A Novel Multi-Stage Stochastic Estimation Algorithm for Estimating the Parameters of the Extended Single Particle Model of a Lithium-Ion Battery
Authors: Wickramanayake Toshan , Iqbal Saqib , Mehran Kamyar 
Published: August 2024

Lithium-ion batteries (LiBs) are commonly used as energy storage for many applications due to LiBs’ high energy density, long cycle life, and reliable manufacturing processes. Hence, effective performance management of LiBs is important and is implemented via Battery Management Systems (BMS)[1]. A BMS also maximises the safety and lifetime of battery packs; and therefore, needs to estimate the internal battery states including State of Charge (SoC) and State of Safety (SoS). Accurate estimation of battery SoX can be realized with the use of an accurate battery model [2].
From the different types of battery models, electrochemical models (EMs) are considered the most accurate type of cell-level model [2]. EMs are sets of coupled-partial differential equation systems, where each equation is derived from the physics occurring in the cell. Of the different EMs, the partial two-dimensional model (P2D) is considered the most accurate cell-level model. However, the P2D model is complex and has no analytical solution, thus requiring computationally expensive iterative solvers for implementation. This makes practical implementation of the P2D model on BMS platforms challenging. Hence, most approaches seen in the literature prefer the use of reduced order versions (ROM) of the P2D model, that require far less computational power for implementation [1].
One such ROM is the extended Single Particle Model (eSPM) which differs from the P2D model in considering each electrode to be modelled by a single spherical particle and charge conservation in the electrolyte to follow a simplified linear relationship. These assumptions help the model be accurate up to a 2C operational current, and enable an analytical solution to be derived using direct solving of the eSPM equations [2]. Due to fast-forward computation and reasonable accuracy, we’ve chosen the eSPM model for parameter estimation (PE). For accurate measurements of the model parameters, expensive and invasive experimental techniques are typically required. Non-destructive PE methods have a growing interest in the literature as they are not only non-invasive but also enable real-time dynamic PE to be done on a BMS [3].
PE of EMs can be categorized under two approaches: deterministic methods and stochastic methods. Deterministic methods require a form of gradient descent to be used, wherein partial derivatives of the parameters of the model need to be calculated. This approach tends to be computationally expensive and tends to converge to local minima instead of searching the parameter space thoroughly. Hence, stochastic approaches incorporate randomness, thus searching the entire parameter space for the global optima [3]. Multiple different stochastic approaches have been employed for PE of EMs [4]. In this work, we compare and contrast some of the commonly used approaches and propose a novel multi-stage approach based on a combination of different stochastic PE algorithms.
The proposed algorithm (see Figure (1)) requires three sets of data as inputs: the cell chemistry, constant current discharge data and the parameter boundaries of the eSPM model. Using these definitions, the first stage of the algorithm estimates a parameter set that has a reasonable level of accuracy i.e. less than 4% root mean square error (RMSE). The second stage of the algorithm does region-by-region optimisation, where parameters sensitive to different regions of the discharge curve are varied until the overall error is reduced to less than 1% RMSE. The proposed approach significantly reduces the estimation processing time compared to the ‘brute-force’ approach and is flexible in selecting different algorithms for each stage.
Thus, contributions of this paper are briefly explained as follows:

A performance comparison of the proposed algorithm with different stochastic estimation algorithms including the genetic algorithm, the particle swarm optimisation algorithm and the simulated annealing algorithm.

A fast and adaptable multi-stage PE algorithm for the LiB eSPM model, with average estimation speeds of less than fifteen minutes for a typical BMS processor, with accuracy of less than 1% RMSE error.

Validation of the algorithm with dynamic drive-cycle test data.

[1] L. Lu et al, “A review on the key issues for lithium-ion battery management in electric vehicles,” Journal of Power Sources, vol. 226. pp. 272–288, Mar. 15, 2013. doi: 10.1016/j.jpowsour.2012.10.060.
[2] S. Abada et al, “Safety focused modeling of lithium-ion batteries: A review,” J Power Sources, vol. 306, pp. 178–192, Feb. 2016, doi: 10.1016/J.JPOWSOUR.2015.11.100.
[3] E. Miguel et al, “Review of computational parameter estimation methods for electrochemical models,” Journal of Energy Storage, vol. 44. Elsevier Ltd, Dec. 15, 2021. doi: 10.1016/j.est.2021.103388.
[4] A. Jokar et al, “An Inverse Method for Estimating the Electrochemical Parameters of Lithium-Ion Batteries,” J Electrochem Soc, vol. 163, no. 14, p. A2876, Oct. 2016, doi: 10.1149/2.0191614JES.

Figure 1