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© 2011 ANSYS, Inc. June 7, 2012 1
Model Order Reduction for Battery Simulation
Xiao Hu, PhD
Confidence by Design
Detroit June 5, 2012
© 2011 ANSYS, Inc. June 7, 2012 2
Outline
ANSYS model order reduction (MOR) technique
Different types of battery simulation
Transfer function based MOR (aka LTI method)
System matrix based MOR
Conclusion
© 2011 ANSYS, Inc. June 7, 2012 3
ANSYS MOR Techniques
Model order reduction (MOR) for linear problems
Transfer function based (or LTI method)
System matrix based
Model order reduction (MOR) for non-linear problems
Proper orthogonal decomposition (POD)
© 2011 ANSYS, Inc. June 7, 2012 4
ANSYS MOR Techniques
Model order reduction (MOR) for linear problems
Transfer function based (or LTI method)
System matrix based
Model order reduction (MOR) for non-linear problems
Proper orthogonal decomposition (POD)
© 2011 ANSYS, Inc. June 7, 2012 5
Different Types of Battery Simulations
Electrical circuit model
Battery system level thermal management simulation
Battery P2D electrochemistry model
Battery 3-D micro-scale electrochemistry model
Mechanical, aging, …
Li+
e
Li+
Li+ Li+
LixC6 Lix-Metal-oxide
e
Jump
© 2011 ANSYS, Inc. June 7, 2012 6
MOR for Battery Simulations
Battery system level thermal management simulation
Battery P2D electrochemistry model
Li+
e
Li+
Li+ Li+
LixC6 Lix-Metal-oxide
e
Jump
Li+
e Li+
Li+ Li+
LixC6 Lix-Metal-oxide
e
Jump
© 2011 ANSYS, Inc. June 7, 2012 7
Motivation of Using MOR for Battery System Level Thermal Management
CFD as a general thermal analysis tool is accurate
Can be expensive for large system level repeated transient CFD analysis
MOR can significantly reduce the model size for system level thermal analysis
© 2011 ANSYS, Inc. June 7, 2012 8
Full Hybrid Electrical Vehicle Battery Pack System Design, CFD Simulation and Testing
Velocity contours of airflow through the brick, inlet and outlet plenum
Airflow path into the battery pack Airflow path and air outlet
Front view of the pack with the two bricks assembled and inlet busbar
Battery Thermal Management Using CFD
1. D. Ghosh, P. D. Maguire, and D. X. Zhu, " Design and CFD Simulation of a Battery Module for a Hybrid Electric Vehicle Battery Pack,” SAE 2009-01-1386 2. D. Ghosh, K. King, B. Schwemmin, D. Zhu, “Full Hybrid Electrical Vehicle Battery Pack System Design, CFD Simulation and Testing,” SAE 2010-01-1080
© 2011 ANSYS, Inc. June 7, 2012 9
Introduction to Transfer Function Based MOR
LTI
Impulse Input
t
1
0
Impulse response
t
2
1
0 1 2 3
Input t
1
LTI 0 1
Input t
2
LTI 1
0
Input t
1
LTI 0 1
2
Output t
2
1
4
3
1 2 3 0
Output t 1 2 3 4
2
1
4
3
0 Output t
2
1
0 1 2 3 4
DT LTI Output Input
This is a Discrete Time example.
© 2011 ANSYS, Inc. June 7, 2012 10
LTI Output Input
LTI
Input t
1
0
Input t
1
LTI 0 1
Input t
2
LTI 1
0
LTI
Output t
2
1
4
3
1 2 3 0
Output t
2
1
0 1 2 3 4
Impulse response
t
2
1
0 1 2 3
Output t 1 2 3 4
2
1
4
3
0
Input t
1
0 1
2
Output of a LTI system is completely characterized by its impulse response*!
If two LTI systems have the same impulse
response, the two systems are equivalent.
* Under the condition of initial rest
Make a small LTI model have the same impulse response as the
modeled large LTI system!
Introduction to Transfer Function Based MOR
© 2011 ANSYS, Inc. June 7, 2012 11
Why Is It Called Transfer Function Based?
Impulse response, step response, or system transfer function contains the same amount of information.
Laplace transfer of impulse response is system transfer function.
Time derivative of step response is impulse response.
© 2011 ANSYS, Inc. June 7, 2012 12
Three Approaches Are Possible
Make sure the reduced model has the same impulse response as the original model
Make sure the reduced model has the same step response as the original model
Make sure the reduced model has the same transfer function as the original model
© 2011 ANSYS, Inc. June 7, 2012 13
Two Approaches Are Implemented
Step response of the reduced model is curve-fitted to that of the original model in the time-domain
Used in current Simplorer 10
Transfer function of the reduced model is curve-fitted to that of the original model in the frequency-domain
Will be added in future Simplorer
© 2011 ANSYS, Inc. June 7, 2012 14
LTI Modeling Using Simplorer 10
1. Create step responses • From CFD / Test
2. Generate .simpinfo file • Automatic using Icepak
4. Simulate inside Simplorer
3. Extract LTI model • Use Simplorer
© 2011 ANSYS, Inc. June 7, 2012 15
MOR is Used to Drastically Reduce Runtime of Long Transient Simulations
The LTI model gives the same results as CFD. The LTI model runs in less than a few seconds while the CFD runs 2 hours on one single CPU.
1. X. Hu, S. Lin, S. Stanton, and W. Lian, "A Foster Network Thermal Model for HEV/EV Battery Modeling," IEEE Trans. on Industry Appl., vol. 47, no. 4, p. 1692-1699, 2011 2. X. Hu, S. Lin, S. Stanton, W. Lian, “A State Space Thermal Model for HEV/EV Battery Modeling", SAE 2011-01-1364 3. X. Hu, S. Lin, S. Stanton, and W. Lian, “A State Space Thermal Model for HEV/EV Non-Linear and Time-Varying Battery Thermal Systems,” IMECE2011-62022, 2011
GM Battery Module Example Using LTI Modeling
© 2011 ANSYS, Inc. June 7, 2012 16
LTI Modeling for Non-Linear Problems – GM Module
Non-linear CFD: Ideal gas law plus temperature dependent properties are used. Full Navier-Stokes equations are solved
LTI: Assumes the system is linear and time invariant.
A speed-up factor of 10,000 is observed. Huge time saving if the error, which is about 1.4%, is acceptable.
© 2011 ANSYS, Inc. June 7, 2012 17
Battery Example with Prismatic Cells
The battery module has 20 prismatic cells.
Water cooling is used.
Average temperature of each cell is monitored.
The LTI model has 1 input and 20 outputs.
Velocity magnitude
© 2011 ANSYS, Inc. June 7, 2012 18
Three sets of LTI models are identified at flow rates of 0.06, 0.075, and 0.09 kg/s.
Interpolation used for intermediate flow rates.
Transient heat source and mass flow rate are applied for testing the model with changing flow rate.
One Cell Battery Example with Changing Flow Rate
© 2011 ANSYS, Inc. June 7, 2012 19
Six Cell Battery Example with Changing Flow Rate
Power dissipation inputs are sinusoidal functions
Flow rate changes at time of 1000 second.
Results are excellent for the entire duration. A small difference is seen during transition period.
Cell 1 Cell 2
Cell 3 Cell 4
© 2011 ANSYS, Inc. June 7, 2012 20
GM Battery Module Example with Changing Flow Rate
The model gives similar results as CFD. The model runs in less than 20 seconds while the CFD runs a couple of days on 6 CPUs.
X. Hu, S. Asgari, S. Lin, S. Stanton, “A Linear Parameter-Varying Model for HEV/EV Battery Thermal Modeling,“ IEEE ECCE Conference, paper no. 1041, 2012.
© 2011 ANSYS, Inc. June 7, 2012 21
Electro-Themeral Coupled Analysis
0
0
0
0
0
00
CONST
H00RT_LRT_S
CT_LCT_SIBattCcapacity
Rseries
VOC
E1
R1
C1 I7C2 C3
R2 R3
CONST
H01
E2
R5
C4 I8C5 C6
R6 R7
CONST
H02
E3
R9
C7 I9C8 C9
R10 R11
CONST
H03
E4
R13
C10 I10C11 C12
R14 R15
CONST
H04
E5
R17
C13 I11C14 C15
R18 R19
CONST
H05
RLoad
SIMPARAM1
Qcell1
Qcell2
Qcell3
Qcell4
Qcell5
Qcell6
Tambien
Temp_block_1
Temp_block_2
Temp_block_3
Temp_block_4
Temp_block_5
Temp_block_6
0.00 2000.00 4000.00 6000.00 8000.00Time [s]
300.00
310.00
320.00
330.00
340.00
350.00
Y1 [k
el]
Curve InfoU1.Temp_block_1
TR
U1.Temp_block_3TR
U1.Temp_block_5TR
Voc=f(SOC, U1.Temp_block_1)
© 2011 ANSYS, Inc. June 7, 2012 22
Lithium Ion Batteries
• Electrochemical Kinetics • Solid-State Li Transport • Electrolytic Li Transport
• Charge Conservation/Transport • (Thermal) Energy Conservation
Li+
e
Li+
Li+ Li+
LixC6 Lix-Metal-oxide
e
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eeee j
F
tcD
t
c
1)(
Newman P2D Electrochemistry Model
Negative Electrode
Positive Electrode
Separator
δs δn δp
x=0 x=L
X. Hu, S. Lin, S. Stanton, “Simulating Rechargeable Lithium-Ion Battery Using VHDL-AMS,“ SAE paper 2012-01-0665, 2012.
© 2011 ANSYS, Inc. June 7, 2012 23
Motivation of Using MOR for Newman P2D Electrochemistry Model
Negative Electrode
Positive Electrode
Separator
δs δn δp
x=0 x=L
h
x=0
2 3 4 5 1
φ1,1 φ2,1 i1,1
i2,1
c1
3
2
1 cs1,1
c2 c3 c4 c5
φ1,2 φ2,2 i1,2,
i2,2
φ1,3 φ2,3 i1,3
i2,3
φ1,4 φ2,4 i1,4
i2,4
φ1,5 φ2,5 i1,5
i2,5
cs1,2
cs1,3
3
2
1 cs2,1
cs2,2
cs2,3
3
2
1 cs3,1
cs3,2
cs3,3
3
2
1 cs4,1
cs4,2
cs4,3
3
2
1 cs5,1
cs5,2
cs5,3
φ1,6 φ2,6 i1,6
i2,6
2 3 4 5 1 6
x=δn
1. M. Doyle, T.F. Fuller, and J. Newman, Journal of Electrochem. Soc., 140, 1526 (1993) 2. C. R. Pals and J. Newman Journal of Electrochem. Soc., 142 (10), 3274-3281 (1995) 3. M. Doyle, J. Newman, Journal of Electochem. Soc., 143, 1890 (1996)
The model is pseudo 2D due diffusion equation solved for each particle.
Many particles exist
© 2011 ANSYS, Inc. June 7, 2012 24
Observations of the Newman P2D Model
Equations are highly non-linear overall.
However, solid-phase diffusion equations are linear. And solid-phase diffusion is the most time consuming part of the P2D model.
Use transfer function based MOR to model the diffusion process and keep the rest non-linear equations intact.
© 2011 ANSYS, Inc. June 7, 2012 25
CFD Geometry/Mesh and Solution for the Diffusion Process in a Spherical Particle
Use CFD to solve the diffusion process ONCE and then use model order reduction to characterize the diffusion process.
X. Hu, S. Stanton, L. Cai, R. White, “A Linear Time-Invariant Model for Solid-Phase Diffusion in Physics-Based Lithium Ion Cell Models,“ J. of Power Sources, vol. 214, p. 40-50, 2012
© 2011 ANSYS, Inc. June 7, 2012 26
Reduced Order Model Validation
Even the 3rd order model gives good accuracy.
3rd order model solves only 3 equations for diffusion
Log scale shows that dynamics near time of zero is captured accurately by the reduced order model
X. Hu, S. Stanton, L. Cai, R. White, “A Linear Time-Invariant Model for Solid-Phase Diffusion in Physics-Based Lithium Ion Cell Models,“ J. of Power Sources, vol. 214, p. 40-50, 2012
© 2011 ANSYS, Inc. June 7, 2012 27
Validation for P2D Electrochemistry Model
The reduced order model runs approximately 6 times faster.
Excellent accuracy is retained by using the reduced order model
Reduced order model allows for non-spherical particles
x=Ln+Ls+Lp
x=Ln+Ls
x=Ln
x=0
Rate 0.1C 0.5C 1C 3C 5C 10C
X. Hu, S. Stanton, L. Cai, R. White, “A Linear Time-Invariant Model for Solid-Phase Diffusion in Physics-Based Lithium Ion Cell Models,“ J. of Power Sources, vol. 214, p. 40-50, 2012
© 2011 ANSYS, Inc. June 7, 2012 28
Impact of Particle Shapes on Cell Performance
X. Hu, S. Stanton, L. Cai, R. White, “A Linear Time-Invariant Model for Solid-Phase Diffusion in Physics-Based Lithium Ion Cell Models,“ J. of Power Sources, vol. 214, p. 40-50, 2012
Impact of different particle shapes only show up at high discharge rates.
© 2011 ANSYS, Inc. June 7, 2012 29
Introduction to System Matrix Based MOR
What is a state?
x1
x2
Initial state at t=t0
x1
x2
x1
x2
x1
x2
x1
x2
What is the main goal of system matrix based MOR?
What is a state trajectory?
What is change of basis?
What is a projection?
Find x1’
© 2011 ANSYS, Inc. June 7, 2012 30
Introduction to System Matrix Based MOR
Obtain the original system matrix
zx V
uxx BKE
uzz BVKVVEVV TTT
E
Er x V
z
≈
.
Substitute and solve the reduced system matrix
Find and project onto low-dimensional subspace
© 2011 ANSYS, Inc. June 7, 2012 31
Workflow of System Matrix Based MOR
E
Er
x V
z
≈
.
© 2011 ANSYS, Inc. June 7, 2012 32
Example: Battery Pack Thermal FE Model
4 Cells
Cooling by 3 air channels
3D FEM model in ANSYS • Uniform heat generation in each cell
• 1D fluid channels
Coupling by heat transfer coefficient
Fixed velocity
FLUID116
© 2011 ANSYS, Inc. June 7, 2012 33
Reduced vs Full Solution
Reduced Full
Simulation time [s] 4000 4000 (100 time steps)
Dimension 40 (10 x Input) 48500 elements
CPU time [s] <1 ~20 min
0000
Cell1
Cell2
Cell3
Cell4 T_Ref
HVALUE=0
HVALUE=0
HVALUE=0
Q
VALUE=20HVALUE=10
0.00 1000.00 2000.00 3000.00 4000.00 5000.00Time [s]
293.00
294.25
295.50
296.75
297.50
Y1 [k
el]
Curve InfoTHM1.T
TRTHM2.T
TRTHM3.T
TRTHM4.T
TR
0.00 1000.00 2000.00 3000.00 4000.00Time [s]
0.00
0.50
1.00
Tem
pera
ture
[cel
]
Curve InfoTCell1.T
TR
ansys_Cell1Imported
Error <1%
Step response of 1W per cell
© 2011 ANSYS, Inc. June 7, 2012 34
Example: Electro-Thermal Response
Heat Temperature
Battery Pack
© 2011 ANSYS, Inc. June 7, 2012 35
Recover the Original Solution
From the reduced results back to the original results whenever desired
Expansion pass Get reduced vectors z(t) and multiply by the matrix V
x1
x2
x V
z
≈
.
© 2011 ANSYS, Inc. June 7, 2012 36
Recover the Temperature Field
REDUCED
FULL
© 2011 ANSYS, Inc. June 7, 2012 37
More Complex Battery Pack Model
3 layers of 33 cells = 99 inputs! • Each cell is a source of heat
• Outputs: Temperature of the cell center
Numerics • FEM: 68000 DOFs
• 4000s, adaptive time step
Δtmax:400s
Δtmin : 1ms
• Dimension of reduced order model
10*Number of inputs
• MOR time is less than one transient simulation
Results • 1W/per cell step response
• CPU time in Simplorer: 5mim 48s
© 2011 ANSYS, Inc. June 7, 2012 38
Comparison
Transfer Function Based System Matrix Based
Linear systems (linearized non-linear systems, small non-linearity)
Need to create the CFD or FEA model and run flow solution
Needs transient step response Needs system matrices
Could use testing data Can not use testing data
Only specified outputs Field solution is possible
Extremely short run time (sec) Short run time (sec, min)
Can handle very large problems Challenging for large problems
Extremely accurate Very accurate
Most efficient with SISO systems (can be extended to MIMO systems)
MIMO systems can be handled efficiently
© 2011 ANSYS, Inc. June 7, 2012 39
Both transfer function based and system matrix based ROM are available with ANSYS
MOR can significantly reduce the transient simulation time while retaining excellent accuracy.
MOR can be applied to battery system level thermal simulation.
MOR can also be used for battery Newman P2D electrochemistry modeling.
Conclusion
© 2011 ANSYS, Inc. June 7, 2012 40
Ralph E. White (University of South Carolina)
Long Cai (University of South Carolina)
Wenyu Lian (General Motors)
Lucas Kostetzer (ESSS)
Evgeny Rudnyi (CADFEM)
Acknowledgement
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