Case Study in Mathematical Modelling

This is a group project from the MSc in MMSC, focused on battery modelling.

Here is a screenshot to get a taste:

A Journal of the Journey

Week 1:

• introductory meeting and presentation of available projects

Week 2:

• Tuesday: first meeting with Robert
• Write code to solve the ODE with a current profile as input
• Derive equations (fitting or approximate functions) for all variables to be included in the code later
• write the exact solution of the equation by hand (check if code solution is similar)
• find out how usually input is given and find a few input profiles that are interesting to examine

Status updates:

• Zella/Nick MATLAB code takes in current input and outputs Voltage over time for ECM. Zella uses polynomial fit to find Voc as a function of state of charge [U(s)]
• Need to model efficiency
• Peter looking at PyBamm--does everything. Could we get our model to do the same thing? Compare our results to PyBamm's results? Also--LTSpice circuits might be helpful for checking results/simulating behavior
• Aoibheann has looked into analytic solution--not too complicated to derive.
• Jad has looked into temperature, charge, and state of health dependence of R0, R1, and C1. 2021 paper--Jad looked at extending parameter fits from linear models to higher order models

Q's for Robert:

• How do we extend the model to incorporate varying parameters such as R0 R1 C1 etc.
• Are linear fits sufficient for parameter finding in 2021 paper? What about quadratic?
• PyBamm comparison?

Week 3:

Monday: meeting cancelled.

Wednesday meeting with Robert:

• How to find R, C from pulse data--window V vs. t data by state of charge in order to create plots for R vs. state of charge and C vs. state of charge. See picture of board. HPPC or GITT data?
• Make comparisons to PyBamm? Need to make sure that are comparing same battery parameters
• Robert's ideas:
  • Calendar aging --> Q (capacity) now a function of time inside integral for state of charge. Cycle aging --> Q a function of current (high currents mean that Q is lower?
• Can we use more than linear fits for parameter search? Compare to 2021 paper if we do quadratic fitting?
• Battery Genome Project? For good battery data
• Google "battery data and where to find it" paper from edinburgh with links to repositories with data. Ideally, we have small pulse tests at various temperatures and states of health.
• Possible end goal: if I throw different usage profiles at the model can I make recommendations about how the battery should be used? For example, if I have a weekly use profile for a car, when should I charge it? What can I optimize over?
• Can possibly break resistor up into cycle resistor and calendar resistor--one a function of time (long times) and other a function of "state of health"
• Robert's recommendation: use model to get different fits for various states of charge (R0, R1, C1 at different states of charge). Want a function that takes in parameters and outputs error difference between model and experimental values of Voltage for a specific current profile.

Thursday:

• Trying to figure out a short-term, medium-term and long-term goals.
• Much going on in general, 2 hour meeting.
• Find "surfaces" R0(s, h, T), R1(s, h, T) and C1(s, h, T) from data.
• Find optimal charging of electric car on a graph?
• Estimate SOC, SOH from V(t), I(t), etc.
• Game?

Week 4 Feb 7:

Meeting with Robert

• Parameter vs SOC curves are nice.
• Hinch Perturbation Methods for multiple timescale analysis -- maybe can say something about how we get a speedup for our computations?
• Heating of the battery may be an interesting question
• Looking at optimization on a graph could be an interesting avenue--where to charge, how fast, etc. (performance)
• Can also look at questions about battery death (from modeling capacity)

Big Recommendations:

• Should also look at generating different controls (i.e. what if we have voltage instead of current, or have power instead of current/voltage, and should be able to generate each of these)
• Look at continuous time version and solve in matlab
• Look at modeling capacity (linearly as a function of time and cycle use?)

Week 5:

Meeting on Tuesday Feb 14:

• Need to start thinking about final product
• Find some question or questions that we can answer in our presentation
• Optimization: model gives V, s, h = f(I). Problem: Find I(t) such that some target (power, integral of current, etc.) is maximized/minimized under constraint on state of health or something
• Tips: search "functional optimization" or "ode constrained optimization". Run model in simplest case (all parameters constant), come up with very simple optimization question that has obvious answer to make sure optimization routine works, then add back in complexity from parameters

Repository Structure and Setup:

To use and sustain a Python virtual environment, install poetry, which works with the pyproject.toml file. After installing poetry (and subsequently after pulling, each time), run

poetry install

in the project folder. To install PyBamm as well (which has 1/epsilon number of dependencies), run

poetry install --with=pybamm

instead or additionally. This sadly requires Python 3.8 based on a pybamm restriction. Without pybamm, 3.11 should be fine too.

Having all dependencies installed, the main interface may be launched up by

python3 main.py

which starts a graphical user interface (with looks depending on your operating system).

The relevant code structure is:

• The folder simulator/ is responsible for the (numerical) simulation itself, which may be invoked without any interface at all.
  • simulation.py features the Simulation class with an iterate() method that represents a numerical integration step in time by an amount of dt.
  • batgraph.py exports a class BatGraph that represents a graph (a tuple of sets of edges and vertices) that the car will drive on.
  • batmobile.py contains the BatMobile class that represents our battery mobile i.e. car. Much of the simulation takes place in this file!
  • battery.py is the central file for our battery modelling project, which exports a Battery class, also featuring an iterate() method. Most of the battery simulation takes place in this file!
  • optimiser.py takes care of the optimisation part of the routing problem. It implements the Metropolis-Hastings (Monte-Carlo Markov Chain) method and defines the graph perturbations.
• The interface code is contained within the interface/ folder.
  • mainwindow.py defines the general layout and actions in the user interface.
  • batmap.py exports the central widget that renders / animates the BatMobile car on the BatGraph.
  • graphs.py handles the connection of the interface and (live) plots. The plots are handled by matplotlib and are very intuitive to use, further almost all commands are the same as they are in MatLab.
• main.py creates a MainWindow and runs the simulator GUI.

Data Used

• Dataset used - Z (this has discharge pulses in case we don't find anything else that's more suitable)