Modeling and Control Problems in Sustainable Transportation and Power Systems
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Jianping Gao, Hongwen He, "Comparison of Nonlinear Filtering Methods for Estimating the State of Charge of Li_{4}Ti_{5}O_{12} LithiumIon Battery", Mathematical Problems in Engineering, vol. 2015, Article ID 485216, 12 pages, 2015. https://doi.org/10.1155/2015/485216
Comparison of Nonlinear Filtering Methods for Estimating the State of Charge of Li_{4}Ti_{5}O_{12} LithiumIon Battery
Abstract
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.
1. Introduction
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 [1–3]. Battery management system (BMS) is essential for the lithiumion battery to maximize its performance, ensure its safety, and extend its life. Estimation for battery state of charge (SoC) is one of the most key techniques in the BMS. Nevertheless, it is difficult to accurately estimate SoC, because SoC is an inner state of each battery cell which cannot be directly measured and is greatly influenced by many factors, including ambient temperature, discharging current, and battery aging [4, 5]. Therefore, the battery SoC has to be estimated with specific estimation techniques according to measured battery parameters, such as voltage, current, and temperature.
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 [6–21]. In terms of the lookingup table based methods, the measurements of battery impedance, open circuit voltage, and so forth are commonly used to infer the SoC of batteries [6–9]. References [10, 11] put forward methods to determine battery SoC based on impedance measurements. Unfortunately, due to the uncertainty of driving cycles and the variable application environment, it is hard to measure these characteristic parameters accurately in realtime. The second type is the amperehour integral method, which has been widely applied to BMS and battery simulations of PHEVs [12]. The amperehour integral method acquires the SoC by integrating the current over the time. This method is simple and can be easily implemented onboard, so that it has been widely used in practice. However, as an openloop estimation algorithm, it cannot deal with problems caused by measurement noise and inaccurate initial SoC [13]. The third type is the datadriven estimation methods. Datadriven control methods merely use inputoutput data of the system to develop a controller. Since these methods do not require an accurate plant model, the estimations and assumptions introduced in the plant modeling step are omitted. For instance, the battery state estimations based on artificial neural network and fuzzy logic models are demonstrated with good accuracy [14, 15]. However, these algorithms are very sensitive to their parameters and they may even be not convergent with bad parameters selection when the train data cannot completely cover the present operating conditions. The last type is modelbased estimation method [16–20]. Plett used an extended Kalman filter (EKF) to identify unknown parameter and adaptively estimate the battery’s SoC [16–18]. An inappropriate matrix of the system noise may lead to remarkable errors and even divergence [19]. As an alternative approach, SoC estimation via SoCOCV lookup table is put forward and the battery open circuit voltage (OCV) is estimated in realtime with a recursive least squares algorithm [21]. Additionally, the unscented Kalman filter (UKF) is also investigated to estimate the lithiumion battery (LiB) SoC [20]. However, the OCVbased SoC estimation method can hardly safeguard the estimation accuracy dynamically for its openloop characteristics.
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: accurate parametric modeling approach, reliable and robust state estimation algorithm, and systematic modeling and estimation scheme.
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 [13, 16–18] have been widely used and the estimation accuracies are appropriate for BMS application in a limited operating condition. For overcoming the last difficulties, the offline model and adaptive filters are widely used. However, it is limited by the unupdated battery model; the SoC estimation will be influenced by the unavoidable uncertainties from battery which aged and operating conditions varied.
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 2. The verification experiments and results are described in Section 3. A detailed comparison among the EKFbased, UKFbased, and AEKFbased SoC estimation approaches is illustrated in Section 4. Finally, conclusions are drawn in Section 5.
2. ModelBased SoC Estimator
2.1. General ModelBased Battery SoC Estimation Scheme
Figure 1 presents the modelbased battery SoC estimation scheme. It contains three steps.
Step 1 (realtime measurement). 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.
Step 2 (parametric modeling). Based on the previous research experience on battery model selection [21, 22], one RC network based lumpedparameter battery model (shown in Figure 2) has been chosen for describing the dynamic behavior of the LiB cell. In parametric modeling step, the battery model has been divided into two parts: OCV and dynamic voltage. Considering that OCV varies with battery SoC, it is common to use variation of OCV to achieve an accurate feedback of SoC. Therefore, we use a nonlinear battery OCV model to describe battery voltage source. With the estimated SoC, the OCV will be updated accordingly. For dynamic voltage part, we use recursive least squares with optimal forgetting factor to identify the parameter in realtime. The description of recursive least squares can be found in [22]. Through the process of Step , we can get online model parameter with realtime measurements of battery current and voltage.
Step 3 (modelbased approach). With the realtime battery model from Step , we can build the statespace function of the battery accordingly. With the statespace function, the adaptive filters can be used to estimate the hidden state (SoC) through operation processes of prior estimation, posterior estimation, and feedback control. In this study, we use three kinds of Kalman filters to verify this approach [21, 23, 24]. The three kinds of advanced Kalman filters have been employed to develop the SoC estimator and their performance has been analyzed and evaluated in Section 4.
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.
2.2. Battery Model
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 [21], one RC network based lumpedparameter model was chosen for this study. However, to improve the model’s accuracy and enhance the relationship between the SoC with battery voltage, a Gaussian model [17] is developed to improve its prediction precision; the model is illustrated in Figure 2. The main dynamic voltage behavior is described bywhere and are the OCV and load current, respectively. and denote the terminal voltage and ohmic resistance of battery, respectively. The RC network is employed to describe the dynamic voltage behavior including polarization resistance and polarization capacitance , and represents the polarization voltage across . It is noted that the positive value of battery current is assumed for discharging operation while the negative value is assumed for charging operation.
To describe the nonlinear characteristic of battery open circuit voltage, the Gaussian model has been selected:where , , () are the constants chosen to make the model fit the SoCOCV well and is the abbreviation of battery SoC.
2.3. Parameters Identification
For identifying the parameters of the lumpedparameter battery model, a regression equation in discretization form for (1) is needed which has been rewritten as follows [3]:where denotes the sampling interval and can be equivalent to at the th sampling times. denotes the time constant and equals . is the current at the time index , and and are terminal voltage and OCV at the time index , respectively.
With (3),
Defining , then (4) can be rewritten by
The difference equation of (5) can be used to eliminate ; then,
Thus,where
From (8), we can obtain
We assume , wherewhere and denote matrixes of input data and parameters at time index , respectively.
We use the recursive least squares to implement the parameters estimating process; the parameter estimates are updated at each sampling intervals. The forgetting factor is optimized by the genetic algorithm and the optimization objective is to achieve the minimum sum of squares of the estimated voltage error. After presetting initial vector and its error covariance matrix , the parameter vector can be determined.
2.4. SoC Estimation with the Kalman Filter Algorithm
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:where denotes the SoC at the th sampling interval and denotes the columbic efficiency of battery. is the sampling interval. denotes the maximum available capacity of battery.
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:where the function is used to describe the state transition of the nonlinear system, the parameter denotes the system state, and the denotes the input of this system. The function is used to describe the measurement process of this nonlinear system, denotes the measurement of the system. To describe the unavoidable noise information, a Gaussian white noise was assumed to describe the process noise () and measurement noise (). These two types of noise possess zero of mean values, and their covariance is described by vectors and , respectively.
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 [16–18, 21]. Based on [16–18], we can conclude the detailed computational process for EKF algorithm and AEKF algorithm, which have been listed below. And the algorithm of UKF algorithm can be found in [13].
Summary of the EKF algorithm is as follows.
Step 1 (initialization). For , set
Step 2 (computation). For , compute the following.
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:wherewhere is Kalman gain matrix at the th sampling time; is the priori estimate of before the measurement is taken into account, is the estimate of after the measurement is taken into account, which is called posteriori estimate; and are the terminal voltage measured and estimated by the battery model at the th sampling time, respectively.
Summary of the AEKF algorithm is as follows.
Step 1 (initialization). For , set:
Step 2 (computation). For , compute the following.
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:where is the innovation covariance matrix based on the innovation sequence inside moving estimation window of size at the th sampling time.
3. Experimental Configurations and Battery Test
A test platform introduced in our previous work [21] was employed to implement the systematic test for evaluating the performance of the proposed modelbased SoC estimation algorithm. Battery used in the test is the Li_{4}Ti_{5}O_{12} lithiumion battery. The battery was placed in a thermal chamber and the test schedule can be found in [21]. The maximum available capacity of the battery is 9.8 Ah [25]. After the static capacity test, an efficiency test has been implemented for determining the coulomb efficiency of the Li_{4}Ti_{5}O_{12}; the results are listed in Table 1 [21].

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 3.
Figure 3 gives the OCV comparison between the model estimates and the test data. It suggests that the proposed model can track the battery OCV behavior closely. The OCV function can be described by the following equation:
In addition to the above three types of test, the hybrid pulse power characteristic (HPPC) [26] and the dynamic stress test (DST) have been carried out for verifying the proposed general modelbased SoC estimation method. The measurements of HPPC test and DST are plotted in Figures 4 and 5, respectively.
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(b)
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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 5(c) will serve as a reference SoC for the evaluation the robust SoC estimation, and its actual initial SoC is 0.8075.
4. Verification Analysis and Discussion
4.1. Parametric Modeling
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 6(a), and the voltage error is shown in Figure 6(b). A statistical analysis on the terminal voltage errors has been conducted and the result is listed in Table 2. It shows that the maximum error of the terminal voltage is less than 2% of its nominal voltage. Additionally, its mean error is 6 mV. It can be concluded that the proposed improved model has good dynamic voltage prediction performance.

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4.2. Analysis on the SoC Estimation Results
The following verification and analysis are based on the AEKF algorithm, and the other two Kalman filters will be discussed in the Section 4.3. The SoC estimation is conducted and the results are shown in Figure 8. Figure 7(a) presents the comparison profiles of the voltages between the estimations and the experimental result. Figure 7(b) presents the voltage error curve for Figure 7(a). Figure 7(c) presents the SoC comparison profiles between the estimations and the reference SoC. Figure 7(d) presents the SoC estimation error.
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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 6(b). The prediction precision depends adaptively on correction performance by adjusting the Kalman filter gain matrix based on the voltage error bound. Figure 7(d) indicates that the SoC estimation error arises with the terminal voltage estimation error, while the AEKF approach can correct the voltage error adaptively and quickly. It also shows that the model precision is important during the estimation for battery SoC.
4.3. Evaluation on the Robust Performance
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 8.
From Figures 8(a) and 8(c), we can observe that the performance of the modelbased SoC estimator cannot be affected by the erroneous initial SoC. In contrast, the convergence trajectories of the SoC estimations are very similar, especially when the initial error in battery SoC has been corrected. We also can observe that the SoC estimation errors are less than 1%. In order to evaluate the estimation performance more intuitively, we select the index of the mean absolute error (MAE) to describe the convergence trajectory. The MAE can be calculated by the following equation:where represents the MAE index of the battery SoC estimations up to and involving the th sample time; denotes the SoC estimation value under the time index .
Figure 9 shows the MAE results. It indicates that the MAE of the SoC estimations which started with incorrect initial SoC values can converge to within 1% in 1 minute (60 sample intervals).
4.4. Discussion
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 10. The estimation errors between the EKFbased SoC estimations and UKFbased estimations are plotted in Figure 11 and the MAE index is plotted in Figure 12.
From Figures 8–12, it can be observed that the AEKFbased approach achieves the best accuracy in these three approaches. Additionally, the AEKFbased SoC estimation is more precise than the EKFbased SoC estimation. The better estimation performance of the AEKF algorithm based method is due to the fact that the adaptively updating for the error covariance greatly improves the estimation performance. The SoC estimation error between the EKFbased method and the UKFbased method is virtually indiscernible.
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.
5. Conclusions
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (51276022) and the National Science & Technology Pillar Program (2013BAG05B00).
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Copyright
Copyright © 2015 Jianping Gao and Hongwen He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.