SOC estimation using electro chemical model and Extended Kalman Filter

Project Report Modeling and Control of Battery Systems Winter 22
TERM PROJECT REPORT
AENG 576 – MODELING AND CONTROL OF BATTERY SYSTEMS
Dr. Youngki Kim
Electrochemical Model-Based Estimation of SOC
Using Extended-Kalman Filter
Nipun Kumar – 31440148
Ratnesh Sharma – 38533793
Satya Patel – 85826429
Varma Jelli – 19000585

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List of Contents
Contents Page No.
1. Motivation/Background………………………………………………………………………4
2. Objective…………………………………..………………………………………….………4
3. Introduction to Electrochemical Models (EM)……………………….…………….………...5
4. Improved Reduced Order Electrochemical Model (iROEM)……………………..…...……..5
Pade Approximation (Negative Electrode)…………………………………………..5
Quadratic Parabolic Polynomial Approximation (Positive Electrode)………………7
Electrolyte Concentration Approximation……………………………………………7
5. Terminal Output Voltage Calculation…………..……………………………………………13
6. Electrochemical Data……………………...………………………………………………….15
7. Terminal Voltage result from iROEM………………………....…………………….……….17
a. Pulse Cycle ………………..……………………………..………………………….17
b. Dynamic Stress Test Cycle..…………………..…………...……..………………….17
8. Terminal Voltage and SOC Estimation using EKF.……… ………...…..……… …………..18
a. Pulse Cycle …………………………………..………………………….…………..21
b. Dynamic Stress Test Cycle..………………………………………… ………..…….24
9. Terminal Voltage from Equivalent Circuit Model considering Electro Chemical
Properties………………………………………………………………………………..……26
10. Conclusion……………………………………………………………………………………31
11. Limitations…………………………………………..………………………………………..31
12. Contributions………………………………………………………………………………….31
13. References…………………………………………...………………………………………..32
List of Figures
Figure Page No.
1. Plot of Pulsed Input current and corresponding Electrode surface concentration comparison
………….………………………………………………………………………………..…….6
2. Plot of Electrode surface concentration from third order pade approach……………………...6
3. Graph of electrolyte concentration along the length of the battery…………………………..12
4. : Graph of electrolyte concentration at different C rates……………………………………..12
5. Block diagram of iROEM model for calculation of Terminal Output Voltage………………14
6. SIMULINK model for TOV calculation using iROEM model………………………………15
7. Reference graphs for Performance evaluation of battery TOVs……………………………...16
8. Terminal Voltage results from iROEM- pulse cycle ……..…………………………………17
9. Terminal Voltage results from iROEM- DST cycle ……………..………………………….18
10. SIMULINK block of EKF written for estimating the new states and SOC………….……….20
11. Pulse current density profile used for SOC estimation……………………………….………21
12. Estimated TOV comparison for pulsed current input………………………………….……..22
13. Estimated TOV error for pulsed current input…………………………………………….….22
14. Estimated SOC for pulsed current input……………………………………………………...23

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Page No
15. Estimated TOV using wrong initial concentration for pulsed current input………………….23
16. DST current density profile used for SOC estimation………………………………………..24
17. Estimated TOV comparison for DST current input…………………………………………..24
18. Estimated TOV error for DST current input………………………………………………….25
19. Estimated SOC for pulsed current input……………………………………………………...25
20. Estimated TOV using wrong initial concentration for DST current input………...…………26
21. Block diagram of second order equivalent circuit RC model………………………………...27
22. Input data for nonlinear RLS method………………………………………………………...28
23. Final voltage fit for double RC using RLS method…………………………………………..28
24. SOC-OCV relationship for LMO battery…………………………………………………….29
25. Terminal voltage of 2nd RC model using electrochemical data……………………………..30
26. Comparison of Terminal voltage of 2nd RC model using electrochemical data with
electrochemical model……………………………………….………………………………30
List of Tables
Table Page No.
1. Electrochemical data for LiyMn2O4 – LixC6 battery …..…………..……….……………...14
2. Table of contributions………………………………………………………………………..31

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Motivation/Background
Lithium-ion batteries are dominating modern EV applications because of their features such as
lightweight, high-energy density, low self-discharge, and long lifespan. Unlike gasoline
vehicles, customers have anxiety about the range provided by an electric vehicle. State of
Charge is an important parameter that indicates the remaining available energy in the battery
at any given point in time. It also helps to protect the battery from overcharging or discharging
and increases the lifespan. An accurate real-time SOC value is a crucial part of a Battery
Management System.
We can directly measure the current and voltage of the battery. However, SOC cannot be
measured directly, and it has to be estimated based on the relationship between different battery
parameters. There are two types of SOC estimation methods [6]: online method (ONM) and
Offline (OFM) method. Coulomb counting method, Fitting estimation method, and Model-
based estimation (MBE) methods come under ONM. Open-circuit voltage method,
electrochemical impedance spectroscopy method, etc., are offline methods. Offline methods
need the battery to be disconnected from regular operation to estimate the SOC, which makes
it not suitable for BMS and hence online methods are preferred.
Equivalent circuit and Electrochemical models are the most used online based MBE methods
for estimating SOC. Equivalent circuit models are less computationally demanding and hence
used widely in BMS applications [6]. On the other hand, Electrochemical models estimate the
battery status through the electrochemical reactions and Li-ion diffusion dynamics. SOC is
calculated based on the Li-ion concentration present in the positive and negative electrodes at
any given time. As mentioned in research by Yuntian Liu, et al. the RMSE of SOC is lower in
electrochemical model compared to equivalent circuit model [6]. The Pseudo-two-dimensional
(P2D) model is extensively used for research on LiB SOC estimation. However, it is not
suitable for BMS application because of the series of coupled and nonlinear partial differential
equations which demand heavy computational power. To improve the practical usability of the
Electrochemical models, reduced order electrochemical models which are simplified versions
of P2D models such as single particle model (SPM) and Single particle model with electrolyte
physics (SPMe) are used. SPM does not consider change in electrolyte concentration which
leads to less accurate results at medium and high discharge rates [1]. So, SPMe has become a
main research topic to reduce its computational complexity and make use in practical
applications.
Objective
Motivated by the above literature study, this project proposes to estimate SOC of the LIB from
a reduced-order electrochemical model (ROEM) using an Extended Kalman Filter (EKF). To
reduce the model complexity, the solid phase equations will be reconstructed by combining the
Pade approximation and quadratic polynomial method. Volume averaging technique will be
used for electrolyte physics simplification. Then an EKF will be used to estimate the SOC of
the battery.
Also, using the transfer function from Pade approximation, we will attempt to obtain a
relationship with the resistances and capacitances of an equivalent circuit model. Estimated
parameters of the ROEM model are then compared against the above enhanced equivalent
circuit model and traditional equivalent circuit model.

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Introduction to Electrochemical Models (EM)
For battery parameter estimation, an EM is quite accurate as it considers the internal chemistry
of the battery to calculate these parameters but computationally it is difficult for BMS
applications without simplifying the model. The model can be simplified by using modified
models such as the pseudo two-dimensional (P2D) model which is one of the more popular
electrochemical models (EM), but it requires large amount of computation and a few limits in
real-time application in BMS. To overcome these drawbacks, many researchers have proposed
reduced-order electrochemical models (ROEMs) such as single particle model (SPM).
However, SPM ignores the electrolyte physics including changes in electrolyte concentration
and potential, which may easily result in the bad accuracy at medium and high C-rate. To
improve on this, an SPM model with electrolyte dynamics has been proposed. The SPMe can
provide better accuracy in the terminal output voltage (TOV) even at high C-rates. This model
focuses on achieving the necessary balances between the model fidelity and computational
complexity
Improved Reduced Order Electrochemical Model (iROEM)
Improved Reduced Order Electro Chemical Model (iROEM) is proposed as an improvement
to SPM by considering the change in electrolyte phase concentration which gives more accurate
results in high discharge rate. The diffusion equation inside the electrode can be solved with
many approaches such as ﬁnite difference method (FDM), parabolic polynomial approximation
(PP), proper orthogonal decomposition (POD) and residue grouping (RG)[1]. To improve the
Observability analysis in iROEM, third order Pade approximation is used in negative electrode
and parabolic polynomial approximation method is used in positive electrode. Electrolyte
diffusion equation is solved by Volume Averaging Technique (VAT) and quadratic
approximation in each phase of negative electrode, separator and positive electrode.
Pade Approximation (Negative Electrode)
From solid diffusion equation, we require 𝐶𝑠,𝑆𝑢𝑟𝑓 (Surface concentration) and 𝐶𝑠
̅ (Average
concentration). 𝐶𝑠,𝑆𝑢𝑟𝑓 is used to calculate the OCV of respective electrode and 𝐶𝑠
̅ is needed
for SOC calculation. In Pade approach, transfer function for 𝐶𝑠,𝑆𝑢𝑟𝑓 is calculated by applying
Laplace transform to the below solid diffusion equation and approximated for a particular
order.
Solid Diffusion Equation:
𝜕𝐶𝑠,𝑖
𝜕𝑡
=
𝐷𝑠,𝑖
𝑟2
𝜕
𝜕𝑟
(𝑟2 𝐶𝑠
𝜕𝑟
)
Boundary Conditions:
𝜕𝐶𝑠,𝑖
𝜕𝑟
|𝑟=0 = 0; 𝐷𝑠,𝑖
𝜕𝐶𝑠,𝑖
𝜕𝑟
|𝑟=𝑅 =
−𝑗𝐿𝑖(𝑥,𝑡)
𝑎𝑠𝐹

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In the main reference paper [1], a comparison of second and third order pade result of 𝐶𝑠,𝑆𝑢𝑟𝑓
with P2D model is made for pulsed input current density. It is observed that error of second
order result during rising edge and falling edge input is approximately 50 mol/𝑚3
, whereas that
of third order it is approximately 20 mol/𝑚3
. The corresponding error and 𝐶𝑠,𝑆𝑢𝑟𝑓 plot from the
reference paper[1] is shown below. So, third order pade approximation is used in iROEM to
solve negative electrode diffusion. We tried to replicate the result of 𝐶𝑠,𝑆𝑢𝑟𝑓 for third order pade
approach using MATLAB and th e corresponding plot is shown below.
The relationship between 𝐶𝑠
̅ (Average concentration) and 𝑗𝐿𝑖
can be expressed using Laplace
transform as below:
𝐶𝑠(𝑠)
̅̅̅̅̅̅̅
𝑗𝐿𝑖(𝑠)
= -
3
𝑅𝑎𝐹𝑠
The transfer function of 𝐶𝑠,𝑆𝑢𝑟𝑓 (Surface concentration) using third order Pade approach by
applying Laplace transform to solid diffusion equation can be found out as:
𝐶𝑠,𝑆𝑢𝑟𝑓
𝑗𝐿𝑖(𝑠)
=
−
3
𝑎𝐹𝑅
−
4𝑅𝑠
11𝑎𝐹𝐷𝑠
−
𝑅3𝑠2
165𝑎𝐹𝐷𝑠
2
𝑠+
3𝑅2𝑠2
55𝐷𝑠
+
𝑅4𝑠3
3465𝐷𝑠
2
Fig. 1: Plot of Pulsed Input current and corresponding Electrode surface concentration comparison between 2nd
order Pade,
third order Pade and P2D model [1].
Fig. 2: Plot of Electrode surface concentration from third order pade approach for given pulsed Input current simulated from MATLAB

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A state space representation in controllable canonical form [2] of above transfer functions are
formed with below mentioned A, B and C matrices considering outputs as 𝐶𝑠,𝑆𝑢𝑟𝑓 (Surface
concentration) and 𝐶𝑠
̅ (Average concentration).
𝑥̇ = 𝐴𝑥 + 𝐵𝑗𝐿𝑖
, 𝑦 = 𝐶𝑥 , 𝑦 = [
𝐶𝑠,𝑆𝑢𝑟𝑓
𝐶𝑠
̅ ]
𝐴 = [
0 1 0
0 0 1
0 −
3465𝐷𝑠
2
𝑅4
−
189𝐷𝑠
𝑅2
], 𝐵 = [
0
0
1
], 𝐶 = [
−10395𝐷𝑠
2
𝑎𝐹𝑅5
−1260𝐷𝑠
𝑎𝐹𝑅3
−21
𝑎𝐹𝑅
−10395𝐷𝑠
2
𝑎𝐹𝑅5
−252𝐷𝑠
𝑎𝐹𝑅3
−3
𝑎𝐹𝑅
]
Initial condition for states is taken as 𝑥0 = [
−𝑎𝐹𝑅5𝐶𝑠,0
10395𝐷𝑠
2
0
0
].
𝐶𝑠,𝑆𝑢𝑟𝑓 and 𝐶𝑠
̅ can be solved from above state space representation and initial conditions
derived from third order pade approximation, in either continuous or discrete time domain. Fig
2 shows the result calculated from shown approach which matches the profile shown in the
reference paper [1].
Quadratic Parabolic Polynomial Approximation (Positive Electrode)
For the positive electrode, quadratic polynomial method has been used. The concentration
profile is assumed as quadratic initially. This profile is given as:
Applying volume averaging and using the below given boundary conditions, we can form the
state-space matrices. The outputs on solving these equations are bulk (𝑐̅(t)) and surface (𝑐𝑆(𝑡))
concentration.
Boundary condition equations:
Electrolyte Concentration Approximation
In this project, electrolyte concentration is approximated with VAT method and considering
the profile of the concentration is quadratic. Electrolyte phase mass balances can be described
by the diffusion equations. Electrolyte concentration field is continuous through the negative
electrode, separator, and positive electrode. Hence, concentration and flux continuities hold at
State-space matrices:
A = 0
B =
−3
𝑅
C = [1 1]𝑇
D = [0
−𝑅
5𝐷𝑠
]
𝑇

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each electrode – separator interface. Considering this, the equations of mass conservation of
electrolyte in different parts of a battery are given below:
Negative electrode region
𝜀𝑒𝑛
𝜕𝐶𝑒
𝜕𝑡
=
𝜕
𝜕𝑥
(𝐷𝑒𝑛
𝜕𝐶𝑒
𝜕𝑥
) + 𝑎𝑛(1 − 𝑡+)𝑗𝑛
With boundary conditions
−𝐷𝑒𝑛
𝜕𝐶𝑒
𝜕𝑥
|
𝑥=0
= 0 , −𝐷𝑒𝑛
𝜕𝐶𝑒
𝜕𝑥
|
𝑥=𝑙𝑛
= 𝑞2𝑖𝑛, 𝐶𝑒(𝑙𝑛,t) =𝐶2𝑖𝑛
Separator region
𝜀𝑒𝑠
𝜕𝐶𝑒
𝜕𝑡
=
𝜕
𝜕𝑥
(𝐷𝑒𝑠
𝜕𝐶𝑒
𝜕𝑥
)
𝐶𝑒(𝑙𝑛,t) =𝐶2𝑖𝑛, 𝐶𝑒(𝑙𝑛+ 𝑙𝑠,t) =𝐶2𝑖𝑝 , −𝐷𝑒𝑠
𝜕𝐶𝑒
𝜕𝑥
|
𝑥=𝑙𝑛
= 𝑞2𝑖𝑛, −𝐷𝑒𝑠
𝜕𝐶𝑒
𝜕𝑥
|
𝑥=𝑙𝑛+𝑙𝑠
= 𝑞2𝑖𝑝
Positive electrode region
𝜀𝑒𝑝
𝜕𝐶𝑒
𝜕𝑡
=
𝜕
𝜕𝑥
(𝐷𝑒𝑝
𝜕𝐶𝑒
𝜕𝑥
) + 𝑎𝑝(1 − 𝑡+)𝑗𝑝
−𝐷𝑒𝑝
𝜕𝐶𝑒
𝜕𝑥
|
𝑥=𝑙𝑛+𝑙𝑠
= 𝑞2𝑖𝑝 , −𝐷𝑒𝑝
𝜕𝐶𝑒
𝜕𝑥
|
𝑥=𝐿
= 0, 𝐶𝑒(𝑙𝑛+ 𝑙𝑠,t) =𝐶2𝑖𝑝
Initial condition across three regions is
𝐶𝑒(𝑥, 0) =𝐶20
According to [1], VAT can be used to describe the electrolyte concentration distribution
Ce(x,t) across the length of the battery.
For example, the volume averaged value of any variable f(x,t) is defined as
〈𝑓(𝑡)〉 =
1
𝐴𝐿𝑖
∫ 𝑓(𝑥, 𝑡)𝐴𝑑𝑥
𝐿𝑖
0
Where A is the surface area, Li denotes the electrode thickness (I =n,s,p). 〈− −〉 Denotes the
volume averaged quantity.

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To begin with, the volume averaged electrolyte concentration of negative electrode region can
be defined as
〈𝐶𝑒𝑛(𝑥, t) 〉 =
1
𝐴𝑙𝑛
∫ 𝐶𝑒𝑛(𝑥, 𝑡)𝐴𝑑𝑥
𝑙𝑛
0
=
1
𝑙𝑛
∫ 𝐶𝑒𝑛(𝑥, 𝑡)𝑑𝑥
𝑙𝑛
0
Then PDE of 𝐶𝑒𝑛(𝑥, t) can be written as
1
𝑙𝑛
∫ [𝜀𝑒𝑛
𝜕𝐶𝑒𝑛
𝜕𝑡
] 𝑑𝑥 =
𝑙𝑛
0
1
𝑙𝑛
∫ [
𝜕
𝜕𝑥
(𝐷𝑒𝑛
𝑒𝑓𝑓 𝜕𝐶𝑒𝑛
𝜕𝑥
) + 𝑎𝑛(1 − 𝑡+)𝐽𝑛] 𝑑𝑥
𝑙𝑛
0
Where, 𝜀𝑒𝑛is electrolyte volume fraction of negative region, 𝐷𝑒𝑛
𝑒𝑓𝑓
is the effective electrolyte
diffusion coefficient
1
𝑙𝑛
(𝐷𝑒𝑛
𝜕𝑥
)
0
𝑙𝑛
= −
𝑞2𝑖𝑛
𝑙𝑛
𝑞2𝑖𝑛 represents the diffusion flux at the interface of the negative electrode and separator.
Finally, the equation can be simplified as
𝑙𝑛𝜀𝑒𝑛
𝑑〈𝐶𝑒𝑛〉
𝑑𝑡
= −𝑞2𝑖𝑛 + 𝑎𝑛(1 − 𝑡+)𝑙𝑛〈𝐽𝑛〉
Another assumption is made as
𝜕
𝜕𝑥
(𝐷𝑒𝑛
𝜕𝑥
) ≈ −
𝑞2𝑖𝑛
𝑙𝑛
Integrating this equation with respect to x gives
𝐷𝑒𝑛
𝜕𝑥
= −
𝑞2𝑖𝑛
𝑙𝑛
𝑥 + 𝑔(𝑡)
g(t) is an arbitrary function. Applying boundary conditions of PDE for 𝐶𝑒𝑛(𝑥, t) gives g(t) = 0
The equation becomes
𝐷𝑒𝑛
𝜕𝑥
= −
𝑞2𝑖𝑛
𝑙𝑛
𝑥
Integrating once more with respect to x gives
𝜕𝐶𝑒𝑛
𝜕𝑥
= −
𝑞2𝑖𝑛
𝐷𝑒𝑛
𝑒𝑓𝑓
𝑙𝑛
𝑥2
2
+ 𝑔(𝑡)
Applying x = Ln into above equation generates
𝐶𝑒𝑛(𝑙𝑛, t) = 𝐶2𝑖𝑛(𝑡) = −
𝑞2𝑖𝑛
𝐷𝑒𝑛
𝑒𝑓𝑓
𝑙𝑛
𝑙𝑛
2
2
+ 𝑔(𝑡)
Where, 𝐶2𝑖𝑛(𝑡) represents the electrolyte concentration at the interface between the negative
electrode and the separator
From equation 1 and 2, we can obtain 𝐶𝑒𝑛(𝑥, t)
(1)
(2)

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𝐶𝑒𝑛(𝑥, t) = 𝐶2𝑖𝑛(𝑡) +
𝑞2𝑖𝑛
2𝐷𝑒𝑛
𝑒𝑓𝑓
𝑙𝑛
(𝑙𝑛
2
− 𝑥2
)
By substituting the averaged length for negative electrode, <x2
> = ln
2
/3, the averaged
electrolyte concentration at negative electrode is obtained as
〈𝐶𝑒𝑛〉 = 𝐶2𝑖𝑛(𝑡) +
𝑞2𝑖𝑛
2𝐷𝑒𝑛
𝑒𝑓𝑓
𝑙𝑛
2
3
𝑙𝑛
2
= 𝐶2𝑖𝑛(𝑡) +
𝑙𝑛𝑞2𝑖𝑛
3𝐷𝑒𝑛
𝑒𝑓𝑓
Using similar approach, the simplified 𝐶𝑒𝑠and 𝐶𝑒𝑝with VAT are given below
𝑙𝑠𝜀𝑒𝑠
𝑑〈𝐶𝑒𝑠〉
𝑑𝑡
= 𝑞2𝑖𝑛(𝑡) − 𝑞2𝑖𝑝(𝑡)
𝑙𝑝𝜀𝑒𝑝
𝑑〈𝐶𝑒𝑝〉
𝑑𝑡
= 𝑞2𝑖𝑝 + 𝑎𝑝(1 − 𝑡+)𝑙𝑝〈𝐽𝑝〉
Also, the 𝐶𝑒𝑠(𝑥, t), and 𝐶𝑒𝑝(𝑥, t) can be obtained as follows
𝐶𝑒𝑠(𝑥, t) = 𝐶2𝑖𝑛(𝑡) −
𝑞2𝑖𝑛
𝐷𝑒𝑠
𝑒𝑓𝑓
(𝑥 − 𝑙𝑛) + (
𝑞2𝑖𝑛 − 𝑞2𝑖𝑝
𝐷𝑒𝑠
𝑒𝑓𝑓
𝑙𝑠
)
(𝑥 − 𝑙𝑛)2
2
𝐶𝑒𝑝(𝑥, t) = 𝐶2𝑖𝑝(𝑡) −
𝑞2𝑖𝑝
2𝑙𝑝𝐷𝑒𝑝
𝑒𝑓𝑓
[𝑙𝑝
2
− (𝐿 − 𝑥)2
]
Solving the averaged <𝐶𝑒𝑠> and <𝐶𝑒𝑝> gives
〈𝐶𝑒𝑠〉 = 𝐶2𝑖𝑛(𝑡) −
𝑙𝑠𝑞2𝑖𝑛
3𝐷𝑒𝑠
𝑒𝑓𝑓
−
𝑙𝑠𝑞2𝑖𝑝
6𝐷𝑒𝑠
𝑒𝑓𝑓
〈𝐶𝑒𝑝〉 = 𝐶2𝑖𝑝(𝑡) −
𝑙𝑝𝑞2𝑖𝑝
3𝐷𝑒𝑝
𝑒𝑓𝑓
Where, 𝐶2𝑖𝑝(𝑡) represents the electrolyte concentration, 𝑞2𝑖𝑝 represents the diffusion flux at the
interface between the positive electrode and the separator region.
The values of 𝐶2𝑖𝑝(𝑡), 𝐶2𝑖𝑛(𝑡), 𝑞2𝑖𝑝(𝑡), 𝑞2𝑖𝑛(𝑡) can be obtained by differentiating the averaged
concentration equations and applying boundary conditions as per [3]
𝐶2𝑖𝑝 = 𝐶20 + 𝛼𝑖𝑛𝑞2𝑖𝑛 + 𝛼𝑖𝑝𝑞2𝑖𝑝
𝐶2𝑖𝑛 = 𝐶2𝑖𝑝 +
𝑙𝑠(𝑞2𝑖𝑛 + 𝑞2𝑖𝑝)
2𝐷2𝑠
where,
𝛼𝑖𝑛 = − (
𝑙𝑛𝑙𝑠𝜀2𝑛
2𝐷2𝑠
+
𝑙𝑠
2
𝜀2𝑠
6𝐷2𝑠
+
𝑙𝑛
2
𝜀2𝑛
3𝐷2𝑛
)
1
(𝑙𝑛𝜀2𝑛 + 𝑙𝑠𝜀2𝑠 + 𝑙𝑝𝜀2𝑝)
And
(5)
(6)
(3)
(4)

11
𝛼𝑖𝑝 = − (
2𝐷2𝑠
+
𝑙𝑠
2
𝜀2𝑠
3𝐷2𝑠
−
𝑙𝑝
2
𝜀2𝑝
3𝐷2𝑝
)
1
(𝑙𝑛𝜀2𝑛 + 𝑙𝑠𝜀2𝑠 + 𝑙𝑝𝜀2𝑝)
(𝑙𝑛𝜀2𝑛𝛼𝑖𝑛 +
2𝐷2𝑠
+
𝑙𝑛
2
𝜀2𝑛
3𝐷2𝑛
)
𝑑𝑞2𝑖𝑛
𝑑𝑡
+ (𝑙𝑛𝜀2𝑛𝛼𝑖𝑝 +
2𝐷2𝑠
)
𝑑𝑞2𝑖𝑝
𝑑𝑡
= −𝑞2𝑖𝑛 + 𝑎𝑛(1 − 𝑡+)𝑙𝑛〈𝐽𝑛〉
𝑙𝑝𝜀2𝑝𝛼𝑖𝑛
𝑑𝑞2𝑖𝑛
𝑑𝑡
+ (𝑙𝑝𝜀2𝑝𝛼𝑖𝑝 −
𝑙𝑝
2
𝜀2𝑝
3𝐷2𝑝
)
𝑑𝑞2𝑖𝑝
𝑑𝑡
= 𝑞2𝑖𝑝 + 𝑎𝑝(1 − 𝑡+)𝑙𝑝〈𝐽𝑝〉
The above equations can be written in state space form and solved for q2in and q2ip. Assuming
an equilibrated initial state, these interfacial fluxes have zero initial condition. Using these
values C2in and C2ip can be calculated from the algebraic equations 5 & 6.
Finally, the electrolyte concentration of the positive and negative electrodes at negative and
positive current collector interfaces can be calculated by substituting x value in the equations
3 & 4.
𝐶𝑒𝑛(0, 𝑡) = 𝐶2𝑖𝑛(𝑡) +
𝑞2𝑖𝑛(𝑡)
2𝐷𝑒𝑛
𝑒𝑓𝑓
𝑙𝑛
𝐶𝑒𝑝(𝐿, 𝑡) = 𝐶2𝑖𝑝(𝑡) −
𝑞2𝑖𝑝(𝑡)
2𝐷𝑒𝑝
𝑒𝑓𝑓
𝑙𝑝
Equations 7 & 8 can be written in state space form as:
[
𝑑𝑞2𝑖𝑛
𝑑𝑡
𝑑𝑞2𝑖𝑝
𝑑𝑡
] = [
−𝑏2
𝑎1𝑏2−𝑎2𝑏1
−𝑏1
−𝑎2
−𝑎1
] [
𝑞2𝑖𝑛
𝑞2𝑖𝑝
] + [
(1−𝑡+)𝑏2𝑎𝑛𝑙𝑛
−(1−𝑡+)𝑏1𝑎𝑝𝑙𝑝
(1−𝑡+)𝑎2𝑎𝑛𝑙𝑛
−(1−𝑡+)𝑎1𝑎𝑝𝑙𝑝
] [
〈𝐽𝑛〉
〈𝐽𝑝〉
]
𝑎1 = 𝑙𝑛𝜀2𝑛𝛼𝑖𝑛 +
𝑙𝑠𝑙𝑛𝜀2𝑛
2𝐷2𝑠
+
𝑙𝑛
2
𝜀2𝑛
3𝐷2𝑛
𝑏1 = 𝑙𝑛𝜀2𝑛𝛼𝑖𝑝 +
𝑙𝑠𝑙𝑛𝜀2𝑛
2𝐷2𝑠
𝑎2 = 𝑙𝑝𝜀2𝑝𝛼𝑖𝑛
𝑏2 = 𝑙𝑝𝜀2𝑝𝛼𝑖𝑝 −
𝑙𝑝
2
𝜀2𝑝
3𝐷2𝑝
𝑦 = [
𝑞2𝑖𝑛
𝑞2𝑖𝑝
] = [
1 0
0 1
] [
𝑞2𝑖𝑛
𝑞2𝑖𝑝
] + [
0 0
0 0
] [
〈𝐽𝑛〉
〈𝐽𝑝〉
]
By solving the state space equations and quadratic equations 9 & 10 electrolyte concentration
can be calculated for x = 0, x=L
(7)
(8)
(9)
(10)

12
Simulations are performed with a discharge rate of 0.5C, 1C and 2C. The figure 4 shows the
electrolyte concentration profile for x = 0, and x = L. It can be seen that at the negative
electrode, the electrolyte concentration increases and stabilizes at a constant value. At the
positive electrode, concentration reduces and stabilizes. The simulation results are comparable
to the reference paper.
The above figure 3 shows the electrolyte concentration across the length of the battery when
discharged with a constant rate of 1C. Results are plotted for time = 50, 150, 300, 1500 seconds.
Concentration profile takes the shape of the parabola that is in line with our assumption made
for approximation. When compared to the reference paper [1], results are comparable.
Electrolyte potential difference
The electrolyte potential difference in lithium-ion batteries can be determined as a function of
electrolyte concentration. Using the equations below, the electrolyte potential difference
between the negative and positive current collectors can be calculated.
For potential in the negative region (0<=x<=Ln)
𝜑𝑒𝑛(𝑥, 𝑡) = 𝜑𝑒𝑛(0, 𝑡) + (1 − 𝑡+)
2𝑅𝑇
𝐹
𝑙𝑛
𝐶𝑒(𝑥, 𝑡)
𝐶𝑒(0, 𝑡)
−
𝑖𝑎𝑝𝑝
2𝑙𝑛𝐾𝑛
𝑒𝑓𝑓
𝑥2
For potential in the separator region (Ln<=x<=Ln+Ls)
𝜑𝑒𝑠(𝑥, 𝑡) = 𝜑𝑒𝑛(0, 𝑡) + (1 − 𝑡+)
2𝑅𝑇
𝐹
𝑙𝑛
𝐶𝑒(0, 𝑡)
−
𝑖𝑎𝑝𝑝
𝐾𝑠
𝑒𝑓𝑓
(𝑥 − 𝑙𝑛) −
𝑖𝑎𝑝𝑝𝑙𝑛
2𝐾𝑛
𝑒𝑓𝑓
For potential in the positive electrode region (Ln+Ls<=x<=Ln+Ls+Lp)
Fig. 3: Graph of electrolyte concentration along the length of the
battery in different regions
Fig. 4: Graph of electrolyte concentration at different C rates
along the length of the battery

13
𝜑𝑒𝑠(𝑥, 𝑡) = 𝜑𝑒𝑛(0, 𝑡) + (1 − 𝑡+)
2𝑅𝑇
𝐹
𝑙𝑛
𝐶𝑒(0, 𝑡)
+
𝑖𝑎𝑝𝑝
2𝑙𝑝𝐾𝑝
𝑒𝑓𝑓
(𝐿 − 𝑥)2
−
𝑖𝑎𝑝𝑝
2
(
𝑙𝑛
𝐾𝑛
𝑒𝑓𝑓
+
2𝑙𝑠
𝐾𝑠
𝑒𝑓𝑓
+
𝑙𝑝
𝐾𝑝
𝑒𝑓𝑓
)
Where, 𝐾𝑖
𝑒𝑓𝑓
is the effective electrolyte conductivity.
The electrolyte potential difference between the battery terminals (x=0 and x=L) can be
calculated as
𝜑𝑒𝑝(𝑥, 𝑡)|𝑥=𝐿
− 𝜑𝑒𝑛(𝑥, 𝑡)|𝑥=0 = 𝜑𝑒𝑝(𝐿, 𝑡) − 𝜑𝑒𝑛(0, 𝑡)
= (1 − 𝑡+)
2𝑅𝑇
𝐹
𝑙𝑛
𝐶𝑒𝑝(𝑥, 𝑡)
𝐶𝑒𝑛(0, 𝑡)
−
𝑖𝑎𝑝𝑝
2
(
𝑙𝑛
𝐾𝑛
𝑒𝑓𝑓
+
2𝑙𝑠
𝐾𝑠
𝑒𝑓𝑓
+
𝑙𝑝
𝐾𝑝
𝑒𝑓𝑓
)
Terminal Output Voltage Calculation
The TOV is then calculated using the following process:
 The applied current is used to calculate the positive (𝑐𝑠,𝑝) and negative (𝑐𝑠,𝑛) surface
concentrations and bulk concentrations using the above explained solid diffusion methods.
 The surface concentrations at both the solid electrodes are used to calculate the open circuit
voltages (OCV) for positive (𝑈𝑝) and negative (𝑈𝑛) electrode using the following
equations:
 The surface concentrations are also used to calculate the overpotentials of both the
electrodes using the following equations:
j is used to represent positive and negative electrodes, and this is replaced by p and n.
Here,
 On solving the electrolyte diffusion, we get electrolyte concentration 𝑐𝑒,𝑝 𝑎𝑛𝑑 𝑐𝑒,𝑛. The
electrolyte ionic conductivity κi for both the electrodes are obtained from the data. These
values are then used to calculate the electrolyte potential using the following equations:

14
For the potential in the negative region (0≤ x ≤ Ln ), we have
For the potential in the positive region (Ln+Ls ≤ x ≤ Ls), we have
Therefore, the electrolyte potential difference across the battery ( x = 0 and x = L ) can be
calculated by
The above equations are then compiled and coded into MATLAB to make the Simulink model
for TOV calculation. This model is showing in the figure below.
Fig. 5: Block diagram of iROEM model for calculation of Terminal Output Voltage (TOV)

15
Electrochemical Data
This project uses LiyMn2O4 – LixC6 battery electrochemical parameters shared in the reference
paper [1,6].
Parameter Value Description
ln 1 × 10 −4
Thickness of the negative electrode (m)
ls 52 × 10 −6
Thickness of the separator (m)
lp 183 × 10 −6
Thickness of the positive electrode (m)
ε2n 0.375 Porosity of the negative electrode
ε2s 1 Porosity of the separator
ε2p 0.444 Porosity of the positive electrode
εfp 0.259 Porosity of filler in positive electrode [6]
εfn 0.172 Porosity of filler in negative electrode[6]
ap 3(1- ε2p- εfp)/Rp Specific surface area of active materials in positive electrode (m−1
)
an 3(1- ε2n- εfn)/Rn Specific surface area of active materials in negative electrode (m−1
)
Brug 1.5 Brugman coefficient
De/D2 7.5 × 10 −11
Diffusion coefficient of electrolyte(m2
s−1
)
D2n D2*( ε2n ^Brug)
Effective Diffusion coefficient of electrolyte in Positive electrode
region (m2
s−1
)
D2s D2*( ε2s ^Brug)
Effective Diffusion coefficient of electrolyte in separator region
(m2
s−1
)
D2p D2*( ε2p ^Brug)
Effective Diffusion coefficient of electrolyte in negative electrode
region (m2
s−1
)
ki 2.344 × 10 −11
Reaction rate constant (m2.5
mol−0.5
s−1
)
F 96,487 Faraday’s constant (C mol−1
)
t+ 0.363 Cationic transport number
iapp 17.5 × C _ rate C-rate times 1C discharge current density (A m−2
)
Fig. 6: SIMULINK model for TOV calculation using iROEM model
Table 1: Electrochemical data for LiyMn2O4 – LixC6 battery

16
Parameter Value Description
c0 2000 Initial concentration of salt (mol m−3
)
Ds,n 3.9 × 10 −14
Solid-phase Li diffusivity/negative electrode (m2
s−1
)
Ds,p 10 −13
Solid-phase Li diffusivity/positive electrode (m2
s−1
)
Rn 12.5 × 10 −6
Particle radius, negative electrode (m)
Rp 8 × 10 −6
Particle radius, positive electrode (m)
R 8.314 Universal gas constant (J mol−1
K−1
)
T 298.15 Ambient temperature (K)
cmax s,p 22,860 Positive maximum concentration (mol m−3
)
cmax s,n 26,390 Positive maximum concentration (mol m−3
)
cs,p 3900 Initial concentration of lithium-ion in solid (mol m−3
)
cs,n 14,870 14,870 Initial concentration of lithium-ion in solid (mol m−3
)
Rf 20 x 10 -4
Current collector contact resistance (Ω m2
)
Electrolyte conductivity can be obtained from
To validate the improved reduced order model in this project, comparison of terminal voltage
is done against the reference paper’s results under two input current conditions, Pulse and DST.
It should be noted that the current profile used in DST test follows positive sign convention for
charging and vice versa. In our project, standard sign convention for current is used, ie. Positive
sign convention for discharging event and vice versa.
Corresponding figures are given below:
SOC estimation using Extended Kalman filter is done with the extrapolated data from these
graphs that is discretized in 5 seconds time steps. Discretized data for both current profiles is
attached in the appendix.
Fig. 7: Reference graphs for Performance evaluation of battery TOVs (a) The pulse current cycle, (b) The battery TOVs of the
SPM/iROEM/P2D in pulse cycle, (c) DST current cycle, (d) The battery TOVs of the SPM/iROEM/P2D in DST

17
Terminal Voltage Results from iROEM
Pulse Cycle
The TOV plot comparison between iROEM and the model using Pade approaches in both
electrodes for pulse current profile is shown below. Initially, when the battery is discharged at
1C rate, the voltage drops followed by a charging phase at 1C rate to which the voltage
increases and the iROEM model shows comparatively less error. There is no significant
difference in TOV by considering polynomial PP approach in one of the electrodes in iROEM.
This is compared with the reference paper, and it seems to comply. Also, there is no significant
difference
Dynamic Stress Test Cycle
The TOV plot comparison between iROEM and the model using Pade approaches in both
electrodes for Dynamic Stress Test (DST) is shown below. In the DST input cycle, current is
changed from 0.5C rate to 4C rate to check the performance of the model at high C rates. As
expected, SPM shows the error at high C rates in comparison with iROEM model [1]. There is
no significant difference observed in TOV by considering polynomial PP approach in one of
the electrodes(iROEM) in comparison to pade approach in both the electrodes.
Fig. 8: Terminal Voltage results from iROEM- pulse cycle (a) Input current density for pulse input (b) Terminal voltage of iROEM
compared with other models
(a)
(b)

18
Terminal Voltage and SOC Estimation result using EKF
As can be seen from calculated TOV values of only iROEM, there is an error wrt actual
measurement value. To estimate the accurate value of SOC from TOV, observability analysis
with Extended Kalman Filter (EKF) is used to predict the correct state values and in turn
electrode concentration values. Using the estimated 𝐶𝑠
̅ , SOC can be predicted accurately.
Change in electrolyte concentration is not considered and estimated in our observability
analysis. So, EKF is used by considering TOV as function of only 𝐶𝑠𝑝,𝑆𝑢𝑟𝑓 , 𝐶𝑠𝑛,𝑆𝑢𝑟𝑓 and 𝑖𝑎𝑝𝑝.
𝑉𝑡 = ℎ(𝐶𝑠𝑝,𝑆𝑢𝑟𝑓, 𝐶𝑠𝑛,𝑆𝑢𝑟𝑓, 𝑖𝑎𝑝𝑝)
By considering both the states of positive and negative electrode solved through third order
Pade approach, we may have six state space variables which can cause difficulty in analysing
the system observability [1]. So, relationship between 𝐶𝑠𝑝
̅̅̅̅ and 𝐶𝑠𝑛
̅̅̅̅ can be obtained from the
SOC definition as below:
𝑐̅𝑠𝑝(𝑡) = 𝐶𝑠𝑝
𝑚𝑎𝑥
[𝜃0%𝑝 +
𝑐̅𝑠𝑛(𝑡) − 𝐶𝑠𝑛
𝑚𝑎𝑥
𝜃0%𝑛
(𝜃100%𝑛 − 𝜃0%𝑛)𝐶𝑠𝑛
𝑚𝑎𝑥 (𝜃100%𝑝 − 𝜃0%𝑝)]
Then by using the quadratic Parabolic polynomial approach and VAT in positive electrode, we
have the relationship between 𝐶𝑠𝑝,𝑆𝑢𝑟𝑓 and 𝐶𝑠𝑝
̅̅̅̅ as below:
𝐶𝑠𝑝,𝑆𝑢𝑟𝑓 = 𝐶𝑠𝑝
̅̅̅̅ −
𝑗𝑝𝑅𝑝
5𝐷𝑠,𝑝
(a)
(b)
Fig. 9: Terminal Voltage results from iROEM- DST cycle (a) Input current density for pulse input (b) Terminal voltage of iROEM
compared with other models

19
So, the relationship between 𝐶𝑠𝑝,𝑆𝑢𝑟𝑓 and 𝐶𝑠𝑛
̅̅̅̅ is obtained as follows:
𝐶𝑠𝑝,𝑠𝑢𝑟𝑓(𝑡) = 𝐶𝑠𝑝
𝑚𝑎𝑥
[𝜃0%𝑝 +
𝑚𝑎𝑥
𝜃0%𝑛
(𝜃100%𝑛 − 𝜃0%𝑛)𝐶𝑠𝑛
𝑚𝑎𝑥 (𝜃100%𝑝 − 𝜃0%𝑝)] −
𝑅𝑝
5𝐷𝑠𝑝
𝑖𝑎𝑝𝑝
𝑎𝑝𝐹𝑙𝑝
By considering above relationship, we may consider TOV as a final function of only negative
electrode concentrations (𝐶𝑠𝑛,𝑆𝑢𝑟𝑓 𝑎𝑛𝑑 𝐶𝑠𝑛
̅̅̅̅) and applied current density. In effective, EKF
considers SPM model for estimation of TOV instead of SPMe.
𝑦 = 𝑉𝑡 = ℎ(𝐶𝑠𝑛,𝑆𝑢𝑟𝑓, 𝐶𝑠𝑛
̅̅̅̅, 𝑖𝑎𝑝𝑝)
After discretization, the three state space variables of negative electrode are considered for
observability analysis and new state space equations are:
𝑥𝑛
̇ =𝐴𝑛
𝑥𝑛 + 𝐵𝑛
𝑖𝑎𝑝𝑝
Over potential and Electrolyte potential differences are assumed to be independent of
𝐶𝑠𝑛,𝑆𝑢𝑟𝑓, 𝐶𝑠𝑛
̅̅̅̅ .
So, 𝐶𝑘 matrix at any time step k for usage in Extended Kalman filter is:
𝐶𝑘 = [
𝜕ℎ(𝑘)
𝜕𝑥1(𝑘)
𝜕ℎ(𝑘)
𝜕𝑥2(𝑘)
𝜕ℎ(𝑘)
𝜕𝑥3(𝑘)
]
where,
𝜕ℎ(𝑘)
𝜕𝑥1(𝑘)
=
𝜕ℎ(𝑘)
𝜕𝜃𝑝(𝑘)
𝜕𝜃𝑝(𝑘)
𝜕𝐶𝑠𝑝,𝑠𝑢𝑟𝑓(𝑘)
𝜕𝑐̅𝑠𝑛(𝑘)
𝜕𝑥1(𝑘)
−
𝜕ℎ(𝑘)
𝜕𝜃𝑛(𝑘)
𝜕𝜃𝑛(𝑘)
𝜕𝐶𝑠𝑛,𝑠𝑢𝑟𝑓(𝑘)
𝜕𝑥1(𝑘)
𝜕ℎ(𝑘)
𝜕𝑥2(𝑘)
=
𝜕ℎ(𝑘)
𝜕𝜃𝑝(𝑘)
𝜕𝜃𝑝(𝑘)
𝜕𝑥2(𝑘)
−
𝜕ℎ(𝑘)
𝜕𝜃𝑛(𝑘)
𝜕𝜃𝑛(𝑘)
𝜕𝑥2(𝑘)
𝜕ℎ(𝑘)
𝜕𝑥3(𝑘)
=
𝜕ℎ(𝑘)
𝜕𝜃𝑝(𝑘)
𝜕𝜃𝑝(𝑘)
𝜕𝑥3(𝑘)
−
𝜕ℎ(𝑘)
𝜕𝜃𝑛(𝑘)
𝜕𝜃𝑛(𝑘)
𝜕𝑥3(𝑘)
The value of each partial differential term calculated is shown as:
𝜕ℎ(𝑘)
𝜕𝜃𝑝(𝑘)
= 0.0556[1 − 𝑡𝑎𝑛ℎ2
(−14.555𝜃𝑝 + 8.609)](−14.55) − 0.0275(−0.492)(0.998 −
𝜃𝑝)
−1.492
(−1) − 0.157 exp(−0.047𝜃𝑝
8
)(−0.047 ∗ 8 ∗ 𝜃𝑝
7
) + 0.810exp(−40(𝜃𝑝 − 0.134))(−40)
;
𝜕𝜃𝑝(𝑘)
=
1
𝐶𝑠𝑝
𝑚𝑎𝑥
= [
𝜃100%𝑝 − 𝜃0%𝑝
(𝜃100%𝑛 − 𝜃0%𝑛)𝐶𝑠𝑛
𝑚𝑎𝑥] 𝐶𝑠𝑝
𝑚𝑎𝑥
𝜕𝑥1(𝑘)
=
−10395𝐷𝑠𝑛
2
𝑎𝑠𝑛𝐹𝑅𝑛
5

20
𝜕𝑥2(𝑘)
=
−252𝐷𝑠
3
𝜕𝑥3(𝑘)
=
−3
𝜕ℎ(𝑘)
𝜕𝜃𝑛(𝑘)
= 1.32 exp(−3𝜃𝑛) (−3) + 10 exp(−2000𝜃𝑛) (−2000)
𝜕𝜃𝑝(𝑘)
=
1
𝐶𝑠𝑛
𝑚𝑎𝑥
𝜕𝑥1(𝑘)
=
−10395𝐷𝑠𝑛
2
5
𝜕𝑥2(𝑘)
=
−1260𝐷𝑠𝑛
3
𝜕𝑥3(𝑘)
=
−21
𝐶𝑘 matrix varies at each time step k due to involvement of 𝐶𝑠𝑝,𝑠𝑢𝑟𝑓 unlike 𝐴𝑛
𝑘 and 𝐵𝑛
𝑘 which
are constant matrices.
Extended Kalman filter is used to estimate the corrected values of 𝐶𝑠𝑛,𝑠𝑢𝑟𝑓 and 𝑐̅𝑠𝑛 with the help
of actual TOV values, Kalman gain and corrected covariance matrix in each time step. The
Simulink block for estimation and the sequence of steps followed for EKF are shown below:
P1 = (A × P0 × A′) + Qw; % P1 is the next time step update for error covariance matrix
𝑥𝑏𝑎𝑟𝑛𝑒𝑤 = 𝐴 × 𝑥0 + 𝐵 × 𝐼𝐼𝑛𝑝𝑢𝑡; % time update for state prediction
𝑉𝑡 = ℎ(𝐶𝑠𝑛,𝑆𝑢𝑟𝑓, 𝐶𝑠𝑛
̅̅̅̅, 𝑖𝑎𝑝𝑝) ; % Output prediction from model
𝐿1 = 𝑃1 × Ck
′
× inv(Ck × 𝑃1 × Ck
′
+ Rw); % Kalman gain calculation
𝑥ℎ𝑎𝑡𝑛𝑒𝑤 = 𝑥𝑏𝑎𝑟𝑛𝑒𝑤 + (𝐿1 × (𝑉
𝑚𝑒𝑎𝑠 − 𝑉𝑡)); % measurement update for the state estimate
𝑃0_𝑛𝑒𝑤 = (𝑒𝑦𝑒(3) − (𝐿1 × 𝐶𝑘)) × 𝑃1; % measurement update for the state error covariance
Fig. 10: SIMULINK block
of EKF written for
estimating the new states and
SOC.

21
The Observability matrix 𝑂(𝑘)at any time step k is given as
𝑂(𝑘) = [
Ck
Ck × 𝐴𝑛
Ck × 𝐴𝑛
× 𝐴𝑛
]
The rank of 𝑂(𝑘) at every time step is evaluated and found out to be ‘THREE’ which is equal
to the number of state space variables considered. This implies that states are observable at
every time step.
Finally, SOC is calculated using the estimated 𝐶𝑠𝑛
̅̅̅̅ value from measurement update for the
state estimate step in EKF.
𝑆𝑂𝐶 =
𝑚𝑎𝑥
𝜃0%𝑛
(𝜃100%𝑛 − 𝜃0%𝑛)𝐶𝑠𝑛
𝑚𝑎𝑥
TOV results are shown below along with error values for only iROEM model and after using
EKF.
Pulse Cycle with correct initial concentration values
Below figures show the input pulsed current density profile, Terminal voltage comparison,
Error in mV and estimated SOC from EKF.
Fig. 11: Pulse current density profile used for SOC estimation

22
It can be evident that terminal voltage estimated using Extended Kalman filter tries to converge
to the actual value. This is because of keeping high w process noise and less v measurement
noise due to which estimated TOV follows Vmeasurement value reducing the error in
comparison from only the iROEM model.
Fig. 12: Estimated TOV comparison for pulsed current input
Fig. 13 Estimated TOV error for pulsed current input

23
Plot of estimated SOC value from corrected Csn value is shown below for Pulse cycle:
Pulse Cycle with wrong initial concentration values
To check the proper working and convergence of EKF model with actual TOV values, wrong
initial concentration (2x) values are entered in the Simulink model. Then TOV plot is shown
with actual values below. It can be seen that predicted TOV converges to actual value at around
1500 secs. With proper tuning of EKF parameters, we can improve or delay the convergence.
Fig. 14 Estimated SOC for pulsed current input
Fig. 15 Estimated TOV using wrong initial concentration for pulsed current input

24
DST Cycle with correct initial concentration values:
Below figures show the input DST current density profile, Terminal voltage comparison, Error
in mV and estimated SOC from EKF.
Fig. 16 DST current density profile used for SOC estimation
Fig. 17: Estimated TOV comparison for DST current input

25
It can be evident that terminal voltage estimated using Extended Kalman filter tries to converge
to the actual value. This is because of keeping high w process noise and less v measurement
noise due to which estimated TOV follows Vmeasurement value reducing the error in
comparison from only the iROEM model.
Plot of estimated SOC value from corrected Csn value is shown below for DST cycle:
DST Cycle with wrong initial concentration values
To check the proper working and convergence of EKF model with actual TOV values, wrong
initial concentration(2x) values are entered in the Simulink model. Then TOV plot is shown
with actual values below. It can be seen that predicted TOV converges to actual value at around
1000 secs. With proper tuning of EKF parameters, we can improve or delay the convergence.
Fig. 18 Estimated TOV error for DST current input
Fig. 19 Estimated SOC for DST current input

26
Equivalent Circuit Model considering Electro Chemical Properties
This section discusses the parameter identification method for equivalent circuit model (ECM)
considering the electrochemical properties with reference to [4]. In an equivalent circuit model
(ECM), resistances, capacitances and voltage sources are used to describe charging and
discharging processes of a Li ion battery and the model is built in frequency- or time-domain.
Based on the accuracy and computational requirements, a second order circuit model) is an
optimal choice due to its low computational complexity and high calculation accuracy. The RC
parameters can be identified via various methods like the nonlinear RLS method. This method
requires an initial guess of RC parameters for optimization. Traditionally, they can be obtained
from the relaxation part of the cell voltage curve after giving a pulse input. Large amount of
dataset is needed for estimating the RC values that can generate an accurate ECM.
Electrochemical model can provide better accuracy compared to ECM but it requires a lot of
computational power. By comparing a unified transfer function expression between cell
voltage and cell current of Electrochemical model and Equivalent circuit model, relationship
between RC parameters and Electro chemical parameters is obtained.
Transfer function of Electrochemical model
Based on the P2D model, the output voltage can be calculated using the open circuit voltage,
the overpotential, the electrolyte potential and the initial voltage drop. The cell voltage of the
electrochemical model is based on the block diagram shown in Fig. 21. Linearization and Pade
approximation is used to simplify the mass conservation and charge conservation expressions
into standard transfer function format. The cell voltage consists of the open circuit voltage part
and impedance part which represents the transient properties of the battery. The second order
transfer function expression of the voltage-current regardless of the open circuit voltage part is
given below.
Fig. 20 Estimated TOV using wrong initial concentration for DST current input

27
Transfer function of ECM
To ensure the comparability between the electrochemical model and the equivalent circuit
model, second order Thevenin model is chosen as shown in the Fig —-. The cell voltage V and
the transient voltage Ud2 can be expressed as
Parameters identification of ECM
By equating the transfer functions of the electrochemical model and equivalent circuit model,
the relationship between the RC parameters can be established
Fig. 21 Block diagram of second order equivalent circuit RC model

28
Simulation results
Using the electrochemical model parameters of LMO battery, the values of RC parameters can
be obtained as
R0 = 6.7 mΩ
Rp = R1 = 0.253 mΩ
Rn = R2 = 0.28 mΩ
Cp = C1 = 621.99 KF
Cn = C2 = 527.65 KF
Similarly, by using the current-voltage data from the reference paper [---] and by using the
nonlinear least squares method, second RC parameter values can be obtained. Figure – shows
the input data and curve fitting of nonlinear RLS method
R0 = 2.9 mΩ
R1 = 2.8 mΩ
R2 = 1.8 mΩ
Fig. 22 Input data for nonlinear RLS method
Fig. 23 Final voltage fit for double RC using RLS method

29
C1 = 6.4 KF
C2 = 88.6 KF
By considering that the relationship between the SOC and OCV is linear, the state space system
of ECM for output voltage can be given by
𝑉𝑡 = 𝑉
𝑜𝑐𝑣 − 𝐼𝑅0 − 𝑉1 − 𝑉2
𝑉1
̇ =
−𝑉1
𝑅1𝐶1
+
𝐼
𝐶1
𝑉2
̇ =
−𝑉2
𝑅2𝐶2
+
𝐼
𝐶2
𝑍̇ =
𝐼
𝑄
[
𝑍̇
𝑉1
̇
𝑉2
̇
] =
[
0 0 0
0
−1
𝑅1𝐶1
0
0 0
−1
𝑅2𝐶2]
[
𝑍
𝑉1
𝑉2
] +
[
−1
𝑄
1
𝐶1
1
𝐶2 ]
𝐼
𝑦 = 𝑉𝑡 − 𝛽 = [𝛼 −1 −1] [
𝑍
𝑉1
𝑉2
] + [𝑅0]𝐼
SOC - OCV relationship graph is take reference from [2]
alpha = 0.0045;
beta = 3.675;
The figure 25 shows terminal voltage of the second order RC model with RC parameters
obtained from test data and nonlinear least squares method, and electrochemical model
parameters data
Fig. 24 SOC-OCV relationship for LMO battery

30
Despite the RC values being completely different initially, it can be seen that the terminal
voltage of both cases is comparable. Model obtained from electrochemical data is not able to
provide a good approximation when the current profile changes direction. This can be because
properties like electrolyte conductivity, which vary with electrolyte concentration are
approximated as constant values in the calculation of RC parameters. This emphasises the
importance of having correct electrochemical model data to approximate its second order
equivalent circuit model. To summarize, this approach can be used to estimate second order
RC parameter values for an equivalent circuit model when the measured data is not available.
Furthermore, comparing these results from the electrochemical model and actual data from
reference paper, it was observed that there is a significant difference in terminal voltage
between the equivalent circuit model and the electrochemical model. It can be due to the
difference in voltage at 100% SOC between the reference [2] and [1]
Fig. 25 Terminal voltage of 2nd
RC model using electrochemical data Fig. 26 Comparison of Terminal voltage of 2nd
RC model using
electrochemical data with electrochemical model

31
Conclusion
1. Compared quadratic polynomial and Pade approximation methods for solid diffusion
dynamics. 3rd
order Pade approximation provided best results for terminal voltage (TOV).
2. Adding electrolyte dynamics improved the terminal voltage (TOV) calculation at high
charge/discharge rate and successfully validated our results with the reference paper.
3. Performed observability analysis with Pade approximation method for negative electrode
solid dynamics
4. The rank of O(k) at every time step is evaluated and found out to be ‘THREE’ which is
equal to the number of state space variables considered. This implies that states are
observable at every time step.
5. Estimated Terminal voltage using Extended Kalman Filter and the estimates are found to
be converging with the actual values.
6. Estimated state of charge (SOC) using Extended Kalman Filter (EKF) with iROEM.
7. Using incorrect initial values, terminal voltage estimation from EKF slowly converges to
the actual value and can be improved by changing the noise parameters.
8. Using electrochemical properties, estimated RC pair values for equivalent circuit model
(ECM), which can be used as an initial non-linear least square regression (LSR) method.
Limitations
1. The iROEM does not include thermal behavior of the battery and needs to be incorporated
for online BMS applications.
2. Calculated RC values from ECM are dependent on electrochemical model parameters. Few
parameters are not time invariant. This affects the accuracy of ECM.
Contributions
Sr.
No.
Name Contribution
1 Nipun
 Pade approximation method for negative electrode and building
Simulink model
 Terminal voltage and SOC estimation using EKF and also developing
Simulink model
2 Ratnesh
 Developing Equivalent Circuit model, Equivalent circuit model with
electrochemical parameters and comparing with the iROEM
Table 2: Table of contributions

32
3 Satya
 Quadratic Parabolic Polynomial method for positive electrode and
building Simulink model
 Terminal voltage results from iROEM and also developing Simulink
model
4 Varma
 Solving electrolyte diffusion equations for developing the iROEM and
subsequent simulink development for electrolyte potential and
integration for TOV. Support in EKF observability matrix preparation
 Literature review, Data gathering - LMO electrochemical parameters,
measurement data generation for SOC estimation using EKF
Everyone has made an equal contribution towards making the presentation and report.
References
1. Longxing Wu, Kai Liu and Hui Pang, "Evaluation And Observability Analysis Of An
Improved Reduced-Order Electrochemical Model For Lithium-Ion
Battery", Electrochimica Acta 368 (2021): 137604, doi:10.1016/j.electacta.2020.137604.
2. Gao J, Zhang Y, He H. A Real-Time Joint Estimator for Model Parameters and State of
Charge of Lithium-Ion Batteries in Electric Vehicles. Energies. 2015; 8(8):8594-8612.
https://doi.org/10.3390/en8088594
3. V. Senthil Kumar , Reduced order model for a lithium-ion cell with uniform reaction rate
approximation, J. Power Sources 222 (2013) 426–441 .
4. Zhang, Xi & Lu, Jinling & Yuan, Shifei & Yang, Jun & Zhou, Xuan. (2017). A novel
method for identification of lithium-ion battery equivalent circuit model parameters
considering electrochemical properties. Journal of Power Sources. 345. 21-29.
10.1016/j.jpowsour.2017.01.126.
5. Yinyin Zhao and Song-Yul Choe, "A Highly Efficient Reduced Order Electrochemical
Model For A Large Format Limn2o4/Carbon Polymer Battery For Real Time
Applications", Electrochimica Acta 164 (2015): 97-107,
doi:10.1016/j.electacta.2015.02.182.
6. Cai, Long & White, Ralph. (2009). Reduction of Model Order Based on Proper Orthogonal
Decomposition for Lithium-Ion Battery Simulations. Journal of The Electrochemical
Society - J ELECTROCHEM SOC. 156. 10.1149/1.3049347.

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