Piyush Tagade

A Generalized Polynomial Chaos-Based Method for Efficient Bayesian Calibration of Uncertain Computational Models

This paper addresses the Bayesian calibration of dynamic models with parametric and structural uncertainties, in particular where the uncertain parameters are unknown/poorly known spatio-temporally varying subsystem models. Independent stationary Gaussian processes with uncertain hyper-parameters describe uncertainties of the model structure and parameters, while Karhunen-Loeve expansion is adopted to spectrally represent these Gaussian processes. The Karhunen-Loeve expansion of a prior Gaussian process is projected on a generalized Polynomial Chaos basis, whereas intrusive Galerkin projection is utilized to calculate the associated coefficients of the simulator output. Bayesian inference is used to update the prior probability distribution of the generalized Polynomial Chaos basis, which along with the chaos expansion coefficients represent the posterior probability distribution. The proposed method is demonstrated for calibration of a simulator of quasi-one-dimensional flow through a divergent nozzle with uncertain nozzle area profile. The posterior distribution of the nozzle area profile and the hyper-parameters obtained using the proposed method are found to match closely with the direct Markov Chain Monte Carlo-based implementation of the Bayesian framework. Efficacy of the proposed method is demonstrated for various choices of prior. Posterior hyper-parameters of the model structural uncertainty are shown to quantify acceptability of the simulator model.

An Efficient Bayesian Calibration Approach Using Dynamically Biorthogonal Field Equations

Present paper proposes new dynamic-biorthogonality based Bayesian formulation for calibration of computer simulators with parametric uncertainty. The formulation uses decomposition of solution field into mean and random field. The random field is represented as a convolution of separable Hilbert spaces in stochastic and spacial dimensions. Both the dimensions are spectrally represented using respective orthogonal bases. In particular, present paper investigates polynomial chaos basis for stochastic dimension and eigenfunction basis for spacial dimension. Dynamic evolution equations are derived such that basis in stochastic dimension is retained while basis in spacial dimension is changed such that dynamic orthogonality is maintained. Resultant evolution equations are used to propagate prior uncertainty in input parameters to the solution output. Whenever new information is available through experimental observations or expert opinion, Bayes theorem is used to update the basis in stochastic dimension. Efficacy of the proposed methodology is demonstrated for calibration of 2D transient diffusion equation with uncertainty in source location. Computational efficiency of the method is demonstrated against Generalized Polynomial Chaos and Monte Carlo method.Copyright

Key Applications of Electrochemical Theory

The solution of the electrochemical thermal model developed in the previous chapters is discussed under various charge–discharge scenarios. The total heat generation is resolved to anode, cathode, and separator sections of the Li-ion cell, and the significant sources identified. The contribution due to irreversible, reversible, phase change processes is identified. Dependence of these multiple components on the operating conditions is studied, and optimal operating conditions are suggested (Figures and discussions reproduced with permissions from Elsevier.).

Key Applications of ROM

The reduced order model developed in the last chapter is solved for academic and realistic scenarios. For the latter, validation against constant-controlled constant current data as well as profiles resembling vehicle drive conditions are considered. For the former, special scenarios like cells with phase change electrodes and cycling studies are analyzed (Figures and discussions reproduced with permissions from Elsevier.).

Theoretical Framework of Electrochemical–Thermal Model (ECT)

The energy balance for a Li-ion cell is derived from principles of nonequilibrium thermodynamics. The equations are integrated into the electrochemical model. The model is extended to special cases like electrodes with phase change. Performance indicators that can be computed from the model that enables an optimal design are developed.

Theoretical Framework of the Electrochemical Model

The basic principles of the electrochemical model for Li ion battery is developed from fundamentals of thermodynamics and transport phenomena. The evolution of the electrochemical model and the inherent assumptions are discussed. The discussions and derivations are self-consistent and complete. Mathematical model for each process in the Li-ion cell is constructed in a stepwise manner to evolve the complete electrochemical model.

Theoretical Framework for Health Estimation Using Machine Learning

Real-time prediction of Remaining Useful Life (RUL) is an essential feature of a robust battery management system (BMS). However, due to the complex nature of the battery degradation, physics-based degradation modeling is often infeasible. Data-driven approaches provide an alternative when physics-based modeling is infeasible. In this chapter, we investigate some of the most popular machine learning-based data-driven approaches used by the Lithium-ion battery community. The chapter first introduces basic concepts of classification and regression, followed by a generic framework for its solution. Finally, we introduce some machine learning algorithms for the solution of this generic framework.

Theoretical Framework of the Reduced Order Models (ROM)

Physically motivated model order reduction of the electrochemical–thermal model is developed. The motivation is to enable the use of the ROM for on-board scenarios. The equations are lighter to solve and are intuitive, with parameters retaining the physical relevance (This approach was first developed by Senthil Kumar (J Power Sources 222:426–441, 2012 [49].).

Theoretical Framework for State Estimation

One of the most important functions of the battery management system is to accurately estimate the battery state using minimal onboard instrumentation. In this chapter, we present a recursive Bayesian filtering framework for onboard battery state estimation by assimilating measurables like cell voltage, current, and temperature with a physics-based model prediction. This framework can be numerically implemented using state-of-the-art filtering/data assimilation algorithms. We first develop a generic framework and then discuss some of the most popular algorithms for its implementation.

Introduction and Perspective

In this introductory chapter, the need for a comprehensive and in-depth understanding of the underlying physics of batteries is discussed. The layout of the book as well as the major themes are briefly introduced to set the tone for the rest of the chapters that follow.