Path-based splitting methods for SDEs and machine learning for battery lifetime prognostics

Abstract

In the first half of this Thesis, we present the numerical analysis of splitting methods for stochastic differential equations (SDEs) using a novel path-based approach. The application of splitting methods to SDEs can be viewed as replacing the driving Brownian-time path with a piecewise linear path, producing a ‘controlled-differential-equation’ (CDE). By Taylor expansion of the SDE and resulting CDE, we show that the global strong and weak errors of splitting schemes can be obtained by comparison of the iterated integrals in each. Matching all integrals up to order p+1 in expectation will produce a weak order p+0.5 scheme, and in addition matching the integrals up to order p+0.5 strongly will produce a strong order p scheme. In addition, we present new splitting methods utilising the ‘space-time’ L´evy area of Brownian motion which obtain global strong Oph1.5q and Oph2q weak errors for a class of SDEs satisfying a commutativity condition. We then present several numerical examples including Multilevel Monte Carlo.

In the second half of this Thesis, we present a series of papers focusing on lifetime prognostics for lithium-ion batteries. Lithium-ion batteries are fuelling the advancing renewable-energy based world. At the core of transformational developments in battery design, modelling and management is data. We start with a comprehensive review of publicly available datasets.

This is followed by a study which explores the evolution of internal resistance (IR) in cells, introducing the original concept of ‘elbows’ for IR. The IR of cells increases as a cell degrades and this often happens in a non-linear fashion: where early degradation is linear until an inflection point (the elbow) is reached followed by increased rapid degradation. As a follow up to the exploration of IR, we present a model able to predict the full IR and capacity evolution of a cell from one charge/discharge cycle. At the time of publication, this represented a significant reduction (100x) in the number of cycles required for prediction. The published paper was the first to show that such results were possible.

In the final paper, we consider experimental design for battery testing. Where we focus on the important question of how many cells are required to accurately capture statistical variation.