DOI: 10.3303/CET23103148 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 09 June 2023; Revised: 27 June 2023; Accepted: 11 July 2023 
Please cite this article as: Angco J.E.T., Rito G.R.R., Domalanta M.R., Paraggua J.A.D., 2023, Multiphysics Modelling, Parameter Sensitivity 
Analysis, and Optimization of a Lithium Polymer Battery, Chemical Engineering Transactions, 103, 883-888  DOI:10.3303/CET23103148 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 103, 2023 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Panos Seferlis, Yee Van Fan, Athanasios Papadopoulos 

Copyright © 2023, AIDIC Servizi S.r.l. 

ISBN 979-12-81206-02-1; ISSN 2283-9216 

Multiphysics Modelling, Parameter Sensitivity Analysis, and 

Optimization of a Lithium Polymer Battery 

Jan Elmo T. Angco, Gio Roman R. Rito, Marcel Roy Domalanta, Julie Anne D.R. 

Paraggua* 

Laboratory of Electrochemical Engineering (LEE), Department of Chemical Engineering, University of the Philippines, 

Diliman, Quezon City 1101, Philippines 

jddelrosario2@up.edu.ph 

Batteries have been a forefront energy storage solution owing to their flexibility to supply power to a broad range 

of applications. Lithium-polymer (LiPo) batteries, a subcategory of the widely adopted lithium-ion batteries, 

exhibit heightened safety properties due to the utilization of a solid or gel polymer that acts as a separator and 

electrolyte. In this study, a pseudo-2D electrochemical coupled thermal multiphysics model of lithium cobalt 

oxide (LCO) cathode, graphite anode, and poly(vinylidene fluoride-hexafluoropropylene) (PVdF-HFP) polymer 

electrolyte LiPo battery was developed using COMSOL Multiphysics®. To improve the model accuracy and 

battery performance, ten parameters for energy density optimization were screened via sensitivity analysis, and 

the five most sensitive parameters were selected for optimization. Simultaneous optimization of these 

parameters through the Constrained Optimization By Linear Approximation (COBYLA) optimization algorithm 

resulted in a ~25 % increase in energy density and a ~21 % increase in power density for both 1 C and 0.5 C 

discharge rates without significantly increasing heat generation during discharge. These increases are attributed 

to the maximization of anode active material and the minimization of electrode and separator thicknesses. The 

developed model can be integrated into the experimental design of batteries to improve target performance 

applications such as energy and power. It can be further employed to optimize various battery chemistries and 

configurations.  

1. Introduction 

Lithium polymer (LiPo) batteries utilize gel polymer electrolytes, improving the resistance to volume variations, 

increasing thermal stability, and making the battery safer overall (Long et al., 2016). Due to the complexity of 

the definition of performance, energy and power densities become major indicators of battery efficiency (Grazioli 

et al., 2016). Energy density describes the battery capacity, while power density describes the output power 

capabilities of the battery. These performance indicators are primarily affected by battery chemistry parameters 

(Domalanta et al., 2022). To simulate battery behavior with different parameters and operating conditions 

without the need for time-consuming experiments, the numerical modeling of batteries raised interest. A well-

established model would entail insights into battery performance optimization by adjusting specific battery 

parameters. Sensitivity analysis of lithium-ion battery models has been the subject of previous studies to provide 

valuable insight into the effects of varying battery design parameters on performance (Jin et al., 2018). 

Optimizing such parameters would entail improving battery capacity, performance, and efficiency, which in turn 

will have knock-on effects on battery manufacturing and usage as more energy can be stored more safely, 

efficiently, and with less material (Pan et al., 2022). 

At present, relatively few studies have tackled the modeling of LiPo batteries (Golmon et al., 2009). Although 

common among lithium-ion batteries, LiPo battery parameter optimization, and sensitivity analyses are relatively 

unexplored territory in research. As such, this study bridges the gap by proposing an electrochemical-thermal 

one-dimensional multiphysics model for LiPo batteries, integrating sensitivity analysis and a subsequent 

parameter optimization algorithm to propose better battery design and optimize performance for energy or power 

use cases. 

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2. Methodology 

The methodology is divided into four parts. The first part is developing the multiphysics model. The second is 

validating the model and determining significant parameters; the third is analyzing its sensitivity, and finally, 

optimizing the parameters for battery energy density. 

2.1 Multiphysics model 

The electrochemical model used was based on the pseudo-2-dimensional (P2D) model of Doyle and Newman 

(1995). It uses a one-dimensional geometry with a combination of porous electrode theory, concentrated 

solution theory for electrolyte mass and charge transfer, Butler-Volmer kinetics, Ohm’s law for solid phase 

current density, and Fick’s law for solid phase mass transfer. The one-dimensional thermal model was based 

on the model developed by Gu and Wang (2002) described in Eq(1). The heat generation term 𝑞 is dependent 

on several phenomena shown in Eq(2), where the first and second terms on the right-hand side represent the 

reversible and irreversible heat generation. The third term describes the Joule heating that occurs because of 

resistance to current flow through the electrode. The physical parameters of the base model were adapted from 

the study of Domalanta et al. (2023). The coupled P2D electrochemical-thermal model described was 

implemented using COMSOL Multiphysics™ 5.5. 

𝜌𝐶𝑝

𝜕𝑇

𝜕𝑡
= 𝜆

𝜕2𝑇

𝜕𝑥2 + 𝑞 (1) 

𝑞 = − ∑ 𝑎𝑠𝑗𝑖�̅�𝑗 (𝑈𝑗 −
𝑇𝜕𝑈𝑗

 𝜕𝑇
)

𝑗

+ ∑ 𝑎𝑠𝑗𝑖�̅�𝑗(�̅�𝑗)

𝑗

− ∑ (⟨𝑖𝑘⟩ ⋅ ∇ ⟨𝜙𝑘⟩𝑘)

𝑘=𝑒&𝑠

 
(2) 

2.2 Model validation and parameter estimation 

The developed multiphysics model was validated by comparing the root mean square error (RMSE) of the 

model's discharge data against the experimental data of Domalanta et al. (2023). To closely match the 

experimental data, modification of the model’s transport and kinetic parameters was performed using the 

parameter estimation function of COMSOL Multiphysics Optimization Module, which minimizes the deviations 

between two discrete data sets by varying one or more model parameters. In this study, the Levenberg-

Marquardt method for parameter estimation was selected as proposed by Santhanagopalan et al. (2007). 

2.3 Sensitivity analysis 

The parameter sensitivity was calculated using a Multiparametric Sensitivity Analysis (MPSA) derived from the 

work of Jiang et al. (2020) described in Eq(3). For this study, the sensitivity of energy and power densities to 

variations in battery design parameters are the basis of assessment. Each parameter will be assessed for two 

objective functions. This study used ten generated values for each parameter. The average of the results will 

then be used to assess the overall sensitivity of energy and power densities to the tested parameter. The five 

most sensitive parameters for energy density at 1 C and 0.5 C discharge rates were considered sensitive and 

subjected to parameter optimization. 

𝑀𝑖 = √
1

𝑚
∑ [

(
𝑔𝑖(𝑥0) − 𝑔𝑖(𝑥𝑛)

𝑔𝑖(𝑥0)
)

(
𝑥0 − 𝑥𝑛

𝑥0
)

]

2
𝑚

𝑛=1

 (3) 

2.4 Parameter optimization 

This study utilized the built-in COBYLA module for optimization in COMSOL Multiphysics as illustrated in Figure 

1. This applies to nonlinear systems and does not require the derivative of the objective function to be known 

(Mei et al., 2018). This algorithm employs iterative linear approximations to optimize the objective function, 

energy density optimization, within given constraints. First, the power density of the modeled battery must not 

fall by more than 10 % in the optimization result to avoid solving with a significantly reduced rate capability. 

Second, the discharge time is limited to the maximum theoretical value for a given C-rate. For instance, the 

maximum discharge time is limited to 3,600 s at 1 C, while the discharge time is limited to 7,200 s at 0.5 C. The 

result of this constraint is that the optimization algorithm will minimize the battery mass such that the battery 

capacity is still 2,000 mAh. Third, the heat generation and the average heat generation rate should not increase 

by more than 20 % to avoid excessive heat generation, which may lead to unsafe battery operation. 

884



 

Figure 1: LiPo battery optimization procedure flowchart 

3. Results and discussion 

3.1 Parameter estimation 

Simultaneous estimation of the rate constants and diffusion coefficients in the anode and cathode via the 

Levenberg-Marquardt method yielded different values depending on the discharge rate, as shown in Table 1. 

Table 1: Estimated parameter values from the Levenberg-Marquardt method 

Parameter Initial Estimated 

  1 C Discharge 0.5 C Discharge 

kcathode (mol2.5/mol0.5•s) 6.67 x 10-11 9.60 x 10-13 9.59 x 10-13 

kanode (mol2.5/mol0.5•s) 1.76 x 10-11 9.48 x 10-13 9.38 x 10-13 

DLi.s,cathode (m2/s) 1.00 x 10-11 1.00 x 10-13 1.07 x 10-13 

DLi.s,anode (m2/s) 5.50 x 10-14 3.90 x 10-14 1.93 x 10-14 

The values of kcathode, kanode, and DLi.s,cathode changed proportionally regardless of the C-rate, describing that the 

C-rate value does not hold a substantial effect on the estimation of the mentioned parameters. Additionally, the 

aforementioned parameters decreased by two orders of magnitude. The kcathode, kanode, and DLi.s,cathode decreased 

by 70-, 18-, and 100-fold in both C-rates. Only the estimated DLi.s,anode value changed with respect to the C-rate, 

decreasing at 60 % of its original value at 1 C and 35 % of its initial value at 0.5 C. The deviation stems from 

the experimental data; two different batteries with the same battery chemistry were used to obtain the discharge 

data at 1 C and 0.5 C to ensure fresh and uncycled batteries and to circumvent degradation and capacity fade 

effects. As such, these two batteries may not be accurately identical in terms of anode structure and solid-

electrolyte interface thickness, affecting the estimated diffusion coefficient. 

  

Figure 2: Comparison between fitted model results (black lines), experimental data (red lines), and reference 

literature model results (blue lines) for (a) 1 C and (b) 0.5 C discharge rates 

Figure 2 displays parameter adjustments' effects during discharge at 1 C and 0.5 C. For both discharge rates, 

significant deviations are observed at the start of the discharge process. Near the end of the discharge, the 

model results almost mirror the experimental data. It was found that the maximum deviation was observed at 

the start of the discharge process for both C-rates having a minimal RMSE of only 35 mV for both C-rates.  This 

accuracy has shown significant improvement from the reference base model of Domalanta et al. (2023), which 

had an RMSE of 60 mV and 70 mV, for 1 C and 0.5 C. Improvement was attributed to the fitted parameter 

values in Table 1 that may have a closer resemblance to the actual property values unlike that of the base model 

reference that relied on literature-reported values. 

3.2 Sensitivity analysis 

Each selected design battery parameter was analyzed for each discharge rate. Energy density sensitivity (MED) 

reveals the two most sensitive parameters are similar for both discharge rates, which are the anode active 

material fraction (εs,anode) and length (Lanode) in order. This suggests that the anode properties are integral to the 

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optimization of energy density. It is critical to note that their MED sensitivity values are several magnitudes higher 

than other values, with 0.985 for εs,anode, and 0.632 for Lanode, (both at 1 C). The next most sensitive parameter 

is Lcathode with a MED value of 0.256 at 1 C, which is around four times less than that of εs,anode, emphasizing their 

impact on optimization. Sensitivity values vary with C-rates due to transport limitations at higher C-rates and 

thermodynamic limitations at lower C-rates. 

Power density sensitivity (MPD) values are lower compared to MED values on average, around five times, and 

there are no notable values from MPD values since they are generally in line with prominent parameters for MED 

in which Lanode, and Lcathode are the most sensitive with an average MPD value of 0.291 at 1 C. Results from this 

sensitivity analysis are aligned with the findings of Hosseinzadeh et al. (2017), where the lengths and the active 

material fraction were determined to be sensitive for both densities; while particle size is more sensitive to energy 

density. The top four parameters in each discharge rate are chosen for optimization. Given that the sensitivity 

of εs,anode, Lanode, and Lcathode overlap for both discharge rates, only Lsep and Rp,anode were added. 

3.3 Parameter optimization 

The five selected parameters were optimized to maximize energy density using the COBYLA optimization 

algorithm with the parameter range, and optimized values are presented in Table 2. 

Table 2: Optimized parameters for the five most sensitive parameters 

Parameter  Range 1 C Discharge Rate 0.5 C Discharge Rate 

  Initial Optimized Initial Optimized 

   Value % Change  Value % Change 

εs,anode 0.4–0.7 0.5052 0.7 +38.6 % 0.5052 0.69645 +37.9 % 

Lanode (μm) 35–105 73.5 55.559 -24.4 % 73.5 55.068 -25.1 % 

Lcathode (μm) 35–105 70 50.134 -28.4 % 70 53.169 -24.0 % 

Lsep (μm) 20–40 25 20 -20.0 % 25 20 -20.0 % 

Rp,anode 
(μm) 5–20 12.5 12.45 -0.4 % 12.5 12.417 -0.7 % 

From the sensitivity analysis, it can be determined that the anode material fraction greatly influences the 

battery's energy density and increases in value after optimization. Increasing anode material enhances lithium-

ion density, improving the energy density. The optimized anode material fraction for 1 C discharge lies at the 

upper boundary (0.7) of the parameter constraint. This upper boundary value was selected based on related 

literature detailing the maximum experimentally obtainable values of anode active material fraction. As such, 

the “true” optimal value may lie beyond the constraints but may be physically limited by the manufacturing of 

porous anode materials. The anode material fraction can be increased by using stable graphite with low-capacity 

fade and higher purity that could be achieved from optimized carbonization of a high carbon source (Du et al., 

2017). Additionally, the anode’s thickness was minimized to provide enough lithium ions to meet the specified 

battery capacity, while the cathode’s thickness was optimized to maximize its saturation with lithium ions by the 

end of the discharge process. The thickness of the separator was minimized to the boundary value of 20 µm, 

the lowest in related literature (Deimede and Elmasides, 2015). Because the separator acts to prevent short-

circuiting, it is recommended to minimize its thickness to a safe value to reduce the proportion of its mass to 

other battery components. Specific thickness can be achieved by carefully selecting physical parameters during 

battery coating/lamination and calendering. 

Table 3: Battery performance indicators of the optimized model 

Parameter  1 C Discharge Rate 0.5 C Discharge Rate 

 Initial Optimized  Initial Optimized  

  Value % Change  Value % Change 

Energy Density (Wh/kg) 137.4 171.12 +24.5 % 142.06 176.81 +24.5 % 

Power Density (W/kg) 145.83 178.49 +22.4 % 75.088 90.642 +20.7 % 

Discharge Time (s) 3,391.7 3,451.3 +1.8 % 6,810.7 7,022.4 +3.1 % 

Qgen (kJ) 7.2819 7.3813 +1.4 % 5.808 6.5171 +12.2 % 

Avg. Q (W) 2.147 2.139 -0.4 % 0.9280 0.9280 +8.8 % 

From Table 3, it was found that when the objective function is energy density, it also significantly improved the 

battery's power density. This result stems from the applied constraints to the optimization algorithm, specifically, 

the discharge time. Since these parameters cannot be adjusted to increase the discharge time and, by 

extension, the total capacity of the battery, the only course that the optimization algorithm can take is to minimize 

the mass of the battery. Given that the discharge rate is also fixed, this inevitably increases power density as 

well. This also explains the lack of change in the anode active particle radius, as sensitivity analysis revealed 

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that adjusting this parameter affects the battery model's discharge time rather than power density. Finally, similar 

energy density improvements were obtained by a similar optimization study from Lee et al. (2020), lending 

credence to the magnitude of improvement obtained in this study. 

The critical difference between the results at 1 C and 0.5 C is the relative increase in heat generation observed 

in the optimized 0.5 C model. In the 1 C optimization, there was barely any change in heat generation, while the 

optimized 0.5 C model yielded an 8.8 % increase in average heat generation. This deviation is attributed to the 

significantly different values of the anode diffusion coefficient obtained during the parameter estimation step. 

Since the estimated 0.5 C anode diffusion coefficient is about half the value of the diffusion coefficient estimated 

at 1 C, the optimized 0.5 C model has significantly greater polarization within its anode because of the 

concentration gradients within the solid particles and between the particles and electrolytes. Consequently, this 

increased resistance increases the ohmic heat generation in the optimized 0.5 C model. 

 

Figure 3: Comparison between the base (solid lines) and optimized (dash lines) battery models for the (a) 

discharge curve and (b) temperature profile for 1 C (red lines) and 0.5 C (black lines) discharge rate 

The optimized model’s discharge voltage is slightly lower than the base model, as illustrated in Figure 3a. This 

decrease is inferred to be the result of decreasing the electrode thicknesses since a fixed amount of current 

must be extracted from less active material. In addition, increasing the active material fraction reduces the 

anode’s porosity, constricting the lithium-ion transport in the electrolyte. This increases the concentration of 

lithium ions in the electrolyte, as these ions cannot quickly migrate to the cathode. This limits the rate of the 

intercalation reaction at the cathode, necessitating an increase in overpotential to compensate. These 

phenomena are theorized to be the major contributors to the slight reduction in discharge voltage.  

The thermal behavior at both discharge rates is illustrated in Figure 3b. Thermal performance is critical to a 

lithium-ion battery’s safety since thermal runaway can occur at excessively high temperatures. The optimized 

model temperature profile is observed to be slightly higher than that of the base model of around 1 K. It can be 

said that the optimized model does not operate at significantly higher temperatures than the base model. In 

addition, the base model temperature increases at around 40 % into the discharge process, then falls, until 

finally rising again continuously until the end of the discharge process. This phenomenon occurs presumably 

due to the endothermicity of the intercalation reaction at the cathode (Domalanta et al., 2023). The optimized 

model’s thermal behavior experiences the same trends, though the changes in the temperature profile onset 

are earlier than in the base model. Such difference arises from the lower amount of cathode active material in 

the optimized model; the endothermic stage of the cathode intercalation process, which occurs when the 

cathode reaches a certain degree of lithium saturation, occurs sooner since the cathode saturates faster in the 

optimized model. Finally, the temperature increase in the 1 C model is greater than that of the 0.5 C model 

because of the greater ohmic heat generation brought about by the increased current density. 

4. Conclusions 

A coupled electrochemical-thermal lithium-ion battery model was developed and fit to actual battery discharge 

data via parameter estimation. The produced model was then used for sensitivity analysis of ten battery design 

parameters for the most effective optimization. Based on sensitivity analysis data, the selected design 

parameters for optimization are the anode active material fraction, the anode active particle size, and the lengths 

of the anode, cathode, and separator. These parameters were optimized towards maximizing energy density 

with given constraints on power density reduction, maximum allowable discharge time, and heat generation 

limitation. In conclusion, optimization models yielded an approximate of about 25 % increase in energy density 

at both 1 C and 0.5 C discharge rates.  

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Nomenclature 

cp – isobaric specific heat, J/(kg·K) 

DLi,s – solid-phase diffusion coefficient, m2/s 

gi – power/energy density parameter for x 

k – rate constant, m2.5/(mol0.5·s) 

Mi – calculated sensitivity of parameter i 

T – local temperature, K 

t – time, s 

xn – generated values for parameter analysis of n 

q – battery heat generation, W/m3 

α – specific surface area, 1/m 

λ – thermal conductivity, W/(m·K) 

ρ – density, kg/m3 

Acknowledgments 

The authors would like to acknowledge the Advanced Batteries Center Program (REBCell Project) funded by 

the Department of Science and Technology (DOST), through the Niche Centers in the Regions for R&D (NICER) 

Program, and the Office of the Chancellor of the University of the Philippines Diliman, through the Office of The 

Vice Chancellor for Research and Development, through the Outright Research Grant (Project No. 222210 

ORG) for funding this research. 

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