Research Article  Open Access
Zheng Liu, Xuanju Dang, Hanxu Sun, "Online State of Charge Estimation for LithiumIon Battery by Combining Incremental Autoregressive and Moving Average Modeling with Adaptive HInfinity Filter", Mathematical Problems in Engineering, vol. 2018, Article ID 7480602, 16 pages, 2018. https://doi.org/10.1155/2018/7480602
Online State of Charge Estimation for LithiumIon Battery by Combining Incremental Autoregressive and Moving Average Modeling with Adaptive HInfinity Filter
Abstract
The state of charge (SOC) estimation is one of the most important features in battery management system (BMS) for electric vehicles (EVs). In this article, a novel equivalentcircuit model (ECM) with an extra noise sequence is proposed to reduce the adverse effect of model error. Model parameters identification method with variable forgetting factor recursive extended least squares (VFFRELS), which combines a constructed incremental autoregressive and moving average (IARMA) model with differential measurement variables, is presented to obtain the ECM parameters. The independent open circuit voltage (OCV) estimator with error compensation factors is designed to reduce the OCV error of OCV fitting model. Based on the IARMA battery model analysis and the parameters identification, an SOC estimator by adaptive Hinfinity filter (AHIF) is formulated. The adaptive strategy of the AHIF improves the numerical stability and robust performance by synchronous adjusting noise covariance and restricted factor. The results of experiment and simulation have verified that the proposed approach has superior advantage of parameters identification and SOC estimation to other estimation methods.
1. Introduction
In the field of electric vehicles (EVs), the Lithiumion batteries have been widely researched and quickly developed in battery management system (BMS) at present [1, 2]. Onboard BMS is used to guarantee the safety of the work and reliable battery application in high energy EVs. An accurate SOC is one of the core functions in the BMS, which provides the fundamental principle of the battery available capacity for maintaining a longer lasting time. In many cases, other additional functions in the BMS also rely on precise SOC estimation [3, 4]. Beyond that, the accurate SOC estimation is dramatically impacted on the time scale for charge and discharge; otherwise overcharge and overdischarge will have serious and negative impact on battery life time and EVs driving safety. Therefore, a reliable and highprecision SOC prediction method is significant for the optimization of BMS in EVs control strategy [5, 6].
Various efforts focusing on the estimation of SOC have been made in the last few years. In general, the known estimation methods could be classified into two categories: modelfree methods and modelbased methods. The classical modelfree approach is the AmpereHour integral (AH) [7, 8] that is the remainder charge is equal to the integral of charge or discharge current. Nevertheless, two preconditions should be satisfied simultaneously, which are exact initialization SOC data and the highprecision sensor. In the process of current integration, AH suffers from the accumulative error and rounding error in disturbances condition, such as current or temperature fluctuations, etc. The OCV measurement [9–11] also is commonly used technique for SOC estimation based on the nonlinear OCVSOC fitting curve under incremental OCV test or lowcurrent OCV test. However, it is a poor choice for realtime SOC estimation in EVs ongoing application because the battery's steady state is not achieved until the long rest time. In addition, the machine learning approach (MLA), such as the artificial neural network (ANN) [12, 13], the fuzzy logic [14, 15], and the support vector machine (SVM) [16, 17], takes a Lithiumion battery as a blackbox model. Although MLA does not require the exact currentvoltage information of battery dynamics, it is almost impossible to cover all the battery loading data through studying the training dataset. And because of large amount of time consumption, it is not a better design for BMS.
In the modelbased estimation methods, the choice of model should be firstly determined. Equivalentcircuit model (ECM) is the most widely used model for battery applications. The ECM can be classed as one reduced form of complex electrochemical model, which is applied to approximate the battery dynamic behavior with identified parameters. In [18], the dynamic response of battery can be simulated by oneorder ECM, and a recursive least squares (RLS) method is used for parameters identification. Other types of ECM which introduce two or three order RC network are presented in [19, 20]. Although multiorder ECM can lead to better performance in representing battery dynamic behavior than the oneorder ECM, the more ECM parameters need to be identified with deepening of algorithm complexity. In [21, 22], an additional noise is added to oneorder ECM, but the RLS with constant forgetting factor may encounter the difficulties of compromise between stability and convergence. Therefore, the effective ECM is largely dependent on the balance between complexity and precision of corresponding algorithm.
In the ECMbased state observers, the flexible usage of Kalman filter has been generally accepted for its better performance of optimal estimation, which assumes some special properties of the problem formulation, such as model errorfree and known noise statistics. For example, the extended Kalman filter (EKF) algorithm is applied for Lithiumion battery state estimation to obtain a desired resistance estimation result [23, 24]. But with the strong fluctuating current problems and the undeniable error of the firstorder Taylor series approximation, the convergence speed of EKF is slow or even cannot get a convergence solution in time. The unscented Kalman filter (UKF) with unscented transformation (UT) method is utilized to handle the SOC estimation problem in strong nonlinear battery systems [25, 26], and the estimation results show that the UKF has better robustness and higher precision than the EKF. The adaptive EKF (AEKF) and adaptive UKF (AUKF) based on the dynamic adaptive strategy of adjusting noise were adopted in state estimator to achieve the goal of highprecision and better stability than EKF and UKF [27, 28]. In [29–33], an HInfinity Filter (HIF) based on the minimax residual error criterion observer is designed for model parameters identification and state estimation, which has more robust to model error uncertainty without assumptions of accurate battery model. Generally, the HIFs are more robust than KFs [34, 35], but they do not sufficiently consider the effect of the error of process noise and measurement noise. In other words, the uncertain covariance of noise can result in unavoidable estimation error of the HIF. One common feather of the aforementioned methods is that OCV can be calculated from OCVSOC fitting model [23–35]. However, due to fitting error in OCVSOC curve [9], any small perturbation in OCV calculation by OCVSOC fitting model may cause a large deviation in SOC estimation, especially a wide flat area existing in the OCVSOC curve with a strong nonlinear relationship.
The main contributions of this paper include the following: (1) a novel IARMA battery model based on VFFRELS method is designed to identify the battery model parameters; (2) an error compensation based independent OCV estimator is adopted to accurately capture the OCV; (3) with design and application of adaptive HIF (AHIF) including noise covariance and restricted factor adjustment to the proposed model, the SOC is directly estimated with reasonable performance. Compared with the wide usage of AEKF, the proposed method has the features of accurate model, strong robustness, and high precision.
The remainder of this paper is arranged as follows. Section 2 presents details of the IARMAbased battery ECM and the model parameters identification based on VFFRELS algorithm. The error feed forward compensation based independent OCV estimator is analyzed in Section 3. Then based on the IARMA battery model and identified parameters, an AHIFbased SOC estimator is proposed in Section 4. Section 5 shows the experimental and simulation results under the different conditions. Finally, some conclusions are drawn in Section 6.
2. Battery Model and Parameters Identification
2.1. Battery Model Derivation
In a simplified model sense, the dynamic electrical characteristic of the Lithiumion battery can be commonly represented by the oneorder ECM [18], as shown in Figure 1.
Oneorder ECM is expressed as follows:where is used to describe the SOC dependent OCV which is associated with OCVSOC fitting curve, indicates the ohmic resistance, the parallel composed of a polarization resistance and a polarization capacitance, represents polarization voltage across parallel , is the current in the charging and discharging status, and represents the terminal voltage.
The oneorder ECM cannot completely describe battery dynamic nature, since the model random noise characteristic is not included in (1). As illustrated in Figure 2, a colored noise has been added as a compensation module in oneorder ECM, and is to cover the random error which is not considered in the oneorder ECM [21, 22].
Based on Figure 2, the mathematical model of oneorder ECM with additive colored noise can be derived as
In (2), the colored noise is a noise sequence driven by a series of white noise sequence, and are its regression of data; thus,
The nonlinear time series model is adopted to analyze battery ECM; let u and y be the charge/discharge current and the terminal voltage of battery ECM, respectively. The ECM in (2) to be identified can be redefined as a multivariable regression expression: where and are the orders of regression expression; and are the undetermined coefficient.
2.2. Parameters Identification
By comparing (2) with (4), the polarization voltage is firstly needed to eliminate from (2). The discretization form of (2) by bilinear transformation method: ( is the discretization operator) is given as follows:where , and is the OCV, terminal voltage, and load current at the kth sampling time, respectively, and the coefficient can be obtained by , , . Then , , and can be obtained according to the inverse equations of , , and ; thus, , , .
By comparing (5) with (4), the immeasurable part is assumed as the residual model error (RME) of autoregression (AR) model. Since the above colored noise sequence can be used as a MA process, (5) can be considered as an autoregressive moving average model (ARMA) with external input system.
Similarly, the terminal voltage of the previous step could be expressed as follows:
By subtracting (6) from (5), the incrementalbased ARMA (IARMA) equation will be derived as follows:where , , and are calculated as , , and , respectively. Meanwhile the immeasurable part is also assumed as the RME of IARMA model [36, 37].
If the parameters identification algorithm is based on (5) or (7), whether the RME can be ignored compared with terminal voltage in (5) or (7) is the first consideration. The detailed property of two RME of ARMA and IARMA will be analyzed as a quantifiable form in Section 5.
From (7), it is seen that and of the IARMA model are and , respectively. In accordance with nonlinear regression mode principle, (7) can be rewritten as leastsquares form: where
In (8), is the unknown parameters set, and is the coefficient determined by known variable. Since the time sequence of is immeasurable, cannot be got directly; therefore some traditional least squares (LS) series online identification algorithms are unable to identify . Accordingly, the RELS algorithm [21, 22] is introduced to identify the immeasurable noise terms first. Based on (5) and (8), can be approximated to the difference between measured terminal voltage and identified terminal voltage; meanwhile, can be approximated to the difference between measured differential terminal voltage and identified differential terminal voltage, as follows.where
Thereafter, can be replaced with , and we define the following: where , is a given initial value of the noise sequence.
With the updated by (11), the regression data of colored noise can be obtained by (5)(10). The procedures of IARMA and RELS based identification method are illustrated in Figure 3.
The RELS is used to solve the regression model described in Figure 3. However, the RELS with constant forgetting factor may encounter the difficulties of balancing between stability and convergence if the model parameters change with different rates. The battery dynamics nature of oneorder ECM can be distinguished into two slow parameters and one fast parameter [38–40]. The fast parameter represents internal ohmic resistance , while the slow parameters include and . In this regard, a large forgetting factor should be assigned to the parameters changing slowly to guarantee the stability of the algorithm, while a small forgetting factor is more appropriate for the tracking of fast varying parameter. In seeking to address this problem, the VFFRELS with multiple forgetting factors [41–43] for identification is applied in this paper. With the VFFRELS, the forgetting factors can be decoupled and tuned separately to improve the parameters stability and tracking accuracy of SOC estimation.
3. Independent OCV Estimator with Error Compensation
3.1. OCV Estimator
In this section, an independent OCV estimator is introduced. The OCV is observed with the aid of battery ECM characterization. in (1) can be rewritten in the discretetime form as follows:
From (1), obviously can be expressed in the following discretetime relationship as follows:
Substituting (13) into (12) yields the following:
Since the OCV can be described as a slow timevarying variable, then the OCV estimation can be solved by the expression of , , and , as follows:
3.2. OCV Error Compensation
The OCVSOC curve can be easily obtained under incremental OCV test and lowcurrent OCV test [35]. The nonlinear characteristic relationship of OCVSOC fitting model is built in processing the experimental data by the polynomial curve fitting methods. where is the polynomial fitting coefficient, N is the order of fitting model, and N is set to 8.
The direct calculation of OCV by OCVSOC fitting model contains perturbations due to the existence of fitting error. Therefore, the OCV error will largely decline the accuracy of SOC estimation. Besides, there exists a flat area in the OCVSOC curve with a strong nonlinear relationship when the value of SOC varies from 0.2 to 0.8 [18]. Even if the measurement error of voltage or current is relatively slight, the OCV error can lead to a relatively large OCVbased SOC estimation error. To solve this problem, OCV is corrected with two compensations by considering the intrinsic OCV variation.
First, the residual error of terminal voltage is introduced to approximately describe the behaviors of deviation of OCV. A microadjustment factor is adopted for compensating error caused by nonequivalence behavior between terminal voltage and OCV. The simple correction formula is shown as follows:where is used as first compensation, microadjustment factor is below a certain range ( in this paper), and the proportion coefficient is 10^{2}. The microadjustment factor is determined by the slope of OCVSOC curve.
Next, due to the fitting error in OCVSOC fitting model, the second appropriate compensation is necessary to further eliminate OCV error. The second compensation is adopted to reduce the fitting error in OCVSOC fitting model. for OCV is equal to the behaviors of the deviation between by OCV estimator and by OCVSOC fitting model.
The block diagram of two OCV compensation methods is shown in Figure 4. The terminal voltage estimation value () and OCV estimation value () are utilized to the get corresponding compensation factor ( and ) which are employed to correct the OCV by the feed forward method.
4. Adaptive HIF Based SOC Estimator
4.1. HIF with Tunable Restricted Factor
In the strict sense, the accurate ECM and noise statistics quantity is the precondition to bring into playing KFs advantages. However, the KFs cannot be guaranteed to work stably all the time when these conditions cannot be satisfied. The HInfinity Filter (HIF) based on the game theory which is designed to minimize the maximum estimation error can make up for the lack of the precondition; thus it is common in resolving the problem of system identification in [29–31]. The SOC estimation in [32–35] shows that the HIF can provide the impressive results with inexact or unknown initialized state. The cost function of HIF is defined as follows: where is the state value and is state estimation; and are the initial states value. and are the process noise vector and measurement noise vector, respectively. , , , and are the symmetric positive definite matrixes.
Further, since it is difficult to directly minimize cost function , a userspecified restricted factor which has important influence on the precision and robustness is presetting to guarantee an optimized boundary constrained condition, making sure that satisfies . In [32, 33], the restricted factor is set to a fixed value based on the engineering practice experience, and three restricted factors of 1.531, 1.048, and 1.003 are adopted in state estimation. For ensuring both robustness and precision, the selfadapting restricted factor is incorporated into HIF by innovation mechanism.
By applying the matrix inversion lemma to , we obtain
As is a covariance matrix, it therefore should be positive definiteness. This means that
In other words, the restricted factor should satisfywhere denotes the greatest eigenvalue of matrix ; therefore, selfadapting method for restricted factor is where is used to correct and ; the quadratic sum of innovation error from (27) is used as estimation error . As the restricted factor is inversely proportional to the estimation error, the correction coefficient expression is shown as follows:where is an unknown coefficient associated with experiments and , once is determined, the correction coefficient is only dependent on innovation error . That is, the relationship between restricted factor and estimation error can be quantitatively depicted by (22)(23).
The calculation process of the discretetime HIF with tunable restricted factor is shown as follows:
4.2. HIF with Adaptive Strategy
Multiple adaptive ways have integrated into original KFs such as the Bayes method, maximum likelihood criterion, correlation method, and covariance matching method. As mentioned before, the HIF approaches to KF when . Hence it can be concluded that the noise covariance matrix and in HIF play the similar role in the KFs which should be adjusted properly. Therefore, an adaptive HIF with the maximum likelihood criterion [34] is designed to update the noise covariance at each stage of measurement correction to strengthen its jamming immunity of SOC estimation.
According to (2), the innovation error can be written as an expression of difference between measurement estimation and true measurement value:
The measurement innovation covariance is deduced through the use of probability theory.where denotes the covariance of , then the definition of innovation estimation variance is introduced to describe through using moving window method of innovation.
By combining (28) and (29), the following observation noise covariance equation can be drawn:
Similarly, the state noise can be expressed as follows:
Substituting the prior state estimation into (31) yields
Based on the principle of orthogonality between the innovation and the residual, the state noise covariance is taken on both sides of (32).
Based on the adaptive strategy analysis, the adaptive iteration approach which is detailed in (30) and (33) is utilized to compensate for process and measurement noises uncertainties in HIF algorithm.
4.3. SOC Estimator Based on the AHIF
First, the ECM parameters and OCV are identified as two independent parameters set by using VFFRELS and compensation method in improved oneorder ECM. Next, the HIF accompanied with tunable restricted factor and adaptive strategy is applied to SOC estimation. The formulas of state and observation are as follows:where , is the terminal voltage, represents a linear combination of the state, and the userdefined matrix . The system matrixes which describe the dynamics of the ECM are given as follows:
With the equations of state and observation above, the battery SOC is estimated by using the improved AHIF. The process of the HIF with tunable restricted factor for the SOC estimation can be seen in (24)(26), the noise covariance adaptive adjusted strategy is given in (30) and (33), and the general flowchart of the IARMAAHIF based parameters and SOC joint estimation method is illustrated in Figure 5. The SOC estimation is obtained by recursion procedure based on identified ECM parameters and OCV.
5. Results and Discussion
To evaluate the performance of parameters identification and SOC estimation, the 20Ah/24V Lithiumion phosphate battery is selected as the test objects, in which the cathode and anode are formed from Lithium iron phosphate and graphite, respectively. In order to acquire highprecision experimental data such as charge/discharge current, voltage, available capacity, etc., the battery experimental platform is set up for the validation experiments. The reliable test platform is composed of a host computer which is used for data collection and data processing, a high precise power battery test system (Arbin Corp, EVTS) with the maximum charging/discharging current of 300A and maximum voltage of 400V, and a programmable temperature chamber to provide controllable temperature.
The data of two operating conditions including Dynamic Stress Test (DST) and Federal Urban Driving Schedule (FUDS) at a constant temperature 25°C for the battery are collected to evaluate the performance of model parameters identification and SOC estimation. In addition, the statistical index such as maximum absolute error (MAE) and average absolute error (AAE) are used to qualify performance of the identification and estimation algorithms.
5.1. Model Parameters Identification Results and Comparative Analysis
First, the Lithiumion battery is fully charged by standard charging method, and then it is left to rest for one hour before being discharged in working condition. The discharge currenttime distribution of the DST and FUDS cycle in which the duration of uninterrupted working condition is 35000 seconds is shown in Figure 6. With the data flows of collected discharge current and terminal voltage, the VFFRELS algorithm and IARMA model are combined to identify the ECM parameters and terminal voltage. The identification results of , , and are shown in Figure 7.
(a)
(b)
(a)
(b)
As shown in Figures 7(a) and 7(b), the identification values of ECM parameters is able to converge to stable values rapidly in the DST and FUDS cycle, respectively. Among them, the ohmic resistance and the polarization resistance exhibit the similar tendencies except for a little fluctuation. Specifically, keeps a higher stability and for the ohmic resistance is equal to the ratio of terminal voltage variation to transient current when the variable current is turned off. By contrast to and , the polarization capacitance varies significantly because and are direct correlation to intricate electrochemical activity. Although the introduction of multiorder ECM can lead to better stable and in representing battery dynamic characteristics than the oneorder ECM in theory, the more ECM parameters need to be identified with increasing of algorithm complexity which is not suitable for realtime application. To summarize, and oneorder of ECM have to realtime change with limited amplitude during operating condition to keep closer track of electrochemical activity.
With the identified , , and , the terminal voltage as ECM observation is recursive obtained in each sampling period. Figures 8(a) and 8(b) demonstrate the two modelbased measured and identified battery terminal voltages under DST cycle. The red line in Figure 8(a) is the ARMAbased estimation value and the blue line denotes the IARMAbased estimation value. As indicated in Figure 8(b), the AAE of ARMA and IARMA are 0.048% and 0.0012%, respectively, correspondingly the MAE are 0.42% and 0.075%, respectively. Figures 9(a) and 9(b) are shown for the measured and estimated battery terminal voltages under FUDS cycle, the red line in Figure 9(a) is the ARMAbased estimation value and the blue line denotes the IARMAbased estimation value. In Figure 9(b), the AAE of ARMA and IARMA are 0.058% and 0.0021%, respectively; correspondingly the MAE are 0.52% and 0.10%, respectively. As illustrated in Figures 8 and 9, the two modelbased error stabilization of the terminal voltage has been achieved as continuous discharge, but it has small amplitude fluctuations. It means that the IARMA ECM and the ARMA ECM can both stably track the measurement value of the terminal voltage. The estimation error of the IARMA ECM is obviously decreased compared with the ARMA ECM. As summarized in Tables 1 and 2, the maximum terminal voltage error (MTVE) of the ARMA ECM is up to 109mV and 138mV under the DST and FUDS cycle, respectively; correspondingly the MTVE of the IARMA ECM is only 18 mV and 25mV under the DST and FUDS cycle, respectively. The average terminal voltage error of the IARMA ECM is only 0.35 mV and 1.3mV under the DST and FUDS cycle, respectively, which can be neglected regarding the sensors precision of the EVTS.


(a)
(b)
(a)
(b)
Besides, from (5) and (7), it can be inferred that the assumed RME indicates the effect of the OCV in the ARMA ECM is . Similarly, the RME which defines the distribution of OCV with incremental analysis method in the IARMA ECM can be represented as . As the simulation results in Figure 10, the RME value of ARMA ECM is in a range from 0.6943 to 0.6941, and the maximum RME value of IARMA ECM is less than 0.03. By comparison with the Figures 10(a)–10(d), the RME in IARMA ECM is far smaller than that in ARMA ECM. Therefore, the proposed IARMA ECM can provide much better performance in parameters identification.
(a)
(b)
(c)
(d)
According to the above analysis, these comparing estimated results of the RME and terminal voltage error fully verify the capacity of the IARMA ECM to describe battery electric property. And the good performance further verifies the authenticity of the parameters identification strategy in practice.
5.2. SOC Estimation Results and Comparative Analysis
Two operation conditions including DST and FUDS are applied to evaluate the performance of SOC estimation method at 25°C. The reference SOC is calculated by coulomb counting (CC) method in EVTS and the reference SOC is used to approximate true value. As the testing battery in nonlab operation condition, it is difficult to obtain an accurate initial SOC value; thus, it is assumed that the initial SOC is set to 90% (with 10% initial error).
The two different models are compared in regard to the estimation result and error in Figure 11. The ARMAbased AHIF and IARMAbased AHIF are used to estimate the battery SOC and their tracking results are shown in Figures 11(a)–11(d), where the black dashedline presents the reference SOC with CC method and the red dottedline represents the SOC estimation with ARMAAHIF algorithm, while the blue solidline indicates the SOC estimation with IARMAAHIF algorithm. Among them, Figures 11(a) and 11(c) compare the observed SOC with the reference SOC and Figures 11(b) and 11(d) show the SOC estimation error in which the MAE is less than 1% except at the beginning of discharge. It shows that the two models based estimated SOC can both fast track the reference SOC at the inaccurate initial SOC. In particular, it is clear that the estimated SOC by proposed IARMAbased estimator has lower error in comparison with ARMAbased estimator since the former model can more accurately estimate the terminal voltage, and the difference between the measured and estimated terminal voltage causes a correction term to eliminate the SOC estimation error in a feedback control loop. In short, it is accordingly demonstrated that the IARMAbased AHIF algorithm can get a better SOC estimation than ARMAbased AHIF algorithm.
(a)
(b)
(c)
(d)
To further explore the promoting effects by error compensation strategy, the IARMAbased AHIF and corresponding error compensation (ΔOCV) are utilized to estimate the battery SOC and their tracking results are shown in Figures 12(a)–12(d), respectively, where the black dashedline shows the reference SOC and the red dashedline represents the SOC estimation with single AHIF algorithm, while the blue solidline means the SOC estimation using AHIFΔOCV. Among them, Figures 12(a) and 12(c) compare the observed SOC with the reference SOC under DST and FUDS cycles, respectively; Figures 12(b) and 12(d) show the SOC estimation error between single AHIF and AHIFΔOCV under DST and FUDS cycles, respectively. The result shows that the above two algorithms can both rapidly track the reference SOC at the inaccurate initial SOC. It is worth noting that the AHIFΔOCV reduces the SOC estimation error significantly in comparison with single AHIF, in which the AAE of SOC estimation is decreased to 0.11% under DST cycle and to 0.18% under FUDS cycle, respectively. It can be concluded that the error compensation strategy is conducive to improving SOC estimation precision in AHIF algorithm.
(a)
(b)
(c)
(d)
Combining IARMA model and error compensation method in AHIF estimator, strengthening the robustness of SOC estimation is even more obvious. The AHIFΔOCV is compared with the AEKFΔOCV in terms of estimation precision and convergence. The comparison results, including tracking trajectory and estimation error under DST and FUDS cycles, are shown in Figure 13, where the black dasheddotedline shows the reference SOC, the red dashedline represents the SOC estimation using AEKFΔOCV, and the green dotedline denotes the SOC estimation using single AEKF, while the blue solidline means the SOC estimation using AHIFΔOCV. As indicated in Figures 13(a) and 13(c), the SOC estimation results and reference SOC are subjected to DST and FUDS profile, respectively. The SOC estimation errors by those three algorithms are drawn in Figures 13(b) and 13(d). The estimated SOC by AHIFΔOCV tracks the reference SOC with a smoother path than by AEKF, which reflects the reliable correction capacity of the error compensation. The simulation results show that the three algorithms can always track the SOC reference trajectory after the convergence. Moreover, the comparison estimation results of the three algorithms mentioned above are summarized in Tables 3 and 4. From the new perspective on combining IARMA model and error compensation in AHIF and AEKF, an improvement of SOC estimation accuracy plays a certain significant role in the SOC estimation by Kalman filter family state observers.


(a)
(b)
(c)
(d)
6. Conclusion
As we all know that battery model uncertainty and OCV error have negative impacts on the battery SOC estimation. To overcome these shortcomings, the improved oneorder ECM based on IARMA analysis, independent OCV estimator, and adaptive HInfinity Filter are proposed for SOC estimation. These methods include the following: (1) the RELS with variable forgetting factor for parameters identification is applied in IARMA model; (2) the independent OCV estimator with error compensation is adopted to accurately capture the OCV; (3) the highprecision SOC estimation is recursively obtained with improved AHIF including noise covariance and restricted factor adjustment. A highprecision experimental platform has been established to gather reliable data of charge/discharge current and terminal voltage. Two typical operation conditions (DST and FUDS) are adopted to evaluate the performance of parameters identification and state estimation. The simulation results indicate the favorable performances whose average SOC estimation error is only 0.11% (DST) and 0.18% (FUDS) by the proposed integrated method. As a modelbased approach, the proposed OCVbased SOC estimator is sensitive to the change of OCVSOC curve. Therefore, the OCVSOC will be considered as a normalized function including temperature and SOH in the future work. The OCVSOC normalized function effectively updating is the key factor in the improvement of the precision and robustness of SOC estimator.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
Xuanju Dang and Zheng Liu proposed the original idea. Zheng Liu designed novel algorithm. Hanxu Sun, Xuanju Dang, and Zheng Liu performed and analyzed the experiments together. Zheng Liu wrote the original manuscript. Zheng Liu and Xuanju Dang revised the final manuscript.
Acknowledgments
This work was financially supported by the National Natural Science Foundation Project of China (61263013 and 61603107), Information Science Project of Guangxi Experiment Center (20130110), Guangxi Natural Science Foundation (2015GXNSFAA139297 and 2016GXNSFDA380001), and Fundamental Ability Enhancement Project for Young and MiddleAged University Teachers in Guangxi Province (2017KY0865).
References
 A. Farmann, W. Waag, A. Marongiu, and D. U. Sauer, “Critical review of onboard capacity estimation techniques for lithiumion batteries in electric and hybrid electric vehicles,” Journal of Power Sources, vol. 281, pp. 114–130, 2015. View at: Publisher Site  Google Scholar
 S. Han, B. Zhang, X. Sun, S. Han, and M. Höök, “China’s Energy Transition in the Power and Transport Sectors from a Substitution Perspective,” Energies, vol. 10, no. 5, p. 600, 2017. View at: Publisher Site  Google Scholar
 M. A. Hannan, M. M. Hoque, P. J. Ker, R. A. Begum, and A. Mohamed, “Charge equalization controller algorithm for seriesconnected lithiumion battery storage systems: Modeling and applications,” Energies, vol. 10, no. 9, 2017. View at: Google Scholar
 J. Li, Q. Lai, L. Wang, C. Lyu, and H. Wang, “A method for SOC estimation based on simplified mechanistic model for LiFePO4 battery,” Energy, vol. 114, pp. 1266–1276, 2016. View at: Publisher Site  Google Scholar
 K. Uddin, A. D. Moore, A. Barai, and J. Marco, “The effects of high frequency current ripple on electric vehicle battery performance,” Applied Energy, vol. 178, pp. 142–154, 2016. View at: Publisher Site  Google Scholar
 B. Nastasi and G. Lo Basso, “Hydrogen to link heat and electricity in the transition towards future Smart Energy Systems,” Energy, vol. 110, pp. 5–22, 2016. View at: Publisher Site  Google Scholar
 X. Q. Zhang, W. P. Zhang, H. Y. Li, and M. Zhang, “Review on state of charge estimation methods for liion batteries,” Transactions on Electrical and Electronic Materials, vol. 18, pp. 136–140, 2017. View at: Google Scholar
 K. S. Ng, C. S. Moo, Y. P. Chen, and Y. C. Hsieh, “Enhanced coulomb counting method for estimating stateofcharge and stateofhealth of lithiumion batteries,” Applied Energy, vol. 86, no. 9, pp. 1506–1511, 2009. View at: Publisher Site  Google Scholar
 C. Weng, J. Sun, and H. Peng, “A unified opencircuitvoltage model of lithiumion batteries for stateofcharge estimation and stateofhealth monitoring,” Journal of Power Sources, vol. 258, pp. 228–237, 2014. View at: Publisher Site  Google Scholar
 F. Zheng, Y. Xing, J. Jiang, B. Sun, J. Kim, and M. Pecht, “Influence of different open circuit voltage tests on state of charge online estimation for lithiumion batteries,” Applied Energy, vol. 183, pp. 513–525, 2016. View at: Publisher Site  Google Scholar
 H. Matsui, T. Nakamura, Y. Kobayashi, M. Tabuchi, and Y. Yamada, “Opencircuit voltage study on LiFePO4 olivine cathode,” Journal of Power Sources, vol. 195, no. 19, pp. 6879–6883, 2010. View at: Publisher Site  Google Scholar
 L. Kang, X. Zhao, and J. Ma, “A new neural network model for the stateofcharge estimation in the battery degradation process,” Applied Energy, vol. 121, pp. 20–27, 2014. View at: Publisher Site  Google Scholar
 J. Kim, S. Lee, and B. H. Cho, “Discrimination of Liion batteries based on Hamming network using dischargingcharging voltage pattern recognition for improved stateofcharge estimation,” Journal of Power Sources, vol. 196, no. 4, pp. 2227–2240, 2011. View at: Publisher Site  Google Scholar
 T. Wu, M. Wang, Q. Xiao, and X. Wang, “The SOC Estimation of Power LiIon Battery Based on ANFIS Model,” Smart Grid and Renewable Energy, vol. 03, no. 01, pp. 51–55, 2012. View at: Publisher Site  Google Scholar
 S. Sepasi, L. R. Roose, and M. M. Matsuura, “Extended Kalman filter with a fuzzy method for accurate battery pack state of charge estimation,” Energies, vol. 8, no. 6, pp. 5217–5233, 2015. View at: Publisher Site  Google Scholar
 H. Sheng and J. Xiao, “Electric vehicle state of charge estimation: Nonlinear correlation and fuzzy support vector machine,” Journal of Power Sources, vol. 281, pp. 131–137, 2015. View at: Publisher Site  Google Scholar
 V. Klass, M. Behm, and G. Lindbergh, “Capturing lithiumion battery dynamics with support vector machinebased battery model,” Journal of Power Sources, vol. 298, pp. 92–101, 2015. View at: Publisher Site  Google Scholar
 X. Dang, L. Yan, H. Jiang, X. Wu, and H. Sun, “Opencircuit voltagebased state of charge estimation of lithiumion power battery by combining controlled autoregressive and moving average modeling with feedforwardfeedback compensation method,” International Journal of Electrical Power & Energy Systems, vol. 90, pp. 27–36, 2017. View at: Publisher Site  Google Scholar
 X. Dang, L. Yan, K. Xu, X. Wu, H. Jiang, and H. Sun, “OpenCircuit VoltageBased State of Charge Estimation of Lithiumion Battery Using Dual Neural Network Fusion Battery Model,” Electrochimica Acta, vol. 188, pp. 356–366, 2016. View at: Publisher Site  Google Scholar
 X. Lai, Y. Zheng, and T. Sun, “A comparative study of different equivalent circuit models for estimating stateofcharge of lithiumion batteries,” Electrochimica Acta, vol. 259, pp. 566–577, 2018. View at: Publisher Site  Google Scholar
 T. Feng, L. Yang, X. Zhao, H. Zhang, and J. Qiang, “Online identification of lithiumion battery parameters based on an improved equivalentcircuit model and its implementation on battery stateofpower prediction,” Journal of Power Sources, vol. 281, pp. 192–203, 2015. View at: Publisher Site  Google Scholar
 Y. Li, L. Wang, C. Liao, L. Wang, and D. Xu, “Recursive modeling and online identification of lithiumion batteries for electric vehicle applications,” Science China Technological Sciences, vol. 57, no. 2, pp. 403–413, 2014. View at: Publisher Site  Google Scholar
 Y. Wang, C. Zhang, and Z. Chen, “An adaptive remaining energy prediction approach for lithiumion batteries in electric vehicles,” Journal of Power Sources, vol. 305, pp. 80–88, 2016. View at: Publisher Site  Google Scholar
 C. Song, Y. Shao, S. Song et al., “Insulation resistance monitoring algorithm for battery pack in electric vehicle based on extended kalman filtering,” Energies, vol. 10, no. 6, p. 714, 2017. View at: Publisher Site  Google Scholar
 Z. W. He, M. Y. Gao, C. S. Wang, L. Y. Wang, and Y. Liu, “Adaptive state of charge estimation for Liion batteries based on an unscented kalman filter with an enhanced battery model,” Energies, vol. 6, no. 8, pp. 4134–4151, 2013. View at: Publisher Site  Google Scholar
 P. Tagade, K. S. Hariharan, P. Gambhire et al., “Recursive Bayesian filtering framework for lithiumion cell state estimation,” Journal of Power Sources, vol. 306, pp. 274–288, 2016. View at: Publisher Site  Google Scholar
 Y. Li, C. Wang, and J. Gong, “A multimodel probability SOC fusion estimation approach using an improved adaptive unscented Kalman filter technique,” Energy, vol. 141, pp. 1402–1415, 2017. View at: Publisher Site  Google Scholar
 F. Sun, R. Xiong, and H. He, “Estimation of stateofcharge and stateofpower capability of lithiumion battery considering varying health conditions,” Journal of Power Sources, vol. 259, pp. 166–176, 2014. View at: Publisher Site  Google Scholar
 S. P. Bhattacharyya, “Robust control under parametric uncertainty: An overview and recent results,” Annual Reviews in Control, vol. 44, pp. 45–77, 2017. View at: Publisher Site  Google Scholar
 O. Gazi, Understanding digital signal processing, vol. 13 of Springer Topics in Signal Processing, Springer, Singapore, 2018. View at: Publisher Site  MathSciNet
 H. K. Sahoo and P. K. Dash, “Robust estimation of power quality disturbances using unscented H1 filter,” Electrical Power and Energy Systems, vol. 73, pp. 438–447, 2015. View at: Publisher Site  Google Scholar
 Y. Zhang, R. Xiong, H. He, and W. Shen, “LithiumIon Battery Pack State of Charge and State of Energy Estimation Algorithms Using a HardwareinTheLoop Validation,” IEEE Transactions on Power Electronics, vol. 32, no. 6, pp. 4421–4431, 2017. View at: Publisher Site  Google Scholar
 C. Chen, R. Xiong, and W. Shen, “A LithiumIon BatteryintheLoop Approach to Test and Validate Multiscale Dual H Infinity Filters for StateofCharge and Capacity Estimation,” IEEE Transactions on Power Electronics, vol. 33, no. 1, pp. 332–342, 2018. View at: Publisher Site  Google Scholar
 M. Charkhgard and M. H. Zarif, “Design of adaptive H_{∞} filter for implementing on stateofcharge estimation based on battery stateofchargevarying modeling,” IET Power Electronics, vol. 8, pp. 1825–1833, 2015. View at: Google Scholar
 C. Lin, Q. Yu, R. Xiong, and L. Y. Wang, “A study on the impact of open circuit voltage tests on state of charge estimation for lithiumion batteries,” Applied Energy, vol. 205, pp. 892–902, 2017. View at: Publisher Site  Google Scholar
 H. Wu, S. Yuan, X. Zhang, C. Yin, and X. Ma, “Model parameter estimation approach based on incremental analysis for lithiumion batteries without using open circuit voltage,” Journal of Power Sources, vol. 287, pp. 108–118, 2015. View at: Publisher Site  Google Scholar
 Z. Wei, K. J. Tseng, N. Wai, T. M. Lim, and M. SkyllasKazacos, “Adaptive estimation of state of charge and capacity with online identified battery model for vanadium redox flow battery,” Journal of Power Sources, vol. 332, pp. 389–398, 2016. View at: Publisher Site  Google Scholar
 Y. Wang, C. Zhang, and Z. Chen, “Online battery stateofcharge estimation based on an integrated estimator,” Applied Energy, vol. 185, pp. 2026–2032, 2017. View at: Publisher Site  Google Scholar
 C. Zhang, W. Allafi, Q. Dinh, P. Ascencio, and J. Marco, “Online estimation of battery equivalent circuit model parameters and state of charge using decoupled least squares technique,” Energy, vol. 142, pp. 678–688, 2018. View at: Publisher Site  Google Scholar
 H. Dai, T. Xu, L. Zhu, X. Wei, and Z. Sun, “Adaptive model parameter identification for large capacity Liion batteries on separated time scales,” Applied Energy, vol. 184, pp. 119–131, 2016. View at: Publisher Site  Google Scholar
 X. Chen, W. Shen, M. Dai, Z. Cao, J. Jin, and A. Kapoor, “Robust adaptive slidingmode observer using RBF neural network for lithiumion battery state of charge estimation in electric vehicles,” IEEE Transactions on Vehicular Technology, vol. 65, no. 4, pp. 1936–1947, 2016. View at: Publisher Site  Google Scholar
 Y. Naderahmadian, M. A. Tinati, and S. Beheshti, “Generalized adaptive weighted recursive least squares dictionary learning,” Signal Processing, vol. 118, pp. 89–96, 2016. View at: Publisher Site  Google Scholar
 M. Badoni, A. Singh, and B. Singh, “Variable forgetting factor recursive least square control algorithm for DSTATCOM,” IEEE Transactions on Power Delivery, vol. 30, no. 5, pp. 2353–2361, 2015. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2018 Zheng Liu et al. 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.