# Lithium-Ion Battery Parameters and State of Charge Joint Estimation Using Bias Compensation Least Squares and the Alternate Algorithm.

1. Introduction

2. BCLS-ALT-Based SOC Joint Estimation Algorithm

2.1. Battery Model. Compared with the existing battery models and considering the calculation's complexity, the first-order ECM is recognized as a better option for modeling lithium-ion batteries [44-46]. It is widely used in related research of lithium-ion batteries [47]. In order to ensure accuracy and simplicity, the first-order ECM has been selected in this article as shown in Figure 1.

The first-order ECM can be represented as

[mathematical expression not reproducible], (1)

where [U.sub.OC] is the voltage source, [R.sub.P] is the polarization resistance, [C.sub.P] is the polarization capacitance, [R.sub.0] is the ohmic impedance, [U.sub.OC] is the open circuit voltage, [U.sub.T] is the terminal voltage, [I.sub.L] is the current, [C.sub.n] is the nominal capacity, and [eta] is Coulombic efficiency. Equation (1) can be discretized, and the discrete system equation can be expressed as follows:

[mathematical expression not reproducible]. (2)

On the basis of known knowledge, OCV is the function of the SOC and temperature, which can be expressed as the following equation:

[U.sub.OC,T] = [K.sub.1,T] [SOC.sup.5] + [K.sub.2,T] [SOC.sup.4] + [K.sub.3,T] [SOC.sup.3] + [K.sub.4,T] [SOC.sup.2] + [K.sub.5,T] SOC + [K.sub.6,T], (3)

where [K.sub.1,T] (i = 0, ..., 6) are the coefficients, which can be fit based on an experimental database. In addition, T is the temperature of the battery working environment.

2.2. BCLS-Based Parameter identification Algorithm. The model-based SOC estimation method is highly dependent on the model parameters. In this section, the online-identified parameters of the battery model using the BCLS are introduced. This method uses the square norm of the discrete function as a metric to get the identification parameters. When the system error is considered, the discrete expression of the system is to be identified from equation (1). According to equations (1) and (2), the Laplace equation of the battery model can be obtained as

[mathematical expression not reproducible], (4)

G(s) = [R.sub.p]/1 + [R.sub.p][C.sub.p] s + [R.sub.0]. (5)

Equation (5) can be discretized by bilinear transformation [48]. Substituting s = ((2/T)(1 - [Z.sup.-1]/1 + [Z.sup.-1])) into equation (5), G(s) can be described as

G([Z.sup.-1]) = [[a.sub.2] + [a.sub.3] [Z.sup.-1]]/[1 - [a.sub.1] [Z.sup.-1]], (6)

[mathematical expression not reproducible], (7)

where [a.sub.1], [a.sub.2], and [a.sub.3] are the system coefficients. Equation (4) can be transformed into the following difference equation:

[U.sub.[DELTA]](k) = [U.sub.OC](k) - [U.sub.T](k), [U.sub.[DELTA]](k) = [a.sub.1] [U.sub.[DELTA]] (k - 1) + [a.sub.2] I(k) + [a.sub.3]I (k - 1), (8)

where I(k) is the system input; [U.sub.[DELTA]](k) is the system output, and [U.sub.[DELTA]](k) can be represented as

[U.sub.[DELTA]](k) = [phi](k)[theta] + e(k), (9)

where e (k) is the systematic error. [phi] (k) and [theta] can be written as

[phi](k) = [[U.sub.[DELTA]] (k - 1)I(k)I(k - 1)], [theta] = [[[a.sub.1] [a.sub.2] [a.sub.3]].sup.T]. (10)

Extending [phi](k) to N-dimensional, [PHI](k), [U.sub.[DELTA]], and e can be written as

[mathematical expression not reproducible. (11)

Defining the function J([theta]) [40], J([theta]) can be described as

J([theta]) = [N.summation over (i=1)] [([U.sub.[DELTA]] - [PHI][theta]).sup.2] = [N.summation over (i=1)] [(e(i + 1)).sup.2]. (12)

The extreme value of J([theta]) can be got as

[partial derivative]J ([theta])/[partial derivative] ([theta]) = [partial derivative]/[partial derivative] ([theta]) ([U.sub.[DELTA]] - [PHI][theta]) [([U.sub.[DELTA]] - [PHI][theta]).sup.T] = 0. (13)

Least squares estimation result can be obtained as

[[??].sub.LS] = [[[PHI].sup.T] [PHI]].sup.-1] [[PHI].sup.T] [U.sub.[DELTA]]. (14)

However, the least square method is only effective for white noise. If the noise is not white noise, the parameter estimation by the least square method is not unbiased and consistent estimation. To solve the above problems, online estimation of noise variance and real-time bias compensation are applied. The noise variance [[??].sup.2] [39] can be described as

[mathematical expression not reproducible], (15)

where [[??].sub.BCLS](k - 1) is the result of the bias compensation least squares at time k - 1, [LAMBDA] is the correlation matrix, and F(k) is the error criterion function. A and F(k) [42] can be represented as

[mathematical expression not reproducible], (16)

F(k) = F(k - 1) + [E.sup.2] (k)/1 + [[phi].sup.T] (k) Q(k - 1) [phi] (k), (17)

where Q is the covariance matrix, and E(k) is the error of least squares estimation. Q(k) and E(k) [41] can be written as

Q(k) = Q(k - 1)- [K.sub.c](k)/(k) [[phi].sup.T] (k) Q (k - 1), (18)

E(k) = [U.sub.[DELTA]] (k) - [phi](k)[[??].sub.RLS] (k - 1), (19)

where [K.sub.c] is the gain matrix. [K.sub.c] [43] can be calculated as

[K.sub.c] (k) = Q(k - 1) [phi] (k)/1 + [[phi].sup.T](k)Q(k - 1) [phi] (k). (20)

The bias compensation least squares [40] estimation of [theta] is obtained by

[mathematical expression not reproducible]. (21)

With the computational formula of the BCLS, we can estimate the parameters of the battery model online, and the general process of the identification is listed in Table 1.

2.3. BCLS-ALT-Based SOC Joint Estimation. In order to achieve accurate SOC estimation and low computational cost, a BCLS-ALT algorithm for SOC estimation is proposed in this paper. With the parameters identified online by the BCLS, the ALT algorithm is then applied to estimate the SOC. The ALT algorithm is consisted of the AHI method and the HIF. The flowchart of the ALT algorithm is shown in Figure 2. The HIF is used to obtain accurate initial values of the AHI method, and it is also applied to correct the errors produced by the AHI method. The SOC estimation can be switched between the AHI method and the HIF in the proposed BCLS-ALT algorithm.

2.3.1. AHI Method and the H-Infinity Filter Algorithm. The AHI method is widely used in most EVs [11] and is the most efficient method. However, the estimation accuracy of this method is affected by the initial error and cumulative error produced by measurement errors of battery current. The AHI method [7] is formulated as

SOC(t) = SOC(0) - [[integral].sup.t.sub.0] [eta][i.sub.L] dt/[C.sub.n], (22)

where SOC(0) represents the SOC at the initial time 0, [C.sub.n] is the nominal capacity, [eta] denotes the coulombic efficiency, and [i.sub.L] is the battery working current. The HIF is popular due to high robustness and accuracy. Similar to other model-based SOC estimation methods, the computational efficiency of the HIF is much lower than that of the AHI method. This can be known through its process of calculation. To implement the HIF to estimate the SOC, the discrete-time system equation can be derived as

[mathematical expression not reproducible]. (23)

where [w.sub.k] is the process noise of [u.sub.k]. [v.sub.k] is the measurement noise of observation [y.sub.k]. [x.sub.k] is the system state at time k. From equation (2), A, B, and D can be written as

[mathematical expression not reproducible]. (24)

Since the relationship between the SOC and OCV is nonlinear, this results in equation (23) exhibiting nonlinear behavior. However, this problem can be solved according to the related theory of Burgos et al. [49] as shown in equation (25). Then, the cost function J can be constructed using game theory as in equation (26) [50].

[mathematical expression not reproducible], (25)

[mathematical expression not reproducible], (26)

where [z.sub.k] is denoted for the SOC, and [[??].sub.k] is defined for the [z.sub.k]'s estimated value; [x.sub.0] denotes the initial SOC, and [[??].sub.0] represents the [x.sub.0]'s estimation value. [S.sub.k], [P.sub.0], [W.sub.k], and [V.sub.k] are the weighting matrices in equation (26). They are selected based on the specific situation [24].

[s.sup.2.sub.M] = [s.sup.T] Ms. (27)

For simplicity, [S.sub.k], [W.sub.k], and [V.sub.k] were set as the identity matrices, and their dimensions were determined using equation (26). [P.sub.0] was determined by the initial error. The cost function J can be regarded as a contest between nature and engineers. Nature always tries to maximize the estimation error by introducing errors (current error [w.sub.k], voltage noise [v.sub.k], and the initial error in the denominator) [19]. However, appropriate methods can be applied to minimize the estimation error, so that the value of the function J is as small as possible to obtain an accurate SOC. However, it is difficult to minimize J directly; thus, a bound value [theta] that can be easily satisfied was determined. That is, a value for [[??].sub.k] should be the satisfied condition [19].

J < 1/[epsilon]. (28)

Equations (26) and (28) can be integrated and expressed

[mathematical expression not reproducible]. (29)

From equation (23), the following can be derived:

[mathematical expression not reproducible], (30)

where [bar.[S.sub.k]] is defined as

[bar.[S.sub.k]] - [L.sub.k][S.sub.k][L.sub.k]. (31)

Applying these results, [J.sub.1] can be written as equation (32). Thus, the discrete H-infinity filter can be considered as a minimax problem as shown in equation (33). To solve this problem, Dan et al. derived the equations and analyzed related theories [51]. Their findings demonstrated that when the function J has a maximum or a minimum, [x.sub.0], [w.sub.k], [y.sub.k], and [[??].sub.k] are determined. Satisfying equation (34) can guarantee that there is a solution for the estimator [19].

[mathematical expression not reproducible], (32)

[mathematical expression not reproducible], (33)

[P.sup.-1.sub.k] - [epsilon] [bar.[S.sub.k]] + [C.sup.T.sub.k] [v.sup.-1.sub.k] [C.sub.k] > 0. (34)

2.3.2. ALT Algorithm. To develop an SOC estimation algorithm with both high accuracy and low computational cost that can be applied in on-board BMS, the alternate algorithm combining the AHI method and HIF is proposed. When the BMS is started, the initial value of the SOC will be set by the HIF. Once the SOC converges to the true value, the ALT algorithm will switch the computational logic to the AHI method to improve the computational efficiency. The algorithm-switch condition is that the amplitude of the SOC is less than 1%, which means that the HIF has found the true value and ended the convergence process. The cumulative errors of the SOC estimation using the AHI method will be produced by measurement errors of battery current. Therefore, in order to ensure an accurate estimation of the SOC, the proposed algorithm switches back to the HIF algorithm to correct the SOC. This algorithm-switch condition is the increment of the SOC, which is written by the [DELTA]SOC. From the start of the AHI method, if the [DELTA]SOC is more than the [SOC.sub.ALT], the ALT algorithm will switch back to the HIF. The [SOC.sub.ALT] is the switching threshold set by the on-board BMS. The specific calculation steps of the ALT algorithm are shown in Table 2.

Considering the principle of the HIF, to get a more accurate algorithm, (1/e) should be set lower. However, if (1/e) is set too low, the HIF algorithm will fail to converge. Therefore, e is set to 0.01 in this work. Generally, the initial estimation error and the measured noise statistics cannot be known and set in advance in the application of the algorithm. In order to make the calculation simple, the initial state of all matrices is set as the identity matrices. The dimensions of the matrices, such as [S.sub.k], [W.sub.k], and [V.sub.k], were determined by equation (15), and [P.sub.0] was determined by the initial error [19].

3. Experimental Verifications

In order to verify the efficiency and accuracy of the proposed SOC estimator, the low-current OCV test and dynamic cycles test were conducted at 0[degrees]C, 25[degrees]C, and 40[degrees]C, respectively. The schematic of the test bench is shown in Figure 3. It consists of Bitrode MCV12-100 for the battery test, a thermal chamber for environment control, and a host computer for operation control and data display/ storage. The test sample is the A123 lithium-ion cell with remaining capacity 1.1 Ah. The battery tester can charge/ discharge a battery according to the designed program on the host computer. The acquired data are used to determine model parameters and verify the proposed SOC estimator.

In the low-current OCV test, the cell was charged and discharged at a constant rate of C/20. The cut-off voltage for charging was 3.6 V, and the cut-off current was 0.01 C. The cut-off voltage for discharge was 2 V. The OCV-SOC curve can be obtained by using the average value of the charge-discharge equilibrium potential [52, 53] as shown in Figure 4. The coefficients of equation (3) are presented in Table 3 for obtaining the OCV. In the dynamic cycles test, the dynamic stress test (DST) and federal urban driving schedule (FUDS) are used to simulate the actual driving cycles of EVs. The DST and FUDS load profiles are used to verify the performance of parameter identification and SOC estimation of the proposed algorithm.

The BCLS is applied to identify the parameters of the ECM under DST condition at 25[degrees]C. The results of the [R.sub.0], [R.sub.P], and [C.sub.P] are shown in Figures 5(a)-5(c), respectively. The ohmic resistance [R.sub.0] and polarization resistance [R.sub.P] are stable at the beginning of the discharge and increases at the end of the discharge. The polarization capacitance [C.sub.P] decreases slightly with the depth of the discharge.

The BCLS algorithm applies online estimation of noise variance and performs bias compensation in real time to have a better estimation effect when there is colored noise. To verify that the BCLS algorithm has better performance, we compare the accuracy of the RLS algorithm and the BCLS algorithm in model parameter identification. As shown in Figure 5, under DST condition, the model parameters are identified by the RLS algorithm. The results show that the [R.sub.0] and [R.sub.P] do not track the change of the DST current and is constant during most of the discharge time. The [C.sub.P] is gradually increasing. The curve of [C.sub.P] is smooth and does not seem to be affected by the changes of current. Comparing the measured terminal voltage of the battery with the simulated terminal voltage of the battery model, the model error caused by the two parameter identification algorithms is obvious, as shown in Figure 5(d). Using the parameters identified by the BCLS algorithm, the error of the battery model terminal voltage is smaller. Figure 6 shows the results of the states by using the BCLS-ALT algorithm under the DST test at 25[degrees]C. Figures 6(a) and 6(c) show the observed terminal voltage, the reference voltage, and the voltage error where the error is less than 0.03 V except at the end of discharge. Figure 6(b) compares the estimated SOC by using the BCLS-ALT algorithm and the AHI method with the reference SOC. It shows that despite the fluctuations at the start of using HIF, the estimated SOC by applying the proposed algorithm can quickly converge to the reference SOC with a 20% initial SOC error and a 10% current drift. However, with a 10% current drift, the error of the estimated SOC by using the AHI method increases with the depth of the discharge as shown in Figure 6(d). From Figure 6, it can be seen that the proposed algorithm can estimate the SOC under the DST test at 25[degrees]C effectively.

4. Discussions

In the following, the accuracy and efficiency of the proposed BCLS-ALT algorithm are discussed. Section 4.1 analyzes the influence of different switching thresholds on the accuracy and efficiency of the BCLS-ALT algorithm and gives the optimal switching threshold. Section 4.2 discusses the performances of the proposed BCLS-ALT algorithm under different dynamic load profiles. Section 4.2 evaluates the adaptability of the proposed algorithm at different ambient temperatures.

4.1. SOC Estimations Using Different [SOC.sub.ALT]. Due to that the measurement error of current cannot be eliminated, the errors of the SOC estimation by using the AHI method will increase with the depth of the discharge. The proposed BCLS-ALT algorithm can deal with the above issue by switching the computational logic to the HIF algorithm to correct errors caused by the AHI method. The [SOC.sub.ALT] is the switching threshold for switching the AHI method to the HIF. From the start of the AHI method, if the increment of the SOC ([DELTA]SOC) is more than the [SOC.sub.ALT], the ALT algorithm will switch to the HIF. To ensure the accuracy and efficiency of the proposed BCLS-ALT algorithm, we compare the accuracy of SOC estimation using different [SOC.sub.ALT] as shown in Figure 7. Figures 7(a) and 7(b) show the effects of different switching thresholds the [SOC.sub.ALT] on the accuracy and efficiency of SOC estimation under DST and FUDS conditions, respectively. The computational efficiency increases with the increase of the switching threshold [SOC.sub.ALT]. The RMSE, which represents the accuracy of the calculation, increases first and then decreases with the increase of the switching threshold [SOC.sub.ALT]. If the [SOC.sub.ALT] is small, the HIF algorithm cannot converge to the true value when correcting the SOC; and if the [SOC.sub.ALT] is large, the mean absolute error of SOC estimation will increase. In the above two cases, the accuracy of the SOC estimation will decrease. When the switching threshold [SOC.sub.ALT] is 25%, the RMSE reaches the minimum value, which means that the accuracy of the SOC estimation is the highest. When the switching threshold [SOC.sub.ALT] is set to 25%, the comparison of computational time under different dynamic load profiles is shown in Figure 7(c), which reveals that the proposed BCLS-ALT algorithm can greatly reduce the computational time of the model-based SOC estimation method. This algorithm is very suitable for the on-board BMS with limited computing power. In addition, the computational time is obtained by MATLAB R2014b software on a Lenovo E40 PC with Intel Core i5-4210U CPU produced by Intel at 1.70 GHz and 8.0 GB RAM.

4.2. SOC Estimations under Different Dynamic Tests. In the study by Liu et. al. [11], the adaptability of the alternate algorithm is only verified under DST condition, and it is not enough to prove the pros and cons of the algorithm. In this paper, to verify that the proposed algorithm can adapt to different dynamic conditions, the BCLS-ALT algorithm and the SOC estimation method based on the recursive least squares and the alternate (RLS-ALT) algorithm are, respectively, applied to estimate the SOC under the DST test and the FUDS test at 25[degrees]C. To make the simulation closer to the EVs operation conditions, we set the initial error of the SOC to 20% and the drift of the current to 10%. Using the optimal switching threshold, the SOC results are shown in Figures 8 and 9. Despite the fluctuations at the algorithm switching, the two methods are robust and can converge to the reference SOC with an initial SOC error. Nevertheless, the BCLS-ALT algorithm converges more quickly than the RLS-ALT algorithm. Due to the current drift, the error of the SOC estimated by the AHI method increases with the depth of the discharge and even close to 10% at the end of discharge. While the BCLS-ALT algorithm and the RLS-ALT algorithm can correct the error of the SOC estimation, the accuracy of the SOC estimation by the BCLS-ALT algorithm is higher after the error-correction. Figure 10 further analyzes the errors of the two algorithms. At 25[degrees]C, the RMSE of the proposed algorithm is smaller, which indicates that the BCLS-ALT algorithm can obtain accurate battery model parameters and is more robust and adaptable under different dynamic load profiles than the RLS-ALT algorithm.

4.3. SOC Estimations at Different Temperatures. The BCLS-ALT algorithm can update model parameters online with the ambient temperature changes and maintain accurate SOC estimation. In order to verify this, the DST tests were also performed at 40[degrees]C and 0[degrees]C, respectively. To study the influence of model parameters on SOC estimation at different temperatures, case 1 and case 2 are defined for the RLS-ALT algorithm. Case 1: SOC estimation through the ALT algorithm with offline parameters identified by the RLS algorithm at 25[degrees]C. Case 2: SOC estimation through the ALT algorithm with offline parameters identified by the RLS algorithm at 40[degrees]C and 0[degrees]C, respectively. At different ambient temperatures, the two algorithms also can converge quickly to the reference SOC with 20% initial error and 10% current drift as shown in Figures 11 and 12. The error analysis of the SOC is shown in Figure 10. Figure 10 shows that smaller RMSE and MAE can be provided by the BCLS-ALT algorithm; due to that, it can update the model parameters in real time with the ambient temperature changes, nevertheless, the RLS-ALT algorithm cannot respond to changing operation conditions. The RMSE of case 1 at 40[degrees]C is smaller than that at 0[degrees]C, compared the error analysis as shown in Figure 10. At the same time, it also reveals that the parameter change caused by the temperature rise from 25[degrees]C to 40[degrees]C is smaller than that caused by the temperature drop from 25[degrees]C to 0[degrees]C. At 40[degrees]C and 0[degrees]C, the RMSE of two algorithms are almost the same. This shows that although the ambient temperature changes of the EVs are inevitable, the algorithm with real-time updating of battery parameters can provide a highly accurate SOC estimation. Thus, the proposed BCLS-ALT algorithm can provide accurate and efficient estimation of the SOC, which is more suitable for EVs.

5. Conclusions

In this paper, the BCLS-ALT-based SOC joint estimation algorithm with high accuracy and low computational cost is proposed. The robustness and accuracy of the proposed algorithm were verified under the DST test and FUDS test at different ambient temperatures of 40[degrees]C, 25[degrees]C, and 0[degrees]C, respectively. The experimental results show that the accuracy and the computational efficiency of the BCLS-ALT algorithm are high, using the optimal switching threshold. The proposed algorithm converges faster with 20% initial error and 10% current drift compared with the RLS-ALT algorithm. Despite the current drift, due to updating model parameters in real-time, the BCLS-ALT algorithm is more robust under different dynamic load profiles and different ambient temperatures. Therefore, the proposed BCLS-ALT algorithm is more suitable for on-board BMS with limited computing power but requiring high estimation accuracy.

https://doi.org/10.1155/2020/1757384

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

Wei Xiong and Yimin Mo proposed the original idea. Wei Xiong designed the novel algorithm. Wei Xiong, Yimin Mo, and Cong Yan performed and analyzed the experiments together. Wei Xiong wrote the original manuscript. Wei Xiong and Yimin Mo revised the final manuscript.

Acknowledgments

This work was supported by the Manufacturing Development Research Center of Wuhan City Circle (No. WZ2017Y14).

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Wei Xiong, (1,2) Yimin Mo [ID], (1) and Cong Yan (1)

(1) School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China

(2) School of Mechanical Engineering, Hubei Engineering University, Xiaogan 432000, China

Correspondence should be addressed to Yimin Mo; 174181461@qq.com

Received 4 April 2020; Revised 12 July 2020; Accepted 28 July 2020; Published 14 August 2020

Caption: Figure 1: The first-order ECM for a lithium-ion cell.

Caption: Figure 2: Flowchart of the proposed method for SOC estimation.

Caption: Figure 3: Schematic of the test bench.

Caption: Figure 4: OCV-SOC curves.

Caption: Figure 5: Results of parameters identification by using the BCLS and RLS at 25[degrees]C: (a) [R.sub.0]. (b) [R.sub.P]. (c) [C.sub.P]. (d) Model error.

Caption: Figure 6: Results of the states by using the BCLS-ALT algorithm at 25[degrees]C: (a) [U.sub.T]. (b) SOC. (c) [U.sub.L] error. (d) SOC error.

Caption: Figure 7: Computational efficiency and accuracy. (a) Computational efficiency and accuracy under the DST test. (b) Computational efficiency and accuracy under the FUDS test. (c) Computational time.

Caption: Figure 8: SOC results under the DST test at 25[degrees]C: (a) SOC estimation based on the BCLS-ALT algorithm. (b) SOC estimation based on the RLS-ALT algorithm. (c) SOC error.

Caption: Figure 9: SOC results under the FUDS test at 25[degrees]C: (a) SOC estimation based on the BCLS-ALT algorithm. (b) SOC estimation based on the RLS-ALT algorithm. (c) SOC error.

Caption: Figure 10: Error analysis: (a) MAEs. (b) RMSEs.

Caption: Figure 11: SOC results at 0[degrees]C: (a) SOC by the BCLS-ALT algorithm. (b) SOC by the RLS-ALT algorithm at case 1. (c) SOC by the RLS-ALT algorithm at case 2. (d) SOC error.

Caption: Figure 12: SOC results at 40[degrees]C: (a) SOC by the BCLS-ALT algorithm. (b) SOC by the RLS-ALT algorithm at case 1. (c) SOC by the RLS-ALT algorithm at case 2. (d) SOC error.
```Table 1: BCLS for parameters identification.

Step 1: initialization: for [mathematical expression
not reproducible].
Step 2: time update: for k = 1,2, ....
Observation data regression vector: [PHI](k) via (11).
Gain matrix: [K.sub.c] via (20).
Covariance matrix: Q(k) via (18).
Error criterion function F(k) via (17).
Noise variance [[??].sup.2] (k) via (15).
Step 3: measurement update:
Update the posterior state: [[??].sub.LS] (k) via (14).
Update original system state: [[??].sub.BCLS] (k) via (21).
Step 5: the state vector output [[a.sub.1] [a.sub.2] [a.sub.3]].
Calculate model parameters: [R.sub.0], [R.sub.P], and [C.sub.P]
via (7).

Table 2: Alternate algorithm for state of charge (SOC) estimation.

Initialization
Current [u.sub.k], terminal voltage [y.sub.k] at each time, and
initial SOC [x.sub.0]
L = [1 0]
Weighting matrices: [S.sub.k], [P.sub.k], [w.sub.k], and [v.sub.k]
k = 1
Set [SOC.sub.ALT]

Estimation process
For k = 1: 1: length ([u.sub.k])
Step 1: HIF algorithm
Determine
[bar.[S.sub.k]] = [L.sup.T.sub.k][S.sub.k][L.sub.k] (35)
Step 2: linearize [C.sub.k] via (25)
Step 3: calculate gain matrix [K.sub.k]
[K.sub.k] = A[P.sub.k] [[I - [epsilon] [bar.[S.sub.k]] [P.sub.k]
+ [C.sup.T.sub.k][V.sup.-1.sub.k][C.sub.k][P.sub.k]].sup.-1]
[C.sup.T.sub.k][V.sup.-1.sub.k] (36)
Step 4: calculate estimation of [mathematical expression not
reproducible] (37)
Step 5: state estimation at time k + 1
[mathematical expression not reproducible] (38)
Step 6: update covariance matrix
[mathematical expression not reproducible] (39)
Step 7: output SOC estimation at time k + 1
[SOC.sub.k+1] = [LX.sub.k+1] (40)
If the amplitude of the SOC < 1%
Step 8: AHI method
[SOC.sub.k+2] = [SOC.sub.k+1] + [LBu.sub.k] (41)
If [DELTA]SOC = [SOC.sub.k+2+n] - [SOC.sub.k+2] > [SOC.sub.ALT]
Switch to Step 1
Else
k = k + 1,
Back to Step 8
End
End
Step 9: update time
k = k + 1
End
Output
SOC (k)

Table 3: The coefficients of equation (3).

[K.sub.1]         [K.sub.2]        [K.sub.3]

0[degrees]C    1.98[e.sup.-9]   -5.35[e.sup.-7]   5.44[e.sup.-5]
25[degrees]C   1.88[e.sup.-9]   -5.07[e.sup.-7]   5.13[e.sup.-5]
40[degrees]C   2.02[e.sup.-9]   -5.42[e.sup.-7]   5.44[e.sup.-5]

[K.sub.4]   [K.sub.5]   [K.sub.6]

0[degrees]C    -0.002569   0.056590    2.806504
25[degrees]C   -0.002408   0.052822    2.845685
40[degrees]C   -0.002529   0.055013     2.83002
```