As we accelerate the energy transition towards carbon neutrality, power systems worldwide are undergoing a fundamental transformation characterized by a significantly increased penetration of distributed renewable energy sources. Rooftop photovoltaic (PV) systems, typically connected to low-voltage distribution networks via power electronic inverters, are at the forefront of this shift. A key metric for evaluating this transformation in a regional grid is the renewable energy penetration level, often defined as the ratio of the rated power of new energy generation to the total rated power of all generation units. However, the large-scale integration of these distributed solar inverter systems introduces new operational challenges. The inherent intermittency of renewable generation and the fast, multi-loop dynamic response of power electronic interfaces lead to complex interactions within the source-grid system. One critical emerging issue is the phenomenon of harmonic resonance, which becomes increasingly prevalent as penetration levels rise, ultimately limiting the hosting capacity of the grid for rooftop PV. This paper delves into this challenge, analyzing the coupled relationship between grid strength and penetration, and proposes a novel distributed control scheme to suppress harmonic resonance, thereby enabling higher and more economical penetration of rooftop solar power.
The system under consideration consists of multiple three-phase, current-controlled grid-connected solar inverter units operating in parallel, a common configuration for aggregated rooftop PV installations. Each unit employs an LCL output filter and uses a Phase-Locked Loop (PLL) for grid synchronization. The collective output of these inverters is connected to the point of common coupling (PCC), which then links to the main utility grid through an equivalent grid impedance, primarily inductive.
To analyze the system dynamics, particularly the source-grid interaction, the impedance-based modeling approach is adopted. Each grid-connected solar inverter can be represented by a Norton equivalent circuit: an ideal controlled current source in parallel with its output admittance \(Y_{Ceq}\). The grid side is represented by a Thevenin equivalent: an ideal voltage source in series with the grid impedance \(Z_g\). For a system with \(n\) identical parallel inverters, the aggregate behavior can be studied through the total equivalent admittance seen from the grid side. The stability of the interconnection is assessed using the generalized Nyquist criterion or, more directly, by examining the impedance ratio \(L(s)\):
$$ L(s) = Z_g(s) \cdot Y_{sys}(s) = Z_g(s) \cdot \left( \sum_{i=1}^{n} Y_{Ceq,i}(s) \right) = \frac{Z_g(s)}{Z_{sys}(s)} $$
where \(Z_{sys}(s)\) is the equivalent parallel impedance of the \(n\) inverters. A system is prone to harmonic resonance if, at the frequency \(f_c\) where \(|Z_g(j2\pi f_c)| = |Z_{sys}(j2\pi f_c)|\), the phase difference exceeds 180°, indicating a negative damping margin.
The output impedance of a single grid-connected solar inverter \(Z_{Ceq}\) is a complex function derived from its control loops (current regulator, PLL, feedforward) and main circuit parameters. Its expression in the dq-domain is given by:
$$ Z_{Ceq} = \left[ I + (G_{ci} + k_v G_{PLLv})G_{del}G_{id} \right]^{-1} \left[ G_{del}G_{id}Z_{out}^{-1} – (G_{ci}G_{PLLi} + k_v G_{PLLv})k \right] $$
where \(G_{ci}\) is the current controller matrix, \(k_v\) is the voltage feedforward coefficient, \(G_{PLLv}, G_{PLLi}\) are PLL coupling transfer matrices, \(G_{del}\) models computation delay, \(G_{id}\) is the transfer matrix from duty cycle to current, \(Z_{out}\) is the passive LCL filter impedance, and \(k\) is the sampling filter matrix. The grid impedance is typically \(Z_g = sL_g + j\omega L_g\) in the dq-frame. A critical observation is that the magnitude of the inverter’s output impedance \(|Z_{Ceq}|\) decreases as its output current (or power) increases. Consequently, for a fixed grid impedance \(Z_g\), increasing the number \(n\) of parallel inverters or their individual output power reduces the aggregate system impedance \(Z_{sys} = Z_{Ceq}/n\). This effectively makes the grid appear “weaker” from the perspective of the inverter control systems, a phenomenon known as “grid strength weakening.”
This leads to a fundamental coupling: the renewable penetration level \(\delta_{penetration}\) directly influences the perceived grid strength. The penetration is defined as:
$$ \delta_{penetration} = \frac{P_{renewable}}{P_{renewable} + P_{grid}} \times 100\% = \frac{n \alpha P_{inv}}{n \alpha P_{inv} + P_{grid}} \times 100\% $$
where \(P_{inv}\) is the rated power of a single solar inverter, \(n\) is the number of units, \(\alpha\) is the power dispatch coefficient (\(0 < \alpha \leq 1\)), and \(P_{grid}\) is the grid’s rated capacity. The short-circuit ratio (SCR), a traditional measure of grid strength, can be reformulated to show its inverse relationship with penetration and grid impedance \(L_g\):
$$ k_{SCR} \propto \frac{1}{L_g \cdot \delta_{penetration}} $$
This equation succinctly captures the “strong grid weakens” trend: for a given physical grid impedance (\(L_g\)), the effective SCR decreases as the penetration \(\delta_{penetration}\) increases. As the SCR falls, the impedance intersection point between \(Z_g\) and \(Z_{sys}\) moves into a frequency region where the phase difference is more likely to exceed 180°, triggering harmonic resonance and limiting further penetration growth. The following table summarizes key system parameters for a typical case study.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (Phase) | \(U_{PCC}\) | 311 V |
| Grid Rated Power | \(P_{grid}\) | 100 kW |
| Inverter Rated Power | \(P_{inv}\) | 18.66 kW |
| Inverter-side Filter Inductor | \(L_f\) | 3 mH |
| Grid-side Filter Inductor | \(L_t\) | 0.5 mH |
| Filter Capacitor | \(C_f\) | 14.1 µF |
| Damping Resistor | \(R_d\) | 2 Ω |
| Switching Frequency | \(f_s\) | 20 kHz |
To quantify the penetration boundary imposed by harmonic resonance, we analyze the impedance Bode plots for varying \(n\) and \(\alpha\). With all solar inverter units operating at full rated power (\(\alpha=1\)), the system remains stable with \(n=3\) inverters (\(\delta_{penetration} \approx 35.89\%\)). However, when \(n\) increases to 4 (\(\delta_{penetration} \approx 42.74\%\)), the magnitude of \(Z_{g,qq}\) and \(Z_{sys,qq}\) intersect at approximately 351 Hz with a phase difference of 201°, indicating instability due to harmonic resonance. This sets the initial penetration boundary.
One straightforward grid-support method is to curtail the active power output of the solar inverter units, i.e., reducing \(\alpha\). This increases the individual inverter impedance \(Z_{Ceq}\), thereby raising the aggregate \(Z_{sys}\) and improving the damping margin at the impedance intersection. The table below shows how power curtailment affects the stable number of inverters and the maximum achievable penetration.
| Dispatch Power (\(\alpha\)) | Max Stable Units (\(n_{max}\)) | Penetration \(\delta_{penetration}\) | Economic Index \(\lambda_{pu} (= \alpha)\) |
|---|---|---|---|
| 1.0 (Full Power) | 3 | 35.89% | 1.00 |
| 0.8 | 4 | 37.39% | 0.80 |
| 0.6 | 7 | 43.94% | 0.60 |
While power curtailment can raise the penetration boundary, it comes at a direct cost to economic yield, as indicated by the proportional drop in \(\lambda_{pu}\). Therefore, an alternative solution that maintains full inverter utilization (\(\alpha=1\)) is highly desirable.
To enable higher penetration without power curtailment, we propose a Distributed Damping Reconstruction (DDR) control scheme. The core idea is to embed a virtual damping element within the control law of each individual solar inverter. Instead of adding a physical resistor, which would cause losses, we synthesize an equivalent damping effect through a modified control signal. The principle originates from block diagram manipulation: a physical damping feedback path with impedance \(Z_{damp}\) can be equivalently transformed into a control loop that adds a signal to the modulator. The equivalent damping control voltage \(\Delta v_{damp}\) is derived from the inverter output current \(\Delta i_o^s\):
$$ \Delta v_{damp} = Z_{damp} \cdot G_{EQU} \cdot \Delta i_o^s $$
where \(G_{EQU}\) is a fixed transformation matrix dependent on the inverter’s LCL filter parameters and delay. \(Z_{damp}\) is a simple proportional gain matrix \(k_{damp}I\). This damping voltage is then added directly to the modulation signal. This approach is “distributed” because each solar inverter autonomously implements this damping feature, and the system’s overall damping capability scales naturally with the number of units. It requires no additional hardware sensors or centralized damping equipment.
The modified output impedance of a solar inverter with DDR control, \(Z_{Ceq\_revised}\), becomes:
$$ Z_{Ceq\_revised} = \left[ I + (G_{ci} + k_v G_{PLLv} + Z_{damp}G_{EQU}G_{PLLi})G_{del}G_{id} \right]^{-1} \cdot \left[ G_{del}G_{id}Z_{out}^{-1} – (G_{ci}G_{PLLi} + k_v G_{PLLv})k – Z_{damp}G_{EQU}k \right] $$
The key effect of the added term \(Z_{damp}G_{EQU}G_{PLLi}\) is to increase the magnitude of the inverter’s output impedance, particularly in the mid-to-high frequency range, thereby reshaping the \(Z_{sys}\) characteristic to avoid unfavorable intersections with \(Z_g\).
The damping gain \(k_{damp}\) is the crucial tuning parameter. Its selection is governed by a dual-stability constraint:
- System-Level Resonance Suppression Constraint (Lower Bound): \(k_{damp}\) must be large enough to ensure sufficient positive damping margin (phase difference < 180°) at the impedance intersection for the target penetration level. For our base case targeting penetration with \(n=4\) at full power, analysis shows \(k_{damp} > 0.09\) is required to achieve critical stability.
- Inverter-Level Internal Stability Constraint (Upper Bound): Excessively large \(k_{damp}\) can over-compensate the internal feedback, pushing the closed-loop poles of the individual solar inverter into the right-half plane, causing it to become unstable on its own. Analyzing the closed-loop characteristic equation reveals the critical upper limit.
The closed-loop transfer function from reference to output current for a single inverter with DDR control is complex but can be simplified for pole-zero analysis:
$$ F_r(s) \approx \frac{k_{pwm}(k_ps + k_i)e^{-sT_d}}{D(s) + k_{damp} \cdot N(s)} $$
where \(D(s)\) is the original characteristic polynomial and \(N(s)\) is a polynomial introduced by the DDR path. Root locus analysis as a function of \(k_{damp}\) determines the stability boundary. For the given system parameters, this analysis establishes an upper bound of \(k_{damp\_critical} \approx 5.2\). Therefore, the feasible range for effective and stable operation is \(k_{damp} \in [0.09, 5.2]\). To maximize the resonance suppression capability for penetration enhancement, we select a value near the upper limit, e.g., \(k_{damp} = 5\).
With the DDR control implemented (\(k_{damp}=5\)) and inverters operating at full power (\(\alpha=1\)), the penetration boundary is significantly extended. Impedance analysis now shows that the system remains stable with up to \(n=6\) inverters connected, corresponding to a penetration level of:
$$ \delta_{penetration} = \frac{6 \times 1 \times 18.66}{6 \times 1 \times 18.66 + 100} \times 100\% \approx 52.82\% $$
The resonance only reappears when \(n\) increases to 7. This represents a substantial improvement from the initial boundary of 35.89%, achieved while maintaining optimal economic dispatch (\(\lambda_{pu}=1\)). The following table contrasts the performance of different strategies.
| Control Strategy | Dispatch (\(\alpha\)) | Max Stable Units (\(n\)) | Penetration (\(\delta\)) | Economic Index (\(\lambda_{pu}\)) |
|---|---|---|---|---|
| Conventional Control | 1.0 | 3 | 35.89% | 1.00 |
| Conventional + Power Curtailment | 0.6 | 7 | 43.94% | 0.60 |
| Proposed DDR Control | 1.0 | 6 | 52.82% | 1.00 |
Time-domain simulations and experimental results on a scaled-down hardware platform validate the theoretical analysis. The conventional control system exhibits severe harmonic oscillation in grid current when the fourth solar inverter is connected at full power. In contrast, under the same conditions, the system with the proposed DDR control maintains stable, sinusoidal current waveforms. The experimental platform successfully demonstrated stable operation with six parallel solar inverter units at full rated power, confirming the predicted penetration enhancement.
In conclusion, the large-scale integration of rooftop photovoltaic systems presents a specific challenge: the increase in penetration level inherently weakens the perceived grid strength at the point of interconnection, fostering conditions for harmonic resonance that cap further integration. While active power curtailment of solar inverter units offers a basic solution, it undermines economic efficiency. The Distributed Damping Reconstruction control scheme proposed herein provides an elegant and effective alternative. By embedding a virtual damping element into the control loop of each individual solar inverter, the scheme actively reshapes the aggregate output impedance of the inverter cluster. Crucially, the damping gain is designed within a dual-stability framework that ensures both system-wide resonance suppression and internal inverter stability. This method enables a significant increase in the permissible penetration level—from 35.89% to 52.82% in the studied case—while allowing all inverters to operate at their full, economically optimal power rating. The scheme is inherently scalable, does not require extra hardware, and offers a practical pathway to support higher penetration of distributed rooftop solar resources, facilitating a more secure and economical transition towards a sustainable power grid.