Passive Tracking of the Electrochemical Impedance of a Hybrid Electric Vehicle Battery and State of Charge Estimation through an Extended and Unscented Kalman Filter
Abstract
:1. Introduction
2. Impedance Estimation Method
2.1. Linear and Time-Invariant Hypothesis
2.2. Coherence
2.3. Impedance Estimation in Frequency Domain
2.4. Impedance Estimation in Time Domain
2.5. Experimental Protocol
2.6. Results and Discussion
3. SoC Estimation through EKF and UKF
3.1. Overview
3.2. EKF Algorithm
- (1)
- Initialize the original parameters
- (2)
- Estimate the predicted state
- (3)
- Update the estimated covariance
- (4)
- Compute the near-optimal Kalman gain
- (5)
- Update the estimated state
- (6)
- Predict the estimated covariance
- (7)
- Repeat the recursive filter calculation from step 2 to 6.
3.3. UKF Algorithm
- (1)
- Initialize the original parameters are the same as Equations (22) and (23).
- (2)
- For k calculate the sigma points for the state model
- (3)
- Propagate the sigma points through the state model
- (4)
- Calculate the propagated mean
- (5)
- Calculate the propagated covariance
- (6)
- For k calculate the sigma points for the measurement function
- (7)
- Propagate sigma points through the measurement function
- (8)
- Calculate the propagated mean
- (9)
- Calculate the estimated covariance
- (10)
- Compute the Near-Optimal Kalman gain
- (11)
- Update the estimated state
- (12)
- Predict the estimated covariance
- (13)
- D the recursive filter calculation from step 2 to 12.
3.4. Results and Discussion
4. Conclusions
5. Future Work
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Symbol | Name | Units |
Frequency domain | - | |
Laplace domain | - | |
Continuous time domain | - | |
k | Discrete-time domain | - |
m | mth element of a vector | - |
Order of the battery impedance model | - | |
ns | Selected order of the battery impedance model | - |
^ | Estimate | - |
* | Complex conjugate | - |
Impedance estimation notation | Units | |
Cross power spectral density | (W) | |
Power spectral density of the current | (W) | |
Power spectral density of the voltage | (W) | |
Battery impedance | (Ω) | |
Spectral coherence | - | |
Battery voltage | (V) | |
Voltage of the mth RC node of the battery model | (V) | |
Battery current | (A) | |
Cross periodogram between the current and voltage | - | |
A | Normalization factor | - |
Forgetting factor | - | |
Numerator coefficient of the battery impedance | - | |
Denominator coefficient of the battery impedance | - | |
Residues of the partial fraction expansion | - | |
Poles of the partial fraction expansion | - | |
Direct term of the partial fraction expansion | - | |
Series resistance of the battery impedance model | (Ω) | |
mth resistance of the battery impedance model | (Ω) | |
mth capacity of the battery impedance model | (F) | |
Dimension of the estimated impedance | - | |
Phase root mean square error | (°) | |
Modulus root mean square error | (Ω) | |
Sampling time | (s) | |
Kalman filter notation | ||
State variable | ||
Measured variable | ||
Input variable | ||
State function | ||
Measurement function | ||
System noise | ||
Measurement noise | ||
System noise covariance matrix | ||
Measurement noise covariance matrix | ||
State estimation error covariance matrix | ||
State function matrix | ||
Jacobian matrix | ||
Kalman gain matrix | ||
Sigma points vector | ||
Dimension of x | ||
Scaling parameter | ||
Mean sigma points weights | ||
Covariance sigma points weights | ||
β | Scaling parameter | |
Scaling parameter determining the spread of sigma points |
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Power-Train Components | Name | Characteristics |
---|---|---|
Energy Storage System (ESS) | Lithium iron phosphate (LFP) prismatic cells from A123 | Capacity = 39.2 Ah; nominal voltage = 340 V; nominal energy = 13.3 kWh; configuration: 7 × 15s2p. |
Internal Combustion Engine (ICE) | Model MPE850 from Weber | 41 kW, 2 cylinders, 850 cc. |
Electric Generator | Model YASA-400 | 93 kW, axial flux permanent magnet. |
Electric Motors Unit | Model GVK210-100L6 from Linamar | 2 × 80 kW, unit ratio = 8.49. |
Vehicle dynamics | 2015 Subaru BRZ Limited | Drag coefficient = 0.28; frontal area = 1.9695 m2; PHEV mass = 1300 kg; wheel radius = 0.3 m. |
Parameter | UDDS | HWFET |
---|---|---|
0.0873 Ω | 0.0865 Ω | |
0.0014 Ω | 0.0026 Ω | |
0.4187 kF | 0.2467 kF | |
0.2743 Ω | 0.2621 Ω | |
0.410 kF | 2.065 kF |
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Sockeel, N.; Ball, J.; Shahverdi, M.; Mazzola, M. Passive Tracking of the Electrochemical Impedance of a Hybrid Electric Vehicle Battery and State of Charge Estimation through an Extended and Unscented Kalman Filter. Batteries 2018, 4, 52. https://doi.org/10.3390/batteries4040052
Sockeel N, Ball J, Shahverdi M, Mazzola M. Passive Tracking of the Electrochemical Impedance of a Hybrid Electric Vehicle Battery and State of Charge Estimation through an Extended and Unscented Kalman Filter. Batteries. 2018; 4(4):52. https://doi.org/10.3390/batteries4040052
Chicago/Turabian StyleSockeel, Nicolas, John Ball, Masood Shahverdi, and Michael Mazzola. 2018. "Passive Tracking of the Electrochemical Impedance of a Hybrid Electric Vehicle Battery and State of Charge Estimation through an Extended and Unscented Kalman Filter" Batteries 4, no. 4: 52. https://doi.org/10.3390/batteries4040052