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A Repeated Measurements Approach to $SoH$ Battery Modelling of Cyclic Aged Data in a Laboratory Environment

This paper presents a novel first-order linearised nonlinear repeated measurements model using regularised iterative generalised least squares to accurately predict battery State of Health (SoH) with a precision of ±0.191% by distinguishing between measurement noise and cell-to-cell variation in laboratory-aged cyclic data.

Original authors: Mark Cary, Charles Bokor

Published 2026-08-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Mark Cary, Charles Bokor

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Batteries are the silent engines of the modern world, powering everything from the phones in our pockets to the cars on our roads. Yet, like all living things, they age. Over time, the chemical reactions inside a battery cell degrade, causing it to hold less energy than it did when new. Scientists call this the "state of health," a measure of how much life remains in a battery before it fails. Understanding exactly how and why this happens is crucial for building better, safer, and longer-lasting energy storage. In a laboratory setting, researchers can watch this process unfold by subjecting battery cells to repeated cycles of charging and discharging, much like a heart beating. However, watching a single battery age is not enough; to understand the true nature of battery life, scientists must look at many batteries at once, tracking how they change over time and how they differ from one another. The challenge lies in the data itself: it is not just a collection of random points, but a series of connected stories, where each battery has its own unique path of decline, influenced by both the conditions of the test and its own internal quirks.

In a recent study, researchers from Loughborough University and Oxford Brookes University tackled this complexity by developing a new way to model how batteries age. They worked with ten specific battery cells, all made with a nickel-cobalt-aluminum cathode and a graphite-silicon anode, which are common in high-performance applications. These cells were placed in a controlled environment at a steady temperature of 25 degrees Celsius and subjected to a rigorous testing protocol. Every cell was discharged at a constant rate, but the charging current was varied, with some cells charged at 1 ampere and others at higher rates up to 5 amperes. Every fifty cycles, the researchers paused to measure the cell's capacity, calculating exactly how much energy it had lost compared to its original state. This process continued until the cells failed, generating a rich set of data that showed not just how much capacity was lost, but how the rate of loss changed over time for each individual cell.

The researchers found that the aging of these batteries followed a predictable pattern that could be described by a simple mathematical relationship known as a power law. This means that the loss of capacity does not happen in a straight line; instead, it accelerates or decelerates in a specific way depending on how the battery is used. However, a simple model that looks at each battery in isolation misses a critical piece of the puzzle. It fails to account for the fact that while every battery is unique, they all share a common history and are subject to the same physical laws. The team realized that the data contained two distinct types of variation. The first was the small, random noise that occurs during every measurement, like a slight tremor in a ruler. The second was the genuine difference between cells, where one battery might age slightly faster or slower than another even under identical conditions. To capture this, they built a hierarchical model, a structure that treats the data as a set of nested stories: the individual story of each cell sitting within the broader story of the entire group.

To make sense of this complex structure, the researchers employed a sophisticated statistical approach that allowed them to separate the noise from the signal. They used a method that essentially smooths out the random errors while preserving the true differences between the cells. This technique, which involves a process called regularization, helps prevent the model from overreacting to minor fluctuations in the data, ensuring that the final picture is clear and reliable. They also developed a way to automatically spot and remove data points that were clearly wrong or outliers, such as a measurement that suddenly jumped far off the expected path, which could skew the results. By doing this, they ensured that their model was built on the most accurate representation of reality possible.

The results of this approach were strikingly precise. The model was able to predict the state of health of the batteries with an accuracy of within 0.191 percent for the specific training data presented. This level of precision is significant because it demonstrates the model's ability to fit the observed laboratory data effectively. Furthermore, the model provided a way to calculate confidence intervals, which are ranges that show how certain the prediction is. For example, the researchers could say with high confidence that a battery's remaining capacity would fall within a very narrow band, giving engineers a reliable tool for planning. The study also revealed that the differences between the cells were quite small, suggesting that when manufactured under controlled conditions, these batteries are remarkably consistent. However, the model also highlighted that the rate of aging is heavily influenced by the charging current, with higher currents leading to different patterns of degradation.

This work represents a significant step forward in how we understand battery aging. By treating the data as a collection of repeated measurements rather than isolated points, the researchers created a model that reflects the true structure of the aging process. It acknowledges that while every battery has its own personality, they all follow the same fundamental rules. The model is a practical tool for analyzing controlled laboratory data, but the researchers note that applying it to real-world scenarios requires modification. In real-world applications, such as electric vehicles, the ageing stress applied to the battery pack varies with environmental conditions, charging practices, and customer usage. Consequently, the model structure needs to be adapted to handle these time-dependent variables. While the study was conducted in a controlled laboratory with constant conditions, the researchers suggest that extensions to time-dependent covariate protocols may address the real-world in-service analysis problem. The ability to accurately predict when a battery will fail is a powerful capability, one that could lead to safer, more efficient, and longer-lasting energy systems for the future. The study confirms that with the right mathematical tools, the complex and often unpredictable journey of a battery's life can be mapped with remarkable clarity, provided the model is appropriately adjusted for the specific conditions of its use.

Technical Summary: A Repeated Measurements Approach to SoH Battery Modelling of Cyclic Aged Data

Problem Statement
Laboratory studies assessing the cyclic ageing of battery cells generate longitudinal data, characterized by repeated State of Health (SoH) measurements taken over the life of individual cells. A critical challenge in analyzing this data is the presence of two distinct sources of variation: intra-individual variation (measurement noise within a single cell) and inter-individual variation (cell-to-cell or test-to-test variability). Traditional approaches often fail to explicitly model these hierarchical variance components simultaneously, particularly when dealing with nonlinear ageing profiles. Furthermore, standard two-stage regression methods require complete ageing profiles for each cell to estimate individual parameters before modeling population trends, which can be inefficient and sensitive to outliers.

Methodology
The authors propose a hierarchical nonlinear statistical model based on a first-order linearised repeated measurements approach. The methodology is structured in two stages:

Stage 1 (Intra-cell Model): The ageing profile for the $i$ -th cell is modeled as a nonlinear power law function of ageing time ( $x$ ), specifically $y_{ij} = B_i x_{ij}^{1/A_i} + e_{ij}$ . The random error term $e_{ij}$ represents measurement noise and is assumed to be normally distributed with a common variance scale parameter $\sigma^2$ .
Stage 2 (Inter-cell Model): The variation in the power law parameters ( $A_i$ and $B_i$ ) across different cells is modeled as a function of the applied charging current ( $a_i$ ). This relationship is captured using a single-knot cubic B-spline basis. The parameters are treated as mixed effects, allowing for both systematic trends driven by the charging current and random cell-to-cell deviations.

Parameter Identification and Regularization
To estimate the model parameters, the authors employ a novel Regularised Iterative Generalised Least Squares (RIGLS) scheme. Because the marginal likelihood integral for nonlinear mixed-effects models lacks a closed-form solution, the authors utilize a first-order Taylor series expansion (linearization) of the model function about the expected value of the random effects.

Key technical features of the identification process include:

Hyper-parameter Optimization: A ridge regression term ( $\lambda \theta^T \theta$ ) is introduced to the negative log-likelihood function to prevent overfitting. The regularization parameter $\lambda$ is dynamically re-estimated at each iteration using a fixed-point iteration formula derived from the Bayesian Information Criterion (BIC).
Outlier Detection: Prior to the main fitting procedure, a preliminary fit is conducted using the Fast Least Trimmed Squares (FAST-LTS) algorithm. This robust method identifies and excludes aberrant observations (outliers) that could unduly influence the parameter estimates.
Confidence and Prediction Intervals: The Delta method is applied to derive analytical expressions for confidence and prediction intervals, accounting for both parameter uncertainty and random measurement variability.

Case Study and Results
The methodology was applied to experimental data from 10 Nickel-Cobalt-Aluminum (NCA) cylindrical cells aged under controlled laboratory conditions (25°C). The cells were subjected to constant discharge currents (2 A) and varying charge currents (1 A to 5 A).

Data Processing: One cell was identified as an outlier during the Level-2 analysis and excluded. Additionally, the FAST-LTS procedure identified specific aberrant data points within individual cell profiles (e.g., early-life anomalies or end-of-life rapid decay) which were removed prior to final modeling.
Model Performance: The full hierarchical model converged rapidly (within 2–3 iterations). The training Root Mean Square Error (RMSE) for the capacity loss prediction was 0.191% for SoH values in the range [0, 20].
Variance Analysis: The estimated level-1 variance scale parameter ( $\sigma^2$ ) was 0.00772, indicating high accuracy in the power law fit. The level-2 covariance matrix revealed a strong negative correlation (-0.743) between the power law parameters $A$ and $B$ , validating the necessity of the joint estimation approach over assuming independence.
Interval Estimation: The model successfully generated narrow 95% confidence intervals and wider prediction intervals, demonstrating the ability to quantify uncertainty in both parameter estimates and future observations.

Significance and Claims
The paper claims to present the first application of a nonlinear repeated measurements approach specifically for cyclic ageing SoH modelling. The primary significance lies in the model's ability to:

Reflect Data Structure: Explicitly account for both intra-cell (noise) and inter-cell (variability) variance components within a unified hierarchical framework.
Efficient Identification: Utilize the RIGLS approach with automatic hyper-parameter tuning to identify complex nonlinear models without requiring complete ageing profiles for every cell (though complete profiles were available in this study).
Robustness: Integrate robust outlier detection (FAST-LTS) and regularization to handle noisy experimental data and prevent overfitting.
Interpretability: Provide a fully parametric model that is scientifically interpretable, offering realistic confidence and prediction intervals essential for battery end-of-life prediction.

The authors note that while the model is effective for controlled laboratory data, future work would need to address time-dependent covariates to adapt the framework for real-world electric vehicle applications where ageing stresses vary dynamically.

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