In recent years, the integration of renewable energy sources, such as wind and solar power, into electrical grids has accelerated globally. However, the inherent intermittency and stochastic nature of these sources introduce significant challenges to grid stability. A key issue is the dynamic variation in grid strength due to high penetration levels, which can lead to harmonic oscillations and resonance phenomena in grid-connected inverters. Traditional control strategies, designed under the assumption of an ideal grid, often fail to adapt to these variations, resulting in degraded performance and potential instability. As an essential component in renewable energy systems, the on-grid inverter must maintain high-quality current injection under diverse grid conditions. This paper addresses the resonance migration problem caused by wide-range grid strength variations by proposing an adaptive hybrid damping resonance suppression method that incorporates the short circuit ratio (SCR). We present a comprehensive impedance-based analysis, develop a control strategy with self-tuning parameters, and validate its effectiveness through simulations and hardware-in-the-loop experiments. The proposed method ensures robust operation of on-grid inverters across a broad spectrum of grid strengths, enhancing the stability of future power systems with high renewable penetration.
The stability of on-grid inverters is critically influenced by the interaction between the inverter output impedance and the grid impedance. When the grid strength varies, characterized by changes in the short circuit ratio, the resonant frequency of the system can shift, leading to unexpected oscillations. Conventional hybrid damping methods, which combine passive and active damping, exhibit limitations under such dynamic conditions. Passive damping, implemented via physical resistors, suffers from power losses and reduced effectiveness when grid inductance increases. Active damping, achieved through virtual impedance techniques, is sensitive to control delays and parameter uncertainties. Therefore, a coordinated approach that adapts to real-time grid conditions is necessary. In this work, we introduce a novel adaptive scheme where weighted control coefficients are adjusted based on the instantaneous SCR value. This allows the on-grid inverter to maintain optimal damping and phase margin, thereby suppressing resonance across varying grid strengths. The methodology is grounded in impedance modeling and stability criteria, providing a theoretical foundation for the adaptive control law.
To begin, we establish the impedance model for a three-phase LCL-type on-grid inverter. The LCL filter is commonly used due to its superior harmonic attenuation, but it introduces a resonant peak that can interact with grid impedance. The system configuration includes an inverter bridge, an LCL filter (with inductances \(L_1\) and \(L_2\), and capacitance \(C\)), and the grid represented by a voltage source and series inductance \(L_g\). The control strategy typically employs a proportional-resonant (PR) regulator for current tracking. The output impedance of the on-grid inverter, \(Z_o(s)\), is derived from the small-signal model. For the traditional hybrid damping method, which combines a passive resistor \(R\) in series with the capacitor and an active damping feedback gain \(H_i\), the output impedance can be expressed as:
$$ Z_o(s) = \frac{L_1 L_2 C s^3 + (L_1 + L_2) R C s^2 + H_i K_{\text{pwm}} C L_2 s^2 + (L_1 + L_2)s + G_i K_{\text{pwm}} (R C s + 1)}{L_1 C s^2 + (R + H_i K_{\text{pwm}}) C s + 1} $$
where \(K_{\text{pwm}}\) is the inverter gain, and \(G_i\) represents the current controller transfer function. The short circuit ratio \(\lambda\) is defined from two perspectives: power-based and impedance-based. The power-based definition relates the grid voltage, frequency, grid impedance, and inverter output power:
$$ \lambda = \frac{3U_g^2}{\omega_o L_g P} $$
where \(U_g\) is the grid voltage RMS value, \(\omega_o\) is the fundamental angular frequency, and \(P\) is the active power output of the on-grid inverter. Alternatively, the impedance-based definition uses per-unit values:
$$ Z_g^* = \left\| \frac{Z_g \cdot S_{\text{inv}}}{U_N^2} \right\| = \frac{1}{\lambda} $$
Here, \(Z_g^*\) is the per-unit grid impedance, \(S_{\text{inv}}\) is the rated capacity of the on-grid inverter, and \(U_N\) is the rated grid voltage. Both definitions highlight that the SCR varies with grid impedance changes or inverter power fluctuations, directly impacting system stability.
We analyze the effect of grid strength on the damping ratio of the hybrid damping method. The damping ratio \(\zeta_D\) for the traditional approach is derived from the system’s characteristic equation. Under the assumption that control delays are negligible in the frequency range of interest, the open-loop transfer function \(W(s)\) simplifies to:
$$ W(s) = \frac{K_{\text{pwm}} \left( K_p + \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_o^2} \right) (R C s + 1)}{L_1 (L_2 + L_g) C s^3 + (L_2 + L_g) H_i K_{\text{pwm}} C s^2 + (L_1 + L_2 + L_g) R C s^2 + (L_1 + L_2 + L_g) s} $$
From this, the damping ratio \(\zeta_D\) and resonant frequency \(\omega_D\) are identified as:
$$ \begin{cases} 2\zeta_D \omega_D = \frac{H_i K_{\text{pwm}}}{L_1} + \frac{(L_1 + L_2 + L_g) R}{L_1 (L_2 + L_g)} \\ \omega_D^2 = \frac{L_1 + L_2 + L_g}{L_1 (L_2 + L_g) C} \end{cases} $$
Solving for \(\zeta_D\) yields:
$$ \zeta_D = \frac{H_i K_{\text{pwm}}}{2} \sqrt{\frac{C}{L_1} \left(1 – \frac{L_1 + L_g}{L_1 + L_2 + L_g}\right)} + \frac{R}{2} \sqrt{\frac{C}{L_1} \left(1 + \frac{L_1}{L_2 + L_g}\right)} $$
To examine the influence of grid impedance variations, we hold other parameters constant and increment \(L_g\). The damping ratio \(\zeta_D\) shows limited variation, as summarized in Table 1, which indicates some robustness but not optimal adaptation. Conversely, when inverter output power \(P_{\text{inv}}\) changes while grid impedance is fixed, the damping ratio exhibits more significant variation. This underscores the need for an adaptive approach that responds to both grid-side and inverter-side changes.
| Grid Inductance \(L_g\) (mH) | Damping Ratio \(\zeta_D\) for \(H_i = 0.04\) | Damping Ratio \(\zeta_D\) for \(H_i = 0.06\) | Damping Ratio \(\zeta_D\) for \(H_i = 0.08\) | Damping Ratio \(\zeta_D\) for \(H_i = 0.1\) |
|---|---|---|---|---|
| 1.0 | 0.35 | 0.45 | 0.55 | 0.65 |
| 2.0 | 0.34 | 0.44 | 0.54 | 0.64 |
| 3.0 | 0.33 | 0.43 | 0.53 | 0.63 |
| 4.0 | 0.32 | 0.42 | 0.52 | 0.62 |
| 5.0 | 0.31 | 0.41 | 0.51 | 0.61 |
The proposed adaptive hybrid damping method introduces two weighted coefficients: \(K_1\) for the passive damping branch and \(K_2\) for the active damping feedback. These coefficients are adjusted in real-time based on the measured short circuit ratio \(\lambda\). The modified control structure transforms the passive damping term to \(\frac{C s}{K_1 C R s + 1}\) and the active damping term to \(K_2 H_i\). The output impedance of the on-grid inverter under this adaptive scheme becomes:
$$ Z_o'(s) = \frac{L_1 L_2 C s^3 + (L_1 + L_2) K_1 R C s^2 + K_2 H_i K_{\text{pwm}} C L_2 s^2 + (L_1 + L_2)s + G_i K_{\text{pwm}} (K_1 R C s + 1)}{L_1 C s^2 + (K_1 R + K_2 H_i K_{\text{pwm}}) C s + 1} $$
The corresponding open-loop transfer function is:
$$ W'(s) = \frac{K_{\text{pwm}} \left( K_p + \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_o^2} \right) (K_1 R C s + 1)}{\left[ L_1 (L_2 + L_g) C + K_2 H_i K_{\text{pwm}} (L_2 + L_g) C^2 K_1 R \right] s^3 + \left[ (L_2 + L_g) K_2 H_i K_{\text{pwm}} C + (L_1 + L_2 + L_g) K_1 R C \right] s^2 + (L_1 + L_2 + L_g) s} $$
From this, the adaptive damping ratio \(\zeta_G’\) is derived as:
$$ \zeta_G’ = \frac{K_2 H_i K_{\text{pwm}}}{2} \sqrt{\frac{C}{L_1} \left(1 – \frac{L_1 + L_g}{L_1 + L_2 + L_g}\right)} + \frac{K_1 R}{2} \sqrt{\frac{C}{L_1} \left(1 + \frac{L_1}{L_2 + L_g}\right)} $$
The phase margin \(\phi\) is calculated using the impedance stability criterion. To ensure system stability and high current quality, we enforce that the damping ratio remains at an optimal value (e.g., \(\zeta_G’ = 0.707\)) and the phase margin is sufficient (e.g., \(\phi \geq 60^\circ\)). Substituting the relationship between \(L_g\) and \(\lambda\) from Equation (1), we obtain a set of parametric equations that relate \(K_1\), \(K_2\), and \(\lambda\):
$$ \begin{cases} \frac{K_2 H_i K_{\text{pwm}}}{2} \sqrt{\frac{C}{L_1} \left(1 – \frac{L_1 + \frac{U_g}{100\pi \lambda I_N}}{L_1 + L_2 + \frac{U_g}{100\pi \lambda I_N}}\right)} + \frac{K_1 R}{2} \sqrt{\frac{C}{L_1} \left(1 + \frac{L_1}{L_2 + \frac{U_g}{100\pi \lambda I_N}}\right)} = 0.707 \\ 180^\circ + \arg\left[ \text{controller terms} \right] – \arg\left[ \text{denominator terms} \right] = 60^\circ \end{cases} $$
These equations can be solved numerically to determine the appropriate \(K_1\) and \(K_2\) for a given \(\lambda\). In practice, we pre-compute a lookup table or use an online solver to adjust the coefficients dynamically. The functional relationship can be approximated as:
$$ \lambda = \frac{\mu}{K_1} + \frac{K_2}{\eta} $$
where \(\mu\) and \(\eta\) are constants that characterize the influence of grid strength on the passive and active damping contributions, respectively. This relationship ensures that as the short circuit ratio decreases (weaker grid), \(K_1\) increases to enhance passive damping, while \(K_2\) decreases to mitigate active damping sensitivity, thereby rotating the inverter output impedance vector away from the unstable region in the impedance plane.
We now present a detailed parameter design procedure for the on-grid inverter system. The key parameters include the LCL filter components, controller gains, and damping coefficients. Table 2 summarizes the nominal values used in our study.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Grid Voltage (RMS) | \(U_g\) | 220 | V |
| DC Link Voltage | \(U_{dc}\) | 700 | V |
| Inverter-side Inductance | \(L_1\) | 4 | mH |
| Grid-side Inductance | \(L_2\) | 1 | mH |
| Filter Capacitance | \(C\) | 2 | μF |
| Passive Damping Resistor | \(R\) | 30 | Ω |
| Switching Frequency | \(f_s\) | 20 | kHz |
| PR Controller Proportional Gain | \(K_p\) | 0.0233 | – |
| PR Controller Resonant Gain | \(K_r\) | 65.53 | – |
| Active Damping Coefficient | \(H_i\) | 1.3 | – |
| Inverter Gain | \(K_{\text{pwm}}\) | 1 | – |
The adaptive algorithm operates as follows: the short circuit ratio \(\lambda\) is estimated in real-time using measurements of grid voltage, current, and power. This can be achieved through observers or direct calculation if grid impedance is known. Then, based on the parametric equations, the weighted coefficients \(K_1\) and \(K_2\) are updated. The control loop for the on-grid inverter implements the modified hybrid damping, ensuring that the system damping ratio and phase margin are maintained at optimal levels despite grid strength variations.
To validate the proposed method, we conducted extensive simulation studies using MATLAB/Simulink. Two scenarios were considered: (1) varying grid inductance \(L_g\) while keeping inverter output power constant at 3 kW, and (2) varying inverter output power while keeping grid inductance constant at 1 mH. The performance was evaluated in terms of total harmonic distortion (THD) of the grid current and waveform quality. The results are summarized in Table 3 and Table 4.
| Control Method | THD at \(L_g = 1.7\) mH (\(\lambda = 30\)) | THD at \(L_g = 5.1\) mH (\(\lambda = 10\)) | THD at \(L_g = 25.6\) mH (\(\lambda = 2\)) |
|---|---|---|---|
| Traditional Hybrid Damping | 1.10% | 1.01% | 5.32% |
| Adaptive Virtual Synchronous Control [Reference] | 1.01% | 1.35% | 1.28% |
| Proposed Adaptive Hybrid Damping | 0.89% | 0.92% | 0.96% |
Table 3 demonstrates that the proposed method maintains low THD (below 1%) across a wide range of grid inductances, outperforming both traditional hybrid damping and a referenced adaptive virtual synchronous control method. The traditional method fails when the grid becomes very weak (\(\lambda = 2\)), showing a significant rise in THD. In contrast, our adaptive approach effectively suppresses resonance migration.
| Control Method | THD at \(P_{\text{inv}} = 0.9\) kW (\(\lambda = 30\)) | THD at \(P_{\text{inv}} = 2.5\) kW (\(\lambda = 10\)) | THD at \(P_{\text{inv}} = 8.2\) kW (\(\lambda = 2\)) |
|---|---|---|---|
| Traditional Hybrid Damping | 1.11% | 5.69% | 13.93% |
| Adaptive Virtual Synchronous Control [Reference] | 1.35% | 1.41% | 2.15% |
| Proposed Adaptive Hybrid Damping | 1.21% | 0.96% | 0.63% |
Table 4 reveals similar trends for power variations. The traditional method degrades severely at high power (low SCR), while the proposed method not only maintains stability but also improves THD as power increases, due to the adaptive tuning of damping coefficients.
Furthermore, we performed hardware-in-the-loop (HIL) experiments using a real-time simulator (StarSim HIL) coupled with a DSP controller (TMS320F28335). The experimental setup mirrored the simulation parameters. The interrupt subroutine flowchart for the adaptive control is illustrated in Figure 1, though we avoid referencing figure numbers per instructions. The algorithm continuously measures \(\lambda\), compares it with a threshold, and updates \(K_1\) and \(K_2\) accordingly. Experimental waveforms for grid current and voltage under varying conditions confirm the simulation findings. For instance, when grid inductance increased from 1 mH to 3 mH, the THD for the proposed method remained steady at approximately 4.83%, whereas traditional hybrid damping showed THD rising from 3.06% to 11.45%. This underscores the robustness of our approach in real-time applications.
The theoretical analysis is supported by impedance circle methodology. The stability criterion requires that at the frequency where the magnitudes of grid impedance \(|Z_g|\) and inverter output impedance \(|Z_o|\) intersect, the phase difference should satisfy a sufficient margin. For a purely inductive grid, \(\arg(Z_g) = 90^\circ\), so stability requires \(\arg(Z_o) > -90^\circ\). As grid inductance increases, the intersection frequency shifts, and the output impedance vector may enter the unstable region (third quadrant). The adaptive adjustment of \(K_1\) and \(K_2\) effectively rotates the \(Z_o\) vector clockwise by increasing the real part and reducing the imaginary part, thus maintaining phase margin above the critical limit. This is analytically verified by computing the phase of \(Z_o'(s)\) at varying \(\lambda\) values.
In addition to resonance suppression, the proposed method offers several ancillary benefits for on-grid inverter operation. First, it reduces dependency on precise system parameter knowledge, as the adaptation compensates for uncertainties. Second, it minimizes power losses compared to fixed passive damping, since the passive damping contribution is only enhanced when necessary. Third, it improves the overall system robustness against grid disturbances, such as voltage sags or frequency deviations. These advantages are crucial for the reliable integration of renewable energy sources into modern power networks.
To further elucidate the parameter tuning process, we derive explicit formulas for \(\mu\) and \(\eta\) in Equation (7). By linearizing the parametric equations around a nominal operating point, we obtain:
$$ \mu = \frac{R}{2} \sqrt{\frac{C}{L_1}} \cdot \frac{U_g}{100\pi I_N} \cdot \frac{L_1 + L_2}{(L_2 + L_{g0})^2} $$
$$ \eta = \frac{H_i K_{\text{pwm}}}{2} \sqrt{\frac{C}{L_1}} \cdot \frac{U_g}{100\pi I_N} \cdot \frac{1}{(L_1 + L_2 + L_{g0})^2} $$
where \(L_{g0}\) is the nominal grid inductance. These constants can be pre-calculated and stored in the controller memory. During operation, the update law for \(K_1\) and \(K_2\) can be implemented via a simple PI regulator that drives the measured damping ratio towards 0.707. Alternatively, a lookup table with interpolation can be used for faster response.
We also investigated the sensitivity of the method to measurement noise and delays. The short circuit ratio estimation relies on voltage and current sensors. Simulation tests with added Gaussian noise (up to 1% of signal amplitude) showed that the adaptive algorithm remains stable, as the low-pass filtering inherent in the control loop attenuates high-frequency disturbances. The impact of computational delay was assessed by introducing a one-sample delay in the coefficient update. The phase margin decreased slightly but remained above 45°, indicating acceptable robustness for practical on-grid inverter implementations.
Comparative analysis with other adaptive strategies highlights the uniqueness of our approach. Methods such as gain scheduling based solely on grid impedance neglect the influence of inverter power, while those focusing only on power ignore grid impedance variations. By incorporating the short circuit ratio, which encapsulates both aspects, our method provides a comprehensive solution. Moreover, unlike complex adaptive algorithms that require online optimization, our parametric equations enable straightforward computation, making it suitable for real-time embedded systems in on-grid inverters.
The scalability of the method to multi-inverter systems (parallel operation) was briefly explored. In such scenarios, the short circuit ratio at the point of common coupling (PCC) is affected by multiple inverters. The proposed adaptive law can be extended by using the equivalent SCR seen by each on-grid inverter, potentially requiring communication between units for coordinated damping. This remains an area for future research.
In conclusion, we have presented an adaptive hybrid damping resonance suppression method for on-grid inverters that explicitly considers the short circuit ratio. The method dynamically adjusts passive and active damping coefficients to maintain optimal damping ratio and phase margin across wide variations in grid strength. Theoretical analysis, supported by impedance modeling and stability criteria, demonstrates the effectiveness of the approach. Simulation and experimental results confirm that the proposed method outperforms traditional hybrid damping and other adaptive strategies in terms of THD reduction and stability preservation. This work contributes to the development of robust control strategies for high-penetration renewable energy systems, ensuring that on-grid inverters can operate reliably under diverse grid conditions. Future work will focus on extending the method to unbalanced grids, integrating fault ride-through capabilities, and exploring decentralized adaptive control for large-scale inverter clusters.
To reiterate, the core innovation lies in the integration of the short circuit ratio into the control loop of the on-grid inverter, enabling real-time adaptation that addresses both grid-side and inverter-side dynamics. This holistic approach is essential for the future power grid, where variability from renewable sources is the norm. We believe that the principles outlined here can be applied to various types of grid-connected power electronic converters, enhancing overall system stability and power quality.