With the rapid expansion of renewable energy integration and smart grid development, the demand for reliable and efficient energy storage systems has become paramount. Among various technologies, lithium-ion based energy storage cells have gained prominence due to their favorable economic characteristics and thermal stability. However, accurate estimation of the State of Charge (SOC) remains challenging due to the inability to directly measure internal electrochemical states. Furthermore, energy storage cells undergo numerous deep charge-discharge cycles throughout their operational lifetime, leading to gradual degradation that significantly impacts SOC estimation accuracy. This deterioration increases system complexity while reducing both economic viability and safety, ultimately limiting the potential of energy storage systems.
Traditional approaches for SOC estimation often struggle to maintain accuracy across the entire lifecycle of energy storage cells. Methods combining health state models with battery models introduce substantial computational complexity, while data-driven approaches require extensive training data and lack generalization capabilities. The unique operational characteristics of energy storage systems, particularly their predominance of constant power discharge profiles, present both challenges and opportunities for innovative estimation techniques.
We present a comprehensive framework for SOC estimation in energy storage cells that maintains high accuracy throughout the complete aging process. Our approach leverages the consistent operational patterns of energy storage systems by utilizing full discharge voltage characteristics to determine cycle aging state through Dynamic Time Warping (DTW), then integrates this information with advanced filtering techniques for precise SOC estimation. This methodology specifically addresses the lifecycle challenges faced by energy storage cells in practical applications.
The foundation of our approach begins with establishing an accurate electrical model for energy storage cells. We selected the second-order equivalent circuit model due to its balance between computational efficiency and representation accuracy. This model effectively captures the dynamic behavior of energy storage cells while remaining suitable for real-time implementation. The model structure comprises a voltage source representing the open-circuit voltage, an ohmic resistance accounting for immediate voltage drops, and two RC parallel networks modeling polarization effects.
The governing equations for the second-order model are derived from fundamental circuit theory and electrochemical principles. The continuous-time representation follows these relationships:
$$ U(t) = E_0 – I(t) \times R_{\Omega} – U_P(t) – U_C(t) $$
$$ \frac{dU_P(t)}{dt} = -\frac{1}{R_P C_P} U_P(t) + \frac{1}{C_P} I(t) $$
$$ \frac{dU_C(t)}{dt} = -\frac{1}{R_C C_C} U_C(t) + \frac{1}{C_C} I(t) $$
$$ \frac{dSOC(t)}{dt} = -\frac{\eta I(t)}{C_q} $$
where \( E_0 \) represents the open-circuit voltage, \( R_{\Omega} \) denotes the ohmic resistance, \( R_P \) and \( C_P \) form the electrochemical polarization RC pair, \( R_C \) and \( C_C \) constitute the concentration polarization RC pair, \( U_P(t) \) and \( U_C(t) \) are the corresponding polarization voltages, \( \eta \) indicates the coulombic efficiency, and \( C_q \) symbolizes the capacity of the energy storage cells.
For practical implementation in digital systems, we discretize these equations using a zero-order hold approximation with sampling period \( \Delta t \):
$$ U_P(t) = e^{-\frac{\Delta t}{\tau_P}} U_P(t-1) + R_P (1 – e^{-\frac{\Delta t}{\tau_P}}) I(t-1) $$
$$ U_C(t) = e^{-\frac{\Delta t}{\tau_C}} U_C(t-1) + R_C (1 – e^{-\frac{\Delta t}{\tau_C}}) I(t-1) $$
$$ SOC(t) = SOC(t-1) – \frac{\eta \Delta t I(t-1)}{C_q} $$
where \( \tau_P = R_P C_P \) and \( \tau_C = R_C C_C \) represent the time constants of the polarization networks.
Parameter identification represents a critical step in ensuring model accuracy. We employ the recursive least squares method with forgetting factor to adaptively track parameter variations in energy storage cells. The transfer function derived from the equivalent circuit model in the s-domain is:
$$ G(s) = \frac{E_0(s) – U(s)}{I(s)} = R_{\Omega} + \frac{R_P}{1 + \tau_P s} + \frac{R_C}{1 + \tau_C s} $$
Applying bilinear transformation with sampling time T = 1 second yields the discrete-time transfer function:
$$ H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}} $$
The corresponding difference equation becomes:
$$ U(t) = -a_1 U(t-1) – a_2 U(t-2) + b_0 I(t) + b_1 I(t-1) + b_2 I(t-2) + (1 + a_1 + a_2) E_0(t) $$
We redefine coefficients for clarity:
$$ U(t) = (1 – k_1 – k_2) E_0(t) + k_1 U(t-1) + k_2 U(t-2) + k_3 I(t) + k_4 I(t-1) + k_5 I(t-2) $$
The recursive estimation updates parameters using:
$$ \hat{\theta}(t) = \hat{\theta}(t-1) + K(t) [U(t) – \phi^T(t) \hat{\theta}(t-1)] $$
$$ K(t) = \frac{P(t-1) \phi(t)}{\lambda + \phi^T(t) P(t-1) \phi(t)} $$
$$ P(t) = \frac{1}{\lambda} [P(t-1) – K(t) \phi^T(t) P(t-1)] $$
where \( \hat{\theta} = [k_1, k_2, k_3, k_4, k_5]^T \) represents the parameter vector, \( \phi(t) = [U(t-1), U(t-2), I(t), I(t-1), I(t-2)]^T \) is the regression vector, and \( \lambda \) denotes the forgetting factor typically set between 0.95 and 0.99 for energy storage cells.
The physical parameters are recovered from the estimated coefficients through the following relationships:
$$ b_1 = \frac{k_3 – k_4 + k_5}{1 + k_1 – k_2} $$
$$ b_2 = \frac{1 + k_1 – k_2}{4(1 – k_1 – k_2)} $$
$$ b_3 = \frac{1 + k_2}{1 – k_1 – k_2} $$
$$ b_4 = \frac{k_3 + k_4 + k_5}{1 – k_1 – k_2} $$
$$ b_5 = \frac{k_3 – k_5}{1 – k_1 – k_2} $$
$$ d_1 = \frac{b_3 + \sqrt{b_3^2 – 4b_2}}{2} $$
$$ d_2 = \frac{b_3 – \sqrt{b_3^2 – 4b_2}}{2} $$
Finally, the equivalent circuit parameters are obtained as:
$$ R_{\Omega} = b_1 $$
$$ R_P = \frac{b_5 – R_{\Omega} d_1 – b_4 d_2}{d_1 – d_2} $$
$$ R_C = b_4 – R_{\Omega} – R_P $$
$$ C_P = \frac{d_1}{R_P} $$
$$ C_C = \frac{d_2}{R_C} $$
To validate the parameter identification process, we conducted constant power discharge tests on commercial energy storage cells with nominal capacity of 100 Ah. The discharge power was maintained at 0.5P (50A equivalent current) until reaching the cutoff voltage. The identification results demonstrate excellent convergence with root mean square error below 2 mV across the entire discharge range.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Ohmic Resistance | \( R_{\Omega} \) | 0.0082 | Ω |
| Polarization Resistance | \( R_P \) | 0.0035 | Ω |
| Polarization Capacitance | \( C_P \) | 1245 | F |
| Concentration Resistance | \( R_C \) | 0.0018 | Ω |
| Concentration Capacitance | \( C_C \) | 2850 | F |
| Time Constant 1 | \( \tau_P \) | 4.36 | s |
| Time Constant 2 | \( \tau_C \) | 5.13 | s |
The core innovation of our approach lies in utilizing the Dynamic Time Warping algorithm to determine the aging state of energy storage cells based on their discharge voltage profiles. As energy storage cells undergo repeated cycles, their voltage characteristics exhibit systematic shifts that correlate with degradation. The DTW algorithm measures similarity between sequences by computing an optimal match between two given sequences, making it ideal for comparing discharge curves at different aging stages.
Given two voltage sequences \( V = [V(1), V(2), …, V(m)] \) from a test cycle and \( V_r = [V_r(1), V_r(2), …, V_r(n)] \) from a reference cycle, we construct a cost matrix \( C \) of size \( m \times n \) where each element represents the local distance between sequence points:
$$ C(i,j) = |V(i) – V_r(j)| $$
The cumulative distance matrix \( D \) is computed recursively:
$$ D(i,j) = C(i,j) + \min \begin{cases} D(i-1,j) \\ D(i,j-1) \\ D(i-1,j-1) \end{cases} $$
with boundary conditions:
$$ D(1,1) = C(1,1) $$
$$ D(i,1) = \sum_{k=1}^{i} C(k,1) $$
$$ D(1,j) = \sum_{k=1}^{j} C(1,k) $$
The DTW distance is defined as \( D(m,n) \), with smaller values indicating higher similarity. For each test discharge curve, we compute DTW distances against a comprehensive database containing reference curves from energy storage cells at known cycle counts, selecting the reference with minimum distance as the matched aging state.
To establish the reference database, we cycled multiple identical energy storage cells through constant power discharge protocols at 25°C, recording complete voltage profiles at regular intervals. The database encompasses cycles from initial activation through 1000 cycles, covering the typical operational lifespan of energy storage cells in stationary applications.
| Actual Cycle Count | Identified Cycle Count | Minimum DTW Distance | Mean Distance to Other Cycles |
|---|---|---|---|
| 20 | 20 | 0.0245 | 0.1583 |
| 200 | 200 | 0.0318 | 0.2347 |
| 800 | 800 | 0.0452 | 0.3829 |
The relationship between cycle count and capacity degradation follows a characteristic pattern for lithium-ion energy storage cells. Through extensive testing, we established an empirical model describing capacity fade:
$$ C_q(N) = C_{q0} – \alpha \cdot \ln(1 + \beta N) $$
where \( C_q(N) \) represents capacity at cycle N, \( C_{q0} \) denotes initial capacity, and \( \alpha \), \( \beta \) are degradation parameters specific to the chemistry and operating conditions of the energy storage cells. For our test cells, parameter values were \( \alpha = 0.85 \) Ah and \( \beta = 0.012 \).
With the aging state determined through DTW and corresponding capacity established, we implement the Unscented Kalman Filter for robust SOC estimation. The UKF addresses nonlinearities in the system without requiring linearization, providing superior performance compared to extended Kalman filters for energy storage cells. Our state vector comprises SOC and both polarization voltages:
$$ \mathbf{x} = [SOC, U_P, U_C]^T $$
The state transition equations are:
$$ SOC_k = SOC_{k-1} – \frac{\eta \Delta t I_{k-1}}{C_q(N)} $$
$$ U_{P,k} = e^{-\frac{\Delta t}{\tau_P}} U_{P,k-1} + R_P (1 – e^{-\frac{\Delta t}{\tau_P}}) I_{k-1} $$
$$ U_{C,k} = e^{-\frac{\Delta t}{\tau_C}} U_{C,k-1} + R_C (1 – e^{-\frac{\Delta t}{\tau_C}}) I_{k-1} $$
The measurement equation models terminal voltage:
$$ U_k = E_0(SOC_k) – I_k R_{\Omega} – U_{P,k} – U_{C,k} $$
The open-circuit voltage \( E_0(SOC) \) relationship is characterized through experimental testing and represented as a ninth-order polynomial:
$$ E_0(SOC) = \sum_{i=0}^{9} p_i SOC^i $$
with coefficients determined through least-squares fitting to experimental data.
The UKF algorithm proceeds through prediction and update steps. First, we generate sigma points around the current state estimate:
$$ \mathcal{X}_{k-1} = [\hat{\mathbf{x}}_{k-1}, \hat{\mathbf{x}}_{k-1} + \gamma \sqrt{\mathbf{P}_{k-1}}, \hat{\mathbf{x}}_{k-1} – \gamma \sqrt{\mathbf{P}_{k-1}}] $$
where \( \gamma = \sqrt{L + \lambda} \), L is the state dimension, and \( \lambda \) is a scaling parameter. The transformed sigma points are propagated through the state transition function:
$$ \mathcal{X}_{k|k-1} = f(\mathcal{X}_{k-1}, I_{k-1}) $$
The predicted state mean and covariance are computed as:
$$ \hat{\mathbf{x}}_{k|k-1} = \sum_{i=0}^{2L} W_i^{(m)} \mathcal{X}_{i,k|k-1} $$
$$ \mathbf{P}_{k|k-1} = \sum_{i=0}^{2L} W_i^{(c)} (\mathcal{X}_{i,k|k-1} – \hat{\mathbf{x}}_{k|k-1}) (\mathcal{X}_{i,k|k-1} – \hat{\mathbf{x}}_{k|k-1})^T + \mathbf{Q} $$
where \( W_i^{(m)} \) and \( W_i^{(c)} \) are weights for mean and covariance calculations, and \( \mathbf{Q} \) represents process noise covariance.
For the update step, we transform the predicted sigma points through the measurement function:
$$ \mathcal{Y}_{k|k-1} = h(\mathcal{X}_{k|k-1}, I_k) $$
The predicted measurement mean and covariance are:
$$ \hat{y}_{k|k-1} = \sum_{i=0}^{2L} W_i^{(m)} \mathcal{Y}_{i,k|k-1} $$
$$ \mathbf{P}_{yy} = \sum_{i=0}^{2L} W_i^{(c)} (\mathcal{Y}_{i,k|k-1} – \hat{y}_{k|k-1}) (\mathcal{Y}_{i,k|k-1} – \hat{y}_{k|k-1})^T + \mathbf{R} $$
$$ \mathbf{P}_{xy} = \sum_{i=0}^{2L} W_i^{(c)} (\mathcal{X}_{i,k|k-1} – \hat{\mathbf{x}}_{k|k-1}) (\mathcal{Y}_{i,k|k-1} – \hat{y}_{k|k-1})^T $$
where \( \mathbf{R} \) represents measurement noise covariance. The Kalman gain and final update are:
$$ \mathbf{K} = \mathbf{P}_{xy} \mathbf{P}_{yy}^{-1} $$
$$ \hat{\mathbf{x}}_k = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K} (y_k – \hat{y}_{k|k-1}) $$
$$ \mathbf{P}_k = \mathbf{P}_{k|k-1} – \mathbf{K} \mathbf{P}_{yy} \mathbf{K}^T $$
We evaluated our DTW-UKF approach under three distinct aging stages corresponding to 20, 200, and 800 cycles. At each stage, we conducted constant power discharge tests while comparing SOC estimation results against reference values obtained through coulomb counting with periodic capacity calibration. The initial SOC was intentionally offset by 20% to assess correction capability.
| Method | 20 Cycles RMSE | 200 Cycles RMSE | 800 Cycles RMSE | Maximum Error | Average RMSE |
|---|---|---|---|---|---|
| Ah Integration | 1.82% | 3.76% | 5.48% | 9.85% | 3.69% |
| UKF Only | 1.63% | 2.34% | 3.12% | 6.92% | 2.36% |
| DTW-UKF | 0.87% | 1.21% | 1.43% | 3.76% | 1.17% |
The results demonstrate significant improvement in SOC estimation accuracy throughout the lifecycle of energy storage cells. Our DTW-UKF method maintains maximum errors below 4% even at advanced aging stages, while traditional methods exhibit substantially larger errors. The adaptive capacity adjustment based on DTW identification proves crucial for sustaining accuracy as energy storage cells degrade.
Further analysis reveals distinct advantages across different operational regions. During high SOC regions (above 80%), all methods perform reasonably well, with errors typically below 2%. However, in the critical mid-to-low SOC range (20-80%), where energy storage cells most frequently operate, our method shows particular superiority. The table below details performance across SOC ranges for 800-cycle aged energy storage cells.
| SOC Range | Ah Integration Error | UKF Only Error | DTW-UKF Error |
|---|---|---|---|
| 80-100% | 1.95% | 1.23% | 0.76% |
| 50-80% | 3.84% | 2.17% | 1.08% |
| 20-50% | 5.92% | 3.45% | 1.52% |
| 0-20% | 7.63% | 4.28% | 1.87% |
The computational requirements of our approach remain manageable for practical implementation. The DTW algorithm operates on complete discharge cycles, which typically occur over several hours, allowing ample time for computation between cycles. The UKF executes in real-time with minimal computational overhead, making the combined approach suitable for embedded systems in energy storage applications.
Several factors contribute to the enhanced performance of our methodology. The DTW-based cycle identification effectively captures the subtle voltage profile changes that correlate with degradation in energy storage cells. By accurately determining the aging state, we establish the appropriate capacity value for SOC calculation, eliminating a primary source of error in traditional approaches. The UKF then provides robust estimation against measurement noise and model uncertainties, with the unscented transformation properly handling the nonlinear OCV-SOC relationship.
The consistency of constant power discharge profiles in energy storage systems creates ideal conditions for DTW application. Unlike automotive applications with highly variable load profiles, energy storage systems typically undergo predictable charge-discharge patterns, enabling reliable cycle state identification through voltage curve comparison. This characteristic makes our approach particularly well-suited for stationary energy storage cells.
Implementation considerations include maintaining a comprehensive reference database specific to the chemistry and format of energy storage cells deployed. The database should encompass the entire expected lifespan under typical operating conditions. Regular updates to the reference database can further enhance long-term accuracy as more field data becomes available.
Our approach demonstrates scalability across different capacities and chemistries of energy storage cells. The model parameters and reference curves adapt to specific cell characteristics, while the core algorithm remains unchanged. This flexibility ensures broad applicability across diverse energy storage installations.
In conclusion, we have developed and validated a comprehensive framework for accurate SOC estimation in energy storage cells throughout their complete lifecycle. The integration of DTW for cycle state identification with UKF for real-time estimation addresses the fundamental challenge of capacity fade in aging energy storage cells. Experimental results confirm superior performance compared to conventional methods, with average accuracy improvements exceeding 47% while maintaining computational efficiency suitable for practical implementation. This advancement contributes significantly to enhancing the reliability, safety, and economic viability of energy storage systems employing lithium-ion technology.
Future work will focus on extending this methodology to accommodate varying operational conditions, including temperature effects and different discharge rates. Additionally, we plan to investigate hybrid approaches combining physical models with machine learning techniques to further enhance estimation accuracy and robustness for next-generation energy storage cells.