State of Charge Estimation and Evaluation of Lithium Battery Using Kalman Filter Algorithms

The accurate and rapid estimation of the state of charge (SOC) is important and difficult in lithium battery management systems. In this paper, an adaptive infinite Kalman filter (AUKF) was used to estimate the state of charge for a 18650 LiNiMnCoO2/graphite lithium-ion battery, and its performance was systematically evaluated under large initial errors, wide temperature ranges, and different drive cycles. In addition, three other Kalman filter algorithms on the predicted SOC of LIB were compared under different work conditions, and the accuracy and convergence time of different models were compared. The results showed that the convergence time of the AUKF algorithms was one order of magnitude smaller than that of the other three methods, and the mean absolute error was only less than 50% of the other methods. The present work can be used to help other researchers select an appropriate strategy for the SOC online estimation of lithium-ion cells under different applicable conditions.


Introduction
Since the 1990s, lithium batteries have become one of the best choices for current consumer-grade electric vehicle power batteries due to their good stability and high energy density. To ensure the safety and reliability of electric vehicles (EVS), the battery management system (BMS) must provide real-time and accurate information about the usage status of the on-board power battery pack. The SOC (state of charge) is one of the most important states tracked in a battery to optimize the performance and extend the lifetime of batteries, so its estimation is an important task for battery management [1,2]. The accurate estimation of the SOC involves many nonlinear effects such as open circuit voltage (OCV), instantaneous current, charge and discharge rate, ambient temperature, battery temperature, parking time, self-discharge rate, Coulomb efficiency, resistance characteristics, SOC initial value, depth of discharge (DOD), etc. [3]. These factors are affected by different materials and processes, and they also interact with each other, so the SOC calculation of power batteries is complex and difficult, which is a challenge that has not been overcome for many years [4].
In recent years, ternary lithium-ion batteries have been considered as power batteries with great application prospects and market competitiveness due to their outstanding specific capacity, good rate performance, and high working voltage [5]. In order to allow the ternary lithium-ion battery to perform better, it is necessary to use appropriate methods to evaluate its SOC. The estimation methods of the SOC are mainly divided into the following categories [6]. (1) The ampere-hour integration method [7], which obtains the battery SOC by integrating the current during the charge and discharge processes in the dimension of time. This method is simple, but it is sensitive to the initial SOC value and is prone to error accumulation. (2) The open-circuit voltage (OCV) method [8], which establishes the relationship between the OCV and SOC based on the off-line open-circuit voltage test [9]. The advantage of this method is its high estimation accuracy, which requires strategy that could effectively manage the SOC while improving the system frequency stability. A dual-input neural network combining gated recurring unit (GRU) layers and fully connected layers (acronymized as a DIGF network) was developed by taking both the time-series voltage and current measurements and the battery's SOH as inputs [41]. The present works in the literature have focused more on the accuracy and speed of the algorithms, with less emphasis on the usage environment, which has important effects for successful estimation.
In this paper, considering the actual working environments of an electric vehicle, response speed, and calculation cost, the AUKF was used to estimate their SOC under large initial errors, wide temperature ranges, and different drive cycles. The results were compared with the EKF, UKF, and AEKF to provide a suitable SOC algorithm for designers.

Subjects
The single battery selected in this paper was a 18,650 LiNiMnCoO 2 , graphite lithiumion cell, and its main parameters are shown in Table 1.

Model Structure
An accurate estimation of the SOC of a battery is strongly dependent on the appropriate battery model. There are three types of commonly used battery models: electrochemical models, black-box models, and equivalent circuit models (ECMs) [42]. Among these models, the ECMs can be analyzed and expressed by a mathematical model, which has a clear physical meaning, in addition to a simple and flexible intuitive structure. The main advantage lies in the convenience of online calculation and computer simulation analysis. Therefore, an equivalent circuit model was used in this paper.
The ECMs use traditional circuit elements such as resistance, capacitance, and a constant voltage source to describe the external characteristics of power batteries, which is convenient for battery characteristic analysis and model parameter identification [43]. The higher the equivalent circuit order, the better the data fitting effect and the more complex the model. According to the trade-off between the structural complexity and the model prediction accuracy, a second-order model was selected, as shown in Figure 1. [41]. The present works in the literature have focused more on the accuracy the algorithms, with less emphasis on the usage environment, which has imp for successful estimation.
In this paper, considering the actual working environments of an elect sponse speed, and calculation cost, the AUKF was used to estimate their SO initial errors, wide temperature ranges, and different drive cycles. The resu pared with the EKF, UKF, and AEKF to provide a suitable SOC algorithm f

Subjects
The single battery selected in this paper was a 18650 LiNiMnCoO2, gra ion cell, and its main parameters are shown in Table 1.

Model Structure
An accurate estimation of the SOC of a battery is strongly dependent priate battery model. There are three types of commonly used battery m chemical models, black-box models, and equivalent circuit models (ECMs these models, the ECMs can be analyzed and expressed by a mathematical has a clear physical meaning, in addition to a simple and flexible intuitive main advantage lies in the convenience of online calculation and comput analysis. Therefore, an equivalent circuit model was used in this paper. The ECMs use traditional circuit elements such as resistance, capacitan stant voltage source to describe the external characteristics of power batte convenient for battery characteristic analysis and model parameter identific higher the equivalent circuit order, the better the data fitting effect and the m the model. According to the trade-off between the structural complexity a prediction accuracy, a second-order model was selected, as shown in Figur  U ocv is equal to the open circuit voltage (OCV) of the battery which is related to the SOC; R 0 represents the contact resistance between the active material, the current collector, the lead electrode, and the active material/current collector. The dynamic characteristics of the power battery are described by the polarization resistance, R pi , and the polarization capacitance, C pi , including the polarization characteristics and the diffusion effect, where i = 1, 2; I is the charge and discharge current, of which the charge is negative, and the discharge is positive; and U 0 is the terminal voltage of the battery.
In the second-order RC battery model shown in Figure 1, the voltages V 1 and V 2 at both ends of the capacitors C p1 and C p2 and the SOC are the state variables; I is the input variable; and U 0 is the output variable. Combined with the ampere-hour integration method [44], the discredited state space model shown in Equation (1) can be obtained. Based on this model, the AUKF was used to estimate the battery SOC.
where ∆t is the sampling interval, taken as ∆t = 1 s; V 1(k) , V 2(k) , and SOC (k) refer to the voltage across the capacitors C p1 , C p2 , and SOC at k sampling time; V 1(k+1) , V 2(k+1) , and SOC (k+1) refer to the voltage across the capacitors, C p1 and C p2 , and the SOC at k + 1 sampling time; η is the Coulomb efficiency; C N is the total capacity of the battery; and U ocv is the open circuit voltage of the battery, which is a function of the SOC.

Introduction of Experimental Platform
The experimental platform is shown in Figure 2, which was used to obtain the terminal voltage and current of the battery. The PC sends commands to an Arbin BT2000 battery test system to simulate the battery working conditions as well as to charge or discharge the battery. The battery test system transmits the collected voltage and current data to the PC in real-time.
capacitance, Cpi, including the polarization characteristics and the diffusion effec i = 1, 2; I is the charge and discharge current, of which the charge is negative, discharge is positive; and U0 is the terminal voltage of the battery.
In the second-order RC battery model shown in Figure 1, the voltages V1 a both ends of the capacitors Cp1 and Cp2 and the SOC are the state variables; I is t variable; and U0 is the output variable. Combined with the ampere-hour int method [44], the discredited state space model shown in Equation (1) can be o Based on this model, the AUKF was used to estimate the battery SOC.
where ∆t is the sampling interval, taken as ∆t = 1s; V1(k), V2(k), and SOC(k) refer to the across the capacitors Cp1, Cp2, and SOC at k sampling time; V1(k+1), V2(k+1), and SOC to the voltage across the capacitors, Cp1 and Cp2, and the SOC at k + 1 sampling t the Coulomb efficiency; CN is the total capacity of the battery; and Uocv is the ope voltage of the battery, which is a function of the SOC.

Introduction of Experimental Platform
The experimental platform is shown in Figure 2, which was used to obtain th nal voltage and current of the battery. The PC sends commands to an Arbin BT2 tery test system to simulate the battery working conditions as well as to charge charge the battery. The battery test system transmits the collected voltage and curr to the PC in real-time.

Measurement of the Relationship Curve between OCV and SOC
Obtaining the OCV of the battery is critical because it can be used for resist pacitance parameter identification and the SOC estimation. The OCV is a single function of the SOC; therefore, the corresponding OCV can be obtained through tionship curve between the OCV and SOC. The low-current OCV test and the incr the OCV test are two common methods for observing the OCV-SOC relationshi

Measurement of the Relationship Curve between OCV and SOC
Obtaining the OCV of the battery is critical because it can be used for resistancecapacitance parameter identification and the SOC estimation. The OCV is a single-valued function of the SOC; therefore, the corresponding OCV can be obtained through the relationship curve between the OCV and SOC. The low-current OCV test and the incremental the OCV test are two common methods for observing the OCV-SOC relationship, while the latter has high tracking accuracy and anti-interference ability [45]; therefore, this paper used the latter to obtain the relationship curve between the OCV and SOC.
The incremental OCV test is shown in Figure 3. First, the cell was discharged until its SOC became 0%. Then, the cell was charged by using a positive pulse current (i.e., C/20) with a width corresponding to a certain amount of charge (i.e., 10% SOC). In the relaxation period, the SOC was allowed to stand for 2 h to eliminate the polarization effects inside the cell. When the terminal voltage rose to the upper cutoff voltage, it entered into the constant voltage charging stage. In this stage, when the charging current dropped below C/20, the charging was completed. Finally, an averaging step and a linear interpolation step were applied to obtain the OCV-SOC curve. The OCV-SOC curve obtained by the linear interpolation method is shown in Figure 4. The curve was fitted into a fifth-order OCV-SOC function using MATLAB, and its OCV-SOC equation is: (2) the latter has high tracking accuracy and anti-interference ability [45]; therefore, this paper used the latter to obtain the relationship curve between the OCV and SOC. The incremental OCV test is shown in Figure 3. First, the cell was discharged until its SOC became 0%. Then, the cell was charged by using a positive pulse current (i.e., C/20) with a width corresponding to a certain amount of charge (i.e., 10% SOC). In the relaxation period, the SOC was allowed to stand for 2 h to eliminate the polarization effects inside the cell. When the terminal voltage rose to the upper cutoff voltage, it entered into the constant voltage charging stage. In this stage, when the charging current dropped below C/20, the charging was completed. Finally, an averaging step and a linear interpolation step were applied to obtain the OCV-SOC curve. The OCV-SOC curve obtained by the linear interpolation method is shown in Figure 4. The curve was fitted into a fifth-order OCV-SOC function using MATLAB, and its OCV-SOC equation is:

Model Parameter Identification Test
In this paper, the first half of the data of the dynamic stress test (DST) was used to identify the model parameters on the test samples. The DST was specified in the USABC (American Advanced Battery Association) test programs as collecting data, simulating the dynamic discharge state, and can be reduced to the utmost required quantity according to the specified performance of the test sample. Figure 5a shows the current section of the DST. Although the DST consists of sorts of current steps with different amplitudes and lengths, and takes into account regenerative charging, it is still a simplification from the actual loading conditions of the battery. Thus, the DST was performed on the test battery to determine the model parameters in this study. In order to evaluate the SOC estimation results (such as the accuracy and robustness), not only the DST, but a more complex dynamic current profile, the federal urban driving schedule (FUDS), was used. Similar to the the latter has high tracking accuracy and anti-interference ability [45]; therefore, this paper used the latter to obtain the relationship curve between the OCV and SOC. The incremental OCV test is shown in Figure 3. First, the cell was discharged until its SOC became 0%. Then, the cell was charged by using a positive pulse current (i.e., C/20) with a width corresponding to a certain amount of charge (i.e., 10% SOC). In the relaxation period, the SOC was allowed to stand for 2 h to eliminate the polarization effects inside the cell. When the terminal voltage rose to the upper cutoff voltage, it entered into the constant voltage charging stage. In this stage, when the charging current dropped below C/20, the charging was completed. Finally, an averaging step and a linear interpolation step were applied to obtain the OCV-SOC curve. The OCV-SOC curve obtained by the linear interpolation method is shown in Figure 4. The curve was fitted into a fifth-order OCV-SOC function using MATLAB, and its OCV-SOC equation is:

Model Parameter Identification Test
In this paper, the first half of the data of the dynamic stress test (DST) was used to identify the model parameters on the test samples. The DST was specified in the USABC (American Advanced Battery Association) test programs as collecting data, simulating the dynamic discharge state, and can be reduced to the utmost required quantity according to the specified performance of the test sample. Figure 5a shows the current section of the DST. Although the DST consists of sorts of current steps with different amplitudes and lengths, and takes into account regenerative charging, it is still a simplification from the actual loading conditions of the battery. Thus, the DST was performed on the test battery to determine the model parameters in this study. In order to evaluate the SOC estimation results (such as the accuracy and robustness), not only the DST, but a more complex dynamic current profile, the federal urban driving schedule (FUDS), was used. Similar to the In this paper, the first half of the data of the dynamic stress test (DST) was used to identify the model parameters on the test samples. The DST was specified in the USABC (American Advanced Battery Association) test programs as collecting data, simulating the dynamic discharge state, and can be reduced to the utmost required quantity according to the specified performance of the test sample. Figure 5a shows the current section of the DST. Although the DST consists of sorts of current steps with different amplitudes and lengths, and takes into account regenerative charging, it is still a simplification from the actual loading conditions of the battery. Thus, the DST was performed on the test battery to determine the model parameters in this study. In order to evaluate the SOC estimation results (such as the accuracy and robustness), not only the DST, but a more complex dynamic current profile, the federal urban driving schedule (FUDS), was used. Similar to the DST, the current sequence of the FUDS was also transmitted from the timespeed profile of industry standard vehicles. The corresponding current profile is shown in Figure 5b. DST, the current sequence of the FUDS was also transmitted from the time-speed profile of industry standard vehicles. The corresponding current profile is shown in Figure 5b.

Model Parameter Identification Method
The MATLAB exponential fitting expression used was: where c0, c1, and c2 are constants. Comparing Equations (3) and (4), we obtain: Since the OCV changes little with the SOC in the platform voltage region, resulting in a large error, the following method can be used to obtain R0: fore, when t approaches 0, Equation (3) can be simplified to , Applying it to the actual circuit as shown in Figure 1, the following equation is obtained:

Model Parameter Identification Method
The exponential fitting method is used to fit the response curve of the battery terminal voltage to the pulse current, and then the parameters in the battery model can be obtained. The equation of discharge current and output voltage in the circuit model shown in Figure 1 can be expressed as: The MATLAB exponential fitting expression used was: where c 0 , c 1 , and c 2 are constants. Comparing Equations (3) and (4), we obtain: Since the OCV changes little with the SOC in the platform voltage region, resulting in a large error, the following method can be used to obtain R 0 : For Equation (3), lim Applying it to the actual circuit as shown in Figure 1, the following equation is obtained: where U 2 and U 1 are the cell terminal voltages of two consecutive sampling points before and after the current suddenly drops to zero (the data used were from 2000 to 2200 corresponding to the abscissa of Figure 5a). Using this method, the terminal voltage response curve when the current suddenly reaches 0 can be obtained, as shown in Figure 6.
where U2 and U1 are the cell terminal voltages of two consecutive sampling points before and after the current suddenly drops to zero (the data used were from 2000 to 2200 corresponding to the abscissa of Figure 5a). Using this method, the terminal voltage response curve when the current suddenly reaches 0 can be obtained, as shown in Figure 6. Referring to the "Freedom CAR Battery Experiment Manual", without considering the charging situation, the model parameters as shown in Table 2 were obtained by using Equations (3)-(6) via the discharge method.

AUKF Algorithm for SOC Estimation
For the Kalman filter algorithms, it must be assumed that the noise is Gaussian white noise. However, the statistical characteristics of noise in the actual BMS during data acquisition are unknown. Adopting the adaptive Kalman filter method, the state variables can be dynamically estimated from the measurement data, and the statistical characteristics of noise can be continuously estimated and corrected, so the SOC of the battery can be accurately estimated. The estimation process of the SOC using the AUKF is shown in Figure 7. Referring to the "Freedom CAR Battery Experiment Manual", without considering the charging situation, the model parameters as shown in Table 2 were obtained by using Equations (3)-(6) via the discharge method.

AUKF Algorithm for SOC Estimation
For the Kalman filter algorithms, it must be assumed that the noise is Gaussian white noise. However, the statistical characteristics of noise in the actual BMS during data acquisition are unknown. Adopting the adaptive Kalman filter method, the state variables can be dynamically estimated from the measurement data, and the statistical characteristics of noise can be continuously estimated and corrected, so the SOC of the battery can be accurately estimated. The estimation process of the SOC using the AUKF is shown in Figure 7. The AUKF algorithm for SOC estimation mainly includes four steps, as shown below.  The AUKF algorithm for SOC estimation mainly includes four steps, as shown below.

Algorithm Initialization
Initial state estimation: where ∧ S 0 is the initial SOC value; Initial posterior error covariance: P 0 ; Initial process noise covariance: Q 0 ; Initial measurement noise covariance: V 0 ; Window size for covariance matching: L.

Sigma Points (UT Transform) Are Generated at the k−1 Moment
Sigma data point sequence is constructed by using a series of sampling points. The sigma point generated at k−1 step can be expressed as: where λ = 3α 2 − 1 is a scale parameter, which can be adjusted to improve the approximation accuracy; α is the scaling coefficient, which determines the distribution of sigma points and is generally set to a very small positive value; and N is the dimension of the extended state variable.

Time Update
Update the sample point: The prior estimation is obtained: wherex K|K−1 is the predicted value of the sigma points at the k|k−1 moment and W (i) m is the weight used to calculate the mean value, which is determined by the following formula: The prior error covariance of the system state is obtained: where W (i) c is the weight used to calculate the covariance, which is determined by the following formula: where the constant β is generally determined by experience, and for the Gaussian distribution, it is generally taken as β = 2.

Generating Sigma Points at the k|k−1 Moment
The sigma point generated at k|k−1 step can be expressed as:

Calculating the Predicted Output Voltage and Covariance
The predicted output voltage for the sigma points at k|k−1 step U The predicted output voltage covariance at the k|k−1 moment D k can be expressed as:

Modifying System State Estimation
The cross-covariance of the state variables and output variables at the k|k−1 moment is calculated by: where U k is the measurement of the battery voltage at the k moment.
The Kalman gain G k is calculated by: State estimation correction: State covariance correction:

Adjustment of Q and V
In this brief, the adaptive estimation of the process noise covariance Q and measurement noise covariance V on the basis of the voltage residual of the battery model and the covariance of the voltage residual were considered. Therefore, Q and V were estimated and updated iteratively from the following where µ k is the voltage residual of the battery model at the k moment, and F k is the approximation to the covariance of the voltage residual at the k moment. Through the iteration of the above steps 2~4, the AUKF is established. The accuracy of the SOC estimation is improved by adaptive adjustment of the process noise and measurement noise.

Simulation with Different Initial Errors
In this paper, a FUDS cycle was used to test the samples, the temperature was set to 25 • C, and the initial SOC value of the experiment was arbitrarily set to 50%. For verification and comparison, the initial SOC value, S 0 , in the simulation was set to 80% and 20%, respectively (the initial error is ±30%). The estimation results based on the AUKF are shown in Figures 8 and 9 in which the reference value of the SOC is calculated by the ampere-hour method. The results show that the AUKF can quickly compensate for the initial SOC error and accurately track the experimental SOC values under different initial values. After correcting the initial error, the difference between the two results was almost indistinguishable. Figures 8b and 9b show that the estimation error was large in the early stage due to inaccurate initial values, but after a certain convergence time, the estimation error was stable, within 2%. In order to evaluate and compare the performance of the algorithm from the two aspects of estimation accuracy and robustness, the mean absolute error (MAE) and convergence rate were used in this paper. The MAE of SOC can be calculated by using the following formula: where S k is the experimental SOC at time k. The convergence rate is the corresponding time when the error converges below 2% error (MAE) and convergence rate were used in this paper. The MAE of SOC can be calculated by using the following formula: where Sk is the experimental SOC at time k. The convergence rate is the corresponding time when the error converges below 2%   Table 3 shows the MAE and convergence rate under two different SOC initial values. The AUKF corrects the SOC prediction via the real-time online prediction and estimation of noise, quickly adjusts the influence of inaccurate initial values, and causes its estimated value to converge to the actual SOC; therefore, it has good robustness.

Simulation at Different Temperatures
In order to evaluate the performance of the AUKF at different temperatures, Figures  10 and 11 show the estimation results and errors when the temperature is 0 °C and the initial value error is ±10%, respectively. Compared with the results at 25 °C, the initial errors can also be compensated for quickly, but the differences between the stable value and the actual SOC were relatively larger than those at 25 °C. The MAE was 0.0132 and 0.0136, respectively, as shown in Table 4. This indicates that the AUKF exhibited excellent performance at low temperature.  Table 3 shows the MAE and convergence rate under two different SOC initial values. The AUKF corrects the SOC prediction via the real-time online prediction and estimation of noise, quickly adjusts the influence of inaccurate initial values, and causes its estimated value to converge to the actual SOC; therefore, it has good robustness.

Simulation at Different Temperatures
In order to evaluate the performance of the AUKF at different temperatures, Figures 10 and 11 show the estimation results and errors when the temperature is 0 • C and the initial value error is ±10%, respectively. Compared with the results at 25 • C, the initial errors can also be compensated for quickly, but the differences between the stable value and the actual SOC were relatively larger than those at 25 • C. The MAE was 0.0132 and 0.0136, respectively, as shown in Table 4. This indicates that the AUKF exhibited excellent performance at low temperature.         Figures 12 and 13 are the estimation results and errors when the temperature was 45 • C and the initial value error was also ±10%, respectively. We can see that the initial errors can be compensated for quickly, and both of the MAE were 0.0091, as shown in Table 5. This result further shows that the AUKF has good temperature characteristics.  Figures 12 and 13 are the estimation results and errors when the temperature was 45 °C and the initial value error was also ±10%, respectively. We can see that the initial errors can be compensated for quickly, and both of the MAE were 0.0091, as shown in Table 5. This result further shows that the AUKF has good temperature characteristics.    Figures 12 and 13 are the estimation results and errors when the temperature was 45 °C and the initial value error was also ±10%, respectively. We can see that the initial errors can be compensated for quickly, and both of the MAE were 0.0091, as shown in Table 5. This result further shows that the AUKF has good temperature characteristics.

Simulation Comparison of Different Algorithms
In order to further evaluate the performance of the AUKF, it was compared with the AEKF, EKF, and UKF algorithms in this paper. In the AEKF, the adaptive adjustment was realized via the covariance matching of residual voltage. The EKF and UKF use constant value of Q k and V k to estimate the SOC, and also use the trial and error method to determine the parameters. Figures 14 and 15 show the estimation results and errors based on the four different Kalman filtering algorithms at an initial SOC of 80% and 20% under the FUDS cycle, respectively. In order to make the results more readable, the statistical table of the MAE and convergence time is shown in Table 6. Compared with the three other algorithms, the MAE value of the AUKF algorithm could be reduced by about one order of magnitude, and the convergence speed could be increased by an order of magnitude; therefore, it could estimate the SOC more quickly and accurately. Meanwhile, it could also be seen that the accuracy of the AUKF and AEKF was higher than that of the EKF and UKF. The results show that adaptive adjustment of the covariance difference between the process noise and the measurement noise can improve the estimation accuracy of the SOC. In addition, the UKF had better performance compared with the EKF. The comprehensive performance ranking of the four calculation methods was AUKF > AEKF > UKF > EKF.

Simulation Comparison of Different Algorithms
In order to further evaluate the performance of the AUKF, it was compared AEKF, EKF, and UKF algorithms in this paper. In the AEKF, the adaptive adjustm realized via the covariance matching of residual voltage. The EKF and UKF use value of Qk and Vk to estimate the SOC, and also use the trial and error method mine the parameters. Figures 14 and 15 show the estimation results and errors b the four different Kalman filtering algorithms at an initial SOC of 80% and 20% u FUDS cycle, respectively. In order to make the results more readable, the statisti of the MAE and convergence time is shown in Table 6. Compared with the thr algorithms, the MAE value of the AUKF algorithm could be reduced by about o of magnitude, and the convergence speed could be increased by an order of ma therefore, it could estimate the SOC more quickly and accurately. Meanwhile, it co be seen that the accuracy of the AUKF and AEKF was higher than that of the E UKF. The results show that adaptive adjustment of the covariance difference betw process noise and the measurement noise can improve the estimation accuracy of t In addition, the UKF had better performance compared with the EKF. The compre performance ranking of the four calculation methods was AUKF > AEKF > UKF >    To further compare the performance of the four Kalman filter algorithms, the ab four algorithms were applied to a DST cycle. Figures 16 and 17 show the estimation re and errors based on different Kalman filtering algorithms with an initial SOC of 80% 20%, respectively. Table 7 is the statistical table of the MAE and convergence time. F the results, one can see that the AUKF had the fastest convergence speed and the sma MAE, which further confirms the superiority of AUKF. In addition, we also found the EKF had the worst performance, while the UKF and AEKF had their own advanta The UKF needed less calculation time but a larger mean absolute error, while the A showed the opposite. This is different from the above result of the FUDS cycle. Never less, the results further show that the idea of adaptive adjustment and nonlinear funct can improve computational accuracy.  To further compare the performance of the four Kalman filter algorithms, the above four algorithms were applied to a DST cycle. Figures 16 and 17 show the estimation results and errors based on different Kalman filtering algorithms with an initial SOC of 80% and 20%, respectively. Table 7 is the statistical table of the MAE and convergence time. From the results, one can see that the AUKF had the fastest convergence speed and the smallest MAE, which further confirms the superiority of AUKF. In addition, we also found that the EKF had the worst performance, while the UKF and AEKF had their own advantages. The UKF needed less calculation time but a larger mean absolute error, while the AEKF showed the opposite. This is different from the above result of the FUDS cycle. Nevertheless, the results further show that the idea of adaptive adjustment and nonlinear functions can improve computational accuracy.

Conclusions
In this paper, the AUKF was used to estimate the SOC of lithium-ion cells online, and its performance was evaluated systematically under large initial errors, wide temperature ranges, and different drive cycles. The results show that the estimation error was stable within 2%, and that the convergence speed was less than 50 s, which illustrates the excellent performance of the AUKF. Moreover, compared with the AEKF, EKF, and UKF, the AUKF was one order of magnitude smaller than that of the other three methods under the same initial value, and its mean absolute error was only 50% of that of the other methods. This is mainly because the AUKF not only has the advantage that UKF is better than EKF, but also extends the idea of covariance matching based on an output voltage residual sequence model to the UKF to realize adaptive regulation. In the process of the SOC estimation, the current SOC was continuously modified according to the mean and variance estimation results of each step of noise, which can correct the initial value error. Therefore, the AUKF had a better accuracy and a faster convergence speed than the other three algorithms, and can estimate the battery SOC more accurately and quickly.
Considering the identification accuracy and calculation time, this paper adopted the off-line exponential fitting method to identify the model parameters. If the identification accuracy needs to be further improved, the online identification method can be used to identify the battery model parameters in future work.

Data Availability Statement:
The data presented in this study are available on request from the corresponding authors.