Research Article The Virtual Resistance Control Strategy ...downloads.hindawi.com/journals/mpe/2014/350367.pdf · When grid voltage increases to % of the nominal voltage, wind turbine

Research ArticleThe Virtual Resistance Control Strategy for HVRT ofDoubly Fed Induction Wind Generators Based on ParticleSwarm Optimization

Zhen Xie and Xue Li

School of Electric Engineering and Automation Hefei University of Technology Hefei 230009 China

Correspondence should be addressed to Zhen Xie ppsd2003xiesinacom

Received 17 January 2014 Revised 9 May 2014 Accepted 25 May 2014 Published 10 July 2014

Academic Editor Marcelo M Cavalcanti

Copyright copy 2014 Z Xie and X Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Grid voltage swell will cause transient DC flux component in the doubly fed induction generator (DFIG) stator windings creatingserious stator and rotor current and torque oscillation which is more serious than influence of the voltage dip It is found thatvirtual resistance manages effectively to suppress rotor current and torque oscillation avoid the operation of crowbar circuit andenhance its high voltage ride through technology capability In order to acquire the best virtual resistance value the excellentparticle library (EPL) of dynamic particle swarm optimization (PSO) algorithm is proposed It takes the rotor voltage and rotorcurrent as two objectives which has a fast convergence performance and high accuracy Simulation and experimental results verifythe effectiveness of the virtual resistance control strategy

1 Introduction

Many studies have focused on the impact of grid voltagesags [1ndash4] on wind turbine generators and low voltage ride-through (LVRT) technology but insufficient attention hasbeen given to the influence of grid voltage swell on thesame generators and high voltage ride-through (HVRT)technology In wind power systems grid voltage swell may becaused by single-phase line-to-ground faults sudden removalof large loads and the use of a large capacitor compensator [56] The commonly used DFIG [7] which is connected to thegrid through the converter follows the principle of grid volt-age sag faults When grid voltage swells its transient processalso causes the current or voltage to swell in the stators androtors of the DFIG To avoid these problems and protect theconverter wind turbine generators can be automatically takenoff-grid and perform grid-connected operation after failurerecovery However this off-grid strategy no longer meets thestandards of the grid connection of modern large-scale windpower generation [8]Mostwind power integration standardsrequire the LVRT ability of wind farms the expansion of windpower generation capacity and the improvement of integra-tion standards will soon make HVRT ability a prerequisite

for wind farms The wind power integration standards ofEON in Germany require the HVRT ability of wind tur-bine generators when grid voltage increases to 12 pu windturbine generators should maintain long-term fault ride-through and absorb a certain amount of reactive power inhigh-voltage conditionsMoreover the ratio between reactivecurrent and the change rate of grid voltage should be 2 1Thisstandard is the control requirement of reactive compensationin high-voltage conditions Australia first established HVRTstandards for grid-connected wind turbine generators [9]When grid voltage increases to 130 of the nominal voltagewind turbine generators should maintain fault ride-throughin 60ms andprovide a sufficiently high fault recovery currentThis standard requires the capability of resisting and runningthrough the high voltage of wind turbine generators

DFIG can achieve HVRT by increasing hardware circuitsand improving system control (Figure 1) A chopper circuiton the DC side of the converter was used by [5] to preventvoltage on the DC side from increasing because of the reverseflow of the grid side converter energy triggered by a grid volt-age swell Reference [10] proposed that the series resistanceof current transformers on the rotor side could prevent rotorovercurrent thus preventing the rotor side converter from

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 350367 11 pageshttpdxdoiorg1011552014350367

2 Mathematical Problems in Engineering

is

+

Rs L1s L1r

s r

minus

Lm120595s 120595r

j120596120595r

+

minus

Rrir

+ minus

Figure 1 DFIG equivalent circuit at static stator-oriented referenceframe

losing control over the generator because crowbar resistancewill be used if a grid fault occurs This series resistancecan also continuously provide reactive power to the gridduring the fault and reduce torque ripple References [611] examined and compared the static synchronous com-pensator (STATCOM) and dynamic voltage restorer (DVR)schemes during a grid voltage swell In the DVR schemegenerator voltage is maintained by compensating for thevoltage difference between normal and fault conditions Inthe STATCOM scheme grid voltage decreases by controllingthe reactive current injected into the power grid Especiallywhen grid impedance is relatively low both schemes signif-icantly increase the cost of using the whole set of hardwareThe STATCOM scheme is uneconomical because it requirescurrent far stronger than the rated current of wind turbinegenerators The enhanced controlling strategy of the DFIGunder a grid voltage swell is proposed in [12] decouplingcontrol in fault conditions is achieved by establishing anaccuratemodel of the dynamic process of theDFIG rotor fieldcurrent This model considers the compensation dosage ofthe rotor field current change on the basis of the conventionalcontroller Proportional-integral control should be used innormal operations and vector hysteresis control in grid faults[13]

In the HVRT of DFIG converter overcurrent can beavoided by using the crowbar on the rotor side thusrapidly increasing the generator torque and generating anuncontrollable output of active and reactive power whenthe grid voltage swells To solve these problems and ensurethe excellent HVRT of the DFIG this study analyzes theelectromagnetic transient course of the DFIG under the gridvoltage swell and examines the control strategy of activedamping in the rotor converter according to the passivedamping scheme of the series resistance of the generator rotor[14 15] Thus rotor overvoltage is managed when the gridvoltage swells the rotor overcurrent is maximally controlledand the HVRT of the DFIG is improved To acquire the bestvirtual resistance value this paper adopts the EPL of dynamicPSO algorithm for the multiobjective optimization design ofparameters It takes the rotor voltage and rotor current astwo objectives for the integrated optimization As the resultof the multiple iterations the algorithm can be caught inlocal optimumThus the algorithm is improved in this paperwhich is on the basis of the excellent library and dynamicimprovement This makes the algorithm more suitable forthe dynamic changes of the actual cases Therefore the localoptimum of PSO algorithm can not only be avoided but alsothe speed and accuracy of the optimum can be improved

PI

PI

SVPWM

DC+

+

minus

abc

ilowastrd

ilowastrq

rdc

rdc

rqc

rqc

lowastrd

lowastrq

lowastrD

lowastrQ

2r2s

3s2r

3s2r

+

120579r

120579r

120579r

120579s120579sr

ird

irq

isd

isq

DFIG

PLLC

AB

Feedforwarddecoupling

Statorflux

calculation

Ra

Ra

d

dt

d

dt

120596r

120596sr

120595sd

120595sq iAiB iC

us

iaib

ic

Figure 2 Block diagram of the DFIG control system based onvirtual resistance

Simulation and experimental results confirm the feasibility ofthis damping scheme

2 Dynamic Analysis of DFIG duringGrid Voltage Swell

DFIG equivalent circuit is shown in Figure 1 to analyze thedynamic behavior during grid voltage swell For simplicityall rotor variables are converted to stator side and linearmagnetic circuits are assumed [16] With the application ofmotor convention the DFIG model can be expressed as

V119904= 119877119904

119894119904+

119889

119889119905

119904 (1)

V119903= 119877119903

119894119903+

119889

119889119905

119903minus 119895120596119903119903 (2)

119904= 119871119904

119894119904+ 119871119898

119894119903 (3)

119903= 119871119898

119894119904+ 119871119903

119894119903 (4)

The symbols and their corresponding meanings are asfollows

V voltage vector119894 current vector120596119903 rotor electrical speed

flux vector119877 resistance119871 inductance

Subscripts 119903 and 119904 are the rotor and stator respectivelyThe rotor voltage can be calculated from (2) as follows

V119903=

119871119898

119871119904

(119901 minus 119895120596119903) 119904+ [119877119903+ 120590119871119903(119901 minus 119895120596

119903)]

119894119903 (5)

where 120590 = 1 minus 1198712119898119871119904119871119903

Mathematical Problems in Engineering 3

Under normal condition the rotor voltage term inducedby the stator flux is expressed as

V1199030= 119895120596119904119897

119871119898

119871119904

119904= 119904

119871119898

119871119904

119881119904119890119890

119895120596119904119905

(6)

where 120596119904119897= 120596119904minus120596119903and 119904 is the slip (119904 = 120596

119904119897120596119904) and119881

119904119890is the

stator voltage amplitudeThe rotor voltage at rotor terminal generated by the

converter is expressed as

V119903=

119871119898

119871119904

119904V119904+ [119877119903+ 120590119871119903(119901 minus 119895120596

119903)]

119894119903 (7)

At 119905 = 1199050 a fault appears in the grid The corresponding

stator side voltage is expressed as

V119904(119905 lt 1199050) = 119881119904119890119890

119895120596119904119905

V119904(119905 ge 1199050) = (1 + 119902)119881

119904119890119890

119895120596119904119905

(8)

where 119902 is the grid voltage swell ratio If the resistance 119877119904is

neglected the steady-state stator flux is proportional to thevoltage

119904(119905 lt 1199050) =

119881119904119890

119895120596119904

119890

119895120596119904119905

119904(119905 ge 1199050) = (1 + 119902)

119881119904119890

119895120596119904

119890

119895120596119904119905

(9)

The stator flux under grid voltage swell is

119904(119905 ge 1199050) = (1 + 119902)

119881119904119890

119895120596119904

119890

119895120596119904119905

minus 119902

119881119904119890

119895120596119904

119890

1198951205961199041199050

119890

minus(119905minus1199050)120591119904

(10)

where 120591119904= 120590119871119904119877119904

The stator flux of (10) can be divided into forced fluxand natural flux The forced flux is proportional to the gridvoltage and rotates at synchronous speed The natural fluxis a transient flux which guarantees that no discontinuityappears in the stator flux during grid voltage swell Thenatural flux is proportional to the grid voltage swell ratio anddoes not rotateThe time constant of stator winding decreasesexponentially the amplitude of the natural flux to zero

If 1199050

= 0 is assumed the induced voltage in rotorreference frame during grid voltage swell can be obtainedby substituting (10) in the first term of the right hand of (5)(neglecting the term 1120591

119904due to its low value)

V1199030=

119871119898

119871119904

119881119904119890[119904 (1 + 119902) 119890

119895120596119904119897119905

+ (1 minus 119904) 119902119890

minus119895120596119903119905

119890

minus(119905minus1199050)120591119904

] (11)

Each component of the stator flux induces a correspond-ing component A small component proportional to the slipis first induced by the forced flux The small componentrotates at slip frequency and achieves someHertzThe secondcomponent induced by natural flux is important because itis proportional to 1 minus 119904 In addition the second componentrotates at rotor electrical speed 120596

119903

3 HVRT Control Strategy of DFIG

Virtual impedance control strategy is used to enhance HVRTcapability and suppress the oscillation of DFIG during gridvoltage swell [17]

31 Stability Analysis of the DFIG System Using stator fluxas state variables rotor current and stator voltage as inputvariables stator flux state equation can be expressed as

119889

119889119905

120595119904119902= minus

119877119904

119871119904

120595119904119902minus 120596119904120595119904119889+

119871119898

119871119904

119877119904119894119903119902+ 119906119904

119889

119889119905

120595119904119889= minus

119877119904

119871119904

120595119904119889+ 120596119904120595119904119902+

119871119898

119871119904

119877119904119894119903119889

(12)

The characteristic equation is expressed as

120582

2

+ 2

119877119904

119871119904

120582 +

119877

2

119904

119871

2

119904

+ 120596

2

119904= 0 (13)

The damping factor 120585 and natural oscillation frequency120596119899are

120585 =

119877119904

120596119899119871119904

120596119899=radic

119877

2

119904

119871

2

119904

+ 120596

2

119904

(14)

Given the relatively low stator resistance of megawattgenerators DFIG have the feature of underdamping Thesystem is prone to oscillation when the grid voltage swellsbecause its natural oscillation frequency is close to thegrid frequency Therefore system damping control includ-ing both passive and active damping controls should beconsidered Thus the virtual resistance control strategy wasemployed to increase the system damping and improve thedynamic HVRT performance of DFIG

32 PassiveDampingControl Strategy In aDFIG electromag-netic transient process motivated by a grid voltage swell therotor inducts strong back electromotive force (EMF) once thegrid voltage swells because the rotor resistance and leakageimpedance of the MW DFIG are relatively low This strongEMF causes rotor overcurrent Therefore passive dampingcontrol is introduced that is dynamic series resistance isallocated to the side of the generator rotor When the gridvoltage swells the passive damping of the dynamic seriesresistance of this generator rotor efficiently increases thesystem damping reduces the peak oscillation of the rotorcurrent decreases the time constant of the rotor circuit andquickly attenuates the rotor current However introducing adynamic resistance increases system loss and reduces systemefficiency thus further complicating the resistance designTherefore a control algorithm that imitates the dynamicresistance (ie active damping control) should be considered


ilowastr+minus minus

+

minusE

irC(s) T(s) G(s)

Ra

r

Figure 3 Block diagram of the current control system based onvirtual resistance

33 Virtual Resistance Control Strategy To address the short-comings of the practical dynamic resistance active dampingcontrol based on virtual resistance can be adopted under agrid voltage swell to increase system resistance and improvethe HVRT of the DFIG A structural chart of the DFIGactive damping control based on virtual resistance is shownin Figure 2

For the analysis the rotor side voltage can be reexpressedas

V119904= (119877119903+ 119877119904

119871

2

119898

119871

2

119904

) 119894119903+ 120590119871119903

119889

119889119905

119894119903+ 119895120596119904119897120590119871119903119894119903+ 119864

119864 =

119871119898

119871119904

120595119904minus

119871119898

119871119904

(

119877119904

119871119904

+ 119895120596119903)120595119904

(15)

where 119864 is the back electromotive force which is a distur-bance component for the rotor current control loop Virtualresistance control strategy is used to reduce the time constantof the rotor current control loop improve its dynamicresponse and inhibit the effect of the disturbance on thedynamic performance of the rotor current The rotor currentcontrol loop with virtual resistance control strategy is shownin Figure 3

The rotor current closed loop controlled object transferfunction 119866(119904) can be expressed as

119866 (119904) =

119894119903(119904)

V119903(119904)

=

119870119892

119879119892119904 + 1

119879119892=

120590119871119903

(119877119903+ 119871

2

119898119877119904119871

2

119904)

119870119892=

1

(119877119903+ 119871

2

119898119877119904119871

2

119904)

(16)

The switching frequency of the rotor side converter isusually high thus the delay of inertia can be neglected tosimplify the analysis

119879 (119904) = 119870119905 (17)

Assume that the virtual resistance is equal to119877119886 the rotor

current closed loop controlled object transfer function canthen be obtained using (16) and (17) as shown in Figure 3

119866

1015840

(119904) =

119894119903(119904)

V119903(119904)

=

119870119892119870119905

119879119892119904 + 1 + 119877

119886119870119892119870119905

(18)

After the introduction of virtual resistance when (16) and(18) are compared the time constant of the rotor currentclosed loop is reduced by 1 + 119877

119886119870119892119870119905times

The PI controller of the rotor current loop is designedby the internal model principle before the introduction ofvirtual resistance the transfer function is expressed as

119862 (119904) =

119870119888119879119892119904 + 119870119888

119870119905119870119892119904

(19)

where119870119888is the bandwidth of the rotor current closed loop

After the introduction of virtual resistance the PI con-troller transfer function is expressed as

119862

1015840

(119904) =

119870119888119879119892119904 + 119870119888+ 119877119886119870119892119870119905119870119888

119870119905119870119892119904

(20)

Before the introduction of virtual resistance the rotorcurrent closed loop transfer function is expressed as

119866119901(119904) =

119894119903(119904)

119894

lowast

119903(119904)

=

119904

119904 + 119870119888

(21)

After the introduction of virtual resistance the rotorcurrent closed loop transfer function is expressed as

119866

1015840

119901(119904) = 119866

119901(119904) (22)

Before the introduction of virtual resistance the distur-bance transfer function is expressed as

119866119864119868(119904) =

119894119903(119904)

119864 (119904)

=

119870119892119904

(119904 + 119870119888) (119879119892119904 + 1)

(23)

After the introduction of virtual resistance the distur-bance transfer function is expressed as

119866

1015840

119864119868(119904) =

119894119903(119904)

119864 (119904)

= minus

119870119892119904

(119904 + 119870119888) (119879119892119904 + 1 + 119877

119886119870119892119870119905)

(24)

from (16) the following expression is obtained

119866 (119904) =

119894119903(119904)

V119903(119904)

=

1

120590119871119903119904 + 119877119903+ (119871

2

119898119871

2

119904) 119877119904

(25)


119866

1015840

(119904) =

119894119903(119904)

V119903(119904)

=

119870119905

120590119871119903119904 + 119877119903+ (119871

2

119898119871

2

119904) 119877119904+ 119870119905119877119886

(26)

From (25) and (26) the introduction of the feedbackcoefficient 119877

119886is equivalent to series resistance 119870

119905119877119886in the

rotor circuit of the DFIG The equivalent circuit of the rotorbased on virtual resistance can be shown in Figure 4

It is known that regulating virtual resistance can changethe dynamic response of the rotor current control loopduring grid voltage swell Figure 5 is bode graph before andafter the introduction of virtual resistance Compared tononvirtual resistance strategy in the low frequency regionactive damping strategy has better inhibitory characteristicon the disturbance In the high frequency region bothhave the same disturbance inhibitory characteristic Figure 6is the step response of the disturbance transfer function


998400r

KtRa120590Lr

r E

Rr +L2mL2s

Rs

Figure 4 The rotor equivalent circuit based on virtual resistance

10minus1 100 101 102 103minus90

minus45

0

45

90

Phas

e (de

g)

0

10

20

30

40

50

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)

Without virtual resistanceWith virtual resistance

Figure 5 Bode graph on disturbance with and without activedamping strategy

with and without virtual resistance compared to nonvirtualresistance strategy the overshoot was obviously inhibited bythe virtual resistance strategy under step load disturbancewhich can improve the capability of disturbance suppressionunder grid voltage fluctuations Figure 7 is the root locusof the disturbance transfer function with and without vir-tual resistance after the introduction of virtual resistancethe system root is near the real axis which can increasethe system damping and suppress the oscillation of thesystem

4 The Selection of Virtual Resistance

41 The Establishment of Objective Function The rotor cur-rent objective function is as follows

1198911(119877119886) = 119894119889119902

119894119889119902= radic119894

2

119903119889119899+ 119894

2

119903119902119899(119899 = 1 2 3 4)

(27)

where

1198941199031198891

= minus

119871119898120596slip1205951199041199020

119871119904(120590119877119903+ 119877119886)

1198941199031199021

=

119871119898120596slip1205951199041198890

119871119904(120590119877119903+ 119877119886)

1198941199031198892

= minus

119871119898120596slip (1205951199041198890 minus 1205951199041198892)

119871119904(120590119877119903+ 119877119886)

0

10

20

30

40

50

60

70

80

90Step response

Time (s)

Am

plitu

de


0 05 1 15 2 25

Figure 6 Step response of the disturbance transfer function withand without virtual resistance

1198941199031199022

= minus

119871119898120596slip (1205951199041199020 minus 1205951199041199022)

119871119904(120590119877119903+ 119877119886)

1198941199031198893

=

119871119898120596119903[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199023

=

119871119898120596119903[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031198894

=

119871119898120596119903119886119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199024

=

119871119898120596119903119887119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

(28)

where119898 = 120590119871119903 119899 = (120590119877

119903+ 119877119886) 119889 = 119898119899 119886 = (119889 minus 120590)(120595

1199041198890minus

1205951199041198892)minus120596119904(1205951199041199020minus1205951199041199022) and 119887 = (dminus120590)(120595

1199041198890minus1205951199041198892) + 120596119904(1205951199041199020minus

1205951199041199022) 119894119903119900

is the initial state of current 120572119904and 120572

119889119902represent

the closed loop and the open loop bandwidth of the controlmodel 120590 denotes the damping time constant 119871

119898 119871119904 and

119871119903are the mutual inductance and the stator and rotor self-

inductances 120596119904is the synchronous speed 120595

1199041198890and 120595

1199041199020are

the steady state stator flux of 119889 and 119902 axes prior to the voltagedrop 120595

1199041198892and 120595

1199041199022are the steady state stator flux of 119889 and 119902

axes after the voltage dropThe rotor voltage objective function is as follows

1198912(119877119886) = V119889119902

V119889119902= radicV2119889119899+ V2119902119899

(119899 = 1 2 3)

(29)

where

V1199031198891

= minus

120596slip119877119886119871119898

119899119871119904

1205951199041199020 V

1199031199021= minus

120596slip119877119886119871119898

119899119871119904

1205951199041198890

V1199031198892

=

120596slip119877119886119871119898 (1205951199041199020 minus 1205951199041199022)

119899119871119904

V1199031199022

=

120596slip119877119886119871119898 (1205951199041198890 minus 1205951199041198892)

119899119871119904


minus60 minus50 minus40 minus30 minus20 minus10 0minus20minus15minus10

minus505

101520

024046064078087093097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)

(a) Without virtual resistance


minus505

101520 024046064078087093

097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)

(b) With virtual resistance

Figure 7 The root locus of the disturbance transfer function with and without virtual resistance

V1199031198893

=

120596119903119877119886119871119898[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

V1199031199023

=

120596119903119877119886119871119898[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

(30)

where119898 119886 and 119887 are the same as the above description 120596slipis the slip frequency and 120596slip = 119904 sdot 120596119904 119904 denotes the slip

Obviously there are two objective functions the currentand voltage This paper adopts the method of importantobjective and satisfied constraints to simply the objectivefunctions Let the current objective function as the firstoptimization goal when the grid voltage swell is less than03 pu Let the voltage objective function as the first goalwhen the grid voltage swell is greater than 03 pu Meanwhilemake the satisfaction function as a constraint condition Theconstraint value is described as follows

max 1198911(119877119886) le 1205721

max 1198912(119877119886) le 1205722

119869 = 12057311198911(119877119886) + 12057321198912(119877119886)

(31)

where 1205721 1205722are the satisfaction function values and equal to

the maximum current allowed to flow through the converterWhen the swell amplitude reduces 120573

1decreases and 120573

2

increases Thus we can seek out the best solution

42 The EPL of Dynamic PSO Algorithm In particle swarmoptimization (PSO) algorithm the particles are optimizedby changing the speed and position of the particles As thealgorithm proceeds the effective distance between particleswill be diminished and the particle concentration increasedThen particles with small Euclidean distance will iteraterepeatedly prone to premature convergence and lead thePSO algorithm to the local best solution which affect thespeed of convergence and the precision of algorithm Henceit is necessary to adjust the particle concentration in PSOalgorithm The particle concentration is associated with the

degree of similarity between particles The degree of similar-ity between particles is decided by two aspects the Euclideandistance between two particles 119889(119909

119894 119909119895) and the difference

between their fitness function value |fit(119909119894) minus fit(119909

119895)| If 119909

119894=

(1199091198941 1199091198942 119909

119894119898) and 119909

119895= (1199091198951 1199091198952 119909

119895119898) are noted as two

particles of PSO algorithm the Euclidean distance betweenthem is

119889 (119909119894 119909119895) = radic

119898

sum

119896=1

(119909119894119896minus 119909119895119896)

2

(32)

When 119889(119909119894 119909119895) le 119863 and |fit(119909

119894) minus fit(119909

119895)| le 119891 the

two particles 119909119894and 119909

119895can be regarded as similarity 119863 is

the threshold value of distance and 119891 is the threshold valueof fitness The concentration of the particle 119894 is defined asfollows

119886119894=

119899

119873

(33)

where 119899 is the particle number which is similar to 119894119873 is thetotal particle number in the particle swarm

The probability of retained particles is

119875 (119909119894) =

fit (119909119894) 119886119894

sum

119873

119895=1[fit (119909

119895) 119886119894]

(34)

The probability of retained particles is determined by theconcentration and the fitness of particle 119894 The probabilityof retained particles is lower if the particles concentrationis larger and the probability is higher if the fitness valueis larger This method controls the evolution of the highparticle concentration and also prevents the occurrence ofldquoprematurerdquo phenomenon Concurrently the particles withlarger fitness value can evolve fully which can keep thepopulation diversity

The particles with larger fitness value inevitably containsome key information Hence this paper adopts probabilisticmethod to keep some excellent particles and then theseexcellent particles iterate into the next generation withlarge probability It not only avoids the blind search of thealgorithm but also can improve the convergence speed


The steps to establish the excellent particle library (EPL)are as follows after each iteration of particle swarm optimiza-tion (PSO) the PSO can produce two best solutions (denotedas fitness) which are 119875

119894best denoted as pbest and 119866best denotedas gbest 119875

119894best is the fitness which has been achieved tillcurrent iteration and 119866best is the global best solution Let thecapacity of the EPL be 119876 make the 119875

119894best sort in terms of thefitness of the particles from small to large and select119866best andthe first119876minus1 of119875

119894best to replace the original excellentmemoryparticles in the EPL Thus in processing a new generationof particles the particles of smaller fitness in PSO will bereplaced by more excellent particles Hence the fitness of thewhole particle groups is improved and the convergence speedof the algorithm is accelerated

In a static environment on the basis of local and globaloptimal solutions the particles can finally find the optimalvalue in the solution space through iterative formula Butbecause the rotor voltage and current of doubly fed inductiongenerator change at any time the dynamic improvement isintroduced to adapt to the changes of the current environ-ment

The achievement of dynamic PSO is as follows first ofall the solution space is divided into 119899

1uniform subspaces

In the subspaces 1198992particles sensitive to the environment

are initialized randomlyThen the fitness of sensitive particlesdenoted as 119891

119894is calculated separately in every iteration and

the difference of the adjacent two particles denoted as Δ119891119894is

calculated by (33) In the end all the absolute values ofΔ119891119894are

summed denoted as 119865 by (34) Consider

Δ119891119894= 119891119894(119896 + 1) minus 119891

119894(119896) (35)

119865 =

119899

sum

119894=1

1003816

1003816

1003816

1003816

Δ119891119894

1003816

1003816

1003816

1003816

(119899 = 1198991times 1198992) (36)

If 119865 is not equal to 0 it can be thought that the externalenvironment has been changed We can set a threshold valuedenoted as 119865

119900 When 119865 is greater than 119865

119900 the dynamic

response is triggered The response mechanism is used toreinitialize the position and velocity of particles proportion-ally which is described as follows

If 119865 gt 119865119900 then

V (119894) = rand (119872) times Vmax119909 (119894) = rand (119872) times 119909max

(37)

where rand(119872) are119872 dimensions random numbers in [0 1]119909119894= (119909

1198941 1199091198942 119909

119894119872) and V

119894= (V

1198941 V1198942 V

119894119872) are

denoted as the position and velocity of the 119894th particle selectedfrom the population of the new initialization119872 denotes thedimension 119883max denotes the maximum position and 119881maxdenotes the maximum velocity

43 Steps of the EPL of Dynamic PSO Algorithm Probabilityto be selected is related to the concentration of the particlesBoth the current and voltage in the objective function are tofind the minimum values so opposite to (34) it is replacedby (38) as follows

119901 (119909119894) =

1 (119886119894fit (119909119894))

sum

119873

119895=1[1 (119886

119894fit (119909119895))]

(38)

The detailed algorithm simulation steps are as follows

Step 1 Initialize the particle parameters which include thesearching space the particle numbers119873 initial position andvelocity All of them are random numbers meanwhile theposition and velocity are between 0 and 1 considering thatthe resistance cannot be negative

Step 2 Establish the EPL according to (31) to calculate thefitness of each particle and acquire the local optimal solution119875119894best and the global optimal solution 119866

119894best of the particle119909119894 Put 119866

119894best and the smaller fitness value into the EPL asexcellent particles Judge whether the particle completes theiterations and if it does the iteration will end

Step 3 Supposing the EPL is made up of 1198731particles and

random initialization 119872 sensitive particles calculate thefitness of sensitive particle 119894 by (31) and 119865 by (36) If 119865 gt 1198651then reinitialize according to (37) and to Step 2

Step 4 The velocity and position of each particle are updatedby the following equation

V119905+1119894119889

= 120596V119905119894119889+ 1198881times rand

1(sdot) times (119901

119894119889minus 119909

119905

119894119889)

+ 1198882times rand


119892119889minus 119909

119905

119892119889)

119909

119905+1

119894119889= 119909

119905

119894119889+ V119905+1119894119889

(39)

where 120596 is used as an inertia weight and the initial value of120596 is set within the range [0 1] 120596 is updated by the followingequation

120596 = 120596max minus120596max minus 120596min

119879

sdot 119905 (40)

where120596max and120596min denote themaximum inertia weight andthe minimum 119879 is the maximum number of iterations and 119905is the current iteration number at the present number

Step 5 Regulate the particle concentration Select119873 particlesfrom the (119873 + 119872) particles generated in Steps 1 and 2 Theprobability of each particle is selected according to (38)

Step 6 Use the excellent particles in EPL to replace theparticles of larger fitness In this way a new generation ofparticle swarm is generated and then go to Step 2

Step 7 End the iteration

5 Results and Analysis

Supposing the initial voltage of the stator 119880119904= 690V the

initial voltage of the rotor 119880119903= 220V when the voltage swell

is 03 and the rotor speed is 1800 rmin the fitness curvesof the ordinary PSO algorithm the changing inertia weightPSO algorithm and the EPL of dynamic PSO algorithm areas Figure 8

From Figure 8 the ordinary PSO algorithm convergesfinally but the convergence speed is slow and the convergencetime is long Comparing with the ordinary PSO algorithmthe changing inertia weight PSO algorithm increases the


0 20 40 60 80 100 120 140 160 180 200700

800

900

1000

1100

1200

1300

Iteration numbers

Fitn

ess

Ordinary PSOChanging inertia weight PSOThe EPL of dynamic PSO

Fitness and iteration numbers = 200

Figure 8 Fitness curve

convergence speed of the algorithm and shortens the con-vergence time The excellent particle library of dynamic PSOalgorithm is on the establishment of excellent particle libraryand a dynamic optimization is added Considering that therotor voltage changes along with the voltage swell the newparticles which can respond to the external environmentchange are used in the EPL of dynamic PSO algorithmThus local optimum of the algorithm will be avoided becauseof the similarity and unnecessary search between particlesand the convergence speed and convergence precision of thealgorithm are guaranteed

Through the comparison of three kinds of PSO algorithmthis paper put forward the EPL of the dynamic PSO algorithmbymodifying the distance between the particles adjusting theparticle concentration avoiding the repeated computation inthe process of PSO algorithm and saving a lot of time tomakethe algorithm fast convergence

In conclusion the EPL of dynamic PSO algorithm issuitable for calculating the virtual resistance values in dif-ferent rotor speeds and different voltage swells In practicalapplication the virtual resistance can be selected accordingto the current rotor speed and voltage swell

6 Simulation Results

Simulations were performed using MATLAB based on thesystem configuration shown in Figure 2 The current studyuses 2MW DFIG as the prototype to conduct simulationstudies to verify the correctness of the aforementionedtheoretical analysisThemain parameters of DFIG in the sim-ulation are as follows the stator inductance 119871

119904= 00125H

the rotor inductance 119871119903= 00126H the mutual inductance

119871119898= 00123H the stator winding resistance 119877

119904= 00043Ω

and the rotor winding resistance 119877119903= 00041Ω

Figures 9ndash11 are simulationwaveforms of rotor current 119902-axis component 119894

119903119902 rotor voltage V

119903119902 A-phase rotor current

119894119903119886

that used damping respectively when the amplitude ofgrid voltage swell are 13 pu and the fault duration is 100msIn Figure 9 the three-phase grid voltage swells to 130 of its

normal value for 100ms and thus induces big dc componentin the stator flux linkage which will result in the high rotorcurrent the oscillation amplitude of the rotor current in thegrid voltage is higher than the amplitude of the rotor currentin the grid voltage swells because the point of grid voltageswells is uncertain It can be seen from Figure 9 to Figure 11that the use of damping control has amore obvious inhibitioneffect on the rotor current and the rotor voltage does notexceed the upper limit voltage when failure amplitude ismaximumWhen the grid voltage swells with the rising of thevalue of virtual resistance the value of rotor current decreasesand the value of rotor voltage increases It can play a role ofsuppression of rotor current and at the same time the rise ofthe rotor voltage would threaten the safety of the converter

7 Experimental Results

Experimental tests of the virtual resistance control strategywere performed on a laboratory setup of the 11 Kw DFIGsystem to further verify the effectiveness of the proposedstrategy A 15 Kw AC motor driving the DFIG at desiredspeed was used to emulate wind turbine A TMS320LF28335DSP was employed to control the rotor side converter andthe grid side converter The switching frequency for bothconverters was set at 5 kHz with a sampling frequency of10 kHz PM100CLA120was used for the switching deviceThewaveforms are acquired by a DPO4032 digital oscilloscope

The main parameters of DFIG are as follows the statorresistance 119877

119904= 02858Ω the rotor resistance 119877

119903= 02983Ω

the stator leakage reactance 119871119904= 0001323H the rotor

leakage reactance 119871119903= 0001781H and excitation reactance

119871119898= 00676HFigures 12ndash14 show the experimental waveforms of the

119902 axis rotor current component 119894119903119902 119902 axis rotor voltage

component V119903119902 and A-phase rotor current 119894

119903119886as the DFIG

undergoes subsynchronization synchronization and super-synchronization When the grid voltage three-phase sym-metrical swell is 30 the fault duration is 100ms andthe virtual resistance is 05 pu and 15 pu Increasing thevirtual resistancemore significantly inhibits the rotor currentoscillation caused by the grid voltage swell on the controlsystem When the grid voltage recovers the oscillationamplitude of the rotor current becomes higher than whenthe grid voltage swells because of the fault duration andthe point of grid voltage recovery If the point is cut offby the fault and the electromagnetic oscillation caused bythe grid voltage swell has not yet stopped the oscillationcaused by grid recovery will have been superimposed onthe previous one The oscillation amplitude of the rotorcurrent under the super-synchronized operation is higherthan that in the subsynchronized operation Because the rotorside induction voltage in the supersynchronized operation ishigher than that in the subsynchronized operation under gridvoltage swell the generator power is greater with the highermotor speed Controlling the virtual resistance increasesthe rotor voltage but reduces the rotor current oscillationTherefore the selection of virtual resistance should considerthe inhibition of the converter rotor voltage


minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)

(a) Rotor current and voltage with 05 pu virtual resistance

2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02

(b) Rotor current and voltage with 15 pu virtual resistance

Figure 9 Rotor current voltage with variable damping control in subsynchronization

02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04


Figure 10 Rotor current voltage with variable damping control in synchronization

minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04


Figure 11 Rotor current and voltage with variable damping control in supersynchronization

8 Conclusion

This paper analyzes the DFIG electromagnetic transientprocess under a grid voltage swell introduces active dampingcontrol to DFIG rotor excitation control based on passivedamping control and proposes an improved control strategyfor variable damping and the acquisition of virtual resistances

on the basis of the EPL of dynamic PSO algorithmDue to therapidity and accuracy of the algorithm the virtual resistancevalue is more accurate The control method based on activedamping inhibited the rotor current oscillation caused bythe grid voltage swell reduced the effect of electromagnetictorque oscillation in the system during HVRT and reducedthe crowbar motion and its adverse effects Variable damping


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


Figure 12 Rotor current and voltage with variable damping coefficient in subsynchronization

ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


Figure 13 Rotor current and voltage with variable damping coefficient in synchronization

ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


Figure 14 Rotor current and voltage with variable damping coefficient in supersynchronization


control over the effect of different rotation speeds and therange of grid voltage swell in the system improved the HVRTcontrol of DFIG

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The project was supported by National Natural ScienceFoundation of China (51277050)

References

[1] H Nian Y Song P Zhou and Y He ldquoImproved direct powercontrol of a wind turbine driven doubly fed induction generatorduring transient grid voltage unbalancerdquo IEEE Transactions onEnergy Conversion vol 26 no 3 pp 976ndash986 2011

[2] J Hu Y He L Xu and B W Williams ldquoImproved control ofDFIG systems during network unbalance using PI-R currentregulatorsrdquo IEEE Transactions on Industrial Electronics vol 56no 2 pp 439ndash451 2009

[3] J Lopez P Sanchis X Roboam and L Marroyo ldquoDynamicbehavior of the doubly fed induction generator during three-phase voltage dipsrdquo IEEE Transactions on Energy Conversionvol 22 no 3 pp 709ndash717 2007

[4] S Xiao G Yang H Zhou and H Geng ldquoAn LVRT controlstrategy based on flux linkage tracking for DFIG-basedWECSrdquoIEEE Transactions on Industrial Electronics vol 60 no 7 pp2820ndash2832 2013

[5] H Geng C Liu and G Yang ldquoLVRT capability of DFIG-based WECS under asymmetrical grid fault conditionrdquo IEEETransactions on Industrial Electronics vol 60 no 6 pp 2495ndash2509 2013

[6] J P da Costa H Pinheiro T Degner and G Arnold ldquoRobustcontroller for DFIGs of grid-connected wind turbinesrdquo IEEETransactions on Industrial Electronics vol 58 no 9 pp 4023ndash4038 2011

[7] O Abdel-Baqi and A Nasiri ldquoA dynamic LVRT solution fordoubly fed induction generatorsrdquo IEEE Transactions on PowerElectronics vol 25 no 1 pp 193ndash196 2010

[8] S Muller M Deicke and R W de Doncker ldquoDoubly fedinduction generator systems for wind turbinesrdquo IEEE IndustryApplications Magazine vol 8 no 3 pp 26ndash33 2002

[9] H Zhou G Yang and JWang ldquoModeling analysis and controlfor the rectifier of hybrid HVdc systems for DFIG-based windfarmsrdquo IEEE Transactions on Energy Conversion vol 26 no 1pp 340ndash353 2011

[10] B C RabeloWHofmann J L da Silva R G deOliveira and SR Silva ldquoReactive power control design in doubly fed inductiongenerators for wind turbinesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 4154ndash4162 2009

[11] G Iwanski and W Koczara ldquoDFIG-based power generationsystem with UPS function for variable-speed applicationsrdquoIEEE Transactions on Industrial Electronics vol 55 no 8 pp3047ndash3054 2008

[12] N Patin E Monmasson and J-P Louis ldquoModeling and controlof a cascaded doubly fed induction generator dedicated to

isolated gridsrdquo IEEE Transactions on Industrial Electronics vol56 no 10 pp 4207ndash4219 2009

[13] A Luna F K de Araujo Lima D Santos P RodrıguezE H Watanabe and S Arnaltes ldquoSimplified modeling of aDFIG for transient studies in wind power applicationsrdquo IEEETransactions on Industrial Electronics vol 58 no 1 pp 9ndash202011

[14] K Rothenhagen and F W Fuchs ldquoCurrent sensor fault detec-tion isolation and reconfiguration for doubly fed inductiongeneratorsrdquo IEEE Transactions on Industrial Electronics vol 56no 10 pp 4239ndash4245 2009

[15] F Bonnet P E Vidal and M Pietrzak-David ldquoDual directtorque control of doubly fed induction machinerdquo IEEE Trans-actions on Industrial Electronics vol 54 no 5 pp 2482ndash24902007

[16] J Lopez E Gubıa E Olea J Ruiz and L Marroyo ldquoRidethrough of wind turbines with doubly fed induction generatorunder symmetrical voltage dipsrdquo IEEE Transactions on Indus-trial Electronics vol 56 no 10 pp 4246ndash4254 2009

[17] Z Xie Q Shi H Song X Zhang and S Yang ldquoHigh voltageride through control strategy of doubly fed induction windgenerators based on active resistancerdquo in Proceedings of theIEEE 7th International Power Electronics and Motion ControlConference (IPEMC 12) pp 2193ndash2196 IEEE Harbin ChinaJune 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is

+

Rs L1s L1r

s r

minus

Lm120595s 120595r

j120596120595r

+

minus

Rrir

+ minus

Figure 1 DFIG equivalent circuit at static stator-oriented referenceframe

losing control over the generator because crowbar resistancewill be used if a grid fault occurs This series resistancecan also continuously provide reactive power to the gridduring the fault and reduce torque ripple References [611] examined and compared the static synchronous com-pensator (STATCOM) and dynamic voltage restorer (DVR)schemes during a grid voltage swell In the DVR schemegenerator voltage is maintained by compensating for thevoltage difference between normal and fault conditions Inthe STATCOM scheme grid voltage decreases by controllingthe reactive current injected into the power grid Especiallywhen grid impedance is relatively low both schemes signif-icantly increase the cost of using the whole set of hardwareThe STATCOM scheme is uneconomical because it requirescurrent far stronger than the rated current of wind turbinegenerators The enhanced controlling strategy of the DFIGunder a grid voltage swell is proposed in [12] decouplingcontrol in fault conditions is achieved by establishing anaccuratemodel of the dynamic process of theDFIG rotor fieldcurrent This model considers the compensation dosage ofthe rotor field current change on the basis of the conventionalcontroller Proportional-integral control should be used innormal operations and vector hysteresis control in grid faults[13]

In the HVRT of DFIG converter overcurrent can beavoided by using the crowbar on the rotor side thusrapidly increasing the generator torque and generating anuncontrollable output of active and reactive power whenthe grid voltage swells To solve these problems and ensurethe excellent HVRT of the DFIG this study analyzes theelectromagnetic transient course of the DFIG under the gridvoltage swell and examines the control strategy of activedamping in the rotor converter according to the passivedamping scheme of the series resistance of the generator rotor[14 15] Thus rotor overvoltage is managed when the gridvoltage swells the rotor overcurrent is maximally controlledand the HVRT of the DFIG is improved To acquire the bestvirtual resistance value this paper adopts the EPL of dynamicPSO algorithm for the multiobjective optimization design ofparameters It takes the rotor voltage and rotor current astwo objectives for the integrated optimization As the resultof the multiple iterations the algorithm can be caught inlocal optimumThus the algorithm is improved in this paperwhich is on the basis of the excellent library and dynamicimprovement This makes the algorithm more suitable forthe dynamic changes of the actual cases Therefore the localoptimum of PSO algorithm can not only be avoided but alsothe speed and accuracy of the optimum can be improved

PI

PI

SVPWM

DC+

+

minus

abc

ilowastrd

ilowastrq

rdc

rdc

rqc

rqc

lowastrd

lowastrq

lowastrD

lowastrQ

2r2s

3s2r

3s2r

+

120579r

120579r

120579r

120579s120579sr

ird

irq

isd

isq

DFIG

PLLC

AB

Feedforwarddecoupling

Statorflux

calculation

Ra

Ra

d

dt

d

dt

120596r

120596sr

120595sd

120595sq iAiB iC

us

iaib

ic

Figure 2 Block diagram of the DFIG control system based onvirtual resistance

Simulation and experimental results confirm the feasibility ofthis damping scheme

2 Dynamic Analysis of DFIG duringGrid Voltage Swell

DFIG equivalent circuit is shown in Figure 1 to analyze thedynamic behavior during grid voltage swell For simplicityall rotor variables are converted to stator side and linearmagnetic circuits are assumed [16] With the application ofmotor convention the DFIG model can be expressed as

V119904= 119877119904

119894119904+

119889

119889119905

119904 (1)

V119903= 119877119903

119894119903+

119889

119889119905

119903minus 119895120596119903119903 (2)

119904= 119871119904

119894119904+ 119871119898

119894119903 (3)

119903= 119871119898

119894119904+ 119871119903

119894119903 (4)

The symbols and their corresponding meanings are asfollows

V voltage vector119894 current vector120596119903 rotor electrical speed

flux vector119877 resistance119871 inductance

Subscripts 119903 and 119904 are the rotor and stator respectivelyThe rotor voltage can be calculated from (2) as follows

V119903=

119871119898

119871119904

(119901 minus 119895120596119903) 119904+ [119877119903+ 120590119871119903(119901 minus 119895120596

119903)]

119894119903 (5)

where 120590 = 1 minus 1198712119898119871119904119871119903



V1199030= 119895120596119904119897

119871119898

119871119904

119904= 119904

119871119898

119871119904

119881119904119890119890

119895120596119904119905

(6)


119904119897120596119904) and119881

119904119890is the



V119903=

119871119898

119871119904

119904V119904+ [119877119903+ 120590119871119903(119901 minus 119895120596

119903)]

119894119903 (7)



V119904(119905 lt 1199050) = 119881119904119890119890

119895120596119904119905

V119904(119905 ge 1199050) = (1 + 119902)119881

119904119890119890

119895120596119904119905

(8)



119904(119905 lt 1199050) =

119881119904119890

119895120596119904

119890

119895120596119904119905

119904(119905 ge 1199050) = (1 + 119902)

119881119904119890

119895120596119904

119890

119895120596119904119905

(9)


119904(119905 ge 1199050) = (1 + 119902)

119881119904119890

119895120596119904

119890

119895120596119904119905

minus 119902

119881119904119890

119895120596119904

119890

1198951205961199041199050

119890

minus(119905minus1199050)120591119904

(10)

where 120591119904= 120590119871119904119877119904


If 1199050



V1199030=

119871119898

119871119904

119881119904119890[119904 (1 + 119902) 119890

119895120596119904119897119905

+ (1 minus 119904) 119902119890

minus119895120596119903119905

119890

minus(119905minus1199050)120591119904

] (11)


119903




119889

119889119905

120595119904119902= minus

119877119904

119871119904

120595119904119902minus 120596119904120595119904119889+

119871119898

119871119904

119877119904119894119903119902+ 119906119904

119889

119889119905

120595119904119889= minus

119877119904

119871119904

120595119904119889+ 120596119904120595119904119902+

119871119898

119871119904

119877119904119894119903119889

(12)


120582

2

+ 2

119877119904

119871119904

120582 +

119877

2

119904

119871

2

119904

+ 120596

2

119904= 0 (13)


120585 =

119877119904

120596119899119871119904

120596119899=radic

119877

2

119904

119871

2

119904

+ 120596

2

119904

(14)





+

minusE

irC(s) T(s) G(s)

Ra

r




V119904= (119877119903+ 119877119904

119871

2

119898

119871

2

119904

) 119894119903+ 120590119871119903

119889

119889119905

119894119903+ 119895120596119904119897120590119871119903119894119903+ 119864

119864 =

119871119898

119871119904

120595119904minus

119871119898

119871119904

(

119877119904

119871119904

+ 119895120596119903)120595119904

(15)



119866 (119904) =

119894119903(119904)

V119903(119904)

=

119870119892

119879119892119904 + 1

119879119892=

120590119871119903

(119877119903+ 119871

2

119898119877119904119871

2

119904)

119870119892=

1

(119877119903+ 119871

2

119898119877119904119871

2

119904)

(16)


119879 (119904) = 119870119905 (17)



119866

1015840

(119904) =

119894119903(119904)

V119903(119904)

=

119870119892119870119905

119879119892119904 + 1 + 119877

119886119870119892119870119905

(18)


119886119870119892119870119905times


119862 (119904) =

119870119888119879119892119904 + 119870119888

119870119905119870119892119904

(19)



119862

1015840

(119904) =

119870119888119879119892119904 + 119870119888+ 119877119886119870119892119870119905119870119888

119870119905119870119892119904

(20)


119866119901(119904) =

119894119903(119904)

119894

lowast

119903(119904)

=

119904

119904 + 119870119888

(21)


119866

1015840

119901(119904) = 119866

119901(119904) (22)


119866119864119868(119904) =

119894119903(119904)

119864 (119904)

=

119870119892119904

(119904 + 119870119888) (119879119892119904 + 1)

(23)


119866

1015840

119864119868(119904) =

119894119903(119904)

119864 (119904)

= minus

119870119892119904

(119904 + 119870119888) (119879119892119904 + 1 + 119877

119886119870119892119870119905)

(24)


119866 (119904) =

119894119903(119904)

V119903(119904)

=

1

120590119871119903119904 + 119877119903+ (119871

2

119898119871

2

119904) 119877119904

(25)


119866

1015840

(119904) =

119894119903(119904)

V119903(119904)

=

119870119905

120590119871119903119904 + 119877119903+ (119871

2

119898119871

2

119904) 119877119904+ 119870119905119877119886

(26)



119905119877119886in the




998400r

KtRa120590Lr

r E

Rr +L2mL2s

Rs


10minus1 100 101 102 103minus90

minus45

0

45

90

Phas

e (de

g)

0

10

20

30

40

50

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)






1198911(119877119886) = 119894119889119902

119894119889119902= radic119894

2

119903119889119899+ 119894

2

119903119902119899(119899 = 1 2 3 4)

(27)

where

1198941199031198891

= minus

119871119898120596slip1205951199041199020

119871119904(120590119877119903+ 119877119886)

1198941199031199021

=

119871119898120596slip1205951199041198890

119871119904(120590119877119903+ 119877119886)

1198941199031198892

= minus

119871119898120596slip (1205951199041198890 minus 1205951199041198892)

119871119904(120590119877119903+ 119877119886)

0

10

20

30

40

50

60

70

80

90Step response

Time (s)

Am

plitu

de


0 05 1 15 2 25


1198941199031199022

= minus

119871119898120596slip (1205951199041199020 minus 1205951199041199022)

119871119904(120590119877119903+ 119877119886)

1198941199031198893

=

119871119898120596119903[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199023

=

119871119898120596119903[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031198894

=

119871119898120596119903119886119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199024

=

119871119898120596119903119887119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

(28)

where119898 = 120590119871119903 119899 = (120590119877

119903+ 119877119886) 119889 = 119898119899 119886 = (119889 minus 120590)(120595

1199041198890minus


1199041198890minus1205951199041198892) + 120596119904(1205951199041199020minus

1205951199041199022) 119894119903119900




119898 119871119904 and



1199041198890and 120595

1199041199020are


1199041198892and 120595



1198912(119877119886) = V119889119902

V119889119902= radicV2119889119899+ V2119902119899

(119899 = 1 2 3)

(29)

where

V1199031198891

= minus

120596slip119877119886119871119898

119899119871119904

1205951199041199020 V

1199031199021= minus

120596slip119877119886119871119898

119899119871119904

1205951199041198890

V1199031198892

=

120596slip119877119886119871119898 (1205951199041199020 minus 1205951199041199022)

119899119871119904

V1199031199022

=

120596slip119877119886119871119898 (1205951199041198890 minus 1205951199041198892)

119899119871119904



minus505

101520

024046064078087093097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



minus505

101520 024046064078087093

097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



V1199031198893

=

120596119903119877119886119871119898[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

V1199031199023

=

120596119903119877119886119871119898[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

(30)



max 1198911(119877119886) le 1205721

max 1198912(119877119886) le 1205722

119869 = 12057311198911(119877119886) + 12057321198912(119877119886)

(31)




2






119895)| If 119909

119894=

(1199091198941 1199091198942 119909

119894119898) and 119909

119895= (1199091198951 1199091198952 119909



119889 (119909119894 119909119895) = radic

119898

sum

119896=1

(119909119894119896minus 119909119895119896)

2

(32)

When 119889(119909119894 119909119895) le 119863 and |fit(119909

119894) minus fit(119909

119895)| le 119891 the




119886119894=

119899

119873

(33)



119875 (119909119894) =

fit (119909119894) 119886119894

sum

119873

119895=1[fit (119909

119895) 119886119894]

(34)











1uniform subspaces







Δ119891119894= 119891119894(119896 + 1) minus 119891

119894(119896) (35)

119865 =

119899

sum

119894=1

1003816

1003816

1003816

1003816

Δ119891119894

1003816

1003816

1003816

1003816

(119899 = 1198991times 1198992) (36)



119900 the dynamic


If 119865 gt 119865119900 then


(37)


1198941 1199091198942 119909

119894119872) and V

119894= (V

1198941 V1198942 V

119894119872) are



119901 (119909119894) =

1 (119886119894fit (119909119894))

sum

119873

119895=1[1 (119886

119894fit (119909119895))]

(38)









V119905+1119894119889

= 120596V119905119894119889+ 1198881times rand


119894119889minus 119909

119905

119894119889)

+ 1198882times rand


119892119889minus 119909

119905

119892119889)

119909

119905+1

119894119889= 119909

119905

119894119889+ V119905+1119894119889

(39)



119879

sdot 119905 (40)











0 20 40 60 80 100 120 140 160 180 200700

800

900

1000

1100

1200

1300

Iteration numbers

Fitn

ess









119904= 00125H



119904= 00043Ω





119894119903119886







119903= 02983Ω






119903119886as the DFIG



minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)


2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02



02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04



minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04



8 Conclusion




ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











V1199030= 119895120596119904119897

119871119898

119871119904

119904= 119904

119871119898

119871119904

119881119904119890119890

119895120596119904119905

(6)


119904119897120596119904) and119881

119904119890is the



V119903=

119871119898

119871119904

119904V119904+ [119877119903+ 120590119871119903(119901 minus 119895120596

119903)]

119894119903 (7)



V119904(119905 lt 1199050) = 119881119904119890119890

119895120596119904119905

V119904(119905 ge 1199050) = (1 + 119902)119881

119904119890119890

119895120596119904119905

(8)



119904(119905 lt 1199050) =

119881119904119890

119895120596119904

119890

119895120596119904119905

119904(119905 ge 1199050) = (1 + 119902)

119881119904119890

119895120596119904

119890

119895120596119904119905

(9)


119904(119905 ge 1199050) = (1 + 119902)

119881119904119890

119895120596119904

119890

119895120596119904119905

minus 119902

119881119904119890

119895120596119904

119890

1198951205961199041199050

119890

minus(119905minus1199050)120591119904

(10)

where 120591119904= 120590119871119904119877119904


If 1199050



V1199030=

119871119898

119871119904

119881119904119890[119904 (1 + 119902) 119890

119895120596119904119897119905

+ (1 minus 119904) 119902119890

minus119895120596119903119905

119890

minus(119905minus1199050)120591119904

] (11)


119903




119889

119889119905

120595119904119902= minus

119877119904

119871119904

120595119904119902minus 120596119904120595119904119889+

119871119898

119871119904

119877119904119894119903119902+ 119906119904

119889

119889119905

120595119904119889= minus

119877119904

119871119904

120595119904119889+ 120596119904120595119904119902+

119871119898

119871119904

119877119904119894119903119889

(12)


120582

2

+ 2

119877119904

119871119904

120582 +

119877

2

119904

119871

2

119904

+ 120596

2

119904= 0 (13)


120585 =

119877119904

120596119899119871119904

120596119899=radic

119877

2

119904

119871

2

119904

+ 120596

2

119904

(14)





+

minusE

irC(s) T(s) G(s)

Ra

r




V119904= (119877119903+ 119877119904

119871

2

119898

119871

2

119904

) 119894119903+ 120590119871119903

119889

119889119905

119894119903+ 119895120596119904119897120590119871119903119894119903+ 119864

119864 =

119871119898

119871119904

120595119904minus

119871119898

119871119904

(

119877119904

119871119904

+ 119895120596119903)120595119904

(15)



119866 (119904) =

119894119903(119904)

V119903(119904)

=

119870119892

119879119892119904 + 1

119879119892=

120590119871119903

(119877119903+ 119871

2

119898119877119904119871

2

119904)

119870119892=

1

(119877119903+ 119871

2

119898119877119904119871

2

119904)

(16)


119879 (119904) = 119870119905 (17)



119866

1015840

(119904) =

119894119903(119904)

V119903(119904)

=

119870119892119870119905

119879119892119904 + 1 + 119877

119886119870119892119870119905

(18)


119886119870119892119870119905times


119862 (119904) =

119870119888119879119892119904 + 119870119888

119870119905119870119892119904

(19)



119862

1015840

(119904) =

119870119888119879119892119904 + 119870119888+ 119877119886119870119892119870119905119870119888

119870119905119870119892119904

(20)


119866119901(119904) =

119894119903(119904)

119894

lowast

119903(119904)

=

119904

119904 + 119870119888

(21)


119866

1015840

119901(119904) = 119866

119901(119904) (22)


119866119864119868(119904) =

119894119903(119904)

119864 (119904)

=

119870119892119904

(119904 + 119870119888) (119879119892119904 + 1)

(23)


119866

1015840

119864119868(119904) =

119894119903(119904)

119864 (119904)

= minus

119870119892119904

(119904 + 119870119888) (119879119892119904 + 1 + 119877

119886119870119892119870119905)

(24)


119866 (119904) =

119894119903(119904)

V119903(119904)

=

1

120590119871119903119904 + 119877119903+ (119871

2

119898119871

2

119904) 119877119904

(25)


119866

1015840

(119904) =

119894119903(119904)

V119903(119904)

=

119870119905

120590119871119903119904 + 119877119903+ (119871

2

119898119871

2

119904) 119877119904+ 119870119905119877119886

(26)



119905119877119886in the




998400r

KtRa120590Lr

r E

Rr +L2mL2s

Rs


10minus1 100 101 102 103minus90

minus45

0

45

90

Phas

e (de

g)

0

10

20

30

40

50

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)






1198911(119877119886) = 119894119889119902

119894119889119902= radic119894

2

119903119889119899+ 119894

2

119903119902119899(119899 = 1 2 3 4)

(27)

where

1198941199031198891

= minus

119871119898120596slip1205951199041199020

119871119904(120590119877119903+ 119877119886)

1198941199031199021

=

119871119898120596slip1205951199041198890

119871119904(120590119877119903+ 119877119886)

1198941199031198892

= minus

119871119898120596slip (1205951199041198890 minus 1205951199041198892)

119871119904(120590119877119903+ 119877119886)

0

10

20

30

40

50

60

70

80

90Step response

Time (s)

Am

plitu

de


0 05 1 15 2 25


1198941199031199022

= minus

119871119898120596slip (1205951199041199020 minus 1205951199041199022)

119871119904(120590119877119903+ 119877119886)

1198941199031198893

=

119871119898120596119903[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199023

=

119871119898120596119903[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031198894

=

119871119898120596119903119886119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199024

=

119871119898120596119903119887119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

(28)

where119898 = 120590119871119903 119899 = (120590119877

119903+ 119877119886) 119889 = 119898119899 119886 = (119889 minus 120590)(120595

1199041198890minus


1199041198890minus1205951199041198892) + 120596119904(1205951199041199020minus

1205951199041199022) 119894119903119900




119898 119871119904 and



1199041198890and 120595

1199041199020are


1199041198892and 120595



1198912(119877119886) = V119889119902

V119889119902= radicV2119889119899+ V2119902119899

(119899 = 1 2 3)

(29)

where

V1199031198891

= minus

120596slip119877119886119871119898

119899119871119904

1205951199041199020 V

1199031199021= minus

120596slip119877119886119871119898

119899119871119904

1205951199041198890

V1199031198892

=

120596slip119877119886119871119898 (1205951199041199020 minus 1205951199041199022)

119899119871119904

V1199031199022

=

120596slip119877119886119871119898 (1205951199041198890 minus 1205951199041198892)

119899119871119904



minus505

101520

024046064078087093097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



minus505

101520 024046064078087093

097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



V1199031198893

=

120596119903119877119886119871119898[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

V1199031199023

=

120596119903119877119886119871119898[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

(30)



max 1198911(119877119886) le 1205721

max 1198912(119877119886) le 1205722

119869 = 12057311198911(119877119886) + 12057321198912(119877119886)

(31)




2






119895)| If 119909

119894=

(1199091198941 1199091198942 119909

119894119898) and 119909

119895= (1199091198951 1199091198952 119909



119889 (119909119894 119909119895) = radic

119898

sum

119896=1

(119909119894119896minus 119909119895119896)

2

(32)

When 119889(119909119894 119909119895) le 119863 and |fit(119909

119894) minus fit(119909

119895)| le 119891 the




119886119894=

119899

119873

(33)



119875 (119909119894) =

fit (119909119894) 119886119894

sum

119873

119895=1[fit (119909

119895) 119886119894]

(34)











1uniform subspaces







Δ119891119894= 119891119894(119896 + 1) minus 119891

119894(119896) (35)

119865 =

119899

sum

119894=1

1003816

1003816

1003816

1003816

Δ119891119894

1003816

1003816

1003816

1003816

(119899 = 1198991times 1198992) (36)



119900 the dynamic


If 119865 gt 119865119900 then


(37)


1198941 1199091198942 119909

119894119872) and V

119894= (V

1198941 V1198942 V

119894119872) are



119901 (119909119894) =

1 (119886119894fit (119909119894))

sum

119873

119895=1[1 (119886

119894fit (119909119895))]

(38)









V119905+1119894119889

= 120596V119905119894119889+ 1198881times rand


119894119889minus 119909

119905

119894119889)

+ 1198882times rand


119892119889minus 119909

119905

119892119889)

119909

119905+1

119894119889= 119909

119905

119894119889+ V119905+1119894119889

(39)



119879

sdot 119905 (40)











0 20 40 60 80 100 120 140 160 180 200700

800

900

1000

1100

1200

1300

Iteration numbers

Fitn

ess









119904= 00125H



119904= 00043Ω





119894119903119886







119903= 02983Ω






119903119886as the DFIG



minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)


2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02



02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04



minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04



8 Conclusion




ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











+

minusE

irC(s) T(s) G(s)

Ra

r




V119904= (119877119903+ 119877119904

119871

2

119898

119871

2

119904

) 119894119903+ 120590119871119903

119889

119889119905

119894119903+ 119895120596119904119897120590119871119903119894119903+ 119864

119864 =

119871119898

119871119904

120595119904minus

119871119898

119871119904

(

119877119904

119871119904

+ 119895120596119903)120595119904

(15)



119866 (119904) =

119894119903(119904)

V119903(119904)

=

119870119892

119879119892119904 + 1

119879119892=

120590119871119903

(119877119903+ 119871

2

119898119877119904119871

2

119904)

119870119892=

1

(119877119903+ 119871

2

119898119877119904119871

2

119904)

(16)


119879 (119904) = 119870119905 (17)



119866

1015840

(119904) =

119894119903(119904)

V119903(119904)

=

119870119892119870119905

119879119892119904 + 1 + 119877

119886119870119892119870119905

(18)


119886119870119892119870119905times


119862 (119904) =

119870119888119879119892119904 + 119870119888

119870119905119870119892119904

(19)



119862

1015840

(119904) =

119870119888119879119892119904 + 119870119888+ 119877119886119870119892119870119905119870119888

119870119905119870119892119904

(20)


119866119901(119904) =

119894119903(119904)

119894

lowast

119903(119904)

=

119904

119904 + 119870119888

(21)


119866

1015840

119901(119904) = 119866

119901(119904) (22)


119866119864119868(119904) =

119894119903(119904)

119864 (119904)

=

119870119892119904

(119904 + 119870119888) (119879119892119904 + 1)

(23)


119866

1015840

119864119868(119904) =

119894119903(119904)

119864 (119904)

= minus

119870119892119904

(119904 + 119870119888) (119879119892119904 + 1 + 119877

119886119870119892119870119905)

(24)


119866 (119904) =

119894119903(119904)

V119903(119904)

=

1

120590119871119903119904 + 119877119903+ (119871

2

119898119871

2

119904) 119877119904

(25)


119866

1015840

(119904) =

119894119903(119904)

V119903(119904)

=

119870119905

120590119871119903119904 + 119877119903+ (119871

2

119898119871

2

119904) 119877119904+ 119870119905119877119886

(26)



119905119877119886in the




998400r

KtRa120590Lr

r E

Rr +L2mL2s

Rs


10minus1 100 101 102 103minus90

minus45

0

45

90

Phas

e (de

g)

0

10

20

30

40

50

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)






1198911(119877119886) = 119894119889119902

119894119889119902= radic119894

2

119903119889119899+ 119894

2

119903119902119899(119899 = 1 2 3 4)

(27)

where

1198941199031198891

= minus

119871119898120596slip1205951199041199020

119871119904(120590119877119903+ 119877119886)

1198941199031199021

=

119871119898120596slip1205951199041198890

119871119904(120590119877119903+ 119877119886)

1198941199031198892

= minus

119871119898120596slip (1205951199041198890 minus 1205951199041198892)

119871119904(120590119877119903+ 119877119886)

0

10

20

30

40

50

60

70

80

90Step response

Time (s)

Am

plitu

de


0 05 1 15 2 25


1198941199031199022

= minus

119871119898120596slip (1205951199041199020 minus 1205951199041199022)

119871119904(120590119877119903+ 119877119886)

1198941199031198893

=

119871119898120596119903[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199023

=

119871119898120596119903[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031198894

=

119871119898120596119903119886119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199024

=

119871119898120596119903119887119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

(28)

where119898 = 120590119871119903 119899 = (120590119877

119903+ 119877119886) 119889 = 119898119899 119886 = (119889 minus 120590)(120595

1199041198890minus


1199041198890minus1205951199041198892) + 120596119904(1205951199041199020minus

1205951199041199022) 119894119903119900




119898 119871119904 and



1199041198890and 120595

1199041199020are


1199041198892and 120595



1198912(119877119886) = V119889119902

V119889119902= radicV2119889119899+ V2119902119899

(119899 = 1 2 3)

(29)

where

V1199031198891

= minus

120596slip119877119886119871119898

119899119871119904

1205951199041199020 V

1199031199021= minus

120596slip119877119886119871119898

119899119871119904

1205951199041198890

V1199031198892

=

120596slip119877119886119871119898 (1205951199041199020 minus 1205951199041199022)

119899119871119904

V1199031199022

=

120596slip119877119886119871119898 (1205951199041198890 minus 1205951199041198892)

119899119871119904



minus505

101520

024046064078087093097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



minus505

101520 024046064078087093

097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



V1199031198893

=

120596119903119877119886119871119898[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

V1199031199023

=

120596119903119877119886119871119898[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

(30)



max 1198911(119877119886) le 1205721

max 1198912(119877119886) le 1205722

119869 = 12057311198911(119877119886) + 12057321198912(119877119886)

(31)




2






119895)| If 119909

119894=

(1199091198941 1199091198942 119909

119894119898) and 119909

119895= (1199091198951 1199091198952 119909



119889 (119909119894 119909119895) = radic

119898

sum

119896=1

(119909119894119896minus 119909119895119896)

2

(32)

When 119889(119909119894 119909119895) le 119863 and |fit(119909

119894) minus fit(119909

119895)| le 119891 the




119886119894=

119899

119873

(33)



119875 (119909119894) =

fit (119909119894) 119886119894

sum

119873

119895=1[fit (119909

119895) 119886119894]

(34)











1uniform subspaces







Δ119891119894= 119891119894(119896 + 1) minus 119891

119894(119896) (35)

119865 =

119899

sum

119894=1

1003816

1003816

1003816

1003816

Δ119891119894

1003816

1003816

1003816

1003816

(119899 = 1198991times 1198992) (36)



119900 the dynamic


If 119865 gt 119865119900 then


(37)


1198941 1199091198942 119909

119894119872) and V

119894= (V

1198941 V1198942 V

119894119872) are



119901 (119909119894) =

1 (119886119894fit (119909119894))

sum

119873

119895=1[1 (119886

119894fit (119909119895))]

(38)









V119905+1119894119889

= 120596V119905119894119889+ 1198881times rand


119894119889minus 119909

119905

119894119889)

+ 1198882times rand


119892119889minus 119909

119905

119892119889)

119909

119905+1

119894119889= 119909

119905

119894119889+ V119905+1119894119889

(39)



119879

sdot 119905 (40)











0 20 40 60 80 100 120 140 160 180 200700

800

900

1000

1100

1200

1300

Iteration numbers

Fitn

ess









119904= 00125H



119904= 00043Ω





119894119903119886







119903= 02983Ω






119903119886as the DFIG



minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)


2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02



02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04



minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04



8 Conclusion




ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










998400r

KtRa120590Lr

r E

Rr +L2mL2s

Rs


10minus1 100 101 102 103minus90

minus45

0

45

90

Phas

e (de

g)

0

10

20

30

40

50

Mag

nitu

de (d

B)

Bode diagram

Frequency (rads)






1198911(119877119886) = 119894119889119902

119894119889119902= radic119894

2

119903119889119899+ 119894

2

119903119902119899(119899 = 1 2 3 4)

(27)

where

1198941199031198891

= minus

119871119898120596slip1205951199041199020

119871119904(120590119877119903+ 119877119886)

1198941199031199021

=

119871119898120596slip1205951199041198890

119871119904(120590119877119903+ 119877119886)

1198941199031198892

= minus

119871119898120596slip (1205951199041198890 minus 1205951199041198892)

119871119904(120590119877119903+ 119877119886)

0

10

20

30

40

50

60

70

80

90Step response

Time (s)

Am

plitu

de


0 05 1 15 2 25


1198941199031199022

= minus

119871119898120596slip (1205951199041199020 minus 1205951199041199022)

119871119904(120590119877119903+ 119877119886)

1198941199031198893

=

119871119898120596119903[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199023

=

119871119898120596119903[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)] 119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031198894

=

119871119898120596119903119886119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

1198941199031199024

=

119871119898120596119903119887119890

minus120590119905

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

(28)

where119898 = 120590119871119903 119899 = (120590119877

119903+ 119877119886) 119889 = 119898119899 119886 = (119889 minus 120590)(120595

1199041198890minus


1199041198890minus1205951199041198892) + 120596119904(1205951199041199020minus

1205951199041199022) 119894119903119900




119898 119871119904 and



1199041198890and 120595

1199041199020are


1199041198892and 120595



1198912(119877119886) = V119889119902

V119889119902= radicV2119889119899+ V2119902119899

(119899 = 1 2 3)

(29)

where

V1199031198891

= minus

120596slip119877119886119871119898

119899119871119904

1205951199041199020 V

1199031199021= minus

120596slip119877119886119871119898

119899119871119904

1205951199041198890

V1199031198892

=

120596slip119877119886119871119898 (1205951199041199020 minus 1205951199041199022)

119899119871119904

V1199031199022

=

120596slip119877119886119871119898 (1205951199041198890 minus 1205951199041198892)

119899119871119904



minus505

101520

024046064078087093097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



minus505

101520 024046064078087093

097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



V1199031198893

=

120596119903119877119886119871119898[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

V1199031199023

=

120596119903119877119886119871119898[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

(30)



max 1198911(119877119886) le 1205721

max 1198912(119877119886) le 1205722

119869 = 12057311198911(119877119886) + 12057321198912(119877119886)

(31)




2






119895)| If 119909

119894=

(1199091198941 1199091198942 119909

119894119898) and 119909

119895= (1199091198951 1199091198952 119909



119889 (119909119894 119909119895) = radic

119898

sum

119896=1

(119909119894119896minus 119909119895119896)

2

(32)

When 119889(119909119894 119909119895) le 119863 and |fit(119909

119894) minus fit(119909

119895)| le 119891 the




119886119894=

119899

119873

(33)



119875 (119909119894) =

fit (119909119894) 119886119894

sum

119873

119895=1[fit (119909

119895) 119886119894]

(34)











1uniform subspaces







Δ119891119894= 119891119894(119896 + 1) minus 119891

119894(119896) (35)

119865 =

119899

sum

119894=1

1003816

1003816

1003816

1003816

Δ119891119894

1003816

1003816

1003816

1003816

(119899 = 1198991times 1198992) (36)



119900 the dynamic


If 119865 gt 119865119900 then


(37)


1198941 1199091198942 119909

119894119872) and V

119894= (V

1198941 V1198942 V

119894119872) are



119901 (119909119894) =

1 (119886119894fit (119909119894))

sum

119873

119895=1[1 (119886

119894fit (119909119895))]

(38)









V119905+1119894119889

= 120596V119905119894119889+ 1198881times rand


119894119889minus 119909

119905

119894119889)

+ 1198882times rand


119892119889minus 119909

119905

119892119889)

119909

119905+1

119894119889= 119909

119905

119894119889+ V119905+1119894119889

(39)



119879

sdot 119905 (40)











0 20 40 60 80 100 120 140 160 180 200700

800

900

1000

1100

1200

1300

Iteration numbers

Fitn

ess









119904= 00125H



119904= 00043Ω





119894119903119886







119903= 02983Ω






119903119886as the DFIG



minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)


2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02



02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04



minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04



8 Conclusion




ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus505

101520

024046064078087093097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



minus505

101520 024046064078087093

097

0992

024046064078087093097

0992

1020304050

Root locus

Real axis (sminus1)

Imag

inar

y ax

is (sminus1)



V1199031198893

=

120596119903119877119886119871119898[119887 cos (120596

119904119905) minus 119886 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

V1199031199023

=

120596119903119877119886119871119898[119886 cos (120596

119904119905) + 119887 sin (120596

119904119905)]

119899119871119904[(119889 minus 120590)

2

+ 120596

2

119904]

119890

minus120590119905

(30)



max 1198911(119877119886) le 1205721

max 1198912(119877119886) le 1205722

119869 = 12057311198911(119877119886) + 12057321198912(119877119886)

(31)




2






119895)| If 119909

119894=

(1199091198941 1199091198942 119909

119894119898) and 119909

119895= (1199091198951 1199091198952 119909



119889 (119909119894 119909119895) = radic

119898

sum

119896=1

(119909119894119896minus 119909119895119896)

2

(32)

When 119889(119909119894 119909119895) le 119863 and |fit(119909

119894) minus fit(119909

119895)| le 119891 the




119886119894=

119899

119873

(33)



119875 (119909119894) =

fit (119909119894) 119886119894

sum

119873

119895=1[fit (119909

119895) 119886119894]

(34)











1uniform subspaces







Δ119891119894= 119891119894(119896 + 1) minus 119891

119894(119896) (35)

119865 =

119899

sum

119894=1

1003816

1003816

1003816

1003816

Δ119891119894

1003816

1003816

1003816

1003816

(119899 = 1198991times 1198992) (36)



119900 the dynamic


If 119865 gt 119865119900 then


(37)


1198941 1199091198942 119909

119894119872) and V

119894= (V

1198941 V1198942 V

119894119872) are



119901 (119909119894) =

1 (119886119894fit (119909119894))

sum

119873

119895=1[1 (119886

119894fit (119909119895))]

(38)









V119905+1119894119889

= 120596V119905119894119889+ 1198881times rand


119894119889minus 119909

119905

119894119889)

+ 1198882times rand


119892119889minus 119909

119905

119892119889)

119909

119905+1

119894119889= 119909

119905

119894119889+ V119905+1119894119889

(39)



119879

sdot 119905 (40)











0 20 40 60 80 100 120 140 160 180 200700

800

900

1000

1100

1200

1300

Iteration numbers

Fitn

ess









119904= 00125H



119904= 00043Ω





119894119903119886







119903= 02983Ω






119903119886as the DFIG



minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)


2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02



02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04



minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04



8 Conclusion




ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















1uniform subspaces







Δ119891119894= 119891119894(119896 + 1) minus 119891

119894(119896) (35)

119865 =

119899

sum

119894=1

1003816

1003816

1003816

1003816

Δ119891119894

1003816

1003816

1003816

1003816

(119899 = 1198991times 1198992) (36)



119900 the dynamic


If 119865 gt 119865119900 then


(37)


1198941 1199091198942 119909

119894119872) and V

119894= (V

1198941 V1198942 V

119894119872) are



119901 (119909119894) =

1 (119886119894fit (119909119894))

sum

119873

119895=1[1 (119886

119894fit (119909119895))]

(38)









V119905+1119894119889

= 120596V119905119894119889+ 1198881times rand


119894119889minus 119909

119905

119894119889)

+ 1198882times rand


119892119889minus 119909

119905

119892119889)

119909

119905+1

119894119889= 119909

119905

119894119889+ V119905+1119894119889

(39)



119879

sdot 119905 (40)











0 20 40 60 80 100 120 140 160 180 200700

800

900

1000

1100

1200

1300

Iteration numbers

Fitn

ess









119904= 00125H



119904= 00043Ω





119894119903119886







119903= 02983Ω






119903119886as the DFIG



minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)


2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02



02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04



minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04



8 Conclusion




ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120 140 160 180 200700

800

900

1000

1100

1200

1300

Iteration numbers

Fitn

ess









119904= 00125H



119904= 00043Ω





119894119903119886







119903= 02983Ω






119903119886as the DFIG



minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)


2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02



02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04



minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04



8 Conclusion




ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus2

0

2

Vs

(pu)

051

15

i rq

(pu)

minus020

0204

Vrq

(pu)

2 25 3 35 4 45 5 55 6minus2

0

2

t (s)

i ra

(pu)


2 25 3 35 4 45 5 55 6

t (s)

0

2

Vs

(pu)

05

1

15

i rq

(pu)

00204

Vrq

(pu)

0

2

i ra

(pu)

minus2

minus2

minus02



02

012

002

2 25 3 35 4 45 5 55 60

1

2

minus2Vs

(pu)

i rq

(pu)

minus02Vrq

(pu)

t (s)

i ra

(pu)


02

012

002

2 25 3 35 4 45 5 55 6012

t (s)

Vs

(pu)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus02minus04



minus04

0

2

0

1

2

002

2 25 3 35 4 45 5 55 6

0

2

minus2

i rq

(pu)

minus02

Vrq

(pu)

Vsq

(pu)

minus2

t (s)

i ra

(pu)


0

2

0

1

2

002

0

2

Vs

(pu)

2 25 3 35 4 45 5 55 6

t (s)

i rq

(pu)

Vrq

(pu)

i ra

(pu)

minus2

minus2

minus02minus04



8 Conclusion




ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)



ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)


ug (300Vdiv)

ird (5Adiv)

Vrq (100Vdiv)

t (200msdiv)

ira (10Adiv)







Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article The Virtual Resistance Control Strategy ...downloads.hindawi.com/journals/mpe/2014/350367.pdf · When grid voltage increases to % of the nominal voltage, wind turbine