Calculation Method and Characteristic Analysis of Short-Circuit Current for Grid-Forming VSGs Under Symmetrical Faults

Abstract

With the increasing penetration of renewable energy, grid-forming inverters that provide voltage and frequency support are receiving significant attention. However, due to the voltage source characteristics of grid-forming Virtual Synchronous Generators (VSGs), they are prone to excessive short-circuit currents during three-phase grid faults. Moreover, conventional short-circuit current analysis methods developed for grid-following inverters cannot be directly applied to VSG-based systems. Consequently, research on fault current calculation for grid-forming VSGs has become critically important. To address this issue, this paper employs mathematical analysis to derive a short-circuit current calculation method for VSG grid-connected systems under three-phase fault conditions. Based on the derived analytical expressions, the short-circuit current characteristics of the VSG system are systematically analyzed. The correctness of the proposed analytical expressions is validated through simulations conducted on the MATLAB/Simulink(R2024b) platform. Simulation results show that the error between the analytical and simulated values remains within 8%, and decreases to below 3% after 2–3 cycles, indicating that the proposed analytical model can effectively capture the dynamic behavior of the fault current.

1. Introduction

The global transition towards green and low carbon energy is accelerating globally. The large-scale integration of renewable generation, primarily wind and photovoltaic power, has led to the emergence of “dual-high” characteristics in power systems [1]. This shift from an electromechanical transient-dominated system to a power electronics-dominated system has resulted in insufficient inertia and damping support for the grid [2,3,4], thereby posing new challenges to power system security and stability. In response, Virtual Synchronous Generator (VSG) control technology, which emulates the external characteristics of synchronous machines, has been proposed. Its ability to provide virtual inertia and damping effectively addresses these challenges and enhances grid stability [5,6,7].

VSG control belongs to grid-forming control technology. Compared with conventional grid-following control based on phase locked loops (PLLs), grid-forming control differs fundamentally in its control philosophy. As a result, the fault current characteristics of VSGs differ significantly from those of grid-following inverters, posing potential challenges to the reliable operation of existing protection systems in renewable energy-dominated power grids [8]. Although VSGs emulate the external characteristics of synchronous generators, the overcurrent capability of grid-connected inverters is inherently limited. Therefore, certain current-limiting measures must be adopted during faults. Current-limiting strategies for grid-forming inverter-based sources can be categorized into direct current-limiting, virtual impedance current-limiting, and voltage reference-limiting [9]. During the low-voltage ride-through (LVRT) process, the short-circuit current characteristics of VSGs are significantly influenced by control strategies. This makes traditional protection coordination and parameter tuning principles inadequate for the operational requirements of new-type power systems [10]. The short-circuit current differs considerably from that of traditional grid-following control. Therefore, investigating the short-circuit current characteristics of VSGs and their influencing factors is of great practical significance for accurately evaluating their impact on grid protection schemes and for improving relay protection design.

Currently, research on the analytical calculation of short-circuit fault current for grid-connected inverters with traditional grid-following control in renewable energy systems is relatively well established. Existing studies can be categorized into two aspects based on whether the dynamic response and transient characteristics of the power system are considered: steady-state transient fault current calculation or steady-state fault current calculation; refs. [11,12,13] derived analytical expressions for steady-state short-circuit current under different fault types based on the controlled current source characteristic of grid-following inverters. For transient fault current calculation, refs. [14,15,16] derived three-phase short-circuit current formulas exhibiting a second-order response in the dq-axis frame based on circuit models and control equations, and analyzed their fundamental characteristics and influencing factors. However, during fault-equivalent modeling, grid-following inverters are typically represented as controlled current sources, which differs significantly from the structure of grid-forming control. Consequently, the guidance provided by the aforementioned literature for calculating the fault current of grid-forming inverter-based power sources is limited.

Regarding the fault current analysis of grid-forming inverters, existing approaches mainly include time domain numerical integration, hybrid methods combining numerical integration with physical modeling, and purely analytical methods. Numerical integration and time domain simulations have been employed to analyze fault current characteristics in [17,18]. Hybrid approaches combining numerical integration and physical models were adopted in [19,20,21,22,23] to calculate fault currents, offering improved efficiency over pure simulation but still requiring iterative computations. For purely analytical methods, several recent studies have attempted to derive closed-form expressions for VSG fault currents. In [24], an analytical expression for steady-state short-circuit current under symmetric fault conditions was briefly derived, and both steady-state current and transient inrush current were limited by adjusting the reactive power loop reference and virtual impedance, respectively. Although some analytical models incorporate the dynamics of control loops, they neglect key engineering factors such as virtual impedance [25,26], which limits the applicability of these models. In [27], singular perturbation theory was used to derive an analytical expression for fault current in grid-forming converters, but the influence of virtual impedance was not considered. References [28,29,30] incorporated virtual impedance into fault current analysis and derived corresponding analytical expressions; however, the specific equivalent modeling process of the virtual impedance was not explicitly presented. In [31], the dynamics of the reactive power–voltage outer control loop were neglected, despite their significant impact on the internal electromotive force and fault current in practical systems. Moreover, none of these studies considered the influence of variations in the virtual internal electromotive force on the analytical calculation of fault current. More recent analytical efforts have made significant progress. The studies in [32,33,34,35] established multiple relationships between internal electromotive force and fault current based on voltage, current, and power loop dynamics, and then employed variable elimination and Laplace transform techniques to derive fault current expressions. However, the influence of proportional–integral terms in the dual-loop voltage control on the analytical short-circuit current expression was not addressed.

Despite these valuable contributions, several critical gaps remain in the analytical calculation of VSG fault currents:

• Lack of methods tailored for grid-forming VSG architecture: Conventional short-circuit current analytical methods developed for grid-following inverters cannot be directly applied to VSG-based systems due to fundamental differences in control philosophy. VSGs emulate synchronous generator behavior through virtual inertia and damping, involving complex coordination among power control loops, voltage–current dual loops, and virtual inertia regulation. This gap makes research on fault current calculation for grid-forming renewable energy/storage systems particularly important and urgent.
• Incomplete consideration of virtual impedance: In existing analytical studies on grid-forming short-circuit current calculation, very few papers have fully incorporated the virtual impedance loop in their derivations. Virtual impedance, which emulates the synchronous reactance of synchronous generators, significantly influences both the operational stability and fault current characteristics of VSGs, yet its effects have not been systematically quantified in analytical expressions.
• Neglect of PI terms in dual-loop control: Previous studies have not fully addressed the influence of proportional–integral terms in the dual-loop voltage control when deriving fault current expressions, limiting the accuracy and comprehensiveness of existing analytical models.

To address the limitations, this paper proposes an analytical method for calculating short-circuit currents in VSG-based grid-connected systems under three-phase fault conditions, with the following key innovations:

• Analytical method specifically tailored for grid-forming VSG architecture: We develop a short-circuit current calculation approach that accounts for the unique characteristics of VSG control.
• Incorporation of virtual impedance: Our derivation explicitly incorporates virtual resistance R_v and virtual inductance L_v in the differential equation model, with detailed presentation of the equivalent modeling process.
• Complete modeling of PI terms in dual-loop control: We fully account for the proportional–integral parameters (K_p, K_i) of both voltage and current loops in the analytical derivation, providing a more comprehensive model.

The proposed method is developed as follows: First, the system topology of the VSG-controlled converter is introduced. Based on the control principles, a differential equation model with the inductor current as the state variable is established and solved in the dq reference frame, yielding a fault current expression applicable to electromagnetic transient timescales and accounting for variations in the virtual internal electromotive force. Subsequently, the steady-state and transient components of the short-circuit current expression are examined. By interpreting their mathematical forms, the corresponding physical mechanisms are revealed, thereby clarifying the roles of different control loops during fault conditions. Finally, the parametric composition of each component in the short-circuit current expression is analyzed. All analytical derivations in this paper are performed using the computational software Wolfram Mathematica 14.

2. The Fundamental Principle of VSG-Controlled Grid-Forming Inverters

Figure 1 illustrates the typical configuration and control block diagram of a VSG-based grid-forming inverter, which is structurally composed of the power stage and the control system. As shown, the DC side of the inverter is connected via a DC bus to a renewable or storage source. Given the inertia of the DC-link capacitor and the ability of the source to maintain a stable DC bus voltage, the inverter can be considered effectively decoupled from the DC source for analysis purposes. Thus, the DC bus and all components on its left can be treated as an ideal voltage source. The control system is primarily divided into three parts: the VSG loop, the voltage and current double loop controller, and the three-phase sinusoidal pulse width modulation (SPWM) stage. The VSG loop emulates the behavior of a synchronous generator to produce a VEMP signal. This signal is transformed via Park transformation into a voltage reference in the dq-frame, which is fed to the dual-loop controller. Using feedback from the output voltage and current, the double loop controller generates a stabilized closed loop signal. This signal then undergoes an inverse Park transformation before being sent to the SPWM stage, where it is converted into the gate driving pulses that control the power switches. A corresponding control block diagram of the grid-forming converter is provided in Figure 2.

The active power control loop of the VSG emulates the rotor swing equation of a synchronous generator (SG), given by the following:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}\theta }{\mathrm{d}\mathrm{t}}}=\Delta \omega =\omega -{\omega }_{\mathrm{n}}\\ \mathrm{J}{\omega }_{\mathrm{n}}{\displaystyle \frac{\mathrm{d}\omega }{\mathrm{d}\mathrm{t}}}={\mathrm{P}}_{\mathrm{ref}}-{\mathrm{P}}_{\mathrm{e}}-\mathrm{D}{\omega }_{\mathrm{n}}(\omega -{\omega }_{\mathrm{n}})\end{array}\right.$$

(1)

where θ is the power angle between the grid voltage U_g and the internal voltage of the grid-forming inverter U_g; ω_n is the rated angular frequency; J is the virtual inertia; P_ref is the active power reference value; P_e is the actual output active power of the grid-forming inverter; and D is the damping coefficient.

The reactive power–voltage loop of the grid-forming VSG inverter emulates the reactive power–voltage droop characteristic of a synchronous generator. It outputs the amplitude of a VEMP to maintain system voltage stability. The specific control equation can be expressed as follows:

$${\displaystyle \frac{d(E_{\mathrm{ref}}-E_{\mathrm{n}})}{dt}}=k_{\mathrm{q}}(Q_{\mathrm{ref}}-Q_{\mathrm{e}})$$

(2)

where k_q is the reactive power integral coefficient; E_ref is the VSG-amplitude reference; E_n is the VSG no-load VEMP amplitude; Q_ref is the VSG reactive power reference value; and Q_e is the VSG actual output reactive power.

Regarding the virtual impedance loop, it emulates the synchronous reactance of an SG. This loop enhances the operational stability of the VSG control. The specific control equation is expressed as follows:

$$\left\{\begin{array}{l}{E}_{\mathrm{ref}\text{\_}d}-(R_{\mathrm{v}}{i}_{\mathrm{d}}-{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{q}})=U_{\mathrm{ref}\text{\_}\mathrm{d}}\\ E_{\mathrm{ref}\text{\_}\mathrm{q}}-(R_{\mathrm{v}}{i}_{\mathrm{q}}+{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{d}})=U_{\mathrm{ref}\text{\_}\mathrm{q}}\end{array}\right.$$

(3)

where E_{ref_d}, E_{ref_q} are dq-axis components of the VEMP; U_{ref_d}, U_{ref_q} are dq-axis components of the point of common coupling (PCC) voltage; R_v is virtual resistance; L_v is virtual inductance; and i_d, i_q are dq-axis components of the current.

For the voltage and current control loops, a composite control strategy is adopted. This strategy combines feedforward decoupling of voltage and current in the dq-frame with double loop correction using PI regulators. This enables the actual voltage output at the PCC to track the reference signal in real time through the inverter system. The specific control equation is expressed as follows:

$$\left\{\begin{array}{cc}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}& = {\mathrm{K}}_{\mathrm{p}}\left[(E_{\mathrm{ref}\text{\_}\mathrm{d}}-u_{\mathrm{d}})-R_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}+{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}\right]\\ & +K_{\mathrm{i}}{\displaystyle \int (E_{\mathrm{ref}\text{\_}\mathrm{d}}-u_{\mathrm{d}})-R_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}+{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}} dt}\\ i_{\mathrm{ref}\text{\_}\mathrm{q}}& = {\mathrm{K}}_{\mathrm{p}}\left[(E_{\mathrm{ref}\text{\_}\mathrm{q}}-u_{\mathrm{q}})-R_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}-{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}\right]\\ & +K_{\mathrm{i}}{\displaystyle \int (E_{\mathrm{ref}\text{\_}\mathrm{d}}-u_{\mathrm{q}})-R_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}-{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}} dt}\end{array}\right.$$

(4)

where K_p is the proportional coefficient of the voltage loop; K_i is the integral coefficient of the voltage loop; and i_{ref_d}, i_{ref_q} is the dq-axis current reference value.

3. Analytical Calculation of VSG Fault Current

3.1. Solving Idea of VSG Short-Circuit Current

Considering that the transient characteristics of both VSG and SG can be divided into an electromagnetic transient process on a short timescale and an electromechanical transient process on a relatively longer timescale, to obtain a referential and applicable analytical expression for the fault current, this paper focuses solely on the analytical calculation of the VSG fault current within the electromagnetic transient timescale. This is because incorporating the influence of the electromechanical transient in fault analysis would necessitate the introduction of the rotor motion equation, leading to a more complex analytical model—at least a fourth-order nonlinear differential equation. Such high-order nonlinear equations make it difficult to derive an explicit analytical expression for the fault current, thereby rendering the analytical calculation of short-circuit current impractical for application.

Based on the above considerations, the following assumptions are made in this chapter when deriving the analytical expression for the short-circuit current:

Assumption 1.

The rotor voltage equation is not considered during the transient process, and the VSG angular frequency ω is assumed to be constant.

Assumption 2.

The transient process of voltage sag at the PCC is ignored. The voltage sag at the PCC is assumed to occur instantaneously and remain constant.

Assumption 3.

The transient change in the voltage power angle is considered negligible, and the voltage power angle is assumed to remain unchanged before and after the transient.

Assumption 4.

Given the fast response speed of the current loop, the impact of its response delay on the short-circuit current is neglected. The current reference value is assumed to be equal to the output value of the actual circuit.

Based on the above assumptions, the rationale for each is as follows, justified by the separation of time scales between electromagnetic and electromechanical transients:

Assumption 1 (constant ω). 

The VSG angular frequency is governed by the virtual inertia and damping, which have time constants on the order of hundreds of milliseconds to seconds. During the initial electromagnetic transient, the variation in ω is negligible. Treating it as constant enables a linear, solvable mode without significant loss of accuracy.

Assumption 2 (Instantaneous and constant voltage sag). 

A three-phase fault causes an almost instantaneous voltage drop at the PCC; the transient of the sag itself occurs on a microsecond scale, far faster than the electromagnetic transient of interest. Moreover, during the fault and before any control actions or protection intervention, the voltage magnitude remains approximately constant. This assumption simplifies the system input to a known constant, facilitating analytical solution.

Assumption 3 (Negligible power angle variation). 

The power angle δ is determined by the active power loop dynamics, which evolve on the electromechanical timescale. Within the first few cycles after the fault, δ changes very little, and its influence on the fault current is limited mainly to the initial phase.

Assumption 4 (Ignored current loop delay). 

The current loop typically has a high bandwidth, corresponding to response times of a few milliseconds. Compared to the tens-of-milliseconds duration of the electromagnetic transient, this delay is relatively small. Neglecting it allows us to approximate the actual current by its reference, which is a common simplification in analytical studies of power electronic systems when focusing on external dynamics.

Under the above assumptions, the analytical expression for the VSG short-circuit current proposed in this chapter can be analyzed and solved step by step according to the following procedure:

Step 1: Establish and solve the equation relating the current reference command to the VEMP:

In the dq coordinate system, first simultaneously combine the d-axis voltage loop control equation and the q-axis voltage loop control equation. Then, determine the initial values of the d-axis and q-axis currents and the excitation relationship between the VEMP and the terminal voltage. Next, solve this system of differential equations with initial value conditions to establish the functional relationship between the fault current reference command and the amplitude of the VEMP. Based on Assumption 4, the complete relationship between the actual short-circuit current output value and the amplitude of the VEMP generated by VSG is obtained, namely:

$$i_{\mathrm{dq}}(t)\approx i_{\mathrm{ref}\text{\_}\mathrm{dq}}(t)=f_1(E_{\mathrm{ref}}(t),{\displaystyle \frac{d}{dt}}{E}_{\mathrm{ref}}(t),{\displaystyle \frac{d^2}{d{t}^2}}{E}_{\mathrm{ref}}(t) , \dots , {\displaystyle \frac{d^{\mathrm{n}}}{d{t}^{\mathrm{n}}}}{E}_{\mathrm{ref}}(t))$$

(5)

By ignoring the derivative terms in the above expression, an approximate expression for the relationship between the actual short-circuit current output value and the amplitude of the VEMP generated by the virtual synchronous machine is obtained, namely:

$$i_{\mathrm{dq}}(t)\approx i_{\mathrm{ref}\text{\_}\mathrm{dq}}(t)\approx f_1(E_{\mathrm{ref}}(t))$$

(6)

It is important to note that E_ref(t) at this point is a function of time, not a constant, and needs to be solved in subsequent steps.

Step 2: Combine and solve the equation for the VEMP amplitude:

First, based on the instantaneous reactive power theory, obtain the expression for the VEMP in terms of reactive power $Q_{\mathrm{e}}(t)=f_2(E_{\mathrm{ref}}(t))$. Then, write the reactive power–voltage control equation of the VSG $g(Q_{\mathrm{e}}(t),Q_{\mathrm{ref}}(t),{\displaystyle \frac{d}{dt}}{E}_{\mathrm{ref}}(t))=0$. Subsequently, substitute the VEMP reactive power expression into the reactive power–voltage control equation to form the combined equation $g(f_2(E_{\mathrm{ref}}(t)),Q_{\mathrm{ref}}(t),{\displaystyle \frac{d}{dt}}{E}_{\mathrm{ref}}(t))=0$, and determine the initial condition $g(f_2(E_{\mathrm{ref}}(0)),Q_{\mathrm{ref}}(0),{\displaystyle \frac{d}{dt}}{E}_{\mathrm{ref}}(0))=0$ for this equation. Finally, solve this initial value problem to obtain the specific expression for the VEMP amplitude E_ref(t).

Step 3: Calculate the analytical expression for the abc three-phase fault current:

First, substitute E_ref(t) back into the functional relationship (1) obtained in Step 1 to get the analytical expression for the short-circuit current i_dq(t). Then, use the inverse Park transform to convert the short-circuit current from the dq coordinate system to the abc coordinates, thus obtaining the three-phase fault current expression i_abc(t).

The solution procedure for the analytical expression of the VSG short-circuit current is illustrated in Figure 3.

3.2. Establishment of the Relationship Equation Between Current and Internal VEMP

First, based on (4), differentiate both sides and neglect the derivative term of the VEMP. After rearranging, the equation yields the following:

$$\left\{\begin{array}{l}{E}_{\mathrm{ref}\text{\_}\mathrm{d}}(t)-u_{\mathrm{d}}(t)={\displaystyle \frac{\left(K_{\mathrm{P}}{R}_{\mathrm{v}}+1\right)}{K_{\mathrm{i}}}}{\displaystyle \frac{d}{dt}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}(t)-{\displaystyle \frac{K_{\mathrm{P}}{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}}{K_{\mathrm{i}}}}{\displaystyle \frac{d}{dt}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}(t)+R_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}(t)-{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}(t)\\ E_{\mathrm{ref}\text{\_}\mathrm{q}}(t)-u_{\mathrm{q}}(t)={\displaystyle \frac{\left(K_{\mathrm{P}}{R}_{\mathrm{v}}+1\right)}{K_{\mathrm{i}}}}{\displaystyle \frac{d}{dt}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}(t)+{\displaystyle \frac{K_{\mathrm{P}}{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}}{K_{\mathrm{i}}}}{\displaystyle \frac{d}{dt}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}(t)+R_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}(t)+{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}(t)\end{array}\right.$$

(7)

Next, determine the initial value and the value of the excitation term for (7). Considering that all current reference commands are continuous quantities before and after the fault, the initial value of the current reference command can be obtained by calculating the current values at the instant immediately before the fault occurs.

At the instant before the fault occurs, the converter is considered to be operating under steady-state conditions. Denote the d-axis and q-axis current commands at this time as i_{ref_d} (0_) and i_{ref_q} (0_), respectively. Under these conditions, the differential terms in (7) can be considered zero, resulting in the following:

$$\left\{\begin{array}{l}{E}_{\mathrm{ref}\text{\_}\mathrm{d}}(0_{-})-u_{\mathrm{d}}(0_{-})=R_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}(0_{-})-{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}(0_{-})\\ E_{\mathrm{ref}\text{\_}\mathrm{q}}(0_{-})-u_{\mathrm{q}}(0_{-})=R_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{q}}(0_{-})+{\omega }_N{L}_{\mathrm{v}}{i}_{\mathrm{ref}\text{\_}\mathrm{d}}(0_{-})\end{array}\right.$$

(8)

The phasor diagrams of the VEMP and output voltage before and after the fault are shown in Figure 4 below.

Before the short-circuit fault occurs, the PCC voltage vector is U_{ref0_}, with a phase angle of α_{0_}. The VEMP vector is E_{ref0_}, which leads the PCC voltage vector by an angle of δ_{0_}. When the short-circuit fault occurs, the PCC voltage abruptly changes to U, and its phase angle jumps from α_{0_} to α. Due to the low-voltage ride-through (LVRT) control, the VEMP changes to E. According to Assumption 1, its phase angle is assumed not to change abruptly due to the virtual inertia, so the resulting phase angle by which it leads the PCC voltage becomes δ.

Substituting the data shown in the figure above into (8) for solving yields the following:

$$\left\{\begin{array}{l}{i}_{\mathrm{ref}\text{\_}d}(0_{-})= {\displaystyle \frac{1}{Z_{\mathrm{v}}}}[E_{\mathrm{ref0\_}}\cos {\phi }_{\mathrm{v}}-U_{\mathrm{ref0\_}}\cos ({\phi }_{\mathrm{v}}+{\delta }_{0\text{\_}})]\\ i_{\mathrm{ref\_q}}(0_{-})=-{\displaystyle \frac{1}{Z_{\mathrm{v}}}}[E_{\mathrm{ref0\_}}\sin {\phi }_{\mathrm{v}}-U_{\mathrm{ref0\_}}\sin ({\phi }_{\mathrm{v}}+{\delta }_{0-})]\end{array}\right.$$

(9)

where $Z_{\mathrm{v}}=\sqrt{R_{\mathrm{v}}^2+{\omega }_{\mathrm{N}}^2{L}_{\mathrm{v}}^2}$, ${\phi }_{\mathrm{v}}=\arctan {\displaystyle \frac{{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}}{R_{\mathrm{v}}}}$;

Defining the time of fault occurrence as t = 0, the excitation term on the left-hand side of (7) at this moment is as follows:

$$\left\{\begin{array}{l}{E}_{\mathrm{ref}\text{\_}\mathrm{d}}(t)-u_{\mathrm{d}}(t)=E_{\mathrm{ref}}(t)-U\cos \delta \\ E_{\mathrm{ref}\text{\_}\mathrm{q}}(t)-u_{\mathrm{q}}(t)=0-(-U\sin \delta )\end{array}\right.$$

(10)

By solving (7) and (10) simultaneously, neglecting all derivative terms of E_ref(t), and further simplifying based on Assumption 4, the final solution is obtained as follows:

$$\left\{\begin{array}{ll}{i}_{\mathrm{d}}(t)& ={\displaystyle \frac{1}{Z_{\mathrm{v}}}}{e}^{-{\displaystyle \frac{t}{\tau }}}[U\cos \left({\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}+\delta \right)-U_{0\_}\cos \left({\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}+{\delta }_{0\_}\right)]\\ & -{\displaystyle \frac{1}{Z_{\mathrm{v}}}}{e}^{-{\displaystyle \frac{t}{\tau }}}\left(E_{\mathrm{ref}}(t)-E_{0\_}\right)\cos \left({\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}\right)\\ & -{\displaystyle \frac{1}{Z_{\mathrm{v}}}}[U\cos \left({\phi }_{\mathrm{v}}+\delta \right)-E_{\mathrm{ref}}(t)\cos \left({\phi }_{\mathrm{v}}\right)]\\ i_q(t)& ={\displaystyle \frac{1}{Z_{\mathrm{v}}}}{e}^{-{\displaystyle \frac{t}{\tau }}}[-U\sin \left({\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}+\delta \right)+U_{0\_}\sin \left({\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}+{\delta }_{0\_}\right)]\\ & +{\displaystyle \frac{1}{Z_{\mathrm{v}}}}{e}^{-{\displaystyle \frac{t}{\tau }}}\left(E_{\mathrm{ref}}(t)-E_{0\_}\right)\sin \left({\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}\right)\\ & +{\displaystyle \frac{1}{Z_{\mathrm{v}}}}[U\sin \left({\phi }_{\mathrm{v}}+\delta \right)-E_{\mathrm{ref}}(t)\sin \left({\phi }_{\mathrm{v}}\right)]\end{array}\right.$$

(11)

where

$$\left\{\begin{array}{l}\tau ={\displaystyle \frac{K_{\mathrm{P}}^2{Z}_{\mathrm{v}}^2+2K_{\mathrm{P}}{R}_{\mathrm{v}}+1}{K_{\mathrm{i}}\left(K_{\mathrm{P}}{Z}_{\mathrm{v}}^2+R_{\mathrm{v}}\right)}}\\ {\omega }_{\mathrm{N}1}={\displaystyle \frac{{\omega }_{\mathrm{N}}{L}_{\mathrm{v}}{K}_{\mathrm{i}}}{K_{\mathrm{P}}^2{Z}_{\mathrm{v}}^2+2K_{\mathrm{P}}{R}_{\mathrm{v}}+1}}\end{array}\right.$$

It can be seen that the dq-axis short-circuit current in (11) is divided into a steady-state component and a transient component. For simplicity of calculation, (11) is approximated here:

①

Neglect the decay of the excitation term in the transient component.

②

Uniformly use the power angle during the fault occurrence stage, taken as δ = δ₀.

After applying the above approximations, (11) can be simplified to the following:

$$\left\{\begin{array}{l}{i}_{\mathrm{d}}(t)={\displaystyle \frac{1}{Z_{\mathrm{v}}}}{e}^{-{\displaystyle \frac{t}{\tau }}}(U-U_{0\_})\cos \left(\delta +{\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}\right)-{\displaystyle \frac{1}{Z_{\mathrm{v}}}}[U\cos \left(\delta +{\phi }_{\mathrm{v}}\right)-E_{\mathrm{ref}}(t)\cos \left({\phi }_{\mathrm{v}}\right)]\\ i_{\mathrm{q}}(t)={\displaystyle \frac{1}{Z_{\mathrm{v}}}}{e}^{-{\displaystyle \frac{t}{\tau }}}(U_{0\_}-U)\sin \left(\delta +{\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}\right)+{\displaystyle \frac{1}{Z_{\mathrm{v}}}}[U\sin \left(\delta +{\phi }_{\mathrm{v}}\right)-E_{\mathrm{ref}}(t)\sin \left({\phi }_{\mathrm{v}}\right)]\end{array}\right.$$

(12)

3.3. The Establishment and Calculation of the VEMP Magnitude

According to Step 2, use the instantaneous power theory to solve for the reactive power, at which point we have the following:

$$Q_{\mathrm{e}}(t)={\displaystyle \frac{3}{2}}(u_{\mathrm{q}}(t)i_{\mathrm{d}}(t)-u_{\mathrm{d}}(t)i_{\mathrm{q}}(t))$$

(13)

In order for the system to be able to inject sufficient reactive power to the external grid to support the PCC point voltage when a fault occurs, the reactive power reference value is typically set to the following:

$$Q_{\mathrm{ref}}=(U_{\mathrm{pcc}}^{\ast }-{(U_{\mathrm{pcc}}^{\ast })}^2)S_{\mathrm{N}}$$

(14)

where $U_{pcc}^{\ast }$ is the per-unit value of the PCC voltage; and S_N is the system rated power.

Solving (2):

$$Q_{\mathrm{e}}(t)={\displaystyle \frac{3U}{2Z_{\mathrm{v}}}}{e}^{-{\displaystyle \frac{t}{\tau }}}\left(U-U_0\right)\sin \left({\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}\right)+{\displaystyle \frac{3U}{2Z_{\mathrm{v}}}}[E_{\mathrm{ref}}(t)\sin \left({\phi }_{\mathrm{v}}-\delta \right)-U\sin {\phi }_{\mathrm{v}}]$$

(15)

At this point, by substituting (15) and (14) into (2):

$$\left\{\begin{array}{l}{\displaystyle \frac{d}{dt}}{E}_{\mathrm{ref}}(t)+d_1{E}_{\mathrm{ref}}(t)=d_3+d_2\sin ({\omega }_{\mathrm{N}1}t+{\phi }_{\mathrm{v}}{)\mathrm{e}}^{-{\displaystyle \frac{t}{\tau }}}\\ d_1={\displaystyle \frac{3U{k}_{\mathrm{q}}}{2Z_{\mathrm{v}}}}\sin ({\phi }_{\mathrm{v}}-\delta )\\ d_2={\displaystyle \frac{3U{k}_{\mathrm{q}}}{2Z_{\mathrm{v}}}}(U_0-U)\\ d_3=k_{\mathrm{q}}\left(U_{\ast }-U_{\ast }^2\right)S_{\mathrm{N}}+{\displaystyle \frac{3U{k}_{\mathrm{q}}}{2Z_{\mathrm{v}}}}U\sin {\phi }_{\mathrm{v}}\end{array}\right.$$

(16)

To simplify the calculation, the method proposed in [36] is adopted to reduce (16) into its envelope form, yielding the following:

$${\displaystyle \frac{d}{dt}}{E}_{\mathrm{ref}}(t)+d_1{E}_{\mathrm{ref}}(t)\approx d_3+d_2{\mathrm{e}}^{-{\displaystyle \frac{t}{\tau }}}$$

(17)

At this point, its initial conditions are $E_{\mathrm{ref}}(0)=E_{\mathrm{n}}$.

Solving (17) yields the following:

$$E_{\mathrm{ref}}(t)={\displaystyle \frac{d_3}{d_1}}+E_{\mathrm{n}}+\left(-{\displaystyle \frac{d_2}{d_1-\frac{1}{\tau }}}-{\displaystyle \frac{d_3}{d_1}}\right){\mathrm{e}}^{-d_{1}t}+\left({\displaystyle \frac{d_2}{d_1-\frac{1}{\tau }}}\right){\mathrm{e}}^{-{\displaystyle \frac{t}{\tau }}}$$

(18)

3.4. Calculation of the Analytical Expression for Fault Current

Performing an inverse Park transformation on (11) converts the short-circuit current from the dq-axis to the abc coordinate system, yielding the following:

$$\begin{array}{l}{i}_{\mathrm{a}}(t)={\displaystyle \frac{1}{Z_{\mathrm{v}}}}[E_{\mathrm{ref}}(t)\cos ({\omega }_{\mathrm{N}}t-{\phi }_{\mathrm{v}})-U\cos ({\omega }_{\mathrm{N}}t-\delta -{\phi }_{\mathrm{v}})]\\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}\left(U-U_0\right)\cos ({\omega }_{\mathrm{N}}t-{\omega }_{\mathrm{N}1}t-\delta -{\phi }_{\mathrm{v}})\\ i_{\mathrm{b}}(t)={\displaystyle \frac{1}{Z_{\mathrm{v}}}}[-E_{\mathrm{ref}}(t)\sin ({\displaystyle \frac{\pi }{6}}-{\omega }_{\mathrm{N}}t+{\phi }_{\mathrm{v}})+U\sin ({\displaystyle \frac{\pi }{6}}-{\omega }_{\mathrm{N}}t+\delta +{\phi }_{\mathrm{v}})]\\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}\left(U-U_0\right)\sin (-{\displaystyle \frac{\pi }{6}}+{\omega }_{\mathrm{N}}t-{\omega }_{\mathrm{N}1}t-\delta -{\phi }_{\mathrm{v}})\\ i_{\mathrm{c}}(t)={\displaystyle \frac{1}{Z_{\mathrm{v}}}}[-E_{\mathrm{ref}}(t)\sin ({\displaystyle \frac{\pi }{6}}+{\omega }_{\mathrm{N}}t-{\phi }_{\mathrm{v}})+U\sin ({\displaystyle \frac{\pi }{6}}+{\omega }_{\mathrm{N}}t-\delta -{\phi }_{\mathrm{v}})]\\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}\left(U-U_0\right)\sin (-{\displaystyle \frac{\pi }{6}}-{\omega }_{\mathrm{N}}t+{\omega }_{\mathrm{N}1}t+\delta +{\phi }_{\mathrm{v}})\end{array}$$

(19)

Finally, substituting (18) into (19) gives the complete expression for the three-phase short-circuit current:

$$\begin{array}{ll}{i}_{\mathrm{a}}(t)& ={\displaystyle \frac{H_{\mathrm{s}1}}{Z_{\mathrm{v}}}}\cos ({\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t)+{\displaystyle \frac{H_{\mathrm{s}2}}{Z_{\mathrm{v}}}}\cos (\delta +{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t)\\ & +H_{\mathrm{t}1}\cos (\delta +{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t+{\omega }_{\mathrm{N}1}t){\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}+H_{\mathrm{t}2}\cos ({\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t){\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}\\ & -H_{\mathrm{t}2}\cos ({\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t){\displaystyle \frac{{\mathrm{e}}^{-d_{1}t}}{Z_{\mathrm{v}}}}\\ i_{\mathrm{b}}(t)& =-{\displaystyle \frac{H_{\mathrm{s}1}}{Z_{\mathrm{v}}}}\sin ({\displaystyle \frac{\pi }{6}}+{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t)-{\displaystyle \frac{H_{\mathrm{s}2}}{Z_{\mathrm{v}}}}\sin ({\displaystyle \frac{\pi }{6}}+\delta +{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t)\\ & \hspace{1em}-H_{\mathrm{t}1}\sin ({\displaystyle \frac{\pi }{6}}+\delta +{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t+{\omega }_{\mathrm{N}1}t){\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}-H_{\mathrm{t}2}\sin ({\displaystyle \frac{\pi }{6}}+{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t){\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}\\ & \hspace{1em}+H_{\mathrm{t}2}\sin ({\displaystyle \frac{\pi }{6}}+{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t){\displaystyle \frac{{\mathrm{e}}^{-d_{1}t}}{Z_{\mathrm{v}}}}\\ i_{\mathrm{c}}(t)& =-{\displaystyle \frac{H_{\mathrm{s}1}}{Z_{\mathrm{v}}}}\sin ({\displaystyle \frac{\pi }{6}}-{\phi }_{\mathrm{v}}+{\omega }_{\mathrm{N}}t)-{\displaystyle \frac{H_{\mathrm{s}2}}{Z_{\mathrm{v}}}}\sin ({\displaystyle \frac{\pi }{6}}-\delta -{\phi }_{\mathrm{v}}+{\omega }_{\mathrm{N}}t)\\ & -H_{\mathrm{t}1}\sin ({\displaystyle \frac{\pi }{6}}-{\delta }_1-{\phi }_{\mathrm{v}}+{\omega }_{\mathrm{N}}t-{\omega }_{\mathrm{N}1}t){\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}-H_{\mathrm{t}2}\sin ({\displaystyle \frac{\pi }{6}}-{\phi }_{\mathrm{v}}+{\omega }_{\mathrm{N}}t){\displaystyle \frac{{\mathrm{e}}^{-\frac{t}{\tau }}}{Z_{\mathrm{v}}}}\\ & +H_{\mathrm{t}2}\sin ({\displaystyle \frac{\pi }{6}}-{\phi }_{\mathrm{v}}+{\omega }_{\mathrm{N}}t){\displaystyle \frac{{\mathrm{e}}^{-d_{1}t}}{Z_{\mathrm{v}}}}\end{array}$$

(20)

where $H_{\mathrm{s}1}=E_{\mathrm{n}}+{\displaystyle \frac{d_3}{d_1}};H_{\mathrm{s}2}=-U;H_{\mathrm{t}1}=U-U_0;H_{\mathrm{t}2}={\displaystyle \frac{d_2}{\left(-\frac{1}{\tau }+d_1\right)}}$.

3.5. Analysis of Fault Short-Circuit Current Characteristics in VSG

To elucidate the composition of the fault current and its representation in the dq-frame, the terms in (20) are compared and summarized in

Table 1. Correspondences of components in the abc coordinate system.

abc Coordinate System	Three-Phase Current Components
${\displaystyle \frac{H_{\mathrm{s}1}}{Z_{\mathrm{v}}}}\cos ({\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t)+{\displaystyle \frac{H_{\mathrm{s}2}}{Z_{\mathrm{v}}}}\cos (\delta +{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t)$	Steady-state power frequency component is(t)
${\displaystyle \frac{H_{\mathrm{t}1}}{Z_{\mathrm{v}}}}\cos (\delta +{\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t+{\omega }_{\mathrm{N}1}t){\mathrm{e}}^{-{\displaystyle \frac{t}{\tau }}}$	Inherent non-power frequency decaying component it1(t)
${\displaystyle \frac{H_{\mathrm{t}2}}{Z_{\mathrm{v}}}}\cos ({\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t){\mathrm{e}}^{-{\displaystyle \frac{t}{\tau }}}$	Inherent power frequency periodic decaying component it2(t)
${\displaystyle \frac{H_{\mathrm{t}2}}{Z_{\mathrm{v}}}}\cos ({\phi }_{\mathrm{v}}-{\omega }_{\mathrm{N}}t){\mathrm{e}}^{-d_{1}t}$	Natural power frequency periodic decaying component it3(t)

.

According to the table above, the three-phase fault current consists of one steady-state power frequency component and three transient components: an inherent non-power frequency decaying component i_t1(t) with time constant τ, an inherent power frequency periodic decaying component i_t2(t) with time constant τ, and a natural power frequency periodic decaying component i_t3(t) with time constant 1/d1.

It should be noted that, since the value of time constant τ is solely determined by the internal parameters of the VSG and depends on the inherent characteristics of the grid-side converter, the decaying component dominated by τ is referred to as the “inherent” decaying component. In contrast, the value of time constant 1/d1 depends not only on the internal impedance but also on the voltage dip magnitude and the voltage dip power angle. Therefore, the decaying component dominated by 1/d1 is referred to as the “natural” decaying component.

As can be seen from (20), the amplitude and decay coefficient of the transient component of the short-circuit current are jointly influenced by multiple factors, primarily including fault parameters, control parameters, and variations in angular parameters. Specifically, fault parameters such as the voltage amplitude U during the fault have a direct impact on the magnitude of the short-circuit current. Control parameters encompass the reactive power integral coefficient k_q, virtual resistance R_v, virtual inductance L_v, etc., which determine the decay characteristics and response speed of the short-circuit current.

Among these factors, the voltage amplitude U during the fault significantly affects the initial value of the short-circuit current. A higher voltage dip typically leads to a larger short-circuit current amplitude, thereby influencing the transient response characteristics of the current. The reactive power integral coefficient k_q reflects the system’s reactive power regulation capability. A larger value of k_q makes the system more sensitive to voltage disturbances, resulting in a faster decay process of the short-circuit current. The virtual resistance R_v and virtual inductance L_v, as internal electrical parameters of VSG, primarily affect the damping and response speed of the short-circuit current. Specifically, the virtual resistance mainly controls the decay rate of the short-circuit current, while the virtual inductance influences the phase and waveform characteristics of the current.

Based on the derivation of the specific analytical expression for the short-circuit current, it is evident that the short-circuit current consists of four components: one steady-state fundamental frequency component and three transient components. By analyzing this expression, the extent to which the control parameters amplify or suppress the amplitudes and time decay coefficients of the three transient components of the short-circuit current can be evaluated, thereby revealing the dynamic characteristics of the grid-forming VSG short-circuit current. The specific characteristics are summarized in

Table 2. Impact of parameter on the amplitude of components in the analytical expression.

Parameters	U	kq	Rv	Lv
it1(t)	Decrease	Unchanged	Unchanged	Unchanged
it2(t)	Decrease	Increase	Decrease	Decrease
it3(t)	Decrease	Increase	Decrease	Decrease

and

Table 3. Impact of parameter on the time constant of components in the analytical expression.

Parameters	U	kq	Rv	Lv
it1(t)	Unchanged	Unchanged	Decrease	Increase
it2(t)	Unchanged	Unchanged	Decrease	Unchanged
it3(t)	Decrease	Decrease	Unchanged	Increase

.

4. Simulation Verification of the Fault Current Analytical Expression

4.1. Validation Under 0.45 pu Voltage Dip

To validate the correctness of the three-phase symmetrical short-circuit current analytical expression established in this paper and the parameter influence analysis, a grid-connected simulation model, as shown in Figure 1, was built in Matlab/Simulink, and relevant simulation verification was conducted. VSG has a rated active power of 66 kW and a rated reactive power of 0 kVar, with a rated voltage and frequency of 380 V/50 Hz. The fault scenario is as follows: A three-phase short-circuit fault occurs in the grid at 0 s, causing a voltage dip to 0.45 pu, with a fault duration of 0.2 s. The main simulation parameters are listed in

Table 4. Parameters of the VSG grid-connected simulation model.

Parameters	Values	Parameters	Values
Rg (Ω)	0.005	Lv (mH)	6
Lg (mH)	0.8	Rv (Ω)	0.2
J (kg/m2)	0.1	Kp	5.5
U (pu)	0.45	Ki	5000
kq	0.2	D	100

.

Under the aforementioned simulation parameter settings, a simulation analysis of a three-phase short-circuit fault was conducted on the VSG system, and the corresponding short-circuit current waveforms were obtained. Figure 5a–c shows the comparison curves between the simulated and calculated values of the short-circuit current at the moment the three-phase short-circuit fault occurs. It can be observed from the figures that the simulation waveforms and calculation results show good agreement in both amplitude and phase variations, particularly during the decay process after the first two to three cycles following the fault occurrence, where the waveforms match significantly well.

Figure 5a–e present an analysis of the relative error between the analytical calculation and the simulation results of the three-phase short-circuit current. As shown in the figure, the relative errors of the three-phase currents are all within 10%, indicating that the derived short-circuit current calculation expression in this paper can effectively simulate the actual dynamic characteristics of the fault current in most cases. Figure 5d illustrates the comparison of the virtual internal electromotive force E_ref. The high degree of fit between the analytical and simulated waveforms demonstrates that the proposed method accurately captures the dynamics of the internal EMF during the fault transient, which is crucial for correct fault current characterization. This close agreement further validates the accuracy of the reactive power–voltage loop modeling and the envelope simplification method adopted in Section 3.3.

Error Analysis: In the initial stage of the fault, certain errors exist between the analytical calculation results and the simulation values. This is primarily because the influence of the initial power angle was not considered during the derivation process, and the exponential decay sine term was neglected in the excitation term derivation, with only the envelope form being used for simplification. Specifically, the relative error in the initial fault stage remains within 8%, which falls within the acceptable range for engineering applications.

As the fault evolves, the discrepancy between the analytically calculated and simulated results gradually diminishes. During the middle-to-late stages of the fault, the relative errors for the three-phase short-circuit currents all fall below 5%. This close agreement not only validates the constructed calculation model but also confirms that the analytical method accurately captures the dynamic behavior of the fault current.

4.2. Validation Under 0.2 pu Voltage Dip and Robustness Analysis

To further verify the effectiveness and robustness of the proposed analytical method under different fault conditions, an additional simulation scenario with a more severe voltage dip to 0.2 pu was conducted. All other parameters remain the same as in Section 4.1. The results are presented in Figure 6.

Figure 6a–c show the comparison curves for phase A, phase B, and phase C currents under the 0.2 pu voltage dip. Despite the more severe fault condition, the analytical calculations still demonstrate excellent agreement with the simulation waveforms. The current amplitudes, phase shifts, and decay characteristics are all accurately captured, confirming the robustness of the proposed method across different fault severities.

Figure 6d shows the comparison of the virtual internal electromotive force under the 0.2 pu dip. Similar to the previous case, the analytical and simulated EMF waveforms exhibit high consistency. This indicates that the modeling of the reactive power–voltage loop remains valid even under deeper voltage sags, and the envelope simplification method maintains good accuracy.

Figure 6e presents the relative error analysis for the 0.2 pu case. The errors remain within 8% initially and drop below 5% after 2–3 cycles, consistent with the 0.45 pu case.

5. Conclusions

Currently, research on the fault current of grid-forming VSG primarily focuses on the proposal of current-limiting control strategies. In contrast, studies on the calculation of fault currents in grid-forming renewable energy/storage systems, particularly the analytical computation of fault currents under VSG control architectures, remain relatively scarce. This is mainly due to the complexity of the electromagnetic and electromechanical transient characteristics within VSG systems during faults, which requires comprehensive consideration of multiple influencing factors such as voltage dips, rotor angle variations, and the adjustment of virtual inertia and damping. Therefore, there is an urgent need for rigorous analytical methods to accurately determine the fault characteristics and short-circuit currents of VSG. To address this, this paper employs numerical computations via Mathematics software alongside appropriate simplifications to derive a precise analytical expression for the grid-forming short-circuit current under the VSG framework.

Based on the analytical expression, we conclude that the grid-forming short-circuit current formula consists of one steady-state component and three decaying components. The decaying terms include an inherent non-power frequency periodic component decaying with time t, an inherent power frequency periodic decaying component, and a free power frequency periodic component governed by 1/d1. Furthermore, simulation case studies demonstrate that the relative error between the simulated and actual values remains within 7%, and this error reduces to below 5% after two to three cycles. These results indicate that the derived analytical expression effectively captures the dynamic behavior of the short-circuit current, further validating the proposed method’s effectiveness for fault analysis in VSG systems. This work thus offers theoretical support and a technical basis for fault current calculation in practical engineering applications. Future research may utilize the explicit analytical expression to identify key control parameters that influence the short-circuit current in grid-forming VSGs and explore current limitation through targeted parameter adjustment.

This work thus offers theoretical support and a technical basis for fault current calculation in practical engineering applications. Several directions for future research are identified:

Application to complex power systems: The proposed method will be extended to more complex network models, such as the IEEE 39-bus system, to validate its effectiveness under various fault scenarios and system configurations, and to analyze the interaction among multiple VSG units.

Experimental validation: Physical experiments on a laboratory testbed will be conducted to further confirm the model’s adequacy under real operating conditions, accounting for factors not captured in simulations.

Parameter optimization for current limitation: The explicit analytical expression can be utilized to identify key control parameters for current limitation through targeted parameter adjustment.

Extension to longer time scales: Electromechanical transient effects such as power angle variation and frequency dynamics will be incorporated to extend the model’s validity beyond the electromagnetic transient period, enabling multi-timescale fault analysis.

Practical implementation and measurement system requirements: Future work will also address the practical implementation aspects of the proposed method on physical hardware platforms. This includes defining the requirements for electrical parameter measurement systems that are essential for real-world application. Unlike in the MATLAB/Simulink simulation environment where parameters are directly accessible, physical implementation requires careful consideration of measurement uncertainties and their impact on the accuracy of fault current calculation. These factors will be particularly important during hardware prototyping and parameter optimization processes.