Reliability-based design optimization of a pouch battery module using Gaussian process modeling in the presence of cell swelling

Abstract

A pouch battery pack includes multi-stacked battery module structures that protect the inner pouch battery cells from external hazards and deformation that may arise due to swelling effects. Recent research has found that the stack pressure, which is the suppressing force on the battery cells inside the battery module structure, has a significant impact on the degree to which the state-of-health (SOH) degrades and amount that the mechanical properties of pouch batteries change. Consequently, it is important to optimize the battery module structure design with consideration of the SOH and the structural reliability. To identify how significantly design affect the SOH and the mechanical properties, experiments under different levels of initial stack pressure and uncertainty quantification using Gaussian process are explored in this research. Reliability-based design optimization for the pouch battery module optimize the structural design that minimizes volume while satisfying structural reliability and SOH requirements. This work suggests a data-driven approach for achieving reliability-based design using experiment. Further, this research suggests formulations to calculate the performance functions, which are significant factors for reliable design of pouch battery modules.

Appendices

Appendix 1: Statistical moments of SOH

Derivation of the statistical moments of the SOH model by the statistical moments of the capacity fade function.

$$\begin{aligned} {\text{E}} [{\text{SOH}} (t;\;SP_{initial} )] & = \{ 1 - {\text{E}} [F_{Cap} (t;\;SP_{initial} )]\} \times 100\% = \left\{ {1 - {\text{E}} \left[ {\int_{0}^{t} {f_{Cap} (t;\;SP_{initial} )dt} } \right]} \right\} \times 100\% \\ & = \left\{ {1 - \int_{0}^{t} {{\text{E}} [f_{Cap} (t;\;SP_{initial} )]} dt} \right\} \times 100\% = \left\{ {1 - \sum\limits_{t = 1}^{n} {{\text{E}} [f_{Cap} (t;\;SP_{initial} )]} } \right\} \times 100\% \\ \end{aligned}$$

$$\begin{aligned} {\text{E}} [SOH^{2} (t;\;SP_{initial} )] & = \{ {\text{E}} [1 - 2F_{Cap} (t;\;SP_{initial} ) + F_{Cap}^{2} (t;\;SP_{initial} )]\} \times (100\% )^{2} \\ & = \{ 1 - 2{\text{E}} [F_{Cap} (t;\;SP_{initial} )] + {\text{E}} [F_{Cap}^{2} (t;\;SP_{initial} )]\} \times (100\% )^{2} \\ & = \left\{ {1 - 2{\text{E}} \left[ {\int_{0}^{t} {f_{Cap} (t;\;SP_{initial} )dt} } \right] + E\left[ {\int_{0}^{t} {f_{Cap}^{2} (t;\;SP_{initial} )dt} } \right]} \right\} \times (100\% )^{2} \\ & = \left\{ {1 - 2\sum\limits_{t = 1}^{n} {{\text{E}} [f_{Cap} (t;\;SP_{initial} )]} + \sum\limits_{t = 1}^{n} {{\text{E}} [f_{Cap}^{2} (t;\;SP_{initial} )]} } \right\} \times (100\% )^{2} , \\ \end{aligned}$$

$$\begin{aligned} {\text{Var}} [SOH(t;\;SP_{initial} )] & = {\text{E}} [SOH^{2} (t;\;SP_{initial} )] - {\text{E}} [SOH(t;\;SP_{initial} )]^{2} \\ & = \left\{ {1 - 2\sum\limits_{t = 1}^{n} {{\text{E}} [f_{Cap} (t;\;SP_{initial} )]} + \sum\limits_{t = 1}^{n} {{\text{E}} [f_{Cap}^{2} (t;\;SP_{initial} )]} } \right\} \times (100\% )^{2} \\ &\quad - \left( {1 - \sum\limits_{t = 1}^{n} {{\text{E}} [f_{Cap} (t;\;SP_{initial} )]} } \right)^{2} \times (100\% )^{2} \\ & = \left\{ {\sum\limits_{t = 1}^{n} {{\text{E}} [f_{Cap}^{2} (t;\;SP_{initial} )]} - \left( {\sum\limits_{t = 1}^{n} {{\text{E}} [f_{Cap} (t;\;SP_{initial} )]} } \right)^{2} } \right\} \times (100\% )^{2} \\ & = \sum\limits_{t = 1}^{n} {{\text{Var}} [f_{Cap} (t;\;SP_{initial} ) \times 100\% ]} . \\ \end{aligned}$$

Appendix 2: R²

The formulation of the R² consists of the variability in the response and the outputs. The sum of the variability of the response (S_yy) has a variability of the y_i (SS_R) and the residual of the variability left (SS_E).

$$R^{2} = \frac{{SS_{R} }}{{S_{yy} }} = 1 - \frac{{SS_{E} }}{{S_{yy} }} = \frac{{\sum\limits_{i = 1}^{n} {(\hat{y}_{i} - \overline{y})^{2} } }}{{\sum\limits_{i = 1}^{n} {(y_{i} - \overline{y})^{2} } }}.$$

Mean absolute error (MAE) and root mean square error (RMSE) is a measure that quantify the error with mean of absolute differences and root mean square of differences.

$$MAE = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {\hat{y}_{i} - y_{i} } \right|} \quad {\text{and}}\quad RMSE = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {\hat{y}_{i} - y_{i} } \right)}^{2} } .$$