The continuous research and development of new energy technologies have led to the emergence of new technical characteristics in the next-generation power system: wind, solar, and other renewable sources are being integrated into the grid on a large scale through power electronic converters. As the final link in new energy power generation, grid-tied inverter technology is crucial. The quality of control performance of the inverter system often directly determines the power quality fed into the grid. To address the stability challenges posed by the declining inertia of traditional power systems, researchers have proposed the concept of the grid-forming (GFM) inverter by drawing inspiration from the principles of synchronous generators. Various GFM control methods exist, including droop control and virtual synchronous generator (VSG) control. Recently, virtual oscillator control (VOC) has gained attention as a novel GFM control technique. It controls the inverter by emulating the dynamic characteristics of a weakly nonlinear oscillator. Since VOC operates using instantaneous current feedback signals in the time domain, it offers superior transient performance compared to droop control. Common virtual oscillators include the Dead-Zone oscillator, the Van der Pol oscillator, and the Andronov-Hopf (AH) oscillator. All can self-synchronize and converge to a steady-state limit cycle from any initial point. However, the Dead-Zone and Van der Pol oscillators lack inherent power regulation capability. Furthermore, the Van der Pol oscillator’s output contains significant harmonics. In contrast, the Andronov-Hopf VOC (AH-VOC) offers the advantage of harmonic-free output, enabling precise control objectives for grid-tied inverters.
Much of the existing research on VOC focuses on operation under ideal grid conditions. However, in practical power systems, grid faults can lead to excessive currents. Due to the inherent limitation of traditional VOC in current limiting, system instability can occur during grid faults. To address this limitation, literature has proposed an AH-VOC strategy based on a voltage-current double closed-loop structure, with a preliminary analysis of system stability, yet without a detailed examination of system behavior under grid faults. Currently, a common approach for managing the transient response of grid-forming inverters during grid faults is the use of virtual impedance control methods. This method can effectively suppress overcurrent during faults. However, introducing virtual impedance also alters the equivalent impedance seen by the controller, necessitating continuous adjustment of the preset voltage reference values. Moreover, the selection of the virtual impedance value itself requires in-depth research, adding complexity to the controller design and tuning.
To tackle the aforementioned issues, this work begins by developing a mathematical model for a system based on AH-VOC, analyzing its control principles. Subsequently, by studying the swing equation of a synchronous generator, an enhanced AH-VOC strategy incorporating virtual inertia control is proposed. An analysis of parameter selection within this strategy is conducted, leading to the development of an adaptive virtual inertia control strategy. Finally, the effectiveness of the proposed strategy is validated by building a simulation model in MATLAB/Simulink.
System Control Structure Based on AH-VOC
The control structure of the grid-tied inverter system adopted in this work is illustrated in the figure above. The AH-VOC generates the voltage reference signal based on the set voltage and frequency references. This signal is then fed to the inner voltage and current control loops, which subsequently produce the necessary control signals for the inverter’s pulse-width modulation (PWM). This section provides a detailed analysis and modeling of this structure.
Virtual Oscillator Control
The structure of the AH-VOC is central to the control scheme. The core oscillator circuit consists of a resonant inductor \( L \) and a resonant capacitor \( C \), with a natural resonant frequency \( \omega_n = 1/\sqrt{LC} \). A constant \( \varepsilon = \sqrt{L/C} \) is defined. The state variables are chosen as the capacitor voltage and the scaled inductor current:
$$ \mathbf{x_v} = [x_1, x_2]^T = [v_C, \varepsilon i_L]^T $$
where \( x_1 \) and \( x_2 \) are the state variables, \( v_C \) is the voltage across the resonant capacitor, and \( i_L \) is the current through the resonant inductor.
The nonlinear state-dependent voltage source \( v_m \) and current source \( i_m \) are given by:
$$ v_m = \frac{\xi}{\omega_n}(2X_n^2 – ||\mathbf{x_v}||^2)\varepsilon i_L, $$
$$ i_m = \frac{\xi}{\varepsilon \omega_n}(2X_n^2 – ||\mathbf{x_v}||^2)v_C $$
where \( \xi \) is a positive constant controlling the convergence speed, and \( X_n \) is the amplitude of the oscillator’s circular limit cycle. The sources \( v_m \) and \( i_m \) interact with the LC tank to drive the state norm \( ||\mathbf{x_v}|| \to \sqrt{2} X_n \), maintaining a circular limit cycle at the constant frequency \( \omega_n \).
As shown in the control diagram, the oscillator connects to the input and output through current and voltage scaling factors \( K_i \) and \( K_v \), respectively. The output voltage is obtained by scaling the circuit states:
$$ \mathbf{v_{\alpha\beta}} = K_v [v_C, \varepsilon i_L]^T = K_v \mathbf{x_v} $$
where \( \mathbf{v_{\alpha\beta}} \) represents the output control voltage vector in the stationary \( \alpha\beta \)-frame.
Similarly, the input current error is scaled by \( K_i \). A key step is introducing a rotation matrix \( \mathbf{R}(\psi) \) to the scaled current error to establish the proper power coupling. For a grid-forming inverter that should exhibit \( P-f \) and \( Q-V \) droop characteristics in steady state, the angle is set to \( \psi = \pi/2 \). The input to the oscillator is thus:
$$ \mathbf{c} = [c_1, c_2]^T = K_i \mathbf{R}(\pi/2) (\mathbf{i_{\alpha\beta}} – \mathbf{i_{\alpha\beta}^*}) = K_i \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \Delta\mathbf{i_{\alpha\beta}} $$
where \( \mathbf{c} \) is the state vector input to the AH-VOC, \( \mathbf{i_{\alpha\beta}} \) is the measured inverter output current, \( \mathbf{i_{\alpha\beta}^*} \) is its reference, and \( \Delta\mathbf{i_{\alpha\beta}} = \mathbf{i_{\alpha\beta}} – \mathbf{i_{\alpha\beta}^*} \).
The current reference \( \mathbf{i_{\alpha\beta}^*} \) is calculated from the active and reactive power references \( P_{ref} \) and \( Q_{ref} \):
$$ \begin{bmatrix} i_{\alpha}^* \\ i_{\beta}^* \end{bmatrix} = \frac{2}{3||\mathbf{v_{\alpha\beta}}||^2} \begin{bmatrix} v_{\alpha} & v_{\beta} \\ v_{\beta} & -v_{\alpha} \end{bmatrix} \begin{bmatrix} P_{ref} \\ Q_{ref} \end{bmatrix} $$
The dynamic response of the state variables in the virtual oscillator circuit is derived from the capacitor voltage and inductor current equations:
$$ C \frac{d v_C}{dt} = -i_L + i_m – c_1 $$
$$ L \frac{d i_L}{dt} = v_C + v_m – \varepsilon c_2 $$
Substituting equations (1) to (4) into (6) and performing algebraic manipulation yields the complete dynamic model of the AH-VOC in the \( \alpha\beta \)-frame:
$$ \dot{v}_{\alpha} = \frac{\xi}{K_v^2}(2V_n^2 – ||\mathbf{v_{\alpha\beta}}||^2)v_{\alpha} – \omega_n v_{\beta} + \frac{K_v K_i}{C} \Delta i_{\beta} $$
$$ \dot{v}_{\beta} = \omega_n v_{\alpha} + \frac{\xi}{K_v^2}(2V_n^2 – ||\mathbf{v_{\alpha\beta}}||^2)v_{\beta} – \frac{K_v K_i}{C} \Delta i_{\alpha} $$
where \( V_n = \sqrt{2} X_n \) is the desired rated voltage amplitude of the grid-tied inverter output, typically set to the grid’s nominal voltage.
The voltage and current scaling coefficients are defined as:
$$ K_v = V_n, \quad K_i = \frac{3V_n}{S_{rated}} $$
where \( S_{rated} \) represents the system’s rated apparent power.
The output voltage magnitude \( V_L \) and phase angle \( \theta_L \) from the AH-VOC are:
$$ V_L = \sqrt{(v_{\alpha}^2 + v_{\beta}^2)/2}, \quad \theta_L = \arctan\left(\frac{v_{\alpha}}{v_{\beta}}\right) $$
From the dynamics, the dynamic response model for the voltage magnitude and phase (frequency) can be derived, revealing the inherent droop characteristics:
$$ \dot{V}_L = \frac{\xi}{K_v^2}(2V_n^2 – 2V_L^2)V_L – \frac{K_v K_i}{3C V_L} (Q – Q_{ref}) $$
$$ \dot{\theta}_L = \omega = \omega_n – \frac{K_v K_i}{3C V_L^2} (P – P_{ref}) $$
where \( \omega \) is the instantaneous frequency output of the AH-VOC, and \( P \) and \( Q \) are the instantaneous active and reactive power outputs of the grid-tied inverter. Equation (10) clearly shows that the AH-VOC exhibits \( Q-V \) and \( P-f \) droop behavior in its steady-state operation, making it a true grid-forming inverter control strategy.
Voltage-Current Inner Loop Control
The voltage and current inner loop control structure is employed to ensure accurate tracking of the voltage reference \( \mathbf{v_{\alpha\beta}^*} \) generated by the AH-VOC. A cascaded control structure is typically used, where an outer voltage loop generates the current reference for a faster inner current loop. Both loops commonly utilize PI controllers for zero steady-state error tracking. The control is often implemented in the synchronous rotating \( dq \)-frame for simplicity, where the AC quantities become DC values at steady state. The voltage loop PI controller processes the error between the reference and measured voltages to produce the reference current \( i_{dq}^* \). This current reference is then tracked by the inner current loop’s PI controller, which outputs the voltage command \( v_{dq}^{**} \) to the PWM modulator. The transformation between the \( \alpha\beta \) and \( dq \) frames requires a phase-locked loop (PLL) or, in the case of a GFM inverter, the angle \( \theta_L \) generated by the AH-VOC itself can be used. The parameters of the PI controllers (\( K_{p,v}, K_{i,v}, K_{p,i}, K_{i,i} \)) are designed based on the system’s filter parameters (\( L_f, C_f, R_f \)) to achieve desired bandwidth and stability margins. This dual-loop structure enhances the disturbance rejection capability and improves the dynamic performance of the grid-tied inverter.
Proposed Virtual Inertia Control Strategy
AH-VOC Integrated with Virtual Inertia Control
To enhance the system inertia and improve the transient frequency response of the grid-tied inverter during disturbances, a virtual inertia control method inspired by synchronous generators is integrated with the AH-VOC. The core idea is for the inverter to inject or absorb a specific amount of power to counteract frequency deviations caused by power imbalances. This is analogous to the swing equation of a synchronous generator:
$$ P_{gen} – P_{load} = J \omega \frac{d\omega}{dt} + D (\omega – \omega_{ref}) $$
where \( P_{gen} \) and \( P_{load} \) are the generated and load powers, \( J \) is the moment of inertia, \( D \) is the damping coefficient, and \( \omega_{ref} \) is the system reference frequency (e.g., \( 2\pi \times 50 \) rad/s).
Let \( P_{VI} = P_{gen} – P_{load} \) represent the power provided by the virtual inertia control. Rewriting the equation gives the virtual inertia control law:
$$ P_{VI} = K_d \frac{d\Delta\omega}{dt} + K_r \Delta\omega $$
where \( K_d = J\omega \) is the virtual inertia constant, \( K_r = D \) is the virtual damping constant, and \( \Delta\omega = \omega – \omega_{ref} \) is the frequency deviation. The term \( K_d \frac{d\Delta\omega}{dt} \) provides inertial response by opposing the rate of change of frequency (RoCoF), smoothing the frequency trajectory. The term \( K_r \Delta\omega \) provides damping, helping to restore the frequency to its nominal value. Crucially, in the AH-VOC framework, the real-time frequency \( \omega \) is readily available from the oscillator’s internal state, as defined in Equation (10). This eliminates the need for a separate frequency measurement or estimation algorithm, simplifying implementation.
The virtual inertia power \( P_{VI} \) is then added to the original active power reference \( P_{ref} \) to form a new, dynamically adjusted power reference for the AH-VOC:
$$ P_{ref,new} = P_{VI} + P_{ref} $$
This modified reference is fed into the current reference calculation block (Equation 5) and influences the oscillator’s dynamics via Equation (10). The complete control block diagram of the AH-VOC with integrated virtual inertia control for the grid-tied inverter is shown conceptually in the figure, where the block labeled “Virtual Inertia Control” implements Equation (12) and sums its output with \( P_{ref} \).
Adaptive Virtual Inertia Control for Fault Conditions
During grid faults of varying severity, the required level of virtual inertia support from the inverter differs. Using constant coefficients \( K_d \) and \( K_r \) may lead to suboptimal performance—too low inertia during severe faults fails to stabilize the system, while too high inertia during mild faults or normal operation can cause sluggish response and unnecessary power oscillations. Therefore, an adaptive virtual inertia control strategy is proposed where the coefficients adjust dynamically based on the severity of the grid disturbance. The proposed adaptation law is:
$$ K_d = 0.05 K_r, \quad K_r = \frac{V_{PCC}}{V_{ref}} \cdot \frac{\Delta P}{\omega \Delta\omega} $$
where \( V_{PCC} \) is the measured voltage magnitude at the point of common coupling (PCC), \( V_{ref} \) is the nominal PCC voltage, and \( \Delta P = P – P_{ref} \) is the instantaneous active power imbalance.
Rationale Behind the Adaptation Law:
- Voltage Factor (\( V_{PCC}/V_{ref} \)): During a voltage sag (a common fault), the current for a given power output increases. To prevent overcurrent, the inverter’s power output capability is effectively reduced. This term reduces the virtual inertia gain proportionally to the voltage dip, ensuring the control effort remains within the inverter’s safe operating area during faults.
- Power-Frequency Deviation Factor (\( \Delta P / (\omega \Delta\omega) \)): This term inherently adjusts the damping gain based on the system’s transient state. A large power imbalance (\( \Delta P \)) or a large frequency deviation (\( \Delta\omega \)) calls for stronger damping action. The division by \( \omega \Delta\omega \) provides a normalization effect. The inertia constant \( K_d \) is set as a fraction of \( K_r \) to maintain a consistent ratio between inertial and damping responses.
This adaptive strategy allows the grid-tied inverter to provide strong inertia support during initial fault transients when \( \Delta P \) and \( d\Delta\omega/dt \) are large, and then gradually reduce the support as the system moves towards a new equilibrium, enhancing both transient stability and steady-state performance.
Simulation Verification and Analysis
To validate the effectiveness of the proposed control strategies for the grid-tied inverter, a detailed simulation model was built in MATLAB/Simulink. The system parameters are listed in the table below.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Apparent Power | \( S_{rated} \) | 2000 VA |
| Rated Active Power | \( P_{rated} \) | 2000 W |
| Rated Reactive Power | \( Q_{rated} \) | 0 var |
| Grid Voltage (RMS, line-to-line) | \( V_g \) | 110 V |
| Resonant (Nominal) Frequency | \( \omega_n / (2\pi) \) | 50 Hz |
| Resonant Capacitor | \( C \) | 0.4679 F |
| Convergence Coefficient | \( \xi \) | 15 \( \text{s}^{-1}\text{V}^{-2} \) |
| Resonant Inductor | \( L \) | 21.68 μH |
| Inverter-side Filter Inductor | \( L_1 \) | 3.6 mH |
| Grid-side Filter Inductor | \( L_2 \) | 1.2 mH |
| Filter Capacitor | \( C_f \) | 6 μF |
| Line Resistance | \( R_g \) | 0.4 Ω |
| Line Inductance | \( L_g \) | 1.2 mH |
A pre-synchronization strategy was implemented to avoid large inrush currents during the grid connection of the voltage-controlled grid-tied inverter. The inverter starts in islanded mode, synchronizing its capacitor voltage with the grid voltage. After synchronization is achieved, the static transfer switch is closed at t=0.2s, and control seamlessly transitions to the standard AH-VOC power control mode.
Case 1: 50% Grid Voltage Sag. A symmetrical three-phase fault causing a 50% voltage sag at the PCC is applied at t=0.5s and cleared at t=0.8s. The active power responses under three control strategies are compared:
- Standard AH-VOC: Shows significant oscillation and a relatively slow settling time after the fault.
- AH-VOC with Fixed Virtual Inertia (Constant \( K_d, K_r \)): Performance improves, with reduced oscillation amplitude and faster settling compared to the standard AH-VOC.
- AH-VOC with Proposed Adaptive Virtual Inertia: Demonstrates the best transient performance. The settling time is the shortest, and the power overshoot during fault recovery is minimized.
Quantitative analysis shows the proposed adaptive strategy for the grid-tied inverter reduces the settling time after fault inception and limits the peak power during fault recovery more effectively than the other two methods.
Case 2: 60% Grid Voltage Sag. A more severe fault is simulated. The performance of the fixed-virtual-inertia strategy degrades noticeably due to the inappropriate constant gains for the heightened disturbance. The system exhibits larger oscillations and a longer recovery period. In contrast, the proposed adaptive virtual inertia control for the grid-tied inverter automatically increases the support gains in response to the deeper voltage sag and larger power imbalance. Consequently, it maintains significantly better transient stability, damping oscillations more effectively and leading to a more stable and faster recovery. This comparative result clearly validates the necessity and effectiveness of the adaptive parameter adjustment mechanism for the grid-tied inverter under varying fault conditions.
Conclusion
This work focused on enhancing the fault ride-through capability of a grid-tied inverter system based on Andronov-Hopf Virtual Oscillator Control. An adaptive virtual inertia control strategy was proposed and integrated into the AH-VOC framework. The strategy dynamically adjusts the virtual inertia and damping coefficients based on real-time measurements of PCC voltage and system power/frequency deviation. Simulation studies under symmetrical grid voltage sags of 50% and 60% were conducted. The results demonstrated that the proposed control strategy consistently improves the transient performance of the grid-tied inverter during and after grid faults. Key improvements include faster system regulation speed during the fault, reduced power overshoot during fault recovery, and enhanced overall stability. Most importantly, the adaptive nature of the strategy allows the grid-tied inverter to tailor its support level to the severity of the disturbance, proving its effectiveness and robustness over non-adaptive virtual inertia approaches. Future work may involve experimental validation and extension of the strategy to unbalanced grid fault conditions.