In our research, we focus on the critical challenge of low-quality, fragmented, and multi-source heterogeneous operation data collected by battery management systems (BMS) for lithium-ion energy storage cells. The data often lack complete charge-discharge cycles due to complex operating conditions such as peak shaving and frequency regulation. To address this, we study the splicing and reconstruction mechanism of energy storage cell operation data using gradient descent optimization. We propose a novel data-driven method that integrates both steady-state and transient behavior analysis, enabling the restoration of continuous and reliable charge-discharge sequences from scattered data fragments. Our approach is validated through extensive experiments using HPPC, RPT, and real-world operational data from grid-scale energy storage stations. Furthermore, we demonstrate an engineering application where the reconstructed data are combined with incremental capacity analysis (ICA) for accurate state-of-health (SOH) estimation of energy storage cells.
Introduction
The operation data of energy storage cells form the foundation for data-driven state estimation and fault prediction. However, current BMS data acquisition often results in poor quality datasets, including incomplete cycles, varying current rates, and temporal discontinuities. Traditional methods such as polynomial fitting, equivalent circuit models, or machine learning approaches introduce manual errors and lack robustness. In our work, we systematically analyze the underlying physical and electrochemical characteristics of lithium-ion energy storage cells during operation. By combining statistical analysis with gradient descent optimization, we derive explicit empirical equations and boundary conditions for data splicing and reconstruction. Our methodology ensures that the reconstructed data preserve continuity in voltage, current, capacity, and voltage change rate, thereby meeting the stringent requirements for subsequent modeling tasks such as capacity increment analysis and SOH estimation.
Mechanism of Splicing and Reconstruction
We define the splicing and reconstruction problem as follows: given two consecutive data fragments Xfront and Xback collected under similar operating conditions, we aim to merge them into a single coherent sequence. The core mechanism relies on satisfying a set of constraints derived from the physics of energy storage cells. We employ the gradient descent algorithm to iteratively minimize a loss function that quantifies the mismatch at the splicing point.
Gradient Descent-based Optimization
Let X = {x1, x2, …, xm} be the set of all points from the two fragments. The loss function J(X) is defined as:
$$
J(X) = \sum_{i=2}^{m} \max(0, |x_i – x_{i-1}| – p)^2
$$
where p is the boundary tolerance at the splicing point. The gradient with respect to xi is computed as:
$$
\frac{\partial J}{\partial x_i} = 2(|x_i – x_{i-1}| – p) \cdot \text{sign}(x_i – x_{i-1})
$$
We then update the parameters using the gradient descent rule:
$$
x_i^{(t+1)} = x_i^{(t)} – \alpha \frac{\partial J}{\partial x_i}
$$
where α is the learning rate. Through iterative optimization, we find the optimal boundary condition p that minimizes the overall discontinuity.
Empirical Equations and Boundary Conditions
Through extensive statistical analysis of laboratory and field data, we established the following key constraints for energy storage cell data splicing:
1. Current Magnitude Condition
The difference between the current of the front fragment (Ifront) and the back fragment (Iback) must be within 5 A:
$$
|I_{\text{front}} – I_{\text{back}}| \leq 5\ \text{A}
$$
2. Cumulative Capacity Condition
Let Cfront and Cback be the cumulative charge/discharge capacity sets for the fragments. The difference at the splicing point must be zero:
$$
C_{\text{d}} = C_{\text{front}}^{\text{end}} – C_{\text{back}}^{\text{start}} = 0
$$
3. Voltage Condition
The voltage difference at the splicing point must not exceed 5 mV:
$$
|U_{\text{front}}^{\text{end}} – U_{\text{back}}^{\text{start}}| \leq 0.005\ \text{V}
$$
4. Voltage Change Rate Condition
The difference in voltage change rate (k = dU/dt) must be less than 0.0001 V/s:
$$
|k_{\text{front}}^{\text{end}} – k_{\text{back}}^{\text{start}}| \leq 0.0001\ \text{V/s}
$$
5. Transient Duration Condition
When switching between charging, discharging, and resting states, the energy storage cell exhibits a transient period. From our statistics, the required settling time T must exceed the maximum observed transient duration:
$$
T \geq T_{\max} = 96\ \text{s}
$$
The statistics of transient durations from HPPC tests are summarized in the table below.
| Order | Duration (s) | Order | Duration (s) |
|---|---|---|---|
| 1 | 61 | 7 | 82 |
| 2 | 67 | 8 | 85 |
| 3 | 69 | 9 | 86 |
| 4 | 72 | 10 | 87 |
| 5 | 77 | 11 | 88 |
| 6 | 80 | 12 | 96 |
| Minimum | 61 | ||
| Mean | 79.82 | ||
| Maximum | 96 | ||
Methodology for Splicing and Reconstruction
Based on the mechanism above, we developed a systematic procedure for energy storage cell operation data splicing and reconstruction:
- Data Acquisition: Collect time-series measurements of current, voltage, and cumulative capacity from BMS or monitoring sensors of energy storage cells.
- Fragment Segmentation: Partition the raw data into homogeneous segments where each segment corresponds to a single operating mode (charging, discharging, or resting) with constant current rate.
- Data Cleaning and Preprocessing: Remove outliers, interpolate missing values, and filter noise using median or low-pass filters.
- Splicing and Reconstruction: For each pair of consecutive fragments that satisfy the boundary conditions, apply the gradient descent algorithm to find the optimal alignment and merge them. The detailed flow includes:
- Check current equality (within 5 A).
- Verify cumulative capacity continuity (difference = 0).
- Ensure voltage and voltage-change-rate continuity.
- Skip the transient period (≥96 s after mode change).
- Perform gradient descent to minimize the loss function at the junction.
- Curve Optimization: Further refine the reconstructed curve using physical models (e.g., Kalman filter, RC equivalent circuit) or machine learning (e.g., support vector regression, XGBoost).
- Validation and Evaluation: Assess reconstruction accuracy using statistical metrics such as root mean square error (RMSE), R-squared (R²), and mean absolute error (MAE).
The following figure illustrates a typical setup of an energy storage cell used in our tests.
Validation Using Experimental Data
We validated our mechanism using two benchmark datasets: Hybrid Pulse Power Characterization (HPPC) test and Reference Performance Test (RPT). The objective was to demonstrate that fragmented voltage curves can be accurately restored to continuous charging or discharging profiles.
HPPC Test Validation
We used HPPC discharge data from a 280 Ah energy storage cell at a current of 0.5 C (140 A). The original voltage curve contained multiple interruptions due to pulse sequences. By applying our splicing mechanism (transient condition, voltage continuity), we reconstructed a smooth discharge voltage profile. The reconstructed curve overlapped with the original continuous segments with an RMSE less than 0.1 mV.
RPT Test Validation
We selected two RPT charging curves from a 75 Ah energy storage cell at 0.2 C. Each curve was artificially divided into five fragments. Using our method, we successfully reassembled the fragments back into the original charging profile. The results for two different cycles (cycle 1 at 12°C and cycle 1000 at 16°C) are summarized in the table below.
| Curve | Number of Fragments | Reconstructed RMSE (V) | R² | MAE (V) |
|---|---|---|---|---|
| RPT curve 1 (cycle 1, 12°C) | 5 | 0.0032 | 0.9998 | 0.0021 |
| RPT curve 2 (cycle 1000, 16°C) | 5 | 0.0041 | 0.9997 | 0.0028 |
These results confirm that our splicing and reconstruction mechanism works effectively under controlled experimental conditions, preserving both the shape and the statistical properties of the original energy storage cell data.
Validation Using Real-World Operation Data
To assess the robustness of our method, we applied it to real operational data from a grid-scale energy storage station. Two typical operating scenarios were considered: peak shaving and frequency regulation.
Peak Shaving Scenario
Data from a 271 Ah energy storage cell under 0.5 C (135 A) charging conditions were used. Two continuous charging voltage curves were each split into five random fragments. Our method successfully reconstructed the original continuous curves. The RMSE for the reconstructed peaks was below 0.005 V. The empirical range of parameters encountered in these real fragments is given in the following table.
| Parameter | Range |
|---|---|
| Voltage (V) | 3.307 – 3.680 |
| Current (A) | 2.5 – 136.4 |
| State of Energy (%) | 41 – 83 |
| Duration per fragment (s) | 100 – 1220 |
Frequency Regulation Scenario
We analyzed two battery clusters (cluster 0 and cluster 1) each consisting of 240 energy storage cells (271 Ah) operating under frequency regulation. The current was 10 A for cluster 0 and 30 A for cluster 1, with a sampling interval of 5 s. Multiple fragments that satisfied our boundary conditions were selected from a large dataset and spliced into continuous voltage sequences. The reconstructed curves exhibited smooth transitions, and the overall voltage profiles matched the expected behavior of energy storage cells under dynamic grid support.
Engineering Application: SOH Estimation via Incremental Capacity Analysis
One major practical benefit of our splicing and reconstruction method is enabling incremental capacity analysis (ICA), which requires complete charge-discharge cycles for accurate SOH estimation. In real applications, such full cycles are rarely available. By applying our method, we can construct continuous data sequences from partial fragments.
The combined procedure for SOH estimation is as follows:
- Data reconstruction: Use our splicing method to create a full charge or discharge curve from multiple fragments.
- ICA curve generation: Compute the incremental capacity curve from the reconstructed voltage-capacity data. The IC curve is obtained by differentiating capacity with respect to voltage: dQ/dU.
- Define mid-capacity range: Identify the voltage range between the second peak (u1) and the cutoff voltage (u2) on the IC curve. The area under this segment is defined as the mid-capacity Qmid.
- Compute SOH: Compare the current mid-capacity Qnow with the initial mid-capacity Qstart:
$$
\text{SOH} = \frac{Q_{\text{now}}}{Q_{\text{start}}} \times 100\%
$$
The ICA-based SOH estimation using our reconstructed data achieved an error below 2% when validated against reference test results. This demonstrates that the splicing and reconstruction mechanism not only improves data quality but also enables practical battery health monitoring in energy storage systems.
Conclusion
In this work, we have presented a comprehensive study on the splicing and reconstruction of energy storage cell operation data. The key contributions are as follows:
- We revealed the splicing and reconstruction mechanism based on gradient descent optimization and derived empirical equations for current, voltage, capacity, and voltage change rate boundaries.
- We proposed a step-by-step methodology that ensures both continuity and physical consistency of reconstructed data.
- Through extensive validation using HPPC, RPT, and real peak-shaving/frequency-regulation data, we demonstrated the high accuracy and robustness of our method for various energy storage cell types and operating conditions.
- The engineering application of combining our reconstruction with incremental capacity analysis enabled reliable SOH estimation, highlighting the practical value of the approach.
Our findings provide a solid foundation for future data-driven modeling, fault diagnosis, and life prediction of energy storage cells. The methods can be integrated into BMS software to enhance data quality and support advanced analytics without requiring additional hardware modifications.