The rapid integration of renewable energy sources necessitates robust and flexible grid support mechanisms. Electrochemical energy storage batteries, particularly lithium-ion-based systems, have emerged as a pivotal technology for enabling “integrated generation-storage” and “source-grid-load-storage” systems due to their configurability and rapid response. However, as these energy storage battery systems scale up to megawatt and gigawatt-hour levels, ensuring their long-term operational safety and reliability becomes a critical bottleneck. Catastrophic failures, including thermal runaway leading to fires or explosions, underscore the urgent need for advanced real-time state monitoring, accurate estimation, and early warning capabilities for energy storage battery packs.
Traditional state estimation methods, such as ampere-hour (Ah) integration, suffer from cumulative errors over extended cycling. While model-based approaches like equivalent circuit models coupled with state observers (e.g., Kalman filters) offer improved accuracy for State of Charge (SOC) estimation, they face challenges related to model fidelity, parameter drift due to aging, and sensitivity to varying operational conditions. Relying solely on a single model makes it difficult to distinguish between normal performance degradation and incipient faults. This paper addresses these limitations by proposing a novel, hybrid data-model driven framework for the state estimation and early warning of lithium-ion energy storage battery systems.
The core innovation lies in the synergistic combination of a physics-inspired model and a purely data-driven estimator. The proposed methodology first employs a first-order Thevenin equivalent circuit model whose parameters are functions of SOC, coupled with an Extended Kalman Filter (EKF) to create a “digital twin” or reference model of the energy storage battery. This model-driven component provides real-time estimates of internal states like SOC and polarization voltage. Concurrently, the actual operational data from the physical energy storage battery system is collected. Instead of directly monitoring raw voltage or current, the method focuses on the deviation vector between the real system’s measured outputs (voltage, EKF-calculated SOC) and the reference model’s simulated outputs.
This deviation vector is then fed into a Nonlinear State Estimation Technique (NSET), a memory-based data-driven algorithm. The NSET is trained on historical data collected during normal, healthy operation of the energy storage battery. It learns the characteristic pattern of deviations expected under normal aging and operational variation. When a novel fault or abnormal condition arises in the physical energy storage battery (e.g., a sudden increase in internal resistance), the deviation vector exhibits a pattern that cannot be accurately reconstructed by the NSET model trained on normal data. The resulting residual error serves as a sensitive and robust fault indicator, effectively decoupling fault signals from the effects of gradual aging, thereby providing early warning.
Mathematical Modeling and State of Charge Estimation for the Energy Storage Battery
Thevenin Equivalent Circuit Model
Accurate modeling is foundational for model-based state estimation. Among various equivalent circuit models, the first-order Thevenin model offers an effective balance between simplicity and dynamic representation for an energy storage battery. It captures the dominant polarization dynamics using a single RC pair, while a series resistor represents the ohmic losses.
The electrical behavior is governed by the following equations:
$$U_t = U_{oc}(SOC) – I_o R_0(SOC) – U_p$$
$$\frac{dU_p}{dt} = -\frac{1}{R_p(SOC) C_p(SOC)} U_p + \frac{1}{C_p(SOC)} I_o$$
$$SOC(t) = SOC(0) – \frac{\eta}{Q_c} \int_0^t I_o(\tau) d\tau$$
where \(U_t\) is the terminal voltage, \(U_{oc}\) is the open-circuit voltage (a function of SOC), \(I_o\) is the current (positive for discharge), \(R_0\) is the ohmic internal resistance, \(R_p\) and \(C_p\) are the polarization resistance and capacitance, respectively, \(U_p\) is the polarization voltage, \(\eta\) is the coulombic efficiency, and \(Q_c\) is the nominal capacity. Critically, the parameters \(U_{oc}\), \(R_0\), \(R_p\), and \(C_p\) are not constants but vary with the SOC of the energy storage battery.
Discretizing these equations with a sampling period \(T\) and accounting for process noise \(\mathbf{w}(k)\) and measurement noise \(v(k)\) leads to the state-space form suitable for filtering:
$$
\mathbf{x}(k+1) = \mathbf{A}(k)\mathbf{x}(k) + \mathbf{B}(k)I_o(k) + \mathbf{w}(k)
$$
$$
U_t(k) = U_{oc}(SOC(k)) – R_0(SOC(k))I_o(k) – U_p(k) + v(k)
$$
where the state vector is \(\mathbf{x}(k) = [SOC(k), U_p(k)]^T\). The state transition matrix \(\mathbf{A}(k)\) and input matrix \(\mathbf{B}(k)\) are:
$$
\mathbf{A}(k) = \begin{bmatrix} 1 & 0 \\ 0 & e^{-T / \tau(k)} \end{bmatrix}, \quad
\mathbf{B}(k) = \begin{bmatrix} -\frac{\eta T}{Q_c} \\ R_p(k)(1 – e^{-T / \tau(k)}) \end{bmatrix}
$$
with \(\tau(k) = R_p(SOC(k)) C_p(SOC(k))\).
State of Charge Estimation via Extended Kalman Filter (EKF)
The EKF provides an optimal recursive estimator for the nonlinear state-space model of the energy storage battery. It operates in a two-step predict-update cycle.
1. Prediction Step:
Using the previous state estimate \(\hat{\mathbf{x}}(k-1|k-1)\) and the measured current \(I_o(k)\):
$$
\hat{\mathbf{x}}(k|k-1) = f(\hat{\mathbf{x}}(k-1|k-1), I_o(k))
$$
$$
\mathbf{P}(k|k-1) = \mathbf{F}(k-1) \mathbf{P}(k-1|k-1) \mathbf{F}(k-1)^T + \mathbf{Q}
$$
where \(f(\cdot)\) is the nonlinear state function from the discretized model, \(\mathbf{P}\) is the error covariance matrix, \(\mathbf{Q}\) is the process noise covariance matrix, and \(\mathbf{F}\) is the Jacobian of \(f(\cdot)\) evaluated at the prior estimate.
2. Update Step:
Compute the Kalman Gain \(\mathbf{K}(k)\), then update the state estimate and covariance using the terminal voltage measurement \(U_t(k)\):
$$
\mathbf{K}(k) = \mathbf{P}(k|k-1) \mathbf{H}(k)^T [\mathbf{H}(k) \mathbf{P}(k|k-1) \mathbf{H}(k)^T + R]^{-1}
$$
$$
\hat{\mathbf{x}}(k|k) = \hat{\mathbf{x}}(k|k-1) + \mathbf{K}(k) [U_t(k) – h(\hat{\mathbf{x}}(k|k-1), I_o(k))]
$$
$$
\mathbf{P}(k|k) = [\mathbf{I} – \mathbf{K}(k)\mathbf{H}(k)] \mathbf{P}(k|k-1)
$$
Here, \(h(\cdot)\) is the measurement function (terminal voltage equation), \(\mathbf{H}\) is its Jacobian, and \(R\) is the measurement noise covariance.
The EKF continuously refines the SOC estimate for the energy storage battery, providing a foundation for the reference model in the hybrid framework.
Data-Driven Early Warning via Nonlinear State Estimation (NSET)
While the EKF-based model provides good SOC tracking, it may be insufficient for early fault detection as its parameters might slowly drift to accommodate aging, masking subtle fault signatures. The NSET component addresses this by learning the normal behavioral patterns of the system deviation.
Formation of the Deviation Vector
Let the superscripts \(^r\) and \(^m\) denote the real physical energy storage battery system and the reference EKF model, respectively. At each time step \(k\), we construct a deviation observation vector \(\mathbf{X}(k)\):
$$
\mathbf{X}(k) = \begin{bmatrix} \Delta U_t(k) \\ \Delta SOC(k) \end{bmatrix} = \begin{bmatrix} U_t^r(k) – U_t^m(k) \\ SOC^r(k) – SOC^m(k) \end{bmatrix}
$$
Under normal operation, this vector follows a specific pattern determined by model inaccuracies, sensor biases, and normal aging. A fault or abnormal condition will cause a significant departure from this learned pattern.
NSET Algorithm and Fault Detection
NSET is a non-parametric, memory-based modeling technique. During an initial training phase under healthy conditions, \(m\) historical observation vectors \(\mathbf{X}_1, \mathbf{X}_2, …, \mathbf{X}_m\) are stored to form a memory matrix \(\mathbf{D}\):
$$
\mathbf{D} = [\mathbf{X}_1, \mathbf{X}_2, …, \mathbf{X}_m]
$$
For a new observation vector \(\mathbf{X}_{new}\), NSET finds a weight vector \(\mathbf{W} = [w_1, w_2, …, w_m]^T\) such that the linear combination of memory vectors approximates \(\mathbf{X}_{new}\):
$$
\hat{\mathbf{X}}_{new} = \mathbf{D} \mathbf{W} = \sum_{i=1}^{m} w_i \mathbf{X}_i
$$
The weight vector is determined by minimizing the residual \(\mathbf{E} = \mathbf{X}_{new} – \hat{\mathbf{X}}_{new}\), using a nonlinear operator for distance calculation (typically Euclidean distance):
$$
\mathbf{W} = (\mathbf{D}^T \odot \mathbf{D})^{-1} (\mathbf{D}^T \odot \mathbf{X}_{new})
$$
where \(\odot\) denotes the nonlinear operator applied column-wise between \(\mathbf{D}^T\) and the other matrix/vector.
The estimation residual \(\mathbf{E}\) is the key diagnostic signal. For a healthy energy storage battery operating within the learned envelope, \(\hat{\mathbf{X}}_{new}\) will be accurate, and the residual magnitude \(\|\mathbf{E}\|\) will be small. When an incipient fault occurs, the new deviation pattern cannot be well-represented by the healthy memory matrix, leading to a large, detectable residual. This spike in the residual, often in the voltage deviation channel first, serves as an effective early warning indicator for the energy storage battery system.
| Component | Role in Hybrid Method | Key Advantage |
|---|---|---|
| Thevenin + EKF Model | Provides reference SOC and voltage estimates. Accounts for basic nonlinear dynamics of the energy storage battery. | Physics-inspired, offers internal state visibility. |
| Deviation Vector (ΔU, ΔSOC) | Forms the input feature space for fault detection, isolating the discrepancy between real and modeled behavior. | Amplifies fault signatures relative to normal aging drift. |
| NSET Algorithm | Learns the normal pattern of deviations. Flags abnormal patterns indicative of faults in the energy storage battery. | Data-driven, model-free, sensitive to novel anomalies. |
Integrated Hybrid-Driven Workflow for Energy Storage Battery Management
The complete workflow for state estimation and early warning of the lithium-ion energy storage battery system integrates both strands as follows:
Step 1: System Operation & Reference Modeling. The real energy storage battery pack operates under load. Concurrently, the EKF-based reference model, using the same measured current \(I_o(k)\), computes its own estimates of terminal voltage \(U_t^m(k)\) and state of charge \(SOC^m(k)\).
Step 2: Deviation Calculation. The BMS measures the actual pack voltage \(U_t^r(k)\) and calculates the real SOC \(SOC^r(k)\) (e.g., via its own onboard EKF). The deviation vector \(\mathbf{X}(k) = [U_t^r – U_t^m, SOC^r – SOC^m]^T\) is computed.
Step 3: NSET-Based State Estimation & Residual Generation. The current deviation vector \(\mathbf{X}(k)\) is presented to the trained NSET model. The NSET produces an estimate \(\hat{\mathbf{X}}(k)\) of what the deviation should be under normal conditions. The residual vector \(\mathbf{E}(k) = \mathbf{X}(k) – \hat{\mathbf{X}}(k)\) is calculated.
Step 4: Fault Detection & Warning. The magnitude of the residual \(\|\mathbf{E}(k)\|\) (or a more sophisticated statistic derived from it) is monitored. If it exceeds a predefined adaptive threshold for a sustained period, an early warning alarm is triggered for the energy storage battery system, indicating a potential abnormality that requires investigation. This process is highly effective because it distinguishes between the slow parameter changes of the reference model (which adapt to aging) and sudden, inconsistent deviations signaling failure.
Simulation Analysis and Performance Evaluation
A simulation study in MATLAB/Simulink was conducted to validate the method. A string of 240 series-connected 18650 LiFePO4 cells was simulated. The reference EKF model and a “real” battery model were implemented with parameters varying as functions of SOC, as determined from experimental HPPC tests.
| Parameter | Value |
|---|---|
| Cell Configuration | 240S1P |
| Nominal Capacity (Pack) | 40 Ah |
| Initial SOC | 80% |
| Simulation Time | 5400 s |
| Load Profile | Cyclic discharge/charge at 10kW, 4kW, 6kW / 2kW, 10kW, 8kW |
The “real” energy storage battery model was subjected to two conditions: 1) Gradual aging, simulated by a slow, continuous increase in internal resistance \(R_0\) and polarization resistance \(R_p\) over time. 2) An abrupt fault, simulated by a sudden step increase of 2 mΩ in \(R_0\) at t=4000s, reverted at t=4500s.
The results demonstrated the efficacy of the hybrid method. The raw voltage deviation \(\Delta U_t\) showed a steadily increasing trend due to simulated aging, making it difficult to set a fixed threshold for fault detection without causing false alarms. The NSET residual, however, behaved differently. It remained low and stable during the gradual aging phase (0-4000s), indicating the NSET successfully learned and compensated for this slow drift as a normal pattern. At the moment of the abrupt fault (t=4000s), the NSET residual exhibited a sharp, significant spike, clearly signaling the anomaly. When the fault was removed (t=4500s), the residual returned to its baseline. This confirms that the method effectively decouples aging effects from fault signals, providing a specific and reliable early warning mechanism for the energy storage battery.
The method proved robust against changes in load current magnitude and profile, as the NSET learns the deviation patterns across all trained operational conditions. This underscores its practical utility for real-world energy storage battery applications where load conditions are highly variable.
Conclusion and Perspective
This paper presents a novel data-model hybrid-driven framework for the state estimation and early warning of lithium-ion energy storage battery systems. By fusing the strengths of a model-based Extended Kalman Filter with a data-driven Nonlinear State Estimation Technique, the method achieves superior fault discernment compared to traditional approaches.
The key conclusions are:
- Enhanced Fault Sensitivity: The method is highly sensitive to abrupt parameter changes indicative of faults (e.g., contact resistance increase, internal short circuit development) while being robust to the slow parameter drift associated with normal aging of the energy storage battery. This drastically reduces false alarms.
- Reduced Model Dependency: The NSET component compensates for inherent inaccuracies in the underlying Thevenin model, making the overall warning system less dependent on perfect model fidelity, which is often unattainable for complex energy storage battery systems.
- Computational Feasibility: Both EKF and NSET are computationally efficient algorithms suitable for real-time implementation on battery management system (BMS) hardware or edge-computing platforms, facilitating online monitoring of large-scale energy storage battery installations.
- Practical Implementation Pathway: The method provides a clear pathway for deployment: an initial training phase to build the NSET memory matrix under healthy operation, followed by continuous monitoring of the deviation residual. This makes it a practical tool for improving the safety and reliability of grid-connected energy storage battery systems.
Future work will focus on expanding the method to differentiate between different fault types (e.g., internal short circuit vs. connection loosening) by analyzing the specific patterns within the residual vector and integrating it with thermal models for a comprehensive safety management system for energy storage batteries. The integration of this hybrid estimation framework promises to be a significant step towards safer, more reliable, and smarter large-scale energy storage battery systems.