In the context of advancing modern power systems, the role of cell energy storage system has become increasingly critical for integrating renewable energy and providing grid flexibility. Accurate estimation of the State of Charge (SOC) for battery packs within these systems is essential to ensure safety, prevent overcharge or over-discharge, and optimize operational efficiency. However, SOC estimation under dynamic grid conditions such as peak shaving and frequency regulation poses significant challenges due to complex current profiles and battery inconsistencies. This paper addresses these challenges by proposing an enhanced data-driven model that combines Kernel Principal Component Analysis (KPCA), Pelican Optimization Algorithm (POA), and Bidirectional Gated Recurrent Unit (BiGRU) for robust SOC estimation. Our focus is on cell energy storage system applications, where high voltage and current environments demand precise monitoring.
The SOC of a battery pack, consisting of N series-connected cells, can be derived from individual cell SOC values. The pack SOC is defined as:
$$SOC_{pack} = \frac{Q_{dis}^{pack}}{Q_{pack}} = \frac{\min_{1 \leq k \leq N} (SOC_k C_k)}{\min_{1 \leq k \leq N} (SOC_k C_k) + \min_{1 \leq k \leq N} ((1 – SOC_k) C_k)}$$
where \(Q_{dis}^{pack}\) is the remaining discharge capacity, \(Q_{pack}\) is the current pack capacity, \(SOC_k\) is the SOC of the k-th cell, and \(C_k\) is its maximum available capacity. To capture pack inconsistencies under different grid services, we introduce voltage and temperature extremes as features. The voltage extreme \(\Delta U\) and temperature extreme \(\Delta T\) are given by:
$$\Delta U = U_{max} – U_{min}, \quad \Delta T = T_{max} – T_{min}$$
where \(U_{max}, U_{min}\) and \(T_{max}, T_{min}\) are the maximum and minimum values among cells. These metrics are vital for characterizing SOC variations in cell energy storage system operations.
Our proposed model leverages KPCA for feature dimensionality reduction, which maps original data to a high-dimensional space to retain essential information while eliminating redundancy. Given a dataset \(X = \{x_1, x_2, \dots, x_n\}\) with each sample \(x_j\) being a d-dimensional vector, the covariance matrix in feature space \(F\) is:
$$C_F = \frac{1}{n} \sum_{j=1}^{n} \Phi(x_j) \Phi^T(x_j)$$
where \(\Phi(x)\) is a nonlinear mapping function. By solving the eigenvalue problem, we obtain principal components. The q-th principal component \(T_q\) is computed as:
$$T_q = v_q \Phi(x) = \sum_{j=1}^{n} \alpha_{q,j} K(x_j, x)$$
Here, \(v_q\) is the eigenvector, \(\alpha_{q,j}\) are coefficients, and \(K(x_j, x)\) is the kernel matrix. This process reduces input dimensions, enhancing model efficiency for cell energy storage system SOC estimation.
For temporal modeling, we employ BiGRU, a variant of recurrent neural networks that captures both forward and backward dependencies in sequences. The BiGRU output at time t is:
$$\vec{c}_t = G(X_t, \vec{c}_{t-1}), \quad \overleftarrow{c}_t = G(X_t, \overleftarrow{c}_{t-1})$$
$$h_t = \vec{w}_t \vec{c}_t + \overleftarrow{c}_t \overleftarrow{w}_t + b_t$$
where \(\vec{c}_t\) and \(\overleftarrow{c}_t\) are hidden states from forward and backward passes, \(\vec{w}_t\) and \(\overleftarrow{w}_t\) are weights, \(b_t\) is bias, and \(G(\cdot)\) is the GRU function. This bidirectional structure is adept at handling the time-series nature of SOC data in cell energy storage system.
To optimize BiGRU hyperparameters, we use POA, a metaheuristic algorithm inspired by pelican hunting behavior. POA exhibits superior convergence and search efficiency. In a m-dimensional space with n pelicans, positions are initialized as:
$$x_{ij} = l_j + \alpha (u_j – l_j), \quad i=1,2,\dots,n, \quad j=1,2,\dots,m$$
where \(\alpha\) is a random number in [0,1], and \(u_j, l_j\) are bounds. The algorithm proceeds in two phases: exploration and exploitation. In Phase 1, position update is:
$$x_{p1_{ij}} = \begin{cases} x_{ij} + \sigma (P_j – a x_{ij}), & F_P < F_i \\ x_{ij} + \sigma (x_{ij} – P_j), & F_P \ge F_i \end{cases}$$
where \(\sigma\) is random, \(P_j\) is prey position, \(a\) is 1 or 2, \(F_P\) is prey fitness, and \(F_i\) is candidate fitness. If fitness improves, the position is updated. In Phase 2, for local refinement:
$$x_{p2_{ij}} = x_{ij} + R \left(1 – \frac{t_s}{T}\right) (2\beta – 1) x_{ij}$$
with \(R=0.2\), \(t_s\) as current iteration, \(T\) as max iterations, and \(\beta\) random in [0,1]. Again, updates occur if fitness improves. POA fine-tunes BiGRU parameters, crucial for accurate SOC estimation in cell energy storage system.
We designed experiments to simulate real-world grid conditions for a cell energy storage system. A lithium iron phosphate battery pack with 8 series-connected cells was used, with specifications summarized in Table 1.
| Parameter | Value |
|---|---|
| Nominal Voltage | 25.6 V |
| Nominal Capacity | 220 Ah |
| Dimensions | 555 mm × 430 mm × 154 mm |
| Operating Temperature | -20°C to 55°C |
| Mass | 60 kg |
The experimental setup included a battery testing system, thermal chamber, and monitoring software. Peak shaving and frequency regulation profiles were applied, with current commands shown in Figure 5 of the original content. Peak shaving involved near-constant current at 0.5C, while frequency regulation involved frequent shallow cycles with currents up to 0.5C. Data on voltage, current, temperature, and SOC were collected at 15-second intervals.
Data preprocessing involved extracting features: pack terminal voltage, current, highest/lowest cell voltages, voltage extreme, temperature extreme, and average pack temperature. These were combined with historical SOC values to form input sequences. KPCA was applied to reduce dimensionality, with contribution rates and eigenvalues for each scenario presented in Table 2.
| Scenario | Principal Components | Cumulative Contribution Rate | Eigenvalues >1 |
|---|---|---|---|
| Peak Shaving | First 3 | 90.231% | Yes |
| Frequency Regulation | First 3 | 92.9% | Yes |
Thus, the first three principal components were selected as inputs for the BiGRU model. The model architecture included 22 input neurons (combining historical SOC and KPCA features), optimized hyperparameters via POA, and a single output for SOC estimation. Data was split into training, validation, and test sets in a 6:1:3 ratio.
Performance was evaluated using Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and coefficient of determination (R²), defined as:
$$E_{RMS} = \sqrt{\frac{1}{M} \sum_{i=1}^{M} (SOC_{M_i} – SOC_{P_i})^2}$$
$$E_{MA} = \frac{1}{M} \sum_{i=1}^{M} |SOC_{M_i} – SOC_{P_i}|$$
$$R^2 = 1 – \frac{\sum_{i=1}^{M} (SOC_{M_i} – SOC_{P_i})^2}{\sum_{i=1}^{M} (SOC_{M_i} – \overline{SOC}_Q)^2}$$
where \(SOC_{M_i}\) and \(SOC_{P_i}\) are measured and predicted SOC, \(\overline{SOC}_Q\) is the mean measured SOC, and M is the number of samples.
We compared our KPCA-POA-BiGRU model against alternatives: Kernel Extreme Learning Machine (KELM), BiLSTM, and standard BiGRU. All models used the same KPCA-processed features. Results for peak shaving and frequency regulation are shown in Table 3.
| Scenario | Model | RMSE | MAE | R² |
|---|---|---|---|---|
| Peak Shaving | KPCA-POA-BiGRU | 0.008645 | 0.008527 | 0.9987 |
| KELM | 0.024090 | 0.019680 | 0.9654 | |
| BiLSTM | 0.011050 | 0.012310 | 0.9892 | |
| BiGRU | 0.011360 | 0.012440 | 0.9875 | |
| Frequency Regulation | KPCA-POA-BiGRU | 0.009983 | 0.010410 | 0.9930 |
| KELM | 0.037650 | 0.022560 | 0.9532 | |
| BiLSTM | 0.015780 | 0.011020 | 0.9785 | |
| BiGRU | 0.016620 | 0.012330 | 0.9771 |
Our model achieved the lowest errors and highest R² values, demonstrating its superiority. For instance, in peak shaving, RMSE reduced by 23.9% compared to standard BiGRU, highlighting the benefit of POA optimization. In frequency regulation, the complex current profiles were better handled by the bidirectional architecture, with RMSE as low as 0.009983. This underscores the model’s robustness for cell energy storage system applications under varying grid demands.
To further validate adaptability, we tested on mixed scenarios combining peak shaving and frequency regulation. Two separate models were trained for each scenario and switched based on grid commands, compared to a single model trained on mixed data. Errors are summarized in Table 4.
| Model Type | RMSE | MAE | R² |
|---|---|---|---|
| Single Model | 0.02872 | 0.025460 | 0.9755 |
| Dual Models (Switched) | 0.01097 | 0.009862 | 0.9936 |
The dual-model approach reduced RMSE by 61.8%, proving that scenario-specific modeling enhances accuracy and robustness for cell energy storage system SOC estimation. This flexibility is crucial for real-world operations where grid conditions fluctuate.
In conclusion, our KPCA-POA-BiGRU framework effectively addresses SOC estimation challenges in cell energy storage system under peak shaving and frequency regulation. By integrating feature fusion, dimensionality reduction, bidirectional temporal learning, and metaheuristic optimization, the model achieves high precision and adaptability. Future work could extend to other grid services or battery chemistries, further advancing the reliability of cell energy storage system in modern power networks. The methodologies presented here contribute to safer and more efficient energy storage management, essential for sustainable grid integration.
The cell energy storage system is a cornerstone of grid stability, and accurate SOC estimation is paramount. Our model’s ability to handle dynamic conditions through advanced data-driven techniques sets a foundation for improved battery management systems. As the deployment of cell energy storage system expands, such innovations will play a key role in ensuring operational safety and longevity, ultimately supporting the transition to resilient power systems.