## 1. Introduction

The creation of battery models is essential for better understanding of the impact of design variables as well as operating conditions on battery performance, in terms of efficiency, degradation and safety. They are also cost effective tools for determination of an optimised battery design that can reduce the need for experimental trial and error. Beyond that, most variables that underpin the operation of a battery are not directly measurable, whereas they can be predicted through a validated model [

1].

Battery models can be broadly categorised into four groups, namely: empirical models (equivalent circuit and neural network), electrochemical engineering models, multiphysics models, and molecular/atomistic models [

1,

2]. These battery models include different levels of details and differ in terms of complexity, computational cost and reliability. They can be chosen based on the particular needs for a specific application. Due to the complexity of batteries, electrochemical models are often viewed as the best approach for investigating the impact of design variables on battery performance during a charge and discharge process.

The most common electrochemical models in the literature [

3] are single particle models (SPM) [

4,

5] porous-electrode models [

6,

7] and pseudo-two-dimensional models (P2D) [

8,

9]. The main difference is the level of complexity and the computational time associated with their use. Once an efficient model is developed, it can be employed to address a number of real-world challenges, such as the identification of transport and kinetic parameters, the occurrence of capacity fade, improving life time, and improving energy/power density [

1]. The energy/power density can be improved by manipulating either the design parameters or operating protocols [

1]. Much of the previous work on improving the performance of batteries has focused on battery packs rather than a single battery cell design [

10]. The important design parameters which can be varied within the manufacturing process to achieve the optimal battery performance are known to be: electrode thickness [

11,

12,

13,

14], porosity [

6,

10,

11,

12,

15], particle size [

14,

16], electrode surface area, geometry and the dimensions of current collectors [

6,

17,

18].

Given the importance of battery design, Newman et al. and co-workers [

11,

12] developed an analytical model of a lithium ion battery to optimise porosity and thickness of the positive electrode for maximum specific energy, while holding other parameters constant. In another study, Singh et al. [

13] investigated experimentally the amount of energy that may be extracted from a cell manufactured using thick electrodes (320 μm) compared to cells that employ a thinner electrode (70 μm) for a graphite/lithium nickel manganese cobalt oxide (Gr/NMC) chemistry. They observed a significant capacity loss when using the thicker cells at C-rates of C/2 due to poor kinetics. The authors suggest that the proposed thick electrodes could be advantageous for certain applications where a continuous low C-rate is required. Research presented by Wu et al. [

14], employed an electrochemical-thermal model of a lithium ion battery for a Li

_{y}Mn

_{2}O

_{4} chemistry. The model was used to investigate the impact of particle size and electrode thickness on heat generation and the performance of the battery. They found that a battery containing a thin electrode showed a better performance in terms of temperature rise and material utilisation, but the effect of particle size was not monotonic across the discharge rates. In a recent study published by Ramadesigan et al. [

6] a multi-layered porosity distribution was investigated rather than optimising a uniform porosity in a lithium ion battery design. They developed a simple electrochemical porous electrode model, which did not include the solid-phase intercalation mechanism. The model was applied for a cathode made of lithium cobalt oxide. For a fixed value of active material, optimal multi-layered porosity distribution across the positive electrode was found. The authors managed to decrease the ohmic resistance by circa: 15–33%, employing the optimal porosity distribution. Golmon et al. [

15] applied a gradient-based optimisation method on a multi-scale battery model to maximise the usable capacity. The electrochemical-mechanical multi-scale model was an extension of Doyle and Newman’s electrochemical battery model [

19,

20]. The variables were particle size and porosity, with constraints placed on the stress levels in the cathode particles. Other examples can be seen in [

16,

18]. Darling et al. [

16], developed a 1D theoretical model to investigate the influence of different particle size distributions on the operation of porous intercalation electrodes. Chen et al. [

18] developed a technique to enhance the achieved capacity of Li-based batteries through improvements in both electronic and ionic conductivity of materials.

Parameter sensitivity study is another area of research performed in modelling approaches for finding the influential design variables which could potentially improve battery performance [

14,

21,

22,

23,

24,

25]. Zhang et al. [

21,

23] developed a coupled P2D electrochemical-thermal model for a 2.3 Ah cylindrical Li-ion battery by improving the open source FORTRAN code maintained by Newman’s research group. They established a parameter sensitivity matrix and used clustering theory to group the parameters according to their average sensitivity. They performed a sensitivity study of up to 30 parameters under different operating conditions. Out of the 30 parameters, 10 were found to be highly sensitive, seven of those were sensitive, along with 10 low sensitivity and three insensitive parameters. The most sensitive parameters were found to be anode particle radius, diffusion coefficient in the negative electrode, stoichiometry of the anode, volume fraction of the electrolyte and active material in the negative electrode, contact resistance, reaction rate of the negative and positive electrode as well as the activation energy of the electrolyte ionic conductivity. Moreover, it was found that the sensitivity of the parameters is strongly dependent on the operating conditions. Edouard et al. [

22] developed a single particle (SP) electrochemical-thermal model applying the pseudo-2D mathematical structure. The model was applied to evaluate the sensitivity of the key parameters that are involved in battery aging. They pointed out that the limiting mechanism in a battery can switch depending on the operating conditions and battery design [

14]. Moreover, the combined effect of design parameters can be even more significant than their individual effect. This presents a great challenge for experimentally optimising a battery design. Du et al. [

24] introduced a surrogate modelling framework to map the effect of design parameters, such as cathode particle size, diffusion coefficient and electrical conductivity on battery performance. They quantified the relative impact of various parameters through global sensitivity analysis employing a cell-level model in conjunction with tools such as kriging, polynomial response, and radial-basis neural networks. Ghaznavi et al. [

25] applied a mathematical approach to conduct a sensitivity study on a lithium-sulfur cell. They focused on the effects of discharge current and conductivity of the positive electrode over a wide range of values.

To date, much progress has been made in modelling and design optimisation of lithium ion batteries to map the trade-off between power and energy density. However, in most studies only a few design variables have been selected and the optimisation is limited to a narrow range of operating conditions. Moreover, the interaction effect of influential parameters has not been comprehensively studied. Therefore, there exists a critical need to stablish a framework to access both the individual and the interaction effect of various parameters on the energy and power of a cell. This study attempts to quantify the strength of design factors and the combined effect of variables on specific energy as well as specific power of a battery.

Within this paper, a 1D electrochemical-thermal model of an electrode pair of a lithium ion battery is developed in Comsol Multiphysics. Each pair is assumed to be a sandwiched model of different layers, a negative current collector, a negative electrode, a separator, a positive electrode and a positive current collector. The anode is made of graphite and the cathode material is lithium phosphate (LFP). The mathematical model is validated against the literature data for a 10 Ah LFP pouch cell operating under 1 C to 5 C electrical load at 25 °C ambient temperature. It is a commercial cell and can be employed for vehicle applications as it is a large format cell. The validated model is used to conduct statistical analysis of the most influential parameters that dictate cell performance, i.e., particle size (${r}_{p}$), electrode thickness (${L}_{pos}$), volume fraction of the active material (${\epsilon}_{s,pos}$) and C-rate, and their interaction on the two main responses, namely; specific energy and specific power. This is to achieve an optimised window for energy and power within the defined range of design variables. The design factors are chosen in a way that they can be varied during the manufacturing process of a cell, in order to make the developed statistical model more applicable for industry.

In

Section 2, the mathematical modelling approach is explained and contains the derivation of the electrochemical-thermal model along with the statistical analysis.

Section 3 presents the model validation accompanied by simulation results of analysis of variance (ANOVA), which elaborates the main and combined effect of the factors as well as the optimised design of the cell. Further work and conclusions are discussed in

Section 4 and

Section 5, respectively.

## 4. Further Work

In this paper, we employed a 3-level full factorial design on the numerical results obtained from an electrochemical-thermal model. The influence of four design variables were investigated to achieve the optimal responses (here specific energy and power). However, the developed model is not limited to the defined design variables and responses. By changing the factors, or adding new design variables to the existing model, a new set of simulation studies can be run and used for further analysis. As an example, it would be of interest to investigate whether the optimal design for achieving the highest energy and power is influenced by the operating temperature of the cell. Another area of interest is to investigate the impact of particle size as well as porosity distribution on the obtained energy and power.

Moreover, battery degradation rate is another interesting system response to be considered. However, studying the degradation process, needs improvements to the current model by adding the impact of solid-electrolyte interface (SEI) growth and its effect on the capacity fade within the battery.

Finally, as mentioned earlier in this study, the optimised design of the cell was achieved through analysing 81 test cases. Given this number, it seems quite unlikely to get all the required data through experiments. On the other hand, to make a comparable test case, all of the materials (including the separator and electrolyte) must be similar to those of the commercial cell. Furthermore, the production procedure for the cell itself has to be similar. Understanding such detailed design information for a commercial cell is known to be difficult because of issues of confidentiality. However, manufacturing a new proto-type cell (manufactured by the University) with optimised design accompanied by an experimental evaluation is another step to be considered to ultimately validate the proposed optimised cell design and simulation framework.

## 5. Conclusions

In this study a 1D electrochemical-thermal model of an electrode pair of a lithium ion battery is developed in Comsol Multiphysics. The mathematical model is validated against the literature data for a 10 Ah LFP pouch cell operating under 1 C to 5 C electrical load at 25 °C ambient temperature. The validated model is used to conduct statistical analysis of the most influential parameters that dictate cell performance, i.e., particle size (${r}_{p})$, electrode thickness (${L}_{pos}$), volume fraction of the active material (${\epsilon}_{s,pos})$ and C-rate, and their interaction on the two main responses, namely; specific energy and specific power. This is to achieve an optimised window for energy and power within the defined range of design variables. The range of variation of the design variables for LFP lithium ion battery is determined based on data from literature (${r}_{p}$: 30–100 $\mathrm{nm}$, ${L}_{pos}$: 20–100 $\mathsf{\mu}\mathrm{m}$, ${\epsilon}_{s,pos}$: 0.3–0.7, C-rate: 1–5). A statistical model is developed by ANOVA of the numerical data in a full factorial design frame work. A full factorial design methodology is carried out to analyse the obtained results of the 1D electrochemical-thermal model and to determine the optimum energy and power by manipulating key design variables of the positive electrode. The summary of the statistical results are as follows:

The significant factors for the specific energy are ranked as:

Similarly, for the specific power it is defined as:

In conclusion, the main effect and the interaction effect of all design variables on the energy and power, it is observed that the optimum energy can be achieved when (${r}_{p}<40$ $\mathrm{nm}$), ($75\mathsf{\mu}\mathrm{m}{L}_{pos}100\mathsf{\mu}\mathrm{m}$), ($0.4<{\epsilon}_{s,pos}<0.6$) and while the C-rate is below 4 C. The optimum power is achieved for a thin electrode (${L}_{pos}<30\mathsf{\mu}\mathrm{m}$), with high porosity and high C-rate (5 C). It is clear that the optimum energy and power cannot be achieved at the same time, hence the battery should be designed so that the power to energy ratio for a specific application is satisfactory. Finally, it should be mentioned that the developed model is not limited to the defined design variables and the responses. By changing the factors, or adding new design variables to the existing model, a new set of simulation can be run and used for further analysis.