CN107070270A - A kind of output impedance bearing calibration for improving LCL type combining inverter stability - Google Patents

A kind of output impedance bearing calibration for improving LCL type combining inverter stability Download PDF

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CN107070270A
CN107070270A CN201710086350.8A CN201710086350A CN107070270A CN 107070270 A CN107070270 A CN 107070270A CN 201710086350 A CN201710086350 A CN 201710086350A CN 107070270 A CN107070270 A CN 107070270A
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mrow
omega
msubsup
grid
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CN107070270B (en
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杜燕
崔林波
杨向真
苏建徽
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Hefei University of Technology
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Hefei University of Technology
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
    • H02M7/00—Conversion of ac power input into dc power output; Conversion of dc power input into ac power output
    • H02M7/42—Conversion of dc power input into ac power output without possibility of reversal
    • H02M7/44—Conversion of dc power input into ac power output without possibility of reversal by static converters
    • H02M7/48—Conversion of dc power input into ac power output without possibility of reversal by static converters using discharge tubes with control electrode or semiconductor devices with control electrode
    • 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

Abstract

The invention discloses a kind of output impedance bearing calibration for improving LCL type combining inverter stability, it is characterized in that：In the LCL type combining inverter containing active damping, add the phase compensator based on grid-connected current, build the virtual impedance of series connection, impedance angle for correcting inverter output impedance, by the Phase margin for improving light current combining inverter off the net, improve grid-connected inverters electric current waveform quality, improve light current grid-connected inverter system stability off the net.

Description

A kind of output impedance bearing calibration for improving LCL type combining inverter stability
Technical field
The invention belongs to combining inverter control technology field, combining inverter is improved more particularly to one kind steady Qualitatively output impedance bearing calibration, for improving grid-connected inverters electric current waveform quality, improves light current combining inverter off the net The stability of a system.
Background technology
Based on regenerative resource, the distributed generation technology of such as wind energy, solar energy is mankind's reply energy crisis and ring One of important channel of border pollution.Combining inverter as the key interface equipment between distributed generation system and power network, its Performance quality directly decides grid-connected current quality.
With the grid-connected power of distributed power source increase and access mains position it is widely distributed, it is contemplated that longer transmission ＆ distribution Electric line, more isolating transformer, substantial amounts of Distributed-generation equipment are mounted on the factors such as grid entry point PCC points, and power network is more next The characteristic of light current net is more shown, electric network impedance can not ignore.Under weak grid conditions, electric network impedance and inverter output resistance It is anti-to produce reciprocation, cause the combining inverter of forceful electric power stable operation off the net to produce vibration, unstability.Therefore, parallel network reverse The design of device is, it is necessary to weaken the influence for eliminating electric network impedance to inverter performance, it is ensured that combining inverter is in all kinds of electric network impedances Under the conditions of stable operation.Under weak grid conditions, the stability of combining inverter is determined by two parts：One is that forceful electric power is off the net, and The stability (hereafter referred to collectively as inverter body stability) of net inverter；Two be that light current is off the net, and electric network impedance and inverter are defeated Go out stability caused by the reciprocation of impedance (hereafter referred to collectively as light current combining inverter stability off the net).When inverter body When stable, light current combining inverter stability off the net can be improved by changing the output impedance of inverter.However, grid-connected inverse at present Become in device impedance correction method, one is to need accurate measurement electric network impedance, and measurement error causes compensation inaccurate；Two be to need to draw Enter higher differentiation link, Project Realization is difficult.
The content of the invention
The present invention is that there is provided one kind raising LCL type combining inverter is steady to avoid the deficiency present in above-mentioned prior art Qualitatively output impedance bearing calibration.By introducing the phase compensator based on grid-connected current, correction inverter output impedance Impedance angle, improves the Phase margin of light current combining inverter off the net, improves grid-connected inverters electric current waveform quality, improves light current net Lower combining inverter stability.
The present invention adopts the following technical scheme that to solve technical problem：
The present invention improve LCL type combining inverter stability output impedance bearing calibration the characteristics of be：Containing active resistance In the LCL type combining inverter of Buddhist nun, the phase compensator based on grid-connected current is added, the virtual impedance of series connection is built, for school The impedance angle of positive inverter output impedance, by improving the Phase margin of light current combining inverter off the net, improves grid-connected inverters Current waveform quality, improves light current grid-connected inverter system stability off the net.
The characteristics of present invention improves the output impedance bearing calibration of LCL type combining inverter stability lies also in：The side Method is to carry out as follows：
Step 1：By grid entry point voltage sampling signal upcc_s, grid-connected current sampled signal ig_sWith capacitance current sampled signal ic_sBy current controller, output signal u is obtainedi1=(iref-ig_s)(Kp+Ki/s)-ic_s；
The capacitance current sampled signal ic_sRefer to the current sampling signal for constituting the filter capacitor C in LCL filter； irefIt is the command signal of grid-connected current, wherein irefAmplitude be setting value, irefPhase by grid entry point voltage sampling signal upcc_sObtained by phaselocked loop；KpFor grid-connected current adjuster Gi(s) proportionality coefficient, KiFor grid-connected current adjuster Gi(s) Integral coefficient, s is multifrequency domain variable.
Step 2：By grid-connected current sampled signal ig_sPass through phase compensator Gfc(s) output signal, is obtained for ui2；
Step 3:Modulated signal VmFor：Vm=ui1-ui2, by the modulated signal VmSPWM makers are input to, PWM is produced Signal controls the switching tube of inverter.
The characteristics of impedance angle bearing calibration of LCL combining inverters output impedance of the present invention, lies also in：
The phase compensator Gfc(s) it is expressed as：
In formula (1), K is phase compensator Gfc(s) gain, p is pole angular, and z is zero point angular frequency；The gain K, pole angular p and zero point angular frequency z parameter value are by following conditional decisions：
(1), gain K value is determined by high frequency Immunity Performance, as expressed by formula (2)：
|Gfc(s)|max=K≤Kmax (2)
In formula (2), | Gfc(s)|maxFor phase compensator Gfc(s) maximum modulus value, KmaxFor K maximum occurrences, in order to carry High Gfc(s) high frequency Immunity Performance, takes KmaxFor 1；
(2), pole angular p is not less than its minimum value pmin, pminAs expressed by formula (3)：
pmin=max { pcmin,pomin} (3)
In formula (3), pcminIt is the minimum p value determined by current tracking cut-off frequency, pominIt is by inverter body stability The minimum p value determined；
pcminObtained by formula (4)：
In formula (4)：
Each parameter set by LCL grid-connected inverter systems is respectively：Inverter side inductance L in LCL filter1And net side Inductance L2, the inverter bridge gain G for the inverter bridge being made up of switching tubeinv, grid-connected current downsampling factor Ki2, grid-connected current adjuster Gi(s) Proportional coefficient Kp, and grid-connected current adjuster Gi(s) integral coefficient Ki；
ωcThe cut-off angular frequency tracked for desired inverter current；
pominObtained by formula (5)：
In formula (5)：
Electric capacity C in LCL filter, and filter capacitor current sample COEFFICIENT Ki1For the system of LCL combining inverters Setup parameter；
PMosFor desired Zos/ZoPhase margin；
ωosFor virtual series impedance ZosWith inverter output impedance ZoIntersection point angular frequency, ωosObtained by formula (6)：
In formula (6), j is imaginary unit；
In Gfc(s) j ω are used inosReplace s operators and obtain Gfc(jωos)；
In Gi(s) j ω are used inosReplace s operators and obtain Gi(jωos)；
(3), zero point angular frequency z is not more than its maximum zmax, zmaxObtained by formula (7)：
In formula (7), ω0For current first harmonics angular frequency；GM is expectation current first harmonics gain；
(4), zero point angular frequency z is not less than its minimum value zmin, zminObtained by formula (8)：
In formula (8)：
ZgSet according to Grid-connection standards；
PM is desired impedance ratio Zg/ZoPhase margin；
ωgFor inverter output impedance ZoWith electric network impedance ZgIntersection point angular frequency, ωgObtained by formula (9)：
In formula (9), j is imaginary unit, Gi(jωg) it is in Gi(s) j ω are used ingS operators are replaced to obtain；
(5), pole angular p is not more than its maximum pmax, pmaxObtained by formula (10)：
Compared with the prior art, the present invention has the beneficial effect that：
The present invention adds the phase compensator based on grid-connected current, by remolding the impedance angle of inverter output impedance, has Effect improves the Phase margin of light current combining inverter off the net；The inventive method need not change original controller parameter, also not Need to measure electric network impedance value in real time, in the case of electric network impedance large-scope change, can guarantee that the stabilization of combining inverter Work, and the THD numerical value of grid-connected current declines to a great extent.
Brief description of the drawings
Fig. 1 is LCL grid-connected inverter system structured flowcharts in the present invention.
Fig. 2 is LCL combining inverter control block diagrams in the present invention.
Fig. 3 is the Bode diagram of inverter output impedance before and after addition phase compensator.
Fig. 4 is not added with grid-connected current simulation waveform during phase compensator for light current is off the net.
Grid-connected current simulation waveform when Fig. 5 adds phase compensator for light current is off the net.
Label in figure：1 DC source, 2 inverter bridges, 3 be LCL filter, 4 public electric wire nets, by ideal voltage source UgAnd power network Impedance ZgComposition；5 current controllers；6 phase compensators；7 be SPWM makers.
Embodiment
Referring to Fig. 1, light current single-phase LCL grid-connected inverter systems off the net include in the present embodiment：DC source 1, by four bands The inverter bridge 2 of the switching tube composition of fly-wheel diode, the LCL filter being made up of filter inductance L1, L2 and filter capacitor C, by Ideal voltage source UgWith electric network impedance ZgThe public electric wire net 4 of composition, current controller 5, phase compensator 6 and SPWM makers 7.
As depicted in figs. 1 and 2, the output impedance bearing calibration of LCL type combining inverter stability is improved in the present embodiment It is：In the LCL type combining inverter containing active damping, the phase compensator based on grid-connected current is added, the void of series connection is built Intend impedance, the impedance angle for correcting inverter output impedance, by improving the Phase margin of light current combining inverter off the net, changes Kind grid-connected inverters electric current waveform quality, improves light current combining inverter stability off the net.
The output impedance bearing calibration that LCL type combining inverter stability is improved in the present embodiment is carried out as follows：
Step 1：By grid entry point voltage sampling signal upcc_s, grid-connected current sampled signal ig_sWith capacitance current sampled signal ic_sBy current controller, output signal u is obtainedi1=(iref-ig_s)(Kp+Ki/s)-ic_s；
The capacitance current sampled signal ic_sRefer to the current sampling signal for constituting the filter capacitor C in LCL filter； irefIt is the command signal of grid-connected current, by grid entry point voltage sampling signal upcc_sObtained by phaselocked loop；KpAdjusted for grid-connected current Save device Gi(s) proportionality coefficient, KiFor grid-connected current adjuster Gi(s) integral coefficient, s is multifrequency domain variable.
Step 2：By grid-connected current sampled signal ig_sPass through phase compensator Gfc(s) output signal, is obtained for ui2；
Step 3:Modulated signal VmFor：Vm=ui1-ui2, by the modulated signal VmSPWM makers are input to, PWM is produced Signal controls the switching tube of inverter.
Phase compensator G proposed in the present embodimentfc(s), its expression formula is formula (1)：
In formula (1), K is phase compensator Gfc(s) gain, p is pole angular, and z is zero point angular frequency；Gain K, pole Point angular frequency p and zero point angular frequency z parameter value is by following conditional decisions：
(1), gain K value is determined by high frequency Immunity Performance, as expressed by formula (2)：
|Gfc(s)|max=K≤Kmax (2)
In formula (2), | Gfc(s)|maxFor phase compensator Gfc(s) maximum modulus value, KmaxFor K maximum occurrences, in order to carry High Gfc(s) high frequency Immunity Performance, takes KmaxFor 1.
(2), pole angular p is not less than its minimum value pmin, pminAs expressed by formula (3)：
pmin=max { pcmin,pomin} (3)
In formula (3), pcminIt is the minimum p value determined by current tracking cut-off frequency, pominIt is by inverter body stability The minimum p value determined.
pcminDetermined by current tracking cut-off frequency, p is bigger, and cut-off frequency is bigger.Because z has little influence on cut-off frequency, Make z=0, it is assumed that ωcThe cut-off angular frequency tracked for desired inverter current, pcminObtained by formula (4)：
In formula (4)：
Each parameter set by LCL grid-connected inverter systems is respectively：Inverter side inductance L in LCL filter1And net side Inductance L2, the inverter bridge gain G for the inverter bridge being made up of switching tubeinv, grid-connected current downsampling factor Ki2, grid-connected current adjuster Gi(s) Proportional coefficient Kp, and grid-connected current adjuster Gi(s) integral coefficient Ki。
pominDetermined by inverter body stability.The inverter body stability added after phase compensator can be by Zos/ ZoJudge, and its bigger stability of p is better.Because z has little influence on inverter body stability, z=0 is made, it is assumed that PMosBy a definite date The Z of prestigeos/ZoPhase margin, pominObtained by formula (5)：
In formula (5)：
Electric capacity C in LCL filter, and filter capacitor current sample COEFFICIENT Ki1For the system of LCL combining inverters Setup parameter；ωosFor virtual series impedance ZosWith inverter output impedance ZoIntersection point angular frequency, ωosObtained by formula (6)：
In formula (6), j is imaginary unit；
In Gfc(s) in, j ω are usedosReplace s operators and obtain Gfc(jωos)；
In Gi(s) in, j ω are usedosReplace s operators and obtain Gi(jωos)；
(3), phase-compensatory contro can influence system fundamental wave gain, to ensure preferable fundamental wave tracking performance, fundamental wave gain Design requirement should be met；Assuming that GM is desired current first harmonics gain, zero point angular frequency z is not more than its maximum zmax, zmaxBy Formula (7) is obtained：
In formula (7), ω0For current first harmonics angular frequency；
(4), light current combining inverter stability off the net is by Zg/ZoJudge, it is assumed that PM is desired impedance ratio Zg/ZoPhase Angle nargin, zero point angular frequency z is not less than its minimum value zmin, zminObtained by formula (8)：
In formula (8)：
ZgSet according to Grid-connection standards；
ωgFor inverter output impedance ZoWith electric network impedance ZgIntersection point angular frequency, ωgObtained by formula (9)：
In formula (9), j is imaginary unit, Gi(jωg) it is in Gi(s) j ω are used ingS operators are replaced to obtain；
(5), pole angular p is not more than its maximum pmax, pmaxObtained by formula (10)：
Parameter K, p, z span are obtained according to above formula (1)~(10).
To verify the validity of impedance angle bearing calibration proposed by the present invention, built in Matlab/simulink specified Capacity is 6kw single-phase grid-connected inverter models.Fig. 3 is the Bode diagram of inverter output impedance before and after addition phase compensator, its Middle ZoTo be added without the inverter output impedance of phase compensator, Z'osTo add the inverter output impedance of phase compensator, Zg For electric network impedance.From figure 3, it can be seen that when electric network impedance is 2.6mH, inverter output impedance and electric network impedance intersection point A The Phase margin at place brings up to 31.3 ° by original -0.6 °, and the stability of system is greatly improved.Fig. 4 is added without for light current is off the net Grid-connected current waveform during phase compensator, now electric network impedance is 2.6mH, it can be seen that grid-connected current is vibrated.Fig. 5 is The grid-connected current waveform that light current is off the net when adding phase compensator, because the impedance angle of the now output impedance of inverter is obtained very Good compensation, the stability of system is improved, and grid-connected current waveform quality also reaches very big improvement.Base proposed by the invention The combining inverter improved stability method corrected in impedance angle, can improve the stability of light current system off the net well.

Claims (3)

1. a kind of output impedance bearing calibration for improving LCL type combining inverter stability, it is characterized in that：Containing active damping In LCL type combining inverter, the phase compensator based on grid-connected current is added, the virtual impedance of series connection is built, it is inverse for correcting Become the impedance angle of device output impedance, by improving the Phase margin of light current combining inverter off the net, improve grid-connected inverters electric current Waveform quality, improves light current grid-connected inverter system stability off the net.
2. the output impedance bearing calibration according to claim 1 for improving LCL type combining inverter stability, it is characterized in that Methods described is to carry out as follows：
Step 1：By grid entry point voltage sampling signal upcc_s, grid-connected current sampled signal ig_sWith capacitance current sampled signal ic_sIt is logical Overcurrent controller, obtains output signal ui1=(iref-ig_s)(Kp+Ki/s)-ic_s；
The capacitance current sampled signal ic_sRefer to the current sampling signal for constituting the filter capacitor C in LCL filter；irefIt is The command signal of grid-connected current, wherein irefAmplitude be setting value, irefPhase by grid entry point voltage sampling signal upcc_sIt is logical Cross phaselocked loop acquisition；KpFor grid-connected current adjuster Gi(s) proportionality coefficient, KiFor grid-connected current adjuster Gi(s) integration system Number, s is multifrequency domain variable.
Step 2：By grid-connected current sampled signal ig_sPass through phase compensator Gfc(s) output signal, is obtained for ui2；
Step 3:Modulated signal VmFor：Vm=ui1-ui2, by the modulated signal VmSPWM makers are input to, pwm signal is produced Control the switching tube of inverter.
3. the output impedance bearing calibration according to claim 1 for improving LCL type combining inverter stability, its feature It is：The phase compensator Gfc(s) it is expressed as：
<mrow> <msub> <mi>G</mi> <mrow> <mi>f</mi> <mi>c</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>K</mi> <mfrac> <mrow> <mi>s</mi> <mo>+</mo> <mi>z</mi> </mrow> <mrow> <mi>s</mi> <mo>+</mo> <mi>p</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
In formula (1), K is phase compensator Gfc(s) gain, p is pole angular, and z is zero point angular frequency；The gain K, pole Point angular frequency p and zero point angular frequency z parameter value is by following conditional decisions：
(1), gain K value is determined by high frequency Immunity Performance, as expressed by formula (2)：
|Gfc(s)|max=K≤Kmax (2)
In formula (2), | Gfc(s)|maxFor phase compensator Gfc(s) maximum modulus value, KmaxFor K maximum occurrences, in order to improve Gfc (s) high frequency Immunity Performance, takes KmaxFor 1；
(2), pole angular p is not less than its minimum value pmin, pminAs expressed by formula (3)：
pmin=max { pcmin,pomin} (3)
In formula (3), pcminIt is the minimum p value determined by current tracking cut-off frequency, pominIt is to be determined by inverter body stability Minimum p value；
pcminObtained by formula (4)：
<mrow> <msub> <mi>p</mi> <mrow> <mi>c</mi> <mi>min</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>&amp;omega;</mi> <mi>c</mi> <mn>2</mn> </msubsup> <msub> <mi>KG</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;</mi> <mi>c</mi> </msub> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>K</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mrow> </msqrt> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mi>c</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
In formula (4)：
Each parameter set by LCL grid-connected inverter systems is respectively：Inverter side inductance L in LCL filter1With net side inductance L2, the inverter bridge gain G for the inverter bridge being made up of switching tubeinv, grid-connected current downsampling factor Ki2, grid-connected current adjuster Gi(s) Proportional coefficient Kp, and grid-connected current adjuster Gi(s) integral coefficient Ki；
ωcThe cut-off angular frequency tracked for desired inverter current；
pominObtained by formula (5)：
In formula (5)：
Electric capacity C in LCL filter, and filter capacitor current sample COEFFICIENT Ki1For the default of LCL combining inverters Parameter；
PMosFor desired Zos/ZoPhase margin；
ωosFor virtual series impedance ZosWith inverter output impedance ZoIntersection point angular frequency, ωosObtained by formula (6)：
<mrow> <mo>|</mo> <msub> <mi>G</mi> <mrow> <mi>f</mi> <mi>c</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>j&amp;omega;</mi> <mrow> <mi>o</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>|</mo> <mo>=</mo> <mo>|</mo> <msub> <mi>j&amp;omega;</mi> <mrow> <mi>o</mi> <mi>s</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mrow> <mi>o</mi> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>L</mi> <mn>2</mn> </msub> <mi>C</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mrow> <mi>o</mi> <mi>s</mi> </mrow> <mn>2</mn> </msubsup> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <msub> <mi>G</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>j&amp;omega;</mi> <mrow> <mi>o</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
In formula (6), j is imaginary unit；
In Gfc(s) j ω are used inosReplace s operators and obtain Gfc(jωos)；
In Gi(s) j ω are used inosReplace s operators and obtain Gi(jωos)；
(3), zero point angular frequency z is not more than its maximum zmax, zmaxObtained by formula (7)：
<mrow> <msub> <mi>z</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>K</mi> <mi>i</mi> </msub> <msub> <mi>p</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>&amp;omega;</mi> <mn>0</mn> </msub> <mi>K</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mn>20</mn> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
In formula (7), ω0For current first harmonics angular frequency；GM is expectation current first harmonics gain；
(4), zero point angular frequency z is not less than its minimum value zmin, zminObtained by formula (8)：
<mrow> <msub> <mi>z</mi> <mi>min</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>KG</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mo>&amp;lsqb;</mo> <mfrac> <mrow> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>+</mo> <mi>tan</mi> <mi> </mi> <mi>P</mi> <mi>M</mi> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mi>tan</mi> <mi> </mi> <mi>P</mi> <mi>M</mi> <mo>)</mo> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <msub> <mi>K</mi> <mi>i</mi> </msub> <msub> <mi>p</mi> <mi>min</mi> </msub> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>p</mi> <mi>min</mi> </msub> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msub> <mi>KG</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <msub> <mi>K</mi> <mi>p</mi> </msub> <msub> <mi>p</mi> <mi>min</mi> </msub> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
In formula (8)：
ZgSet according to Grid-connection standards；
PM is desired impedance ratio Zg/ZoPhase margin；
ωgFor inverter output impedance ZoWith electric network impedance ZgIntersection point angular frequency, ωgObtained by formula (9)：
<mrow> <mo>|</mo> <msub> <mi>Z</mi> <mi>g</mi> </msub> <mo>|</mo> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <msub> <mi>j&amp;omega;</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>L</mi> <mn>2</mn> </msub> <mi>C</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <msub> <mi>G</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>j&amp;omega;</mi> <mi>g</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>j&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msub> <mi>L</mi> <mn>1</mn> </msub> <mi>C</mi> </mrow> </mfrac> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
In formula (9), j is imaginary unit, Gi(jωg) it is in Gi(s) j ω are used ingS operators are replaced to obtain；
(5), pole angular p is not more than its maximum pmax, pmaxObtained by formula (10)：
<mrow> <msub> <mi>p</mi> <mi>max</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>KG</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mi>tan</mi> <mi> </mi> <mi>P</mi> <mi>M</mi> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mi>max</mi> </msub> <mo>+</mo> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <mi>tan</mi> <mi> </mi> <mi>P</mi> <mi>M</mi> <mo>+</mo> <msubsup> <mi>&amp;omega;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>(</mo> <mi>tan</mi> <mi> </mi> <mi>P</mi> <mi>M</mi> <mo>+</mo> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mo>)</mo> <mo>(</mo> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <msub> <mi>K</mi> <mi>i</mi> </msub> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mi>g</mi> <mn>2</mn> </msubsup> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>-</mo> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>K</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;omega;</mi> <mi>g</mi> </msub> <msub> <mi>CK</mi> <mrow> <mi>i</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>G</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>v</mi> </mrow> </msub> <mi>tan</mi> <mi> </mi> <mi>P</mi> <mi>M</mi> <mo>)</mo> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow> 2
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