Novel Topological Aggregation Method for Grid-Forming Converters of Renewable Energy with Threshold Correction

Abstract

With the increasing integration of renewable energy sources into modern power grids, the dynamic and operational complexity of power systems has grown significantly, rendering traditional control and modeling approaches insufficient. Virtual Synchronous Generator (VSG) technology has emerged as a critical solution, replicating the dynamics of synchronous generators to enhance the stability of inverter-based systems. Aggregation of VSG units is essential for efficient modeling and analysis of large-scale systems, especially in addressing challenges like power oscillations, dynamic stability, and control interactions. This paper proposes an improved homological equivalent method tailored for VSGs, incorporating a precise definition of critical parameters such as the time window and threshold to enhance its applicability. Additionally, aggregation methods for VSG control and circuit parameters are developed to facilitate the representation of complex systems. The proposed methods are validated through Matlab/Simulink simulations, where the performance of the aggregated VSG model is compared with the original VSG group. Results confirm the reliability and effectiveness of the approach in retaining dynamic characteristics while reducing modeling complexity. This research provides a theoretical foundation and practical methodology for VSG aggregation, contributing to enhanced stability analysis and operational efficiency in high-renewable-penetration power systems.

1. Introduction

With the depletion of traditional fossil fuels, the rapid development and utilization of renewable energy technologies, such as solar and wind power, are transforming the energy structure and grid supply modes. However, the large-scale integration of renewable energy has increased power system complexity, making traditional linear control models inadequate. Virtual Synchronous Generator (VSG) technology has emerged as a promising solution, emulating the dynamics of synchronous generators to enhance the stability of inverter-based grid-connected systems [1]. Research on VSG homological equivalent methods provides valuable insights for equivalence analysis in local grids. Power plants must meet grid standards to ensure reliable interconnection, with performance evaluation focused on the point of interconnection, making aggregated modeling a practical approach. Dynamic aggregated equivalent models simplify the representation of individual units while retaining essential system dynamics, enabling efficient stability analysis [2]. As renewable energy penetration grows, aggregating grid-forming units like VSGs becomes essential but presents challenges, such as exacerbated power oscillations and complex interactions in parallel operations. Dynamic aggregated models for VSG-based systems simplify multi-unit representation while providing a foundation for analyzing collective dynamics and interactions in modern grids.

In traditional large AC grid systems, equivalent studies on synchronous generators have been extensively conducted. With the increasing integration of renewable energy and inverter-based resources, VSG technology has emerged as a critical solution to emulate the dynamics of conventional synchronous generators. Aggregation of VSG units is essential for efficient modeling and system-level analysis in high-renewable penetration grids [3], addressing challenges such as power oscillations, dynamic stability, and control interactions.

A generalized composite load aggregation method based on an improved K-means algorithm [4] was proposed for distribution networks with VSG and PQ control strategies, but it was validated only under a single fault scenario. In contrast, this paper focuses on VSG unit stability and uses threshold corrections to classify and aggregate multiple VSG units, offering better adaptability to complex fault scenarios. Reference [5] introduces a coherency-based method for MMC aggregation, which may introduce dynamic errors. This paper addresses VSG stability directly through threshold corrections, ensuring more accurate aggregation without dynamic errors. Transient stability analysis and aggregation of grid-forming (GFM) and grid-following (GFL) converters are discussed in Reference [6], though the assumption of uniform control parameters limits real-world applicability. Reference [7] presents a control approach for aggregated VSGs but lacks thorough analysis of communication delays. This paper incorporates threshold corrections, considering communication delays and ensuring more reliable VSG aggregation under dynamic conditions. Weighted dynamic aggregation for parallel DC-DC converters is presented in Reference [8], focusing on DC microgrids, with limited extension to hybrid AC-DC systems. By contrast, our work focuses on VSG units and extends the aggregation method to hybrid AC-DC systems under dynamic conditions. Reference [9] proposes a capacity-reduction-based method for variable frequency motors, but its robustness under parameter variations requires further study. Reference [10] proposes a novel coherency identification method based on Hamilton’s action principle for dynamic aggregation modeling of grid-connected inverters. This approach effectively reduces the computational complexity of large-scale models while preserving the key dynamic characteristics of the original system. It is primarily designed for aggregating inverter clusters with similar dynamic behaviors. In contrast, this paper uses threshold corrections to focus on VSG stability, providing a more practical solution for large-scale VSG aggregation.

Reference [11] apply the aggregation equivalence methods of synchronous generators to their studies. However, due to significant differences in control strategies between power-synchronization-based converters and phase-locked-loop-based converters, the aggregation methods for the former are challenging to directly apply to the latter. Reference [12] proposes a coherency criterion for multi-converter systems based on impedance voltage drop and derived a calculation method for equivalent control parameters under power constraint conditions. Reference [13] revealed that the unbalanced voltage input of the phase-locked loop during faults is the root cause of transient synchronization instability. However, the approach of directly aggregating large-scale grid-connected renewable energy generation units into a single inverter model, without considering the interactions between inverters, inevitably introduces significant errors in transient synchronization stability analysis. Reference [14] analyzes the impact of interactions between inverters on transient synchronization stability but lacks a study on the aggregation and equivalence process for multi-inverter systems. Reference [15] aggregates only the inverters connected to the same bus, lacking research on multi-bus topologies. Furthermore, the dynamics of phase-locked loops are neglected during the aggregation process, making it incapable of accurately describing the transient synchronization behavior of inverters. Reference [16] uses the inverter control type as the clustering criterion to perform dynamic equivalence of the full-order inverter models. Reference [17] proposes an aggregation model for hybrid systems containing GFL and GFM converters, which significantly reduces the number of system state variables from 2(M + N) to 4, improving the analysis efficiency.

This paper makes the following three contributions: (1) it improves the traditional homological equivalent method for synchronous generators and proposes a criterion tailored fto VSGs, with more precise definitions of the time window and threshold parameters, thereby enhancing the method’s universality; (2) it develops aggregation methods for VSG control parameters and circuit parameters; and (3) it verifies the reliability of the proposed aggregation method through Matlab simulations by comparing the performance of the aggregated VSG model with the original VSG group. This research provides new theoretical foundations and practical methods for VSG modeling and aggregation in high-renewable-penetration power grids, contributing to improved efficiency in system-level stability analysis and operational optimization.

2. Principle of Virtual Synchronous Generator (VSG)

The VSG emulates the dynamic characteristics of a synchronous generator by simulating rotational inertia and damping properties through control algorithms. This enables VSGs to provide stability and inertia support similar to traditional synchronous generators in distributed generation systems, thereby enhancing the system inertia of the power grid.

The control components of the VSG primarily include the power calculation module, active power–frequency control module, reactive power–voltage control module, and dual-loop voltage–current control module. The topology of the VSG’s main circuit and control loops is shown in Figure 1. In the figure, $L_f$ and $C_f$ denote the filtering inductance and capacitance on the DC side of the VSG, respectively; $L_g$ and $R_g$ correspond to the equivalent inductance and resistance of the three-phase grid, respectively; $U_{dc}$ represents the DC-side voltage of the VSG; $C_{dc}$ refers to the stabilizing capacitor on the DC side; $U_g$ indicates the grid voltage; $i_{pcc}$ denotes the current injected into the grid by the VSG; and $u_{pcc}$ represents the output voltage of the VSG.

2.1. Power Control Loop of the VSG

2.1.1. Active Power–Frequency Control

The active power–frequency control of the VSG mimics the inertia and frequency response of traditional synchronous generators. This allows renewable energy equipment to support the grid with functions similar to inertia support and primary frequency regulation. The rotor motion equation is as follows:

$$\left\{\begin{array}{c}J{\omega }_n{\displaystyle \frac{d\omega }{dt}}=P_m-P_e-D(\omega -{\omega }_n)\hfill \\ {\displaystyle \frac{d\delta }{dt}}=\omega -{\omega }_n\hfill \end{array}\right.$$

(1)

In the equation, $\omega$ and ${\omega }_n$ represent the actual rotor angular velocity and the rated angular velocity, respectively; J and D denote the moment of inertia and the damping coefficient, respectively; $P_m$ and $P_e$ correspond to the mechanical power and the electromagnetic power, respectively; $\delta$ represents the power angle.

2.1.2. Reactive Power–Voltage Control

Reactive power–voltage control emulates the excitation control system of a synchronous generator to regulate output voltage and reactive power. It first adjusts the voltage by modulating the difference between the reference voltage and the actual output voltage, thereby stabilizing the inverter’s output voltage. Additionally, an integral module is designed to ensure that the output voltage remains consistent with the reference voltage. The control equation is as follows:

$$E_u=U_n+{\displaystyle \frac{1}{K_{s}s}}\Delta U$$

(2)

In the equation, $K_s$ is the integral coefficient; $E_u$ represents the output voltage amplitude; $\Delta U$ denotes the difference between the reference voltage and the output voltage; and s is the Laplace operator.

The reactive power–voltage regulation equation is as follows:

$$Q_{\mathrm{ref}}-Q_e=K_q\left(U-U_n\right)$$

(3)

In the equation, $Q_{\mathrm{ref}}$ and $Q_e$ represent the reactive power reference value and the actual output value, respectively; $K_q$ is the reactive power–voltage droop coefficient; and U and $U_n$ denote the actual output voltage and the reference voltage, respectively.

Combining Equations (2) and (3), the overall model for reactive power–voltage control is expressed as follows:

$$E_u=U_n+{\displaystyle \frac{1}{K_{s}s}}\left[K_q\left(U_n-U\right)+\left(Q_{\mathrm{ref}}-Q_e\right)\right]$$

(4)

2.1.3. Voltage–Current Dual-Loop Control

After going through the power control module, three-phase modulation waves are created. These waves are then converted into DC values using the Park transformation. The DC values are then input into the voltage–current dual-loop control system. The block diagram of the voltage–current dual-loop control system is shown in Figure 2.

The open-loop transfer function of the VSG current inner loop is as follows:

$$G_{ki}\left(s\right)={\displaystyle \frac{p{K}_{ip}}{L_{f}s}}$$

(5)

In the equation, $K_{ip}$ is the proportional coefficient of the current loop for the VSG; p is the power distribution coefficient.

The closed-loop transfer function of the VSG current inner loop is as follows:

$$G_{bi}\left(s\right)={\displaystyle \frac{G_{ki}\left(s\right)}{1+G_{ki}\left(s\right)}}={\displaystyle \frac{\frac{p{K}_{ip}}{L_{f}s}}{1+\frac{p{K}_{ip}}{L_{f}s}}}$$

(6)

The open-loop transfer function of the VSG voltage outer loop is the following:

$$G_{ku}\left(s\right)={\displaystyle \frac{K_{vp}+\frac{K_{vi}}{s}}{s{C}_f}}{G}_{bi}\left(s\right)$$

(7)

In the equation, $K_{vp}$ and $K_{vi}$ represent the proportional coefficient and integral coefficient of the VSG voltage outer loop, respectively.

The closed-loop transfer function of the VSG voltage outer loop is as follows:

$$G_{bu}\left(s\right)={\displaystyle \frac{G_{ku}\left(s\right)}{1+G_{ku}\left(s\right)}}={\displaystyle \frac{\frac{\left(K_{vp}+\frac{K_{vi}}{s}\right)G_{bi}\left(s\right)}{s{C}_f}}{s{C}_f+\left(K_{vp}+\frac{K_{vi}}{s}\right)G_{bi}\left(s\right)}}$$

(8)

3. Coherent Identification Method Based on Stability Margin Correction

The coherent equivalence process of conventional AC systems generally consists of five steps: (1) dividing the system into a study area and an external system area, where the equivalence process retains the study area while simplifying the external system; (2) identifying coherent generator groups in the external area; (3) simplifying and merging generator buses; (4) simplifying the network; and (5) aggregating the dynamics of generators to obtain equivalent machine parameters. Since the VSG simulates the characteristics of a synchronous generator, it exhibits an observable power angle when the external system is subjected to disturbances. Based on this, the coherent identification method for synchronous generators can be adapted to the VSG. Coherent identification methods are traditionally applicable to conventional synchronous generators. However, the control structure of VSGs incorporates matrix equations analogous to those of synchronous generators, allowing the adoption of corresponding relationships when applying coherence criteria to achieve the desired identification outcome. Nevertheless, traditional coherent identification methods rely on fixed time windows and thresholds, which pose limitations. To address this, the influence of stability margin factors is incorporated into the selection of thresholds, enhancing the robustness of the method.

3.1. Homological Equivalence Method for VSGs

In power systems, two generators are considered quasi-coherent if their power angle difference remains stable within a specific time frame following a disturbance. This indicates that the generators exhibit similar oscillatory behavior and their power angle variations tend to converge. The determination of quasi-coherence is based on the analysis of relative power angle dynamics between generators. Specifically, the identification criterion evaluates whether the maximum deviation of the relative power angle from its final steady-state value over a given time window stays within a predefined threshold. If this condition is satisfied, the generators are classified as quasi-coherent. This approach effectively captures the dynamic consistency of generators during transient conditions.

Based on the above theory, the criterion for quasi-coherence identification can be expressed as follows:

$$\underset{t\in [0,\tau ]}{max}\left|\Delta {\delta }_i-\Delta {\delta }_j\right|\le \epsilon$$

(9)

In the equation, $\Delta {\delta }_i$ and $\Delta {\delta }_j$ represent the differences in power angles of the ith and jth VSG units relative to their respective initial values over a given time interval. $\epsilon$ is the preset threshold. If the result is less than or equal to $\epsilon$, the two generators can be considered coherent.

3.2. Time Window Determination

In AC systems, $\tau$ reflects the time during which the synchronous generator power angle changes after experiencing a disturbance, representing the time constant of the synchronous generator inertia. The inertia time constant of the synchronous generator, $T_{SG}$, is expressed as follows:

$$T_{SG}={\displaystyle \frac{J_{SG}{\mathrm{\Omega }}_0^2}{S_{SG}}}={\displaystyle \frac{J_{SG}}{S_{SG}}}{\left({\displaystyle \frac{2\pi n_0}{60}}\right)}^2$$

(10)

In the equation, $J_{SG}$ represents the rotational inertia of the synchronous generator, $S_{SG}$ denotes the rated capacity of the synchronous generator, and $n_0$ is the rated rotational speed of the synchronous generator. The inertia time constant of hydro generator units is typically in the range of 1–3 s, while that of thermal generator units is generally in the range of 7–8 s.

The active power control loop of the VSG emulates the mechanical equation of a generator, and its virtual inertia time constant is expressed as follows:

$$T_{VSG}={\displaystyle \frac{J}{D_P}}$$

(11)

$D_P$ in distribution networks or microgrids typically ranges from 1 to 10. The coefficient a is used to represent the ratio of the inertia time constant between synchronous generators and VSGs, defined as:

$$a={\displaystyle \frac{T_{SG}}{T_{VSG}}}\approx 10\sim 100$$

(12)

Based on this ratio, the time window for coherence identification can be adjusted accordingly. According to the estimation of a, $\tau$ = 0.1∼0.6 s can be selected as the time window for quasi-coherence identification. In this paper, $\tau$ is chosen as 0.4 s.

3.3. Threshold Determination

In multi-VSG systems, the stability of the system reflects its ability to resist disturbances. Therefore, evaluating the stability margin of the system under different operating points provides a basis for determining the threshold, freeing its selection from the limitations of specific systems and traditional experience-based settings, thereby expanding the applicability of the coherence criteria.

The stability margin of conventional synchronous generators is commonly evaluated using small-signal analysis. This method involves linearizing the system’s state-space equations and analyzing the eigenvalue distribution to assess stability. When subjected to small disturbances, the system is considered to have a robust stability margin if all eigenvalues possess negative real parts, allowing the system to gradually return to a stable operating point. Conversely, the presence of any eigenvalue with a positive real part indicates system instability. VSG, which emulates the inertia and damping characteristics of synchronous generators, adopts an analogous analytical framework by leveraging small-signal stability analysis to evaluate the stability margin of the system.

For a single VSG injecting power into the grid at the point of common coupling (POC) through grid impedance, its output active power and reactive power can be expressed as follows:

$$P_e={\displaystyle \frac{3}{2}}{\displaystyle \frac{U_{g}Usin\delta }{X_g}}$$

(13)

$$Q_e={\displaystyle \frac{3}{2}}{\displaystyle \frac{U^2-U_{g}Ucos\delta }{X_g}}$$

(14)

In the equations, U and $U_g$ represent the terminal voltage of the VSG and the grid voltage, respectively, while $X_g$ denotes the equivalent reactance.

From Equations (1), (4), (14) and (15), the state-space equations of a single VSG are constructed as follows:

$$\left[\begin{array}{c}\dot{\delta }\\ \dot{\omega }\\ \dot{U}\end{array}\right]=\left[\begin{array}{c}\omega -{\omega }_n\\ {\displaystyle \frac{P_m-\frac{3U_{g}Usin\delta }{2X_g}-D(\omega -{\omega }_n)}{J{\omega }_n}}\\ {\displaystyle \frac{1}{K_q}}\left(Q_{\mathrm{ref}}-{\displaystyle \frac{\frac{3}{2}\left(U^2-U_{g}Ucos\delta \right)}{X_g}}\right)\end{array}\right]$$

(15)

To rewrite Equation (16) in state-space form, the system can be expressed as follows:

$$\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$$

(16)

In the equation, $\mathbf{x}$ represents the state variables, and $\mathbf{u}$ denotes the external inputs. The matrix $\mathbf{A}$ describes the dynamic relationships among the internal state variables of the system, while the matrix $\mathbf{B}$ characterizes how the input variables influence the system’s dynamic behavior.

For small disturbance analysis of nonlinear equations, it is assumed that the system oscillates with small amplitudes around the equilibrium point. The equilibrium point satisfies the following:

$$\left\{\begin{array}{c}{\omega }_0={\omega }_n\hfill \\ P_m={\displaystyle \frac{3}{2}}{\displaystyle \frac{U_g{U}_{0}sin{\delta }_0}{X_g}}\hfill \\ Q_{\mathrm{ref}}={\displaystyle \frac{3}{2}}{\displaystyle \frac{U_0^2-U_g{U}_{0}cos{\delta }_0}{X_g}}\hfill \end{array}\right.$$

(17)

By performing a Taylor expansion of the nonlinear equations around the equilibrium point, the linearized state-space equations can be expressed as follows:

$$\dot{\Delta \mathbf{x}}=\mathbf{A}\Delta \mathbf{x}+\mathbf{B}\Delta \mathbf{u}$$

(18)

By taking partial derivatives of the state-space equations term by term, the specific expression for the matrix $\mathbf{A}$ can be obtained as follows:

$$\mathbf{A}=\left[\begin{array}{ccc}0& 1& 0\\ -{\displaystyle \frac{3U_g{U}_{0}cos{\delta }_0}{2J{\omega }_n{X}_g}}& -{\displaystyle \frac{D}{J{\omega }_n}}& -{\displaystyle \frac{3U_{g}sin{\delta }_0}{2J{\omega }_n{X}_g}}\\ -{\displaystyle \frac{3U_{0}sin{\delta }_0}{2K_q{X}_g}}& 0& -{\displaystyle \frac{1}{K_q}}\left({\displaystyle \frac{3}{2}}{\displaystyle \frac{2U_0-U_{g}cos{\delta }_0}{X_g}}\right)\end{array}\right]$$

(19)

The stability margin of the system depends on the eigenvalues of the matrix $\mathbf{A}$. If all the eigenvalues of $\mathbf{A}$ have negative real parts, the system is stable at the equilibrium point.

The solution can be obtained through eigenvalue decomposition as follows:

$$det(\mathbf{A}-\lambda \mathbf{I})=0$$

(20)

Based on the eigenvalues of the state matrix, the stability of the system can be determined. If all eigenvalues have negative real parts, the system is stable. The more negative the real parts, the faster the system recovers to a stable state, indicating a higher stability margin. The dominant eigenvalues typically have the greatest impact on the dynamic behavior of the system; therefore, analyzing the real parts of the dominant eigenvalues directly reflects the system’s worst stability margin.

For VSGs, the range of J is significantly smaller compared to traditional synchronous generators, meaning that VSGs have weaker inertia responses and their dynamic characteristics are more sensitive. As a result, the selection of an appropriate threshold for the quasi-coherence identification process becomes crucial. Traditional synchronous generators, with their larger inertia time constants, exhibit slower power angle variations, allowing for the use of a larger threshold for quasi-coherence determination. However, VSGs, which emulate the characteristics of synchronous generators, have smaller inertia, and their power angle differences can change more rapidly. Consequently, using a larger threshold may lead to incorrect identification and compromise the accuracy of stability analysis. In this paper, 1° is chosen as the baseline value for quasi-coherence identification. This choice is made considering the dynamic characteristics of VSGs, ensuring that the power angle variations are accurately captured while maintaining sensitivity to the system’s dynamic behavior. Although 1° serves as the baseline, it can be adjusted based on the stability conditions of the system. Systems with faster dynamic responses, for example, may require a lower threshold, while more stable systems may benefit from a slightly larger threshold. The selection of 1° as the baseline strikes a balance between maintaining the system’s sensitivity and avoiding excessive aggregation, thus providing an effective and flexible method for dynamic aggregation of VSG models in high-renewable-penetration grids.

The threshold adjustment model can be expressed as follows:

$$\epsilon ={\epsilon }_0·\rho$$

(21)

$$\rho ={\displaystyle \frac{\left|{\sigma }_c\right|}{\left|{\sigma }_t\right|}}$$

(22)

In the equation, ${\epsilon }_0$ represents the baseline threshold, ${\sigma }_t$ denotes the real part of the target eigenvalue, and ${\sigma }_c$ represents the real part of the dominant eigenvalue of the system.

For the selection of the target eigenvalue, this paper adopts the mean of the real parts of all system eigenvalues as the target eigenvalue’s real part. The smallest absolute value of the real parts of the eigenvalues of each unit is selected as the dominant eigenvalue of the system, expressed as follows:

$${\sigma }_t={\displaystyle \frac{1}{n}}\sum _{i=1}^n\mathrm{Re}\left({\lambda }_i\right)$$

(23)

$${\sigma }_c=\underset{i=1,\dots ,n}{min}\mathrm{Re}\left({\lambda }_i\right)$$

(24)

4. VSG Equivalent Aggregation Modeling

The aggregation of virtual synchronous generator (VSG) homology primarily involves two stages: the aggregation of VSG control parameters and the aggregation of VSG main circuit parameters.

4.1. Aggregation of Control Parameters for Homological VSGs

The control parameters of homological VSGs directly reflect the virtual rotor motion equations. Consequently, the aggregation of homological VSG control parameters precisely corresponds to the aggregation of the rotor motion equations in traditional synchronous generators. When aggregating the parameters of the virtual rotor motion equations, it is essential to preserve the inherent dynamic characteristics of the virtual rotor. This requires the superposition-based aggregation of specific control parameters within the control system that characterize the motion equations.

4.1.1. Aggregation of Active Power Loop Control Parameters

The active power control loop of the xth VSG can be expressed as follows:

$$D_{px}={\displaystyle \frac{D}{{\omega }_{nx}}}$$

(25)

$$J_x{\displaystyle \frac{d\omega }{dt}}=\left(P_{\mathrm{ref}x}-P_x\right){\displaystyle \frac{1}{{\omega }_{nx}}}-D_{px}\left({\omega }_x-{\omega }_{nx}\right)$$

(26)

In the equation, $J_x$ represents the rotational inertia of the xth VSG; ${\omega }_x$ and ${\omega }_{nx}$ denote the angular velocity and nominal angular velocity of the xth VSG, respectively; and $D_{px}$ is the damping coefficient of the xth VSG.

Each VSG converges and operates at the common bus, with their output voltages having the same nominal frequency.

$${\omega }_{n1}={\omega }_{n2}=\cdots ={\omega }_{nn}$$

(27)

In the equation, ${\omega }_{nn}$ represents the nominal angular frequency of the nth VSG.

When the system is in the steady state, the differential term equals zero. We obtain the following:

$$0=\left(\sum _{x=1}^n{P}_{\mathrm{ref}x}-\sum _{x=1}^n{P}_x\right){\displaystyle \frac{1}{{\omega }_{nx}}}-\sum _{x=1}^n{D}_{px}\left({\omega }_x-{\omega }_{nx}\right)$$

(28)

When multiple VSGs are equivalently aggregated into a single VSG equivalent model, the aggregated parameters can be expressed as follows:

$$\left\{\begin{array}{c}{P}_{\mathrm{refs}}=\sum _{x=1}^n{P}_{\mathrm{ref}x}\hfill \\ P_s=\sum _{x=1}^n{P}_x\hfill \\ D_{ps}=\sum _{x=1}^n{D}_{px}\hfill \end{array}\right.$$

(29)

In the equation, $P_{\mathrm{refs}}$ and $P_s$ represent the active power reference value and the actual output value of the aggregated VSG, respectively; $D_{ps}$ denotes the aggregated damping coefficient.

The active power loop of the aggregated VSG model can be expressed as follows:

$$J_s{\displaystyle \frac{d\omega }{dt}}=\left(P_{\mathrm{refs}}-P_s\right){\displaystyle \frac{1}{{\omega }_n}}-D_{ps}\left(\omega -{\omega }_n\right).$$

(30)

In the equation, $J_s$ represents the rotational inertia of the aggregated VSG, and ${\omega }_n$ denotes the nominal angular frequency of the aggregated VSG. The rotational inertia $J_s$ of the aggregated VSG can be expressed as follows:

$$J_s=\sum _{x=1}^n{J}_x$$

(31)

4.1.2. Aggregation of Reactive Power Loop Control Parameters

From Equation (4), the reactive power control loop of the xth VSG can be expressed as follows:

$$E_{ux}={\displaystyle \frac{K_{qx}\left(U_{nx}-U_x\right)+Q_{\mathrm{ref}x}-Q_x}{s{K}_{sx}}}$$

(32)

In the equation, $E_{ux}$ represents the output voltage amplitude of the xth VSG; $K_{qx}$ and $K_{sx}$ denote the reactive droop coefficient and integral coefficient of the xth VSG, respectively; s is the Laplace operator; $U_{nx}$ and $U_x$ are the reference and actual output voltages of the xth VSG, respectively. $Q_{\mathrm{ref}x}$ and $Q_x$ denote the reactive power reference value and actual value of the xth VSG, respectively.

Since each VSG operates convergently on the AC bus and considering the approximate relationship $E_{ux}\approx U_x$, the output voltage amplitudes of all VSGs are approximately equal. Thus, we can obtain the following:

$$\left\{\begin{array}{c}{U}_{n1}=U_{n2}=\cdots =U_{nx}=U_{ns}\hfill \\ U_1=U_2=\cdots =U_x=U_s\hfill \\ E_{u1}=E_{u2}=\cdots =E_{ux}=E_{us}\hfill \end{array}\right.$$

(33)

By combining Equations (32) and (33), we obtain the following:

$$E_{us}={\displaystyle \frac{{\sum }_{x=1}^n\left[K_{qx}\left(U_{nx}-U_x\right)\right]+{\sum }_{x=1}^n{Q}_{\mathrm{ref}x}-{\sum }_{x=1}^n{Q}_x}{s{\sum }_{x=1}^n{K}_{sx}}}$$

(34)

Thus, the aggregated reactive power loop parameters of the VSG model can be expressed as follows:

$$\left\{\begin{array}{c}{K}_{qs}=\sum _{x=1}^n{K}_{qx},\hfill \\ K_s=\sum _{x=1}^n{K}_{sx}\hfill \\ Q_{\mathrm{refs}}=\sum _{x=1}^n{Q}_{\mathrm{ref}x}\hfill \\ Q_s=\sum _{x=1}^n{Q}_x\hfill \end{array}\right.$$

(35)

In the equation, $K_{qs}$ and $K_{ss}$ represent the reactive droop coefficient and integral coefficient of the aggregated VSG, respectively; $Q_{\mathrm{refs}}$ and $Q_s$ denote the reactive power reference value and the actual reactive power output of the aggregated VSG, respectively.

4.1.3. Aggregation of Voltage–Current Dual-Loop Control Parameters

By superimposing the open-loop transfer functions of the current loops for each VSG, the open-loop transfer function expression for the equivalent single VSG current loop can be obtained:

$$G_{kis}\left(s\right)=\sum _{x=1}^n{\displaystyle \frac{p_x{K}_{ipx}}{L_{fx}s}}$$

(36)

By superimposing the transfer functions of the current loops for each VSG, the transfer function expression for the equivalent single VSG current loop can be obtained as follows:

$$G_{bis}\left(s\right)={\displaystyle \frac{\frac{K_{ips}}{L_{fs}s}}{1+\frac{K_{ips}}{L_{fs}s}}}={\displaystyle \frac{{\sum }_{x=1}^n\left[\frac{p_x{K}_{ipx}}{s{L}_{fx}}\right]}{1+{\sum }_{x=1}^n\left[\frac{p_x{K}_{ipx}}{s{L}_{fx}}\right]}}$$

(37)

In the equation, $L_{fs}$ represents the AC filter inductance of the aggregated VSG, and $K_{ips}$ denotes the proportional coefficient of the current loop for the aggregated VSG.

The expressions for the current loop control parameters between individual VSGs in parallel and the aggregated VSG model are given as follows:

$$K_{ips}=L_{fs}\sum _{x=1}^n{\displaystyle \frac{K_{ipx}{p}_x}{L_{fx}}}$$

(38)

Similarly, by superimposing the open-loop transfer functions of the voltage loops for each parallel VSG, the open-loop transfer function expression for the aggregated VSG voltage loop can be obtained as follows:

$$G_{kus}\left(s\right)=\sum _{x=1}^n\left(K_{vpx}+{\displaystyle \frac{K_{vix}}{s}}\right){\displaystyle \frac{G_{bis}\left(s\right)}{s{C}_{fs}}}$$

(39)

In the equation, $C_{fs}$ represents the AC filter capacitance of the aggregated VSG.

By superimposing the transfer functions of the voltage loops for each parallel VSG, the transfer function expression for the aggregated VSG voltage loop can be obtained as follows:

$$G_{bus}\left(s\right)={\displaystyle \frac{\left(K_{vps}+\frac{K_{vis}}{s}\right)G_{bis}\left(s\right)}{C_{fs}s+\left(K_{vps}+\frac{K_{vis}}{s}\right)G_{bis}\left(s\right)}}={\displaystyle \frac{{\sum }_{x=1}^n\left(K_{vpx}+\frac{K_{vix}}{s}\right)G_{bis}\left(s\right)}{C_{fs}s+{\sum }_{x=1}^n\left(K_{vpx}+\frac{K_{vix}}{s}\right)G_{bis}\left(s\right)}}$$

(40)

In the equation, $K_{vps}$ and $K_{vis}$ represent the proportional and integral coefficients of the voltage loop for the aggregated VSG, respectively.

The voltage loop control parameters for each VSG in parallel and the aggregated VSG model are expressed as follows:

$$K_{vps}=\sum _{x=1}^n{K}_{vpx},K_{vis}=\sum _{x=1}^n{K}_{vix}.$$

(41)

4.2. Aggregation of Main Circuit Parameters

The expressions for the circuit parameters between multiple parallel VSGs and the aggregated VSG model are given as follows:

$${\displaystyle \frac{1}{L_{fs}}}=\sum _{x=1}^n\left({\displaystyle \frac{1}{L_{fx}}}\right),C_{fs}=\sum _{x=1}^n{C}_{fx}.$$

(42)

5. Simulation Verification

To verify the correctness of the improved homological equivalence method proposed in this paper, a simulation model of four VSG-based renewable energy systems with different parameters was built in Matlab/Simulink. These systems were connected in parallel to the same AC bus A. For the purpose of the study, bus A was defined as the common connection point between the VSGs and the power grid. Additionally, to emphasize the research focus, the parallel VSG region was aggregated, as shown in Figure 3.

It is assumed that under steady-state operating conditions, the voltages and frequencies of the individual VSGs are synchronized, enabling them to work collaboratively on the common bus of the same power grid without significant phase deviations or frequency mismatches. Furthermore, it is assumed that the control algorithms between the VSGs interact stably without causing oscillations or instability.

5.1. Original System Parameters

The basic parameters of the system are set as shown in

Table 1. Parameters. The table shows the main parameters of the system.

Parameter (Unit)	Value
Nominal Voltage U (V)	380
Equivalent Inductance L (mH)	0.0032
Equivalent Resistance R ($\mathrm{\Omega }$)	0.1

and

Table 2. VSG system parameters with units.

Parameter (Unit)	VSG1	VSG2	VSG3	VSG4
$U_{dc}$ (V)	600	600	600	600
$P_{\mathrm{ref}}$ (kW)	6	5	5	5
$Q_{\mathrm{ref}}$ (kvar)	0	0	0	0
$R_s$ ($\mathrm{\Omega }$)	0.7	0.75	0.8	0.82
$X_s$ (mH)	18	19	20	30
$R_g$ ($\mathrm{\Omega }$)	0.05	0.04	0.04	0.01
$X_g$ (mH)	4.4	4.5	4.6	4.4
C (μF)	20	20	20	30
J	0.1	0.12	0.18	0.17
$D_p$	4	4.2	2.6	2.5
$K_q$	950	960	1009	1000
$K_s$	155	157	160	165

This table lists the VSG system parameters along with their units.

. The system eigenvalues were analyzed, and the correction coefficients were determined, as shown in

Table 3. Analysis of system eigenvalues.

VSG	${\mathit{\delta }}_0$ (°)	${\mathit{U}}_0$ (V)	$\mathbf{Re}\mathbf{\left(}{\mathit{\lambda }}_{\mathit{i}}\mathbf{\right)}$
VSG1	0.038	379.721	−0.064
VSG2	0.033	379.798	−0.056
VSG3	0.033	379.788	−0.023
VSG4	0.032	379.806	−0.023

This table summarizes the eigenvalues and related parameters for each VSG.

.

Based on Equations (21)–(24), the corrected threshold results are shown in

Table 4. Correction coefficient values.

Parameter	Value
${\sigma }_c$	−0.023
${\sigma }_t$	−0.042
$\rho$	0.54

This table presents the correction coefficient values used in the analysis.

.

5.2. Corrected Homological Determination and Parameter Aggregation

In the simulation, all four VSG units reached a stable state at 3 s. At 3 s, a grid-side voltage drop fault was applied, with a grounding resistance of 0.1 m$\mathrm{\Omega }$. The fault was cleared after 0.05 s. The power angle curves of VSG₁ to VSG₄ obtained from the simulation are shown in Figure 4.

From Figure 5, the relative virtual power angles of each VSG unit can be obtained, as shown in Table 4.

After considering the stability margin, the corrected threshold for homological determination is set to 0.54°. The parameters of the aggregated VSG_A and VSG_B are shown in

Table 5. Relative virtual power angles of each VSG unit (50% voltage sag).

VSG	VSG1	VSG2	VSG3	VSG4
VSG1	0	0.2192	0.82266	0.60499
VSG2	0.2192	0	1.0419	0.82418
VSG3	0.82266	1.0419	0	0.21767
VSG4	0.60499	0.82418	0.21767	0

This table lists the relative virtual power angles of each VSG unit during a 50% voltage sag scenario.

.

The parameters of aggregated VSGs are shown in

Table 6. Parameters of aggregated VSGs.

Parameter	VSGA	VSGB
$U_{dc}$ (V)	600	600
$P_{\mathrm{ref}}$ (kW)	6	5
$Q_{\mathrm{ref}}$ (kvar)	0	0
$R_f$ ($\mathrm{\Omega }$)	0.7	0.75
$L_f$ (mH)	18	19
$R_g$ ($\mathrm{\Omega }$)	0.05	0.04
$L_g$ (mH)	4.4	4.5
C (μF)	20	20
J	0.1	0.12
$D_p$	4	4.2
$K_q$	950	960
$K_s$	155	157

This table lists the parameters of the aggregated VSG_A and VSG_B.

.

5.3. Case Validation

According to the corrected homological determination results based on the threshold, VSG₁ and VSG₂ are aggregated into VSG_A, while VSG₃ and VSG₄ are aggregated into VSG_B, connected in parallel to the same bus, as shown in Figure 6. Once the system stabilizes at 3 s, different faults are introduced and cleared after 0.05 s. The changes in active and reactive power after aggregation are compared with those before aggregation.

5.3.1. Case 1: Single-Phase Ground Fault

As shown in Figure 7 and Figure 8, during the single-phase ground fault, the output active power experiences a drop. Meanwhile, the reactive power–voltage control loop comes into effect, outputting reactive power to the system to provide voltage support. A quantitative analysis of the errors before and after aggregation reveals that the transient average error of active power is 0.22%, and the transient average error of reactive power is 0.33%. The errors in active and reactive power before and after aggregation are minimal and fall within the acceptable range for power grid analysis.

5.3.2. Case 2: 50% Voltage Sag

For the voltage sag fault, a quantitative analysis of the errors before and after aggregation reveals that the transient average error of active power is 0.34%, and the transient average error of reactive power is 0.52%. The transient processes of active and reactive power before and after aggregation are very similar, falling within the acceptable range for power grid analysis, as shown in Figure 9 and Figure 10. The above analysis demonstrates the correctness of the proposed homological equivalence method based on stability threshold correction.

The analysis confirms that the proposed homological equivalence method effectively simplifies the model without compromising the accuracy of the system’s dynamic behavior. The minimal errors observed in both active and reactive power during the transient process indicate that the method retains the essential characteristics of the original system. Furthermore, the approach provides an efficient way to reduce computational complexity, making it wellsuited for large-scale grid simulations. Overall, the stability threshold correction ensures that the aggregation maintains the integrity of the system’s performance while optimizing simulation efficiency. This method is not only effective for voltage sag faults but can be adapted for various grid disturbances, further enhancing its applicability in practical grid analysis.

6. Conclusions

This paper investigates the equivalent aggregation method for VSGs. By drawing an analogy with the traditional synchronous homological equivalence method, a homological determination criterion for VSGs is designed. On this basis, the impact of stability margins on the threshold of the criterion is considered, broadening the aggregation standards under different operating conditions. This provides an aggregation approach for studying the hybrid homological aggregation of VSG groups and traditional units. A parameter aggregation method based on VSG control is proposed. The feasibility of the aggregation method is demonstrated through theoretical analysis. System simulation results show that for VSG systems with different stability margins, the improved homological determination method with optimized threshold selection is more applicable. The aggregated model can maintain a highly consistent dynamic response with the parallel model, with an equivalent error of less than 1%.

Although the method has proven effective in smaller systems, its scalability to larger power grids presents certain challenges. As the number of VSG units grows, the computational load of the aggregation method may increase, potentially affecting real-time performance. However, parallel computing and optimization techniques can help distribute this load, ensuring the accuracy of the results. Furthermore, large-scale grids with high renewable energy penetration are subject to dynamic disturbances, which require the aggregation method to be robust and adaptable. Incorporating adaptive thresholds and refining stability margin assessments can help address these challenges. By tackling these scalability issues, the proposed method can be extended to larger grids, enhancing the efficiency and stability of grids with high renewable energy integration.