Lithium-ion batteries serve as the core energy storage and release component in a traction battery pack. The real-time and accurate estimation of their State of Charge (SOC) is an essential premise for the Battery Management System (BMS) to perform effective monitoring, optimal energy allocation, and safety warning. In my research, I propose a robust estimation method that works under multiple operating conditions, thereby helping to improve the overall performance of electric vehicles, prolong the service life of the traction battery pack, reduce energy consumption, and enhance safety.
In this thesis, I focus on the nonlinear characteristics of a lithium-ion traction battery pack, which are strongly affected by many coupled factors. I first studied the working principle and electrical characteristics, including voltage characteristics, internal resistance characteristics, and capacity characteristics. Then, I designed charge/discharge experiments on a lithium-ion cell to collect data, and through data processing, I revealed the relationship between battery capacity and temperature, aging degree, and discharge rate. Because SOC estimation normally requires an accurate battery equivalent model, I compared several battery models, such as electrochemical models, neural network models, and equivalent circuit models. I selected a second-order RC equivalent circuit model, which effectively represents the working characteristics of a traction battery pack with limited computational load. Then, the Hybrid Pulse Power Characteristic (HPPC) test was designed to gather dynamic voltage data. Based on these data, I identified all model parameters and generated the open-circuit voltage versus SOC curve. After curve fitting, I built a function expression to link open-circuit voltage with SOC.
The Extended Kalman Filter (EKF) is widely used in SOC estimation for a traction battery pack. However, when the internal state changes severely, or when strong measurement noise appears, the performance of EKF can degrade significantly, causing unacceptable SOC estimation error. In addition, EKF is very sensitive to the initial state value, while the initial SOC of a traction battery pack is often difficult to obtain accurately. To solve these problems, I propose a Correct Covariance Strong Tracking Extended Kalman Filter (CSTEKF). This algorithm combines the advantages of the Strong Tracking Extended Kalman Filter (STEKF) and a corrected covariance technique. It can effectively overcome the adverse effects of strong noise and drastic state changes, and can provide higher estimation accuracy. The strong tracking filter improves the resistance to sudden changes by adjusting the update mechanism. The corrected covariance improves the adaptability to noise and model uncertainties by modifying the covariance update strategy. Finally, I verified the algorithm performance in simulations. The results show that CSTEKF can overcome the influence of inaccurate initial SOC values, and compared with EKF and STEKF, CSTEKF significantly improves the SOC estimation accuracy of a traction battery pack, thus offering a more reliable SOC estimation result for BMS.
1 Lithium-Ion Battery Principles and SOC-Influencing Factors
1.1 Working Principle of a Traction Battery Pack
The lithium-ion cells that compose a traction battery pack use lithium metal oxide as the positive electrode and graphite as the negative electrode. A polymer separator with selective ion permeability is placed between the electrodes; this separator only allows lithium ions to pass through while blocking electrons. During charging, lithium ions are extracted from the positive electrode lattice, move through the electrolyte, and insert into the graphite negative electrode. Meanwhile, electrons travel through the external circuit to the negative electrode. During discharging, the reverse reaction occurs: lithium ions escape from the negative electrode and return to the positive electrode, while electrons pass through the load circuit to form a closed loop. This lithium-ion intercalation/deintercalation mechanism is the core principle for energy storage and release in a traction battery pack.
1.2 Voltage, Internal Resistance, and Capacity Characteristics
The voltage characteristics of a lithium-ion cell in a traction battery pack include open-circuit voltage (OCV), working voltage, and voltage plateau. The OCV has a deterministic, nonlinear relationship with SOC. However, because of polarization effects, the working voltage is always lower than the theoretical electromotive force during discharge and higher during charge. The voltage plateau results from the coexistence of two phases inside electrode materials, leading to a nearly constant terminal voltage in a specific SOC range.
The internal resistance of a cell can be divided into ohmic resistance, polarization resistance, and contact resistance. The ohmic resistance originates from the resistance of the electrode materials, electrolyte, and current collectors. The polarization resistance is related to electrochemical reaction kinetics and concentration gradients. The internal resistance is highly dependent on temperature, SOC, and aging level. For instance, at low temperature, the electrolyte viscosity increases, so the migration rate of lithium ions decreases, causing elevated internal resistance. During cycling, the continuous growth of the solid electrolyte interphase and the loss of active material produce an irreversible increase in internal resistance. Therefore, the impedance characteristics must not be ignored when estimating SOC of a traction battery pack.
Capacity is a fundamental parameter for SOC calculation. The theoretical capacity is based on the complete reaction of active materials; the rated capacity is defined under standard test conditions; and the actual available capacity strongly depends on temperature, discharge rate, and state of health. My experiments quantified these effects. I used a cell with a rated capacity of 72 Ah and measured the effective capacity at temperatures from –20 °C to 55 °C. The normalized actual capacity as a function of temperature is shown in Table 1.
| Temperature (°C) | Available capacity (Ah) | Capacity retention (%) |
|---|---|---|
| –20 | 63.0 | 87.5 |
| –10 | 66.5 | 92.4 |
| 0 | 68.8 | 95.6 |
| 15 | 71.0 | 98.6 |
| 25 | 72.0 | 100 |
| 55 | 73.2 | 101.7 |
Figure 1 shows a typical electric vehicle battery pack architecture. The image is placed here to illustrate the physical construction of a modern traction battery pack.
I also conducted accelerated aging experiments on a 2.5 Ah commercial cell. The cell was charged at 1 C constant current to 4.2 V, then held at constant voltage until the current decayed to 0.05 C. After resting, the cell was discharged at 1 C to 2.8 V. This cycle was repeated 200 times. The measured capacity retention at different cycle numbers is summarized in Table 2.
| Cycle number | Capacity retention (%) |
|---|---|
| 0 | 100 |
| 50 | 99.2 |
| 100 | 98.5 |
| 150 | 97.8 |
| 200 | 97.0 |
Discharge rate also affects the available capacity. I discharged a 72 Ah cell at 0.3 C, 0.5 C, 1 C, 2 C, 3 C, and 5 C. The terminal voltage versus discharged capacity curves clearly show that lower discharge rates lead to a higher voltage at the same discharged capacity and a flat voltage plateau. Under high discharge rates, the voltage drops rapidly near the end of discharge. In a traction battery pack, the discharge rate is typically less than 1.5 C; therefore, I did not need to correct the actual capacity for the discharge-rate effect when estimating the SOC.
1.3 Definition and Influencing Factors of SOC
The SOC of a traction battery pack is defined as the ratio of the remaining charge to the rated capacity:
$$
SOC(t) = \frac{Q_t}{Q_N} \times 100\%
$$
or equivalently,
$$
SOC(t) = \left( \frac{Q_0 – \int_{0}^{t} \eta I(\tau) d\tau}{Q_N} \right) \times 100\%
$$
Here, \(Q_0\) is the initial charge, \(Q_N\) is the rated capacity, \(I(\tau)\) is the instantaneous current, and \(\eta\) is the coulombic efficiency. Since the actual usable capacity depends on temperature and aging, I use the following expression for the real-time calculation of SOC:
$$
SOC(t) = \frac{Q_c}{Q_N} = \frac{\eta Q_N}{Q_N} = \eta
$$
Actually, the last expression is oversimplified; in practice, the capacity \(Q_c\) in the real SOC model is obtained by correcting \(Q_N\) with the temperature factor, aging factor, and rate factor. These corrected capacity terms are the basis for accurate SOC estimation of a traction battery pack across various operating conditions.
2 Modeling and Parameter Identification of the Traction Battery Pack
2.1 Selection of Battery Equivalent Circuit Model
Several types of battery models exist: electrochemical models, neural network models, and equivalent circuit models. Electrochemical models are based on first-principles partial differential equations and have very high accuracy, but they require intense computation and many parameters that are difficult to obtain in an embedded BMS. Neural network models can learn arbitrary nonlinear relations; however, they need extensive training data and are difficult to use in a traction battery pack when the operating conditions vary over a wide range. Equivalent circuit models use circuit components to describe the electrical dynamic behavior of a cell. Among them, the Rint model is too simple and cannot represent the polarization effect. The Thevenin model introduces one RC network and can describe the transient voltage response, but is insufficient for the long-time-constant diffusion process. The PNGV model adds a series capacitor to simulate the open-circuit voltage variation due to current accumulation; however, the error increases with time because of the integral effect. The GNL model has multiple RC networks and a self-discharge resistor; it is accurate but has too many parameters to identify easily online.
Considering the tradeoff between model precision and computational workload, I selected a second-order RC equivalent circuit model for the traction battery pack. The model consists of a voltage source \(U_{OCV}\), an ohmic resistance \(R_0\), and two RC parallel networks: \(R_1 C_1\) and \(R_2 C_2\). The first RC network represents the electrochemical polarization, and the second RC network represents the concentration polarization. The terminal voltage satisfies Kirchhoff’s voltage law:
$$
\begin{cases}
U_T = U_{OCV} – U_1 – U_2 – I_T R_0,\\[4pt]
U_1 = I_T R_1 – C_1 R_1 \frac{d U_1}{dt},\\[4pt]
U_2 = I_T R_2 – C_2 R_2 \frac{d U_2}{dt}.
\end{cases}
$$
In addition, the SOC dynamics is given by
$$
SOC_t = SOC_0 – \frac{\int_{0}^{t} \eta I(\tau) d\tau}{C_N},
$$
where \(C_N\) is the available capacity of the cell at the current state. Combining these equations, the state-space representation of the equivalent circuit model is
$$
\begin{bmatrix}
\dot{U}_{OCV} \\[4pt]
\dot{U}_1 \\[4pt]
\dot{U}_2
\end{bmatrix}
=
\begin{bmatrix}
0 & 0 & 0 \\[4pt]
0 & -\frac{1}{R_1 C_1} & 0 \\[4pt]
0 & 0 & -\frac{1}{R_2 C_2}
\end{bmatrix}
\begin{bmatrix}
U_{OCV} \\[4pt]
U_1 \\[4pt]
U_2
\end{bmatrix}
+
\begin{bmatrix}
-\frac{1}{C_N} \\[4pt]
\frac{1}{C_1} \\[4pt]
\frac{1}{C_2}
\end{bmatrix}
I_T ,
$$
$$
U_T =
\begin{bmatrix} 1 & -1 & -1 \end{bmatrix}
\begin{bmatrix}
U_{OCV} \\[4pt]
U_1 \\[4pt]
U_2
\end{bmatrix}
– I_T R_0 .
$$
2.2 HPPC Test and OCV-SOC Relationship
To identify the model parameters of a traction battery pack, I carried out a Hybrid Pulse Power Characteristic (HPPC) test at 25 ± 1 °C. The test procedure was:
- Charge the cell at 1 C constant current to 4.2 V, then continue with constant voltage charging until the current is lower than 0.05 C.
- Rest the cell for 120 min.
- Discharge at a constant current of 1 C for 180 s, then rest for 120 min.
- Repeat the discharge–rest cycle until the terminal voltage reaches 2.7 V.
- At the end, rest for another 120 min.
The entire HPPC test lasted about 33 h. I sampled the terminal voltage every 0.1 s and obtained 16 pulse cycles. For each pulse cycle, I recorded the voltage response. Figure 3 in my original thesis shows a typical one-pulse voltage curve: The voltage drops abruptly from point A to point B when the current is applied because of the ohmic resistance. Then the voltage decreases slowly from B to C because the RC networks are being charged. When the current is interrupted, the voltage jumps up from C to D due to the disappearance of the ohmic drop, followed by a gradual relaxation from D to E as the polarization voltages decay.
After each discharge pulse and a sufficient rest, I recorded the OCV value for the corresponding SOC. The measured OCV-SOC points are given in Table 3.
| SOC (%) | OCV (V) |
|---|---|
| 10 | 3.328 |
| 20 | 3.416 |
| 30 | 3.462 |
| 40 | 3.498 |
| 50 | 3.539 |
| 60 | 3.603 |
| 70 | 3.705 |
| 80 | 3.812 |
| 90 | 4.033 |
| 100 | 4.189 |
Using the least-squares regression technique in MATLAB, I fitted a polynomial of order seven to the discrete OCV-SOC data. The resulting expression is:
$$
U_{OCV}(SOC) = 34.7 SOC^7 – 117.6 SOC^6 + 161.8 SOC^5 – 119.2 SOC^4 + 54.3 SOC^3 – 16.2 SOC^2 + 3.3 SOC + 3.116 .
$$
This polynomial describes the OCV-SOC relation for the cell within a traction battery pack precisely enough for subsequent online SOC estimation.
2.3 Identification of Resistors and Capacitors
To determine \(R_0\), I used the instantaneous voltage jumps across the ohmic resistance. In the pulse response, the voltage drop from A to B and the voltage rise from C to D are both related to \(R_0\). Since the same ohmic resistance acts during current application and removal, I calculated:
$$
R_0 = \frac{(V_A – V_B) + (V_D – V_C)}{2I} .
$$
For the RC parameters, I analyzed the relaxation stage D-E. During this stage, both RC networks are in zero-input response. The terminal voltage satisfies
$$
U_T = U_{OCV} – U_{10} e^{-t/\tau_1} – U_{20} e^{-t/\tau_2} ,
$$
where \(\tau_1 = R_1 C_1\) and \(\tau_2 = R_2 C_2\). The time constants can be estimated using exponential fitting of the voltage-relaxation curve. At the B-C stage, the RC networks are in zero-state response, and the terminal voltage is:
$$
U_T = U_{OCV} – I R_0 – I R_1 \left(1 – e^{-t/\tau_1}\right) – I R_2 \left(1 – e^{-t/\tau_2}\right) .
$$
Combining the fitted B values and time constants, I obtained \(R_1\), \(R_2\), \(C_1\), and \(C_2\). Table 4 presents the identified model parameters at ten different SOC points for a cell in the traction battery pack.
| SOC | \(R_0\) / mΩ | \(R_1\) / mΩ | \(R_2\) / mΩ | \(C_1\) / F | \(C_2\) / F |
|---|---|---|---|---|---|
| 0.1 | 22.8 | 27.8 | 4.7 | 1011.8 | 456.42 |
| 0.2 | 23.7 | 36.1 | 5.2 | 967.71 | 420.79 |
| 0.3 | 24.2 | 28.4 | 5.5 | 962.91 | 355.44 |
| 0.4 | 24.7 | 31.1 | 6.1 | 935.51 | 321.05 |
| 0.5 | 24.8 | 31.5 | 6.7 | 887.07 | 271.71 |
| 0.6 | 25.3 | 37.3 | 7.8 | 863.13 | 238.11 |
| 0.7 | 25.7 | 40.0 | 9.4 | 836.13 | 210.55 |
| 0.8 | 26.8 | 53.9 | 11.8 | 834.82 | 202.01 |
| 0.9 | 28.7 | 104.7 | 6.4 | 754.42 | 214.51 |
The results show that \(R_0\) generally increases as the SOC decreases. Also, the \(R_1 C_1\) network has a much larger capacitance than the \(R_2 C_2\) network, which confirms that concentration polarization dominates the voltage transient over a long time scale in a traction battery pack.
2.4 Model Accuracy Verification
After obtaining all model parameters, I built a simulation model of the traction battery pack cell in MATLAB/Simulink. I applied a variable-current test and compared the simulated terminal voltage with the experimentally measured voltage. The maximum absolute voltage error remained within 0.05 V over the entire test. Therefore, the identified second-order RC model properly represents the dynamic behavior of the cell in a traction battery pack.
3 SOC Estimation Based on CSTEKF
3.1 Problem with the Extended Kalman Filter
The discrete nonlinear system of a lithium-ion cell can be expressed as:
$$
\begin{cases}
x_{k+1} = f(x_k, u_k) + w_k, \\
y_k = g(x_k, u_k) + v_k,
\end{cases}
$$
where \(w_k\) and \(v_k\) are zero-mean Gaussian white noises with covariance matrices \(Q_k\) and \(R_k\), respectively. The EKF linearizes both \(f\) and \(g\) by first-order Taylor expansion around the current estimate. It defines the state-transition Jacobian \(A_k\) and measurement Jacobian \(C_k\). For the second-order RC model, using the state vector \(x = [SOC, U_1, U_2]^T\) and input current \(I_T\), the Jacobians are:
$$
A_k =
\begin{bmatrix}
1 & 0 & 0\\
0 & e^{-\frac{\Delta t}{R_1 C_1}} & 0\\
0 & 0 & e^{-\frac{\Delta t}{R_2 C_2}}
\end{bmatrix},
\quad
B_k =
\begin{bmatrix}
-\frac{\Delta t}{C_N}\\
R_1\left(1-e^{-\frac{\Delta t}{R_1 C_1}}\right)\\
R_2\left(1-e^{-\frac{\Delta t}{R_2 C_2}}\right)
\end{bmatrix},
$$
$$
C_k =
\begin{bmatrix}
\frac{\partial U_{OCV}}{\partial SOC} & -1 & -1
\end{bmatrix}.
$$
The classical EKF algorithm executes time update and measurement update as follows:
$$
\hat{x}_k^- = f(x_{k-1},u_{k-1}),
$$
$$
P_k^- = A_{k-1} P_{k-1} A_{k-1}^T + Q_{k-1},
$$
$$
K_k = P_k^- C_k^T \left(C_k P_k^- C_k^T + R_k\right)^{-1},
$$
$$
\hat{x}_k = \hat{x}_k^- + K_k \left(z_k – g(\hat{x}_k^-,u_k)\right),
$$
$$
P_k = (I – K_k C_k) P_k^-.
$$
Using EKF to estimate the SOC of a traction battery pack, I found three serious limitations. First, when the measurement noise is strong, the EKF cannot quickly reject the noise effect, resulting in a long transient error. Second, the EKF is sensitive to an inaccurate initial SOC; if the initial SOC is very different from the actual value, the convergence time may be long. Third, the first-order linearization error of the nonlinear measurement equation becomes large when the system states change rapidly, so the SOC estimation error increases.
3.2 Strong Tracking Extended Kalman Filter
To overcome the limitations of EKF, I adopted the Strong Tracking Extended Kalman Filter (STEKF). The goal is to keep the innovation sequence orthogonal and to track sudden changes. The STEKF introduces a time-varying fading factor \(\lambda_k\) into the prior error covariance:
$$
P_k^- = \lambda_k A_{k-1} P_{k-1} A_{k-1}^T + Q_{k-1}.
$$
The fading factor is computed from the innovation sequence:
$$
r_k = z_k – g(\hat{x}_k^-, u_k),
$$
$$
E_k =
\begin{cases}
r_1 r_1^T, & k=1,\\[4pt]
\dfrac{\rho E_{k-1} + r_k r_k^T}{1+\rho}, & k>1,
\end{cases}
$$
where \(\rho\) is the forgetting factor (\(0 \le \rho \le 1\)). Then I compute the scalar parameter:
$$
e_k = \frac{\operatorname{tr}\left(E_k – \beta R_k – C_k Q_{k-1} C_k^T\right)}
{\operatorname{tr}\left(C_k A_{k-1} P_{k-1} A_{k-1}^T C_k^T\right)},
$$
and the fading factor is selected as:
$$
\lambda_k =
\begin{cases}
e_k, & e_k \ge 1,\\
1, & e_k < 1.
\end{cases}
$$
Subsequently, the Kalman gain and updates are calculated in the same way as in EKF, but using the modified \(P_k^-\). This fading-factor mechanism reduces the influence of historical data and ensures that the filter puts larger weight on the latest innovation, thereby improving the ability to track state jumps in a traction battery pack.
3.3 Corrected Covariance Technique
In a standard Kalman filter, by virtue of the orthogonal projection theorem for linear Gaussian systems, the following relation is true:
$$
C_k P_k^- = 0.
$$
However, when the measurement equation is nonlinear and is approximated by a curve-fitted polynomial, the actual \(\hat{x}_k^-\) may not be consistent with the true state. Therefore, the above orthogonality is violated. A direct consequence is that the calculated Kalman gain is suboptimal. The corrected covariance technique overcomes this by re-evaluating the measurement Jacobian at the posterior estimate \(\hat{x}_k\). Let
$$
\tilde{C}_k = \frac{\partial g(x_k,u_k)}{\partial x_k}\bigg|_{x_k = \hat{x}_k}.
$$
Then the Kalman gain is corrected using \(\tilde{C}_k\):
$$
\tilde{K}_k = \frac{P_k \tilde{C}_k^T}{\tilde{C}_k P_k \tilde{C}_k^T + R_k}.
$$
Finally, the state and covariance are corrected with this improved gain:
$$
\tilde{x}_k = \hat{x}_k^- + \tilde{K}_k r_k,
$$
$$
\tilde{P}_k = (I – \tilde{K}_k \tilde{C}_k) P_k (I – \tilde{K}_k \tilde{C}_k)^T + \tilde{K}_k R_k \tilde{K}_k^T.
$$
This modified covariance update makes the estimate less sensitive to inaccurate linearization and model mismatch in a traction battery pack SOC estimation process.
3.4 The Proposed CSTEKF Algorithm
My proposed CSTEKF integrates the strong tracking fading factor and the corrected covariance method within the EKF framework. The complete algorithm is summarized in the following steps:
| Step | Formula | Description |
|---|---|---|
| 1 | \(\hat{x}_k^- = f(x_{k-1}, u_{k-1})\) | State prior prediction |
| 2 | \(r_k = z_k – g(\hat{x}_k^-, u_k)\) | Innovation sequence |
| 3 | \(E_k\) computed recursively | Innovation covariance estimate |
| 4 | \(e_k\) computed from \(E_k\) | For the fading factor |
| 5 | \(\lambda_k = \max(1, e_k)\) | Fading factor |
| 6 | \(P_k^- = \lambda_k A_{k-1}P_{k-1}A_{k-1}^T + Q_{k-1}\) | Prior covariance with fading factor |
| 7 | \(K_k = P_k^- C_k^T (C_k P_k^- C_k^T + R_k)^{-1}\) | Original Kalman gain |
| 8 | \(\hat{x}_k = \hat{x}_k^- + K_k r_k, \quad P_k = (I-K_k C_k) P_k^-\) | Interim posterior state and covariance |
| 9 | \(\tilde{C}_k = \left.\partial g/\partial x_k\right|_{x_k=\hat{x}_k}\) | Corrected measurement Jacobian |
| 10 | \(\tilde{K}_k = P_k \tilde{C}_k^T (\tilde{C}_k P_k \tilde{C}_k^T + R_k)^{-1}\) | Corrected Kalman gain |
| 11 | \(\tilde{x}_k = \hat{x}_k^- + \tilde{K}_k r_k\) | Corrected state estimate |
| 12 | \(\tilde{P}_k = (I – \tilde{K}_k \tilde{C}_k) P_k (I – \tilde{K}_k \tilde{C}_k)^T + \tilde{K}_k R_k \tilde{K}_k^T\) | Corrected covariance estimate |
The CSTEKF algorithm provides more reliable SOC estimates for a traction battery pack because it can quickly correct the initial SOC error, suppress strong-noise influences, and remain robust under rapidly changing states.
4 Simulation Results and Discussion
4.1 Simulation Setup
I performed simulations in MATLAB/Simulink. The battery cell used in the traction battery pack had a rated capacity \(C_N = 4.9\) Ah. The working condition was an intermittent discharge process with a nominal discharge current of 4 A. The true SOC was initialized at 0.9. In the first comparison, the filter state was initialized as \(x_0 = [0.8, 0, 0]^T\). This initial SOC error of –0.1 allowed me to examine the convergence behavior. The process noise covariance was set to \(Q = \mathrm{diag}(0.001, 0.001, 0.001)\), the measurement noise covariance was \(R = 1\), the weakening factor \(\beta = 1.2\), and the forgetting factor \(\rho = 0.9\). The sampling time was \(\Delta t = 1\) s.
4.2 Initial Convergence Comparison
Table 5 reports the convergence time for the three filters when the initial state was \(x_0 = [0.8,0,0]^T\). CSTEKF converges to the actual SOC within about 40 seconds, while STEKF takes approximately 52 seconds, and EKF requires about 130 seconds. This demonstrates that the fading factor plus covariance correction accelerates the initial SOC recovery of a traction battery pack.
| Algorithm | Convergence time (s) |
|---|---|
| EKF | ~130 |
| STEKF | ~52 |
| CSTEKF | ~40 |
4.3 Performance under Strong Noise and State Changes
In the intermittent discharge simulation, I deliberately injected a strong noise disturbance near 3400 s. The EKF displayed a substantial SOC estimation error and required a long time to recover. The STEKF recovered more quickly, but the CSTEKF produced the smallest error and returned to the true value almost immediately. In addition, between 2000 and 2200 s, I doubled the discharge current to test the tracking of sudden state changes. The CSTEKF followed the SOC trajectory with minimal overshoot and lower bias than the other two filters.
4.4 Quantitative Error Indices
To objectively evaluate the three algorithms, I calculated the Mean Absolute Error (MAE), Mean Relative Error (MRE), and Root Mean Square Error (RMSE). The results are listed in Table 6 for the scenario with initial state \(x_0 = [0.8,0,0]^T\).
| Algorithm | MAE | MRE | RMSE |
|---|---|---|---|
| EKF | 0.012679 | 0.049104 | 0.01834 |
| STEKF | 0.0064869 | 0.028341 | 0.0093169 |
| CSTEKF | 0.0036344 | 0.015256 | 0.0051752 |
Table 7 shows the results for a more challenging initial condition \(x_0 = [0.1,0,0]^T\), where the true SOC was 0.9. This simulates an extremely inaccurate initial SOC in a traction battery pack.
| Algorithm | MAE | MRE | RMSE |
|---|---|---|---|
| EKF | 0.029589 | 0.068345 | 0.086876 |
| STEKF | 0.016489 | 0.039685 | 0.069241 |
| CSTEKF | 0.011672 | 0.024413 | 0.052004 |
In both cases, CSTEKF achieves the lowest MAE, MRE, and RMSE. The results confirm that the CSTEKF algorithm enhances the robustness and accuracy of SOC estimation for a traction battery pack even in the presence of strong noise and large initial error.
4.5 Discussion
The improvement of CSTEKF over EKF is mainly due to two essential mechanisms. First, the fading factor \(\lambda_k\) makes the filter automatically increase its gain when the innovation deviates from its theoretical range. This produces faster convergence when the SOC estimate is far from the actual value or when a sudden load change occurs. Second, the corrected covariance technique re-linearizes the nonlinear measurement model at the updated state estimate. This reduces the linearization error caused by the first-order approximation, thereby improving the SOC estimation of a traction battery pack under large current steps and temperature-dependent parameter uncertainties.
5 Conclusion and Outlook
In my research, I investigated accurate SOC estimation of a traction battery pack from three aspects: battery characterization, equivalent circuit modeling and parameter identification, and robust filtering algorithm design. I analyzed the working mechanism and SOC-influencing factors of lithium-ion batteries. Experimental results quantified the temperature effect and aging effect on capacity, emphasizing how these factors should be considered when estimating SOC of a traction battery pack.
I constructed a second-order RC equivalent circuit model and identified the model parameters using the HPPC test. The OCV-SOC relationship was fitted by a seventh-order polynomial with sufficient accuracy. The model was validated by comparing the simulated and measured terminal voltages; the error was below 0.05 V.
To overcome the limitations of the EKF, I proposed the CSTEKF algorithm. The algorithm incorporates a strong tracking fading factor to react quickly to state changes and applies a corrected covariance update to improve the state estimate when linearization error exists. Simulation results demonstrated that CSTEKF outperforms both EKF and STEKF regarding MAE, MRE, and RMSE, particularly when the initial SOC is uncertain or when strong measurement noise is present. Thus, CSTEKF provides a reliable and accurate SOC estimation solution for a traction battery pack.
There are several directions for future work. First, more complex models that adapt to multiple temperatures and aging levels should be developed to improve the universality of CSTEKF. Second, online parameter identification methods can be integrated with the proposed algorithm for real-time adaptation to cell degradation. Third, joint SOC and state-of-health estimation may be explored to build a comprehensive monitoring framework for a lithium-ion traction battery pack. These improvements will help to make the proposed algorithm even more suitable for practical electric vehicle applications.