The pervasive integration of renewable energy sources into the modern power grid is fundamentally enabled by power electronic interfaces, primarily on-grid inverters. Among various filter topologies, the LCL filter is widely adopted for on-grid inverters due to its superior high-frequency harmonic attenuation capability compared to a simple L filter. However, the performance and stability of an LCL-type on-grid inverter are highly sensitive to parameter variations. Uncertainties in grid-side impedance, aging of passive components leading to deviations in the inductance and capacitance values from their nominal ratings, and fluctuations in the DC input voltage can significantly degrade power quality and even induce operational instability. Traditional stability analysis methods, often based on impedance models or detailed state-space representations, can be computationally intensive and less suited for rapid, online assessment under varying conditions. This necessitates the development of intelligent, data-driven approaches for robust stability evaluation.
This work addresses the challenge of stability analysis for LCL-type single-phase on-grid inverters under parameter perturbations. We propose a novel framework that leverages a neural network trained on data derived from a parameterized system model. The core innovation lies in using a relatively small, strategically generated dataset based on the statistical distribution characteristics of key system parameters to train a neural network. This network effectively learns the complex mapping between circuit parameters and system stability, offering a fast and accurate tool for stability assessment.
The structure of this article is as follows: First, we establish a comprehensive state-space model for a digitally controlled LCL-type single-phase on-grid inverter. The controller incorporates a quasi-Proportional Resonant (quasi-PR) scheme combined with Capacitor Current Feedback (CCF) active damping to mitigate the inherent resonance of the LCL filter without the losses associated with physical resistors. Subsequently, we detail the methodology for neural network-based stability analysis, including the design of the network architecture, the formulation of the training dataset based on parameter distributions, and the optimization process. Finally, the effectiveness of the proposed method is rigorously validated through time-domain simulations in MATLAB/Simulink and experimental results obtained from a hardware prototype.
System Modeling of the LCL-Type Single-Phase On-Grid Inverter
The system under study consists of a full-bridge inverter connected to the grid through an LCL filter. The control objective is to ensure the grid current \(i_g\) accurately tracks its sinusoidal reference \(i_{ref}\). The controller structure is depicted in the following schematic, which illustrates the power stage and the digital control loop. The quasi-PR controller ensures zero steady-state error at the fundamental frequency, while the CCF loop provides active damping. A standard Phase-Locked Loop (PLL) extracts the grid phase angle for synchronization.
The dynamics of the LCL filter are described by the following equations in the Laplace domain:
$$ I_c(s) = C s U_c(s) = I_1(s) – I_g(s) $$
$$ I_1(s) = \frac{U_{inv}(s) – U_c(s)}{L_1 s} $$
$$ I_g(s) = \frac{U_c(s) – U_g(s) – R_g I_g(s)}{L_2 s + L_g s} $$
where \(U_{inv}\) is the inverter output voltage, \(U_c\) is the capacitor voltage, \(I_1\) is the inverter-side inductor current, \(I_g\) is the grid current, and \(U_g\) is the grid voltage. \(L_1\), \(L_2\), and \(C\) are the filter parameters, while \(R_g\) and \(L_g\) represent the grid impedance.
The transfer function of the quasi-PR controller is given by:
$$ G_{PR}(s) = K_p + \frac{2K_r\omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where \(K_p\) and \(K_r\) are the proportional and resonant gains, \(\omega_0\) is the fundamental angular frequency, and \(\omega_c\) is the cutoff frequency. The CCF path is modeled as a gain \(K_c\). The computational and PWM delay is approximated by a first-order Padé approximation: \(G_d(s) = 1 / (1 + s T_d)\), where \(T_d\) is the delay time.
Defining the state vector as \(\mathbf{x} = [u_c, i_1, i_g, x_1, x_2, v_{ref}]^T\), where \(x_1\) and \(x_2\) are the state variables of the quasi-PR controller, and the input vector as \(\mathbf{u} = [u_g, i_{ref}]^T\), the system can be represented in the standard state-space form:
$$ \dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} $$
$$ y = \mathbf{C} \mathbf{x} $$
where the output \(y\) is the grid current \(i_g\). The system matrix \(\mathbf{A}\) is a function of the physical and control parameters:
$$ \mathbf{A} = f(L_1, L_2, C, K_p, K_r, K_c, T_d, …) $$
For stability analysis based on the Lyapunov indirect method, the eigenvalues \(\lambda_i\) of matrix \(\mathbf{A}\) are decisive. The system is asymptotically stable if and only if the real part of all eigenvalues lies in the left-half of the complex plane:
$$ \text{Re}(\lambda_i(\mathbf{A})) < 0 \quad \forall i $$
Conversely, if any eigenvalue has a non-negative real part, the system is unstable. The stability of this on-grid inverter is thus a function of its parameters, which can be conceptually expressed as:
$$ S = \mathcal{F}(L_1, L_2, C, …) $$
where \(S\) represents the stability status. This complex, nonlinear relationship forms the basis for our neural network modeling approach.
Neural Network Design for Stability Assessment
The primary goal is to construct a model that can predict the stability status \(S\) given a set of uncertain parameters, most critically the filter components \(L_1\), \(L_2\), and \(C\). We employ a feedforward Back-Propagation Neural Network (BPNN) due to its proven capability as a universal function approximator and its suitability for this moderate-complexity mapping task.
Input and Output Design
The inputs to the neural network are the key parameters prone to variation: the inverter-side inductance \(L_1\), the grid-side inductance \(L_2\), and the filter capacitance \(C\). This choice directly targets the parameter uncertainty problem in on-grid inverters.
The output is designed for intuitive interpretation. Instead of directly predicting the eigenvalues, we formulate the problem as a binary classification task. The network is trained to output a value close to 0 if the system is stable and a value close to 1 if it is unstable. This is achieved by setting the target values during training based on the eigenvalue analysis of the state matrix \(\mathbf{A}\): if \(\max(\text{Re}(\lambda_i)) < 0\), the target is 0; otherwise, it is 1. This binary output provides a clear and immediate stability verdict for the on-grid inverter system.
Network Architecture and Training
A BPNN with a single hidden layer is chosen to balance model complexity, training speed, and the risk of overfitting, given the relatively small expected dataset. The Sigmoid function is selected as the activation function for its smooth gradient, which is suitable for the binary classification output. The network structure is illustrated below:
Input Layer (3 neurons): \(L_1\), \(L_2\), \(C\)
Hidden Layer (n neurons): Sigmoid activation
Output Layer (1 neuron): Sigmoid activation (Binary output: Stable~0, Unstable~1)
The training process minimizes a loss function, typically the Mean Squared Error (MSE) between the network’s predicted output and the binary target. The optimization is performed using the gradient descent algorithm, which is computationally efficient for this scale of problem. The weights and biases are updated iteratively through forward propagation and error backpropagation.
Dataset Generation and Network Training Process
A significant challenge in data-driven methods for engineering systems is the acquisition of large, realistic datasets. We circumvent this by generating a compact but representative dataset based on the known statistical characteristics of the parameters.
Parameter Characterization and Data Generation
We assume the parameters follow certain distributions based on practical considerations:
- \(L_1\) (Inverter-side Inductor): Subject to numerous independent factors like temperature and current, its variation is modeled as a Normal distribution around its nominal value.
- \(C\) (Filter Capacitor): Aging processes, often linked to applied reactive power, can be characterized by a Weibull distribution.
- \(L_2\) (Grid-side Inductor): Grid impedance measurements suggest a Normal distribution.
Three sets of 50 data points each are randomly generated for \(L_1\), \(L_2\), and \(C\) according to their respective distributions, centered around their nominal values. For each parameter combination \((L_1, L_2, C)\), the system matrix \(\mathbf{A}\) is constructed, its eigenvalues are computed, and a binary stability label (0 or 1) is assigned. This results in a combined training dataset of 150 samples. An example subset is shown in Table 1.
| C (μF) | L1 (mH) | L2 (mH) | max(Re(λ)) | Stability Label |
|---|---|---|---|---|
| 11.90 | 1.10 | 0.44 | 83.80 | 1 (Unstable) |
| 1.06 | 1.10 | 0.44 | -52.30 | 0 (Stable) |
| 11.29 | 1.10 | 0.44 | 25.88 | 1 (Unstable) |
| 11.00 | 1.07 | 0.44 | -7.10 | 0 (Stable) |
Training and Validation
The dataset is divided into a training set (e.g., 70%) and a test set (30%). The training set is used to adjust the network’s weights, while the test set evaluates its generalization performance. The training loop continues until the loss function converges to a minimum or a predefined number of epochs is reached. A successful training is indicated by a decreasing loss curve and a high classification accuracy on the test set, confirming that the network has effectively learned the stability boundary in the parameter space of the on-grid inverter.
Simulation and Experimental Verification
To validate the proposed neural network-based stability analysis method, we conduct simulations and experiments based on a system with the nominal parameters listed in Table 2.
| Parameter | Symbol | Nominal Value |
|---|---|---|
| Rated Power | S_inv | 3000 W |
| DC Link Voltage | U_dc | 400 V |
| Grid Voltage (RMS) | U_g | 220 V |
| Inverter-side Inductance | L_1 | 1.0 mH |
| Grid-side Inductance | L_2 | 0.2 mH |
| Filter Capacitance | C | 10 μF |
| PR Controller Gains | K_p / K_r | 0.11 / 11.0 |
| CCF Gain | K_c | 0.1 |
Neural Network Prediction Results
After training, the neural network is used to predict stability for specific parameter variations. Table 3 shows the network’s output for changes in individual parameters while others are held at boundary values. A threshold of 0.5 is applied to the sigmoid output for final classification.
| Fixed Parameters | Variable Param. & Value | NN Output | Predicted Stability |
|---|---|---|---|
| L1=1.1mH, L2=0.44mH | C = 8 μF | 0.0596 | Stable (0) |
| L1=1.1mH, L2=0.44mH | C = 11 μF | 0.0424 | Stable (0) |
| L1=1.1mH, L2=0.44mH | C = 12 μF | 0.9620 | Unstable (1) |
| C=11μF, L2=0.44mH | L1 = 1.0 mH | 0.0499 | Stable (0) |
| C=11μF, L2=0.44mH | L1 = 1.2 mH | 0.9554 | Unstable (1) |
| C=11μF, L1=1.1mH | L2 = 0.4 mH | 0.0440 | Stable (0) |
| C=11μF, L1=1.1mH | L2 = 0.6 mH | 0.9620 | Unstable (1) |
The network predicts instability when capacitance \(C\) exceeds approximately 11.5 μF, or when inductance \(L_1\) exceeds approximately 1.15 mH, or when \(L_2\) exceeds approximately 0.5 mH, under the given fixed conditions.
Simulation Verification
Time-domain simulations in MATLAB/Simulink are performed to verify these predictions. The grid current \(i_g\) and capacitor voltage \(u_c\) are monitored. With nominal parameters, both waveforms are clean sinusoids. When parameters are adjusted to the predicted unstable regions—for instance, setting \(C = 12 \mu F\) or \(L_1 = 1.3 mH\)—the simulation results exhibit clear and growing oscillations in both \(i_g\) and \(u_c\), confirming instability. The simulated waveforms for the stable and unstable cases align perfectly with the neural network’s binary classification, validating its accuracy. The critical parameter values for instability observed in simulation are within a reasonable margin of the network’s predicted boundaries, accounting for modeling approximations.
Experimental Validation
Further validation is conducted on a scaled-down three-phase inverter experimental platform configured for single-phase operation. Due to voltage limitations, parameters are adjusted: \(U_{dc} = 50V\), \(U_g = 25V\), \(K_p=0.2\), \(K_r=5\), \(K_c=0.04\). The neural network, retrained with a dataset reflecting this new nominal point, provides stability predictions for parameter changes. For example, it predicts stability for \(L_2 = 0.505 mH\) (output ~0.042) and instability for \(L_2 = 0.473 mH\) (output ~0.963). The experimental results, capturing \(u_c\) and \(i_g\), show a perfectly stable operation in the first case and distinct oscillatory instability in the second case. Similarly, varying capacitance \(C\) from a stable value of \(8.0 \mu F\) to an unstable value of \(10.0 \mu F\) produced experimental waveforms that transitioned from stable to unstable, again corroborating the neural network’s judgment. These experimental results conclusively demonstrate the practical efficacy of the proposed neural network-based method for assessing the stability of an on-grid inverter under parameter uncertainty.
Conclusion
This article has presented a novel, data-driven approach for stability analysis of LCL-type single-phase on-grid inverters. By constructing a parameterized state-space model and leveraging the function approximation capability of a neural network, we developed a tool that can rapidly predict stability based on key circuit parameters (\(L_1\), \(L_2\), \(C\)). The method innovatively uses a compact training dataset generated from the statistical distributions of these parameters, making it practical for applications where large operational datasets are unavailable.
The core contributions are threefold: First, the integration of a quasi-PR controller with CCF active damping provides a high-performance control framework for the on-grid inverter. Second, the formulation of stability assessment as a neural-network-based binary classification problem offers a fast and intuitive alternative to complex analytical derivations. Third, the validation through both detailed simulation and hardware experiment confirms the accuracy and robustness of the method.
The proposed framework is directly applicable to the online monitoring and stability assessment of on-grid inverters. By inputting measured or estimated parameters, system operators can obtain an immediate stability verdict. Future work will focus on expanding the parameter space to include controller gains and grid impedance variations, employing larger datasets, and exploring more advanced neural network architectures to further improve prediction accuracy and robustness for complex on-grid inverter systems operating in evolving grid conditions.