The ongoing global shift towards low-carbon energy systems has led to a rapid and significant increase in the penetration of renewable energy generation within power grids. This transformation is characterized by the “double-high” feature: a high proportion of new energy sources and a high proportion of power electronic devices. Unlike traditional synchronous generators (SGs), power electronic grid-connected interfaces inherently lack inertia and damping, and do not possess inherent frequency regulation or voltage control capabilities. This deficiency increases the difficulty of grid voltage and frequency regulation, posing substantial challenges to the safe and stable operation of modern power systems. To address this, control strategies that emulate the characteristics of traditional SGs have been proposed, most notably the Virtual Synchronous Generator (VSG) control. By mimicking the rotor swing equation and governor characteristics of an SG, VSG control enables on-grid inverters to provide virtual inertia and damping, allowing them to actively participate in system frequency and voltage regulation, thereby enhancing grid stability. As a fundamental grid-forming control strategy, VSG holds broad application prospects.
While VSG control improves stability by emulating SG behavior, it also inherits oscillation-related challenges associated with SGs. Recent research has extensively explored oscillation issues in VSG-controlled on-grid inverters. However, many studies employ linearized small-signal models that neglect the nonlinear limiting links (limiters) within the VSG control structure. In practice, because a VSG-controlled inverter exhibits voltage-source characteristics, it is highly susceptible to overcurrent during large grid disturbances. The inclusion of current and other limiters is therefore essential for protecting the inverter power devices. Ignoring these limiters does not accurately represent the actual operating conditions of a VSG during large disturbances and fails to correctly describe its oscillation characteristics. The impact of limiter saturation on oscillation phenomena has garnered significant attention, particularly in studies of grid-following inverters and renewable energy generation systems. It has been identified that limiter saturation under large disturbances can lead to a specific type of instability known as “switched oscillation.” This phenomenon, observed in systems like hydro-turbines, permanent magnet synchronous generator-based wind turbines, and doubly-fed induction generators, is characterized by a periodic switching of the control mode between saturated and non-saturated states. Its underlying mechanism is fundamentally linked to non-smooth bifurcations in the system’s dynamics. For VSG-controlled on-grid inverters, which contain multiple limiters emulating SG protections, such switched oscillations are also highly probable under severe transients. A deep understanding of this mechanism is crucial for system design and for developing effective oscillation suppression strategies, which are vital for maintaining power system security.
The core structure of a VSG-controlled on-grid inverter comprises the power stage and the control system. The power stage typically includes a DC voltage source, a three-phase inverter bridge, an LC filter, and grid-connecting impedance. The control system is primarily divided into the active power-frequency loop and the reactive power-voltage loop. The active-power frequency control emulates the rotor motion and primary frequency regulation of an SG, described by:
$$
\begin{cases}
P_m = P_{ref} + m(\omega_0 – \omega) \\
J \frac{d\omega}{dt} = P_m – P_e – D(\omega – \omega_0) \\
\frac{d\delta}{dt} = \omega – \omega_0
\end{cases}
$$
where \(P_m\) is the virtual mechanical power, \(P_{ref}\) and \(P_e\) are the reference and measured active power, \(m\) is the active power-frequency droop coefficient, \(\omega\) and \(\omega_0\) are the VSG output angular frequency and its rated value, \(J\) is the virtual inertia, \(D\) is the virtual damping coefficient, and \(\delta\) is the virtual power angle. The reactive power-voltage control emulates the automatic voltage regulator (AVR), given by:
$$
E = E_{ref} + n(Q_{ref} – Q_e)
$$
where \(E\) and \(E_{ref}\) are the output voltage amplitude and its reference, \(n\) is the reactive power-voltage droop coefficient, and \(Q_{ref}\) and \(Q_e\) are the reference and measured reactive power.
To ensure safe operation, multiple limiters are embedded within the VSG control structure for the on-grid inverter: 1) Angular Frequency Limiter: Constrains the frequency within statutory limits (e.g., 49.5-50.5 Hz for a 50 Hz system). 2) Voltage Amplitude Limiter: Constrains the output voltage reference within permissible bounds (e.g., ±7%). 3) Current Limiter: Protects the inverter switches from overcurrent. A common method is d-axis priority limiting. 4) Modulation Signal Limiter
Under a large disturbance, such as a three-phase short-circuit fault on the grid side, a VSG-controlled on-grid inverter with these limiters can exhibit sustained low-frequency oscillations post-fault clearance. Time-domain simulation shows that after fault removal, the active power, voltage, and current at the point of common coupling (PCC) enter a persistent oscillatory state rather than settling to a steady-state equilibrium. Frequency-domain analysis of the oscillation period reveals that the active power oscillates at a dominant low frequency (e.g., 0.36 Hz). Consequently, the PCC voltage and current spectra contain sideband components at frequencies \(f_0 \pm f_{osc}\) (e.g., 49.64 Hz and 50.36 Hz).
The dynamical essence of this oscillation is revealed through phase portrait analysis. The trajectory in the \(\omega\)-\(\delta\) plane forms a non-smooth limit cycle, continuously crossing the switching boundary \(\Sigma_1: \{(\omega, \delta) | \omega = \omega_{min} \}\). On crossing, the trajectory’s topological structure changes, indicating the occurrence of a C-bifurcation (a type of non-smooth bifurcation). This corresponds to the periodic saturation and release of the angular frequency limiter. Similarly, analysis in the \(\phi_q\)-\(\delta\)-\(I_{d,ref}\) space (where \(\phi_q\) is a virtual flux component) shows the trajectory repeatedly crossing the boundary \(\Sigma_2: \{(\phi_q, \delta, I_{d,ref}) | I_{d,ref} = -I_{max} \}\), forming a non-smooth limit cycle via C-bifurcation, linked to the saturation of the current limiter. Most notably, the trajectory in the \(\delta\)-\(I_{q,ref}\)-\(E_{r,ref}\) space spends significant time on the boundary \(\Sigma_3: \{(\delta, I_{q,ref}, E_{r,ref}) | E_{r,ref} = E_{r,max} \}\), corresponding to prolonged saturation of the modulation signal limiter, with repeated C-bifurcations. Thus, the switched oscillation in the on-grid inverter is fundamentally caused by non-smooth bifurcations occurring at the boundaries defined by the saturated limiters.
The relationship between the non-smooth bifurcation and the specific limiters is critical. Investigation via simulation, by selectively disabling limiters, reveals that the switched oscillation phenomenon persists only when both the current limiter and the modulation signal limiter are active. Disabling either of these two limiters causes the system to damp out oscillations and converge to a stable equilibrium point. The angular frequency limiter influences the oscillation amplitude and frequency but is not essential for the bifurcation’s occurrence. The voltage amplitude limiter has minimal impact in the studied scenario. Crucially, if only a single limiter (any one) is active, the system does not oscillate. Therefore, the non-smooth bifurcation leading to switched oscillation in this on-grid inverter is a result of the combined action of multiple limiters, primarily the current and modulation signal limiters.
The values of the limiter thresholds significantly affect the bifurcation characteristics. As the current limit value \(I_{max}\) decreases, the system undergoes a transition from a stable equilibrium to a non-smooth limit cycle (i.e., the onset of oscillations). A smaller \(I_{max}\) makes the on-grid inverter more susceptible to switched oscillation. Furthermore, as \(I_{max}\) decreases, the average oscillation amplitude and the oscillation frequency also decrease. Similarly, reducing the DC-link voltage \(U_{dc}\) (which lowers the effective modulation signal limit \(E_{r,max}\)) induces the same bifurcation, making oscillation more likely and reducing its amplitude and frequency.
| Limiter Parameter | Value (pu) | Avg. Osc. Amplitude (MW) | Osc. Frequency (Hz) | System State |
|---|---|---|---|---|
| \(I_{max}\) | 1.12 | 0.479 | 0.203 | Non-smooth Limit Cycle |
| 1.14 | 0.480 | 0.251 | ||
| 1.16 | 0.481 | 0.289 | ||
| 1.18 | 0.482 | 0.360 | ||
| \(I_{max}\) | 1.20 | – | – | Stable Equilibrium |
| \(U_{dc}\) | 3.40 | 0.431 | 0.149 | Non-smooth Limit Cycle |
| 3.45 | 0.443 | 0.190 | ||
| 3.50 | 0.454 | 0.221 | ||
| 3.55 | 0.464 | 0.260 | ||
| \(U_{dc}\) | 4.55 | – | – | Stable Equilibrium |
System parameters, including both VSG control parameters and external grid parameters, profoundly influence the non-smooth bifurcation characteristics of the on-grid inverter. The oscillation characteristics can be theoretically linked to an equivalent circuit when the modulation limiter is saturated. In this state, the on-grid inverter acts as a voltage source of fixed amplitude \(E_{r,max}\) with an angle \(\alpha\). The delivered active power \(P_e\) is then a function of \(E_{r,max}\), grid voltage \(U_g\), line impedance \(Z_g\), impedance angle \(\phi\), and the angles \(\delta\) and \(\alpha\):
$$
P_e = \frac{E_{r,max}}{Z_g} \left[ \frac{E_{r,max}}{2} \cos(\phi) + U_g \cos(\delta + \alpha – \phi) \right] + \frac{E_{r,max} U_g}{2 Z_g} \sin(\delta + \alpha – \phi)
$$
This relation indicates that oscillation amplitude is directly affected by \(E_{r,max}\), \(U_g\), \(Z_g\), and \(\phi\), and indirectly by parameters affecting the switching point dynamics (like \(J\), \(D\), \(I_{max}\), fault severity). Oscillation frequency is directly influenced by the droop coefficient \(m\) which governs the rate of frequency change, and also by other dynamic and switching point factors.
Simulation analysis of single-parameter variations confirms and quantifies these effects. Key findings are summarized below:
| Parameter | Trend (Increasing) | Effect on Bifurcation/Oscillation | Impact on Avg. Amplitude | Impact on Frequency |
|---|---|---|---|---|
| Active Droop \(m\) | ↑ | Promotes bifurcation (oscillation) | ~ Constant | Decreases |
| Reactive Droop \(n\) | ↑ | Suppresses bifurcation | ~ Constant | Increases |
| Virtual Inertia \(J\) | ↑ | Oscillation persists; limit cycle shape changes | Increases slightly | Decreases then increases |
| Damping \(D\) | ↑ | Suppresses bifurcation | Decreases slightly | Increases |
| Short-Circuit Ratio (SCR) | ↑ | Promotes then suppresses bifurcation (window exists) | Increases | Decreases |
| Impedance Angle \(\phi\) | ↑ | Promotes bifurcation | Increases | Decreases |
| Fault Duration \(t_f\) | ↑ | Promotes bifurcation | Increases slightly | Decreases |
The influence degree of key parameters on the power oscillation characteristics can be ranked. Parameters with a major influence on the average oscillation amplitude include the Short-Circuit Ratio (SCR), DC-link voltage \(U_{dc}\), current limit \(I_{max}\), damping coefficient \(D\), and line impedance angle \(\phi\). Parameters with a major influence on the oscillation frequency include the active droop coefficient \(m\), reactive droop coefficient \(n\), current limit \(I_{max}\), SCR, and damping coefficient \(D\).
In conclusion, this analysis reveals that the VSG-controlled on-grid inverter is susceptible to a distinct form of instability under large disturbances: switched oscillation induced by limiter saturation. The dynamical essence of this phenomenon is non-smooth bifurcation, fundamentally different from oscillations in smooth systems. For the grid-forming on-grid inverter, this manifests as a periodic switching of control mode, representing a complex instability mechanism. This oscillation can lead to loss of synchronism, severe voltage fluctuations causing generator trips, and large power swings damaging equipment.
The non-smooth bifurcation is primarily a result of the combined saturation of the current limiter and the modulation signal limiter. The system is more prone to switched oscillation when limiter thresholds are smaller and fault severity is greater. Key external parameters like grid strength (SCR) and line impedance angle, along with control parameters like droop coefficients and damping, significantly influence the onset and characteristics of the oscillation. This understanding is vital for the robust design and operation of VSG-based on-grid inverters in future power systems. Future work should focus on developing formal non-smooth dynamical models for the on-grid inverter to precisely delineate stability boundaries and on devising control strategies to mitigate this switched oscillation risk.