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## Lithium-ion battery modelling

The main purpose of modeling is to develop a mathematical representation able to simulate a system behavior. In lithium ion batteries, many complex phenomena are involved such as: the mass transport, migrations of ions, red-ox reactions (i.e. transformations of the chemical compounds when they react with electrodes) and 14 side reactions. The current collectors can be neglected since their conductivity is orders of magnitude higher than the values of active materials or electrolytes as reported in § 3.5. Thus, the potential drop in the collectors may be reasonably neglected. For these reasons, any proposed model can partially describe its behavior. In recent years, many models are developed for different purposes [31], [32], such as the integration of a battery in a more complex electrical system (e.g. a EV powertrain) or to focus into the internal physics (e.g. mass transport and chemical reactions). Some niche models are based on stochastics, artificial neural networks or the fuzzy logic [33]–[36]. These approaches are not based on the physics of the system, but they still can reproduce its behavior. Thus, it is possible to divide all these models in two families based on: the phenomenology or the physics of the system. The phenomenological models can reproduce the battery behavior (i.e. the voltage drop under an external load) after a test campaign aiming to characterize electrically the cell performances in several operating conditions (e.g. temperature, state of charge, degradation, etc.). These models includes, as an example, empirical equations (Shepherd in 1965 [37]) and equivalent electric circuits. Instead, the electrochemical model describes the physics of the involved phenomena, such as (the list is not exhaustive): diffusion of ions, mechanical strain, charge transfer and migrations of ions.

However, some hybrid models containing elements of both families can also be found, as an example: the model developed by Rakhmatov & Vrudhula 2001 [38] contains the diffusion of lithium in the solid phase and the empirical Peukert’s law, or the transmission line model that uses electric lumped elements to simulate the porous electrodes [39], [40].

Today, when real time computations are required (e.g. in the battery management systems, BMS), the simple approach with the equivalent electrical circuit is usually preferred [41], [42]. In fact, in EVs is important to know, the battery state of charge, power fade, capacity fade, and instantaneous available power, that are used by the battery management systems (BMS) to estimate the present operating condition of the battery packs. This is achieved by adding the control theory to an equivalent electrical circuit. The equivalent electrical circuit is composed of several lumped circuital elements (e.g. voltage generators, resistors, capacitors). The values of this elements are estimated with experiments and electrical tests [43], [44]. However, these components are usually functions of the state-of-charge (SOC), state of health (SOH) and temperature[45]–[48]. It should be mentioned that many researches are implementing simplified electrochemical models (e.g. single particle or linearizing the charge transfer relationship, cf. § 1.4) with a control systems in the BMS [49]–[52]. The transmission line model (TLM), uses circuital elements disposed in the configuration illustrated in Figure 3 [39], [53]–[57]. In this schematics, the resistances are attributed to electrode and electrolyte conductivity, while the lithium diffusion in the electrodes and the charge transfer kinetics are represented by non-constant and frequency dependent impedances [58]. Like in phenomenological models, these elements are influenced by the battery state of charge, the temperature and the state of degradation.

Figure 3 – The transmission line model (TLM) is a representation of the porous electrode where horizontal resistors represent the ohmic resistance in the solid phase and liquid phase while the

impedances 1, 2, , … , +1, −1, represent the kinetic resistance and the diffusion in the solid phase [59]. The arrows indicate where the electrons and ions circulate.

While the phenomenological based models are used to characterize existing cells, the electrochemical models could be used to design new cells [15], [60], [61]. In the electrochemical models, the mass transport and reaction kinetic are described in a system of partial differential equations. The objective is to predict the internal variables such as the lithium concentrations and the electrical potentials. At least one spatial dimension is required, such as the cell cross section, as indicated in Figure 5. The physical and mathematical background was established by J. Newman and its research group at Berkeley University in the ’90 [62], [63]. The mathematical system of equations is available in several books such as: Advances in Lithium-Ion Batteries, 2002 [16]. The original formulation is based on a pseudo two-dimensional geometry, where one dimension is used for the lithium diffusion inside the active material and the other for the transport of charge species in the cell cross section. In the last decades, many variants of that model have been produced, extending the equations system to 3D (three dimensional) geometries to simulation the thermal fluxes, as an example [64].

More complex and simpler models than the Newman’s P2D are the so-called multi-scales and the single particle, respectively. The multi-scales model introduce several spatial and time dimensions where different physical phenomena acts [32], [65], [66]. On the other side, the single particle represents the porous electrode with a bulky spherical particle [67].

While research group of J. Newman created a free software in Fortran called Dualfoil, many commercial tools are now proposed. Among them we can mention the Batteries & Fuel cells package for Comsol Multiphysics®, Battery Design Studio® developed by CD-Adapco and Fire® from AVL [9]. The large number of parameters that is required in the original model is maintained and even increased. A critical review on the models is proposed in § 2.2.

In summary, in the last years, most of the effort from scientist and engineers, for the electrochemical models was to: (i) increase the complexity to describe more phenomena, (ii) simplify the physics to reduce the calculation time for real time applications. Consequently, we decided to work to simplify the Newman’s model, whitout any changes in its electrochemical foundamentals.

### Review on electrochemical models

In recent years, many works dealt with variants and improvements of the P2D original model from J. Newman and his co-workers [15], [61], [68]–[77]. The initial set of equations is extended by including thermal effects by Bernardi et al. 1985 [78], [79], the ageing by Darling et al. 1998 [80], [81] or the mechanical deformations/swelling by Christensen et al. 2006 [82], [83].

A synthetic review of the recent advances on electrochemical models representing the state of the art is reported:

o Barai 2015 [84]: effect of the particles sizes on the mechanical degradation of the active material in the negative electrode;

o Miranda 2015 [85]: effects of thickness, porosity and tortuosity of the separator membrane to battery performances;

o Suthar 2015 [86]: mechanical stress induced by the lithium intercalation considers the capacity fade for different values of porosity, SEI growth, lithium plating;

o Zhao 2015 [87]: P2D electrochemical model is coupled with a double layer capacitance and a 3D thermal model;

o Ecker 2015 [88] : simulations are based on the measurement of the parameters in prismatic cells made by Kokam® ;

o Cobb 2014 [89]: effect of porosity and tortuosity;

o Chandrasekaran 2014 [90], [91]: performance of a graphite-NMC cell and lithium plating induced by fast galvanostatic charge;

o Kim 2014 [92]: The performance of prismatic GS-Yuasa LEV50 50-Ah NMC with a 3D thermal model;

o Zhang 2014 [93]: degradation of a graphite/LCO during the cycling at high temperature;

o Legrand 2014 [94]: How to maximize the charging rate and avoiding the lithium plating;

o Legrand 2014 [95]: proposed electrochemical model includes the double layer capacitance;

o Sikha et al. 2014 [96]: original model uses a 2D geometry including the strain and stress effects to study a nanowire electrode;

o Cobb 2014 [97]: evaluation of performances for a 3D printed electrode; o Mao 2014 [98]: simulations of short circuits in a MCMB/ LCO cell;

o Ferrese 2014 [99], [100]: The PDE equations system for Li-Metal/LCO cells is solved with COMSOL to investigate the concentration of lithium along the negative electrode/separator interface during the cell cycling.

o Northrop 2014 [101]: computational time efficiency during the simulations;

o Fu 2013 [102]: mechanical stresses and heat fluxes for pouch C/LMO cells are studied during cycling;

o Guo 2013 [103], [104]: The electrochemical model and the thermal model are solved in a decoupled equation system to study a battery module;

o Ji 2013A [105]: A model containing a double layer capacitance is used to investigate different heating strategies for sub-zero temperatures to predict Lithium-plating;

o Ji 2013B [106]: A thermal-electrochemical model implemented in a commercial software is used to validate the discharge rates of 18650 type cells for temperatures ranging from −20°C to 45°C;

o Christensen 2013 [107]: performances of 18650 cells cooled by natural and forced convection are studied;

o Lin 2013[108]: A degradation model including SEI growth, manganese dissolution and electrolyte decomposition for a C/LMO cell is developed;

o Awarke 2013 [81]: A P3D-thermal model including the SEI growth for a 40 Ah Li-ion during cycling of pouch cells;

o Reimers 2013 [109]: electrochemical model PDE equations system is proposed in a decoupled form;

o Zavalis 2012 [110]: The short-circuits are investigated in a prismatic cell having C/NCA electrodes;

o Less 2012 [111]: The study is focused on the correlation between the geometrical scales of the structure, the material anisotropic properties, and the geometrical morphology of the electrodes compared with the macroscopic battery performances;

o Ferrese 2012 [112]: The growth of dendrites in a 2D geometry of Li-Metal/LCO; o Chandrasekaran 2011[113]: The performances of cell having Li-Metal and a blend of graphite and silicon as negative and positive electrode, respectively; o Jannesari 2011 [114]: The effect of SEI thickness variation across electrodes depth;

o Martínez-Rosas 2011 [72]: An equation system with dimensionless spatial coordinates and algebraic approximations is developed;

o Christensen 2010 [115] : diffusion equation induced by mechanical stress is introduced in Dualfoil®;

o Stephenson 2007 [116]: The study is focused on the transfer of electrons between particles having different: sizes, materials and contact resistances of carbon additive;

o Stewart 2008 [117]: The different results associated to the salt activity between the concentrated solution and dilute solutions are compared;

o Nyman 2010 [73]: A The current load profile “EUCAR” is simulated for NCA cells;

The new electrochemical models are likely to consider: thermal balance, side reactions and solid mechanics. In additions, as the computational power increases, the dimensions of the geometry increase from 1D to 3D or considering a real electrode scanned with the tomography. Consequently, the number of parameters required are increasing but many of them are still not accurately measured. In fact, most these studies and models uses the fitting over many parameters. However, as the number parameters increases if they are poorly measured, they can be compensated with additional terms obtained from the fitting. In conclusion, it is not guaranteed that the fitted parameters are physically consistent. This reflect the conclusions previously discussed by Hemery 2013 in his Ph.D. manuscript [118]. Thus, as the mathematical complexity increases the battery voltage is simulated more precisely but is not assured that the other variables are correct such as the gradients of concentrations [119]. The aim of this work is to simplify as much as possible the Newman’s model without losing its physical nature in order to understand meaning of each parameter and their role in the overall system.

**Tests and simulations**

In the literature, the majority of real EVs driving cycles are simulated with equivalent electrical circuits or simplified physical models (i.e. ODE instead of PDE) [49], [120]. In fact, a real driving profile contains charges (during the regenerative brake) and pulse-rest periods (acceleration). Thus, the thermal fluxes generated can be consistent and the values of the parameters used in the Newman’s model are not constant. For this reason, more than 25 parameters must be measured for each cell. Another difficulty is to deal with a non-accurate estimation of the state of charge (SOC). In fact, the most common estimator is based on the coulomb counting method (affected by measurement errors due to current sensor and integrations errors) that are amplified when it is applied to dynamic current profiles.

However, the diffusion of Newman’s based models are rapidly rising in recent years, but they still represent a minority [73], [121]–[123].

In addition, only few studies are focused on galvanostatic charges or pulse-rest periods [80], [124]. In fact, the constant voltage charge phase is difficult to simulate because it requires different boundary conditions [125], or as an alternative, a complex control feedback able to deal with current while it maintains the voltage constant could be used but it complexifies the study[63].

In conclusion, most studies are based on galvanostatic discharges, as example reported in Figure 4 (A), where different discharge rates are simulated at 273 K and compared with experiments for a graphite/NCA cell. The figure illustrates that the higher is the C-rate, the higher the errors are over the voltage while at 0.1C the rated capacity deviates from the experiment. In fact, at high current rate and low temperature the kinetic limitations are very important and the simulations are more complex (Doyle et al. 1996 [126]).

Figure 4 – (A) The discharge voltages as function of the specific capacity are measured (dashed curves) and simulated (solid curves) 273 K for several C-rates [127]: 0.01C, 0.1C, 0.5C and 1 C. (B) The voltage during the GITT as a function of the test time (in hours) is are measured (dashed line) and simulated (solid line) [80].

The pulse rest sequences are studied by Darling et al. 1998 using the Newman’s model [80], for a Li-metal/LMO cell. The simulations and experiments reported in Figure 4 (B) are performed with four C/2-rate pulses followed by 1 hour of rest period. The deviations of the simulated voltages from the experiments, evidences the complexity in the battery modeling. In fact, large voltage errors after the relaxation are found even in recent papers [88]. These voltage mismatches after the relaxation could be attributed to a non-accurate estimation of time constants of the model.

In the next chapter, the literature is critically reviewed to identify some non-physical features or some methodologies that are in contrast with the Newman’s P2D model.

#### Electrochemical equations system

In this chapter, the Newman’s model is introduced, and the related literature is critically reviewed. The main features contrasting with the original model and rickety assumptions on the involved electrochemical phenomena are discussed. Then, a dimensionless PDE model is proposed aiming to reduce as much as possible the number of parameters. Finally, the dimensional and the dimensionless models are compared.

The basis for modeling the porous electrodes has been reviewed by Newman and Tiedemann in 1975 [128]. The porous electrodes are represented as a superposition of two macro-homogenous and continuous phases that coexist in every point of the cell. These phases are either solid or liquid. Furthermore, the solid phase is represented with spheres, attributed to particles of the active materials.

**Table of contents :**

**1 The state of the art on electrochemical modelling for Lithium-ion batteries **

1.1. Framework and objectives

1.2. Lithium-ion working principles

1.3. Lithium-ion battery modelling

1.4. Review on electrochemical models

1.5. Tests and simulations

**2 Electrochemical equations system **

2.1. Newman’s PDE equations system (diluted solutions)

2.2. Critical review of the literature

2.3. The system of equations proposed in this study

2.3.1 Comparison with Newman

2.3.2 Non-dimensional PDE equations system (positive and negative porous electrodes) 39

2.3.3 Non-dimensional PDE equations system for the Li-metal foil negative electrode

**3 Analysis of the parameters from literature **

3.1. Electrolyte conductivity

3.2. Electrolyte diffusivity

3.3. Transport number

3.4. Solid phase diffusivity

3.5. Solid phase conductivity

3.6. Kinetic reaction rate constant

3.7. Dimensional design parameters

3.8. Dimensionless parameters

3.9. The ageing effect on parameters

**4 Electrical and physicochemical characterizations **

4.1. Electrical characterization

4.1.1 Reproducibility analysis of a test protocol for galvanostatic discharges

4.1.2 Voltage dip during galvanostatic discharges

4.1.3 Galvanostatic discharge to 0.05 V

4.2. LGC INR18650MH1 chemical characterization

**5 Electrode balancing **

5.1. Introduction to electrode balancing

5.2. Introduction to isotherms and the states of lithiation in either complete cell and “half-cell” configurations

5.3. How the shape of the isotherms influences the accuracy on the initial states of lithiation 97

5.4. Identification of the state of lithiation in LGC INR18650MH1 half cell

5.5. The aging scenarios

**6 Numerical simulations with COMSOL **

6.1. Galvanostatic discharges

6.1.1 Kinetic redox limitation

6.1.2 Electrolyte mass transport

6.1.3 Electronic transport

6.1.4 Solid phase diffusivity

6.1.5 Mixed case: solid phase diffusivity and electronic transport

6.1.6 Mixed case: solid phase diffusivity and electrolyte diffusivity

6.1.7 Mixed case: electronic transport and electrolyte mass transport diffusivity

6.1.8 Conclusions and perspectives

6.2. Pulse-rest sequences

6.2.1 Introduction

6.2.2 Time constants

6.2.3 GITT

6.2.4 First steps to an appropriate interpretative framework of GITT

**7 Conclusions and perspectives **

**8 Appendix**

8.1. Mesoscopic 1D porous electrode model

8.2. Estimation of the state of charge at 4.3 V

8.3. Differential voltage

8.4. PDE equations system in COMSOL® (half-cell)

8.5. PDE equations system in COMSOL® (full-cell)

8.6. Simulation of the LGC INR18650MH1 with COMSOL®

8.7. C-rate profile used in galvanostatic discharges in § 4.1.2

8.8. C-rate profile used in galvanostatic discharges in § 4.1.3

**References**

Résumé (long)

Abstract (grand public)

Résumé (grand public)

Acknowledgments