MC 5417

## Candidate

Sepideh Afshar

Applied Mathematics, University of Waterloo

## Title

Lithium-Ion battery SOC estimation.

## Abstract

Lithium-ion batteries are frequently used in Hybrid electric vehicles (HEVs) which taking the place of gas-engine vehicles. An important but not measurable quantity in HEVs is the amount of charge remaining in the battery in a drive cycle. The remaining charge is normally identified by a variable called state of charge (SOC). A potential way of estimating SOC is to relate this variable with the state of a dynamical system. Afterwards, SOC can be estimated through an observer design. As a precise model, electrochemical equations are chosen in this research to estimate the SOC.

The first part of this thesis considers comparison studies of commonly-used finite-dimensional estimation methods for different distributed parameter systems (DPSs). In this part, the system is first approximated by a finite-dimensional representation; the observer dynamics is a copy of finite-dimensional representation and a filtering gain obtained through observer design. The main outcome of these studies is comparing the performance of different observers in state estimation of different types of DPSs after truncation. The studies are then expanded to investigate the effect of truncated model by increasing the order of finite-dimensional approximation of the system numerically. The simulation results are also compared with mathematical properties of the systems.

A modified sliding mode observer is improved next to take care of system's nonlinearity and compensate the estimation error due to disturbances coming from an external input. It is proved that the modified SMO provides exponential convergence of the estimation error in the existence of an external input. Simulations results of comparison studies indicate the improved performance of the modified SMO observer in most cases.

Approximating and well-posedness of two general classes of nonlinear DPSs are studied next. The main concern of these studies is to produce a low-order model which converges to the original equation as the order of approximation increases. The available results in the literature are limited to specified classes of systems. These classes does not cover the lithium-ion cell model; however, the general forms presented here include the electrochemical equations as specific version.

In order to facilitate the electrochemical model for observer design, simplification of the model is considered in the next step. The original electrochemical equations are composed of both dynamical and constraint equations. They are simplified such that a fully dynamical representation can be derived. The fully dynamical representation is beneficial for real-time application since it does not require solving the constraint equation at every time iteration while solving the dynamical equations. The electrochemical equations can next be transformed into the general state space form studied in this thesis.

Finally, an adaptive EKF observer is designed via the low-order model for SOC estimation. The electrochemical model employed here is a variable solid-state diffusivity model. Compared to other models, VSSD model provides more accuracy for cells with Lithium ion phosphate positive electrode which are considered here. The adaptive observer is constructed based on considering an adaptive model for open circuit potential term in the electrochemical equations. The parameters of this model are identified simultaneously with the sate estimation. Compared to experimental data, simulation results show the efficiency of the designed observer in existence of modeling inaccuracy.