The global push towards decarbonization is fundamentally reshaping our energy infrastructure. As the integration of intermittent renewable sources like solar and wind accelerates, the role of energy storage battery systems has become paramount. These systems are critical for grid stability, load leveling, and ensuring a reliable power supply. Within this context, Lithium-ion (Li-ion) batteries have emerged as the dominant technology due to their high energy density, long cycle life, and decreasing costs. Consequently, the precise management and monitoring of these energy storage battery units are essential for safety, longevity, and optimal performance. The accurate and reliable estimation of the State of Charge (SOC), defined as the ratio of remaining capacity to maximum available capacity, is a cornerstone function of any advanced Battery Management System (BMS) for energy storage battery applications. SOC provides crucial information for preventing over-charge and over-discharge, optimizing charge/discharge strategies, and predicting the remaining operational time of the energy storage battery system.
However, SOC is an internal state that cannot be measured directly. It must be inferred indirectly from measurable parameters such as terminal voltage, current, and temperature. This indirect estimation poses a significant challenge due to the complex, nonlinear, and time-varying electrochemical dynamics within a Li-ion energy storage battery. The quest for a robust, accurate, and computationally efficient SOC estimation algorithm is therefore a central research focus in the field of energy storage battery management.
Broadly, existing SOC estimation methods can be categorized into two paradigms: model-based methods and data-driven methods. Model-based approaches, such as the Coulomb Counting (Ampere-hour integration) method and the Open-Circuit Voltage (OCV) method, rely on a physical or equivalent circuit model of the energy storage battery. While Coulomb counting is simple, it suffers from cumulative errors due to sensor inaccuracies and unknown initial conditions. The OCV method can be accurate but requires the battery to be at rest for a prolonged period, making it unsuitable for real-time applications. Advanced model-based filters, like the Kalman Filter and its variants (EKF, UKF), offer dynamic estimation capabilities but their performance is heavily dependent on the accuracy of the underlying battery model and the precise knowledge of noise statistics. Any model mismatch or parameter drift in the energy storage battery can lead to significant estimation errors.
Data-driven methods, on the other hand, circumvent the need for an explicit physical model. They learn the complex nonlinear mapping between measurable inputs (voltage, current, temperature) and the target output (SOC) directly from historical operational data of the energy storage battery. Artificial Neural Networks (ANNs) are a powerful class of data-driven tools known for their universal approximation capabilities. Traditional feedforward networks like the Back-Propagation (BP) Neural Network have been widely applied. However, they are prone to getting trapped in local minima during training and can be sensitive to initial weight settings. To overcome these limitations, probabilistic frameworks and hybrid architectures have been explored.
This article presents a novel data-driven approach for SOC estimation of Li-ion energy storage battery systems. The core contribution is the design and application of a Modified Probabilistic Neural Network (MPNN). This architecture ingeniously combines the global optimization characteristics of a probabilistic layer with a compensatory learning mechanism, enhancing the network’s inference capability and robustness. The proposed method aims to deliver high estimation accuracy across various operating conditions, making it particularly suitable for the dynamic environment of a grid-scale energy storage battery facility.
Fundamentals and Challenges of SOC Estimation
The State of Charge for an energy storage battery is formally defined as:
$$ S_{LiB}(t) = \frac{Q_c(t)}{Q_{max}} \times 100\% $$
where \( Q_c(t) \) is the remaining charge at time \( t \), and \( Q_{max} \) is the maximum charge capacity of the battery. In practice, \( Q_c(t) \) is not measurable. A common model-based starting point is the equivalent circuit model (ECM), which uses electrical components like resistors and capacitors to approximate the battery’s dynamic behavior. A first-order RC model is often used:
$$
\begin{aligned}
V_t &= V_{oc}(S_{LiB}) – I \cdot R_0 – V_p \\
\dot{V_p} &= -\frac{1}{R_p C_p} V_p + \frac{1}{C_p} I
\end{aligned}
$$
where \( V_t \) is the terminal voltage, \( V_{oc} \) is the open-circuit voltage (a nonlinear function of SOC), \( I \) is the current (positive for discharge), \( R_0 \) is the ohmic resistance, and \( R_p \) and \( C_p \) model the polarization dynamics with \( V_p \) as the polarization voltage. Estimating SOC from this model requires jointly solving these equations, often using filters, which is complex and model-dependent.
The data-driven perspective reframes this as a nonlinear regression problem. The goal is to find a function \( \mathcal{F} \) such that:
$$ \hat{S}_{LiB}(k) = \mathcal{F}( V(k), I(k), T(k), \varphi(k-1) ) $$
where \( \hat{S}_{LiB}(k) \) is the estimated SOC at time step \( k \), \( V(k) \), \( I(k) \), and \( T(k) \) are the measured voltage, current, and temperature, and \( \varphi(k-1) \) may represent a short history of past measurements or states. The function \( \mathcal{F} \) is what a neural network learns from data. This approach is inherently adaptable to different energy storage battery chemistries and aging states, as long as representative training data is available.
Architecture of the Modified Probabilistic Neural Network (MPNN)
The proposed MPNN is an 8-layer feedforward network designed to learn the mapping for SOC estimation with high fidelity. Its structure introduces key innovations over standard Probabilistic Neural Networks (PNN) and Fuzzy Neural Networks (FNN). The complete architecture is visualized below, and each layer’s function is detailed subsequently.
Layer 1: Input Layer
This layer receives the measurable parameters from the energy storage battery. For a comprehensive estimate, we typically use voltage, current, and temperature. Let \( m \) be the number of input features.
$$
\mathbf{X} = [X_1, X_2, …, X_m]^T = [V, I, T, …]^T
$$
$$
f^1_k = x^1_k = X_k, \quad k=1,2,…,m
$$
where \( f^1_k \) is the output of the k-th node in the input layer.
Layer 2: Normalization Layer
To ensure stable and efficient training, all input features are normalized to a [0, 1] range. This mitigates the influence of differing scales among inputs like voltage (volts) and current (amperes).
$$
x^2_k = f^1_k
$$
$$
f^2_k = \frac{x^2_k – \min(X_k)}{\max(X_k) – \min(X_k)}, \quad k=1,2,…,m
$$
Here, \( \max(X_k) \) and \( \min(X_k) \) are the maximum and minimum values of the k-th feature from the training dataset.
Layer 3: Probabilistic (Radial Basis) Layer
This is the core layer that differentiates the network from standard BP networks. It employs Gaussian Radial Basis Functions (RBFs) to transform the normalized inputs into a probabilistic feature space. This approach helps in capturing the distribution of the input data and aids in achieving a global optimum during training. We define \( n \) pattern nodes (or centers) for each input feature.
$$
x^3_{kj} = f^2_k, \quad \text{for all } j
$$
$$
f^3_{kj} = \exp\left( – \frac{(x^3_{kj} – c^3_{kj})^2}{2 (b^3_{kj})^2} \right), \quad k=1,…,m; \ j=1,…,n
$$
where \( c^3_{kj} \) is the center of the j-th Gaussian node for the k-th input, and \( b^3_{kj} \) is its width (or spread). The centers are typically spaced evenly across the [0,1] input domain, and the width is set proportionally (e.g., \( b^3_{kj} = 1/n \)).
Layer 4: Threshold Layer
This layer acts as a filter to enhance feature selection. It passes only the significant probabilistic activations, reducing noise from weakly activated nodes. A threshold \( th \) is predefined.
$$
x^4_{kj} =
\begin{cases}
0, & \text{if } f^3_{kj} \leq th \\
f^3_{kj}, & \text{if } f^3_{kj} > th
\end{cases}
$$
$$
f^4_{kj} = x^4_{kj}
$$
This step helps the network focus on the most relevant patterns in the data, improving generalization for the energy storage battery SOC estimation task.
Layer 5: Fuzzy Rule Layer
This layer aggregates the filtered probabilistic information from all input features for each pattern node \( j \). It implements a fuzzy rule: the output of each node \( j \) is a weighted sum of all thresholded activations corresponding to that pattern across all \( m \) inputs.
$$
x^5_j = \sum_{k=1}^{m} w^4_{kj} f^4_{kj}, \quad j=1,…,n
$$
$$
f^5_j = x^5_j
$$
The weight \( w^4_{kj} \) is a positive trainable parameter that determines the contribution of the k-th input’s j-th pattern to the final rule output \( j \). It is parameterized as \( w^4_{kj} = \exp(wi^4_{kj}) \) to ensure positivity.
Layer 6: Compensation Layer
This layer introduces a critical innovation. It applies a compensatory transformation to each rule layer output, providing additional degrees of freedom to optimize the fuzzy inference mechanism. This allows the network to fine-tune the contribution of each rule, leading to a more accurate and flexible model for the complex behavior of the energy storage battery.
$$
c_j = \frac{1}{1 + \exp(-ci_j)}, \quad j=1,…,n
$$
$$
x^6_j = (x^5_j)^{1-c_j} \cdot (c_j / m), \quad j=1,…,n
$$
$$
f^6_j = x^6_j
$$
Here, \( c_j \in (0, 1) \) is the compensation degree for rule \( j \), governed by the trainable parameter \( ci_j \). When \( c_j \to 0 \), the output approximates the original \( x^5_j \); when \( c_j \to 1 \), it tends towards a constant \( 1/m \). The network learns the optimal \( c_j \) values during training.
Layer 7: Output Layer
This layer produces the normalized SOC estimate by linearly combining the compensated rule outputs.
$$
x^7_j = f^6_j
$$
$$
f^7 = \sum_{j=1}^{n} w^7_j x^7_j
$$
where \( w^7_j \) are the final layer’s trainable weights.
Layer 8: Denormalization Layer (Virtual Layer)
The final step is to convert the normalized output \( f^7 \) back to the actual SOC scale (0% to 100%). This requires the minimum and maximum SOC values (\( S_{min}, S_{max} \)) from the training data, which correspond to \( Q_{min}/Q_{max} \) and \( Q_{max}/Q_{max} \).
$$
\hat{S}_{LiB} = f^7 \cdot (S_{max} – S_{min}) + S_{min}
$$
This \( \hat{S}_{LiB} \) is the final estimated State of Charge for the energy storage battery.
Training Methodology for SOC Estimation
The MPNN is trained in a supervised manner using historical data collected from the target energy storage battery under various charge/discharge profiles. The dataset consists of tuples \(\{ \mathbf{X}_p, S_{LiB, p} \}\), where \( \mathbf{X}_p \) is a vector of measured inputs (V, I, T) at sample \( p \), and \( S_{LiB, p} \) is the corresponding reference SOC (often obtained via precise laboratory Coulomb counting under controlled conditions).
Loss Function: The primary objective is to minimize the difference between the estimated SOC and the reference SOC. We define the Mean Squared Error (MSE) over a batch of \( N \) samples as the loss function \( E \):
$$ E = \frac{1}{2} \sum_{p=1}^{N} e_p^2 $$
$$ \text{where } e_p = S_{LiB, p} – \hat{S}_{LiB, p} $$
Parameter Update: The network’s trainable parameters are the weight factors \( wi^4_{kj} \) (in the fuzzy rule layer), the compensation factors \( ci_j \), and the output weights \( w^7_j \). These are updated using the gradient descent algorithm with backpropagation. The learning rates \( \eta_{wi} \) and \( \eta_{w7} \) are set to small positive values (e.g., 0.01). The update rules derived via the chain rule are:
For the fuzzy rule layer weights:
$$
\Delta wi^4_{kj} = \eta_{wi} \cdot e_p \cdot \frac{S_{max}-S_{min}}{Q_{max}} \cdot w^7_j \cdot \left(1 – c_j + \frac{c_j}{m}\right) \cdot \frac{x^6_j}{x^5_j} \cdot f^4_{kj} \cdot w^4_{kj}
$$
For the output layer weights:
$$
\Delta w^7_j = \eta_{w7} \cdot e_p \cdot \frac{S_{max}-S_{min}}{Q_{max}} \cdot x^7_j
$$
The parameters of the probabilistic layer (centers \( c^3_{kj} \) and widths \( b^3_{kj} \)) are typically initialized on a fixed grid and may be held constant or finely tuned depending on the implementation.
The complete workflow for applying the MPNN to energy storage battery SOC estimation is as follows:
- Data Collection: Gather comprehensive charge/discharge cycle data from the target energy storage battery, measuring voltage, current, temperature, and accurate reference SOC.
- Data Preprocessing: Segment the data into training and testing sets. Normalize all input features and the target SOC based on the training set’s extrema.
- MPNN Initialization: Construct the network with chosen dimensions (\( m, n \)). Initialize centers, widths, thresholds, and weights as described.
- Offline Training: Iteratively present training data batches to the MPNN, compute the loss, and update the trainable parameters until convergence (e.g., until the Sum of Absolute Errors, SAE, stops decreasing significantly).
- Online Estimation: Deploy the trained MPNN model in the BMS. For each new set of real-time measurements (\( V, I, T \)), normalize them using the stored training min/max values, feed them through the network, and output the denormalized SOC estimate \( \hat{S}_{LiB} \).
Comparative Analysis of Neural Network Approaches for SOC
To contextualize the MPNN, it is useful to compare its characteristics with other common neural network approaches used for energy storage battery SOC estimation. The table below summarizes key aspects:
| Method | Key Principle | Advantages for energy storage battery SOC | Potential Limitations |
|---|---|---|---|
| Back-Propagation (BP) NN | Multi-layer perceptron trained with gradient descent to minimize output error. | Strong nonlinear fitting ability; well-understood. | Prone to local minima; sensitive to initial weights and learning parameters. |
| Recurrent NN (RNN/LSTM/GRU) | Contains internal loops to retain memory of past inputs, modeling temporal dynamics. | Excels at capturing time-series dependencies in battery voltage/current. | More complex training (vanishing/exploding gradients); computationally heavier. |
| Fuzzy Neural Network (FNN) | Combines fuzzy logic (linguistic rules) with neural network learning. | Can incorporate expert knowledge; handles uncertainty well. | Rule base design can be subjective; inference mechanism may be rigid. |
| Probabilistic NN (PNN) | Uses Bayesian strategy with a radial basis function layer for pattern classification/regression. | Fast training; converges to global optimum; robust. | Fixed inference mechanism; requires sufficient representative data for each pattern node. |
| Proposed MPNN | PNN enhanced with a threshold filter and a compensatory learning mechanism. | Global optimization property of PNN + adaptive inference via compensation; high accuracy and robustness. | Increased structural complexity; more hyperparameters (threshold, compensation factors) to configure. |
Simulation Experiments and Performance Evaluation
To validate the effectiveness of the proposed MPNN for energy storage battery SOC estimation, comprehensive tests were conducted using publicly available battery datasets. The cell used for demonstration is a commercial 18650-20R Li-ion battery with a nominal capacity of 2000 mAh. The input feature vector was chosen as \( \mathbf{X} = [V, I, T]^T \). The MPNN was configured with \( m=3 \), \( n=100 \) pattern nodes, a threshold \( th=0.001 \), and initial compensation factors \( ci_j = 0 \).
Performance Metrics: Two standard metrics were used to quantitatively evaluate estimation accuracy:
1. Mean Absolute Error (MAE):
$$ EMA = \frac{1}{N} \sum_{p=1}^{N} |e_p| $$
2. Root Mean Square Error (RMSE):
$$ ERMS = \sqrt{ \frac{1}{N} \sum_{p=1}^{N} e_p^2 } $$
Test 1: Constant Low-Current Charge/Discharge
This test evaluates the MPNN’s accuracy under steady, mild conditions typical of some energy storage battery operational scenarios. The battery was charged at a constant 0.1A (C/20 rate) from 0% to 80% SOC and then discharged at the same rate from 80% down to 10% SOC. The MPNN estimates closely tracked the reference SOC curve. The relative estimation error remained consistently within a ±1% band throughout both the charge and discharge processes, demonstrating excellent baseline accuracy and stability for the energy storage battery.
Test 2: Dynamic Current Profile (Federal Urban Driving Schedule – FUDS)
This is a more rigorous and realistic test, simulating the highly variable current demands seen in applications like buffering renewable fluctuations in an energy storage battery plant. The battery was subjected to a dynamic current profile with rapid charge and discharge pulses. The performance of the MPNN was compared against a standard FNN and a basic PNN.
The results were compelling. While all three methods could generally follow the SOC trend, the MPNN exhibited superior precision. The estimation error for the MPNN was confined within ±1%, significantly tighter than the PNN (±3%) and the FNN, which showed larger deviations. This highlights the benefit of the compensatory mechanism in adapting to the complex nonlinearities induced by dynamic operating conditions of the energy storage battery.
Test 3: Performance Across Different Temperatures
The robustness of an SOC estimation algorithm across a range of operating temperatures is crucial for real-world energy storage battery systems deployed in varying climates. The MPNN, FNN, and PNN were tested on dynamic current data collected at temperatures from 0°C to 50°C. The table below summarizes the average MAE and RMSE for each method across these temperatures:
| Temperature (°C) | FNN | PNN | MPNN | |||
|---|---|---|---|---|---|---|
| MAE | RMSE | MAE | RMSE | MAE | RMSE | |
| 0 | 0.0048 | 0.0098 | 0.0041 | 0.0062 | 0.0018 | 0.0022 |
| 10 | 0.0032 | 0.0077 | 0.0030 | 0.0045 | 0.0011 | 0.0014 |
| 20 | 0.0027 | 0.0072 | 0.0026 | 0.0041 | 0.0009 | 0.0012 |
| 25 | 0.0023 | 0.0069 | 0.0021 | 0.0040 | 0.0009 | 0.0012 |
| 30 | 0.0020 | 0.0064 | 0.0018 | 0.0042 | 0.0010 | 0.0011 |
| 40 | 0.0021 | 0.0065 | 0.0019 | 0.0045 | 0.0012 | 0.0013 |
| 50 | 0.0028 | 0.0068 | 0.0024 | 0.0047 | 0.0013 | 0.0015 |
The data clearly shows that the MPNN consistently outperforms both the FNN and the basic PNN across the entire temperature spectrum. Its MAE and RMSE are consistently below 0.002 (0.2% in SOC) for moderate temperatures and remain excellent even at extremes (0.18% MAE at 0°C, 0.13% at 50°C). This demonstrates the MPNN’s strong generalization capability and robustness, which are essential for dependable SOC estimation in diverse environmental conditions for energy storage battery systems.
Conclusion and Future Perspectives
Accurate State-of-Charge estimation is a non-negotiable requirement for the safe, efficient, and reliable operation of modern energy storage battery systems. This article has presented a novel data-driven solution based on a Modified Probabilistic Neural Network. The MPNN architecture successfully integrates the strengths of probabilistic neural networks—namely, their tendency to converge to a global optimum—with an adaptive compensatory learning mechanism that refines the inference process. The incorporation of a threshold layer further enhances feature selectivity.
Extensive simulation experiments on real Li-ion battery data have validated the MPNN’s superior performance. It achieves high-precision SOC estimation (with errors consistently below 1%) not only under constant low-current conditions but, more importantly, under dynamic load profiles that mimic real-world energy storage battery applications. Its robustness across a wide temperature range further underscores its practical viability. The method’s data-driven nature liberates it from the constraints and potential inaccuracies of fixed electrochemical models, making it adaptable to different energy storage battery types and aging states through retraining.
Future work will focus on enhancing the MPNN’s capabilities for real-world energy storage battery management. Key directions include: 1) Investigating adaptive mechanisms for the probabilistic layer parameters (centers and widths) to further optimize the feature space; 2) Extending the framework to simultaneously estimate other critical states like State of Health (SOH) and Remaining Useful Life (RUL), creating a comprehensive battery analytics tool; 3) Developing efficient online learning algorithms to allow the MPNN to adapt to the gradual aging and capacity fade of the energy storage battery throughout its operational life without requiring full retraining; and 4) Implementing the algorithm on embedded BMS hardware to evaluate its computational efficiency and real-time performance in field-deployed energy storage battery systems.