An improved design for virtual synchronous generator control loop based on synergy theory

Virtual synchronous generator (VSG) technology has achieved some results in distributed generation networks and enhanced system stability. For problems that may occur in the system due to ignoring parameter linkages, this paper designs a synergistic controller that establishes the connection between the active frequency and the reactive voltage control loop based on synergy theory. As the output of the synergetic controller, the derived control law u is added to the power frequency control in the form of negative feedback. Two disturbance forms, a step disturbance


I. INTRODUCTION
In recent years, renewable energy sources such as wind and solar have been increasingly utilized as the energy crisis continues to intensify.However, most of these new energy generation units lack the essential capabilities of conventional energy generation, and the inverters composed of power electronics cannot provide inertia and damping to the system to ensure grid stability. 1,2Especially, they have intermittency and volatility that cannot be ignored, exacerbating the difficulty of control and bringing greater challenges, 3,4 as well as seriously affecting the quality of power supply and the stable and safe operation of the power grid.
Hence, the investigation about the incorporation of new energy into the grid holds considerable value and significance.Since renewable energy sources constantly enter the grid in large numbers, it is imperative to provide the necessary frequency and voltage support to the grid and provide integral damping for the system in the event of disturbances.6][7] This technique has been extensively applied and has achieved considerable results, particularly in the field of automatic generation control (AGC). 8,9From the application characteristics of VSG, it is clear that it will be the key to making full use of new energy sources and ensuring stability in the power system.
Due to the increasing number of new energy sources entering the grid and the diversification of power generation forms, it has become harder to control VSG.Although VSG possesses the capability to support system inertia and improve frequency stability, it still has other stability issues under different types of disturbance.The inability to attain a more optimal control of VSG will have a detrimental impact on the utilization and consumption of new energy sources such as wind and solar.Therefore, it is necessary to find better forms to achieve optimized control.Current research on power system control mainly focuses on dynamic response and stability control.In detail, it explicitly includes two aspects, which are (i) model optimization and (ii) application of new control forms and methods.The second form of research is more diversified.7][18] Some researchers relied on strategies to provide virtual inertia for VSG, thereby improving the frequency stability of the system. 19,20Simulation results have shown the effectiveness of the proposed approach.There has also been some research on the improvement of control. 21,22Based on the analysis of the amplitude-frequency characteristics of virtual power, 23 we propose an improved VSG control strategy.The study focused on inertia and damping inertia but did not involve the influence of output state parameter changes on the control process.A simplified VSG internal controller was developed to modify the frequency control loop by removing the governor component and directly setting the active power. 24The proportional-integral (PI) regulator was used as the voltage part, and the improved controller was properly pre-synchronized.It can be concluded that this method was superior to that of the droop control method.A new control strategy was presented based on traditional control and combined with the synchronous generator model. 25The research focused on power frequency and excitation control, demonstrating that the microgrid inverter using a VSG control strategy could reasonably simulate the output characteristics of the actual synchronous generator.However, the above-mentioned studies mainly focus on virtual inertia and damping, and the active and reactive power parts are controlled individually without involving linkage analysis between the parameters or the interplay between the control loops.These neglected influences may be the root cause of the inadequate control effect of the system, and their existence is also susceptible to affecting the system's stability negatively.
This paper proposes a state-parameter collaborative control strategy based on synergetic theory to achieve unified control of the system and fully consider the interactions between them.7][28] It follows the directional self-organization principle of the system and is a nonlinear method used in control systems.The design of the controller based on the synergetic control theory does not require linearization, and the nonlinear model of the system can be used directly.The characteristics of synergetic control make it highly suitable for digital control implementation.Its controller bandwidth requirements are relatively low, and it can also reduce power filtering problems in power electronics applications.Reference 29 proposed decentralized improved synergetic excitation controllers for synchronous generators.Simulation results show its effectiveness.Reference 30 proposed a fractional nonlinear synergetic controller for wind turbines.The experimental results also verify that the method has a good control effect on system stability.Reference 31 designed a fixedtime synergetic controller for a hydraulic turbine governing system.Compared with some classical control forms, it shows certain advantages.
In response to the above-mentioned issues, this paper proposes a control method based on synergetic theory to improve the control loop.This research introduces a synergetic controller to regulate VSG on the basis of conventional droop control and appends a negative feedback loop to enhance the output stability of VSG and solve the problem of parameter interaction.Consequently, the contribution of this paper can be summarized as follows: 1.A synergetic control theory suitable for nonlinear models is introduced, and the interconnectivity between state variables and systems that are not studied by existing controls is analyzed.2. The original VSG model is improved by incorporating a negative feedback setting into frequency control.It fully considers the interaction between parameters and control loops.3. The source of negative feedback is the output of the synergetic controller.Power frequency control and reactive voltage control were established by the synergetic controller to further minimize the impact of parameter changes during the control process.4. Based on the new model obtained under this method, simulation research was conducted on its effectiveness under load disturbances and three-phase short circuit faults.The results showed that introducing the synergetic controller can reduce oscillations and improve output stability compared to the original droop control.
The rest of the parts are organized as follows: Sec.II briefly describes the basic principles and mathematical model of the VSG.Section III introduces synergy theory, gives the design method of the controller, and deduces the control law of the new model.The simulation in Sec.IV demonstrates the control effect of the technology.The sensitivity analysis of the system and synergetic controller parameters is carried out in Sec.V. Section VI conducted a simple test on the control effectiveness of a multi-machine VSG system.Finally, Sec.VII presents the conclusions of this study.

II. THE BASIC PRINCIPLE AND MATHEMATICAL MODEL OF VSG
A. Basic principle of VSG the prime motor, and the inverter is the traditional synchronous generator.In the synchronous generator, the mechanical shaft and winding provide the necessary moment of inertia and damping for the system's stable operation.It is essential to use the energy storage system to establish virtual inertia and damping under VSG.The effects of resistance and capacitance are ignored in this work.The inverter control loop (including the VSG control loop) controls the frequency and voltage output of the system.

B. Mathematical model of VSG
The mathematical model of a synchronous generator reflects its own mechanical and electromagnetic characteristics, including inertia and damping, which are at the core of VSG technology.Therefore, the synchronous generator model is used in the study of VSG.The mathematical model of the rotor motion equation in the form of a normalized value is as follows: where J is the virtual moment of inertia of the synchronous generator, Tm is the normalized value of the mechanical torque, Te is the normalized value of the electromagnetic torque, D is the damping coefficient, and ω is the normalized value of the actual electrical angular velocity.
In the rotor acceleration equation, Tm − Te is the rotor acceleration torque, which is expressed as The speed change in the transient process is tiny.Therefore, the base value of the angular velocity is ω * ≈ 1, providing Therefore, (1) can be reduced to Synchronous generators generally have second-order, third-order, and fifth-order models.The third-order model is adopted in this paper.
The block diagram illustrating the active power-frequency and reactive voltage control of VSG is depicted in Fig. 2.
VSG control can be divided into two parts: an active power control loop and a reactive power control loop.The power-frequency control of the VSG simulates the governor to characterize the droop characteristics of the active power and system frequency.This control adjusts the frequency by detecting the power difference ΔP to control the virtual mechanical torque output, as shown in Fig. 2(a).The VSG damping coefficient is used to describe the variation in the output power when the frequency changes in units.The reactive power-voltage control of the VSG simulates the excitation regulation function of synchronous generators to realize the droop characteristics of reactive power and voltage amplitude.It changes the reactive power output to achieve the function of voltage stabilization.Reactive voltage control mainly adjusts the output voltage according to the VSG output voltage amplitude deviation ΔU and the reactive power difference ΔQ, which is shown in Fig. 2(b).According to Ref. 32, the model of the reactive power control part in Fig. 2 can be expressed as follows: where Pm is the mechanical power, Pe is the electromagnetic power, δ is the virtual power angle, ωB is the basic value of the electrical angular velocity, E is the virtual voltage, E 0 is the rated output voltage, Dn is the reactive power voltage droop coefficient, Q ref is the reference value of the reactive power, and Qe is the actual value of the reactive power.
To simplify the analysis, considering only the system reactance, the active power and reactive power output of the VSG to the power grid are expressed as follows: where U is the output voltage and XL is the equivalent reactance between the VSG and the power grid.
The system is required to maintain a specific frequency to guarantee stability in microgrids under fluctuating loads.The synchronous generator set regulates the active power and frequency of the system through the prime mover governor.The governor adjusts the mechanical power Pm according to the angular frequency deviation.The mechanical adjustment part can simulate rotor dynamics with a virtual mechanical phase angle.The prime mover-governor equation is where P ref is the given active power and Dm is the adjustment coefficient of the active power.

III. THE SYNERGETIC CONTROL OF VSG A. Synergy theory
The core content of synergetic control is to design a control manifold composed of system state variables.The control system converges to an equilibrium state along the designed control manifold.The control manifold contains the expected performance index of the system and reflects the relationship between the state variables that constitute the control manifold.
The controlled nonlinear system is where x is the system state variable, u is the control variable, and t is time.
In order to obtain the control law of the control variable u, it is necessary to construct an appropriate macro variable ψ.The purpose of the control is to make the system converge to the control manifold ψ(x, t) = 0 in a finite time.
After selecting the appropriate macro-variable, to ensure that the system can converge to the control manifold under the action of synergetic control, the dynamic process of macro-variable convergence is defined as follows: where T is the controller parameter, which represents the converging speed of the closed-loop system to the manifold specified by the macro-variable.
Assuming that the value of ψ is ψ(t 0 ) at the initial time t 0 , then (10) can be solved, From the above-mentioned equation, when the time t approaches infinity, the macro variable gradually decays to 0 with the attenuation index.That is, the system will move from a random initial state to a stable state.
From the above-mentioned analysis, it can be seen that the design parameter T determines the time to converge to the manifold after the system disturbance occurs.The smaller the T is, the greater the attenuation rate of macro-variables and the shorter the time for the system to converge to the manifold.The larger the T, the smaller the system's damping is, and the longer it will take for the system to reach stability.Therefore, the value of T should be less than the internal dynamic oscillation time of the system as far as possible.The system can quickly converge to the manifold and maintain the system's stability in the process of material or energy input and output.
The control rate of control variable u can be obtained by the above-mentioned formula to ensure that the system approaches and converges to the control manifold ψ(x, t) = 0 and moves along the control manifold.

B. Synergetic controller design
The controllers are designed using synergy theory.A reasonable macro-variable must be created when designing control schemes using synergetic control theory.A macro variable is a function containing state variables.The purpose of this paper is to explore the effectiveness of a new method, so the macro variable is selected as a linear combination of state variables, which is in line with previous research.

Macro variable selection
In Fig. 2, it can be seen that the active power frequency and reactive power voltage loop control are carried out separately, so it is easy to ignore the interaction between parameters.Therefore, to establish a connection between active and reactive power, this paper takes the linear combination of the virtual power angle δ, angular velocity ω, and excitation electromotive force E of the VSG as a macro variable to keep VSG stable in the transient process.The form is as follows: where K 1 , K 2 , and K 3 are control parameters, δ ref , ω ref , and E ref are the reference values of the virtual power angle, angular velocity, and excitation electromotive force, respectively.

Control law derivation
The control form in this paper is focused on the feedback effect because it can feed the output signal back to regulate the input, thereby affecting the control process.The negative feedback is achieved by introducing the difference between the output and expected value, which is transmitted in a reversed manner to the input so that the output plays the opposite role to the input, achieving stable and precise control of the system.
This section establishes the connection between the active power frequency and reactive power voltage controlled by the designed synergetic controller.The output u of the synergetic controller is added to the active power frequency control.A control block diagram of this process is shown in Fig. 3.As can be seen from Fig. 3, the introduction of the synergetic controller makes (5) transform into The following results can be obtained by substituting (12) into the convergence Eq. ( 10): Equation ( 15) can then be obtained by combining ( 13) and ( 14).
The control law u can be obtained as From Eq. ( 16), we can see that the first part is the active power reference value P ref , the second part is the active power output, the third part is the power angle rate of change, the fourth part is the macro-variable, and the fifth part is the voltage rate of change.A control block diagram of the VSG with synergetic control is shown in Fig. 4. As can be seen from Fig. 4, the improvement of this paper is reflected in two points: 1.The feedback loop is designed to feed back the controller output to the active power frequency control loop, achieving steady control with gradually decreasing errors; 2. It established a control connection between active and reactive power and realized centralized control of the two loops by considering the interaction between state parameters and systems.The controller designed in this paper is named synergetic droop control.

IV. SIMULATION RESULTS
The effectiveness of the VSG synergetic control strategy proposed in this study was verified by simulation in MATLAB.The reference value in the synergetic controller is given according to the calculated value of the AGC program.Droop control is the most mature and widely used form of control, and this paper is based on the original droop control to improve the control mode.Therefore, the comparison object is the conventional droop control method. 33n the simulation, two distinct types of disturbances were simulated, namely step disturbances and three-phase short circuit faults.The control results were analyzed by referring to the parameter setting form of the synchronous generator.voltage drops from 1 to 0.9.The output results and response curves of the two forms are shown in Fig. 5.
Figure 5 shows that, during power regulation and terminal voltage regulation, the system exhibits significant fluctuations in the active power, reactive power, power angle, and angular velocity of conventional droop control, and the time it takes to attain a stable state is longer.In contrast, the improved synergetic droop control regulation process is smooth, especially the active power and power angle, which do not show obvious overshooting.The time to reach a steady state is greatly shortened.The overshoot of angular velocity is reduced to a certain extent and can reach a stable state within 0.3s.The output variation in E under the two control forms is essentially the same, and the synergistic droop changes are more stable without significant fluctuations.In the synergetic droop mode, the control signal u and the macro-variable ψ of the system undergo changes in accordance with the system adjustment and return to a zero-value state within 0.5 s of the adjustment, which meets the system's requirements for control rate and macro-variable changes.

B. Three-phase short-circuit fault
As shown in the figure below, a short circuit occurs at the position shown in Fig. 6.When t = 1 s, a short circuit fault occurs at the side of the high-voltage line, and when t = 1.1, the fault line is cut off.
The basic parameter settings are the same as in the case of a load disturbance.The system response is shown in Fig. 7.
From Fig. 7, both control forms generate regulating oscillations when a short-circuit fault occurs in the system.The four state variables under conventional droop control have obvious overshoots, and the time to reach equilibrium is long, about 2.5s.However, there is no overshoot of active power, reactive power, or virtual voltage under synergetic droop control.The overshoot of angular velocity and power angle is much smaller than conventional droop control, and the time to reach equilibrium is less than 0.5s.The control signal u and macro-variable ψ output of the synergetic droop method are in accordance with the response of the state variables under its control, achieving a zero-value state within a brief period of time.Therefore, compared with conventional droop control, the designed synergetic controller has a better effect on three-phase short circuit faults.
From the simulation results, it can be seen that the introduction of the synergetic controller can significantly improve the stability of the system.It includes two aspects: one is to reduce the oscillation amplitude, and the other is to shorten the adjustment time.

V. SENSITIVITY STUDY
Based on the research content of this paper, we think that the uncertainty of the new model obtained in this article includes two aspects: the uncertainty of the VSG system parameters and the uncertainty of the synergetic control parameters.Therefore, we performed a sensitivity study to check the robustness of the synergetic controller under the system operating conditions.

A. System parameter
As shown in Eq. ( 5), parameters J and D are key factors that affect the transmission of the system.Therefore, it is necessary to study and analyze them.
Figures 8 and 9 show that alterations in system parameters have a certain effect on the efficiency of synergetic droop control.There exists a corresponding trend in the impact of parameters J and D on the system.The amplitude of the oscillation decreases as J and D increase within limits.The time it takes for the angular velocity to reach equilibrium is obviously shortened under the three-phase short circuit fault.Overall, the effect of a change in J on the system is significantly greater than that of D, and the control effect exhibits more remarkable fluctuation as it changes.The aforementioned findings indicate that synergetic droop control exerts a better control effect when the system parameters are larger.It is consistent with the actual situation.

B. Synergetic parameters
Considering the many limitations of optimization algorithms, such as the selection of objective functions and the determination of spatial dimensions, optimizing parameters using algorithms is a complex task.Therefore, this study provided a brief understanding and analysis of the impact of collaborative parameters, laying the foundation for the algorithm optimization work of later parameters.The system's sensitivity to the change in K value was also analyzed.The control parameters selected for the collaborative control macro variables included K 1 , K 2 , and K 3 .Therefore, this paper analyzes the influence of the value change of the three synergy coefficients on system stability.Table I and Fig. 10 show the overshoot and adjustment time of the active power, reactive power, power angle, and frequency under power step conditions when K 1 is 500, 1000, 2000, 3000, and 4000, respectively.Figures 11 and 12 analyze the changes in the active power, reactive power, power angle, and frequency with different K 2 and K 3 under the condition of a three-phase short-circuit fault.
It can be seen from Fig. 10 that the positive overshoot of frequency is more sensitive to the change of K 1 .With the increase in K 1 , the positive overshoot of frequency decreases to 0, and the other control objects are unaffected.In terms of negative overshoot, the power angle is more sensitive to the increase of K 1 , showing a gradually decreasing trend, especially when it changes from 500 to 1000.The influence of adjustment time is mainly reflected in active power and frequency.The change in active power shows a trend of first decreasing and then increasing.These research results provide a reference for the subsequent selection of K 1 .
It can be seen from Fig. 11 that with the increase of K 2 , the oscillation amplitude of the state variable decreases, and the stabilization time is short, so the control effect is better.The response of the active power and frequency shows that the control effect of K 2 = 100 is better than that of K 2 = 120.Therefore, when K 2 increases to a certain extent, the control effect will worsen, so there may be an optimal K 2 .In Fig. 12, with the decrease of K 3 , the oscillation amplitude of the state variable decreases, and the time for reactive power to reach the equilibrium state becomes longer.The time required for active power and power angle to reach equilibrium initially increases and then decreases.The effect of K 3 = 10 and K 3 = 1 is basically the same, with little change.
The optimal value can be roughly determined by comparing different control coefficients K, which provides a reference for the later system optimization control.

VI. MULTI-MACHINES ENVIRONMENT TESTING
There is a coupling relationship between multiple VSGs in a multi-machine system due to the shared bus, and the regulation of one unit will affect the operation of other units.To further verify the method's effectiveness, simulation tests were conducted on the joint power supply of two VSGs.The system structure is shown in Fig. 13.Settings: in the initial working condition, the active power of VSG 1 was 0.8, while that of VSG 2 was 0.6.The active power of VSG 1 was reduced to 0.5 when t = 1 s and then decreased again to 0.8 when t = 4 s.P e1 denotes the active power output of VSG 1 , and P e2 denotes  Figures 14(c), 14(g), and 14(h) show that when VSG 1 undergoes power regulation, VSG 2 also changes.This indicates that in a system with two VSGs, the regulation of one unit will affect the operation of the other unit, illustrating a coupling relationship between the two units.The output active power, power angle, and frequency of the two units show that the system under collaborative droop control has better control over power changes than conventional droop control.The state variable can achieve stable control within 0.5s and reduce output overshoot.From the overall output of the system, the overshoot of the bus frequency output does not decrease, but the regulation time is shortened.The time for the reactive power to reach a steadystate is not significantly diminished, but its overshoot is significantly reduced, and no regulation oscillation occurs.In addition, the reduction in the overshoot of the joint active power is accompanied by a reduction in the time to equilibrium state.Figures 14(a) and 14(b) indicate that the larger the power regulation amplitude of the unit, the greater the variation amplitude of the control signal and macro-variable output.Then, both values become 0 when the system reaches a stable state, which is consistent with the theory.
From the above-mentioned simulation results, it can be seen that the control method proposed in this paper is effective and can realize better control based on the original control.

VII. CONCLUSION
This work presents a synergetic controller based on synergy theory to improve the control loop.The interaction between parameters is fully considered, and the connection between active and reactive power links is established.This form is conducive to improving the comprehensiveness of the loop and parameter control.The simulation results show that the designed controller has better regulation performance than the traditional control, slight overshoot, and short regulation time, regardless of step disturbance, three-phase short circuit, or joint operation of multiple units.Moreover, the influence of system and controller parameters on control performance is also analyzed, which provides a reference for subsequent parameter selection.In the follow-up study, we will adopt advanced algorithms to optimize the parameters involved in the system and highlight control advantages compared to other better control methods.

Figure 1 FIG. 1 . 2 ©
Figure1shows a simplified schematic diagram of the gridconnected VSG.The inverter is composed of the battery, capacitor, and power semiconductor.The DC power source is regarded as

FIG. 2 .
FIG. 2. Control loops of a VSG: (a) active power control loop and (b) reactive power control loop.

FIG. 3 .
FIG. 3. Control block diagram of the active power-frequency and reactive power-voltage loops.

FIG. 4 .
FIG. 4. The central topology and control block of a VSG with the synergetic controller.

FIG. 8 .
FIG. 8. Response of active power and angular velocity under different parameters during power regulation in the system: (a) different D and (b) different J.

FIG. 9 .
FIG. 9. Response of active power and angular velocity under different parameters when a three-phase short circuit fault occurs in the system: (a) different D and (b) different J.

FIG. 11 .
FIG. 11.Change of the active power, reactive power, power angle, and frequency under different K 2 : (a) active power, (b) reactive power, (c) frequency, and (d) power angle.

FIG. 12 .
FIG. 12. Change of the active power, reactive power, power angle, and frequency under different K 3 : (a) active power, (b) reactive power, (c) frequency, and (d) power angle.

FIG. 13 .
FIG. 13.The structure of the multi-machine system.

FIG. 14 .
FIG. 14.Comparison of control effects between two control methods under multi-machine joint operation: (a) control signal u 1 and u 2 , (b) macro-variable ψ 1 and ψ 2 , (c) active power variation of VSG 1 and VSG 2 , (d) bus angular velocity, (e) active power joint output, (f) reactive power joint output, (g) power angle variation of VSG 1 and VSG 2 , and (h) angular velocity variation of VSG 1 and VSG 2 .

TABLE I .
Overshoot and adjustment time of the control object under different K 1 .