CN112751364B - Virtual synchronous machine grid-connected control method based on linear/nonlinear active disturbance rejection control - Google Patents
Virtual synchronous machine grid-connected control method based on linear/nonlinear active disturbance rejection control Download PDFInfo
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
- H02J3/00—Circuit arrangements for ac mains or ac distribution networks
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- H—ELECTRICITY
- H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
- H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
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Abstract
The invention provides a virtual synchronous machine grid-connected control method based on linear/nonlinear active disturbance rejection control, which comprises the following steps: establishing a three-phase-locked loop controller which comprises a linear active disturbance rejection control part, a nonlinear active disturbance rejection control part and switching control of the linear active disturbance rejection control part and the nonlinear active disturbance rejection control part; setting relevant control parameters in the three-phase-locked loop according to the stability analysis and control performance of the three-phase-locked loop controller and the advantages of a comprehensive bandwidth method and an empirical method; designing a switching mechanism of a three-phase-locked loop controller based on the requirements of improving the dynamic performance, the steady-state filtering capability and the anti-interference performance of the three-phase-locked loop; and introducing the voltage amplitude, the frequency and the phase of the grid-connected point obtained by the three-phase-locked loop controller into a virtual synchronous machine power control loop through a logical relation to obtain a grid-connected control strategy.
Description
Technical Field
The invention relates to the field of grid-connected control of virtual synchronous generators, in particular to a grid-connected control strategy.
Background
In recent years, the contradiction between energy crisis, environmental pollution, climate warming and rapid economic development is increasingly prominent, and accelerating the development and efficient utilization of renewable energy sources becomes a major issue to be solved in the world energy field. The distributed power supply is connected to the urban power grid in various forms according to local conditions, so that local renewable energy can be fully utilized, and fossil energy consumption is replaced and reduced to the maximum extent. In areas with high proportion of renewable energy access, high load density and high power supply reliability requirements, the requirements of novel urban power grid architectures such as double rings, grids and petals and novel urban power grids accessed in a micro-grid mode are increasingly enhanced. Therefore, the distributed power supply taking full use of renewable energy sources as a core becomes an important component in a novel urban power grid in the future, and meanwhile, the novel urban power grid also becomes an important carrier for high-proportion and distributed access of renewable energy sources.
In recent years, a Virtual Synchronous Generator (VSG) concept is widely applied to distributed power supply control, so that a distributed power supply can simulate rotation inertia, frequency modulation and voltage regulation and excitation regulation characteristics of the synchronous generator, and adverse effects of distributed energy on a power grid are reduced. However, the VSG can only simulate the external characteristics of the synchronous generator in terms of control, and its main circuit is still composed of fragile power electronics and has a weak overload capability. In order to avoid grid-connected current impact, the distributed power supply based on VSG control needs to be synchronized with a power grid through a pre-synchronization unit before being merged into the power grid, and even if the voltage amplitude, the phase and the frequency of a grid-connected point change, the distributed power supply can be smoothly merged into the power grid.
The grid-connected control in the current practical engineering application generally adopts a phase-locked loop to track the voltage information of a locking power grid, and a presynchronization controller is designed based on the phase-locked loop to realize the synchronization process. The existing commonly used control strategies suitable for VSG grid connection include methods such as pre-synchronization control based on virtual impedance or grid-connected/off-grid mode switching of a multi-droop control inverter, and the like, and the methods are all used for smoothing the mode switching process by modifying a phase-locked loop, so that seamless switching between VSG grid connection and VSG grid disconnection is realized.
However, the method does not consider the influence of the LC filter, and only synchronizes the output voltage of the bridge arm with the voltage of the power grid, rather than synchronizes the output voltage of the inverter with the power grid. In addition, if the amplitude, phase and frequency of the grid voltage change to different degrees during the grid connection process of the VSG, it is also necessary to discuss whether the control strategy is feasible or not.
The Nonlinear active interference rejection control (NLADRC) has the advantages of high tracking accuracy, strong anti-interference capability and the like, and the Linear active interference rejection control (LADRC) has strong engineering applicability in the aspects of parameter setting and stability analysis. Therefore, the combination of linear active disturbance rejection/nonlinear active disturbance rejection control (L/NLADRC) can solve the above-mentioned problems encountered by the VSG during the grid-connection process well.
Disclosure of Invention
The invention aims to provide a virtual synchronous machine grid-connected control method based on linear/nonlinear active disturbance rejection control, which is used for realizing the rapid grid connection of VSG (voltage source generator) under the conditions of grid-connected point voltage amplitude, phase and sudden change and effectively reducing impact current. The purpose of the invention is realized by adopting the following technical scheme:
a virtual synchronous machine grid-connected control method based on linear/nonlinear active disturbance rejection control is characterized by comprising the following steps:
step 1: establishing a three-phase-locked loop controller which comprises a linear active disturbance rejection control part, a nonlinear active disturbance rejection control part and switching control of the linear active disturbance rejection control part and the nonlinear active disturbance rejection control part;
the linear active disturbance rejection control part comprises a linear Extended State Observer (ESO) and a linear state error feedback control rate (LSEF), wherein an expression of the linear state error feedback control rate is determined according to the following formula:
in the above formula, z1Is an estimate of the output of the phase locked loop, z2Is an estimate of the total disturbance, β1,β2E is the corresponding error gain, e is the feedback error, b is the system gain, and u is the intermediate variable of the system;
the linear state error feedback control rate is determined as follows:
in the above formula, kpIs a constant of proportionality, uqrefIs a reference value of the q-axis component, u0Is a state variable;
the nonlinear active disturbance rejection control part comprises a Nonlinear Extended State Observer (NESO) and a nonlinear state error feedback control rate (NLSEF), wherein the expression of the nonlinear state error feedback control rate can be determined according to the following formula:
in the above formula, fal is a nonlinear function, alpha1、α2、δ1Is a parameter in fal, λ1、λ2Are each z1And z2δ is the selected maximum error range;
the nonlinear state error feedback control rate is determined as follows:
in the above formula, KpIs a constant of proportionality, α3、δ2Also a parameter in fal, uqrefIs a reference value of the q-axis component, u0Is a state variable.
Step 2: setting relevant control parameters in the three-phase-locked loop according to the stability analysis and control performance of the three-phase-locked loop controller and the advantages of a bandwidth method and an empirical method;
and step 3: designing a switching mechanism of a three-phase-locked loop controller based on the requirements of improving the dynamic performance, the steady-state filtering capability and the anti-interference performance of the three-phase-locked loop; the switching mechanism is as follows: in the initial control stage, the reference input is roughly tracked by utilizing linear active disturbance rejection control, and then the nonlinear active disturbance rejection control is switched to improve the tracking precision and disturbance rejection capability; in the process of nonlinear active disturbance rejection control, when disturbance is large, or output state estimation error is large, or input signals and differential signals of each order deviate from corresponding output state estimation of the extended state observer to a far distance, in order to ensure system stability and control performance, switching is performed to linear active disturbance rejection control, and the switching mechanism is quantitatively expressed as: when setting M as the total disturbance threshold, | z2Less than or equal to M, less than or equal to | e |, and less than or equal to 1, and | u |qref-z1When the | is less than or equal to 1, switching the linear extended state observer to a nonlinear extended state observer, and switching the linear state error feedback control rate to the nonlinear state error feedback control rate; when z2|>M or | e |>1 or | uqref-z1|>1, when the linear state error feedback control rate is satisfied, switching from a nonlinear extended state observer to a linear extended state observer, and switching from the nonlinear state error feedback control rate to the linear state error feedback control rate;
and 4, step 4: and introducing the voltage amplitude, the frequency and the phase of the grid-connected point obtained by the three-phase-locked loop controller into a virtual synchronous machine power control loop through a logical relation to obtain a grid-connected control strategy.
In step 2, beta in linear active disturbance rejection control1、β2、kpThe setting method comprises the following steps: parameter beta1、β2By arranging the ESO pole of the linear extended state observer at-omegaoIs obtained; parameter kpBy arranging the ESO pole of the linear extended state observer at-omegacIs obtained. Wherein ω isoFor linear expansion of the ESO bandwidth, omega, of the state observercThe controller bandwidth of the linear error state feedback law.
In step 2, lambda in the nonlinear active disturbance rejection control1、λ2、α1、α2、α3、δ1、δ2、KpThe setting method comprises the following steps: the method is obtained by an empirical method and has the following principle: delta1And delta2The filter factor of the fal function filter is excessively small, so that the nonlinear active disturbance rejection controller is easy to generate a flutter phenomenon; increasing delta can make the filtering effect better, but at the same time, the delay of tracking is also increased; alpha is alpha1、α2The smaller the value is, the faster the tracking is, but the filtering effect is deteriorated; alpha is alpha3The smaller the value is, the stronger the anti-interference capability is, but the high-frequency flutter of the control quantity can be caused; kp=ωcn;λ1=3ωon、λ2=0.6ωon 2Corresponding, ωonFor non-linear extended state observer ESO bandwidth, omegacnController bandwidth for nonlinear error state feedback law
In step 4, the grid-connected control strategy is as follows: the grid-connected switch is in a disconnected state, and the initial active power given value PrefIf the frequency is zero, a PI controller is adopted to enable the output frequency and the phase of the virtual synchronous machine to quickly track the frequency and the phase of the power grid; given value of reactive power QrefWhen the voltage amplitude is zero, the three-phase-locked loop controller is adopted to enable the virtual synchronous machine to output the voltage amplitude to track the voltage amplitude and the phase of the power grid; and the grid-connected switch of the grid-connected point is closed, the active given value and the reactive given value of the virtual synchronous machine are increased, and the virtual synchronous machine transmits power to the power grid and participates in voltage regulation and frequency modulation of the power grid.
Compared with the prior art, the invention has the following beneficial effects:
according to the technical scheme provided by the invention, the VSG grid-connected control method based on the L/NADRC is realized, through designing a three-phase-locked loop based on linear/nonlinear active disturbance rejection control switching, integrating the advantages of a bandwidth method and an empirical method, setting relevant control parameters of the three-phase-locked loop, embedding grid-connected point voltage information acquired by the phase-locked loop into an active loop and a reactive loop of the VSG through certain logic control, realizing the rapid grid connection of the VSG under the conditions of amplitude, phase and sudden change of the grid-connected point voltage, and effectively reducing grid-connected impact current. And (4) carrying out simulation in a semi-physical simulation platform, and verifying the grid-connected control method.
Drawings
Fig. 1 is a schematic structural diagram of a three-phase-locked loop based on linear/nonlinear active disturbance rejection control switching;
FIG. 2 is a block diagram of active power/frequency control for L/NADRC based grid-tie control;
FIG. 3 is a control block diagram of an active loop in a grid connection process;
FIG. 4 is a reactive loop control block diagram during grid connection;
FIG. 5 is a partial simulation verification result diagram of a grid-connected control strategy based on L/NADRC.
Detailed Description
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention provides a VSG grid-connected control method based on L/NADRC, which comprises the following steps:
(1) the power control adopted by the VSG is active power/frequency control and reactive power/voltage control respectively;
the active power/frequency control includes frequency droop control and virtual inertia control, as shown by the following formula, by introducing a frequency droop systemNumber mpThe damping coefficient D and the virtual moment of inertia J simulate the frequency modulation and inertia links of the synchronous generator; reactive power/voltage control induced voltage droop coefficient nqAnd simulating the voltage regulation characteristic of the synchronous generator to generate virtual internal potential of the synchronous generator.
In the above formula, Pm、PeMechanical power and electromagnetic power (output power) for the VSG; prefIs a reference value of active power; omega, omeganThe angular frequency of the internal potential of the VSG and the rated angular frequency; J. d is respectively a rotational inertia and a damping coefficient; qref、QeOutputting reactive power for the reactive power reference value and the VSG; eoE is VSG no-load electromotive force and a calculated voltage reference value; m isp,nqThe frequency droop coefficient and the voltage droop coefficient are obtained; θ is the VSG output voltage phase.
(2) The controller for establishing the three-phase-locked loop comprises a linear active disturbance rejection control part, a nonlinear active disturbance rejection control part and switching control of the linear active disturbance rejection control part and the nonlinear active disturbance rejection control part, wherein the linear active disturbance rejection control part comprises a linear Extended State Observer (ESO) and a linear state error feedback control rate (LSEF), and the nonlinear active disturbance rejection control part comprises a Nonlinear Extended State Observer (NESO) and a nonlinear state error feedback control rate (NLSEF). And according to the stability analysis and control performance of the three-phase-locked loop controller, the advantages of a bandwidth method and an empirical method are integrated, and relevant control parameters in the three-phase-locked loop are set.
As shown in fig. 1, the three-phase-locked loop controller is composed of lacrc and NADRC together.
U in the figureabcTo the grid-connected point voltage umIs uabcAmplitude, ωgIs uabcAngular frequency, θ is PLL output phase, ud,uqIs uabcThe dq axis component of (a).
When the switch is arranged at the position 1, the controller selected by the three-phase-locked loop is LADRC; when the switch is placed in position 2, the controller selected by the three-phase-locked loop is NADRC.
LADRC consists of a Linear Extended State Observer (LESO) and a linear state error feedback control rate (LSEF);
wherein the expression of LESO can be determined as follows:
in the above formula, z1Is an estimate of the output of the phase locked loop, z2Is an estimate of the total disturbance, β1,β2For the corresponding error gain, e is the feedback error, b is the system gain, and u is the intermediate variable of the system.
The expression of LSEF may be determined as follows:
in the above formula, kpIs a constant of proportionality, uqrefIs a reference value of the q-axis component, u0Is a state variable.
Further, the narro consists of a nonlinear extended state observer (NLESO) and a nonlinear state error feedback control rate (NLSEF);
where the expression for NLESO can be determined as follows:
in the above formula, fal is a nonlinear function, alpha1、α2、δ1Is a parameter in fal, z1,z2Also an estimate of the output and disturbance, lambda1、λ2δ is the selected maximum error range for the corresponding error gain.
The expression for NLSEF can be determined as follows:
in the above formula, fal is a nonlinear function, alpha1、α2、δ1Is a parameter in fal, z1,z2Also an estimate of the output and disturbance, lambda1、λ2δ is the selected maximum error range for the corresponding error gain.
The parameters required to be adjusted comprise: beta in LADRC1、β2、b、kpThe specific method comprises the following steps:
the parameter b is obtained by establishing a system model for calculation; parameter beta1、β2By placing the observer pole at- ωoIs obtained; parameter kpBy arranging the observer pole at- ωcIs obtained. Wherein ω isoFor linear expansion of the ESO bandwidth, omega, of the state observercController bandwidth for linear error state feedback law.
The parameters required to be adjusted comprise: lambda in NADRC1、λ2、α1、α2、α3、δ1、δ2、KpThe specific method comprises the following steps:
the method is obtained by an empirical method, the principle is that delta is a filtering factor of a fal function filter, and the flutter phenomenon easily occurs to a nonlinear active disturbance rejection controller due to the fact that the value is too small; increasing delta can make the filtering effect better, but at the same time, the delay of tracking is also increased; alpha (alpha) ("alpha")1、α2The smaller the value is, the faster the tracking is, but the filtering effect is deteriorated; alpha is alpha3The smaller the value, the stronger the anti-interference capability, but the high-frequency flutter of the control quantity can be caused; k isp=ωcn;λ1=3ωon、λ2=0.6ωon 2Corresponding, ωonFor non-linear extended state observer ESO bandwidth, omegacnController bandwidth for the nonlinear error state feedback law.
(3) The switching mechanism of LADRC and NADRC in the three-phase-locked loop based on the linear/nonlinear active disturbance rejection control switching is as follows:
in the initial control stage, the LADRC is utilized to approximately track the reference input and then is switched to the NADRC so as to improve the tracking precision and the anti-interference capability; when the disturbance is large, or the output state estimation error is large, or the input signal and each order differential signal thereof deviate from the output state estimation of the corresponding extended state observer to a far extent, the LADRC is switched to ensure the system stability and the control performance.
When z2M is less than or equal to I, e is less than or equal to 1 and uqref-z1When the | is less than or equal to 1, switching from LESO to NLESO and from LSEF to NLSEF; when | z2|>M or | e | |)>1 or | uqref-z1|>When 1 is established, the signal is switched from NLESO to LESO and from NLSEF to LSEF.
(4) The grid-connected control strategy is shown in FIG. 2, wherein S1、S2、S3、S4Being a switch, ωg、θgU calculated for three phase locked loop based on linear/nonlinear active disturbance rejection control switchingabcFrequency and phase.
In the grid connection process, the grid connection switch is in a disconnected state, and the initial active power given value PrefIs zero, switch S1、S2Breaking, S3、S4Closing, as shown in FIG. 3, enables the VSG output frequency omega to track omega quickly and accurately by using a simple PI controllerg. Further, θ will begSubtracting the VSG phase theta, sending the difference to a PI controller, and adding the difference to omega to adjust thetagThe difference value of theta is zero, so that the theta is ensured to quickly and accurately track the thetag;
Given value of reactive power QrefIs zero, switch S5Put in the linear active disturbance rejection/nonlinear active disturbance rejection control three-phase-locked loop, as shown in FIG. 4, the calculated uabcAmplitude VgSide, make the VSG output voltage amplitude Vg;
Ensure VSG output voltage amplitude, frequency and phase to track u fast without errorabcAfter that, the grid connection process is finished, and the switch S is turned on1、S2Closure, S3、S4Breaking, S5Is placed in EoAnd (3) side. The grid-connected point grid-connected switch is closed, the VSG active given value and the VSG reactive given value are increased, and power is transmitted to the power gridAnd participate in voltage regulation and frequency modulation of the power grid.
In order to verify the effectiveness of the VSG grid-connected control method based on the L/NADRC, the strategy is verified on a semi-physical simulation platform, and partial results are shown in FIG. 5. The grid voltage is an ideal situation. Closing a grid-connected switch for 0.1s, and merging the VSG into a power grid; 0.2sVSG power output (P: 20kW, Q: 1 kVar); 0-0.1s is a pre-synchronous tracking stage, and 0.1s-0.2s is used for observing the impact current.
The result proves that the VSG grid-connected control method provided by the invention can quickly and stably track the frequency and the phase of the voltage of the power grid, meets the requirements of quick grid connection and small impact current at the grid-connected time, and can well realize the grid-connected process of the VSG when the amplitude, the frequency and the phase of the voltage of the power grid change.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting the same, and although the present invention is described in detail with reference to the above embodiments, those of ordinary skill in the art should understand that: modifications and equivalents may be made to the embodiments of the invention without departing from the spirit and scope of the invention, which is to be covered by the claims.
Claims (2)
1. A virtual synchronous machine grid-connected control method based on linear/nonlinear active disturbance rejection control is characterized by comprising the following steps:
step 1: establishing a three-phase-locked loop controller which comprises a linear active disturbance rejection control part, a nonlinear active disturbance rejection control part and switching control of the linear active disturbance rejection control part and the nonlinear active disturbance rejection control part;
the linear active disturbance rejection control part comprises a linear Extended State Observer (ESO) and a linear state error feedback control rate (LSEF), wherein an expression of the linear state error feedback control rate is determined according to the following formula:
in the above formula, z1Is an estimate of the output of the phase locked loop, z2Is an estimate of the total disturbance, β1,β2E is the corresponding error gain, e is the feedback error, b is the system gain, and u is the intermediate variable of the system;
the linear state error feedback control rate is determined as follows:
in the above formula, kpIs a constant of proportionality, uqrefIs a reference value of the q-axis component, u0Is a state variable;
the nonlinear active disturbance rejection control part comprises a Nonlinear Extended State Observer (NESO) and a nonlinear state error feedback control rate (NLSEF), wherein the expression of the nonlinear state error feedback control rate can be determined according to the following formula:
in the above formula, fal is a nonlinear function, alpha1、α2、δ1Is a parameter in fal, λ1、λ2Are each z1And z2δ is the selected maximum error range;
the nonlinear state error feedback control rate is determined as follows:
in the above formula, KpIs a constant of proportionality, α3、δ2Also a parameter in fal, uqrefIs a reference value of the q-axis component, u0Is a state variable;
and 2, step: according to the stability analysis and control performance of the three-phase-locked loop controller and the advantages of a comprehensive bandwidth method and an empirical method, relevant control parameters in the three-phase-locked loop are set, and the method comprises the following steps:
beta in linear active disturbance rejection control1、β2、kpThe setting method comprises the following steps: parameter beta1、β2By arranging the ESO pole of the linear extended state observer at-omegaoIs obtained; parameter kpBy arranging the ESO pole of the linear extended state observer at-omegacIs obtained; wherein ω isoFor linear expansion of the ESO bandwidth, omega, of the state observercController bandwidth for linear error state feedback law;
λ in nonlinear active disturbance rejection control1、λ2、α1、α2、α3、δ1、δ2、KpThe setting method comprises the following steps: the method is obtained by an empirical method and has the following principle: delta1And delta2The filter factor of the fal function filter is excessively small, so that the nonlinear active disturbance rejection controller is easy to generate a flutter phenomenon; increasing delta can make the filtering effect better, but at the same time, the delay of tracking is also increased; alpha is alpha1、α2The smaller the value is, the faster the tracking is, but the filtering effect is deteriorated; alpha is alpha3The smaller the value, the stronger the anti-interference capability, but the high-frequency flutter of the control quantity can be caused; kp=ωcn;λ1=3ωon、λ2=0.6ωon 2Corresponding, ωonFor non-linear extended state observer ESO bandwidth, omegacnController bandwidth for the nonlinear error state feedback law;
and step 3: designing a switching mechanism of a three-phase-locked loop controller based on the requirements of improving the dynamic performance, the steady-state filtering capability and the anti-interference performance of the three-phase-locked loop; the switching mechanism is as follows: in the initial control stage, the reference input is roughly tracked by utilizing linear active disturbance rejection control, and then the nonlinear active disturbance rejection control is switched to improve the tracking precision and disturbance rejection capability; in the process of nonlinear active disturbance rejection control, when the disturbance is large, or the error of the output state estimation is large, or the input signal and each order differential signal thereof deviate from the output state estimation of the corresponding extended state observer to a long distance, the stability and the controllability of the system are ensuredThe switching can be linear active disturbance rejection control, and the switching mechanism is quantitatively expressed as: when setting M as the total disturbance threshold, | z2Less than or equal to M, less than or equal to | e |, and less than or equal to 1, and | u |qref-z1When the | is less than or equal to 1, switching the linear extended state observer to a nonlinear extended state observer, and switching the linear state error feedback control rate to the nonlinear state error feedback control rate; when z2|>M or | e | |)>1 or | uqref-z1|>1, when the linear state error feedback control rate is satisfied, switching from a nonlinear extended state observer to a linear extended state observer, and switching from the nonlinear state error feedback control rate to the linear state error feedback control rate;
and 4, step 4: and introducing the voltage amplitude, the frequency and the phase of the grid-connected point obtained by the three-phase-locked loop controller into a virtual synchronous machine power control loop through a logical relation to obtain a grid-connected control strategy.
2. The virtual synchronous machine grid-connection control method according to claim 1, wherein in step 4, the grid-connection control strategy is as follows: the grid-connected switch is in a disconnected state, and the initial active power given value PrefIf the frequency is zero, a PI controller is adopted to enable the output frequency and the phase of the virtual synchronous machine to quickly track the frequency and the phase of the power grid; given value of reactive power QrefWhen the voltage amplitude is zero, the three-phase-locked loop controller is adopted to enable the virtual synchronous machine to output the voltage amplitude to track the voltage amplitude and the phase of the power grid; and the grid-connected switch of the grid-connected point is closed, the active given value and the reactive given value of the virtual synchronous machine are increased, and the virtual synchronous machine transmits power to the power grid and participates in voltage regulation and frequency modulation of the power grid.
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