Multi-index nonlinear robust virtual synchronous generator control method for microgrid inverters

Yanan Guo

doi:10.1515/nleng-2025-0123

Article Open Access

Multi-index nonlinear robust virtual synchronous generator control method for microgrid inverters

Yanan Guo

Published/Copyright: June 25, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

New energy is widely used in power systems due to its advantages of being green and sustainable. However, it requires microgrid inverters (Mis) to connect to the power grid, and a large number of connections will cause the system to lack damping and inertia, which are not conducive to the stability of the power grid. To improve the control performance of MIs, the study first constructs a nonlinear virtual synchronous generator (VSG) control method based on differential geometry theory, and then combines multi-index nonlinear robust control to design the control method. The research results indicate that the nonlinear VSG control method can achieve frequency oscillations from 51.24 to 50.01 Hz in grid connected environments, with small fluctuations in the response of active and reactive outputs, as well as output phase currents. In off grid environments, it can quickly reach steady-state values without significant overshoot. In contrast with the traditional second-order VSG control method, the multi-index nonlinear robust VSG control method can achieve stability within the range of (49.98–50.04 Hz) and (0.1–1.0 kW) in the event of a fault in the MI system. In a parameter perturbation environment, frequency, active power, and output phase current can all quickly and smoothly reach stable values without static errors. The above outcomes indicate that the raised control method can effectively alleviate the adverse effects of external disturbances and parameter disturbances on MIs.

Keywords: microgrid inverter; nonlinear control; multiple indicators; virtual synchronous generator; robustness

1 Introduction

Energy security is a major foundation of national security, a global and strategic issue related to national economy and social development, and a dependence on security in other areas. It is crucial for the country’s prosperous development, promoting people’s livelihood, and long-term social stability [1,2]. However, the pressure on energy demand is enormous. There are a lot of constraints on the supply and production of energy. Consumption causes heavy ecological damage, and the overall energy technology level is backward [3]. Microgrids can address the issues of energy crisis and ecological pollution, and can use new energy for power generation in more efficient ways, while also combining with traditional power generation methods [4,5]. But it requires the application of microgrid inverter (MI) to connect to the grid, which cannot provide the damping and inertia characteristics of synchronous generators (SG). Therefore, when a large number of connections are made to the grid, it will greatly threaten the security of the grid [6,7]. Nonlinear control is widely used in nonlinear systems and has achieved many results in optimization design of static and transient performance. In recent years, scholars have also been exploring its application design in MI system control. At present, virtual synchronous generator (VSG) technology can enhance the inertia of systems by enhancing virtual inertia, so many studies have designed and optimized this technology from stability, control design, and other methods. Chen et al. designed an improved VSG control method based on proportional integral (PI) control to strengthen small signal stability in response to the negative impact of VSG transient synchronization stability. With semi physical experimental outcomes, the effect and superiority of the method were verified [8]. Li et al. designed the transient dynamic characteristics of VSG based on the signal model to address the transient instability issue of VSG technology. The research results showed that the main factor affecting the transient stability of VSG was the active power control loop, and the experimental results of the loop verified the effectiveness of this method [9]. Yu et al. proposed an adaptive neural network control method for a class of multi-input multi-output nonlinear systems. The stability of the system was demonstrated through embedded backstepping based on Lyapunov synthesis, and simulation outcomes denoted the effect of this method [10]. Zhai et al. designed an observability-based scheme to achieve the identification of unknown nonlinearities by fuzzy logic systems. Analysis showed that all signals of the system were optimal in finite time, and this method had superiority [11].

The continuous reduction in fossil fuels has made microgrids the backbone of the power grid. But with the massive input of new energy, the control strategy of MI, an important technology, is facing a huge threat. Yorino et al. developed a novel grid connected inverter based on a non-interference core, and evaluated the optimization effect of frequency and transient stability. Simulation results showed that the proposed inverter had significant grid stability capability [12]. Jha et al. designed an adaptive vector control method to reduce the deviation between MI voltage and frequency parameters caused by classical droop. Simulation experiments confirmed the effectiveness of this method [13]. Nguyen et al. proposed a harmonic suppression-based MI controller design to address the challenge of voltage ride through capability to the MI controller. They conducted experiments on a real-time simulator with hardware settings, and the results verified its effectiveness [14]. Enes et al. designed a voltage reactive power control method using a combination of static voltage regulator and photovoltaic inverter to address the issues of reverse power flow, grid instability, and voltage in microgrids. The outcomes indicated that this method improved the performance of microgrids [15]. Elik and Meral compensated for harmonic and unbalanced voltage and current as needed, and based on this, studied and designed a coordinated virtual impedance control method for a three-phase four wire inverter from vehicles to the power grid. They also proposed an independent neutral current control method to regulate the hysteresis link and have neutral current control capability. The research results showed that the control scheme achieved the operational capability of grid connected mode or independent mode, and smooth transitions between operational modes. Finally, comprehensive testing in various scenarios verified that the proposed control scheme was superior to existing methods [16].

Based on the above content, it can be concluded that current microgrids and nonlinear control methods can make the system have small damping and inertia, but they also bring about power oscillation problems. Moreover, the current solutions have long adjustment times and complex control structures, which cannot achieve good results. Therefore, the study first introduces VSG control and differential geometry theory (DGT) to construct a nonlinear VSG control method. Based on this, a multi-index nonlinear robust VSG control method (MINR-VSG) is designed. The research aims to improve the damping and inertia of the MI system, as well as enhance its control and anti-interference capabilities after being connected to the power grid, ensuring the stability and safety of a large number of new energy resources connected to the power grid, and thus contributing to energy security and sustainable development. The novelty of the research mainly lies in two aspects. On the one hand, it combines traditional feedback linear nonlinear control and VSG control, and introduces them into the design of MI control methods. On the other hand, the MINR-VSG control method is proposed to solve the problems of MI interference and uncertain parameters, enabling the MI system to achieve better transient stability and dynamic and static performance. The research structure contains four sections. Section 1 is a review of relevant research outcomes. Section 2 describes the construction of a nonlinear VSG control method based on DGT, and designing the MINR-VSG control method. Section 3 is to prove the effect of the raised research methods. Section 4 is the summary of the research.

2 Construction of MINR-VSG control methods

Although VSG control can provide certain damping and inertia in the response of MI systems, it also leads to power oscillation problems. Moreover, the current power oscillation suppression methods have problems such as long debugging time and complex control structure, which cannot achieve satisfactory results. Therefore, how to further improve the performance of VSG control methods is currently a research focus. In response to the above issues, the study first establishes a nonlinear VSG control method based on DGT, and then designs an MINR-VSG control method to address the problem of system disturbance.

2.1 Nonlinear VSG control method based on DGT

To address the problems of damping and lack of inertia in MI, most research on control methods is limited to traditional PI control and improvements to linear control. They all ignore nonlinear higher-order terms to achieve model linearization, and finally establish feedback control based on voltage and current dual loops [17]. Only applying the above control methods for control design will affect the restoration accuracy of the system. When the MI system is disturbed, the above methods will cause system description deviation, thereby affecting control performance and causing instability in system frequency and power control. Therefore, the study first designs a nonlinear VSG control method based on DGT, and then selects MI output current as the output function to establish a corresponding radiative nonlinear model. First, the proposed overall control strategy framework for microgrid transformers is studied, as shown in Figure 1.

Figure 1

Schematic diagram of the overall control strategy framework for microgrid transformers.

In Figure 1, MI is connected to the power grid or supplied to the load through a three-phase filtering circuit and an inductor circuit, while the DC side supplies power to the system through an equivalent distributed power source. In addition, the control strategy framework includes outer loop nonlinear VSG control and inner loop voltage and current proportional resonant (PR) control. After obtaining the reference value in the former, the modulation wave is controlled by the latter to drive the switching transistor. In the nonlinear VSG control section, the corresponding active power control simulates the rotor motion equation, expressed as Eq. (1):

(1) ω = d ϑ d t , Δ ω = ω − ω s , J d ω d t = P m − P e ω s − k ω ⋅ Δ ω , .

where ω and ω s correspond to the actual angular velocity and grid angular frequency. ϑ represents the generator power angle. ϑ and k ω represent the virtual moment of inertia and damping coefficient, respectively. P m and P e correspond to the mechanical and electromagnetic powers, respectively. If MI works during grid connection, there is no need for frequency modulation processing; If MI operates while off grid, frequency modulation needs to be achieved through active frequency droop control, as shown in Eq. (2).

(2) P m − P r = 1 A P ( ω r − ω s ) ,

where P r and ω r represent the reference value of active power and the set value of angular frequency, respectively, while A P represents the active droop coefficient. Eq. (3) is the combination of Eqs. (1) and (2).

(3) J dΔ ω d t = 1 ω s P r − P e + 1 A P ( ω r − ω s ) − k ω ⋅ Δ ω .

The reactive power control of nonlinear VSG control is achieved through excitation control of traditional SG, and the expression is shown in Eq. (4).

(4) E 1 = E 0 − A q Q ,

where E 1 represents the voltage reference value when working off the grid, while E 0 , A q , and Q represent the no-load electromotive force, reactive power regulation coefficient, and MI output reactive power, respectively. The inverter power supply provides reactive power to the grid when working on grid connection, while when working off grid, it only needs to provide load reactive power. The calculation is shown in Eq. (5).

(5) E = E 1 + E 2 , E 2 = k E ( Q r − Q ) ,

where k E and Q _r represent the demand coefficient and reactive power reference values, respectively. From the above content, the active frequency control and reactive voltage control framework of VSG control are illustrated in Figure 2.

Figure 2

Schematic diagram of active frequency control framework for VSG control. (a) Active power-frequency control and (b) reactive power-voltage control.

The state equation expression of the VSG model based on the MI system is shown in Eq. (6).

(6) x ̇ 1 = − R L x 1 + ω x 2 + u sd0 − u sd L , x ̇ 2 = − ω x 1 − R L x 2 + u sq0 − u sq L , x ̇ 3 = 1 − k ω A P ω s J A P ω s x 3 − 1.5 A P u sd J ω s x 1 − 1.5 A P u sq J ω s x 2 + A P P r + k ω A P ω s 2 − ω s J A P ω s ,

where u sd and u sq , u sd0 and u sd 0 correspond to the dq axis components of the output voltage and voltage reference value, respectively. R and L represent the virtual resistance and virtual reactance, respectively. For Eq. (6), by selecting the output functions y 1 = Δ i sd , y 2 = Δ i sq , and control variables u 1 = u sd and u 2 = u sq , the radiative nonlinear control system can be obtained, as shown in Eq. (7).

(7) x ̇ = f ( x ) + M 1 ( x ) u 1 + M 2 ( x ) u 2 , y 1 = Δ i sd = I 1 ( x ) , y 2 = Δ i sq = I 2 ( x ) , f ( x ) = − R L x 1 + ω x 2 + u sd0 L − ω x 1 − R L x 2 + u sq0 L 1 − k ω A P ω s J A P ω s x 3 + A P P r + k ω A P ω s 2 − ω s J A P ω s T M 1 ( x ) = − 1 L 0 − 1.5 A P x 1 J ω s T , M 2 ( x ) = 0 − 1 L − 1.5 A P x 2 J ω s T .

According to the theory of differential geometry, the above system can realize state feedback linearization, and then calculate the relative order of systems I 1 ( x ) and I 2 ( x ) . From this, a 2 × 2-order matrix M ˜ ( x ) can be obtained, as shown in Eq. (8).

(8) M ˜ ( x ) = − 1 L − R L x 1 + ω x 2 + u sd0 L 0 0 − 1 L − ω x 1 − R L x 2 + u sq0 L .

If Eq. (8) is non-singular, then the relative order of the system is ζ 1 + ζ 2 = 2 < 3 , and the system can achieve partial exact linearization. In addition, it is necessary to select a function with state variables for nonlinear transformation, as shown in Eq. (9).

(9) ψ ( x ) = − L ( x 1 2 − x 2 2 ) 2 + J ω s x 3 1.5 A P .

If it conforms to L M i ψ ( x ) = 0 , i = 1 , 2 , it can be transformed into τ ( x ) , thereby verifying the full rank of the corresponding Jacobian matrix J τ ( x ) = ∂ τ ( x ) ∂ X . Combined with Eq. (7) and simplified, the transformed linear system can be obtained as Eq. (10).

(10) z ̇ = B z + G v , z ̇ L = q ( z ) , y = D z , B = 0 0 0 0 , G = 1 0 0 1 , D = 1 0 0 1 , v = v 1 v 2 T , y = y 1 y 2 T ,

By using linear optimal control theory, the quadratic performance index and corresponding feedback control law can be selected, as shown in Eq. (11).

(11) J = 1 2 ∫ 0 ∞ ( z T Q z + v T R v ) d t , v 1 = − l 1 ( x 1 − x 10 ) , v 2 = − l 2 ( x 2 − x 20 ) ,

where x 10 = i sdo and x 20 = i sqo are the reference values for the output current of the MI system on the dq axes, respectively. By substituting Eq. (11) in Eq. (10), the partially exact linearized control law in nonlinear space x can be obtained, as shown in Eq. (12).

(12) u 1 = L 1 + l 1 ( i sd − i sd0 ) − R i sd L + ω i sq + u sd0 L , u 2 = L 1 + l 2 ( i sq − i sq0 ) − R i sq L − ω i sd + u sq0 L ,

After the precise linearization control law design of the MI system with the aforementioned nonlinear VSG control is completed, to further quickly track the given values of the upper layer VSG model controller, the use of voltage and current PR control with better response speed and accuracy is studied. Schematic diagram of inner loop voltage and current control based on PR controller is shown in Figure 3.

Figure 3

Schematic diagram of inner loop voltage and current control based on PR controller.

In Figure 3, the three-phase circuit system has the following relationship, as shown in Eq. (13).

(13) L d I L d t = 1 2 m U dc − U s , C d U s d t = I L − I s ,

where U dc and U s represent the DC voltage source and capacitor voltage. I L and I s represent the inductor current and grid connected current. m represents the modulation signal, and C represents the capacitor. The transfer function expression of the PR controller is shown in Eq. (14).

(14) ς ( a ) = k p + 2 k r ω c a a 2 + 2 ω c a + ω 0 2 , ω 0 = 314 ,

where k p , k r , and ω c are all control parameters.

2.2 Design of MINR-VSG control method

Due to the current focus of research on VSG control, traditional linear control MI systems lack robustness and cannot suppress the effects of parameter perturbations and external disturbances on MI systems [18,19,20]. Therefore, based on the above content, this study combines multiple indicators of nonlinear robust control, designs nonlinear control laws through a fourth-order SG model, and uses linear combinations of several state variables to design output functions. Disturbance and linearity are also designed to obtain the worst-case disturbance and optimal control law. First, it assumes that the system follows Eq. (15).

(15) X ̇ = F X + S 1 w + S 2 u ′ , u ′ ∈ R m 1 , w ∈ R m 2 , y ' = Z X , X ∈ R n , y ′ ∈ R ,

where w , u ' , and y ' correspond to interference, control, and output scalars, respectively, while ω c represents the n dimensional state variable. The state feedback linear control H ∞ requires finding the state feedback u ' = W X of Eq. (15) and making the l 2 gain of the closed-loop system less than the real number c > 0 , while the necessary and sufficient conditions for the existence of W are shown in Eq. (16).

(16) P F + F T P − 1 C 2 P S 1 S 1 T P + P S 2 S 2 T P + Z 1 Z 1 T = 0 .

There exists a semi definite solution P 0 , and F + 1 C 2 P S 1 S 1 T P + S 2 S 2 T P is a stable matrix. The structural diagram of an MINR-VSG control method for MIs designed for research is shown in Figure 4.

Figure 4

Structure diagram of an MINR-VSG control method for MIs.

In Figure 4, first, the angular velocity ϖ and the reference voltage amplitude u 0 are calculated based on the given initial value. Second, the reference value is obtained through the outer loop fourth-order VSG control model. Then, the modulation wave is obtained through the inner loop control. Finally, the trigger pulse is obtained through pulse width modulation, which drives the switching transistor. In the construction of the fourth-order VSG model, the study simulates the characteristics of SG using a fourth-order SG model with excitation system and valve opening. The algebraic equation of the system can be obtained, as shown in Eq. (17) [21,22,23].

(17) P e = p 1 E q ′ sin θ + p 2 sin 2 θ , E q = e 1 E q ′ − e 2 cos θ , U f = U fd 2 + U fq 2 , U fd = h sin θ , U fq = f 1 E q ′ + f 2 cos θ , p 1 = U x dΣ ' , p 2 = U 2 ( x dΣ ' − x qΣ ) 2 x dΣ ' x qΣ , e 1 = x dΣ x dΣ ' , e 2 = x dΣ − x dΣ ' x dΣ ' , f 1 = x e x dΣ ' , f 2 = x d ' x dΣ ' , h = x q x qΣ ,

where x q ' and E q correspond to the no-load potential and transient potential of the generator. x d , x d ′ , and x e are the direct axis synchronous reactance, transient reactance, and equivalent resistance of the line, respectively. x dΣ and x dΣ ' correspond to the sum of x d , x d ′ , transformer x e . U represents infinite system voltage, and θ represents the generator power angle. To ensure the linearization and performance of the system’s state feedback, selecting the output function is crucial, as expressed in Eq. (18).

(18) y 1 ′ = c 1 Δ U f + c 2 Δ P e + c 3 Δ θ , y 2 ′ = c 4 Δ θ + c 2 Δ ϕ + c 3 Δ ϕ ̇ ,

where ϕ represents the rotor angular velocity. Δ U f , Δ P e , and Δ ϕ represent the deviation of voltage, active output, and angular velocity. Δ θ and Δ ϕ ̇ represent the deviation of power angle and the presence of advanced control for diagonal velocity, respectively. y 1 ′ and y 2 ′ represent the output functions of excitation control and valve control, respectively. By selecting the state variables and control variables, an affine nonlinear control system can be got, as expressed in Eq. (19).

(19) x ̇ = f ′ ( x ) + M 1 ' ( x ) u 1 ' + M 2 ' ( x ) u 2 ' + d ( x ) w , y 1 ' = I 1 ' ( x ) , y 2 ' = I 2 ' ( x ) , f ′ ( x ) = ϕ − ϕ 0 ϕ 0 T j ( P m − P m ) − A T j ( ϕ − ϕ 0 ) − c E q − d P m , M 1 ' ( x ) = 0 0 c 0 T , M 1 ' ( x ) = 0 0 0 d T , d ( x ) = 0 d 1 d 2 d 3 T ,

where A and T j correspond to the damping coefficient and rotor inertia time constant of the generator, respectively. T j means the grid synchronization angular velocity, and w means the amount of disturbance generated by system parameter disturbances and external disturbances. By calculating the total relative order of the system, the matrix y 2 ′ can be obtained, which belongs to a non-singular matrix and corresponds to the relative order 2 of the system, indicating that the system can be partially precisely linearized. By selecting ψ 1 ( x ) = Δ θ and ψ 2 ( x ) = Δ ϕ , which comply with L M i ψ j ( x ) = 0 , i , j = 1 , 2 , a nonlinear transformation can be performed to obtain τ ′ ( x ) . From this, it can be verified that the Jacobian matrix J τ ( x ) = ∂ τ ( x ) ∂ X is full rank, and the system can be partially precisely linearized. The disturbance part is designed using a robust nonlinear controller. First, some precise linearization formulas are sorted out, as shown in Eq. (20).

(20) z ̇ L = F z L + S 1 w + S 2 u , z ̇ N = q ( z L , z N ) , y = Z z L , z L = z 1 z 2 T , z N = z 3 z 4 T ,

where z 1 and z 2 represent linear subsystems and nonlinear subsystems, respectively. Then, Eq. (16) is solved and the semi positive definite solution P 0 is found, where F + 1 C 2 P S 1 S 1 T P + S 2 S 2 T P is a stable matrix, worst-case disturbance, and optimal control law. Finally, it is combined with the exact linearization formula of the system part to obtain the final multi-index nonlinear robust control law. Because the above method is to make the output of MI’s grid connection point have SG characteristics, the grid connection voltage and the E q output by the controller need to meet the following conditions, as shown in Eq. (21).

(21) u sd = x qΣ ' u sq , u sq = E q ' − x dΣ ′ i sd ,

where x dΣ ″ and x qΣ ′ must meet the conditions of Eq. (22).

(22) x d Σ ″ = x d Σ ′ + x v , x q Σ ′ = x q ' + x e + x v ,

x v represents virtual impedance. For the convenience of subsequent simulation experiments, a schematic diagram of the simulation structure of the MI system is designed, as denoted in Figure 5.

Figure 5

Schematic diagram of simulation structure for MI system.

In Figure 5, a three-phase ground short circuit is set at the end of the circuit and processed 0.1 s after it occurs.

3 Result analysis of MINR-VSG control methods

To assess the performance of MINR-VSG, the study first set up preliminary experiments to assess the rationality and correctness of the nonlinear VSG control algorithm, and then conducted simulation analysis on the MINR-VSG control method.

3.1 Pre experiment of MINR-VSG control method

To prove the correctness of the nonlinear VSG, simulation experiments were conducted using the software MATLAB. The parameter settings for the MI system are indicated in Table 1.

Table 1

Parameter settings for MI system

Parameter	Set value	Parameter	Set value	Parameter	Set value
U dc /V	800	A P	0.05	k p	100
C/μF	30	A q	0.001	k E	0.01
L/mH	20	ω c	5	k ω	100
J	0.5	k r	500	l 1 , l 2	3*107

To scientifically verify the performance of nonlinear VSG control methods, comparative experiments were conducted using traditional second-order VSG control, and both methods used voltage–current PR control in their inner loops. Research set three simulation modes for simulation, including grid connection, off grid, and parameter perturbation. First, the active and reactive power outputs of the MI system under initial operating conditions were set to 10 kW and 5 kvar, respectively. To better analyze the response of MI system to active output adjustment under different control methods, the study increased the active output from 10 to 20 kW in the first and second settings, and restored it to 10 kW after the second setting.

Figure 6(a) and (b) shows the response curves of the frequency, active output, reactive output, and output phase current of the MI system under various control methods. From Figure 6, the frequency, it is observed that active output, reactive output, and output phase current of the MI system based on the nonlinear VSG control method had small amplitude changes, with the maximum frequency fluctuation decreasing from 51.24 to 50.01 Hz. The overshoot of the curve corresponding to the active output was small and there was no static error. The frequency response curve of the MI system based on second-order VSG control fluctuated greatly and required longer time to recover to a stable state. In addition, active output, reactive output, and output phase current all required a long time to return to steady-state values. To evaluate the dynamic response of the control method to load changes during the operation of the MI system, experiments were conducted in an off-grid environment. The specific settings were as follows: the initial active load and reactive load were set to 20 kW and 3 Kvar, respectively. At the 1.5th second, the active load and reactive load were further increased by 20 kW and 2 Kvar.

Figure 6

Response curve of MI system based on different control methods in grid-connected environment. (a) Frequency and active power output and (b) reactive power output and output phase current.

Figure 7(a) and (d) shows the response curves of frequency and active output, reactive output, and output phase current of MI systems based on different control methods in offline environments. From Figure 7, in contrast with the second-order VSG control method, the raised control method was 66.7% higher in the initial stage, indicating that the nonlinear VSG control method promoted the performance of the MI system. The control method proposed in the study had almost no overshoot and could quickly and smoothly reach the equilibrium point. When the load point changed, its active output could still efficiently and smoothly reach the steady-state value. Finally, in the response results of reactive power output and output phase current, the nonlinear VSG control method could quickly reach a stable value when the load changed, and there was no obvious overshoot. In the second-order VSG method, the corresponding active power and reactive power had overshoots of 1.7 and 0.28 kW, respectively. When the load changed, the corresponding active and reactive power had overshoots of 4.7 and 0.19 kW, respectively. To prove the anti-interference ability of the research method, three perturbation cases of inductance and capacitance parameters were set up in the study.

Figure 7

Response curve of MI system based on different control methods in off-line environment. (a) Frequency and active power output and (b) reactive power output and output phase current.

Figure 8(a) and (b) shows the response results of the frequency and active output, as well as the reactive output and output phase current of the MI system using the nonlinear VSG control method under the corresponding parameter perturbation environment. From Figure 8, when the parameters were perturbed, the response curves of the active output, reactive output, and output phase current of the nonlinear VSG control method did not fluctuate. Only the adjustment time and overshoot change, and the corresponding frequency control change range was −0.01 to 0.01%. The overshoot of the active power was less than 0.3 kW, indicating that this method was suitable for practical applications. In summary, the results validated the correctness of the nonlinear VSG control method, which had excellent control effectiveness and anti-interference ability. To further verify the denoising effect of research methods on noise signals with harmonic components, comparative experiments were conducted with mainstream methods currently available, including Second Harmonic Denoising Based on Variational Mode Decomposition and Savitzky Golay Filtering (SHD-VMD-SGF), Porous Fractional Wavelet Transform (PFWT), and Trigonometric Chain Network (TCN).

Figure 8

The frequency and response results of the active output, reactive output, and output phase current of MI system with nonlinear VSG control method under parameter perturbation environment. (a) Frequency and active power output and (b) output phase current.

Figure 9 shows a comparison of the noise reduction effects of different methods. All methods could achieve good noise reduction effects, and the noise generally show a downward trend. However, with the increase in iteration times, the noise reduction effect of the research method was the best and more stable. Significant noise reduction performance was achieved at around 300 iterations, and the corresponding change curve area was balanced at 400 iterations. The performance of the TCN method was better than that of the SHD-VMD-SGF method and PFWT, because of its small computational complexity and fast effectiveness.

Figure 9

Comparison of noise reduction effects of different methods.

3.2 Result analysis of MINR-VSG control method

To assess the effect and feasibility of MINR-VSG control method, a comparative experiment was conducted using the traditional second-order VSG control method, and three environments were set: active power step (environment A), three-phase grounding short circuit (environment B), and parameter perturbation (environment C). Environment A was set as follows: the initial active output was 1.3 kW, and in the third second, the active output was set to increase to 2.2 kW.

Figure 10(a) and (b) shows the frequency and active output, as well as the response curves of the output phase current, respectively. Results in Figure 10 show that when controlled by the MINR-VSG control method, the peak value of the frequency response curve was 50.003 Hz. After that, the curve fluctuation rapidly decreased, and its active power could achieve a stable state in only 3 s. Finally, in the output phase current response result, the method quickly and smoothly reached the steady-state value, indicating that it could effectively simulate the characteristics of SG. However, in the traditional second-order VSG control method, there was a significant fluctuation in the frequency response curve at the beginning and the third second, and the corresponding active power increased to a stable value within 0.5 s, resulting in overshoot. In comparison with conventional techniques, the research method demonstrated superior stability in frequency response, the capacity to rapidly suppress frequency fluctuations, and the ability to maintain grid stability. At the same time, the research method showed a faster stable speed in active power response, achieving a steady state in only 3 s, which is conducive to quickly respond to changes in power grid demand. However, mainstream methods may experience overshoot, which may cause instantaneous overload of the secondary power grid. To test the anti-interference effect of the MINR-VSG control method, a study was conducted to set the occurrence of a three-phase ground short circuit fault at the first second.

Figure 10

Response curve of MI system based on different control methods in environment A. (a) Frequency and active power output and (b) output phase current.

Figure 11(a) and (b) denotes the frequency, active output, and output phase current response curves of the MI system based on different control methods under environment B. From Figure 11, when the MI system malfunctioned, the frequency and active power fluctuation range of the MINR-VSG control method were (49.98–50.04 Hz) and (0.1–1.0 kW), respectively, and the fluctuation was achieved within 0.2 s. The frequency fluctuation range of the traditional second-order VSG control method was (49.84–50.19 Hz), and the highest fluctuation value of active power was 2.2 kW, with a fluctuation time of more than 0.3 s. In addition, its output phase current amplitude was three times that of the stationary state. The above results showed that compared with other methods, the research method could quickly recover to steady-state values when a fault occurs, which is beneficial for reducing the stress on power grid equipment during the fault period. However, the output current amplitude of the mainstream method reached three times that of a steady state, which may cause the power grid equipment to bear excessive current and increase the risk of equipment damage. In addition, the research method could demonstrate better anti-interference performance in the face of three-phase grounding short-circuit faults, quickly suppress the fluctuations caused by faults, and better simulate the characteristics of SGs. These advantages make it of great application value in power grid control. If the filtering inductance and capacitance of the control method experience parameter perturbations, it will increase the harmonic content in the output curve, resulting in poor power quality and leading to grid imbalance. Therefore, to test the effectiveness of the MINR-VSG control method in suppressing parameter perturbations, three types of capacitors and inductors were studied.

Figure 11

Response curve of MI system based on different control methods in environment B. (a) Frequency and active power output and (b) output phase current.

Figure 12(a) and (b) corresponds to the frequency, active output, and output phase current response results of the MI system based on the MINR-VSG control method in environment C. Results in Figure 12 show that when the inductance or capacitance parameters were perturbed, the stable frequency value of the MI system based on the MINR-VSG control method only underwent a small change in dynamic time without any deviation, and the corresponding active power could quickly and smoothly reach the stable value without any static error. The output phase current response of the MI system showed excellent tracking performance, which could completely eliminate steady-state errors. In summary, the experiment findings demonstrated the correctness and feasibility of the MINR-VSG, which could effectively suppress the adverse effects of external disturbances and parameter disturbances on the MI system.

Figure 12

Response results of MI system based on MINR-VSG control method in environment C. (a) Frequency and active power output and (b) output phase current.

4 Conclusion

In microgrid applications, the use of VSG control can achieve seamless switching between grid connected and off grid, but it will bring resonance phenomena and have adverse effects on the practical application of MI systems. To solve the above problems, the study first constructed a nonlinear VSG control method based on DGT, and then combined multi-index nonlinear robust control, introducing disturbance and linear parts. Finally, the MINR-VSG control method was designed. The experimental results showed that in the MI system controlled by the nonlinear VSG control method in a grid connected environment, the maximum frequency fluctuation decreased from 51.24 to 50.01 Hz, and the amplitude of changes in active and reactive output, and output phase current, was relatively small. When there was a fault in the MI system, the frequency and active power fluctuation range of the MINR-VSG control method were (49.98–50.04 Hz) and (0.1–1.0 kW), respectively, and the fluctuation was achieved within 0.2 s. The frequency fluctuation range of the traditional second-order VSG control method was (49.84–50.19 Hz), and the highest fluctuation value of active power was 2.2 kW, with a fluctuation time exceeding 0.3 s. In the case of parameter disturbance, the stable value did not deviate, and the corresponding active power could quickly and smoothly reach the stable value without static error. In summary, the experimental findings verified the correctness and feasibility of the use of the MINR-VSG control method in the MI system. It could achieve good coordination ability in both grid connected and off-grid states, and effectively suppress the adverse effects of parameter perturbations and external interference. However, there were still certain limitations in the research. The research method only simulated under ideal conditions. In practical applications, the component composition of MIs and the surrounding environment are not always in an ideal state. Therefore, in future research, larger and more complex microgrid systems can be constructed for testing, and the control strategy of parallel VSG systems can be analyzed in depth.

Funding information: The research was supported by Henan Province Science and Technology Research Project: Science and Technology Research Project of Henan Province, Research on a multi-level unit & master-slave random PWM selective harmonic elimination inverter (No. 232102240098).
Author contributions: Yanan Guo: Conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing – original draft, and writing – review and editing. Author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Author states no conflict of interest.
Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-09-04

Revised: 2025-03-10

Accepted: 2025-03-16

Published Online: 2025-06-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2025-0123

Keywords for this article

microgrid inverter; nonlinear control; multiple indicators; virtual synchronous generator; robustness

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