In the era of rapid renewable energy development and smart grid construction, the demand for stability and sustainability in energy storage systems has significantly increased. Among various energy storage technologies, lithium-ion batteries have garnered widespread attention due to their economic efficiency and favorable thermal stability. However, the real-time measurement of internal electrochemical states in lithium-ion batteries is challenging, making it difficult to accurately estimate the State of Charge (SOC). For cell energy storage systems, which often undergo deep-cycle charge-discharge operations, battery aging over repeated cycles affects SOC estimation accuracy over long time scales. This complexity increases the operational difficulty of battery control systems, reducing overall economic efficiency and safety, thereby hindering the further advancement of cell energy storage systems. In this paper, we address this issue by proposing a novel algorithm for lifecycle SOC estimation under constant power discharge conditions, leveraging the unique operational characteristics of cell energy storage systems.
The SOC of a battery is a critical parameter that reflects its remaining capacity, directly impacting the performance and lifespan of cell energy storage systems. Accurate SOC estimation is essential for optimizing energy management, preventing overcharge or over-discharge, and ensuring safe operation. Traditional methods, such as the Ampere-hour (Ah) integration method, suffer from cumulative errors due to measurement inaccuracies and initial value uncertainties. Moreover, as cell energy storage systems age, their capacity degrades, leading to increased SOC estimation errors if not properly accounted for. Therefore, developing algorithms that can adapt to aging effects is crucial for the reliable operation of cell energy storage systems throughout their lifecycle.
To tackle these challenges, we focus on constant power discharge conditions, which are common in cell energy storage systems. These conditions provide consistent operational data that can be leveraged for state estimation. Our approach combines a second-order Resistor-Capacitance (RC) equivalent circuit model for battery modeling, a Dynamic Time Warping (DTW) algorithm for cycle aging state identification, and an Unscented Kalman Filter (UKF) algorithm for SOC estimation. This integrated method, referred to as the DTW-UKF algorithm, aims to achieve high-precision SOC estimation across the entire lifecycle of cell energy storage systems. By utilizing full discharge voltage data as feature identification data, we can dynamically track aging effects and adjust SOC calculations accordingly.
The structure of this paper is as follows: First, we detail the development of the second-order RC model and parameter identification using the recursive least squares method with a forgetting factor. Next, we explain the DTW algorithm for determining the cycle aging state based on discharge voltage curves. Then, we derive the control equations for the RC model and integrate them with the aging state to obtain battery capacity, followed by SOC estimation using the UKF algorithm. Finally, we validate the proposed algorithm under different aging stages (20, 200, and 800 cycles) and compare the results with traditional methods. Throughout this discussion, we emphasize the application to cell energy storage systems, highlighting how our approach enhances their performance and reliability.
The importance of this work lies in its potential to improve the economic and safety aspects of cell energy storage systems. By enabling accurate SOC estimation over the battery’s lifecycle, we can optimize charging and discharging strategies, extend battery lifespan, and reduce maintenance costs. This is particularly relevant for large-scale deployments of cell energy storage systems in renewable energy integration, grid stabilization, and backup power applications. In the following sections, we delve into the technical details, supported by mathematical formulations, tables, and experimental results.
Battery Model Development and Parameter Identification
For cell energy storage systems, an accurate battery model is fundamental to SOC estimation. Among various modeling approaches, equivalent circuit models are widely used due to their simplicity and ability to capture external characteristics with low computational requirements. Given the need for potential microcontroller implementation in practical cell energy storage systems, we adopt a second-order RC equivalent circuit model. This model balances complexity and accuracy, effectively representing battery nonlinearities such as polarization effects.
The second-order RC model consists of an open-circuit voltage source (E0), an ohmic resistor (RΩ), and two RC parallel networks representing electrochemical polarization (RP, CP) and concentration polarization (RC, CC). The output voltage U(t) and current I(t) are measurable, while other parameters require identification. The governing equations of the model are derived from Kirchhoff’s laws and SOC dynamics. The discrete-time state-space representation is as follows:
$$ U(t) = E_0 – I(t) \times R_{\Omega} – U_P(t) – U_C(t) $$
$$ U_P(t) = e^{-\frac{\Delta t}{\tau_P}} \times U_P(t-1) + (1 – e^{-\frac{\Delta t}{\tau_P}}) R_P \times I(t-1) $$
$$ U_C(t) = e^{-\frac{\Delta t}{\tau_C}} \times U_C(t-1) + (1 – e^{-\frac{\Delta t}{\tau_C}}) R_C \times I(t-1) $$
$$ SOC(t) = SOC(t_0) – \frac{\eta \int_{t_0}^{t} I(\tau) d\tau}{C_q} $$
where Δt is the sampling period, τP = RPCP and τC = RCCC are time constants, η is the charge-discharge efficiency, and Cq is the battery capacity. For cell energy storage systems under constant power discharge, these equations form the basis for SOC estimation.
To identify the model parameters (RΩ, RP, CP, RC, CC), we use the recursive least squares method with a forgetting factor. This approach minimizes the error between measured and simulated voltages while adapting to time-varying parameters. The transfer function 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 z-transform with a sampling period of 1 second, we obtain the difference equation:
$$ U(t) = (1 – k_1 – k_2) E_0(t) + k_1 U(t-1) + k_2 U(t-2) + k_3 I(t) + k_4 I(t-1) + k_5 I(t-2) $$
where k1 to k5 are coefficients identified through iterative least squares. The relationships between these coefficients and the physical parameters are:
$$ b_1 = \frac{k_3 – k_4 + k_5}{1 + k_1 – k_2}, \quad b_2 = \frac{1 + k_1 – k_2}{4(1 – k_1 – k_2)}, \quad b_3 = \frac{1 + k_2}{1 – k_1 – k_2}, \quad b_4 = \frac{k_3 + k_4 + k_5}{1 – k_1 – k_2}, \quad b_5 = \frac{k_3 – k_5}{1 – k_1 – k_2} $$
$$ d_1 = \frac{b_3 + \sqrt{b_2^3 – 4b_2^2}}{2}, \quad d_2 = \frac{b_3 – \sqrt{b_2^3 – 4b_2^2}}{2} $$
$$ R_{\Omega} = b_1, \quad R_P = \frac{b_5 – R_{\Omega} d_1 – b_4 d_2}{d_1 – d_2}, \quad R_C = b_4 – R_{\Omega} – R_P, \quad C_P = \frac{d_1}{R_P}, \quad C_C = \frac{d_2}{R_C} $$
We conducted parameter identification under constant power discharge conditions typical for cell energy storage systems. The current and voltage data were recorded, and the identification results showed stable convergence after initial iterations. The root mean square error (RMSE) between simulated and measured voltages was 1.9 mV, indicating high model accuracy. This robust parameter identification process is crucial for reliable SOC estimation in cell energy storage systems.
To summarize the parameter identification results, we present the following table showing average values for key parameters across multiple cycles in a cell energy storage system:
| Parameter | Symbol | Average Value | Unit |
|---|---|---|---|
| Ohmic Resistance | RΩ | 0.025 | Ω |
| Polarization Resistance (Electrochemical) | RP | 0.015 | Ω |
| Polarization Capacitance (Electrochemical) | CP | 1200 | F |
| Polarization Resistance (Concentration) | RC | 0.010 | Ω |
| Polarization Capacitance (Concentration) | CC | 3000 | F |
| Time Constant τP | τP | 18 | s |
| Time Constant τC | τC | 30 | s |
These parameters form the foundation for the SOC estimation algorithm. However, as cell energy storage systems age, these parameters may change. Therefore, integrating aging state identification is essential for lifecycle management.
Dynamic Time Warping Algorithm for Cycle Aging State Identification
In cell energy storage systems, batteries undergo repetitive charge-discharge cycles, leading to gradual capacity fade and voltage curve shifts. To account for aging effects in SOC estimation, we need to accurately determine the cycle aging state. Given the relatively consistent operational patterns in cell energy storage systems, such as constant power discharge, we can leverage historical voltage data for this purpose. The Dynamic Time Warping (DTW) algorithm is particularly suitable for comparing time series data with non-linear distortions, making it ideal for identifying similarities between discharge voltage curves at different aging stages.
The DTW algorithm calculates the optimal alignment between two sequences by minimizing the cumulative distance between them. For cell energy storage systems, we define the measured discharge voltage sequence as V = [V(1), V(2), …, V(n)] and a reference sequence from a database as Vm = [Vm(1), Vm(2), …, Vm(m)]. The algorithm constructs a distance matrix D of size m × n, initialized with infinity values, and computes the DTW distance as follows:
$$ D(0,0) = 0 $$
$$ D(i,1) = |V(i) – V_m(1)| + D(i-1,1) \quad \text{for} \quad i = 1,2,…,n $$
$$ D(1,j) = |V(1) – V_m(j)| + D(1,j-1) \quad \text{for} \quad j = 1,2,…,m $$
For other elements, the recurrence relation is:
$$ D(i,j) = |V(i) – V_m(j)| + \min \left[ D(i-1,j), D(i,j-1), D(i-1,j-1) \right] $$
The final DTW distance, D(m,n), represents the similarity between the sequences. Lower distances indicate higher similarity, allowing us to match the measured voltage curve with the closest reference curve in the database, thereby estimating the cycle aging state (e.g., equivalent cycle count).
To validate the DTW algorithm for cell energy storage systems, we built a database of discharge voltage curves from 0 to 1000 cycles under constant power discharge at 25°C. The battery capacity degraded to 92% of its initial value after 1000 cycles. We then tested the algorithm with voltage data from a separate battery of the same type at 20, 200, and 800 cycles. The DTW distances were computed against the database, and the minimum distances consistently occurred at the corresponding cycle counts, confirming the algorithm’s accuracy. For instance, at 200 cycles, the DTW distance was lowest near the 200-cycle reference, with adjacent cycles showing slightly higher distances, demonstrating sensitivity to aging.
The effectiveness of the DTW algorithm in cell energy storage systems can be summarized using the following table, which shows DTW distances for test cycles against reference cycles:
| Test Cycle Count | Reference Cycle Count (Minimum Distance Highlighted) | DTW Distance (Normalized) |
|---|---|---|
| 20 | 10 | 0.15 |
| 20 | 0.02 | |
| 30 | 0.18 | |
| 200 | 190 | 0.12 |
| 200 | 0.03 | |
| 210 | 0.14 | |
| 800 | 790 | 0.25 |
| 800 | 0.05 | |
| 810 | 0.28 |
This table illustrates that the DTW algorithm reliably identifies the cycle aging state by finding the reference with the minimum distance. This capability is vital for adapting SOC estimation to aging in cell energy storage systems. Once the cycle aging state is determined, we can correlate it with battery capacity fade using empirical data. For example, capacity as a function of cycle count can be modeled as:
$$ C_q(N) = C_{q0} \times \left(1 – \alpha N^{\beta}\right) $$
where Cq0 is the initial capacity, N is the cycle count, and α and β are degradation parameters specific to the cell energy storage system. By updating Cq based on the DTW-derived cycle count, we can improve SOC estimation accuracy over the battery’s lifecycle.
SOC Estimation Using the Unscented Kalman Filter Algorithm
With the battery model parameters identified and the cycle aging state determined via DTW, we now focus on SOC estimation. The Unscented Kalman Filter (UKF) is a powerful non-linear estimation algorithm that overcomes limitations of the standard Kalman Filter by using unscented transformations to approximate state distributions. For cell energy storage systems, the UKF provides robust SOC estimates even in the presence of measurement noise and model uncertainties.
We define the state vector as x(t) = [SOC(t), UP(t), UC(t)]T, the control input as u(t) = I(t), and the measurement as y(t) = U(t). The state-space model derived from the second-order RC equations is:
$$ \mathbf{x}(t) = \mathbf{f}(\mathbf{x}(t-1), u(t-1)) + \mathbf{\omega}(t-1) $$
$$ y(t) = g(\mathbf{x}(t-1), u(t-1)) + \nu(t) $$
where ω(t-1) and ν(t) are process and measurement noise, assumed to be Gaussian with zero mean and covariances Q and R, respectively. The function f(·) represents the state transition:
$$ \begin{bmatrix} SOC(t) \\ U_P(t) \\ U_C(t) \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & e^{-\frac{1}{\tau_P}} & 0 \\ 0 & 0 & e^{-\frac{1}{\tau_C}} \end{bmatrix} \begin{bmatrix} SOC(t-1) \\ U_P(t-1) \\ U_C(t-1) \end{bmatrix} + \begin{bmatrix} -\frac{\eta \Delta t}{C_q} \\ R_P (1 – e^{-\frac{1}{\tau_P}}) \\ R_C (1 – e^{-\frac{1}{\tau_C}}) \end{bmatrix} I(t-1) $$
and g(·) is the observation function:
$$ y(t) = E_0(SOC(t)) – I(t) R_{\Omega} – U_P(t) – U_C(t) $$
Here, E0(SOC(t)) is the open-circuit voltage (OCV) as a function of SOC, which we obtain from experimental data. For cell energy storage systems, we perform a Hybrid Pulse Power Characterization (HPPC) test to measure OCV at different SOC points. The data is fitted with a polynomial equation, such as:
$$ E_0(SOC) = a_0 + a_1 SOC + a_2 SOC^2 + a_3 SOC^3 + a_4 SOC^4 $$
where coefficients a0 to a4 are determined via least squares fitting. In our tests, the fitting achieved a correlation coefficient R² = 0.99, ensuring accurate OCV representation for the UKF algorithm.
The UKF algorithm proceeds through prediction and update steps. First, sigma points are generated around the current state estimate to capture its mean and covariance. These points are propagated through the non-linear functions f(·) and g(·), and the transformed statistics are used to compute the predicted state and measurement. Then, the Kalman gain is calculated to update the state estimate based on the actual measurement. This process is repeated recursively, providing real-time SOC estimates for cell energy storage systems.
A key advantage of our DTW-UKF approach is the integration of aging awareness. The capacity Cq in the state transition equation is not fixed but updated based on the DTW-derived cycle count. This allows the UKF to adapt to capacity fade, thereby maintaining estimation accuracy over the lifecycle of the cell energy storage system. We initialize the SOC with a deliberate error (e.g., 20% deviation) to simulate practical scenarios where initial SOC is unknown, and evaluate the algorithm’s convergence.
To quantify the performance, we compare the DTW-UKF algorithm against two baseline methods: the Ah integration method and a standalone UKF algorithm without aging adaptation. The Ah integration method computes SOC as:
$$ SOC(t) = SOC(t_0) – \frac{\eta \sum_{k=t_0}^{t} I(k) \Delta t}{C_{q,\text{nominal}}} $$
where Cq,nominal is the nominal capacity, which does not account for aging. The standalone UKF uses a fixed capacity, typically the initial value. In contrast, our DTW-UKF algorithm dynamically adjusts Cq using the DTW output, enabling lifecycle SOC estimation.
We validate the algorithms under constant power discharge at 20, 200, and 800 cycles, representing early, mid, and late lifecycle stages of a cell energy storage system. The SOC estimation errors are computed relative to a reference SOC obtained from high-precision current integration with known initial conditions. The results demonstrate that DTW-UKF consistently outperforms the other methods, especially as aging progresses.
The following table summarizes the Root Mean Square Error (RMSE) of SOC estimation for each method across different cycle counts in cell energy storage systems:
| Cycle Count | Ah Integration Method RMSE (%) | Standalone UKF RMSE (%) | DTW-UKF Algorithm RMSE (%) |
|---|---|---|---|
| 20 | 1.8 | 1.6 | 0.9 |
| 200 | 3.8 | 2.3 | 1.2 |
| 800 | 5.5 | 3.1 | 1.4 |
| Average | 3.7 | 2.3 | 1.2 |
From this table, we observe that the DTW-UKF algorithm reduces the average SOC estimation error by approximately 67.6% compared to the Ah integration method and 47.8% compared to the standalone UKF. This significant improvement highlights the importance of incorporating aging state identification in SOC estimation for cell energy storage systems. Moreover, the DTW-UKF algorithm maintains low errors across all cycle counts, with a maximum error below 4% and average error below 2%, ensuring reliable performance throughout the battery’s lifecycle.
To further illustrate the SOC estimation process, we can express the UKF algorithm steps mathematically. Let the state estimate mean be \(\hat{\mathbf{x}}_{t-1|t-1}\) and covariance Pt-1|t-1. The sigma points χi are generated as:
$$ \chi_{i} = \hat{\mathbf{x}}_{t-1|t-1} \pm \sqrt{(n+\lambda) \mathbf{P}_{t-1|t-1}}_i \quad \text{for} \quad i=1,…,2n $$
where n is the state dimension (3 in our case), and λ is a scaling parameter. These sigma points are propagated through f(·):
$$ \chi_{i}^* = \mathbf{f}(\chi_{i}, u(t-1)) $$
The predicted state mean and covariance are:
$$ \hat{\mathbf{x}}_{t|t-1} = \sum_{i=0}^{2n} w_i \chi_{i}^* $$
$$ \mathbf{P}_{t|t-1} = \sum_{i=0}^{2n} w_i (\chi_{i}^* – \hat{\mathbf{x}}_{t|t-1})(\chi_{i}^* – \hat{\mathbf{x}}_{t|t-1})^T + \mathbf{Q} $$
where wi are weights. Similarly, the measurement sigma points are computed via g(·), and the predicted measurement mean and covariance are obtained. The Kalman gain Kt is then calculated, and the state is updated with the actual measurement y(t). This iterative process, combined with DTW-based capacity updates, forms the core of our lifecycle SOC estimation framework for cell energy storage systems.
Experimental Validation and Discussion
To comprehensively validate our DTW-UKF algorithm for cell energy storage systems, we conducted extensive experiments under controlled conditions. The test setup involved lithium-ion batteries typically used in cell energy storage systems, subjected to constant power discharge at 0.5P (half of rated power) in a 25°C environment. We collected voltage, current, and temperature data over 1000 cycles, simulating long-term operation. The battery capacity was monitored periodically, showing a degradation trend from 100% to 92% of initial capacity after 1000 cycles, consistent with typical aging in cell energy storage systems.
For SOC estimation validation, we selected three representative aging stages: 20 cycles (early life), 200 cycles (mid life), and 800 cycles (late life). At each stage, we ran constant power discharge tests and applied the DTW-UKF algorithm, along with the baseline methods. The initial SOC was intentionally set with a 20% error to evaluate convergence. The results, as shown in Table 3, confirm the superiority of our approach. Additionally, we analyzed the SOC estimation error profiles over time. For example, at 800 cycles, the Ah integration method exhibited a steadily increasing error, reaching nearly 10% by the end of discharge, while the standalone UKF error peaked around 7%. In contrast, the DTW-UKF algorithm kept errors within 2% throughout, demonstrating its robustness against aging effects.
The DTW algorithm’s performance in identifying cycle aging states was also verified. We constructed a reference database with voltage curves from 0 to 1000 cycles at 10-cycle intervals. When presented with test data from unseen cycles, the algorithm accurately matched them to the closest reference, with DTW distances monotonically increasing as the cycle difference grew. This reliability ensures that capacity updates are timely and accurate, which is critical for SOC estimation in aging cell energy storage systems.
We further explored the sensitivity of our algorithm to variations in operating conditions, such as temperature and discharge power. While this study focuses on constant power discharge at 25°C, cell energy storage systems may experience fluctuations. To address this, we can extend the DTW database to include multiple temperatures and power levels, or incorporate temperature compensation into the RC model. However, even under nominal conditions, our algorithm proves effective for lifecycle management.
The computational complexity of the DTW-UKF algorithm is manageable for real-time implementation in cell energy storage systems. The DTW algorithm, though O(mn) in time, operates on a per-cycle basis rather than per time step, reducing overhead. The UKF algorithm has O(n³) complexity due to matrix operations, but with n=3, it is efficient for embedded systems. Thus, the combined algorithm is feasible for deployment in battery management systems (BMS) for cell energy storage systems.
To provide deeper insight, we present a formula for the overall SOC estimation error as a function of cycle count N, incorporating both model inaccuracies and aging effects:
$$ \text{Error}_{\text{SOC}}(N) = \sqrt{ \left( \frac{\Delta C_q(N)}{C_q(N)} \right)^2 + \sigma_{\text{model}}^2 + \sigma_{\text{measurement}}^2 } $$
where ΔCq(N) is the capacity estimation error from DTW, σmodel is the RC model error, and σmeasurement is the sensor noise. By minimizing these terms—through accurate DTW matching, precise parameter identification, and robust filtering—we achieve low errors across the lifecycle of cell energy storage systems.
In practice, the integration of DTW and UKF requires careful synchronization. We propose a dual-time-scale strategy: DTW runs periodically (e.g., after each full discharge) to update the cycle count and capacity, while UKF runs at each sampling interval (e.g., 1 second) for real-time SOC estimation. This decouples the slow aging dynamics from fast SOC changes, optimizing computational resources. For cell energy storage systems with regular operational patterns, this approach is highly effective.
We also considered the impact of battery variability within cell energy storage systems. In large-scale deployments, batteries may age unevenly due to manufacturing tolerances or environmental differences. Our algorithm can be applied individually to each battery module, allowing for personalized SOC estimation. The DTW database can be generalized across batteries of the same type, or calibrated per unit during initial cycles. This flexibility enhances the scalability of our method for diverse cell energy storage systems.
Conclusion
In this paper, we have presented a comprehensive framework for lifecycle SOC estimation in cell energy storage systems under constant power discharge conditions. By leveraging the operational consistency of cell energy storage systems, we developed a DTW-UKF algorithm that integrates cycle aging state identification with advanced filtering for accurate SOC estimation. The second-order RC model provides a robust foundation, with parameter identification achieved through recursive least squares. The DTW algorithm effectively determines the cycle aging state by comparing discharge voltage curves, enabling dynamic capacity updates. The UKF algorithm then estimates SOC in real-time, compensating for measurement noise and model uncertainties.
Experimental validation across 20, 200, and 800 cycles demonstrated that our DTW-UKF algorithm achieves an average SOC estimation error of 1.2%, with maximum errors below 4%. Compared to the Ah integration method and standalone UKF, it improves average relative accuracy by approximately 67.6% and 47.8%, respectively. These results underscore the algorithm’s capability to maintain high precision throughout the battery’s lifecycle, addressing a critical challenge in cell energy storage systems management.
The implications of this work are significant for the advancement of cell energy storage systems. Accurate lifecycle SOC estimation enhances energy management, prolongs battery lifespan, and improves system safety and economics. As renewable energy integration expands, reliable cell energy storage systems will play a pivotal role, and our algorithm contributes to their optimal operation. Future work may extend the approach to variable power profiles, temperature effects, and multi-battery systems, further solidifying its applicability. Ultimately, by bridging aging awareness with real-time estimation, we pave the way for smarter and more sustainable cell energy storage systems.
In summary, the DTW-UKF algorithm represents a practical and effective solution for SOC estimation in aging cell energy storage systems. Its combination of model-based and data-driven techniques offers a balanced approach that can be implemented in real-world BMS hardware. We encourage further research and adoption of such methods to unlock the full potential of cell energy storage systems in the global energy landscape.