From d60e763366bc002d854d8a5377e5bb8cfcebd44b Mon Sep 17 00:00:00 2001 From: Brady Planden Date: Thu, 28 Nov 2024 18:10:02 +0000 Subject: [PATCH] update heuristic parameter convergence plot, add tex preprint format. --- joss/figures/contour_heuristic_0.png | Bin 102716 -> 88531 bytes joss/figures/contour_heuristic_1.png | Bin 73773 -> 73717 bytes joss/figures/contour_heuristic_2.png | Bin 81706 -> 82723 bytes joss/figures/heuristic_parameters.png | Bin 91565 -> 97420 bytes joss/figures/joss/contour_evolution.png | Bin 141039 -> 141039 bytes joss/figures/joss/contour_gradient.png | Bin 94135 -> 94135 bytes joss/figures/joss/contour_heuristic.png | Bin 130726 -> 113656 bytes joss/figures/joss/contour_total.png | Bin 369062 -> 351857 bytes joss/figures/joss/converge.png | Bin 53426 -> 53426 bytes joss/figures/joss/design.png | Bin 132350 -> 132350 bytes joss/figures/joss/optimisers_parameters.png | Bin 248240 -> 254747 bytes joss/figures/joss/sim-landscape.png | Bin 100042 -> 100042 bytes 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optimisation and parameterisation' tags: - - Python + - python - batteries - battery models - parameterisation @@ -192,7 +192,7 @@ As an example, we consider the problem of maximising the gravimetric energy dens We gratefully acknowledge all [contributors](https://github.com/pybop-team/PyBOP?tab=readme-ov-file#contributors-) to this package. This work was supported by the Faraday Institution Multiscale Modelling (MSM) project (ref. FIRG059), UKRI's Horizon Europe Guarantee (ref. 10038031), and EU IntelLiGent project (ref. 101069765). -[//]: # (# Discussion Points) +[//]: # (# Open Discussion Points) [//]: # (EIS numerical identification) diff --git a/joss/paper.pdf b/joss/paper.pdf index 4d1e8dadfe219a5f28d13c775380fb128d4407fc..8294081b99d54ed34e9ad2d48936b292c82827d8 100644 GIT binary patch delta 263438 zcmZ6zWl$Afv_32$-Q9-}X^;}6yQEWEIz$1bk#j)0LqfV60SV~_DN$0orMux!@5bM~ z_doMGjvt&64tuY?)>F&RYZhCn7QO)p0f+%e0muQ515g4`1JDA{126(G1F&kag|A_J zFJAD{iiy!a{KEL)^4`kqEr!R;Zi}IowqDzLOKS)CeM043r7Gg3TT+%*e@bh$t(wT? z7>g0j7Wy>yT6Yo8=p8`_CViL&I&B#JJ9SAUdJIG> zJAF>)y9IU>vMo^l(l_Uu+`aF9q0y|0mw!vn=h`%pb#D?Pr!fUScv;hGr%^uM_JN^|Q=Ua<$yV4fth`OlDfA4lT@z zH{WD7Qs!Sko`2uw{G&L#qt@Wg%`Mi&77@civ8X#LBS6aLzGGfan0 zP`7;gbZ{iXx7R$qhVBryQnfu9lN+@!)?Ev=$b0WOenZ4*`mt_tP>1Q)#(L5mw)P6| z2gl{OOq`78w-fI&e^e+;NL7>2xYjh(cDaimvFJk_r)v9-Zw|-49y=6SWYaChUDv9Z z>`#6VL1r|3{A7s#jANFbKB4Tgbf#+ft>|@J^w!7Tl?nUvA 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\defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1} +\fi +% Use upquote if available, for straight quotes in verbatim environments +\IfFileExists{upquote.sty}{\usepackage{upquote}}{} +\IfFileExists{microtype.sty}{% use microtype if available + \usepackage[]{microtype} + \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts +}{} +\makeatletter +\@ifundefined{KOMAClassName}{% if non-KOMA class + \IfFileExists{parskip.sty}{% + \usepackage{parskip} + }{% else + \setlength{\parindent}{0pt} + \setlength{\parskip}{6pt plus 2pt minus 1pt}} +}{% if KOMA class + \KOMAoptions{parskip=half}} +\makeatother +\usepackage{xcolor} +\usepackage{longtable,booktabs,array} +\usepackage{calc} % for calculating minipage widths +% Correct order of tables after \paragraph or \subparagraph +\usepackage{etoolbox} +\makeatletter +\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{} +\makeatother +% Allow footnotes in longtable head/foot +\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}} +\makesavenoteenv{longtable} +\usepackage{graphicx} +\makeatletter +\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi} +\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi} +\makeatother +% Scale images if necessary, so that they will not overflow the page +% margins by default, and it is still possible to overwrite the defaults +% using explicit options in \includegraphics[width, height, ...]{} +\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio} +% Set default figure placement to htbp +\makeatletter +\def\fps@figure{htbp} +\makeatother +\setlength{\emergencystretch}{3em} % prevent overfull lines +\providecommand{\tightlist}{% + \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} +\setcounter{secnumdepth}{-\maxdimen} % remove section numbering +% definitions for citeproc citations +\NewDocumentCommand\citeproctext{}{} +\NewDocumentCommand\citeproc{mm}{% + \begingroup\def\citeproctext{#2}\cite{#1}\endgroup} +\makeatletter + % allow citations to break across lines + \let\@cite@ofmt\@firstofone + % avoid brackets around text for \cite: + \def\@biblabel#1{} + \def\@cite#1#2{{#1\if@tempswa , #2\fi}} +\makeatother +\newlength{\cslhangindent} +\setlength{\cslhangindent}{1.5em} +\newlength{\csllabelwidth} +\setlength{\csllabelwidth}{3em} +\newenvironment{CSLReferences}[2] % #1 hanging-indent, #2 entry-spacing + {\begin{list}{}{% + \setlength{\itemindent}{0pt} + \setlength{\leftmargin}{0pt} + \setlength{\parsep}{0pt} + % turn on hanging indent if param 1 is 1 + \ifodd #1 + \setlength{\leftmargin}{\cslhangindent} + \setlength{\itemindent}{-1\cslhangindent} + \fi + % set entry spacing + \setlength{\itemsep}{#2\baselineskip}}} + {\end{list}} +\usepackage{calc} +\newcommand{\CSLBlock}[1]{\hfill\break\parbox[t]{\linewidth}{\strut\ignorespaces#1\strut}} +\newcommand{\CSLLeftMargin}[1]{\parbox[t]{\csllabelwidth}{\strut#1\strut}} +\newcommand{\CSLRightInline}[1]{\parbox[t]{\linewidth - \csllabelwidth}{\strut#1\strut}} +\newcommand{\CSLIndent}[1]{\hspace{\cslhangindent}#1} +\ifLuaTeX +\usepackage[bidi=basic]{babel} +\else +\usepackage[bidi=default]{babel} +\fi +\babelprovide[main,import]{american} +% get rid of language-specific shorthands (see #6817): +\let\LanguageShortHands\languageshorthands +\def\languageshorthands#1{} +\ifLuaTeX + \usepackage{selnolig} % disable illegal ligatures +\fi +\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}} +\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available +\urlstyle{same} % disable monospaced font for URLs +\hypersetup{ + pdftitle={PyBOP: A Python package for battery model optimisation and +parameterisation}, + pdfauthor={Brady Planden, Nicola E. Courtier, Martin Robinson, Ferran +Brosa Planella, David A. Howey}, + pdflang={en-US}, + colorlinks=true, + linkcolor={Maroon}, + filecolor={Maroon}, + citecolor={Blue}, + urlcolor={Blue}, + pdfcreator={LaTeX via pandoc}} + +\title{PyBOP: A Python package for battery model optimisation and +parameterisation} + +\definecolor{c53baa1}{RGB}{83,186,161} +\definecolor{c202826}{RGB}{32,40,38} +\def \rorglobalscale {0.1} +\newcommand{\rorlogo}{% +\begin{tikzpicture}[y=1cm, x=1cm, yscale=\rorglobalscale,xscale=\rorglobalscale, every node/.append style={scale=\rorglobalscale}, inner sep=0pt, outer sep=0pt] + \begin{scope}[even odd rule,line join=round,miter limit=2.0,shift={(0,3.0889892500000005)},xscale=0.9943,yscale=0.9894] + \path[fill=white,even odd rule,line join=round,miter limit=2.0] (0.0, 3.1221) rectangle (4.364, -0.0336); + \end{scope} + \begin{scope}[even odd rule,line join=round,miter limit=2.0,shift={(-0.025, 0.0216)}] + \path[fill=c53baa1,nonzero rule,line join=round,miter limit=2.0] (1.8164, 3.012) -- (1.4954, 2.5204) -- (1.1742, 3.012) -- (1.8164, 3.012) -- cycle; + \path[fill=c53baa1,nonzero rule,line join=round,miter limit=2.0] (3.1594, 3.012) -- (2.8385, 2.5204) -- (2.5172, 3.012) -- (3.1594, 3.012) -- cycle; + \path[fill=c53baa1,nonzero rule,line join=round,miter limit=2.0] (1.1742, 0.0669) -- (1.4954, 0.5588) -- (1.8164, 0.0669) -- (1.1742, 0.0669) -- cycle; + \path[fill=c53baa1,nonzero rule,line join=round,miter limit=2.0] (2.5172, 0.0669) -- (2.8385, 0.5588) -- (3.1594, 0.0669) -- (2.5172, 0.0669) -- cycle; + \path[fill=c202826,nonzero rule,line join=round,miter limit=2.0] (3.8505, 1.4364).. controls (3.9643, 1.4576) and (4.0508, 1.5081) .. 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(3.8015, 1.7777) -- cycle; + \end{scope} +\end{tikzpicture} +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Authors and Affiliations + +\usepackage[affil-it]{authblk} +\usepackage{orcidlink} +%% \renewcommand\Authsep{, } +\setlength{\affilsep}{1em} +\author[1% + % + \ensuremath\mathparagraph]{Brady Planden% + \,\orcidlink{0000-0002-1082-9125}\,% + } +\author[1,2% + % + ]{Nicola E. Courtier% + \,\orcidlink{0000-0002-5714-1096}\,% + } +\author[3% + % + ]{Martin Robinson% + \,\orcidlink{0000-0002-1572-6782}\,% + } +\author[2,4% + % + ]{Ferran Brosa Planella% + \,\orcidlink{0000-0001-6363-2812}\,% + } +\author[1,2% + % + ]{David A. Howey% + \,\orcidlink{0000-0002-0620-3955}\,% + } + +\affil[1]{Department of Engineering Science, University of Oxford, +Oxford, UK% + } +\affil[2]{The Faraday Institution, Harwell Campus, Didcot, UK% + } +\affil[3]{Research Software Engineering Group, University of Oxford, +Oxford, UK% + } +\affil[4]{Mathematics Institute, University of Warwick, Coventry, UK% + } +\affil[$\mathparagraph$]{Corresponding author: % + % +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\date{28 November 2024} + +\begin{document} +\maketitle + +\section{Summary}\label{summary} + +The Python Battery Optimisation and Parameterisation (\texttt{PyBOP}) +package provides a set of methods for the parameterisation and +optimisation of battery models, offering both Bayesian and frequentist +approaches with example workflows to assist the user. \texttt{PyBOP} has +been developed to enable parameter identification of various battery +models, including the electrochemical and equivalent circuit models +provided by the popular open-source \texttt{PyBaMM} package +(\citeproc{ref-Sulzer:2021}{Sulzer et al., 2021}). + +Similarly, \texttt{PyBOP} can be used for parameter design optimisation +under user-defined operating conditions across a variety of model +structures. \texttt{PyBOP} allows battery model parameterisation using a +range of methods with diagnostics on the performance and convergence of +the identified or optimised parameters. The identified parameters can be +used for prediction, on-line control and design optimisation, all of +which support improved battery utilisation and development. + +\section{Statement of need}\label{statement-of-need} + +\texttt{PyBOP} is a Python package designed to provide a user-friendly, +object-oriented interface for optimising battery models. \texttt{PyBOP} +leverages the open-source \texttt{PyBaMM} package +(\citeproc{ref-Sulzer:2021}{Sulzer et al., 2021}) to formulate and solve +of these battery models. \texttt{PyBOP} is intended to serve a broad +audience of students, engineers, and researchers in both academia and +the battery industry by enabling the use of predictive battery models +where previously this was not possible. \texttt{PyBOP} emphasises clear +and informative diagnostics and workflows for users of varying +expertise, by providing advanced optimisation and sampling algorithms. +These methods are provided through interfaces to \texttt{PINTS} +(\citeproc{ref-Clerx:2019}{Clerx et al., 2019}), \texttt{SciPy} +(\citeproc{ref-SciPy:2020}{Virtanen et al., 2020}), in addition to the +PyBOP's own algorithms such as Adaptive Moment Estimation with Weight +Decay (AdamW), and Cuckoo search. + +\texttt{PyBOP} supports the Battery Parameter eXchange (BPX) standard +(\citeproc{ref-BPX:2023}{Korotkin et al., 2023}) for sharing battery +parameter sets. As these parameter sets are costly to obtain due to: the +equipment and time required for characterisation experiments, the need +for battery domain knowledge, and the computational cost of parameter +estimation. \texttt{PyBOP} reduces these costs by providing fast +computational estimation with parameter set interoperability. + +This package complements other lithium-ion battery modelling packages +built around \texttt{PyBaMM}, such as \texttt{liionpack} for battery +pack simulation (\citeproc{ref-Tranter2022}{Tranter et al., 2022}) and +\texttt{pybamm-eis} for numerical impedance spectroscopy, as the +identified parameters from \texttt{PyBOP} are easily exportable. + +\section{Architecture}\label{architecture} + +\texttt{PyBOP} has a layered data structure designed to compute and +process the forward model predictions and to package the necessary +information for the optimisation and sampling algorithms. The forward +model is solved using the popular battery modelling package, +\texttt{PyBaMM}, with construction, parameterisation, and discretisation +managed by PyBOP's model interface to PyBaMM. This approach provides a +robust object construction process with a consistent interface between +the models and optimisers. The statistical methods and optimisation +algorithms are then constructed to interface cleanly with the forward +model predictions. Furthermore, identifiability metrics are provided +alongside the estimated parameters through Hessian approximation of the +cost functions in the frequentist workflows and posterior moments in the +Bayesian workflows. + +\begin{figure} +\centering +\includegraphics[width=0.8\textwidth,height=\textheight]{figures/PyBOP-high-level.pdf} +\caption{PyBOP's interface to supporting funding agencies, alongside a +visualisation of the general workflow for parameterisation and +optimisation \label{fig:high-level}} +\end{figure} + +\texttt{PyBOP} formulates the inference process into four key classes, +namely the model, the problem, the cost, and the optimiser/sampler, as +shown in \autoref{fig:classes}. Each of these objects represents a base +class with child classes constructing specialised functionality for +inference or optimisation workflows. The model class constructs a +\texttt{PyBaMM} forward model for a given set of model equations +provided by \texttt{PyBaMM}, initial conditions, spatial discretisation, +and numerical solver. By composing \texttt{PyBaMM} directly into +\texttt{PyBOP}, specialised models can be constructed alongside the +standard models, which can be modified, and optimally constructed for +the inference tasks. One such example is spatial rediscretisation, which +is performed when geometric parameters are optimised. In this situation, +\texttt{PyBOP} minimally rediscretises the \texttt{PyBaMM} model while +maintaining the problem, cost, and optimiser objects, providing improved +performance benefits to users. Alongside construction of the forward +model, \texttt{PyBOP}`s model class provides methods for obtaining +sensitivities from the prediction, enabling gradient-based optimisation +algorithms. This prediction, along with it's corresponding +sensitivities, is provided to the problem class for processing and +exception control. A standardised data structure is then provided to the +cost classes, providing a distance, design, or likelihood-based metric +for optimisation. For deterministic optimisation, the optimisers +minimise the corresponding cost function or the negative log-likelihood +if a likelihood class is provided. Bayesian inference is provided by +Monte Carlo sampling classes, which accept the LogPosterior class and +sample from it using Pints' based Monte Carlo algorithms at the time of +submission. In the typical workflow, the classes in +\autoref{fig:classes} are constructed in sequence. + +In addition to the core architecture, \texttt{PyBOP} provides several +specialised inference and optimisation processes. One such instance is +numerical electrochemical impedance spectroscopy predictions by +discretising the forward model into sparse mass matrix form with +accompanying auto-differentiated Jacobian. These objects are then +translated into the frequency domain with a linear solution used to +compute the battery model impedance. In this situation, the forward +models are constructed within the spatial rediscretisation workflow, +allowing for geometric parameter inference from EIS forward model +predictions. Furthermore, \texttt{PyBOP} builds on the JAX +(\citeproc{ref-jax:2018}{Bradbury et al., 2018}) numerical solvers +provided by \texttt{PyBaMM} by providing JAX-based cost functions for +automatic forward model differentiation with respect to the parameters. +This functionality provides a performance improvement alongside an +interface to JAX-based inference packages, such as Numpyro +(\citeproc{ref-numpyro:2019}{Phan et al., 2019}), BlackJAX +(\citeproc{ref-blackjax:2024}{Cabezas et al., 2024}), and Optax +(\citeproc{ref-optax:2020}{DeepMind et al., 2020}). + +\begin{figure} +\centering +\includegraphics[width=0.8\textwidth,height=\textheight]{figures/PyBOP_components.drawio.png} +\caption{The core \texttt{PyBOP} architecture, showcasing the base class +interfaces. Each class provide direct mapping to a classical step in the +optimisation workflow. \label{fig:classes}} +\end{figure} + +The currently implemented subclasses for the model, problem, and cost +classes are listed in \autoref{tab:subclasses}. The cost functions are +grouped by problem type, while the model and optimiser classes can be +selected in combination with any problem-cost pair. + +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.4458}} + >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.2048}} + >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3494}}@{}} +\caption{List of available model, problem and cost (or likelihood) +classes. \label{tab:subclasses}}\tabularnewline +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Battery Models +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Problem Types +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Cost / Likelihood Functions +\end{minipage} \\ +\midrule\noalign{} +\endfirsthead +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Battery Models +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Problem Types +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Cost / Likelihood Functions +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Single particle model (SPM) & Fitting problem & Sum squared error \\ +SPM with electrolyte (SPMe) & & Root mean squared error \\ +Doyle-Fuller-Newman (DFN) & & Minkowski \\ +Many particle model (MPM) & & Sum of power \\ +Multi-species multi-reaction (MSMR) & & Gaussian log likelihood \\ +Weppner Huggins & & Maximum a posteriori \\ +Equivalent circuit model (ECM) & Observer & Unscented Kalman filter \\ +& Design problem & Gravimetric energy density \\ +& & Volumetric energy density \\ +\end{longtable} + +Similarly, the current algorithms available for optimisation tasks are +presented in \autoref{tab:optimisers}. From now on, the point-based +parameterisation and design optimisation tasks will simply be referred +to as optimisation tasks. This simplification can be justified by +examining \autoref{eqn:parameterisation} and \autoref{eqn:design} and +confirming that deterministic parameterisation can be viewed as an +optimisation task to minimise a distance-based cost function. + +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.4062}} + >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3333}} + >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.2604}}@{}} +\caption{The currently supported optimisation algorithms classified by +candidate solution type, including gradient information. (*) Scipy +minimize has gradient and non-gradient methods. +\label{tab:optimisers}}\tabularnewline +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Gradient-based +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Evolutionary +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +(Meta)heuristic +\end{minipage} \\ +\midrule\noalign{} +\endfirsthead +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Gradient-based +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Evolutionary +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +(Meta)heuristic +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Weight decayed adaptive moment estimation (AdamW) & Covariance matrix +adaptation (CMA-ES) & Particle swarm (PSO) \\ +Improved resilient backpropagation (iRProp-) & Exponential natural +(xNES) & Nelder-Mead \\ +Gradient descent & Separable natural (sNES) & Cuckoo search \\ +SciPy minimize (*) & SciPy differential evolution & \\ +& & \\ +\end{longtable} + +As discussed above, \texttt{PyBOP} provides Bayesian inference methods +such as maximum a posteriori (MAP) alongside the point-based methods in +\autoref{tab:subclasses}; however, for a full Bayesian framework, Monte +Carlo sampling is implemented within \texttt{PyBOP}. These methods +construct a posterior distribution on the inference parameters, which +can be used for uncertainty and practical identifiability. The +individual sampler classes are currently composed within \texttt{PyBOP} +from the \texttt{PINTS} library, with a base sampling class implemented +for interoperability and direct integration with the \texttt{PyBOP} +model, problem, and likelihood classes. The currently supported samplers +are listed in \autoref{tab:samplers}. + +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.1810}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2155}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.1379}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2069}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2586}}@{}} +\caption{Sampling methods supported by PyBOP, classified according to +the proposed method. \label{tab:samplers}}\tabularnewline +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Gradient-based +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Adaptive +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Slicing +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Evolutionary +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Other +\end{minipage} \\ +\midrule\noalign{} +\endfirsthead +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +Gradient-based +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Adaptive +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Slicing +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Evolutionary +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +Other +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +Monomial Gamma & Delayed Rejection Adaptive & Rank Shrinking & +Differential Evolution & Metropolis Random Walk \\ +No-U-Turn & Haario Bardenet & Doubling & & Emcee Hammer \\ +Hamiltonian & Haario & Stepout & & Metropolis Adjusted Langevin \\ +Relativistic & Rao Blackwell & & & \\ +\end{longtable} + +\section{Background}\label{background} + +\subsection{Battery models}\label{battery-models} + +In general, battery models can be written in the form of a +differential-algebraic system of equations: \begin{equation} +\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t} = f(t,\mathbf{x},\mathbf{y},\mathbf{u}(t),\mathbf{\theta}), +\label{dynamics} +\end{equation} \begin{equation} +\mathbf{y}(t) = g(t,\mathbf{x},\mathbf{y},\mathbf{u}(t),\mathbf{\theta}), +\label{output} +\end{equation} with initial conditions \begin{equation} +\mathbf{x}(0) = \mathbf{x}_0(\mathbf{\theta}). +\label{initial_conditions} +\end{equation} + +Here, \(t\) is time, \(\mathbf{x}(t)\) are the (spatially discretised) +states, \(\mathbf{y}(t)\) are the outputs (e.g.~the terminal voltage), +\(\mathbf{u}(t)\) are the inputs (e.g.~the applied current) and +\(\mathbf{\theta}\) are the unknown parameters. + +Common battery models include various types of equivalent circuit models +(e.g.~the Thévenin model), the Doyle--Fuller--Newman (DFN) model +(\citeproc{ref-Doyle:1993}{Doyle et al., 1993}; +\citeproc{ref-Fuller:1994}{Fuller et al., 1994}) based on porous +electrode theory and its reduced-order variants including the single +particle model (SPM) (\citeproc{ref-Planella:2022}{Planella et al., +2022}), and the multi-species, multi-reaction (MSMR) model +(\citeproc{ref-Verbrugge:2017}{Verbrugge et al., 2017}). Simplified +models that retain acceptable predictive capabilities at a lower +computational cost are widely used, for example in battery management +systems, while physics-based models are required to understand the +impact of physical parameters on battery performance. This separation of +complexity traditionally results in multiple parameterisations for a +single battery type, depending on the model structure. + +\section{Examples}\label{examples} + +\subsection{Parameterisation}\label{parameterisation} + +The parameterisation of battery models is challenging due to the large +number of parameters that need to be identified compared to the +measurable outputs (\citeproc{ref-Andersson:2022}{Andersson et al., +2022}; \citeproc{ref-Miguel:2021}{Miguel et al., 2021}; +\citeproc{ref-Wang:2022}{Wang et al., 2022}). A complete +parameterisation often requires a stepwise identification of smaller +sets of parameters from a variety of different data sets +(\citeproc{ref-Chen:2020}{Chen et al., 2020}; +\citeproc{ref-Chu:2019}{Chu et al., 2019}; \citeproc{ref-Kirk:2022}{Kirk +et al., 2023}) and excitations. + +A generic data fitting optimisation problem may be formulated as: +\begin{equation} +\min_{\mathbf{\theta}} ~ \mathcal{L}_{(\mathbf{y}_i)}(\mathbf{\theta}) ~~~ +\textrm{subject to equations (\ref{dynamics})\textrm{-}(\ref{initial_conditions})} +\label{eqn:parameterisation} +\end{equation} + +where \(\mathcal{L} : \mathbf{\theta} \mapsto [0,\infty)\) is a cost (or +likelihood) function that quantifies the agreement between the model and +a sequence of observations \((\mathbf{y}_i)\) measured at times \(t_i\). +For gradient-based optimisers, the Jacobian of the cost function with +respect to the unknown parameters, +\((\frac{\partial \mathcal{L}}{\partial \theta})\) is computed for step +size and directional information. + +Next, we demonstrate the fitting of synthetic data where the system +parameters are known. In this example problem, we use \texttt{PyBaMM}'s +implementation of the single particle model (SPM) with an added contact +resistance submodel. We assume that the battery model is already +parameterised except for two dynamic parameters, namely the lithium +diffusivity of the negative electrode active material particle (denoted +``negative particle diffusivity'') and the contact resistance. We +generate synthetic data from a one-hour discharge from 100\% state of +charge, to 0\% (denoted as 1C rate), followed by 30 minutes of +relaxation. This data is then corrupted with zero mean Gaussian noise of +amplitude 2mV, shown by the dots in +\autoref{fig:inference-time-landscape} (left). The initial states are +assumed known, although such an assumption is not generally necessary. +The underlying cost landscape explored by the optimiser is shown in +\autoref{fig:inference-time-landscape} (right). + +\begin{figure} +\centering +\includegraphics[width=1\textwidth,height=\textheight]{figures/joss/sim-landscape.png} +\caption{The cost landscape for the parameterisation problem with a root +mean squared error cost function. \label{fig:inference-time-landscape}} +\end{figure} + +As gradient information is available for this problem, the choice of +distance-based cost function and optimiser is not constrained. Due to +the different magnitudes of the two parameters, we apply the logarithmic +parameter transformation offered by \texttt{PyBOP}. This transforms the +optimisers search space of the optimiser to allow for a common step size +between the parameters, which is generally is not required, but improves +convergence in this problem. As a demonstration of the parameterisation +capabilities of \texttt{PyBOP}, \autoref{fig:convergence-min-max} (left) +shows the rate of convergence for each of the distance-minimising cost +functions, while \autoref{fig:convergence-min-max} (right) shows +analogous results for maximising a likelihood. The optimisation is +performed with SciPy Minimize using the gradient-based L-BFGS-B method. + +\begin{figure} +\centering +\includegraphics[width=1\textwidth,height=\textheight]{figures/joss/converge.png} +\caption{Convergence in the likelihood functions obtained using various +likelihood functions and the L-BFGS-B algorithm. +\label{fig:convergence-min-max}} +\end{figure} + +Next, the performance of the various optimisation algorithms is +presented by category: gradient-based in +\autoref{fig:optimiser-inference} (left), evolutionary strategies in +\autoref{fig:optimiser-inference} (middle) and (meta)heuristics in +\autoref{fig:optimiser-inference} (right) for a mean squared error cost +function. Note that the performance of the optimiser depends on the cost +environment, prior information and corresponding hyperparameters for +each specific problem. + +\begin{figure} +\centering +\includegraphics[width=1\textwidth,height=\textheight]{figures/joss/optimisers_parameters.png} +\caption{Convergence in the parameter values obtained for the various +optimisation algorithms provided by \texttt{PyBOP}. +\label{fig:optimiser-inference}} +\end{figure} + +\begin{figure} +\centering +\includegraphics[width=1\textwidth,height=\textheight]{figures/joss/contour_total.png} +\caption{Cost landscape contour plot with corresponding optimisation +traces. The top row represents the gradient-based optimisers, the middle +row is the evolution-based, and the bottom row is the (meta)heuristics. +The order from left to right corresponds to the entries in +\autoref{tab:optimisers}. \label{fig:optimiser-inference}} +\end{figure} + +This parameterisation task can also be approached from a Bayesian +perspective, which we will present below using PyBOP's sampler methods. +The optimisation equation presented in equation +\autoref{eqn:parameterisation} does not represent the Bayesian parameter +identification task, and as such we introduce the Bayes theorem as, + +\begin{equation} +P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)} +\label{eqn:bayes_theorem} +\end{equation} + +where, \(P(\theta|D)\) is the posterior and represents the probability +density function of the parameter. \(P(D|\theta)\) is the likelihood +function and assesses the parameter values alongside a noise model. +\(P(\theta)\) encapsulates the prior knowledge about the parameters, and +finally \(P(D)\) is the model evidence and acts as a normalising +constant so that the final posterior is a correctly scaled density +function. Our goal in parameter inference is to identify the parameter +values with the highest probability, which can be represented as a +point-based metric or as the posterior distribution, which provides +additional information about the uncertainty of the identified +parameters. Monte Carlo sampling methods are available to obtain this +posterior distribution. These methods sample from the posterior using a +variety of methods, including gradient-based methods such as No-U-Turn +(\citeproc{ref-NUTS:2011}{Hoffman \& Gelman, 2011}) and Hamiltonian +(\citeproc{ref-Hamiltonian:2011}{Brooks et al., 2011}), as well as +heuristic methods such as Differential Evolution +(\citeproc{ref-DiffEvolution:2006}{Braak, 2006}), and finally +conventional methods based on random sampling with rejection criteria +(\citeproc{ref-metropolis:1953}{Metropolis et al., 1953}). PyBOP offers +a sampling class that provides an interface to these samplers, which are +supported by the Probabilistic Inference of Noise Time-Series (PINTS) +package. \autoref{fig:posteriors} below shows the sampled posterior for +the synthetic workflow described above, using an adaptive +covariance-based sampler, Haario Bardenet +(\citeproc{ref-Haario:2001}{Haario et al., 2001}). + +\begin{figure} +\centering +\includegraphics[width=1\textwidth,height=\textheight]{figures/joss/posteriors.png} +\caption{Posterior distributions for model parameters alongside +identified noise on the observations. Shaded area denotes confidence +bounds for each parameter. \label{fig:posteriors}} +\end{figure} + +\subsection{Design optimisation}\label{design-optimisation} + +Design optimisation is supported within \texttt{PyBOP} to guide future +battery design development by identifying parameter sensitivities that +can unlock improvements in battery performance. This problem can be +viewed in a similar way to the parameterisation workflows described +above, but with the aim of maximising a distance metric rather than +minimising it. In the case of design optimisation to maximise +gravimetric energy density, \texttt{PyBOP} minimises the negative of the +cost function, where the cost metric is no longer a distance between two +time series vectors, but the integrated energy of the vector normalised +by with the corresponding cell mass. This is typically quantified for +operating conditions such as a 1C discharge, at a given temperature. + +Design optimisation can be written in the form of a constrained +optimisation problem as: \begin{equation} +\min_{\mathbf{\theta} \in \Omega} ~ \mathcal{L}(\mathbf{\theta}) ~~~ +\textrm{subject to equations (\ref{dynamics})\textrm{-}(\ref{initial_conditions})} +\label{eqn:design} +\end{equation} + +where \(\mathcal{L} : \mathbf{\theta} \mapsto [0,\infty)\) is a cost +function that quantifies the desirability of the design and \(\Omega\) +is the set of allowable parameter values. + +As an example, we consider the problem of maximising the gravimetric +energy density subject to constraints on two of the geometric electrode +parameters (\citeproc{ref-Couto:2023}{Couto et al., 2023}). For this +example, we use the\texttt{PyBaMM} implementation of the single particle +model with electrolyte (SPMe) to investigate the effect of the positive +electrode thickness and the active material volume fraction on the +target cost. Since the active material volume fraction is related to the +electrode porosity, the porosity is defined with a driven constraint +from the volume fraction. In this problem, we estimate the 1C rate from +the theoretical capacity for each iteration of the design. For this +example, we employ the Particle Swarm Optimisation (PSO) algorithm. + +\begin{figure} +\centering +\includegraphics[width=1\textwidth,height=\textheight]{figures/joss/design.png} +\caption{The gravimetric landscape alongside the corresponding initial +and optimised voltage profiles for a 1C discharge. +\label{fig:design_gravimetric}} +\end{figure} + +\autoref{fig:design_gravimetric} (left) shows the optimiser's search +over the gravimetric energy density parameter space. The predicted +improvement in the discharge profile between the initial and optimised +parameter values (right) for their respective applied 1C current. + +\section{Acknowledgements}\label{acknowledgements} + +We gratefully acknowledge all +\href{https://github.com/pybop-team/PyBOP?tab=readme-ov-file\#contributors-}{contributors} +to this package. This work was supported by the Faraday Institution +Multiscale Modelling (MSM) project (ref. FIRG059), UKRI's Horizon Europe +Guarantee (ref. 10038031), and EU IntelLiGent project (ref. 101069765). + +\section*{References}\label{references} +\addcontentsline{toc}{section}{References} + +\phantomsection\label{refs} +\begin{CSLReferences}{1}{0} +\bibitem[\citeproctext]{ref-Andersson:2022} +Andersson, M., Streb, M., Ko, J. Y., Löfqvist Klass, V., Klett, M., +Ekström, H., Johansson, M., \& Lindbergh, G. (2022). 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Review of parameterisation and a +novel database {(LiionDB)} for continuum {Li-ion} battery models. +\emph{Progress in Energy}, \emph{4}(3), 032004. +\url{https://doi.org/10.1088/2516-1083/ac692c} + +\end{CSLReferences} + +\end{document} diff --git a/joss/param_plots.py b/joss/param_plots.py index 8c41d422..ac4f5372 100644 --- a/joss/param_plots.py +++ b/joss/param_plots.py @@ -18,9 +18,9 @@ create_plot["landscape"] = False create_plot["minimising"] = False create_plot["maximising"] = False -create_plot["gradient"] = True +create_plot["gradient"] = False create_plot["evolution"] = False -create_plot["heuristic"] = False +create_plot["heuristic"] = True create_plot["posteriors"] = False @@ -441,10 +441,10 @@ optim = optimiser( cost, verbose=True, - sigma0=0.02, - max_iterations=500, - max_unchanged_iterations=25, - max_evaluations=300, + sigma0=0.05, + max_iterations=150, + max_unchanged_iterations=150, + max_evaluations=150, **kwargs, ) @@ -474,6 +474,7 @@ parameter_fig.data = [] parameter_fig.add_traces(parameter_traces) parameter_fig = plotly.subplots.make_subplots(figure=parameter_fig, rows=2, cols=1) + parameter_fig.update_layout(yaxis1=dict(range=[1e-14, 10e-14])) parameter_fig.show() parameter_fig.write_image("joss/figures/heuristic_parameters.png")

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