Download pdf - Elektrochemische Modelle für Lithium-Ionen Batterien MES 10 - Elektrochemische Modelle LiB...Elektrochemische Modelle für Lithium-Ionen Batterien André Weber Institut für Angewandte

KIT – Universität des Landes Baden-Württemberg undnationales Forschungszentrum in der Helmholtz-Gemeinschaft www.iam.kit.edu/wet

Elektrochemische Modelle für Lithium -Ionen Batterien

André Weber

Institut für Angewandte Materialien - Werkstoffe der Elektrotechnik IAM-WETAdenauerring 20b, Geb. 50.40 (FZU), Raum 314

phone: 0721/608-7572, fax: 0721/[email protected]/wet

Modellbildung elektrochemischer Systeme

Quelle:

Institut für Angewandte Materialien Werkstoffe der Elektrotechnik

Vorlesung MES 10 - Elektrochemische Modelle LiB - Zellen.pptx, Folie: 2, 10.07.2018

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Elektrochemisches ModellStand der Technik & Motivation

IAM-WET

Verhaltensmodell

geringe örtliche AuflösungechtzeitfähigParametrierung einfach

R1

UO

CV

R0 C1

Uba

tt

Ibatt

grobe Verhaltensprädiktion im vermessenen Parameterraum

Differentialgleichungen & FEM

hohe örtliche Auflösungnicht echtzeitfähigParametrierung aufwendig

Simulation von unbekannten Strukturen und MaterialienInter-/Extrapolation des Parameterraums

( )cDt

c ∇⋅∇=∂∂

( ) ( )

−

−−= …00

1exp UU

RT

zFjj

α

physikalisch motiviertes Ersatzschaltungsmodell

geringe örtliche AuflösungechtzeitfähigParametrierung mittel

Diagnose durch Simulation Fehler-/DegradationsfälleInter-/Extrapolation des Parameterraums

UO

CV

Uba

tt

Ibatt

Quelle:



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Distribution of Relaxation Times as Basis for Time-Domain Models of Li-Ion Batteries

IAM-WET

Impedance Measurement

Physically MotivatedModel

Time-DomainSimulation

Model Targets:

Good Real-time Capabilities - model structure deployable on microcontroller

Scalability - accuracy and computational effort easy adaptable

Automization of Parameterization - model quality does not depend on human expertise

Quelle:



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IntroductionBattery Modeling Today

IAM-WET

EIS Measurement

electrochemical impedance measurements

non parametric representation of linear dynamics

variation of operating points (SOC,T)

Quelle:



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Model selection and fit

a priori knowledge necessary

stabilization of fit procedure critical

subtraction of differential capacity

R0 RQ1 RQ2 RQ3 ZGFLWC0

differential capacity


IAM-WET

EIS Measurement

R0 RQ1 RQ2 RQ3 ZGFLWC0 fractional impedance elements

Quelle:



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IAM-WET

EIS Measurement


Approximation of fractional impedance elements

Fractional impedance elements cannot be simulated directly in time domain

Approximation with sum of RC-Elements:

empirical relation [1]

analytical expansion to sum

discretization of the distribution of relaxation times

RQ

RC1 RC2 RC3

Analytical expression for the measured impedance ha s to be known!

[1] Stephan Buller, Phd-Thesis, Aachener Beitrage des ISEA, Shaker Verlag GmbH, 2002

Quelle:



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IAM-WET

EIS Measurement


Time Domain Model and Simulation

OCV

SOC

UO

CV …+

PLECSt

U

t

I

Loss ofaccuracy

time effort

Quelle:



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g(τ): Dirac pulses

g

t

1

p o l

RR

2

p o l

RR

τ1 τ2

ideal process: RC-Elements

RC1 RC2

The Distribution of Relaxation Times

IAM-WET

pol0

( )( )

1polg

Z j R djτω τωτ

∞=

+∫ g(τ): distribution function of relaxation times (DRT)

H. Schichlein, A.C. Müller, M. Voigts, A. Krügel and E. Ivers-Tiffée, Journal of Applied Electrochemistry, Vol. 32, No. 8, pp. 875-882, 2002

The Continuous Distribution of Relaxation Times

g

tτ1 τ2general distribution function

RQ1 RQ2

real process: RQ-Elements

Calculation of the Discrete Distribution of Relaxat ion Times

N is the Number of RC-Elementsdiscrete distribution: � is logarithmical equally spacedg(�n) = 0 at highest and lowest �n

g

log(τ)

ττω δωτ=

=+∑

1

( )( )

1

Nn

polnn

gZ j

j

…

Quelle:



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IAM-WET

EIS Measurement



OCV

SOC

UO

CV …+

PLECSt

U

t

I

Loss ofaccuracy

time effort

Quelle:



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IAM-WET

EIS Measurement

Method of the Distribution of Relaxation Times


OCV

SOC

UO

CV +

PLECSt

U

t

I

…Re(Zth)

Im(Z

th)

deconvolution of EIS

…

τωωτ=

=+∑

1

( )( )

1

Nn

polpolnn

Z jg

Rj

distribution ofrelaxation times

…

Quelle:



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Verification of the DRT ModelExperimental

IAM-WET

2 Ah Li-ion cell from KOKAM

Pouch-cell

Anode: Graphite

Cathode: NCA/LCO-Blend

REM: Anode

2µm 2µm

Anode: Graphite Cathode: NCA / LiCoO2 Blend

LiCoO2

NCA

Quelle:



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pulse height 1C

pulse length = 10s

f = [100kHz … 5mHz]

Uampl = 10mV / 5mV

Verification of the DRT ModelExperimental

IAM-WET, D. Klotz, M. Schönleber, J. P. Schmidt, and E. Ivers-Tiffée, Electrochim. Acta, 56, 8763 (2011).

Electrochemical Measurements

quasi stationary OCV

discharge/charge at C/40

Variation of Imax = C/4 … 2C � 0.5A � 4A

average discharge with Imax/2

transformationto frequencydomain1

merger of high andlow frequency range

impedance spectraf = [100kHz …10µHz]

completely autom

ated process

Quelle:



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impedance spectroscopy time domain measurements

Impedance Measurements in the Frequency and Time Domain

IAM-WET

0.1 0.2 0.3 0.40

0.05

0.1

0.15

Z´ [Ω]

-Z´´

[Ω]

1 2 3

5

10

15

20

25

30

35

40

45

50

55

Z´ [Ω]

-Z´´

[Ω]

1 2 3

5

10

15

20

25

30

35

40

45

50

55

Z´ [Ω]

-Z´´

[Ω]

0.1 0.2 0.3 0.40

0.05

0.1

0.15

Z´ [Ω]

-Z´´

[Ω]

cell

00

00

SOC = 100 %

Û = 10 mV

fmin = 5,88 µHz

fmax = 10 kHzΔSOC = 1,95 %

tMess = 19 days

nIntegration = 3...8

LiCoO2 / C

C = 120 mAh

cellLiCoO2 / C

C = 120 mAh

SOC = 100 %

ΔU = 50 mV

fmin = 6,84 µHz

fmax,ZB = 8,63 Hzfmax = 10 kHz

ΔSOC = 5,01 %

tMess = 4 days

Quelle:



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Measurement ResultsEIS

IAM-WET

f �0

f � ∞∞∞∞

SOC [%]

f �0

10mHz

extended frequency range via pulse measurements

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Measurement ResultsDRT

IAM-WET

SOC ↓

SOC ?

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Measurement ResultsDRT

IAM-WET

complex SOC-dependence

beneficial use of the DRT for identification and se paration of Processes 1,2

interpretation not necessary for parameterization of time domain model

[1] J.P. Schmidt, T. Chrobak, M. Ender, J. Illig, D. Klotz, and E. Ivers-Tiffée, J. Power Sources, 196 (2010) 5342-5348.

[2] J. Illig, T. Chrobak, D. Klotz, and E. Ivers-Tiffée, ECS Trans., 33 (2011) 3-15.

Quelle:



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Measurement ResultsComparison: EIS / DRT-Model

IAM-WET

SOC [%]

DRT Model

Measurement

Very good accordance of measurement and DRT-model

Quelle:



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Verification of the DRT-ModelScalability

IAM-WET

EIS Measurement / DRT-Model Variation of N

τωωτ=

=+∑

1

( )( )

1

Nn

polnn

gZ j

j

Results

N = 100 � 11.6 RC / decade

20 RC-elements show no significant difference to 100 RC-elements

N = 20 � 2.3 RC / decade

10 RC-elements show slight deviations

N = 10 � 1.1 RC / decade

4 RC-elements result in a bad fit of the impedance data

N = 4 � 0.5 RC / decade

Quelle:



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Δ

Verification of the DRT-ModelScalability – Time Domain Simulation

IAM-WET

Comparison of dynamic behavior

seconds-timescale shows significant differences for 4 RC-model

slight increase of accuracy between 10 and 20 RC-model

no visible difference for 20 and 100 RC-model

10 RC-Model shows good accordance and modest calcul ation effort

Imax

-Imax Currrent

Quelle:



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Comparison of Simulation Time

DRT-ModelReal-Time Capability

IAM-WET

Model Order

100

20

10

4

RCs / decade

11.6

2.3

1.1

0.5

Tsim / Tmeas [%]

9%

0.8%

0.5%

0.3%

Simulation Environment

PLECS

Matlab R2009awww.mathworks.com

PLECS 2.1www.plexim.com

Hardware:Intel® Core™ i7 CPU4GB RAM

��

��

��

��

Quelle:



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Physically based models Behavioral models

Electrochemical models Equivalent circuit models Time -domain models

+ charge- and mass transport+ understand electrochemical

reactions+ high accuracy

+ dependence of battery characteristics on operating parameters

+ optimal battery design according to performance and capacity

+ estimation of the current battery condition

+ fast computation+ battery management systems

- time consuming computation - only applicable in the frequency domain

- no physical interpretation possible

Battery Models

IAM-WET

[1] Ma, Y. et al., RSC Advances, 6 (2016), 25435-25443

anode separator cathode

curr

entc

olle

ctor

curr

entc

olle

ctor

[1]

Quelle:



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Physically based models Behavioral models

Electrochemical models Equivalent circuit models Time -domain models

Battery Models

IAM-WET

� New approach combines advantages

� High-resolution electrochemical characterisation needed for physical interpretability

physical interpretability fast computation in the time domain

Quelle:



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Methodology

IAM-WET

Voltage behavior of a battery when applying a curre ntLoss processes that determine the overvoltage

< �� Open circuit voltage ��

Quelle:



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Methodology

IAM-WET

< �� Open circuit voltage

�� Instantaneous voltage drop

��

��


Lithium transport in the electrolyte, electron transportin the electrodes and current collectors

1

Quelle:



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Methodology

IAM-WET



�� Fast kinetic processes

��

�� ,�/��,�



Interface losses at anode and cathode2

1

Quelle:



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Methodology

IAM-WET




�� Slow diffusion processes

��

�� ,�/��,�

�� ,�/�



Interface losses at anode and cathode

Lithium solid state diffusion in anode and cathode3

2

1

Quelle:



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Methodology

IAM-WET





��

�� ,�/��,�

�� ,�/�




Lithium solid state diffusion in anode and cathode

Contact losses included in 4 1

3

2

1

Quelle:



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Methodology

IAM-WET






��

�� ,�/��,�

�� ,�/�



Lithium solid state diffusion in anode and cathode

Contact losses included in 4 1

3

2

1

Quelle:



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Methodology

IAM-WET

How to determine the operating voltage

„fast“ lossprocesses (< 1 Hz)

„slow “ lossprocesses (> 1 Hz)

UOCV

Uop

ηloss

ηDiff

η0

ηCT/SEI

Quelle:



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Methodology

IAM-WET


Z´´

[]

g(f

) [

s]

Electrochemical Impedance Spectroscopy

DRT-Analysis:

Description of the loss processes over the frequency

Equivalent Circuit Modelling

Fast loss processes

P1

P2P3

� �� !" #$�%1 ' �� (�

)

*



UOCV

Uop

ηloss

ηDiff

η0

ηCT/SEI

�� Measurement of the frequency dependent resistance

�process identification

�� process quantification

Quelle:



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Methodology

IAM-WET


Current interruption method in the time domain

Time domain model fit

Slow loss processes

η � , ∙ �. ∙ 1 exp 2�.

', ∙ �3 ∙ 1 exp 2�3



UOCV

Uop

ηloss

ηDiff

η0

ηCT/SEI

�� Measurement of the low-frequent loss contributions

�� process quantification

U [

V]

Quelle:



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Investigated Cell

IAM-WET

anode:

graphite

2 µm

LiNixCoyAl1-x-yO2LiCoO2carbon black

10 µm

SEM imagesHigh power pouch cell

Manufacturer KOKAM

Nominal capacity 350 mAh

Nominal voltage 3,7 V

Max. charge current 700 mA (≙ 2C)

Max. discharge current 7 A (≙ 20C)

Voltage limits (Umin- Umax) 2,7 - 4,2 V

cathode:

Quelle:



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UOCV

Uop

ηloss

ηDiff

η0

ηCT/SEI

Experimental

IAM-WET

Fast loss processes

Quelle:



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ExperimentalAnalysis of high and middle frequency Electrochemical Processes

IAM-WET

Electrochemical impedance spectroscopy: measurement of the frequency-dependent resistance

Excitation and response signal

0

0

( )( ) Re( ) Im( ) ' ''

( )juu t

Z e Z j Z Z j Zi t i

ϕω ⋅= = ⋅ = + ⋅ = + ⋅

T = 25 °CSOC = 60 %

Quelle:



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IAM-WET

Electrochemical impedance spectroscopy: measurement of the frequency-dependent resistance

Excitation and response signal Impedance in a Nyquist diagram

0

0

( )( ) Re( ) Im( ) ' ''

( )juu t

Z e Z j Z Z j Zi t i

ϕω ⋅= = ⋅ = + ⋅ = + ⋅

T = 25 °CSOC = 60 %

Quelle:



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Requirements for validity


IAM-WET

Kramers Kronig validity test to ensure the validity o f the measurement

http://www.iam.kit.edu/wet/Lin-KK.php

If all these requirements are fulfilled, the Kramers Kronigrelations can be derived:

Linearity of the measured system

Time invariance of the measured system

Causality of the measured system

2 20

2 ' Im( ( '))Re( ( )) '

'

ZZ d

ω ωω ωπ ω ω

∞ ⋅=−∫

2 20

2 ' Re( ( '))Im( ( )) '

'

ZZ d

ω ωω ωπ ω ω

∞ ⋅= −−∫

If Re(Z) is known, Im(Z) can be computed.

� Comparison of computed part and measured partreveals the Kramers Kronig residuals .

Quelle:



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Requirements for validity


IAM-WET

Kramers Kronig validity test to ensure the validity o f the measurement

Example: change of SOC

http://www.iam.kit.edu/wet/Lin-KK.php

If all these requirements are fulfilled, the Kramers Kronigrelations can be derived:

Linearity of the measured system

Time invariance of the measured system

Causality of the measured system

2 20

2 ' Im( ( '))Re( ( )) '

'

ZZ d

ω ωω ωπ ω ω

∞ ⋅=−∫

2 20

2 ' Re( ( '))Im( ( )) '

'

ZZ d

ω ωω ωπ ω ω

∞ ⋅= −−∫

If Re(Z) is known, Im(Z) can be computed.

� Comparison of computed part and measured partreveals the Kramers Kronig residuals .

1. Approach the desired SOC

2. Wait for x hours until the cell is in stationary mode

� What is x ?

Quelle:



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IAM-WET

Distribution of relaxation times for the clear sepa ration of loss processes

T = 25 °CSOC = 60 %

Z´´

[]

15 Hz100 Hz

MHz

µHz

Quelle:



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IAM-WET


T = 25 °CSOC = 60 %

Z´´

[]

15 Hz100 Hz

MHz

µHz

relaxation time of process n

2,0

2

1,0

1

22

2

11

1

1111)(

ωτωτωωω

jR

jR

CRjR

CRjR

jZ+

++

=+

++

=

:nnn0, CR=τ

distribution function of relaxation timespol

0

( )( )

1g

Z j R djτω τωτ

∞

=+∫

# � :[2]

Quelle:



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IAM-WET


T = 25 °CSOC = 60 %

Z´´

[]

15 Hz100 Hz

MHz

µHz


2,0

2

1,0

1

22

2

11

1

1111)(

ωτωτωωω

jR

jR

CRjR

CRjR

jZ+

++

=+

++

=

:nnn0, CR=τ


0

( )( )

1g

Z j R djτω τωτ

∞

=+∫

# � :[2]

process 1 process 2

ggeneraldistributionfunction

�*,3�*,. �

Quelle:



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IAM-WET


T = 25 °CSOC = 60 %

Z´´

[]

15 Hz100 Hz

MHz

µHz


2,0

2

1,0

1

22

2

11

1

1111)(

ωτωτωωω

jR

jR

CRjR

CRjR

jZ+

++

=+

++

=

:nnn0, CR=τ


0

( )( )

1g

Z j R djτω τωτ

∞

=+∫

# � :[2]

P1

P2 P3

15 Hz

100 Hz

process 1 process 2


�*,3�*,. �

Quelle:



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process 1 process 2


�*,3�*,. �


2,0

2

1,0

1

22

2

11

1

1111)(

ωτωτωωω

jR

jR

CRjR

CRjR

jZ+

++

=+

++

=

:nnn0, CR=τ


0

( )( )

1g

Z j R djτω τωτ

∞

=+∫

# � :[2]


IAM-WET


T = 25 °CSOC = 60 %

Z´´

[]

15 Hz100 Hz

MHz

µHz

P1

P2 P3

15 Hz

100 Hz

Physical meaning of P1, P2 and P3 ?Physical meaning of P1, P2 and P3 ?

Quelle:



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IAM-WET

Experimental cells for the assignment of loss proce sses

3-electrode setup

IAM-WET-housing

6$2%78!!

6$2%98:3 ;$2%78!!6$2%98:.

Working electrode: Anode or Cathode

Reference electrode: Al-mesh with LTO coating

Counter electrode: Lithium

T = 25 °CSOC = 40 %

• inhouse development• 18 mm electrode diameter• 3-electrode setup[3,4]

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IAM-WET

Experimental cells for the assignment of loss proce sses

3-electrode setup

IAM-WET-housing

6$2%78!!

6$2%98:3 ;$2%78!!6$2%98:.

Working electrode: Anode or Cathode

Reference electrode: Al-mesh with LTO coating

Counter electrode: Lithium

Measurement results

P1 P2 P3

T = 25 °CSOC = 40 %

• inhouse development• 18 mm electrode diameter• 3-electrode setup[3,4]

Charge transfer of the

cathode PCT,C

SEI-process / charge transfer

of the anode PSEI/CT,A: side

peaks due to the porous

electrode structure

Quelle:



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IAM-WET

T = 25 °CSOC = 60 %

Quantification of loss processes by equivalent circ uit modeling

� Frequency dependent processes can be simplified simulated

with RQ elements

<=>,= <?@A/=>,B

Quelle:



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IAM-WET

T = 25 °CSOC = 60 %

Quantification of loss processes by equivalent circ uit modeling

� Frequency dependent processes can be simplified simulated

with RQ elements

� Good fit quality despite simplified model

� Fit delivers resistance and time constant of each process

Im(Z

) [

]

<=>,= <?@A/=>,B

Quelle:



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Experimental

IAM-WET

Slow loss processes



UOCV

Uop

ηloss

ηDiff

η0

ηCT/SEI

Quelle:



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ExperimentalAnalysis of low frequency Electrochemical Processes

IAM-WET

I = 350 mAT = 25 °C

Slow diffusion process PDiff,A/C takes place in the low frequency range

� EIS not applicable


Quelle:



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IAM-WET

I = 350 mAT = 25 °C



� Time domain measurement

1. Start at OCV-level

2. Apply a current pulse

3. Analyze the voltage curve after cut-off of the current

� cell in stationary mode


Quelle:



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IAM-WET

I = 350 mAT = 25 °C



� Time domain measurement

1. Start at OCV-level

2. Apply a current pulse

3. Analyze the voltage curve after cut-off of the current

� cell in stationary mode


4. Subtract η0, ηCT,C and ηCT/SEI,A from relaxing overvoltage

� diffusion overvoltage ηDiff,A/C remains

Quelle:



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IAM-WET

I = 350 mAT = 25 °C

At least two diffusion processes in the cell: anode and cathode

� Fit delivers resistance and time constant :

CDE:: � , ∙ �DE::,. ∙ 1 exp GHIJKK,L ' , ∙ �DE::,3 ∙ 1 exp G

HIJKK,M

Time domain fit model

[5] Pop, V. et al., J. Electrochemical Society, 153 (2006), A2013-A2022

� Relative error below 0.1%

� good fit quality

[5]

Quelle:



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Battery Model

IAM-WET

NOPQQ,B/=(SOC,T)ROPQQ,B/=(SOC,T)

OCV by linear interpolation ofcharge and discharge curve

EISDRT-AnalysisECM TDM

OCV(SOC,T) N (SOC,T)N=>,=(SOC,T)

R=>,=(SOC,T)

N?@A/=>,B(SOC,T)

R?@A/=>,B(SOC,T)

curr

ent

S � TUV, W, , �SXYZ TUV, W C* TUV, W, , C[\]/Y^,_ TUV, W, , CY^,Y$TUV, W, ,% CDE::,_/Y$TUV, W, ,%

Battery Model

Quelle:



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Results

IAM-WET

�`a � �� /��,� ��,� ��,�/�Simulation of a cell discharge

I = 350 mAT = 25 °C

U [V

]

Quelle:



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Results

IAM-WET

Simulation of a cell discharge

� Absolute residuals below 30 mV in 90% of the SOC range

� Only at SOCs < 10% the residuals are higher than 30 mV

I = 350 mAT = 25 °C

U [V

]

resid

uals

[m

V]

�`a � �� /��,� ��,� ��,�/�