Electrochemical-Thermal Modelling and Optimisation of Lithium Ion Battery Design Parameters Using Analysis of Variance (ANOVA)
Authors: Hosseinzadeh Elham , Marco James , Jennings Paul 
Published: February 2020

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 1C to 5C 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 (rp), electrode thickness (Lpos), volume fraction of the active material (εs,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. The range of variation of the design variables for LFP lithium ion battery is determined based on the data from literature which is, namely: rp:30-100 nm, Lpos:20-100 μm, εs,pos:0.3-0.7, C-rate:1-5.
A statistical model is developed by analysis of variance (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 number of required simulations in a 3 level full factorial design for four design variables is equal to 34, i.e. 81 numerical case studies.
The statistical results from the study show that:
At εs,pos=0.3, by increasing Lpos, higher energy can be achieved, whereas the energy doesn’t follow the same pattern at higher εs,pos values. For (εs,pos>0.5), increasing both εs,pos and Lpos at the same time leads to energy drop and the reduction rate is higher for higher active volume fraction. rp has a negative impact while Lpos has a positive impact on the energy. But their interaction does not have a monotonic effect. At small particle size, rp=30 nm, increasing the electrode thickness from 60 µm to 100 µm reduces the energy. However, for the larger particles even though the energy doesn’t drop, but the achieved improvement is quite low. At higher C-rates the energy is more sensitive to the particle size. Meaning that for high C-rate application small particles are more desirable while at lower C-rates there is a bigger window for the optimum rp.
Lpos has a negative effect on the power, whereas C-rate has a positive effect. Their interaction shows that the thickness of the electrode becomes a more influential factor as the C-rate increases, whereas at 1C the thickness doesn’t have a significant impact on the power. The behaviour of the power to the interaction of εs,pos and C-rate is not monotonic. For example by varying εs,pos from 0.3 to 0.5 the power doesn’t change regardless of the C-rate. But their combined effect will be more pronounced at high active volume fraction when the C-rate is high.
In conclusion, considering 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 (rp<35 nm), (55 µm<Lpos<85 µm), (0.55<εs,pos<0.7) and while the C-rate is low, (1C). The optimum power is achieved for thin electrode (Lpos<30 μm), with high porosity (0.3<εs,pos<0.43) and high C-rate (5C). The particle size becomes important only if (0.3<εs,pos<0.43). In that case the optimum particle size is (rp<40 nm). 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 should be run and be used for further analysis.

Figure 1