Experimental Investigation of State and Parameter Estimation within Reconfigurable Battery Systems
Abstract
:1. Introduction
2. Enhanced State and Parameter Estimation
2.1. Battery Model
2.2. Cell characterization
2.3. Kalman Filtering
Algorithm 1 Extended Kalman filter. 

2.4. Switching
3. Input Signal Analysis
4. Implementation
4.1. Simulation
4.2. Experimental Section
5. Results and Discussion
5.1. Simulation
5.2. Experimental Results
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
BEV  Battery electric vehicle 
BMS  Battery management system 
CCCV  Constant current/constant voltage 
DST  Dynamic stress test 
ECM  Equivalent circuit model 
EKF  Extended Kalman filter 
HPPC  Hybrid pulse power characterization 
nRMSE  Normalized rootmeansquared error 
OCV  Opencircuit voltage 
PRBS  Pseudorandom binary sequence 
PWM  Pulsewidth modulation 
RBS  Reconfigurable battery system 
RC  Resistor–capacitor 
RMS  Root mean square 
RMSE  Rootmeansquared error 
SISO  Singleinput, singleoutput 
SOC  State of charge 
SOH  State of health 
UDDS  Urban Dynamometer Driving Schedule 
WLTP  Worldwide Harmonized LightDuty Vehicles Test Procedure 
Nomenclature  
$\mathit{A}$  Statespace system matrix 
$\mathit{B}$  Statespace input matrix 
${C}_{1/2}$  Capacity of the RC elements (F) 
$\mathit{C}$  Statespace output matrix 
D  Statespace feedthrough scalar 
${D}_{PWM}$  Duty cycle of the PWM signal (%) 
f  Nonlinear system function 
${f}_{PWM}$  Frequency of the PWM signal (Hz) 
h  Nonlinear measurement function 
I  Cell current (A) 
$\mathit{I}$  Identity matrix 
k  Time step 
K  Number of time steps 
$\mathit{K}$  Kalman gain 
$\mathit{P}$  State covariance matrix 
$\mathit{q}$  Process noise 
$\mathit{Q}$  Process noise covariance matrix 
r  Measurement noise 
R  Measurement noise covariance 
${R}_{0}$  Internal resistance ($\Omega $) 
${R}_{1/2}$  Resistance of the RC elements ($\Omega $) 
S  Switch 
SOC  State of charge (%) 
${t}_{off}$  Time the cell is disconnected (s) 
u  Statespace input 
v  Terminal voltage (V) 
${v}_{1/2}$  Voltage of the RC elements (V) 
${v}_{\mathrm{OCV}}$  Open circuit voltage (V) 
$\mathit{w}$  Statespace parameter vector 
$\mathit{x}$  Statespace state vector 
$\mathit{y}$  Statespace measurement 
$\Delta t$  Step size (s) 
$\eta $  Coulombic efficiency 
${\sigma}_{I}$  Standard deviation of the current (A) 
${\sigma}_{v}$  Standard deviation of the voltage (V) 
${\tau}_{1/2}$  Time constant of the RC elements (s) 
$\mathbb{R}$  Real numbers 
$\mathbb{E}\left\{\xb7\right\}$  Expectation 
$\mathrm{diag}\{\xb7\}$  Diagonal matrix 
${\xb7}^{T}$  Transpose 
$\widehat{\xb7}$  Estimation 
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Manufacturer  Samsung 

Type  INR1865025R 
Format  18650 
Chemistry  NCA/graphite 
Charge cutoff voltage  $4.2\phantom{\rule{0.166667em}{0ex}}$V 
Discharge cutoff voltage  $2.5\phantom{\rule{0.166667em}{0ex}}$V 
Maximum constant charge current  $4\phantom{\rule{0.166667em}{0ex}}$A 
Maximum constant discharge current  $20\phantom{\rule{0.166667em}{0ex}}$A 
Nominal voltage  $3.6\phantom{\rule{0.166667em}{0ex}}$V 
Nominal capacity @ $0.2\phantom{\rule{0.166667em}{0ex}}$C  $2.5\phantom{\rule{0.166667em}{0ex}}$Ah 
Energy density  216 Wh/kg 
Power density  1.7 kW/kg 
${\mathit{D}}_{\mathbf{PWM}}$  $70\%$  $80\%$  $90\%$ 

${\mathit{f}}_{\mathbf{PWM}}$  
$400\phantom{\rule{0.166667em}{0ex}}$mHz  ×∘  ×  ×∘ 
$318\phantom{\rule{0.166667em}{0ex}}$mHz  ×∘  ×  
$212\phantom{\rule{0.166667em}{0ex}}$mHz  ×∘  ×∘  ×∘ 
$106\phantom{\rule{0.166667em}{0ex}}$mHz  ×∘  ×  
$53\phantom{\rule{0.166667em}{0ex}}$mHz  ×∘  ×  ×∘ 
Quantity  SOC  ${\mathit{v}}_{1}$  ${\mathit{v}}_{2}$  ${\mathit{R}}_{0}$  ${\mathit{R}}_{1}$  ${\mathit{R}}_{2}$ 

WLTP  0.90  0.50  0.89  0.64  0.57  0.55 
Highway  0.21  0.51  0.56  0.04  0.48  0.55 
Quantity  SOC  ${\mathit{v}}_{1}$  ${\mathit{v}}_{2}$  ${\mathit{R}}_{0}$  ${\mathit{R}}_{1}$  ${\mathit{R}}_{2}$ 

WLTP  0.99  0.37  1.13  0.51  0.32  0.98 
Highway  0.54  1.17  0.38  0.22  1.10  0.36 
${\mathit{D}}_{\mathbf{PWM}}$  Passive  $90\%$  $80\%$  $70\%$ 

WLTP  $1.859\phantom{\rule{0.166667em}{0ex}}$A  $1.989\phantom{\rule{0.166667em}{0ex}}$A  $2.114\phantom{\rule{0.166667em}{0ex}}$A  $2.237\phantom{\rule{0.166667em}{0ex}}$A 
Highway  $4.864\phantom{\rule{0.166667em}{0ex}}$A  $5.200\phantom{\rule{0.166667em}{0ex}}$A  $5.533\phantom{\rule{0.166667em}{0ex}}$A  $5.852\phantom{\rule{0.166667em}{0ex}}$A 
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Theiler, M.; Schneider, D.; Endisch, C. Experimental Investigation of State and Parameter Estimation within Reconfigurable Battery Systems. Batteries 2023, 9, 145. https://doi.org/10.3390/batteries9030145
Theiler M, Schneider D, Endisch C. Experimental Investigation of State and Parameter Estimation within Reconfigurable Battery Systems. Batteries. 2023; 9(3):145. https://doi.org/10.3390/batteries9030145
Chicago/Turabian StyleTheiler, Michael, Dominik Schneider, and Christian Endisch. 2023. "Experimental Investigation of State and Parameter Estimation within Reconfigurable Battery Systems" Batteries 9, no. 3: 145. https://doi.org/10.3390/batteries9030145