In the context of rapidly expanding electric mobility, the safety and reliability of an EV battery pack are of paramount importance. The complex electrochemical behavior inside a lithium-ion battery, especially under varying thermal conditions, poses significant challenges for accurate state estimation and health monitoring. As a result, building a high-fidelity battery model and obtaining its parameters with high precision has become a critical research topic. This paper focuses on the modeling of power batteries using electrochemical impedance spectroscopy (EIS) data at different temperatures. My work is divided into two major parts: first, a comprehensive sensitivity analysis and parameter identification of a pseudo-two-dimensional (P2D) electrochemical model for an EV battery pack operating at different temperatures; second, the development of an improved fractional-order impedance model that combines the distribution of relaxation times (DRT) technique and constant phase elements (CPE), followed by accurate parameter identification using a recently proposed metaheuristic algorithm. Throughout this study, I emphasize the applicability of EIS as a non-destructive diagnostic tool for extracting physically meaningful parameters from the EV battery pack without disassembly. The proposed methodologies are expected to contribute to more reliable battery management systems and digital twin technologies.
EIS is a powerful experimental method that probes the dynamic response of an EV battery pack across a wide frequency range. By applying a small sinusoidal perturbation and measuring the complex impedance, it is possible to separate different electrochemical processes such as charge transfer, solid-phase diffusion, double-layer capacitance, and inductive effects. The key advantage of EIS lies in its non-invasiveness and its sensitivity to internal state changes. In this study, I generate EIS data using a one-dimensional P2D model built in COMSOL Multiphysics. The model follows the well-known Doyle-Fuller-Newman formulation, where the governing equations describe solid-phase diffusion, electrolyte-phase diffusion, charge conservation, and Butler-Volmer kinetics. For a given temperature and state of charge (SOC), the model outputs the impedance spectrum from 1 kHz down to 0.001 Hz in the sensitivity analysis, and from 100 Hz to 0.01 Hz during identification.
The first part of this research targets the issue of parameter identifiability in electrochemical models. The P2D model typically involves dozens of parameters, many of which are difficult to measure directly. To reduce the identification burden and to understand how parameter sensitivity changes with temperature, I perform a one-factor-at-a-time (OFAT) sensitivity analysis on 25 important model parameters. The parameter values are varied within physically meaningful ranges, and the resulting changes in the real and imaginary parts of the impedance are quantified by a sensitivity index. The OFAT procedure is performed at three temperatures (-10 °C, 25 °C, and 35 °C) and five SOC levels (5%, 20%, 50%, 80% and 100%). For each parameter, I compute the average sensitivity index over three frequency regions: high frequency (1 kHz–10 Hz), medium frequency (10 Hz–0.1 Hz), and low frequency (0.1 Hz–0.001 Hz). This frequency decomposition is essential because different electrochemical processes dominate in different frequency bands, and the sensitivity of certain parameters may be frequency-dependent.
The sensitivity index for the real part of the impedance is defined as:
$$SI_R(T, \text{SOC}, f) = \sqrt{\frac{1}{N_s} \sum_{j=1}^{N_s} \left[ R(Z_j) – \overline{R(Z)} \right]^2 }$$
and for the imaginary part:
$$SI_I(T, \text{SOC}, f) = \sqrt{\frac{1}{N_s} \sum_{j=1}^{N_s} \left[ I(Z_j) – \overline{I(Z)} \right]^2 }$$
where \(N_s\) is the number of sampled values for each parameter (set to six), \(Z_j\) is the simulated impedance when the parameter is set to the \(j\)-th sample, and the overline denotes the average over all samples. The average sensitivity indices over a frequency band are obtained by simply averaging the per-frequency values within that band. In order to observe the overall parameter sensitivity and its temperature dependency, I further propose a comprehensive sensitivity score (CSS) for each parameter \(v\) and temperature \(T\):
$$CSS_{v,T} = \sqrt{\frac{(ASI_{R}^{v,T})^2 + (ASI_{I}^{v,T})^2}{2}}$$
where \(ASI_R\) and \(ASI_I\) are the average sensitivity indices computed from the real and imaginary parts of the impedance, respectively. This CSS value is used to classify parameters into high, medium, and low sensitivity classes. The threshold values are set as: \(CSS > 0.22\) means high sensitivity, \(0.05 < CSS \le 0.22\) means medium sensitivity, and \(CSS \le 0.05\) means low sensitivity.
The sensitivity analysis requires a large number of simulations. I simulate 2250 EIS spectra in total (25 parameters × 6 values × 3 temperatures × 5 SOC levels). The results reveal that the parameters with the highest sensitivity are generally the negative electrode film resistance, the negative electrode reaction rate coefficient, the particle radii, and the electronic conductivities. Meanwhile, parameters related to the separator, such as separator thickness and separator Bruggeman coefficient, show very low sensitivity and can be safely fixed to their nominal values during identification. According to the CSS classification, the parameter sets for each temperature are summarized in the following table.
| Temperature | High sensitivity | Medium sensitivity | Low sensitivity |
|---|---|---|---|
| 35 °C | \(r^+\), \(r^-\), \(R_f^-\), \(A\), \(k^-\), \(\sigma^+\), \(R_f^+\), \(b^+\), \(\varepsilon^-\), \(b^-\), \(\kappa_e\), \(\varepsilon^+\) | \(c_{dl}^-\), \(L^+\), \(L^-\), \(\sigma^-\), \(c_0\), \(D_s^+\), \(D_s^-\), \(D_e\) | \(k^+\), \(c_{dl}^+\), \(L_s\), \(b_s\), \(\varepsilon_s\), |
| 25 °C | \(r^+\), \(r^-\), \(R_f^-\), \(A\), \(k^-\), \(\sigma^+\), \(R_f^+\), \(L^+\), \(\varepsilon^-\), \(b^-\), \(\kappa_e\), \(D_s^-\) | \(c_{dl}^-\), \(L^-\), \(\varepsilon^+\), \(\sigma^-\), \(c_0\), \(D_s^+\), \(b^+\), \(D_e\) | \(k^+\), \(c_{dl}^+\), \(L_s\), \(b_s\), \(\varepsilon_s\) |
| -10 °C | \(r^+\), \(r^-\), \(R_f^-\), \(A\), \(k^-\), \(\sigma^+\), \(R_f^+\), \(b^+\), \(\varepsilon^-\), \(b^-\), \(\kappa_e\), \(\varepsilon^+\) | \(c_{dl}^-\), \(L^+\), \(L^-\), \(\sigma^-\), \(c_0\), \(D_s^+\), \(D_s^-\), \(D_e\) | \(k^+\), \(c_{dl}^+\), \(L_s\), \(b_s\), \(\varepsilon_s\) |
The CSS analysis reveals an interesting temperature dependency. For instance, the negative electrode active material volume fraction \(\varepsilon^-\) and the negative electrode particle radius \(r^-\) become more sensitive when the temperature decreases. In contrast, the sensitivity of the electrode surface area \(A\), the negative electrode conductivity \(\sigma^-\), and the double-layer capacitance \(c_{dl}^-\) decreases at lower temperatures. This information is valuable for selecting appropriate operating conditions when identifying specific parameters. It also helps in designing experiments that maximize the information content from EIS measurements of an EV battery pack.
After the sensitivity screening, seven parameters with consistently low sensitivity are fixed to their nominal values. The remaining eighteen parameters are identified using the Gray Wolf Optimizer (GWO), a metaheuristic inspired by the hunting behavior of gray wolves. The GWO algorithm maintains a population of candidate solutions, where the three best solutions are designated as alpha, beta, and delta wolves, and the rest are omega wolves. The position update equation is:
$$\vec{X}(t+1) = \frac{\vec{X}_\alpha + \vec{X}_\beta + \vec{X}_\delta}{3}$$
where the individual positions are updated based on the distance vectors:
$$\vec{D}_\alpha = |\vec{C}_1 \cdot \vec{X}_\alpha – \vec{X}|, \quad \vec{D}_\beta = |\vec{C}_2 \cdot \vec{X}_\beta – \vec{X}|, \quad \vec{D}_\delta = |\vec{C}_3 \cdot \vec{X}_\delta – \vec{X}|$$
and the corresponding position updates are:
$$\vec{X}_1 = \vec{X}_\alpha – \vec{A}_1 \cdot \vec{D}_\alpha, \quad \vec{X}_2 = \vec{X}_\beta – \vec{A}_2 \cdot \vec{D}_\beta, \quad \vec{X}_3 = \vec{X}_\delta – \vec{A}_3 \cdot \vec{D}_\delta$$
The coefficient vectors \(\vec{A}\) and \(\vec{C}\) are computed as:
$$\vec{A} = 2\vec{a} \cdot \vec{r}_1 – \vec{a}, \quad \vec{C} = 2\vec{r}_2$$
with \(\vec{a}\) linearly decreasing from 2 to 0 over iterations, and \(\vec{r}_1, \vec{r}_2\) being random vectors in \([0,1]\). The parameter identification objective is to minimize the root mean square error between the measured and simulated EIS spectra:
$$F(x) = \frac{1}{N_m} \sum_{i=1}^{N_m} \left[ \left( R(Z_i) – R(Z_i^{\text{sim}}) \right)^2 + \left( I(Z_i) – I(Z_i^{\text{sim}}) \right)^2 \right]$$
where \(N_m\) is the number of frequency points, \(R\) and \(I\) denote the real and imaginary parts, and \(x\) is the vector of identified parameters. The lower and upper bounds for the parameters are taken from the sensitivity analysis table, and the P2D model is run in COMSOL with MATLAB coupling to generate the simulated impedance.
I perform identification experiments using the public dataset of a commercial NMC811 21700 cell with EIS data at 25 °C and 35 °C at several SOC levels. Five different parameter sets are obtained for the combinations: (35 °C, 95% SOC), (35 °C, 50% SOC), (35 °C, 20% SOC), (25 °C, 95% SOC), and (25 °C, 50% SOC). The identified values for the most important parameters are presented in the following two tables.
| Parameter | SOC=95% | SOC=50% | SOC=20% |
|---|---|---|---|
| \(\varepsilon^-\) | 0.751 | 0.750 | 0.750 |
| \(\varepsilon^+\) | 0.616 | 0.668 | 0.624 |
| \(r^-\) (μm) | 5.730 | 5.530 | 1.980 |
| \(r^+\) (μm) | 3.970 | 1.320 | 5.220 |
| \(D_s^+\) (m²/s) | 5.19×10⁻¹⁵ | 3.02×10⁻¹⁵ | 1.94×10⁻¹⁵ |
| \(D_s^-\) (m²/s) | 8.34×10⁻¹⁵ | 3.40×10⁻¹⁴ | 5.38×10⁻¹⁴ |
| \(b^+\) | 2.60 | 3.40 | 2.70 |
| \(b^-\) | 3.00 | 3.40 | 3.40 |
| \(k^+\) (m/s) | 3.29×10⁻⁹ | 7.28×10⁻¹⁰ | 4.96×10⁻⁹ |
| \(\sigma^+\) (S/m) | 3.84 | 3.32 | 14.71 |
| \(\sigma^-\) (S/m) | 338.8 | 392.2 | 32.8 |
| \(\kappa_e\) (S/m) | 1.08 | 1.08 | 1.00 |
| \(c_{dl}^-\) (F/m²) | 2.83 | 1.85 | 2.91 |
| \(R_f^-\) (Ω·m²) | 0.0055 | 0.0143 | 0.0147 |
| \(k^-\) (m/s) | 1.20×10⁻⁸ | 1.09×10⁻⁸ | 4.71×10⁻⁹ |
| \(R_f^+\) (Ω·m²) | 0.0377 | 0.0053 | 0.0457 |
| \(D_e\) (×10⁻¹⁰ m²/s) | 4.55 | 5.17 | 3.83 |
| \(c_{dl}^+\) (F/m²) | 25.87 | 2.53 | 28.38 |
| Parameter | SOC=95% | SOC=50% |
|---|---|---|
| \(\varepsilon^-\) | 0.750 | 0.800 |
| \(\varepsilon^+\) | 0.714 | 0.615 |
| \(r^-\) (μm) | 1.95 | 2.52 |
| \(r^+\) (μm) | 3.64 | 2.61 |
| \(D_s^+\) (m²/s) | 2.27×10⁻¹⁵ | 5.91×10⁻¹⁵ |
| \(D_s^-\) (m²/s) | 5.33×10⁻¹⁴ | 5.50×10⁻¹⁴ |
| \(b^+\) | 3.20 | 3.40 |
| \(b^-\) | 3.40 | 3.40 |
| \(\sigma^+\) (S/m) | 14.59 | 3.33 |
| \(\sigma^-\) (S/m) | 182.4 | 13.10 |
| \(\kappa_e\) (S/m) | 1.01 | 1.06 |
| \(c_{dl}^-\) (F/m²) | 2.75 | 1.61 |
| \(R_f^-\) (Ω·m²) | 0.0014 | 0.0043 |
| \(k^-\) (m/s) | 8.50×10⁻⁹ | 7.05×10⁻⁹ |
| \(R_f^+\) (Ω·m²) | 0.0405 | 0.0407 |
| \(D_e\) (×10⁻¹⁰ m²/s) | 4.65 | 3.53 |
| \(c_{dl}^+\) (F/m²) | 23.30 | 15.61 |
| \(k^+\) (m/s) | 2.11×10⁻⁹ | 8.41×10⁻¹⁰ |
After identification, I feed the obtained parameter sets back into the P2D model to simulate EIS for each operating condition. The simulated spectra are then compared with the corresponding measured spectra. The root mean square error (RMSE) and mean absolute error (MAE) are computed as:
$$\text{RMSE} = \sqrt{ \frac{1}{N_m} \sum_{i=1}^{N_m} \left[ \left( R(Z_i)-R(Z_i^{\text{sim}}) \right)^2 + \left( I(Z_i)-I(Z_i^{\text{sim}}) \right)^2 \right] }$$
$$\text{MAE} = \frac{1}{N_m} \sum_{i=1}^{N_m} \sqrt{ \left( R(Z_i)-R(Z_i^{\text{sim}}) \right)^2 + \left( I(Z_i)-I(Z_i^{\text{sim}}) \right)^2 }$$
The computed errors are summarized in the following table. For all five cases, the RMSE is below \(2.2\times10^{-4}\,\Omega\), and the MAE is below \(1.9\times10^{-4}\,\Omega\). These values are exceptionally low, confirming that the GWO-identified parameters allow the P2D model to accurately reproduce the EIS response of the EV battery pack over the considered frequency range.
| Temperature | SOC | RMSE (Ω) | MAE (Ω) |
|---|---|---|---|
| 35 °C | 95% | 1.3366×10⁻⁴ | 9.2137×10⁻⁵ |
| 35 °C | 50% | 1.1411×10⁻⁴ | 9.9974×10⁻⁵ |
| 35 °C | 20% | 2.1356×10⁻⁴ | 1.8474×10⁻⁴ |
| 25 °C | 95% | 1.6946×10⁻⁴ | 1.3907×10⁻⁴ |
| 25 °C | 50% | 1.0759×10⁻⁴ | 9.1592×10⁻⁵ |
In addition to the validation against experimental data, I compare the performance of GWO with four other widely used optimization algorithms: particle swarm optimization (PSO), genetic algorithm (GA), simulated annealing (SA), and cuckoo search (CS). All algorithms are executed under identical conditions (25 °C, 95% SOC) with the same number of fitness evaluations. The resulting RMSE values are shown in the table below. GWO achieves the smallest RMSE, demonstrating its superior balance between exploration and exploitation in this high-dimensional parameter identification problem.
| Algorithm | RMSE (Ω) |
|---|---|
| GWO | 1.6946×10⁻⁴ |
| PSO | 3.0521×10⁻⁴ |
| GA | 4.1187×10⁻⁴ |
| CS | 3.7742×10⁻⁴ |
| SA | 5.2345×10⁻⁴ |
The first part of this work demonstrates that the combination of sensitivity analysis and GWO is a powerful approach for parameterizing a P2D model from EIS data of an EV battery pack. The proposed CSS method not only provides a clear classification of parameter sensitivity but also reveals the temperature dependence of sensitivity, which is rarely explored in previous studies. Such temperature-dependent sensitivity information can guide the design of identification experiments and reduce computational cost by focusing on the most influential parameters at the operating temperature of interest.
Moving to the second part of this research, I address the practical limitations of electrochemical models, such as their high complexity and the difficulty of online implementation. Equivalent circuit models (ECMs) are more suitable for online battery management systems because they are computationally lightweight. However, conventional second-order RC models often fail to accurately reproduce the broadband EIS of different batteries, especially at high frequencies and under different aging states. Furthermore, the ideal capacitor element cannot capture the non-ideal behavior caused by surface roughness, porosity, and distributed time constants. To overcome these issues, I develop a novel fractional-order impedance model that leverages the DRT technique and CPE elements.
The proposed modeling framework begins with a DRT analysis of the EIS data. The DRT method transforms the impedance spectrum from the frequency domain into a distribution of relaxation times, which allows a clear separation of overlapping electrochemical processes without any prior assumption of the equivalent circuit topology. I apply DRT to the EIS data of a commercial LG 50 cell (NMC811 chemistry) at 35 °C, 100% SOH, and 70% SOC over the frequency range from 0.01 Hz to 10 kHz. The Nyquist plot shows a semi-circular region from about 0.5 Hz to 400 Hz. The DRT analysis of this region reveals three distinct peaks, indicating that the semi-circle is actually composed of three separate relaxation processes. Based on this finding, a traditional one-time-constant model would be insufficient, and I instead construct a fractional-order impedance model with three R-CPE branches in series.
In addition, the high-frequency inductive behavior is modeled by two RL networks, the ohmic resistance by a pure resistor \(R_0\), and the low-frequency diffusion tail by a Warburg-like element with a variable fractional exponent. The CPE impedance is defined as:
$$Z_{\text{CPE}}(j\omega) = \frac{1}{T (j\omega)^P}$$
where \(T\) is a pseudo-capacitance and \(P\) is the fractional order (\(0<p<1\)). as:
$$Z_W(j\omega) = \frac{1}{T_W (j\omega)^{P_W}}$$
with \(P_W\) allowed to vary between 0 and 1, thus generalizing the classical Warburg element. The total impedance of the proposed fractional-order model is:
$$Z_b = R_0 + Z_{RL1} + Z_{RL2} + \frac{R_3}{1+R_3 T_1(j\omega)^{P_1}} + \frac{R_4}{1+R_4 T_2(j\omega)^{P_2}} + \frac{R_5}{1+R_5 T_3(j\omega)^{P_3}} + Z_W$$
where \(Z_{RL1}\) and \(Z_{RL2}\) are the impedances of the two RL branches:
$$Z_{RL} = \frac{j\omega L_1 R_1}{R_1 + j\omega L_1} + \frac{j\omega L_2 R_2}{R_2 + j\omega L_2}$$
This model is implemented in ZSimpWin software for validation. The model is first validated by fitting EIS data from the same battery at 35 °C, 80% SOH, and five different SOC levels. The fitting results show excellent agreement between the measured and modeled impedance for both the real and imaginary parts. A graphical comparison of the Nyquist plots, real-part spectra, and imaginary-part spectra confirms that the proposed model accurately captures all the features of the measured EIS, including the high-frequency inductive loop, the three overlapping semi-circles, and the low-frequency diffusion tail.
Once the model structure is established, I proceed to identify its 16 parameters using the Sinh Cosh Optimizer (SCHO). SCHO is a recently developed metaheuristic inspired by the properties of hyperbolic sine and cosine functions. It aims to maintain a proper balance between exploration and exploitation through adaptive switching between phases. The core position update equations are summarized as follows. During the exploration phase, the candidate solution is updated using:
$$X_{i,j}^{t+1} = \begin{cases} X_{\text{best},j} + W_1 \cdot r_2 \cdot X_{i,j}^{t}, & r_1 > 0.5 \\ X_{\text{best},j} – W_1 \cdot r_3 \cdot X_{i,j}^{t}, & r_1 < 0.5 \end{cases}$$
where \(W_1 = r_4 \cdot a_1 \cdot (\cosh(r_5)+1) – r_5 \cdot u\) and \(a_1 = 3 \cdot \left( \frac{t}{Max\_iter} \cdot 1.3 \right)\). The second exploration sub-phase uses:
$$X_{i,j}^{t+1} = \begin{cases} X_{i,j}^{t} + W_2 \cdot (X_{\text{best},j} – X_{i,j}^{t}) + \epsilon, & r_6 > 0.5 \\ X_{i,j}^{t} – W_2 \cdot (X_{\text{best},j} – X_{i,j}^{t}) – \epsilon, & r_6 < 0.5 \end{cases}$$
with \(W_2 = r_7 \cdot a_2\), \(a_2 = 2 \cdot \left( \frac{t}{Max\_iter} – 0.5 \right)\), and \(\epsilon = 0.003\). In the exploitation phase, the first sub-phase is:
$$X_{i,j}^{t+1} = \begin{cases} X_{\text{best},j} + W_3 \cdot r_8 \cdot X_{i,j}^{t}, & r_9 > 0.5 \\ X_{\text{best},j} – W_3 \cdot r_9 \cdot X_{i,j}^{t}, & r_9 < 0.5 \end{cases}$$
where \(W_3 = r_{10} \cdot a_1 \cdot (\cosh(r_{11}) – 1) + r_{11} \cdot u\). The second exploitation sub-phase is:
$$X_{i,j}^{t+1} = X_{i,j}^{t} + W_4 \cdot (X_{\text{best},j} – X_{i,j}^{t}) \cdot \frac{\cosh(r_{12})}{\sinh(r_{12})} \cdot r_{13}$$
Specifically, \(W_4 = r_{12} \cdot W_3\). The algorithm also includes a bounded search strategy that dynamically adjusts the search space around the current best solution. The pseudocode of SCHO for parameter identification is presented below.
Algorithm: SCHO-based parameter identification Input: bounds ub, lb; measured EIS data Output: optimal parameter vector x 1. Initialize population, Max_iter, and control parameters 2. While t < Max_iter do 3. For each candidate x, evaluate fitness F(x) 4. If t equals a bounded-search trigger, update the search bounds 5. Compute A = p * cosh(t/Max_iter) / (sinh(t/Max_iter)) + ... 6. If A >= 1 then perform exploration (either sub-phase) 7. Else perform exploitation (either sub-phase) 8. Update candidates and clip to bounds 9. t = t + 1 10. End While 11. Return the best solution
I use SCHO to identify the parameters of the fractional-order model under 20 distinct working conditions: two temperatures (15 °C and 35 °C), two SOH levels (100% and 90%), and five SOC levels (95%, 70%, 50%, 20%, and 5%). The bounds for each parameter are determined from preliminary ZSimpWin fits and literature values. The identified parameters for SOH=100% are listed in the following table.
| Parameter | 15 °C | 35 °C | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 95% | 70% | 50% | 20% | 5% | 95% | 70% | 50% | 20% | 5% | |
| \(R_0\) (Ω) | 0.003 | 0.008 | 0.017 | 0.022 | 0.012 | 0.019 | 0.012 | 0.010 | 0.011 | 0.004 |
| \(R_1\) (Ω) | 0.154 | 0.431 | 0.437 | 0.010 | 0.019 | 0.235 | 0.227 | 0.227 | 0.235 | 0.014 |
| \(L_1\) (H) | 4.40e-7 | 3.48e-7 | 3.47e-7 | 1.97e-7 | 2.31e-7 | 3.84e-7 | 3.00e-7 | 3.17e-7 | 3.94e-7 | 2.78e-7 |
| \(R_2\) (Ω) | 0.0067 | 0.0106 | 0.0110 | 0.4267 | 1.1751 | 0.0079 | 0.0142 | 0.0147 | 0.0104 | 2.7000 |
| \(L_2\) (H) | 2.58e-7 | 2.05e-7 | 2.10e-7 | 3.52e-7 | 3.10e-7 | 2.24e-7 | 2.84e-7 | 2.34e-7 | 2.66e-7 | 3.63e-7 |
| \(R_3\) (Ω) | 0.0114 | 0.0137 | 0.0085 | 0.3889 | 0.1092 | 0.0083 | 0.0065 | 0.0077 | 0.0123 | 0.0240 |
| \(T_1\) (F·s^(P-1)) | 0.761 | 1.19e-13 | 1.400 | 721.9 | 932.3 | 1.998 | 5.42e-13 | 1.381 | 1.104 | 1.690 |
| \(P_1\) | 0.496 | 0.678 | 0.627 | 0.712 | 0.999 | 0.012 | 0.725 | 0.640 | 0.369 | 0.212 |
| \(R_4\) (Ω) | 0.001 | 0.008 | 0.010 | 0.005 | 0.021 | 0.009 | 0.009 | 0.008 | 0.003 | 0.506 |
| \(T_2\) (F·s^(P-1)) | 1.00e-10 | 1.268 | 6.66e-11 | 0.500 | 24.325 | 1.902 | 1.334 | 2.25e-7 | 16.69 | 1399 |
| \(P_2\) | 0.603 | 0.647 | 0.222 | 0.787 | 0.426 | 0.280 | 0.521 | 0.623 | 0.864 | 0.981 |
| \(R_5\) (Ω) | 0.006 | 0.033 | 0.566 | 0.009 | 0.011 | 0.002 | 0.002 | 0.005 | 0.010 | 0.009 |
| \(T_3\) (F·s^(P-1)) | 11.75 | 11950 | 12780 | 10.80 | 4.44e-11 | 8.867 | 12.634 | 10.887 | 13.254 | 20.87 |
| \(P_3\) | 0.590 | 1.000 | 1.000 | 0.626 | 0.926 | 0.495 | 0.642 | 0.166 | 0.323 | 0.744 |
| \(T_W\) (F·s^(P-1)) | 446.1 | 379.8 | 544.8 | 977000 | 40300 | 1721 | 428.4 | 558.3 | 735.8 | 421.7 |
| \(P_W\) | 0.501 | 0.536 | 0.512 | 0.492 | 0.456 | 0.491 | 0.569 | 0.548 | 0.713 | 0.500 |
Similarly, the identified parameters for SOH=90% are reported in the following table.
| Parameter | 15 °C | 35 °C | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 95% | 70% | 50% | 20% | 5% | 95% | 70% | 50% | 20% | 5% | |
| \(R_0\) (Ω) | 0.010 | 0.009 | 0.017 | 0.023 | 0.015 | 0.016 | 0.007 | 0.016 | 0.011 | 0.021 |
| \(R_1\) (Ω) | 0.1775 | 4.8340 | 0.2222 | 0.2802 | 0.1924 | 1.7120 | 0.1730 | 0.3267 | 0.2351 | 0.1915 |
| \(L_1\) (H) | 3.48e-7 | 3.70e-7 | 3.64e-7 | 2.79e-7 | 3.79e-7 | 3.70e-7 | 3.86e-7 | 4.31e-7 | 3.94e-7 | 3.70e-7 |
| \(R_2\) (Ω) | 0.00869 | 0.0143 | 0.0086 | 0.0146 | 0.0106 | 0.01503 | 0.0077 | 0.008326 | 0.0104 | 0.0087 |
| \(L_2\) (H) | 2.09e-7 | 2.50e-7 | 2.59e-7 | 2.23e-7 | 2.48e-7 | 2.29e-7 | 2.27e-7 | 1.81e-7 | 2.66e-7 | 1.90e-7 |
| \(R_3\) (Ω) | 0.00876 | 0.01495 | 0.0049 | 0.0085 | 0.0104 | 0.005898 | 0.0069 | 0.01639 | 0.0123 | 0.9607 |
| \(T_1\) (F·s^(P-1)) | 2.129 | 6.804 | 19.976 | 15.140 | 0.0867 | 11.79 | 5.41e-10 | 8374 | 1.104 | 1406.6 |
| \(P_1\) | 0.513 | 0.375 | 0.619 | 0.569 | 0.626 | 0.519 | 0.0009 | 1.000 | 0.369 | 0.323 |
| \(R_4\) (Ω) | 0.01168 | 0.01271 | 2.4581 | 0.0035 | 0.0091 | 0.01156 | 0.0126 | 0.00365 | 0.0034 | 0.0050 |
| \(T_2\) (F·s^(P-1)) | 9.874 | 2.0e-13 | 841.7 | 0.3612 | 5.1373 | 0.000207 | 3.746 | 3.967 | 16.69 | 2.4789 |
| \(P_2\) | 0.863 | 0.468 | 0.680 | 0.886 | 0.602 | 0.518 | 0.259 | 0.593 | 0.864 | 0.473 |
| \(R_5\) (Ω) | 0.01018 | 0.0208 | 0.0092 | 0.1563 | 0.0140 | 0.01239 | 0.5016 | 0.007327 | 0.0102 | 0.0061 |
| \(T_3\) (F·s^(P-1)) | 3.28e-8 | 3143 | 0.8323 | 927.2 | 12.625 | 3603 | 18271 | 2.6e-12 | 13.254 | 17.062 |
| \(P_3\) | 0.754 | 0.945 | 0.548 | 0.243 | 0.995 | 0.322 | 0.570 | 0.882 | 0.323 | 0.687 |
| \(T_W\) (F·s^(P-1)) | 493.4 | 524.7 | 554.8 | 1791.5 | 319.5 | 582.8 | 706.4 | 651.0 | 735.8 | 607.9 |
| \(P_W\) | 0.526 | 0.502 | 0.000 | 0.905 | 0.619 | 0.518 | 0.620 | 0.500 | 0.713 | 0.764 |
To assess the accuracy of the SCHO-based identification, I use the identified parameter sets as inputs to the fractional-order impedance model and simulate the EIS for each condition. The simulated and measured spectra are compared by plotting the Nyquist diagrams, and the RMSE is computed for every case. For SOH=100%, the RMSE values are all below \(6.0\times10^{-4}\,\Omega\), with the lowest error being around \(1.5\times10^{-4}\,\Omega\) at 35 °C and high SOC. For SOH=90%, the maximum RMSE is also below \(9.0\times10^{-4}\,\Omega\), confirming that the proposed identification method reliably estimates the model parameters across a wide range of temperatures, SOC, and aging states. These results validate that the new fractional-order impedance model, coupled with the SCHO algorithm, achieves high precision in reproducing the broadband EIS of an EV battery pack.
I further compare the proposed model with two classical second-order ECMs: a conventional 2-RC model and a 2-R-CPE model. Both models are implemented in ZSimpWin and fitted to an independent public dataset from a coin-type LR2032 lithium-ion battery at 25 °C, 0% SOC, over 0.025 Hz to 20 kHz. Three cycles (1, 175, and 300) are selected to cover different aging levels. The fitting results clearly show that the 2-RC model fails to capture the high-frequency and low-frequency features, while the 2-R-CPE model performs better but still deviates noticeably in the low-frequency region. In contrast, the proposed model produces the closest agreement with the experimental data across the entire frequency range. This cross-validation using a different battery demonstrates that the developed model is not limited to a single cell chemistry or format, but can be generally applied to the broadband EIS simulation of various EV battery packs.
Finally, the performance of SCHO is compared with GWO, PSO, and SA for the parameter identification of the proposed fractional-order model at 35 °C, 100% SOH, and 95% SOC. All algorithms are run with the same population size (100) and the same number of iterations (1000). The resulting RMSE values are shown in the following table. SCHO achieves the lowest RMSE, outperforming the other three algorithms. This confirms that SCHO provides an excellent balance between exploration and exploitation, making it a robust choice for identifying parameters in complex fractional-order impedance models.
| Algorithm | RMSE (Ω) |
|---|---|
| SCHO | 1.42×10⁻⁴ |
| GWO | 2.88×10⁻⁴ |
| PSO | 3.15×10⁻⁴ |
| SA | 4.76×10⁻⁴ |
In summary, this research presents a systematic approach to power battery modeling and parameter identification using EIS data at different temperatures. In the first part, I analyzed the sensitivity of 25 P2D model parameters and proposed a comprehensive sensitivity score to classify the parameters and reveal their temperature dependence. Using GWO, I successfully identified 18 parameters from EIS data under five different operating conditions, achieving very low RMSE and MAE values. In the second part, I developed a new fractional-order impedance model based on DRT and CPE elements. The model was validated using both the training battery and an independent battery dataset, showing superior fitting accuracy compared to traditional second-order ECMs. The SCHO algorithm was applied to identify the model parameters in 20 different conditions, and the results confirmed the accuracy and reliability of the identification approach. This work provides a clear and practical framework for parameterizing both physical and equivalent-circuit battery models from EIS, which is highly relevant for the development of advanced battery management systems and digital twins of EV battery packs.
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