International Journal of Automotive Technology, Vol. 15, No. 1, pp. 89−96 (2014) DOI 10.1007/s12239−014−0010−1
Copyright © 2014 KSAE/ 075−10 pISSN 1229−9138/ eISSN 1976−3832
DATA-DRIVEN STATE-OF-CHARGE ESTIMATOR FOR ELECTRIC VEHICLES BATTERY USING ROBUST EXTENDED KALMAN FILTER R. XIONG*, F.-C. SUN and H.-W. HE National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China (Received 25 October 2010; Revised 12 August 2013; Accepted 21 August 2013) ABSTRACT−An accurate battery State-of-Charge (SoC) estimation method is one of the most significant and difficult techniques to promote the commercialization of electric vehicles. This paper tries to make two contributions to the existing literatures through a robust extended Kalman filter (REKF) algorithm. (1) An improved lumped parameter battery model has been proposed based on the Thevenin battery model and the global optimization-oriented genetic algorithm is used to get the optimal polarization time constant of the battery model. (2) A REKF algorithm is employed to build an accurate data-driven based robust SoC estimator for a LiFePO4 lithium-ion battery. The result with the Federal Urban Driving Schedules (FUDS) test shows that the improved lumped parameter battery model can simulate the dynamic performance of the battery accurately. More importantly, the REKF based SoC estimation approach makes the SoC estimation with high accuracy and reliability, it can efficiently eliminate the problem of accumulated calculation error and erroneous initial estimator state of the SoC. KEY WORDS : Robust extended Kalman filter, State-of-Charge, Data-driven, LiFePO4 lithium-ion battery, Electric vehicles
1. INTRODUCTION
approach and et al. However, due to the uncertain driving conditions and application environment, these characteristic parameters can hardly be measured and applied in practice (Caumont et al., 2000). (2) The SoC estimation approaches using the ampere-hour counting approach and the battery model which is built through empirical or physical equations. These models are built on the basis of the archived data collected in advance. However, a decaying battery performance resulting from the invariable parameters characteristic in these models and the cumulative error caused by the initial estimated SoC (Lei et al., 2008; Shi et al., 2008). (3) The SoC estimation approaches using the extended Kalman filter (EKF) algorithm. The EKF approach used to estimate the SoC was studied by Plett et al. (2004), which can not only implement the parameters identification for the battery model but can also be employed for online SoC estimation. And therefore it is well suitable for dynamic driving conditions. However, it still strongly depends on the accuracy of the model, as well on the accurate priori noises statistic information. Nevertheless, the system and observation noise statistic information usually are unknown. Generally, improper settings of the noise information will result in remarkable estimation errors and even divergence (Qu et al., 2003). In considering that the system with uncertain model parameters and unknown priori noises information, some approaches with minimum errors for ensuring the robustness of the state estimator have been proposed in (Xi, 1994; Chen et al., 1992). However, these approaches are
To address the two urgent goals nowadays of protecting the environment and achieving energy sustainability, it is of strategic significance on a global scale to replace oildependent vehicles with electric vehicles (EVs) (Suh et al., 2010; Guezennec et al., 2012; Xiong et al., 2013). However, to satisfy the operation voltage and traction power requirements of electric vehicles, battery packs have to be made up of hundreds of cells connected in series or parallel to overcome the limitations of low energy density, low cell capacity and cell voltage (Xiong et al., 2013). But how to avoid the adverse effect of cell inconsistency on battery pack performance and prolong the service life of both the pack and the cells are posing tremendous technological challenges to battery State-of-Charge (SoC) estimation techniques (Zhang et al., 2012; Xiong et al., 2013). A number of researches have previously been proposed for calculating battery SoC and each one has its own advantages. The main SoC estimation approaches can be divided into three kinds (Xiong et al., 2013): (1) The SoC estimation approaches using the measurements of the characteristic parameter of the battery, which includes the open-circuit voltage (OCV) approach, the discharge test approach, the dynamic voltage approach, the resistance *Corresponding author. e-mail:
[email protected]
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unable to obtain the optimal state estimation. To obtain an optimal and robust SoC estimation under uncertain system parameters, such as model parameters or noise statistic characters, a robust extended Kalman filter (REKF) algorithm is proposed for the SoC estimation of a LiFePO4 lithium-ion battery (LiB). This method regards that the model error as a constant value, and employs the separated-bias estimation approach (Na and Zhang, 1998) to estimate the model state and error. Afterwards it uses the error estimation to correct the model state estimation, and hereby obtains the optimal state estimation of the actual system. A LiFePO4 LiB with a nominal voltage of 3.2 V and the nominal capacity of 12 Ah is used as a practical case to verify and evaluate the proposed REKF algorithmbased SoC estimation method.
2. MODELING FOR THE LIB 2.1. Lumped Parameter Battery Model The battery equivalent circuit models mainly include the Rint model, the Thevenin model, the RC model and the PNGV model, where the Thevenin model is commonly used to simulate a LiB cell (Xiong et al., 2012). However, in order to improve the model accuracy and describe the dynamic performance of the LiFePO4 LiB closely, the polarization characteristics of the LiFePO4 LiB is especially considered and described in detail. An improved Thevenin model (lumped parameter battery model) is proposed in this paper, which is implemented by adding an extra RC branch to the Thevenin model as shown in Figure 1 and the relaxation effect of the LiB has been redefined with separated electrochemical polarization and concentration polarization, which is helpful of the dynamic voltage estimation of the battery model. Figure 1 shows that the schema of the lumped parameter battery model for the LiFePO4 LiB, which is mainly composed of three parts including OCV-Uoc, internal resistances and equivalent capacitances. The internal resistances include the ohmic resistance Ro and the electrochemical polarization resistance Rpa as well the concentration polarization resistance Rpc. The equivalent capacitances, which include Cpa and Cpc, are used to describe the transient response during the charging and discharging state. IL and UL are the charging and discharging current and the terminal voltage respectively.
Figure 1. Lumped parameter battery model.
The electrical behavior of the lumped parameter battery model can be expressed as follows: ⎧· Upa - -----I + L⎪ Upa = –------------R C pa C pa pa ⎪ Upc - -----IL⎨ U· = ------------⎪ pc Rpc Cpc + Cpc ⎪ ⎩ UL = OCV – Upa – Upc – IL Ro
(1)
Where Upa and Upc are the voltages crossing on Cpa and Cpc respectively. 2.2. Parameter Identification 2.2.1. Battery test bench The test bench setup is shown in Figure 2. It consists of Arbin BT2000 cycler, a thermal chamber to regulate the operation temperature, a computer to the program and store experimental data and LiFePO4 LiB cells. The eight independent channels battery cycler is responsible for loading the currents or powers profiles on the test cells with the voltage range of 0-5 V and current range of ±100 A. The measurement inaccuracy of the current and voltage sensors inside the Arbin BT2000 system is less than 0.1%. The measured data is transmitted to the host computer through TCP/IP ports. Test cells are connected with the Arbin BT2000 cycler and then placed inside the thermal chamber to maintain the desired temperatures to perform special behavior. The temperature operation range of the thermal chamber is between -55oC and 85oC (Xiong et al., 2013). 2.2.2. Parameters identification In order to acquire data to identify the parameters of the improved battery model, a Hybrid Pulse Power Characterization (HPPC) test was conducted on the LiFePO4 LiB at 10% SoC intervals (constant current 1/3 C=4A discharge segments) starting from 100% to 10%, where the used data in this study was controlled at 25oC in a thermal chamber and each HPPC test was followed by a 2 hour rest period allowing the cell to return to its electrochemical and thermal equilibrium. The sampling interval was set to 1s in this paper. It is furthermore assumed that the current is positive when the battery in discharge operation and negative when the battery in charge
Figure 2. Configuration of battery test bench.
DATA-DRIVEN STATE-OF-CHARGE ESTIMATOR FOR ELECTRIC VEHICLES BATTERY USING ROBUST
operation. The identifying principles can be obtained by discretion equation of the Equation (2). ⎧UL,k = Uoc,k – RoIL, k – Rpa Ipa, k – RpaIpc,k ⎪ ⎧ ⎧ [1 – exp(–∆t ⁄ τpa)] ⎫ 1 – exp(–∆t ⁄ τpa)]- ⎫ ⎪Ipa, k = ⎨1 – [--------------------------------------- – exp(–∆t ⁄ τpa) ⎬ ⎬ × IL, k + ⎨1 – --------------------------------------( ∆ ⁄ ) ( ∆ ⁄ ) t τ t τ ⎪ pa pa ⎩ ⎭ ⎩ ⎭ ⎪ × IL,k – 1 + exp(–∆t ⁄ τpa) × Ipa, k – 1 ⎨ ⎪ ⎧ [1 – exp(–∆t ⁄ τpc)] ⎫ ⎧ [1 – exp(–∆t ⁄ τpc)] ⎫ ⎪ - ⎬ × IL, k + ⎨1 – --------------------------------------- – exp(–∆t ⁄ τpc) ⎬ ⎪Ipc, k = ⎨1 – --------------------------------------(∆t ⁄ τpc) (∆t ⁄ τpc ) ⎩ ⎭ ⎩ ⎭ ⎪ ⎩ × IL,k – 1 + exp(–∆t ⁄ τpc) × Ipc, k – 1
(2)
( g)
(3)
Where χˆ k is the estimation value of the current population χk at generation g, χk is the current individuals k of the population χ, where χ =diag(τpa, τpc), and Uˆ L, k is the estimation value of the terminal voltage UL at individuals k. N is the estimation length, the bigger, the wider search range and the genetic manipulation time of each generation will also become longer. With N smaller, the genetic manipulation time of each generation will become shorter, and the search range will become narrower. N is set based on the duration of each hybrid pulse test, in this paper we use four groups of currents to execute the LiB cell and the N is set to 600s (Xiong et al., 2013). Note that the stopping criterion is determined by the limit of the generation and in this paper it is set to 100. Providing a random value to parameter τpa, τpc and setting limits of the lower and upper boundary, we can calculate the f ( χˆ j ) on the basis of (2), (3) and genetic algorithm model with the measured current and the voltage, then we can get ( g)
Table 1. Identification results of the improved battery model (SoC=0.8 and 0.9) in charging process. SOC
Cpa(F)
Rpa(mΩ)
Cpc(F)
Rpc(mΩ) Ro(mΩ)
0.8
632.73
0.706
3821.40
4.40
2.323
0.9
918.72
0.685
2653.19
4.90
2.842
Table 2. Identification results of the improved battery model (SoC=0.8 and 0.9) in discharging process.
Where ∆t indicates a fixed sampling interval between two adjacent measurement points and the discrete time index is denoted by an integer variable k. Ipa,k and Ipa,k-1 are the load current over the electrochemical polarization resistance Rpa at the kth sampling interval and (k-1)th sampling interval respectively. Ipc,k and Ipc,k-1 are the load current over the concentration polarization resistance Rpc at the kth sampling interval and (k-1)th sampling interval respectively. IL,k and IL,k-1 are the load current at the kth sampling interval and (k-1)th sampling interval respectively. Uoc,k is the OCV at the kth sampling interval. Before the identification process of parameters, the time constants of electrochemical polarization (τpa=RpaCpa) and concentration polarization (τpc=RpcCpc) need to be given in advance based on the battery characteristics. This study uses the genetic algorithm to get the optimal time constants with the Equation (3) to identify the time constant parameters. More importantly, the model parameters are obtained at charge and discharge separately. Note that OCV-Uoc is determined by the experiment, we use the OCV test to get the Uoc values of several different SoC points. The objectives function of the genetic algorithm is as follows: ⎧ min{f( χˆ k ) } ⎪ ⎨ ˆ ( g) ( g) 2 ˆ 1- N ( UL, k – UL, k( χˆ k ) ) ⎪ f( χk ) = --∑ N ⎩ i=1
91
SOC
Cpa(F)
Rpa(mΩ)
Cpc(F)
Rpc(mΩ) Ro(mΩ)
0.8
738.99
0.748
5183.62
4.01
2.638
0.9
968.56
0.693
2283.74
4.54
2.201
the minimal f ( χˆ j ) and the optimal value for τpa and τpc. The identification results of the improved model are shown in Table 1 and Table 2, where we only list the identification result of the SoC at 0.8 and 0.9. From the OCV test, we can establish the OCV function between the open circuit voltage Uoc and the SoC as follows: ⎧ Uoc ( z ) = 2z4 – 3.8z3 + 2.2z2 – 0.18z + 3.2( IL > 0 ) ⎨ 4 3 2 ⎩ Uoc ( z ) = 2.1z – 4z + 2.3z – 0.19z + 3.2( IL < 0 )
(4)
Where z is the abbreviation of the SoC. 2.2.3. Model evaluation The Federal Urban Driving Schedules (FUDS) cycle is a typical working cycle that lasts for 1372 s, which is commonly used to validate or evaluate the SoC estimation approach, the dynamic model of the battery and the fuel economy of hybrid electric vehicles (INEEL, 1996; Smith and Wang, 2006; Michael et al., 2000; Xiong et al., 2013). In this study the FUDS test is applied to evaluate the improved lumped parameter battery model and the REKF algorithm based SoC estimation approach. Four driving cycles are performed with a sampling interval of 1s and a total duration time of 5488s. Figure 3 shows the current profiles of several FUDS cycles. To evaluate the prediction performance of the improved lumped parameter battery model, we have carried out the
Figure 3. Current profile of the FUDS test cycles.
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of the improved model is 23.5 mV and less than 1% of its nominal voltage, which suggest that the improved battery model is more accurate than the Thevenin battery model, and also shows that the parameters identification method based on the genetic algorithm and is accurate and reliable.
3. SOC ESTIMATION USING THE ROBUST EXTENDED KALMAN FILTER 3.1. SoC Definition The SoC is defined as a ratio of the remaining capacity over the maximum available capacity. For time-series process and state estimation application, the discrete-time form of the SoC function is built here. zk = zk – 1 – ηi IL, k ∆t ⁄ CN
(5)
Where zk and zk-1 are the SoC at kth and (k-1)th sampling time respectively, hi is the coulomb efficiency, which is the function of current, temperature and capacity (see in the Table 4); CN is the nominal capacity. Figure 4. Voltage and voltage error profiles under the FUDS test cycles. Table 3. Statistic list of the absolute voltage error with the FUDS cycles. Model
Maximum (V)
Mean (V)
Variance (V2)
Thevenin
0.0750
0.0172
4.99e-05
Improved
0.0235
0.0043
1.08e-05
comparison on the voltage estimation accuracy between the Thevenin battery model and improved lumped parameter battery model, the comparison of the voltage between the experiment data and the estimated voltages are plotted in Figure 4 (a), and their estimation errors are plotted in Figure 4 (b). Figure 4 (a) shows that both the two battery models have desired simulation accuracy in voltage prediction. Figure 4 (b) shows that the maximum estimation error of the battery voltage is less than ±75 mV. However, for the improved battery model, the peak voltage estimation error is less than ±30 mV, where the nominal voltage is 3.20V, thus this peak error is within 1% of its nominal voltage. To make the comparison more clearly, the statistic list of the absolute error of the voltage for the Thevenin battery model and improved battery model is summarized in Table 3, where the result is got under the FUDS cycles in Figure 4. Table 3 shows that the voltage estimation error between the experimental data and the estimation of the battery model of the Thevenin model is around 17.2 mV (0.54% of the nominal voltage) and has a maximum value of 75 mV (2.34% of the nominal voltage). While the maximum error
3.2. Robust Extended Kalman Filter The key idea of robust state estimation is that the modeling errors are regarded as constant bias state vectors. The dynamic states of the system and bias states are estimated separately, and the dynamic states are subsequently corrected using the bias states estimated values. In considering that the nonlinear behavior of the battery model for LiB, we propose a robust SoC estimation approach using the REKF algorithm in this study, which is applicable for nonlinear systems. If modeling errors are neglected, the nonlinear system of interest can be defined by: ⎧ x k = A k – 1 x k – 1 + F k – 1 u k – 1 + B k – 1 b k – 1 + wk – 1 ⎨ ⎩ yk = C k x k + D k b k + vk
(6)
Where, bk ∈ Rn(n + q + m) is defined by a constant bias vector at the kth sampling time. xk is n × 1 system state vector at the kth sampling time; yk is m × 1 observe matrix at the kth sampling time; Wk is process noise with zero mean at the kth sampling time and its covariance is Qk; Vk is measurement noise with zero mean and covariance of Rk. Ak-1 and Fk-1 are system matrixes at the (k-1)th sampling time; Ck is system measurement matrixes at the kth sampling time. Then the state and observer matrixes are defined as: Table 4. Coulombic efficiency of the LiFePO4 LiB. IL (A)
-4
-6
-12
-18
-24
η/%
100
99.56
99.05
93.33
91.64
IL(A)
4
6
12
18
24
η/%
100
99.67
96.59
94.12
92.65
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algorithm, we need a state transition and measurementupdate model that relates the SoC to the voltage. We firstly obtain the discretization equation of model equation in Equation (1) and the result is presented as follows. From the above equations, the system with model error can be transferred to the system with constant bias value, and the state estimation also is changed to be based on the system with constant bias value. Afterwards we can obtain the SoC estimation method based on the REKF algorithm and separated-bias estimation. The completely time-series format based calculating process of the REKF algorithm can be divided to four steps and which is presented as follows. Step 1. Initializing the state estimator. P0, Q0, R0, x0, A0, F0, C0, b0, V0, M0. Step 2. Time update (prior predict without y) + xˆ k = Ak – 1xˆ k + Fk – 1uk – 1 + Bk – 1bk – 1
(7)
P-k = Ak – 1Pk – 1ATk – 1 + Qk – 1
(8)
Sk = CkP-k CTk + Rk
(9)
Kx, k = PTk – 1CTk S–k 1
(10)
Uk = Ak-1Vk-1+Bk
(11)
Gk = CkUk+Dk
(12)
(21)
We then obtain the state transition and measurement equations as follows.
(22)
Step 3. Measure update (posterior predict with y) -
ex, k = yk – Ckxˆ k
(13)
+ xˆ k = xˆ k + Kx, kex, k
(14)
Pk = [I – Kx, k ]P-k
(15)
Kb, k = Mk – 1GTk [Gk Mk – 1GTk + Sk ]
–1
(16)
Vk = Uk −Kx,kGk
(17)
bk = bk-1+Kb,keb,k
(18)
Mk =[I−Kb,kGk]Mk-1
(19)
⎧ 8z3 – 11.4z2 + 4.4z – 0.18 dU oc ( z ) ----------------=⎨ 3 2 dz ⎩ 8.4z – 12z + 4.6z – 0.19
( IL ≥ 0 ) ( IL < 0 )
(23)
After initializing the state estimator of the SoC, we can achieve the SoC estimate with the time-series calculation process of the REKF.
Step 4. The optimal estimate of the state. + xˆ k = xˆ k + Vkbk
Where the calculation of dUoc(z)/dz is calculated as follows on the bases of Equation (4):
(20)
Where I is the unit matrix, Pk is the covariance matrix of the estimation error at the kth sampling time. The other matrixes, including S, U, Kx, Kb, M, G and V, are used for calculating the estimates for the REKF algorithm. 3.3. Data-driven Based Robust SoC Estimator In order to achieve the SoC estimate with the REKF
4. VERIFICATION AND DISCUSSION 4.1. Evaluation of REKF Approach The FUDS test is performed with the power profiles and terminated when the cells reach their lower cutoff voltage level. The measured current, voltage of the LiFePO4 LiB are shown in Figure 3 and Figure 4, its SoC is plotted in Figure 5.
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Figure 5. Reference SoC profiles of the FUDS test. It is noted that, in all cases, “true” SoC is calculated from the Arbin data-logger using Coulomb counting on measured data. And the “true” SoC is only approximately accurate since current sensor error accumulated over time, which causes any estimate computed using coulomb counting to eventually diverge. In order to get the reference SoC profiles by an experimental approach, firstly, the battery is fully charged to make sure that the initial SoC is 1, then it is discharged by nominal current with 0.1 of its present maximum available capacity; after the several cycle test is finished, a further discharge experiment with nominal current is conducted until the battery is fully discharged, and then the true value of the terminal SoC can be calculated according to the definition of SoC. Since the values of the initial SoC and the terminal SoC are accurate, the Coulomb counting method is used to calculate the experimental SoC based on the load current profiles and the Coulomb efficiency map, also a proper adjustment coefficient, which is calculated based on the true values of the initial SoC and terminal SoC, is applied during the calculation. The Coulomb counting method with an adjustment approach based on a further discharge experiment can only be used in laboratory, since it is difficult to keep the battery as standing state frequently in practical EVs application. To evaluate the performance of the proposed SoC estimator with the REKF algorithm and EKF algorithm, we have carried out a comparison between these two estimators, the initial estimator states of the SoC are
Figure 6. SoC estimation results of the EKF and REKF approach.
Figure 7. SoC estimation errors of the EKF and REKF approach.
accurately initialized to 0.899, and the results are presented in Figure 6. After initializing the initial estimator states, the datadriven based SoC estimator can estimate the SoC and voltage in real-time. Figure 6 shows that both of the two SoC estimators have good prediction precision, the estimated SoCs follow the referenced SoC trajectory closely. To compare the two SoC estimators in detail, the SoC estimation error of the two estimators is plotted in Figure 7. It is seen that the estimated SoC using the REKF algorithm converges faster to the reference SoC trajectory, and the estimation accuracy is significantly improved compared to SoC estimate using EKF algorithm. From the zoom figure of the Figure 7, we find that the SoC estimation errors of the two estimator in the first 10s are approaches to the same, this is because that in this time range the load currents are zero, the SoC estimations are mainly calculated by the fitting accuracy of OCV presented in Equation (4). However, when the load current is far from zero, the two estimators can continually optimize the predicted SoCs. Thus, the model inaccuracy in OCV cannot affect the SoC estimation accuracy and the proposed REKF algorithm estimator has a superior performance than the EKF algorithm based method. This emphasizes the advantages of REKF and why this algorithm is suitable for a LiFePO4 LiB, mainly because of its good accuracy and fast convergence performance. In considering that a reliable SoC estimator depends on two factors, the first is the accurate calculation performance, which has been verified in the above analysis; the second is the robustness against erroneous initial estimator state. In other words, provided an erroneous initial estimator state of SoC, the SoC estimator is required to correct it quickly and converge to the reference SoC profiles after several sampling intervals. Thus the following section focuses on the evaluation on the robust performance of the proposed REKF algorithm based SoC estimation approach. 4.2. Discussion If the ampere-hour counting approach is used to estimate
DATA-DRIVEN STATE-OF-CHARGE ESTIMATOR FOR ELECTRIC VEHICLES BATTERY USING ROBUST
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Table 5. Statistic list of the SoC estimation errors.
Figure 8. Converge results for different initial SoC values.
the SoC of the battery, the battery management system generally gets the initial SoC based on the record of the last operation or from the OCV and with the table of OCV vs. SoC. However, due to the polarization, self-recovery, selfdischarge and other factors, it is hard to get the accurate initial SoC, which will over time result in accumulated errors. As a result, the SoC approach requires having the potential to solve the problems of this accumulated error and erroneous initial estimator state of the SoC. To investigate that whether the REKF based SoC estimation approach can efficiently solve these problems or not, a systematically verification experiment a systematically verification experiment has been conducted. Eight different of erroneous initial estimator states of the SoC are initialized to the REKF-based SoC estimator, which includes eight erroneous initial SoCs those are 0.50, 0.84, 0.86, 0.88, 0.90, 0.92, 0.94 and 0.96. The correct initial SoC in Figure 8 is 0.899. Figure 8 describes that the convergence behavior of the REKF based SoC estimation approach for the erroneous initial SoCs of the first 150 sampling intervals. Figure 8 describes the convergence performance of the proposed REKF algorithm–based SoC estimator under different erroneous initial estimator states of the SoC. Even if provided a big erroneous initial SoC and the initial error closes to 0.4 (initial SoC, z0=0.5), the stable SoC estimation error is less than 2% after several sampling intervals for correcting the initial SoC error. It shows that the proposed SoC estimator can correct the initial estimator states of the SoC quickly and accurately. Moreover, a higher precision has been achieved. This performance can well reach the requirements of the battery management system used in the electric vehicles. Note that with big initial SoC error, the convergence speed is a little slow, but the initial SoC error is hard to exceed to 0.1 in the practical electric vehicles application, thus the convergence performance is better than this case. To make the evaluation more clearly, Table 5 represents the detailed statistical list of the absolute estimation errors for different initial SoC values. The errors describe the difference between the estimated SoC and the reference
z0
Maximum
Mean
Variance
Terminal error
0.50
0.0352
0.0144
3.68e-004
0.0156
0.84
0.0279
0.0105
7.40e-005
0.0103
0.86
0.0258
0.0101
6.18e-005
0.0100
0.88
0.0239
0.0097
5.14e-005
0.0095
0.90
0.0098
0.0051
2.65e-005
0.0070
0.92
0.0196
0.0089
3.38e-005
0.0087
0.94
0.0176
0.0085
2.59e-005
0.0082
0.96
0.0181
0.0080
1.79e-005
0.0073
SoC. Note that the estimation results in the statistical list starting from120s, namely the result in the correcting process of the erroneous initial SoCs is removed when making the comparison. Table 5 shows that the SoC estimation error is less than 4% against different erroneous initial SoCs for the LiFePO4 lithium-ion battery who has flatter OCV behavior, and the terminal error is less than 2%. This indicates that the proposed REKF-algorithm based SoC estimation method is accurate enough for electric vehicles application.
5. CONCLUSION To improve the dynamic prediction accuracy of the battery model for LiFePO4 lithium-ion battery, an improved lumped parameter battery model is proposed. The optimal polarization time constants of the model are achieved through a genetic algorithm. The result shows that the maximum estimation error of the improved model is less than 1% of its nominal voltage, which suggests that the proposed improved battery model has ideal prediction precision and can describe the dynamic performance of LiFePO4 lithium-ion battery accurately. To improve the prediction precision and robustness of the SoC estimation method, a REKF algorithm is employed to eliminate the SoC estimation error resulting from the uncertainty of the battery model and the unknown priori noises information. The results based on the FUDS test show that the REKF algorithm-based SoC estimation approach has a better prediction precision than the EKF algorithm-based approach. More importantly, the proposed method has a good convergence performance against different erroneous initial SoC values. It can efficiently solve the problem of SoC estimation accuracy from the accumulated error and inaccurate initial SoC. ACKNOWLEDGEMENT−This work was supported by the National Natural Science Foundation of China (51276022) and the Higher school discipline innovation intelligence plan
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(“111”plan) of China.
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