Modern electric vehicles and portable electronic devices necessitate high-performance and reliable battery systems. To achieve this, it is crucial to monitor the battery's state continuously and accurately. In response, there is ongoing development of remote monitoring and history management system for lithium-ion batteries. Against this context, the authors are engaged in the development of a new Battery Management System that integrates real-time battery state estimation capabilities using the Extended Kalman Filter with functions for estimating internal parameters. This paper presents a report on the development of an internal parameter estimation feature for battery state monitoring under various operational conditions, utilizing the Recursive Least Squares method.
voltage drops due to internal resistance and polarization, making high-precision estimation a challenge. Therefore, several new methods have been proposed to address these shortcomings, such as the Kalman filter method [ 6, 7], the neural network (NN) method [8], and the sliding mode observer [9]. Among these methods, the Kalman filter is particularly eficient and simple. Kalman filters are suitable for operation on the edge computer side because they can cope with large current fluctuations. Edge computer operation reduces delays in data transmission to the central system and enables real-time battery state estimation. In particular, the Extended Kalman Filter (EKF) provides excellent state estimation for nonlinear systems and accurately captures the battery’s real-time behavior, ofering robustness against system uncertainties and external noise.
The authors have already incorporated an Extended Kalman Filter (EKF) into the edge computing side for real-time state estimation of batteries. This technology enables the capture of actual battery behavior in real-time, providing robustness against system uncertainties and external noise. However, the accuracy of estimations under specific conditions such as battery degradation and temperature changes may decrease. Therefore, by also integrating the Recursive Least Squares (RLS) method [10], we have enabled real-time updates of internal parameters, achieving high accuracy in state estimation across all operational conditions. This paper reports on the battery remaining capacity estimation functionality that incorporates internal parameter estimation using RLS integrated with EKF.
This section describes a detailed explanation of Equivalent Circuit Models (ECM), Extended Kalman Filters (EKF), and Recursive Least Squares (RLS). The ECM represents the electrical behavior of batteries through a combination of simple electronic components, helping to understand the dynamic characteristics of batteries. The EKF is used for state estimation of nonlinear systems and accurately tracks the charge state and health of batteries. The RLS method continuously updates the internal parameters of the battery, enabling precise modeling according to changing conditions.
The single-stage RC equivalent circuit model commonly used for parameter estimation in storage batteries is shown in Figure 1 [11]. This model can be easily converted into a state-space model and a regression model for parameter estimation. In this model, the battery’s electromotive force is represented by a voltage source called Open Circuit Voltage (OCV). The model includes internal resistance, which is divided into two parts: a direct current resistance that represents the charge transfer resistance within the electrolyte, and a parallel circuit consisting of a resistor and a capacitor that represents the slow reaction associated with difusion. The terminal voltage of the battery is denoted by , and the terminal current is denoted by . The current lfowing into the battery is considered positive. During discharge, is negative, and during charging, is positive. RRC
By using the forward Euler method and the equation L = RRC + OCV, equation (
L() = a() + ︂) ︂( s
︂) b − a ( − 1) +
bb − 1 OCV() − ︂( ︂( sa + bb s s ︂) bb − 1 L( − 1) + OCV()
Through the variable transformation shown in equation (
To estimate the parameters of a battery online (during charge or discharge mode), the Recursive
Least Squares (RLS) method is a conventional identification method. Figure 2 shows the block
diagram of parameter estimation with system identification. System identification uses statistical
methods to build mathematical models of dynamic systems from measured data. Therefore, the
mathematical regression model of the battery must be built from the equivalent circuit model.
The diferential equation of (the voltage of and the RC circuit , ) is given by
equation (
bb 1 =
s , = (1 + 1)OCV
0 = a, 1 =
sa +
bb
s
b − a
(
In equation (
⎡ () ⎤ () = ⎢⎢⎣− ((− − 1)1)⎥⎦⎥ , = ⎢⎢⎣11⎥⎦⎥
1 = 0, = =
1 − 10 , = ⎡0 ⎤
1 − 10 1 + 1
1 + 1
Following the RLS identification theory, the evaluation function with a forgetting factor for
the RLS is given by equation (
() = ∑︁ − 2()
=1
The RLS algorithm to minimize the equation (
In recent studies, the method for estimating SOC based on the extended Kalman filtering
(EKF) technique has been proposed. The EKF is an approximately optimal state estimator for a
(
Forgetting factor Recursive Process: () = () − ⊺()̂︀( − 1) 1 {︃ () = ( − 1) − ( − 1) () ⊺() ( − 1) }︃
+ ⊺() ( − 1) () ̂︀() = ̂︀( − 1) +
( − 1) () + ⊺() ( − 1) () () zero-mean white Gaussian noise processes with covariance 2 and 2 respectively. For battery equivalent circuit model, the detail of state space model is shown in Table 1.
The EKF algorithm is described as Algorithm 2. This algorithm consists of three steps:
Initialization, Prediction, and Filtering. Here, ^− is the one-step prediction vector, ^ is the
ifltered estimate vector, − is the prediction error covariance matrix, and is the lfitering
error covariance matrix. In this paper, is diferent from
. is the error covariance matrix
of RLS, is the covariance matrix of EKF. As represented in equation (
In this paper, we have added a battery internal parameter estimation function using RLS to a BMS with SOC estimation capabilities based on EKF. To verify its proper functioning, we examined SOC estimation using EKF with and without RLS. The estimation experiments utilized discharge waveforms of 18650-type lithium-ion batteries. Various patterns of discharge waveforms were prepared, and the measured terminal voltage and current were input into the RLS-EKF algorithm to simultaneously estimate circuit parameters and SOC. Figure 6 shows the terminal voltage, terminal current, our SOC estimation results, the true SOC values, and their absolute errors. Additionally, Figure 7 shows the estimation results of the internal parameters using RLS. Figure 8 shows the Comparison of SOC Estimation Results Using EKF with and without RLS. By adding RLS, it is confirmed that SOC can be estimated with higher accuracy.
In this study, a function for estimating internal parameters by RLS was added to a real-time battery condition monitoring system. This function is expected to enable more accurate estimation of the state of the battery, which depends on the state of degradation and temperature changes. At the present stage, the functionality of the BMS has not been evaluated in a real environment, so the accuracy of the SOC estimation and the influence of the internal parameter estimation will be evaluated in a real environment in the future.
This work is partly commissioned by NEDO (Project Number JPNP10020). circuit battery models part 2. parameter and state estimation, Journal of Power Sources 262 (2014) 457–482. [11] M. S. Grewal, A. P. Andrews, Kalman filtering: Theory and Practice with MATLAB, John Wiley & Sons, 2014.