Lithium-ion batteries, as high-density energy storage carriers, are widely used in numerous fields such as electric vehicles and renewable energy systems. Accurate state assessment and power capability prediction are crucial for the Battery Management System (BMS) to ensure safety, optimize performance, and prolong battery life. Among these, the State of Power (SOP), or peak power, which defines the maximum instantaneous power a battery can safely deliver or absorb within a specified short time horizon, is a core parameter for real-time power allocation and overcharge/discharge prevention.
However, predicting the SOP of a lithium-ion battery with high accuracy is challenging. The peak power is not directly measurable and is influenced by various factors including State of Charge (SOC), temperature, and, critically, the battery’s aging state. As a lithium-ion battery undergoes charge-discharge cycles, irreversible side reactions cause capacity fade and increase internal impedance. This aging phenomenon not only shortens the lifespan but also significantly deteriorates the dynamic response, making peak power prediction for aged cells even more difficult.
Existing SOP prediction methods can be categorized into three groups: experimental lookup-table methods, data-driven approaches, and model-based methods. While lookup-table methods are simple, they lack adaptability to aging. Data-driven methods, such as those employing machine learning, can capture complex nonlinearities but require extensive data and suffer from high computational costs and poor interpretability. Model-based methods, particularly those using Equivalent Circuit Models (ECM), offer a good balance between accuracy and computational efficiency, making them prevalent in BMS applications.
Traditional integer-order ECMs, however, often exhibit significant prediction errors, especially under dynamic loads and varying aging conditions. This limitation stems from their inability to fully characterize the diffusion processes and dynamic polarization behaviors within the lithium-ion battery. Fractional-order calculus, with its non-local memory property, provides a powerful tool for modeling such electrochemical processes with distributed time constants. Consequently, the Fractional-Order Equivalent Circuit Model (FECM) has emerged as a superior alternative, offering higher modeling accuracy and robustness across a wider frequency range.
This paper addresses the problem of low-accuracy peak power prediction for lithium-ion batteries under different aging states. We propose a comprehensive method based on a refined FECM. The main contributions are: (1) Proposing a second-order FECM incorporating a Warburg impedance element to accurately capture the dynamic and steady-state polarization characteristics of the lithium-ion battery. Model parameters are identified using a Particle Swarm Optimization (PSO) algorithm. (2) Developing an aging-adaptive SOC estimation algorithm by integrating a decay coefficient into a Fractional-order Adaptive Extended Kalman Filter (FAEKF) framework. This enhances the algorithm’s ability to track parameter variations caused by aging. (3) Formulating a multi-parameter-constrained peak power prediction algorithm based on the identified FECM and the estimated SOC. The constraints include terminal voltage limits, SOC limits, and design current limits, ensuring safe and accurate SOP prediction throughout the battery’s life.
1. Fractional-Order Modeling of Lithium-ion Battery
The proposed Fractional-Order Equivalent Circuit Model (FECM) for the lithium-ion battery is depicted below. It consists of an ohmic resistor \(R_0\), two Constant Phase Elements (CPE1 and CPE2) in parallel with resistors \(R_1\) and \(R_2\) to model short-term and mid-term dynamics, and a Warburg element (\(W\)) to represent the diffusion process at low frequencies. The Open Circuit Voltage (OCV) is represented as a function of SOC, \(U_{OCV}(SOC)\).
$$U_{CPE1}$$
$$U_{CPE2}$$
$$I \rightarrow R_0 \rightarrow \begin{array}{c} R_1 \\ | \\ CPE1 \end{array} \rightarrow \begin{array}{c} R_2 \\ | \\ CPE2 \end{array} \rightarrow W \rightarrow U_{OCV}(SOC)$$
$$U_d$$
The impedance of a CPE is given by \(Z_{CPE}(s) = 1/(C s^{\alpha})\), where \(\alpha\) is the fractional order (\(0 < \alpha < 1\)). Similarly, the Warburg impedance is \(Z_W(s) = W / \sqrt{s^{\gamma}}\). The use of fractional orders \(\alpha\), \(\beta\) (for CPE2), and \(\gamma\) allows the model to accurately represent the distributed time-constant behavior inherent in the electrochemical processes of the lithium-ion battery.
The discrete state-space representation of the FECM is derived using the Grünwald–Letnikov definition for fractional derivatives. The system state vector is defined as \(x_k = [U_{CPE1,k}, U_{CPE2,k}, U_{W,k}, SOC_k]^T\). The input is the current \(I_k\) (discharge positive), and the output is the terminal voltage \(U_{d,k}\). The discrete equations are:
$$
\begin{aligned}
x_{k+1} &= A_k x_k + B_k u_k + \sum_{h=1}^{N_\delta} \mathbf{\Gamma}_h x_{k+1-h} + w_k \\
y_k &= U_{OCV}(SOC_k) + C_k x_k + D_k u_k + v_k
\end{aligned}
$$
where \(w_k\) and \(v_k\) are process and measurement noise, respectively. The matrices are defined as follows:
$$
A_k = \begin{bmatrix}
1 – \frac{T}{R_1 C_1^{\alpha}} & 0 & 0 & 0 \\
0 & 1 – \frac{T}{R_2 C_2^{\beta}} & 0 & 0 \\
0 & 0 & 1 – \frac{T}{W^{\gamma}} & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}, \quad B_k = \begin{bmatrix}
\frac{T}{C_1^{\alpha}} \\
\frac{T}{C_2^{\beta}} \\
\frac{T}{W^{\gamma}} \\
-\frac{\eta T}{3600 Q_n}
\end{bmatrix}
$$
$$
C_k = [-1, -1, -1, 0], \quad D_k = -R_0
$$
The coefficient matrix for the fractional-order memory terms \(\mathbf{\Gamma}_h\) is a diagonal matrix containing the discrete binomial coefficients \(w^{(\alpha)}(h)\), \(w^{(\beta)}(h)\), and \(w^{(\gamma)}(h)\) for the respective states, which are calculated recursively: \(w^{(m)}(h) = (1 – \frac{m+1}{h}) w^{(m)}(h-1)\), with \(w^{(m)}(0)=1\).
2. Parameter Identification and Adaptive SOC Estimation
2.1 Parameter Identification via PSO
The parameter vector for the FECM of the lithium-ion battery is \(\theta = [R_0, R_1, C_1, \alpha, R_2, C_2, \beta, W, \gamma]\). These parameters are identified offline using experimental data from a Hybrid Pulse Power Characterization (HPPC) test. The PSO algorithm is employed to minimize the Root Mean Square Error (RMSE) between the model output voltage and the measured voltage. The fitness function is:
$$ J(\theta) = \sqrt{\frac{1}{N} \sum_{k=1}^{N} (U_{d,k}^{meas} – U_{d,k}^{model}(\theta))^2} $$
2.2 SOC Estimation with Aging Adaptation
Battery aging leads to capacity fade and parameter drift, which degrades the accuracy of model-based SOC estimation. To address this, we propose an adaptive strategy combining online parameter identification with a decay coefficient (\(\lambda\))-based correction within a Fractional-order Adaptive Extended Kalman Filter (FAEKF).
A. Decay Coefficient for Model-Error Compensation:
During operation, the terminal voltage error \(e_k = z_k – \hat{y}_k\) is monitored. A decay coefficient \(\lambda\) is dynamically adjusted to compensate for persistent model errors indicative of parameter mismatch due to aging:
$$
\lambda =
\begin{cases}
1 – \zeta \times \frac{|e_k|}{|e_k|} \times 100\%, & \text{if } |e_k| \leq \epsilon \\
1 – \zeta \times 100\%, & \text{if } |e_k| > \epsilon
\end{cases}
$$
where \(\zeta\) is an error amplification factor (typically 0.3), and \(\epsilon\) is an error threshold. This adjustment scale the Kalman gain, making the estimator more responsive to measurements when the model error is large.
B. Capacity Calibration:
The actual usable capacity \(Q_n\) of the lithium-ion battery is calibrated periodically using the coulomb counting method between two SOC estimation points:
$$ Q_n = \frac{\int_{t_q}^{t_p} I(t) dt}{SOC_p – SOC_q} $$
The effective capacity used for SOC calculation is then \(Q_{eff} = \lambda \cdot Q_n\), providing adaptation to capacity fade.
C. FAEKF Algorithm with Decay Coefficient:
The standard FAEKF time-update and measurement-update steps are modified. The Kalman gain calculation incorporates \(\lambda\) to adjust the reliance on the measurement noise covariance:
$$ K_k = P_k^- C_k^T (C_k P_k^- C_k^T + \lambda R)^{-1} $$
Furthermore, the process noise covariance \(Q\) and measurement noise covariance \(R\) are adapted online using an innovation sequence, enhancing the filter’s robustness to changing noise statistics.
3. Peak Power Prediction Based on Multi-Parameter Constraints
The peak power \(SOP\) for a time horizon of \(L \cdot T\) seconds (where \(T\) is the sampling period) is defined as:
$$
\begin{aligned}
SOP_{L}^{ch} &= U_{d, k+L}^{ch} \times I_{L, min}^{ch} \quad \text{(Charging)} \\
SOP_{L}^{dis} &= U_{d, k+L}^{dis} \times I_{L, max}^{dis} \quad \text{(Discharging)}
\end{aligned}
$$
where \(I_{L, min}^{ch} (<0)\) and \(I_{L, max}^{dis} (>0)\) are the peak currents limited by multiple constraints, and \(U_{d, k+L}\) is the predicted terminal voltage at the end of the horizon.
3.1 Calculation of Constrained Peak Currents
We derive the future terminal voltage \(U_{d,k+L}\) as a function of a constant future current \(I_L\) over the horizon \(L\), based on the FECM state-space model. Using recursive derivation and a first-order Taylor expansion for \(U_{OCV}(SOC_{k+L})\), we obtain a linear relationship:
$$ U_{d,k+L} = U_{OCV}(SOC_k) – a_1 U_{CPE1,k} – a_2 U_{CPE2,k} – a_3 U_{W,k} – F(L) \cdot I_L $$
where \(a_1, a_2, a_3\) are constants derived from the model parameters, and \(F(L)\) is a complex function of \(L\), the model parameters (\(R_0, R_1, C_1, \alpha, R_2, C_2, \beta, W, \gamma\)), and the OCV-SOC curve slope \(\frac{dU_{OCV}}{dSOC}\).
The peak currents are then determined by solving for \(I_L\) under three separate constraints, and taking the most restrictive one.
1. Voltage Limit Constraint:
The terminal voltage must remain within safe limits: \(U_{d,min} \leq U_{d,k+L} \leq U_{d,max}\).
$$
\begin{aligned}
I_{L,U}^{ch} &= \frac{U_{OCV}(SOC_k) – a_1 U_{CPE1,k} – a_2 U_{CPE2,k} – a_3 U_{W,k} – U_{d,max}}{F(L)} \\
I_{L,U}^{dis} &= \frac{U_{OCV}(SOC_k) – a_1 U_{CPE1,k} – a_2 U_{CPE2,k} – a_3 U_{W,k} – U_{d,min}}{F(L)}
\end{aligned}
$$
2. SOC Limit Constraint:
The SOC must remain within limits (e.g., 0.1 and 1.0). Assuming a constant current \(I_L\) over \(L\) steps:
$$
\begin{aligned}
I_{L,SOC}^{ch} &= \frac{(SOC_{max} – SOC_k) \cdot 3600 Q_n}{\eta L T} \\
I_{L,SOC}^{dis} &= \frac{(SOC_{min} – SOC_k) \cdot 3600 Q_n}{\eta L T}
\end{aligned}
$$
3. Design Current Limit:
The current must respect the battery’s specified maximum charge (\(I_{min}^{ch}\)) and discharge (\(I_{max}^{dis}\)) currents.
The final feasible peak currents for the horizon \(L\) are the most restrictive among all constraints:
$$
\begin{aligned}
I_{L,min}^{ch} &= \max\{ I_{min}^{ch}, I_{L,SOC}^{ch}, I_{L,U}^{ch} \} \\
I_{L,max}^{dis} &= \min\{ I_{max}^{dis}, I_{L,SOC}^{dis}, I_{L,U}^{dis} \}
\end{aligned}
$$
3.2 Peak Power Prediction
With the peak currents \(I_{L,min}^{ch}\) and \(I_{L,max}^{dis}\) determined, the corresponding future terminal voltages \(U_{d,k+L}^{ch}\) and \(U_{d,k+L}^{dis}\) are calculated using the voltage equation. The peak charging and discharging powers are then:
$$
\begin{aligned}
SOP_{L}^{ch} &= U_{d,k+L}^{ch} \times I_{L,min}^{ch} \\
SOP_{L}^{dis} &= U_{d,k+L}^{dis} \times I_{L,max}^{dis}
\end{aligned}
$$
4. Experimental Results and Analysis
Tests were conducted on a 32.5Ah ternary lithium-ion battery at 25°C. The test matrix included HPPC tests for model parameter identification and Urban Dynamometer Driving Schedule (UDDS) tests for validation and SOP prediction under dynamic profiles. The battery was cycled over 300 times to collect data at different aging states (0, 100, 200, and 300 cycles).
4.1 Model Validation and SOC Estimation
The FECM parameters identified via PSO from HPPC data are listed in Table 1.
| Parameter | Value |
|---|---|
| \(R_0\) | 0.0035 Ω |
| \(R_1\) | 0.0026 Ω |
| \(C_1\) | 14710 Fα |
| \(\alpha\) | 0.87 |
| \(R_2\) | 0.00045 Ω |
| \(C_2\) | 2387 Fβ |
| \(\beta\) | 0.78 |
| \(W\) | 5011 Ω sγ/2 |
| \(\gamma\) | 0.41 |
The model’s voltage estimation performance under a UDDS profile is excellent. The FECM achieves a maximum voltage error of less than 20 mV and an RMSE of 7.95 mV, outperforming a standard second-order integer-order ECM.
The SOC estimation results using the proposed adaptive FAEKF with decay coefficient are shown in Figure 5. The method effectively combines the FECM’s accuracy with the adaptivity of the decay coefficient, achieving a maximum absolute SOC error of less than 0.02 and rapid convergence. The decay coefficient mechanism successfully compensates for model inaccuracies due to aging, as evidenced by the superior performance of “DC FECM” compared to the standard “FECM” and integer-order counterparts (“ECM”, “DC ECM”).
4.2 Peak Power Prediction Results
Figure 6 illustrates the peak currents under different single constraints and the final combined constraint for prediction horizons of 30s and 60s. The dominant constraint shifts dynamically: from the design current limit at high SOC, to the voltage limit at mid-to-low SOC, and finally to the SOC limit near the cutoff. This demonstrates the necessity of multi-constraint consideration for accurate lithium-ion battery power prediction.
The peak power prediction results for a 30s horizon under the UDDS profile are shown in Figure 7(a). The proposed method (“DC FECM”) closely follows the reference power (calculated offline using the actual limits). The corresponding prediction error is significantly lower than that of other methods. A similar trend is observed for the 60s horizon in Figure 7(b).
4.3 Performance Under Different Aging States
The proposed method’s robustness against aging is validated using data from different cycle counts. Table 2 summarizes the peak power prediction statistics (maximum/average power and error) for four models at 0, 100, 200, and 300 cycles.
| Cycle | Model | Max Power (W) | Avg Power (W) | Max Error (W) | Avg Error (W) |
|---|---|---|---|---|---|
| 0 | ECM | 236.5 | 207.3 | 33.6 | 5.50 |
| FECM | 240.4 | 210.9 | 23.2 | 2.24 | |
| DC ECM | 238.2 | 208.5 | 29.9 | 4.37 | |
| DC FECM | 241.2 | 211.6 | 15.2 | 1.45 | |
| 100 | ECM | 225.9 | 205.7 | 28.8 | 5.63 |
| FECM | 224.2 | 204.9 | 22.8 | 4.57 | |
| DC ECM | 228.5 | 205.1 | 25.5 | 5.08 | |
| DC FECM | 223.6 | 195.1 | 10.7 | 1.69 | |
| 200 | ECM | 207.0 | 192.8 | 26.8 | 5.36 |
| FECM | 211.2 | 188.9 | 20.4 | 3.84 | |
| DC ECM | 209.4 | 190.3 | 22.6 | 4.56 | |
| DC FECM | 212.0 | 187.1 | 8.43 | 1.46 | |
| 300 | ECM | 166.5 | 128.3 | 28.3 | 6.16 |
| FECM | 169.4 | 131.6 | 19.1 | 4.86 | |
| DC ECM | 167.3 | 130.5 | 25.5 | 5.20 | |
| DC FECM | 170.4 | 138.8 | 8.33 | 2.32 |
Key observations are:
- Aging Impact: The maximum and average available peak power of the lithium-ion battery decrease significantly with cycling. At 300 cycles (~80% state of health), the maximum power is only about 70% of its initial value.
- Method Superiority: The proposed “DC FECM” method consistently achieves the lowest prediction error (both maximum and average) across all aging states compared to the integer-order ECM and the standard FECM without the decay coefficient.
- Error Reduction: The maximum prediction error of the proposed method is kept below 16W (less than 6.25% of the typical power range), which represents an average reduction of over 5% compared to the integer-order model across different aging cycles.
Figure 8 visually compares the predicted peak power and error for the “DC FECM” and standard “FECM” methods at 100, 200, and 300 cycles. The “DC FECM” method shows superior tracking accuracy and lower error magnitude, particularly in the mid-to-late stages of discharge where aging effects and dynamic constraints are most pronounced.
5. Conclusion
This paper presented a novel method for predicting the peak power of lithium-ion batteries under different aging states. The core of the method is a Fractional-Order Equivalent Circuit Model (FECM) that accurately captures the battery’s dynamic behavior. An aging-adaptive SOC estimation algorithm was developed by integrating a decay coefficient into a Fractional-order Adaptive Extended Kalman Filter, which effectively compensates for capacity fade and parameter drift. Finally, a multi-parameter-constrained peak power prediction algorithm was formulated based on the identified FECM and the estimated SOC.
Experimental validation using dynamic UDDS profiles at various cycle lives demonstrated the effectiveness of the proposed approach:
- The FECM achieves high voltage modeling accuracy with errors below 20 mV.
- The adaptive SOC estimation maintains a maximum error of less than 0.02.
- The peak power prediction error is kept within 15W (under 6.25%), significantly outperforming traditional integer-order model-based methods. The proposed method reduces the maximum prediction error by approximately 5% across different aging cycles.
The proposed framework provides a accurate and robust solution for real-time peak power prediction in battery management systems, enhancing the safety and performance utilization of lithium-ion batteries throughout their operational life.