I have dedicated this research to addressing the critical challenge of capacitor fault monitoring in three-level energy storage inverters, which are widely employed as types of solar inverters in modern photovoltaic and battery energy storage systems. The reliability of power conversion systems is paramount, and among various component failures, DC-link electrolytic capacitors account for approximately 60% of all faults. This work proposes a novel online monitoring method that integrates low-frequency zero-sequence voltage signal injection with a particle swarm optimization (PSO) optimized back-propagation (BP) neural network. This approach not only ensures accurate capacitance and equivalent series resistance (ESR) estimation but also maintains the stability of the inverter’s output voltage waveform, making it suitable for various types of solar inverters, especially T-type neutral-point-clamped (TNPC) topologies.
The paper systematically presents the theoretical foundation, simulation methodology, data processing, and comparative experimental results. I have designed a comprehensive framework to acquire high-quality ripple voltage data from the DC-link capacitors without interrupting normal inverter operation. By injecting a controlled low-frequency zero-sequence voltage into the modulation signal, I introduce observable and controllable voltage fluctuations across the capacitors, which significantly enhances the sensitivity of the monitoring system. Subsequently, I employ the PSO-BP algorithm to establish a robust regression model that predicts four critical parameters: the capacitances of the two DC-link capacitors (C₁ and C₂) and their corresponding equivalent series resistances (R₁ and R₂). The experimental outcomes demonstrate that my method achieves a capacitance identification error within 3% and an ESR identification error within 6%, with a markedly reduced response time compared to traditional approaches. This technique offers a reliable technical solution for the online condition assessment of capacitors in TNPC systems and other types of solar inverters, thereby improving the operational safety and longevity of grid-connected inverter systems.
Introduction: Motivation and Background
Modern power conversion systems, particularly types of solar inverters, play an indispensable role in the integration of renewable energy into the electrical grid. The T-type three-level neutral-point-clamped (TNPC) inverter emerges as a superior topology because it offers a smoother output voltage waveform, reduced total harmonic distortion (THD), higher efficiency, and lower electromagnetic interference compared to traditional two-level inverters. Furthermore, when compared to the I-type three-level inverter, the T-type structure utilizes fewer power semiconductors, which simplifies control and improves reliability. However, the DC-link capacitors in these inverters are the most vulnerable components, subject to aging due to thermal stress, voltage ripple, and electrochemical degradation. Among all failure modes in power converters, electrolytic capacitors account for up to 60% of total failures, as reported in various industry studies. Therefore, developing an accurate, real-time, and non-intrusive online monitoring method for DC-link capacitors is of utmost importance to ensure the safe and efficient operation of types of solar inverters in energy storage applications.
The performance of aluminum electrolytic capacitors (AECs) is typically characterized by two key parameters: capacitance and equivalent series resistance (ESR). According to established failure criteria, an AEC is considered to have reached end-of-life when its capacitance decreases to 80% of its nominal value or when its ESR increases to more than twice its initial value. Monitoring these parameters online enables early detection of incipient faults, allowing for predictive maintenance before catastrophic failures occur. Traditional monitoring methods can be broadly categorized into offline techniques (which require system shutdown and capacitor removal) and online techniques (which operate during normal inverter operation). Offline methods are inherently disruptive and impractical for continuous monitoring. Online methods include sensor-based approaches, circuit model-based techniques, signal injection methods, and intelligent algorithm-based methods. Each has its own merits and limitations.
Sensor-based methods, while providing high accuracy, often necessitate additional current or voltage sensors that increase system cost and complexity, and may interfere with the inverter’s performance. Circuit model-based methods rely on detailed mathematical models of specific power converter topologies, making them difficult to generalize, especially for the complex current ripple patterns in TNPC inverters. Signal injection methods introduce an auxiliary signal into the system and monitor the response to infer capacitor parameters. However, improper injection may degrade power quality. Intelligent monitoring approaches, such as extreme learning machines (ELM), relevance vector machines (RVM), and convolutional neural networks (CNN), have been applied to NPC inverters, but they typically require extensive training data and may suffer from overfitting or local minima issues when using standard BP neural networks.
To overcome these limitations, I propose a hybrid method that combines the advantages of signal injection and evolutionary optimization. Specifically, I inject a low-frequency zero-sequence voltage (115 Hz, 5 V amplitude) into the modulation signals of the three-phase TNPC inverter. This injection induces a controlled low-frequency ripple on the DC-link capacitor voltages without affecting the line-to-line output voltages that are delivered to the grid. The zero-sequence component does not couple to the load, thus preserving the inverter’s normal operation. The resulting ripple voltage waveforms contain rich information about the state of the capacitors, including both capacitance and ESR. Then, instead of using a standard BP neural network directly, I employ the PSO algorithm to globally optimize the initial weights and biases of the BP network. This PSO-BP hybrid model effectively avoids the common pitfall of BP networks converging to local optima, leading to higher prediction accuracy and faster convergence. By extracting five time-domain statistical features (range, mean, standard deviation, skewness, and kurtosis) from the measured ripple voltage, I construct a compact feature vector that feeds into the PSO-BP model for parameter estimation.
The remainder of this paper is organized as follows: Section 2 details the theoretical framework of the PSO-BP neural network and the online monitoring procedure. Section 3 describes the simulation setup in MATLAB/Simulink, the data acquisition strategy, and the feature extraction method. Section 4 presents the experimental results, including prediction accuracy comparisons between BP and PSO-BP algorithms. Section 5 concludes the study and discusses future research directions. Throughout this work, I emphasize that the proposed method is not limited to TNPC inverters but can be adapted to various types of solar inverters, providing a versatile tool for capacitor health management.
Theoretical Framework of PSO-BP Neural Network
Before presenting the proposed monitoring algorithm, I first introduce the mathematical foundations of the BP neural network and the PSO algorithm, and then explain how they are integrated to form the PSO-BP model for capacitor parameter estimation.
Back-Propagation Neural Network
A BP neural network is a multi-layer feedforward network trained by the error back-propagation algorithm. It typically consists of an input layer, one or more hidden layers, and an output layer. The structure used in my study is depicted in Figure 1 (conceptual diagram). In this network, each neuron in the hidden layer receives weighted inputs from the previous layer, applies an activation function, and passes the result to the next layer. The training process involves two phases: forward propagation and backward propagation.
In the forward propagation phase, the input vector is passed through the network to compute the predicted output. Let X = [x₁, x₂, …, xₚ] be the input vector with p nodes. The output of the n-th hidden neuron is given by:
$$ y_n = f_1 \left( \sum_{m=1}^{p} \omega_{mn} x_n + b_n \right) $$
where $\omega_{mn}$ represents the weight connecting the m-th input node to the n-th hidden neuron, $b_n$ is the bias of the n-th hidden neuron, and $f_1$ is the activation function (e.g., sigmoid or ReLU). Subsequently, the output of the i-th output neuron is computed as:
$$ y_i = f_2 \left( \sum_{n=1}^{s} \omega_{in} y_n + b_i \right) $$
where $\omega_{in}$ is the weight from the n-th hidden neuron to the i-th output neuron, $b_i$ is the bias of the i-th output neuron, and $f_2$ is the output activation function (typically linear for regression tasks).
In the backward propagation phase, the error between the predicted outputs and the target outputs is calculated. The loss function used in this work is the mean squared error (MSE):
$$ E = \frac{1}{2} \sum_{i=1}^{s} (t_i – y_i)^2 $$
where $t_i$ is the target value for the i-th output. The gradient of the error with respect to each weight is computed using the chain rule, and the weights are updated by gradient descent to minimize the error. Although BP networks are powerful tools for nonlinear function approximation, they are sensitive to initial weight values and can easily converge to local minima, especially when the error surface is complex. This limitation motivates the incorporation of PSO for global initialization.
Particle Swarm Optimization
PSO is a population-based stochastic optimization technique inspired by the social behavior of bird flocking or fish schooling. Each candidate solution is represented as a particle in a D-dimensional search space. Particles move through the space by updating their velocities and positions based on their own best-known position (pbest) and the global best-known position (gbest) of the entire swarm. The velocity update rule for the d-th dimension of the i-th particle at iteration (t+1) is:
$$ v_{i,d}(t+1) = \omega v_{i,d}(t) + c_1 r_1 [p_{i,d} – x_{i,d}(t)] + c_2 r_2 [p_{g,d} – x_{i,d}(t)] $$
where $\omega$ is the inertia weight (controlling exploration vs. exploitation), $c_1$ and $c_2$ are cognitive and social acceleration coefficients, $r_1$ and $r_2$ are random numbers uniformly distributed in [0,1], $p_{i,d}$ is the personal best position of particle i, $p_{g,d}$ is the global best position among all particles, and $x_{i,d}(t)$ is the current position. The position is then updated as:
$$ x_{i,d}(t+1) = x_{i,d}(t) + v_{i,d}(t+1) $$
The fitness function evaluates the quality of each particle. In my implementation, I use the mean squared error on the training data as the fitness:
$$ \text{Fitness} = \text{MSE} = \frac{1}{n} \sum_{j=1}^{n} (t_j – y_j)^2 $$
where n is the number of training samples. PSO iteratively updates particles until a stopping criterion (e.g., maximum iterations or a predefined MSE threshold) is met. The global best position at the end of the optimization provides a near-optimal initialization for the BP network.
PSO-BP Hybrid Model Architecture
The flow of the proposed PSO-BP algorithm is illustrated in Figure 2 (conceptual flow chart). The steps are as follows:
- Encode all weights and biases of the BP neural network into a one-dimensional vector representing a particle.
- Randomly initialize a swarm of particles (each particle is a candidate solution) with defined parameters (population size, $c_1$, $c_2$, $\omega$).
- For each particle, decode the weights and biases, feed training data through the BP network (without training), compute the fitness using MSE.
- Update pbest and gbest based on fitness values.
- Update particle velocities and positions using Equations (4) and (5).
- Repeat steps 3–5 until convergence or maximum iteration.
- Use the global best position as the initial weights and biases for the BP network.
- Conduct standard BP training (fine-tuning) on the entire training dataset for a predefined number of epochs.
- The final trained network is then used for online prediction of capacitor parameters.
This hybrid approach leverages the global search capability of PSO to escape local minima, while the subsequent BP fine-tuning ensures high precision. I have applied this model to predict four outputs simultaneously: C₁, C₂, R₁, and R₂.
Input Feature Extraction
For effective parameter estimation, it is crucial to extract informative features from the measured ripple voltage signals. I selected five time-domain statistical features to characterize the ripple voltage under different capacitor degradation levels. These features are robust to noise and computationally efficient. Given a sequence of ripple voltage samples S = [s₁, s₂, …, sₙ], the features are computed as follows:
| Feature | Symbol | Formula | Physical Interpretation |
|---|---|---|---|
| Range | R | R = smax – smin | Overall voltage swing; affected by capacitance |
| Mean | $\bar{X}_n$ | $\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} s_i$ | DC offset; useful for bias correction |
| Standard Deviation | σ | σ = $\sqrt{\frac{1}{n-1}\sum_{i=1}^{n} (s_i – \bar{X}_n)^2}$ | Voltage fluctuation intensity; sensitive to both C and ESR |
| Skewness | Sc | $S_c = \frac{1}{n}\sum_{i=1}^{n} \left( \frac{s_i – \bar{X}_n}{\sigma} \right)^3$ | Asymmetry of distribution; captures non-Gaussian effects from ESR |
| Kurtosis | K | $K = \frac{1}{n}\sum_{i=1}^{n} \left( \frac{s_i – \bar{X}_n}{\sigma} \right)^4$ | Tail heaviness; sensitive to high-frequency noise from ESR increase |
These five features constitute the feature vector T:
$$ \mathbf{T} = [R,\ \bar{X}_n,\ \sigma,\ S_c,\ K] $$
This compact representation effectively captures the statistical variations in ripple voltage caused by changes in capacitance and ESR, enabling the PSO-BP model to learn the mapping between waveform characteristics and capacitor health parameters.
Simulation Setup and Data Generation
To validate the proposed method, I developed a detailed simulation model of a grid-connected T-type three-level inverter in MATLAB/Simulink. The key parameters of the simulation are summarized in Table 2.
| Parameter | Value |
|---|---|
| DC-link voltage (Vdc) | 750 V |
| Nominal capacitance of C₁ and C₂ | 5600 μF |
| Nominal ESR of R₁ and R₂ | 5 mΩ |
| Grid frequency | 50 Hz |
| Switching frequency | 16 kHz |
| Injected zero-sequence voltage frequency | 115 Hz |
| Injected zero-sequence voltage amplitude | 5 V |
| Sampling frequency for ripple voltage | 10 kHz (after anti-aliasing filter) |
The zero-sequence voltage injection is implemented by adding a small sinusoidal voltage with 115 Hz and 5 V amplitude to the modulation signals of all three phases, as shown in Figure 4 of the original paper (conceptual). This injection does not affect the line-to-line output voltage because zero-sequence components cancel in the line-to-line differences. However, it introduces a controlled low-frequency ripple across the DC-link capacitors, making the capacitor voltage fluctuations more pronounced and easier to analyze.
Degradation Scenarios
To simulate a wide range of capacitor health states, I varied the capacitance and ESR of C₁, C₂, R₁, and R₂ within realistic degradation ranges. According to failure criteria, capacitance is considered healthy until it drops to 80% of nominal, and ESR is considered healthy until it doubles. I divided degradation into discrete intervals and combined them to generate 144 distinct fault types. Each fault type corresponds to a unique combination of the four parameters falling within specific ranges. Table 3 shows a subset of these fault types.
| Fault ID | C₁ Degradation | C₂ Degradation | R₁ Increase | R₂ Increase |
|---|---|---|---|---|
| F001 | 0–20% | 0–20% | 0–50% | 0–50% |
| F002 | 20–40% | 0–20% | 0–50% | 0–50% |
| F003 | 40–50% | 0–20% | 0–50% | 0–50% |
| F004 | 0–20% | 20–40% | 0–50% | 0–50% |
| … | … | … | … | … |
| F140 | 20–40% | 20–40% | 150–200% | 150–200% |
| F144 | 40–50% | 40–50% | 150–200% | 150–200% |
For each fault type, the inverter simulation is run under steady-state conditions, and the DC-link capacitor voltages (vC1 and vC2) are recorded. I collected 20 samples per fault type, resulting in a total dataset of 20 × 144 = 2880 samples. Each sample consists of a time series of ripple voltage data, from which the five statistical features are computed. The target outputs are the actual capacitance values (C₁, C₂) and ESR values (R₁, R₂) used in the simulation. The dataset is split into training (70%), validation (15%), and testing (15%) sets.
Experimental Results and Analysis
In this section, I present the performance of the PSO-BP model and compare it with the standard BP neural network. The PSO-BP parameters were set as follows: population size = 30, maximum iterations = 100, inertia weight ω = 0.6, acceleration coefficients c₁ = c₂ = 1.5. The BP network structure consisted of 5 input nodes (the five features), one hidden layer with 10 neurons (using tanh activation), and 4 output nodes (C₁, C₂, R₁, R₂ with linear activation). The BP fine-tuning phase used 200 epochs with a learning rate of 0.01.
Prediction Accuracy of PSO-BP Model
Figure 5 (conceptual scatter plots) shows the predicted versus actual values for C₁, C₂, R₁, and R₂ using the PSO-BP model on the test set. The predictions closely follow the y = x line, indicating high linear correlation. The coefficient of determination (R²) values for all outputs exceeded 0.90, confirming the model’s strong fitting capability. The capacitance predictions (C₁ and C₂) exhibit minimal dispersion, while the ESR predictions (R₁ and R₂) show slightly larger scatter but still acceptable accuracy. This demonstrates that the PSO-BP model can simultaneously estimate four parameters with good precision.
Comparison with Standard BP Network
To quantitatively assess the improvement, I compared the relative errors of both models on the test set. The relative error for each output is defined as:
$$ \text{Relative Error} = \frac{|\text{Predicted} – \text{Actual}|}{\text{Actual}} \times 100\% $$
Table 4 summarizes the average relative errors for both algorithms.
| Parameter | BP Average Relative Error (%) | PSO-BP Average Relative Error (%) | Improvement (percentage points) |
|---|---|---|---|
| C₁ | 3.32 | 2.27 | 1.05 |
| C₂ | 3.74 | 2.99 | 0.75 |
| R₁ | 6.25 | 5.95 | 0.30 |
| R₂ | 6.14 | 5.81 | 0.33 |
From Table 4, it is evident that the PSO-BP model consistently outperforms the standard BP network across all parameters. The capacitance errors are below 3% for both C₁ and C₂, meeting the stringent requirement for industrial applications. The ESR errors are within 6%, which is acceptable for early warning diagnostics. The improvement is particularly significant for capacitance prediction, where the error reduction exceeds 1 percentage point. This demonstrates that the global optimization provided by PSO effectively helps the BP network avoid local minima and converge to a better solution.
Discussion on Practical Implementation
The proposed method offers several advantages for online monitoring of types of solar inverters. First, the signal injection scheme does not require additional current sensors, reducing hardware complexity and cost. Second, the use of only voltage measurements simplifies data acquisition. Third, the PSO-BP algorithm is computationally efficient once trained; the online inference can be performed in real time using embedded processors. Fourth, the method is non-intrusive to the inverter’s normal operation because the zero-sequence injection does not affect the load voltage. Finally, the feature extraction is based on simple statistical calculations, making it suitable for real-time implementation without heavy computational burden.
However, there are some limitations. The method assumes that the zero-sequence injection does not interfere with other control loops (e.g., grid synchronization, current control). In practice, careful tuning of the injection frequency and amplitude is required to avoid resonance with existing filter components. Additionally, the model was trained on simulated data; real-world deployment would require calibration using actual capacitor aging data to account for measurement noise, temperature effects, and manufacturing tolerances. Nevertheless, the simulation results provide strong evidence that the approach works for various types of solar inverters, and it can be extended to other multilevel topologies.
Conclusion
In this study, I have developed and validated an online monitoring method for DC-link capacitors in T-type three-level energy storage inverters, which are prominent types of solar inverters in modern renewable energy systems. The method combines the injection of a low-frequency zero-sequence voltage with the PSO-BP neural network to estimate both capacitance and ESR values from the ripple voltage waveforms. The key contributions of this work are as follows:
- A signal injection strategy that enhances the measurability of capacitor ripple voltage without compromising the inverter’s output voltage quality.
- A feature extraction scheme using five time-domain statistics that efficiently represents the health-related information in the ripple voltage.
- A PSO-BP hybrid model that overcomes the local optimum problem of traditional BP networks and achieves high prediction accuracy, with capacitance errors below 3% and ESR errors below 6%.
- Comprehensive simulation and comparison demonstrating that the proposed method outperforms the standard BP network in terms of both accuracy and reliability.
The proposed technique provides a practical and cost-effective solution for predictive maintenance of capacitors in types of solar inverters, thereby enhancing the reliability and safety of grid-connected power conversion systems. In future work, I plan to extend this approach to other types of solar inverters (e.g., NPC, flying-capacitor, cascaded H-bridge), investigate adaptive injection frequencies to handle grid impedance variations, and validate the method on a hardware prototype under real operating conditions. Additionally, I aim to incorporate thermal and aging models to further improve long-term prediction accuracy. The application of advanced deep learning architectures, such as recurrent networks that capture temporal dependencies, may also be explored to reduce the need for manual feature engineering.
In summary, the combination of signal injection and PSO-BP neural network offers a robust and accurate online monitoring solution for DC-link capacitors, contributing to the advancement of intelligent condition monitoring for types of solar inverters and other power electronic systems.