Control Algorithm for an Inverter-Based Virtual Synchronous Generator with Adjustable Inertia

Abstract

This paper presents the design and implementation of a control algorithm for power converters in a microgrid, with the main objective of providing the flexibility to adjust the system inertia. The increasing integration of renewable energy sources in microgrids has driven the development of advanced control techniques to ensure stability and power quality. The proposed algorithm combines droop control, synchronverter dynamics, and virtual impedance to achieve a robust and efficient control strategy. Simulations were conducted to validate the algorithm’s performance, demonstrating its capability to maintain voltage within acceptable limits and improve the inertial response of the microgrid. The results contribute to the advancement of intelligent and resilient microgrid development, which is essential for the transition towards a more sustainable energy system.

1. Introduction

Synchronous machines have traditionally been essential for electrical power generation systems, particularly in hydroelectric power plants and other forms of energy generation. They not only transform mechanical energy into electrical energy but also maintain synchronization with the grid frequency, thereby assisting in the stability and quality of the power system [1,2,3]. The increasing integration of renewable energy sources through electronic power inverters introduces a new concern in the power generation system. Unlike synchronous machines, which inherently store energy in their rotating inertia, helping to preserve system stability, inverters decouple generation from the load by missing the inertia link, leading to frequency fluctuations larger than in the traditional system, which may compromise stability; microgrids are especially vulnerable to that effect [4,5].

Voltage and frequency fluctuations pose a challenge in power systems, particularly in microgrids, where there are usually no synchronous generators. Frequency regulation is essential to ensure system stability, as imbalances between generation and demand can negatively impact the operation of connected equipment and lead to system instability [6,7]. Likewise, voltage regulation is fundamental for protecting loads, preventing unwanted activation of protection devices, and avoiding potential blackouts. Addressing these issues requires advanced control strategies capable of mimicking the inertial response of traditional synchronous machines while leveraging the flexibility of power electronic inverters [8,9].

This challenge has been addressed from the control point of view; one approach that can be considered traditional is the so-called droop control, which adjusts active and reactive power based on frequency and voltage variations, respectively. One of the most recent contributions to this field is the proposal of virtual synchronous machines (VSMs), which are inverters equipped with a controller that attempts to emulate the behavior of traditional synchronous machines through power inverters and advanced control algorithms. By replicating the inertia and damping characteristics of synchronous generators through a controller, VSM-based approaches can significantly improve frequency and voltage stability, particularly in microgrid applications [10,11,12,13,14,15,16].

This article proposes and tests the design of a control strategy for an inverter to become a virtual synchronous machine (VSM). The primary objective is to achieve VSM operation with the capability to adjust inertia, thereby enhancing dynamic behavior and voltage regulation in a microgrid, while ensuring stable operation under varying conditions. While previous works have independently explored adaptive inertia strategies [17] and PLL-less synchronization techniques [18], the main contribution of this paper lies in the novel and synergistic integration of these concepts into a unified control framework. The key technical innovations presented are as follows:

• A Holistic PLL-less Control Architecture: A comprehensive control structure is proposed that joins synchronverter dynamics with a virtual impedance-based self-synchronization mechanism, eliminating the need for a dedicated Phase-Locked Loop (PLL). This inherently improves the controller’s robustness and stability margin, particularly under weak grid conditions, where PLLs are a known source of instability [19].
• Synergistic Integration of Dynamic Inertia: This demonstrates how an adjustable virtual inertia can be seamlessly incorporated into a PLL-less framework, allowing the system to dynamically modulate its inertial response to minimize frequency deviations (nadir/overshoot) and the Rate of Change of Frequency (RoCoF), without the stability constraints imposed by a traditional PLL.
• Improved Stability Margin and Computational Efficiency: The proposed controller achieves a superior stability margin by eliminating the adverse dynamics of the PLL. Furthermore, the direct emulation of the synchronous machine equations, without the computational overhead of a complex PLL algorithm, offers a pathway towards a more efficient implementation on digital signal processors (DSPs).

Simulation results are provided to validate the effectiveness of the proposed approach. This work contributes to the development of more stable and resilient microgrids, facilitating the integration of renewable energy sources and supporting the transition toward a more sustainable energy system.

2. The System Under Study

The system analyzed in this study employs a common three-phase inverter topology connected to the electrical grid; however, the proposed operational algorithm is also applicable to other topologies. The system under study consists of a full-bridge three-phase inverter, which is usually synthesized with an IGBT six-pack. An LCL filter is used to connect the inverter to the grid or microgrid, as shown in Figure 1.

The objective is to derive an algorithm that allows the system in Figure 1 to operate as a traditional synchronous generator. The Virtual Synchronous Machine (VSM) operates based on a mathematical model that defines its dynamic response to grid conditions. The model incorporates key electromechanical principles, such as the swing and power balance equations that govern the exchange of active and reactive power. As proposed in [14], the VSM must mimic the following Equations (1)–(5).

$$J\ddot{\theta }=T_m-T_e-D_p\dot{\theta }$$

(1)

$$T_e=M_f{i}_f〈i,\overrightarrow{\sin }\left(\theta \right)〉$$

(2)

$$e=\dot{\theta }{M}_f{i}_f\overrightarrow{\sin }\left(\theta \right)$$

(3)

$$P=\dot{\theta }{M}_f{i}_f〈i,\overrightarrow{\sin }\left(\theta \right)〉$$

(4)

$$Q=-\dot{\theta }{M}_f{i}_f〈i,\overrightarrow{\cos }\left(\theta \right)〉$$

(5)

where T_m and T_e are the mechanical torque applied to the rotor and the electromagnetic torque, respectively; e is the generated three-phase voltage, θ is the rotor angle, P and Q represent the active and reactive power, respectively, J is the moment of inertia of all rotating components in the system, and i_f is the field excitation current. M_f is the mutual inductance between the stator windings and the field windings, $\dot{\theta }$ is the angular velocity of the machine, and D_p is the mechanical friction coefficient, which accounts for energy dissipation due to rotational losses. Figure 2 shows a block diagram of the system with the control described by (1)–(5).

3. Previous Works on VSM Derivations

The concept of Virtual Synchronous Machines (VSMs) has gained significant attention in recent years as a promising approach for improving the stability and reliability of modern power grids. Numerous studies have focused on developing and refining VSM control strategies to enhance grid inertia, frequency regulation, and voltage support.

Researchers have explored different implementations, including droop-based control, synchronverter models, and hybrid approaches. Given its potential to bridge the gap between traditional synchronous generators and power electronics-based systems, the development of VSMs remains an active area of research.

Table 1. Comparison of previous works.

Ref.	Year	Control Strategy	Analysis	Controller Function	Conclusion	Disadvantage	P+	P−	T
[20]	2019	VSG droop with auto-tuning	Numerical and experimental	Reduces overshoot (+5) Regulates frequency (+4)	Reduces the ROCOF Reduces frequency deviation	Response capability directly depends on the number of batteries (−3) Islanded mode only (−2)	9	5	4
[21]	2020	VSG with droop control	Numerical and experimental	Reduces overshoot (+5) Regulates frequency (+4)	Significantly reduces response time by mitigating overshoot during active power injection	Only step response tested with fixed parameters Not evaluated with multiple generation sources Islanded mode only (−2)	9	2	7
[22]	2021	VSG with droop and current control	Numerical	Reduces overshoot (+5) Regulates frequency (+5)	Controls sudden frequency changes	Stabilization time greater than 10 s (−5)	10	5	5
[23]	2021	VSG with virtual impedance	Numerical	Reduces response time to a disturbance (+3) Mitigates overshoot (+5)	Very fast frequency response without offset	Control tested only for one generator and one load High computational cost (−3)	8	3	5
[24]	2019	VSG with droop for frequency and voltage	Numerical and experimental	Dynamic inertia is directly dependent on frequency (+4) Smooths the ROCOF (+3) Reduces overshoot (+5)	Mitigates the ROCOF Reduces overshoot	Significant computational burden (−3) Multiple nested control loops (−3) Very long stabilization time (15 s) (−5) Secondary-level control	12	11	1
[25]	2019	VSG with droop for frequency and voltage	Numerical	Controls overshoot (+5) Regulates voltage (+4) Grid connection support (+4) Regulates frequency (+4)	Fast response time Dynamic behavior Increases inertia Reduces the ROCOF Mitigates frequency nadir Very low complexity	Reactive power loop and voltage regulation not analyzed	17	0	17
[26]	2021	VSG with optimized dynamic virtual inertia and damping control	Numerical	Mitigates frequency nadir Reduces the ROCOF (+3) Reduces overshoot (+5)	Distributes inertia evenly across each microgrid node	Very high computational cost (−3)	8	3	5
[27]	2021	VSG with per-node droop control. Nearest node to node compensation.	Numerical and experimental	Mitigates the Rocof (+5) Mitigates RocoV.	Reduced battery implementation Uniform energy distribution among nodes	High computational cost (−3) High implementation cost due to communication equipment (−2) Secondary level control	5	5	0
[28]	2022	VSG with droop adapted according to damping coefficient	Numerical	Reduces overshoot (+5) Reduces response time(+3)	Coordinated inertia increase with damping coefficient	Stabilization time greater than 3 s (−5)	8	5	3
[29]	2020	Microgrid control using VSG based on graph theory	Numerical	Mitigates dependence on battery banks Regulates frequency (+4) Regulates voltage (+4) Mitigates power drop with energy from the nearest node.	Reduces implementation costs No need for external sources; self-managed via distributed generation	Secondary level control Response time greater than 3 s (−5) High computational cost (−3)	8	8	0
[30]	2022	VSG with optimization method	Numerical	Reduces frequency response time (+3) Reduces the ROCOF (+3)	Stabilization time below 1 s	Significant computational burden (−3) Difficult implementation (−2)	6	5	1
[31]	2022	VSG with virtual inertia and damping	Numerical and experimental	Reduces response time to small-scale frequency fluctuations (+3)	Reduces response time Increases inertia with the help of additional storage	Uncontrolled overshoot Fast frequency response, but with fluctuations; stabilization time exceeds 3 s (−5)	11	5	6
[32]	2022	VSG with increased damped inertia	Numerical	Reduces response time (+3) Reduces overshoot (+5)	Mitigates overshoot Improves response time	High cost due to batteries and supercapacitors (−2)	8	2	6
[33]	2022	VSG with PI and virtual inertia	Numerical	Reduces the ROCOF (+3) Decreases response time (+3)	Mitigates sudden frequency changes	Stabilization time greater than 1 s (−5)	6	5	1
[34]	2022	VSG with droop for frequency and voltage	Numerical	Compensates for active (+4) and reactive (+4) power variations Decreases the ROCOF (+3)	Increases inertia Response time under 1 s	Active power overshoot during frequency drops (−5)	11	5	6
[35]	2022	VGS with virtual inertia stored in supercapacitors	Numerical	Bidirectional control for the supercapacitor and compensate voltage imbalances (+1) Regulates voltage (+4) Regulates frequency (+4) Grid connection analysis (+4)	Increases inertia Reduces the number of electronic control components	High cost considering implementation with supercapacitors (−2) High computational cost (−3)	13	5	8
[36]	2022	VSG with segmented virtual inertia	Numerical	Improves reliability Reduces the ROCOF (+3) Improves response time (+3)	Decreases response time Controls overshoot	Significant computational burden (−3)	6	3	3
[37]	2022	VSG with virtual impedance control	Numerical	Reduces the ROCOF (+3) Enhanced response time (+3) Controlled overshoot (+5)	Distributes virtual inertia uniformly in the microgrid to respond faster to active power variations	Stabilization time greater than 6 s (−5)	11	5	6
[38]	2022	VSG with virtual impedance control	Numerical	Voltage imbalance reduction (+4) Harmonic distortion correction Frequency regulation (+4) Supports both linear and non-linear loads Stabilization time below 1 s (+3) Analyzed in grid-connected operation (+4)	Adapts impedance according to load behavior Few control elements required	High computational cost (−3)	15	3	12
[39]	2022	VSG with virtual impedance control	Numerical	Optimizes control parameters based on load variations Regulates frequency (+4) Regulates voltage (+4) Significantly reduced stabilization time (+3)	Reduces maximum transmission power Reduces transient stability margin Maintains voltage levels within established limits	Significantly increases overshoot compared to other control methods Complex implementation (−2)	11	2	9
[17]	2024	Novel VSG with Adaptive Virtual Inertia and Adaptive Damping Coefficient	Numerical	Reduce overshoot (+4) Regulate frequency (+4) Significantly reduced stabilization time (+3)	Improve the transient frequency response,	High computational cost (−3) Depends on a dispatchable power source for frequency support, (−3)	11	6	5

presents a comparative evaluation of recent research studies focused on improving inertia in microgrids through strategies based on Virtual Synchronous Generators (VSGs) [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. The studies are analyzed based on a set of positive and negative criteria, including mitigation of overshoot, reduction in the ROCOF, voltage and frequency regulation, and computational cost, among others. Each paper was assigned a score according to its strengths (P+), limitations (P−), and a total score (T) reflecting its overall effectiveness in enhancing inertia in microgrid environments.

The literature review indicates that most existing studies focus on islanded microgrid scenarios and tend to address frequency or voltage regulation issues separately, highlighting a research gap: the need for integrated control strategies that simultaneously address voltage regulation and inertial support. The three highest-scoring strategies—droop control, synchronverter, and virtual impedance control—serve as the foundation for the algorithm proposed in this work.

4. The Derived Algorithm

Considering the synchronverter model as an emulation of a conventional synchronous machine, its mathematical representation is developed by analyzing three main parts.

The internal electromechanical part is given by the dynamics of currents and fluxes in both the stator and rotor.

$$i_a+i_b+i_c=0$$

(6)

$$\Phi =Lsi+M_f{i}_f\cos \left(\theta \right)$$

(7)

$${\Phi }_f=L_f{i}_f+M_f〈i_f,\cos \left(\theta \right)〉$$

(8)

The mechanical part is given as:

$$J\ddot{\theta }=T_m-T_e-D_p\theta$$

(9)

$$T_e=M_f{i}_f〈i_f,\cos \left(\theta \right)〉$$

(10)

The external electrical can be written in any of the following equations:

$$V=-Rsi-{\displaystyle \frac{d\Phi }{dt}}$$

(11)

$$V=-Rsi-Ls{\displaystyle \frac{di}{dt}}+e$$

(12)

Deriving the flux in (7) yields the following:

$$e=M_f{i}_f\dot{\theta }\overrightarrow{\sin }\left(\theta \right)$$

(13)

The acceleration of the synchronous generator can be expressed as follows:

$$\ddot{\theta }={\displaystyle \frac{1}{J}}\left(T_m-T_e-D_p\dot{\theta }\right)$$

(14)

To represent the mechanical equation of the synchronous generator using a block diagram, the Laplace transform is applied to Equation (14) and solved for the angular velocity as follows:

$$\dot{\theta }={\displaystyle \frac{1}{Js}}\left(T_m-T_e-D_p\dot{\theta }\right)$$

(15)

To obtain the rotor angle θ, it is sufficient to integrate the angular velocity (its derivative). Figure 3 illustrates the block diagram representation of the synchronous generator’s mechanical equation.

4.1. Droop Control for Frequency and Active Power

In a synchronous generator with frequency regulation, the active power regulation system must ensure that torque equilibrium is maintained by adjusting the mechanical torque appropriately and then deliver the active power to the grid based on the grid frequency. This action is carried out through a control loop known as frequency droop control.

According to [14], the frequency droop mechanism can be implemented in a synchronverter by comparing the virtual angular velocity with a reference, which can be equal to the nominal grid angular frequency. The difference between these values, multiplied by a gain, is added to the mechanical torque T_m. The equations show that the effect of the droop control loop is equivalent to a significant increase in the mechanical friction coefficient Dp. Thus, the constant Dp now represents the sum of the imaginary mechanical friction coefficient and the frequency droop coefficient. Therefore, denoting the change in total torque acting on the imaginary rotor as ΔT and the change in angular frequency as Δ$\dot{\theta }$, and considering typical values of frequency droop—where a 100% increase in power corresponds to a frequency drop of 3% to 5% from the nominal values—the following relationship holds the following:

$$Dp={\displaystyle \frac{\Delta T}{\Delta \dot{\theta }}}$$

(16)

The mechanical torque T_m can be obtained by dividing the reference active power P_set by the nominal mechanical speed. For example, if we use the 60 Hz grid frequency, the angular velocity is 377 rad/s, and the power is P = 5000 W, and we consider a 5% frequency variation relative to the maximum torque.

$$Dp={\displaystyle \frac{\raisebox{1ex}{$5000$}\!\left/ \!\raisebox{-1ex}{$377$}\right.}{0.05\times 377}}=0.703$$

(17)

The advantage of the synchronverter is that the moment of inertia J can be chosen to suit specific requirements to have a desired time constant of the frequency drop control loop; for example, if we want the time constant to be one-tenth of the frequency (1/600 = 1.6 mS). J = DpT_f = 0.00117.

4.2. Droop Control for Voltage and Reactive Power

To implement the reactive power regulation system, the voltage control loop of e is first implemented using an integral control with a constant 1/K_q to generate M_f i_f. Recalling the expression for the electromotive force in the synchronverter, given by (3), the constant K_q can be chosen as 377 to cancel the angular velocity.

The next step in voltage droop control is the implementation of the reactive power regulation system, for which a control scheme, as shown in Figure 4, is proposed. The difference between the desired reactive power and the reactive power delivered by the synchronverter represents the reactive power tracking error. This error is multiplied by the voltage droop coefficient Dq and then fed into an integrator with a time constant, Tq. The output of the integrator serves as the voltage reference for the synchronverter, ensuring appropriate reactive power compensation and stable operation.

4.3. Control Loop for Grid Synchronization

For the inverter to operate as a synchronous generator, it must be synchronized with the grid before connection. Synchronization requires matching the frequency, voltage magnitude, phase, and phase sequence of the synchronverter with those of the grid. When these conditions are met and the connection is just established, there is no exchange of active or reactive power between the synchronverter and the grid.

Most grid-connected three-phase converters use a dedicated synchronization unit. The most well-known synchronization method is the Phase-Locked Loop (PLL), which provides the amplitude, frequency, and phase of the grid voltage.

In [14], a radical approach is proposed to enhance the synchronverter by transforming it into a self-synchronized synchronverter through the elimination of the synchronization unit, enabling the synchronverter to automatically synchronize with the grid before connection without the need for an external PLL. To achieve automatic synchronization, the proposed method incorporates an additional control block in the synchronverter, which is responsible for emulating the current flowing through the grid inductors L_g without requiring a direct connection to the grid (virtual impedance). The control block responsible for emulating the current through the grid inductors L_g is derived using the following expression:

$$i_{virtual}={\displaystyle \frac{V{o}_{abc}-V{g}_{abc}}{L_g+R_g}}$$

(18)

A Ctrl switch is implemented to introduce the virtual current into the synchronverter equations during the synchronization phase while allowing the actual current measured by the power electronics sensors to be used during normal operation. The inclusion of virtual impedance loops and damping algorithms minimizes transient currents at the instant of connection to the grid, as shown in recent works [40,41].

By utilizing a virtual current and setting the active and reactive power references to zero, synchronization can be achieved.

There is a natural delay in the system, which causes a difference between the reference voltage Vg_abc and the voltage synthesized by the converter Vo_abc. Consequently, a discrepancy also exists between the estimated power and the measured power at the connection terminals. To compensate for this, a power meter has been included at the terminals to provide feedback to the system.

Feedback is obtained through T_e for the active power control loop. By referring to the synchronous generator equations for electrical torque, T_e is equal to the active power divided by the angular speed or can be found using (10).

Figure 4 illustrates the complete control diagram for the three-phase inverter, which incorporates both voltage and frequency droop control, as well as the system synchronization mechanism.

Table 2. Control System Parameters and Definitions.

Parameter	Symbol	Description	Value	Unit
Power and Grid
Nominal Power	Snom	Nominal apparent power of the inverter	5000	VA
Nominal Grid Freq.	fn	Nominal grid frequency	60	Hz
Nominal Grid Ang. Freq.	ωn	Nominal grid angular frequency (2π fn)	377	rad/s
Line-to-Line Voltage	VLL	RMS line-to-line grid voltage	220	V
DC Bus Voltage	VDC	Inverter DC link voltage	500	V
Active Power Loop
Virtual Inertia	J	Moment of inertia of the virtual rotor	0.00117	kg·m2
Freq. Droop Coeff.	Dp	Damping/droop coefficient for the P-ω loop	0.703	N·m·s/rad
Active Power Filter TC	Tf	Time constant of the active power measurement filter	0.01	s
Reactive Power Loop
Voltage Droop Coeff.	Dq	Droop coefficient for the Q-V loop	0.001	V/VAR
Voltage Loop Integrator TC	Tq	Time constant of the reactive power integrator	0.1	s
Voltage Controller Gain	Kv	Proportional gain for internal voltage regulation	1/377	-
Synchronization
Virtual Inductance	Lg	Inductive part of the virtual impedance	1	mH
Virtual Resistance	Rg	Resistive part of the virtual impedance	0.1	Ω
LCL Filter
Inverter-Side Inductor	Ls	Inverter-side inductance of the LCL filter	2	mH
Inverter-Side Resistor	Rs	Resistive part of inductor at the inverter-side LCL filter	0.1	Ω
Grid-Side Inductor	Lg	Grid-side inductance of the LCL filter	1	mH
Grid-Side Resistor	Rg	Resistive part of inductor at the grid-side LCL filter	0.1	Ω
Filter Capacitor	C	Capacitance of the LCL filter	10	µF
Damping Resistor	Rd	Damping resistance in series with C (not shown in Figure 4)	1	Ω

lists the parameters and definitions used in the control system.

Additionally, to guarantee the proper operation of the inverter, the DC bus voltage must be maintained at a level higher than the peak phase voltage of the AC source:

$$V_{DC\min }>\sqrt{6}\left(V_{phase}\right)$$

(19)

For a phase voltage of 120 V, a DC voltage larger than 294 V is required; therefore, considering (16), a 500 V DC bus was selected to ensure an adequate voltage margin.

From Figure 4, the calculation of the virtual current is observable. This current is obtained from the potential difference between the generated voltage and the grid voltage. When this difference is passed through a virtual impedance, it is converted into a virtual current, which is then used to compute the power values and, subsequently, the torque required for the synchronization process. The virtual impedance includes both inductive and resistive components, as the connection between the grid and the load is inherently inductive.

To bring the synchronverter system online, synchronization must first be performed as follows:

(a)

The Ctr P and Ctr Q switches are initially set to use “0” as the reference. This configuration enables the Voltage Source Converter (VSC) to generate an output voltage, Vo, at a reference frequency (θ_n) and a reference magnitude (Vo-ref).

(b)

Subsequently, the switches are commutated to use Pest and Qest as the new references. This step is critical, as it utilizes the virtual impedance to synchronize the VSC’s frequency and voltage with the grid prior to connection, ensuring that the transient upon connection is minimal. The quality of the synchronization will depend on how accurately the values of Lg and Rg are determined in the real implementation.

(c)

During the synchronization process, the synchronization network must have the reference current i_Virtual selected. The inverter’s connection to the grid is only established when the value of i_Virtual approaches zero.

(d)

Once the VSC inverter is connected to the grid, the synchronization network’s switch selects the actual current being delivered by the inverter. This current, which should be very close to zero at the moment of connection, is then used to calculate the active and reactive power that the VSC supplies to the point of common coupling (PCC). At this stage, the power references, Pref and Qref, are set by the user or by another grid control algorithm.

5. Simulation Results

To validate the performance of the proposed control algorithm, simulations were carried out using MATLAB/Simulink R2020a. A simulation of a three-phase inverter operating as a synchronous generator under dynamic conditions was developed, as shown in Figure 5. The modeled system consists of a three-phase inverter with a 500 V DC bus, an LCL filter specifically designed to attenuate switching harmonics, and a three-phase power grid with 220 V (line-to-line) at 60 Hz. The control structure integrates droop control loops for both frequency and voltage, as well as an automatic synchronization mechanism based on virtual impedance.

The inverter is implemented using a three-phase H-bridge with six IGBT transistors (Figure 6), controlled by a sinusoidal pulse-width modulation (SPWM) signal. The inverter’s output voltage is directly connected to the LCL filter, which consists of two symmetrical inductors, a central capacitor, and a damping resistor that enhances the system’s response to resonances. This filter ensures that the output signal complies with power quality standards.

The control system is based on a synchronverter model that integrates two droop-type control loops: one for frequency and active power regulation and another for voltage and reactive power regulation. The frequency controller adjusts the virtual mechanical torque T_m by comparing the virtual angular frequency with the reference frequency. On the other hand, the voltage controller generates the virtual field excitation M_fi_f based on the required reactive power, which allows the voltage profile to be maintained under both steady-state and transient conditions.

Additionally, a grid synchronization mechanism based on the estimation of a virtual current is incorporated. This current is calculated as the difference between the inverter-generated voltage and the grid voltage, divided by a virtual impedance consisting of resistive and inductive elements. From this estimated current, the active and reactive power are calculated. Along with zero reference values, this enables the inverter to synchronize with the grid without the need for a phase-locked loop (PLL). Grid connection occurs automatically once the voltage, phase, and frequency conditions are matched.

To evaluate the behavior of the inverter operating in synchronous generator mode, it is necessary to apply signals in sequence so that the controllers activate in the correct order; for this, the Simulink Signal Builder block is used. During the simulation, the control blocks are activated sequentially: frequency droop starts at t = 0.5 s, followed by voltage droop at t = 1.0 s, and grid connection at t = 1.5 s. Subsequently, controlled disturbances are introduced in the active and reactive power references, as well as load changes, to evaluate the system’s robustness and response capability. This simulation establishes the initial conditions required for result analysis, demonstrating the proposed system’s stability, accuracy, and speed under dynamic operating conditions.

Once the three-phase inverter model was implemented as a virtual synchronous machine, various tests were conducted to evaluate the system’s dynamic performance under load disturbances and changes in active and reactive power references. The simulation demonstrates that the developed control strategy enables stable operation, rapid response, and compliance with the NTC 1340 [42] and IEEE 1547-2018 [43] standard (58.8–61.2 Hz). Voltage variations remain within the limits defined by the ITIC curve, ensuring compliance with power quality standards in distribution systems.

5.1. Frequency Response

Figure 7a shows the evolution of the frequency at the PCC node under various events, comparing the frequency response of a power control algorithm in the dq reference frame with that of the synchronverter control algorithm presented in this paper. The system is initially stable at approximately 59.96 Hz with a 3 kW demand. The power converter connects to the PCC at t = 2 s, where a minor frequency variation—caused by the virtual impedance effect and slight mismatches with the actual grid impedance (L_g and R_g)—is quickly compensated.

Exploring Figure 7a,b, at t = 3.0 s, a 3000 W active power reference is introduced, causing a slight deviation to 60.12 Hz and 60.11 Hz for the traditional dq-frame control and the synchronverter control, respectively, with the system stabilizing within 296 ms. When the power injection ends at t = 5.5 s, the frequency temporarily drops to 59.82 Hz and 59.83 Hz, stabilizing again within 349 ms. At t = 8.0 s, an additional 2 kW load is connected, resulting in a controlled frequency drop to 59.81 Hz and 59.82 Hz, with full recovery occurring in 150 ms. In all cases, the system remains within the limits established by the NTC 1340 and IEEE 1547-2018 standard, demonstrating an effective inertial response.

As a primary observation, the response times of both control strategies are similar; however, the synchronverter technique exhibits a more damped response compared to the synchronous dq-frame control. Although the stabilization time is not substantially improved, a clear enhancement is observed in the reduction in frequency deviations and in the overall frequency transient.

5.2. Active Power Transfer

The active power behavior shows proper coordination between the inverter and the grid, as illustrated in Figure 8. At t = 3.0 s, the inverter starts injecting 3000 W, supplying 100% of the active load while the grid reduces its contribution to zero. At t = 5.5 s, the reference returns to zero, and the grid resumes supplying the load. At t = 8.0 s, as the load increases to 5000 W, the grid responds immediately, and the inverter attempts to stabilize its contribution, demonstrating effective load sharing and reference tracking.

5.3. Voltage Regulation

The inverter’s output voltage remains stable around 179 V peak (127 Vrms), even during transient reactive power events. Small amplitude variations are observed, as shown in Figure 9, all within the limits defined by the ITIC curve, ensuring adequate power quality for sensitive connected loads.

5.4. Synchronization Without PLL

One of the most significant achievements of the simulation is the ability to synchronize automatically without the use of a phase-locked loop (PLL), as shown in Figure 10. Thanks to the use of a virtual current estimated through a virtual impedance, the inverter can match its frequency, amplitude, and phase with the grid before connection. This synchronization is completed within the first 1.5 s of the simulation, and the connection is established without abrupt power transfer.

In Figure 10, the inverter and grid voltages are shown at the beginning of the virtual machine operation, with a clear phase and amplitude mismatch. Both voltages are matched in amplitude, phase, and frequency at 2 s, the moment when the connection occurs. Additionally, no variations are observed in either voltage, confirming that the controller is operating properly.

5.5. Simulation of DC/AC Converter Under Weak Grid Conditions and Frequency Variation

To test the grid synchronization and frequency stabilization functionality in a weak grid scenario, the supply system from previous simulations is slightly modified. For this new scenario, the main source is modeled as a synchronous generator with limited power capacity, making it sensitive to load disturbances. The simulation used the following parameters for the main source: 10.2 kVA, 1800 RPM, 220 Vac, and 60 Hz.

The test begins with a 5 kW load connected to the PCC. At t = 1.0 s, the synchronverter is connected to the electrical grid but does not supply power to the PCC, establishing the grid frequency at 59.36 Hz, which is within the continuous operating range specified by IEEE 1547-2018. At t = 2.0 s, an 8 kW load is connected to the PCC, causing the generator’s power output to increase and the grid frequency to drop to 58.45 Hz. At t = 3 s, the DC/AC converter is commanded to inject power into the grid. The amount of injected power is determined by the DC/AC converter’s droop control, which aims to re-establish the grid frequency within the permissible limits of 58.8 Hz and 61.2 Hz, eventually settling at 59.5 Hz. At t = 5.5 s, the sudden load is removed, returning the grid to its initial 5 kW load, with the key difference that the grid frequency is maintained at approximately 59.5 Hz. At t = 7.5 s, the 8 kW load is reconnected, causing a small frequency transient that remains within the permissible range of the standard. In the sudden load events at t = 2 s and t = 7.5 s, the effect of the synchronverter-driven droop control on the grid frequency is observed. Operation and synchronization times are illustrated in Figure 11, Figure 12 and Figure 13. The aforementioned events are detailed below:

• Grid synchronization of the DC/AC converter occurs at t = 0.4 s, and its connection to the grid is executed at t = 1 s. The connection is performed manually (Figure 11). This connection step can be automated, a task planned for future experiments.
• A sudden load at the PCC at t = 2 s exceeds the generator’s nominal capacity (Figure 12—green line), forcing the frequency to drop below the 58.8 Hz limit established by the IEEE 1547-2018 standard. Figure 12 shows this frequency drop, and Figure 13 shows the increase in power delivered by the generation to the PCC, reaching approximately 13 kW (red line).
• The DC/AC converter with the synchronverter begins operation to restore the frequency to normal ranges. Here, the power contribution from each generation source is 8.9 kW from the DC/AC converter (Figure 13—blue line) and 3.86 kW from the generator connected to the PCC (Figure 13—red line). This corresponds to a slight drop in the load’s power, attributed to voltage regulation. Figure 12 shows the restoration of the PCC frequency to permissible operating levels (f = 59.5 Hz).
• Sudden load shedding occurs at t = 5.5 s to verify the frequency behavior and power sharing while the DC/AC converter is injecting power into the grid, prioritizing frequency regulation. The frequency remains constant after a short operating transient, which lasts 0.5 s and causes a frequency increase of 0.24 Hz without exceeding permissible limits (see Figure 12).
• The previously described sudden load event is repeated at 7.5 s, identical to the load applied at 2.5 s. This time, the test is performed with the synchronverter’s power droop control active in the DC/AC converter. This event is executed to verify the droop control’s operation in the converter, acting as an inertia compensator with a correction time of 0.6 s.

5.6. Simulation of DC/AC Converter Under Weak Grid Conditions and Non-Linear Load

To test the DC/AC converter system with the synchronverter under non-linear load conditions, the system is evaluated with a THDv of 28.14%. The load is divided into a 2 kW active component and an approximately 5 kW non-linear load, which are kept fixed throughout the test. Additionally, an 8 kW linear load is connected to and disconnected from the PCC, producing an overload at the node and frequency deviation, resulting in a total approximate load power of approximately 15 kW. The voltage waveforms at the PCC node are shown in Figure 14, and their harmonic components are depicted in Figure 15.

The timing for the transient events is kept the same as that used in Section 5.6 to directly compare the events with a linear load versus a linear + non-linear load. A description of the simulation events follows:

• Figure 14a shows the synchronization process of the DC/AC inverter’s waveform with the PCC signal, which contains harmonic content; synchronization is achieved at t = 0.4 s. Figure 14b shows the instant the inverter is connected to the PCC node, where the dominance of the node’s voltage is observed. Figure 15 displays the harmonic content of the voltage signal, which causes oscillations in the power exchanged between the converter and the grid and between the grid and the load. Despite the voltage’s harmonic content, synchronization is possible because the virtual impedance used in this process is equivalent to the coupling impedance (comprising Lg and Rg), which is small, thereby minimizing the power oscillations generated by voltage harmonics during synchronization. As a next step, the THD limit at which synchronization is no longer possible should be established, and the level of interaction and coupling between the power oscillations and the control loops must also be determined, as these topics are subjects for investigation and presentation in future work.
• A sudden load at the PCC exceeds the generator’s nominal capacity, causing the frequency to drop below the 58.8 Hz limit, as shown in Figure 16. In this case, the frequency drop is slightly larger due to the power difference compared to the previous test, as shown in Figure 17.

All subsequent events remain the same as in the test performed in Section 5.5: the start of operation of the DC/AC converter with the synchronverter to restore the frequency to normal ranges, the load shedding to verify frequency behavior and power contribution to the grid from the converter, and the load reconnection to verify the converter’s operation as an inertia compensator.

Since the synchronverter control algorithm uses power measurement for frequency compensation, the presence of harmonics causes sustained power oscillations that will affect the synchronverter’s performance. Papers such as [44,45,46] present studies and potential solutions for the interaction of control loops and grid impedances, which generate oscillations that compromise the stability of the synchronverter.

6. Conclusions

This work presented the design and simulation-based validation of a controller grounded in the concept of a virtual synchronous machine (synchronverter), complemented with droop strategies for frequency and voltage regulation in microgrids. The results demonstrate that the proposed system effectively emulates the inertial behavior of synchronous generators, enabling stable active power transfer and reactive power compensation, even under dynamic load disturbances. The inclusion of a damping resistor in the LCL filter proved essential for improving the transient response of active power. Additionally, the implementation of automatic synchronization without a PLL, using estimated virtual current, reduced system complexity and ensured a safe connection to the grid.

Moreover, the study identified relevant effects of line impedance on voltage control and emphasized the importance of proper phase synchronization to avoid connection errors. The proposed architecture proved to be efficient, computationally lightweight, and suitable for real-world implementation in power converters. While the MATLAB/Simulink results presented in this article validate the fundamental performance of the proposed control algorithm, it is recognized that real-world implementation presents additional challenges, such as computational delays, sensor noise, non-linear loads, and hardware non-linearities. To bridge the gap between simulation and physical deployment, an experimental validation plan is contemplated as the next phase of this research.

The immediate next step will involve validation using Power Hardware-in-the-Loop (PHIL). This methodology enables testing the real-time performance of the control algorithm, implemented on a DSP or similar device, while the power stage (inverter, filter, grid) is simulated in real-time on a real three-phase DC/AC converter, supported by a dSPACE DS1104 system. The objectives of the PHIL study will be as follows:

• Verify the real-time execution and computational feasibility of the PLL-less synchronverter algorithm integrated into a hardware controller.
• Evaluate the controller’s robustness against simulated sensor noise and parameter variations in the grid model.
• Quantify the impact of discrete-time implementation and processing delays on the system’s stability and dynamic response.
• Validate seamless synchronization and power transfer performance under various initially simulated testbeds, including voltage sags, frequency deviations, and phase jumps.

This experimental validation will provide the definitive proof of the algorithm’s practical applicability and performance in a real electrical environment. This phased approach, from simulation to PHIL, ensures a rigorous and low-risk validation pathway for the proposed control strategy.

It must also be considered to establish the THD limit for synchronization and to determine the coupling between power oscillations and control loops, as these are important topics for the stability of the synchronverter.