My research focuses on the state estimation and fault diagnosis of power batteries, which are critical components in electric vehicles and various energy storage systems. As the core energy storage device, the EV battery pack directly determines the vehicle’s driving range, power output, and safety performance. The accurate estimation of the State of Charge (SOC) and reliable fault diagnosis of the EV battery pack have become essential research topics in the field of battery management systems (BMS). In my work, I have developed several filtering algorithms to address the challenges arising from the complex operating conditions and inherent uncertainties of the EV battery pack.
Introduction and Research Motivation
The rapid development of new energy vehicles has led to an increasing demand for high-performance and safe EV battery packs. Unlike traditional vehicles, electric vehicles rely entirely on the EV battery pack for energy storage and supply, making the battery’s state estimation and fault diagnosis vital for ensuring operational safety and efficiency. The SOC represents the remaining capacity of the battery, and its precise estimation is crucial for predicting driving range, preventing overcharge/discharge, and optimizing energy management strategies. However, SOC is a physical quantity that cannot be measured directly; it must be estimated indirectly through measurable external parameters such as current, voltage, and temperature. The internal electrochemical reactions within the EV battery pack are highly nonlinear, environmentally sensitive, and subject to aging, making accurate SOC estimation a formidable challenge.
In addition to SOC estimation, fault diagnosis of the EV battery pack is equally important. Battery faults, such as internal parameter anomalies, increased internal resistance, or short circuits, can lead to performance degradation and even catastrophic failures such as fire or explosion. Traditional fault diagnosis methods often struggle with the interference of system noise and the complexity of the battery’s operating environment. Therefore, my research aims to develop robust filtering algorithms capable of handling unknown-but-bounded noise and parameter uncertainties to improve both SOC estimation accuracy and fault diagnosis reliability for the EV battery pack.
My work is structured into four main parts: first, the modeling and offline parameter identification of the battery; second, the online estimation of time-varying parameters using an orthotope space search filtering algorithm; third, the SOC estimation using an extended zonotopic Kalman filter; and fourth, a fault diagnosis method based on measurement signal constraint-oriented extended zonotopic filtering.
Modeling and Parameter Identification of the EV Battery Pack
The foundation of my research lies in establishing an accurate model for the EV battery pack. I selected a lithium-ion battery as the research subject due to its high energy density, long cycle life, and wide application in electric vehicles. The internal structure of a lithium-ion battery primarily consists of positive and negative electrode materials, electrolyte, separator, and casing. During the charging process, lithium ions are extracted from the positive electrode and embedded into the negative electrode; during discharge, the reverse process occurs. This intercalation/deintercalation mechanism is the basis of the “rocking-chair” battery principle.
For the purpose of state estimation, I analyzed several equivalent circuit models (ECMs), including the Rint model, the N-order Thevenin model, and the PNGV model. Considering the balance between model accuracy and computational complexity, I chose the first-order Thevenin model, which effectively captures the battery’s dynamic response through an RC network to represent polarization effects. The first-order Thevenin equivalent circuit model can be expressed as:
$$
\begin{cases}
U = U_{oc} – U_p – I_c \times R_0, \\
\dot{U}_p = -\frac{U_p}{R_p \times C_p} + \frac{I_c}{C_p},
\end{cases}
$$
where \(U_{oc}\) is the open-circuit voltage, \(R_0\) is the ohmic internal resistance, \(R_p\) and \(C_p\) represent the polarization resistance and capacitance, \(U_p\) is the polarization voltage, \(I_c\) is the operating current, and \(U\) is the measurable terminal voltage of the EV battery pack.
To incorporate the battery’s SOC, which is defined as the ratio of the remaining charge to the nominal capacity, I employed the ampere-hour integration method. The discrete-time state-space representation of the EV battery pack is given by:
$$
\begin{cases}
\begin{bmatrix}
SOC(k+1) \\
U_p(k+1)
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 \\
0 & e^{-T/\tau_p(k)}
\end{bmatrix}
\begin{bmatrix}
SOC(k) \\
U_p(k)
\end{bmatrix}
+
\begin{bmatrix}
-\eta T/Q_0 \\
(1-e^{-T/\tau_p(k)})R_p(k)
\end{bmatrix}
I_c(k) + w(k), \\
U(k) = U_{oc}(SOC(k)) – U_p(k) – I_c(k)R_0 + v(k),
\end{cases}
$$
where \(T\) is the sampling time, \(\tau_p = R_p \times C_p\) is the time constant of the RC network, \(\eta\) is the Coulombic efficiency, \(Q_0\) is the nominal capacity, and \(w(k)\) and \(v(k)\) represent the state disturbance and observation noise, respectively.
Offline Parameter Identification
To obtain the initial parameters for the EV battery pack model, I conducted offline identification experiments using a battery testing platform. This platform consists of a power supply, AC/DC adapter, constant current discharge controller, the battery under test, a data acquisition unit, and a host computer. The experiments were performed on an 18650-type cylindrical lithium-ion battery with a nominal capacity of 1400 mAh and cut-off voltages of 2.8 V and 4.2 V.
To establish the OCV-SOC relationship, I performed a stepwise constant current discharge test. The battery was discharged at a 1/2C rate in stages, with each stage followed by a two-hour rest period to allow the battery to reach equilibrium. The terminal voltage was then measured as the open-circuit voltage. The OCV-SOC data points are listed in the following table:
| SOC (%) | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| OCV (V) | 2.79 | 3.63 | 3.71 | 3.76 | 3.79 | 3.83 | 3.87 | 3.94 | 4.01 | 4.09 | 4.19 |
From the experimental data, I observed that the OCV-SOC curve exhibits a nearly linear relationship within the SOC range of 10% to 90%. Through linear fitting, I obtained the following approximation:
$$
OCV = f(SOC) \approx k_{SOC}SOC + b_{SOC} = 0.53 \times SOC + 3.58.
$$
The off-line parameter identification of the ohmic internal resistance was conducted through the pulse discharge experiment. By analyzing the instantaneous voltage drop at the start and end of the discharge current, the ohmic resistance was identified as \(R_0 = 89m\Omega\). The polarization parameters were identified using the least-squares method based on the terminal voltage response during the relaxation period. The offline parameter identification results are summarized in the table below:
| Temperature (°C) | R0 (Ω) | Rp (Ω) | Cp (F) |
|---|---|---|---|
| 25 | 0.0890 | 0.0315 | 2138.7 |
It is important to note that these offline-identified parameters are fixed values, but in reality, the model parameters of the EV battery pack exhibit time-varying characteristics due to changes in operating conditions, temperature, and aging. Therefore, I further explored online parameter estimation methods to track these variations.
Online Time-Varying Parameter Estimation Using Orthotope Space Search Filtering
To address the issue that fixed model parameters can lead to significant errors in SOC estimation, I proposed an online time-varying parameter estimation algorithm based on an orthotope space search strategy. This method is designed to handle the unknown-but-bounded noise present in the EV battery pack system, which is a common challenge in real-world applications where the noise distribution cannot be precisely characterized.
Based on the first-order Thevenin equivalent circuit model, I derived a regression model suitable for parameter identification. Through Laplace transformation and discretization, the model can be expressed in the following form:
$$
U_{oc}(k) – U(k) = \varphi^T(k)\theta(k) + e(k),
$$
where \(\varphi(k) = [h(k-1), I_c(k), I_c(k-1)]^T\) is the data vector, \(\theta(k) = [\theta_1(k), \theta_2(k), \theta_3(k)]^T\) is the parameter vector, and the parameters are defined as:
$$
\theta_1(k) = \frac{\tau_p}{\tau_p + T}, \quad \theta_2(k) = R_0 + \frac{T R_p}{\tau_p + T}, \quad \theta_3(k) = \frac{-\tau_p R_0}{\tau_p + T}.
$$
The discrete-time control system for parameter estimation can be established as:
$$
\begin{cases}
\theta(k) = \theta(k-1) + \delta(k), \\
h(k) = \varphi^T(k)\theta(k) + e(k),
\end{cases}
$$
where \(|\delta(k)|_\infty \leq \gamma\) represents the parameter variation bound, and \(|e(k)|_\infty \leq \sigma\) represents the bounded measurement noise.
In my approach, I used an orthotope, which is a rectangular box aligned with the coordinate axes, to enclose the parameter vector. The measurement set at time \(k\) is defined by the noise bound as:
$$
S(k) = \{\theta(k) : |h(k) – \varphi^T(k)\theta(k)| \leq \sigma\}.
$$
The orthotope can be expressed as:
$$
O(k) = O(\bar{\theta}(k), d(k)) = \{\theta(k) : \theta(k) = \bar{\theta}(k) + \text{diag}(d(k))\omega, \|\omega\|_\infty \leq 1\},
$$
where \(\bar{\theta}(k)\) is the center of the orthotope, and \(d(k)\) contains the radial lengths of the box along each dimension.
To effectively track the time-varying parameters, I proposed a prediction-update filtering algorithm. In the prediction step, the orthotope from the previous time step is expanded in \(2^{n_\theta}\) directions to account for possible parameter changes. This space search process can be expressed as:
$$
\begin{aligned}
\tilde{O}^{(j)}(k) &= O(\tilde{\theta}^{(j)}(k), \tilde{d}^{(j)}(k)), \\
\tilde{d}^{(j)}_i(k) &= d^*_i(k-1) + \Gamma^{(j)}_k \gamma, \\
\tilde{d}^{(j)}_{i+n_\theta}(k) &= d^*_i(k-1) + (\Gamma^{(j)}_k – I_{n_\theta})\gamma,
\end{aligned}
$$
where \(\Gamma^{(j)}_k = \text{diag}(r^{(j)}_i)\) with \(r^{(j)}_i \in \{0,1\}\), and \(j = 1, \ldots, 2^{n_\theta}\) denotes the expansion direction.
In the update step, the correct expansion direction is determined by checking the intersection of each expanded orthotope with the measurement strip. The optimal direction matrix is selected by maximizing the hypervolume of the resulting orthotope:
$$
\Gamma_k = \arg\max \sum_{k_\gamma=k}^{k+n_\gamma} V^{(j)}(k_\gamma),
$$
where \(V^{(j)}(k_\gamma)\) represents the hypervolume of the orthotope in the \(j\)-th direction at time \(k_\gamma\). By updating the constraints with the measurement strip, I obtained the tightest orthotope containing the true parameter vector.
To validate the effectiveness of my proposed algorithm, I performed a simulation experiment using the following initial conditions: the initial parameter vector was set to the offline identification result \(\theta(0) = [0.9311, 0.0912, -0.0829]^T\), the time-varying parameter bound was \(\gamma = [1, 1, 1] \times 10^{-5}\), and the measurement noise was bounded by \(|e(k)|_\infty \leq 10^{-3}\). The sampling interval was \(T = 5s\), and the total number of iterations was \(N = 600\). I compared my algorithm (Orthotope Search Expansion, OSE) with a traditional global expansion method (Orthotope Expansion, OE). The results showed that my proposed OSE algorithm could effectively track the true time-varying parameters, and compared with the OE algorithm, it significantly reduced the uncertainty range of the parameter estimates, thereby improving the estimation accuracy. This improved parameter estimation provides a solid foundation for accurate SOC estimation of the EV battery pack.
SOC Estimation of the EV Battery Pack Using Extended Zonotopic Kalman Filter
Accurate SOC estimation is crucial for the EV battery pack to ensure safe and efficient operation. In my research, I considered the challenges of both unknown-but-bounded state disturbances and observation noises, as well as the parameter uncertainties in the battery model. To address these challenges, I proposed an Extended Zonotopic Kalman Filter (EZKF) approach for SOC estimation.
The zonotope is a convex polytope that can be represented as a Minkowski sum of line segments. A zonotope \(Z \subset \mathbb{R}^n\) can be defined as:
$$
Z = p \oplus GB^s = \langle p, G \rangle = \{z \in \mathbb{R}^n : z = p + Gb, b \in B^s\},
$$
where \(p \in \mathbb{R}^n\) is the center, \(G \in \mathbb{R}^{n \times s}\) is the generator matrix, and \(B^s = [-1, 1]^s\) is a hypercube. Zonotopes possess useful algebraic properties, such as linear maps, Minkowski sums, and bounding boxes, which make them attractive for set-membership filtering.
Based on the orthotope estimation results of the time-varying parameters, I extended the state-space model of the EV battery pack to account for parameter uncertainties:
$$
\begin{cases}
x(k+1) = (A(\theta(k)) + \Delta A(k))x(k) + (B(\theta(k)) + \Delta B(k))u(k) + Ew(k), \\
y(k) = Cx(k) + Fv(k),
\end{cases}
$$
where \(\Delta A(k)\) and \(\Delta B(k)\) represent the parameter uncertainty ranges obtained from the orthotope feasible sets. The initial state, process disturbance, and measurement noise are assumed to be bounded by zonotopes:
$$
x(0) \in Z(0) = \langle p(0), B(0) \rangle, \quad w(k) \in W = \langle 0, G_w \rangle, \quad v(k) \in V = \langle 0, G_v \rangle.
$$
I designed a zonotopic Kalman filter with the following structure:
$$
\hat{x}(k+1) = A(\theta(k))\hat{x}(k) + B(\theta(k))u(k) + L(k)[y(k) – C\hat{x}(k) – Fv(k)],
$$
where \(\hat{x}(k) \in \mathbb{R}^{n_x}\) is the estimated state, and \(L(k)\) is the filter gain matrix to be designed.
Through a series of derivations, I obtained the prediction form of the state estimation zonotope:
$$
\begin{aligned}
\hat{x}(k+1) &\in \hat{X}(k+1) = \langle \hat{p}(k+1), \hat{G}(k+1) \rangle, \\
\hat{p}(k+1) &= (A(k) – L(k)C)\hat{p}(k) + B(k)u(k) + L(k)y(k), \\
\hat{G}(k+1) &= [(A(k) – L(k)C)\hat{G}(k) \quad M_1(k) \quad M_2(k) \quad EG_w \quad -L(k)FG_v],
\end{aligned}
$$
where \(M_1(k) = [d_A(k)\hat{p}(k)]^+ + [d_A(k)\hat{G}^{\downarrow 1}(k)]^+\) and \(M_2(k) = [d_B(k)u(k)]^+\) are bounding vectors for the parameter uncertainties.
To achieve optimal estimation performance, I derived the optimal gain matrix \(L(k)\) by minimizing the F-radius of the zonotope, which represents the size of the uncertainty set:
$$
L(k) = A(k)\hat{G}(k)\hat{G}^T(k)C^T[C\hat{G}(k)\hat{G}^T(k)C^T + FG_vG_v^TF^T]^{-1}.
$$
The final SOC estimation result and its upper and lower bounds can be computed as:
$$
\begin{aligned}
x(k+1) &= p_i(k+1) – \sum_{j=1}^s |\hat{G}^{\downarrow s}(k+1)|_{i,j}, \\
x(k+1) &= p_i(k+1) + \sum_{j=1}^s |\hat{G}^{\downarrow s}(k+1)|_{i,j}.
\end{aligned}
$$
I validated my proposed EZKF algorithm through experiments with the following settings: initial state \(x(0) = [1, 0.1]^T\), generator matrix \(G(0) = 0.12I_2\), process disturbance bound \(|w(k)|_\infty \leq 10^{-4}\), measurement noise bound \(|v(k)|_\infty \leq 10^{-3}\), sampling interval \(T = 5s\), and \(N = 600\) sampling steps. I compared my algorithm with an ellipsoid filtering (EF) approach. The experimental results, as shown in the figures, demonstrated that my proposed EZKF algorithm, combined with the OSE parameter estimation, provided SOC estimates with significantly tighter bounds compared to the EF algorithm, indicating higher estimation accuracy and lower conservatism. This enhanced precision is essential for the practical management and safety of the EV battery pack.
Fault Diagnosis of the EV Battery Pack Using Measurement Signal Constraint-Oriented Extended Zonotopic Filter
Fault diagnosis is another critical aspect of my research on the EV battery pack. Internal faults, such as abnormal changes in battery parameters due to overcharge/discharge, aging, or internal short circuits, can lead to severe consequences if not detected timely. To address the challenges of fault detection, isolation, and estimation under the influence of noise, I proposed a comprehensive fault diagnosis method based on a measurement signal constraint-oriented extended zonotopic filter (MCO-EZF).
The fault diagnosis state-space model can be formulated as:
$$
\begin{cases}
\theta(k+1) = \theta(k) + \Delta\theta_f(k) + E\delta(k), \\
h(k) = \varphi^T(k)\theta(k) + Fe(k),
\end{cases}
$$
where \(\Delta\theta_f(k)\) represents the parameter change vector when a fault occurs, and it is bounded by \(| \Delta\theta_f(k)|_\infty \leq \zeta\). The fault parameter zonotope is defined as \(\Delta\theta_f(k) \in Z_f(k) = \langle 0, G_f(k)\rangle\) with \(G_f(k) = \text{diag}(\zeta)\).
Fault Detection Based on Residual Zonotope
For fault detection, I designed a residual-based method. The residual signal between the measured output and the model prediction is defined as:
$$
\begin{aligned}
\varepsilon(k) &= h(k) – \varphi^T(k)\hat{\theta}(k) – Fe(k) \\
&\in \langle h(k) – \varphi^T(k)\hat{p}(k), [-\varphi^T(k)\hat{G}(k), -FG_\sigma] \rangle \\
&= \langle p_\varepsilon(k), G_\varepsilon(k) \rangle.
\end{aligned}
$$
When the EV battery pack operates normally, the residual zonotope contains zero: \(0 \in Z_\varepsilon(k)\). However, when a fault occurs, the residual zonotope shifts away from the origin, and the fault can be detected by checking the condition:
$$
0 \notin \left[ p_{\varepsilon,i}(k) – \sum_{j=1}^s |G_{\varepsilon,i,j}(k)|, \quad p_{\varepsilon,i}(k) + \sum_{j=1}^s |G_{\varepsilon,i,j}(k)| \right], \quad i \in \{1, \ldots, n_h\}.
$$
This approach effectively avoids false alarms caused by system noise and measurement disturbances, ensuring reliable fault detection of the EV battery pack.
Fault Isolation Based on Measurement Signal Space Movement
Once a fault is detected, I used a fault isolation method based on the measurement signal’s spatial movement. The measurement strips over time build a convex space, which can be enclosed by a tight orthotope \(O(\theta_c(k), d(k))\). By comparing the center of the parameter estimation zonotope with this orthotope, I can isolate the faulty parameters:
$$
\Gamma_{i,i}(k) =
\begin{cases}
0, & \theta_i^c(k) – d_i(k) \leq \hat{p}_i(k) \leq \theta_i^c(k) + d_i(k), \\
1, & \hat{p}_i(k) < \theta_i^c(k) – d_i(k) \text{ or } \hat{p}_i(k) > \theta_i^c(k) + d_i(k),
\end{cases}
$$
where \(\Gamma(k) = \text{diag}(\Gamma_{i,i}(k))\) is the fault parameter indication matrix. If \(\Gamma_{i,i}(k) = 1\), the \(i\)-th parameter is considered faulty.
Fault Estimation Based on Constraint-Oriented Expansion
For fault estimation, I designed an extended zonotopic filter to track the abnormal parameter changes. The fault estimator is constructed as:
$$
\hat{\theta}(k+1) = \hat{\theta}(k) + \Delta\hat{\theta}_f(k) + E\delta(k) + L(k)[h(k) – \varphi^T(k)(\hat{\theta}(k) + \Delta\hat{\theta}_f(k)) – Fe(k)].
$$
The parameter estimation can be decomposed into the sum of two zonotopes, one representing the impact of system disturbances and the other representing the impact of faults:
$$
\hat{\theta}(k+1) \in \hat{Z}(k+1) = \hat{Z}^{(\delta)}(k) \oplus \hat{Z}^{(f)}(k),
$$
$$
\begin{aligned}
\hat{Z}^{(\delta)}(k) &= \langle (I – L(k)\varphi^T(k))\hat{p}(k) + L(k)h(k), \\
&\quad [(I – L(k)\varphi^T(k))\hat{G}(k), \, EG_\delta, \, -L(k)FG_\sigma] \rangle, \\
\hat{Z}^{(f)}(k) &= \langle 0, [(I – L(k)\varphi^T(k))G_f(k)\Gamma(k)] \rangle.
\end{aligned}
$$
To optimize the fault monitoring effect, I derived the optimal observer gain matrix \(L(k)\) by maximizing the ratio of the fault-related covariance to the disturbance-related covariance:
$$
J(L(k)) = \frac{\text{tr}(\text{cov}(G^{(f)}(k)))}{\text{tr}(\text{cov}(G^{(\delta)}(k)))}.
$$
This optimization can be solved to obtain the optimal gain matrix \(L^*(k)\), which maximizes the fault-related impact while suppressing the disturbance-related impact, thereby improving the accuracy and sensitivity of fault estimation.
To validate the proposed fault diagnosis algorithm, I conducted simulation experiments with the following settings: the initial parameter vector \(\theta(0) = [0, 0]^T\), the generator matrix \(G(0) = I_3\), the disturbance bound \(\gamma = 10^{-4}\), the noise bound \(\sigma = 10^{-3}\), the fault boundary \(\zeta = [1, 1, 1] \times 10^{-2}\), and the sampling interval \(T = 5s\) with \(N = 200\) steps. At time \(k = 100\), I simulated an internal resistance increase fault, where the ohmic resistance changed from \(R_0 = 0.089\Omega\) to \(R_0 = 0.589\Omega\).
I compared my MCO-EZF algorithm with an ellipsoid filtering (EF)-based method. The experimental results, as shown in the figures, demonstrated that:
1. The residual-based fault detection method could effectively detect the occurrence of the fault, and the residual bounds from my MCO-EZF algorithm were tighter than those from the EF algorithm, reducing both false alarms and missed detections.
2. The fault isolation method successfully identified the faulty parameters (\(\theta_2\) and \(\theta_3\) in this case), enabling targeted fault tracking.
3. The MCO-EZF algorithm could accurately track the post-fault parameter values with a faster convergence rate and tighter estimation bounds compared to the EF algorithm, providing more reliable fault estimation results for the EV battery pack.
Conclusion
In this thesis, I have conducted a systematic study on the state estimation and fault diagnosis of power batteries based on filtering algorithms. My work addresses four key aspects of the EV battery pack management:
1. I established a first-order Thevenin equivalent circuit model for the EV battery pack and performed offline parameter identification experiments to obtain the initial battery parameters.
2. I proposed a time-varying parameter online estimation algorithm based on an orthotope space search strategy, which effectively handles unknown-but-bounded noise and improves the accuracy of parameter tracking for the EV battery pack. Compared with traditional methods, my algorithm provides more precise parameter feasible sets with lower conservatism.
3. I developed an extended zonotopic Kalman filter for SOC estimation of the EV battery pack. By leveraging the parameter estimation results and using zonotopes to process dual disturbances, my algorithm achieved more accurate SOC estimates with significantly tighter bounds compared to the ellipsoid filtering method.
4. I designed a comprehensive fault diagnosis method based on a measurement signal constraint-oriented extended zonotopic filter. This method enables fault detection, isolation, and estimation under noise interference and parameter uncertainties, significantly improving the safety and reliability of the EV battery pack.
The proposed filtering algorithms provide effective solutions to the challenges of SOC estimation and fault diagnosis for the EV battery pack. The research contributes to the development of more reliable battery management systems, which is of great significance for the advancement of electric vehicles and energy storage technologies. Future work may focus on extending these methods to more complex battery models, considering temperature effects, and validating the algorithms under diverse real-world operating conditions.
In summary, my research findings highlight the critical importance of robust filtering techniques in ensuring the performance and safety of the EV battery pack. By improving state estimation accuracy and fault diagnostic capabilities, we can promote the sustainable and reliable development of electric transportation and energy systems.