In my research on advanced energy storage systems, I have focused on developing a rapid and accurate technique for detecting the state of energy storage cells. The ability to quickly assess the health and power capability of these cells is critical for ensuring the safety and efficiency of large-scale energy storage stations. Traditional methods, such as capacity measurement and open-circuit voltage estimation, often suffer from long detection times and limited accuracy. To address these challenges, I have designed a novel approach that integrates AC impedance spectroscopy with mathematical analysis, the Levenberg-Marquardt (L-M) algorithm, back propagation (BP) neural networks, and the dichotomy method. This paper presents my work in detail, including the construction of an impedance parameter identification model, the design of the detection technique, and a comprehensive performance evaluation.
The core motivation behind my work is the need for a real-time, non-destructive, and rapid state detection method for energy storage cells. AC impedance spectroscopy is particularly attractive because it can capture detailed electrochemical information about the cell without requiring full charge-discharge cycles. However, the interpretation of impedance data requires accurate modeling and parameter extraction. I have therefore developed a hybrid impedance parameter identification model that combines the efficiency of mathematical analysis with the precision of the L-M algorithm. Furthermore, I employ a BP neural network to estimate the state of health (SOH) of the energy storage cell, and the dichotomy method to estimate the state of power (SOP). The integration of these techniques enables rapid and precise state detection, as demonstrated by my experimental results.
1. Construction of the Impedance Parameter Identification Model
Accurate identification of impedance parameters is a fundamental step in using AC impedance for energy storage cell state detection. The impedance spectrum of an energy storage cell typically exhibits distinct features at different frequency ranges. At high frequencies, inductive behavior becomes significant due to the cell’s internal wiring and electrode geometry. To capture this effect, I introduced an inductive element in the equivalent circuit model, specifically in both second-order and third-order fractional-order impedance models. By incorporating a series inductance, I could more accurately represent the high-frequency characteristics of the energy storage cell.
The process begins with acquiring the AC impedance data of the energy storage cell using an impedance analyzer. I then select an appropriate equivalent circuit model to fit this data. Common models include simple RC circuits, Warburg elements, and constant phase elements (CPE). The initial parameter values are obtained through a mathematical analysis of the impedance spectrum. The initialization formula I used is given by:
$$ Q_w = \frac{1}{N} \sum_{i=1}^{N} \frac{1}{\omega_i} \sqrt{ \left[ Z_{\text{Re}}(\omega_i) – Z_{\text{Re-dl}}(\omega_i) – R_{\text{ohm}} \right]^2 + \left[ Z_{\text{Im}}(\omega_i) – Z_{\text{Im-dl}}(\omega_i) \right]^2 } $$
where \( Q_w \) is the parameter value, \( N \) is the number of data points, \( \omega_i \) is the angular frequency, \( Z_{\text{Re}}(\omega_i) \) and \( Z_{\text{Im}}(\omega_i) \) are the real and imaginary parts of the impedance, \( Z_{\text{Re-dl}}(\omega_i) \) and \( Z_{\text{Im-dl}}(\omega_i) \) are the corresponding components from the parallel branch, and \( R_{\text{ohm}} \) is the ohmic resistance. This formula provides a good initial estimate but is limited in accuracy because the impedance data points are not uniformly distributed across the frequency range.
To improve accuracy, I fused the mathematical analysis with the L-M algorithm. The L-M algorithm is a powerful method for solving nonlinear least-squares problems and provides a numerical approximation to the optimal solution. I constructed an objective function based on the least-squares principle:
$$ F(P) = \frac{1}{2} \| f(P) \|^2 = \frac{1}{2} \sum_{j=1}^{n} \left[ f_{\text{Re},j}^2(P) + f_{\text{Im},j}^2(P) \right] $$
Here, \( F(P) \) is the objective function to minimize, \( P \) is the parameter vector, \( n \) is the number of measured frequencies, and \( f_{\text{Re},j}(P) \) and \( f_{\text{Im},j}(P) \) are the error functions for the real and imaginary parts, respectively. The L-M update rule is:
$$ P_{k+1} = P_k – \left[ J_f^T J_f + \mu \operatorname{diag}(I) \right]^{-1} J_f^T f(P_k) $$
where \( J_f \) is the Jacobian matrix, \( \mu \) is a damping parameter that controls convergence speed and direction, and \( \operatorname{diag}(I) \) is the identity matrix. The flowchart of this hybrid identification model is shown conceptually in the integrated figure below. The model proceeds through data collection, preprocessing, mathematical analysis, initial parameter estimation, and iterative optimization using the L-M algorithm. This combined approach yields highly accurate impedance parameters for the energy storage cell.
Through this model, I can precisely identify nine key impedance parameters that are nonlinearly related to the state of health (SOH) of the energy storage cell. These parameters include ohmic resistance, charge transfer resistance, double-layer capacitance, Warburg impedance coefficients, and others. The accurate extraction of these parameters lays the foundation for subsequent state estimation.
2. Design of the Rapid State Detection Technique
With the impedance parameter identification model established, I designed a comprehensive rapid state detection technique for energy storage cells. The technique estimates both SOH and SOP using machine learning and numerical optimization, respectively.
2.1 State of Health (SOH) Estimation Using BP Neural Network
The relationship between the nine impedance parameters and SOH is highly nonlinear and difficult to model with traditional analytical methods. Therefore, I chose a BP neural network to learn this mapping. The BP network is a multi-layer feedforward network trained with error backpropagation. I used impedance parameters extracted from energy storage cells at different aging stages as input features, and the corresponding true SOH values as targets. The output of a neuron is given by:
$$ y_i = f\left( \sum_{i=1}^{n} w_{ji} x_i + b_j \right) $$
where \( y_i \) is the output of the \( i \)-th neuron, \( f \) is the activation function (e.g., sigmoid or ReLU), \( w_{ji} \) is the weight from the \( i \)-th input to the \( j \)-th neuron, \( x_i \) is the input value (e.g., specific impedance parameters), and \( b_j \) is the bias. The network is trained to minimize the mean squared error between predicted and actual SOH. In my experiments, I used four energy storage cells with different health levels: 100%, 93.54%, 82.31%, and 69.85%. The network achieved excellent accuracy, as shown in Table 1.
| Cell Type | True SOH (%) | Predicted SOH (AC Impedance) (%) | Error (AC Impedance) (%) | Predicted SOH (Traditional Capacity) (%) | Error (Traditional) (%) |
|---|---|---|---|---|---|
| A | 100.00 | 99.8 | 0.2 | 98.5 | 1.5 |
| B | 93.54 | 93.1 | 0.44 | 92.0 | 1.54 |
| C | 82.31 | 81.9 | 0.41 | 80.0 | 2.31 |
| D | 69.85 | 69.3 | 0.55 | 68.0 | 1.85 |
As shown in Table 1, the average absolute error of my AC impedance-based SOH prediction is only 0.4%, whereas the traditional capacity measurement method yields an average error of 1.8%. This significant improvement demonstrates the superior accuracy of the BP neural network approach, which leverages the rich electrochemical information contained in the impedance data.
2.2 State of Power (SOP) Estimation Using the Dichotomy Method
The state of power (SOP) is another crucial parameter for energy storage cells, indicating the maximum power that the cell can safely deliver or absorb. SOP is influenced by both the state of charge (SOC) and SOH. To estimate SOP rapidly, I employed the dichotomy method, which is a simple yet effective numerical search algorithm. I initialize the search interval with the current lower bound \( I_{\text{min}} = 0 \) and the upper bound \( I_{\text{max}} = I_{\text{safe}} \), where \( I_{\text{safe}} \) is the maximum safe current. The target current is iteratively computed as:
$$ I_{\text{target}} = \frac{I_{\text{min}} + I_{\text{max}}}{2} $$
The power output is then calculated as:
$$ P_{\text{current}} = V \cdot I_{\text{target}} $$
where \( V \) is the cell voltage. The algorithm checks whether \( P_{\text{current}} \) meets the target power requirement. If not, the search interval is halved based on the power comparison. This process continues until the desired accuracy is achieved. The estimated SOP value is then used together with the SOH estimate to provide a comprehensive assessment of the energy storage cell’s state. In experiments, the dichotomy-based SOP estimation achieved an average absolute error of 15.38, compared to 38.75 for traditional methods.
2.3 Integrated Detection Workflow
The overall workflow of my rapid state detection technique is as follows: First, I measure the AC impedance spectrum of the energy storage cell along with voltage and current data. Second, I apply the impedance parameter identification model (mathematical analysis + L-M algorithm) to extract accurate impedance parameters. Third, these parameters are fed into the trained BP neural network to estimate the SOH. Fourth, I use the dichotomy method together with the estimated SOH and current measurements to estimate the SOP. Finally, the SOH and SOP results are combined to provide a complete state assessment of the energy storage cell. The entire process takes on average about 1.68 minutes, which is a substantial improvement over traditional methods.
3. Experimental Setup and Performance Analysis
To validate the effectiveness of my proposed technique, I conducted a series of experiments using a constant-current constant-voltage power supply (Agilent 6614C), an electronic load (BK Precision 8500), and an AC impedance analyzer (BioLogic VMP3). The test subjects were 18650 lithium-ion energy storage cells. The algorithms were implemented in Python. I used four cells with different health statuses as mentioned earlier. I also tested eight different battery states for detection time comparison.
3.1 Accuracy of SOH and SOP Estimation
Figure 3 (conceptually described) shows the SOH and SOP estimation results. For SOH, the BP neural network predictions closely match the true values, with a maximum absolute error of 0.55% and an average of 0.4%. In contrast, the traditional capacity method shows errors up to 2.31% and an average of 1.8%. For SOP, the dichotomy method produced predictions that are very close to the true values, with an average absolute error of 15.38, while the traditional method’s error was 38.75. These results confirm that my integrated approach significantly improves the accuracy of both SOH and SOP estimations for energy storage cells.
3.2 Detection Time Comparison
One of the key advantages of my technique is its speed. I compared the detection time of the AC impedance method with two traditional approaches: the capacity measurement method and the open-circuit voltage method. The results for eight different battery states are summarized in Table 2.
| Battery State | AC Impedance Method | Traditional Capacity Method | Open-Circuit Voltage Method |
|---|---|---|---|
| 1 | 1.62 | 5.45 | 9.02 |
| 2 | 1.70 | 5.60 | 9.15 |
| 3 | 1.65 | 5.50 | 9.08 |
| 4 | 1.73 | 5.70 | 9.20 |
| 5 | 1.68 | 5.55 | 9.10 |
| 6 | 1.66 | 5.62 | 9.12 |
| 7 | 1.71 | 5.58 | 9.18 |
| 8 | 1.64 | 5.48 | 9.05 |
| Average | 1.68 | 5.58 | 9.12 |
The average detection time for the AC impedance method is 1.68 minutes, which is 69% faster than the traditional capacity method (5.58 minutes) and 81.5% faster than the open-circuit voltage method (9.12 minutes). This significant reduction in detection time is attributed to the fact that AC impedance measurements can be performed quickly without requiring a full charge-discharge cycle or waiting for voltage stabilization. The impedance spectrum captures comprehensive electrochemical information in a matter of seconds, and the subsequent data processing using the L-M algorithm and BP neural network is computationally efficient.
3.3 Robustness Under Different Conditions
I also evaluated the robustness of my technique by testing energy storage cells at different temperature conditions (though primarily at room temperature in this study). The results indicate that the accuracy remains high as long as the impedance parameters are properly extracted. However, I note that temperature variations can affect the impedance spectrum, and future work will incorporate temperature compensation to further enhance robustness.
4. Discussion
My proposed technique offers several advantages over existing methods. First, the combination of mathematical analysis and the L-M algorithm provides a reliable and accurate identification of impedance parameters, even when the frequency distribution is uneven. Second, the use of a BP neural network effectively captures the nonlinear relationship between impedance parameters and SOH, achieving an average error of only 0.4%. Third, the dichotomy method offers a simple and fast estimation of SOP without requiring complex iterative computations. Finally, the overall detection time is less than 2 minutes, making it suitable for real-time online monitoring of energy storage cells in large-scale battery energy storage systems.
However, there are limitations. My experiments were conducted under controlled laboratory conditions. In practical applications, energy storage cells may experience varying temperatures, load profiles, and aging patterns. I plan to extend my work by incorporating environmental temperature as an input factor and by training the BP neural network on a broader dataset that includes diverse operating conditions. Additionally, I aim to implement the technique on an embedded microcontroller for on-board battery management systems.
5. Conclusion
In this paper, I have presented a rapid AC impedance-based state detection technique for energy storage cells. The technique integrates a hybrid impedance parameter identification model (mathematical analysis + L-M algorithm) with BP neural network-based SOH estimation and dichotomy-based SOP estimation. Experimental results demonstrate that the proposed method achieves an average SOH prediction error of 0.4% and an average detection time of 1.68 minutes, which are significantly better than traditional capacity measurement and open-circuit voltage methods. This work provides key technical support for the development of real-time online monitoring and intelligent management systems for energy storage batteries. Future research will focus on incorporating temperature effects and deploying the technique in practical battery management systems.