Pseudo-Two-Dimensional (P2D) Reformulation Modeling of Li-Ion Batteries Using Kalman Filter (KF), Particle Filter (PF), and Hybrid Particle Filter (HPF) Algorithms
Authors: Morris Larry , Weatherspoon Mark H , Stephens Jamal Frederon , Moss Pedro L. 
Published: February 2020

Accurate prediction of cell potential is necessary to extend the replacement interval of Lithium ion (Li-ion) batteries. This requires developing models that are capable of estimating battery state of charge (SOC), state of health (SOH), and remaining useful life (RUL). Previous studies have been completed to realize SOC, SOH, and RUL using equivalent circuit models (ECMs) with the KF technique. These ECMs have been used because they are computationally faster than physical based models. However, in their mathematical equations, ECMs do not fully account for the electrochemical material that makes up the battery or isothermal characteristics that continuously change during charging and discharging of the battery cells. Physical based battery modeling permits continuous monitoring of Li-ion cells during simple and complex isothermal changes for charging and discharging profiles. This paper examines multiple P2D models of Li-ion batteries using KF, PF, and HPF for a series, parallel, and series/parallel combination battery switching microcontroller based battery management system (BMS).

Physical Models

Physical based models are computationally complex, but they have the advantage of performing comprehensive analysis on the effects of both the solid state and the liquid state of the Li-ion battery, by modelling, porous electrode theory coupled with transport phenomena, and electrochemical reactions represented by coupled nonlinear partial differential equations in one or two dimensions. Efforts in [1], as related to battery degradation, focused on a physical based P2D reformulated model characterized by solid-electrolyte interface layer growth, constant current–constant voltage, and capacity fade to determine the manner in which extreme temperature and high charging rates affect the battery “cycle-life” and “calendar life.” This paper will simulate a reformulated P2D SOC and SOH convergence based model using the governing equations of a Single Particle Model (SPM) with lithium active particles as in [2] and [3].

Kalman Filter Methodology

The KF is a linear quadratic estimation algorithm which uses observation measurements during a given time period to achieve a greater estimation of the battery system. EB (1) is the KF state model used in [4] and [5] to implement a SPM to estimate SOC of Li-ion cells. KF is an effective recursive prediction algorithm that works in two-steps: the prediction step (using the previous time step (k-1), measurement data, and Gaussian distributed white noise with covariance to forecast the current state) and the weighted average of the approximate next measurement.
EB (1)

Particle Filter Methodology

The PF is a non-linear recursive Monte Carlo algorithm which estimates the posterior density of the state variable given a set of observation variables. The recursive Bayesian filter is implemented by the sequential importance sampling (SIS) algorithm, by associating weights to random samples representing the required posterior density. The SIS features are then adapted by the sequential importance resampling (SIR) algorithm in order to complete the resampling phase. The posterior PDF approximation can be defined by EB 2 and in [6].
EB (2)

Hybrid Particle Filter (HPF) Methodology

The HPF is a combination of the KF and PF algorithms for a given set of observation variables. In the HPF model, instead of a proposal and optimal random sample, the KF is implemented to get the initial values for Xk (proposal).
EB (3)

Results

Battery discharged data is presented using the KF, PF, and HPF algorithms in Fig (1). Compared to the KF and PF models, the HPF model has a quicker initial convergence and better approximates the measured data across the discharge profile.
Figure 1 Filter Comparison: Kalman, Particle, and Hybrid Particle
Future work will include multiple reformulated physical based P2D PF battery technology models to simulate SOC, SOH, and RUL of a BMS.

References

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Santhanagopalan, Shriram, et al. "Review of Models for Predicting the Cycling Performance of Lithium Ion Batteries." JPS 156.2 (2006): 620-8. Print.

Rahimian, S.K., S. Rayman, and R.E. White. “State of Charge and Loss of Active Material Estimation of a Lithium Ion Cell under Low Earth Orbit Condition Using Kalman Filtering Approaches.” ECS (2014): A860-A872.

Arulampalam, M. S., et al. "A Tutorial on Particle Filters for Online nonlinear/non-Gaussian Bayesian Tracking." Signal Processing, IEEE Transactions on 50.2 (2002): 174-88. Print.

Figure 1