Estimate the States of Pseudo-Two-Dimensional (P2D) Reformulation Model Li-Ion Battery Management System Using Non-Linear Particle Filter (PF) Compared to Linear Kalman Filter (KF)

Today’s battery cell environment, with the steady advances in technology and limited global resources, makes it imperative to know the approximate battery potential.   In order to reduce cost and extend replacement interval of lithium ion (Li-ion) battery by knowing the best estimation of the battery cell 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 Model (ECM) with the Kalman Filter technic.  ECMs results have been used in previous studies, because they are computationally faster than physical based modelling, however, ECMs do not fully account for the electrochemical material that makes up the battery in their mathematical equations or isothermal characteristics that continuously change during charging and discharging of the battery cells.  Physical based battery modelling permits continuous monitoring of Li-ion cells during simple and complex isothermal changes for charging and discharging profiles.  This paper examines multiple P2D Li-ion technologies using PF versus KF 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 by Subramanian et al. [1] focused on a physical based P2D reformulated model characterized by solid-electrolyte interface layer growth, constant current–constant voltage, and capacity fade as it related to battery degradation 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 Northrop [2] and Prada [3].  Equations 1-3 establish the boundary condition and Butler-Volmer kinetics formula for the Li-ion battery.

Eq (1), Eq (2), Eq (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.  Equation 4 is the Kalman Filter state model; Santhanagopalan [4] and Rahimian [5] used to implement a SPM with Kalman filtering methods 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.

Eq (4)

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 equation 5 and in [5].

Eq (5)

Results

Figure 1 Simulated Measured (noisy), SIS, and SIR filtered battery discharge data.   

Figure 1 Battery Voltage Profile

Future work will include multiple reformulated physical based P2D PF battery technology models to simulate SOC, SOH, and RUL of a BMS.

References

1. Subramanian, Venkat R., et al. "Mathematical Model Reformulation for Lithium-Ion Battery Simulations: Galvanostatic Boundary Conditions." ECS 156.4 (2009): A260-71. Print.
2. Northrop, Paul W. C., et al. "Efficient Simulation and Reformulation of Lithium-Ion Battery Models for Enabling Electric Transportation." ECS 161.8 (2014): E3149-57. Print.
3. Prada, E., et al. “Simplified Electrochemical and Thermal Model of LiFePO4-Graphite Li-Ion Batteries for Fast Charge Applications.”  ECS 159.9 (2012): A1508-19. Print
4. Santhanagopalan, Shriram, et al. "Review of Models for Predicting the Cycling Performance of Lithium Ion Batteries." JPS 156.2 (2006): 620-8. Print.
5. 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.
6. 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.