The transition from traditional power systems dominated by synchronous generators to new power systems centered on renewable energy has led to a highly power-electronic-based grid with high penetration of distributed generation. This evolution weakens the grid’s inertia and damping capabilities, resulting in a progressively weaker grid characteristic. In weak grids, the interaction between the Phase-Locked Loop (PLL) and grid impedance can destabilize current-controlled utility interactive inverters. Conversely, voltage-controlled utility interactive inverters can provide necessary damping and inertia. The grid impedance parameter, especially at the fundamental power frequency, is a key indicator of grid strength and is crucial for ensuring the stable operation of grid-following utility interactive inverters, controller design, and adaptive control mode switching. Therefore, accurate online measurement of grid impedance is of significant importance.
Existing online impedance measurement methods are categorized into passive and active methods. Active methods, which inject specific perturbation signals, are widely used. Among these, injecting a single non-characteristic harmonic frequency is common for power-frequency impedance estimation. However, selecting the perturbation frequency involves a trade-off. Using a high-frequency perturbation (e.g., 200 Hz, 500 Hz) is less affected by grid frequency fluctuations but suffers from inaccuracy due to the skin effect and the nonlinear nature of grid components, making the extrapolation to the fundamental frequency error-prone. Using a frequency close to the fundamental (e.g., 60 Hz, 55 Hz) improves the extrapolation accuracy but makes the measurement highly susceptible to spectral leakage caused by grid frequency fluctuations and background harmonics during the Discrete Fourier Transform (DFT) process. This leakage severely degrades measurement precision.
This article analyzes the underlying mechanism of how grid frequency fluctuation and perturbation frequency choice affect impedance measurement accuracy. We then propose a novel online measurement method based on complementary current signal injection. This method significantly mitigates the impact of fundamental frequency spectral leakage, allowing the use of perturbation frequencies much closer to the power frequency, thereby achieving a more accurate estimation of the fundamental grid impedance.
Theory and Impact of Frequency Fluctuation
The conventional active measurement scheme using a utility interactive inverter involves superimposing a sinusoidal perturbation current command at frequency $f_h$ onto the fundamental current reference. The grid impedance at the perturbation frequency $Z_g(h)$ is calculated from the measured perturbation voltage $V_g(h)$ and current $I_g(h)$ at the Point of Common Coupling (PCC):
$$ Z_g(h) = \frac{V_g(h)}{I_g(h)} = R_g(h) + jX_g(h) $$
The fundamental frequency impedance $Z_{g1}$ is then estimated by scaling the reactance:
$$ Z_{g1} \approx R_{g1} + j \frac{\omega_1}{\omega_h} X_g(h) $$
where $\omega_1$ and $\omega_h$ are the fundamental and perturbation angular frequencies, respectively. The core challenge is the accurate extraction of $V_g(h)$ and $I_g(h)$ from sampled signals containing the strong fundamental component and other noise, typically using DFT.
The analysis using Fourier Transform (FT) reveals the root of the problem. A finite-time measurement is equivalent to multiplying the time-domain signal $x(t)=x_1(t)+x_h(t)$ (fundamental plus perturbation) by a rectangular window function $w(t)$. The FT of the windowed signal is:
$$ V(\omega) = \mathcal{F}\{[x_1(t) + x_h(t)] w(t)\} = \frac{1}{2\pi}[X_1(\omega) * W(\omega) + X_h(\omega) * W(\omega)] $$
where $*$ denotes convolution. The term $X_1(\omega) * W(\omega)$ represents the spread of the fundamental component’s energy across the frequency spectrum due to the window. For a perfect measurement with no spectral leakage, we require $W(\omega_h – \omega_1) + W(\omega_h + \omega_1) = 0$. For a rectangular window of width $\tau_R$, this condition is ideally met when:
$$ \tau_R = \frac{2\pi}{|\omega_1 – \omega_h|} $$
However, in practice, the grid frequency $f_g = \omega_1 / 2\pi$ is not constant and fluctuates (e.g., ±0.05 Hz, ±0.2 Hz). Let $\Delta f = f_g – 50\text{Hz}$ be the frequency deviation. The actual fundamental frequency is $f_1 = 50 + \Delta f$ Hz. The previous condition is violated because $\omega_1$ in the equation is now the actual, fluctuating frequency. Consequently, $W(\omega_h – \omega_1) + W(\omega_h + \omega_1) \neq 0$, meaning the fundamental component “leaks” into the DFT bin corresponding to the perturbation frequency $f_h$. This leakage corrupts the extracted $V_g(h)$ and $I_g(h)$, leading to erroneous impedance calculation.
The severity of this error depends on two key factors:
- Magnitude of Grid Frequency Fluctuation ($\Delta f$): A larger $\Delta f$ causes greater spectral leakage.
- Proximity of Perturbation Frequency to Fundamental ($|f_h – 50|$): The closer $f_h$ is to 50 Hz, the narrower the main lobe of the window’s spectrum needs to be to separate the signals, making the measurement more sensitive to any shift $\Delta f$. The leakage percentage increases significantly as $f_h$ approaches 50 Hz for a given $\Delta f$.
The following table summarizes the relationship between perturbation frequency and required measurement window parameters for a nominal 50 Hz grid.
| Perturbation Frequency $f_h$ (Hz) | Required Window Width $\tau_R$ (for ideal 50 Hz) | Relative Sensitivity to Frequency Fluctuation |
|---|---|---|
| 75 | 0.04 s (2 cycles of 50 Hz) | Lower |
| 60 | 0.10 s (5 cycles of 50 Hz) | Higher |
| 55 | 0.20 s (10 cycles of 50 Hz) | Much Higher |
Furthermore, the phase relationship between the fundamental and perturbation signals at the start of the measurement window also affects the measured impedance value due to the time-shift property of the FT. As grid frequency fluctuates, this initial phase difference varies, causing cyclical errors in the measured impedance. Common mitigation techniques like using a Hanning window reduce sidelobe leakage but double the main lobe width, requiring even longer measurement times and still struggling with perturbations near the fundamental frequency.
Proposed Complementary Current Injection Method
To overcome the limitations described above, we propose a novel online measurement method for utility interactive inverters based on Complementary Current Signal Injection. The core idea is to inject two consecutive perturbation current segments that are complementary (180° out-of-phase) at identical fundamental current phase angles. By post-processing the two sets of sampled data, the fundamental component can be substantially canceled, thereby drastically reducing its leakage effect and allowing the use of perturbation frequencies very close to 50 Hz.
The implementation involves two stages within one measurement cycle, as shown in the following sequence diagram:
| Stage | Time | Action | Sampled Signal (Example: Current) |
|---|---|---|---|
| 1 | $t_0 \rightarrow t_0+\tau_R$ | Inject $+I_h \sin(\omega_h t)$ at phase $\theta_0$ of grid voltage. | $i_1(t) = I_1 \sin(\omega_1 t + \phi_1) + I_h \sin(\omega_h t + \phi_h)$ |
| Buffer | $t_0+\tau_R \rightarrow t_0+T_1$ | Wait until grid voltage phase again reaches $\theta_0$. | No injection, wait. |
| 2 | $t_0+T_1 \rightarrow t_0+T_1+\tau_R$ | Inject $-I_h \sin(\omega_h t)$ at phase $\theta_0$ of grid voltage. | $i_2(t) = I_1 \sin(\omega_1 t + \phi_1′) – I_h \sin(\omega_h t + \phi_h’)$ |
Here, $T_1$ is the period of the actual, fluctuating grid frequency. The key assumption is that the grid frequency $f_1$ and the fundamental component’s magnitude and phase ($I_1, \phi_1$) remain approximately constant over the short duration of one measurement cycle (two injection windows plus the waiting period). Therefore, $I_1 \approx I_1’$ and $\phi_1 \approx \phi_1’$. Similarly, due to the controlled injection, $I_h \approx I_h’$ and $\phi_h \approx \phi_h’$.
The fundamental elimination is performed in the digital signal processor. The stored samples from Stage 1 and Stage 2 are subtracted (for complementary injection):
$$ i_{diff}[n] = i_1[n] – i_2[n] \approx 2 I_h \sin(\omega_h nT_s + \phi_h) + \underbrace{(i_1^{fund}[n] – i_2^{fund}[n])}_{\approx 0} $$
$$ v_{diff}[n] = v_1[n] – v_2[n] \approx 2 V_h \sin(\omega_h nT_s + \psi_h) + \underbrace{(v_1^{fund}[n] – v_2^{fund}[n])}_{\approx 0} $$
where $T_s$ is the sampling period. The difference signals $i_{diff}[n]$ and $v_{diff}[n]$ now contain the perturbation component amplified by a factor of two, while the fundamental component is largely canceled. A standard DFT is then applied to these difference signals to extract the perturbation phasors $2I_h(h)$ and $2V_h(h)$. The grid impedance at $f_h$ is calculated as:
$$ Z_g(h) = \frac{V_h(h)}{I_h(h)} = \frac{2V_h(h) / 2}{2I_h(h) / 2} $$
This “fundamental cancellation” method offers two major advantages:
- Dramatic Reduction of Spectral Leakage Error: By canceling the dominant fundamental component before the DFT, the influence of its spectral leakage on the perturbation frequency bin is minimized, even if the grid frequency has drifted slightly.
- Increased Effective Signal-to-Noise Ratio (SNR): The perturbation signal in the processed data is effectively doubled in amplitude relative to any residual fundamental or background noise, improving measurement robustness.
These advantages allow the perturbation frequency $f_h$ to be chosen very close to the fundamental frequency (e.g., 55 Hz, 52.5 Hz) without suffering from large errors due to grid frequency fluctuations. This proximity leads to a more accurate scaling of reactance to the fundamental frequency, as per the equation $Z_{g1} \approx R_{g1} + j \frac{\omega_1}{\omega_h} X_g(h)$.
Simulation and Experimental Verification
To validate the proposed method, comprehensive simulations and experiments were conducted using a utility interactive inverter system with an LCL filter. The system parameters are listed below.
| Parameter | Value |
|---|---|
| DC-Link Voltage ($V_{dc}$) | 700 V |
| Grid Voltage (Line-to-Line, RMS) | 380 V |
| Rated Power | 10 kVA |
| Inverter-side Inductor ($L_1$) | 2 mH |
| Grid-side Inductor ($L_2$) | 0.5 mH |
| Filter Capacitor ($C_f$) | 10 μF |
| Damping Resistor ($R_d$) | 1 Ω |
| Switching / Sampling Frequency ($f_s$) | 10 kHz |
| Current Controller | PI in dq-frame |
| Nominal Grid Impedance (for simulation) | $R_g = 0.3 \Omega$, $L_g = 0.4 \text{ mH}$ |
Simulation Results: The simulation tested various scenarios. With grid frequency fixed at 50.05 Hz and using a 75 Hz perturbation, the traditional method showed oscillating errors in measured $R_g$ and $L_g$. The oscillation period matched the frequency deviation cycle. When the perturbation frequency was changed to 60 Hz with the same grid drift, the error amplitude increased significantly, confirming the theory that closer perturbations are more sensitive. In contrast, the proposed complementary injection method with 75 Hz perturbation produced constant and accurate measurements of $R_g = 0.3 \Omega$ and $L_g = 0.4 \text{ mH}$, with no oscillating error. Crucially, the method was also tested with perturbation frequencies of 60 Hz and 55 Hz under the same fluctuating grid condition (50.05 Hz). The results remained accurate and stable, demonstrating the method’s capability to enable very-near-fundamental frequency perturbations.
Experimental Results: Experiments were performed on both a programmable grid simulator and a real laboratory grid.
- Grid Simulator Tests: With the simulator frequency set to 50.05 Hz and 50.20 Hz, the traditional 75 Hz injection method yielded measured impedance values that cyclically fluctuated around the true value. The fluctuation magnitude was larger for the 50.20 Hz case. The proposed method, under the same conditions, provided stable, non-fluctuating readings for both resistance and inductance.
- Real Grid Tests: Testing on the actual laboratory grid (with inherent slow frequency fluctuations) provided the most practical validation. Over a 100-second period with 75 Hz injection:
- The traditional DFT method showed clear fluctuations in measured $R_g$ and $L_g$.
- The Hanning window method (with the same total injection time as the proposed two-injection method) reduced but did not eliminate fluctuations, as the main lobe leakage was still significant.
- The proposed complementary injection method yielded essentially constant values ($R_g \approx 0.28 \Omega$, $L_g \approx 0.27 \text{ mH}$), effectively eliminating the fluctuation error.
- Validation via Known Impedance Change (“Incremental Method”): To further verify accuracy on the real grid, a known 1 mH inductor was inserted in series with the grid at the midpoint of a measurement. The proposed method with 75 Hz injection immediately showed a step increase in the measured inductance of approximately 1.08 mH (the measured 75 Hz value of the added inductor), while the resistance showed a minor increase. This confirmed the method’s ability to track real impedance changes accurately.
- Near-Fundamental Frequency Tests on Real Grid: The proposed method was successfully tested with perturbation frequencies of 60 Hz, 55 Hz, and even 40 Hz on the real laboratory grid, producing stable impedance measurements. This confirms the method’s practical utility in enabling accurate power-frequency impedance estimation using perturbation frequencies very close to 50 Hz.
The following table summarizes the key comparative advantages of the proposed method against traditional approaches.
| Feature / Method | Traditional Single Injection | Hanning Window | Proposed Complementary Injection |
|---|---|---|---|
| Sensitivity to Grid Freq. Fluctuation | High, causes cyclic error | Reduced, but not eliminated for $f_h$ near $f_1$ | Very Low (fundamental canceled) |
| Minimum $|f_h – f_1|$ for Reliable Measurement | Large (~25 Hz) | Moderately Large | Very Small (can be a few Hz) |
| Required Injection Duration for Given SNR | $\tau_R$ | $2\tau_R$ or more | $2\tau_R$ |
| Accuracy of Fundamental ($f_1$) Impedance Estimation | Lower (due to extrapolation from higher $f_h$) | Moderate | Higher (can use $f_h$ very close to $f_1$) |
| Computational Load | Low (Direct DFT) | Moderate (Windowing + DFT) | Low (Subtraction + DFT) |
Conclusion
This article analyzed the critical challenge of grid frequency fluctuation-induced spectral leakage in the online measurement of power-frequency grid impedance using utility interactive inverters. The analysis confirmed that the error intensifies as the perturbation frequency approaches the fundamental frequency. To solve this, we proposed a novel online impedance measurement method based on complementary current signal injection. The method works by injecting two consecutive, complementary perturbation currents at the same fundamental phase point. Subsequent signal processing cancels the dominant fundamental component before applying the DFT, thereby drastically reducing spectral leakage and effectively increasing the SNR for the perturbation signal.
The primary benefit of this approach is that it allows the use of perturbation frequencies very close to the power frequency (e.g., 55 Hz) without suffering from large measurement errors caused by typical grid frequency drifts. This leads to a more accurate estimation of the fundamental grid impedance, which is vital for the stability assessment, controller design, and adaptive operation of utility interactive inverters in weak and fluctuating grids. Both simulation and experimental results on simulated and real grids validated the method’s superior accuracy and robustness compared to traditional single-injection and Hanning window methods. The method is practical, does not significantly increase computational burden, and can be readily implemented in the digital controllers of modern utility interactive inverters.