Accurate state of charge (SoC) estimation is of great significance for the lithiumion battery to ensure its safety operation and to prevent it from overcharging or overdischarging. To achieve reliable SoC estimation for Li_{4}Ti_{5}O_{12} lithiumion battery cell, three filtering methods have been compared and evaluated. A main contribution of this study is that a general threestep modelbased battery SoC estimation scheme has been proposed. It includes the processes of battery data measurement, parametric modeling, and modelbased SoC estimation. With the proposed general scheme, multiple types of modelbased SoC estimators have been developed and evaluated for battery management system application. The detailed comparisons on three advanced adaptive filter techniques, which include extend Kalman filter, unscented Kalman filter, and adaptive extend Kalman filter (AEKF), have been implemented with a Li_{4}Ti_{5}O_{12} lithiumion battery. The experimental results indicate that the proposed modelbased SoC estimation approach with AEKF algorithm, which uses the covariance matching technique, performs well with good accuracy and robustness; the mean absolute error of the SoC estimation is within 1% especially with big SoC initial error.
To address the two urgent tasks nowadays of protecting the environment and achieving energy sustainability, it is of a strategic significance on a global scale to replace the oildependent vehicles with electric vehicles. Lithiumion batteries are currently considered to be the development trends of traction batteries and have been widely used in plugin hybrid electric vehicles (PHEVs) due to its high power and energy density, its high voltage, being pollutionfree, having no memory effect, its long cycle life, and its low selfdischarge [
A wide variety of SoC estimation methods have been put forward to improve battery SoC determination, each one having its own advantage, most of which can be divided into four categories: lookingup table based methods, amperehour integral method, datadriven estimation methods, and modelbased estimation methods [
In view of battery behavior and performance being relatively vulnerable to operating conditions and aging levels, what is important for us to do is to achieve the accurate SoC estimation in the longterm. In this point, there are three difficulties that should be considered seriously for achieving efficient and reliable battery SoC estimation for BMS:
In solving the first problem, different kinds of battery models have been proposed and applied to their application field. However, if one applied them to BMS, they can hardly achieve desired performance. It is because of that the adaptive parameter update technique has been neglected; as a result, the model error will be larger as the battery aged or operated conditions changed. In dealing with the second problem, several advanced Kalman filters [
The Li_{4}Ti_{5}O_{12} which can release lithium ions repeatedly for recharging and quickly for high current has been accepted as a novel anode material in Li_{4}Ti_{5}O_{12} LiB. Nevertheless, its dynamic behavior is very different from other LiBs. Traditional battery model fails to ensure high prediction precision in its voltage prediction. A key contribution of this study is that a general threestep modelbased battery SoC estimation scheme, which includes adaptive model parameter updating technique for improving the parametric modeling performance, an open interface for employing adaptive filters to solve the hidden states from strong timevarying dynamic system and series structure based systematic modeling and estimation approach. With the proposed scheme, we have compared the performance of three commonly used filters, which include EKF algorithmbased SoC observer, UKF algorithmbased SoC observer, and AEKF algorithmbased SoC observer. In addition, the Gaussian model describing the open circuit voltage behavior has been developed to improve the performance of the battery model, and then the improved lumpedparameter battery model was applied to the three filters based SoC estimator to improve their prediction performance for Li_{4}Ti_{5}O_{12} LiB (its voltage bounds are 2.7 V and 1.5 V, resp.).
This paper is organized as follows. A description of the general modelbased battery SoC estimation scheme is given in Section
Figure
General modelbased battery SoC estimation scheme.
After the driving cycles loaded on the battery, the datasampling module can measure battery current and voltage in realtime. The measured current and voltage are served as input data for the next modeling and state estimation steps.
Based on the previous research experience on battery model selection [
The schematic diagram of the lumpedparameter model.
With the realtime battery model from Step
It is noted that in essence the Kalman filter based SoC estimation approach is a fusion method. It fuses the estimation results of OCVbased lookup table method and amperehour counting method through the statespace function of battery. SoC serves the communication link between OCVbased lookup table method and amperehour counting method; an inaccurate amperehour counting method will lead to inaccurate SoC estimate and results in a larger erroneous OCV, which will bring bigger voltage error in turn. Thus, the OCV can be served as a feedback control link to regulate the SoC estimation error in amperehour counting method. Therefore, the fusion method has optimal SoC estimation from its capacities of correction, weighting, and filtering. Lastly, together with the realtime measurements of battery current and voltage for online identification of the parameter of battery model, the proposed scheme in fact is modelbased fusion method.
In order to execute the state estimation with statespacebased filter algorithm, we need a model to describe relationship between SoC with battery voltage. According to our previous experience in battery modeling [
To describe the nonlinear characteristic of battery open circuit voltage, the Gaussian model has been selected:
For identifying the parameters of the lumpedparameter battery model, a regression equation in discretization form for (
With (
Defining
The difference equation of (
Thus,
From (
We assume
We use the recursive least squares to implement the parameters estimating process; the parameter estimates are updated at each sampling intervals. The forgetting factor
As mentioned in the introduction section, SoC is a necessary index for ensuring the safety operation of the batteries. SoC can be calculated by the following equation:
Based on the dynamic voltage model of battery and the SoC computational equation, we can build the state equation for recursive prediction and the state equation is described by the following equation:
A few of nonlinear filtering methods have been applied to determining the SoC for electric vehicles batteries, especially of Kalman filtering methods. They are extensively used not only for parameter identification and stats estimation, but also for other typical engineering problems such as global positioning system [
Summary of the EKF algorithm is as follows.
For
For
State estimate time update is as follows:
Error innovation is as follows:
Error covariance time update is as follows:
Kalman gain matrix is as follows:
State estimate measurement update is as follows:
Noise and error covariance measurement update is as follows:
Summary of the AEKF algorithm is as follows.
For
For
State estimate time update is as follows:
Error innovation is as follows:
Adaptive lawcovariance matching is as follows:
Error covariance time update is as follows:
Kalman gain matrix is as follows:
State estimate measurement update is as follows:
Noise and error covariance measurement update is as follows:
A test platform introduced in our previous work [
Results of the coulomb efficiency.
Current/A  3  10  20  40  60  80 

Coulombic efficiency in discharging process  1  1  0.99  0.98  0.95  0.91 
Coulombic efficiency in charging process  1  1  0.99  0.97  —  — 
When we know the exact capacity and coulomb efficiency, we can carry out the OCV test to calibrate the relationship between battery SoC and OCV, and we use 3 A to charge and discharge the cell. In this paper, the hysteresis is ignored. The OCV curves are plotted in Figure
The OCV curves of the test data and the model prediction.
Figure
In addition to the above three types of test, the hybrid pulse power characteristic (HPPC) [
Profiles of the HPPC test: (a) battery current of one cycle; (b) battery terminal voltage (SoC = 0.8); (c) battery current for ten cycles; and (d) complete voltage profile.
Profiles of the DST cycles: current and voltage are plotted in (a) and (b); SoC is plotted in (c).
To achieve an exact SoC, we first charged the battery with CCCV charge mode at the nominal current. Then we discharged some capacity of the battery with the nominal current to achieve an accurate initial SoC. Afterwards, the DST test was loaded and executed. Lastly, a further nominal current discharge experiment was conduct to gain an accurate terminal SoC. Based on the known exact SoC and accurate coulomb efficiency, we can determine the reference SoC profiles with the SoC definitionbased amperehour counting method. The SoC profiles of DST test shown in Figure
Through the online parameter identification operation, we can get the realtime battery model. The voltage profiles of the experimental data and observer are presented in Figure
The statistics list of the terminal voltage error.
Error  Maximum/V  Minimum/V  Mean/V  Covariance/V^{2} 

Value 




The comparisons profiles of the observer and experiment: (a) voltage; (b) voltage error.
The following verification and analysis are based on the AEKF algorithm, and the other two Kalman filters will be discussed in the Section
SoC estimation with AEKF approach: (a) voltage; (b) voltage error; (c) SoC; (d) SoC error.
Selfcorrecting capability for erroneous initial SoC: (a) estimation with SoC_{0} = 0.95; (b) SoC estimation error for (a); (c) estimation with SoC_{0} = 0.50; and (d) SoC estimation error for (c).
We can observe that the prediction inaccuracy of the battery terminal voltage is below 1%, which is lower than the prediction result plotted in Figure
With an accurate initial SoC, most of the SoC estimators can achieve desired estimation performance in a period of time, such as amperehour integral method. However, the estimation accuracy against different unknown initial SoC makes lots of methods unacceptable for electric vehicles application. In this section, we will discuss whether the AEKFbased SoC estimation can achieve accurate SoC estimation with the erroneous initial SoC. Two types of erroneous initial SoC, 0.95 and 0.50, are applied to implement the evaluation. The estimation results are plotted in Figure
From Figures
Figure
MAE results of SoC estimation with AEKF approach.
Based on the above analysis, we can find that the AEKF algorithm is suitable for applying to proposed general modelbased battery SoC estimation scheme, and it can reach high estimation accuracy. To discuss the suitability of the proposed general SoC estimation scheme and compare the AEKF algorithm with other widely used methods, UKF algorithm and EKF algorithm, we have made systematical analysis. The realtime modelbased SoC estimation results using the EKF and UKF are plotted in Figure
SoC results of EKFbased estimation and UKFbased estimation with two different types of SoC_{0}.
Errors of EKFbased estimation and UKFbased estimation.
MAE results of EKFbased estimation and UKFbased estimation.
From Figures
The comparisons between the estimation inaccuracies and MAE index of the AEKFbased, EKFbased and UKFbased methods, show that the maximum MAE of the EKFbased and UKFbased approaches are around 3%, which is higher than the AEKFbased approach. In conclusion, the proposed general modelbased battery SoC estimation scheme can be applied to estimate the SoC of batteries accurately with good robust performance. More importantly, the performance of the proposed scheme is not sensitive with the operated nonlinear filtering methods. For the algorithms of AEKF, EKF, and UKF, all of their estimation errors are less than 5%. It is acceptable for the current requirements of the battery management system. Furthermore, the AEKF algorithm, which can update the error covariance matrix adaptively, has the best estimation accuracy when applied to the proposed scheme and Li_{4}Ti_{5}O_{12} lithiumion battery cell.
This paper presents a comparison of nonlinear filtering methods for estimating the SoC of Li_{4}Ti_{5}O_{12} lithiumion battery. The Gaussian model has been selected to improve the prediction precision of the dynamic battery model. With the new battery model, general modelbased battery SoC estimation has been proposed. It contains the adaptive model parameter updating technique for improving the parametric modeling performance, an open interface for employing adaptive filters to solve the hidden states from strong timevarying dynamic system, and series structure based systematic modeling and estimation approach. Three Kalman filters are employed to build modelbased SoC estimator. With the proposed modelbased scheme, three advanced Kalman filters, which include extended Kalman filter, unscented Kalman filter, and adaptive extended Kalman filter, have been employed to develop the SoC estimator.
The detailed evaluation and comparison are made for modelbased SoC estimator. A comparison for the SoC estimation approach among the AEKFbased, EKFbased, and UKFbased algorithms with the Li_{4}Ti_{5}O_{12} lithiumion battery shows that the proposed method has superior performance, which indicates that the covariance matching approach for EKF is a useful way to improve its filter performance. Adaptive extended Kalman filter is an optimal choice for battery SoC estimation. Experimental results show that the AEKFbased approach can estimate the battery SoC accurately. Further, for different SoC initial values with big error, the mean absolute errors of the SoC estimation are all within 1%; more importantly, the AEKFbased approach can ensure the estimates converge to true values quickly, less than 60 sample intervals.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (51276022) and the National Science & Technology Pillar Program (2013BAG05B00).