With the rapid integration of renewable energy sources like photovoltaic and wind power, utility interactive inverters have become critical interfaces for converting DC power to AC and connecting to the grid. However, the increasing penetration of power electronic devices, along with transformer and line impedances, has led to the emergence of weak grid characteristics. These are primarily manifested as large grid impedance and high background harmonic content. In such environments, utility interactive inverters face significant challenges: the increased grid impedance enhances the interaction between inverters and the grid, potentially leading to resonance issues, while background harmonics further distort grid-connected voltage and current waveforms, compromising power quality and system stability. This article addresses these problems by developing an impedance remodeling strategy that combines improved grid voltage feedforward with adaptive active damping, specifically tailored for multi-inverter systems in weak grids. We will explore the impedance modeling of such systems, analyze the mechanisms of harmonic distortion and resonance, propose a control strategy, and validate its effectiveness through simulations.
The proliferation of utility interactive inverters in modern power systems underscores their importance in enabling renewable energy integration. A utility interactive inverter not only converts power but also must maintain stability and power quality under varying grid conditions. In weak grids, where the grid impedance is non-negligible and background harmonics are prevalent, the dynamics of utility interactive inverters become more complex. Multiple utility interactive inverters operating in parallel can interact with each other and with the grid, leading to coupled resonances that may destabilize the system. This article focuses on mitigating these issues through impedance reshaping, ensuring that utility interactive inverters can operate reliably even in challenging weak grid scenarios.
To understand the behavior of utility interactive inverters in weak grids, we first establish an impedance model for a multi-inverter grid-connected system. Consider a three-phase utility interactive inverter with an LCL filter, as commonly used in grid-tied applications. The inverter employs sinusoidal pulse width modulation (SPWM) in the αβ coordinate system and phase-locked loop (PLL) control. The system parameters include: DC-link voltage \(U_{dc}\), reference current amplitude \(I^*\), grid voltage phase angle \(\theta\), capacitor current feedback coefficient \(H_{i1}\), inverter-side inductor \(L_1\), grid-side inductor \(L_2\), filter capacitor \(C\), grid voltage \(u_g\), point of common coupling (PCC) voltage \(u_{PCC}\), grid current \(i_g\), grid inductance \(L_g\), and grid current feedback coefficient \(H_{i2}\). The current regulator is a quasi-proportional resonant controller with the transfer function:
$$G_i(s) = K_p + \frac{K_r \omega_i s}{s^2 + 2\omega_i s + \omega_0^2}$$
where \(K_p\) is the proportional coefficient, \(K_r\) is the resonant coefficient, \(\omega_i\) is the cutoff frequency, and \(\omega_0\) is the fundamental angular frequency. For a single utility interactive inverter, the control block diagram can be simplified, and using Mason’s gain formula, the grid current can be expressed as:
$$i_g(s) = \frac{T_0(s)}{1 + T_0(s)} \frac{1}{H_{i2}} i_{ref} – Y_0(s) u_{PCC}$$
where \(T_0(s)\) is the open-loop transfer function and \(Y_0(s)\) is the output admittance of the utility interactive inverter. Specifically:
$$T_0(s) = G_i(s) G_{x1}(s) G_{x2}(s) H_{i2}$$
$$Y_0(s) = \frac{s^2 L_1 C + s C H_{i1} K_{PWM} + 1}{s^3 L_1 L_2 C + s^2 L_2 C H_{i1} K_{PWM} + s(L_1 + L_2) + G_i(s) H_{i2} K_{PWM}}$$
Here, \(G_{x1}(s)\) and \(G_{x2}(s)\) are transfer functions related to the control loop, and \(K_{PWM}\) is the gain of the SPWM module. This model forms the basis for analyzing the impedance characteristics of a utility interactive inverter.
For a system with multiple utility interactive inverters connected in parallel, the equivalent Norton model can be derived. Each inverter \(k\) (where \(k = 1, 2, \dots, n\)) is represented by its admittance \(Y_k(s)\) and equivalent current source \(I_{s,k}(s)\). The line admittance for each inverter is \(Y_{l,k}\), and the PCC is node \(n+1\). The total grid current is the sum of currents from all utility interactive inverters, and the interaction between them can lead to resonance when coupled with grid impedance. The equivalent admittance of the multi-inverter system is crucial for stability analysis. We define the total admittance \(Y_T(s) = \sum_{k=1}^n Y_k(s)\), which influences the system’s response to grid disturbances.
The presence of background harmonics in weak grids exacerbates waveform distortion. From the grid current expression, it is evident that \(i_g(s)\) depends on \(u_{PCC}\), meaning that harmonic voltages at the PCC can induce harmonic currents. To mitigate this, grid voltage feedforward is often employed in utility interactive inverters. By adding a feedforward path, the controller compensates for grid voltage disturbances, effectively reducing the inverter’s output admittance magnitude at harmonic frequencies. The ideal feedforward transfer function \(G_f(s)\) that minimizes admittance is:
$$G_f(s) = \frac{1 + s C H_{i1} K_{PWM} + s^2 L_1 C}{K_{PWM}}$$
However, practical implementations often use only the proportional term \(1/K_{PWM}\) due to difficulties in realizing derivative terms. In multi-inverter systems, applying feedforward to each utility interactive inverter increases complexity and cost. Therefore, we propose an improved feedforward strategy that targets a specific inverter to reshape the overall system admittance.
Beyond background harmonics, the large grid impedance in weak grids reduces system stability margins. The interaction between the utility interactive inverter’s output admittance \(Y_0(s)\) and the grid admittance \(Y_g(s)\) must satisfy stability criteria. The phase margin (PM) at the intersection frequency \(f_0\) is given by:
$$PM = 180^\circ – \angle Y_0(j2\pi f_0) + \angle Y_g(j2\pi f_0)$$
For a purely inductive grid impedance, \(\angle Y_g(j2\pi f_0) = -90^\circ\), so:
$$PM = 90^\circ – \angle Y_0(j2\pi f_0)$$
A phase margin of at least \(30^\circ\) is typically required to avoid resonance. As grid impedance increases or more utility interactive inverters are added, the intersection frequency shifts lower, and the phase margin decreases, raising the risk of resonance instability. This underscores the need for active damping mechanisms in utility interactive inverter systems.
To address both background harmonic distortion and resonance, we propose an impedance remodeling strategy that combines improved grid voltage feedforward with a parallel adaptive active damper. This approach aims to reshape the admittance of the multi-inverter system, ensuring stability and power quality for utility interactive inverters in weak grids.
The improved grid voltage feedforward focuses on a target utility interactive inverter within the multi-inverter system. Instead of applying feedforward to all inverters, we introduce a virtual admittance \(Y_{f,k}(s)\) in the target inverter \(k\) such that it cancels the total admittance of the system. This is achieved by setting:
$$Y_{f,k}(s) = -\sum_{k=1}^n Y_k(s) = -Y_T(s)$$
By doing so, the overall system admittance as seen from the PCC is minimized, reducing the impact of background harmonics on grid current. The required feedforward transfer function \(G_{f2}(s)\) for the target utility interactive inverter can be derived as:
$$G_{f2}(s) = \frac{\sum_{k=1}^n Y_{f,k}(s)}{Y_{f,k}(s)} G_f(s)$$
This method simplifies control complexity and lowers costs, as only one utility interactive inverter needs modification. The target inverter can be selected based on redundancy or capacity considerations, enhancing economic feasibility.
In parallel, we employ an adaptive active damper connected at the PCC to suppress resonance between the utility interactive inverter system and the weak grid. The active damper operates similarly to a resistive active power filter, synthesizing a virtual resistor to damp resonances. Its control structure in the αβ coordinate system includes a harmonic extraction unit, a virtual resistor adjustment algorithm, and a current controller. The active damper’s output current is given by:
$$i_{g,a}(s) = I_{s,a}(s) – \frac{u_{PCC}}{Z_{o,a}(s)} – \frac{u_{PCC}}{Z_{v,a}(s)}$$
where \(Z_{o,a}(s)\) is the original port impedance, and \(Z_{v,a}(s)\) is the virtual resistor impedance. The virtual impedance is designed as:
$$Z_{v,a}(s) = \frac{1 + T_A(s)}{T_A(s)} \times \frac{R_V}{G_{NA}(s)}$$
Here, \(T_A(s)\) is the loop gain of the active damper, \(R_V\) is the virtual resistance value, and \(G_{NA}(s)\) is a notch filter to remove fundamental and low-frequency harmonic components from the PCC voltage. The notch filter is defined as:
$$G_{NA}(s) = \prod_{h=1,3,5,7} \frac{s^2 + (h\omega_0)^2}{s^2 + s h\omega_0 / Q + (h\omega_0)^2}$$
where \(Q\) is the quality factor. The active damper adaptively adjusts \(R_V\) based on the harmonic content of the PCC voltage, ensuring effective damping across varying grid conditions.
The virtual resistance adaptation algorithm uses the squared PCC harmonic voltage components \(u_{PCC h,\alpha}\) and \(u_{PCC h,\beta}\) compared to a threshold \(u_{lim}\). A proportional-integral (PI) controller adjusts \(1/R_V\) to maintain \(u_{PCC h} = u_{lim}\), with the threshold set to 3% of the rated voltage to comply with harmonic standards. The PI controller parameters are designed for rapid response. Additionally, to compensate for the current loop gain reduction at high frequencies, a virtual resistor compensation loop \(G_{TR}(s)\) is added. A simplified compensation function is:
$$G_{TR} = \frac{s(L_1 + L_2) + K_{pa} K_{PWM} H_{i2}}{K_{pa} K_{PWM} H_{i2}}$$
where \(K_{pa}\) is the proportional coefficient of the active damper’s current controller. To realize the derivative term \(s\), a first-order high-pass filter with gain \(K\) is used:
$$F(s) = K \frac{s}{s + K}$$
with \(K = \pi f_s\) and \(f_s\) as the sampling frequency. This ensures the virtual resistor exhibits near-pure resistive characteristics over a wide frequency range, enhancing damping performance for utility interactive inverter systems.
The current controller of the active damper employs a multi-resonant controller with phase compensation to suppress its own harmonics and reduce admittance at background harmonic frequencies. The transfer function is:
$$G_{HR}(s) = K_{pa} + \sum_{h=1,3,5,7} \frac{A_h \omega_b (s \cos \theta_h – \omega_h \sin \theta_h)}{s^2 + \omega_b s + \omega_h^2}$$
where \(A_h\) is the gain coefficient, \(\omega_b\) is the bandwidth, \(\omega_h\) is the resonant angular frequency, and \(\theta_h\) is the compensation angle. This design improves stability and harmonic rejection for the utility interactive inverter environment.
To validate the proposed strategy, simulations were conducted under various weak grid conditions. The parameters for the utility interactive inverters and active damper are summarized in the following tables:
| Parameter | Value |
|---|---|
| Grid Voltage \(u_g\) | 380 V |
| Rated Power | 20 kVA |
| Inverter-side Inductor \(L_1\) | 0.8 mH |
| Filter Capacitor \(C\) | 10 μF |
| Grid-side Inductor \(L_2\) | 0.2 mH |
| Line Inductor \(L_{line}\) | 0.1 mH |
| Proportional Coefficient \(K_p\) | 0.68 |
| Resonant Coefficient \(K_r\) | 85 |
| Capacitor Current Feedback \(H_{i1}\) | 0.15 |
| Grid Current Feedback \(H_{i2}\) | 0.05 |
| Parameter | Value |
|---|---|
| Inverter-side Inductor \(L_1\) | 1.2 mH |
| Filter Capacitor \(C\) | 5 μF |
| Grid-side Inductor \(L_2\) | 0.3 mH |
| Rated Power | 3 kVA |
| Proportional Coefficient \(K_p\) | 3 |
| Resonant Coefficient \(K_r\) | 100 |
| Capacitor Current Feedback \(H_{i1}\) | 3 |
| Grid Current Feedback \(H_{i2}\) | 1 |
First, we consider a weak grid with varying grid impedance but no background harmonics. The grid inductance \(L_g\) is changed from 0.1 mH to 1.5 mH and then to 2.5 mH. Without the proposed strategy, the grid current total harmonic distortion (THD) increases significantly with higher impedance, reaching 28.07% at \(L_g = 2.5\) mH, indicating resonance instability. With the improved feedforward and active damper applied, the THD remains below 3% (2.54%, 2.84%, and 2.89% respectively), demonstrating robust performance for utility interactive inverters across impedance variations.
Next, we introduce background harmonics: 2.5% 5th harmonic and 1% 7th harmonic in the grid voltage, with \(L_g = 2\) mH. Without mitigation, the PCC voltage and grid current show severe harmonic amplification, with a THD of 27.08%, failing grid requirements. With the proposed strategy, the THD reduces to 2.69%, and harmonic components are suppressed. This highlights the effectiveness of the impedance remodeling in handling both background harmonics and resonance for utility interactive inverter systems.
For a multi-inverter system with three utility interactive inverters in parallel, under weak grid conditions (\(L_g = 1\) mH) with background harmonics, the system exhibits resonant instability without control, with a THD of 33.91%. After applying the proposed strategy to one target utility interactive inverter and the active damper, the THD drops to 3.09%, validating the approach for multi-inverter scenarios. Compared to methods that require modifying all inverters, this strategy reduces complexity while maintaining performance.
We also compare the proposed strategy with using only an active damper (without improved feedforward). With only the active damper, the grid current THD is 3.30%, with 5th and 7th harmonic current ratios of 1.78% and 2.44%, respectively. With the combined strategy, the THD improves to 2.67%, and harmonic ratios decrease to 0.77% and 1.96%. This shows that the improved feedforward enhances background harmonic suppression, complementing the active damper’s resonance damping for utility interactive inverters.
The impedance reshaping effect can be analyzed through Bode plots. The output admittance of the utility interactive inverter system with the proposed strategy shows reduced magnitude at harmonic frequencies, indicating better harmonic rejection. The virtual resistor synthesized by the active damper exhibits nearly pure resistive characteristics over a wide frequency range, as confirmed by frequency response analysis. This ensures effective damping of resonances between utility interactive inverters and the weak grid.
In conclusion, the impedance remodeling strategy combining improved grid voltage feedforward and adaptive active damping effectively addresses the challenges faced by utility interactive inverters in weak grids. The improved feedforward, applied to a target inverter, reshapes the system admittance to mitigate background harmonic effects, while the active damper suppresses resonance through adaptive virtual resistance. This approach ensures grid current THD remains below 3% under varying grid impedances and background harmonics, enhancing the stability and power quality of multi-inverter systems. The strategy offers a cost-effective and通用 solution for utility interactive inverters, promoting reliable renewable energy integration into weak grids. Future work could explore real-time implementation and optimization for larger-scale systems with diverse utility interactive inverter topologies.
The adaptability of the proposed method is further evidenced by its performance under dynamic grid conditions. For instance, when the grid impedance changes abruptly due to switching events, the active damper’s adaptive algorithm quickly adjusts the virtual resistance value, maintaining damping effectiveness. Similarly, variations in background harmonic levels, such as those caused by nonlinear loads, are compensated by the improved feedforward in the target utility interactive inverter. This dual mechanism ensures that utility interactive inverters remain compliant with grid codes even in highly distorted weak grids.
Moreover, the strategy’s scalability makes it suitable for large-scale deployments of utility interactive inverters, such as in solar farms or wind parks. By focusing control modifications on a subset of inverters, operational costs are minimized without sacrificing performance. The active damper can be implemented as a standalone device, offering flexibility in retrofitting existing utility interactive inverter installations. These advantages underscore the practical relevance of the impedance remodeling approach for modern power systems dominated by utility interactive inverters.
From a theoretical perspective, the impedance models derived in this article provide insights into the interaction dynamics of multiple utility interactive inverters. The use of virtual admittance and virtual resistance concepts aligns with impedance-based stability criteria, offering a systematic framework for designing robust control strategies. The mathematical formulations, including transfer functions and stability margins, serve as a foundation for further research on utility interactive inverter integration in weak grids.
In summary, this article presents a comprehensive solution for enhancing the operation of utility interactive inverters in weak grids. Through impedance remodeling, we address both harmonic distortion and resonance instability, ensuring that utility interactive inverters can reliably support grid stability and power quality. The proposed strategy, validated through simulations, offers a pathway toward more resilient renewable energy systems, where utility interactive inverters play a pivotal role in the energy transition.