We propose a robust and precise diagnostic strategy for both single and double open-switch faults in grid-connected inverters. The core of our method is a dedicated Super-Twisting Sliding Mode Observer (STSMO) designed for accurate current estimation, combined with a novel fault detection and localization scheme that ensures reliability under grid fluctuations. The stability and effectiveness of the proposed framework are rigorously validated through extensive experimental results.
1. Introduction
The grid-connected inverter is a pivotal component in modern renewable energy systems, responsible for converting and conditioning DC power from sources like solar panels or batteries into AC power synchronized with the utility grid. Its operational stability and reliability are paramount for ensuring power quality and system availability. However, power semiconductor switches within the inverter are subjected to high electrical and thermal stresses, making them vulnerable to failures. Statistics indicate that failures of these switches account for a significant portion of inverter malfunctions. Among these, open-circuit faults, often resulting from bond wire lift-off, solder fatigue, or the activation of protection circuits in response to short-circuit events, are particularly common. An undetected open-circuit fault in a grid-connected inverter leads to distorted output currents, increased stress on healthy components, unbalanced grid currents, and a potential loss of generated power. Therefore, developing fast, accurate, and robust fault diagnosis techniques is essential for condition monitoring, predictive maintenance, and enabling fault-tolerant operation of the grid-connected inverter.
Existing fault diagnosis methods can be broadly categorized into signal-based, data-driven (intelligent algorithm-based), and model-based approaches. Signal-based methods analyze the characteristic changes in measured voltages or currents post-fault. While voltage-based methods can be fast, they often require additional voltage sensors, increasing cost and complexity. Current-based methods are more commonly employed due to the inherent availability of current measurements for control purposes. However, their performance can be significantly degraded by grid disturbances and load variations. Data-driven methods, such as those employing machine learning, can handle complex patterns but require extensive training data from various fault and operating conditions, and their real-time implementation and interpretability can be challenging.
Model-based methods offer a compelling balance, providing real-time capability, reliability, and relatively low computational cost. Techniques like Kalman filters, extended state observers, and traditional Sliding Mode Observers (SMOs) have been applied. The SMO is particularly attractive for fault diagnosis in power electronic systems due to its inherent robustness against parameter variations and external disturbances. However, conventional first-order SMOs suffer from the well-known chattering phenomenon—high-frequency oscillations around the sliding surface—which can degrade estimation accuracy and introduce noise. While various approaches, such as using continuous approximations of the sign function or adaptive reaching laws, have been developed to mitigate chattering, they often involve compromises in convergence speed or disturbance rejection capability.
To address these limitations, we introduce a second-order sliding mode technique: the Super-Twisting Algorithm (STA). This algorithm inherently provides higher-order sliding precision while effectively suppressing chattering. We design a dedicated STSMO for the grid-connected inverter system. A key enhancement in our observer design is the replacement of the discontinuous sign function with a smooth hyperbolic tangent function, further smoothing the control action and improving performance under grid disturbances. For fault detection, we move beyond simple residual comparison and propose a “Normalized Difference Current” method based on the Probability Density Function (PDF) difference between the actual and observed currents. This approach, combined with an adaptive threshold, significantly enhances robustness against normal system fluctuations, reducing false alarms. A straightforward yet effective fault localization scheme is then employed to pinpoint the faulty switch(es). The combined method is experimentally validated to diagnose all 21 possible single and double switch open-circuit faults in a three-phase grid-connected inverter rapidly and reliably, even under various grid disturbance scenarios.
2. Hybrid Logical Dynamic Model of a Grid-Connected Inverter
The topology of a standard two-level three-phase Voltage Source Inverter (VSI) interfacing a DC source with the AC grid is considered. The system includes a DC-link capacitor, six insulated-gate bipolar transistor (IGBT) switches T1 to T6 with anti-parallel diodes, and an L-type output filter. The grid is represented by three-phase voltage sources. To develop a model suitable for observer design, we derive a Hybrid Logical Dynamic (HLD) model that incorporates the discrete switching states and the continuous current dynamics.
The voltage equations for the three phases can be written as:
$$ u_{xn} = R i_x + L \frac{di_x}{dt} + u_x, \quad x \in \{a, b, c\} $$
where $u_{xn}$ is the voltage between the inverter output terminal $x$ and the load neutral point $n$, $i_x$ is the phase current, $u_x$ is the grid phase voltage, $R$ is the equivalent resistance, and $L$ is the filter inductance.
The voltage $u_{xn}$ can be expressed in terms of the inverter pole voltages $u_{xg}$ (relative to the DC-link midpoint ‘g’) and the neutral point voltage $u_{ng}$:
$$ u_{xn} = u_{xg} – u_{ng}. $$
Assuming a balanced three-phase system, the sum of the phase voltages is zero: $u_{an} + u_{bn} + u_{cn} = 0$. Combining these equations allows us to eliminate $u_{ng}$ and express $u_{xn}$ directly as a function of the pole voltages:
$$
\begin{bmatrix} u_{an} \\ u_{bn} \\ u_{cn} \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix} \begin{bmatrix} u_{ag} \\ u_{bg} \\ u_{cg} \end{bmatrix}.
$$
The pole voltage $u_{xg}$ depends on the switching state $S_x$ (where $S_x=1$ implies the upper switch in leg $x$ is ON, and $S_x=0$ implies the lower switch is ON) and the direction of the phase current $i_x$ due to the conduction of the freewheeling diodes. Defining an auxiliary variable $\sigma_x$ such that $\sigma_x = 1$ if $i_x > 0$ and $\sigma_x = 0$ if $i_x \le 0$, the pole voltage can be modeled as:
$$ u_{xg} = U_{dc} \cdot [\bar{s}_{x_l}(s_{x_u} + \bar{s}_{x_u} \cdot \bar{\sigma}_x)], $$
where $s_{x_u}$ and $s_{x_l}$ are the switching signals for the upper and lower switches of leg $x$, respectively, and the overbar denotes logical negation. This formulation accurately represents the voltage for all conduction modes.
For simplicity in observer design, we define a composite input vector $\mathbf{u}_{\sigma} = [u_{a\sigma}, u_{b\sigma}, u_{c\sigma}]^T$, where $u_{x\sigma} = \bar{s}_{x_l}(s_{x_u} + \bar{s}_{x_u} \cdot \bar{\sigma}_x)$. The system’s continuous-time state-space model with state vector $\mathbf{i} = [i_a, i_b, i_c]^T$ and input vector $\mathbf{u}_g = [u_a, u_b, u_c]^T$ is then:
$$ \dot{\mathbf{i}} = \mathbf{A} \mathbf{i} + \mathbf{B}_1 \mathbf{u}_{\sigma} + \mathbf{B}_2 \mathbf{u}_g, $$
with
$$ \mathbf{A} = -\frac{R}{L} \mathbf{I}_3, \quad \mathbf{B}_1 = \frac{U_{dc}}{3L} \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix}, \quad \mathbf{B}_2 = -\frac{1}{L} \mathbf{I}_3, $$
where $\mathbf{I}_3$ is the 3×3 identity matrix. This HLD model forms the basis for designing the state observer for the grid-connected inverter currents.
3. Super-Twisting Sliding Mode Observer Design
To account for unmodeled dynamics and disturbances present in a practical grid-connected inverter system, we augment the nominal model. The system used for observer design is expressed as:
$$ \begin{aligned}
\dot{\mathbf{i}} &= \mathbf{A} \mathbf{i} + \mathbf{B} \mathbf{u} + \mathbf{D} \mathbf{v}, \\
\mathbf{y} &= \mathbf{C} \mathbf{i},
\end{aligned} $$
where $\mathbf{u} = \mathbf{B}_1 \mathbf{u}_{\sigma} + \mathbf{B}_2 \mathbf{u}_g$ is the known input, $\mathbf{v}$ represents bounded unknown disturbances (e.g., parameter mismatches, grid harmonics), $\mathbf{y}$ is the measured output (the three-phase currents), and $\mathbf{C} = \mathbf{I}_3$, $\mathbf{D} = \mathbf{I}_3$.
The core of our observer is the Super-Twisting Algorithm (STA), a second-order sliding mode control law known for its robustness and chattering attenuation properties. The standard STA form for a scalar system is:
$$ \begin{aligned}
\dot{\hat{x}}_1 &= -h_1 |\tilde{x}_1|^{1/2} \text{sign}(\tilde{x}_1) + \hat{x}_2 + \rho_1, \\
\dot{\hat{x}}_2 &= -h_2 \text{sign}(\tilde{x}_1) + \rho_2,
\end{aligned} $$
where $\tilde{x}_1 = \hat{x}_1 – x_1$ is the estimation error, and $\rho_i$ are perturbation terms. The terms $|\tilde{x}_1|^{1/2}$ and the integral action of $\hat{x}_2$ contribute to finite-time convergence and smooth control action.
We design a STSMO for the three-phase grid-connected inverter system. A key modification is replacing the discontinuous $\text{sign}(\cdot)$ function with the continuous $\tanh(\cdot)$ function to further smooth the control signal and reduce high-frequency components. The proposed observer structure is:
$$ \begin{aligned}
\dot{\hat{\mathbf{i}}} &= \mathbf{A} \hat{\mathbf{i}} + \mathbf{B} \mathbf{u} – \frac{1}{L} \mathbf{K}_1 |\mathbf{e}|^{1/2} \tanh(\mathbf{e}) + \frac{1}{L} \mathbf{m}, \\
\dot{\mathbf{m}} &= -\mathbf{K}_2 \tanh(\mathbf{e}),
\end{aligned} $$
where $\hat{\mathbf{i}}$ is the estimated current vector, $\mathbf{e} = \mathbf{C} \hat{\mathbf{i}} – \mathbf{y} = \hat{\mathbf{i}} – \mathbf{i}$ is the output estimation error vector, $\mathbf{m}$ is an auxiliary state vector introducing integral action, and $\mathbf{K}_1 = k_1 \mathbf{I}_3$, $\mathbf{K}_2 = k_2 \mathbf{I}_3$ are positive definite diagonal gain matrices. The function $|\mathbf{e}|^{1/2}$ and $\tanh(\mathbf{e})$ are applied element-wise.
The stability of the proposed STSMO is analyzed using Lyapunov theory. Defining the error dynamics as $\dot{\mathbf{e}} = \mathbf{A} \mathbf{e} – \mathbf{D} \mathbf{v} – \frac{1}{L} \mathbf{K}_1 |\mathbf{e}|^{1/2} \tanh(\mathbf{e}) + \frac{1}{L} \mathbf{m}$, we construct a suitable Lyapunov function candidate. Through analysis, we derive sufficient conditions for the observer gains $k_1$ and $k_2$ to guarantee finite-time convergence of the estimation error $\mathbf{e}$ to zero in the absence of disturbances, and to remain bounded in their presence. These conditions ensure the observer is robust and stable for the grid-connected inverter application.
4. Proposed Fault Diagnosis Methodology
The fault diagnosis strategy consists of three sequential stages: 1) Accurate current estimation using the STSMO, 2) Fault detection based on statistical analysis of the estimation residual, and 3) Fault localization using logical phase information.
4.1. Fault Detection via Normalized Difference Current Method
Instead of directly monitoring the current residual $\mathbf{e}(t)$, which is sensitive to noise and transients, we propose a method based on the statistical distribution of the current signal over a short window. Under healthy conditions, the estimated current $\hat{i}_x(t)$ closely tracks the actual current $i_x(t)$. Therefore, their instantaneous values should have very similar statistical distributions. An open-circuit fault causes the actual current waveform to distort (e.g., losing a half-cycle), while the observer, driven by the controller’s commands, continues to produce a near-sinusoidal estimate. This creates a significant divergence in their probability distributions.
We model the sampled data within a moving window for each phase as normally distributed. The Probability Density Function (PDF) for a signal $x$ is given by:
$$ f(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right), $$
where $\mu$ and $\sigma$ are the mean and standard deviation calculated over the window. The window length is chosen as one-quarter of the fundamental period to balance sensitivity and response time.
For each phase $x \in \{a, b, c\}$, we compute the fault detection factor $R_x$ as the absolute difference between the PDFs of the measured and estimated currents:
$$ R_x = | f(i_x) – f(\hat{i}_x) |. $$
Under normal operation, $R_x$ remains near zero. When an open-circuit fault occurs in a switch affecting phase $x$, the distributions diverge significantly during the affected half-cycle, causing a sharp peak in $R_x$.
To make the detection robust against minor asymmetries and disturbances inherent in a grid-connected inverter, we employ an adaptive threshold $T_x$:
$$ T_x = k_3 \cdot \left( \zeta + \min(R_a, R_b, R_c) \right), $$
where $k_3 > 1$ is a gain factor and $\zeta$ is a small safety margin. The use of the minimum value among all phases makes the threshold sensitive to the overall noise floor of the system. A fault in phase $x$ is detected when $R_x > T_x$ for a predefined duration. The binary fault detection variable $G_x$ is then set:
$$ G_x = \begin{cases} 1, & \text{if } R_x > T_x \\ 0, & \text{otherwise} \end{cases}. $$
This adaptive mechanism effectively prevents false alarms during grid sags, swells, or frequency variations.
4.2. Fault Localization
Once a fault is detected ($G_x=1$ for any phase), the next step is to identify the specific faulty switch(es). The location is determined by identifying which half-cycle (positive or negative) of the current is missing. We construct a fault location variable $D_x$ using a phase-locked loop (PLL) or a simulated ideal current reference synchronized with the grid voltage. This reference signal, delayed by a small time $t_d$ to align with the detection moment, is processed:
$$ D_x = \begin{cases}
\ \ 1, & \text{if } I_{x,ref}(t-t_d) \ge +\varepsilon \\
\ \ 0, & \text{if } |I_{x,ref}(t-t_d)| < \varepsilon \\
-1, & \text{if } I_{x,ref}(t-t_d) \le -\varepsilon
\end{cases}, $$
where $\varepsilon$ is a small positive constant to define a dead zone around zero crossing. The value $D_x = 1$ corresponds to the period when the upper switch (T1, T3, T5) should be conducting, and $D_x = -1$ corresponds to the period for the lower switch (T4, T6, T2).
4.3. Comprehensive Fault Diagnosis Variable
The final fault diagnosis variable $B_x$ for each phase is obtained by combining the detection and location variables:
$$ B_x = G_x \cdot D_x. $$
The value of $B_x$ provides direct diagnostic information:
- $B_x = 0$: No fault detected in phase $x$.
- $B_x = +1$: Open-circuit fault in the upper switch of phase $x$ leg.
- $B_x = -1$: Open-circuit fault in the lower switch of phase $x$ leg.
For double-switch faults, the variables $B_a$, $B_b$, and $B_c$ will simultaneously indicate the status of multiple phases, enabling the identification of faults like T1&T2 (same leg), T1&T4 (complementary switches), or T1&T3 (different legs). The overall diagnostic logic for the grid-connected inverter is summarized below.
| Faulty Switch(es) | Expected $B_a$ | Expected $B_b$ | Expected $B_c$ |
|---|---|---|---|
| T1 (Upper A) | +1 | 0 | 0 |
| T2 (Lower A) | -1 | 0 | 0 |
| T3 (Upper B) | 0 | +1 | 0 |
| T4 (Lower B) | 0 | -1 | 0 |
| T5 (Upper C) | 0 | 0 | +1 |
| T6 (Lower C) | 0 | 0 | -1 |
| T1 & T3 | +1 | +1 | 0 |
| T1 & T4 | +1 | -1 | 0 |
| T1 & T5 | +1 | 0 | +1 |
| … (Other combos) | … | … | … |
5. Experimental Validation and Performance Analysis
A laboratory-scale prototype of a three-phase grid-connected inverter was built to validate the proposed method. The system parameters are: Grid phase voltage (RMS): 100 V, Grid frequency: 50 Hz, DC-link voltage: 400 V, Filter inductance (L): 10 mH, Filter resistance (R): 0.18 Ω, Switching frequency: 10 kHz. The control and diagnosis algorithms were implemented on a digital signal processor (DSP). The observer gains were set to $k_1=2$, $k_2=7 \times 10^4$, and the diagnostic parameters were $k_3=1.3$, $\zeta=0.03$, $\varepsilon=0.2$, and $t_d=5$ ms.
5.1. Diagnosis of Open-Circuit Faults
The method was tested for all single-switch and various double-switch open-circuit faults. The table below summarizes the diagnosis results for key fault scenarios, confirming the method’s ability to correctly identify the fault type and location. The diagnostic time is defined as the interval between the fault occurrence and the sustained assertion of the correct $B_x$ signal.
| Fault Scenario | Detected Phase(s) ($G_x=1$) | Diagnosis Variable ($B_a, B_b, B_c$) | Interpretation | Avg. Diagnosis Time (ms) |
|---|---|---|---|---|
| Healthy | None | (0, 0, 0) | No fault | N/A |
| T1 Open | Phase A | (+1, 0, 0) | Upper switch A fault | 4.2 |
| T4 Open | Phase B | (0, -1, 0) | Lower switch B fault | 4.5 |
| T1 & T3 Open | Phases A & B | (+1, +1, 0) | Upper switch A & B fault | 5.1 |
| T1 & T4 Open | Phases A & B | (+1, -1, 0) | Upper A & Lower B fault | 5.3 |
| T1 & T2 Open | Phase A | (0*, 0, 0)* | Same leg fault (Special handling)** | 4.8 |
* For same-leg faults, both half-cycles are affected. The detection factor $R_a$ still peaks, but $D_a$ oscillates. A secondary rule ($G_a=1$ and sustained near-zero current) identifies this case.
** Minimum diagnosis time recorded was 3.87 ms (19% of a 50Hz cycle).
5.2. Robustness Under Grid Disturbances
A critical requirement for a grid-connected inverter fault diagnosis system is robustness against normal grid fluctuations. We tested the proposed method under three common disturbance conditions: voltage variation (±4%), frequency variation (±0.5 Hz), and phase voltage unbalance (4%). In all cases, the adaptive threshold mechanism successfully prevented false detection during the disturbance-only periods. When an open-circuit fault (T1) was intentionally introduced during these disturbances, the system reliably detected and located the fault without error. The table below quantifies the performance.
| Disturbance Type | Disturbance Magnitude | False Alarm? | Fault Correctly Diagnosed? | Remarks |
|---|---|---|---|---|
| Voltage Swing | +4% (Swell) | No | Yes | Threshold adapted to increased $R_x$ noise floor. |
| Voltage Swing | -4% (Sag) | No | Yes | Detection factor peak slightly reduced but still > T. |
| Frequency Shift | +0.5 Hz | No | Yes | Transient in $R_x$ observed, but below adaptive threshold. |
| Voltage Unbalance | 4% (on Phase A) | No | Yes | Unbalance caused asymmetry in $R_b$, $R_c$, but min($R_x$) increased $T$, maintaining reliability. |
5.3. Observer Performance Comparison
The superiority of the proposed STSMO was verified by comparing its current estimation performance against a conventional SMO and an adaptive SMO from the literature. Key performance metrics are convergence speed (time to settle after a transient) and chattering magnitude in steady state. The STSMO with the tanh function demonstrated significantly faster convergence and markedly reduced chattering amplitude, leading to a cleaner and more accurate residual signal $e(t)$ for the subsequent fault detection stage in the grid-connected inverter.
| Observer Type | Key Feature | Convergence Speed | Chattering Amplitude (Steady State) | Suitability for Diagnosis |
|---|---|---|---|---|
| Conventional SMO | Discontinuous sign() function | Slow | High | Poor (Noisy residual) |
| Adaptive SMO [Ref] | Adaptive reaching law | Medium | Medium | Moderate |
| Proposed STSMO | STA with tanh() function | Fast | Very Low | Excellent (Clean residual) |
5.4. Comparative Analysis with Other Diagnostic Methods
The overall performance of our proposed diagnostic method for the grid-connected inverter is compared against several recent methods reported in the literature. The comparison highlights advantages in diagnostic coverage, speed, robustness, and hardware requirements.
| Method / Feature | Extra Hardware Needed? | Robustness to Grid Disturbances | Threshold Type | Min. Diagnosis Time | Fault Types Covered |
|---|---|---|---|---|---|
| Voltage Pattern [Ref] | Yes (Voltage Sensors) | Medium | Fixed | ~1.0 ms | Single-switch only |
| Current Average [Ref] | No | Low | Fixed | ~10 ms | Single & Double |
| Extended State Observer [Ref] | No | Medium | Adaptive | ~6 ms | Single & Double |
| Proposed STSMO-PDF Method | No | High | Adaptive | ~3.87 ms | All Single & Double (21 types) |
6. Conclusion
This paper has presented a comprehensive model-based fault diagnosis scheme for open-switch faults in three-phase grid-connected inverters. The proposed method integrates a high-performance Super-Twisting Sliding Mode Observer with a statistical fault detection mechanism and a logical localization procedure. The STSMO, enhanced by a hyperbolic tangent function, provides fast and chatter-free estimation of the grid currents, forming a reliable basis for fault detection. The novel Normalized Difference Current method, which utilizes the difference in the probability density functions of measured and observed currents coupled with an adaptive threshold, effectively distinguishes fault-induced distortions from those caused by normal grid fluctuations like voltage sags, swells, frequency shifts, and unbalance. This results in a highly robust detection system. The subsequent localization step is simple and effective.
Experimental validation on a laboratory prototype confirms the method’s capability to accurately diagnose all 21 possible single and double open-switch faults. The method demonstrates a rapid response, with a minimum diagnostic time of 3.87 ms (19% of a fundamental cycle), and maintains perfect diagnostic accuracy under various grid disturbance conditions. Compared to existing methods, the proposed approach offers a superior combination of diagnostic coverage, speed, robustness, and practical implementability without requiring additional hardware sensors. This makes it a highly attractive solution for enhancing the reliability and maintainability of grid-connected inverter systems in renewable energy applications.