Efficient Global Sensitivity Analysis of 3D Multiphysics Model for Li-Ion Batteries

The electrode-level model is used to describe the local electrochemical performance of one electro-active element in a single cell as shown in Figure 2. Its domains include an anode, a cathode, and a separator. Two current collectors are defined as boundaries. The intercalation reactions occurring at the electrolyte-electrode interfaces are denoted by

where the backward and forward reactions represent the discharge and charge processes, respectively. Li_yMO₂ represents the functional metallic oxide material for cathodes. The electrode-level model is based on Newman's P2D model^11,37,38 and is simplified with the volume average method.^39,40 Here, the lumped pore-wall flux can be defined as equal to

where δ_i is the electrode thickness, a_i is the specific area of the electrode, F is the Faraday constant, i_{x, i} is the current density on the current collector, flowing in or out of the electro-active layer and j_{x, i} is the intercalation reaction current density. By using this lumped variable , the lithium intercalation and deintercalation processes in active particles are assumed to be spatially independent along the x direction in good accordance with the full-order model.⁴¹ In this way, the solid diffusion process can be derived analytically.⁴² The electrochemical kinetics can be expressed via the Butler-Volmer equation as

where R_u is the ideal gas constant, T is the operating temperature, and α_{a, i} and α_{c, i} are the charge transfer coefficients. Equation 4 denotes the overpotential η_i on the electrode which is defined as the deviation between the difference between the potential in solid and solution phases and the open circuit potential (OCP). Following the model simplification,⁴¹ we define η_ca = η_ca(L) and η_an = η_an(0) to evaluate according to Equations 2 and 3. i_{0, i} is the exchange current density which is given by

where k_{0, i} is the kinetic rate coefficient, and c_e is the salt concentration of the electrolyte at the electrolyte–electrode interfaces. On the cathode, c_{e, ca} = c_e(L) and when on the anode, it denotes c_{e, an} = c_e(0). c_{s, surf, i} is the lithium concentration at the surface of the active solid particles, and c_{s, max, i} denotes the maximum lithium concentration in active particles. For the derivation of c_{s, surf, i}, we refer to Guo's work.⁴² When assuming α_{a, i} = α_{c, i} = 0.5, most of the charge and mass conservation equations in the full-order P2D model can be solved analytically.⁴³ Based on the approximations and simplifications above, the full-order P2D model results in our electrode-level model. The governing equations of the electrode-level model are listed in Table I.

Table I. Governing equations of the electrode-level model.

Cathode and anode Dependent variables with i ∈ {an, ca}	Equation
cs, surf, i(t)
	λn − tan λn = 0, n = 1, 2, 3, ...
ηi(t)
Φs, i(x, t)
	σeffi∇Φs, i|x = 0, L = −ix
	Φs, i|x = 0 = Φ−
Φe(x, t)
	∇Φe|x = 0, L = 0
ce(x, t)
	Deffe, i∇ce|x = 0, L = 0 with L = δan + δsep + δca
Separator
Φe(x, t)
ce(x, t)

The cell-level model simulates the cell temperature T and a pair of electric potentials, Φ₊ and Φ₋, on the current collectors and tabs. At the cell level, there are six computational domains, including the electro-active layers, two current collectors, two electrode tabs, and the protective polymer cover as shown in Figure 2. The thermal and electric properties of each cell component are considered to be anisotropic between the XY plane and the Z direction. The energy conservation of the cell yields

where material density ρ, specific heat capacity C_p, and thermal conductivity k are attributed to every domain. The volumetric heat sources is defined in the electro-active layers and is evaluated by the electrode-level model as follows⁴⁴

where is the reaction heat, is the joule heat, and is the entropy change of the electrodes. The boundary conditions are on the surfaces of the polymer cover domains and tabs with the air convection are

where is the normal unit vector, and β_h is the heat transfer coefficient between the cell and the environment. The ambient temperature T_amb is assumed to be constant. The charge conservation of the cell on the current collectors and tabs follows the Poisson equation,

where σ_j is the electric conductivity of the current collector, and i_Z, is the volumetric current density between the electro-active layers and the current collectors. On the current collectors, , denoting the extra volumetric current density on electro-active layers, and on tabs, i_{Z, j} = 0. The boundary conditions on the tabs read as

and the tab of the negative electrode is defined as the ground electric potential,

Figure 2 illustrates that the electric conduction of the electro-active layers in the Z direction is assumed to be a nonlinear resistor network.⁴⁵ The local resistance R_node can be evaluated with the electrode-level model with

where I_node is the local current through an elemental resistor. Φ_{+, node} and Φ_{−, node} are the local electric potentials on the nodes of the cathode and anode current collectors, respectively. The local current I_node can be evaluated with

where N_elem is the total number of nodes of a meshed current collector. Then we define the difference, Φ_{+, node} − Φ_{−, node}, as an inter-level variable, which couples the electrode-level model with the cell-level model. The resulting coupling expression reads as

The set of physical parameters for a 10 Ah pouch cell (anode: graphite, cathode: NMC) that is 120 mm wide  × 99 mm high is given in Tables II and III. In this paper, β_h is set to 10 W/m² · K, referred to the natural convection value.⁴⁶ The thickness of the cell is varied and determined by specified parameter ranges. All parameters used to initialize the model are set to the literature values taken from Guo et al.¹⁵ The kinetic coefficients k_{0, i}, the solid diffusion coefficients D_{s, i}, and the diffusion coefficient in the electrolyte D_e are denoted as temperature-dependent functions in the Arrhenius type. The ionic conductivity κ is expressed as a temperature-dependent empirical equation. Parameters not listed in Tables II and III are given in the Tables AI and AII. The parameter ranges are defined from the Sauer group's results.⁴⁷ Technically, Gaussian and uniform distributions are used to describe the variability of the parameters and provide the key information for global parameter sensitivities as described in the following.

Table II. Electrochemical properties and geometry data for 1D electrochemical model in global sensitivity analysis.

Parameters	Anode	Range	Separator	Range	Cathode	Range
Design parameters for cell structure
Particle radius, R(μm)	12.5a,u	7–13a,u			5a,u	3–7a,u
Thickness, δ(μm)	91.2c,u		25c,u	20–30c,u	60c,u	30–100c,u
Porosity,	0.4a,u	0.3–0.5a,u	0.4a,u	0.4–0.6a,u	0.4a,u	0.3–0.5a,u
Volume fraction of binder, b	0.19a,u	0.01–0.2a,u			0.09a,u	0.01–0.2a,u
Bruggeman factor, b	2a,u	1.5–4a,u	2a,u	1.5–4a,u	2a,u	1.5–4a,u
Specific surface area, a(m2/m3)
Thickness of current collector, δcc(μm)	11a,g	50% a,g			16a,g	50%a,g
Solid phase
Specific capacity, ρe(mAh/g)	368c,u				178c,u	153–178c,u
Maximum Li+ concentration, cs, max(mol/m3)	28700a				36224a
Initial Li+ concentration, cs, in(mol/m3)	26130				3984.6
Solid diffusion coefficient, Ds(m2/s)	9.0 × 10−14 a,b,u	3.0 × 10−15– 3.0 × 10−13 a,b,u			3.0 × 10−15 a,b,u	10−16–10−14 a,b,u
Electric conductivity, σ, (S/m)	13.991a,b,u	10.571– 17.411a,b,u			23.797a,b,u	19.370– 28.224a,b,u
Reference kinetic coefficient, k0(m2.5/mol0.5 · s)	7.733 × 10−10 a,c,u	10−11–10−9 a,c,u			4.966 × 10−11 a,c,u	10−12–10−10a,c,u
Activation energy of solid diffusion, ED(kJ/mol)	35a,b,u	35–40.8a,b,u			40a,b,u	30–80a,b,u
Activation energy of reference kinetic coefficient, Er(kJ/mol)	53.4a,b,u	45–60a,b,u			30a,b,u	30–50a,b,u
Transfer coefficient, α	0.5				0.5
Solution phase
Initial electrolyte concentration, ce, 0(mol/m3)			1200a,g	10%a,g
Transference number, t+			0.363a	10%a
Activation energy of electrolyte diffusion, Ee(kJ/mol)			17.120a,b,u	16.99– 17.25a,b,u

^aRef. 12. ^bRef. 36. ^cEstimated: based on Ref. 46. ^gGaussian distribution, value represents standard deviation. ^uUniform distribution. Note: Percentage number represents the standard deviation from the expected value, and all physical constants are listed in Appendix A.

It is noticed that the anode thickness δ_an is not considered to be an identified parameter in Table II due to an assumption of the design capacity in the manufacturing process according to:

The reason that we choose a specific capacity ratio between cathode and anode in Equation 18 is to prevent lithium plating, resulting from the overpolarization at the anodes under accidental overcharging, adverse charging conditions after manufacturing, and this is always a self-defined standard in industry.⁴⁸ Based on Equation 18, we can derive a relationship between the cathode and anode thickness as:

Electro-active layer numbers N_layer can be described as

Thus, according to Equations 19 and 20, N_layer, δ_an and δ_ca has correlated equations. It means both N_layer and δ_an will change as well, when we change δ_ca with a fixed design capacity of large-format pouch cell with parallel layers, for example 10Ah. Meanwhile, we motivate to include manufacturing factors, such as the capacity balancing consideration, we keep this relationship in the following GSA. Therefore, in the following, we only choose δ_ca as a parameter, since it can represent the impacts of N_layer and δ_an.

In this work, we simulate a 1C discharge process with an initial and ambient temperature of 25°C. The cell discharge capacity and the local maximum cell temperature are defined as the outputs in the global sensitivity analysis. In the following we use a categorization strategy for the investigated parameter sets referred to the identification results of Zhang et al.,¹⁹ In detail, we categorize them regarding electrode and cell production relevant parameters and physical parameters. The cell-level model is implemented by ANSYS APDL 15.0, and the electrode-level model is executed by MATLAB with the SUNDIALS TB solver.⁴⁹ All simulation studies are run parallelized on three PCs with an 8-core I7–2600 processor with 16GB memory.

The parameter global sensitivities are quantified by analyzing the variation of the quantity of interest (QoI) on the entire parameter space through different methods. The Sobol' indexes,³⁰ well-defined and robust measures for GSA, are the percentages of the total variance of the QoI related to a single parameter or parameter interaction. For a more detailed illustration, the total variance of the QoI, e.g., the discharge capacity or the maximum cell temperature, is decomposed as:³⁰

This is a unique decomposition derived based on the assumption of independence for the parameters, which is the case in this work. V(QoI) is the total variance of the QoI, and the summands, with n_pbeing the number of parameters, are partial variances which are defined as variances of conditional expectations with respect to certain parameters and read as:

Here, Y and X represent the model output, i.e., the QoI, and the model parameters respectively. E( · ) and V( · ) are the mean and the variance of the model output in different parameter spaces. According to Equation 21, the last summand can be directly obtained by taking out other summands from V(QoI).

Finally, these partial variances are normalized by the total variance of the QoI resulting in:

where are then the Sobol' indexes which sum up to one.⁵³ The first-order index S_i represents the effect of a single parameter X_i on the model output. The higher-order index (e.g.,S_ij) describes the effect of the parameter interaction on the simulation results. The total number of first- and higher-order sensitivity indexes for the considered Li-ion battery model with 46 parameters are 2⁴⁶ − 1 = 7 × 10¹³, which leads to unaffordable computation costs. Alternatively, the total sensitivity index is typically used to describe the contribution of the single parameter X_i and joint effects with all other parameters and is defined as:

Frequently, only the total and first-order sensitivity indexes are calculated to keep the computational load manageable, which only have a total number of 2 × 46 = 92. However, the computational burden for implementing MC simulations to calculate the sensitivities is prohibitive, because a single simulation of the battery model is extremely expensive. Thus, the PCE concept is utilized to replace the CPU-intensive model and to calculate the sensitivities efficiently.

PCE has gained much attention in the field of sensitivity analysis due to its efficiency.^33,53 The general idea of PCE is to approximate a second-order model response, e.g., Y = G(X), by a linear combination of orthogonal polynomial basis functions⁵⁴ according to:

where {Ψ_k(X)}^{P − 1}_{k = 0} and {α_k}^{P − 1}_{k = 0} are multivariate polynomials and polynomial coefficients, respectively. P denotes the number of multivariate polynomials used to approximate the model response. It depends on the maximum order of the polynomials p and the size of the analyzed model parameters n_p as:

Moreover, the multivariate polynomials {Ψ_k(X)}^{P − 1}_{k = 0} are constructed by the product of the univariate polynomials⁵⁵ as:

where {ki}npi = 0 denotes the order of the univariate polynomials {ϕkii(Xi)}npi = 0. The univariate polynomials can be built based on the probability distribution of parameter Xi with different methods.55–57 The univariate and multivariate polynomials are mutual orthogonal leading to:

where m and n are the notation for the univariate and multivariate polynomials, respectively. δ_mn is the Kronecker delta function, which is 1 for identical values of m and n and 0 for the others. Based on the orthogonality, the statistical values of the model response, e.g., mean E( · ) and variance V( · ), can directly be calculated by the polynomial coefficients {α_k}^{P − 1}_{k = 0} according to:

Finally, the Sobol' indexes derived from the variance of the conditioned expectation can be obtained immediately as:

where the sets A_i, A_ij, A_Ti are defined as:

We included the PCE-based formula for the second-order indexes S_ij as they will be part of the following parameter studies. Note that Y has to be of finite variance which holds true for most engineering problems.

As the PCE coefficients {α_k}^{P − 1}_{k = 0} fully characterize the relation between the polynomials {Ψ_k(X)}^{P − 1}_{k = 0} and the model response Y, the determination of the coefficients plays an essential role in PCE-based global sensitivity analysis. Various methods are available to compute the PCE coefficients and are typically classified into two main groups: intrusive and non-intrusive methods. The intrusive method, i.e., stochastic Galerkin projection,⁵⁸ has the optimal accuracy compared to the non-intrusive methods. The Galerkin method, however, requires a tedious adaptation of the finite element code of the 3D battery model. Alternatively, non-intrusive methods, e.g., the spectral projection method⁵⁹ and the regression-based method,⁶⁰ have advantages in implementation as they do not change the structure of the numerical model. Blatman⁶¹ compared the efficiency for different non-intrusive methods and concluded that the conventional methods also have problems with complex systems regarding the number of model evaluations. To confront the situation in which a single simulation of the 3D battery model requires intensive computation, other approaches with higher efficiency could be an alternative. Blatman et al.⁶⁰ proposed an adaptive sparse PCE inspired by Efron's work.⁶² Here, the key idea is to select a small number of polynomial functions with strong relevance to the model response, followed by linear regression to calculate the active coefficients. Details regarding the computation of the adaptive sparse PCE are given in Algorithm 1. First, the corresponding model outputs of the N samples are evaluated and provided for the following selection and estimation procedure. Note that the size of the samples can be increased when the accuracy of the PCE model is not satisfied. For each iteration, the size of the multivariate polynomials is increased while the maximum order of polynomials increases from 1 to p_max. Within the iteration, the correlation ratios between the N sample outputs and the corresponding polynomials are calculated. In steps 6 and 7, the most correlated polynomial is selected and moved to the active set. The relevant coefficients of the polynomials in the active set are adapted in an equiangular direction.⁶² The optimal active set for each iteration is determined via cross validation. Finally, the coefficients for the optimal PCE model (Step 17), i.e., the optimal structure of the basic functions selected by least angle regression (LAR), are estimated through a linear regression approach. Note that the cardinality of the active set, i.e., the number of multivariate polynomials used for the final approximation, is typically much smaller than P for ordinary PCE (Equation 29). This can render the computational costs of the proposed sensitivity analysis affordable. The ratio between the cardinality of the active set and the original set (Equation 29) is defined as the index of sparsity (IS) which has an average value of 1.71 × 10^{− 7} in this work.

Algorithm 1. Hybrid least angle regression (LAR) based adaptive sparse PCE⁶⁰

1: Provide S = {X1, ⋅⋅⋅, XN} and calculate Y = G(S), {Ψk(S)}P − 1k = 0 for initialization
Estimation of polynomial coefficients
2: for p = 1: pmax do
3: Set a = {α0, ⋅⋅⋅, αP − 1} = 0, R = Y, Active set = {}, Basic set = {Ψk(X)}P − 1k = 0, m = 0
4: while m ⩽ min(N, P) do
5: k* = argmax |Corr(R, Ψk(S))|
6: Move basis function from the basic set to the active set
7: Calculate a correction term Δ
8: Update polynomial coefficients, a = a + Δ
9: Update residual function
10: Recalculate the coefficients for the active set with linear regression
11: Get mean approximation error via cross–validation procedure
12: m = m +1
13: end while
14: Store and the corresponding optimal active set
15: Stop if satisfies either the required accuracy or increases for the last two iterations
16: end for
17: Estimate relevant polynomial coefficients for the last optimal active set via linear regression
S = {X1, ⋅⋅⋅, XN}: Sample set with number of N for the parameters
pmax: Maximum order of the multivariate polynomials allowed for approximation
active set/basic set: Set of multivariate polynomials which is used/not used for approximation
Corr: Correlation between the residual R and the multivariate polynomials for sample set S = {X1, ⋅⋅⋅, XN}
Δ: The direction and size of the moving step for the polynomial coefficients, which calculated by62
Y = G(S), {Ψk(S)}P − 1k = 0 is the evaluation of the function and multivariate polynomials on sample set S

The main motivation of this work is to quantify the global sensitivities of cell material properties and cell design factors on the battery performance with a significant reduction in the computational cost by utilizing PCE combined with LAR. However, it might still be challenging to run global sensitivity analyses simultaneously for all 46 parameters and to derive meaningful inferences. The reason is that LAR aims at selecting significant basis functions gradually based on the number of provided samples. In other words, we might receive sensitivity values only for parameters that have significant impacts on the QoI. As the information about the parameters with low sensitivity might be concealed by the parameters with high sensitivity, the sensitivities of all 46 parameters (the material properties and the design factors) cannot be easily distinguished. A stepwise design for the sensitivity analysis is utilized to overcome this limitation, where a hierarchical analysis is carried out to categorize different groups of parameter sets a priori based on the results of the previous sensitivity analysis. According to the information about these sensitivities, if the design parameters and the physical property parameters do not have evident interactions but have different sensitivity magnitudes, we categorize them into two groups, i.e., the design and physical properties, and study them individually. If a sensitivity index of a parameter has an extremely high magnitude which leads to unmeasurable sensitivity indexes for the other parameters, then, to avoid the concealment of information about other parameters, this parameter is set to its nominal value in subsequent sensitivity studies.

In this work, a sample set S is generated according to the parameter specifications from Tables II and III by using Latin hypercube sampling. The battery model mentioned in the last section is evaluated for this sample set S. Based on the evaluated outputs, the PCE model is derived, and the global sensitivity analysis is calculated by running MATLAB with the toolbox UQLab.⁶³

The global sensitivity can easily evaluate the sensitivities of performance to a group of cell parameters based on the variation ranges for specific outputs, in this section, the output is utilizable discharge capacity. The utilizable discharge capacity is defined as the discharge capacity when cell voltage reaches the cutoff value 3 V. As shown in Figure 3, for the given parameter set and parameter range in Table II, an increase of the cathode thickness to the upper bound leads to a decrease of utilizable discharge capacity, and a decrease of cathode thickness to the lower bound as used in this work results in an increase of capacity. Figure 3 illustrates a significant capacity variation by changing the cathode thickness. This is not only due to effects of the cathode thickness itself, but also due to the previously mentioned correlation of δ_an, N_layer and δ_ca. When the cathode thickness δ_ca varies in the given range, the anode thickness δ_an and the number of active layers N_layer also changes dramatically according to Equation 41 and 42 to maintain a proper balancing of the cell and the required cell capacity. The layer number N_layer can change from 17 to 64. Since the pouch cell is made of stacked layers as shown in Figure 1, N_layer strongly affects the distributed current on each current collector, thereby the current density on each current collector. Consequently, it affects the lithium transport and cell capacity. The strong negative effect of thick electrodes also needs to be attributed to strong transport limitation, as reported also in the reference.⁶⁴ Understanding the sensitivities of parameters for cell performance is thus crucially important for optimization of cell design.

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In Figure 4, we show the global sensitivities of all parameters for the utilizable capacity after galvanostatic discharge at 1C. Here, a subset of 10 sensitivities is significant, and the sensitivity of the cathode thickness is much larger than the others as it determines the design capacity of the cell. The specific capacity of the cathode material and the solid diffusion coefficients in both electrodes have the next strongest effects on the discharge capacity. This indicates that the discharge capacity at 1C is controlled by the solid diffusion processes, because the ability of lithium ion transport in solid phases determines the electrochemical performance according to the previous model development section. From these identified parameters, three cathode design parameters obtain higher sensitivities. The cathode thickness mainly determines the methodology of cell design, since the variation of cathode thickness also causes strong variation of layer number and anode thickness. To better evaluate the sensitivities of the other parameters, the cathode thickness is fixed to its standard value in the following studies because the cathode thickness has the highest sensitivity of all for its strong correlated effect on battery performance.

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Next, after fixing the cathode thickness, all the other 45 parameters are divided into two groups for individual sensitivity studies. The first group represents the design parameters and includes 13 parameters as listed in the first part of Table II. They cover manufacturing factors and geometries, such as particle radius, porosity, current collector thickness, etc. In design parameters, we use Bruggeman coefficient b to represent tortuosity .^65,66 As the tortuosity of porous electrode can be described by Bruggeman relation, the Bruggeman coefficient can quantify tortuosity effect. The second group (32 parameters) contains the physical properties of the cell component materials, e.g., material densities, thermal conductivity, reaction kinetic parameters, etc. In Table IV, we illustrate the global sensitivities of the design parameters for the utilizable discharge capacity at 1C. Here, the current collector thickness and the cathode pore properties i.e., _ca, b_ca and _{s, ca} dominate the simulation outcome. An explanation of these sensitivities is that current collectors determine the local current density at the electro-active layers and the reaction kinetics. Furthermore, the ionic transport and the interface reactions in the electrodes are determined by the porosity and the pore structure. Thus, the cathode pore properties and its current collector thickness have more impact on the discharge capacity. In the investigated cell, the pore properties of the anode also show visible but weak effects on the discharge capacity, compared with the cathode pores. Therefore, on the utilizable discharge capacity at 1C of the investigated cell, the pore properties of both electrodes play an important role, and the cathode pores are more influential. A comparison of the first-order and total sensitivities of each design parameter reveals the apparent differences which indicates strong interactions with other parameters, especially of the current collector thickness and pore geometries.

Table IV. Global sensitivities of 13 design parameters for the utilizable discharge capacity at 1C after the cathode thickness δ_ca and the physical property parameters are fixed. S^C_i denotes first-order Sobol' sensitivity of utilizable capacity. S^C_Ti denotes total Sobol' sensitivity of utilizable capacity.

Parameter	SCi	SCTi
Ran	0.003	0.005
an	0	0.001
s, an	0.002	0.003
ban	0.003	0.007
δcc, an	0.117	0.129
δsep	0	0.003
sep	0.006	0.007
bsep	0.004	0.010
Rca	0	0.003
ca	0.271	0.401
s, ca	0.031	0.034
bca	0.139	0.269
δcc, ca	0.271	0.280

The parameter interactions are revealed in Figure 5. It shows that the Bruggeman coefficients b_ca, b_sep, porosities _ca, _sep, _an, and current collector thickness δ_ca, δ_an have strong interactions with other parameters. The current collector thickness of the cathode has a strong interaction only with the current collector thickness of the anode. Due to the high thermal conductivities of the battery components, the heat transfer inside the battery proceeds via them mostly. This heat affects the temperature homogeneity inside the battery, thus it affects the local electrochemical performance on each electrode. The pore geometries of the cathode, anode, and separator are the main variables that affect the ionic transport in the electrodes together. According to Equations 4 and 5, the electrochemical performance is influenced by ionic transport in every cell domain. Therefore, the pore geometries in the cathode, anode, and separator interact via the discharge capacity. The cathode particle radius R_ca also shows a clear interaction with the Bruggeman coefficient in the separator b_sep. This is because both parameters strongly determine the overpotentials from solid lithium diffusion process and ionic transport in the electrolyte. As the total overpotential of the cell is related to discharge capacity, R_ca and b_sep have a strong interaction for utilizable discharge capacity. Similarly an interaction between R_an and b_sep exists. The impact of interaction between R_ca and b_sep on utilizable discharge capacity is shown in Figure B1 in Appendix B.

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Figure 6 demonstrates the parameter dependency of the physical properties, in turn. The utilized discharge capacity of the cell is mainly determined by the solid diffusion coefficients of both electrode particles and the specific capacity of the cathode material, with smaller interactions compared with the design parameter group. According to Equations 4 and 5, the solid diffusion process directly affects the electrochemical performance. Based on our assumed manufacturing process, the specific capacity of the cathode determines the electro-active layer numbers and thus determines the current density of each current collector. Therefore, among the parameters in the physical property group, these three parameters have a significant impact on the cell discharge capacity. In addition to the design parameters, the remaining parameters of the physical property group do not have a significant influence on the discharge capacity as shown in Figure 4. Thus, we do not show the calculation of the parameter interactions for the physical property parameters. It is interesting to observe that the discharge capacity at low C-rates is less sensitive to the thermal properties of the cell component materials. It means that the temperature does not evidently affect the discharge process at low C-rates. Thus, for the investigated cell, the discharge capacity at 1C is quite sensitive to the electrode material design, including the structure and the shape.

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For a large-format pouch cell, the investigation of thermal behavior is also critical because the heat in larger cells may not be removed sufficiently fast, finally causing local overheating. Therefore, we consider the maximum cell temperature during operation as a second case for sensitivity analysis. It is a good candidate for safety management in large cells. The maximum temperature of the cell is located close to the tab in the center layer as shown in Figure 7. Based on temperature ranges in Figure 7, we can also conclude that the variation of cathode thickness has a strong impact on cell temperature evolution and variation.

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We evaluate the maximum cell temperature on position A displayed in Figure 7, at the end of the 1C discharge followed by the proposed stepwise design. In Figure 8, we show the global sensitivity results for all 46 parameters for the maximum cell temperature at 1C discharge. Compared with the utilizable discharge capacity, more parameters have significant impacts on the maximum cell temperature. Cathode thickness and pore size strongly affect the maximum temperature of the analyzed cell. The difference between the first-order and total sensitivities indicates significant parameter interactions, due to the strong correlated effect of cathode thickness. The maximum cell temperature is also apparently influenced by the anode pore size, the particle size, the electric conductivities of active materials, and the electrolyte transport properties. Based on Equations 7 to 10, the ionic transport in the electrolyte, the electron conduction in electrodes, and interface reactions mainly contribute to the ohmic and reaction heat, respectively, during operation. These physical processes are affected by the electrodes and their pore structures, the electrolyte properties, and the electric properties of the electrodes. In contrast to the discharge capacity, more parameters evidently affect the thermal behavior of the cell. Physical property parameters still play a minor role in the thermal performance of the cell. Note that most of the sensitive parameters significantly interact, as shown in Figure 8. Therefore, it is necessary to evaluate parameter interactions in both groups as well.

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For a first step, for avoiding strongly correlated parameter and studying design parameter set, we fixed the physical property parameters and the cathode thickness and vary the design parameters exclusively. In Table VtA1tA2, we illustrate the global sensitivities of 13 design parameters for the maximum cell temperature at the 1C discharge. In contrast to the sensitivities for the discharge capacity, all 13 design parameters for the maximum cell temperature have visible total sensitivities, and the cathode pore structure and the particle size have the largest effects among them. It indicates that all design factors are involved in the thermal behavior of the cell individually or indirectly, and the cathode design factors dominate. The pore geometries and the particle radius in cathodes indicate that the generated heat of the battery mainly results from the ohmic and reaction heat during operation. When the first-order and total sensitivities are compared, there are significant parameter interactions for the maximum cell temperature.

Table V. Global sensitivities of 13 design parameters for the maximum cell temperature at 1C after the cathode thickness δ_ca and physical property parameters are fixed. S^T_i denotes first-order Sobol' sensitivity of the maximum cell temperature. S^T_Ti denotes total Sobol' sensitivity of the maximum cell temperature.

Parameter	STi	STTi
Ran	0.005	0.012
an	0	0.015
s, an	0.005	0.023
ban	0.0246	0.039
δcc, an	0	0.010
δsep	0	0.003
sep	0.006	0.014
bsep	0.011	0.024
Rca	0.333	0.346
ca	0.178	0.294
s, ca	0.250	0.279
bca	0.015	0.133
δcc, ca	0	0.020

Note that the cathode thickness has a strong correlation with the other parameters regarding capacity sensitivities; see Figure 4 and Figure 9. This strong correlation might result in different sensitivity measures, when we divide the model parameters into two groups and assume a fixed value for cathode thickness. We, therefore, used the expected value of the cathode thickness to provide a representative sensitivity analysis of the individual group of the design parameters. A better understanding of the effect of the design parameters, in turn, is mandatory for an effective cell design and is discussed in more detail below.

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The sensitivity matrix in Figure 9 reveals the parameter interactions for the maximum cell temperature. It is evident that there are more interacting parameters compared with the discharge capacity output in Figure 5. The porosities and the Bruggeman coefficients determine the pore structure and the pore volume. As mentioned earlier, at the electrode level, the heat generation is determined by the ionic transport and the electrode–electrolyte interfacial reactions. The reason is that the pore size and the structure determine the lithium ion transport in the solid and solution phases, and the electrode particle size influences the specific area and the reaction kinetics at the interfaces. The combined action finally affects the heat generation of the cell during discharge. Moreover, the thickness of the current collectors also interacts with the pore size and the electrode microstructure. This results from the current density and the heat transport through the current collector. As the current collectors distribute and collect the current to the solid and solution phases, the thickness of the two current collectors influences local heat generation and heat transfer, and this directly affects the maximum cell temperature.

At the cell level, it is interesting to see that the pore structure of the anode interacts with that of the cathode in Figure 9. It is because the pore geometries of both electrodes determine the solid volume of the whole battery, and thus, the number of electro-active layers and the heat conductivity of the battery, based on a fixed nominal capacity of the cell. If the pores are bigger and have more space, the total number of layers should be larger, and this results in a worse heat transfer from the inside to the outside, and it finally determines the maximum temperature of the cell.

Unlike the design parameters, significant sensitivities of the physical property group concentrate on only three parameters, as shown in Figure 10. The maximum cell temperature is most sensitive to the solid diffusion coefficient in the cathode particles. It indicates that the overpotential from the solid diffusion process in the cathode mainly contributes to the heat generation of the cell according to Equations 7 to 10. The specific capacity of the cathode material shows a visible but smaller sensitivity. This indicates that specific capacity of cathode material can also affect the accumulation of heat in the cell.

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GSA reliably works for assigning confidence in model prediction and investigation of less know parameters that result in large uncertainties with small parametric deviations in highly non-linear systems.²³ In non-linear Li-ion battery systems, it includes many uncertain parameters, such as electrode particle sizes, porosity, electrode structures and thermodynamics. In the case of complex, non-linear systems with large amount of parameters, GSA can effectively and efficiently quantify:

• (1)  
  the relationship between parameters and objective outputs, for example battery performance,
• (2)  
  parameter uncertainties for more reliable and accurate model predictions,
• (3)  
  the parameter identifiability for simplification of complex model with high computational cost.

With these functions, more advanced models for large-format Li-ion batteries can be developed and improved by effectively identifying relevant uncertain parameters, thereby investigating and predicting detailed physical processes such as local temperature and discharge capacity, etc. In engineering perspectives, it will contribute much to design and optimization of different lithium-ion battery systems, and development of efficient on-line monitoring models for fast prediction and estimation of practical battery systems, for example, electric vehicles, large-format consumer electronic cells, battery management systems, etc.

A global sensitivity analysis of a 3D multiphysics model for a Li-ion cell was performed to analyze impacts of 46 parameters on cell performances including discharge capacity and temperature variation. For this, an efficient PCE method was implemented with a stepwise procedure. To overcome the difficulties of large computational costs due to the number of parameters, a sparsity-based and highly efficient version of PCE was applied. The number of random cases was drastically decreased, and the computational efficiency was improved. In this vein, 46 parameters of the 3D multiphysics model were efficiently identified and investigated in two studies, namely sensitivity to thermal performance by analyzing the maximum local cell temperature and sensitivity to electrochemical performance by analyzing the utilizable discharge capacity at 1C. Through the stepwise design approach, the cathode thickness was identified as the most sensitive parameter for the discharge capacity and the maximum cell temperature. Moreover, it was found that the discharge capacity and the maximum cell temperature at 1C were sensitive to nine design parameters, including the electrode particle sizes and the pore geometries of the investigated cell, and the parameters strongly interact with each other. For the discharge capacity, the cathode design parameters play an important role. The design parameters of all cell components are visibly sensitive to the maximum cell temperature as well. Within the physical property cluster, only the solid diffusion coefficient of the electrode particles and the specific capacity of the cathode material were sensitive to the utilizable discharge capacity. It shows that the utilizable capacity is related to electrode geometries. In contrast, the thermal behavior of the cell is affected not only by the electrode geometry but also by the cell geometry and the manufacturing process, which determine the thermal management of the battery.

To sum up, a global sensitivity study using the Sobol' indexes with a PCE approach is successfully and efficiently used for a 3D multiphysics model of a large-format Li-ion battery. This work also inspires future research. One aspect is the investigation of the performance of other efficient approaches for constructing CPU-friendly surrogate models, e.g., low rank approximation concepts. The other aspect is effective improvement in the efficiency of the cell design and battery testing process, i.e., only to concentrate varying highly sensitive parameters for cell performances, it is easy to optimize a well-managed cell design for higher energy/power densities and evaluate hot spots inside the cell to reduce safety risks. Moreover, battery tests can be more effectively designed based on monitoring outputs and highly sensitive parameters.