Using Pseudo-Two-Dimensional (P2D) Reformulation Model with a Particle Filter to Estimate State of Charge, State of Health, and Remaining Useful Life in Li-Ion Battery Management System (BMS)
Authors: Morris Larry , Weatherspoon Mark H. , Nelson Ruben , Stephens Jamal Frederon , Moss Pedro L. 
Published: February 2020

Discovering the precise state of charge (SOC), state of health (SOH), and remaining useful life (RUL) while monitoring depth of discharge (DOD) for Lithium ion (Li-ion) batteries can be computational enormous. Electrical engineers using equivalent circuit models (ECM) or enhanced equivalent circuit models (EECM) have achieved an acceptable level of accuracy modelling the current, voltage, and temperature behaviors to get SOC, SOH, RUL, and DOD. ECMs and EECMs are computationally fast but 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. Chemical engineers have achieved greater levels of accuracy using physical-based models over ECMs and EECMs. Unfortunately, the number of factors and variables needed to compute the system of differential algebraic equations for the nonlinear physical-based models to achieve this level of precise data is computationally too expensive for real-time systems. In today’s environment, the use of filters has provided a bridge between empirical and physical based models. This paper examines an ECM and a porous electrode pseudo two-dimensional (P2D) model using particle filter (PF) technology for a series, parallel, and series/parallel combination battery switching microcontroller based battery management system (BMS).

Mathematical and Physical Models

BMSs are developed to manage the charging and discharging profile, and basic knowledge of all the states that the battery topology goes through is paramount in order to design a safe and more sustainable battery system. ECMs are used in BMSs because they can model battery characteristics for a desired state using resistors and capacitors to emulate the internal battery cell resistance and capacitance behaviors. Because ECMs are linearly based mathematical equations, simulation of these models can be performed quickly. Additionally, Kalman filtering techniques (KFT) have been used extensively with ECMs to assist in the estimation of the SOC and SOH for BMSs. However, the effectiveness of ECMs and KFTs are not optimal for the non-linear behavior of batteries.
Physical based models have the advantage of performing comprehensive analysis on the effects of both the solid phase and the liquid phase, but they can be computationally complex. Modelling, porous electrode theory coupled with transport phenomena, and electrochemical reactions represented by coupled nonlinear partial differential equations in one or two dimensions give physical based models an advantage over ECMs. Santhanagopalan [1] and Rahimian [2] have used the single particle model (SPM) implementing Kalman filtering methods to estimate SOC of Li-ion cells. Efforts by Subramanian et al. [3] 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 “calendar life.” This paper will simulate a reformulated P2D SOC and SOH convergence based model using the governing equations of a SPM with lithium active particles as in Northrop [4]:
Eq (1), Eq (2), Eq (3)

Particle Filter Methodology

The PF is an algorithm based on a recursive Monte Carlo technique of estimating the posterior density of the state variable given a set of observation variables. The sequential importance sampling (SIS) algorithm implements the recursive Bayesian filter by associating weights to random samples representing the required posterior density. The sequential importance resampling (SIR) filter adapts the SIS features and adds a resampling phase. The posterior PDF approximation can be defined as in [5]:
Eq (4), Eq (5)

Results

Figure 1(a) shows the simulated discharge (at 1C rate) of a 1.2 Volt Li-ion battery at 100% SOC using an empirical ECM in MatLab Simulink. Figure 1(b) includes simulated measured (noisy), SIS, and SIR filtered battery discharge data.
Figure 1 Battery Voltage Profile
Future work will include a reformulated physical based P2D PF battery model to simulate SOC, SOH, and RUL of a BMS.

References

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.

Subramanian, Venkat R., et al. "Mathematical Model Reformulation for Lithium-Ion Battery Simulations: Galvanostatic Boundary Conditions." ECS 156.4 (2009): A260-71. Print.

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.

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