CN112072649A - Proportional integral frequency-locked loop based on synchronous coordinate system and modeling method thereof - Google Patents

Proportional integral frequency-locked loop based on synchronous coordinate system and modeling method thereof Download PDF

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CN112072649A
CN112072649A CN202010902464.7A CN202010902464A CN112072649A CN 112072649 A CN112072649 A CN 112072649A CN 202010902464 A CN202010902464 A CN 202010902464A CN 112072649 A CN112072649 A CN 112072649A
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frequency
phase angle
observation
error
voltage
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CN112072649B (en
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全相军
吴在军
唐成红
陈永华
王智
胡秦然
窦晓波
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Southeast University
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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
    • 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
    • H02J3/38—Arrangements for parallely feeding a single network by two or more generators, converters or transformers
    • H02J3/381—Dispersed generators
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03L—AUTOMATIC CONTROL, STARTING, SYNCHRONISATION, OR STABILISATION OF GENERATORS OF ELECTRONIC OSCILLATIONS OR PULSES
    • H03L7/00—Automatic control of frequency or phase; Synchronisation
    • H03L7/06—Automatic control of frequency or phase; Synchronisation using a reference signal applied to a frequency- or phase-locked loop
    • H03L7/08—Details of the phase-locked loop
    • H03L7/085—Details of the phase-locked loop concerning mainly the frequency- or phase-detection arrangement including the filtering or amplification of its output signal
    • 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
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]

Abstract

The invention discloses a proportional integral frequency locking ring based on a synchronous coordinate system and a modeling method thereof, wherein the proportional integral frequency locking ring under the synchronous coordinate system, namely PI-SRF-FLL, is provided, a nonlinear error dynamic system is established, a Lyapunov function is defined, and the stability of the Lyapunov function is proved; a small signal transfer function of the system is established through an auxiliary variable method, so that the influence of phase angle transformation, frequency change and amplitude change on the observation of the system dq axis voltage and frequency observation phase angle is given, and an accurate FLL parameter design rule is given. The method disclosed by the invention can provide a more accurate model to guide the parameter design of the frequency-locked loop system and improve the performance of the frequency-locked loop.

Description

Proportional integral frequency-locked loop based on synchronous coordinate system and modeling method thereof
Technical Field
The invention relates to a distributed power grid-connected inverter control technology, in particular to a proportional-integral frequency locking ring based on a synchronous coordinate system and a modeling method thereof.
Background
When the distributed power supply is connected to a large power grid through a grid-connected converter, the grid-connected converter needs to control the power grid synchronization system to accurately estimate amplitude phase information of fundamental wave positive sequence voltage of the power grid. At present, the most widely used power grid synchronization systems are mainly phase-locked loops (PLLs) and frequency-locked loop (FLL) technologies.
The PLL generally uses a synchronous coordinate transformation as a Phase Detector (PD) unit to obtain an error between a power grid phase angle and an estimated phase angle as a feedback, so as to implement power grid phase angle synchronization, and therefore, a mainstream PLL technology is mostly a synchronous reference frame PLL (SRF-PLL) based on a synchronous coordinate system. Unlike PLL, FLL detects an error between the grid frequency and the estimated frequency as a feedback to achieve synchronization of the grid frequency, a grid phase angle is generally obtained by direct calculation, and acquisition of the frequency error is generally achieved in a stationary coordinate system, so FLL is mostly implemented in the stationary coordinate system. In recent years, the phase-locked loop technology is continuously improved and developed, so that the performance of the phase-locked loop is continuously improved, while the frequency-locked loop technology is relatively slowly developed, and the reason for the slow development of the frequency-locked loop technology is the coordinate system implemented by the frequency-locked loop technology. The PLL technique is mostly implemented based on a synchronous coordinate system, and in the synchronous coordinate system, ac is converted into dc, so that a small signal model can be conveniently established, and thus a PLL loop filter (in-loop filter) can be easily designed, thereby improving the PLL filtering performance. The FLL is generally implemented in a stationary coordinate system (related to frequency error detection), and each physical quantity in the stationary coordinate system is an alternating-current signal, so that it is difficult to establish a small-signal model, so that the design of a filter becomes difficult, and the development of a frequency-locked loop is hindered.
Disclosure of Invention
The purpose of the invention is as follows: an object of the present invention is to provide a proportional-integral frequency-locked loop based on a synchronous coordinate system.
The invention also aims to provide a modeling method of the proportional-integral frequency-locked loop based on the synchronous coordinate system, and aims to provide a more accurate model to guide the parameter design of a frequency-locked loop system and improve the performance of the frequency-locked loop.
The technical scheme is as follows: the proportional integral frequency locking loop based on the synchronous coordinate system comprises a pre-low pass filter under the synchronous coordinate system and a frequency observation loop based on the proportional integral, wherein the input grid voltage uαβObtaining the voltage u under the dq coordinate system after dq transformation (the phase angle of the dq transformation comes from a frequency-locked loop)dq，udqObtaining an observed value of dq axis voltage after passing through a low-pass filter
Figure BDA0002660230640000021
Observed value
Figure BDA0002660230640000022
And the observation error edqAfter calculation, a signal x is obtainedaIAfter the signal passes through a proportional-integral controller, a frequency observed value is obtained
Figure BDA0002660230640000023
Frequency observation
Figure BDA0002660230640000024
After differential operation, the phase angle is adjusted to the initial phase angle
Figure BDA0002660230640000025
Obtaining the phase angle generated by the FLL after operation
Figure BDA0002660230640000026
Feedback on input grid voltage uαβDq transformation of (1).
A modeling method of a proportional-integral frequency-locked loop based on a synchronous coordinate system comprises the following steps:
(1) providing a proportional-integral frequency locking ring based on a synchronous coordinate system, namely PI-SRF-FLL, establishing a nonlinear error dynamic system of the proportional-integral frequency locking ring, defining a Lyapunov function, and proving the stability of the proportional-integral frequency locking ring;
(2) and establishing a small signal transfer function of the PI-SRF-FLL nonlinear error dynamic system by an auxiliary variable method, thereby giving the influence of phase angle transformation, frequency change and amplitude change on dq axis voltage and frequency observation phase angle observation of the system and giving an accurate FLL parameter design criterion.
Further, the method for establishing the nonlinear error dynamic system of the proportional-integral frequency-locked loop based on the synchronous coordinate system in the step (1) comprises the following steps:
proportional integral frequency-locked loop based on a synchronous coordinate system, a pre-low-pass filter under the synchronous coordinate system and a frequency observation loop based on proportional integral, wherein the pre-low-pass filter is expressed by the following differential equation:
Figure BDA0002660230640000027
wherein, the superscript ". cndot.represents differentiation, udq＝ud+juqIs a complex variable of dq-axis voltage in a synchronous coordinate system, udIs d-axis voltage, uqIs the q-axis voltage; e.g. of the typedq＝ed+jeqAs dq-axis observation error, edAs d-axis observation error, eqThe q-axis observation error;
Figure BDA0002660230640000028
for the dq-axis voltage observations,
Figure BDA0002660230640000029
as an observed value of the d-axis voltage,
Figure BDA00026602306400000210
is a q-axis voltage observation;
Figure BDA00026602306400000211
is the dq-axis voltage observed differential, k is the coefficient; the frequency observation loop is:
Figure BDA00026602306400000212
wherein the content of the first and second substances,
Figure BDA00026602306400000213
in order to differentiate the frequency observations,
Figure BDA00026602306400000214
for frequency observation value, D is proportional gain coefficient of frequency observation loop, design
Figure BDA00026602306400000215
V is the input voltage amplitude, kfIs an adjustable coefficient; x is the number ofaIThe calculation formula is as follows:
Figure BDA00026602306400000216
im () takes an imaginary part, "+" denotes a complex conjugate,
Figure BDA00026602306400000217
is the complex conjugate of the dq-axis voltage observations;
wherein k is greater than 0, D is greater than 0, the formula (1) is a complex variable low-pass filter, and k is the cut-off frequency of the complex variable low-pass filter; meanwhile, the equation (1) is also regarded as a dq axis voltage observer, and k is an observation gain at the moment; considering the dq transformation, the dq axis voltage is expressed as:
Figure BDA0002660230640000031
wherein u isαβ＝uα+juβ＝Vcosθ+jV sinθ＝Vejθuαβ＝uα+jvβ＝V cosθ+jV sinθ＝VejθIs a voltage complex variable u under a static alpha beta coordinate systemαIs a complex variable of the alpha-axis voltage, uβIs a beta axis voltage complex variable;
Figure BDA0002660230640000032
then for the dq transformation operator to be the one,
Figure BDA0002660230640000033
to observe the phase angle error between the voltage and the input voltage,
Figure BDA0002660230640000034
in order to be a frequency observation value,
Figure BDA0002660230640000035
is the phase angle generated by the FLL, which is not necessarily the same as the grid phase angle theta,
Figure BDA0002660230640000036
is an adjustable initial phase angle; and (4) obtaining the derivation of two sides of the formula (3):
Figure BDA0002660230640000037
wherein the content of the first and second substances,
Figure BDA0002660230640000038
to differentiate the dq-axis voltage in the synchronous coordinate system,
Figure BDA0002660230640000039
is the differential of the phase angle error between the observed voltage and the input voltage;
Figure BDA00026602306400000310
is the frequency error, the error dynamics from which equation (1) is derived is:
Figure BDA00026602306400000311
wherein the content of the first and second substances,
Figure BDA00026602306400000312
for differentiation of the dq-axis observation error, the grid frequency is assumed to be a direct current signal
Figure BDA00026602306400000313
The frequency error dynamic equation is：
Figure BDA00026602306400000314
Wherein the content of the first and second substances,
Figure BDA00026602306400000315
is the differential of the frequency error; the real number form of the PI-SRF-FLL nonlinear error dynamic system error dynamic equation is as follows:
Figure BDA00026602306400000316
wherein the content of the first and second substances,
Figure BDA00026602306400000317
is edIs the differential of the d-axis observation error,
Figure BDA00026602306400000318
is the differential of the d-axis voltage,
Figure BDA00026602306400000319
is the differential of the q-axis voltage.
Further, the phase angle generated by PI-SRF-FLL
Figure BDA00026602306400000320
Static error exists between the phase angle of the real power grid and the phase angle of the real power grid, and the static error passes through the initial phase angle
Figure BDA00026602306400000321
Adjustment, phase angle error
Figure BDA00026602306400000322
Obtained by the following formula:
Figure BDA00026602306400000323
thus, there are:
Figure BDA0002660230640000041
wherein the content of the first and second substances,
Figure BDA0002660230640000042
the observed value of the voltage amplitude is regarded as the observed value, and meanwhile, the accurate observed value of the power grid phase angle is represented as:
Figure BDA0002660230640000043
consider that:
Figure BDA0002660230640000044
the two sides of the formula (8) are derived:
Figure BDA0002660230640000045
wherein the content of the first and second substances,
Figure BDA0002660230640000046
is the differential of the phase angle error,
Figure BDA0002660230640000047
as a differential of the observed value of the q-axis voltage,
Figure BDA0002660230640000048
is the differential of the d-axis voltage observations;
general formula
Figure BDA0002660230640000049
Substitution (12) with (11) yields:
Figure BDA00026602306400000410
equation (13) describes the dq-axis voltage observed when the input voltage phase angle changes
Figure BDA00026602306400000411
The change of the phase angle is obtained by substituting the equations (11) and (13) into the equation (10), and the phase angle transfer function is as follows:
Figure BDA00026602306400000412
wherein the content of the first and second substances,
Figure BDA00026602306400000413
is a power grid phase angle observed value;
dq-axis voltage observation oriented with grid voltage
Figure BDA00026602306400000414
Then obtained from the following formula:
Figure BDA00026602306400000415
from equation (16), the observed dq-axis voltage of the grid voltage orientation is the magnitude of the input voltage, since its imaginary part is always zero.
Further, the lyapunov function is defined in the step (1):
Figure BDA00026602306400000416
wherein p is1，p2，p3Is an arbitrary positive number, let p1＝p2>0 and p3＝k/D>0, then there are:
Figure BDA0002660230640000051
obviously, if:
Figure BDA0002660230640000052
then
Figure BDA0002660230640000053
The system is stable, all errors converge to zero, and the observation target of the system is realized; since the frequency error on the right side of the above equation monotonically increases, k only needs to satisfy the following condition:
Figure BDA0002660230640000054
wherein, ω isemaxIndicating the frequency observation error maximum.
Further, the step (2) is specifically as follows:
definition of
Figure BDA0002660230640000055
Wherein the content of the first and second substances,
Figure BDA0002660230640000056
is the complex conjugate of the dq-axis observation error, xaRIs xaReal part of (x)aIIs xaImaginary part of, then xaThe dynamic equation of (a) is:
Figure BDA0002660230640000057
the time constant is determined by k only, and the transfer function matrix of the above formula is:
Figure BDA0002660230640000058
wherein s is a Laplacian variable;
let s be 0 in the above formula to obtain xaThe steady state values are:
Figure BDA0002660230640000059
wherein the content of the first and second substances,
Figure BDA00026602306400000510
denotes xaThe steady-state average value of (a) is,
Figure BDA00026602306400000511
denotes xaRThe steady-state average value of (a) is,
Figure BDA00026602306400000512
denotes xaIA steady state average value of;
considering general ωeK, therefore
Figure BDA00026602306400000513
Thus, there are:
Figure BDA0002660230640000061
if the parameter D is selected as:
Figure BDA0002660230640000062
and d ═ kfk, then the system is normalized to a typical second order system with a closed loop transfer function:
Figure BDA0002660230640000063
Figure BDA0002660230640000064
for frequency observation, the design rule of the PI-SRF-FLL parameter is as follows:
(a) designing a pre-filter parameter k according to a preset time constant or a filtering requirement;
(b) d is selected according to the time requirement.
Has the advantages that: compared with the prior art, the invention has the following advantages:
(1) the invention creatively designs the proportional integral frequency-locked loop and the modeling method thereof based on the synchronous coordinate system, each physical quantity in the model is a direct current quantity, the modeling analysis and the filtering design of the system are convenient, and the simulation and the test prove that the PI-SRF-FLL designed by the invention can successfully observe the frequency, the phase angle and the amplitude information of the power grid, and has better performance.
(2) The transfer function obtained by derivation can be used for analyzing the influence of the power grid synchronization system on the inverter control, and the analysis method adopted by the invention can be used for analyzing the traditional FLL technology.
(3) The method is also applicable to the reduced-order-generalized-integrator (ROGI) based FLL under the static coordinate system.
Drawings
FIG. 1 is a schematic diagram of a proportional-integral frequency-locked loop PI-SRF-FLL based on a synchronous coordinate system designed by the present invention;
FIG. 2 is a PI-SRF-FLL equivalent small signal block diagram designed by the present invention.
Detailed Description
The invention is described in detail below with reference to the figures and specific embodiments.
As shown in fig. 1, the proportional-integral frequency-locked loop (PI-SRF-FLL) based on the synchronous coordinate system includes a pre-low-pass filter under the synchronous coordinate system and a frequency observation loop based on proportional-integral. Input network voltage uαβObtaining the voltage u under the dq coordinate system after dq transformation (the phase angle of the dq transformation comes from a frequency-locked loop)dq，udqObtaining an observed value of dq axis voltage after passing through a low-pass filter
Figure BDA0002660230640000071
Observed value
Figure BDA0002660230640000072
And the observation error edqAfter calculation, a signal x is obtainedaIThe signal passes through a proportional integral controller to obtain the frequencyObserved value
Figure BDA0002660230640000073
Frequency observation
Figure BDA0002660230640000074
After differential operation, the phase angle is adjusted to the initial phase angle
Figure BDA0002660230640000075
Obtaining the phase angle generated by the FLL after operation
Figure BDA0002660230640000076
Feedback on input grid voltage uαβDq transformation of (1).
As shown in fig. 2, it is a PI-SRF-FLL equivalent small signal block diagram for analyzing and designing frequency-locked loop parameters.
In the following formulas, "·" above the letter represents a differential, "+" represents a complex conjugate, and "-" above the letter represents a steady-state average value.
The invention relates to a modeling method of a proportional-integral frequency-locked loop based on a synchronous coordinate system, which comprises the following steps:
(1) providing a proportional-integral frequency-locked loop (PI-SRF-FLL) based on a synchronous coordinate system, establishing a nonlinear error dynamic system of the proportional-integral frequency-locked loop, defining a Lyapunov function, and proving the stability of the proportional-integral frequency-locked loop;
the main implementation steps are as follows:
the proposed PI-SRF-FLL of the present invention shown in fig. 1 comprises a pre-low pass filter under a synchronous coordinate system, which is represented by the following differential equation (where "·" on the letter in the following equation represents a differential), and a proportional-integral-based frequency observation loop:
Figure BDA0002660230640000077
wherein, the superscript ". cndot.represents differentiation, udq＝ud+juqIs a complex variable of dq-axis voltage in a synchronous coordinate system, udIs d-axis voltage, uqIs the q-axis voltage; e.g. of the typedq＝ed+jeqAs dq-axis observation error, edAs d-axis observation error, eqThe q-axis observation error;
Figure BDA0002660230640000078
for the dq-axis voltage observations,
Figure BDA0002660230640000079
as an observed value of the d-axis voltage,
Figure BDA00026602306400000710
is a q-axis voltage observation;
Figure BDA00026602306400000711
is the dq-axis voltage observed differential, k is the coefficient; the frequency observation loop is:
Figure BDA00026602306400000712
wherein the content of the first and second substances,
Figure BDA00026602306400000713
in order to differentiate the frequency observations,
Figure BDA00026602306400000714
for frequency observation value, D is proportional gain coefficient of frequency observation loop, design
Figure BDA00026602306400000715
V is the input voltage amplitude, kfIs an adjustable coefficient; x is the number ofaIThe calculation formula is as follows:
Figure BDA00026602306400000716
im () takes an imaginary part, "+" denotes a complex conjugate,
Figure BDA00026602306400000717
is the complex conjugate of the dq-axis voltage observations;
wherein k >0 and D > 0. Formula (1) is a complex variable low-pass filter, and k is the cut-off frequency of the complex variable low-pass filter; it can also be considered as a dq-axis voltage observer, where k is the observed gain. Considering the dq transformation, the dq axis voltage can be expressed as:
Figure BDA0002660230640000081
wherein u isαβ＝uα+juβ＝V cosθ+jV sinθ＝VejθIs a voltage complex variable u under a static alpha beta coordinate systemαIs a complex variable of the alpha-axis voltage, uβIs a beta axis voltage complex variable;
Figure BDA0002660230640000082
then for the dq transformation operator to be the one,
Figure BDA0002660230640000083
to observe the phase angle error between the voltage and the input voltage,
Figure BDA0002660230640000084
in order to be a frequency observation value,
Figure BDA0002660230640000085
is the phase angle generated by the FLL, which is not necessarily the same as the grid phase angle theta,
Figure BDA0002660230640000086
is an adjustable initial phase angle; and (4) obtaining the derivation of two sides of the formula (3):
Figure BDA0002660230640000087
wherein the content of the first and second substances,
Figure BDA0002660230640000088
to differentiate the dq-axis voltage in the synchronous coordinate system,
Figure BDA0002660230640000089
for observing electricityDifferentiation of the phase angle error between the voltage and the input voltage;
Figure BDA00026602306400000810
is the frequency error, the error dynamics from which equation (1) is derived is:
Figure BDA00026602306400000811
wherein the content of the first and second substances,
Figure BDA00026602306400000812
for the differentiation of the dq-axis observation error, the frequency error dynamic equation is (assuming that the grid frequency is a direct current signal)
Figure BDA00026602306400000813
)：
Figure BDA00026602306400000814
The real number form of the nonlinear error dynamic equation of the PI-SRF-FLL nonlinear error dynamic system is as follows:
Figure BDA00026602306400000815
wherein the content of the first and second substances,
Figure BDA00026602306400000816
is edIs the differential of the d-axis observation error,
Figure BDA00026602306400000817
is the differential of the d-axis voltage,
Figure BDA00026602306400000818
is the differential of the q-axis voltage.
Defining the Lyapunov function:
Figure BDA00026602306400000819
p1，p2，p3is an arbitrary positive number, let p1＝p2>0 and p3＝k/D>0, then there are:
Figure BDA00026602306400000820
obviously, if:
Figure BDA0002660230640000091
then
Figure BDA0002660230640000092
The system is stable, all errors are converged to zero, and the observation target of the system is realized. Since the frequency error on the right side of the above equation monotonically increases, k only needs to satisfy the following condition:
Figure BDA0002660230640000093
wherein, ω isemaxIndicating the frequency observation error maximum.
(2) Through an auxiliary variable method, a small signal transfer function of the system is established, so that the influence of phase angle transformation, frequency variation and amplitude variation on the observation of the system dq axis voltage and frequency observation phase angle is given, and an accurate FLL parameter design rule is given. The main implementation steps are as follows:
definition of
Figure BDA0002660230640000094
Wherein the content of the first and second substances,
Figure BDA0002660230640000095
is the complex conjugate of the dq-axis observation error, xaRIs xaReal part of (x)aIIs xaImaginary part of, then xaThe dynamic equation of (a) is:
Figure BDA0002660230640000096
the time constant of which is determined only by k. The transfer function matrix of the above formula is (s is a labella variable):
Figure BDA0002660230640000097
wherein s is a Laplacian variable;
let s be 0 in the above formula, x can be obtainedaSteady state values are (the "-" on the letter indicates the steady state average):
Figure BDA0002660230640000098
wherein the content of the first and second substances,
Figure BDA0002660230640000099
denotes xaThe steady-state average value of (a) is,
Figure BDA00026602306400000910
denotes xaRThe steady-state average value of (a) is,
Figure BDA00026602306400000911
denotes xaIA steady state average value of;
considering general ωeK, therefore
Figure BDA00026602306400000912
Thus, there are:
Figure BDA00026602306400000913
the above equation is of paramount importance for FLL systems because it describes the mathematical relationship between the frequency error to the frequency observation input variable. If the parameter D is selected as:
Figure BDA0002660230640000101
and d ═ kfk, then the system can be normalized to a typical second order system with a closed loop transfer function:
Figure BDA0002660230640000102
the equation describes the closed loop dynamic characteristic of frequency observation, and is a series connection of two first-order inertia links. The design rule of the PI-SRF-FLL parameter is as follows: 1) designing a pre-filter parameter k according to a preset time constant or a filtering requirement; 2) d is selected according to the time requirement.
Like the traditional FLL, the phase angle generated by the PI-SRF-FLL is not subjected to closed loop feedback of the grid phase angle
Figure BDA0002660230640000103
Static error exists between the phase angle of the real power grid and the phase angle of the real power grid, and the static error can pass through the initial phase angle
Figure BDA0002660230640000104
Adjustment, value of phase angle error
Figure BDA0002660230640000105
Can be obtained by the following formula:
Figure BDA0002660230640000106
wherein the content of the first and second substances,
Figure BDA0002660230640000107
in order to be able to correct the phase angle error,
Figure BDA0002660230640000108
as an observed value of the d-axis voltage,
Figure BDA0002660230640000109
is a q-axis voltage observation;
thus, there are:
Figure BDA00026602306400001010
wherein the content of the first and second substances,
Figure BDA00026602306400001011
can be considered as an observed value of the voltage amplitude. Meanwhile, the accurate observed value of the phase angle of the power grid can be expressed as follows:
Figure BDA00026602306400001012
consider that:
Figure BDA00026602306400001013
the two-sided derivation of equation (18) can be found:
Figure BDA00026602306400001014
wherein the content of the first and second substances,
Figure BDA00026602306400001015
is the differential of the phase angle error,
Figure BDA00026602306400001016
as a differential of the observed value of the q-axis voltage,
Figure BDA00026602306400001017
is the differential of the d-axis voltage observations;
substituting equations (15) and (21) into (22) yields:
Figure BDA0002660230640000111
equation (23) describes the dq-axis voltage observed when the input voltage phase angle changes
Figure BDA0002660230640000112
A change in phase angle. Substituting equations (21) and (23) into (20) yields a phase angle transfer function as:
Figure BDA0002660230640000113
wherein the content of the first and second substances,
Figure BDA0002660230640000114
is a power grid phase angle observed value;
Figure BDA0002660230640000115
wherein u isdqestA dq axis voltage oriented for the grid voltage;
for a grid voltage oriented dq axis value, equation (15) represents u as the input voltage angle changesdqestChange of (1), obviously udqestDirected to the grid voltage, i.e. uqest0; and dq axis voltage observation oriented with grid voltage
Figure BDA0002660230640000116
Then obtained from the following formula:
Figure BDA0002660230640000117
from equation (26), the observed dq-axis voltage of the grid voltage orientation is the magnitude of the input voltage, since its imaginary part is always zero.

Claims (6)

1. A proportional integral frequency-locking loop based on a synchronous coordinate system is characterized by comprising a pre-low-pass filter under the synchronous coordinate system and a proportional integral frequency-locking loop based on the proportional integralFrequency observation loop, in which the mains voltage u is inputαβObtaining the voltage u under the dq coordinate system after dq transformationdq，udqObtaining an observed value of dq axis voltage after passing through a low-pass filter
Figure FDA0002660230630000011
Observed value
Figure FDA0002660230630000012
And the observation error edqAfter calculation, a signal x is obtainedaIAfter the signal passes through a proportional-integral controller, a frequency observed value is obtained
Figure FDA0002660230630000013
Frequency observation
Figure FDA0002660230630000014
After differential operation, the phase angle is adjusted to the initial phase angle
Figure FDA0002660230630000015
Obtaining the phase angle generated by the FLL after operation
Figure FDA0002660230630000016
Feedback on input grid voltage uαβDq transformation of (1).
2. A modeling method of a proportional-integral frequency-locked loop based on a synchronous coordinate system is characterized by comprising the following steps:
(1) providing a proportional-integral frequency locking ring based on a synchronous coordinate system, namely PI-SRF-FLL, establishing a nonlinear error dynamic system of the proportional-integral frequency locking ring, defining a Lyapunov function, and proving the stability of the proportional-integral frequency locking ring;
(2) and establishing a small signal transfer function of the PI-SRF-FLL nonlinear error dynamic system by an auxiliary variable method, thereby giving the influence of phase angle transformation, frequency change and amplitude change on dq axis voltage and frequency observation phase angle observation of the system and giving an accurate FLL parameter design criterion.
3. The modeling method of the proportional-integral frequency-locked loop based on the synchronous coordinate system as claimed in claim 2, wherein the establishing method of the nonlinear error dynamic system of the proportional-integral frequency-locked loop based on the synchronous coordinate system in step (1) comprises:
proportional integral frequency-locked loop based on a synchronous coordinate system, a pre-low-pass filter under the synchronous coordinate system and a frequency observation loop based on proportional integral, wherein the pre-low-pass filter is expressed by the following differential equation:
Figure FDA0002660230630000017
wherein, the superscript ". cndot.represents differentiation, udq＝ud+juqIs a complex variable of dq-axis voltage in a synchronous coordinate system, udIs d-axis voltage, uqIs the q-axis voltage; e.g. of the typedq＝ed+jeqAs dq-axis observation error, edAs d-axis observation error, eqThe q-axis observation error;
Figure FDA0002660230630000018
for the dq-axis voltage observations,
Figure FDA0002660230630000019
as an observed value of the d-axis voltage,
Figure FDA00026602306300000110
is a q-axis voltage observation;
Figure FDA00026602306300000111
is the dq-axis voltage observed differential, k is the coefficient; the frequency observation loop is:
Figure FDA00026602306300000112
wherein the content of the first and second substances,
Figure FDA00026602306300000113
in order to differentiate the frequency observations,
Figure FDA00026602306300000114
for frequency observation value, D is proportional gain coefficient of frequency observation loop, design
Figure FDA0002660230630000021
kfFor adjustable coefficients, V is the input voltage amplitude, xaIThe calculation formula is as follows:
Figure FDA0002660230630000022
im () takes an imaginary part, "+" denotes a complex conjugate,
Figure FDA0002660230630000023
is the complex conjugate of the dq-axis voltage observations;
wherein k is greater than 0, D is greater than 0, the formula (1) is a complex variable low-pass filter, and k is the cut-off frequency of the complex variable low-pass filter; meanwhile, the equation (1) is also regarded as a dq axis voltage observer, and k is an observation gain at the moment; considering the dq transformation, the dq axis voltage is expressed as:
Figure FDA0002660230630000024
wherein u isαβ＝uα+juβ＝Vcosθ+jVsinθ＝VejθIs a voltage complex variable u under a static alpha beta coordinate systemαIs a complex variable of the alpha-axis voltage, uβIs a beta axis voltage complex variable;
Figure FDA0002660230630000025
then for the dq transformation operator to be the one,
Figure FDA0002660230630000026
for observing the phase between the voltage and the input voltageThe error in the angle is a function of,
Figure FDA0002660230630000027
in order to be a frequency observation value,
Figure FDA0002660230630000028
is the phase angle generated by the FLL, which is not necessarily the same as the grid phase angle theta,
Figure FDA0002660230630000029
is an adjustable initial phase angle; and (4) obtaining the derivation of two sides of the formula (3):
Figure FDA00026602306300000210
wherein the content of the first and second substances,
Figure FDA00026602306300000211
to differentiate the dq-axis voltage in the synchronous coordinate system,
Figure FDA00026602306300000212
is the differential of the phase angle error between the observed voltage and the input voltage;
Figure FDA00026602306300000213
is the frequency error, the error dynamics from which equation (1) is derived is:
Figure FDA00026602306300000214
wherein the content of the first and second substances,
Figure FDA00026602306300000215
for differentiation of the dq-axis observation error, the grid frequency is assumed to be a direct current signal
Figure FDA00026602306300000216
Then the frequency errorThe dynamic equation is:
Figure FDA00026602306300000217
wherein the content of the first and second substances,
Figure FDA00026602306300000218
is the differential of the frequency error; the real number form of the PI-SRF-FLL nonlinear error dynamic system error dynamic equation is as follows:
Figure FDA00026602306300000219
wherein the content of the first and second substances,
Figure FDA00026602306300000220
is edIs the differential of the d-axis observation error,
Figure FDA00026602306300000221
is the differential of the d-axis voltage,
Figure FDA00026602306300000222
is the differential of the q-axis voltage.
4. The modeling method of proportional-integral frequency-locked loop based on synchronous coordinate system of claim 3, wherein the phase angle generated by PI-SRF-FLL
Figure FDA0002660230630000031
Static error exists between the phase angle of the real power grid and the phase angle of the real power grid, and the static error passes through the initial phase angle
Figure FDA0002660230630000032
Adjustment, phase angle error
Figure FDA0002660230630000033
Obtained by the following formula:
Figure FDA0002660230630000034
thus, there are:
Figure FDA0002660230630000035
wherein the content of the first and second substances,
Figure FDA0002660230630000036
the method can be regarded as an observed value of the voltage amplitude, and meanwhile, the accurate observed value of the power grid phase angle is represented as follows:
Figure FDA0002660230630000037
consider that:
Figure FDA0002660230630000038
the two sides of the formula (8) are derived:
Figure FDA0002660230630000039
wherein the content of the first and second substances,
Figure FDA00026602306300000310
is the differential of the phase angle error,
Figure FDA00026602306300000311
as a differential of the observed value of the q-axis voltage,
Figure FDA00026602306300000312
is the differential of the d-axis voltage observations;
general formula
Figure FDA00026602306300000313
Substitution (12) with (11) yields:
Figure FDA00026602306300000314
equation (13) describes the dq-axis voltage observed when the input voltage phase angle changes
Figure FDA00026602306300000315
The change of the phase angle is obtained by substituting the equations (11) and (13) into the equation (10), and the phase angle transfer function is as follows:
Figure FDA00026602306300000316
wherein the content of the first and second substances,
Figure FDA00026602306300000317
is a power grid phase angle observed value;
dq-axis voltage observation oriented with grid voltage
Figure FDA00026602306300000318
Then obtained from the following formula:
Figure FDA0002660230630000041
from equation (16), the observed dq-axis voltage of the grid voltage orientation is the magnitude of the input voltage, since its imaginary part is always zero.
5. The modeling method of proportional-integral frequency-locked loop based on synchronous coordinate system as claimed in claim 2, wherein the lyapunov function is defined in step (1):
Figure FDA0002660230630000042
wherein p is1，p2，p3Is an arbitrary positive number, let p1＝p2>0 and p3＝k/D>0, then there are:
Figure FDA0002660230630000043
obviously, if:
Figure FDA0002660230630000044
then
Figure FDA0002660230630000045
The system is stable, all errors converge to zero, and the observation target of the system is realized; since the frequency error on the right side of the above equation monotonically increases, k only needs to satisfy the following condition:
Figure FDA0002660230630000046
wherein, ω isemaxIndicating the frequency observation error maximum.
6. The modeling method of the proportional-integral frequency-locked loop based on the synchronous coordinate system as claimed in claim 2, wherein the step (2) is specifically:
definition of
Figure FDA0002660230630000047
Wherein the content of the first and second substances,
Figure FDA0002660230630000048
is the complex conjugate of the dq-axis observation error,xaRis xaReal part of (x)aIIs xaImaginary part of, then xaThe dynamic equation of (a) is:
Figure FDA0002660230630000049
the time constant is determined by k only, and the transfer function matrix of the above formula is:
Figure FDA00026602306300000410
wherein s is a Laplacian variable;
let s be 0 in the above formula to obtain xaThe steady state values are:
Figure FDA0002660230630000051
wherein the content of the first and second substances,
Figure FDA0002660230630000052
denotes xaThe steady-state average value of (a) is,
Figure FDA0002660230630000053
denotes xaRThe steady-state average value of (a) is,
Figure FDA0002660230630000054
denotes xaIA steady state average value of;
considering general ωeK, therefore
Figure FDA0002660230630000055
Thus, there are:
Figure FDA0002660230630000056
if the parameter D is selected as:
Figure FDA0002660230630000057
and d ═ kfk, then the system is normalized to a typical second order system with a closed loop transfer function:
Figure FDA0002660230630000058
Figure FDA0002660230630000059
for frequency observation, the design rule of the PI-SRF-FLL parameter is as follows:
(a) designing a pre-filter parameter k according to a preset time constant or a filtering requirement;
(b) d is selected according to the time requirement.
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