In this work, a statespace battery model is derived mathematically to estimate the stateofcharge (SoC) of a battery system. Subsequently, Kalman filter (KF) is applied to predict the dynamical behavior of the battery model. Results show an accurate prediction as the accumulated error, in terms of rootmeansquare (RMS), is a very small value. From this work, it is found that different sets of
Battery Management System (BMS) [
Typical Lithiumion battery packs for electric vehicle.
Battery modeling is performed in many ways depending on the types of battery. In general, the resulting battery model is a mathematical model comprising numerous mathematical descriptions [
The stateestimation process usually leads to some state variables in a dynamical system. The SoC is a measure of a battery’s available power and thus it is important to calculate this value accurately from BMS by the cell voltage, temperature, and polarization effect caused by the electrolyte concentration gradient during high rate charging/discharging cycle [
Schematic of
From this mathematical expression, it is noted that the SoC cannot be explicitly measured. In the literature, there is a myriad of methods dealing with predicting and estimating of SoC. The most popular of these methods are described in the following paragraphs.
At present, Coulombcounting [
Another prominent SoC estimator is the wellknown Kalman filter (KF), invented by Kalman in 1960. Although his popular work was published almost 54 years ago in [
Various artificial intelligence (AI) methods, mainly the neural networks and fuzzy logic, are being applied in the estimation of battery’s SoC [
In this work, a mathematical derivation leading to a statespace model is presented. The basic schematic model is adopted from [
Many modelbased stateestimations have been proposed in [
In this derivation, we aim to form a statespace model consisting of the state variables
Following the voltages and currents illustrated in Figure
By substituting all capacitor and resistor values from Table
Parameters for cell model [











By defining matrix
and
As such (
Based on control theories, a lumped linear network can be written in the form
Further, the above statespace variables are transformed to a transfer function,
Output response of RC model due to constant input.
In control theory, observability is the degree to which the internal states of a system can be predicted via its external outputs. As such, for an observable system, the behavior of the entire system can be predicted via the system’s outputs. On the other hand, if a system is not observable, the current values of some of its states cannot be estimated through the output signal. This means the controller does not know the states’ values. In theory, the observability of a system can be determined by constructing an observability matrix
Further, substituting
A continuous timeinvariant linear system can be described in the state variable form as
If we assume that the applied input
Both
By inclusion of these noises, the resulting system can now be described by
Discrete system model with noises
An important property of the KF is that it minimizes the sumofsquared errors between the actual value
For the case of a battery, it is well understood that only the terminal quantities can be measured (terminal voltage
Parallel connection of plant and Kalman filter.
The genetic algorithm (GA), introduced by John Holland, is an approach based on biological evolution [
The pseudocode of the GA is presented in Algorithm
(1) BEGIN
(2) Initialize population
(3)
(4)
(5)
(6)
(7)
(8)
(9) END
GA flowchart.
In the context of Kalman filter, GA is applied to tune the
The program, implemented in MATLAB, is given in the appendix to clarify the results obtained in this work. Take note that the
RMS error recorded during charging operation.
Output 

RMS error [V] 

Measurement, 
1, 1 



Estimated (KF), 
1, 1 



Estimated (KF), 
0.012697316315642, 

(After GA tuning)  2.303940992875865 
The convergence characteristic of GA.
The time plot of the estimated error from 0 s to 60000 s is shown in Figure
The voltage error recorded and depicting measured (green color) and estimated (blue color) errors.
The charging characteristic is illustrated in Figure
Response of RC battery model in terms of
For the discharging process, the initial value of terminal voltage,
Response of RC battery model in terms of
In this work, we successfully obtained the state variables of the RC model representing a battery in terms of mathematical derivations. The derivations lead to the conclusion that there exist three state variables relevant to a battery’s statespace model. With this stateestimation model, a prominent technique known as the Kalman filter is applied in the aim of estimating stateofcharge for a Battery Management System. From numerical results, the KF is shown to be accurate in predicting the dynamic behavior of the system. This is shown by a very small RMS error of the estimation in comparison to its measurement. The estimated error is further reduced after incorporating the optimized values of
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The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to acknowledge the support from Xi’an JiaotongLiverpool University under RDF130113. The authors would also like to thank their colleague, Sanghyuk Lee, for the sharing of his expertise, contributing to the success of this project.