The integration of large-scale renewable energy sources (RES) into the power grid, predominantly through power electronic inverters, has introduced significant stability challenges, with wide-frequency oscillatory instability being a prominent concern. These oscillations, spanning sub-synchronous, super-synchronous, and higher harmonic bands, can be triggered by dynamic interactions between multiple on grid inverter units and the grid impedance. Traditional oscillation mitigation strategies often focus on reshaping the impedance of a single, aggregated on grid inverter model or rely on costly external damping devices. These approaches overlook the complex coupling effects among multiple, physically distinct inverters within a plant, leading to suboptimal global damping performance and limited effectiveness across a wide frequency spectrum.
This article presents a systematic, hierarchical strategy for wide-frequency oscillation suppression based on the coordinated impedance reshaping of multiple on grid inverters. The core innovation lies in actively and collaboratively modifying the impedance characteristics of individual inverter units to achieve an optimal station-level impedance profile, thereby enhancing system damping across a broad frequency range without requiring additional hardware.
1. Frequency-Domain Impedance Modeling and Coupling Analysis
1.1 Impedance Model of a Grid-Connected Inverter
The impedance of a single voltage-source converter (VSC)-based on grid inverter, including its current control loop, phase-locked loop (PLL), and filter, can be derived using frequency-domain linearization. The terminal impedance $Z_{\text{inv}}(s)$ seen from the grid side is expressed as:
$$Z_{\text{inv}}(s) = \frac{-\omega L – G_{I,\text{tf},u}(s)}{1 + G_{U,\text{tf},u}(s) – G_{\theta,\text{tf},u}(s)G_{\text{PLL}}(s)}$$
where $L$ is the filter inductance, $G_{I,\text{tf},u}(s)$, $G_{U,\text{tf},u}(s)$, and $G_{\theta,\text{tf},u}(s)$ are transfer functions representing the influence of current, voltage, and PLL angle perturbations on the converter output voltage, and $G_{\text{PLL}}(s)$ is the PLL transfer function.
1.2 Aggregate Impedance of a Multi-Inverter Station
A wind or solar plant comprises numerous on grid inverters connected via collection networks. For stability analysis, inverters with similar dynamics can be aggregated. Considering a station with $n$ parallel inverter equivalents (each equivalent includes the inverter output impedance and its dedicated line impedance), the total equivalent station impedance $Z_{\text{eq}}$ is not a simple sum but a parallel combination:
$$Z_{\text{eq}} = \frac{1}{\sum_{i=1}^{n} Y_i} = \frac{1}{\sum_{i=1}^{n} \frac{1}{Z_i \angle \theta_i}}$$
where $Z_i \angle \theta_i$ is the impedance of the $i$-th equivalent branch.
1.3 Analysis of Magnitude and Phase Coupling
The magnitude and phase of $Z_{\text{eq}}$ result from complex coupling among all units. The magnitude can be derived as:
$$|Z_{\text{eq}}| = \frac{1}{\sqrt{\sum_{i=1}^{n} Z_i^{-2} + 2\sum_{1 \leq i < j \leq n} (Z_i Z_j)^{-1} \cos(\theta_i – \theta_j)}}$$
This shows that the station’s impedance magnitude depends on both the individual magnitudes $Z_i$ and the cosine of the phase differences between units. Improving the magnitude of one unit does not guarantee a proportional improvement in the station’s magnitude if phase differences are unfavorable.
The phase angle $\phi_{\text{eq}}$ of the station impedance is given by:
$$\phi_{\text{eq}} = -\arg\left(\sum_{i=1}^{n} \frac{1}{Z_i} e^{-j\theta_i}\right)$$
It is a nonlinear function of all $Z_i$ and $\theta_i$. The phase of the aggregate impedance is a “neutralized” result of all individual phases, weighted by their admittance magnitudes. Crucially, improving the phase characteristic of one inverter (e.g., making it more inductive) might be counteracted by the phase characteristics of others, potentially reducing the overall station’s phase margin. This highlights the necessity for coordinated control.
2. Inverter-Level Impedance Shaping Control Loops
To facilitate coordinated reshaping, two distinct supplemental control loops are designed for each on grid inverter, each targeting a specific aspect of its impedance.
2.1 Supplementary Damping Control (SDC) for Magnitude Enhancement
This loop aims to increase the inverter’s impedance magnitude in critical frequency bands, thereby improving its damping contribution. A feedback path is added, typically using a measured signal rich in oscillation information (e.g., terminal power). The block diagram of an on grid inverter with SDC is shown below, where $G_{\text{extra}}(s)$ represents the SDC compensator, often comprising a washout filter, gain, and phase compensator.
The modified current perturbation transfer function becomes:
$$G’_{I,\text{tf},u}(s) = G_{I,\text{tf},u}(s) + \Delta G(s)$$
where $\Delta G(s)$ is the contribution from the SDC path. This modification primarily increases $|Z_{\text{inv}}(j\omega)|$ in selected bands (e.g., 0-300 Hz) with minimal impact on its phase, simplifying coordination.
2.2 Active Damping Control (ADC) for Phase Reshaping
This loop aims to reshape the phase characteristic of the on grid inverter, making it more inductive (positive phase) in frequency regions where it might be capacitive or have negative damping. This is often implemented by inserting a filtering block (e.g., a combination of a notch and a low-pass filter) into the current feedback path within the existing controller.
The modified inner current controller transfer function is:
$$H’_{\text{in}}(s) = H_{\text{in}}(s) \cdot G_{\text{se}}(s)$$
where $G_{\text{se}}(s)$ is the series filter. This filter introduces phase lead or lag at specific frequencies. A properly designed ADC can effectively rotate the inverter’s impedance phase towards the inductive region in the mid-to-high frequency range, mitigating resonance risks with the inductive grid. However, it may slightly reduce the impedance magnitude in the sub/super-synchronous range, necessitating careful parameter tuning.
3. Hierarchical Coordinated Impedance Reshaping Strategy
The proposed strategy operates on two levels: the unit level optimizes control parameters, and the station level determines the optimal placement of the SDC and ADC loops among the available inverters.
3.1 Unit-Level Parameter Tuning
For inverters selected to employ ADC, its parameters (e.g., filter frequencies $f_{\text{low}}$, $f_{\text{trap}}$, damping ratio $\zeta_f$, and gain $k$) are optimized. The goal is to achieve a desirable phase profile while minimizing adverse effects on magnitude. An optimization problem is formulated:
$$\max_{\mathbf{T}} \sum_{f=1}^{300} |Z_{\text{AD}}(j\omega_f, \mathbf{T})|$$
$$\text{subject to: } \arg(Z_{\text{AD}}(j\omega_{\text{res}})) > \theta_{\text{ref}}, \quad \omega_{\text{res}} \in \text{Critical Bands}$$
where $\mathbf{T} = [f_{\text{low}}, k, f_{\text{trap}}, \zeta_f]$ is the parameter vector, and $\theta_{\text{ref}}$ is a target phase angle (e.g., $45^\circ$) ensuring sufficient distance from the -90° (capacitive) boundary. This ensures the ADC-enhanced inverter provides good damping across a wide band.
3.2 Station-Level Optimal Compensation Placement
Not all inverters need to be equipped with supplemental controls. The station-level optimizer decides where to deploy the limited number of control “actions” (SDC or ADC) to maximize the station’s overall damping. A decision vector $\mathbf{Z}$ is defined for $n$ inverters:
$$Z_i = \begin{cases}
0 & \text{No additional control} \\
1 & \text{Apply SDC} \\
2 & \text{Apply ADC}
\end{cases}$$
The optimization aims to maximize a composite objective function $G$ reflecting improved impedance characteristics at identified resonance frequencies $\omega_{\text{res}}^{(m)}$:
$$\max_{\mathbf{Z}} G = \sum_{m=1}^{M} \left( \text{MOR}(\mathbf{Z}) + \text{PBD}(\mathbf{Z}) \right)$$
where:
- $\text{MOR}(\mathbf{Z})$ (Magnitude Over Resonance) measures the normalized impedance magnitude increase at resonance: $$\text{MOR} = \frac{|Z_{\text{eq}}(\omega_{\text{res}})| – |Z_{\text{eq0}}(\omega_{\text{res}})|}{|Z_{\text{eq,max}}(\omega_{\text{res}})| – |Z_{\text{eq0}}(\omega_{\text{res}})|}$$
- $\text{PBD}(\mathbf{Z})$ (Phase Boundary Distance) measures how far the phase is from the unstable boundary: $$\text{PBD} = \frac{\min(|\arg(Z_{\text{eq}}(\omega_{\text{res}})) – (-90^\circ)|, |\arg(Z_{\text{eq}}(\omega_{\text{res}})) – 90^\circ|)}{90^\circ}$$
The optimization is subject to a constraint on the maximum number of controlled inverters: $\sum_{i=1}^n \mathbb{I}(Z_i \neq 0) \leq N_{\text{max}}$.
4. Simulation Verification
A system with six 3-MW on grid inverters connected through a realistic collector and grid network was modeled in PSCAD/EMTDC to validate the strategy. The network topology includes both radial and star-connected feeders.
4.1 System Parameters and Optimized Configuration
Key parameters for the on grid inverters are listed below:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Rated Power | 3 MW | Filter Inductance (L) | 8.5 mH |
| DC Voltage | 2 kV | PLL PI Gains | 50, 100 |
| Terminal Voltage | 1.14 kV | Current Control PI Gains | 0.6, 20 |
The coordinated optimization process yielded the following station-level configuration and unit-level ADC parameters:
| Setting | Value | Result | Value |
|---|---|---|---|
| Number of Inverters (n) | 6 | Decision Vector $\mathbf{Z}$ | [0, 0, 0, 1, 2, 0] |
| Max Controlled Units ($N_{\text{max}}$) | 3 | ADC Gain ($k$) | 0.01 |
| Target Phase ($\theta_{\text{ref}}$) | 45° | Low-Pass Filter Freq ($f_{\text{low}}$) | 300 Hz |
| – | – | Notch Filter Freq ($f_{\text{trap}}$) | 50 Hz |
| – | – | Notch Damping ($\zeta_f$) | 1 |
This result indicates that the optimal strategy is to apply SDC to inverter #4 and ADC to inverter #5, while leaving others in their original state.
4.2 Impedance and Stability Analysis
The Bode plots of the total station impedance before and after coordination show significant improvement. The coordinated reshaping strategy successfully increases the impedance magnitude in the mid-to-high frequency range (e.g., above 50 Hz) and shifts the phase from capacitive towards inductive, effectively reducing the risk of harmonic resonance. The Nyquist plot of the impedance ratio $Z_g / Z_{\text{eq}}$ confirms the enhanced stability margin, as the curve moves away from the critical (-1, j0) point.
4.3 Time-Domain Oscillation Suppression
Two oscillation scenarios were tested:
Scenario 1 – Super-Synchronous Oscillation (80 Hz): A grid disturbance was applied. The effectiveness of different control combinations was compared.
| Strategy # | Inverter #4 | Inverter #5 | Oscillation Damping Performance |
|---|---|---|---|
| 1 | Original | Original | Persistent, slow decay |
| 2 | ADC only | Original | Improved but not fully suppressed |
| 3 | SDC only | Original | Moderate improvement |
| 4 | ADC | ADC | Effective suppression |
| 5 | SDC | SDC | Effective suppression |
| 6 (Proposed) | SDC | ADC | Fastest decay, minimal initial overshoot |
The proposed coordinated strategy (Strategy 6) demonstrated superior performance, rapidly suppressing the 80Hz oscillation component and reducing its magnitude to less than 10% of the uncontrolled case.
Scenario 2 – 5th Harmonic Resonance (250 Hz): A background voltage harmonic was introduced. The proposed coordinated strategy reduced the current Total Harmonic Distortion (THD) at the point of common coupling from 3.76% to 2.05%. While a dual-SDC strategy showed slightly better harmonic suppression, the coordinated strategy offers a more balanced performance across the entire wide-frequency spectrum (0-300 Hz), effectively managing both sub/super-synchronous and higher harmonic oscillations.
5. Conclusion
This article has presented a comprehensive, two-layer strategy for suppressing wide-frequency oscillations in power systems with high penetration of on grid inverters. The method is grounded in a detailed analysis of the impedance coupling effects among multiple inverter units, revealing that uncoordinated local impedance shaping can lead to suboptimal global damping. To address this, dedicated control loops for magnitude enhancement (SDC) and phase reshaping (ADC) were designed at the unit level. A hierarchical coordination framework was then established, where station-level optimization determines the most effective placement of these control actions, and unit-level tuning ensures their parameters are optimized for wide-band performance. Time-domain simulations on a detailed six-inverter system model confirmed that the proposed coordinated impedance reshaping strategy effectively dampens oscillations across a broad frequency range (from sub-synchronous to 5th harmonic), demonstrating its potential as a cost-effective and robust solution for enhancing the stability of modern renewable-rich power grids.