In modern power systems, the increasing integration of renewable energy sources has led to a significant rise in the use of power electronic devices, such as solar inverters. These devices, while enabling efficient energy conversion, often introduce harmonic distortions due to their nonlinear characteristics. Accurate measurement of grid harmonic impedance is crucial for harmonic mitigation, stability analysis, and resolving power quality disputes. Traditional methods for harmonic impedance measurement, whether passive or active, face limitations such as inherent biases, measurement errors, or grid disturbances. In this article, we propose a novel approach that leverages solar inverters to estimate grid harmonic impedance without additional equipment or grid disruptions, while simultaneously mitigating harmonics.
The core idea involves controlling solar inverters to operate as virtual harmonic resistors at specific harmonic frequencies. Under fundamental wave conditions, the solar inverter functions normally to generate power, but at harmonic frequencies, it behaves as a programmable resistor. By adjusting the equivalent virtual harmonic resistance values and measuring the resulting changes in harmonic voltage amplitudes at the point of common coupling (PCC), we can estimate the system’s harmonic impedance using mathematical models and least squares fitting. This method not only provides high measurement accuracy but also contributes to harmonic suppression, making it a dual-purpose solution for modern power systems.
We begin by modeling the solar inverter in a d-q synchronous rotating coordinate system. The state equations for a three-level solar inverter can be expressed as follows:
$$ \dot{x} = \begin{bmatrix} -\frac{R_C}{L_1} + j\omega_g R_C & \frac{R_C}{L_1} – \frac{1}{L_1} \\ \frac{R_C}{L_2} & -\frac{R_C}{L_2} + j\omega_g – \frac{1}{L_2} \\ \frac{1}{C} & -\frac{1}{C} – j\omega_g \end{bmatrix} x + \begin{bmatrix} \frac{1}{L_1} \\ 0 \\ 0 \end{bmatrix} K_{PWM} u_{rdq} + \begin{bmatrix} 0 \\ -\frac{1}{L_2} \\ 0 \end{bmatrix} u_{gdq} $$
where \( x = [i_{1dq}, i_{2dq}, u_{cdq}]^T \) represents the inverter’s bridge-side current, grid-side current, and capacitor voltage in the d-q frame, \( \omega_g \) is the synchronous angular velocity, \( K_{PWM} \) is the modulation coefficient, \( u_{rdq} \) is the modulation signal, and \( u_{gdq} \) is the grid voltage disturbance. This model forms the basis for designing control loops, including DC-link voltage control and harmonic current regulation using Fast Fourier Transform (FFT)-based techniques.
The virtual harmonic resistor concept is implemented by detecting harmonic voltages at the PCC and controlling the solar inverter to inject harmonic currents in phase with these voltages. For a given harmonic order \( h \), the inverter acts as a resistor \( r_h \), absorbing harmonic power and reducing the PCC voltage. The equivalent circuit for the h-th harmonic is shown below, where \( U_{sh} \) is the grid equivalent harmonic voltage, and \( Z_{sh} = R_{sh} + jX_{sh} \) is the system harmonic impedance.
When the virtual resistor \( r_h \) is connected, the harmonic voltage at the PCC changes from \( U_{pcch1} \) to \( U_{pcch2} \), given by:
$$ U_{pcch1} = U_{sh} $$
$$ U_{pcch2} = \frac{r_h}{\sqrt{(R_{sh} + r_h)^2 + X_{sh}^2}} U_{sh} $$
By varying \( r_h \) and measuring \( U_{pcch} \), we can derive the harmonic impedance. For multiple measurements, the relationship can be linearized to estimate \( R_{sh} \) and \( |Z_{sh}| \) using least squares fitting. Specifically, for pairs of virtual resistance values \( r_{hi} \) and corresponding voltage measurements, we define:
$$ x_i = -\frac{2(r_{h(i+1)} – A_i r_{hi})}{A_i – 1} $$
$$ y_i = \frac{r_{h(i+1)}^2 – A_i r_{hi}^2}{A_i – 1} $$
where \( A_i = \frac{U_{pcch_i}^2 r_{h(i+1)}^2}{U_{pcch(i+1)}^2 r_{hi}^2} \). The linear equation \( y = kx + b \) is fitted, with \( \hat{k} = R_{sh} \) and \( \hat{b} = |Z_{sh}|^2 \). The least squares estimates are:
$$ \hat{k} = \frac{\sum_{i=1}^n (x_i – \bar{x})(y_i – \bar{y})}{\sum_{i=1}^n (x_i – \bar{x})^2} $$
$$ \hat{b} = \bar{y} – \hat{k} \bar{x} $$
This approach minimizes errors from measurement noise and inverter limitations. To validate the method, we conducted simulations and real-time experiments using a hardware-in-the-loop platform. The solar inverter was configured with parameters as listed in Table 1.
| Parameter | Value |
|---|---|
| Grid Inductance | 1 mH |
| Grid Resistance | 0.001 Ω |
| Switching Frequency | 19.2 kHz |
| Fundamental Voltage | 220 V (RMS) |
| Harmonic Voltages (5th, 7th, 11th, 13th) | 10 V, 7 V, 5 V, 5 V (RMS) |
In the simulation, we applied virtual harmonic conductance values ranging from 0.05 S to 0.325 S, as detailed in Table 2. Each set of conductance values was used to measure the PCC harmonic voltages, and the data was processed to fit the linear model.
| Set | Conductance (S) | Set | Conductance (S) |
|---|---|---|---|
| 1 | 0.05 | 7 | 0.20 |
| 2 | 0.08 | 8 | 0.225 |
| 3 | 0.10 | 9 | 0.25 |
| 4 | 0.125 | 10 | 0.275 |
| 5 | 0.15 | 11 | 0.30 |
| 6 | 0.17 | 12 | 0.325 |
The fitted linear equations for different harmonics are summarized in Table 3. For instance, for the 5th harmonic, the equation was \( y = 0.05108x + 2.482 \), indicating a high correlation and accurate estimation.
| Harmonic Order | Linear Equation |
|---|---|
| 5th | y = 0.05108x + 2.482 |
| 7th | y = 0.08454x + 5.097 |
| 11th | y = 0.3621x + 13.16 |
| 13th | y = 0.4154x + 20.95 |
The estimated harmonic impedances were compared with actual values, as shown in Table 4. The results demonstrate that the method achieves high accuracy for lower harmonics, with errors below 5% for the 5th and 7th orders. However, for higher harmonics like the 13th, the error increases due to faster dynamics and limitations in proportional-integral (PI) control. Advanced control strategies, such as active disturbance rejection or hysteresis control, could improve performance for higher frequencies.
| Harmonic Order | Actual Impedance (Ω) | Estimated Impedance (Ω) | Error (%) |
|---|---|---|---|
| 5th | 1.5708 | 1.5756 | 0.306 |
| 7th | 2.1991 | 2.2577 | 2.665 |
| 11th | 3.4558 | 3.6277 | 4.973 |
| 13th | 4.0841 | 4.5772 | 12.070 |
In terms of harmonic suppression, the virtual resistor approach significantly reduces PCC harmonic voltages. For example, with a virtual conductance of 0.3 S, the 5th harmonic voltage decreased by 11%, and the 7th by 15.96%, as per simulation results. This contrasts with traditional harmonic injection methods, which often exacerbate harmonic levels. The dual functionality of solar inverters in this context—power generation and harmonic control—makes them invaluable assets in grid management.
For practical implementation, the control algorithm involves transforming measured voltages and currents into α-β coordinates, performing FFT to extract harmonic components, and generating reference currents for the inverter. The reference current for the h-th harmonic is \( i_{h}^* = \frac{u_{h}}{r_h} \), where \( u_h \) is the harmonic voltage. This ensures the inverter injects currents proportional to the harmonic voltages, emulating a resistor. The overall control structure integrates this with fundamental wave control, ensuring seamless operation.
In conclusion, the proposed method using solar inverters for grid harmonic impedance measurement offers a non-intrusive, accurate, and multifunctional solution. It leverages existing solar inverter infrastructure without additional costs, providing harmonic mitigation while estimating impedance. Future work could focus on enhancing control techniques for higher harmonics and extending the approach to multi-inverter systems. This technology aligns with the evolving needs of modern power systems, promoting stability and power quality in renewable-rich grids.