As the electric car market continues to expand at an unprecedented pace, the safe and reliable operation of traction batteries has become one of the primary technical challenges facing the automotive industry. In this study, I focus on the Prognostics and Health Management (PHM) technology for electric car lithium batteries, aiming to address the critical issues of State of Health (SOH) estimation and Remaining Useful Life (RUL) prediction. Throughout the design process of electric cars, the battery system serves as the core energy storage component, and its degradation directly affects the vehicle’s driving range, power performance, and overall safety. I first analyzed the electrochemical working principles and degradation mechanisms of lithium-ion batteries, establishing a comprehensive experimental platform to collect voltage, current, and capacity data under cyclic charge-discharge conditions. A multi-source database combining laboratory testing with open-access datasets was constructed to support model development. To overcome the difficulty of directly measuring battery capacity, I proposed a multi-feature health factor extraction framework based on charge-discharge characteristic parameters. Six indirect health indicators were selected and carefully validated using Pearson and Spearman correlation coefficients. A t-Distributed Stochastic Neighbor Embedding (t-SNE) technique was further used to reduce the dimensionality of extracted features, eliminating information redundancy while preserving the degradation-related content necessary for subsequent modeling. For SOH estimation, I built an integrated deep learning architecture combining a convolutional neural network (CNN), gated recurrent unit (GRU), and attention mechanism, whose hyperparameters were automatically tuned by an improved Northern Goshawk Optimization (INGO) algorithm developed with multiple enhancement strategies. Extensive ablation experiments on both public and self-collected datasets demonstrated that the proposed method achieved superior estimation accuracy and robustness compared to conventional baselines. In the realm of RUL prediction, I proposed a novel hybrid method integrating an optimized Variational Mode Decomposition (VMD) driven by the RIME algorithm with an optimized Transformer-Bidirectional Long Short-Term Memory (BiLSTM) network. The RIME-VMD preprocessing effectively suppressed capacity regeneration and noise interference, while the Transformer-BiLSTM backbone captured both global contextual information and local sequential dependencies. The INGO algorithm was again employed to optimize all critical hyperparameters of the predictive network. Comprehensive experiments at various prediction starting points confirmed that the RUL prediction framework could deliver high precision and strong generalization ability. Finally, I developed an interactive software platform for lithium battery PHM applications. This system supports data import, health factor extraction, dimensionality reduction, SOH evaluation, and RUL forecasting through user-friendly interfaces, providing practical decision support for electric car battery maintenance strategies.
1. Introduction
In response to the national strategic goals of carbon peak and carbon neutrality, the rapid development of the new energy vehicle industry has become a global consensus. Over the past seven years, the production and sales of electric cars in China have demonstrated a remarkable growth trajectory. As the statistics published by the China Association of Automobile Manufacturers show, the annual production of electric cars reached millions of units, with the market share continuously rising. Electric cars primarily rely on high-performance battery systems for propulsion, and lithium battery technology has become the dominant choice due to its superior energy density, low self-discharge rate, and long cycle life, as shown in Table 1.
| Parameter | Value |
|---|---|
| Nominal Capacity (mAh) | 2600 |
| Nominal Voltage (V) | 3.7 |
| Charge Cut-off Voltage (V) | 4.2 |
| Discharge Cut-off Voltage (V) | 2.7 |
| Operating Temperature (°C) | -20~60 |
Because the lithium battery is a complex electrochemical system whose performance degrades over time and usage cycles, the Battery Management System (BMS) in an electric car plays an essential role in monitoring, protecting, and optimizing the operation of the battery pack. The Prognostics and Health Management (PHM) technology has emerged as a crucial component of a modern BMS, bridging the gap between traditional fault diagnosis and future maintenance planning. Battery PHM enables a transition from “post-fault maintenance” and “time-based maintenance” to “condition-based predictive maintenance.” Two central tasks constitute the core of battery PHM research: the estimation of State of Health (SOH) and the prediction of Remaining Useful Life (RUL). These two metrics reflect the battery condition on different timescales. While SOH estimation indicates the present maximum available capacity and the short-term performance trend, RUL prediction aims at projecting the battery’s future trajectory until it reaches the End of Life (EOL) criterion, which is commonly defined as the moment when the capacity declines to a threshold (typically 70% to 80% of its rated value). Both tasks are of vital importance to guarantee the reliable and safe operation of electric cars, helping drivers plan charging strategies and prompting battery replacement actions before catastrophic failures occur.
Based on their methodological foundations, the SOH estimation and RUL prediction approaches can be broadly classified into three categories: model-based methods, data-driven methods, and hybrid fusion methods. Model-based methods, including electrochemical models, equivalent circuit models, and empirical models, aim at building mathematical representations of the internal electrochemical dynamics and degradation mechanisms of the battery. Although these models can, under ideal conditions, offer good interpretability and prediction performance, they often require extensive prior knowledge and complicated parameter identification procedures. Their accuracy is highly sensitive to variations in operating conditions and cell chemistry. Moreover, establishing an accurate and comprehensive electrochemical model is quite challenging due to the complex interplay of multiple physical and chemical processes, such as the loss of lithium inventory, active material degradation, SEI film growth, and conductive network deterioration. Consequently, the application of model-based methods alone is not always practical for real-world electric car fleets, where operating profiles vary greatly.
Data-driven methods, in contrast, do not require searching for the explicit degradation mechanism. Instead, they directly learn from the monitored operational data, such as voltage, current, and temperature time series, and build an implicit mapping from features to SOH or capacity. Various statistical and machine-learning techniques have been reported in the literature, including Gaussian process regression, support vector regression, relevance vector machines, and autoregressive models, as shown in Table 2.
| Category | Representative Algorithms | Characteristics |
|---|---|---|
| Statistical Learning | Wiener Process, Autoregression | Simple but limited by assumptions |
| Classical Machine Learning | GPR, SVM, RVM | Good for small samples |
| Deep Learning | CNN, GRU, LSTM, BiLSTM, Transformer | Powerful feature representation |
| Data-driven Fusion | CNN-LSTM, Transformer-BiLSTM | High accuracy and robustness |
Despite the progress made, there are still several critical research gaps in lithium battery PHM technology. First, in most practical electric car applications, the battery capacity cannot be directly measured at every sampling instant because the measurements require specific charge-discharge conditions that are not always available during routine driving. The conventional coulomb-counting methods may induce accumulated errors over time. Therefore, extracting indirect health indicators that exhibit a strong correlation with battery capacity becomes a necessary and meaningful undertaking. A high-quality health indicator should contain enough degradation information, reflect the underlying aging direction, and be robust to measurement uncertainty.
Secondly, the quality of input features directly determines the ceiling of the learning algorithm. A single health factor cannot adequately describe the full picture of capacity degradation, whereas overly redundant high-dimensional features may confuse the model and degrade its prediction accuracy. Consequently, a systematic method for constructing a set of complementary yet compact health factors is needed. Thirdly, battery degradation is a highly nonlinear, time-varying, and uncertain process, which makes it extremely difficult to be perfectly modeled by a single technique. Individual machine learning models often exhibit specific inductive biases, so a carefully designed fusion of several complementary architectures is likely to achieve better predictions. Network hyperparameters, such as learning rate and hidden-layer sizes, have significant impacts on deep learning performance. To further release the potential of the network model, I propose to use an enhanced swarm-intelligence optimizer to automatically search for the best hyperparameter combination. In view of the above analyses, this dissertation conducts a comprehensive study on the PHM technology for electric car lithium batteries, proposes high-precision SOH estimation and RUL prediction frameworks, verifies them using both self-collected laboratory data and publicly accessible battery data, and ultimately designs a prototype PHM software system.
2. Lithium Battery Characteristics and Degradation Mechanisms
2.1 Basic Working Principle
Lithium-ion batteries are secondary batteries that convert chemical energy into electrical energy through reversible electrochemical reactions. The main components include a positive electrode, a negative electrode, an electrolyte, and a separator. The positive electrode material is usually a lithium metal oxide such as lithium cobalt oxide or lithium iron phosphate, whereas the negative electrode is typically made of graphite or other carbonaceous material. During the charging process, lithium ions are extracted from the positive electrode, move through the electrolyte and the separator, and then intercalate into the negative electrode. Simultaneously, electrons travel through the external circuit and generate a charging current. The reverse process occurs during discharge, in which the lithium ions are released from the negative electrode and travel back to the positive electrode, while electrons flow through the external load and thus supply power to the electric car. This “rocking-chair” mechanism is highly reversible. However, side reactions and structural transformations that accompany repeated cycles lead to a gradual fade in capacity and an increase in internal resistance, which are the primary manifestations of battery aging. A typical half-reaction can be described on the positive electrode as:
$$ \mathrm{LiCoO_2} \xrightarrow[\text{discharge}]{\text{charge}} \mathrm{Li_{1-n}CoO_2} + n \mathrm{Li^+} + n e^- $$
with the corresponding negative-electrode reaction,
$$ 6C + n \mathrm{Li^+} + n e^- \xrightarrow[\text{discharge}]{\text{charge}} \mathrm{Li_n C_6} $$
2.2 Degradation Mechanisms
The degradation of lithium batteries is attributed to both internal and external factors. Internally, the primary causes of capacity fading include the loss of lithium inventory, the loss of active materials in the electrodes, and the loss of electrical conductivity. Lithium inventory loss occurs through the formation and growth of the solid electrolyte interphase (SEI) film, which consumes cyclable lithium ions. Metallic lithium deposition can also occur on the negative electrode surface, leading to the formation of dendrites and further loss of active lithium. The loss of active positive and negative materials may be induced by structural fatigue, particle cracking, or dissolution of transition metals. Conductivity losses are often caused by the degradation of the conductive additive network, the decomposition of binder materials, and the corrosion of current collectors. Externally, the battery degradation speed is significantly influenced by temperature, overcharge or overdischarge, high C-rate cycling, and mechanical stress. High temperatures accelerate parasitic side reactions, while low temperatures increase the electrolyte viscosity and hinder lithium-ion transport. Mechanical vibrations, which are common in the electric car operating environment, are also capable of damaging the internal structure and accelerating capacity fade. These internal and external mechanisms are coupled in a complex way, giving rise to nonlinear capacity degradation trajectories and stochastic capacity regeneration events.
2.3 Experimental Setup and Data Collection
To support the development and evaluation of data-driven PHM techniques, I designed a cycle-life experiment for 18650-format commercial lithium batteries. The experiments were conducted on a battery test system (GN-CD30V15A) that can perform programmable charge and discharge operations. The test bench consists of a host computer, the battery cycler, four-wire Kelvin connections, and dedicated battery holders. During each cycle, the battery was first charged using the constant current-constant voltage protocol at a current of 1.5 A up to a voltage of 4.2 V, followed by a constant voltage phase until the current dropped below 0.1 A. After a rest period of 20 minutes, the battery was discharged at 2 A to a cut-off voltage of 2.7 V, and then rested again for 20 minutes. The measured capacity values from each discharge step were recorded as the reference capacity for that cycle. The repeated cycling process continued until the capacity fell below the failure threshold of 1.8 Ah. In total, 213 charge-discharge cycles were recorded for the experimental cell, which I denote as the A1 dataset. Although the measurement equipment was set up in a laboratory environment at ambient temperature, the capacity-fade curves clearly exhibit fluctuations, a phenomenon that is often referred to as capacity regeneration.
In addition to the laboratory-generated dataset, I used the well-known NASA battery open-source dataset in this study. I selected four battery cells, designated B0005, B0006, B0007, and B0018, all of which were cycled under similar constant current constant voltage charging and constant-current discharging protocols but at different discharge cut-off voltages. Table 3 lists the operating conditions of these battery cells.
| Cell | Charge Cut-off (V) | Temperature (°C) | Discharge Cut-off (V) | Charge Current (A) | Discharge Current (A) |
|---|---|---|---|---|---|
| B0005 | 4.2 | 24 | 2.7 | 1.5 | 2 |
| B0006 | 4.2 | 24 | 2.5 | 1.5 | 2 |
| B0007 | 4.2 | 24 | 2.2 | 1.5 | 2 |
| B0018 | 4.2 | 24 | 2.5 | 1.5 | 2 |
The gradual decrease in discharge capacity of these cells as the cycle number progresses confirms the nonlinear degradation patterns of lithium batteries, which are influenced by capacity regeneration and measurement noise.
3. Health Factor Extraction and Dimensionality Reduction for Electric Car Batteries
3.1 Feature Extraction
To enable real-time SOH monitoring of electric car batteries without performing costly direct capacity measurements, I extracted multiple indirect health factors (HIs) from the routinely recorded charge-discharge profiles. Each HI should satisfy two basic requirements: strong correlation with the battery capacity degradation state and computational simplicity from raw voltage, current, and temperature data. Based on the analysis of the battery’s charge curves, discharge curves, and temperature dynamics, I selected six representative health factors. In the following, I denote them as HI1 through HI6.
3.1.1 Equal-voltage-rise charging time (HI1)
During the constant-current charging phase, the terminal voltage rises at a rate that is influenced by the battery’s internal resistance and polarization effects. As the battery ages, the time interval needed to traverse the voltage window from 3.8 V to 4.2 V gradually shortens. I defined this interval as the equal-voltage-rise charging time. This feature is closely related to the cell’s capacity retention because the degraded cell displays higher internal resistance and lower effective electrode capacity, both of which accelerate the voltage response during charging.
3.1.2 Root mean square of discharge voltage (HI2)
For each charge-discharge cycle, the discharge voltage profile contains rich information about the state of health. I computed the root mean square (RMS) value of the discharge voltage over the full discharge period. The RMS voltage tends to decline with the aging of the cell because the polarization resistance increases and the duration of the discharge plateau is shortened. This feature provides a compressed representation of the discharge-voltage dynamics.
3.1.3 Equal-voltage-drop discharge time (HI3)
Selecting the discharge voltage window between 3.0 V and 4.0 V, I computed the time required for the terminal voltage to drop by a fixed amount, such as from 4.2 V to 3.2 V. This equal-voltage-drop time gradually decreases as the battery degrades and thus constitutes an indicator that can track the performance fade. An advantage of this feature is that it is relatively insensitive to the absolute voltage level and can be extracted from partial discharge data.
3.1.4 Equal-current-drop charging time (HI4)
During the constant voltage charging phase of the standard charging protocol, the current decays from an initial high value to the termination threshold of 0.1 A. The interval over which the current falls from 0.5 A to 0.1 A is denoted as the equal-current-drop charging time. As the battery ages, this time interval typically shortens. The change arises from the modified kinetic parameters of the aged electrodes.
3.1.5 Constant-current charging duration (HI5)
The constant current phase duration is defined as the elapsed time during which the charging current is held at 1.5 A until the battery voltage reaches its upper limit. This phase is governed by the charge-acceptance capability of the battery. With aging, the polarization increases and the time before the cell reaches the upper voltage cutoff is reduced.
3.1.6 Time to peak discharge temperature (HI6)
Because heat generation during discharge is closely associated with the internal resistance and irreversible side reactions, the time for the surface temperature of the battery to reach its peak value was also recorded. This temporal feature decreases with battery aging due to the higher heat generation rates of aged cells. The time-to-peak-temperature can be a useful complement to the purely electrical features.
3.2 Correlation Analysis
To quantitatively verify that the chosen health factors are linearly and monotonically associated with battery capacity, I computed both the Pearson and Spearman correlation coefficients between each HI and the measured capacity. The Pearson coefficient is defined as:
$$ r_P = \frac{\sum_{i=1}^{n}\left(x_i – \bar{x}\right)\left(y_i – \bar{y}\right)}{\sqrt{\sum_{i=1}^{n}\left(x_i – \bar{x}\right)^2 \sum_{i=1}^{n}\left(y_i – \bar{y}\right)^2}} $$
whereas the Spearman coefficient is calculated after ranking the two variables, which makes it a robust measure of monotonic association:
$$ r_S = 1 – \frac{6\sum_{i=1}^{n} d_i^2}{n(n^2 – 1)} $$
with \(d_i\) representing the difference between the ranks of the \(i\)-th observation pair. The calculated absolute values are listed in Table 4.
| Dataset | Statistic | HI1 | HI2 | HI3 | HI4 | HI5 | HI6 |
|---|---|---|---|---|---|---|---|
| B0005 | Pearson | 0.9948 | 0.9834 | 0.9998 | 0.9956 | 0.9985 | 0.9998 |
| Spearman | 0.9892 | 0.9554 | 0.9992 | 0.9890 | 0.9969 | 0.9995 | |
| B0006 | Pearson | 0.9504 | 0.9644 | 0.9998 | 0.9282 | 0.9946 | 0.9998 |
| Spearman | 0.9575 | 0.9445 | 0.9998 | 0.9571 | 0.9967 | 0.9998 | |
| B0007 | Pearson | 0.9869 | 0.9652 | 0.9997 | 0.9931 | 0.9984 | 0.9992 |
| Spearman | 0.9873 | 0.9447 | 0.9991 | 0.9900 | 0.9972 | 0.9986 | |
| B0018 | Pearson | 0.8589 | 0.9716 | 0.9944 | 0.8631 | 0.8641 | 0.9941 |
| Spearman | 0.9512 | 0.9703 | 0.9845 | 0.9507 | 0.9447 | 0.9845 | |
| A1 | Pearson | 0.8672 | 0.9765 | 0.9924 | 0.9241 | 0.8542 | – |
| Spearman | 0.9561 | 0.9562 | 0.9923 | 0.9042 | 0.9286 | – |
It can be seen that the absolute values of both correlation coefficients are above 0.85 in the vast majority of cases, indicating that the extracted health factors are strongly correlated with battery capacity. Therefore, I employed HI1 to HI6 as the raw feature inputs for the subsequent SOH estimation and RUL prediction tasks.
3.3 t-SNE Dimensionality Reduction
Although all the selected health factors are highly correlated with capacity, there exists a certain amount of information overlap among them. Feeding a redundant feature set into the model may increase the computational burden and potentially confuse the learning algorithm. To generate a high-quality compact representation, I adopted the t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm. t-SNE performs a nonlinear transformation from the high-dimensional feature space into a low-dimensional embedding space by preserving the local neighborhood structure of the original data points. The procedure begins by converting the Euclidean distances between points into conditional probabilities using a Gaussian kernel in the high-dimensional space. The joint probability between samples \(x_i\) and \(x_j\) is denoted as \(p_{ij}\). In the low-dimensional map, the corresponding probability \(q_{ij}\) is modeled by Student’s t-distribution with one degree of freedom. The objective is to minimize the Kullback-Leibler divergence between \(p_{ij}\) and \(q_{ij}\):
$$ C = \sum_i \sum_j p_{ij} \log \frac{p_{ij}}{q_{ij}} $$
Through gradient-descent iterations based on the expression,
$$ \frac{\partial C}{\partial y_i} = 4\sum_j \left(p_{ij} – q_{ij}\right)\left(y_i – y_j \right)\left(1 + \lVert y_i – y_j \rVert^2\right)^{-1} $$
the low-dimensional coordinates are gradually refined. During practical implementation, the perplexity parameter must be carefully determined. Through a comparative study of perplexity values of 5, 15, 25, 40, and 50, I found that the perplexity of 40 yields the most informative and visually clear reduction in which the two-dimensional representative points form a smooth trajectory that closely mirrors the capacity fade process. Consequently, I selected the t-SNE configuration with a perplexity of 40 as the default input transformer for all subsequent experiments. The resulting lower-dimensional health indicators therefore avoid redundant information and remain strongly representative of battery degradation.
4. Lithium Battery SOH Estimation Based on INGO-CNN-GRU-Attention
4.1 Architecture of the CNN-GRU-Attention Network
To deliver accurate SOH estimation results for electric car lithium batteries, I constructed a multi-level deep learning architecture. The model is composed of an input layer, a CNN feature extraction module, a GRU temporal modeling module, an attention layer, and an output layer.
The CNN module contains two one-dimensional convolutional layers and two pooling layers. The first convolutional layer applies a set of learnable filters along the time axis of the input health-factor matrix, revealing local patterns. The pooling layers reduce the resolution and provide translation invariance. The features learned by the CNN layers are flattened and fed into the downstream GRU layer. The GRU network processes sequential features using only two gates, i.e., the reset gate and the update gate, thereby reducing the computational complexity compared with LSTM while maintaining comparable capability for modeling long-term dependencies. The mathematical operations of the GRU can be expressed as follows:
$$ r_t = \sigma\left(W_r \cdot [h_{t-1}, x_t] + b_r\right) $$
$$ z_t = \sigma\left(W_z \cdot [h_{t-1}, x_t] + b_z\right) $$
$$ \tilde{h}_t = \tanh\left(W_h \cdot [r_t \odot h_{t-1}, x_t] + b_h\right) $$
$$ h_t = (1 – z_t) \odot h_{t-1} + z_t \odot \tilde{h}_t $$
where \(x_t\) is the current input vector, \(h_{t-1}\) is the previous hidden state, \(r_t\) is the reset gate output, and \(z_t\) is the update gate.
The attention mechanism is attached to the output layer of the GRU. It first calculates an attention score for every time-step based on the output vector of the GRU, and then derives a weighted sum. This design grants the network the capability to focus on the most important temporal segments, such as the onset of a strong capacity regeneration event or the period when the battery capacity crosses a critical threshold. The formula of the attention layer can be given as:
$$ \alpha_t = \frac{\exp\left(\tanh\left(h_t\right)^T u_a\right)}{\sum_{j=1}^{T}\exp\left(\tanh\left(h_j\right)^T u_a\right)} $$
$$ s = \sum_{t=1}^{T} \alpha_t h_t $$
Finally, a fully connected layer with a Sigmoid activation maps the weighted feature vector to the normalized SOH value.
4.2 Improved Northern Goshawk Optimization
The hyperparameters of the above deep network have a decisive influence on the final estimation result. Manually tuning them is a labor-intensive task that often relies on the researcher’s experience. To this end, I designed the Improved Northern Goshawk Optimization (INGO) algorithm. The baseline Northern Goshawk Optimization algorithm simulates two stages of the goshawk’s hunting behavior: prey identification and attack, and chase and escape. In the first stage,
$$ x_{i,j}^{new,P1} = x_{i,j} + r \cdot \left(p_{i,j} – I x_{i,j}\right), \quad \text{if } F_p < F_i $$
$$ x_{i,j}^{new,P1} = x_{i,j} + r \cdot \left(x_{i,j} – p_{i,j}\right), \quad \text{if } F_p \ge F_i $$
whereas the second stage drives the local exploitation by updates around the present best solution and a time-dependent radius.
Three improvement strategies were introduced to overcome the limitations of the original algorithm. First, the SPM chaotic map generates the initial population so as to distribute the initial solutions evenly across the search domain. This avoids the loss of population diversity caused by random initialization. Second, the sine cosine algorithm is integrated into the exploration stage to reduce the risk of premature convergence:
$$ x_{i,j}^{new,P1} = x_{i,j} + r_1 \cdot \sin\left(w_1\right) \cdot \left|r_2 p_{i,j} – r_3 x_{i,j}\right| $$
for the first exploration condition, with an alternative expression using \(\cos(w_1)\) in the other case. Here the parameter \(r_1\) linearly decreases with the number of iterations, while \(w_1\) is an adaptive weight. Third, the opposition-based learning strategy based on pinhole imaging is adopted, which establishes a dynamic reflection point and thus broadens the search space. The mathematical relationship of the pinhole-imaging technique is described by:
$$ X_{i,j} = \frac{c_i + d_i}{2} + \frac{c_i + d_i}{2v} – \frac{x_{i,j}}{v} $$
where \(v\) is the scaling factor that adjusts the degree of opposition.
To evaluate the performance of the INGO algorithm, I conducted a comprehensive benchmark test on three representative functions, including the unimodal sphere function and the multimodal Rastrigin and Ackley functions. The results shown in Table 5 indicate that INGO converges more rapidly and more stably than the standard NGO, Whale Optimization Algorithm, and Particle Swarm Optimization in most cases.
| Function | Algorithm | Best Value | Average | Std. |
|---|---|---|---|---|
| Sphere | INGO | 0 | 0 | 0 |
| NGO | 1.10e-101 | 9.24e-98 | 4.50e-97 | |
| WOA | 2.42e-88 | 4.64e-78 | 1.85e-77 | |
| PSO | 7.11e-06 | 2.56e-03 | 4.55e-03 |
The INGO algorithm was used to optimize three critical hyperparameters of the CNN-GRU-Attention model, i.e., the number of hidden units of the GRU, the initial learning rate, and the L2 regularization coefficient. The optimal combination found by INGO was \(\{L_1, \mathrm{Lr}, \mathrm{K}\} = \{55, 0.001, 1.044\times10^{-4}\}\). With these hyperparameters assigned, the final INGO-CNN-GRU-Attention model was trained and evaluated.
4.3 Experimental Results and Ablation Study
To verify the effectiveness of each module in the proposed method, I designed an ablation study with several model variants including CNN-GRU-Attention, INGO-GRU-Attention, INGO-CNN-GRU, and the complete INGO-CNN-GRU-Attention. For all experiments, the data were divided into a training set containing the first 60% of the cycle series, and the test set contains the remaining 40%. Two representative metrics, namely root mean square error (RMSE) and mean absolute error (MAE), are used for quantitative comparison:
$$ \mathrm{RMSE} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(x(i) – y(i)\right)^2} $$
$$ \mathrm{MAE} = \frac{1}{N}\sum_{i=1}^{N}\left|x(i) – y(i)\right| $$
The summarized results on the NASA cells and the A1 laboratory cell are reported in Table 6. It is evident that the full proposed model achieves the smallest RMSE and MAE on all the battery datasets. The CNN and the attention mechanism both contribute to better estimation, while the INGO hyperparameter optimizer further improves the performance relative to manually configured networks.
| Battery | Algorithm | RMSE | MAE |
|---|---|---|---|
| B0005 | CNN-GRU-Attention | 0.0213 | 0.0177 |
| INGO-GRU-Attention | 0.0204 | 0.0183 | |
| INGO-CNN-GRU | 0.0161 | 0.0128 | |
| INGO-CNN-GRU-Attention | 0.0071 | 0.0058 | |
| B0006 | CNN-GRU-Attention | 0.0157 | 0.0124 |
| INGO-GRU-Attention | 0.0180 | 0.0157 | |
| INGO-CNN-GRU | 0.0204 | 0.0164 | |
| INGO-CNN-GRU-Attention | 0.0087 | 0.0075 | |
| B0007 | CNN-GRU-Attention | 0.0207 | 0.0180 |
| INGO-GRU-Attention | 0.0179 | 0.0159 | |
| INGO-CNN-GRU | 0.0291 | 0.0247 | |
| INGO-CNN-GRU-Attention | 0.0077 | 0.0065 | |
| B0018 | CNN-GRU-Attention | 0.0121 | 0.0095 |
| INGO-GRU-Attention | 0.0129 | 0.0112 | |
| INGO-CNN-GRU | 0.0219 | 0.0164 | |
| INGO-CNN-GRU-Attention | 0.0086 | 0.0073 | |
| A1 | CNN-GRU-Attention | 0.0112 | 0.0089 |
| INGO-GRU-Attention | 0.0152 | 0.0122 | |
| INGO-CNN-GRU | 0.0324 | 0.0281 | |
| INGO-CNN-GRU-Attention | 0.0066 | 0.0054 |
The visualization of SOH estimation results for the B0005, B0006, B0007, B0018, and A1 batteries reveals that the estimation curves of the proposed model closely track the actual SOH trajectory even in the regions where other models show considerable deviations. In particular, on the A1 dataset, which has a larger number of cycles and more frequent capacity regenerations, the proposed method remains stable and generates the lowest estimation error. The results therefore demonstrate that the fusion of CNN, GRU, and attention modules in conjunction with the INGO hyperparameter tuning strategy constitutes a powerful and robust solution to the SOH estimation problem of electric car batteries.
5. Lithium Battery RUL Prediction Based on RIME-VMD and Optimized Transformer-BiLSTM
5.1 RIME-Optimized Variational Mode Decomposition
For the RUL prediction task, I focus on the first dimension of the t-SNE-reduced health indicators, denoted as HIF, which captures the dominant degradation trend. Although the HIF sequence inherits the influence of capacity regeneration and noise, direct modeling of this noisy signal may adversely affect prediction accuracy. To address this issue, I first applied the Variational Mode Decomposition algorithm to separate the HIF sequence into multiple intrinsic mode functions (IMFs). The decomposition solves the following variational problem:
$$ \min_{\{u_k\},\{\omega_k\}} \sum_k \Big\Vert \partial_t \Big[ \big(\delta(t) + \frac{j}{\pi t}\big) * u_k(t)\Big] e^{-j\omega_k t} \Big\Vert_2^2 $$
$$ \text{s.t.} \quad \sum_k u_k(t) = f(t) $$
The augmented Lagrangian expression is then minimized iteratively using the alternate direction method of multipliers, yielding the optimal center frequencies and decomposed modes. The number of modes \(K\) and the penalty factor \(\alpha\) are the two key hyperparameters that greatly influence the decomposition quality. Instead of selecting them manually through trial and error, I applied the RIME optimization algorithm to search for the best combination using the minimum envelope entropy as the fitness function:
$$ \mathrm{RIME}\left(K, \alpha\right) \rightarrow \min\ E_e $$
The RIME algorithm iteratively updates the candidate solutions through soft-rime and hard-rime search mechanisms, and the best parameters are fed to the VMD procedure. Taking the B0005 cell as an example, the optimal parameters determined by RIME were \(K = 6\) and \(\alpha = 420\). The correlations between the resulting IMFs and the original HIF were computed, and the results showed that the first three IMFs had much larger correlation coefficients than the high-frequency components. Therefore, I retained the first three IMFs and reconstructed a de-noised HIF sequence, thus suppressing the disturbances caused by the capacity-regeneration effect.
5.2 Transformer-BiLSTM Architecture
The Transformer model has a powerful global feature extraction capability using the self-attention mechanism. The normalized HIF data is first embedded with positional encoding, and then a multi-head attention mechanism is used to capture the global interactions between time steps. The scaled dot-product attention is defined as:
$$ \mathrm{Attention}(Q, K, V) = \mathrm{softmax}\left(\frac{Q K^T}{\sqrt{d_k}}\right) V $$
and the multi-head attention is the concatenation of the outputs from \(h\) different attention heads. To adapt the Transformer structure to the limited data scale of battery datasets, I designed a simplified encoder structure whose output feature sequence is passed to a BiLSTM network. The BiLSTM layer processes the sequence in both the forward and the backward directions, thus enabling the network to capture long-term contextual dependencies from both past and future information. The backward and forward hidden states are then combined to generate the final representation at every output time step as:
$$ \hat{y}_t = W_y \overrightarrow{h_t} + W_y^{‘} \overleftarrow{h_t} + b_y $$
As a consequence, the Transformer-BiLSTM model is able to exploit the advantages of the attention mechanism and the bidirectional recurrent structure simultaneously. Its hyperparameters include the number of attention heads, the hidden dimension of the Transformer, the number of hidden BiLSTM units, the initial learning rate, and the L2 regularization strength. To reach the global optimum, I again used the INGO algorithm to jointly tune all five hyperparameters. The final optimal combination was obtained as BiLSTM hidden size of 120, Transformer head number of 4, Transformer hidden dimension of 42, learning rate of \(1.5\times10^{-3}\), and regularization rate of \(1.1\times10^{-4}\).
5.3 Evaluation of RUL Prediction
I performed a series of experiments on the NASA B0005, B0006, and B0007 cells and on the laboratory A1 cell. The EOL thresholds were set to 1.4 Ah for B0005 and B0006, 1.45 Ah for B0007, and 1.85 Ah for the A1 cell. The prediction starting points were set to cycle 101 for the NASA cells and cycle 128 for the A1 cell. The performance evaluation was based on both the RMSE of the capacity trajectory and the absolute error of the final RUL value. The metrics are defined as follows:
$$ \mathrm{RMSE} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(c(i) – q(i)\right)^2} $$
$$ \mathrm{AE} = \lvert \mathrm{RUL}_r – \mathrm{RUL}_p \rvert $$
The results of the proposed model were compared against three other variants: RIME-VMD-BiLSTM, RIME-VMD-Transformer, and Transformer-BiLSTM without pre-processing. The recorded values are listed in Table 7.
| Battery | Algorithm | RMSE | RUL Ground Truth | RUL Predicted | AE |
|---|---|---|---|---|---|
| B0005 | Transformer-BiLSTM | 0.0275 | 23 | 30 | 7 |
| RIME-VMD-Transformer | 0.0240 | 23 | 16 | 7 | |
| RIME-VMD-BiLSTM | 0.0166 | 23 | 18 | 5 | |
| Proposed method | 0.0081 | 23 | 23 | 0 | |
| B0006 | Transformer-BiLSTM | 0.0258 | 7 | 10 | 3 |
| RIME-VMD-Transformer | 0.0307 | 7 | 2 | 5 | |
| RIME-VMD-BiLSTM | 0.0195 | 7 | 9 | 2 | |
| Proposed method | 0.0099 | 7 | 8 | 1 | |
| B0007 | Transformer-BiLSTM | 0.0272 | 42 | 57 | 15 |
| RIME-VMD-Transformer | 0.0181 | 42 | 39 | 3 | |
| RIME-VMD-BiLSTM | 0.0310 | 42 | – | – | |
| Proposed method | 0.0080 | 42 | 43 | 1 | |
| A1 | Transformer-BiLSTM | 0.0312 | 65 | – | – |
| RIME-VMD-Transformer | 0.0205 | 65 | 60 | 5 | |
| RIME-VMD-BiLSTM | 0.0306 | 65 | – | – | |
| Proposed method | 0.0097 | 65 | 66 | 1 | |
From these experiments, several conclusions can be drawn. The proposed method yields the lowest RMSE values for all the batteries and produces RUL prediction errors not greater than one charge-discharge cycle, suggesting very precise tracking of the battery capacity to the EOL. The comparative models without the full RIME-VMD pre-processing or without the joint Transformer-BiLSTM structure show larger deviations, and in some cases do not even reach the failure threshold before the end of the measured data.
To evaluate the robustness of the proposed RUL prediction method with respect to the amount of available historical data, I conducted additional experiments at distinct prediction starting points. The NASA cells were tested with starting points of 71, 85, and 101 cycles, while the A1 cell was tested with starting points of 86, 107, and 128 cycles. The results summarized in Table 8 demonstrate that the proposed method maintains low prediction error for every starting point. The maximum value of the AE metric is only 2 cycles and the maximum RMSE is 0.0131, proving the strong adaptability of the method for electric car battery health prediction even in data-limited online applications.
| Battery | Prediction Start | RMSE | True RUL | Predicted RUL | AE |
|---|---|---|---|---|---|
| B0005 | 71 | 0.0094 | 53 | 53 | 0 |
| 85 | 0.0090 | 39 | 39 | 0 | |
| 101 | 0.0081 | 23 | 23 | 0 | |
| B0006 | 71 | 0.0112 | 37 | 37 | 0 |
| 85 | 0.0115 | 23 | 23 | 0 | |
| 101 | 0.0099 | 7 | 8 | 1 | |
| B0007 | 71 | 0.0095 | 72 | 71 | 1 |
| 85 | 0.0090 | 58 | 57 | 1 | |
| 101 | 0.0080 | 42 | 43 | 1 | |
| A1 | 86 | 0.0131 | 127 | 129 | 2 |
| 107 | 0.0101 | 86 | 88 | 2 | |
| 128 | 0.0097 | 65 | 66 | 1 |
6. Development of the Lithium Battery PHM System
To translate the above algorithmic advances into a useful engineering tool for the operation and maintenance of electric car batteries, I developed a prototype Lithium Battery PHM software system. The system was designed with a hierarchical architecture, separating the complex underlying algorithms from the intuitive graphical user interface. The application platform is based on an interactive environment that supports event-driven programming. Users can navigate through simple screens to load data sets, extract health factors, perform dimensionality reduction, estimate SOH, and predict RUL without requiring any programming knowledge.
The system begins with a user login screen that authenticates the operator and grants access to the main software. The next screen is the data processing interface. By clicking the “battery dataset” button, the user can choose from among various data sources, e.g., a NASA cell or the laboratory A1 data. The key electrical parameters are shown in a table on the right panel. The extracted HI1 through HI6 are displayed as curves in a plot area once the “HI extraction” operation is carried out. When the user further clicks the “HI reduction” button, the t-SNE algorithm is invoked and the resulting reduced-dimensional features are displayed as well. From this function-selection screen, the user may enter the SOH estimation module or the RUL prediction module.
Within the SOH estimation page, the user selects the estimation starting point from a drop-down list and then clicks “start estimation.” The real SOH curve and the estimated SOH curve are shown in the same coordinate system. The status pane below the coordinate system displays the current SOH value and changes the state of a lamp from green to red when the SOH is below 0.70. This graphical status encourages the user to replace the battery in a timely manner. The “save result” button exports the estimated SOH curve and its value for subsequent report generation.
Within the RUL prediction page, the first operation is to apply the RIME-VMD algorithm. The system automatically searches for the optimal decomposition parameters, decomposes the HIF data, and plots all relevant IMF components. The optimum \(K\) and \( \alpha \) values are displayed in the status bar. After the user performs the “HIF reconstruction” operation, the reconstructed sequence is shown together with the original capacity curve. In the final step, the user chooses the prediction starting point and clicks “start prediction”. The real capacity trajectory and the predicted capacity trajectory are then displayed, and the RUL value appears in the dedicated status bar. In an example with the B0005 cell starting from cycle 101, the software reports a RUL of 23 cycles, which provides actionable guidance for maintenance planners to schedule a battery replacement before the end of battery life.
The PHM system therefore provides an integrated and transparent workflow for battery health management. Because the backend architecture encapsulates all the data-driven algorithms as callable functions, the complete processing pipeline can be seamlessly reused or updated in later developments. The system was developed in a modular way, so other health indicators, newly added data sources, or improved network structures could be integrated with minimal modifications.
7. Conclusions
This dissertation presents a comprehensive study on the Prognostics and Health Management technology for the electric car lithium battery, with the goal of improving the accuracy and reliability of both SOH estimation and RUL prediction. The main conclusions are drawn as follows.
(1) Six health indicators were extracted from the routinely recorded charge-discharge data and verified via Pearson and Spearman coefficients. Their strong correlation with capacity shows that the indicators can be used as effective substitutes for direct capacity measurements, which is especially useful in an electric car where direct capacity estimation is not always feasible. The t-SNE algorithm was applied with a carefully tuned perplexity to construct a compact but stable input set that improves the quality and efficiency of the downstream models.
(2) The INGO algorithm was developed by combining the SPM chaotic map, sine cosine algorithm, and opposition-based learning. The benchmark tests indicate that INGO has superior convergence speed and search accuracy relative to the original NGO and other common swarm intelligence algorithms. An integrated model consisting of CNN, GRU, and attention mechanism was designed to estimate SOH. The CNN extracted localized degradation features, the GRU modeled temporal dependencies, and the attention layer focused on the most informative sequence segments. With INGO determining the optimal hyperparameters, the fusion model continuously achieves the smallest RMSE and MAE values in the ablation studies on both the NASA cells and the laboratory A1 cell, thus demonstrating its high estimation accuracy and robustness.
(3) I proposed a new method for battery RUL prediction based on RIME-VMD decomposition and optimized Transformer-BiLSTM network. The RIME-VMD technique effectively removed the capacity regeneration component and measurement noise, yielding a smooth reconstructed health factor that reflects the capacity degradation tendency well. By combining the global attention of the Transformer with the bidirectional time series modeling capability of the BiLSTM, the network accurately captures the long-term degradation pattern and short-term fluctuations. Extensive experiments at different prediction starting points show that the proposed method provides reliable and highly accurate RUL predictions with AE no greater than 2 cycles across all tested cells. This promising generalization ability is of particular significance for real electric car battery management scenarios in which operating conditions vary widely.
(4) A prototype PHM software system was developed. The integration of data loading, feature extraction, t-SNE reduction, SOH estimation, and RUL forecasting into one platform makes it convenient for end-users to evaluate battery condition and determine appropriate replacement timing. This helps to improve the safety and economy of electric car operation.
While the methods proposed in this work have achieved satisfactory results through simulation and experimental verification, several limitations and possible directions for future research remain. On one hand, the ablation study and comparison experiments were based on laboratory data and public datasets, which may not fully cover the dynamic driving profiles of actual electric cars. To improve the applicability of the models, I plan to collect real-world field data from in-service electric cars and validate the proposed algorithms under various temperature, vibration, and load conditions. On the other hand, the deep integrated model has a relatively complex structure and requires considerable computational resources during training. To support embedded deployment in the battery management unit of an electric car, model compression and edge-computing techniques could be explored in future research. Finally, the current study concentrates on individual cells. For a battery pack composed of many cells in series and parallel, the cell-to-cell inconsistency adds another layer of complexity to the health estimation problem. Extending the SOH and RUL approaches to the pack level and developing methods to balance cell degradation rates will be an important research topic in the future.