An accurate estimation of the state of charge (SOC) of the battery is of great significance for safe and efficient energy utilization of electric vehicles. Given the nonlinear dynamic system of the lithiumion battery, the parameters of the secondorder RC equivalent circuit model were calibrated and optimized using a nonlinear least squares algorithm in the Simulink parameter estimation toolbox. A comparison was made between this finite difference extended Kalman filter (FDEKF) and the standard extended Kalman filter in the SOC estimation. The results show that the model can essentially predict the dynamic voltage behavior of the lithiumion battery, and the FDEKF algorithm can maintain good accuracy in the estimation process and has strong robustness against modeling error.
In the context of countries vigorously promoting energy conservation and low carbon economy to solve energy crisis and mitigate global warming, the solar photovoltaic power generation is emerging as the technology of choice for energysaving and environmentally sustainable transportation. It is suggested that the storage battery is second only to the photovoltaic modules as the most important part of solar photovoltaic system; thus its performance will directly affect the operational state and reliability of the system. It highlights the need to quickly and accurately estimate the state of charge (SOC) of the battery. A battery management system is required to ensure safe and reliable operation of the battery. One of its basic functions is to measure the SOC, which indicates the remaining charge of the battery so that the driver can be reminded to charge the battery prior to its depletion.
SOC is usually estimated indirectly by some measurable quantities [
We introduced an alternative nonlinear Kalman filtering technique known as finite difference extended Kalman filter (FDEKF) in this study and used the finite difference method instead of the Taylor series expansions to estimate the covariance matrix. It has theoretical advantages that manifest themselves in more accurate predictions and also strong robustness against modeling uncertainty by making full use of the error information generated by model linearization.
The remainder of this paper is arranged as follows. Section
The commonly used battery models include the electrochemical model and equivalent circuit model (ECM) [
Equivalent circuit of secondorder RC model.
Here
The model parameters need to be accurately estimated from the test data to establish a battery model with good performance. In this paper, the model is parameterized using a semiautomatic process that can satisfy the constraints on the optimized parameters. This process uses a number of measured data sets under a variety of conditions. The parameters are optimized by minimizing the error between measured and simulated results using the nonlinear least squares algorithm in the Simulink parameter estimation toolbox. The battery model can be established in Simulink based on the secondorder RC model, as shown in Figure
Secondorder RC model in Simulink.
In order to use EKF methods for battery SOC estimation, the cell should be modeled in a discretetime statespace form. Specifically, we model the nonlinear battery system by a state equation and an output equation below:
At each time step,
The state of the ECM in Figure
Obviously, the relationship between OCV and SOC is nonlinear. The EKF approach is to linearize the equations at each sample point using Taylor series expansions. The specific steps are as follows.
Define
Initialization is
Recursive calculation is
There are two problems for EKF in SOC estimation:
Schei first conceived the thought of finite difference [
The first two terms on the right hand side of (
First we introduce the following four square Cholesky factorizations:
The factorization of the noise covariance matrices
Then we calculate the partial derivative of the nonlinear function by the firstorder polynomial approximation
Define
Let the
We use (
To verify the effectiveness and performance of the FDEKF, we applied the identification and SOC estimation algorithms to the experimental data obtained on the LP2770102AC lithiumion battery. This is a lithium iron phosphate battery that can be used in portable high power devices, grid stabilization energy storage, and electric vehicles and hybrid electric vehicles. Its nominal capacity is 12.5 Ah and nominal voltage is 3.3 V. For the tests, we used a DigatronMCT 300540 cell cycler with a measurement accuracy of ±5 mV for voltage and ±50 mA for current. The battery temperature was kept at room temperature (20 ± 2°C) throughout the experiment.
In this section, we first estimated the values of ECM parameters using an iterative numerical optimization algorithm implemented by Simulink parameter estimation and then compared the FDEKFbased and EKFbased SOC estimation.
The battery was fully charged so that SOC = 100%, and then the constantcurrent discharge pulse test was performed (18 min discharging and 60 min resting). The discharge lasted 790 min, and the sample time was 1 s. Figure
Terminal current/voltage curves by constantcurrent discharge pulse.
The parameters
Parameter constraints.
Parameter  Initial value  Minimum  Maximum 


3.3 V  2 V  3.6 V 

0.01 
0 
1 

0.005 
0 
1 

0.005 F  0 F  1 F 

10000 
1 
100000 Ω 

10000 F  1 F  100000 F 
The current and voltage data were used as the input to the identification algorithm described in Section
The parameters estimation results with 1/3C discharge.
Measured and calculated battery voltage with 1/3C discharge
Calculated battery voltage error with 1/3C discharge
Trajectories of the optimization variables
Cost function of the optimization variables
We used the dynamic stress test (DST) data to validate the accuracy of the model. Single DST working condition includes 14 steps and it is repeated five times; the process is as follows:
The results of model validation with DST data.
True data of DST and its estimated data
Estimated voltage error of DST data
We then applied the experimental data obtained in Section
Comparison of SOC estimation and error curve of constantcurrent discharge pulse.
True SOC and its experiment results with EKF and FDEKF
SOC experiment errors with EKF and FDEKF
Figure
To evaluate the comprehensive performance of the two filters in the quantitative analysis, we define the root mean square error (RMSE) and single mean computation time
Comparison of SOC estimation.
Algorithm  Single mean computing time  RMSE 

EKF 

0.0174 
FDEKF 

0.0018 
In order to confirm the robustness of FDEKF, we performed another two tests with changing current and DST data using the parameters in Table
Comparison of SOC estimation and error curve of changingcurrent discharge pulse.
Terminal current/voltage curves of changingcurrent discharge pulse
SOC estimation results of changingcurrent discharge pulse
SOC estimation errors of changingcurrent discharge pulse
Comparison of SOC estimation and error curve of DST data with EKF and FDEKF.
Terminal current/voltage curves of DST data
SOC estimation results of DST data with EKF and FDEKF
SOC estimation errors of DST data with EKF and FDEKF
The test bench is shown in Figure
Test bench.
The extended structure diagram.
We tested the single battery under DST working condition on this test bench using the EKF and FDEKF algorithms. Table
Comparison of the simulated and experimental SOC results.
Comparison point  True SOC  Simulation error of SOC  Experimental error of SOC  

EKF  FDEKF  EKF  FDEKF  
First Loop end point  0.172  0.8%  0.4%  1.7%  0.9% 
Second Loop end point  0.352  0.9%  0.3%  2.6%  1.1% 
Third Loop end point  0.532  1.0%  0.4%  4.0%  1.3% 
Forth Loop end point  0.712  1.4%  0.5%  3.7%  1.3% 
Fifth Loop end point  0.893  1.5%  0.7%  4.9%  1.5% 


Average error  —  1.12%  0.46%  3.38%  1.22% 
In this study, we proposed a robust and powerful realtime SOC estimator for the lithiumion batteries, and the parameters of the secondorder ECM were estimated using the nonlinear least squares algorithm. This new linearization technique for SOC estimation is known as the finite difference extended Kalman filter. Compared to the EKF method, the FDEKF method is able to track the realtime SOC more quickly and accurately with the accurate model. When the model parameters change, it also has stronger robustness against modeling uncertainties and maintains good accuracy in the estimation process.
The authors declare that there is no conflict of interests regarding the publication of this paper.