A New Method for State of Charge Estimation of Lithium-Ion Batteries Using Square Root Cubature Kalman Filter
Abstract
:1. Introduction
1.1. Review of Methods
1.2. Contribution of This Study
1.3. Organization of This Paper
2. Model of the Battery
Lumped-Parameter Model of the Battery
3. Cubature Integral Approximation Method and SRCKF Algorithm
3.1. Cubature Using Numerical Integral Approximation Method
3.2. SRCKF Algorithm
- Step 1. Initialization of state estimation. We set the initial value of state to and obtain
- Step 2. State prediction ().
- Step 2.1 n cubature points for battery states at time () are calculated.
- Step 2.2: The battery state Formula (7) is used to propagate cubature points and generate new points.
- Step 2.3: After the newly generated points are weighted, the sum of the weighted points is determined and then the prediction value of the battery state at time can be estimated (SRCKF uses equal weights).
- Step 2.4: The square root of the covariance matrix of the prediction value is estimated.
- Step 3.1: A set of equal-weight cubature points by the square root matrix of the state prediction value and its covariance matrix is generated using the spherical radial rule ().
- Step 3.2: The battery observation Formula (8) is used to propagate the cubature points.
- Step 3.3: The prediction value of observation at time is generated.
- Step 3.4: The square root of the covariance matrix of the estimated prediction value of observation is estimated.
- Step 3.5: The observation and prediction values of the square root of the covariance matrix of each other are estimated.
- Step 3.6: The SRCKF filter gain matrix is solved.
- Step 3.7: The best state value estimate of the battery at time is calculated.
- Step 3.8: The square root of the error covariance matrix of the optimal state estimation of the battery at time is calculated.
4. Experimental Configurations, HPPC Test and Off-Line Identification of Relevant Parameters
5. Results and Discussion
5.1. Experiment A: Test with Constant Discharge Current
5.2. Experiment B: DST
5.3. Experiment C: DST with Noise Measurement
6. Conclusions
- (1)
- The shortcomings of several common algorithms for battery SOC are investigated, and the SRCKF algorithm is recommended for its accuracy and speed. To balance between model accuracy and computation cost, the Thevenin model was applied to simulate the dynamic characteristics of the LIB, based on which Equations (9) and (10) were derived. With this model, experimental and parameter identification methods for off-line identification of battery parameters were proposed, and four Simulink algorithm models were established according to the EKF, UKF, CKF, and SRCKF algorithms, respectively.
- (2)
- Constant current discharge tests and DST cycles were conducted to verify the performance of the proposed method against other three methods (i.e., EKF, UKF, and CKF). The results of experiments showed that the estimation SOC error with the proposed method quickly converged to 2% within approximately 25 s, whereas the initial SOC error reached 50%. The RMSEs (0.00892), maximum error (0.02469), and mean error (−0.00704) of the proposed method were the best among all the methods in DST. Moreover, the maximum error was less than 5.5% even when the measurement noise of the voltage and the current was up to 2.5% in DST with noise.
- (3)
- Given the insufficient sensor accuracy and inaccurate hardware circuit, the signal of voltage and current are often interfered. Hence, a real EV running in a complex working condition (such as DST) with noises is very important. The method should be able to handle the nonlinear changes and random disturbance. In comparison with other methods (EKF, UKF, and CKF), only the proposed method avoided the filtering divergence in DST with noise (up to 5%).
Acknowledgments
Author Contributions
Conflicts of Interest
Nomenclature
i | load current |
Uocv | open-circuit voltage |
Ut | terminal voltage |
Up | polarization voltage |
Rp | polarization resistance |
Cp | polarization capacitance |
U0 | internal resister voltage |
Rchg | internal resistance when discharging |
Rdchg | internal resistance when discharging |
Rn | integral region |
uk | known input |
E(·) | expectation |
Nmax | max voltage or the max current |
Greek Symbol
σ(·) | spherical metric unit |
ω(x) | nonnegative weight function |
(ωi,ζi) | weight point set |
ωk−1 | system noise |
νk | noise of measurement |
θ | arithmetic mean |
α | scaling factor |
Acronyms and Abbreviations
OCV | open-circuit voltage |
SOC | state of charge |
SRCKF | square root cubature Kalman filter |
EKF | Extended Kalman filter |
UKF | unscented Kalman filter |
CKF | cubature Kalman filter |
EV | electric vehicle |
LIB | lithium-ion battery |
BMS | battery management system |
PF | particle filter |
UT | unscented transform |
LECU | local electronic control unit |
BMU | battery management unit |
HPPC | hybrid pulse power characteristic |
SOH | state of health |
DST | dynamic stress test |
EMI | electromagnetic interference |
RMSE | root mean square error |
Ah | ampere-hour |
C | discharge rate |
MATLAB | MATrix LABoratory |
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Item | Value |
---|---|
Cell Voltage | 3.7 V |
Cell Capacity | 35 Ah |
Maximum charging rates | 2 C |
Maximum discharging rates | 4 C |
Item | Value |
---|---|
Max value of discharging current | 200 A |
Max value of charging current | 200 A |
Range of voltage measurement | 0~5 V |
Current measurement error | 0.1%F.S. but not better than ±20 mA |
Voltage measurement error | 0.1%F.S. but not better than ±20 mV |
Temperature measurement error | ±1 °C |
Coefficients | g | f | e | d | c | b | a |
---|---|---|---|---|---|---|---|
Rp | 0.004107 | −0.052842 | 0.316150 | −0.917690 | 1.395537 | −1.062447 | 0.318105 |
Rdchg | 0.001812 | −0.006602 | 0.041589 | −0.144026 | 0.258410 | −0.226639 | 0.076809 |
Uocv | 2.971124 | 7.396502 | −37.015633 | 101.27299 | −143.46866 | 100.40782 | −27.381834 |
Cp | 3839 | 86,475 | −502,083 | 2,080,696 | −4,107,296 | 3,696,315 | −1,244,161 |
Coefficients | g | f | e | d | c | b | a |
---|---|---|---|---|---|---|---|
Rp | 0.008551 | −0.108954 | 0.621934 | −1.758040 | 2.593975 | −1.910763 | 0.554538 |
Rdchg | 0.005575 | −0.027267 | 0.171668 | −0.538437 | 0.840745 | −0.636656 | 0.187370 |
Uocv | 2.712023 | 10.728256 | −55.006330 | 149.858773 | −213.2116 | 151.254438 | −42.189757 |
Cp | 2677 | −51,640 | 397,937 | −773,697 | 312,761 | 400,225 | −282,581 |
Methods | EKF | UKF | CKF | SRCKF |
---|---|---|---|---|
Computation cost (s) | 0.0023 | 0.01788 | 0.006775 | 0.00774 |
Methods | Convergence Time (s) | RMSE | Max Error | Mean Error |
---|---|---|---|---|
EKF | 25 | 0.01211 | 0.02727 | 0.00270 |
UKF | 110 | 0.00970 | 0.01701 | −0.00268 |
CKF | 60 | 0.00602 | 0.01233 | 0.00008 |
SRCKF | 25 | 0.00599 | 0.01308 | 0.00015 |
Methods | Convergence Time (s) | RMSE | Max Error | Mean Error |
---|---|---|---|---|
EKF | 65 | 0.01560 | 0.05064 | −0.01443 |
UKF | 85 | 0.00864 | 0.03058 | −0.01047 |
CKF | 60 | 0.00912 | 0.02665 | −0.00705 |
SRCKF | 25 | 0.00892 | 0.02469 | −0.00704 |
SRCKF | RMSE | Max Error | Mean Error |
---|---|---|---|
1% noise | 0.01081 | 0.03489 | −0.00273 |
2.5% noise | 0.01693 | 0.05347 | 0.00055 |
5% noise | 0.02004 | 0.07971 | 0.00279 |
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Cui, X.; Jing, Z.; Luo, M.; Guo, Y.; Qiao, H. A New Method for State of Charge Estimation of Lithium-Ion Batteries Using Square Root Cubature Kalman Filter. Energies 2018, 11, 209. https://doi.org/10.3390/en11010209
Cui X, Jing Z, Luo M, Guo Y, Qiao H. A New Method for State of Charge Estimation of Lithium-Ion Batteries Using Square Root Cubature Kalman Filter. Energies. 2018; 11(1):209. https://doi.org/10.3390/en11010209
Chicago/Turabian StyleCui, Xiangyu, Zhu Jing, Maji Luo, Yazhou Guo, and Huimin Qiao. 2018. "A New Method for State of Charge Estimation of Lithium-Ion Batteries Using Square Root Cubature Kalman Filter" Energies 11, no. 1: 209. https://doi.org/10.3390/en11010209