VSG technology plays a pivotal role in improving power system stability by emulating the inertial and damping properties of traditional synchronous generators. As depicted in Fig. 1, the primary circuit topology and control architecture of the VSG are illustrated.

The diagram can be divided into three main components. First, the main power circuit employs a three-phase bridge inverter configuration, which generates three-phase sinusoidal voltage through an LC filter circuit and can operate in both standalone and grid-connected modes. Second, the VSG power loop serves as the core of the virtual synchronous generator, comprising both active power-frequency control and reactive power-voltage control. Notably, the active power-frequency control incorporates equations for both the rotor motion and the governor dynamics. Third, the dual-loop voltage and current control section completes the system architecture. The core objective of this control strategy is to enable the inverter to mimic the operational dynamics of a synchronous generator, thereby providing analogous functional attributes. To systematically evaluate the VSG’s control performance in mitigating frequency instability and power fluctuations caused by renewable generation variability, this study initially adopts an ideal DC voltage source assumption. While this simplification allows for focused analysis of the control dynamics under standardized conditions, it should be noted that practical implementations require additional energy storage devices to suppress DC-side voltage fluctuations, particularly in high-penetration scenarios [22,23].

In the schematic diagram, U_g represents the grid phase voltage, with its phase angle fixed at 0°. The inverter generates a phase voltage denoted as E, characterized by a phase angle δ. U_dc symbolizes the DC-link voltage. On the inverter side, L₁, R₁, and C correspond to the inductance, resistance, and capacitance of the filter components, respectively. Similarly, L₂ and R₂ represent the impedance on the grid side. The voltages at the midpoint of each inverter bridge arm are labeled as e_a, e_b, and e_c. The three-phase inductor currents are indicated by i_1x (x = a, b, c), while the port voltages are denoted as u_ox (x = a, b, c). The three-phase currents injected into the load and grid after filtering are represented by i_2x (x = a, b, c), and the three-phase capacitor voltages are expressed as u_cx (x = a, b, c). The active power reference value is designated as P_ref, and the real-time active power output of the inverter is P_e. Similarly, the reactive power reference value is Q_ref, and the real-time reactive power output is Q_e. The VSG power control loop generates a reference voltage amplitude E_ref and virtual internal potential angle θ, which are combined to form the three-phase reference voltage u_refx (x = a, b, c). This reference voltage is fed into the dual-loop voltage-current control system, which adjusts the output of the three-phase modulating wave u_rx (x = a, b, c). During the pulse-width modulation process, the modulating wave is compared with a carrier wave to generate pulse signals. These signals drive the switching devices, thereby controlling the inverter’s operation.

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Active-frequency loop control

The VSG active-frequency loop control is composed of two main components: the virtual governor module and the rotor motion equation. The VSG virtual governor module expression is: (1){Pm=Pref+ΔPΔP=Kp(ω0−ω)where:

P_m signifies the mechanical power input to the system, ΔP captures the deviation or variation in active power, and K_p stands for the primary frequency regulation coefficient, which determines the system’s response to frequency changes. Additionally, ω₀ denotes the nominal or reference angular frequency, and ω corresponds to the actual angular frequency output.

The virtual governor within the VSG emulates the operational characteristics of a traditional synchronous generator’s governor, playing a critical role in maintaining frequency stability under varying load conditions. In parallel, the virtual rotor component is responsible for managing transient frequency dynamics, providing rapid and accurate response to sudden changes in system conditions. The mathematical formulation describing the motion of the virtual rotor is detailed in the following formula. (2){Jdωdt=Pm−Peω−D(ω−ω0)dθdt=ωwhere:

J corresponds to the rotational inertia of the synchronous machine, which determines its ability to resist changes in rotational speed. The parameter D represents the damping coefficient, reflecting the system’s capacity to dissipate energy and stabilize oscillations. Additionally, θ denotes the phase angle of the virtual internal electromotive force, which plays a pivotal role in synchronizing the VSG with the grid.

The block diagram of the VSG’s active power-frequency control loop, derived by combining the virtual governor expression and the rotor motion equation, is shown in Fig. 2.

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Reactive power-voltage loop control

When the reactive power load increases in the system, synchronous generators increase their reactive power output to maintain system balance, resulting in a corresponding voltage drop. Conversely, when the reactive load decreases, the generator voltage rises.

The mathematical relationship between a synchronous generator’s reactive power output and its terminal voltage can be expressed by the following equation: (3)U−U0Q−Q0=−1Kqwhere:

U₀ is the rated voltage amplitude, Q₀ is the reactive power set value, and K_q is the synchronous generator reactive-voltage sag coefficient.

The reactive-voltage loop control of the VSG is formulated based on Eq. (3). To prevent voltage step changes during the regulation process, an inertial element 1/(k_is) must be incorporated into the existing VSG excitation control system. This modification makes the dynamic response of the excitation regulation system more closely emulate the characteristics of a synchronous generator, ultimately generating the voltage reference signal E_ref.

Fig. 3 illustrates the block diagram of the VSG reactive-voltage loop control.

The expression for the voltage reference E_ref can be derived from the above figure: (4)Eref=1kis[Qref−Qe+Kq(U0−U)]

In the control architecture of the VSG, E_ref represents the reference value of the virtual internal EMF, while k_i denotes the integral gain within the reactive power control loop. Additionally, K_q is the reactive-voltage droop coefficient, U₀ is the nominal voltage, and U represents the actual system output voltage. These control parameters collectively ensure that the voltage output accurately tracks the predefined reference value.

The active-frequency control loop generates a real-time phase angle signal, which is combined with the voltage reference signal from the reactive-voltage control loop to form the voltage command signal for the inner voltage-current double closed-loop control. The mathematical expression for the voltage command signal is provided below: (5)[urefaurefburefc]=[Erefsin⁡θErefsin⁡(θ−120∘)Erefsin⁡(θ+120∘)]

To enhance the system’s control performance, a voltage-current dual-loop control strategy is implemented. In the voltage outer loop, a quasi-proportional resonant control method is employed, enabling effective tracking of AC signals and achieving zero steady-state error control. For the current inner loop, proportional control is utilized to expedite the system’s dynamic response.

The voltage-current dual-loop control structure is illustrated in Fig. 4. In this configuration, the reference voltage u_ref from the outer loop is compared with the feedback voltage u_c obtained from the filter capacitor. The resulting error signal is processed by the voltage controller G_u(s) to produce the reference current i_ref for the inner current loop. This reference current is then compared with the measured inductor current i_L, and the error is fed into the proportional current controller G_i(s). The output of G_i(s) generates the necessary modulation signals to control the switching devices, ensuring accurate regulation of the switching process. The proportional controller G_i(s) plays a central role in enabling this precise and dynamic control mechanism.

The expressions for the active power (P_e) and reactive power (Q_e) output by the VSG are presented as follows: (6){Pe=3UgZg(Ecos⁡(θg−δ)−Ugcos⁡θg)Qe=3UgZg(Esin⁡(θg−δ)−Ugsin⁡θg)

In Eq. (6), δ corresponds to the power angle of the VSG. Additionally, Z_g and θ_g denote the equivalent impedance and its phase angle of the VSG filter circuit, respectively. These parameters are critical for characterizing the electrical behavior of the VSG, as they influence the power flow dynamics and stability of the system. The power angle δ plays a key role in determining the active power transfer, while the impedance Z_g and its angle affect the voltage regulation and reactive power compensation capabilities of the VSG.

The expressions of Z_g and θ_g are as follows: (7){Zg=(ωL1)2+R2θg=arctan⁡(ωL1/R)where:

L₁ denotes the inductance of the filter circuit, and R represents the output resistance of the VSG.

Adopting the analytical methodology used for small-signal modeling of synchronous generators [24], the output resistance of the VSG is assumed to be negligible. By combining the relationships described in Eqs. (1), (2) and (6), the small-signal model for the active power control loop of the VSG is developed. This model, which captures the dynamic behavior of the VSG under small disturbances, is graphically represented in Fig. 5.

During the modeling process, the primary FM coefficient K_p shares a similar mathematical form with the damping coefficient D. To integrate these concepts, the notion of a damping droop coefficient D₁ is introduced, defined as D₁ = D + K_p/ω₀. Based on Fig. 5, the closed-loop transfer function G(s) can be derived, which characterizes the response of the VSG to changes in the active power command value. (8)G(s)=ΔPeΔPref=3EUg/XgJω0s2+D1ω0s+3EUg/Xg

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Impact on Active Power Output Characteristics

The dynamic characteristics of the second-order system, including its natural oscillation frequency ω_n and damping ratio ξ, are determined from the closed-loop transfer function G(s). This transfer function describes the system’s response to variations in the active power command of the VSG, as expressed in Eq. (8). (9){ωn=3EUgJω0Xgξ=D1ω0Xg12JEUg

When the system exhibits underdamped behavior (0 < ξ < 1) and an error band of ±5% is specified, the system’s overshoot and settling time can be determined accordingly: (10){σ%=e−πξ/1−ξ2×100%ts=3.5ξωn=7JD1

Eqs. (9) and (10) reveal that the dynamic response of the second-order system is primarily governed by two key parameters: the virtual inertia J and the effective damping coefficient D₁. The latter, D₁, is a composite parameter that combines the inherent damping coefficient D and the primary FM coefficient K_p. These parameters collectively define the system’s oscillatory behavior and its ability to attenuate disturbances. Specifically, the virtual inertia J influences the system’s natural frequency, while the effective damping coefficient D₁ determines the rate of oscillation decay.

When the primary frequency modulation coefficient K_p and damping coefficient D are held constant, an increase in the virtual inertia parameter J leads to a higher overshoot (σ%) and an extended settling time (t_s). On the other hand, if J remains unchanged, increasing both D and K_p reduces the overshoot (σ%) and decreases the settling time (t_s). Fig. 6 demonstrates the active dynamic response curves of the system under different virtual regulation coefficients, where the arrows indicate the variation trend of active power as the parameters increase.

According to the analysis above, the following conclusions can be drawn: The virtual inertia J primarily governs the oscillation frequency of the system, while the damping coefficient D determines the rate at which these oscillations diminish. Specifically, a larger J results in more frequent oscillations, whereas a higher D accelerates the decay of these oscillations. By carefully selecting appropriate values for J and D, the dynamic response performance of the active output can be significantly improved.

The following section further examines the influence of virtual inertia and damping coefficients on the dynamic response of the system’s active power output. Under a specified power command, the characteristic variation curve of the active power output dynamic response is illustrated in the accompanying figure. As depicted, an increase in J amplifies both the overshoot and settling time of the active power output. Additionally, under underdamped conditions, elevating D significantly reduces the overshoot and settling time of the power output, consistent with the simulation findings presented in Fig. 6. It is important to note that Fig. 7 demonstrates the ideal variation pattern of the active dynamic response influenced by virtual inertia and damping coefficients. Since virtual inertia and damping coefficients are affected by rotor angular frequency fluctuations and require continuous adjustment to suppress system frequency oscillations—necessitating consideration of their impact on system frequency—the overshoot and settling time cannot be reduced to zero.

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The Effect on the Frequency Output Characteristics

According to Eq. (2) can be obtained: (11)Δω=Pm−Peω−JdωdtD (12)dωdt=Pm−Peω−D(ω−ω0)J

Assuming the numerator on the right-hand side of Eqs. (11) and (12) remain constant, the damping coefficient D and the virtual inertia J exhibit inverse relationships with the angular velocity deviation Δω and the rate of change of angular velocity dω/dt, respectively. Specifically, a higher D results in a smaller Δω, while a larger J reduces dω/dt. By adjusting J and D, the system can effectively suppress angular frequency fluctuations, thereby improving its stability.

As demonstrated in the previous section, the inherent contradiction between transient and steady-state characteristics of a VSG cannot be resolved solely through parameter optimization. Specifically, while increasing the virtual inertia reduces angular frequency deviation, it simultaneously exacerbates power output overshoot and oscillations. Consequently, a mathematical framework must be developed to enable autonomous adjustment of key VSG parameters.

By emulating the electrical behavior of synchronous generators, VSGs exhibit power-angle curves and frequency oscillation curves analogous to their physical counterparts. When subjected to disturbances, the VSG control system determines its steady-state frequency according to the power-frequency droop curve. During transient regulation, two critical metrics govern system stability: transient settling time (t_s) and overshoot magnitude (σ%) in the dynamic response. Fig. 8 illustrates the power angle characteristic curve and the angular frequency oscillation curve of the synchronous generator.

In Fig. 8a, when the active input increases from P₁ to P₂, the system transitions from the initial steady-state point A to point B. This shift occurs through several decaying oscillation cycles. Driven by inertia, the system continues to evolve from point B to point C, eventually reaching a new steady state after additional decaying oscillations. According to the characteristics of the variation of the angular frequency in the figure, the first decay oscillation cycle is divided into four intervals: (I) t₁–t₂, (II) t₂–t₃, (III) t₃–t₄, and (IV) t₄–t₅.

During the first interval, the virtual rotor speed of the virtual synchronous generator is higher than the grid frequency and continues to increase. The rate of change of angular speed (dω/dt) initially experiences a sudden rise before gradually decreasing. To prevent excessive dω/dt and speed deviation (Δω) during this interval, it is necessary to increase both the virtual inertia (J) and the damping coefficient (D).

In the second interval, the virtual rotor speed enters a deceleration phase (dω/dt < 0), where the speed gradually decreases from its peak value while remaining above the grid reference frequency (ω > ω₀). During this stage, it is advisable to reduce the virtual inertia to accelerate the restoration of angular velocity to its rated value, while simultaneously increasing the damping coefficient when Δω remains large to suppress angular velocity deviation further.

The selection criteria for the virtual inertia and damping coefficient parameters in Intervals III and IV follow the same principles as those for Intervals I and II, and thus will not be reiterated here.

As illustrated in Section 2.3.1, the virtual inertia value is determined by both the rate of change of angular frequency and its deviation, while the damping coefficient is solely dependent on the angular frequency deviation. Table 2 summarizes the adjustment principles for the virtual inertia and damping coefficient across different operational stages.

According to the principles of adaptive control parameter adjustment in Table 2, the computational expressions for determining the magnitude of the values of the virtual inertia and damping coefficient are given: (13)J={J0+Kj|dωdt|,Δω(dωdt)>0∩|dωdt|>TjJ0,Δω(dωdt)≤0∪|dωdt|≤Tj (14)D={D0+Kd|Δω|,|Δω|>TdD0,|Δω|≤Tdwhere:

J₀ and D₀ represent the virtual inertia and damping coefficients of the VSG under stable operating conditions, respectively. K_j and K_d are the adjustment coefficients for virtual inertia and damping, while T_j and T_d denote the threshold limits for the rate of change of angular frequency and angular frequency deviation, respectively. The flowchart of the control strategy is shown in Fig. 9.

The selection of these thresholds is crucial for the system’s steady-state performance. Appropriately chosen thresholds ensure that the system remains robust against frequent variations in control parameters. Through adaptive tuning of these key parameters, the dynamic performance of the system can be significantly improved. The principle of inertia damping adaptive control is shown in Fig. 10.

The characteristic root of the transfer function of the equivalent second-order system is obtained from Eq. (8) as: (15)Go(s)=3EUg/Zs(D+Kpω0+Js)ω0

The open-loop transfer function of the system can be obtained from Fig. 5: (16)s2+(DJ+KpJω0)s+1Jω03EUgZ=0

According to Eqs. (15) and (16), the distribution of system poles under parameter variation is derived, and the pole distribution is shown in Fig. 11, where the arrows indicate the movement trajectories of the conjugate poles s₁ and s₂.

When D is held constant and J is increased from 0.05 to 15, the distribution of the system’s poles is depicted in Fig. 11a. According to Liapunov stability theory, system stability is ensured when the poles are located in the left half-plane. As observed from the figure, the conjugate poles gradually approach the imaginary axis, with a faster movement near the imaginary axis compared to the real axis. This trend indicates a degradation in system stability. When the virtual inertia exceeds a certain threshold, the pole distribution becomes more concentrated. At this stage, further adjustments to the virtual inertia have minimal impact on system stability.

When J is held constant and D is increased from 0 to 50, the pole distribution of the system is shown in Fig. 11b. As the damping coefficient increases, the conjugate poles shift towards the negative real axis in the left half-plane. During this phase, the damping ratio increases, leading to a reduction in overshoot and an improvement in the system’s dynamic response performance. However, when the damping coefficient D exceeds a certain threshold, the system transitions from underdamped to overdamped. The response time increases, and the original conjugate poles separate along the negative real axis. The dominant pole moves closer to the imaginary axis, resulting in deteriorated system stability.

Building on KU Leuven’s virtual synchronous generator architecture, during the initial grid frequency deviation, virtual inertial power becomes the dominant part, during which the inverter primarily supplies inertial power. Compliance with the following formulation enables full utilization of the inverter’s capacity: (17)J⩽Pmaxmax{ω(dω/dt)}

The stability margin of the system can be evaluated by analyzing the location of its poles. Specifically, the greater the distance of the poles from the imaginary axis, the higher the system’s stability margin. To guarantee system stability, the pole locations must adhere to the following equation: (18)Re(s)=−ω0ξ=−D2J⩽−10

In this study, the virtual inertia J₀ is configured as 0.2 kg·m², and the damping coefficient D₀ is set to 10 N·m·s/rad. The adjustment coefficients K_j and K_d, along with the threshold values T_j for the rate of change of angular velocity and T_d for angular velocity deviation, are determined according to specific application needs.

This study conducts a comparative analysis of the power and frequency responses of the VSG system under four distinct control strategies: fixed-parameter control, adaptive virtual inertia control, adaptive damping coefficient control, and coordinated adaptive control of both inertia and damping.

The VSG grid-connected model is developed in the MATLAB/Simulink environment, and relevant simulation experiments are carried out. The simulation parameters are provided in detail in Table 3.

The simulation experiment is conducted over 1 s. Initially, the active power output of the VSG is set at 15 kW. At 0.2 s, the active power abruptly increases to 30 kW, and then sharply drops to 10 kW at 0.6 s.

Fig. 13 displays the current output waveforms for the four control methods under study. The traditional control method, which emulates the inertia and damping characteristics of synchronous generators, exhibits current overshoot and oscillations during power command transitions. The adaptive inertia control method effectively suppresses these overshoots and oscillations. While the adaptive damping control eliminates overshoot and oscillation completely, it results in less smooth current transitions. The inertia-damping adaptive control combines the benefits of both approaches, delivering smooth current variations during power command changes without noticeable overshoot or oscillation, demonstrating superior dynamic response performance.

Fig. 14 presents the dynamic response curves of the active output of the VSG under different control strategies. Facing the sudden increase of the input power, the overshoot of the output active power of the three control strategies, fixed parameter control, adaptive rotational inertia control, and coordinated adaptive control, decreases in order, which are 36%, 15.6%, and 3.4%, respectively. Accordingly, the time required for these control strategies to reach the steady state is also different, about 0.21, 0.19 and 0.12 s. The overshooting amount of the adaptive damping coefficient control is 3.5%, and the settling time is about 0.08 s. Compared with the inertia-damping coordinated adaptive control, the response is faster, which is the optimal performance among the four control strategies, and is analyzed in conjunction with the effect on the frequency of the system.

Fig. 15 shows the comparative analysis of the output frequency of the VSG with different control strategies. The maximum frequency amplitudes of the four control strategies are 50.46, 50.21, 50.25, and 50.17 Hz, respectively. As shown in the figure, the adaptive virtual inertia control and adaptive damping coefficient control are better than the traditional fixed parameter control. Further comparison reveals that the synergistic combined control strategy of virtual inertia and damping coefficient can significantly improve the system performance. Specifically, this combined control approach not only shortens the time required for the system to reach the steady state condition but also minimizes the frequency deviation.

Based on the simulation results of Figs. 14 and 15, the quantitative metrics of different control strategies can be concluded in Table 4.

Fig. 16 illustrates the dynamic changes in virtual inertia and damping coefficient during the adaptive control process. Once the system achieves stability, the adjusted parameters revert to their initial values. The proper selection of control parameters effectively mitigates excessive virtual inertia regulation to some extent.

In summary, while adaptive damping coefficient control excels in enhancing the dynamic response performance of the system’s active output, the adaptive control strategy that integrates both virtual inertia and damping coefficient offers a more comprehensive improvement. This integrated approach not only mitigates system overshoot and enhances dynamic performance but also reduces the amplitude of frequency fluctuations and frequency deviations.

Fig. 17 demonstrates the frequency regulation performance under load step changes. With the active power command maintained at 20 kW, the system experiences a load step increase from 5 to 10 kW at 0.3 s, followed by a step decrease back to 5 kW at 0.7 s, while keeping all other parameters consistent with Table 3. Under these load disturbances, the conventional VSG control exhibits significant power oscillations, resulting in grid frequency fluctuations of approximately 0.2 Hz. In contrast, the proposed adaptive control strategy substantially improves system stability. The adaptive parameters dynamically adjust to suppress power overshoot and mitigate frequency oscillations. When a load step change occurs, the adaptive controller rapidly compensates for the active power imbalance, ensuring smooth frequency regulation without oscillatory behavior. Consequently, the grid frequency fluctuation amplitude is effectively reduced to less than 0.1 Hz, demonstrating superior dynamic performance compared to conventional control methods.

Fig. 18 shows the dynamic response curve of the VSG under a large grid voltage sag condition. Initially, the inverter operates normally in grid-connected mode. At 0.5 s, a grid fault causes a symmetrical voltage drop to 0.5 p.u. As can be seen from the figure, the voltage sag has a significant impact on the power output of the VSG, resulting in a severe transient power overshoot within a short period. The adaptive control strategy smooths the power output of the VSG to some extent, but its response speed is slower compared to the other three control methods. In terms of system stability, the adaptive control strategy is more effective in suppressing frequency fluctuations than the other three control strategies, significantly improving grid stability.

Fig. 19 shows the voltage and current waveforms output by the inverter. Under large grid voltage sags, maintaining constant active power output from the inverter can lead to output currents far exceeding the rated value. In practical applications, this may trigger protective devices to disconnect the inverter from the grid, thereby losing its grid support capability—and in severe cases, potentially damaging the inverter. While enhancing the low-voltage ride-through (LVRT) capability of virtual synchronous generators represents a separate research direction beyond the scope of this study, the simulation results demonstrate that the proposed adaptive control strategy effectively improves grid frequency stability during severe voltage sags.