[ Paper Review ] Electrochemical–mechanical coupled model for computationally efficient prediction of long-term capacity fade of lithium-ion batteries

🔗 Electrochemical–mechanical coupled model for computationally efficient prediction of long-term capacity fade of lithium-ion batteries

• Journal : Journal of Energy Storage ‘24 (IF 8.9)

 

[ summary ]

• 제안 모델 : ISIF model, SVD-ISIF model
  • Aging에 영향을 주는 mechanical fatigue fracture of cathode particles 모델링
  • ISIF ( inhomogeneous stress-induced fracture)
    • Electrochemical–mechanical coupled capacity fade model
    • Knee point 포함한 long-term capacity degradation 예측 가능
    • Electrochemical-based model과 다르게 time-consuming calculation 필요 없음
  • SVD-ISIF : (hybrid model) ISIF model + Sparse variational dropout Bayesian neural network (SVDBNN)
• Long-term capacity fading data (3000 to 20,000 cycles)의 실험 데이터로 검증

 

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1. Problem Statement

[ Existing method ]

1. Electrochemical model

• Strength : Reflect various physical phenomena using physical parameters and principles
• Weakness : Cannot be generalization, Difficult to solve quickly
  • It composed of many physical parameters & complex coupled nonlinear PDE

2. Empirical model

: Designed to reflect empirically obtained key features of capacity fade trends

• Strength : Small number of parameters & simple equations
• Weakness : Too simple to reflect the effects of capacity fade patterns (e.g. knee point)

3. Data-driven capacity model

• Strength : Capture complex aging dynamics through implicit analysis of data
• Weakness : Inability to predict the entire lifespan, Require large amounts of data

 

 

[ Proposed method ]

1. ISIF (Inhomogeneous stress induced fracture) model

• Electrochemical–mechanical coupled capacity-fade model
• Model mechanical fatigue fracture of cathode particles
• Predict long-term capacity-fade including knee-point

2. SVD-ISIF model (hybrid model)

• (SVDBNN) Sparse Variational Dropout Bayesian neural network → data-driven model
• Improve accuracy & data efficiency, especially when data is abundant

 

 

2. Contribution

• Ability to predict the capacity fade over the entire battery lifespan, long-term (3000 to 20,000 cycle) capacity fade including the ‘‘knee point’’
• Ability to predict the capacity fade using only the operating conditions, without early cycle experimental data
• Smaller computational burden and fewer model parameters than electrochemical models

 

 

3. Methodology

Total capacity fade = capacity fade in cathode + capacity fade in anode

• In cathode, Fatigue fracture of cathode particles → LAM (Loss of Active Material)
• In anode, SEI layer formation, lithium plating on the surface of the anode particles → LLI (Loss of Lithium Inventory)

In this paper, express ‘Amount of capacity fade’ using ‘RCL (relative capacity loss)’

 

(2) Total RCL (n-th aging cycle) = RCL by cathode degradation + anode degradation

 

 

[ ISIF model scheme ]

• Input : Aging cycle condition & the number of aging cycles (N)
• Output : RCL

 

3.1  Capacity fade caused by cathode degradation ( $\Delta RCL_n^+$ )

: In cathode, Fatigue fracture of cathode particles → lead LAM (Loss of Active Material)

: Fatigue fracture is the result of cyclic stress (due to lithium intercalation / deintercalation)

 

 

3.1.1  Fatigue fracture in solid materials

• (3) S-N curve equation

• (4) Fatigue damage fraction

• (5) Cumulative damage (from 1 to n-th aging cycle)

 

 

 

3.1.2 Fatigue damage of individual cathode particles

• (6) Surface tangential stress (at time t with radius R) on cathode particles

• (7) Maximum tangential stress that causes crack generation (set $ k_s^+$ as variable)

• (8) Define function $ f(C_{chg,n})$ that fits 2-dimensional polynomial of $ C_{chg,n} $

• (9) Update stress range applied to the particle of radius R at the n-th aging cycle

• (10) Update fatigue damage fraction for radius R particle due to the n-th single aging cycle

• ​(11) Update Cumulative damage during aging cycle

 

 

 

3.1.3 Fatigue fracture of entire cathode particles

 

1️⃣ time before fatigue fracture / 2️⃣ pre-knee point / 3️⃣ post-knee point / knee point (n=k)

 

• Assume : radius R of cathode particles within range $[R_{min}, R_{max}]$ (Fig.5 (b)
  • The larger electrode particles suffer from more fatigue damage → $D_n(R)$ is monotonically increasing function (Eq.11)
  • $D_n(R)$ lead to the update of $R_{th,n}$(the smallest fractured cathode particle in the n-th aging cycle)

 

• (12) Particle number distribution & (13) Normal distribution of particle number according R

 

 

3.1.4 Capacity loss due to cathode degradation

 

• (14) Reduced proportion of room for storing lithium in cathode particles

• (15) RCL caused by cathode degradation

• (16) Total active surface area of the cathode particles

 

 

 

As aging : With decrease $R_{th,n}$ → decrease $A_{act, n+1}$ → increase $S_n(R)$ in Fig.5

• $R_{th,n}$ : n-th cycle에서 cathode particle Radius
• $A_{act, n+1}$ : n+1 cycle에서 active particle의 Area
• $S_n(R)$ : n-th cycle에서 cyclic stress range (S)

 

 

 

3.2 Capacity fade caused by anode degradtion ( $\Delta RCL_n^-$ )

: In anode, SEI layer formation, lithium plating on the surface of the anode particles

• (17) Aging density function

• Change in RCL during (18) charging, (19) discharging, (20) rest

• (21) Total change in the RCL due to anode degradation in the n-th aging cycle

 

 

 

3.3 SVD-ISIF model : A hybrid capacity fade model

 

Hybrid model : ISIF + Sparse Variational Dropout Bayesian Neural Network (SVD)

• Improve accuracy
• Provide uncertainty of the prediction

Sparse Variational Dropout Bayesian Neural Network (SVDBNN)

• Strength : better generalization performance than traditional Bayesian Neural Networks (BNNs) due to its sparser network structure
• Network size : 6-400-300-1
• Batch size : 200 / Epoch of learning : 10

 

[ SVD-ISIF model scheme ]

• Input : Aging cycle condition & the number of aging cycles (N)
• Output : RCL & uncertainty of the prediction result

 

4. Experiment Result Anaylsis

[ Experimental Setup ]

• Long-term (>3000 cycles) experimental data of 2170 NCM batteries
• 25 ◦C inside a constant temperature and humidity chamber
• Use RCL (relative capacity loss) to express capacity fade
• Profile : 20 types of different aging cycle (CCCV)
  • Cycle (#8-9. #18-20) has knee points (Fig. 2)

20 Types of Aging cycle

 

• Configuration of Profile
  : Consisted of repeated aging cycles and periodic reference performace tests (RPTs)
  
  • RPTs: To measure capacity periodically

RPTs in a test profile

• Aging cycle test (single cycle )
  • charging in constant current(CC)–constant voltage(CV) mode
  • 10-min rest
  • discharging in CC mode
  • 10-min rest

[ Result Analysis ] 

1. Computational Time

• #12 aging cycle condition, which had a total length of 14,613 cycle
• Compare simulation time
  • P2D+SEI : P2D model incorporating capacity fade due to SEI layer formation
  • P2D (SEI+CRACK) : P2D model incorporating capacity fade due to SEI layer formation and particle cracking
  • ISIF model

• ISIF model has a lower computational burden
  • ISIF model focus on key aging mechanisms for predicting capacity fade

 

2. Prediction Accuracy

 

[ Experiment Setup ]

• K-fold cross-validation (K=20)
  • Among 20 time-series experimental datasets, train dataset : 19 / test dataset : 1
• Model : ISIF model, SVDBNN, SVD-ISIF model
• Metric : MAE(Mean Average Error) & MxAE(maximum absolute error)

 

1. Comparing ISIF model and SVDBNN

• Both MAE and MxAE show similar values
• ISIF shows uniform estimation performance across all aging cycle conditions
• SVDBNN shows very low prediction accuracy for data with specific aging cycle condition or knee points → Fig. 8 ( Aging cycle condition #9, #19 )
• Only SVDBNN fails to predict the tendency of knee point → impractical to use SVDBNN

 

2. SVD-ISIF model

• SVD-ISIF model shows high estimation performance at both MAE, MxAE
• Demonstrate that combining ISIF and BNN can reduce overfitting

 

5. Conclusion

• ISIF model
  • Predict long-term capacity fade with small computation burden
  • Not require early cycle data → only require the aging cycle condition

ISIF model	SVD-ISIF model
small computation burden	higher computation burden, better generalization
lower prediction accuracy	higher prediction accuracy

 

Limitation

• Not reflect the effects of temperature, different cathode materials, transient particle stresses