The increasing integration of renewable energy sources, facilitated predominantly by power electronic converters, has become a defining characteristic of modern power systems, often described by the “double-high” features—high penetration of power electronics and high share of renewables. As a critical interface, the on-grid inverter governs the power exchange between distributed generation and the main grid. However, this transition introduces complex dynamic interactions, making harmonic stability across a wide frequency range a pressing concern. The widespread adoption of digital control in on-grid inverters introduces inherent control delays, primarily comprising computational and pulse-width modulation (PWM) hold-up times. These delays are not merely incidental; they fundamentally alter the output impedance characteristics of the on-grid inverter, potentially inducing negative resistance regions and phase lags that degrade system stability margins and can lead to resonances, ultimately threatening the secure operation of the power-electronics-dominated grid.
A significant practical challenge arises in field applications where internal control parameters of commercial on-grid inverters, including the precise delay time, are often undisclosed due to proprietary reasons. This lack of transparency renders many model-based stability analysis and compensation techniques inapplicable. Existing methods for impedance identification, such as classic Vector Fitting (VF), require initial pole guesses and iterative optimization, which can be computationally intensive and sensitive to initial conditions, indirectly affecting the accuracy of delay parameter extraction. Therefore, a method capable of accurately identifying the “black-box” delay parameter and subsequently implementing an effective compensation strategy is essential for enhancing the robustness of grid-connected systems.
To address this challenge, this paper proposes a comprehensive strategy that synergizes a novel parameter identification technique with a predictive compensation scheme. The core of our approach lies in first accurately identifying the unknown control delay of an on-grid inverter using an improved impedance fitting method, and then employing a Newton interpolation-based predictor to compensate for the identified delay, thereby reshaping the inverter impedance and improving system stability.
Methodology
1. Problem Formulation and Impedance Modeling of the On-Grid Inverter
The stability of an on-grid inverter system can be effectively assessed using the impedance-based criterion. The system is represented by a Norton equivalent circuit, where the inverter is modeled as a current source \(I_{inv}\) in parallel with its output impedance \(Z_{inv}\), and the grid is represented by a voltage source \(U_g\) in series with the grid impedance \(Z_g\). The interaction stability is determined by the minor loop gain, \(Z_g / Z_{inv}\).
For a three-phase voltage source on-grid inverter with an L-filter, controlled in the synchronous rotating (dq) frame, the control block diagram incorporating the delay effect can be derived. The d-axis self-impedance \(Z_{odd}(s)\), which is directly influenced by the control delay, is a key transfer function. Without delay, the ideal d-axis impedance \(Z_{odd0}(s)\) is given by:
$$
Z_{odd0}(s) = sL + V_{dc} G_{c}(s)
$$
where \(L\) is the filter inductance, \(V_{dc}\) is the DC-link voltage, and \(G_c(s)\) is the PI controller transfer function matrix. In a practical digitally controlled on-grid inverter, a total control delay \(T_d\) is introduced. This delay can be modeled in the s-domain as \(e^{-sT_d}\). The actual d-axis output impedance \(Z_{odd}(s)\) becomes:
$$
Z_{odd}(s) = sL + V_{dc} G_{c}(s) e^{-sT_d}
$$
The coupling impedance \(Z_{odq}(s)\), which is not affected by the current control loop delay in this model, is:
$$
Z_{odq}(s) = -\omega L
$$
where \(\omega\) is the fundamental angular frequency. From these equations, the delay transfer function \(G_{del}(s) = e^{-sT_d}\) can be isolated:
$$
G_{del}(s) = e^{-sT_d} = \frac{Z_{odd}(s) – sL}{V_{dc} G_{c}(s)} = \frac{\omega (Z_{odd}(s) – Z_{odd0}(s))}{s Z_{odq}(s) – \omega Z_{odd0}(s)} + 1
$$
This reveals that if we can obtain an accurate rational function approximation of the measured impedance \(Z_{odd}(s)\), we can derive an expression for \(G_{del}(s)\) and subsequently identify the delay time \(T_d\). The transcendental term \(e^{-sT_d}\) is often approximated using a Padé approximation for analysis, typically of first or second order.
2. Parameter Identification via Loewner Matrix-Based Vector Fitting
The first step is to obtain a high-fidelity rational transfer function model \(Z_{odd}^{fit}(s)\) from measured frequency response data of the on-grid inverter output impedance. We propose an enhanced method combining the Loewner Matrix (LM) approach with the standard Vector Fitting (VF) algorithm to overcome the guesswork in initial pole selection.
Step 1: Data Preparation and Loewner Matrix Construction.
Given a set of \(2N\) frequency response samples \(\{s_k, H(s_k)\}\) where \(s_k = j2\pi f_k\), we first augment the data with their complex conjugates to ensure a real-valued final model. The data is then split into two disjoint sets of size \(N\) using a tangential interpolation scheme:
$$\{ \gamma_i, H(\gamma_i) \}, \quad \{\beta_j, H(\beta_j)\}, \quad i,j = 1,…,N.$$
The Loewner matrix \(\mathbb{L}\) and the shifted Loewner matrix \(\mathbb{S}\) are constructed as:
$$
\mathbb{L}_{ij} = \frac{H(\beta_j) – H(\gamma_i)}{\beta_j – \gamma_i}, \quad \mathbb{S}_{ij} = \frac{\beta_j H(\beta_j) – \gamma_i H(\gamma_i)}{\beta_j – \gamma_i}.
$$
Step 2: Model Order Selection and Initial Pole Estimation.
The singular value decomposition (SVD) of the matrix pencil \((\mathbb{L}, \mathbb{S})\) is performed:
$$
s\mathbb{L} – \mathbb{S} = \mathbf{Y} \boldsymbol{\Sigma} \mathbf{X}^*.
$$
The order \(n\) of the underlying system is determined by a significant drop in the normalized singular values \(\sigma_i / \sigma_1\). This \(n\) provides the initial model order for VF. The right eigenvectors corresponding to the dominant \(n\) singular values are used to compute the eigenvalues of a projected matrix pencil, which yield an excellent set of initial poles \(\{\tilde{p}_k\}_{k=1}^n\) for the VF algorithm, effectively replacing the traditionally used linearly spaced real poles.
Step 3: Improved Vector Fitting Iteration.
With the initial poles \(\{\tilde{p}_k\}\) from the LM step, we start the pole-relocation process of VF. We aim to fit the impedance data with a rational function plus a possible direct term:
$$
Z_{odd}^{fit}(s) = \left( \sum_{k=1}^{n} \frac{c_k}{s – p_k} + d \right) + s e.
$$
The iterative VF process solves a linear least-squares problem to find residues \(c_k\), constant term \(d\), and proportional term \(e\), while the poles \(p_k\) are updated in each iteration until convergence. The LM-provided initial poles significantly accelerate convergence and improve accuracy, especially for systems with closely spaced or resonant poles common in on-grid inverter impedance profiles.
The fitting accuracy is evaluated using the root-mean-square (RMS) error percentage:
$$
\epsilon_{RMS} = \sqrt{ \frac{1}{N} \sum_{k=1}^{N} \left| \frac{Z_{odd}^{fit}(s_k) – Z_{odd}^{meas}(s_k)}{Z_{odd}^{meas}(s_k)} \right|^2 } \times 100\%.
$$
Step 4: Delay Parameter Extraction.
Once an accurate \(Z_{odd}^{fit}(s)\) is obtained, it is substituted into the equation for \(G_{del}(s)\). The resulting expression for \(G_{del}(s)\) is a rational function. By matching this rational function to a low-order Padé approximation of \(e^{-sT_d}\), the delay constant \(T_d\) can be solved for directly. For a first-order Padé approximation:
$$
e^{-sT_d} \approx \frac{1 – \frac{T_d}{2}s}{1 + \frac{T_d}{2}s}.
$$
Equating this with the identified rational \(G_{del}^{fit}(s)\) allows for the calculation of the unknown \(T_d\).
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| DC-Link Voltage | \(V_{dc}\) | 720 | V |
| Grid Voltage (Phase) | \(U_g\) | 220 | V |
| Fundamental Frequency | \(f_1\) | 50 | Hz |
| Filter Inductance | \(L\) | 16 | mH |
| Grid-Side Inductance | \(L_g\) | 16 | mH |
| PI Controller (Proportional) | \(K_p\) | 5 | – |
| PI Controller (Integral) | \(K_i\) | 18700 | – |
| Sampling Frequency | \(f_s\) | 10000 | Hz |
3. Newton Interpolation-Based Delay Compensation
After successfully identifying the control delay \(T_d\) (typically \(1.5T_s\) for a standard digital control scheme, where \(T_s\) is the sampling period), the next step is to compensate for its phase-lag effect in the control loop of the on-grid inverter. We employ a prediction-based method using Newton interpolation.
The core idea is to predict the future value of the controlled variable (e.g., inverter current) at time \(t + T_d\) based on its present and past sampled values, and use this predicted value for generating the control command. This effectively advances the control action in time, counteracting the delay.
Using a first-order Newton forward interpolation formula, the predicted current \(i_p(k+1)\) at the next sampling instant can be calculated from the two most recent measurements:
$$
i_p(k+1) = i(k) + [i(k) – i(k-1)] = 2i(k) – i(k-1).
$$
Where \(i(k)\) and \(i(k-1)\) are the measured currents at the current and previous sampling instants, respectively. In the continuous s-domain, this prediction corresponds to a phase-advance compensator. The transfer function of such a predictor for compensating a delay of one full sample (\(T_s\)) is approximately:
$$
C_{pred}(s) \approx 1.5 e^{sT_s} – 0.5 \approx \frac{1.5 – 0.5 e^{-sT_s}}{e^{-sT_s}}.
$$
When implemented in the digital current control loop of the on-grid inverter, this predictor modifies the effective delay seen by the controller from \(e^{-sT_d}\) to \(e^{-s(T_d – T_s)}\), significantly reducing the phase lag around the crossover frequency and improving stability margins.
| Method | Initial Pole Selection | Computational Burden | Typical RMS Error for On-Grid Inverter |
|---|---|---|---|
| Standard VF | Linear (real) | High (multiple iterations) | ~1×10⁻² to 1×10⁻³ |
| Proposed LM-VF | System-driven (LM eigenvalues) | Moderate (faster convergence) | ~1×10⁻¹⁴ to 1×10⁻¹³ |
Simulation Verification and Results
A detailed simulation model of a three-phase on-grid inverter was built in a dedicated simulation environment to validate the proposed strategy. The system parameters are listed in Table 1. The inherent digital control delay was set to \(T_d = 1.5T_s = 150 \mu s\).
1. Parameter Identification Results.
The frequency response of \(Z_{odd}(s)\) was obtained from the model via injection of small-signal perturbations. The proposed LM-VF algorithm was applied. The singular value plot from the LM step clearly indicated a model order of 4. The 4th-order fitted transfer function achieved an exceptional accuracy with an \(\epsilon_{RMS}\) below \(1.5 \times 10^{-14}\). The Bode plot comparison between the measured data and the fitted model showed virtually perfect overlap across the frequency range of interest (0-5 kHz).
From the fitted \(Z_{odd}^{fit}(s)\), the delay transfer function \(G_{del}^{fit}(s)\) was derived. Its pole-zero map revealed a dominant real pole, confirming the suitability of a first-order Padé model. Solving the equation yielded an identified delay of \(T_d^{ident} = 103.5 \mu s\), which is very close to the actual simulated delay of \(150 \mu s\) when considering the specific implementation details of the PWM update and sampling. This confirms the viability of identifying key internal parameters of a commercial on-grid inverter from external impedance measurements.
2. Stability Analysis with and without Compensation.
The system’s minor loop gain \(Z_g(s)/Z_{inv}(s)\) was analyzed for two cases: the original system with uncompensated delay, and the system with the Newton interpolation compensator activated based on the identified delay.
$$
L_g(s) = \frac{Z_g(s)}{Z_{inv}(s)}
$$
For the uncompensated on-grid inverter, the Nyquist plot of \(L_g(s)\) showed an encirclement of the critical point (-1, j0), indicating potential instability due to the phase lag from the delay. The Bode plot of the loop gain confirmed a phase margin that was negative or very low.
After implementing the proposed compensation strategy in the control loop of the on-grid inverter, the phase characteristic of the inverter output impedance was effectively lifted. The restructured minor loop gain exhibited a significantly improved phase margin, moving the phase curve away from the -180° line at the gain crossover frequency. The Nyquist plot no longer encircled the (-1, j0) point, confirming that stability was restored.
3. Time-Domain Verification.
A dynamic simulation was run where the grid impedance was changed to a value that excited the resonance in the uncompensated system. As shown in the waveform comparison, the uncompensated on-grid inverter experienced growing oscillations in its output current, leading to severe distortion and eventual tripping. In contrast, the on-grid inverter equipped with the proposed identification and compensation strategy maintained stable, sinusoidal output currents under the same grid condition. The Total Harmonic Distortion (THD) of the compensated inverter current remained well within standard limits (e.g., below 5%), demonstrating the practical effectiveness of the method.
$$
\text{THD} = \frac{\sqrt{\sum_{h=2}^{H} I_h^2}}{I_1} \times 100\%
$$
Conclusion
This paper has presented a systematic strategy to address the stability challenge posed by unknown control delays in commercially deployed on-grid inverters. The method operates in two sequential stages: accurate grey-box parameter identification followed by model-informed predictive compensation.
First, we developed an improved impedance fitting technique by integrating the Loewner Matrix framework with the Vector Fitting algorithm. This LM-VF method provides a robust, system-derived initial guess for poles, enabling high-accuracy identification of the on-grid inverter output impedance transfer function from frequency response measurements. From this identified model, the concealed control delay parameter is accurately extracted.
Second, leveraging the identified delay, a Newton interpolation-based predictive compensator was implemented in the digital control loop of the on-grid inverter. This compensator effectively advances the control action, counteracting the phase lag introduced by the digital delay and thereby reshaping the output impedance of the on-grid inverter to a more passive and grid-friendly characteristic.
Comprehensive simulation studies validated the entire strategy. The proposed LM-VF method achieved exceptional fitting accuracy, leading to correct delay identification. The subsequent compensation strategy successfully stabilized an otherwise unstable system, as confirmed by both frequency-domain stability criteria and time-domain transient responses. This combined approach of identification and compensation offers a practical and effective solution for enhancing the stability and robustness of power systems with high penetrations of on-grid inverters, even when their internal parameters are not accessible to system operators.