The precise characterization of the terminal admittance of a utility interactive inverter is a cornerstone for stability analysis in modern power systems with high penetration of renewable energy. Conventional measurement techniques, often reliant on the Fast Fourier Transform (FFT), suffer from significant inaccuracies when the grid frequency deviates from its nominal value due to spectral leakage and the picket-fence effect. This work presents a novel, high-precision method for measuring the frequency-coupled admittance matrix of grid-connected inverters. The core innovation lies in reframing the admittance measurement process—inherently based on the injection of deterministic perturbation signals—as a problem of estimating deterministic components within a broadband signal. A polynomial-fitting-based algorithm is developed to accurately extract these components, followed by a frequency correction model for rapid iterative refinement. Finally, a complete measurement framework for frequency-coupled admittance is established. Simulation and experimental results demonstrate that the proposed method achieves superior accuracy in broadband signal measurement compared to standard and improved FFT techniques. It reliably extracts the admittance characteristics of a utility interactive inverter across various grid frequency scenarios, proving its robust applicability for deterministic component measurement.
Introduction and Problem Formulation
The large-scale integration of renewable generation, primarily interfaced through power electronic converters like the utility interactive inverter, is reshaping power systems. While enabling the energy transition, this shift introduces new stability challenges, particularly wide-band oscillations arising from dynamic interactions between inverters and weak grids or among inverters themselves. Impedance-based stability analysis has become a prevalent tool for studying these small-signal stability issues. This method treats the grid and the utility interactive inverter as two separate subsystems, analyzes their individual terminal characteristics (impedance or admittance), and applies stability criteria to the interconnected system. Consequently, accurate in-situ measurement of the admittance of power electronic-based generation is a critical step.
Existing methods often inject a sinusoidal voltage perturbation in series with the grid connection and measure the resulting voltage and current responses at the point of common coupling (PCC). For a utility interactive inverter, due to control dynamics such as Phase-Locked Loops (PLLs) and synchronous-frame current controllers, a perturbation at a given frequency \(f_p\) will typically excite not only a response at \(f_p\) but also at a coupled frequency \(f_{p1} = 2f_n – f_p\), where \(f_n\) is the grid fundamental frequency. This phenomenon is described by the frequency-coupled admittance matrix \(\mathbf{Y}(f_p)\):
$$
\begin{bmatrix}
I_p \\
I_{p1}
\end{bmatrix}
=
\begin{bmatrix}
Y_{11}(f_p) & Y_{12}(f_p) \\
Y_{21}(f_p) & Y_{22}(f_p)
\end{bmatrix}
\begin{bmatrix}
V_p \\
V_{p1}
\end{bmatrix}
$$
where \(V_p, V_{p1}\) and \(I_p, I_{p1}\) are the complex phasors of the voltage and current at frequencies \(f_p\) and \(f_{p1}\), respectively. The elements \(Y_{11}\) and \(Y_{22}\) are the self-admittances, while \(Y_{12}\) and \(Y_{21}\) are the cross-coupling admittances. Solving for this 2×2 matrix requires two linearly independent perturbation injections, leading to the system:
$$
\begin{bmatrix}
I_p & I’_p \\
I_{p1} & I’_{p1}
\end{bmatrix}
=
\begin{bmatrix}
Y_{11} & Y_{12} \\
Y_{21} & Y_{22}
\end{bmatrix}
\begin{bmatrix}
V_p & V’_p \\
V_{p1} & V’_{p1}
\end{bmatrix}
$$
The accuracy of the extracted admittance matrix \(\mathbf{Y}\) is therefore directly contingent upon the precision with which the eight complex phasors (\(V_p, V_{p1}, I_p, I_{p1}, V’_p, V’_{p1}, I’_p, I’_{p1}\)) can be measured. The FFT, while computationally efficient, is highly sensitive to off-nominal grid frequencies. A slight deviation in \(f_n\) causes non-integer period sampling, resulting in spectral leakage that severely corrupts the amplitude and phase estimates of the deterministic perturbation components. This is illustrated in the following comparison for a simple multi-tone signal when \(f_n = 49.9\) Hz. Advanced methods like windowing and spectral interpolation offer limited improvement. Recognizing that the perturbation frequencies \(f_p\) and \(f_{p1}\) are known a priori, the measurement challenge is fundamentally one of high-precision estimation of deterministic components within a noisy, multi-frequency signal. This work addresses this core problem.
Deterministic Component Measurement via Polynomial Fitting
The measured signal \(x_w(t)\) at the PCC after perturbation injection can be modeled as a sum of deterministic sinusoidal components plus noise:
$$
x_w(t) = \sqrt{2}X_0(t)\cos(2\pi f_n t + \phi_0(t)) + \sum_{m=1}^{M} \sqrt{2}X_m \cos(2\pi f_m t + \phi_m) + \eta(t)
$$
Here, the first term represents the fundamental component with potentially slowly varying amplitude \(X_0(t)\) and phase \(\phi_0(t)\). The summation includes \(M\) deterministic perturbation components (e.g., at \(f_p\) and \(f_{p1}\)) with constant amplitudes \(X_m\) and phases \(\phi_m\). \(\eta(t)\) is additive noise. The core task is to estimate \(X_m\) and \(\phi_m\) for \(m = 1,…,M\), given the known frequencies \(f_m\).
The proposed method is based on approximating the time-varying envelope of each component using low-order polynomials. For a single component at frequency \(f_m\), its contribution \(x_m(t)\) can be expressed as:
$$
x_m(t) = \sqrt{2}X_m\cos(2\pi f_m t + \phi_m) = A_m \cos(2\pi f_m t) – B_m \sin(2\pi f_m t)
$$
where \(A_m = \sqrt{2}X_m \cos \phi_m\) and \(B_m = \sqrt{2}X_m \sin \phi_m\). If the component’s amplitude and phase were slowly varying, they could be approximated as polynomials in time. We extend this idea by treating \(A_m\) and \(B_m\) themselves as polynomials, even for constant components, as this provides a flexible fitting model. For a polynomial order \(n\), we have:
$$
\begin{aligned}
A_m(t) &\approx A_{m0} + A_{m1}t + … + A_{mn}t^n \\
B_m(t) &\approx B_{m0} + B_{m1}t + … + B_{mn}t^n
\end{aligned}
$$
The complete signal model for \(x_w(t)\) becomes a linear combination of basis functions:
$$
x_w(t) \approx \sum_{m=0}^{M} \left[ \left( \sum_{k=0}^{n} A_{mk}t^k \right) \cos(2\pi f_m t) – \left( \sum_{k=0}^{n} B_{mk}t^k \right) \sin(2\pi f_m t) \right]
$$
where index \(m=0\) corresponds to the fundamental component at \(f_n\). This can be written compactly in matrix form as:
$$
x_w(t) \approx \mathbf{p}(t) \cdot \mathbf{H}
$$
Here, \(\mathbf{p}(t)\) is a row vector containing all the basis functions (\(t^k \cos(2\pi f_m t)\), \(t^k \sin(2\pi f_m t)\)), and \(\mathbf{H}\) is a column vector of all polynomial coefficients \(A_{mk}\) and \(B_{mk}\). Given sampled data, we construct the regression matrix \(\mathbf{P}\) where each row is \(\mathbf{p}(t)\) at a sample instant. The least-squares estimate of the coefficient vector is:
$$
\hat{\mathbf{H}} = (\mathbf{P}^T\mathbf{P})^{-1} \mathbf{P}^T \mathbf{X_w}
$$
where \(\mathbf{X_w}\) is the vector of signal samples. From the estimated coefficients for the \(m\)-th component, its amplitude \(X_m’\), phase \(\phi_m’\), and a refined frequency estimate \(f_m’\) are calculated:
$$
\begin{aligned}
X_m’ &= \frac{1}{\sqrt{2}} \sqrt{A_{m0}^2 + B_{m0}^2}, \\
\phi_m’ &= \arctan\left( \frac{B_{m0}}{A_{m0}} \right), \\
f_m’ &= f_m + \Delta f_m = f_m + \frac{1}{2\pi} \frac{A_{m0}B_{m1} – B_{m0}A_{m1}}{A_{m0}^2 + B_{m0}^2}.
\end{aligned}
$$
The term \(\Delta f_m\) acts as a frequency correction. The process is iterative: after solving for \(\hat{\mathbf{H}}\), the frequencies are updated (\(f_m^{(k+1)} = f_m^{(k)} + \Delta f_m\)), the basis matrix \(\mathbf{P}\) is recalculated, and a new least-squares solution is found. Iteration continues until the frequency corrections fall below a threshold or a maximum count is reached. This algorithm is summarized below.
| Step | Action |
|---|---|
| 1 | Input signal \(x_w(t)\), initial frequencies \(f_m\), threshold \(\epsilon\), max iterations \(K\). |
| 2 | Construct time vector and basis matrix \(\mathbf{P}\) using current \(f_m\). |
| 3 | Solve \(\hat{\mathbf{H}} = (\mathbf{P}^T\mathbf{P})^{-1} \mathbf{P}^T \mathbf{X_w}\). |
| 4 | Calculate frequency corrections \(\Delta f_m\) from \(\hat{\mathbf{H}}\). |
| 5 | If \(\max|\Delta f_m| < \epsilon\) or iterations \(= K\), go to Step 6. Else, update \(f_m = f_m + \Delta f_m\), \(k=k+1\), go to Step 2. |
| 6 | Output final component amplitudes \(X_m’\) and phases \(\phi_m’\). |
Frequency-Coupled Admittance Measurement Framework
Integrating the polynomial-fitting measurement core into a complete procedure for characterizing a utility interactive inverter yields the following framework:
- Sweep Definition: Define a set of \(K\) perturbation frequencies \(\{f_p\}\) covering the range of interest (e.g., 10 Hz to 2000 Hz).
- First Perturbation Injection: Inject a balanced, positive-sequence sinusoidal voltage disturbance at frequency \(f_p^{(i)}\).
- First Measurement: After transients settle, sample the three-phase PCC voltages and currents. Use the polynomial-fitting algorithm to estimate the complex phasors \(V_p, I_p\) at \(f_p\) and \(V_{p1}, I_{p1}\) at \(f_{p1} = 2f_n – f_p\).
- Second Perturbation Injection: Inject a balanced, negative-sequence sinusoidal voltage disturbance at the coupled frequency \(f_{p1}\).
- Second Measurement: Sample the voltages and currents again. Use the fitting algorithm to estimate the new set of phasors \(V’_p, I’_p\) and \(V’_{p1}, I’_{p1}\).
- Admittance Calculation: With two linearly independent vector pairs, solve for the admittance matrix at \(f_p^{(i)}\):
$$\begin{bmatrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{bmatrix} = \begin{bmatrix} I_p & I’_p \\ I_{p1} & I’_{p1} \end{bmatrix} \begin{bmatrix} V_p & V’_p \\ V_{p1} & V’_{p1} \end{bmatrix}^{-1}$$ - Iteration: Repeat steps 2-6 for all frequencies \(f_p^{(i)}\) in the sweep.
This framework ensures that the critical phasor measurements are obtained with high accuracy, even under off-nominal grid frequency conditions, leading to a reliable frequency-coupled admittance model for the utility interactive inverter.
Simulation and Experimental Validation
Algorithm Performance with Synthetic Signals
The performance of the polynomial-fitting (PF) method was first benchmarked against standard FFT and an improved FFT method (I-FFT) using synthetic signals. A test signal containing a 50 Hz fundamental and six deterministic perturbation components (forming three coupled pairs, e.g., 10 & 110 Hz) was generated. Measurements were conducted under different grid frequencies (\(f_n\)) and noise levels. Key results are summarized below.
| Grid Freq. (Hz) | Method | Max |Freq. Error| (Hz) | Max |Amp. Error| (%) | Max |Phase Error| (°) |
|---|---|---|---|---|
| 50.0 | FFT | 0 | 1.08×10⁻¹⁴ | 2.15×10⁻¹¹ |
| I-FFT | 1.07×10⁻⁵ | 1.02×10⁻⁶ | 5.82×10⁻³ | |
| Proposed PF | 0 | 6.51×10⁻¹⁴ | 9.83×10⁻¹² | |
| 49.9 | FFT | 0.2 | 0.22 | 45.18 |
| I-FFT | 9.92×10⁻⁷ | 7.64×10⁻⁴ | 0.13 | |
| Proposed PF | 0 | 3.64×10⁻¹⁴ | 9.72×10⁻¹² | |
| 50.5 | FFT | 0 | 0.48 | 74.76 |
| I-FFT | 3.24×10⁻³ | 1.78×10⁻³ | 0.64 | |
| Proposed PF | 0 | 4.31×10⁻¹⁴ | 9.72×10⁻¹² |
| SNR | Method | Max |Amp. Error| (%) | Max |Phase Error| (°) |
|---|---|---|---|
| 40 dB | FFT | 2.80×10⁻³ | 0.25 |
| I-FFT | 3.98×10⁻³ | 2.29 | |
| Proposed PF | 7.23×10⁻³ | 0.35 | |
| 70 dB | FFT | 1.22×10⁻⁴ | 3.12×10⁻³ |
| I-FFT | 1.04×10⁻⁴ | 3.66×10⁻² | |
| Proposed PF | 4.70×10⁻⁴ | 1.54×10⁻² |
The results clearly show the proposed PF method’s immunity to grid frequency variation, maintaining machine-level precision, while FFT errors become unacceptable. Under noise, all methods show gracefully degrading performance, with PF maintaining competitive, often superior, accuracy especially in phase estimation.
Admittance Measurement of a Utility Interactive Inverter
The framework was tested in a detailed Simulink model of a three-phase utility interactive inverter with LCL filter and standard vector current control. Admittance was measured from 10 Hz to 2000 Hz. The theoretical admittance was derived using an analytical model. Measurements were taken at nominal (50 Hz) and off-nominal (49.9 Hz, 50.1 Hz) grid frequencies. The results for the self-admittance \(Y_{11}\) are exemplary and summarized below.
| Grid Freq. (Hz) | Method | Max |\(Y_{11}\) Amp. Error| (dB) | Max |\(Y_{11}\) Phase Error| (°) |
|---|---|---|---|
| 50.0 | FFT | 0.25 | 6.39 |
| I-FFT | 0.25 | 6.45 | |
| Proposed PF | 0.29 | 6.41 | |
| 49.9 | FFT | 5.14 | 259.75 |
| I-FFT | 0.30 | 13.31 | |
| Proposed PF | 0.28 | 6.44 | |
| 50.5 | FFT | 17.45 | 195.14 |
| I-FFT | 0.45 | 30.87 | |
| Proposed PF | 0.43 | 6.45 |
The proposed method consistently produces results aligning with the theoretical model across all frequency scenarios. The FFT method fails catastrophically when the grid frequency shifts. The I-FFT method shows significant phase errors at off-nominal frequencies, whereas the PF method’s accuracy remains stable and high.
Experimental Verification
The method was implemented on a 380V hardware platform comprising a grid emulator, a series voltage disturbance injector, measurement sensors, and a real-time controller. The admittance of a passive RL load (known impedance) was measured. The results confirmed the simulation findings: the standard FFT produced large errors in the negative-sequence admittance phase due to grid frequency drift, while the proposed polynomial-fitting-based measurement accurately matched the theoretical impedance across the frequency sweep. This practical validation underscores the method’s readiness for real-world characterization of utility interactive inverters and other grid-connected power electronic devices.
Conclusion
This paper has presented a high-precision measurement method for the frequency-coupled admittance of utility interactive inverters. By conceptualizing the perturbation-based measurement as a deterministic signal component estimation problem, a robust polynomial-fitting algorithm was developed. This algorithm iteratively refines its frequency estimates to accurately extract the amplitude and phase of known perturbation components, even under significant grid frequency deviations and in the presence of noise. The complete measurement framework, built around this core algorithm, enables the accurate characterization of the full 2×2 frequency-coupled admittance matrix. Extensive simulation and experimental results demonstrate that the proposed method significantly outperforms conventional FFT-based techniques in accuracy and robustness when the grid frequency is not precisely nominal. This advancement provides a reliable tool for the black/grey-box impedance modeling essential for stability analysis in future renewable-rich power systems.