Internal resistance is modeled as a time-varying condition-monitoring signal whose trajectory reveals battery degradation. For battery (i) at aging time (t), a common model is
[ CM_i(t)=R_i(t)=b_{i0}+b_{i1}t+b_{i2}t^2+e_i(t) ]
where (R_i(t)) is the measured internal resistance, the coefficients describe the battery’s degradation trajectory, and (e_i(t)) represents measurement noise.
The key idea is to model both the average resistance-growth trend and the cell-specific deviations from that trend. A mixed-effects regression captures differences between cells while allowing the fitted resistance trajectory to support life, reliability, and hazard analysis.
How Internal Resistance Becomes a Monitoring Signal
Resistance is measured from a controlled electrical response
Internal resistance is not usually observed directly; it is estimated from the voltage response to a known current perturbation. For a brief discharge pulse,
[ R_i(t)=\frac{V_{\text{rest},i}(t)-V_{\text{loaded},i}(t)}{I_i(t)} ]
where (V_{\text{rest}}) is the voltage before the pulse, (V_{\text{loaded}}) is the voltage during the pulse, and (I) is the applied current.
A typical test applies a high-current pulse after the cell has rested sufficiently for the chosen test protocol. The resulting resistance estimate becomes one observation in the cell’s degradation history.
Resistance growth indicates aging
As a battery degrades, its internal resistance commonly increases. This increase reduces power capability because a larger voltage drop occurs under load:
[ \Delta V \approx I R ]
Thus, resistance provides a condition-monitoring signal that is directly related to the cell’s ability to deliver current without excessive voltage loss.
Testing conditions must be controlled
Resistance depends not only on degradation, but also on temperature, state of charge, pulse duration, current level, and rest time. Calendar-life testing therefore measures resistance under defined temperature and SoC conditions so that changes over time are not incorrectly attributed to aging.
The Mixed-Effects Resistance Model
Fixed effects describe the population trend
A quadratic condition-monitoring model can be written as
[ CM_i(t)=b_{i0}+b_{i1}t+b_{i2}t^2+e_i(t) ]
The coefficients have practical interpretations:
- (b_{i0}): estimated initial resistance or baseline level.
- (b_{i1}): approximately linear resistance-growth rate.
- (b_{i2}): curvature in the degradation trajectory.
- (e_i(t)): random measurement or model error.
The quadratic term allows the resistance-growth rate to change with age rather than assuming constant degradation.
Random effects describe cell-to-cell variability
The coefficient vector for cell (i) is
[ \mathbf b_i= \begin{bmatrix} b_{i0}\ b_{i1}\ b_{i2} \end{bmatrix} ]
and is commonly modeled as
[ \mathbf b_i \sim \mathcal N(\boldsymbol\mu_b,\mathbf\Sigma_b) ]
Here, (\boldsymbol\mu_b) represents the average battery trajectory, while (\mathbf\Sigma_b) represents variability between cells.
This variability may arise from manufacturing differences, including electrode pressing and cell assembly. Two nominally identical cells can therefore begin with different resistances or degrade at different rates.
Measurement noise completes the model
The error term is often assumed to follow a normal distribution:
[ e_i(t)\sim \mathcal N(0,\sigma_e^2) ]
This separates random measurement variation from the underlying resistance trajectory. In a more general implementation, the noise variance can be allowed to change with operating condition or aging time if the data show nonconstant variance.
How the Model Supports Degradation Analysis
Fitted trajectories smooth noisy observations
Individual pulse measurements can fluctuate because of test repeatability and operating conditions. Fitting the mixed-effects model estimates the latent resistance trajectory rather than treating every measured point as a definitive change in health.
The result is a smoother, statistically interpretable condition-monitoring signal for each cell.
The coefficients provide degradation features
The fitted coefficients can be used as health-related features:
- A high (b_{i0}) indicates elevated initial resistance.
- A high (b_{i1}) indicates rapid early resistance growth.
- A positive (b_{i2}) indicates accelerating resistance growth.
- The full fitted curve estimates resistance at unobserved aging times within the model’s valid range.
These features allow analysts to compare cells and battery formulations on a common mathematical basis.
The model enables population-level inference
Because the coefficients have a population distribution, the analysis does more than fit one cell at a time. It estimates both the typical degradation behavior and the uncertainty caused by cell-to-cell differences.
That information can be used to construct reliability or hazard functions describing the probability that a cell reaches a defined end-of-life condition over time.
End of life requires a defined threshold
Resistance modeling alone does not define failure. A study must specify an end-of-life criterion, such as a resistance limit, a power-performance limit, or a combined capacity-and-resistance requirement.
Once a threshold is defined, the fitted trajectory can be evaluated to estimate when a cell is expected to cross it. The resulting crossing-time distribution can support life and reliability analysis.
Extending the Model to Test Conditions
Temperature and SoC can be treated as experimental factors
Resistance measurements taken at different temperatures or SoC levels should not be pooled without accounting for those conditions. A practical approach is to fit separate trajectories for each test condition or include temperature and SoC as explanatory variables.
The basic time-dependent structure remains:
[ R_i(t)=\text{baseline and condition effects}+\text{aging trend}+\text{cell-specific effects}+\text{noise} ]
This prevents reversible operating-condition effects from being confused with irreversible degradation.
Capacity and resistance provide complementary signals
Capacity retention measures how much charge the battery can deliver, whereas resistance growth measures how effectively it can deliver current. Monitoring both can distinguish different degradation manifestations more effectively than relying on either signal alone.
The resistance trajectory is therefore best viewed as one condition-monitoring channel within a broader battery health model.
Understanding the Trade-offs
A quadratic model is useful but not universally valid
The quadratic form is flexible enough to represent curvature while remaining simple to estimate. However, it may extrapolate unrealistically outside the observed aging interval, especially if the true degradation process changes regime.
It should therefore be validated against held-out data and used cautiously for long-range life prediction.
Measurement conditions can dominate apparent aging
A resistance increase caused by lower temperature or a different SoC is not necessarily permanent degradation. Inconsistent pulse protocols, rest periods, or voltage sampling can produce apparent resistance changes unrelated to cell aging.
Standardized test conditions are essential for meaningful trajectory comparisons.
Normality assumptions are modeling choices
The multivariate normal distribution for random effects and the normal measurement-error assumption are convenient statistical approximations. They should be checked against residuals and fitted coefficient distributions rather than accepted automatically.
If the data show strong skew, outliers, or nonconstant variance, the model may require robust errors or another distributional formulation.
Resistance is not a complete health diagnosis
An elevated resistance can signal reduced power capability, but it does not uniquely identify the underlying degradation mechanism. Capacity, temperature response, impedance characteristics, and other diagnostic signals may be required to explain why resistance is changing.
Making the Right Choice for Your Goal
The appropriate use of the model depends on whether the priority is measurement, comparison, or prediction.
- If your primary focus is resistance measurement: Apply a controlled current pulse and calculate (R=(V_{\text{rest}}-V_{\text{loaded}})/I) under repeatable temperature, SoC, and rest conditions.
- If your primary focus is cell-to-cell comparison: Fit a mixed-effects model with normally distributed random intercepts and trajectory coefficients to separate population behavior from manufacturing variability.
- If your primary focus is degradation-rate estimation: Use (b_{i1}) and (b_{i2}) to quantify linear and accelerating resistance growth, while validating the quadratic form over the intended aging range.
- If your primary focus is life or reliability prediction: Define an explicit resistance or performance end-of-life threshold, then use fitted trajectory distributions to estimate threshold-crossing times and hazard functions.
A well-controlled resistance measurement combined with a validated mixed-effects trajectory model turns raw voltage-pulse data into an interpretable signal of battery degradation.
Summary Table:
| Model Component | Formula/Description | Purpose |
|---|---|---|
| Resistance Measurement | (R_i(t) = \frac{V_{rest,i}(t) - V_{loaded,i}(t)}{I_i(t)}) | Estimates internal resistance from voltage response to current pulse. |
| Quadratic Mixed-Effects Model | (CM_i(t) = b_{i0} + b_{i1}t + b_{i2}t^2 + e_i(t)) | Models resistance trajectory with fixed (population) and random (cell-specific) effects. |
| Fixed Effects | (b_{i0}, b_{i1}, b_{i2}) | Describe population baseline, linear growth rate, and curvature. |
| Random Effects | (\mathbf{b}_i \sim \mathcal{N}(\boldsymbol{\mu}_b, \mathbf{\Sigma}_b)) | Capture cell-to-cell variability in coefficients. |
| Measurement Noise | (e_i(t) \sim \mathcal{N}(0, \sigma_e^2)) | Accounts for random measurement error. |
| End-of-Life Criterion | Defined threshold (e.g., resistance limit) | Used with fitted trajectory to predict failure time. |
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