As the global energy landscape shifts towards decarbonization under strategic goals like “Dual Carbon,” the integration of renewable energy sources such as solar and wind power has accelerated. These sources typically interface with the power grid through power electronic converters, notably on grid inverters, which facilitate energy conversion and transfer. However, unlike traditional synchronous generators, on grid inverters inherently lack inertia and damping, leading to a “low-inertia, weak-damping” characteristic that threatens grid stability, especially during disturbances. Frequency fluctuations in AC buses can become severe, compromising system reliability. To address this, Virtual Synchronous Generator (VSG) technology has emerged as a promising solution, enabling on grid inverters to emulate the inertial and damping properties of conventional generators. My research focuses on an improved adaptive control strategy for VSG parameters in on grid inverters, aiming to suppress frequency and power oscillations effectively without introducing noise from derivative operations. This article delves into the mathematical modeling, control design, stability analysis, and simulation validation of this approach, emphasizing the role of on grid inverters in modern power systems.
The core of VSG control lies in mimicking the rotor dynamics of a synchronous generator. For a three-phase voltage-source on grid inverter, the active power-frequency control is derived from the swing equation. Let $$J_v$$ represent the virtual inertia in kg·m², $$D_v$$ the virtual damping in (N·m·rad)/s, $$\delta$$ the power angle in degrees, $$P_m$$ the mechanical power in kW, $$P_e$$ the actual output power in kW, $$\omega$$ the output angular frequency in rad/s, and $$\omega_0$$ the reference angular frequency (typically 314 rad/s for 50 Hz). The rotor motion equation is:
$$ J_v \frac{d\Delta\omega}{dt} = \frac{P_m}{\omega_0} – \frac{P_e}{\omega_0} – D_v (\omega – \omega_0) $$
$$ \frac{d\delta}{dt} = \omega $$
Where $$\Delta\omega = \omega – \omega_0$$. The frequency regulation is modeled using a droop characteristic: $$P_m = k_p (\omega_0 – \omega) + P_{ref}$$, with $$k_p$$ as the active power droop coefficient. By defining $$D = \omega_0 D_v$$ and $$J = \omega_0 J_v$$, the combined VSG active-power control equation becomes:
$$ J \frac{d\omega}{dt} + D (\omega – \omega_0) = P_{ref} – P_e + k_p (\omega_0 – \omega) $$
$$ \frac{d\delta}{dt} = \omega $$
For reactive power-voltage control, the equation is: $$U_m = U_{ref} + k_q (Q_{ref} – Q)$$, where $$U_m$$ is the calculated terminal voltage, $$U_{ref}$$ the reference voltage, $$k_q$$ the reactive droop coefficient, $$Q_{ref}$$ the reference reactive power, and $$Q$$ the output reactive power. These equations form the basis for VSG implementation in on grid inverters, allowing them to participate in grid frequency and voltage support. The control structure typically involves synthesizing reference voltages from $$\delta$$ and $$U_m$$, followed by voltage-current double-loop control and space vector PWM generation to drive the inverter switches.
However, fixed VSG parameters ($$J$$ and $$D$$) often fail to balance rapid response and stability during transients. To enhance performance, I propose an adaptive control strategy where $$J$$ and $$D$$ are adjusted based on system dynamics. The transient response of an on grid inverter under VSG control can be divided into four stages after a disturbance, as shown in the power-angle curve. Let $$\Delta P = P_{ref} – P_e$$ and $$\Delta\omega = \omega – \omega_0$$. From the swing equation, we derive:
$$ \frac{d\omega}{dt} = \frac{\Delta P – (k_p + D)\Delta\omega}{J} $$
$$ \Delta\omega = \frac{\Delta P – J \frac{d\omega}{dt}}{k_p + D} $$
These relations indicate that increasing $$J$$ reduces the rate of change of frequency ($$|d\omega/dt|$$), making frequency smoother, while increasing $$D$$ or $$k_p$$ decreases frequency deviation ($$|\Delta\omega|$$). In the four stages—acceleration with $$\Delta P > 0, d\omega/dt > 0, \Delta\omega > 0$$; deceleration with $$\Delta P < 0, d\omega/dt < 0, \Delta\omega > 0$$; reverse acceleration with $$\Delta P < 0, d\omega/dt < 0, \Delta\omega < 0$$; and recovery with $$\Delta P > 0, d\omega/dt > 0, \Delta\omega < 0$$—the adaptive rules for $$J$$ and $$D$$ vary. Traditionally, these rules rely on both $$\Delta\omega$$ and $$d\omega/dt$$, but using $$d\omega/dt$$ introduces noise through differentiation. I observe that $$\Delta P$$ and $$d\omega/dt$$ share the same sign in each stage, allowing $$\Delta P$$ to replace $$d\omega/dt$$ in the adaptation logic. This eliminates derivative noise, simplifies control, and improves transient waveforms for on grid inverters.
The proposed adaptive expressions for virtual inertia $$J$$ and damping $$D$$ are:
$$ J = \begin{cases}
J_0, & |\Delta\omega| < K_{J0} \\
J_0 + k_1 h|\Delta P|, & |\Delta\omega| \geq K_{J0}, \Delta P \Delta\omega > 0 \cap |\Delta P| \geq h K_{J1} \\
J_0 – k_2|\Delta\omega|, & |\Delta\omega| \geq K_{J0}, \Delta P \Delta\omega \leq 0 \cap |\Delta P| \geq h K_{J1} \\
J_0, & |\Delta\omega| > K_{J0}, |\Delta P| < h K_{J1}
\end{cases} $$
$$ D = \begin{cases}
D_0, & |\Delta\omega| < K_D \\
D_0 + k_3|\Delta\omega| + k_4 h|\Delta P|, & |\Delta\omega| \geq K_D, \Delta P \Delta\omega > 0 \\
D_0 + k_5|\Delta\omega|, & |\Delta\omega| \geq K_D, \Delta P \Delta\omega \leq 0
\end{cases} $$
Here, $$J_0$$ and $$D_0$$ are nominal values, $$K_{J0}, K_{J1}, K_D$$ are thresholds, $$k_1$$ to $$k_5$$ are tuning coefficients, and $$h$$ is a conversion factor between $$d\omega/dt$$ and $$\Delta P$$. This strategy ensures that $$J$$ and $$D$$ respond to both $$\Delta\omega$$ and $$\Delta P$$, mitigating frequency swings without differentiation. The control flow involves sampling $$\Delta\omega$$ and $$\Delta P$$, checking thresholds, and adjusting parameters accordingly before feeding them into the VSG control loop. This enhances the adaptability of on grid inverters to disturbances.
To validate the strategy, stability analysis is conducted via the closed-loop transfer function. Assuming a small power angle, $$P_e \approx (U U_g / X) \delta$$, where $$U$$ and $$U_g$$ are inverter and grid voltages, and $$X$$ is equivalent reactance. Combining with the VSG equation, the transfer function from $$P_{ref}$$ to $$P_e$$ is:
$$ G(s) = \frac{P_e(s)}{P_{ref}(s)} = \frac{\frac{U U_g}{JX}}{s^2 + \frac{1}{J}(D + k_p)s + \frac{U U_g}{JX}} $$
Using parameters like $$U = U_g = 311 \, \text{V}$$, $$k_p = 1000$$, $$X = 0.942 \, \Omega$$, root locus plots illustrate stability. As $$J$$ increases from 0.1 to 2.0 kg·m² with fixed $$D = 15 \, \text{(N·m·rad)/s}$$, poles move closer to the imaginary axis, reducing stability margin and slowing convergence. Conversely, with $$J = 0.5 \, \text{kg·m}^2$$ and $$D$$ varying from 5 to 100 (N·m·rad)/s, poles initially move away from the axis, improving stability, but excessive $$D$$ leads to overdamping and longer settling times. Thus, constraints are imposed: $$J \leq P_{\max} / \max\{|\omega (d\omega/dt)|\}$$ and $$D < 40 \, \text{(N·m·rad)/s}$$ for effective control in on grid inverters.
Simulation studies verify the efficacy of the adaptive control for on grid inverters. A single VSG-connected system is modeled with parameters: $$P_{ref} = 12 \, \text{kW}$$, $$\omega_0 = 314 \, \text{rad/s}$$, DC voltage $$U_{dc} = 800 \, \text{V}$$, switching frequency $$f_c = 5 \, \text{kHz}$$, filter inductance $$L_1 = 5 \, \text{mH}$$, $$k_p = 1500$$, $$J_0 = 0.2 \, \text{kg·m}^2$$, $$D_0 = 10 \, \text{(N·m·rad)/s}$$, thresholds $$K_D = 0.5$$, $$K_{J0} = 0.5$$, $$K_{J1} = 0.2$$, and tuning coefficients $$k_1 = 0.05$$, $$k_2 = 0.4$$, $$k_3 = k_5 = 0.5$$, $$k_4 = 0.1$$, $$h = 0.02$$. Three scenarios are compared: fixed VSG parameters, adaptive control using only $$\Delta\omega$$, and the proposed adaptive control with $$\Delta P$$ replacement.
| Control Method | Maximum Power Overshoot (kW) | Maximum Frequency Deviation (rad/s) | Rate of Change |
|---|---|---|---|
| Fixed VSG Parameters | 1.5 | 2.44 | High |
| Adaptive (Δω only) | 0.8 | 1.50 | Moderate |
| Proposed Adaptive (ΔP-based) | 0.4 | 1.06 | Low |
When the active power reference steps from 10 kW to 7 kW at t=1 s, the proposed method shows superior performance, minimizing overshoot and smoothing transients. Further tests involve power steps from 5 kW to 12 kW and then to 7 kW, comparing conventional adaptive control (with $$d\omega/dt$$) and the proposed method. The results confirm that using $$\Delta P$$ avoids noise, reduces frequency fluctuations, and maintains low power change rates. For instance, frequency deviations are 1.04 rad/s versus 0.59 rad/s, highlighting the benefit for on grid inverter operations.
The conversion factor $$h$$ plays a key role in balancing response speed and smoothness. Simulations with $$h = 0.00, 0.02, 0.04, 0.06$$ under power reference changes demonstrate that as $$h$$ increases, frequency and power variations become smoother, but settling time extends. For on grid inverters, selecting $$h = 0.02$$ offers a good compromise, ensuring effective oscillation damping without excessive delay. This adaptability is crucial for integrating renewable sources, as on grid inverters must respond dynamically to grid conditions while maintaining stability.
In conclusion, my research presents an improved adaptive VSG control strategy for on grid inverters that simultaneously adjusts virtual inertia and damping based on angular frequency deviation and power imbalance, replacing the derivative term to eliminate noise. Stability analysis via root locus guides parameter selection, and simulations validate significant reductions in frequency and power oscillations compared to fixed-parameter and derivative-based adaptive methods. The strategy enhances the inertial support and damping capabilities of on grid inverters, contributing to grid stability in high-renewable penetration scenarios. Future work could explore multi-VSG coordination and real-time implementation in microgrids, further advancing the role of on grid inverters in sustainable energy systems.