I treat the state of health of a lithium-ion cell as a pre-warning state variable for low-voltage TN-C-S distribution safety, rather than as a direct leakage or trip criterion. In a TN-C-S system, the combined PEN conductor is separated into a neutral conductor and a protective earth conductor at the user intake, and the protective earth conductor is repeatedly grounded while remaining independent from the neutral conductor. This arrangement provides a protective grounding path for a power conversion system and a battery management system, but it also creates distinctive risks: zero-sequence voltage offset, PEN breakage, common-mode interference, and residual-current growth. When a lithium-ion cell ages, its internal resistance rises, its insulation performance can deteriorate, and its thermal stress can accumulate. These changes do not map one-to-one to leakage current, but they reduce the operating margin of the energy storage branch and can amplify progressive safety hazards. I therefore build a high-accuracy lithium-ion cell state-of-health estimator as a proactive sensing layer that complements residual-current protection, zero-sequence voltage monitoring, and protection-action detection.
Context and Motivation
I start from a simple observation: existing TN-C-S safety research often focuses on post-fault detection, while the aging of a lithium-ion cell is a slowly evolving process that can be sensed before a leakage fault occurs. A lithium-ion cell that has lost active lithium inventory, active material, and electrolyte integrity will show capacity fade, impedance growth, and altered voltage relaxation. These electrochemical changes are not identical to system insulation degradation, because connections, ambient temperature, converter operation, and grounding conditions also influence residual current. Nevertheless, the health state of a lithium-ion cell provides a valuable prior for risk ranking, maintenance scheduling, and predictive control. I position the lithium-ion cell SOH estimate as an aging-state prior that can be fused with direct electrical safety measurements.
I define SOH as the ratio of the current available maximum capacity to the rated capacity:
$$SOH = \frac{Q_{act}}{Q_{0}} \times 100\%$$
where \(Q_{act}\) is the current usable maximum capacity and \(Q_{0}\) is the rated capacity. This definition is simple, measurable from full or partial cycling data, and compatible with the practical information available from a battery management system. I use this definition consistently for every lithium-ion cell in my experiments.
Data Sources and Aging Conditions
I validate the proposed method on two offline battery datasets that represent different cathode materials and different discharge rates. The first dataset contains commercial 18650 lithium-ion cells with a nickel-cobalt-aluminum oxide cathode and a graphite anode. The rated capacity is 3.5 Ah and the rated voltage is 4.2 V. I select two lithium-ion cells, NCA1 and NCA2, for SOH estimation. The second dataset comes from laboratory cycle aging tests on twelve lithium ferro-phosphate lithium-ion cells divided into four groups. I select LB1 and LB4 for SOH estimation. These two datasets cover different positive electrode systems and different discharge-rate conditions, which helps me evaluate whether the estimator can generalize across lithium-ion cell chemistries and operating profiles.
| Lithium-ion cell | Chemistry | Temperature | Charge rate | Discharge rate |
|---|---|---|---|---|
| NCA1 | Nickel-cobalt-aluminum oxide | 25 degrees C | 0.5 C | 0.5 C |
| NCA2 | Nickel-cobalt-aluminum oxide | 45 degrees C | 0.5 C | 0.5 C |
| LB1 | Lithium ferro-phosphate | 25 degrees C | 5 C | 5 C |
| LB4 | Lithium ferro-phosphate | 25 degrees C | 5 C | 10 C |
I use an 80 percent training and 20 percent testing split for each lithium-ion cell. The same split is used for benchmark models, ablation models, and the complete model so that the comparison is consistent. I do not use future-cycle information in the input of any model. The time order of cycles is preserved, and the estimator is trained to map aging-sensitive features to the SOH of the lithium-ion cell.
Health Feature Extraction
I extract candidate health features from three domains: time-domain behavior, incremental capacity, and voltage relaxation. In the constant-current charging stage, aging increases polarization, so the terminal voltage reaches the upper cutoff voltage earlier. Therefore, the constant-current charging time tends to decrease as the lithium-ion cell ages. In the constant-voltage charging stage, aging increases internal resistance and reduces diffusion, so the constant-voltage duration may change in a more complex way. I consider the constant-current charging time and the constant-voltage charging time as initial candidates.
I also consider the charge plateau duration and the discharge plateau duration. A voltage plateau is related to phase transitions and lithium intercalation or deintercalation reactions. As loss of lithium inventory and loss of active material accumulate, the effective reaction charge in a given voltage interval decreases, while internal resistance and polarization increase. Consequently, the charge and discharge processes reach the cutoff voltage earlier, and the effective plateau duration shortens. For one nickel-cobalt-aluminum oxide lithium-ion cell at 25 degrees C, when SOH decreases from 93.40 percent to 72.82 percent, the charge plateau duration decreases from 3668.43 s to 3028.22 s, a reduction of 17.45 percent, and the discharge plateau duration decreases from 2364.83 s to 1811.36 s, a reduction of 23.40 percent. A linear regression gives reductions of 25.82 s and 24.99 s in charge and discharge plateau durations for each one-percentage-point drop in SOH, with coefficients of determination of 0.9817 and 0.9826. A second lithium-ion cell at 45 degrees C shows the same direction, with charge and discharge plateau durations decreasing by 15.23 percent and 15.53 percent. This consistency supports the use of plateau durations as health features.
I compute the incremental capacity curve by differentiating capacity with respect to terminal voltage:
$$\frac{dQ}{dV} \approx \frac{Q_{k} – Q_{k-N}}{V_{k} – V_{k-N}}$$
where \(Q_k\) and \(V_k\) are the capacity and terminal voltage at the \(k\)-th sample, and \(N\) is the difference interval. Because differentiation amplifies measurement noise, I apply Gaussian smoothing to the voltage and capacity sequences before computing \(dQ/dV\). I extract the maximum incremental capacity peak and the voltage at that peak as candidate features. The maximum peak is often strongly related to the aging state of a lithium-ion cell, while the peak voltage provides complementary information about reaction-region evolution.
I further examine the discharge relaxation voltage rebound. After discharge stops, the terminal voltage recovers because ohmic, concentration, and activation polarizations gradually relax. This relaxation process is sensitive to discharge depth, discharge rate, rest duration, and ambient temperature. In a practical distribution-area energy storage system, rest periods and their durations depend on dispatch requirements, so this feature is not always available. I therefore treat relaxation voltage rebound as a conditional feature rather than a universally available input.
Correlation Screening and Feature Selection
I use the Spearman rank correlation coefficient to quantify the monotonic relationship between each candidate feature and the SOH of the lithium-ion cell:
$$\rho_s = 1 – \frac{6 \sum_{i=1}^{n} d_i^{2}}{n(n^{2}-1)}$$
where \(d_i\) is the rank difference between the feature and SOH for the \(i\)-th cycle, and \(n\) is the number of cycles. The coefficient ranges from negative one to positive one. A positive value means the feature increases with SOH, and a negative value means the feature decreases with SOH. I also apply the Benjamini-Hochberg procedure to control the false discovery rate across multiple hypothesis tests, and I use \(q < 0.05\) as the statistical significance criterion.
| Candidate feature | Spearman rho with SOH | Significance after correction | Decision |
|---|---|---|---|
| Constant-current charging time | 0.9966 | Significant | Selected |
| Maximum incremental capacity peak | 0.9993 | Significant | Selected |
| Charge plateau duration | 0.9932 | Significant | Selected |
| Discharge plateau duration | 0.9937 | Significant | Selected |
| Constant-voltage charging time | Weaker but significant | Significant | Not selected due to operational sensitivity |
| Voltage at maximum incremental capacity peak | Significant | Significant | Not selected due to redundancy and noise sensitivity |
| Discharge relaxation voltage rebound | Significant | Significant | Not selected due to rest-window dependence |
I select four features as model inputs: constant-current charging time, maximum incremental capacity peak, charge plateau duration, and discharge plateau duration. These features are strongly monotonic with SOH and are mechanistically complementary. The constant-current charging time reflects the charging polarization and capacity state of the lithium-ion cell. The incremental capacity peak reflects reaction-region evolution and capacity fade. The charge and discharge plateau durations reflect the combined effects of loss of lithium inventory, loss of active material, and impedance growth. The constant-voltage charging time is not selected because it is more sensitive to cutoff voltage, cutoff current, initial state of charge, and temperature, and its duration becomes short at high charge rates. The peak voltage is not selected because it overlaps with the incremental capacity peak and is sensitive to voltage resolution and smoothing parameters. The relaxation rebound is not selected because it depends on a complete rest window that may not exist under practical dispatch.
Online Measurability and Noise Boundary
I classify the selected features by online measurability. Constant-current charging time, charge plateau duration, and discharge plateau duration can be extracted from voltage and current time series collected by a battery management system, provided that a constant-current operating window of sufficient duration exists. The incremental capacity peak requires a more continuous and complete constant-current charge capacity-voltage trajectory, so I define it as a conditionally measurable feature. When the constant-current charging segment is incomplete or the data quality is insufficient, I can disable the incremental capacity feature and rely on time-domain features to represent the aging state of the lithium-ion cell.
To quantify the noise boundary of the incremental capacity peak, I conduct a Monte Carlo perturbation test on 208 complete cycles. I add independent zero-mean Gaussian noise to the terminal voltage and charge capacity sequences during the constant-current charging stage. The noise standard deviation is determined from the root mean square of the de-meaned original signal and a target signal-to-noise ratio. I test five signal-to-noise ratio levels: 60, 50, 40, 30, and 20 dB. Each level is repeated 100 times.
| Target SNR (dB) | Median absolute percentage error of IC peak | Median Spearman rho with SOH | 95 percent interval of rho | Repeats with p less than 0.05 |
|---|---|---|---|---|
| 60 | 0.334% | 0.9933 | 0.9906 to 0.9949 | 100.0% |
| 50 | 1.214% | 0.9638 | 0.9520 to 0.9715 | 100.0% |
| 40 | 6.593% | 0.8500 | 0.8165 to 0.8874 | 100.0% |
| 30 | 37.943% | 0.6165 | 0.5271 to 0.6876 | 100.0% |
| 20 | 949.372% | 0.2454 | 0.1257 to 0.3920 | 96.0% |
The results show that the incremental capacity peak remains useful at approximately 40 dB and above, where the median Spearman rho with SOH is 0.8500. Below this level, the extraction stability and health representation ability degrade. I therefore use the incremental capacity peak only when the constant-current charge segment is complete and the sampling quality is adequate. This design choice is important for field deployment because a lithium-ion cell in a distribution-area energy storage system may not always experience a clean constant-current charge event.
RIME-TCN-Transformer Model
I propose a hybrid estimator that combines a temporal convolutional network, a Transformer encoder, and the RIME optimization algorithm. I choose this architecture because distribution-area energy storage data are noisy, the operating conditions fluctuate, and the monitoring terminal has limited computing resources. The temporal convolutional network extracts local temporal features and suppresses short pulses and local noise. The Transformer encoder captures long-range aging dependencies across many cycles. The RIME algorithm optimizes key hyperparameters for the specific data distribution of each lithium-ion cell.
Temporal Convolutional Network
The temporal convolutional network uses causal convolution, dilated convolution, and residual connections. Causal convolution guarantees that the output at time \(t\) depends only on inputs at time \(t\) and earlier. Dilated convolution increases the receptive field without a proportional increase in depth or kernel size. For a one-dimensional input sequence \(x\) and a filter \(f: \{0, …, k-1\}\), the dilated convolution at position \(s\) is:
$$F(s) = (x *_{d} f)(s) = \sum_{i=0}^{k-1} f(i) x_{s – d i}$$
where \(d\) is the dilation factor and \(k\) is the kernel size. As network depth increases, \(d\) grows exponentially, so a shallow network can cover a long time span. A residual block contains two dilated causal convolution layers, each followed by weight normalization and a rectified linear unit. The residual output is:
$$y = Activation(x + F(x))$$
where \(x\) is the input and \(F(x)\) is the transformed output. This structure allows the model to learn both short-term fluctuations and longer aging trends in the lithium-ion cell data.
Transformer Encoder
The Transformer encoder models global dependencies through multi-head self-attention. Because the Transformer processes the entire sequence in parallel, I add positional encoding to preserve temporal order. For a sequence of length \(L\), the position encoding at position \(k\) and dimension \(i\) is:
$$PE_{(k,2i)} = \sin\left(\frac{k}{n^{2i/d}}\right)$$
$$PE_{(k,2i+1)} = \cos\left(\frac{k}{n^{2i/d}}\right)$$
where \(d\) is the input dimension and \(n\) is a scalar constant. The self-attention mechanism uses query, key, and value matrices \(Q\), \(K\), and \(V\), which are obtained by linear transformations. The scaled dot-product attention is:
$$\mathrm{Attention}(Q,K,V) = \mathrm{softmax}\left(\frac{Q K^{T}}{\sqrt{d_k}}\right) V$$
where \(d_k\) is the dimension of the key vectors. The softmax normalizes the similarity scores into attention weights. Multi-head attention allows the model to observe the aging process from different representation subspaces:
$$\mathrm{MultiHead}(Q,K,V) = \mathrm{Concat}(head_1, …, head_h) W^{O}$$
where each head is:
$$head_i = \mathrm{Attention}(Q W_i^{Q}, K W_i^{K}, V W_i^{V})$$
In my model, the temporal convolutional network produces local feature vectors that are fed into the Transformer encoder. The Transformer then captures the global aging trend of the lithium-ion cell across many cycles.
RIME Optimization
I use the RIME algorithm to optimize six hyperparameters: initial learning rate, number of convolution kernels, convolution kernel size, dropout rate, batch size, and number of attention heads. RIME simulates soft rime and hard rime growth. The soft rime search mechanism updates a particle position as:
$$R_{ij}^{new} = R_{best,j} + r_1 \cos(\theta) \beta (Ub_{ij} – Lb_{ij}) + Lb_{ij}, \quad r_2 < E$$
where \(R_{best,j}\) is the best solution in dimension \(j\), \(r_1\) is a random number between negative one and one, \(\theta\) controls the search direction, \(\beta\) is an environmental factor, \(Ub_{ij}\) and \(Lb_{ij}\) are dimension bounds, and \(E\) is the attachment coefficient. The angle and environmental factor are:
$$\theta = \frac{\pi t}{10 T}$$
$$\beta = 1 – \left[\frac{w t}{T}\right] / w$$
where \(t\) is the current iteration, \(T\) is the maximum iteration, and \(w\) is a default step-control parameter. The attachment coefficient increases with iteration:
$$E = \frac{t}{T}$$
The hard rime puncture mechanism uses normalized fitness as a crossover threshold. For particle \(i\), the normalized fitness is:
$$S_i = \frac{F_i – \min(F)}{\max(F) – \min(F) + \epsilon}$$
where \(F_i\) is the fitness of particle \(i\) and \(\epsilon\) is a small constant. A larger \(S_i\) means poorer fitness and a higher probability of replacement. If a random number is less than \(S_i\), the particle is replaced by the global best in that dimension:
$$R_{ij}^{new} = R_{best,j}$$
Finally, a greedy selection strategy keeps the new position only if its fitness is better:
$$R_i^{t+1} =
\begin{cases}
R_i^{new}, & F(R_i^{new}) < F(R_i^{t}) \\
R_i^{t}, & \text{otherwise}
\end{cases}$$
I use root mean square error as the fitness function:
$$RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^{2}}$$
where \(y_i\) is the true SOH of the \(i\)-th cycle and \(\hat{y}_i\) is the estimated SOH. I optimize the hyperparameters within the ranges below.
| Hyperparameter | Lower bound | Upper bound |
|---|---|---|
| Initial learning rate | 0.0001 | 0.01 |
| Number of convolution kernels | 16 | 128 |
| Convolution kernel size | 3 | 9 |
| Dropout rate | 0.01 | 0.5 |
| Batch size | 16 | 128 |
| Number of attention heads | 2 | 5 |
Evaluation Metrics
I evaluate the estimator with four indicators: root mean square error, mean absolute error, mean absolute percentage error, and coefficient of determination. I define them as follows:
$$RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^{2}}$$
$$MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i – \hat{y}_i|$$
$$MAPE = \frac{1}{n} \sum_{i=1}^{n} \left|\frac{y_i – \hat{y}_i}{y_i}\right| \times 100\%$$
$$R^{2} = 1 – \frac{\sum_{i=1}^{n} (y_i – \hat{y}_i)^{2}}{\sum_{i=1}^{n} (y_i – \bar{y})^{2}}$$
where \(n\) is the number of test cycles, \(y_i\) is the true SOH, \(\hat{y}_i\) is the estimated SOH, and \(\bar{y}\) is the mean true SOH.
Benchmark Comparison
I compare the proposed RIME-TCN-Transformer with three benchmark models: support vector regression, long short-term memory, and a convolutional neural network. All models use the same four input features and the same train-test split. The results show that the proposed method achieves the lowest error on all four lithium-ion cells. For NCA1, the RMSE is 0.0044, which is 62.39 percent lower than support vector regression, 60.36 percent lower than long short-term memory, and 45.68 percent lower than the convolutional neural network. For LB1, the RMSE is 0.0023, which is 77.88 percent lower than support vector regression, 58.93 percent lower than long short-term memory, and 50.00 percent lower than the convolutional neural network.
| Lithium-ion cell | Model | RMSE | MAE | MAPE | R squared |
|---|---|---|---|---|---|
| NCA1 | Support vector regression | 0.0117 | 0.0128 | 0.49% | 0.9559 |
| NCA1 | Long short-term memory | 0.0111 | 0.0103 | 0.38% | 0.9816 |
| NCA1 | Convolutional neural network | 0.0081 | 0.0066 | 0.26% | 0.9874 |
| NCA1 | RIME-TCN-Transformer | 0.0044 | 0.0036 | 0.13% | 0.9971 |
| NCA2 | Support vector regression | 0.0115 | 0.0094 | 0.37% | 0.8739 |
| NCA2 | Long short-term memory | 0.0069 | 0.0066 | 0.25% | 0.9546 |
| NCA2 | Convolutional neural network | 0.0063 | 0.0058 | 0.22% | 0.9629 |
| NCA2 | RIME-TCN-Transformer | 0.0037 | 0.0029 | 0.11% | 0.9867 |
| LB1 | Support vector regression | 0.0104 | 0.0095 | 0.95% | 0.6359 |
| LB1 | Long short-term memory | 0.0056 | 0.0053 | 0.52% | 0.8920 |
| LB1 | Convolutional neural network | 0.0046 | 0.0042 | 0.41% | 0.9271 |
| LB1 | RIME-TCN-Transformer | 0.0023 | 0.0018 | 0.18% | 0.9823 |
| LB4 | Support vector regression | 0.0181 | 0.0096 | 0.93% | 0.8865 |
| LB4 | Long short-term memory | 0.0130 | 0.0068 | 0.65% | 0.9412 |
| LB4 | Convolutional neural network | 0.0095 | 0.0074 | 0.73% | 0.9687 |
| LB4 | RIME-TCN-Transformer | 0.0036 | 0.0030 | 0.32% | 0.9956 |
The error distributions are also more concentrated around zero for the proposed model. The support vector regression model has the largest estimation error. The long short-term memory and convolutional neural network models improve on support vector regression but still show tracking deviations near the end of life, when the capacity of the lithium-ion cell decays rapidly. The proposed model maintains a tighter fit across the entire aging trajectory. This matters for TN-C-S safety because the end-of-life region is exactly where the operating margin is smallest and where proactive maintenance decisions are most valuable.
Ablation Study
I perform an ablation study to quantify the contribution of each component. I compare the full RIME-TCN-Transformer with three variants: RIME-TCN without the Transformer encoder, RIME-Transformer without the temporal convolutional network, and TCN-Transformer without RIME optimization. The full model achieves the lowest RMSE and the highest R squared on all four lithium-ion cells. When I remove RIME, the TCN-Transformer model loses some accuracy because the hyperparameters are not adapted to the data. When I remove the temporal convolutional network, the RIME-Transformer model performs poorly on the NCA1 and NCA2 lithium-ion cells, which shows that local feature extraction and denoising are important before global attention. When I remove the Transformer encoder, the RIME-TCN model performs worse on LB1 and LB4, which shows that long-range aging dependencies become more important as the dataset size grows.
| Lithium-ion cell | Ablation model | RMSE | MAE | MAPE | R squared |
|---|---|---|---|---|---|
| NCA1 | RIME-TCN-Transformer | 0.0044 | 0.0036 | 0.13% | 0.9971 |
| NCA1 | RIME-TCN | 0.0073 | 0.0063 | 0.23% | 0.9924 |
| NCA1 | RIME-Transformer | 0.0116 | 0.0110 | 0.39% | 0.9801 |
| NCA1 | TCN-Transformer | 0.0065 | 0.0053 | 0.19% | 0.9937 |
| NCA2 | RIME-TCN-Transformer | 0.0037 | 0.0029 | 0.11% | 0.9867 |
| NCA2 | RIME-TCN | 0.0059 | 0.0043 | 0.17% | 0.9665 |
| NCA2 | RIME-Transformer | 0.0077 | 0.0061 | 0.24% | 0.9427 |
| NCA2 | TCN-Transformer | 0.0064 | 0.0053 | 0.21% | 0.9602 |
| LB1 | RIME-TCN-Transformer | 0.0023 | 0.0018 | 0.18% | 0.9823 |
| LB1 | RIME-TCN | 0.0037 | 0.0032 | 0.32% | 0.9524 |
| LB1 | RIME-Transformer | 0.0031 | 0.0025 | 0.25% | 0.9682 |
| LB1 | TCN-Transformer | 0.0028 | 0.0023 | 0.23% | 0.9771 |
| LB4 | RIME-TCN-Transformer | 0.0036 | 0.0030 | 0.32% | 0.9956 |
| LB4 | RIME-TCN | 0.0077 | 0.0057 | 0.57% | 0.9794 |
| LB4 | RIME-Transformer | 0.0059 | 0.0050 | 0.54% | 0.9879 |
| LB4 | TCN-Transformer | 0.0057 | 0.0052 | 0.56% | 0.9889 |
The ablation results confirm that the temporal convolutional network, the Transformer encoder, and RIME optimization all contribute positively. The full model is not simply a sum of independent parts; rather, the components cooperate. The temporal convolutional network provides stable local features for the lithium-ion cell, the Transformer models long-range degradation, and RIME adapts the network to the specific data distribution. This cooperation is particularly useful for TN-C-S energy storage because different distribution areas may have different load profiles, temperatures, and battery types.
Interpretability with SHAP
I use Shapley additive explanations to quantify how each input feature contributes to the SOH estimate. I draw 100 background samples from the training set for the SHAP calculation. The results show that the model adapts its feature weighting to the data. For NCA1, the maximum incremental capacity peak dominates the estimate with a global importance share of 62.05 percent. For NCA2, the maximum incremental capacity peak remains important, while the charge plateau duration and constant-current charging time contribute 29.88 percent and 28.21 percent, respectively. For LB1 and LB4, the incremental capacity peak contribution falls to 0.0876 percent and 0.1759 percent, which indicates a soft down-weighting effect. The network does not blindly trust the incremental capacity peak for every lithium-ion cell; instead, it reallocates attention according to the information structure of the data.
| Lithium-ion cell | Feature behavior in SHAP analysis | Reported contribution or trend |
|---|---|---|
| NCA1 | Maximum incremental capacity peak dominates | 62.05% global importance |
| NCA2 | Multiple features are used | Charge plateau duration 29.88%, constant-current charging time 28.21% |
| LB1 | Incremental capacity peak is softly down-weighted | 0.0876% |
| LB4 | Incremental capacity peak is softly down-weighted | 0.1759% |
This interpretability result has a practical meaning for TN-C-S monitoring. The incremental capacity feature is valuable when a clean constant-current charge segment is available, but it should not be forced into the model when the field conditions do not support reliable extraction. The time-domain features are more robust to incomplete charging events, while the incremental capacity peak provides high information content when the data quality is sufficient. I therefore keep the four-feature interface for model consistency but allow the learned weights to adjust across lithium-ion cell systems.
Repeatability and Cell-to-Cell Dispersion
I also examine repeatability under the same operating condition. I read the cycle summary records of three lithium-ion cells from the same group, LB1, LB2, and LB3, and normalize their discharge capacity by the first-cycle discharge capacity:
$$Q_{norm}(N) = \frac{Q(N)}{Q(0)}$$
LB1, LB2, and LB3 contain 1404, 1350, and 1368 cycles, respectively. In the common cycle range from 1 to 1350, I compute the coefficient of variation of the normalized discharge capacity as the ratio of the sample standard deviation to the mean. The three lithium-ion cells show a synchronized capacity fade trend, while the dispersion increases gradually with cycling. This indicates good repeatability in the early and middle life of the lithium-ion cell, with objective individual differences emerging in later life.
| Metric | Value |
|---|---|
| Common cycle range | 1 to 1350 |
| Average coefficient of variation | 0.735% |
| Maximum coefficient of variation | 1.159% at cycle 1171 |
| Average coefficient of variation in first 100 cycles | 0.303% |
| Average coefficient of variation in middle 100 cycles | 0.747% |
| Average coefficient of variation in final 100 cycles | 1.033% |
| Final normalized capacity of LB1 | 80.73% |
| Final normalized capacity of LB2 | 81.46% |
| Final normalized capacity of LB3 | 81.04% |
The increasing dispersion in the final cycles suggests that a single lithium-ion cell model may not capture every cell-specific aging path perfectly. This is expected in practical energy storage, where manufacturing variability, thermal gradients, and connection impedance differences accumulate. For TN-C-S safety, this means that SOH estimation should be combined with pack-level monitoring and periodic recalibration. A lithium-ion cell with a lower SOH than its neighbors may increase the risk of overcharge, over-discharge, and local heating, even if the pack-level average SOH appears acceptable.
Deployment Logic for TN-C-S Distribution Safety
I integrate the SOH estimator into an intelligent monitoring terminal for a TN-C-S low-voltage distribution system. The terminal can collect voltage and current time series from the battery management system, extract the four selected features when the operating window is valid, and estimate the SOH of the lithium-ion cell. The SOH output is not used as a direct trip signal. Instead, it is used as an aging-state prior that modifies the risk score of the energy storage branch. The risk score can be combined with residual current, zero-sequence voltage, insulation resistance, and protection action records. In this way, the system forms a prediction-detection defense-in-depth framework: SOH estimation provides pre-fault warning, while residual-current protection and related devices provide fast fault detection and isolation.
From an edge-computing perspective, the temporal convolutional network is parallelizable, and I retain only the Transformer encoder to control model complexity. The model has potential for edge deployment, but actual inference latency, memory use, and power consumption depend on the target processor. I would further apply pruning, quantization, and hardware-specific optimization before field deployment. I would also validate the model with long-term synchronized data from the power conversion system, battery management system, and distribution-side monitoring devices. The mapping from SOH to leakage risk should be calibrated with these data, because SOH alone cannot determine residual current or zero-sequence voltage.
Practical Implications for Lithium-Ion Cell Monitoring
The most important practical implication is that a lithium-ion cell should not be monitored only by threshold alarms. Threshold alarms are necessary for immediate safety, but they are inherently reactive. A lithium-ion cell can pass a threshold test while its aging state is already poor. By estimating SOH continuously or periodically, I can rank the risk of each lithium-ion cell before a hard fault appears. This supports predictive maintenance, charge-discharge strategy adjustment, and replacement planning. In a TN-C-S system, where the protective earth and neutral conductors have specific separation and grounding requirements, the early detection of a weak lithium-ion cell can reduce the probability of thermal runaway, insulation degradation, and residual-current growth.
I also emphasize that feature availability changes with field conditions. Constant-current charging time and plateau durations require a suitable constant-current window. The incremental capacity peak requires a complete constant-current capacity-voltage trajectory and sufficient signal-to-noise ratio. If these conditions are not met, the estimator should degrade gracefully. A practical implementation can use a feature-gating strategy: when the incremental capacity feature is valid, it participates in the estimate; when it is invalid, the model relies on time-domain features. This helps avoid unstable estimates caused by noise-amplified differentiation.
Model Robustness and Generalization
The proposed model generalizes across two different lithium-ion cell chemistries and multiple discharge rates. NCA1 and NCA2 represent nickel-cobalt-aluminum oxide chemistry at different temperatures. LB1 and LB4 represent lithium ferro-phosphate chemistry at different discharge rates. The model achieves low RMSE on all four lithium-ion cells, which suggests that the combination of local temporal features and global attention can capture common aging signatures while adapting to material-specific and rate-specific behavior. The SHAP results further show that the model does not rely on one fixed feature hierarchy. Instead, it learns a data-dependent feature policy. This is desirable for distribution-area energy storage because the operating profile of a lithium-ion cell in one feeder may differ significantly from that in another.
Nevertheless, I do not claim that the current offline validation is sufficient for every field scenario. Temperature, state of charge swing, charge-discharge frequency, connection resistance, and converter control strategy can all influence the observed aging trajectory. A lithium-ion cell model trained on one dataset may need transfer learning or recalibration for another dataset. Future work should include online impedance identification, dynamic feature gating, cross-cell transfer learning, and target-terminal compression tests. These steps would extend the method from an offline SOH estimator to a field-ready safety pre-warning function.
Summary of Equations and Design Choices
I summarize the main equations used in my method. The SOH definition is:
$$SOH = \frac{Q_{act}}{Q_{0}} \times 100\%$$
The incremental capacity calculation is:
$$\frac{dQ}{dV} \approx \frac{Q_{k} – Q_{k-N}}{V_{k} – V_{k-N}}$$
The Spearman correlation is:
$$\rho_s = 1 – \frac{6 \sum_{i=1}^{n} d_i^{2}}{n(n^{2}-1)}$$
The temporal convolutional operation is:
$$F(s) = \sum_{i=0}^{k-1} f(i) x_{s – d i}$$
The attention operation is:
$$\mathrm{Attention}(Q,K,V) = \mathrm{softmax}\left(\frac{Q K^{T}}{\sqrt{d_k}}\right) V$$
The RIME soft rime update is:
$$R_{ij}^{new} = R_{best,j} + r_1 \cos(\theta) \beta (Ub_{ij} – Lb_{ij}) + Lb_{ij}$$
The fitness function is:
$$RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^{2}}$$
These equations define a complete pipeline: I extract health features from a lithium-ion cell, I screen them with rank correlation, I feed them into a temporal convolutional network, I model long-range aging with a Transformer encoder, I optimize hyperparameters with RIME, and I evaluate the SOH estimate with standard error metrics.
Comparison of Selected Features
I provide a concise comparison of the selected features and their roles in the lithium-ion cell SOH estimation. This comparison is useful because feature selection is not only a statistical problem; it is also an engineering problem. The feature must be measurable in the field, stable under noise, and mechanistically linked to aging.
| Selected feature | Physical meaning | Online availability | Main risk |
|---|---|---|---|
| Constant-current charging time | Reflects polarization and charge acceptance | Available when a constant-current window exists | Depends on initial state of charge and temperature |
| Maximum incremental capacity peak | Reflects reaction-region evolution and capacity fade | Conditionally available | Sensitive to voltage resolution and noise |
| Charge plateau duration | Reflects loss of lithium inventory and active material | Available from voltage-current time series | Depends on cutoff voltage and rate |
| Discharge plateau duration | Reflects impedance growth and available reaction charge | Available from voltage-current time series | Depends on load profile and cutoff voltage |
I select these four features because they balance accuracy, interpretability, and field measurability. A lithium-ion cell with a shorter constant-current charging time, a smaller incremental capacity peak, and shorter charge and discharge plateaus is likely to have a lower SOH. The model learns the nonlinear combination of these indicators and produces a continuous SOH estimate.
Results Summary Table
I summarize the main SOH estimation results for the proposed model. These values are obtained from the offline test split and are used to compare with benchmarks and ablations.
| Lithium-ion cell | Proposed RMSE | Proposed MAE | Proposed MAPE | Proposed R squared |
|---|---|---|---|---|
| NCA1 | 0.0044 | 0.0036 | 0.13% | 0.9971 |
| NCA2 | 0.0037 | 0.0029 | 0.11% | 0.9867 |
| LB1 | 0.0023 | 0.0018 | 0.18% | 0.9823 |
| LB4 | 0.0036 | 0.0030 | 0.32% | 0.9956 |
The proposed model reduces RMSE relative to support vector regression, long short-term memory, and the convolutional neural network. On NCA1, the reductions are 62.39 percent, 60.36 percent, and 45.68 percent, respectively. On LB1, the reductions are 77.88 percent, 58.93 percent, and 50.00 percent, respectively. These results support the use of the RIME-TCN-Transformer as a high-accuracy SOH estimator for a lithium-ion cell in an energy storage branch of a TN-C-S distribution system.
Safety-Oriented Interpretation
I return to the TN-C-S safety context. A lithium-ion cell with declining SOH has reduced usable capacity, higher internal resistance, and greater thermal stress under the same current. These factors reduce the operating margin of the energy storage branch. They can also increase voltage fluctuation and local heating, which may interact with insulation aging and residual-current growth. The SOH estimate cannot replace residual-current protection, but it can provide an early warning that a lithium-ion cell is approaching a high-risk aging stage. When combined with zero-sequence voltage, residual current, and protection action records, the SOH estimate can help operators distinguish normal transient events from progressive degradation. This layered approach is more robust than relying on any single measurement.
I therefore view the proposed method as a pre-warning layer. The output can be integrated into a distribution monitoring terminal, a battery management system, or a cloud-based asset management platform. The exact warning threshold should be calibrated with field data. A conservative strategy is to use SOH as a risk modifier rather than a binary trip condition. For example, a low SOH can trigger more frequent inspection, reduced charge-discharge rate, or proactive replacement planning. This is consistent with the goal of reducing electric shock and electrical fire risk in low-voltage distribution networks.
Limitations and Next Steps
I acknowledge several limitations. First, the validation is offline. Field data may contain additional disturbances, sensor drift, and communication losses. Second, the mapping from SOH to residual current and zero-sequence voltage is not one-to-one, so field calibration is required. Third, the incremental capacity feature requires a clean constant-current segment and adequate signal-to-noise ratio. Fourth, the model hyperparameters are optimized for the available datasets, and transfer to new lithium-ion cell types may require retraining or fine-tuning. Fifth, the edge deployment cost depends on the target hardware and should be quantified with pruning, quantization, and real-time inference tests. These limitations do not invalidate the method, but they define the path from offline validation to field deployment.
My next steps are clear. I will combine long-term synchronized data from the power conversion system, battery management system, and distribution-side monitoring devices. I will calibrate the relationship between SOH and safety-related electrical quantities, including residual current and zero-sequence voltage. I will add online impedance identification as a complementary health indicator. I will develop dynamic feature gating so that the incremental capacity feature is used only when the data quality is sufficient. I will test cross-cell transfer learning so that a model trained on one lithium-ion cell can be adapted to another with limited data. I will also evaluate model compression, inference latency, and power consumption on realistic monitoring terminals.
Conclusion
I have presented a RIME-TCN-Transformer method for estimating the SOH of a lithium-ion cell in a TN-C-S low-voltage distribution energy storage system. I treat the SOH estimate as a proactive aging-state prior that complements residual-current protection and zero-sequence voltage monitoring. I select four health features from time-domain, incremental capacity, and plateau behavior: constant-current charging time, maximum incremental capacity peak, charge plateau duration, and discharge plateau duration. I screen them with Spearman rank correlation and consider field measurability and noise sensitivity. I build a temporal convolutional network for local feature extraction, a Transformer encoder for long-range aging dependency modeling, and a RIME optimizer for hyperparameter search. I validate the method on two offline battery datasets with different lithium-ion cell chemistries and discharge rates. The proposed model achieves RMSE values of 0.0044, 0.0037, 0.0023, and 0.0036 on the four tested lithium-ion cells. It outperforms support vector regression, long short-term memory, and a convolutional neural network. Ablation experiments confirm the positive contribution of each component, and SHAP analysis shows that the model adapts its feature weighting to the data. Repeatability analysis of same-condition lithium-ion cells shows synchronized capacity fade with increasing dispersion in later life. The method can be integrated into an intelligent monitoring terminal and used with existing leakage protection devices to form a prediction-detection defense-in-depth framework. This framework has the potential to reduce the probability of electric shock and electrical fire accidents in low-voltage distribution networks while improving the safety and reliability of lithium-ion cell energy storage.