1. Introduction
Lithium batteries have been widely used in electric vehicles due to their high energy density, high power density, high efficiency, and long lifespan [
1]. Overcharge/discharge should be strictly avoided in the use of lithium batteries. Otherwise, the battery service life will be significantly reduced [
2,
3]. Thus, accurate estimation of battery state of charge (SOC) becomes an important issue in lithium battery applications.
According to the research literature on SOC estimation algorithms, the algorithm can be classified into four types: Coulomb counting method, intrinsic parameter direct measurement method, computer intelligence method, and model-based method.
The coulomb counting method is the simplest and most widely used method. However, as the counting time increases, the estimation error is accumulated. Therefore, the coulomb counting method requires additional error correction. Chen [
4], Liu [
5], Kutluay [
6] and others have proposed many correction methods for the coulomb counting method.
The intrinsic parameter measurement method is to select suitable battery parameters that can directly characterize the SOC for direct measurement, and then estimate the battery SOC through these parameters. The commonly used intrinsic parameters include open circuit voltage (OCV), impedance spectrum [
7], electromotive force (EMF), and internal resistance (IR). The advantage of this method is that the battery SOC can be directly obtained through the mapping relationship between the SOC and the battery intrinsic parameter, and the disadvantage is that the estimation accuracy relies excessively on the accuracy of the mapping relationship between SOC and intrinsic parameters of battery. Holger [
8] summarized the SOC estimation method based on the impedance spectrum. Waag [
9] confirmed that the impedance parameter of a Li-ion battery changes negatively as the battery ages.
Computer intelligence methods found in literature mainly include the neural network model, genetic algorithm and particles swarm optimization. The advantage of these methods is that they do not require a thorough understanding of battery internal mechanism, but they do require large amounts of data for the machine learning process. Sheikhan et al. [
10] used an artificial neural network to realize feedback correction of the traditional time-integration method: Considering the aging effect of power battery, they used an optimized multi-layer structure perceptron and radial basis function (RBF) neural network as corrective measures of the coulomb counting method to achieve accurate estimation of SOC. Wei et al. [
11] used the neural network model to estimate SOC and improved the estimation accuracy through the Kalman filter.
The model-based SOC estimation method is the most popular method in recent years [
12]. It can be divided into two categories: Electrochemical and equivalent circuit.
The advantage of the electrochemical model is that it can reflect the internal mechanism of the battery. Its disadvantages mainly include two aspects: difficulty and cost. The high complexity model requires a high performance processor, which is costly, while the difficulty of online parameters identification increases with the aging of batteries [
13,
14].
The equivalent circuit model-based SOC estimation method mainly include the Kalman filter algorithm, the Gauss–Hemitian orthogonal filter algorithm, and the particle filter algorithm. Haykin [
15] first used Kalman filtering and battery model for parameter estimation in his book
Kalman Filtering and Neural Networks. Since then, many scholars have improved and optimized the SOC estimation of Kalman filtering [
16]. Ambient temperature has a negative influence on the parameters of the battery equivalent circuit model. To solve this problem, Xing et al. [
17] revised the equivalent circuit model with open circuit voltage (OCV)-SOC curves at different ambient temperatures, and they obtained better SOC estimation accuracy by the unscented Kalman filter (UKF).
Unlike the Kalman filter, the particle filter (PF) is no longer restricted by the Gaussian distribution of process noise and observation noise, so it is more suitable for the application of battery SOC estimation in a complex environment affect [
18]. PF is a Bayesian filtering based on the sequential Monte Carlo method, and the basic idea of it is to approximate the posterior density of the estimating variable by discrete random samples [
19]. In order to solve the problem of particle degradation, sequential importance resampling (SIS) and auxiliary particle filter (APF) are widely applied [
20,
21]. Many others use techniques, such as adaptive filter [
22], Kalman filtering [
23], and the Markov-chain [
24]. In reference [
25], the authors put forward a ‘tether’ particle to circumvent the degeneracy problems.
Although there are many SOC estimation algorithms, the improved coulomb counting method and the equivalent circuit model-based method are still the two most frequently used methods in practical system applications. However, the equivalent circuit model-based method has higher precision and wider applicability than the coulomb counting method. Therefore, the application and test results of the equivalent circuit model-based method in a soft-packed lithium iron phosphate battery are studied in this paper. In order to overcome the negative impact of Kalman filtering using Gaussian distribution to estimate SOC, we used the particle filter algorithm to estimate SOC and put forward an auxiliary particle filter based on improved importance density function. The estimation accuracy of SOC had also been further improved.
This paper is organized as follows. In
Section 2, an equivalent circuit model is used and its parameters are identified with typical discharge test data on different SOC value. In
Section 3, the battery model is discretized and standardized, and the algorithm flow and steps of battery SOC estimation using standard particle filter algorithm are given. In
Section 4, the performance of SOC estimation algorithm based on standard particle filter is tested by experiments. In
Section 5, the auxiliary particle filter algorithm which improves the importance density function is proposed and deduced, and the algorithm flow and implementation steps are given. In
Section 6, the performance of the auxiliary particle filter algorithm is tested and compared with that of the standard particle filter algorithm.
2. Battery Model and Parameter Identification
The battery sample used is a 3.22 V/2.3 Ah lithium iron phosphate battery (as shown in
Figure 1), and a uC-ZS08 battery testing system is used as the battery testing experimental equipment with an accuracy of 0.1% (as shown in
Figure 2).
The first-order Thévenin equivalent circuit model is applied, which has lower complexity and takes the relationship between electromotive force of battery and SOC into account. In this model, the dynamic performance of the battery, such as rebound voltage characteristics, can be accurately simulated in the process of battery charging and discharging. A schematic diagram of the battery model is shown in
Figure 3.
In
Figure 1,
VOCV(
SOC) represents the open circuit voltage of the battery which is equal to the electromotive force in numerical value, and it has a fixed mapping relationship with battery SOC at a certain temperature. This mapping relationship can be obtained by the charge/discharge test of the battery and expressed by
VOCV =
f(
SOC). The influence of ambient temperature is not discussed in this paper. For the sample batteries in this paper, the relationship we measured can be expressed as Equation (1).
The equivalent circuit model of battery can be expressed by the following equations
where
SOC0 is the initial
SOC value of battery.
QN is the battery rated capacity.
Rb and
RΩ are polarized and ohmic internal resistance of the battery respectively, while
Vb and
VΩ are their voltages.
Cb is battery polarized capacitance.
η is the coulomb efficiency, which can be obtained by the battery charge/discharge test.
Η < 1 (when battery is discharging)
η = 1 (when battery is charging).
I and
Vout are the working current and the terminal voltage of the battery separately, which can be measured directly by sensors in real time.
The state and output equations can be deduced from Equation (2). Meanwhile, we set
and
as the process noise and measurement noise.
The parameters in the battery model cannot be directly obtained by theoretical derivation analysis. The data of the output voltage value with the input current are tested by applying different current excitation to the test battery in various initial SOC. This data is used to identify the parameters in the battery model.
For a typical discharge cycle as shown in
Figure 4, the terminal voltage of the battery
Vout will suddenly drop instantaneously at the beginning of the discharge current, which is the time of 60 s.
This sudden voltage drop is caused by the ohmic internal resistance in the battery model. It can be identified by the ratio of the falling voltage difference to the discharge current. The calculation expression is
At the end of discharge, the time of 120 s, the internal electron-chemical reaction of the battery does not stop, which causes a rebound voltage phase after the discharge stops. The rebound voltage phase can be divided into the abrupt phase (A-B) and the gradual phase (B-C). The reason for the abrupt phase is the same as the instantaneous voltage drop at the beginning of discharging, which is caused by the ohmic internal resistance in the battery model. The gradual phase of the rebound voltage curve is caused by the polarized internal resistance
Rb and the polarized internal capacitance
Cb. The curve of the rebound voltage phase can be approximated by Equation (6),
Theoretically, the value of the polarized resistance
Rb and the polarized capacitance
Cb in the battery model will change along with different
SOC. The identification results of the experimental data also verify this point. The identification results of
Rb and
Cb under different SOC are shown in
Table 1.
The battery parameters of other SOC states are obtained by using the quadratic polynomial difference. The result is shown in
Figure 5 and the equations are as follows
After the model parameter identification is completed, according to Equation (3) and the identification result, the battery Simulink model can be established in MATLAB/Simulink as shown in
Figure 6.
In order to verify the accuracy of the battery model and parameter identification, the battery model of
Figure 6 is applied with a user-defined operating cycle as shown in
Figure 7. The charge/discharge current switches frequently in this operating cycle. At the same time, the same operating cycle was applied to the sample battery test equipment uC-ZS08. The battery terminal voltage data can be obtained from both model simulation and battery test, and the comparison is shown in
Figure 8. The voltage error between them is shown in
Figure 9.
In
Figure 9, the maximum error of the terminal voltage and the experimental terminal voltage is less than 60 mV and the average error is 24.1 mV. The model and parameters identification are proved to be accurate.
3. The SOC Estimation Algorithm of the Standard Particle Filter
The basic idea of PF is to use discrete random sample points to approximate the probability density function of system state variables. The minimum variance estimate is obtained by replacing the integral operation with the mean of the samples. It is no longer limited to the theoretical framework of Kalman filter and has a good solution to the state estimation of nonlinear and non-Gaussian distribution.
According to Equations (3) and (4), the state equations and output equation of the battery can be rewritten in the following form after discretion,
where Δ
t is sampling time.
The state variable is set at
and the output variable is
. and standardize Equations (3) and (4) as follows:
According to the theory of the standard particle filter algorithm combined with the established battery model, the combination of the particle filter and the battery model for estimating the SOC is shown in following summary of standard particle filter operation.
Step 1: Initialization.
Generate the initial particles set from a Gaussian distribution and
and
number of particles M = 100
the process noise r and measurement noise v by another two Gaussian distribution, with a variance of for R and for v.
For k = 1, 2, ..., n, loop from Step 2 to Step 6.
Step 2: State prediction.
The particles
at the current moment is predicted according to the state equation using the particles
at the previous moment.
Step 3: Set the particle weights involved in the estimation iteration.
Update and normalize the weight of each particle and calculate the weight of the
ith particle by the equation as following
Step 4: Normalize the particles weight to
,
Step 5: Perform a resampling algorithm.
The principle of resampling algorithm is to discard the particles with smaller weights and copy the particles with larger weights. The copying times is determined by the weight of the particle. Firstly, generate
M numbers
between [0, 1] randomly. Then, the weight values of the particles are sequentially added until the sum of the weights is greater than
λi. If the following formula is satisfied
then set
Step 6: Estimation state update
5. Auxiliary Particle Filter on Improved Importance Density Function
The principle of auxiliary particle filter (APF) is to make full use of particles with high predictive likelihood from the previous moment to the current moment. In this way, sample particles can be generated. Comparing the sample particles generated by the standard particle filter, the auxiliary particle filter can obtain a more satisfactory effect when the likelihood function of the sampled particle is narrower or is at the tail of the prior probability distribution.
Introduce an auxiliary variable
m to represent the order number of particles at time (
k−1). By using the Bayes Theorem, the joint probability density function
is expressed as
Generate the importance probability density function
of
is
where
is the character related with
predicted by
, which can be
or
.
Define
. Because
then
The operation of the APF SOC estimation is summarized in the following summary of auxiliary particle filter operation.
Step 1: Initialization
Generate the initial particles set from a Gaussian distribution
Introducing a new auxiliary variable m, which represents the order list of particles at the previous sample time, and the new particles set can be written as
Set
,
,
M,
Q and
R as in the summary of standard particle filter operation before in
Section 3.
For k = 1, 2,..., n, loop from Step 2 to Step 6.
Step 2: For i = 1:M, calculate or generate from
Step 3: Calculate the particles weight by Equation (22)
Step 4: Normalize the weights as in Step 4 in the summary of standard particle filter operation before in
Section 3.
Step 5: Perform a resampling algorithm as in Step 5 in the summary of standard particle filter operation below, and record the order number of the resampling particles to mi. Then new particles are set for the time of (k−1),
Step 6: State prediction as in Step 2 in the summary of standard particle filter operation before in
Section 3 and get the particles of current time
.
Step 7: Calculate the particles weight by Equation (20), and normalize them.
Step 8: Estimation state update.
Calculate the mean of the particles and update the state estimation
,
7. Conclusions
The particle filter algorithm breaks through the limitation that the process noise and observed noise of the Kalman filter used in Gaussian distribution, which is suitable for the SOC estimation problem of the battery.
The particle filter algorithm was selected as the main algorithm for estimating SOC, and we studied and compared the problem of adaptability and accuracy of PF and APF algorithms based on the model we established and the variable we selected. In addition, a more detail algorithm implementation was discussed in this paper.
Firstly, the Thévenin battery model was established considering the changes of the internal resistance and the capacitance with different SOC, and the particle filter algorithm was combined to complete the simulation analysis of the battery SOC. At the same time, as a comparison, this paper shows how to improve the importance density function of the standard particle filter algorithm. The APF of the importance density function was combined with the battery model to perform SOC estimation under the recursive action of the algorithm. Finally, the two estimation algorithms of PF and APF were compared comprehensively. Although the SOC estimation error of the APF algorithm was only 0.5% less than that of PF, RMSE of the APF algorithm was much better than the PF algorithm, which was reduced by 0.9%.