In the context of modern energy systems, the integration of intermittent renewable sources like wind and solar necessitates robust and reliable energy storage solutions. The cell energy storage system has emerged as a critical component for grid stability, peak shaving, and backup power. The accurate and real-time estimation of the State-of-Charge (SOC) for individual cells within such a system is paramount. SOC is a fundamental metric, akin to a fuel gauge, representing the available capacity relative to its maximum. Precise SOC knowledge prevents harmful operations such as overcharge and over-discharge, which can lead to accelerated degradation, thermal runaway, and catastrophic failure, thereby ensuring the safety, longevity, and efficiency of the entire cell energy storage system. This article presents a comprehensive model-driven approach for online SOC estimation, addressing key challenges in parameter accuracy and nonlinear system dynamics.
The foundation of a model-driven SOC estimation strategy is a precise mathematical representation of the battery cell’s electrical behavior. Among various models, equivalent circuit models (ECMs) offer a favorable balance between accuracy and computational complexity, making them suitable for real-time applications in a cell energy storage system. We adopt the first-order Thevenin model, which effectively captures the dominant dynamic characteristics, including the open-circuit voltage (OCV), ohmic resistance, and polarization effects. The model consists of a voltage source representing the OCV, a series resistor $R_0$ representing the ohmic resistance, and a parallel $R_1C_1$ network representing the polarization dynamics.
The governing equations for the Thevenin model in continuous time are:
$$ \dot{U_1}(t) = -\frac{1}{R_1 C_1} U_1(t) + \frac{1}{C_1} I(t) $$
$$ U_t(t) = V_{OCV}(SOC(t)) – I(t)R_0 – U_1(t) $$
where $U_1$ is the voltage across the polarization $RC$ branch, $I$ is the load current (positive for discharge), $U_t$ is the terminal voltage, and $V_{OCV}(SOC)$ is the nonlinear relationship between the open-circuit voltage and SOC. A crucial step is to empirically determine this $V_{OCV}(SOC)$ function. Through an incremental constant-current discharge test with long relaxation periods, OCV points at various SOC levels are obtained. A high-order polynomial is typically used for fitting:
$$ V_{OCV}(SOC) = \alpha_6 SOC^6 + \alpha_5 SOC^5 + \alpha_4 SOC^4 + \alpha_3 SOC^3 + \alpha_2 SOC^2 + \alpha_1 SOC + \alpha_0 $$
where $\alpha_0 … \alpha_6$ are the polynomial coefficients identified from experimental data. This accurate OCV-SOC map is essential for both parameter identification and the observer algorithm.
The parameters $R_0$, $R_1$, and $C_1$ are not constants; they vary with SOC, temperature, and aging. Therefore, online identification is necessary for a reliable cell energy storage system. We employ the Forgetting Factor Recursive Least Squares (FFRLS) method for this purpose. By applying the bilinear transform to discretize the model’s frequency-domain transfer function, we obtain a linear-in-parameters form:
$$ y_k = U_{oc,k} – U_{t,k} = \theta_k^T \phi_k $$
where $y_k$ is the known output, $\phi_k = [-y_{k-1}, I_k, I_{k-1}]^T$ is the regressor vector, and $\theta_k = [c_1, c_2, c_3]^T$ is the parameter vector to be identified online by FFRLS. The original model parameters are then recovered from $c_1$, $c_2$, and $c_3$ at each time step $k$:
$$ R_{0,k} = \frac{c_2 – c_3}{1 – c_1}, \quad R_{1,k} = \frac{c_2 + c_3}{1 + c_1} – R_{0,k}, \quad C_{1,k} = \frac{T(1 – c_1)}{2R_{1,k}(1 + c_1)} $$
where $T$ is the sampling period. The forgetting factor ($\lambda$, typically 0.95-0.99) allows the algorithm to discount old data, enabling it to track slow parameter variations within the cell energy storage system.
| Model | Structure | Advantages | Disadvantages | Suitability for Cell Energy Storage System |
|---|---|---|---|---|
| Rint | OCV + $R_0$ | Extremely simple, low computation | Poor dynamic accuracy | Low, for very rough estimates only |
| Thevenin (1st-order RC) | OCV + $R_0$ + ($R_1$//$C_1$) | Good balance of accuracy/complexity, captures main dynamics | May not capture very long-term relaxation | High, optimal for most real-time BMS applications |
| Dual Polarization (2nd-order RC) | OCV + $R_0$ + 2($R_i$//$C_i$) | Higher accuracy, captures short & long-term transients | More parameters, higher computation, risk of overfitting | Medium, for applications requiring very high precision offline |
| PNGV | Thevenin + series capacitor | Models capacity effect under current | Complex, more parameters to identify | Low, specific to automotive profiles |
With a dynamically identified model, the SOC estimation problem can be formulated within a state-space framework. The system state vector is defined as $\mathbf{x}_k = [SOC_k, U_{1,k}]^T$. The state transition and measurement equations are derived from the discretized Thevenin model:
$$ \mathbf{x}_{k+1} = \begin{bmatrix} 1 & 0 \\ 0 & \exp(-\frac{T}{R_{1,k}C_{1,k}}) \end{bmatrix} \mathbf{x}_k + \begin{bmatrix} -\frac{T}{Q_n} \\ R_{1,k}(1 – \exp(-\frac{T}{R_{1,k}C_{1,k}})) \end{bmatrix} I_k + \mathbf{w}_k $$
$$ y_k = U_{t,k} = V_{OCV}(SOC_k) – U_{1,k} – I_k R_{0,k} + v_k $$
where $Q_n$ is the nominal cell capacity, and $\mathbf{w}_k$ and $v_k$ represent process and measurement noise, respectively. The $V_{OCV}(SOC_k)$ term makes the measurement equation highly nonlinear. While the Extended Kalman Filter (EKF) linearizes this nonlinearity, it can introduce significant errors. Therefore, we employ the Unscented Kalman Filter (UKF), which uses a deterministic sampling approach (the Unscented Transform) to better handle the nonlinearities inherent in a cell energy storage system.
The UKF algorithm operates as follows. First, a set of $2n+1$ sigma points $\boldsymbol{\chi}_{i,k}$ (where $n$ is the state dimension) are generated around the current state estimate $\hat{\mathbf{x}}_k$ with covariance $\mathbf{P}_k$:
$$ \boldsymbol{\chi}_{0,k} = \hat{\mathbf{x}}_k $$
$$ \boldsymbol{\chi}_{i,k} = \hat{\mathbf{x}}_k + \left( \sqrt{(n+\lambda)\mathbf{P}_k} \right)_i, \quad i=1,…,n $$
$$ \boldsymbol{\chi}_{i+n,k} = \hat{\mathbf{x}}_k – \left( \sqrt{(n+\lambda)\mathbf{P}_k} \right)_i, \quad i=1,…,n $$
where $\lambda$ is a scaling parameter. These sigma points are then propagated through the nonlinear state and measurement functions ($f$ and $h$):
$$ \boldsymbol{\chi}_{i,k+1|k} = f(\boldsymbol{\chi}_{i,k}, I_k), \quad \mathcal{Y}_{i,k+1|k} = h(\boldsymbol{\chi}_{i,k+1|k}, I_k) $$
The predicted mean and covariance are computed using weighted sums:
$$ \hat{\mathbf{x}}_{k+1|k} = \sum_{i=0}^{2n} W_i^{(m)} \boldsymbol{\chi}_{i,k+1|k}, \quad \hat{\mathbf{P}}_{k+1|k} = \sum_{i=0}^{2n} W_i^{(c)} (\boldsymbol{\chi}_{i,k+1|k} – \hat{\mathbf{x}}_{k+1|k})(\boldsymbol{\chi}_{i,k+1|k} – \hat{\mathbf{x}}_{k+1|k})^T + \mathbf{Q} $$
$$ \hat{y}_{k+1|k} = \sum_{i=0}^{2n} W_i^{(m)} \mathcal{Y}_{i,k+1|k} $$
Finally, the Kalman gain $\mathbf{K}_{k+1}$, state update, and covariance update are performed using the standard formulas based on the predicted and cross-covariance matrices. This approach provides a more accurate estimation of the posterior mean and covariance of the states for nonlinear systems compared to EKF, leading to superior SOC tracking in the dynamic environment of a cell energy storage system.
The complete SOC estimation framework, which we term the FFR-UKF (Forgetting Factor Recursive – Unscented Kalman Filter) strategy, synergistically combines online parameter identification and nonlinear state estimation. The block diagram illustrates this closed-loop process: measured current $I_k$ and terminal voltage $U_{t,k}$ are the inputs. The FFR block continuously identifies the model parameters ($R_0$, $R_1$, $C_1$). These updated parameters are fed into the state-space model used by the UKF block. The UKF algorithm produces an optimal SOC estimate $\hat{SOC}_k$, which is also used to evaluate $V_{OCV}(\hat{SOC}_k)$ for the next iteration, creating a robust and self-correcting estimation loop.
| Evaluation Metric | Ampere-Hour Integration | Extended Kalman Filter (EKF) | FFR-UKF (Proposed) |
|---|---|---|---|
| Mean Absolute Error (MAE) | ~2.5% – 5% (diverges) | ~0.8% – 1.5% | ~0.3% – 0.6% |
| Root Mean Square Error (RMSE) | High | Moderate | Low |
| Convergence Speed from Initial Error | No convergence | Slow to Moderate (600+ steps) | Fast (100-150 steps) |
| Dependence on Initial SOC | Extremely High | High | Low |
| Robustness to Parameter Variation | Poor | Moderate (if not co-estimated) | High (online identification) |
| Computational Load | Very Low | Moderate | Moderate to High |
The proposed method’s efficacy is validated through extensive simulation and experimental analysis using real data from Li-ion cells typical of a cell energy storage system. Under a complex dynamic discharge profile, the FFR-UKF method demonstrates remarkable accuracy. Compared to the traditional Ampere-Hour (AH) method—which accumulates error due to current sensor inaccuracies and unknown initial SOC—and the standard EKF, the FFR-UKF estimates closely follow the reference SOC trajectory. The absolute error remains consistently below 1%, significantly outperforming the other methods. A key test for any SOC algorithm in a cell energy storage system is robustness to initial condition errors. Tests with initial SOC errors of +25% and -50% show that while both EKF and FFR-UKF eventually converge, the FFR-UKF achieves convergence much faster (within 2-3% of the true SOC in under 150 sampling steps) thanks to the accurate model provided by online parameter identification and the superior nonlinear estimation of UKF.
In conclusion, the accurate estimation of State-of-Charge is a non-negotiable requirement for the safe, efficient, and reliable management of a cell energy storage system. The model-driven approach combining online parameter identification via Forgetting Factor Recursive Least Squares with nonlinear state estimation via the Unscented Kalman Filter presents a powerful solution. This FFR-UKF framework directly addresses the core challenges of time-varying battery parameters and nonlinear dynamics. By providing high estimation accuracy (often above 99%), rapid convergence from erroneous initial conditions, and inherent robustness through continuous model adaptation, this method is well-suited for the demanding real-time operational environment of large-scale cell energy storage systems. Future work may focus on integrating aging models and temperature effects to further enhance the estimation lifecycle performance across the entire operational range of the cell energy storage system.