear
attention is not apparent as usual. Furthermore, V1 gets 3.45/8.83%, which
further reveals that the absence of positional embedding will bring specific
degradation effects.

On the other hand, we can get the same conclusions in terms of the ctc greedy
search scenario.

####  4.3.2 Effect of Gated Linear Units

Table 3:  Effect of GLU |  |  |  |  |  |
---|---|---|---|---|---|---
Model |  with GLU |  GLU Acti-vation | Test Clean | Test Other
|  |  ctc greedy |  attention rescore |  ctc greedy |  attention rescore
Conformer |  $\times$ |  - |  3.51 |  3.31 |  9.12 |  8.51
LMEC-E |  $\times$ |  - |  3.70 |  3.32 |  9.37 |  8.47
LMEC-E-$S_{w}$ |  $\checkmark$ |  Swish |  3.61 |  3.24 |  9.34 |  8.50
LMEC-E-E |  $\checkmark$ |  ELU |  3.70 |  3.24 |  9.24 |  8.41
LMEC-E-R |  $\checkmark$ |  ReLU |  3.62 |  3.30 |  9.14 |  8.39
LMEC-E-G |  $\checkmark$ |  GeLU |  3.57 |  3.17 |  9.18 |  8.38

Then, we test the effectiveness of the GLU module and the result is
illustrated in Table3. In this set of comparative experiments, the head number
of all models is set to 8 as well.

In Table3, four kinds of GLU modules with different activation functions are
evaluated. It can be seen that when applying the GLU module, the WER of our
proposed LMEC model can be reduced, and LMEC-E-G with GeLU activation function
achieves the best results. Specifically, when we use ctc greedy search to
evaluate the models, LMEC-E-G achievements 3.51/2.03% relative WER reduction
on test-clean/test-other compared with LMEC-E without GLU module. Meanwhile,
through the attention rescoring, LMEC-E-G outperforms LMEC-E 4.52/1.06%
relative WER reduction on test-clean/test-other. In addition, our final
proposed LMLA-E-G obtains a comparable WER compared with Conformer.

###  4.4 Effect of Training and Inference Method

Table 4:  Differentiated Training Method

|  |  |  |  |
---|---|---|---|---|---
Model | Train Method | Test Clean | Test Other
ctc greedy |  attention rescore |  ctc greedy |  attention rescore
|  |  |  |  |
LMEC-E | L-Prod |  3.70 |  3.32 |  9.37 |  8.47
LMEC-E | R-Prod |  3.81 |  3.34 |  9.54 |  8.69
|  |  |  |  |
LMEC-E-G | L-Prod |  3.57 |  3.17 |  9.18 |  8.38
LMEC-E-G | R-Prod |  3.61 |  3.17 |  9.26 |  8.45

  * •

[L]MLA-*-{$S_{w}$/E/R/G}: GLU block with different activation functions.

It is not difficult to see from the Table4 that compared with R-Prod training,
L-Prod training effect and model convergence stability are better. Because the
calculation result of L-Prod and R-Prod are equal in the forward pass of
cosFormer, LBLA and LMLA, the comparison results in this paper are all
obtained by L-Prod training and R-Prod inference.

##  5 Conclusions

This paper demonstrates that multiplicative RPE is not cost-effective in ASR
tasks under the paradigm of linear attention, and it is recommended to use APE
that can be learned to replace cosFormer kernel. Furthermore, the authors
recommend using GLU module with GeLU activation function to replace FFN to
further improve model performance. In order to illustrate the effectiveness of
the proposed points, the authors mainly conduct experiments on the encoder
part, and in fact, the innovative points of the experiments is completely
orthogonal to the decoder part.

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# BattX: An Equivalent Circuit Model for Lithium-Ion Batteries Over Broad
Current Ranges

Nikhil Biju  Huazhen Fang  Department of Mechanical Engineering, University of
Kansas, Lawrence, KS 66045, USA

###### Abstract

Advanced battery management is to lithium-ion battery systems as the brain is
to the human body. Its performance rests on the use of battery models that are
both fast and accurate. However, mainstream equivalent circuit models and
electrochemical models have yet to meet this need well, due to struggle with
either predictive accuracy or computational complexity. This problem has
acquired urgency as some emerging battery applications running across broad
current ranges, e.g., electric vertical take-off and landing aircraft, can
hardly find usable models from the literature. Motivated to address the
problem, we develop an innovative model in this study. Called BattX, the model
is an equivalent circuit model but draws comparisons to a single particle
model with electrolyte and thermal dynamics, thus combining their respective
merits to be computationally efficient, accurate, and physically
interpretable. The model design pivots on leveraging multiple circuits to
approximate major electrochemical and physical processes in
charging/discharging. Given the model, we develop a multipronged approach to
design experiments and identify its parameters in groups from experimental
data. Experimental validation proves that the BattX model is capable of
accurate voltage prediction for charging/discharging across low to high
C-rates.

##  1 Introduction

Lithium-ion batteries (LiBs) are a key power source for consumer electronics,
electrified transportation, smart grids, and renewable energy. Compared with
alternative battery electrochemistries, they provide a set of outstanding
features, including high energy/power density, high nominal voltage, no memory
effect, low self-discharge rates, and long cycle life [1, 2, 3]. Recent
technological advances have further improved their power performance and cost
efficiency for a wider application spectrum. High-quality dynamic models are
foundational to monitoring and control of LiBs for guaranteed operational
safety and performance. While the growing research has led to a variety of
useful models, the literature still lacks fast and accurate models for
applications involving charging/discharging from low to high current rates. To
fill this gap, we propose a first-of-its-kind equivalent circuit model named
BattX and demonstrate its predictive fidelity over broad C-rate ranges.

Literature Review. Research on LiB dynamic modeling has flourished in the past
decades to produce a vast literature. The mainstream models generally fall
into two categories: electrochemical models, and equivalent circuit models
(ECMs). Electrochemical models explicitly describe electrochemical reactions,
transport of lithium ions, and distribution of charge and potential inside a
LiB cell. Depending on the need for accuracy, they exist in diverse scales,
from atomic/molecular to species level, and in different dimensions, from 1D
to 3D and beyond, and often coupled with different physical processes, e.g.,
thermodynamics and stress/strain [4]. Generally, electrochemical models
present high mechanistic fidelity as well as high computational complexity.
Battery management researchers hence must selectively focus on those that
offer desirable accuracy-computation trade-off, due to practical demands for
fast computation. A favorable choice is the pseudo-2D Doyle-Fuller-Newman
(DFN) model, which describes the diffusion of lithium ions and charge transfer
across the electrodes, electrolyte, and separator of a sandwich cell [5]. The
search for more efficient models has led to the single particle model (SPM),
which represents each electrode by a single spherical particle and neglects
the electrolyte dynamics [6]. The simplification enhances computational
efficiency to a great extent but also limits the SPM model to low to moderate
C-rates (around or less than 1 C). Subsequent studies have emerged to expand
the SPM model by adding characterization of a cell’s thermal behavior [7, 8],
electrolyte dynamics [9, 10, 11, 12, 13], stress buildup [13], or degradation
[14], to elevate its prediction capability. The literature has also presented
a few computational methods to speed up simulation of the SPM model or its
improved versions [15, 16].

ECMs represent another important pathway to modeling LiBs. They are circuit
analogs composed of electrical components to simulate a cell’s dynamic
behavior, capture phenomena in charging/discharging, and track state-of-charge
(SoC) and power capability. With simple structures, they are accessible to
interpretation, easy to calibrate, and scalable to large LiB systems composed
of many cells. Also, they are governed by low-order ordinary differential
equations, thus allowing for very fast computation. These benefits combine to
make them popular candidates for real-world battery management systems with
limited computing resources [17]. A basic ECM, called the Rint model, cascades
an open-circuit voltage (OCV) source with an internal resistor, in which the
voltage source is SoC-dependent [18]. One can add to the Rint model a set of
serially connected RC pairs to describe the transient behavior in a cell’s
voltage response, leading to the so-called Thevenin’s model [19, 20].
Depending on the number of RC pairs used, one can set the model to capture
transients at multiple time scales [21]. The literature has presented a few
approaches to modify the Thevenin’s model for better accuracy. For example,
the study in [20, 3] incorporates hysteresis in charging/discharging; in [22,
23, 24, 25], different circuit parameters (e.g., the internal resistance) are
made dependent on the SoC, temperature, or current loads, and the OCV is
parameterized using different function forms for higher fitting accuracy. Even
though phenomenological ECMs and electrochemical models were largely two
disparate threads of research, a growing number of studies have explored to
develop ECMs drawing upon electrochemical modeling. The work in [26, 27]
proposes the nonlinear double capacitor model to approximate the ion diffusion
in the electrodes of a cell and characterizes the nonlinear voltage behavior
simultaneously. This model is interpretable as a reduced-order version of the
SPM, and it is further supplemented in [28] with a data-based voltage
hysteresis model to attain better accuracy. The study in [29] derives an ECM
using circuit elements to characterize charge transfer and diffusion
potentials; the derivation also helps explain some conventional ECMs from an
electrochemical perspective. In [30], an ECM is coupled with diffusion
dynamics to attain higher prediction accuracy.

Structural simplicity underlies the wide use of ECMs in battery management,
but also restricts their accuracy. Most of today’s ECMs are accurate enough
for only low C-rates, and recent progress has led to ECMs that are provably
suitable for about 1 C [26, 27]. However, the literature still faces an
absence of ECMs capable of predicting a cell’s voltage behavior from low to
high C-rate ranges. This gap will pose potential barriers for emerging
battery-powered applications that must operate across wide current ranges. One
example is electric vertical take-off and landing (eVTOL), which requires
discharging of up to 5 C in the take-off and landing phases and necessitates
precise models to fulfill high-stakes safety requirements [31].

Statement of Contributions. To address the above challenge, we develop a new
ECM that promises to predict over broad current ranges. Our work takes
inspirations from electrochemical modeling to design and conjoin circuits to
simulate a LiB cell’s electrode, electrolyte, and thermal dynamics as well as
their effects on the terminal voltage. Here, we select the SPM with
electrolyte and thermal dynamics (SPMeT) as the reference benchmark. The
obtained ECM, called BattX, hence is physically comparable and represents a
reduced-order analog to the SPMeT model. By design, the BattX model
comprehensively accounts for the aforementioned different types of dynamics
that have a phenomenologically appreciable impact at high C-rates. This endows
it with not only excellent prediction capability, but also considerable
physical fidelity and interpretability. Desirably, the model still retains
relatively compact structures to present high computational efficiency,
carrying a potential to facilitate embedded battery management systems. To sum
up, this paper delivers the following specific contributions.

  * 1.

We propose the principled design of the BattX model and further elucidate the
underlying rationale by showing its connections with the SPMeT model in
detail.

  * 2.

We develop a multipronged parameter identification approach to extract the
parameters of the BattX model from measurement data made on LiBs. The approach
will make the model readily available in practice.

  * 3.

We provide experimental evaluation results to validate the effectiveness and
accuracy of the BattX model. The experiments involve charging/discharging at
high C-rates and consider operation profiles of eVOTL as a case study.

Organization. The rest of the paper is organized as follows. Section 2
presents the BattX model design as a whole. Section 3 proceeds to elucidate on
the model’s correspondence to the SPMeT model. Section 4 develops the
parameter identification pipeline of the model. Section 5 evaluates the model
using experimental data. Finally, Section 6 offers concluding remarks.

##  2 The BattX Model

This section presents the structure and governing equations of the BattX
model. We will provide the detailed rationale for the model design
subsequently in Section 3.

Figure 1: The BattX model comprising: sub-circuit A to simulate the lithium-
ion diffusion in the electrode phase; sub-circuit B to simulate the lithium-
ion diffusion in the electrolyte phase; sub-circuit C to simulate heat
conduction and convection; and sub-circuit D to simulate the terminal voltage.

At the core, the BattX model attempts to characterize the multiple major
dynamic processes innate to a LiB cell in order to capture the cell’s behavior
from low to high current rates. This is akin to electrochemical modeling to a
certain extent, but a main difference is that the BattX model leverages
circuit analogs to simulate the processes. Fig. 1 shows the overarching
structure of the model. As is seen, it consists of four coupled sub-circuits,
which are labeled as A to D. These sub-circuits are designed to approximate
the cell’s electrode-phase diffusion, electrolyte-phase diffusion, thermal
evolution, and voltage response, respectively.

To begin with, sub-circuit A uses a chain of resistors and capacitors to
approximate the lithium-ion diffusion in the electrode phase. Its governing
equations are

$\displaystyle\dot{V}_{s,1}(t)$
$\displaystyle=\frac{V_{s,2}(t)-V_{s,1}(t)}{C_{s,1}R_{s,1}}+\frac{I(t)}{C_{s,1}},$
(1a) $\displaystyle\dot{V}_{s,i}(t)$
$\displaystyle=\frac{V_{s,i-1}(t)-V_{s,i}(t)}{C_{s,i}R_{s,i-1}}+\frac{V_{s,i+1}(t)-V_{s,i}(t)}{C_{s,i}R_{s,i}},\;\;i=2,\ldots,N-1$
(1b) $\displaystyle\dot{V}_{s,N}(t)$
$\displaystyle=\frac{V_{s,N-1}(t)-V_{s,N}(t)}{C_{s,N}R_{s,N-1}},$ (1c)

where $I$ is the applied current, with $I>0$ for charging and $I<0$ for
discharging, $V_{s,j}$ for $j=1,\ldots,N$ are the voltages across the
individual capacitors $C_{s,j}$, $R_{s,j}$ are the resistors that the current
must flow through, and the subscript $s$ refers to the solid phase. We set
$0\leq V_{s,j}\leq 1$ for the purpose of normalization and then define the SoC
as the percentage ratio of the currently available charge over the total
charge capacity, which is

$\displaystyle\mathrm{SoC}=\frac{\sum_{j=1}^{N}C_{s,j}V_{s,j}}{\sum_{j=1}^{N}C_{s,j}}\times
100\%.$

That is, $\mathrm{SoC}=100\%$ when $V_{s,j}=1$ for all $j$, and
$\mathrm{SoC}=0$ when $V_{s,j}=0$ for all $j$. A brief interpretation of sub-
circuit A is as follows, with more details to be shown in Section 3. Overall,
the charge transfer between the capacitors in the circuit mimics the diffusion
of lithium ion in the solid phase or electrode. Then, $V_{s,j}$ for
$j=1,\ldots,N$ correspond to the lithium-ion concentrations at $N$ different
locations, from the surface to the center, that spread along the radius of an
electrode sphere; $C_{s,j}$ for $j=1,\ldots,N$ are analogous to the volumes of
the subdomains if one subdivides the electrode sphere at these discrete
locations; $R_{s,j}$ for $j=1,\ldots,N-1$ resist the charge transfer or
equivalently, the solid-phase diffusion in the SPMeT model, and are hence
inversely proportional to the diffusivity.

Along similar lines to sub-circuit A, sub-circuit B uses a resistor-capacitor
chain to approximate the lithium-ion diffusion in the electrolyte. Its
dynamics is governed by

$\displaystyle\dot{V}_{e,1}(t)$
$\displaystyle=\frac{V_{e,2}(t)-V_{e,1}(t)}{C_{e}R_{e}}+\frac{I(t)}{C_{e}},$
(2a) $\displaystyle\dot{V}_{e,2}(t)$
$\displaystyle=\frac{V_{e,1}(t)-2V_{e,2}(t)+V_{e,3}(t)}{C_{e}R_{e}},$ (2b)
$\displaystyle\dot{V}_{e,3}(t)$
$\displaystyle=\frac{V_{e,2}(t)-V_{e,3}(t)}{C_{e}R_{e}}-\frac{I(t)}{C_{e}},$
(2c)

wehre the notations in above have similar meanings as in (1), and the
subscript $e$ refers to the electrolyte. We let $0\leq V_{e,j}\leq 1$ for
$j=1,2,3$ as in the case of $V_{s,j}$, and further assume that $V_{e,j}=0.5$
for $j=1,2,3$ when the cell is at equilibrium. One can interpret sub-circuit B
as analogous to the one-dimensional electrolyte-phase diffusion that is
discretized along the spatial coordinate. In particular, $V_{e,j}$ for
$j=1,2,3$ can be associated with the lithium-ion concentrations at the
locations of the anode, separator, and cathode, and $R_{e}$ embodies
resistance to the diffusion. The spatial discretization is assumed to be
uniform, thus leading to the same values of $R_{e}$ and $C_{e}$ for each
region as shown in (2).

Sub-circuit C is a lumped circuit model for the thermal dynamics, with the
design inspired by [32]. Here, we consider the cell to be a cylindrical one
without loss of generality and concentrate its spatial dimensions into two
singular points that represent the surface and core, respectively. This
simplification allows to describe the evolution of the temperatures at these
two points, $T_{\mathrm{surf}}$ and $T_{\mathrm{core}}$, by

$\displaystyle\dot{T}_{\mathrm{core}}(t)$
$\displaystyle=\frac{Q(t)}{C_{\mathrm{core}}}+\frac{T_{\mathrm{surf}}(t)-T_{\mathrm{core}}(t)}{R_{\mathrm{core}}C_{\mathrm{core}}},$
(3a) $\displaystyle\dot{T}_{\mathrm{surf}}(t)$
$\displaystyle=\frac{T_{\mathrm{amb}}(t)-T_{\mathrm{surf}}(t)}{R_{\mathrm{surf}}C_{\mathrm{surf}}}-\frac{T_{\mathrm{surf}}(t)-T_{\mathrm{core}}(t)}{R_{\mathrm{core}}C_{\mathrm{surf}}},$
(3b)

where $T_{\mathrm{amb}}$ is the ambient temperature, $C_{\mathrm{surf/core}}$
and $R_{\mathrm{surf/core}}$ represent the thermal capacitance and resistance
at the surface and core, respectively, and $Q$ is the internal heat generation
rate accompanying electrochemical reactions inside the cell during
charging/discharging. From a heat transfer perspective, (3a) approximately
describes the heat conduction between the cell’s surface and core, and (3b)
grasps the convection between the surface and the ambient environment.
Further, $Q$ is characterized as

$\displaystyle
Q=-I\left[U_{s}\left(\mathrm{SOC}\right)-U_{s}(V_{s,1})-R_{o,T}I\right],$ (4)

where $U_{s}(\cdot)$ is the nonlinear OCV function, $V_{s,1}$ is defined in
sub-circuit A, and $R_{o,T}$ is the internal resistance.

Finally, sub-circuit D summarizes the effects of the solid-phase and
electrolyte-phase dynamics on the terminal voltage. It contains two voltage
sources, $U_{s}$ and $U_{e}$, in series with an internal resistance $R_{o,T}$.
The terminal $U$ is given by

$\displaystyle
U=U_{s}(V_{s,1}(t))+U_{e}(V_{e,1}(t),V_{e,N_{e}}(t))+R_{o,T}I(t).$ (5)

Here, $U_{s}$ simulates the solid-phase OCV. As the SPMeT model mandates that
the open-circuit potential of solid material relies on the lithium-ion
concentration at the surface of the electrode, $U_{s}$ should come as a
function of $V_{s,1}$, and its exact form will depend on the cell. For the
cell used in our experiments in Sections 4-5, we find the following
parameterization of $U_{s}$ suitable:

$\displaystyle U_{s}(V_{s,1})=h_{1}\left(V_{s,1}\right)\cdot
H\left(0.9-V_{s,1}\right)+h_{2}\left(V_{s,1}\right)\cdot
H\left(V_{s,1}-0.9\right),$

where $H(\cdot)$ is the Heaviside step function, $h_{1}(V_{s,1})$ captures the
behavior when $V_{s,1}\leq 0.9$ as

$\displaystyle h_{1}(V_{s,1})$
$\displaystyle=\alpha_{0}+\alpha_{1}\frac{1}{1+\mathrm{exp}(\alpha_{2}(V_{s,1}(t)-\alpha_{3}))}+\alpha_{4}\frac{1}{1+\mathrm{exp}(\alpha_{5}(V_{s,1}-\alpha_{6}))}+\alpha_{7}\frac{1}{1+\mathrm{exp}(\alpha_{8}(V_{s,1}-\alpha_{9}))}+$
$\displaystyle\quad\alpha_{10}\frac{1}{1+\mathrm{exp}(\alpha_{11}V_{s,1}(t))}+\alpha_{12}V_{s,1}(t),$

and $h_{2}(V_{s,1})$ is for when $0.9<V_{s,1}\leq 1$ with

$\displaystyle
h_{2}(V_{s,1})=\alpha_{13}\mathrm{exp}(\alpha_{14}V_{s,1})+\alpha_{15}\mathrm{exp}(\alpha_{16}V_{s,1}).$

Here, $\alpha_{i}$ for $i=0,\ldots,15$ are constant coefficients. Next, we
need to determine the form of $U_{e}$. In the SPMeT model, the electrolyte
potential depends on the electrolyte concentration at the anode and cathode.
We hence make $U_{e}$ as a function of $V_{e,1}$ and $V_{e,N_{e}}$ and express
it as

$\displaystyle
U_{e}(t)=\beta_{1}\left(\mathrm{ln}\left(\frac{V_{e,1}(t)+\beta_{2}}{V_{e,3}(t)+\beta_{2}}\right)\right),$
(6)

where $\beta_{i}$ for $i=1,2$ are constant coefficients. As the last element
of the model, $R_{o,T}$ is not a constant and instead depends on
$\mathrm{SoC}$ and $T_{\mathrm{core}}$. It is given by

$\displaystyle
R_{o.T}=R_{o}(\mathrm{SoC})\cdot\mathrm{exp}\left(\kappa_{1}\left(\frac{1}{T_{\mathrm{core}}}-\frac{1}{T_{\mathrm{amb}}}\right)\right),$
(7)

where $\kappa_{1}$ is a constant coefficient. In above, the first term
$R_{o}(\mathrm{SoC})$ captures the dependence of $R_{o,T}$ on $V_{s,1}$ and
takes the form

$\displaystyle
R_{o}(\mathrm{SoC})=\gamma_{1}+\gamma_{2}\cdot\mathrm{exp}\left(-\gamma_{3}\mathrm{SoC}\right),$
(8)

where $\gamma_{i}$ for $i=1,2,3$ are coefficients, and the second term shows
the temperature dependence due to the Arrhenius law. Similarly, an Arrhenius
relationship can be used to capture the relationship between the electrode-
phase diffusion constant and temperature:

$\displaystyle
R_{s,1,T}=R_{s,1}\cdot\mathrm{exp}\left(\kappa_{2}\left(\frac{1}{T_{\mathrm{core}}}-\frac{1}{T_{\mathrm{amb}}}\right)\right),$
(9)

Putting together all the above equations, we will obtain a complete
description of the BattX model. This model is the first ECM that can predict
over broad current ranges, due to the integration of the circuits
approximating the electrode, electrolyte, and thermal dynamics into a whole.
The model design also leads to profound comparability with electrochemical
modeling, especially the SPMeT, which will be revealed further in the next
section. We will address the identification of the model parameters in Section
4.

##  3 Rationale for the BattX Model Design

In this section, we will use the SPMeT model as a benchmark to explain the
rationale for the design of the BattX model. We will show that the SPMeT
model, if appropriately discretized, will reduce to a structure that is
approximately equivalent to the proposed circuit analogs of the BattX model.
Our main references about the SPMeT model include [33, 34]. We will focus on
expounding sub-circuits A, B, and D, with the sub-circuit C-based lumped
thermal model well addressed in [32].

###  3.1 Connection between Sub-circuit A and SPMeT

The SPMeT model characteristically couples the SPM model with the electrolyte
and thermal dynamics. What it inherits from the SPM model is the
representation of the electrodes as two spherical particles. The diffusion of
lithium-ions in each particle follows Fick’s second law in spherical
coordinates [7, 8]:

$\displaystyle\frac{\partial c_{s,j}(r,t)}{\partial
t}=\frac{D_{s,j}}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial
c_{s,j}(r,t)}{\partial r}\right),$ (10)

where $c_{s,j}$ is the solid-phase (electrode) lithium-ion concentration,
$D_{s}$ is the constant diffusion coefficient, and $r$ is the radial
coordinate. The subscript $j\in\\{n,p\\}$, where $n$ and $p$ refer to the
anode (negative) and cathode (positive), respectively. The boundary conditions
for (10) are

$\displaystyle\left.\frac{\mathrm{d}c_{s,j}}{\mathrm{d}r}\right|_{r=0}=0,\;\;\left.\frac{\mathrm{d}c_{s,j}}{\mathrm{d}r}\right|_{r=R_{j}}=-\frac{J_{j}}{D_{s,j}},$

where $R$ is the radius of a particle. The molar flux $J$ at the
electrode/electrolyte interface is given by

$\displaystyle J_{p}(t)=\frac{i(t)}{FS_{p}},\;\;J_{n}(t)=-\frac{i(t)}{FS_{n}}$

where $i$ is the applied current density, with $i>0$ for charging and $i<0$
for discharging, $S$ is the surface area of a particle, and $F$ is Faraday’s
constant.

Figure 2: Spherical discretization of an electrode particle.

Next, let us reduce the PDE in (10) into a system of ODE equations using a
finite volume method [35, 36]. The subscript $j\in{n,p}$ will be dropped in
sequel without causing confusion. First, we subdivide the particle into a set
of continuous finite volumes at discrete locations
$r_{1}=R>r_{2}>\ldots>r_{N}>r_{N+1}=0$ that spreads inward from the surface to
the center, as show in Fig. 2. The lithium-ion amount within the $i$th finite
volume is given by

$\displaystyle
Q_{i}(t)=\int_{r_{i+1}}^{r_{i}}c_{s}(r,t)dV=\int_{r_{i+1}}^{r_{i}}c_{s}(r,t)\cdot
4\pi r^{2}dr,$

for $i=1,\ldots,N$. Then, using (10), we have

$\displaystyle\dot{Q}_{i}(t)=\int_{r_{i+1}}^{r_{i}}\dot{c}_{s}(r,t)\cdot 4\pi
r^{2}dr=\int_{r_{i+1}}^{r_{i}}d\left(4\pi D_{s}r^{2}\frac{\partial
c_{s}(r,t)}{\partial r}\right)=4\pi D_{s}r_{i}^{2}\left.\frac{\partial
c_{s}(r,t)}{\partial r}\right|_{r_{i}}-4\pi
D_{s}r_{i+1}^{2}\left.\frac{\partial c_{s}(r,t)}{\partial
r}\right|_{r_{i+1}}.$ (11)

To proceed, we replace $c_{s}(r,t)$ by the average lithium-ion concentration
within the $i$th finite volume, $\bar{c}_{s}(r_{i},t)$:

$\displaystyle\bar{c}_{s}(r_{i},t)=\frac{Q_{i}(t)}{\Delta V_{i}},$ (12)

where $\Delta V_{i}=4\pi(r_{i}^{3}-r_{i+1}^{3})/3$. From (11)-(12), it follows
that

$\displaystyle\dot{\bar{c}}_{s}(r,t)=\frac{4\pi D_{s}r_{i}^{2}}{\Delta
V_{i}}\left.\frac{\partial c_{s}(r,t)}{\partial r}\right|_{r_{i}}-\frac{4\pi
D_{s}r_{i+1}^{2}}{\Delta V_{i}}\left.\frac{\partial c_{s}(r,t)}{\partial
r}\right|_{r_{i+1}}.$

Then, we approximate the concentration gradient along the radial coordinate as

$\displaystyle\left.\frac{\partial c_{s}(r,t)}{\partial
r}\right|_{r_{i}}=\frac{\bar{c}(r_{i-1},t)-\bar{c}(r_{i},t)}{\Delta r_{i}},$

where $\Delta r_{i}=(r_{i-1}-r_{i+1})/2$. Given the boundary conditions, we
further have

$\displaystyle\dot{\bar{c}}_{s}(r_{1},t)$ $\displaystyle=-\frac{4\pi
D_{s}r_{2}^{2}}{\Delta V_{1}\Delta
r_{2}}\left(\bar{c}_{s}(r_{1},t)-\bar{c}_{s}(r_{2},t)\right)+\frac{4\pi
r_{1}^{2}}{\Delta V_{1}FS}i(t),$ (13a)
$\displaystyle\dot{\bar{c}}_{s}(r_{i},t)$ $\displaystyle=\frac{4\pi
D_{s}r_{i}^{2}}{\Delta V_{i}\Delta
r_{i}}\left(\bar{c}_{s}(r_{i-1},t)-\bar{c}_{s}(r_{i},t)\right)-\frac{4\pi
D_{s}r_{i+1}^{2}}{\Delta V_{i}\Delta
r_{i+1}}\left(\bar{c}_{s}(r_{i},t)-\bar{c}_{s}(r_{i+1},t)\right),\;\;i=2,\ldots,N-1,$
(13b) $\displaystyle\dot{\bar{c}}_{s}(r_{N},t)$ $\displaystyle=\frac{4\pi
D_{s}r_{N}^{2}}{\Delta V_{N}\Delta
r_{N}}\left(\bar{c}_{s}(r_{N-1},t)-\bar{c}_{s}(r_{N},t)\right).$ (13c)

The above ODEs show the spatially discretized solid-phase diffusion. Note that
they share the same structure with (1). A closer inspection of (1) and (13)
suggests: 1) $V_{s}$ is a mirror of $\bar{c}_{s}(r,t)$, and its distribution
reflects the distribution of lithium-ion concentrations inside an electrode
particle; 2) $C_{s}$ is a mirror of $\Delta V$, associating the capacitance
with the volume of a finite volume element within the particle; 3) $R_{s}$
roughly corresponds to $\Delta r/(D_{s}\cdot 4\pi r^{2})$ to grasp the effect
of $D_{s}$, $\Delta r$ and $r$ on the diffusion resistance at different
locations. This unveiled connection with the SPMeT model justifies the design
of sub-circuit A.

###  3.2 Connection between Sub-circuit B and SPMeT

Figure 3: Three regions immersed in the electrolyte.

The SPMeT model includes one-dimensional electrolyte diffusion, which also
follows Fick’s second law. The electrolyte diffusion is considered in the
electrode and separator domains that are all immersed in the electrolyte.
Based on the coordinates in each domain as shown in Fig. 3, the governing
equations are

$\displaystyle\frac{\partial c_{e,p}(x,t)}{\partial t}$
$\displaystyle=D_{e}\frac{\partial^{2}c_{e,p}(x,t)}{\partial
x^{2}}+\frac{1-t^{0}_{c}}{\epsilon_{e,p}FL_{p}}i(t),$ (14a)
$\displaystyle\frac{\partial c_{e,\mathrm{sep}}(x,t)}{\partial t}$
$\displaystyle=D_{e}\frac{\partial^{2}c_{e,\mathrm{sep}}(x,t)}{\partial
x^{2}},$ (14b) $\displaystyle\frac{\partial c_{e,n}(x,t)}{\partial t}$
$\displaystyle=D_{e}\frac{\partial^{2}c_{e,n}(x,t)}{\partial
x^{2}}-\frac{1-t^{0}_{c}}{\epsilon_{e,n}FL_{n}}i(t),$ (14c)

where $c_{e,j}$ for $j\in\\{n,p,\mathrm{sep}\\}$ is the lithium-ion
concentration in the electrolyte surrounding the anode, cathode and separator,
$\epsilon_{e,j}$ is the electrolyte volume fraction, $D_{e,j}$ is the
electrolyte diffusion coefficient, and $t^{0}_{c}$ is the constant
transference number. We assume that $\epsilon_{e,j}$ and $D_{e,j}$ are the
same for any $j\in\\{n,p,\mathrm{sep}\\}$. The boundary conditions are given
by

$\displaystyle\frac{\partial c_{e,p}(0_{p},t)}{\partial x}$
$\displaystyle=\frac{\partial c_{e,n}(L_{n},t)}{\partial x}=0,$
$\displaystyle\frac{\partial c_{e,p}(L_{p},t)}{\partial x}$
$\displaystyle=\frac{\partial c_{e,\mathrm{sep}}(0_{\mathrm{sep}},t)}{\partial
x},$ $\displaystyle\frac{\partial
c_{e,\mathrm{sep}}(L_{\mathrm{sep}},t)}{\partial x}$
$\displaystyle=\frac{\partial c_{e,n}(0_{n},t)}{\partial x},$ $\displaystyle
c_{e}(L_{p},t)$ $\displaystyle=c_{e}(0_{\mathrm{sep}},t),$ $\displaystyle
c_{e}(L_{\mathrm{sep}},t)$ $\displaystyle=c_{e}(L_{n},t).$

To convert (14) into ODEs, we concentrate the electrodes and separator into
singular points and further suppose $L_{p}=L_{n}$ and $L_{\mathrm{sep}}$ is
negligible. The singular point that represents the electrodes are located at
the midpoint of each domain, and the average lithium-ion concentration is
denoted as $\bar{c}_{e,j}$. Then, we apply the finite difference to (14) and
obtain

$\displaystyle\dot{\bar{c}}_{e,p}(t)$
$\displaystyle=\frac{4D_{e}}{L^{2}}\left(\bar{c}_{e,\mathrm{sep}}(t)-{\bar{c}}_{e,p}(t)\right)+\frac{1-t^{0}_{c}}{\epsilon_{e}FL}i(t),$
(15a) $\displaystyle\dot{\bar{c}}_{e,\mathrm{sep}}(t)$
$\displaystyle=\frac{4D_{e}}{L^{2}}\left(\bar{c}_{e,p}(t)-2\bar{c}_{e,\mathrm{sep}}(t)+\bar{c}_{e,n}(t)\right),$
(15b) $\displaystyle\dot{\bar{c}}_{e,n}(t)$
$\displaystyle=\frac{4D_{e}}{L^{2}}\left(\bar{c}_{e,\mathrm{sep}}(t)-{\bar{c}}_{e,n}(t)\right)-\frac{1-t^{0}_{c}}{\epsilon_{e}FL}i(t).$
(15c)

As is seen, (14) is structurally similar to (2), and the similarity lends to
interpretation of (2) through the lens of electrochemical modeling.
Specifically, we can associate $V_{e,1}$, $V_{e,2}$ and $V_{e,3}$ with
${\bar{c}}_{e,p}$, ${\bar{c}}_{e,\mathrm{sep}}$ , and ${\bar{c}}_{e,n}$,
respectively. Further, $C_{e}$ can be linked with the spatial lengths of the
electrode domains, which decide the volume of the electrolyte, and $R_{e}$
comes as the inverse of $D_{e}$ to measure the resistance against electrolyte
diffusion.

###  3.3 Connection between Sub-circuit D and SPMeT

In the SPMeT model, the terminal voltage $V$ consists of four terms that
represent the solid-phase OCV, electrolyte-phase voltage, overpotential, and
voltage over the film resistance, respectively. Then, coming back to sub-
circuit D of the BattX model, $U_{s}$ mirrors the solid-phase OCV, $U_{e}$
corresponds to the electrolyte-phase voltage, and $R_{o,T}$ plays a role to
mainly capture the film resistance as well as the overpotential effect. Less
trivially, we elaborate on the form of $U_{e}$ in (6). The electrolyte-phase
voltage is given by

$\phi_{e}(0_{p},t)-\phi_{e}(L_{n},t)=\frac{L_{p}+2L_{sep}+L_{n}}{2\bar{k}}i(t)+k_{\mathrm{conc}}\left(\mathrm{ln}c_{e}(0_{p},t)-\mathrm{ln}c_{e}(L_{n},t)\right),$
(16)

where $\phi_{e}$ is the electrolyte electric potential, and $\bar{k}$ and
$k_{\mathrm{conc}}$ are two coefficients that are related with electrolyte
conductivity and molar activity. The first term in above is accounted by for
$R_{o,T}$. Following the discussion in Section 3.2, we can approximate the
second term as

$\displaystyle
k_{\mathrm{conc}}\left(\ln\bar{c}_{e,p}(t)-\ln\bar{c}_{e,n}(t)\right).$

This form is found to bear equivalence to (6), when making linear projections
of $\bar{c}_{e,p}(t)$ and $\bar{c}_{e,n}(t)$ to $V_{e,1}$ and $V_{e,3}$,
respectively.

##  4 Parameter Identification for the BattX Model

In this section, we investigate how to determine the parameters of the BattX
model. To this end, we separate the model’s parameters into different groups
based on the dynamic processes that they belong to or prominently influence.
We then design experiments accordingly and use different current profiles to
excite different dynamic processes and obtain voltage or temperature data
suitable for the identification of the corresponding parameter groups.
Finally, we extract the parameters from the data, group by group, through data
fitting and some empirical tuning.

To begin with, we set up the following parameter groups for the BattX model:

  * 1.

$\Theta_{U_{s}}=\left\\{\alpha_{i},i=0,1,\ldots,16\right\\}$, which includes
the parameters in $U_{s}$ in sub-circuit D;

  * 2.

$\Theta_{R_{o}}=\left\\{\gamma_{i},i=1,2,3\right\\}$, which includes the
parameters in $R_{o}$ in sub-circuit D;

  * 3.

$\Theta_{s}=\left\\{C_{s,i},i=1,\ldots,N,R_{s,j},j=1,\ldots,N-1\right\\}$,
which includes the parameters of sub-circuit A;

  * 4.

$\Theta_{\mathrm{Th}}=\left\\{C_{\mathrm{surf}},R_{\mathrm{surf}},C_{\mathrm{core}},R_{\mathrm{core}}\right\\}$,
which includes the parameters in the lumped thermal model in sub-circuit C;

  * 5.

$\Theta_{e}=\left\\{C_{e},R_{e},\beta_{1},\beta_{2}\right\\}$, which includes
the parameters in sub-circuit B and the parameters in $U_{e}$ in sub-circuit
D;

  * 6.

$\Theta_{\mathrm{Arr}}=\left\\{\kappa_{1},\kappa_{2}\right\\}$, which includes
the Arrhenius-law-related parameters;.

By grouping the parameters as above, we can design different current input
profiles to stimulate different parts of the cell’s dynamics so as to identify
the parameters group by group. This multi-pronged approach includes the
following steps.

Step 1: Identification of $\Theta_{U_{s}}$. Since $U_{s}$ represents the OCV
source, we can capture it by applying a trickle constant current with a
magnitude of $1/30$ C to fully charge or discharge the cell. As the current is
extremely small, sub-circuit A, which is an analog to the solid
(electrode)-phase diffusion, is almost always at equilibrium, with
$V_{s_{i}}=\mathrm{SoC}$ for $i=1,\ldots,N$ ($\mathrm{SoC}$ can be obtained
via Coulomb counting); meanwhile, sub-circuits B and C, $U_{e}$, and the
voltage across $R_{o,T}$, are all negligible in this case. Hence, $U\approx
U_{s}$, and we can construct the following data fitting problem to identify
$\Theta_{U_{s}}$:

$\displaystyle\hat{\Theta}_{U_{s}}=\arg\min_{\Theta_{U_{s}}}\sum_{t_{k}}\left[U(t_{k})-U_{s}\left(\mathrm{SoC}(t_{k});\Theta_{U_{s}}\right)\right]^{2},$
(17)

where $k$ is the discrete time index in the experiment.

Step 2: Identification of $\Theta_{R_{o}}$. $R_{o}$ is an integral part of the
internal resistance $R_{o,T}$, and $R_{o}=R_{o,T}$ when $T=T_{\mathrm{ref}}$.
To identify $\Theta_{R_{o}}$, we apply a 0.5 C pulse current profile, which
includes long enough rest periods between two consecutive pulses to allow for
sufficient voltage recovery, to discharge the cell from $100\%$ to $0\%$ of
SoC when the ambient temperature is $T_{\mathrm{ref}}$. With discharging at
0.5 C, the cell will see only a negligible increase in its temperature, and
$U_{e}\approx 0$. For the terminal voltage $U$, we will see a sharp drop or
jump at the beginning or end of every pulse, and this is almost solely due to
the voltage change across $R_{o}$. Therefore, using the voltage jump, one can
approximate $R_{o}$ as

$\displaystyle\tilde{R}_{o}(t_{*})=\left|\frac{U(t_{*+1})-U(t_{*})}{I}\right|,$
(18)

where $t_{*}$ is the instant when a pulse stops. Further, the instantaneous
SoC can be readily determined via Coulomb counting. Collecting $R_{o}$ for all
$t_{*}$, we can formulate the following data fitting problem to estimate
$\Theta_{R_{o}}$:

$\displaystyle\hat{\Theta}_{R_{o}}=\arg\min_{\Theta_{R_{o}}}\sum_{t_{*}}\left[\tilde{R}_{o}(t_{*})-R_{o}(\Theta_{R_{o}};t_{*})\right]^{2}.$
(19)

Step 3: Identification of $\Theta_{s}$. The number of parameters in
$\Theta_{s}$ depends on $N$, and when $N$ is large, $\Theta_{s}$ will be
poorly identifiable to defy accurate estimation. To formulate a tractable
identification problem, we assume that

$\displaystyle C_{s,i}=\eta_{i}C_{s,1},\;R_{s,j}=\sigma_{j}R_{s,1},$ (20)

where $\eta_{i}$ and $\sigma_{i}$ for $i=1,\ldots N$ and $j=1,\dots N-1$ are
pre-specified coefficients with $\eta_{1}=\sigma_{1}=1$, and
$\sum_{i=1}^{N}\eta_{i}C_{s,i}$ is the total capacity of the cell. This allows
us to consider only two parameters, i.e.,
$\Theta_{s}=\left\\{C_{s,1},R_{s,1}\right\\}$, greatly facilitating the
parameter estimation. The simplification is also reasonable—the difference
among $C_{s,i}$ and $R_{s,j}$ can be viewed as a result of the selection of
the discretization points as shown in (13), and one can specify $\eta_{i}$ and
$\sigma_{j}$ assuming that they result from a certain selection. The practical
selection of $\eta_{i}$ and $\sigma_{j}$ can be through analysis of the
discretization shown in Section 3.1 and tuning. Going forward, we apply a 0.5
C constant-current profile to discharge the cell from full to zero SoC. In
this setting, sub-circuit A is excited, but the dynamics of sub-circuits B and
C have no appreciable effects. That is, the cell’s temperature remains almost
the same, and $U_{e}\approx 0$. We can conduct data fitting as below to find
out $\Theta_{s}$:

$\displaystyle\hat{\Theta}_{s}=\arg\min_{\Theta_{s}}\sum_{t_{k}}\left[U(t_{k})-R_{o}\left(\hat{\Theta}_{R_{o}};t_{k}\right)I(t_{k})-U_{s}\left(V_{s,1}\left({\Theta_{s}};t_{k}\right);\hat{\Theta}_{U_{s}}\right)\right]^{2},$
(21)

where $\hat{\Theta}_{U_{s}}$ and $\hat{\Theta}_{R_{o}}$ have been obtained in
Steps 1 and 2, and the form of $V_{s,1}({\Theta_{s}},t)$ is derived in
Appendix.A.

Step 4: Identification of $\Theta_{\mathrm{Th}}$. Based on [32], a
straightforward idea to determine $\Theta_{\mathrm{Th}}$ is to fit it to the
measurement data of $T_{\mathrm{surf}}$ and/or $T_{\mathrm{core}}$ given the
lumped thermal model in (3). However, the idea is hard to be applied here,
because $Q$ in our model is dependent on $R_{o,T}$, as shown in (4), and
unavailable before $R_{o,T}$ is identified. To overcome this issue, we choose
to use prior knowledge to guide the estimation of $\Theta_{\mathrm{Th}}$.
Here, we can approximate $R_{\mathrm{core}}$ based on the conductivity of the
cell’s electrode materials and jellyroll structure. Furthermore, we can infer
$R_{\mathrm{surf}}$ and $C_{\mathrm{surf}}$ from the form factors and
specifications, casing material (usually aluminum), and the cooling system.
Finally, $C_{\mathrm{core}}$ can be deduced given the cell’s total heat
capacity. A 2C constant current full discharge profile is used to acquire the
data encompassing significant temperature changes. With the measurement data,
we can begin from the approximate values of the parameters and continually
tune them until achieving sufficient fitting accuracy to finalize
$\hat{\Theta}_{\mathrm{Th}}$.

Step 5: Identification of $\Theta_{e}$ and $\Theta_{\mathrm{Arr}}$. Sub-
circuit B will have substantial effects on $U$ only at high C-rates.
Therefore, we use a 3 C constant current profile to fully discharge the cell
such that large enough $U_{e}$ will result and present itself into the voltage
response. This then allows to identify $\Theta_{e}$. In the meantime, 3 C
discharging will subject the cell to important temperature increases, which,
in turn, will drive down $\Theta_{\mathrm{Arr}}$-dependent $R_{s,T}$ and
$R_{o,T}$ and influence the voltage response. As such, we need to consider the
estimation of $\Theta_{e}$ and $\Theta_{\mathrm{Arr}}$ together. The following
data fitting problem can be formulated:

$\displaystyle\hat{\Theta}_{\mathrm{e}},\hat{\Theta}_{\mathrm{Arr}}=\arg\min_{\Theta_{\mathrm{e}},\Theta_{\mathrm{Arr}}}\sum_{t_{k}}\left[U(t_{k})-R_{o,T}\left(\hat{\Theta}_{R_{o}},\Theta_{\mathrm{Arr}},T_{t_{k}};t_{k}\right)I(t_{k})-U_{s}\left(V_{s,1}\left({\hat{\Theta}_{s}},\Theta_{\mathrm{Arr}},T_{t_{k}};t_{k}\right);\hat{\Theta}_{U_{s}}\right)-U_{e}\left(\Theta_{e};t_{k}\right)\right]^{2}.$
(22)

Here, $U_{e}$ depends on $V_{e,1}$ and $V_{e,3}$ as shown in (6), and the
explicit form of $V_{e,1}$ and $V_{e,3}$ is shown in Appendix.B. Note that no
closed-form expression of $U_{s}$ exists in this step, as the changing
$R_{s,T}$ makes sub-circuit A become a time-varying system. It is thus
impossible to solve the problem in (22) using nonlinear optimization. To
alleviate the difficulty, we suggest to apply some empirical tuning.
Specifically, we can pick a sample of $\Theta_{\mathrm{Arr}}$ using prior
knowledge, then estimate $\Theta_{e}$ by solving the above data fitting
problem, and iterate this procedure until getting the lowest possible fitting
errors. Despite the time and effort needed, this iterative method is often
found effective with a sufficient number of tries.

The above steps together constitute our parameter identification approach for
the BattX model. The following remarks summarize our further insights.

###### Remark 1

We point out that the data fitting problems outlined in Steps 1-5 are non-
trivial to solve, as they entail nonlinear nonconvex optimization. The
nonconvexity can easily get the parameter search stuck in local minima to
produce physically meaningless parameter estimates. To mitigate the issue, it
is sensible to constrain the search within a believably correct parameter
space [21]. Specifically, one can set up approximate lower and upper bounds
for every possible parameter and then limit the numerical optimization within
the resultant parameter space. The prior knowledge used to establish such
bounds can be derived from both experience and observation or analysis of the
measurement data. Other helpful ways to overcome the local minima issue
include adding regularization terms that encode prior knowledge of some
parameters and applying different initial guesses to repeatedly run the
numerical optimization [21].

###### Remark 2

We consider Samsung INR18650-25R cells (see Section 5 for the specifications)
as a baseline when selecting the discharging C-rates in each step of the above
approach, because they are used in the experimental validation of the BattX
model (see Section 5). However, a user or practitioner may need to adjust the
specific C-rates, depending on the cells to apply the model to. The overall
guiding rule is the same—using current profiles of different C-rates to excite
different dynamic processes to obtain data informative for the identification
of the parameters associated with each process.

##  5 Experimental Validation of the BattX Model

This section offers the experimental validation of the proposed BattX model.
All the experiments were conducted on a Samsung INR18650-25R cell with NCA
cathode and graphite anode using a PEC® SBT4050 battery tester. The cell’s
nominal capacity is 2.5 Ah, nominal voltage is 3.6 V, maximum cut-off voltage
is 4.2 V, minimum cut-off voltage is 2.5 V, and maximum continuous discharge
current is 20 A. The tester is able to run charging or discharging tests of up
to 40 V and 50 A under arbitrary current or power load profiles. The
experiments comprised two parts. The first part collected datasets following
the parameter identification approach in Section 4 to identify the model
parameters. In the second part, new datasets were generated to evaluate the
predictive capability of the identified model.

###  5.1 Model Identification

The experiments and model identification procedure are as follows.

Figure 4: SoC/OCV curve fitting based on $\hat{\Theta}_{U_{s}}$.

(a)

(b)

Figure 5: Identification of $\Theta_{R_{o}}$: (a) terminal voltage profile
under intermittent discharging at 0.5 C to identify $\Theta_{R_{o}}$; (b)
fitting of $R_{o}\mathrm{(}\mathrm{SoC})$ with $\tilde{R}_{o}$ based on
$\hat{\Theta}_{R_{o}}$.

  * 1.

Based on Section 4, we first charged the cell using the popular constant-
current/constant-voltage method, let it rest for one hour, and then fully
discharged it using a 1/30 C constant-current load. We calculated the total
capacity $2.55$ Ah using the Coulomb counting method and used the voltage data
to find out $\hat{\Theta}_{U_{s}}$ based on (17), which is given by

$\displaystyle\hat{\Theta}_{U_{s}}$
$\displaystyle=\left\\{-9.048,-2.360,-12.986,0.010,13.036,-32.840,-0.087,2.359,\right.$
$\displaystyle\quad\quad\quad\left.-14.863,0.055,-0.788,-7.136,0.966,31.132,-3.414,0.513,1.816\right\\}.$

The SoC/OCV fitting result under the obtained $\hat{\Theta}_{U_{s}}$ is shown
in Fig. 4.

  * 2.

Next, the cell was charged to full again idled for one hour, and then
discharged under a 0.5 C pulse load profile. Specifically, a load was applied
for five minutes, followed by a two-hour rest, and this cycle continued until
the cut-off voltage was met. Fig. 5 shows the profile, which includes a total
of 12 pulses. With the data, we calculated $R_{o}$ at different SoC via (18)
and then used (19) to compute $\hat{\Theta}_{R_{o}}$ as shown in Table 1. The
reconstructed $R_{o}$ is compared with the measurements in Fig. 5.

  * 3.

Going further, we fully charged the cell again as in the previous steps, and
then fully discharged it using a 0.5 C constant-current load, with the
objective of identifying $\Theta_{s}$. As explained in Section 4, we could
impose a pre-determined relation like (20) to reduce the number of parameters
to estimate. Here, we let the spherical particle be discretized into five
finite volumes, and the resulting $\eta_{i}$ and $\sigma_{j}$ are

$\displaystyle\eta_{i}$
$\displaystyle=\left\\{1,0.6066,0.3115,0.1148,0.0164\right\\},$
$\displaystyle\sigma_{j}$ $\displaystyle=\left\\{1,1.77,4.00,15.98\right\\}.$

Then, (21) was executed to determine $\hat{\Theta}_{s}$. Fig. 6 illustrates a
comparison between the predicted terminal voltage (with the dynamics of sub-
circuits B and C neglected) based on $\hat{\Theta}_{s}$ and the measurements.
Table 1 shows the estimation for $\hat{\Theta}_{s}$.

  * 4.

Then, we ran a 2 C constant-current discharging test to collect the
temperature data. The cell’s surface temperature increased by about 10 K
throughout the test. We leveraged prior knowledge and empirical tuning to
determine $\Theta_{\mathrm{Th}}$, as suggested in Section 4. While the
procedure is coarse-grained, we obtained $\hat{\Theta}_{\mathrm{Th}}$ that
leads to accurate fitting with the surface temperature data and physically
reasonable estimation of the core temperature, as shown in Fig. 7. Table 1
summarizes the numerical estimates of $\hat{\Theta}_{\mathrm{Th}}$ .

  * 5.

Finally, the cell was fully discharged at a constant current of 3 C to excite
the cell’s electrolyte dynamic and thermal behavior more discernible, for the
purpose of identifying $\Theta_{e}$ and $\Theta_{\mathrm{Arr}}$. Following
Section 4, we iteratively tuned $\hat{\Theta}_{\mathrm{Arr}}$ and then ran
(22) to find $\hat{\Theta}_{e}$ until the achievement of both physically sound
estimates and accurate voltage data fitting. Fig. 8 shows that the BattX model
based on all the identified parameters fits well the measured voltage, and
Table 2 shows the estimation results.

Figure 6: Terminal voltage fitting under 0.5 C constant-current discharging based on $\hat{\Theta}_{s}$. Figure 7: Temperature fitting and prediction based on $\hat{\Theta}_{\mathrm{Th}}$. Name | $\gamma_{1}$ | $\gamma_{2}$ | $\gamma_{3}$ | $C_{s,1}$ | $R_{s,1}$ | $C_{\mathrm{surf}}$ | $R_{\mathrm{surf}}$ | $C_{\mathrm{core}}$ | $R_{\mathrm{core}}$
---|---|---|---|---|---|---|---|---|---
Initial Guess | 1 | 1 | 1 | 4391 | 0.090 | 7 | 6 | 20 | 1
Lower Bound | - | - | - | 3600 | 0.054 | 3 | 3 | 5 | 0.5
Upper Bound | - | - | - | 5500 | 0.167 | 12 | 20 | 50 | 7
Final estimate | 0.026 | 0.061 | -14.36 | 4521 | 0.114 | 10 | 7 | 40 | 4
Table 1: Identification summary for $\Theta_{R_{o}}$, $\Theta_{s}$, and $\Theta_{\mathrm{Th}}$: initial guesses, bound limits, and final estimates. Figure 8: Terminal voltage fitting under 3 C constant-current discharging based on $\hat{\Theta}_{e}$. Name | $C_{e}$ | $R_{e}$ | $\beta_{1}$ | $\beta_{2}$ | $\kappa_{1}$ | $\kappa_{2}$
---|---|---|---|---|---|---
Initial Guess | 1032 | 0.028 | 0.53 | 0.31 | 15 | 22
Lower Bound | 500 | 0.002 | 0.42 | 0.19 | 10 | 10
Upper Bound | 5000 | 0.080 | 1.00 | 0.423 | 100 | 100
Final estimate | 3691 | 0.007 | 0.789 | 0.317 | 30 | 70
Table 2: Identification summary for $\Theta_{\mathrm{Arr}}$ and $\Theta_{e}$:
initial guesses, bound limits, and final estimates.

From above, we have come up with an explicit setup of the BattX model for the
cell. Next, we will fit the model to new datasets to assess how well it
predicts.

###  5.2 Model Testing and Validation

To further evaluate the obtained BattX model, we generated new datasets by
applying a variety of current load profiles that span a broad range of
currents. The first tests involved full discharging of the cell at a constant
current of 1 C, 4 C, and 5 C separately. Fig. 9 compares the model’s
prediction of the terminal voltage prediction with the measurement, where a
close match is observed in all the three cases. Note that, even though the
model was identified based on tests of only up to 3 C, it can well predict 4 C
and 5 C. This suggests the model’s high fidelity and interoperability.

Figure 9: Voltage prediction by the BattX model versus the measurements at
constant-current discharging at 1, 4 and 5 C.

Further, we adopted the Urban Dynamometer Driving Schedule (UDDS) as a
variable load profile and scaled it to be between $-8$ C and $5$ C. The
validation of the BattX model over this dataset is shown in Fig. 10. The top
figure in Fig. 10 illustrates the load profile, which includes both charging
and discharging as well as a rest period. The voltage prediction of the model,
as shown in Fig. 10, closely follows the true voltage overall. A slight
discrepancy appears at the end of the test when the cell is about to be
depleted. This is likely because the radical changes of the internal
resistance at low SoC and high temperature are hard to be thoroughly captured.

Figure 10: Terminal voltage prediction by the BattX model versus the
measurements in the UDDS-based test. Top: the UDDS-based current profile;
middle: the voltage prediction in comparison with the measurements; bottom:
magnified views within two time windows. Figure 11: Temperature prediction by
the BattX model versus the measurements in the UDDS-based test.

Fig. 11 then demonstrates the comparison of the predicted surface temperature
with the measurement, showing an acceptable accuracy. The estimation of the
core temperature is also given in Fig. 11, which is reasonable by empirical
knowledge and observation.

Recently, LiB-powered eVTOL has attracted increasing interest as a promising
solution to urban air mobility and decarbonization of aviation. A safety-
critical application, eVTOL must maintain fast and accurate monitoring of its
onboard battery system throughout a mission. Conventional equivalent circuit
models are impossible to meet this need, as eVTOL often requires high-rate
discharging—it must discharge as fast as 5 C in the takeoff and landing
phases. However, the proposed BattX model holds a significant advantage to
eVTOL battery performance modeling. We consider a notional eVTOL flight, which
includes three phases, takeoff, cruising, and landing. The three phases
involve discharging at 5 C, 1.48 C, and 5 C, respectively [31]. We generated a
current load profile sequentially comprising a flight, full discharge, and
another flight. Fig. 12 displays the profile over time. Fig. 12 shows that the
BattX model achieves accurate prediction compared with the measurement.
Especially, the accuracy is found satisfactory at the times of high discharge
rates. The surface temperature prediction in Fig. 13 also well agrees with the
actual temperature, and the core temperature estimation shows a realistic
trend that one can trust to be close enough with the truth.

Figure 12: Terminal voltage prediction by the BattX model versus the
measurements in the test simulating an eVTOL operation cycle. Top: the current
profile; bottom: comparison between the prediction and measurements. Figure
13: Temperature prediction by the BattX model versus the measurements in the
test simulating an eVTOL operation cycle.

##  6 Conclusions

LiBs have found their way into many sectors as a key technology to drive
forward electrification and decarbonization. For LiB applications,
computationally fast and accurate models are a bedrock for real-time
monitoring and simulation to ensure their performance and safety. Although the
literature has presented different dynamic models, few of them are effective
when current loads range from low to high. To overcome the problem, we
proposed the BattX model in this study. This model is an ECM in its form, but
unlike other ECMs, it lends to interpretation as a quasi-electrochemical
model. This is because it is designed to use separate yet coupled circuits to
approximate the lithium-ion diffusion in the electrode and electrolyte phases,
heat transfer, and nonlinear voltage behavior in charging/discharging of a
cell. With the novel design, the model offers high predictive accuracy over
broad current ranges and still retains relatively simple structures for low
computational costs. We also developed a parameter identification approach for
the model. The approach groups the parameters based on the dynamic processes
or components that they belong to, and then identifies the parameters of each
group using experimental data. Finally, the experimental validation showed
that the BattX model has high accuracy and fidelity across low to high
C-rates.

## Acknowledgement

This work was supported in part by the United States National Science
Foundation under Awards CMMI-1763093 and CMMI-1847651.

## Appendix

###  A Derivation of $V_{s,1}$ under Constant Current $I$

In Section 4, the identification of $\Theta_{s}$ in (21) requires the
expression of $V_{s,1}$ when the applied current $I$ is constant. The
derivation is as follows.

Consider the governing equations of sub-circuit A in (1) under the assumption
in (20), and rewrite them compactly into the following form:

$\displaystyle\dot{V}_{s}(t)$ $\displaystyle=A_{s}V_{s}(t)+B_{s}I(t),$ (A.1)

where

$\displaystyle V_{s}$
$\displaystyle=\left[\begin{matrix}V_{s,1}&V_{s,2}&\cdots&V_{s,N}\end{matrix}\right]^{\top},$
$\displaystyle A_{s}$ $\displaystyle=\mu_{s}\Omega_{s},$
$\displaystyle\mu_{s}$ $\displaystyle=\frac{1}{C_{s,1}R_{s,1}},$
$\displaystyle\Omega_{s}$
$\displaystyle=\begin{bmatrix}-1\over\eta_{1}\sigma_{1}&1\over\eta_{1}\sigma_{1}&0&\cdots&\cdots&0\cr\frac{1}{\eta_{2}\sigma_{1}}&-\frac{1}{\eta_{2}\sigma_{1}}-\frac{1}{\eta_{2}\sigma_{2}}&1\over\eta_{2}\sigma_{2}&0&\cdots&0\cr\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\cr
0&\cdots&\cdots&0&\frac{1}{\eta_{N}\sigma_{N-1}}&\frac{-1}{\eta_{N}\sigma_{N-1}}\end{bmatrix},$
$\displaystyle B_{s}$
$\displaystyle=\begin{bmatrix}\frac{1}{C_{s,1}}&0&\cdots&0\end{bmatrix}^{\top}.$

The solution to (A.1) is given by

$\displaystyle
V_{s}(t)=e^{A_{s}t}V_{s}(0)+\int_{0}^{t}e^{A_{s}(t-\tau)}B_{s}I(\tau)d\tau.$

When $I$ is constant, it becomes

$\displaystyle
V_{s}(t)=e^{A_{s}t}V_{s}(0)+\int_{0}^{t}e^{A_{s}(t-\tau)}d\tau\cdot B_{s}I.$
(A.2)

To find the explicit form of $V_{s}(t)$, we must derive the expression of
$e^{A_{s}t}$. To this end, we look at $\Omega_{s}$ first and note that it is
rank-deficient with one zero eigenvalue. Further, assume the other non-zero
eigenvalues to be distinct, and denote the eigenvalues of $\Omega_{s}$ as
$\lambda_{i}$ for $i=1,2,\ldots,N$ with $\lambda_{1}=0$. Then, by the Cayley-
Hamilton theorem, we have

$\displaystyle
e^{A_{s}t}=\left[\Phi^{-1}\phi(\mu_{s},t)\right]\otimes\Omega_{s},$ (A.3)

where

$\displaystyle\Phi$
$\displaystyle=\begin{bmatrix}1&\lambda_{1}&\cdots&\lambda_{1}^{N-1}\\\
1&\lambda_{2}&\cdots&\lambda_{2}^{N-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
1&\lambda_{N}&\cdots&\lambda_{N}^{N-1}\end{bmatrix},$
$\displaystyle\phi(\mu_{s},t)$
$\displaystyle=\begin{bmatrix}1&e^{\mu_{s}\lambda_{2}t}&\cdots&e^{\mu_{s}\lambda_{N}t}\end{bmatrix}^{\top}.$

The operator $\otimes$ is defined as

$\displaystyle a\otimes A=\sum_{i=1}^{n}a_{i}A^{i-1},$

for $a\in\mathbb{R}^{n\times 1}$ and $A\in\mathbb{R}^{n\times n}$. Inserting
(A.3) into (A.6), we obtain

$\displaystyle V_{s}(t)$
$\displaystyle=\left[\Phi^{-1}\phi(\mu_{s},t)\right]\otimes\Omega_{s}\cdot
V_{s}(0)+\int_{0}^{t}\left[\Phi^{-1}\phi(\mu_{s},t-\tau)\right]\otimes\Omega_{s}d\tau\cdot
B_{s}I$
$\displaystyle=\left[\Phi^{-1}\phi(\mu_{s},t)\right]\otimes\Omega_{s}\cdot
V_{s}(0)+\left[\Phi^{-1}\int_{0}^{t}\phi(\mu_{s},t-\tau)d\tau\right]\otimes\Omega_{s}\cdot
B_{s}I$
$\displaystyle=\left[\Phi^{-1}\phi(\mu_{s},t)\right]\otimes\Omega_{s}\cdot
V_{s}(0)+\left[\Phi^{-1}\left(\bar{\phi}(\mu_{s},t)-\bar{\phi}(\mu_{s},0)\right)\right]\otimes\Omega_{s}\cdot
B_{s}I,$ (A.4)

where

$\displaystyle\bar{\phi}(\mu_{s},t)=\begin{bmatrix}t&{e^{\mu_{s}\lambda_{2}t}\over\mu_{s}\lambda_{2}}&\cdots&e^{\mu_{s}\lambda_{N}t}\over\mu_{s}\lambda_{N}\end{bmatrix}^{\top}.$

Given (A), $V_{s,1}$ can be expressed as

$\displaystyle V_{s,1}(t)=\mathbf{e}_{1}^{\top}V_{s}(t),$

where $\mathbf{e}_{1}=\begin{bmatrix}1&0&\cdots&0\end{bmatrix}^{\top}_{N\times
1}$.

###  B Derivation of $V_{e,1}$ and $V_{e,3}$ under Constant Current $I$

The explicit expressions of $V_{e,1}$ and $V_{e,3}$ under a constant current
$I$ are needed to represent $U_{e}$ for the identification of $\Theta_{e}$ in
(22). We can follow similar lines in Appendix.A to find them out. Let us
rewrite the governing equations of sub-circuit B in (2) compactly as

$\displaystyle\dot{V}_{e}(t)=A_{e}V_{e}(t)+B_{e}I(t),$ (A.5)

where

$\displaystyle V_{e}$
$\displaystyle=\begin{bmatrix}V_{e,1}&V_{s,2}&V_{e,3}\end{bmatrix}^{\top},$
$\displaystyle A_{e}$ $\displaystyle=\mu_{e}\Omega_{e},$
$\displaystyle\mu_{e}$ $\displaystyle={1\over C_{e}R_{e}},$
$\displaystyle\Omega_{e}$ $\displaystyle=\begin{bmatrix}-1&1&0\cr 1&-2&1\cr
0&1&-1\end{bmatrix},$ $\displaystyle B_{e}$
$\displaystyle=\begin{bmatrix}\frac{1}{C_{e}}&0&-\frac{1}{C_{e}}\end{bmatrix}^{\top}.$

The solution to (A.5) under a constant current $I$ is

$\displaystyle
V_{e}(t)=e^{A_{e}t}V_{e}(0)+\int_{0}^{t}e^{A_{e}(t-\tau)}d\tau\cdot B_{e}I.$
(A.6)

The eigenvalues of $\Omega_{e}$ are $0,-1,-3$, respectively. By the
Cayley–Hamilton theorem, it follows that

$\displaystyle
e^{A_{e}t}=\left[\Psi^{-1}\psi(\mu_{e},t)\right]\otimes\Omega_{e},$ (A.7)

where

$\displaystyle\Psi$ $\displaystyle=\begin{bmatrix}1&0&0\cr 1&-1&1\cr
1&-3&9\end{bmatrix},$ $\displaystyle\psi(\mu_{e},t)$
$\displaystyle=\begin{bmatrix}1&e^{-\mu_{e}t}&e^{-3\mu_{e}t}\end{bmatrix}^{\top}.$

Based on (A.7), we can derive that

$\displaystyle V_{e}(t)$
$\displaystyle=\left[\Psi^{-1}\psi(\mu_{e},t)\right]\otimes\Omega_{e}\cdot
V_{e}(0)+\left[\Psi^{-1}\left(\bar{\psi}(\mu_{e},t)-\bar{\psi}(\mu_{e},0)\right)\right]\otimes\Omega_{e}\cdot
B_{e}I,$ (A.8)

where

$\displaystyle\bar{\psi}(\mu_{e},t)=\begin{bmatrix}t&-{e^{-\mu_{e}t}\over\mu_{e}}&-{e^{-3\mu_{e}t}\over
3\mu_{e}}\end{bmatrix}^{\top}.$

With (A.8), one can extract $V_{e,1}$ and $V_{e,3}$ from $V_{e}$.

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