Ensuring the safe and stable operation of large-scale cell energy storage systems hinges on the ability to accurately monitor the internal state of its fundamental building blocks—the lithium-ion batteries. Among all state parameters, the State of Charge (SOC) is paramount, representing the available capacity relative to the fully charged state. Accurate SOC knowledge prevents over-charge and over-discharge, mitigates thermal runaway risks, and optimizes system dispatch, thereby extending the lifespan and enhancing the reliability of the entire cell energy storage system. However, achieving precise, real-time SOC estimation is challenging due to the battery’s strong nonlinearity, time-varying parameters, and the complex, uncertain noise environments typical of grid-scale energy storage applications.
Traditional methods exhibit significant limitations for application in a dynamic cell energy storage system. The open-circuit voltage (OCV) method requires long rest periods, which are impractical for continuously operated storage. The ampere-hour (Ah) integration method is simple but suffers from cumulative errors from sensor inaccuracy and unknown initial SOC. While data-driven approaches like neural networks can model nonlinearity, they demand vast amounts of training data and lack interpretability. Model-based filters, particularly the Kalman Filter (KF) family, offer a robust framework. The Extended Kalman Filter (EKF) linearizes nonlinear functions, introducing errors. In contrast, the Unscented Kalman Filter (UKF) uses a deterministic sampling technique (the Unscented Transform) to approximate the state distribution, handling nonlinearities more accurately and reliably, which is crucial for the cell energy storage system management.
The accuracy of any model-based filter, however, is fundamentally tied to the fidelity of the underlying battery model and its parameters. An imprecise model will lead to biased state estimation regardless of the filter’s sophistication. Therefore, this work proposes a synergistic approach combining online parameter identification using the Forgetting Factor Recursive Least Squares (FFRLS) algorithm with state estimation using the UKF. This integrated FFRLS-UKF strategy continuously updates the battery model parameters in real-time, ensuring the filter always operates with a representative model, leading to highly accurate and robust SOC estimation for the cell energy storage system under varying operational conditions.
Battery Modeling and Online Parameter Identification
Establishing the OCV-SOC Relationship
The relationship between a battery’s open-circuit voltage (OCV) and its SOC is a critical foundation for model-based estimation. To characterize this for a target cell energy storage system battery, a controlled intermittent constant-current discharge test is conducted. Starting from a fully charged state (SOC=1.00), the battery is discharged at a 1C rate to release 5% of its nominal capacity, followed by a long rest period (e.g., 900 seconds) to reach a fully relaxed state where the terminal voltage approximates the OCV. This pulse-relax cycle is repeated until the battery is fully discharged.
The recorded OCV points at different SOC levels are presented in the table below:
| SOC | 0.00 | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 |
|---|---|---|---|---|---|---|---|
| OCV (V) | 3.00 | 3.25 | 3.42 | 3.55 | 3.65 | 3.72 | 3.78 |
| SOC | 0.35 | 0.40 | 0.45 | 0.50 | 0.55 | 0.60 | 0.65 |
|---|---|---|---|---|---|---|---|
| OCV (V) | 3.83 | 3.87 | 3.90 | 3.94 | 3.97 | 4.00 | 4.03 |
| SOC | 0.70 | 0.75 | 0.80 | 0.85 | 0.90 | 0.95 | 1.00 |
|---|---|---|---|---|---|---|---|
| OCV (V) | 4.06 | 4.09 | 4.12 | 4.15 | 4.18 | 4.21 | 4.24 |
This OCV-SOC relationship is highly nonlinear. A polynomial function is commonly used for fitting, providing a continuous mapping for the estimation algorithm. A 6th-order polynomial is often sufficient for accuracy:
$$V_{oc}(z) = \alpha_0 + \alpha_1 z + \alpha_2 z^2 + \alpha_3 z^3 + \alpha_4 z^4 + \alpha_5 z^5 + \alpha_6 z^6$$
where $z$ denotes the SOC, $V_{oc}$ is the open-circuit voltage, and $\alpha_0, \alpha_1, …, \alpha_6$ are the polynomial coefficients identified from the test data. This $V_{oc}(z)$ function is essential for constructing the battery model output equation.
Thevenin Equivalent Circuit Model
For the cell energy storage system application, a model must balance accuracy and computational simplicity to enable real-time implementation. The first-order Thevenin model (or RC model) is widely adopted. It consists of an ideal voltage source $V_{oc}(z)$, a series ohmic resistance $R_0$, and an RC parallel network ($R_1$, $C_1$) that captures the dynamic polarization voltage.
The state-space representation of the Thevenin model is derived from its electrical dynamics. Defining the state vector as $x_k = [z_k \quad U_{1,k}]^T$, where $z_k$ is the SOC and $U_{1,k}$ is the voltage across the polarization RC branch, the discrete-time state equation and output equation are:
$$
\begin{aligned}
x_{k+1} &= A_k x_k + B_k i_k + w_k \\
V_{t,k} &= V_{oc}(z_k) – i_k R_{0,k} – U_{1,k} + v_k
\end{aligned}
$$
With the matrices defined as:
$$
A_k = \begin{bmatrix} 1 & 0 \\ 0 & \exp(-\Delta t / (R_{1,k} C_{1,k})) \end{bmatrix}, \quad
B_k = \begin{bmatrix} -\eta \Delta t / Q_n \\ R_{1,k} (1 – \exp(-\Delta t / (R_{1,k} C_{1,k}))) \end{bmatrix}
$$
where $i_k$ is the load current (positive for discharge), $V_{t,k}$ is the terminal voltage, $Q_n$ is the nominal capacity, $\eta$ is the Coulombic efficiency, $\Delta t$ is the sampling period, and $w_k$, $v_k$ are process and measurement noise, respectively. The parameters $R_{0,k}$, $R_{1,k}$, and $C_{1,k}$ are time-varying and must be identified online for an adaptive model in a cell energy storage system.
Online Parameter Identification via FFRLS
The core challenge is obtaining accurate values for $R_0$, $R_1$, and $C_1$ in real-time. The Forgetting Factor Recursive Least Squares (FFRLS) algorithm is ideal for this purpose, as it can track slow parameter variations typical of a cell energy storage system’s aging and thermal changes.
The derivation begins with the model’s transfer function in the frequency domain:
$$ \frac{V_{oc}(s) – V_t(s)}{I(s)} = R_0 + \frac{R_1}{1 + s R_1 C_1} $$
Applying the bilinear transform $s = \frac{2}{\Delta t} \frac{1-z^{-1}}{1+z^{-1}}$ yields the discrete-time transfer function:
$$ G(z^{-1}) = \frac{V_{oc}(z^{-1}) – V_t(z^{-1})}{I(z^{-1})} = \frac{a_2 + a_3 z^{-1}}{1 + a_1 z^{-1}} $$
Transforming this back to the time domain gives the linear regression form:
$$ y_k = V_{oc,k} – V_{t,k} = a_1 y_{k-1} + a_2 i_k + a_3 i_{k-1} $$
This can be written as:
$$ y_k = \phi_k^T \theta_k $$
where the regressor vector is $\phi_k = [-y_{k-1}, i_k, i_{k-1}]^T$ and the parameter vector to be identified is $\theta_k = [a_{1,k}, a_{2,k}, a_{3,k}]^T$.
The FFRLS algorithm updates the parameter estimates recursively:
$$
\begin{aligned}
K_k &= \frac{P_{k-1} \phi_k}{\lambda + \phi_k^T P_{k-1} \phi_k} \\
\hat{\theta}_k &= \hat{\theta}_{k-1} + K_k (y_k – \phi_k^T \hat{\theta}_{k-1}) \\
P_k &= \lambda^{-1} (I – K_k \phi_k^T) P_{k-1}
\end{aligned}
$$
Here, $K_k$ is the gain matrix, $P_k$ is the covariance matrix, and $\lambda$ is the forgetting factor ($0 < \lambda \leq 1$), which discounts older data to allow tracking of time-varying parameters in the dynamic cell energy storage system environment.
Once the parameters $a_1, a_2, a_3$ are identified, the physical model parameters for the cell energy storage system battery are recovered using the following equations:
$$
\begin{aligned}
R_{0,k} &= \frac{a_{2,k} – a_{3,k}}{1 – a_{1,k}} \\
R_{1,k} &= \frac{a_{2,k} + a_{3,k}}{1 + a_{1,k}} – R_{0,k} \\
C_{1,k} &= \frac{\Delta t (1 – a_{1,k})}{2 R_{1,k} (1 + a_{1,k})}
\end{aligned}
$$
This online identification process continuously refreshes the $A_k$, $B_k$ matrices and the output equation, providing the UKF algorithm with an accurate and up-to-date model for every estimation step.
SOC Estimation via the Unscented Kalman Filter
With an adaptively identified model, the Unscented Kalman Filter (UKF) is employed to estimate the SOC. The UKF’s strength lies in its use of the Unscented Transform (UT) to propagate the mean and covariance of the state through the nonlinear system, yielding a more accurate estimate than the EKF’s first-order linearization.
UKF Algorithm Steps
Given the nonlinear state-space model for the cell energy storage system battery:
$$ x_{k+1} = f(x_k, i_k, \theta_k) + w_k $$
$$ V_{t,k} = h(x_k, i_k, \theta_k) + v_k $$
where $f(\cdot)$ is the nonlinear state transition function (defined by the Thevenin model state equations) and $h(\cdot) = V_{oc}(z_k) – i_k R_{0,k} – U_{1,k}$ is the nonlinear measurement function. The UKF proceeds as follows:
1. Initialization:
$$ \hat{x}_0^+ = E[x_0], \quad P_0^+ = E[(x_0 – \hat{x}_0^+)(x_0 – \hat{x}_0^+)^T] $$
2. Sigma Point Calculation: At time step $k-1$, calculate $2n+1$ sigma points $\chi_{k-1}^{(i)}$ around the previous state estimate $\hat{x}_{k-1}^+$, where $n$ is the state dimension ($n=2$):
$$
\begin{aligned}
\chi_{k-1}^{(0)} &= \hat{x}_{k-1}^+ \\
\chi_{k-1}^{(i)} &= \hat{x}_{k-1}^+ + \left( \sqrt{(n+\kappa) P_{k-1}^+} \right)_i, \quad i=1,…,n \\
\chi_{k-1}^{(i)} &= \hat{x}_{k-1}^+ – \left( \sqrt{(n+\kappa) P_{k-1}^+} \right)_{i-n}, \quad i=n+1,…,2n
\end{aligned}
$$
where $\kappa$ is a scaling parameter, and $\left( \sqrt{M} \right)_i$ denotes the $i$-th column of the matrix square root.
3. Time Update (Prediction):
Propagate each sigma point through the state function:
$$ \chi_k^{-(i)} = f(\chi_{k-1}^{(i)}, i_{k-1}, \theta_{k-1}), \quad i=0,…,2n $$
Compute the predicted state mean and covariance:
$$
\begin{aligned}
\hat{x}_k^- &= \sum_{i=0}^{2n} W_m^{(i)} \chi_k^{-(i)} \\
P_k^- &= \sum_{i=0}^{2n} W_c^{(i)} [\chi_k^{-(i)} – \hat{x}_k^-][\chi_k^{-(i)} – \hat{x}_k^-]^T + Q
\end{aligned}
$$
where $W_m^{(i)}$ and $W_c^{(i)}$ are weights for mean and covariance, respectively, and $Q$ is the process noise covariance matrix.
4. Measurement Update (Correction):
Propagate the predicted sigma points through the measurement function:
$$ \gamma_k^{(i)} = h(\chi_k^{-(i)}, i_k, \theta_k), \quad i=0,…,2n $$
Compute the predicted measurement mean, covariance, and cross-covariance:
$$
\begin{aligned}
\hat{y}_k &= \sum_{i=0}^{2n} W_m^{(i)} \gamma_k^{(i)} \\
P_{yy,k} &= \sum_{i=0}^{2n} W_c^{(i)} [\gamma_k^{(i)} – \hat{y}_k][\gamma_k^{(i)} – \hat{y}_k]^T + R \\
P_{xy,k} &= \sum_{i=0}^{2n} W_c^{(i)} [\chi_k^{-(i)} – \hat{x}_k^-][\gamma_k^{(i)} – \hat{y}_k]^T
\end{aligned}
$$
where $R$ is the measurement noise covariance. Then, compute the Kalman gain and update the state estimate:
$$
\begin{aligned}
K_k &= P_{xy,k} P_{yy,k}^{-1} \\
\hat{x}_k^+ &= \hat{x}_k^- + K_k (V_{t,k}^{measured} – \hat{y}_k) \\
P_k^+ &= P_k^- – K_k P_{yy,k} K_k^T
\end{aligned}
$$
The first element of $\hat{x}_k^+$ is the optimally estimated SOC for the cell energy storage system battery at time $k$.
Integrated FFRLS-UKF Framework and Experimental Validation
The proposed framework tightly integrates the FFRLS identifier with the UKF estimator in a closed loop. In each sampling interval of the cell energy storage system’s battery management cycle, the measured terminal voltage $V_{t,k}$ and current $i_k$ are fed into the FFRLS algorithm to update the parameter vector $\theta_k$ (and thus $R_{0,k}, R_{1,k}, C_{1,k}$). These updated parameters are then passed to the UKF’s state transition $f(\cdot)$ and measurement $h(\cdot)$ functions. The UKF subsequently executes its prediction and correction steps using this refreshed model to produce the final SOC estimate. This co-estimation structure ensures the filter’s internal model remains aligned with the battery’s actual dynamic behavior.
Performance Evaluation and Comparison
The proposed FFRLS-UKF method was validated using experimental data from a lithium-ion battery under dynamic discharge profiles. Its performance was compared against two benchmark methods: the standard Ampere-hour Integration (Ah) and the Extended Kalman Filter (EKF). The evaluation metrics were Mean Error (ME), Root Mean Square Error (RMSE), and the convergence behavior under different initial SOC errors, critical for assessing robustness in a cell energy storage system.
The SOC estimation results and errors for a typical test are summarized below:
| Algorithm | ME (%) | RMSE (%) | Max Error (%) |
|---|---|---|---|
| Ampere-hour Integration | 0.934 | 1.082 | 3.5 |
| EKF | 0.705 | 0.602 | 2.1 |
| Proposed FFRLS-UKF | 0.400 | 0.201 | 1.2 |
The proposed method clearly outperforms the others, achieving an accuracy of over 99.2% under ideal conditions. More importantly, its performance under initial SOC error demonstrates its critical advantage for real-world cell energy storage system applications where the initial SOC is often unknown.
Convergence test results under wrong initial SOC conditions:
| Initial SOC Error | Algorithm | Steps to Converge (<2%) | Final RMSE (%) |
|---|---|---|---|
| +0.25 (True: 0.75, Initial: 1.00) | EKF | ~690 | 0.65 |
| FFRLS-UKF | ~104 | 0.22 | |
| -0.25 (True: 0.75, Initial: 0.50) | EKF | ~722 | 0.71 |
| FFRLS-UKF | ~133 | 0.25 |
The FFRLS-UKF method converges to the true SOC value significantly faster than the EKF. This rapid convergence is attributed to the UKF’s more accurate nonlinear estimation and, crucially, the adaptive model provided by FFRLS. When the initial SOC is wrong, the terminal voltage predicted by the model based on the wrong state will have a large error. The FFRLS component quickly adjusts the model parameters ($R_0, R_1, C_1$) to better fit the observed voltage, which in turn provides a more accurate gradient (via $P_{xy}$ and $K_k$) for the UKF to correct the state. This synergy allows the overall algorithm to recover from large initial errors efficiently, a must-have feature for a reliable cell energy storage system battery management system that may experience power cycles or sensor resets.
Conclusion
Accurate and robust State-of-Charge estimation is a cornerstone for the safe, efficient, and reliable operation of modern cell energy storage systems. This work presents a comprehensive online estimation strategy that addresses the core challenges of battery nonlinearity and parameter uncertainty. By integrating an online parameter identification algorithm (FFRLS) with an advanced nonlinear state estimator (UKF), the proposed method ensures the estimation model remains accurate in real-time. Experimental validation confirms its superior performance over conventional methods like Ah integration and EKF, particularly in terms of estimation accuracy, speed of convergence from erroneous initial conditions, and overall robustness. The FFRLS-UKF framework provides a practical, high-performance solution for enhancing the intelligence and reliability of battery management in large-scale cell energy storage systems, contributing directly to grid stability and the effective integration of renewable energy sources.