I begin with the recognition that a Li-ion battery is not merely an electrochemical energy storage device. In a TN-C-S low-voltage distribution system, the Li-ion battery becomes part of the protective and operational architecture that supports backup power, peak shaving, voltage support, and demand response. When the Li-ion battery ages, its internal resistance increases, its available capacity decreases, and its insulation performance may degrade. These changes can contribute to residual current growth, zero-sequence voltage offset, leakage current, and unexpected protection tripping. Therefore, I treat the state of health (SOH) of the Li-ion battery as a pre-warning state variable that complements direct safety measurements such as residual current, zero-sequence voltage, and protection actions.
The central problem I address is that existing TN-C-S safety research often focuses on post-fault detection. Such methods are necessary, but they do not provide early warning before a fault occurs. At the same time, Li-ion battery operating conditions in distribution-area energy storage fluctuate frequently, and the electrochemical aging mechanisms are complex. High-precision SOH estimation for a Li-ion battery under such conditions is difficult. I therefore construct a SOH estimation model called RIME-TCN-Transformer, which combines a temporal convolutional network (TCN), a Transformer encoder, and the RIME optimization algorithm. I use this model to estimate the SOH of a Li-ion battery from features that can be extracted from battery management system (BMS) data and, when conditions permit, from incremental capacity analysis.
My motivation is practical. In a TN-C-S system, the front section uses a PEN conductor, and at the user inlet the PEN conductor is separated into a neutral conductor N and a protective conductor PE. The PE conductor is repeatedly grounded and then laid independently from the N conductor. This arrangement provides a protective grounding path for the power conversion system (PCS), the battery management system (BMS), and other core equipment. However, the shared PEN section creates specific electrical safety risks, including zero-sequence voltage offset, PEN breakage that can energize enclosures, and common-mode interference. Aging of the Li-ion battery increases residual current and can further increase the probability of leakage faults. Consequently, accurate SOH estimation of the Li-ion battery is a foundational element for proactive safety management.
I define SOH as the ratio of the current available maximum capacity to the rated capacity. For a Li-ion battery, this definition is expressed as
$$ SOH = \frac{Q_{act}}{Q_0} $$
where \(Q_{act}\) is the current available maximum capacity and \(Q_0\) is the rated capacity. I use this definition throughout my study because it is directly linked to the usable energy margin of the Li-ion battery and because it can be computed from complete charge-discharge cycles in aging datasets.
To validate my approach, I use two battery datasets. The first dataset is a public aging dataset of commercial 18650 Li-ion battery cells with a nickel-cobalt-aluminum oxide positive electrode and a graphite negative electrode. The rated capacity is 3.5 Ah and the rated voltage is 4.2 V. I select two cells, denoted NCA1 and NCA2, for SOH estimation. The second dataset is obtained from laboratory cycle aging tests on twelve lithium iron phosphate Li-ion battery cells divided into four groups. Cells LB1–LB3 form the first group, LB4–LB6 form the second group, LB7–LB9 form the third group, and LB10–LB12 form the fourth group. The tests are conducted at 25 °C. All Li-ion battery cells are charged at 5 C. The discharge rates are 5 C, 10 C, 15 C, and 20 C for the four groups, respectively. I select LB1 and LB4 for SOH estimation. These two datasets cover different positive electrode material systems and different discharge-rate conditions, which helps me simulate diverse operating scenarios of a Li-ion battery in a TN-C-S energy storage branch.
| Battery cell | Temperature | Charge rate | Discharge rate |
|---|---|---|---|
| NCA1 | 25 °C | 0.5 C | 0.5 C |
| NCA2 | 45 °C | 0.5 C | 0.5 C |
| LB1 | 25 °C | 5 C | 5 C |
| LB4 | 25 °C | 5 C | 10 C |
I extract health features from three perspectives: time-domain characteristics, incremental capacity, and voltage relaxation. Raw charge-discharge data contain rich aging information, but they are high-dimensional and redundant. Using raw data directly increases computational complexity and slows convergence. Therefore, I select features that are physically meaningful and measurable in a distribution-area energy storage system.
In the constant-current charging stage, the terminal voltage reaches the cutoff voltage faster as the Li-ion battery ages because polarization voltage increases. As a result, the constant-current charging time decreases with cycling. In the constant-voltage charging stage, internal resistance increases and diffusion slows down, so the constant-voltage charging time increases. I initially consider both constant-current charging time and constant-voltage charging time as candidate health features. I also consider charge and discharge plateau durations. A voltage plateau or slowly varying region is closely related to lithium intercalation or deintercalation reactions and phase evolution. With cumulative loss of lithium inventory and loss of active material, the effective reaction charge within a specific voltage interval decreases. At the same time, internal resistance and polarization increase, causing the Li-ion battery to reach the cutoff voltage earlier and shortening the effective plateau duration.
For example, in a 25 °C NCA1 Li-ion battery cell over 208 complete cycles, when SOH decreases from 93.40% to 72.82%, the charging plateau duration decreases from 3668.43 s to 3028.22 s, a decline of 17.45%. The discharging plateau duration decreases from 2364.83 s to 1811.36 s, a decline of 23.40%. Linear regression shows that for every 1 percentage point decrease in SOH, the charge and discharge plateau durations shorten by 25.82 s and 24.99 s, respectively, with \(R^2\) values of 0.9817 and 0.9826. At 45 °C, the NCA2 Li-ion battery cell shows the same trend, with charge and discharge plateau durations decreasing by 15.23% and 15.53%. These results indicate that plateau duration shortening is consistent across temperatures, although the rate differs. I therefore use charge plateau duration and discharge plateau duration as candidate health features.
The incremental capacity (IC) curve is obtained by differentiating battery capacity with respect to terminal voltage. The IC calculation is
$$ \frac{dQ}{dV} \approx \frac{Q_k – Q_{k-N}}{V_k – V_{k-N}} $$
where \(Q_k\) and \(V_k\) are the capacity and terminal voltage at the \(k\)-th sample point, and \(N\) is the difference interval. The differentiation amplifies voltage and capacity sampling noise. I apply Gaussian smoothing to the voltage and capacity measurement sequences before computing \(dQ/dV\). I extract the IC curve peak and the voltage corresponding to the IC curve peak as candidate health features. The IC feature is only used when the constant-current charging segment is complete and the sampling quality is sufficient. Under low signal-to-noise ratio (SNR) conditions, the model should down-weight or disable this feature.
Both datasets contain rest data after charge and discharge. I specifically extract the voltage rebound during the discharge relaxation stage as a candidate health feature. Compared with dynamic conditions, the relaxation stage can represent the internal kinetic state of the Li-ion battery under relatively small load current disturbance. However, the voltage rebound during discharge relaxation is a combined result of ohmic polarization, concentration polarization, and activation polarization recovery. Its evolution is sensitive to discharge depth, discharge rate, rest duration, and ambient temperature. It also depends on a relatively clear and complete rest window. In actual distribution-area energy storage operation, charge and discharge scheduling is flexible, and rest periods vary with dispatch requirements. Therefore, the stable acquisition of this feature is constrained by operating conditions.
I initially extract seven candidate health features. To build an accurate and efficient SOH estimation model, I use the Spearman rank correlation coefficient to quantify the correlation between each candidate health feature and the SOH of the Li-ion battery. The Spearman coefficient is
$$ \rho_s = 1 – \frac{6 \sum_{i=1}^{n} d_i^2}{n(n^2-1)} $$
where \(\rho_s \in [-1,1]\). The absolute value indicates correlation strength, and the sign indicates direction. A value close to 1 indicates a strong monotonic relationship. I treat discharge capacity as a positive proportional representation of SOH. Because a positive proportional transformation does not change sample ranks, I perform a two-sided Spearman rank correlation test on all seven candidate health features using complete cycle samples. To reduce the cumulative risk of type I errors from multiple hypothesis testing, I apply the Benjamini-Hochberg method to control the false discovery rate (FDR) and use \(q < 0.05\) as the statistical significance criterion. The significance test is not the only basis for feature selection. I also consider correlation strength, collinearity, electrochemical aging mechanisms, and field measurability.
| Candidate feature | Domain | Mechanistic link to Li-ion battery aging | Spearman correlation with SOH | Selected |
|---|---|---|---|---|
| Constant-current charging time | Time domain | Polarization increase causes earlier cutoff | 0.9966 | Yes |
| Constant-voltage charging time | Time domain | Diffusion limitation and resistance increase | Significant but weaker | No |
| IC curve maximum peak | Incremental capacity | Loss of lithium inventory and active material | 0.9993 | Yes |
| Voltage at IC peak | Incremental capacity | Phase evolution and polarization shift | Significant but redundant | No |
| Charging plateau duration | Time domain | Reduced effective reaction charge | 0.9932 | Yes |
| Discharging plateau duration | Time domain | Reduced effective reaction charge | 0.9937 | Yes |
| Discharge relaxation voltage rebound | Voltage relaxation | Polarization recovery and kinetics | Significant but condition-dependent | No |
After Benjamini-Hochberg FDR correction, all seven candidate health features are statistically significant with \(q < 0.05\). However, their correlation strength, information independence, and engineering applicability differ. Constant-current charging time, IC curve maximum peak, charging plateau duration, and discharging plateau duration have Spearman correlation coefficients of 0.9966, 0.9993, 0.9932, and 0.9937, respectively. They reflect the aging process of the Li-ion battery from different angles: charging process characteristics, incremental capacity features, and plateau evolution. They also have good mechanistic complementarity. I therefore select these four features as model inputs.
The constant-voltage charging time passes the significance test but has a relatively weaker correlation with SOH. It is also affected by cutoff voltage, cutoff current, initial state of charge, and temperature. At higher charging rates, the constant-voltage stage becomes shorter, which limits its cross-condition stability. The voltage at the IC peak is significantly correlated but strongly overlaps with the IC peak and plateau features. It is also sensitive to voltage sampling resolution and smoothing parameters. The additional information it provides beyond the IC peak is limited. The discharge relaxation voltage rebound reflects the internal polarization recovery process after discharge stops, but it is sensitive to discharge depth, discharge rate, temperature, and rest duration. It depends on a relatively complete rest observation window. Considering the random charge and discharge scheduling and rest periods in actual distribution-area energy storage, stable acquisition of this feature is constrained. Thus, the three unselected features are excluded not because of insufficient statistical significance but because of correlation strength, feature redundancy, and engineering application conditions. This selection preserves effective health information while balancing input complementarity and practical realizability.
I further analyze the online measurability of the selected features. Constant-current charging time and charge-discharge plateau durations can be calculated from the voltage-current time series collected by the BMS by identifying the relevant constant-current operating segments. They do not require additional dedicated sensors. However, their effective extraction still depends on the existence of constant-current operating windows with sufficient duration. In contrast, the IC peak requires a relatively continuous and complete constant-current charge capacity-voltage trajectory. It places higher demands on voltage sampling resolution, sampling noise, and data continuity. I therefore define the IC peak as a conditionally measurable feature, distinct from time-domain features that can be directly extracted from conventional operating data. When a complete constant-current charging segment is unavailable or the data quality is insufficient for stable IC curve reconstruction, the IC feature can be temporarily disabled, and time-domain features such as constant-current time and plateau durations can serve as the main health representations.
To evaluate the sensitivity of the IC feature to measurement noise and its applicable boundary, I conduct a noise perturbation experiment on 208 complete cycles from the original data. In the constant-current charging stage of each cycle, I add independent zero-mean Gaussian noise to the terminal voltage and charging capacity sequences. The noise standard deviation is determined from the root mean square (RMS) of the de-meaned original signal and a specified SNR. I set five SNR levels: 60, 50, 40, 30, and 20 dB. Each level is independently repeated 100 times to ensure repeatability. As the noise level increases, the IC peak extraction error and its SOH representation ability decrease. When SNR decreases from 60 dB to 40 dB, the median absolute percentage error (MdAPE) of the IC peak increases from 0.334% to 6.593%, and the median Spearman correlation coefficient \(\rho\) between the IC peak and SOH decreases from 0.9933 to 0.8500. When SNR further decreases to 30 dB and 20 dB, the median \(\rho\) decreases to 0.6165 and 0.2454, respectively. At 20 dB, only 96% of repeated tests still satisfy \(p < 0.05\). These results indicate that at approximately 40 dB and above, the IC peak still maintains strong SOH ranking representation ability. As sampling noise further increases, feature extraction stability and health state representation ability decline. Therefore, the engineering application of the IC feature should be dynamically selected according to actual sampling quality and operating condition completeness, rather than being treated as a fixed input feature without data quality constraints.
| SNR (dB) | MdAPE (%) | Median \(\rho\) | 95% interval of \(\rho\) | Repeats with \(p<0.05\) (%) |
|---|---|---|---|---|
| 60 | 0.334 | 0.9933 | 0.9906–0.9949 | 100.0 |
| 50 | 1.214 | 0.9638 | 0.9520–0.9715 | 100.0 |
| 40 | 6.593 | 0.8500 | 0.8165–0.8874 | 100.0 |
| 30 | 37.943 | 0.6165 | 0.5271–0.6876 | 100.0 |
| 20 | 949.372 | 0.2454 | 0.1257–0.3920 | 96.0 |
I now describe the RIME-TCN-Transformer model. I integrate TCN, Transformer, and RIME with engineering adaptation for TN-C-S low-voltage energy storage monitoring, where data disturbances are obvious, operating conditions are complex, and terminal computing resources are limited. TCN uses causal dilated convolution and residual structure to expand the receptive field while ensuring temporal causality. It extracts local temporal features with low computational cost, making it suitable for short pulse disturbances, rate mutations, and random noise in field monitoring data. It also has good parallel computing capability and edge deployment adaptability. Transformer uses a multi-head self-attention mechanism to model dependencies at different time scales. It captures long-term degradation trends across multiple cycles, compensates for the limitations of local convolution in long-range dependency modeling, and enhances the representation of complex operating condition changes and aging trajectory drift. Because optimal hyperparameter combinations depend on distribution-area operating conditions, Li-ion battery types, and data distributions, RIME uses a gradient-free swarm intelligence search to adaptively optimize key hyperparameters. This reduces manual tuning cost and improves parameter configuration efficiency in different application scenarios. The three modules handle local temporal feature extraction, global degradation dependency modeling, and hyperparameter optimization, forming a complementary SOH estimation framework for the Li-ion battery.
TCN combines causal convolution, dilated convolution, and residual connections. For a one-dimensional sequence input \(x \in \mathbb{R}^n\) and a filter \(f: \{0,…,k-1\} \to \mathbb{R}\), the dilated convolution operation \(F\) at position \(s\) is
$$ F(s) = (x *_{d} f)(s) = \sum_{i=0}^{k-1} f(i) \cdot x_{s – d i} $$
where \(d\) is the dilation factor, \(k\) is the kernel size, and \(s – d i\) indicates the sampled historical time. As network layers increase, \(d\) typically grows exponentially with base 2. This design allows TCN to cover a very long time span with a shallow network depth. The dilated causal convolution structure ensures that the output at time \(t\) depends only on inputs from time \(t\) and earlier. A TCN residual block contains two dilated causal convolution layers, each followed by weight normalization and a ReLU activation function. The residual block output is
$$ y = Activation(x + F(x)) $$
where \(x\) is the input and \(F(x)\) is the output after convolution transformations. For the Li-ion battery SOH estimation task, TCN extracts local temporal features from noisy charge-discharge time series and suppresses short pulse disturbances and local noise, providing a more stable feature sequence for subsequent global aging trend modeling.
The Transformer module captures long-term aging trends across hundreds of cycles. Its structure includes an encoder and a decoder. Unlike CNN, which processes data sequentially, Transformer inputs the entire sequence in parallel. To help the model understand the temporal information of Li-ion battery data, I add positional encoding. Transformer typically uses sine and cosine functions for positional encoding. For an input vector of length \(L\), the positional encoding at the \(k\)-th data position 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 vector dimension and \(n\) is a user-defined scalar. The attention mechanism is central to Transformer. The self-attention calculation relies on three learnable linear transformation matrices \(W_Q\), \(W_K\), and \(W_V\). The input vector is mapped into query \(Q\), key \(K\), and value \(V\). The scaled dot-product attention is
$$ Attention(Q,K,V) = softmax\left( \frac{QK^T}{\sqrt{d_k}} \right)V $$
where \(QK^T\) computes the dot product to measure similarity between each query vector and key vector. \(\sqrt{d_k}\) is a scaling factor that prevents numerical instability when the vector dimension increases. The softmax function ensures that attention weights sum to 1, normalizing similarity into a probability distribution. To allow the model to observe input data from different perspectives, Transformer introduces multi-head attention:
$$ MultiHead(Q,K,V) = Concat(head_1,…,head_h) W_O $$
Each attention head is
$$ head_i = Attention(Q W_i^Q, K W_i^K, V W_i^V) $$
where \(h\) is the number of attention heads, \(W_O\) is the output projection matrix, and \(W_i\) is the projection matrix of the \(i\)-th head.
RIME is a metaheuristic optimization algorithm that simulates soft rime and hard rime growth behaviors to achieve a balance between global search and local exploitation. I use RIME to adaptively optimize key hyperparameters of the network, accommodating data distribution differences across distribution areas and Li-ion battery systems. This reduces reliance on manual experience and improves parameter configuration efficiency and deployment maintainability.
In the soft rime search mechanism, the growth process of soft rime particles simulates random diffusion and condensation of ice crystals in a breeze. For particle \(i\) at iteration \(t\) in dimension \(j\), the position update is
$$ R_{ij}^{new} = R_{best,j} + r_1 \cos\theta \cdot \beta \cdot (h \cdot (Ub_{ij} – Lb_{ij}) + Lb_{ij}) , \quad r_2 < E $$
where \(R_{best,j}\) is the current global best solution in dimension \(j\), \(r_1\) is a random number in \((-1,1)\), \(\theta\) controls the search direction with \(\cos\theta\), \(\beta\) is an environmental factor that follows the iteration number to simulate external environmental influence, \(Ub_{ij}\) and \(Lb_{ij}\) are the upper and lower bounds of the dimension, \(h\) is a random number in \((0,1)\) for controlling the distance between two particles, and \(r_2\) is a random number in \((0,1)\) that controls whether the particle position updates together with \(E\). The angle parameter is
$$ \theta = \frac{\pi t}{10 T} $$
where \(t\) is the current iteration and \(T\) is the maximum iteration. The environmental factor is
$$ \beta = 1 – \left[ \frac{w t}{T} \right] / w $$
where \(w\) is usually 5 and \([\cdot]\) denotes rounding. The adhesion coefficient \(E\) affects the condensation probability and increases with iteration:
$$ E = \frac{t}{T} $$
RIME introduces normalized fitness as the threshold for the hard rime puncture mechanism. If \(F_i\) is the fitness value of the \(i\)-th particle, its normalized value is
$$ S_i = \frac{F_i – \min(F)}{\max(F) – \min(F) + \epsilon} $$
where \(\epsilon\) is a small value to prevent division by zero. A value of \(S_i\) close to 1 indicates poor fitness and a greater need for replacement. A value close to 0 indicates good fitness and a greater need for retention. For particles with poor fitness, RIME uses a position crossover update. For each particle \(i\) in dimension \(j\), a random number \(r_3 \in (0,1)\) is generated. If \(r_3 < S_i\), the particle is considered poor and the hard rime puncture condition is met. The \(j\)-th dimension of particle \(i\) is forced to the global best solution value:
$$ R_{ij}^{new} = R_{best,j} $$
If \(r_3 \geq S_i\), the particle has good fitness and remains in place. To ensure that the new position is better than the original, RIME uses a positive greedy selection strategy. After calculating the fitness of all new positions \(R_i^{new}\), it compares the old and new fitness:
$$ R_i^{t+1} = \begin{cases} R_i^{new} & \text{if } F(R_i^{new}) < F(R_i^t) \\ R_i^t & \text{otherwise} \end{cases} $$
where \(R_i^{t+1}\) is the position of particle \(i\) entering the next iteration.
I use RIME to optimize six hyperparameters of the TCN-Transformer network: initial learning rate, number of convolution kernels, convolution kernel size, dropout, batch size, and number of attention heads. The fitness function is the root mean square error (RMSE):
$$ RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i – \hat{y}_i)^2} $$
where \(\hat{y}_i\) is the predicted SOH of the \(i\)-th cycle and \(y_i\) is the true SOH of the \(i\)-th cycle. The optimization ranges are listed in the following table.
| Hyperparameter | Upper bound | Lower bound |
|---|---|---|
| Initial learning rate | 0.01 | 0.0001 |
| Number of convolution kernels | 128 | 16 |
| Convolution kernel size | 9 | 3 |
| Dropout | 0.5 | 0.01 |
| Batch size | 128 | 16 |
| Number of attention heads | 5 | 2 |
For deployment in a TN-C-S intelligent monitoring terminal, the causal dilated convolution in TCN has good parallel computing characteristics. I retain only the Transformer encoder structure to control model complexity and improve computational efficiency, giving the overall network edge deployment potential. Model parameter count, inference latency, and power consumption are closely related to the target processor architecture, computing resources, and actual load conditions. Subsequent work can combine specific terminal hardware with model pruning, quantization compression, and field load testing to further optimize deployment and evaluate real-time performance and resource adaptation.
I evaluate the RIME-TCN-Transformer model using four metrics: RMSE, mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (\(R^2\)). I compare the proposed method with three widely used baseline algorithms: support vector regression (SVR), convolutional neural network (CNN), and long short-term memory (LSTM). I divide the four Li-ion battery datasets into 80% training and 20% testing and conduct training and testing in the same computing environment. The comparison results are shown in the following table.
| Battery | Model | RMSE | MAE | MAPE | \(R^2\) |
|---|---|---|---|---|---|
| NCA1 | SVR | 0.0117 | 0.0128 | 0.49% | 0.9559 |
| NCA1 | LSTM | 0.0111 | 0.0103 | 0.38% | 0.9816 |
| NCA1 | CNN | 0.0081 | 0.0066 | 0.26% | 0.9874 |
| NCA1 | RIME-TCN-Transformer | 0.0044 | 0.0036 | 0.13% | 0.9971 |
| NCA2 | SVR | 0.0115 | 0.0094 | 0.37% | 0.8739 |
| NCA2 | LSTM | 0.0069 | 0.0066 | 0.25% | 0.9546 |
| NCA2 | CNN | 0.0063 | 0.0058 | 0.22% | 0.9629 |
| NCA2 | RIME-TCN-Transformer | 0.0037 | 0.0029 | 0.11% | 0.9867 |
| LB1 | SVR | 0.0104 | 0.0095 | 0.95% | 0.6359 |
| LB1 | LSTM | 0.0056 | 0.0053 | 0.52% | 0.8920 |
| LB1 | CNN | 0.0046 | 0.0042 | 0.41% | 0.9271 |
| LB1 | RIME-TCN-Transformer | 0.0023 | 0.0018 | 0.18% | 0.9823 |
| LB4 | SVR | 0.0181 | 0.0096 | 0.93% | 0.8865 |
| LB4 | LSTM | 0.0130 | 0.0068 | 0.65% | 0.9412 |
| LB4 | CNN | 0.0095 | 0.0074 | 0.73% | 0.9687 |
| LB4 | RIME-TCN-Transformer | 0.0036 | 0.0030 | 0.32% | 0.9956 |
The results show that SVR has the largest estimation error. LSTM and CNN have lower overall errors than SVR, but they still show tracking deviations during the late-life rapid capacity decay stage. In comparison, RIME-TCN-Transformer achieves higher SOH estimation accuracy on all four Li-ion battery cells under the offline test conditions. This indicates that the integrated feature extraction, long-range dependency modeling, and hyperparameter optimization modules work together effectively under the current datasets and split method. Considering that actual distribution-area operating environments involve more complex load fluctuations, measurement noise, and condition changes, the adaptability and robustness of the model under field conditions still require further evaluation with measured data.
For NCA1, compared with SVR, the proposed model reduces RMSE, MAE, and MAPE by 62.39%, 71.88%, and 73.47%, respectively. Compared with LSTM and CNN, RMSE is reduced by 60.36% and 45.68%. The \(R^2\) of the proposed model on NCA1 is 0.9971, higher than 0.9816 for LSTM and 0.9874 for CNN. The error box plots show that the interquartile range of the proposed model is compact and the median error is close to zero, indicating good error concentration. Due to capacity fluctuations, LB1 and LB4 show some outliers under all models, with SVR having relatively more outliers on LB4. Overall, the proposed model exhibits a stable error distribution under the current test conditions.
The offline benchmark comparison shows that RIME-TCN-Transformer achieves low SOH estimation errors on both Li-ion battery datasets. This provides effective battery health information support for TN-C-S energy storage pre-warning. The current validation focuses on the battery SOH estimation layer. The warning threshold setting and the mapping between SOH degradation and leakage risk in actual distribution areas can be further calibrated using synchronized monitoring data from the PCS, BMS, and distribution side. This would build a complete validation chain extending from Li-ion battery health perception to system safety warning.
I conduct an ablation study to investigate the effectiveness and contribution of each component in the RIME-TCN-Transformer model. I divide the four Li-ion battery datasets into 80% training and 20% testing and construct three control models: RIME-TCN, RIME-Transformer, and TCN-Transformer. I compare them with the full RIME-TCN-Transformer model. The results are listed in the following table.
| Battery | Model | RMSE | MAE | MAPE | \(R^2\) |
|---|---|---|---|---|---|
| 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 |
Comparing TCN-Transformer and RIME-TCN-Transformer on NCA1, RMSE decreases from 0.0065 to 0.0044, a reduction of 32%. MAE and MAPE decrease by 32% and 31.5%, respectively, and \(R^2\) is closer to 1. This demonstrates the direct accuracy improvement brought by RIME hyperparameter optimization. The RIME-Transformer model without the TCN module performs poorly on NCA1 and NCA2. On NCA1, its RMSE is 0.0116, the largest among the four models. This indicates that on small datasets, directly feeding the raw sequence into the Transformer module without initial local feature extraction and denoising by TCN leads to reduced estimation accuracy. The RIME-TCN model without the Transformer module performs poorly on LB1 and LB4. This indicates that when the dataset scale increases, the model cannot handle long-term aging patterns in long-sequence Li-ion battery data without the Transformer module, resulting in increased error.
The error box plots show that the proposed RIME-TCN-Transformer has the flattest and most compact error distribution on all four Li-ion battery cells, concentrated around the zero-error line. RIME-TCN and RIME-Transformer show obvious upward and downward shifts on NCA1 and LB4. In particular, RIME-TCN on LB4 has extreme outliers, indicating that when the global temporal connection provided by Transformer is missing, the TCN model is more susceptible to outlier interference and produces obvious errors in some cycles. The ablation results under the current test conditions show that the full RIME-TCN-Transformer model achieves low SOH estimation error, indicating good synergy among the functional modules under the current datasets and experimental settings. From an engineering perspective, this model helps reduce parameter adaptation costs across different distribution-area operating conditions and Li-ion battery systems, supporting subsequent model transfer, field deployment, and rapid configuration during operation and maintenance.
I introduce SHAP (Shapley Additive Explanations) to analyze the contributions of the four input features: constant-current charging time, IC curve maximum peak, charging plateau duration, and discharging plateau duration. Although the RIME-TCN-Transformer model shows high SOH estimation accuracy and robustness, its complex network structure and nonlinear mapping give it black-box characteristics. I randomly sample 100 points from the training set as the background dataset for SHAP value calculation.
| Battery | Feature | Global importance | Interpretation |
|---|---|---|---|
| NCA1 | IC curve maximum peak | 62.05% | Dominant health indicator for this Li-ion battery |
| NCA1 | Other three features | 37.95% | Complementary aging information |
| NCA2 | IC curve maximum peak | Dominant | Strong aging representation |
| NCA2 | Charging plateau duration | 29.88% | Effective reaction charge reduction |
| NCA2 | Constant-current charging time | 28.21% | Polarization and cutoff behavior |
| LB1 | IC curve maximum peak | 0.09% | Soft down-weighting under this data structure |
| LB4 | IC curve maximum peak | 0.18% | Soft down-weighting under this data structure |
In NCA1 and NCA2 Li-ion battery cells, the IC curve maximum peak dominates SOH estimation. In NCA1, its global importance is as high as 62.05%. This indicates that, from the perspective of the RIME-TCN-Transformer model, the IC curve peak best reflects the SOH of the NCA1 Li-ion battery. The local contribution plot for NCA1 also reflects this phenomenon. I use the average SOH estimate of 100 random training cycles as the baseline. Because the test set Li-ion battery is more aged than the training set, the SHAP values of the test set features are negative, meaning these features cause the model SOH estimate to be lower than the baseline, reflecting battery aging. In NCA1, red points (high peak values) are near the right side and blue points (low peak values) are near the left side. This indicates that a higher IC peak leads the model to give a higher SOH estimate, consistent with the physical rule that a healthier Li-ion battery has a higher IC curve peak. In NCA2, besides the dominant IC curve maximum peak, charging plateau duration and constant-current charging time contribute 29.88% and 28.21%, respectively. This shows that the model comprehensively uses multiple features for SOH estimation.
In the second dataset, SHAP results show that the IC peak global contribution in LB1 and LB4 is about 0.09% and 0.18%, respectively. Compared with the NCA data, the model output on LB1 and LB4 is significantly less sensitive to the IC peak. This indicates that under a fixed four-feature input framework, the network can adaptively allocate weights to different features according to the information structure of the data, which can be regarded as a data-driven soft down-weighting phenomenon. Considering that the IC peak still shows high information contribution in the NCA data, and a unified input structure helps maintain model interface consistency across datasets, I retain this feature as a candidate input. From an engineering application perspective, the activation of the IC feature should still match actual sampling conditions. Feature validity criteria can be set based on constant-current charging segment completeness, SNR, and data continuity. When extraction conditions are met, the feature participates in SOH estimation. When acquisition conditions are limited, its participation is reduced or time-domain features are used for health state representation. The results further indicate that the model response to the four input features changes with Li-ion battery system, operating condition, and data distribution, showing clear data dependence. The current results mainly reveal feature contribution differences under different data conditions and provide a basis for feature configuration across Li-ion battery systems. Their applicability to broader battery types and actual distribution-area operating environments can be further evaluated with more independent cells and field synchronized monitoring data.
To increase repeated samples under the same operating condition and examine cell-to-cell dispersion of aging trajectories, I read the cycle summary records of LB2 and LB3 and analyze them together with LB1. All three Li-ion battery cells belong to the first group of operating conditions. I normalize the discharge capacity as
$$ Q_{norm}(N) = \frac{Q(N)}{Q(0)} $$
where \(Q(0)\) is the first-cycle discharge capacity of each cell. LB1, LB2, and LB3 contain 1404, 1350, and 1368 cycles, respectively. Within the common 1–1350 cycles of the three cells, I compute the capacity dispersion as the ratio of the sample standard deviation to the mean of the normalized discharge capacity, that is, the coefficient of variation. The normalized discharge capacity trajectories show that LB1–LB3 exhibit a generally synchronous capacity decay trend under the same operating condition. The capacity dispersion curve shows that the dispersion among cells gradually increases with cycling. Within the common 1–1350 cycle interval, the average coefficient of variation of the normalized discharge capacity of the three cells is 0.735%, and the maximum is 1.159% at cycle 1171. The average coefficients of variation for the first 100 cycles, the middle 100 cycles, and the last 100 cycles are 0.303%, 0.747%, and 1.033%, respectively. The final normalized capacities of LB1, LB2, and LB3 are 80.73%, 81.46%, and 81.04%, respectively. These results indicate that the three repeated Li-ion battery cells under the same operating condition have similar capacity decay directions, and the experimental data have good repeatability. At the same time, the increased dispersion in the later stage indicates that objective cell-to-cell differences still exist in long-term aging.
| Cell group | Cells | Cycles | Average coefficient of variation | Maximum coefficient of variation | Final normalized capacity |
|---|---|---|---|---|---|
| First group | LB1, LB2, LB3 | 1404, 1350, 1368 | 0.735% | 1.159% | 80.73%, 81.46%, 81.04% |
| Early stage | LB1–LB3 | First 100 | 0.303% | — | — |
| Middle stage | LB1–LB3 | Middle 100 | 0.747% | — | — |
| Late stage | LB1–LB3 | Last 100 | 1.033% | — | — |
I now discuss the engineering implications of my study. The proposed method provides high-precision SOH information for a Li-ion battery in a TN-C-S low-voltage distribution energy storage system. This information can be integrated into an intelligent monitoring terminal. Together with existing residual-current protection devices, it can form a prediction–detection defense-in-depth framework. The Li-ion battery SOH is not a direct criterion for leakage or tripping. Instead, it is a prior aging state variable. It complements zero-sequence voltage, residual current, and protection action measurements. The mapping between SOH and safety risk still needs to be calibrated through synchronized data from the PCS, BMS, and distribution side.
From the perspective of electrochemical mechanisms, solid electrolyte interphase thickening, loss of lithium inventory, and electrolyte side reactions cause capacity decay and internal resistance increase. These processes can exacerbate voltage fluctuations and thermal stress during charge and discharge. However, the SOH of a Li-ion battery does not correspond one-to-one with system insulation, residual current, or protection tripping. It is also affected by connection state, ambient temperature, PCS operating mode, and grounding conditions. Therefore, I position SOH as a pre-perception quantity for aging state, not as a direct criterion for leakage or tripping. The mapping from SOH to safety risk still requires calibration using synchronized data from the PCS, BMS, and distribution side.
My method also has limitations. The current validation is based on offline battery datasets. Field conditions involve more complex noise, load fluctuations, and operating condition changes. The IC feature depends on complete constant-current charging segments and sufficient sampling quality. In distribution-area energy storage, charge and discharge scheduling may not always provide such segments. The time-domain features are more robust in this respect, but they still require sufficient constant-current windows. The RIME optimization is performed offline. Online adaptation of hyperparameters may be needed if the Li-ion battery system or operating condition changes significantly. Future work can address online internal resistance identification, dynamic feature gating, cross-cell model transfer, and model compression, inference latency, and power consumption testing on target monitoring terminals.
In conclusion, I address the need for long-term aging state perception and pre-warning safety risk identification in a TN-C-S low-voltage distribution energy storage system. I treat the SOH of a Li-ion battery as a pre-state quantity reflecting battery degradation. I retain direct safety measurements such as zero-sequence voltage, residual current, and protection actions. I construct the RIME-TCN-Transformer SOH estimation model and validate it in terms of health feature selection, feature measurability, model estimation performance, and interpretability. The main conclusions are as follows.
First, I introduce Li-ion battery SOH estimation into TN-C-S low-voltage distribution safety monitoring as a pre-state perception quantity reflecting battery aging. It complements zero-sequence voltage, residual current, and protection actions, providing a basis for risk warning and predictive management of the energy storage system.
Second, from seven candidate health features, I use Spearman rank correlation tests and comprehensively consider correlation strength, collinearity, electrochemical mechanisms, and engineering measurability. I select constant-current charging time, IC peak, charging plateau duration, and discharging plateau duration as model inputs. The time-domain features can be extracted from conventional BMS voltage-current time series. The IC feature is used when the constant-current capacity-voltage trajectory is complete and sampling quality meets requirements. At 40 dB, the median Spearman correlation coefficient between the IC peak and SOH is 0.8500.
Third, I construct the RIME-TCN-Transformer Li-ion battery SOH estimation model. TCN extracts local temporal features, Transformer models long-term aging dependencies, and RIME performs key hyperparameter optimization. The experimental results show that the RMSE of the proposed model on the public dataset and the laboratory dataset is 0.0044 and 0.0023, respectively, outperforming SVR, LSTM, and CNN baseline models. Ablation experiments and SHAP analysis further show that the modules have synergistic effects, and the model can adaptively use input information according to different data characteristics.
Overall, the proposed method can provide relatively high-precision Li-ion battery aging state information for a TN-C-S low-voltage distribution energy storage system. It can serve as a pre-state perception supplement to existing zero-sequence voltage, residual current, and protection action monitoring. It provides a methodological basis for a layered safety monitoring system consisting of health state perception, electrical safety detection, and protection action. Future research can further combine long-term synchronized operation data from distribution-area PCS, BMS, and distribution side to calibrate the relationship between SOH and safety representation quantities such as residual current and zero-sequence voltage. It can also conduct engineering validation around online internal resistance identification, dynamic feature gating, cross-cell model transfer, and model compression, inference latency, and power consumption testing on target monitoring terminals. This would further improve the pre-perception method for Li-ion battery energy storage safety in actual distribution-area operating environments.
| Symbol | Meaning |
|---|---|
| \(SOH\) | State of health of the Li-ion battery |
| \(Q_{act}\) | Current available maximum capacity |
| \(Q_0\) | Rated capacity |
| \(\rho_s\) | Spearman rank correlation coefficient |
| \(d_i\) | Rank difference between paired observations |
| \(dQ/dV\) | Incremental capacity |
| \(d\) | Dilation factor in TCN |
| \(k\) | Convolution kernel size |
| \(Q,K,V\) | Query, key, and value matrices in attention |
| \(W_O\) | Output projection matrix |
| \(h\) | Number of attention heads |
| \(R_{ij}^{new}\) | New position of particle \(i\) in dimension \(j\) |
| \(R_{best,j}\) | Global best solution in dimension \(j\) |
| \(F_i\) | Fitness value of particle \(i\) |
| \(S_i\) | Normalized fitness of particle \(i\) |
| \(RMSE\) | Root mean square error |
| \(MAE\) | Mean absolute error |
| \(MAPE\) | Mean absolute percentage error |
| \(R^2\) | Coefficient of determination |
| \(SNR\) | Signal-to-noise ratio |
| \(MdAPE\) | Median absolute percentage error |
The model I propose is designed to be compatible with the TN-C-S low-voltage distribution environment. In such an environment, the Li-ion battery is part of a backup energy storage system. The safety of the Li-ion battery affects the residual current, zero-sequence voltage, and protection behavior of the distribution system. By estimating the SOH of the Li-ion battery with high accuracy, I provide an early indication of aging. This early indication can be used to schedule maintenance, adjust charging and discharging strategies, and reduce the probability of electric shock and electrical fire accidents in the low-voltage distribution network. The Li-ion battery SOH estimation is therefore not only a battery management task but also a distribution safety task.
I believe that the combination of TCN, Transformer, and RIME is well suited to the characteristics of Li-ion battery aging data. TCN captures local temporal patterns and suppresses noise. Transformer captures long-range dependencies and global aging trends. RIME optimizes hyperparameters and reduces manual tuning. The SHAP analysis shows that the model can adaptively adjust feature weights according to the data. For a Li-ion battery with a strong IC peak contribution, the model relies on the IC peak. For a Li-ion battery with weaker IC peak contribution, the model down-weights the IC peak and relies more on time-domain features. This adaptive behavior supports cross-battery-system generalization and robustness.
In future work, I plan to integrate the model into a distribution-area intelligent monitoring terminal. I will test the model with field data from PCS, BMS, and distribution-side sensors. I will calibrate the relationship between Li-ion battery SOH and safety quantities such as residual current and zero-sequence voltage. I will also investigate online internal resistance identification and dynamic feature gating to improve the practical applicability of the method. I will evaluate model compression, inference latency, and power consumption on target hardware. These steps will help transform the proposed Li-ion battery SOH estimation method into a practical pre-warning tool for TN-C-S low-voltage distribution safety enhancement.