The integration of large-scale renewable energy sources, such as wind and photovoltaic power, presents significant challenges to the transient stability and disturbance resistance of traditional power systems. The on grid inverter, as the critical interface, must provide not only power conversion but also essential grid-supporting functions traditionally inherent in synchronous generators. The Virtual Synchronous Generator (VSG) control strategy has emerged as a pivotal solution, designed to emulate the mechanical and electromagnetic characteristics of synchronous machines, thereby endowing the on grid inverter with inherent inertia and damping properties. However, conventional VSG control with fixed parameters often exhibits limitations in weak grid conditions—characterized by a low Short-Circuit Ratio (SCR)—where significant output fluctuations and poor transient response during disturbances can compromise system stability.
This article addresses these challenges by proposing a multi-parameter cooperative adaptive control strategy. The core objective is to dynamically and coordinately adjust the virtual inertia, damping coefficient, and active power-frequency droop coefficient of the VSG-based on grid inverter to suppress power oscillations and enhance transient performance, particularly under weak grid scenarios.
Fundamentals and Challenges of VSG Operation
The operational principle of a VSG-based on grid inverter centers on mimicking the swing equation of a synchronous generator. The active power control loop is derived from the rotor motion equation:
$$
J\frac{d\omega}{dt} = P_m – P_e – D(\omega – \omega_0)
$$
$$
P_m = P_{ref} + k_p(\omega_0 – \omega)
$$
$$
\frac{d\theta}{dt} = \omega
$$
where \( J \) is the virtual moment of inertia, \( D \) is the damping coefficient, \( \omega \) and \( \omega_0 \) are the VSG output and grid synchronous angular frequencies, respectively, \( \theta \) is the power angle, \( P_m \) and \( P_e \) are the virtual mechanical and electromagnetic power, \( P_{ref} \) is the active power reference, and \( k_p \) is the active power-frequency droop coefficient.
In a weak grid with high impedance, the system’s natural damping ratio diminishes. Fixed-parameter VSG control faces a dilemma: a large \( J \) can slow the frequency rate-of-change but may exacerbate power overshoot and prolong settling time; a large \( D \) suppresses oscillation but can introduce steady-state error; and an inappropriate \( k_p \) affects the sensitivity of power response to frequency deviations. Therefore, a rigid parameter set cannot optimally handle the diverse dynamic requirements across different stages of a transient event.
Principle of Multi-Parameter Cooperative Adaptive Control
The proposed strategy is founded on analyzing the transient process following a disturbance, such as a step change in power reference. The response can be segmented into intervals based on the signs of the angular frequency deviation (\( \Delta\omega = \omega – \omega_0 \)), the active power deviation (\( \Delta P = P_{ref} – P_e \)), and the rate-of-change of frequency (ROCOF, \( d\omega/dt \)).
The adaptive law for each parameter is designed to fulfill distinct objectives in each interval: to suppress frequency deviation acceleration or to expedite the return to equilibrium. Crucially, to avoid the noise amplification associated with using ROCOF directly, the adaptive functions utilize only \( \Delta\omega \) and \( \Delta P \) as inputs. The Gaussian Error Function (erf) is incorporated to ensure smooth, bounded, and continuous parameter transitions, preventing abrupt changes that could induce instability.
The core adaptive laws are defined as follows:
1. Adaptive Virtual Inertia (\( J \)):
$$ J = J_0 + k_J \cdot \left( \frac{\Delta\omega}{|\Delta\omega|} \right) \cdot \text{erf}\left( \frac{\Delta P \cdot \Delta\omega}{\alpha_J} \right) \quad \text{for} \quad |\Delta\omega| > \delta_\omega $$
where \( J_0 \) is the nominal inertia, \( k_J \) is a tuning gain, \( \alpha_J \) is a scaling factor, and \( \delta_\omega \) is a small threshold to avoid chattering near equilibrium.
2. Adaptive Damping Coefficient (\( D \)):
$$ D = D_0 + k_D \cdot \left( \frac{\Delta\omega}{|\Delta\omega|} \right) \cdot \text{erf}\left( \frac{\Delta P \cdot \Delta\omega}{\alpha_D} \right) \quad \text{for} \quad |\Delta\omega| > \delta_\omega $$
where \( D_0 \) is the nominal damping, and \( k_D, \alpha_D \) are tuning parameters.
3. Adaptive Droop Coefficient (\( k_p \)):
$$ k_p = k_{p0} + u \cdot \text{erf}\left( \frac{\Delta P \cdot \Delta\omega}{\alpha_p} \right) $$
where \( k_{p0} \) is the nominal droop coefficient, and \( u, \alpha_p \) are tuning parameters.
The sign term \( \left( \frac{\Delta\omega}{|\Delta\omega|} \right) \) dictates the direction of change (increase or decrease) for \( J \) and \( D \), while the erf function modulates the magnitude smoothly. The conditions for parameter adjustment in different transient intervals are summarized in the table below.
| Interval | Δω Sign | ΔP Sign | J Trend | D Trend | k_p Trend |
|---|---|---|---|---|---|
| I | > 0 | > 0 | Increase | Increase | Increase |
| II | > 0 | < 0 | Decrease | Increase | Increase |
| III | < 0 | < 0 | Increase | Increase | Increase |
| IV | < 0 | > 0 | Decrease | Increase | Increase |
Stability Analysis and Parameter Constraint Design
To ensure system stability and guide parameter selection, a small-signal model of the VSG-based on grid inverter is analyzed. Considering the power-angle relationship \( P_e \approx (3EU/X)\theta \) for a primarily inductive grid impedance, the linearized swing equation yields the following characteristic equation:
$$
J s^2 \Delta \omega + D s \Delta \omega + \left( k_p + \frac{3EU}{X \omega_0} \right) \Delta \omega = \Delta P_m
$$
The system’s natural frequency \( \omega_n \) and damping ratio \( \zeta \) are:
$$ \omega_n = \sqrt{\frac{3EU}{J X \omega_0} + \frac{k_p}{J}} $$
$$ \zeta = \frac{D}{2 \sqrt{J \left( \frac{3EU}{X \omega_0} + k_p \right) }} $$
These expressions clarify the influence of each parameter. \( J \) primarily affects \( \omega_n \) (oscillation frequency), \( D \) directly controls \( \zeta \) (damping), and \( k_p \) influences both the effective stiffness and the damping ratio. To maintain adequate stability margins and ensure realistic operation, the adaptive parameters are constrained within practical bounds:
$$ J_{min} \leq J \leq J_{max} = \frac{P_{max}}{2\pi \cdot |df/dt|_{max}} $$
$$ D_{min} \leq D \leq D_{max} = \frac{P_{max}}{\Delta \omega_{max}} $$
$$ k_{p,min} \leq k_p \leq k_{p,max} $$
where \( P_{max} \) is the maximum power rating, \( |df/dt|_{max} \) is the maximum allowable frequency slew rate, and \( \Delta \omega_{max} \) is the maximum permissible frequency deviation.
Simulation Verification and Performance Comparison
The effectiveness of the proposed multi-parameter cooperative adaptive control strategy is validated through simulation studies in MATLAB/Simulink, focusing on a VSG-based on grid inverter operating in a weak grid (SCR ≈ 2.5). The performance is compared against three benchmark strategies: (1) Fixed-parameter VSG control, (2) Inertia-only (\(J\)) adaptive control, and (3) A conventional multi-parameter adaptive control using ROCOF.
Case 1: Response to Active Power Step Change (4kW to 16kW)
The proposed strategy demonstrates superior transient performance. Key metrics are summarized below:
| Control Strategy | Max Power Overshoot | Settling Time (2% band) | Max Frequency Deviation |
|---|---|---|---|
| Fixed-Parameter | 19.83% | 264 ms | 1.49 rad/s |
| J-Adaptive Only | 9.41% | 236 ms | 1.09 rad/s |
| ROCOF-based Adaptive | 6.49% | 222 ms | 0.84 rad/s |
| Proposed Strategy | 4.00% | 216 ms | 0.67 rad/s |
The proposed method achieves the lowest overshoot, fastest settling, and smallest frequency swing. Furthermore, the adaptive parameters \( J \), \( D \), and \( k_p \) transition smoothly without the sharp spikes or discontinuities observed in the ROCOF-based method, thanks to the erf function.
Case 2: Response to a Three-Phase Grid Fault
Under a severe 200ms three-phase short-circuit fault, the proposed strategy for the on grid inverter showcases enhanced robustness. Post-fault recovery is significantly faster and smoother compared to other methods. The power and frequency oscillations are damped more effectively, and the system returns to steady-state with minimal lingering deviation, confirming the strategy’s capability to handle large disturbances.
Case 3: Analysis of Droop Coefficient Adaptation
A test involving a grid frequency dip validates the role of the adaptive \( k_p \). By adjusting the tuning gain \( u \), the trade-off between frequency support and power output is managed. A higher \( u \) allows the on grid inverter to provide stronger frequency regulation (smaller frequency nadir) by injecting more active power, demonstrating the strategy’s flexibility in providing grid support services.
Conclusion
This article has presented a comprehensive adaptive control strategy for Virtual Synchronous Generators applied to on grid inverter systems. By formulating a multi-parameter cooperative adaptation mechanism for virtual inertia, damping, and droop coefficients based on power and frequency deviations—and employing the Gaussian Error Function for smooth transitions—the strategy effectively addresses the transient stability challenges in weak grids. Theoretical analysis via a small-signal model provides a foundation for parameter constraint design. Extensive simulation results confirm that the proposed strategy outperforms conventional fixed-parameter and single-parameter adaptive VSG controls, offering significantly reduced power oscillation, improved transient response speed, and enhanced robustness against faults. This approach contributes to developing more resilient and grid-friendly on grid inverter technologies for future renewable-rich power systems.