In order to adapt the energy storage cell to the complex operating conditions of modern power systems, such as peak shaving and frequency regulation, an improved recursive least squares (RLS) algorithm is proposed in this work. This algorithm consists of two mutually constrained RLS estimators. A dedicated charge-discharge pulse current profile is designed for the energy storage cell, and the improved RLS algorithm is employed for parameter identification. A third-order equivalent circuit model of the energy storage cell is constructed and validated in MATLAB/Simulink. The simulation results show that under the FUDS (Federal Urban Driving Schedule) test, the maximum voltage error is less than 2.68%, and under the DST (Dynamic Stress Test) test, the error is less than 3.04%. These results verify the accuracy and effectiveness of the proposed modeling approach, providing a robust method for building energy storage cell models under demanding operational scenarios.
Accurate modeling of the energy storage cell is fundamental to battery management systems. Among various modeling approaches, the equivalent circuit model (ECM) is widely adopted due to its simplicity and real-time capability. The recursive least squares (RLS) algorithm, as an online identification method, has been extensively used for ECM parameter estimation. However, conventional RLS often fails to maintain precision under rapidly changing load profiles typical of grid ancillary services. To address this limitation, we extend the second-order RC model to a third-order RC model and develop an improved RLS structure comprising two interdependent RLS blocks. This design enables simultaneous estimation of multiple time constants and resistances, thereby enhancing the dynamic representation of the energy storage cell.
1. Equivalent Circuit Model of Lithium-Ion Energy Storage Cell
1.1 Third-Order RC Model
The proposed model consists of a controlled voltage source representing the open-circuit voltage (OCV), an ohmic resistor \( R_0 \), and three parallel RC networks in series. Each RC pair captures polarization effects with different time constants. The state equations in the s-domain are given by:
\[
U_{OC}(s) = R_0 I(s) + U_1(s) + U_2(s) + U_3(s) + U(s)
\tag{1}
\]
\[
U_i(s) = \frac{R_i}{\tau_i s + 1} I(s), \quad i = 1,2,3
\tag{2}
\]
where \(\tau_i = R_i C_i\) is the time constant of each RC branch. In the zero-input response, the voltage across each RC network decays exponentially:
\[
U_i(t) = U_{i0} e^{-t/\tau_i}
\tag{3}
\]
1.2 Relationship Between OCV and SOC
The energy storage cell used in this study is a Panasonic NCR18650B with a nominal capacity of 3200 mAh. Open-circuit voltage (OCV) tests were conducted at 1 C rate using a NEWARE BT4000 tester. The discharge OCV data are listed in Table 1.
| SOC (%) | 100 | 90 | 80 | 70 | 60 | 50 | 40 | 30 | 20 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Charge OCV (V) | 4.201 | 4.046 | 3.911 | 3.834 | 3.793 | 3.672 | 3.602 | 3.516 | 3.394 | 3.297 |
| Discharge OCV (V) | 4.184 | 4.035 | 3.896 | 3.815 | 3.714 | 3.618 | 3.540 | 3.466 | 3.384 | 3.251 |
Using curve fitting, the OCV-SOC function is obtained as:
\[
U_{OC}(S) = -67.431 S^7 + 236.074 S^6 – 323.335 S^5 + 218.610 S^4 – 74.523 S^3 + 10.866 S^2 + 0.7983 S + 3.138
\tag{4}
\]
2. Improved Parameter Identification Method for Energy Storage Cell
2.1 Test Profile Design
To emulate the grid-scale frequency regulation duty, a composite current profile is designed. It includes a constant-current discharge phase (0.3 C, 1200 s), a rest phase (3400 s), fast pulse changes (2.5 s interval) simulating primary frequency regulation, and slow pulse changes (10 s interval) simulating secondary frequency regulation. More than 71% of primary regulation power bursts last less than 2 minutes, while 95.52% of secondary regulation bursts last less than 5 minutes. This profile challenges the energy storage cell across wide SOC and dynamic conditions.
2.2 Improved RLS Algorithm Architecture
The conventional RLS algorithm is extended by introducing two interdependent RLS estimators. The first RLS focuses on the slow dynamics (large time constant \(\tau_3\)), while the second RLS focuses on fast dynamics (\(\tau_1\) and \(\tau_2\)). A mutual constraint mechanism is embedded: the error from one RLS updates the forgetting factor and gain of the other. The complete identification procedure is outlined below.
Step 1: Initialize parameters.
Step 2: Using the backward difference approximation \( s = (1 – z^{-1})/T \), transform the continuous-time model into discrete form. Define the voltage difference:
\[
U_{120}(k) = U_k – U_3(k) – U(k)
\tag{5}
\]
\[
U_{120}(k) = a(k) U_{120}(k-1) + b(k) I(k) – c(k) I(k-1)
\tag{6}
\]
where
\[
a = \frac{\tau_2}{T + \tau_2}, \quad b = \frac{T(R_0 + R_1 + R_2) + (R_0 + R_1)\tau_2}{T + \tau_2}, \quad c = \frac{(R_0 + R_1)\tau_2}{T + \tau_2}
\tag{7}
\]
Step 3: Similarly, for the fast dynamics branch, define:
\[
U_{10}(k) = U_k – U_2(k) – U_3(k) – U(k)
\tag{8}
\]
\[
U_{10}(k) = d(k) U_{10}(k-1) + f(k) I(k) – g(k) I(k-1)
\tag{9}
\]
where
\[
d = \frac{\tau_1}{T + \tau_1}, \quad f = \frac{T(R_0 + R_1) + R_0\tau_1}{T + \tau_1}, \quad g = \frac{R_0\tau_1}{T + \tau_1}
\tag{10}
\]
Step 4: From the identified coefficients \(a,b,c,d,f,g\), solve for \(\tau_1, \tau_2, R_0, R_1, R_2\).
Step 5: Using the voltage drop during rest and discharge:
\[
U_G – U_C = I (R_0 + R_1 + R_2 + R_3)
\tag{11}
\]
compute \(R_3\).
Step 6: Compute prediction errors \(e_1(k)\) for \(U_{120}\) and \(e_2(k)\) for \(U_{10}\).
Step 7: Update \(U_2(k)\) and \(U_3(k)\) using:
\[
U_3(k+1) = \frac{\tau_3}{T + \tau_3} U_3(k) + \frac{T R_3(k)}{T + \tau_3} I(k)
\tag{12}
\]
\[
U_2(k+1) = \frac{\tau_2}{T + \tau_2} U_2(k) + \frac{T R_2(k)}{T + \tau_2} I(k)
\tag{13}
\]
Step 8: Adjust the RLS gain matrices based on \(e_1(k)\) and \(e_2(k)\). The algorithm iterates until both errors converge below a predefined threshold.
3. Model Validation
3.1 Identified Parameters
Applying the improved RLS algorithm across the full SOC range (100% to 0%), the identified parameters are summarized in Table 2.
| SOC (%) | \(R_0\) (Ω) | \(\tau_1\) (s) | \(C_1\) (F) | \(R_2\) (Ω) | \(\tau_2\) (s) | \(R_3\) (Ω) | \(\tau_3\) (s) |
|---|---|---|---|---|---|---|---|
| 100 | 0.0839 | 0.0870 | 0.0456 | 0.0048 | 0.6240 | 0.0313 | 49.3261 |
| 90 | 0.1039 | 0.1080 | 0.0458 | 0.0050 | 0.5913 | 0.1135 | 54.8481 |
| 80 | 0.0897 | 0.1010 | 0.0460 | 0.0122 | 0.2604 | 0.1104 | 77.4731 |
| 70 | 0.0868 | 0.1007 | 0.0483 | 0.0139 | 0.2238 | 0.1135 | 82.4747 |
| 60 | 0.0811 | 0.0943 | 0.0458 | 0.0132 | 0.2248 | 0.1115 | 77.4731 |
| 50 | 0.0649 | 0.0793 | 0.0452 | 0.0144 | 0.2228 | 0.1115 | 82.4747 |
| 40 | 0.0715 | 0.0831 | 0.0520 | 0.0116 | 0.2289 | 0.1104 | 97.4786 |
| 30 | 0.0808 | 0.0932 | 0.0457 | 0.0125 | 0.2263 | 0.0865 | 60.3330 |
| 20 | 0.0992 | 0.1019 | 0.0363 | 0.0028 | 1.0890 | 0.0896 | 104.8443 |
| 10 | 0.1038 | 0.1145 | 0.0367 | 0.0038 | 0.3051 | 0.1042 | 101.4598 |
| 0 | 0.0873 | 0.0960 | 0.0459 | 0.0087 | 0.4532 | 0.1208 | 109.9083 |
3.2 Validation Under FUDS and DST
To evaluate the accuracy, the proposed third-order model with improved RLS (denoted as Model 1) is compared with a conventional second-order model identified by standard RLS (Model 2). The FUDS test simulates an urban driving pattern for electric vehicles, while the DST test represents a dynamic stress scenario. The voltage tracking errors are recorded.
| Test | Model | Max absolute error (V) | Max relative error (%) | RMSE (V) |
|---|---|---|---|---|
| FUDS | Model 1 (Proposed) | 0.098 | 2.68 | 0.045 |
| Model 2 (Conventional RLS) | 0.214 | 5.91 | 0.112 | |
| DST | Model 1 (Proposed) | 0.112 | 3.04 | 0.052 |
| Model 2 (Conventional RLS) | 0.245 | 6.78 | 0.129 |
The results demonstrate that the improved RLS-based third-order model significantly reduces voltage estimation errors. The maximum relative error under FUDS is 2.68% and under DST is 3.04%, compared to 5.91% and 6.78% for the conventional method. This improvement is particularly notable during rapid current transients, where the dual-RLS architecture accurately captures the polarization dynamics of the energy storage cell.
4. Conclusion
This paper presents a refined modeling methodology for the energy storage cell based on an improved RLS algorithm. A third-order RC equivalent circuit model is adopted, and a novel dual-RLS structure with mutual constraints is proposed for parameter identification. The OCV-SOC relationship is accurately characterized through polynomial fitting. A realistic test profile mimicking grid frequency regulation is designed, and the model is validated under FUDS and DST conditions. The simulation results confirm that the proposed method achieves high accuracy with maximum voltage errors below 3.04%, outperforming conventional RLS-based models. This work provides a reliable foundation for state-of-charge estimation, health monitoring, and optimal control of energy storage cells in advanced power systems.