The widespread adoption of electric vehicles (EVs) is crucial for mitigating environmental pollution and addressing the global energy crisis. At the heart of every EV lies its energy source: the high-voltage lithium-ion battery pack. Ensuring the safe and reliable operation of these lithium-ion battery systems is paramount, as failures can lead to significant safety hazards, hindering further market penetration. The Battery Management System (BMS) acts as the brain of the lithium-ion battery pack, continuously monitoring critical parameters like voltage, current, and temperature to estimate states such as State of Charge (SOC) and State of Health (SOH). However, the accuracy of these estimations and the safety decisions they inform are entirely dependent on the fidelity of the sensor data. Faults in voltage or current sensors can lead to catastrophic BMS miscalculations. While significant research focuses on state estimation algorithms, comprehensive and efficient methods for diagnosing sensor faults within lithium-ion battery systems remain an urgent need. In this work, I present a novel, model-based fault detection strategy that leverages statistical signal processing theory to provide rapid and reliable diagnosis of sensor faults in automotive lithium-ion battery packs.
My approach is grounded in the model-based fault diagnosis paradigm, which utilizes a mathematical representation of the system to generate residuals—the differences between measured and estimated outputs. A key challenge in this domain is the efficient and adaptive evaluation of these residuals to distinguish between normal noise and genuine fault signals. Traditional methods often employ fixed thresholds or Cumulative Sum (CUSUM) techniques, which can be slow to respond and sensitive to noise. To overcome these limitations, I have developed a fault detection framework centered on the Likelihood Ratio Test (LRT) and Constant False Alarm Rate (CFAR) processing. This method dynamically calculates detection thresholds based on the instantaneous noise characteristics, ensuring a consistent false alarm probability while maximizing the probability of timely fault detection. The core of my model is a second-order Equivalent Circuit Model (ECM) for the lithium-ion battery, whose parameters are identified offline. The state estimation, crucial for residual generation, is performed using the Extended Kalman Filter (EKF).
Methodology: A Model-Based Framework with Statistical Detection
1. Lithium-Ion Battery Modeling
Accurate modeling is the foundation of any model-based diagnosis strategy. For its balance between fidelity and computational complexity, I selected the second-order RC equivalent circuit model to represent the dynamics of the lithium-ion battery. This model effectively captures the major electrical behaviors, including the ohmic drop and the short-term and long-term polarization effects.
The continuous-time state-space representation of this model is given by:
$$
\begin{aligned}
\dot{U}_1 &= -\frac{1}{R_1 C_1} U_1 + \frac{1}{C_1} I \\
\dot{U}_2 &= -\frac{1}{R_2 C_2} U_2 + \frac{1}{C_2} I \\
U_t &= U_{OC}(SOC) – I R_0 – U_1 – U_2
\end{aligned}
$$
where \(U_1\) and \(U_2\) are the voltages across the polarization RC pairs \((R_1, C_1)\) and \((R_2, C_2)\), respectively; \(I\) is the load current; \(U_t\) is the terminal voltage; \(R_0\) is the ohmic internal resistance; and \(U_{OC}(SOC)\) is the open-circuit voltage, which is a known function of the battery’s State of Charge (SOC). The SOC itself is defined by the Coulomb counting method:
$$
SOC(t) = SOC(t_0) + \frac{1}{Q} \int_{t_0}^{t} \eta I(\tau) d\tau
$$
where \(Q\) is the nominal capacity and \(\eta\) is the Coulombic efficiency (assumed to be 1).
For digital implementation and state estimation, the model is discretized with a sampling period \(T\). The discrete-time state vector is chosen as \(\mathbf{x}_k = [U_{1,k}, U_{2,k}, SOC_k]^T\), with the input being the measured current \(I_k\). The discrete nonlinear state and output equations are:
$$
\begin{aligned}
\mathbf{x}_k &= f(\mathbf{x}_{k-1}, I_{k-1}) + \mathbf{w}_{k-1} \\
y_k &= U_{t,k} = g(\mathbf{x}_k, I_k) + v_k
\end{aligned}
$$
where \(\mathbf{w}_k\) and \(v_k\) are independent process and measurement noise sequences, assumed to be zero-mean Gaussian with covariances \(\mathbf{Q}_k\) and \(R_k\), respectively.
2. State Estimation via Extended Kalman Filter (EKF)
To generate the estimated terminal voltage for residual calculation, I employ the EKF to recursively estimate the state vector \(\mathbf{x}_k\). The EKF linearizes the nonlinear functions \(f\) and \(g\) around the current state estimate using their Jacobian matrices \(\mathbf{A}_k\), \(\mathbf{B}_k\), and \(\mathbf{C}_k\). The standard EKF procedure involves two main steps per iteration: prediction and update.
Prediction Step:
State prediction: \(\hat{\mathbf{x}}_{k|k-1} = f(\hat{\mathbf{x}}_{k-1|k-1}, I_{k-1})\)
Error covariance prediction: \(\mathbf{P}_{k|k-1} = \mathbf{A}_{k-1} \mathbf{P}_{k-1|k-1} \mathbf{A}_{k-1}^T + \mathbf{Q}_{k-1}\)
Update Step:
Innovation (residual): \(e_k = y_k – g(\hat{\mathbf{x}}_{k|k-1}, I_k)\)
Kalman Gain: \(\mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{C}_k^T (\mathbf{C}_k \mathbf{P}_{k|k-1} \mathbf{C}_k^T + R_k)^{-1}\)
State estimate update: \(\hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k e_k\)
Error covariance update: \(\mathbf{P}_{k|k} = (\mathbf{I} – \mathbf{K}_k \mathbf{C}_k) \mathbf{P}_{k|k-1}\)
The EKF provides a robust estimate of the battery’s internal states, including the SOC and the polarization voltages, which are not directly measurable.
3. Residual Generation and Fault Detection Strategy
Sensor faults, particularly bias faults common in Hall-effect sensors, are modeled as an additive error on the true signal:
$$
\begin{aligned}
I^m &= I^r + I_f \\
U^m_t &= U^r_t + U_f
\end{aligned}
$$
where the superscript \(m\) denotes the measured (and potentially faulty) value, \(r\) denotes the real value, and \(I_f\), \(U_f\) are the fault magnitudes.
The core idea of my diagnosis scheme is to generate two key residuals by comparing estimates derived under faulty and fault-free assumptions.
1. Terminal Voltage Residual (\(r_o\)): This is the primary and fastest-responding residual. I calculate the difference between the voltage estimated by the EKF (which uses potentially faulty current \(I^m\) as input) and the directly measured terminal voltage \(U^m_t\).
$$ r_o = U^e_t – U^m_t $$
Under normal conditions, \(r_o\) fluctuates around zero with a distribution dominated by sensor and model noise. A sensor fault (in either current or voltage) will cause a statistically significant shift in the mean of \(r_o\).
2. State of Charge Residual (\(r_s\)): This secondary residual provides complementary information. It is the difference between the SOC estimated by the EKF (using \(I^m\)) and the SOC calculated via the Coulomb counting method (using the same \(I^m\)).
$$ r_s = SOC_{EKF} – SOC_{CC} $$
A persistent current sensor fault will cause a growing divergence between these two SOC estimates, while a voltage fault has a more complex, indirect effect on the EKF’s SOC convergence.
My fault detection logic is illustrated in the following conceptual block diagram and proceeds by statistically evaluating these residuals against dynamically computed thresholds.
4. The Core Innovation: Likelihood Ratio Test with CA-CFAR Thresholding
The traditional approach of comparing residuals to a fixed threshold is suboptimal because the noise characteristics can change with operating conditions. My method solves this by formulating the fault detection problem as a statistical binary hypothesis test:
$$
\begin{aligned}
\mathcal{H}_0&: \text{No fault (residual follows noise distribution)} \\
\mathcal{H}_1&: \text{Fault present (residual distribution has shifted)}
\end{aligned}
$$
For a sequence of independent residual samples \(\mathbf{r} = \{r_1, r_2, …, r_n\}\), the optimal test (under certain conditions) is based on the Likelihood Ratio (LR):
$$
\Lambda(\mathbf{r}) = \frac{p(\mathbf{r} | \mathcal{H}_1)}{p(\mathbf{r} | \mathcal{H}_0)}
$$
A decision is made by comparing the log-likelihood ratio to a threshold \(T\):
$$ \log \Lambda(\mathbf{r}) \overset{\mathcal{H}_1}{\underset{\mathcal{H}_0}{\gtrless}} T $$
The critical challenge is setting the threshold \(T\) to maintain a desired performance. I address this using Cell-Averaging Constant False Alarm Rate (CA-CFAR) detection, a renowned technique in radar signal processing. The CA-CFAR algorithm estimates the local noise power (\(\hat{\sigma}^2\)) in real-time from a window of reference cells adjacent to the cell under test (the current residual). This allows the threshold to adapt to changing noise levels.
The detection threshold \(T\) is calculated as:
$$ T = \beta \cdot \hat{\sigma}^2 $$
where the scale factor \(\beta\) is derived from the desired false alarm probability \(P_{FA}\) and the number of reference cells \(N\):
$$ \beta = N (P_{FA}^{-1/N} – 1) $$
For my implementation, I set \(P_{FA} = 10^{-3}\) and \(N=20\) (10 reference cells on each side of the test cell), yielding \(\beta \approx 8.25\). The noise power is estimated as the mean of the squared values in the reference window:
$$ \hat{\sigma}^2 = \frac{1}{N} \sum_{i=1}^{N} x_i^2 $$
where \(x_i\) are the samples in the reference cells. This dynamic threshold ensures that the probability of false alarm remains constant at the prescribed level (\(0.001\)), regardless of changes in the residual noise variance, thereby maximizing the detection probability for any given fault.
Experimental Validation and Results
I validated the proposed fault detection method using publicly available experimental data for a 18650 lithium iron phosphate (lithium-ion battery) cell. The model parameters (\(R_0, R_1, C_1, R_2, C_2, U_{OC}(SOC)\)) were identified offline from Hybrid Pulse Power Characterization (HPPC) test data. The algorithm’s performance was then tested under two dynamic driving cycles: Dynamic Stress Test (DST) and Federal Urban Driving Schedule (FUDS). Sensor faults were simulated by adding bias signals of various magnitudes to the measured current or voltage data streams.
Performance was quantified using standard metrics: Detection Probability (\(P_D\)), False Alarm Probability (\(P_{FA}\)), Missed Detection Probability (\(P_M\)), and most importantly, the Detection Time (the delay between fault onset and algorithm alarm). Monte Carlo simulations with 1000 trials per fault scenario were conducted to obtain statistical results.
1. Voltage Sensor Fault Diagnosis
I injected bias faults of +0.1 V and +0.04 V into the voltage sensor signal. The terminal voltage residual \(r_o\) proved to be an extremely sensitive indicator.
Key Results for Voltage Faults:
- Detection Time: For a 0.1 V fault, the average detection time was ~5 seconds under DST and similarly fast under FUDS. For a smaller 0.04 V fault, the detection time increased only slightly to ~8 seconds. This demonstrates the rapid response capability of the LRT-CFAR method.
- Comparison with CUSUM: The traditional CUSUM method required ~7 seconds and ~11 seconds for the 0.1 V and 0.04 V faults, respectively. My method offers a 20-30% reduction in detection time.
- Multiple Faults: The algorithm successfully detected and tracked multiple occurrences of voltage faults within a single drive cycle, as shown in the results for four sequential fault events.
- Performance via SOC Residual (\(r_s\)): While detectable, voltage faults manifested in the SOC residual with a much longer delay (~100 seconds), confirming that \(r_o\) is the primary and faster indicator for voltage sensor issues.
The following table summarizes the detection performance metrics for voltage sensor faults under the DST cycle, highlighting the high detection rate and controlled false alarm rate.
| Parameter | 0.1 V Fault (Single) | 0.1 V Fault (Multiple) | 0.04 V Fault (Single) | 0.04 V Fault (Multiple) |
|---|---|---|---|---|
| Detection Probability (\(P_D\)) | 1.000 | 1.000 | 1.000 | 0.996 |
| False Alarm Probability (\(P_{FA}\)) | 0.0033 | 0.0020 | 0.0042 | 0.0050 |
| Missed Detection Probability (\(P_M\)) | 0.000 | 0.000 | 0.000 | 0.004 |
2. Current Sensor Fault Diagnosis
Similarly, I injected bias faults of +0.2 A and +0.08 A into the current sensor signal.
Key Results for Current Faults:
- Detection Time (via \(r_o\)): The terminal voltage residual remained effective. Detection times were ~14 seconds for the 0.2 A fault and ~16 seconds for the 0.08 A fault under DST. This is slightly longer than for voltage faults but still very responsive.
- Dramatic Advantage over CUSUM: The CUSUM method was significantly slower, requiring ~159 seconds and ~187 seconds for the same faults. My LRT-CFAR method achieved a >90% reduction in detection time, a critical improvement for safety.
- Performance via SOC Residual (\(r_s\)): For current faults, the SOC residual \(r_s\) becomes a strong secondary indicator. It showed a clear, growing deviation with detection times of ~336 seconds for a 0.2 A fault, which is still vastly superior to the >1000 seconds required by the CUSUM method on the same residual.
The performance metrics for current sensor faults under the FUDS cycle are summarized below. Notably, the method maintains a high detection probability even for smaller faults, with a slight, expected increase in false alarms as the fault signal becomes harder to distinguish from noise.
| Parameter | 0.2 A Fault (Single) | 0.2 A Fault (Multiple) | 0.08 A Fault (Single) | 0.08 A Fault (Multiple) |
|---|---|---|---|---|
| Detection Probability (\(P_D\)) | 1.000 | 1.000 | 1.000 | 0.960 |
| False Alarm Probability (\(P_{FA}\)) | 0.0034 | 0.0029 | 0.0540 | 0.0900 |
| Missed Detection Probability (\(P_M\)) | 0.000 | 0.000 | 0.000 | 0.040 |
Discussion and Conclusions
In this work, I have designed and validated a novel fault detection method for sensors in electric vehicle lithium-ion battery packs. The integration of the Likelihood Ratio Test principle with dynamic CA-CFAR thresholding forms the core innovation. This approach fundamentally shifts the residual evaluation from a simple, static threshold comparison to an adaptive statistical decision process that guarantees a constant, user-defined false alarm rate.
The experimental results conclusively demonstrate the superiority of this method:
- Significantly Reduced Detection Time: Compared to the conventional CUSUM method, my approach reduces fault detection latency by 20% to over 90%, depending on the fault type and magnitude. This rapid response is crucial for preventing BMS actions based on erroneous data.
- High Sensitivity and Robustness: The method reliably detects small-magnitude faults (e.g., 0.04 V, which is ~1% of normal operating voltage) and can successfully track multiple fault occurrences within a single operational cycle.
- Dual-Residual Strategy: Utilizing both the terminal voltage residual (\(r_o\)) and the SOC residual (\(r_s\)) provides a layered detection approach. The voltage residual offers immediate detection for both voltage and current faults, while the SOC residual provides valuable confirmatory evidence, especially for persistent current sensor faults.
- Adaptive Thresholding: The CA-CFAR mechanism ensures the algorithm remains effective across different driving cycles (DST and FUDS) without manual threshold tuning, enhancing its practical applicability.
The primary contributions of this research are therefore: 1) the novel application of LRT/CFAR theory to the problem of lithium-ion battery sensor fault diagnosis, and 2) the demonstration of its clear advantages in speed and adaptive performance over traditional methods like CUSUM.
For future work, several avenues warrant exploration. The current study focuses on a single lithium-ion battery cell; extending this method to a full battery pack with many cells in series and parallel presents challenges in computational scalability and fault isolation that need to be addressed. Furthermore, the impact of significant temperature variations on model parameters and, consequently, on the residual generation and fault detection performance should be investigated to enhance robustness under all real-world conditions. Finally, integrating this diagnostic module with fault-tolerant control and estimation algorithms would create a more resilient BMS architecture, further safeguarding the operation of electric vehicles powered by lithium-ion battery technology.