# Graded lithium-ion battery pouch cells to homogenise current distributions
and mitigate lithium plating

R. Drummond Email<EMAIL_ADDRESS>Department of Automatic
Control and Systems Engineering, University of Sheffield, Sheffield, S1 3JD,
UK E. C. Tredenick  Department of Engineering Science, University of Oxford,
Oxford, OX1 3PJ, UK  The Faraday Institution, Quad One, Harwell Campus,
Didcot, OX11 0RA, UK  T. L. Kirk  Department of Mathematics, Imperial College
London, London, SW7 2AZ, UK  M. Forghani  The Faraday Institution, Quad One,
Harwell Campus, Didcot, OX11 0RA, UK  Department of Materials, University of
Oxford, Oxford, OX1 3PH, UK  P. S. Grant  The Faraday Institution, Quad One,
Harwell Campus, Didcot, OX11 0RA, UK  Department of Materials, University of
Oxford, Oxford, OX1 3PH, UK  S. R. Duncan  Department of Engineering Science,
University of Oxford, Oxford, OX1 3PJ, UK  The Faraday Institution, Quad One,
Harwell Campus, Didcot, OX11 0RA, UK

###### Abstract

Spatial distributions in current, temperature, state-of-charge and degradation
across the plane of large format lithium-ion battery pouch cells can
significantly impact their performance, especially at high C-rates. In this
paper, a method to smooth out these spatial distributions by grading the
electrode microstructure in-the-plane is proposed. A mathematical model of a
large format pouch cell is developed and validated against both temperature
and voltage experimental data. An analytic solution for the optimal graded
electrode that achieves a uniform current distribution across the pouch cell
is then derived. The model predicts that the graded electrodes could
significantly reduce the likelihood of lithium plating in large format pouch
cells, with grading increasing the C-rate at which plating occurs from 2.75C
to 5.5C. These results indicate the potential of designing spatially varying
electrode architectures to homogenise the response of large format pouch cells
and improve their high rate performance.

##  1 Introduction

Lithium-ion battery electrodes are complex non-linear dynamical systems
reacting non-uniformly in space and time, especially when subject to high
C-rate currents. Phenomena such as travelling wave fronts in charging LiFePO4
(LFP) cathodes 1; 2 and localised lithium-plating in graphite anodes 3 are
just some examples of the diverse range of non-linear behaviours that have
been observed in battery electrodes, and it is these localised behaviours that
often govern cell performance. To protect against these spatially localised
electrochemical behaviours, the use of graded electrode structures (as in
those with controlled spatial variations in electrode microstructure) has been
proposed with the idea being that introducing heterogeneity into the electrode
microstructure can reduce the heterogeneity in the electrochemical response.

Recently, experimental data have provided support for the use of graded
electrodes in practical cells 4. A common theme is the potential of graded
electrodes to increase both the capacity at high C-rates 5; 6 and the cycle
life of thick electrodes 7; 8, with other benefits including a reduction in
electrode tortuosity 9. Data for both standard electrodes, such as LFP
cathodes 7; 8, and non-standard electrodes, such as anodes arranged in a
layered structure composed of high power Li4Ti5O12 and high capacity SnO2 10,
have shown similar benefits. However, care has to be taken when designing
graded electrodes, as structuring the electrode incorrectly can deteriorate
performance 11; 8. To prevent this, mathematical models have been developed to
predict the electrochemical response of graded electrodes so as to quickly and
cost-effectively optimise their designs. Several different models have now
been developed covering a range of length scales (including micro-
structural12; 13 and Doyle-Fuller-Newman-type models14; 2) and performance
metrics (such as fast charging capacitance 2 and electrical impedance 15).
These results highlight the growing maturity of battery modelling and
parameterisation, which, in turn, is making it increasingly straightforward to
embed modelling pro-actively in the battery design process 16. An example of
this battery co-design process are the trapezoidal electrodes7 that were
developed using insights 15 on how the distribution of carbon and binder
affects the conductivity of the LiFePO4 electrodes.

The focus of most existing studies on graded electrodes for lithium-ion
batteries has been on controlling the through-thickness local electrode
microstructure 12; 17, including bilayer cathodes2 composed of discrete layers
of both  Li[Ni0.6Co0.2Mn0.2]O2 (NMC622) and LFP. Focusing on the through-
thickness electrode structure (with experiments typically only involving coin
cells 7) is justified by the fact that it is charge mobility through the
electrode thickness between the anode and cathode that governs coin cell
performance. However, in recent years, there has been a general trend in
commercial cells towards larger cell formats (demonstrated, for example, by
the introduction of the larger 4680 cylindrical cells 18). The idea being that
larger cell formats increase the specific energy density, as the ratio of
active material to superfluous materials such as casings and control circuits
is increased. Large format cells have their own problems compared to coin
cells; in particular, they experience new forms of spatial heterogeneities,
which emerge at the macro-scale (as in across their width and length). In
particular, thermal hot spots 19; 20 and spatially localised distributions of
electrode degradation 21; 22; 23, lithium plating 24, state-of-charge 25, and
currents in-the-plane of large-format pouch cells 26; 27 have been identified
as impacting performance, especially in high rate applications.

The significance of in-the-plane spatial heterogeneities on the performance of
large format pouch cells is becoming increasingly apparent. In turn, this has
driven efforts to understand how these heterogeneities develop using
mathematical models. Through these models, predictions for chemical reaction
28 and current distributions along the plane of the current collectors 29 (and
how these distributions evolve with changes in the cell’s design, e.g. its
current collector thickness 30 and tapering 31) have been developed, with
analytical solutions even being proposed 32. The push to understand the planar
dynamics of pouch cells has meant that the number of spatial dimensions in
these models has increased, with 2D 33 and 3D 34; 35; 36; 37; 20; 38 effects
often included, as opposed to the through-thickness models of, for example,
the DFN model and single particle models 39; 40. One of the main drawbacks of
large format battery models is their complexity, a consequence of their multi-
dimensional spatial domains. This model complexity leads to several challenges
including long simulation times and difficulties in model parameterisation, as
discussed in the electro-thermal modelling studies 1920. These challenges
demonstrate a need for easily parameterised and computationally efficient
pouch cell models able to capture the spatial distributions seen in
experimental data. This problem is considered with the model introduced in
Section 2.

(a) Representation of electrode/separator sandwich of the pouch cell. The
cathode resistance is assumed to be graded. A left handed co-ordinate system
is used.

(b) Model simplification for large format pouch cells with the electrode
thickness being significantly thinner than the width and length. Circuit
parameters are found in Table A2.

(c) Spatial domain of the 2D pouch cell model.

Figure 1: Schematic of the 2D model for a lithium-ion battery pouch cell.

The formation of spatial heterogeneities across pouch cells indicates that it
may be possible to grade the local electrode microstructure in-the-plane to
mitigate their effect, similar to what has been achieved with through-
thickness grading 41; 17. This graded electrode problem is illustrated in
Figure 1, with Figure 1(a) indicating the planar and through-thickness
directions of the pouch cell, Figure 1(b) indicates the simplification that
allows the model to be reduced to a 2D problem, and Figure 1(c) indicates the
2D model domain.

##### Contributions:

In this paper, the problem of modelling and optimising graded electrode
microstructures for large format pouch cells with controlled variations in
resistance in-the-plane is considered. The following contributions are
obtained:

  1. 1.

A model for a large format pouch cells is described and validated against
experimental data20.

  2. 2.

An analytic solution for the distribution of carbon black in the electrode
plane that achieves a flat current distribution across the pouch cell is
derived.

  3. 3.

Compared to uniform electrodes, it is predicted that graded electrodes can
increase the charging rate when lithium plating occurs from 2.75C to 5.2C.

It is proposed that a primary advantage of these graded pouch cells is to
mitigate the impact of lithium plating (see Section 4), as the flatter current
distributions reduce the likelihood of negative anode overpotentials. The
results highlight the importance of in-the-plane effects when characterising
lithium plating in large format cells. By contrast, it is shown that the high
thermal/electronic conductivity of the current collectors means that only
minimal benefits of grading are observed when considering the thermal and
voltage response; Section 3.4 shows that the graded cells only delivered a
1.2% increase in stored capacitance for 4C charging compared to the uniform
electrodes, a conclusion in agreement with Hosseinzadeh, et al. 201841. These
validated modelling results indicate the potential of grading to improve the
cycle-life and high-rate performance of large format pouch cells. More
broadly, they indicate how improvements in battery design and manufacturing
could reduce the performance gap observed between coin cells constructed in
laboratories and the large format cells used in commercial high-rate
applications 42.

##  2 Mathematical model

In this section, a mathematical model of a large format pouch cell is
introduced. Specific focus is given to a comparison between the model and
experimental data20, with the presented model being simpler in form
(specifically, here, circuit dynamics are used to capture the through-
thickness electrical response, whereas a simple electrochemical model is used
in Lin, et al, 202220) whilst still being able to accurately capture the in-
plane spatial distributions seen in the experimental data (see Section 2.5).
Keeping the model formulation relatively simple while retaining the important
physics-based mechanisms allows analytic solutions to the optimal graded
electrode design problem (explored in Section 3) to be obtained, which is the
focus of this work.

###  2.1 Model formulation

Figure 1 presents a schematic for the model’s formulation and spatial domain.
The model parameters and variables are defined in Tables A1 and A2. The
following assumptions are made:

  1. A1:

The current collectors’ thicknesses are much thinner than their width and
height.

  2. A2:

The circuit models of Figure 1(b) describe the electrical dynamics at each
point in the plane of the pouch cell.

  3. A3:

The multi-layer pouch cell can be modelled as an effective single thick
layer19; 20.

A1 is justified because large format pouch cells are considered (with typical
current collector thicknesses being $\approx 10\mu$m and their planar
dimensions being $\approx$ 15cm $\times$ 20cm). Defining $i_{x}(y,z,t)$ as the
charging current through the electrode, and $I_{x,k}(x,y,z,t)$ as the current
flowing in the $x$-direction through current collector $k$, then, under this
thin plate assumption, the current gradient in the $x$-direction through the
current collectors can be approximated by

$\displaystyle\frac{\partial I_{x,k}(x,y,z,t)}{\partial
x}\approx\pm\frac{i_{x}(y,z,t)}{L_{ck}},\quad k\in\\{n,p\\},$ (1)

following the ideas of Taheri, et al. 201432. A2 is a simplification of the
electrodes’ electrical dynamics, but, even so, the resulting model is able to
capture the spatial distributions seen in the experimental data (see Section
2.5) while being simple enough to generate an analytic solution to the optimal
electrode design problem (Section 3.1) and parameterise (illustrated by the
relative lack of parameters in Table A2). A3 is justified under the assumption
that each layer has a similar composition and so react similarly.

###  2.2 Model equations

The following equations characterise the proposed mathematical model for
large-format pouch cells, with the models domain described in Figure 1(c).
Regarding the notation, each model variable in Eqn. (3) is defined in the
($y,z$)-plane of the pouch cell, as illustrated in Figure 1(c), and may also
evolve in time $t$. This $(y,z,t)$ dependency of the variables is dropped in
the model equations (3) to ease readability. $V(y,z,t)$ is the voltage
(defined as the potential of the current collector at the cathode minus that
of the anode) at each point in the plane, $T(y,z,t)$ is the cell temperature,
$i_{x}(y,z,t)$ is the local through-thickness current, $R(y,z)$ is the local
resistance of the two electrodes and the separator that is constant with time,
while SoC$(y,z,t)$ is the local state of charge in the plane. Table A1
summarises the model’s variables, while Table A2 gives its parameters. For the
graded cell, the only parameter varying in $(y,z)$ is $R(y,z)$, with the rest
being constants. Using the notation $\Delta f=\frac{\partial^{2}f}{\partial
y^{2}}+\frac{\partial^{2}f}{\partial z^{2}}$ and defining

$\displaystyle\sigma_{\text{eff}}(L_{\text{cn}}+L_{\text{cp}})$
$\displaystyle=\dfrac{\
L_{\scriptscriptstyle\text{cn}}L_{\scriptscriptstyle\text{cp}}\
\sigma_{\scriptscriptstyle\text{cn}}\
\sigma_{\scriptscriptstyle\text{cp}}}{\sigma_{\scriptscriptstyle\text{cn}}L_{\scriptscriptstyle\text{cn}}+\sigma_{\scriptscriptstyle\text{cp}}L_{\scriptscriptstyle\text{cp}}},$
(2)

then

$\displaystyle i_{x}$ $\displaystyle=\frac{V-U(\text{SoC,T})-v_{1}-v_{2}}{R},$
(3a) $\displaystyle\Delta V$
$\displaystyle=-\frac{i_{x}}{\sigma_{\text{eff}}(L_{\text{cn}}+L_{\text{cp}})},\
\ \text{on}\ \ \Omega,\ t>0,$ (3b) $\displaystyle\rho\frac{\partial
T}{\partial t}$ $\displaystyle=\lambda_{\scriptscriptstyle\text{eff}}\Delta
T+R\frac{{i_{x}}^{2}\
}{L_{x}}-h_{\scriptscriptstyle\text{faces}}\left(T-T_{\scriptscriptstyle\text{ref}}\right),\
\ \text{on}\ \ \Omega,\ t>0,$ (3c)
$\displaystyle\frac{\partial\text{SoC}}{\partial t}$
$\displaystyle=\frac{i_{x}}{C_{\scriptscriptstyle\text{cell}}},\ \ \text{on}\
\ \Omega,\ t>0,$ (3d) $\displaystyle\frac{\partial v_{1}}{\partial t}$
$\displaystyle=-\frac{v_{1}}{R_{1}C_{1}}+\frac{i_{x}}{C_{1}},$ (3e)
$\displaystyle\frac{\partial v_{2}}{\partial t}$
$\displaystyle=-\frac{v_{2}}{R_{2}C_{2}}+\frac{i_{x}}{C_{2}},$ (3f)

where $\Omega=[0,L_{y}]\times[0,L_{z}]$ is the projection of the cell onto the
$(y,z)$-plane and $L_{y}$ and $L_{z}$ are the cell width and height,
respectively. Eqn. (3a) defines the through-thickness current (as in the
current flowing across the electrodes charging the active material particles).
Eqn. (3b) defines the distribution of the voltage across the two current
collectors. Eqn. (3c) defines the localised temperature dynamics of the cell.
Using the circuit model of Figure 1(b) to describe the electrical dynamics at
each point ($y,z)$ in the plane, Eqn. (3d) describes the state-of-charge and
Eqns. (3e)-(3f) are the relaxation dynamics associated with Figure 1(b)’s RC
pairs.

###  2.3 Boundary Conditions

Using the notation of $n_{\perp}$ as the two-dimensional unitary vector normal
to the boundary and $\nabla T(y,z,t)=[\partial T/\partial y,\,\partial
T/\partial z]^{\top}$, then the model’s boundary conditions are:

$\displaystyle n_{\perp}\cdot\nabla T$
$\displaystyle=-\frac{h}{\lambda_{\scriptscriptstyle\text{eff}}}\left(T-T_{\scriptscriptstyle\text{ref}}\right),\
\ \text{on}\ \ \partial\Omega_{\scriptscriptstyle\text{ext}},\ t>0,$ (4a)
$\displaystyle n_{\perp}\cdot\nabla T$
$\displaystyle=-\frac{h_{\scriptscriptstyle\text{tab}}}{\lambda_{\scriptscriptstyle\text{eff}}}\left(T-T_{\scriptscriptstyle\text{ref}}\right),\
\ \text{on}\ \ \partial\Omega_{\scriptscriptstyle\text{tab}},\ t>0,$ (4b)
$\displaystyle\diff{\text{V}}z\Big{|}_{z=L_{z}}$
$\displaystyle=\frac{I_{\scriptscriptstyle\text{app}}}{\sigma_{\scriptscriptstyle\text{ck}}A_{\scriptscriptstyle\text{tab}}},\
\ \text{on}\ \
\partial\Omega_{\scriptscriptstyle\text{tab,k}},~{}k\in\\{n,\,p\\},\,t>0,$
(4c) $\displaystyle n_{\perp}\cdot\nabla V$ $\displaystyle=0,\ \ \text{on}\ \
\partial\Omega_{\scriptscriptstyle\text{ext}},\ t>0,$ (4d)

where $\partial\Omega_{\scriptscriptstyle\text{tab,n }}$ and
$\partial\Omega_{\scriptscriptstyle\text{tab,p}}$ are the negative and
positive tabs, $\partial\Omega_{\scriptscriptstyle\text{tab
}}=\partial\Omega_{\scriptscriptstyle\text{tab,n
}}\cup\partial\Omega_{\scriptscriptstyle\text{tab,p }}$, $\partial\Omega$ is
the external boundary region and
$\partial\Omega_{\scriptscriptstyle\text{ext}}$ is the external boundary
region without the tabs,
$\partial\Omega_{\scriptscriptstyle\text{ext}}=\partial\Omega\setminus\partial\Omega_{\scriptscriptstyle\text{tab}}$.

###  2.4 Numerical Solution Procedure

To simulate the model, the partial-differential-algebraic equations of Eqns.
(3) and (4) were discretised in space using Chebyshev spectral collocation43
and solved using “ode15s” in MATLAB® 2022b44. On average, a 5C charge of a
uniform cell took 12.7 s (with 24x24 nodes) when run on a desktop Macintosh
(MacBook Pro, Monterey, v 12.4, M1 2020, 16GB, 256GB, 64 bit). The unknown
model parameters were found by fitting to 4C square-wave-excitation
experiments starting at 30% SOC 20. The open-source code for the pouch cell
model simulations can be found at:
https://github.com/EloiseTredenick/2DBatteryTemperatureGradedModel.

##### Comparison to existing models:

The main differences between the large-format pouch cell model of (3) and
existing models 20 32 are: i) the inclusion of the spatially-varying
resistance, $R(y,z)$, in (3), ii) the model’s relative simplicity, with no
DFN-type through thickness electrochemical modelling being used; instead, the
circuit of Figure 1(b) is distributed in the plane. The relative simplicity of
the model is a deliberate design choice as it significantly simplifies the
model parameterisation problem, which is crucial for capturing the
experimental data.

It is also noted that graded planar cells were also investigated in
Hosseinzadeh, et al. 201841, where a DFN-type model with grading both in-the-
plane and through-the-thickness of the electrodes was developed. The two main
differences between this paper and Hosseinzadeh, et al. 201841 are: i) an
analytic solution for planar graded electrodes to achieve a uniform current
distribution is obtained here (see Eqn. (7)), ii) this paper’s model
predictions are validated against experimental data from Lin, et al, 202220,
iii) it is proposed that the benefits of in-the-plane grading are to reduce
electrode degradation (especially plating), whereas voltage and thermal
considerations were considered in 41.

###  2.5 Comparison to experimental data

The experimental data from Lin, et al, 202220 is used to validate the lithium-
ion battery pouch cell model of (3). Briefly, the data is obtained from
square-wave-excitation cycling experiments starting at 30% state of charge
(SoC), using 20 Ah pouch cells from A123 Systems with LFP positive electrodes
and graphite negative electrodes, which were cycled at 4C (at currents $\pm
80$ A) every 100 seconds. A thermal imaging camera was then used to capture
thermograms of the cell surface and to visualise the cell’s temperature
distribution across the plane during the cycling.

###  2.6 Model validation

Figures 2 and 3 compares the model simulations against the experimental data20
for cycling using the parameters from Table A2. The parameters were estimated
using a combination of fitting and a comparison to values from the literature,
with the notation $R_{0}$ for the resistance of the uniform cell. Model
agreement was observed in both the electrical and thermal response, as shown
in Figures 2(a) and 2(b).

Figure 2(b) shows that the model was able to capture the evolution of the
maximum, minimum, and average values of the cell temperature across the plane
of the pouch cell during the cycling. A main point of difference between the
trajectories of the model and the experimental data is that the model’s
trajectories are much smoother. The increased stochasticity of the data is to
be expected, as the sensing noise and the inherent randomness in the
microstructure introduced by manufacturing limitations are not captured by
Eqn. (3). However, the model is still able to capture the general trends of
the data, with the time constants and steady-state values of the first-order
responses seen in the temperature dynamics being captured.

Figure 3 compares snapshots in the plane of the temperature distribution of
the model and the experimental data 20, where the snapshots were taken at $t=$
100s, 500s, 1000s, and 2500s. Even though the underlying model of Eqn. (3) is
simple, strong agreement was made with the experimental thermal data from Lin,
et al, 202220. In particular, the formation of a thermal hot spot as well as
the general shape of the temperature contours were captured by the model,
although, again, the model’s diffusion dynamics resulted in smoother contours
than those seen in the data.

Figure 4 illustrates how the validated model could be used to infer the impact
of design changes in the pouch cell on the degree of spatial heterogeneity
across the plane. Defining the variation in the current density as the
difference between the maximum and minimum current density across the plane,
Figure 4(a) shows how this variation changes with resistance (as in, with
$R(y,z)=R_{0}$), Figure 4(b) shows how it changes with pouch cell length, and
Figure 4(c) shows how it changes with current collector thickness. The figure
identifies some near linear relationships between these cell parameters and
the variation in current density, with explicit characterisations for these
expressions obtainable from the analytical solution of Taheri, et al, 201432.
As could be expected, the figure indicates that the variation of the current
distribution across the pouch cell increases as: i) the cell resistance
decreases, ii) the aspect ratio increases, and iii) the current collectors get
thinner. Notably, Figure 4(a) indicates that as the cell ages, and so as its
resistance increases, the variation in current distribution across the cell is
likely to decrease, as more current is encouraged to flow along the current
collectors than through the thickness of the cell.

(a) Comparison between the average voltages of the experimental data and the
model of (3) during cycling at 4C.

(b) Minimum, average, and maximum temperatures across the plane of the pouch
cell for the experimental data and the model of (3) at 4C.

Figure 2: Comparison between the pouch cell temperature and voltage responses
of the mathematical model of (3) and the experimental data of Lin, et al,
202220.

(a) Model: $t=100$s.

(b) Data: $t=100$s.

(c) Model: $t=500$s.

(d) Data: $t=500$s.

(e) Model: $t=1000$s.

(f) Data: $t=1000$s.

(g) Model: $t=2500$s.

(h) Data: $t=2500$s.

Figure 3: Temperature distributions across the plane of the pouch cell.
Comparison between the model of (3) and the experimental data. Snapshots at
$t$ = 100s, 500s, 1000s and 2500s are shown.

##  3 In-the-plane electrode grading

The spatial distributions seen in the response of the pouch cell, for example
in the temperature data of Figure 3, presents an opportunity to apply
electrode grading to create a more uniform electrochemical response. In this
section, an analytic solution for the resistance distribution across the plane
of the pouch cell, $R(y,z)$, to achieve a uniform current distribution is
described.

###  3.1 Grading manufacturing

It is assumed that the electrode resistance $R(y,z)$ can be controlled locally
in space. The reasons for focusing on the resistance distribution are: i)
following Eqns. (3a) and (3b), the electrode resistance $R(y,z)$ directly
influences the current distribution across the pouch cell (and it is
variations in the current which are targeted to be smoothed out by the
optimally graded electrode), ii) existing results on through-thickness
electrode grading have demonstrated how this resistance grading can be
achieved, e.g. by controlling the relative mass fraction of carbon black in
LFP cathodes using spray printing techniques 15. It is envisaged that such
spray techniques could be generalised to produce the optimised planar graded
electrodes proposed in this study.

(a) Current variation with resistance.

(b) Current variation with length.

(c) Current variation with current collector thickness.

Figure 4: Variation of the current distribution (difference between its
maximum and minimum) with pouch cell resistance, length, and current collector
thickness.

###  3.2 Resistance distribution for homogeneous current

The problem of distributing the resistance (achieved by controlling the weight
fraction of active carbon in the electrode microstructure) to achieve
homogeneous current distribution across the pouch cell is considered. This
targeted flat current distribution is equal to the average current density,
with

$\displaystyle i_{x}(y,z,t)=\frac{I_{\text{app}}(t)}{A_{\text{cell}}}.$ (5)

Eqn. (3) implies that, under isothermal conditions of the open-circuit
voltage, if a uniform current distribution is achieved at $t=0$, then $v_{1}$,
$v_{2}$ and the SoC, will evolve uniformly in space as well. The implication
is that if the current distribution is uniform at $t=0$, it will remain
uniform during the simulation, following the model equations of (3a) and (3b),
which may not hold exactly in practice as local current spikes might emerge
caused by the variability in the local electrode composition. However, this
can be used to simplify the graded electrode design optimisation problem.
Specifically, it converts a dynamic optimisation problem into a static one. As
in, it changes the optimisation from one of determining how the resistance
should be distributed to achieve a uniform current during the whole charge to
one that only has to consider the distribution at the start of the charge. A
consequence of this problem formulation is that it allows an analytic solution
to the graded electrode design optimisation to be derived (see Eqn (7)).

####  3.2.1 Problem formulation

With the above motivation, the desired uniform current distribution of Eqn.
(5) is substituted into Eqns (3a) and (3b) to give, after simplifying,

$\displaystyle\Delta
R+\frac{1}{\sigma_{\text{eff}}(L_{\text{cn}}+L_{\text{cp}})}=0,\ \ \text{on}\
\ \Omega,$ (6a) which is a non-homogeneous Laplace equation in the spatial
variable $R(y,z)$. Using the same approach, the boundary conditions for the
voltages from Eqns. (4c) and (4d) can be translated into boundary conditions
for the resistance to achieve a uniform current density, as in
$\displaystyle\frac{\partial R}{\partial z}\Big{|}_{z=L_{z}}$
$\displaystyle=-\frac{A_{\text{cell}}}{\sigma_{\scriptscriptstyle\text{ck}}A_{\scriptscriptstyle\text{tab},k}},\
\ \text{on}\ \
\partial\Omega_{\scriptscriptstyle\text{tab,k}},\,k\in\\{n,p\\}\ t>0,$ (6b)
$\displaystyle n_{\perp}\cdot\nabla R$ $\displaystyle=0,\ \ \text{on}\ \
\partial\Omega_{\scriptscriptstyle\text{ext}}.$ (6c)

####  3.2.2 Solution

The solution to Eqn. (6) is

$\displaystyle R(y,z)=R_{\text{quad}}(z)+R_{\text{hyper}}(y,z)$ (7a) composed
of a quadratic part, $R_{\text{quad}}(y,z)$, and a hyperbolic part,
$R_{\text{hyper}}(y,z)$, satisfying $\displaystyle R_{\text{quad}}(z)$
$\displaystyle=d_{0}+d_{1}z+d_{2}z^{2},$ (7b) $\displaystyle
R_{\text{hyper}}(y,z)$
$\displaystyle=\sum_{n=0}^{\infty}c_{n}\cos\left(\frac{n\pi
y}{L_{y}}\right)\cosh\left(\frac{n\pi z}{L_{y}}\right).$ (7c) The coefficients
of this solution are $\displaystyle d_{1}$ $\displaystyle=0,$ (7d)
$\displaystyle d_{2}$
$\displaystyle=-\frac{1}{2\sigma_{\text{eff}}(L_{\text{cn}}+L_{\text{cp}})},$
(7e) $\displaystyle c_{n}$ $\displaystyle=\frac{2}{n\pi\sinh\left(\frac{n\pi
L_{z}}{L_{y}}\right)}\int^{L_{y}}_{0}h(y)\cos\left(\frac{n\pi
y}{L_{y}}\right)~{}dy,$ (7f) with $\displaystyle
h(y)=\begin{cases}\frac{L_{z}}{\sigma_{\text{eff}}(L_{\text{cn}}+L_{\text{cp}})}&y\in\Omega_{\text{ext}},~{}z=L_{z},\\\
-\frac{A_{\text{cell}}}{\sigma_{\scriptscriptstyle\text{c1}}A_{\scriptscriptstyle\text{tab},1}}+\frac{L_{z}}{\sigma_{\text{eff}}(L_{\text{cn}}+L_{\text{cp}})}&y\in\Omega_{\scriptscriptstyle\text{tab,1}},~{}z=L_{z},\\\
-\frac{A_{\text{cell}}}{\sigma_{\scriptscriptstyle\text{c2}}A_{\scriptscriptstyle\text{tab},2}}+\frac{L_{z}}{\sigma_{\text{eff}}(L_{\text{cn}}+L_{\text{cp}})}&y\in\Omega_{\scriptscriptstyle\text{tab,2}},~{}z=L_{z},\end{cases}$
(7g)

and the constant $d_{0}$ being a free variable setting the average value of
the resistance.

(a) Resistance of the uniform cell.

(b) Current density of the uniform cell.

(c) Resistance of the graded cell.

(d) Current density of the graded cell.

Figure 5: Distributions in resistance and current density of the graded and
uniform cells after a 4C charge. The uniform cell has a constant resistance,
but a spatially varying current distribution. The graded cell has a spatially
varying resistance, but a constant current distribution.

The solution of Eqn (7) shows that the resistance should roughly decrease
quadratically away from the tabs in order to flatten the current distribution.
The physical meaning of decreasing the resistance away from the tabs is to
encourage current to flow down the length of the current collectors, instead
of going directly through the electrodes as it enters one of the tabs and
leaves the other. In the particular setup of Figure 1 and Table A2, where the
cell has a relatively high aspect ratio, the variation of the resistance is
much larger in the $z$-direction rather than the $y$-direction, but, following
Eqn. (7), this may change for wider cell formats or for different tab
configurations.

###  3.3 Results: Flat current distribution across pouch cells

Figure 5 compares the current and resistance distributions achieved with both
uniform and graded pouch cells (with the resistance of the graded cells
following the solution derived in Eqn. 7). For this analysis, both pouch cells
had equal average resistances of 1.5 $\times 10^{-3}$ $\Omega$ to ensure a
fair comparison. Significant variation was seen in the current density
distribution of the uniform cell (with a maximum value of 3925 A m-2 at the
tabs and a minimum value of 2138 A m-2 at the opposite end where $z=0$). By
contrast, the graded cell achieved a constant current density of 2667 A m-2
across the plane, as predicted.

###  3.4 Results: Dynamic response of graded pouch cells

Figure 6 compares the optimised graded electrode cell following Eqn (7) and
the uniform cell for a full 4C charge at $I_{\text{app}}$ = 80A. Figure 6
provides justification for the design choice of Section 3.1 to optimise the
spatial resistance of the pouch cells to achieve a uniform initial current
distribution in space, as Figure 6(a) indicates that this uniformity in the
current was retained during the whole charge. By contrast, the current
distribution in the uniform cell is more variable. Initially, large current
gradients exist across the plane of the uniform cell, as shown in Figure 6(a),
with the maximum current density 3925 A/m2 at the tabs and the minimum current
2138 A/m2 at the opposite end. The non-uniform current distribution then
causes the voltage to vary, following Eqns. (3d)-(3f). Simultaneously, there
is a negative feedback effect acting on the voltages from Eqns. (3a) and (3b)
that smoothens this variation. The interactions between these two modes causes
the spatial distribution of the current to evolve in both space and time. In
particular, Figure 6(a) shows that the current does not change significantly
during 100s $\leq t\leq$450s when the OCV curve is flat (since the smoothing
effect of $U$ in Eqn. (3a) is not active). It is this smoothing effect which
causes the region near the tabs to charge first, but then, when it is fully
charged, the majority of the current then flows through the bottom half of the
cell.

(a) Current density.

(b) Temperature.

(c) Voltage

(d) State-of-charge.

Figure 6: Spatial-heterogeneity of the model’s variables, including state-of-
charge, temperature, voltage and current density during a 4C charge. The
difference between these variables at the tabs and opposite the tab is shown.

Figure 7 shows snapshots of the current distribution across the current
collectors of the pouch cell with uniform electrodes at $t$ = 1s, 200s, 500s
and 600s during the 4C charge. At the start of the simulation, the peak
current was at the tabs, with this peak moving down the cell during the
charge. It is noted that the maximum variation in the current distribution in
the plane was recorded near the end of the charge, which is also the time when
lithium-plating is most likely to occur.

Whilst Figure 6(a) indicates that there would be large differences between the
current distributions of the uniform and graded cells, Figures 6(b) and 6(d)
indicate that the impact of these differences on the variable _internal_ to
the current collectors would actually be quite limited. Specifically, Figure
6(b) compares the temperatures, Figure 6(c) the voltages, and Figure 6(d) the
state-of-charges. Although there are some differences between the uniform and
graded electrodes, it is relatively limited compared to the large variations
seen in the current distributions. In particular, even though the profiles of
the two voltage curves in Figure 6(c) are different, they reach the cut-off
voltage of 3.85 V at roughly the same time (607s for the graded cell and 600s
for the uniform one). As such, the capacitance improvement of the graded cell
over the uniform one is only 1.2%. These limited gains in the capacitance are
a result of the large thermal and electrical conductivities of the current
collectors (see Table A2), which smooth out the variations in the electrodes
responses across the plane. Similarly, for the current, Appendix Figure A1
compares the modelled 4C cycling results for the graded and uniform cells. The
current density of the graded cell is close to $I(t)/A_{\text{cell}}$, while
the uniform cell has large variations both in time and across the plane. The
voltage of both cells is similar, while the maximum temperature is slightly
reduced by $0.3\,^{\circ}\mathrm{C}$ for the graded cell.

(a) Current density: $t=1$s.

(b) Current density: $t=200$s.

(c) Current density: $t=500$s.

(d) Current density: $t=600$s.

Figure 7: Current density distributions in the plane of the uniform pouch cell
using the model of (3) during a constant current charge at 4C. Contour lines
indicate current density values in A m-2 going through the electrode sandwich.
The location of maximum current travels from the tabs to opposite them during
the charge.

When only variables defined internally to the current collectors (such as
temperature and voltage) are used to assess the value of graded pouch cells,
their benefits may be lost. This observation agrees with the conclusions of
Hosseinzadeh, et al, 201841 that porosity distribution through the electrode
thickness has the potential to produce superior battery performance instead of
when the porosity is varied along the electrode height. However, in this
paper, it is proposed that to unlock the benefits of graded pouch cells, a
performance metric that focuses on _through-thickness effects_ should instead
be considered. This observation motivates the analysis of the following
section on the potential benefits of graded electrodes to mitigate lithium
plating.

##  4 Results: Plating reduction with graded electrodes

The potential of graded electrodes to mitigate lithium plating in large format
cells is explored in this section. As discussed in relation to Figures
6(b)-6(d) and in the results of Hosseinzadeh, et al, 201841, it was observed
that the high conductivity of the current collectors limits the apparent value
of electrode grading in-the-plane when only variables found “internally” to
the current collectors (with internal variables being understood as those,
such as voltage and temperature, that are smoothed out when flowing in the
plane of the current collectors) are used to assess performance, as opposed to
variables flowing through the electrodes (such as the current distributions of
Figure 7 and 6(a)). However, as lithium plating is a localised degradation
phenomenon dependent upon the charging current and electrochemical state of
the electrode at a given point, it is shown in this section that plating may
be effectively mitigated by in-the-plane electrode grading. The focus towards
mitigating lithium plating is also motivated by experimental studies 24 which
show significant spatial variations in plating across graphite anodes.
Controlling the distribution of active carbon in the electrode microstructure
to mitigate the spatial distribution in plating is proposed as a means to
reduce the accelerated degradation of large format pouch cells in high-rate
applications.

A mathematical modelling approach was used to compare the extent of lithium
plating between the graded and uniform cells. However, modelling lithium
plating is known to be challenging, with existing models typically being
additions to complex electrochemical models, with example models discussed in
45; 46. By contrast, in this paper, a simpler, control-orientated model
adapted from 47 was used, with this model based off of the earlier results of
48. For the state-constrained control problem considered in 47, the following
condition for lithium plating was developed, with the anode regarded as
experiencing plating if the following inequality is satisfied

$\displaystyle
a_{\text{lp}}\log(b_{\text{lp}}\text{SoC}(y,z,t))+c_{\text{lp}}+d_{\text{lp}}i_{x}(y,z,t)\geq
0.$ (8)

The plating inequality47 of Eqn. (8) was determined by fitting a curve when
the anode over-potential in a DFN model went negative. Here,
$a_{\text{lp}}=1.74$, $b_{\text{lp}}=9.32$, $c_{\text{lp}}=-4.46$, following
47, and $d_{\text{lp}}=0.0055$.

(a) C-rate dependency of plating.

(b) Plating onset during 5C charging.

Figure 8: Comparing the onset of lithium-plating between the uniform and
graded pouch cells. Figure 8(a) shows the C-rate dependency of the plating and
Figure 8(b) compares the progression of plating during 5C charging.  Figure 9:
Distribution of plating across the plane at the end of a 5C charge of the
uniform cell. The grey region has experienced plating and the white region is
free of plating.

Figure 8 compares the onset of lithium-ion plating for both the uniform and
graded pouch cells using Eqn. (8). The purpose of this figure is to illustrate
the spatial, temporal and C-rate rate dependence of plating, and how modelling
can capture the heterogeneity of pouch cell degradation. Figure 8(a) shows the
areal percentage of the pouch cell experiencing plating during charging at
different C-rates. The uniform pouch cell sees a gradual increase in the area
of the pouch cell experiencing plating when the C-rate is increased beyond
2.75C, with 100% of the pouch cell experiencing plating at 5.2C. By contrast,
the graded pouch cell experiences no plating until 5.5C whereupon there is a
sudden jump to 100% of its area that is plating. The differences between these
two curves is due to variation in the current distribution of the uniform
cell; this variation leads to higher current spikes and so a greater
likelihood of plating. By contrast, the model predicts that the flatter
current distribution of the graded pouch cell can delay the onset of plating
until a critical point is reached, and, at that point, the flat distributions
cause the whole plane to plate. The significance of this figure is that it
suggests that grading pouch cells can increase the maximum charging C-rate for
which plating can be avoided. In other words, grading in the plane could
increase the operating envelope of pouch cells such that they could be fast
charged without suffering from plating.

Figure 8(b) shows the time when plating is initiated during a 5C charge.
During this charge, the graded cell does not plate whereas the uniform
electrodes experiences plating at the end of charge. The propensity of plating
to occur in uniform cells at the end of charge is one of the main reasons why
optimal fast charging currents are often tapered at the end of charge49; 50.

Figure 9 illustrates the spatial distribution of plating across the plane of
the uniform pouch cell at the end of the 5C charge. The grey area indicates a
region where plating is predicted to occur. The figure suggests that, for
these cells, most of the plating occurs opposite the tabs with the plating
then progressing up the plane. This effect is in response to the current
distribution being highest at the tabs at the start of the charge, causing the
active particles in this region to be charged first. When these particles
become fully charged, they hinder further charging in this region and so cause
current to flow down towards the bottom of the pouch cell later on during the
charge. By restricting the area where the current can flow through, the local
current densities in this region at the bottom of the cell are increased (see
also Figure 6(a)), causing a higher likelihood of plating.

## Conclusions

A model for large format lithium-ion battery pouch cells with spatial grading
across the plane was developed and validated against experimental data. An
analytic solution for the optimal electrical resistance distribution across
the plane of large format pouch cells to achieve a flat current distribution
was derived. The main benefit of graded electrodes identified was to increase
the allowable charging C-rate from which lithium plating could be avoided; the
graded electrodes increased the safe charging rate from 2.75C to 5.5C. By
contrast, only a 1.2% increase in charging capacity was achieved for 4C
charging. These results indicate the potential of in-the-plane graded
electrode microstructures for enabling fast charging of large format pouch
cells that by limiting the spatial heterogeneities.

#### Acknowledgements

Funding for this work was partially provided by the Faraday Institution
project ‘Nextrode - next generation electrodes’ (grant number FIRG015 and
FIRG066). RD acknowledges funding by a UKIC Research Fellowship from the Royal
Academy of Engineering.

### Code Availability

The open-source code for the pouch cell simulations that have been used to
produce the results of this study are available online
(https://github.com/EloiseTredenick/2DBatteryTemperatureGradedModel).

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## Appendix

The open circuit voltage (OCV) function for the LFP cells51; 20 at 0.02 C is
modelled by:

$U(\text{SoC},T)=\frac{dU}{dT}\
\left(T-T_{\scriptscriptstyle\text{ref}}\right)+0.0027\ T\
I_{sign}+3.382+0.0047\ \text{SoC}+1.627\ \exp(-81.163\ \text{SoC}^{1.0138})\\\
+7.6445\times 10^{-8}\ \exp(25.36\ \text{SoC}^{2.469})-8.441\times 10^{-8}\
\exp(25.262\ \text{SoC}^{2.478})-3.3852+3.2585,$ (A1)

where $dU/dT=-10^{-4}$.

The initial conditions for the simulations are:

$\displaystyle\text{SoC}(0)$ $\displaystyle=\text{SoC}_{0},\ \text{on}\
\Omega\ \text{and}\ \partial\Omega,$ (A2a) $\displaystyle U(0)$
$\displaystyle=U_{0}=U(\text{SoC}_{0}),\ \text{on}\ \Omega\ \text{and}\
\partial\Omega,$ (A2b) $\displaystyle V(y,z,0)$ $\displaystyle=V_{0}=U_{0}+\
R_{0}\ I_{\scriptscriptstyle\text{app}}/A_{\text{tab}},\ \text{on}\ \Omega\
\text{and}\ \partial\Omega,$ (A2c) $\displaystyle v_{1}(y,z,0)$
$\displaystyle=0,\ \text{on}\ \Omega\ \text{and}\ \partial\Omega,$ (A2d)
$\displaystyle v_{2}(y,z,0)$ $\displaystyle=0,\ \text{on}\ \Omega\ \text{and}\
\partial\Omega,$ (A2e) $\displaystyle T(y,z,0)$
$\displaystyle=T_{\scriptscriptstyle\text{ref}}=T_{0},\ \text{on}\ \Omega\
\text{and}\ \partial\Omega.$ (A2f)

Table A1: Variables of the model.  Variable |  Description |  Unit
---|---|---
$i_{x}$ |  Current density through electrode thickness |  A$/$m2
$I$ |  Current density in the current collector |  A$/$m2
$I_{\scriptscriptstyle\text{app}}$ |  Applied current |  A
$R$ |  Cell series resistance area |  $\Omega$m2
SoC |  Local state of charge |  -
$T$ |  Temperature |  K
$t$ |  Time |  s
$U$ |  Open circuit voltage (OCV) |  V
$V$ |  Voltage |  V
$v_{1}$ |  Voltage across RC-pair |  V
$v_{2}$ |  Voltage across RC-pair |  V
$x$ |  Direction through cell |  m
$y$ |  Direction along cell width |  m
$z$ |  Direction along cell height |  m
Table A2: Model parameters. Experimental values obtained from 20 for an 80Ah capacity cell. Parameter |  Description |  Units |  Value |  Source
---|---|---|---|---
$A_{\scriptscriptstyle\text{tab}}$ |  Area of a tab |  m2 |  1.56$\times 10^{-4}$ |  Experiment
$A_{\scriptscriptstyle\text{cell}}$ |  Cell area $(L_{y}\times L_{z})$ |  m2 |  $30\times 10^{-3}$ |  Experiment
$C_{\scriptscriptstyle\text{cell}}$ |  Capacity of cell (20 Ah) per cell area |  A s$/$m2 |  20 $\times 3600/A_{\scriptscriptstyle\text{cell}}$ |  Experiment
$C_{1}$ |  Capacitance of RC-pair |  F |  3.49$\times 10^{4}$ |  Fitting
$C_{2}$ |  Capacitance of RC-pair |  F |  1.11$\times 10^{4}$ |  Fitting
$h$ |  Heat transfer coefficient |  W$/$m${}^{2}/$K |  10 |  Fitting
$h_{\scriptscriptstyle\text{tab}}$ |  Heat transfer coefficient of tabs |  W$/$m${}^{2}/$K |  51.58 |  Fitting
$h_{\scriptscriptstyle\text{faces}}$ |  Heat transfer coefficient of current collector faces |  W$/$m${}^{2}/$K |  5 |  Fitting
$L_{\scriptscriptstyle\text{cn}}$ |  Thickness of negative current collector in $x$ direction |  $\mu$m |  $25$ |  Experiment
$L_{\scriptscriptstyle\text{cp}}$ |  Thickness of positive current collector in $x$ direction |  $\mu$m |  $25$ |  Experiment
$L_{\scriptscriptstyle\text{x}}$ |  Electrode thickness including $n=42$ layers (110 $\mu$m without $n$) |  m |  0.0046 |  Experiment
$L_{\scriptscriptstyle\text{y}}$ |  Height of battery |  m |  $0.2$ |  Experiment
$L_{\scriptscriptstyle\text{z}}$ |  Width of battery |  m |  $0.15$ |  Experiment
$R_{0}$ |  Series resistance of Uniform cell |  $\Omega$ m2 |  1.5$\times 10^{-3}\times A_{\scriptscriptstyle\text{cell}}$ |  Fitting
$R_{1}$ |  Resistance of RC-pair |  $\Omega$ m2 |  9.03$\times 10^{-4}\times A_{\scriptscriptstyle\text{cell}}$ |  Fitting
$R_{2}$ |  Resistance of RC-pair |  $\Omega$ m2 |  1.80$\times 10^{-4}\times A_{\scriptscriptstyle\text{cell}}$ |  Fitting
$T_{\scriptscriptstyle\text{ref}}$ |  Constant absolute reference temperature |  K |  298.15 ($25\,^{\circ}\mathrm{C}$) |  Experiment
$\lambda_{\scriptscriptstyle\text{eff}}$ |  Effective thermal conductivity |  W$/$m$/$K |  4.5 |  Fitting
$\rho$ |  Heat capacity |  J$/$K$/$m3 |  125530 |  Fitting
$\sigma_{\scriptscriptstyle\text{cp}}$ |  Electronic conductivity of copper positive current collector |  S/m |  $5.96\times 10^{7}$ |  Experiment
$\sigma_{\scriptscriptstyle\text{cn}}$ |  Electronic conductivity of aluminium negative current collector |  S$/$m |  $3.77\times 10^{7}$ |  Experiment
Figure A1: Comparison between the graded and uniform models during cycling at
4C and an applied current of 80 A. The current and temperature responses with
time are shown, including the minimum, average, and maximum temperatures
across the plane of the pouch cell. The voltage is similar between the two
models.


Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

[orcid=0000-0002-5214-7595, url=https://github.com/bnagy/metronome-paper,
<EMAIL_ADDRESS>]

[orcid=0000-0002-2272-2077, url=https://artjomsh.github.io/web/,
<EMAIL_ADDRESS>]

[orcid=0000-0002-0899-5976<EMAIL_ADDRESS>]

[orcid=0000-0002-1003-4541<EMAIL_ADDRESS>]

# Metronome: tracing variation in poetic meters via local sequence alignment

Ben Nagy  Artjoms Šeļa  Mirella De Sisto  Petr Plecháč  Institute of Polish
Language, Polish Academy of Sciences (IJP PAN), Kraków  Tilburg University
Institute of Czech Literature, Czech Academy of Sciences, Prague

(2024)

###### Abstract

All poetic forms come from somewhere. Prosodic templates can be copied for
generations, altered by individuals, imported from foreign traditions, or
fundamentally changed under the pressures of language evolution. Yet these
relationships are notoriously difficult to trace across languages and times.
This paper introduces an unsupervised method for detecting structural
similarities in poems using local sequence alignment. The method relies on
encoding poetic texts as strings of prosodic features using a four-letter
alphabet; these sequences are then aligned to derive a distance measure based
on weighted symbol (mis)matches. Local alignment allows poems to be clustered
according to emergent properties of their underlying prosodic patterns. We
evaluate method performance on a meter recognition tasks against strong
baselines and show its potential for cross-lingual and historical research
using three short case studies: 1) mutations in quantitative meter in
classical Latin, 2) European diffusion of the Renaissance hendecasyllable, and
3) comparative alignment of modern meters in 18–19th century Czech, German and
Russian. We release an implementation of the algorithm as a Python package
with an open license.

###### keywords:

computational poetics, meters, distant, versification studies

##  1 Introduction

All poetic forms come from somewhere, even if we tend to think they have been
with us forever (as we do with familiar things). Indeed, a prosodic template
can be copied in a culture for centuries or millennia; but it can also be
altered by individuals, imported from foreign traditions, adapted from
vernacular folksongs, or fundamentally changed by the pressures of language
evolution. Yet these relationships are notoriously difficult to trace across
languages and times: a lot of evidence is too fragmentary; it depends too much
on written heritage, or modern ethnographic observations. National schools of
metrical studies are also disparate, and rarely agree with each other, while
literary scholarship over the course of the 20th century distanced itself from
metrics, which are now more closely associated with linguistics and phonology.
Research on the history of versification that was built on top of historical
linguistics in the works of Antoine Meillet [23], Roman Jakobson [19] and,
later, Mikhail Gasparov [16], was in decline, but interest is presently
resurgent, following the expansion of computational and quantitative methods
that allow new angles of inquiry [29, 33, 7].

This work is a continuation of the historical line of thinking about meters
and metrical variation, and aims primarily to describe relationships between
the form of poetic texts. We use local sequence alignment to identify regions
of structural similarities in poems; the method relies on encoding poetic
texts as strings of prosodic features using a simple four-letter alphabet that
can be used with any language and is independent of the organizing verse
principle (quantitative, syllabic, accentual). These sequences are then
aligned to derive a distance measure based on weighted symbol (mis)matches.
The resulting relationships do not inherently imply any evolutionary
connection between texts, but the structural similarity is illuminating: it
might identify and localize emerging metrical derivatives, show continuity in
forms across times and traditions, or signal formal descent, which can be
useful in historical studies. While the prosodic and structural features of
verse today are increasingly used for authorship attribution [24, 26], this
work focuses on detecting patterns that transcend individual variation and
shape cultural history on a large scale: the often unseen, but powerful force
of poetic meters.

Poetic meters organize speech into periodic units, causing utterances to be
perceived in systemic relation to each other [14]. The basis for establishing
a metrical frame of reference can be different: alliteration, rhyme, recurrent
prosodic patterns of vowel quality or quantity, or equivalence in duration.
Meters act as rules, with varying levels of fuzziness and variation,
undoubtedly constrained by cognition and language prosody [31, 10, 11]. These
rules, even when scholars and poets formulate them prescriptively, depend on
shifting social conventions and are always tested against the affordances of a
language [34]. This entails variation in the poetic realization of abstract
patterns: extra-metrical stresses, additional syllables and other
irregularities test the boundaries of metrical inertia and audience
expectation. Metrical patterns are reaffirmed by individual usages that copy
forms from the past: taken together, poems reveal governing tendencies that
are not easily overwritten by individual flair. This important cultural aspect
of metrical forms has prompted research into bottom-up approaches to meter
recognition and verse regularity [30, 32, 27] that supports clustering
according to the shared structural principles that emerge from data. This
descriptive approach might be considered a counterpoint to generative
frameworks that abstract metrical variation under the premise of ‘deep
structures’ and language-specific ‘correspondence rules’ [13], drawing away
from the historical and cultural component of poetic meters.

We evaluate our method on the non-trivial task of meter recognition, using a
labelled cross-linguistic corpus, showing the method’s ability to recognize
similarity in metrical organization simultaneously across different languages.
Classification, however, is not our primary goal: we highlight the method’s
potential for historical research in three case studies that span multiple
languages, versification systems and time periods (from classical Latin
quantitative verse to accentual-syllabic Romantic poetry). In the hope that
the methodology will find further use in the community, we release an
accompanying metronome library for Python.111Available at
https://github.com/bnagy/metronome, installable via pip.

##  2 Method

The guiding metaphor for the development of the Metronome tools was the idea
of the genome, and the four DNA molecules that are the basis for all cellular
life. While consisting of only four symbols, genomes vary hugely in length and
are capable of expressing minor variation, such as among human siblings, or
vast differences—the same symbols define a cactus, a dolphin or a hummingbird.
In the same way, the alphabet for the metronome was chosen to be as simple as
possible, allowing the same kinds of analyses that are performed with DNA;
comparison between very disparate samples, grouping related families into
clades or clusters, and expressing or locating changes over time. We use the
following symbols: for strong and weak syllables, ‘S’ and ‘w’, for word
breaks, ‘.’ and for end of line ‘|’ (the specific glyphs are only mnemonics,
any would work). Defining prosody in this way (strong/weak vs
stressed/unstressed) allows the same alphabet to encode quantitative
traditions, like classical Latin, as well as modern accentual traditions.

In bioinformatics, there are several strategies for sequence alignment,
depending on the task. For genomes (or metronomes) that are very different in
length and possibly very dissimilar, the best way to compare sequences is with
a ‘local sequence alignment’. This approach finds the shared region between
two sequences that is the most similar. In contrast, a ‘global sequence
alignment’ is a better choice where the strings are closely related. We apply
the well-known Smith-Waterman algorithm [35] for local sequence alignment,
which uses a flexible alphabet and allows different bonuses and penalties for
symbol (mis)matches. Smith-Waterman is agnostic regarding the sequence
alphabet—it is commonly applied both to nucleic acid sequences (four symbols)
as well as proteins (20 symbols), and produces a match score that depends on a
bonus/penalty matrix. In our work, the best local match score is normalized
using the self-score of the shorter sequence, producing a comparable distance
(or similarity) in [0,1].

The general intuition behind the bonuses and penalties (which determined the
initial values) was based on empirical understanding of poetic meter—for
example matching line lengths are rewarded more than matching syllable
strengths, and mismatches between word breaks and syllables are not penalized
at all (although matching word breaks are rewarded, which allows the algorithm
to group schemes that have customary caesura positions, discussed more in the
Appendix). To optimize the bonus/penalty matrix, we then performed a series of
iterative classification tasks on the cross-language dataset, and visually
inspected the clustering results of several corpus subsets with known metrical
schemes.

The metronome package leverages methods from the BioPython package [6] which
provides highly optimized routines for the sequence alignments. Our code adds
an assortment of sampling strategies, with the core of the package being a
parallel method that produces an $n\times n$ distance matrix from n input
samples (which have been encoded as metronomes). These distance matrices can
be used directly as inputs to create hierarchical clustering / dendrogram
methods (such as hclust in R) or to spatial clustering methods like UMAP
(using a ‘pre-calculated’ metric).

##  3 Evaluation

We evaluate the performance of the algorithm in a non-trivial cross-linguistic
task by looking at clustering performance on samples of individual poems in
twelve meters, and coming from four languages (Czech, German, Russian, and
Classical Latin). We compare this performance to three baselines: simple edit
distance, local alignment with an unweighted substitution matrix, and a
machine-learning approach based on the frequencies of prosodic $n$-grams.

###  3.1 Data and sampling

Our modern language subset (67,940 works) covers the six most common meters:
iambic and trochaic pentameter, tetrameter, and tetrameter/trimeter
(alternating lines of tetrameter and trimeter, sometimes called ‘common meter’
in English). The Czech subset (44416 works), German subset (15172 works), and
Russian subset (8352 works) come from the PoeTree dataset [28]. Metrical
annotation in Czech was performed using machine learning methods, and the
output was manually verified [25]; in German it was produced using a rule
based algorithm [4], and a similar approach was implemented in Russian, and
used the accentual classes of words to infer rhythm.

The Classical Latin subset (2588 works) covers the six most common meters from
that corpus: elegiac couplets, hendecasyllables and stichic hexameter (making
up the bulk of samples and authors), as well as a few senarii, scazons, and
alcaics. The largest language / meter set is Czech iambic pentameter (over
17,000 works) and the smallest is Latin alcaics (38 works, all from Horace).

Since the samples are of such disparate size, we downsampled to just 10
examples of each language / meter combination, yielding 240 random samples per
evaluation run. We then classified samples by meter (twelve meter labels)
which implies a random baseline accuracy of about 8%.

Figure 1: Classification performance of Metronome vs SVM and two baseline
alignment algorithms. Smoothed distribution of accuracy results over 50 random
subsamples, with median scores.

###  3.2 Classification Methods

For the alignment-based methods, which operate in a similar way to Metronome,
we created an $n\times n$ distance matrix, to which we applied a $k$-Nearest
Neighbours (kNN) classifier ($k$=7), with a simple accuracy metric. These
alignment-based methods are as follows. For the Naïve sequence aligner, we
take the best local alignment, but without permitting indels (insertions or
deletions inside the aligned areas). This is similar to a local Hamming
distance, and is a very unforgiving alignment (since word breaks are
included), used only as a baseline. Next we added indels to the Naïve aligner,
making it more like a local Levenshtein distance but with uniform
match/mismatch bonuses and penalties—this is similar to the way local sequence
aligners would work on DNA sequences. Finally, we used the full Metronome
configuration with a variable penalty matrix. The fourth method, and the only
one that was not alignment-based, was a standard machine-learning approach, in
which we applied an SVM classifier to the frequencies of the 500 most-frequent
symbol 9-grams, using an 80/20 train/test split. 9-grams were chosen because
the symbol alphabet is very small, and so it is advantageous to the classifier
to be able to match almost an entire line. Based on testing, the best results
were with 500 most-frequent 9-grams, reduced by SVD to 50 dimensions and then
$z$-scaled. It must be noted here that the SVM classifier is hampered greatly
by the presence of word-breaks. For most meters, word breaks are not part of
the metrical constraints, they are only relevant to some questions of style
and in a few traditions that have mandatory or customary caesurae in certain
places (it is very important in classical Latin and also in modern
alexandrines, for example). In other words, in terms of predicting the name of
the metrical pattern, the word breaks are noise. If the aim were simply to
create an optimal classifier, machine-learning approaches perform at least as
well as Metronome (and are much faster), at the expense of nuance and
interpretability, by simply discarding the word breaks.

###  3.3 Results

The classification tests were run 50 times each, and the full distributions
and medians are reported in Figure 1. As discussed above, the unexpectedly
poor performance of the SVM classifier (usually a solid benchmark) is mainly
due to the presence of word breaks, but it is also clear that the custom
penalty matrix used in Metronome provides a solid increase in performance
across languages and traditions, compared to the standard string aligner.
Classification accuracy, however, is merely a proxy for the real utility of
Metronome as a tool to understand metrical relationships between poems. In the
following section we provide a few small examples of this potential.

Figure 2: Cladogram of a selection of poems by Catullus. _Carmen_ 55, while
still composed in hendecasyllables, is visibly different to the rest of that
clade. Figure 3: A visual comparison of the metronome strings (formatted to
add line breaks) for the beginning of _Carmina_ 55 (variant with collapsed
choriamb) and 41 (standard hendecasyllable).

##  4 Showcases

### Showcase A: Mutations in Classical Latin meter

The first showcase simply highlights the sensitivity of the metronome to minor
metrical variations. During a manual inspection of the dendrogram of the poems
of Catullus, it was observed that one hendecasyllable poem, Catullus Carmen
55, was quite different to all of the others of that meter. This can be seen
in Figure 2—notice how poem 55 branches off from the rest of the
hendecasyllable group much ‘earlier’ (indicating more difference), and yet is
still clearly part of the overall clade. Thinking that this might be an
algorithmic error, we inspected the text, and discovered that 55 is a metrical
oddity. In 55, Catullus often collapses the central double breve of the
choriamb (_uu _) to form a mollossus (___), yielding a line of ten syllables
instead of the standard 11 (hendecasyllable is, literally, ‘11 syllables’).
This variation can be seen in Figure 3, which shows the start of 55, alongside
its metronome transformation, contrasted with 41, a normal hendecasyllable
with a standard choriamb in every line. This license occurs only twice more in
all of Catullus (in 58b), making 55 a very odd animal indeed.222An
approachable description of the meters of Catullus, including a discussion of
this issue, can be found in Peter Green’s bilingual edition [17] p. 32 ff. It
is precisely this kind of ‘loose’ family grouping, based on the meters as they
are written, not as they are defined by textbooks, that highlights the
exciting potential of the metronome as a tool to provide insights from a
‘distant reading’ of poetic meter.

The raw data for this showcase were provided by David Chamberlain, who
maintains an extensive digital selection of fully scanned Greek and Latin
verse under an open license [5]. Minor post-processing was all that was
required to convert it to the Metronome string format.

### Showcase B: European diffusion of a ‘Renaissance meter’

Figure 4: A metronome-based cladogram of various samples of Renaissance meter.
The inset number is the entropy-based variability from the regular metrical
form (see [32]). Shakespeare is the most regular, de La Torre the least.

The second showcase focuses on a typological comparison of different
implementations of Renaissance meter. During the Renaissance, a new metrical
form spread across Europe by replacing previous forms, or by causing
readjustments to the pre-existing templates. Originally it was a regular
10-syllable line that became 11-syllable in Italian because of its fixed
penultimate stress—this hendecasyllable became widely known through Petrarch’s
poetry. Later, the same meter developed into the staple form of modern
English: the iambic pentameter.

In each tradition that adopted the innovation, the meter underwent some
changes and adaptations to the recipient language and versification style
which led to different implementations of the same poetic form [8]. The new
meter probably originated in the Occitan tradition [2, 12, 3] and first
reached Italy from there. The Europe-wide renown of Petrarch and his
endecasillabo quickly spread the new meter to other Romance traditions, like
Spanish, and vernaculars of the Italian peninsula, like Neapolitan, Venetian
and Sicilian. French poetry was already using decasyllabic verse (amour
courtois) [18], but this form was adjusted to the Italian trend of composing
sonnets in endecasillabo [18, 21], before being replaced by the alexandrine
meter. Occitan verse also influenced Catalan and Portuguese poetry, which
already had a pre-existing decasyllabic form.

Later in the Renaissance, the new metrical form spread independently to
English and to Dutch poetry. Subsequently, the Dutch Renaissance meter
influenced German and Frisian poets. All West Germanic adaptations developed
strict iambic meter from the contact with isosyllabic forms. This was the
beginning of modernity in European poetry: the rise of accentual-syllabic
meters everywhere east of France.

The metrical alignment (see Figure 4) of small samples from many Renaissance
traditions clearly distinguishes between the three previously described
‘directions’ of adoption of the form: 1) Italian peninsula and Spanish; 2)
Occitan, Catalan and French; and, finally, 3) Germanic traditions that pivoted
to regular iambic meter (the values on the graph show how ‘free’ the samples
are with regard to the placement of stresses—note how the Germanic samples are
close to 0). While the finer historical relationships between traditions may
remain opaque to this method (which depends on the individual samples, and
technical issues with word boundaries like synalepha),333 Poetry in many
Romance traditions (Spanish, Italian …) makes extensive use of
synalepha—merging two syllables into one—which complicates syllable counting
and word boundary placement. We manually annotated all instances of synalepha,
and treated cross-word syllable elision as one ‘phonetic word’, placing a word
boundary after the contracted sequence. For metronome analysis, all synalephas
were annotated as a single syllable: “strong” if there was a strong syllable
present on either side of the elision, and “weak” when both syllables were
unstressed.  overall it captures the key structural and linguistic
similarities emerging from the implementation of the same form.

The data employed in this showcase were originally compiled by Mirella De
Sisto [8]; the samples were selected from the works of poets who can be
considered representative of the Renaissance tradition of their respective
languages. Manual annotation was performed, in collaboration with experts in
the phonology and metrics of the respective languages.

### Showcase C: Modern expansion: accentual-syllabic verse

During the 18–19th centuries, foot-based accentual-syllabic verse, which
emerged through interaction with isosyllabic Renaissance meter, rapidly spread
from Germany to the North and East [16, 207-209] [20] and was critical for
establishing newly emerging national literatures. Through highly influential
(and highly edited) folksong collections [1, 22], like the Volkslieder (1786),
regular trochaic meters became closely associated with oral traditions
throughout Europe, while iambic verse, from German courtly odes to
Shakespeare’s dramatic genius, provided a basis for high-status secular
literature. This formed a cornerstone for the enduring cultural—and, as a
result, semantic—opposition of iambic and trochaic meters [33].

As with any technology, conquest and power played a major role in metrical
adoption, and it is not a coincidence that ‘fashionable’ accentual-syllabic
verse spread quickly through the territories, communities, and languages of
the Austro-Hungarian and Russian empires, also following the unification of
literary languages and school systems [16, 238-239]. New verse often disrupted
already existing versification, or weakened local alternatives (for example,
purely syllabic verse in Polish, Czech and Ukrainian).

Figure 5: UMAP cluster of 3222 poems in Czech, German, and Russian from the
PoeTree corpus, in the six most common European meters. Metronome distance is
used as the clustering metric.

The last showcase (Fig. 5) demonstrates the possibility of cross-linguistic
alignment of distinct modern poetic forms in Czech, German, and Russian. We
are able to recognize the structural unity of the same meters (colors), detect
the broader divide between trochee and iamb (inner and outer regions on the
plot), yet preserve language-specific variation (point shapes)—the latter
occurs partly because of differences in rhythm, and partly because word
boundaries also encode differences in word lengths. German verse was a source
for both Slavic literatures at different times, and the evidence here of
structural similarity is a step towards better understanding the individual
metrical lineages, and tracing formal connections through global literary
history.

The data presented in Fig. 5 is a subset of the same data we used in our
evaluation (cross-linguistic meter recognition) and comes from annotated
corpora of Czech, German, and Russian poetry (all parts of the PoeTree).

##  5 Discussion

We move now to consider some limitations, and opportunities for further work.
Our approach was informed by meters that depend on isosyllabicity and
recurrent prosodic patterns, so we expect it to struggle with forms that
emerged around other principles. For example, alliteration patterns that
define verse in Old English and Old Norse (dróttkvætt) cannot be traced and
would require a radically different approach to encoding the ‘alphabet’ for
alignment. The same can be said of poetic forms in the same meter that are
distinguished based on the shape of stanza and/or a pattern of rhymes. For
example, both ottava rima and English sonnets can be written in iambic
pentameter, yet represent distinct traditions. In this case, the expansion of
the alphabet would be straightforward: adding a symbol for a stanza boundary
would allow us to trace basic stanza composition (but make the analysis less
broadly applicable). Additionally, the general methodology of Metronome
(alignment of abstracted schemes) might be applied to rhyme patterns instead
of rhythm sequences: one can envision, for example, a reconstruction of the
history and typology of European sonnets based on classic rhyme-to-letter
encoding (abab|abab|cdc|cdc).

Our current approach is also under-investigated in relation to caesurae—fixed-
position word boundaries that can play a significant role in defining meters,
like the French alexandrine (a meter of 12 syllables, where a caesura
consistently bisects the verse, 6+6). To address this, we performed metronome
analysis on simulated variations of the alexandrine (see the Appendix), to see
if caesura-based forms can be distinguished from plain syllabics in our
current approach. Preliminary results are very promising: even under
conservative restrictions, the metronome tends to distinguish pseudo-poems
that are governed by caesurae vs. pseudo-poems of the same line-length that
are not.

Finally, the method depends on the availability of raw scansions of large
collections of poetry. This can be tricky, since scansion itself depends on
existing metrical theory, shared within a community, a consensus that is
rarely maintained across national traditions of scholarship. On one hand,
theory-agnostic scansion is hardly ever possible, but on the other hand, the
presence of theory in automated scansion systems can create self-fulfilling
cycles, where empirical patterns are forced to fit a few anticipated schemes
(for review see [9]), which regularizes and conceals actual variation.

In the authors’ other work we often advocate for the utility of bottom-up
approaches to meter recognition, since any pattern that is repeated enough
times, any strong organizational principle, will always make itself visible;
the very nature of systematic meter will cause it to leave its mark on any
observation or measurement, be it taken on a text’s prosody, morphology, or
syntax [15]. Meters are, essentially, quite simple technologies of rule
iteration: the more you repeat one, the more obvious it becomes, even for such
crude measuring instruments. This makes even a ‘sub-optimal’ automated
scansion, made without expert knowledge of the underlying theory and
tradition, usable and useful.

In conclusion, this study shows how the alignment of metrical strings can be
enlightening in a variety of contexts—from tracing slight deviations in well-
established meters, to recognizing large groups of similar forms across
languages and times. Our proposed method works well with both long and short
sequences, and is applicable in a wide range of poetic traditions (our
showcases feature three major ones: quantitative, syllabic and accentual-
syllabic). The method’s sensitivity to large and small structural differences
in verse organization highlights its potential in both historical and cross-
linguistic comparative research.

##  6 Availability of Data and Code

The preprint repository may be found at https://github.com/bnagy/metronome-
paper. All code and data is available under CC-BY, except where restricted by
upstream licenses. The code repository includes full reproduction data and
code for the evaluation, as well as various supplemental figures and
explanations.

###### Acknowledgements.

Ben Nagy is supported by Polish Academy of Sciences Grant 2020/39/O/HS2/02931.
Artjoms Šeļa was supported by the project “Large-Scale Text Analysis and
Methodological Foundations of Computational Stylistics” (SONATA-BIS
2017/26/E/HS2/01019). Mirella De Sisto was supported by the research project
CIGE/2022/114 funded by the Valencian Government (GE 2023, Conselleria
d’Innovació, Universitats, Ciència i Societat Digital). Petr Plecháč was
supported by the project “European Poetry: Distant Reading (2023–2025)” (Czech
Science Foundation GA23-07727S). Data visualisations are produced with ggplot
[36] and cladograms use ggtree [37].

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## Appendix. Simulated alexandrines

Figure 6: Clustering of simulated pseudo-poems that represent five conditions
of the alexandrine form: three syllabic and four accentual-syllabic

Our analysis of modern meters in this paper did not require the detection of
caesurae—systemic word boundaries in the line (and often a syntactic pause)
that help to define the meter in some traditions, like syllabic French verse
and classical Latin. The standard modern example is the French 12-syllable
alexandrine with a central caesura following a strong syllable. It is
important to know whether the metronome can distinguish forms based on the
presence, or placement, of caesurae. To test this, we generated pseudo-
scansion sequences under four different conditions that imitate, albeit
naïvely, verse prosody.

Each generated poem has a size drawn from a Poisson distribution with
$\lambda=14$, so that the length of all poems in the synthetic tests will
average 14 lines, the length of a sonnet.

  1. 1.

Iambic hexameters with alexandrine caesura (alex): each line is strictly a
sequence of “wSwSwS.wSwSwS”, with an obligatory word boundary after the 6th
syllable. All other word boundaries in the line are determined randomly from a
distribution of probable word-lengths;

  2. 2.

Plain iambic hexameter (Iamb-6): same as above, but no hard-coded caesura,
word boundaries are assigned randomly;

  3. 3.

Classic French alexandrine (alexFrench): each line has a general structure of
“xxxxxS.xxxxxS”, where x is any (w or S) symbol, and word boundaries are
determined freely in each hemistich.

  4. 4.

‘Romantic’ alexandrine (alexRomantic): has a general structure of
xxxS.xxxS.xxxS; this form is associated with Victor Hugo and Romantic
experiments with the classic meter;

  5. 5.

Plain 12-syllable meter (12syl): a hypothetical meter that does not have any
restrictions, except line length.

We generate lines under simple assumptions and do not follow the structure of
any language in particular, nor do we try to plausibly reproduce word-length
distribution and stress placement within words (so e.g. unlikely words with
prosodic structures like “ww” or “SSS” can occur). To distribute word
boundaries within a line, we draw 1-,2-,3-, and 4-syllable words with the
corresponding probability ratios 1:3:1:0.25. This ratio leads to a probability
of 45% that there will be a word boundary after the 6th syllable in a non-
alexandrine line, simply by chance. This is reasonably high, so the metronome
needs to prove its sensitivity in tracing regular differences.

While this setup is unrealistic, it is transparent, and allows us to control
metrical principles independent of linguistic regularities. Figure 6 shows the
resulting clustering of the ‘poems’: it is clear that the forms that employ
caesurae can be distinguished from those that do not. The alignment works
better for ‘syllabic’ alexandrines because, unlike iambics, they have an
additional distinctive feature—two fixed stressed positions versus an
unregulated 12-syllable line.

Further tests confirm the outcome: for 100 runs with 20 poems generated for
each form, we measure the cluster efficiency via Adjusted Rand Index (ARI).
The median ARI for iambic meters is 0.55, indicating a somewhat noisy
recognition of caesura-based forms. The median ARI for syllabics is 0.95—an
almost perfect clustering of the three varieties. Thus, we expect that the
main structuring principle of caesura-based forms should be discoverable, even
in very noisy settings.