Optimization of a Single Lithium-ion Battery Cell with a Gradient-based Algorithm

Nansi Xuea, Wenbo Du

a, Amit Gupta

c, Wei Shyy

d, Ann Marie Sastry

b, Joaquim R.R.A. Martins

a

aDepartment of Aerospace Engineering, University of Michigan, Ann Arbor, Michigan, USA

bDepartment of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA

cDepartment of Mechanical Engineering, Indian Institute of Technology, New Delhi, India

dDepartment of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water

Bay, Hong Kong

Abstract

This paper presents a numerical framework for automating the design of lithium-ion cells to maximize

cell energy density while meeting specific power density requirements. The battery is modeled using a

physics-based electrochemistry model that considers ion transport and includes mass balance and

interfacial reactions [6 -7]. The design is automatically performed by coupling the battery model with a

gradient-based optimization algorithm. We demonstrate the potential for gradient-based optimization by

applying this framework to optimize the design of a lithium-ion cell with spinel manganese dioxide

cathode and meso-carbon micro beads (MCMB) anode for a range of power requirements. Results

indicate that variations in the electrode thickness and porosity at optimal cell designs can be quantified via

active mass ratios and it is found that the active mass ratios for optimal cell designs are independent of

discharge rate.

Keywords

Lithium ion battery

Porous electrode model

Electrode design

Numerical optimization

Spinel manganese

1. Introduction

Electrochemical energy storage devices are playing an increasing role in everyday life, as they are being

used to power everything from mobile electronic devices to hybrid vehicles. Various lithium-ion

chemistries have been identified as candidates for secondary batteries due to their high theoretical energy

densities and tolerance to cycling. To a large extent, the cell performance is determined by the properties

of the material used. However cell design variables such as electrode thickness and porosity and particle

sizes play critical roles in realizing the full potential of the materials. Conventional cell designs have

relied on ad-hoc rules – such as using thicker electrode for high energy, low power cells and vice versa or

parametric analysis of one or two variables to guide the design process. To identify the optimal designs of

a cell, the design process has to include all relevant design variables, and the interactions between various

parameters on cell performance need to be considered.

Various mathematical models have been developed to examine the physical phenomena occurring within

an electrochemical cell, with varying levels of fidelity and computational cost. Single particle models

represent each electrode as a single spherical particle [1-2]. Liaw et al. [4, 5] have developed an

equivalent circuit model and subsequently used it to examine the impact of variation in cell properties on

overall energy capacity. Newman et al. [6] have developed a pseudo-2D model that uses porous electrode

and concentrated solution theories [7]. This is a macroscopic cell model that treats the electrode as a

homogeneous continuum. It ignores the microscopic structures in the electrode but accounts for the

change in effective properties via Bruggeman’s equation [8]. Subsequently, various authors have studied

at the effects of microstructural variations on transport properties and cell performance using micro-scale

models [3, 39-40]. Additional work has been done to account for various degradation mechanisms and

side reactions occurring within the cell [9-14] to account for differences in performance between ideal

electrochemical cells and practical results.

Several authors have carried out numerical optimizations of lithium-ion cells. Newman [15] optimized the

geometric design of a cell with respect to the positive electrode thickness and porosity using a simplified

reaction zone model. By confining the lithium ion intercalation to a narrow zone in the positive electrode,

he obtained an analytical solution and related the dependence of energy capacity to the following

dimensionless parameter:

(1)

where is the open-circuit potential, the electrolyte conductivity, the discharge time, the

capacity density of the active material, and the separator thickness. Values for optimal electrode

thickness and porosity were found by fixing the discharge time td. Subsequent optimization efforts

involved design of an iron phosphate lithium-ion cell while maintaining constant capacity ratio and

porosity in the negative electrode [16] as well as coupling lithium-ion batteries with capacitors for hybrid

electric vehicle operations [17].

In an optimization study with respect to cycling rate, particle size, diffusivity and conductivity using a

surrogate model formulation, Du et al. [18] quantified the cell performance as a function of ratio of

discharge time to diffusion time. Ramadesigan et al. [19] found that optimizing the spatial distribution of

porosity can lower the internal resistance by 15 to 33% compared to one with uniform distribution.

Golmon et al. [20] went one step further and varied both the spatial porosity and the particle size

distribution using a multi-scale model. An improvement in energy capacity of less than 2% was found by

varying porosity and particle size distribution compared to a cell with optimized constant porosity and

particle size.

Most of the earlier optimization studies have focused on a single electrode, the effects of few design

parameters, or both. Simpler algebraic models or models that neglect mass transport limitations in the

solution phase are used to obtain analytical solutions or to reduce the complexity of analyzing the

electrochemical storage system. While simulations with the simpler models are orders of magnitude faster

than the full physics-based model, their validity is often limited, since not all physical processes are

included [21]. To capture the effects of all relevant design variables on cell performance, both

morphology and material transport properties need to be considered as variables in the design process.

The design space should include all relevant parameters of a full lithium-ion sandwich cell with insertion

materials for both electrodes. With this goal in mind, we present a numerical framework that couples an

optimization algorithm with a detailed electrochemistry model optimize the design of the cell. Our

objectives are as follows:

1. Demonstrate the application of gradient-based optimization algorithm to maximize the energy

density of a lithium-ion cell subject to a power requirement

2. Determine electrode morphological designs at optimal cell designs under various discharge rates

In the following sections we present in detail the optimization framework with its constituent

components, followed by the optimization results of a representative lithium-ion cell and a brief

discussion.

2. Methodology

a. Optimization framework

A general optimization problem has the following form:

minimi e ( ) (2)

sub ect to {

( )

( )

where ( )) is the objective function we want to minimize with respect to bounded variables x, and

subject to inequality constraints ( ) and equality constraints ( ). In this case the objective function is

the gravimetric energy density of the lithium-ion cell; the variables are the electrode design parameters:

thickness, porosity, and particle size.

Various optimization methods can be used to tackle the battery optimization problem [18, 23, 41-44].

Given the nonlinear nature of the problem and the computational cost (each function evaluation requires

several minutes), a gradient based optimization scheme is well suited to handle such a problem due to its

numerical efficiency and accuracy [40] as it checks mathematical optimality conditions to determine

optimum. However it could potentially converge to local minima should the design space be multimodal.

The gradient-based optimization process is iterative and follows the flowchart shown in Fig. 1.

Optimization is initialized at a randomly chosen starting point in the design space. This is to minimize the

chance of algorithm converging to local minima. The optimization algorithm makes use of both the

function evaluation as well as the gradient to converge towards the optimal design point. The optimality

of the design point is determined by the Karush-Kuhn-Tucker (KKT) conditions [22], which are given by:

( ) ∑ ( ) ∑ ( )

(3)

for all ( ) for all

Figure 1: Flowchart of a gradient-based optimization scheme

where μ and λ are the Lagrange multipliers associated with the inequality and equality constraints. There

are three main components to the cell design optimization framework: i) optimization algorithm, ii)

gradient computation, and iii) the function evaluation which is the electrochemistry cell model. These

three aspects of the framework will be explained in greater details in the following sections.

b. Optimization method

For this paper, the gradient-based optimization method used is the sequential quadratic programming

(SQP) [23]. SQP methods have proven to be very effective at solving large-scale constrained optimization

problems with nonlinear functions in the objective and constraints. SQP methods assume the objective

Start Initialize design variables

Function evaluation

Gradient calculation

Optimality

condition

satisfied?

Stop

Optimizer

yes

no

Update variables

function to be convex near the optimal design and hence approximate the problem at each iteration point

as a quadratic sub-problem:

minimi e

(4) sub ect to

In Eqs. (4) gk is the gradient vector of objective function with respect to design variables, Wk is the

estimate of the second-order derivatives, and AT is the Jacobian of the constraints with respect to the

design variables. The quadratic sub-problem is solved with a quasi-Newton approach whereby the second

order derivative is updated with a quasi-Newton approximation rather than computed explicitly to

improve the convergence of the optimization. There are numerous implementations of the SQP method

that differ in a number of details. We use the SNOPT package, developed by Murray et al. [23]. SNOPT

is suitable for solving nonlinear, constrained optimization problems, which is the type of problem in this

work.

c. Gradient computation

Gradients of the objective and all constraints are needed to determine the search direction and step size in

search of the optimal design point. Several methods [45] can be used to obtain gradients, ranging in

accuracy and difficulty of execution. Given the relative small number of design variables and the

complexity in deriving analytic gradients, we apply a complex-step approximation method [24] to

compute the derivatives due to its accuracy and the ease of implementation. The complex-step method is

similar to the more common finite-difference approach, except that a small step is taken in the imaginary

axis instead of the real axis. Taking the Taylor series expansion about a point in the real axis with a pure

complex step, we obtain the following:

( ) ( ) ( ) ( )

( )

( )

( )

(5)

The first-order derivative can be isolated as a second-order approximation from the equation (4):

( ( ))

( ) (6)

Eq. (6) shows that complex step derivative does not involve a subtraction operation, and hence it is not

subject to cancellation errors resulting from finite precision of machine arithmetic, which is a problem

with finite difference derivative approaches. A comparison of the derivatives computed using the

complex-step method and a forward-difference approach is shown in Fig. 2. The normalized errors of the

two approaches are plotted against the step size. From Fig. 2 we can see that the cancellation error

dominates at small step size for the forward difference approach, whereas the error of complex-step

approximation remains at machine-zero (10-16

for double-precision arithmetic).

Figure 2: Comparison of errors between finite-difference and complex-step approximation

While the complex-step approximation offers the advantage of increased accuracy, it only works in cases

when the design variable or objective function of interest is not already complex in the original function.

Application of complex step to computation programs also requires modification to the original source

code, which can be non-trivial in certain instances.

d. Cell model

The cell modeled in this work consists of a full sandwich cell with porous positive and negative

electrodes, as shown in Fig 3. The active materials in both electrodes are modeled as uniformly-sized

spherical particles. The pseudo-2D model with porous electrode and concentrated solution theory [7, 25-

26] is used to examine the energy and power capacity of the lithium-ion cell with respect to the design

variables. The migration of lithium ions from negative electrode to positive electrode during discharge is

modeled via governing equations in the direction perpendicular to the electrode-separator interface, while

the diffusion of the lithium ions in the radial direction in the spherical solid particles is approximated via

the method of superposition [27]. Five state variables are computed in the cell model: ion concentration

in electrolyte (c) and in solid phase (cs), interfacial reaction rate (in), and potentials in liquid phase (φ2)

and solid phase (φ1). The governing equations are listed in Table 1. A detailed derivation of the governing

equations is provided by Fuller et al. [26].

( )

√ |

State

variable

Governing equations

c

(

ln ln

)

(7.1)

cs

(

ln ln

)

(7.2)

in

[e p( ( )

) e p(

( )

)] (7.3)

φ2

(

) ( ln ln

) ln (7.4)

φ1 (7.5) Table 1: Governing equations of state variables in electrodes [6]

3. Problem setup

The numerical optimization framework is used to maximize the gravimetric energy density of an ideal

lithium-ion cell subject to power requirement constraints. The materials of the cell and their respective

properties are listed in Table 2. The cell performance is obtained through one galvanostatic discharge

cycle of the cell until a cutoff voltage of 3.0 V is reached. This cutoff value is selected as further

discharge of cell in real life could lead to irreversible damage to the battery.

The cell model used herein has been validated for a lithium ion cell with manganese dioxide positive

electrode and carbon negative electrode by Doyle et al. [28] via comparison with experimental results.

Negative

electrode

Positive

electrode

Sep

arat

or

L+ L- Ls

Cu

rren

t co

llect

or

(+)

Cu

rren

t co

llect

or

(-)

Electrolyte

Solid particles

Figure 3: Schematic diagram of a lithium-ion sandwich cell

Parameter Value

Cathode material Spinel Mn2O4

Initial stoichiometric parameter (y in LiyMn2O4) 0.01

Coulombic capacity 148 mAh/g

Density 4280 kg/m3

Volume fraction of inert filler 0.1

Inert filler material PVDF

Electrolyte material liPF6 in EC:DMC

Initial salt concentration 1000 mol/m3

Anode material MCMB 2528 graphite

Initial stoichiometric parameter (x in LixC6) 1.0

Coulombic capacity 372 mAh/g

Density 2260 kg/m3

Inert filler material PVDF

Volume fraction of inert filler 0.05

Positive current collector material Aluminum

Current collector thickness 25.0 μm

Current collector density 8954.0 kg/m3

Negative current collector material Copper

Current collector thickness 25.0 μm

Current collector Density 2707.0 kg/m3

Ambient temperature 298 K

Cutoff voltage 3.0 V Table 2: Fixed parameters and properties in lithium-ion cell

To maximize the energy density, we selected the 12 design variables listed in Table 3. While cycling rate

is an operating condition and not a cell design variable, it is necessary to allow it to vary freely within the

design space in order to meet the constrained power requirement. The simulation technique and the

subsequent optimizations are not confined to any particular operating conditions. Allowing cycling rate to

vary freely enables us to obtain optimal cell designs to meet different application requirements. The

remaining 11 variables consist of 5 variables for each of the electrodes and the separator thickness.

Among the 5 electrode variables, porosity, thickness and to some extent particle size can be controlled

during the manufacturing process while the diffusivity and conductivity are material properties. The

bounds on the particle size, diffusivity and conductivity represent the minimum and maximum reported

values from the literature [30-38]. These parameters are treated as bounded variables instead of given

exact values as it is not the intent of this work to optimize the design of a particular lithium-ion cell

subject to a specific pre-fabrication treatment and manufacturing process. Rather these variables are

intended to provide guidelines on what combination of properties and cell designs would produce

batteries of optimized performance, such as maximum achievable cell capacity.

Design variables Lower bound Upper bound

Cycling rate (C) 0.1 10.0

Separator thickness (μm) 10.0 100.0

Cathode particle si e (μm) 0.2 20.0

Cathode thickness (μm) 40.0 250.0

Cathode porosity 0.1 0.6

Cathode diffusivity (m2/s) 10

-16 10

-11

Cathode conductivity (S/m) 1.0 100.0

Anode particle si e (μm) 0.5 50.0

Anode thickness (μm) 40.0 250.0

Anode porosity 0.1 0.6

Anode diffusivity (m2/s) 10

-16 10

-11

Anode conductivity (S/m) 1.0 100.0 Table 3: Design variables and their bounds

The range of the conductivities for both electrodes is high compared to the conductivities of the pure

active materials in each electrode. These values reflect the increased electronic conductivity after the

addition of the carbon additives to the solid matrix. The cell model does not account for the effects of

carbon additives on effective transport explicitly. However using the high conductivity allows us to

design cells that have performance more representative of what can be obtained in practice.

We carried out nine different optimization runs at each constrained power requirement. The optimizations

are initiated from random design points to reduce the possibility of optimization converging to local

optima. Results from various optimization cases are discussed in the following section.

4. Results and discussions

We demonstrate the potential for gradient-based optimization framework in cell design by applying to cell

designs with maximum gravimetric energy density at various specific power requirements. The

constrained power requirement increases from 50 W/kg/m2 at increment of 50 W/kg/m

2 until 1500

W/kg/m2, at which point a further increase in power is only possible by expanding the range of the

cycling rate. The competing effects of increasing power and decreasing energy at higher power form a

Pareto front. A Pareto front contains solutions that are not dominated by any other points in the design

space, i.e. the solutions on the Pareto front have the best objective functions at their respective constraints

[46, 47]. It offers valuable insight into multi-objective optimization in numerous applications [46, 48]. In

this case the Pareto front is formed from the set of points with the maximum achievable cell energy

capacity for each required power level.

Results from the various optimizations show variations in other design variables at the optimal design

points and the variations are presented in the form of box plots. The box represents the interquartile range

while the data in the upper and lower quartiles are represented by ‘+’ and ‘ ’ symbols. Results show that

while there are some variations in the optimal design variables between the various runs, the final

objective functions (energy densities) are remarkably close to one another.

Figure 4: Pareto front of optimal energy density vs. specific power. Also shown in the plot are Ragone plots at three

discharge rates. The Ragone plot shows the variation of energy density with discharge rates for a particular cell

Fig. 4 shows the optimal energy density with respect to power requirements. As expected, the maximum

achievable energy density decreases as the power requirement increases. For all nine optimization runs at

each specified power requirement, the final optimal energy densities converged to within 0.1 % of one

another. We compared the Pareto-front to the Ragone plots. The Ragone plot compares the performance

of the energy storage device at different power ratings by plotting its available energy density against

power density [49]. While the Pareto front consists of cell designs optimized for each power requirement,

the Ragone plot shows the variation of energy density versus power density for a single cell. Three

representative Ragone plots are shown in Fig. 4. The 0.2-C cell corresponds to the power requirement of

50 W/kg/m2, while the 2-C is at 400 W/kg/m

2, and the 9-C is at 1500 W/kg/m

2. Each Ragone plot

intersects the optimal Pareto front at only one point, indicating that each cell design is optimized for only

a specific power requirement and no single cell is the best design for all power requirements. The 0.2-C

cell has the highest energy density at 50W/kg/m2; however, its energy capacity decreases rapidly if its

discharge rate is increased. All three Ragone curves e hibit the typical ‘knee’ observed in most Ragone

plots, indicating rapid deterioration of cell performance away from its specified design condition.

Comparison between the Ragone plots and the optimal Pareto front demonstrates the importance of a

properly designed cell for a specific operating condition. Any deviation of cell design away from its

intended purpose will lead to rapid loss of energy capacity.

Next we quantify the design variables of the optimal cell designs for various power requirements. Optimal

cell designs result in minimal separator thickness and particle sizes in both electrodes. These results are

not surprising, as the separator does not contribute to the physics model of an ideal cell, since no

electrochemical reactions occur there. The purpose of the ions in the separator is solely to transport

charges from one electrode to the other. Therefore the separator thickness only affects the weight of the

cell, which is to be minimized in order to maximize the energy density of the cell. In practical cell design,

there is a minimum separator thickness to prevent short-circuit due to dendritic growths from the

electrodes.

Smaller particle sizes are preferred as they reduce diffusion path length for lithium ions in the electrodes

and also increase interfacial surface area. A large interfacial surface area is important for ensuring an

adequate rate of transport of ions between the two phases. Reduction in particle size results in more

interfacial surface area at a smaller porosity, and hence the electrode is able to pack more active material,

and this results in a cell design with higher energy density. Increase in surface area has the additional

benefits of reducing the current density and subsequently the transport overpotential in the cell. Various

authors have shown improved electrochemical performances of smaller particles in positive electrodes

[51, 52]. Cells made with nano-scale particles have demonstrated high energy capacities and fast

charge/discharge kinetics of lithium ions compared to cells with micro-sized particles. On the other hand,

cycling stability of electrodes with small particles is in question due to the polarization overpotential

associated with small particles and possible increase in manganese dissolution [13, 53]. Future work

should include degradation models to account for capacity fade mechanisms.

Fig. 5 plots the optimal values for electrode thickness and porosity as the discharge rate is varied. As the

discharge rate increases, a thinner, more porous electrode is preferred in order to meet transport rate

requirements. The combination of decreased electrode thickness and increased porosity reduces the

amount of active materials in the cell and hence reduces the energy capacity of the cell.

Figure 5: Variations of electrode thickness and porosity of optimal cell designs with respect to cell power requirement

Fig. 5 also shows that the thicknesses of both positive and negative electrode plateau as the power

requirement increases, indicating that the optimal cell thicknesses are 150μm for the positive electrode

and 70μm for negative electrode. Further increase in electrode thicknesses does not result in any increase

of energy capacity of the cell designed for moderate to high discharge rates. The negative electrode

porosity plateaus with increased power requirement as well, while that of positive electrode continues to

increase.

The changes in electrode thickness and porosity can be quantified by plotting the mass ratio of positive to

negative active material materials, given by:

(7)

where L is the electrode thickness, ε the mass fraction of active materials, ρ the solid density, and + and –

signs denote positive and negative electrodes respectively. The optimal mass ratios range from 2.77 to

2.85, which is consistent with the optimal mass ratio of 2.8 reported by Tarascon and Guyomard [38]. The

optimization results also show that the optimal mass ratio for a well-designed cell is not constant but is a

linear function of the cell power, although the overall variation in the mass ratio is less than 3%.

Figure 6: active material mass ratio for optimal cell designs

Fig. 6 also plots the positive electrode to negative electrode capacity ratios. The capacity ratios of the

optimal designs range from 0.99 to 1.02, even though there is no enforcement of charge balance in the

problem formulation. The balance of capacity ratios indicates that nearly all the active materials have

been consumed when the cutoff potential is reached and there is no excess capacity in the electrode that

adds parasitic weight to the cell. In practical cell designs the negative electrode often has a larger capacity

to ensure full utilization of the positive electrode and to compensate for loss of cyclable lithium due to

side reactions such as solid-electrolyte layer (SEI) formation. Such side reactions are not included in the

current model and their effects on cell behavior are ignored. The proximity of the capacity ratios to unity

for optimal cell designs at all power requirements validates the approach of requiring charge balance in

the electrodes for preliminary cell design.

The results from the various optimization runs initialized from different starting points indicate that the

diffusivity and conductivity values did not converge to specific values at the optimal designs.

Initialization from different starting points resulted in different final design points as well. The

conductivity value range used in this study reflects cathode materials already doped with conduction-

enhancing carbon additives [50]. While higher values are favored, in this range the cell performance is

insensitive to change in conductivity.

The ranges of diffusivity at optimal cell designs are plotted in Fig. 7. For all cases the diffusivity values of

the optimal designs converged closer to the upper bound, with values ranging from 10-12

– 10-11

m2/s.

Given that the diffusivity bound spans 5 orders of magnitude, the ranges of optimal diffusivity are small.

Du et al. [18] have shown that the performance of the lithium manganese batteries in related to a

dimensionless time τ, which is defined as the ratio between the discharge time tdis, and the diffusion time,

tdif, i.e.,

(7)

Where Rs,p and Ds are the solid particle radius and bulk diffusivity in the positive electrode respectively.

Cells for which τ if greater than 1 demonstrate very little increase in energy capacity upon a further

increase in diffusivity. All optimal designs converged to the smallest particle sizes given by the lower

bound, and hence this results in very small diffusion time even at high cycling rates such that the cells are

diffusion-limited only when the bulk diffusivity is close to the lower bound of 10-16

m2/s.

Figure 7: diffusivity variations at optimal design points: while diffusivity ranges are small considering the overall range of

values, there exist some variations in diffusivities between different optimizations (shown in the inserts)

To examine the effects of diffusivity on energy density, a one-dimensional sweep of diffusivity is carried

out at the optimal design points, with all other design parameters remaining fixed. The differences

between the energy densities and the maximum achievable values are plotted as functions of

dimensionless time τ at four representative discharge rates in Fig. 7. The values of dimensionless time, τ,

at the optimal cell designs for all four cells are much greater than 1. As the value of τ decreases as

diffusivity decreases, the energy density of the cell decreases as well. However the reduction is not

significant. As shown in the right plot in Fig. 8, the decrease in energy density is more than 1% only when

anode τ is close to the value of 1. Fig. 8 also indicates that the effects of diffusivity on energy density

increases as cell power requirement increases. This is to be expected as higher power cell requires higher

effective diffusion rate, which is directly proportional to the bulk diffusivity.

Figure 8: variations of energy density as a function of diffusivity at optimal design points

In the current cell model the bulk material properties are treated as constants and the effective properties

are only functions of porosity via Bruggeman’s equation [8]. In effect they are functions of electrode

material compositions and particle sizes as well. Zhu et al. [55] has shown the aggregation of carbon

additives to active material as function of particle size, carbon-to-active-material mass ratio and

temperature. Decreasing particle size was shown to reduce the amount of carbon particles attached to

active materials, hence limiting the conductivity. A more refined treatment should include the dependence

of material properties on other cell variables, and thereby leads to more realistic electrode designs.

5. Conclusions

A numerical framework has been developed to automate the design of lithium ion battery cells using a

gradient-based optimization algorithm. The objective is to design cells with maximum gravimetric energy

densities under different discharge scenarios. The gradient-based optimization framework is able to

efficiently locate the optimal cell designs at various power requirements. Results have shown that the

optimal cell designs should have the smallest possible particle size and separator thickness. The decrease

in electrode thickness and increase in porosity at higher discharge rates can be quantified via an active

material mass ratio at optimal designs. Increasing the discharge rate results in a cell design with a higher

negative to positive mass ratio compared to one designed for lower discharge rate, although the total

difference is less than 3%. The diffusivities and conductivities should be high in optimal cell designs, but

their effects on energy density are limited in an idealized cell. While the current framework is able to

obtain numerically verified optimal designs, its practicality needs to be improved by including additional

constraints due to various capacity fade mechanisms. Factors such as SEI layer formation, electrode

dissolution, and lithium plating on charging affect cell designs and need to be considered as well in future

work.

Acknowledgements

This work was funded by the Department of Energy and the General Motors/University of Michigan

Advanced Battery Coalitions for Drivetrains (GM/UM ABCD).

Nomenclature

Electrochemistry variables τ dimensionless time

L+ thickness of positive electrode Trz dimensionless reaction zone parameter

L- thickness of negative electrode ε+ positive active material porosity

Ls separator thickness ε- negative active material porosity

a Interfacial surface area U open-circuit voltage

R particle size in electrode F Faraday’s constant

c electrolyte concentration R universal gas constant

cs ion concentration in solid matrix T ambient temperature

D electrolyte diffusion coefficient

Ds solid matrix diffusion coefficient Optimization parameters

κ conduction coefficient in electrolyte A derivatives of constraints w.r.t. variables

σ conduction if solid matrix c inequality constraint

φ1 solid matrix potential equality constraint

φ2 electrolyte potential f objective function

i2 current density in liquid phase g gradient vector

ρ+ positive active material density p solution to quadratic subproblem in SQP

ρ- negative active material density μ inequality constraint Lagrangian multiplier

q+ positive electrode capacity λ equality constraint Lagrangian multiplier

q- negative electrode capacity W estimate of second-order derivatives

tdis discharge time Δx step size in derivative approximation

tdif diffusion time

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