Research Article Equivalent Modeling of DFIG-Based Wind Power …downloads.hindawi.com/journals/mpe/2016/8426492.pdf · 2019-07-30 · Research Article Equivalent Modeling of DFIG-Based

Research ArticleEquivalent Modeling of DFIG-Based Wind Power PlantConsidering Crowbar Protection

Qianlong Zhu Ming Ding and Pingping Han

Hefei University of Technology Hefei Anhui 230009 China

Correspondence should be addressed to Qianlong Zhu zhuqianl19163com

Received 24 February 2016 Accepted 30 June 2016

Academic Editor Jason Gu

Copyright copy 2016 Qianlong Zhu et al 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

Crowbar conduction has an impact on the transient characteristics of a doubly fed induction generator (DFIG) in the short-circuitfault condition But crowbar protection is seldom considered in the aggregation method for equivalent modeling of DFIG-basedwind power plants (WPPs) In this paper the relationship between the growth of postfault rotor current and the amplitude of theterminal voltage dip is studied by analyzing the rotor current characteristics of a DFIG during the fault process Then a terminalvoltage dip criterion which can identify crowbar conduction is proposed Considering the different grid connection structuresfor single DFIG and WPP the criterion is revised and the crowbar conduction is judged depending on the revised criterionFurthermore an aggregation model of the WPP is established based on the division principle of crowbar conduction Finallythe proposed equivalent WPP is simulated on a DIgSILENT PowerFactory platform and the results are compared with those ofthe traditional equivalent WPPs and the detailed WPP The simulation results show the effectiveness of the method for equivalentmodeling of DFIG-based WPP when crowbar protection is also taken into account

1 Introduction

With the rapid development of wind power generation andpower electronics the doubly fed induction generator (DFIG)has recently become the most commonly used wind turbinein wind power plant (WPP) based on its characteristicsof maximum power point tracking (MPPT) [1] decoupledcontrol of active and reactive powers [2] the use of a powerconvertor with a rated power of 25 of total system power[3] and so forth To investigate the effect of the ongoingchanges in power systems due to the increasing penetrationof wind power a wide range of studies from steady-state totransient are necessary A detailed WPP model in whichthe dynamics of each wind turbine and the internal networkare fully represented is not suitable as it could significantlyincrease the order of the mathematical model of the powersystem to be solved and thus influence the overall simulationtime As a result the equivalent WPP model is generallyrecommended for reflecting the collective response of thewhole WPP on large power systems

Equivalent WPPs using aggregated wind turbines areclassified as either the single-machine representation or themultiple-machine representation [4] In the first case theactual WPP is modeled as a unique rescale wind turbinein the single-machine representation Accordingly this isrational when all wind turbines are operating in an identicalcondition whereas that might not be the case in real-worldsystem operation Nowadays in order to improve the con-formity of the single-machine representation with the actualWPP numerical identification methods for optimizing theparameters of equivalent models are applied The optimizedobjects include the generator [5 6] power convertor [7]and passive frequency-dependent network [8] The fuzzylogic system is also utilized to compute the mechanicaltorque compensating factor which is integrated into a single-machine aggregated WPP [9]

In the second case study of the multiple-machine repre-sentation focuses on selecting reasonable clustering indicesby quantifying and abstracting key features of wind tur-bines The coherency method is presented in [10 11] by

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8426492 16 pageshttpdxdoiorg10115520168426492

2 Mathematical Problems in Engineering

DFIG Infinitegrid

MV HV

35kV220 kV690V35kV

Figure 1 Schematic of DFIG generation system

DFIG modelGrid code

02

04

06

08

10

Volta

ge (p

u)

0 1 2 3 4minus1

Time (s)

(a)

00

05

10

15

20

Activ

e pow

er (M

W)

0 1 2 3 4minus1

Time (s)

DFIG model(b)

minus4

minus3

minus2

minus1

0

1

2

3

Reac

tive p

ower

(Mva

r)

0 1 2 3 4minus1

Time (s)

DFIG model(c)

Figure 2 (a) PCC voltage of the DFIG model (b) PCC output active power (c) PCC output reactive power

clustering wind turbines with similar wind speed In [12]the wind direction is implemented To represent the powerloss on the collective network within the WPP a method ofcalculating the equivalent impedance is introduced in [13]

Moreover research work in [14] analyzes the impact of theline impedance on the diversity in the voltage profile of eachwind turbine and describes a voltage-profile-based approachto develop multiple-machine aggregated WPP In addition a

Mathematical Problems in Engineering 3

096 098 1 102 104 106 108 11

005

115

2055

06

065

07

075

Steady-state terminal voltage (pu)Active power (MW)

Term

inal

vol

tage

dip

thre

shol

d (p

u)

(a)

096 098 1 102 104 106 108 11

005

115

206

065

07

075

08

Term

inal

vol

tage

dip

thre

shol

d (p

u)

Active power (MW)Steady-state terminal voltage (pu)

(b)

096 098 1 102 104 106 108 11

005

115

2

065

07

075

08

085

Term

inal

vol

tage

dip

thre

shol

d (p

u)

Active power (MW) Steady-state terminal voltage (pu)

(c)

Figure 3 Terminal voltage dip threshold of DFIG (a) power factor = minus095 (b) power factor = 1 and (c) power factor = 095


Z1 ZnZnminus1Z2

WT1 WTnWTnminus1WT2

Figure 4 Daisy-chain physical diagram of the WPP

clustering index considering the influence of fault types oncoherency is depicted in [15]

Compared with conventional power resources windresources are stochastic and fluctuating Equivalent modelsof an actual WPP might change due to different operatingconditions Therefore the actual application should be basedon historical data for the specific WPP and establish aprobabilistic equivalent WPP [13 16 17] This makes theequivalent WPP very practical and easy to use

On the basis of previous equivalentmethods the researchhot-spot has gradually shifted to the validation and anal-ysis of equivalent WPPs using field measurement dataThe primary focus of reference [17] is to explain that themultiple-machine representation should take the diversityin protection conduction into account by investigating thediscrepancies between the simulation results and the actualmeasurement Perdana et al [18] and Singh et al [19] exploremethods for validating the equivalent WPPs for SCIG-basedand DFIG-based WPP respectively Meanwhile reference[20] supports the fact that traditional clustering indicesfail to cluster wind turbines when crowbar protection isenabled Therefore it is necessary to develop an equivalentmethod for modeling WPPs considering system protec-tion

This paper proposes a new equivalent method for mod-eling WPPs when the crowbar protection is taken intoaccount Based on the rotor current of the DFIG afteran external three-phase short-circuit fault the terminalvoltage dip threshold is determined and from this it canbe identified whether the crowbar protection is activatedThen the threshold is revised based on the influence of thechain topology of the WPP The revised terminal voltagedip threshold can be utilized to judge whether the crowbaris conducting Furthermore the equivalent WPP is estab-lished based on the division principle of crowbar conduc-tion Finally the proposed equivalent WPP is simulatedby the DIgSILENT PowerFactory platform and the resultsare compared with those of traditional equivalent WPPsand the detailed WPP The simulation results validate theeffectiveness of the equivalent method for modeling DFIG-based WPP

2 Postfault Rotor Current TransientAnalysis of DFIG

21 Postfault Rotor Current Calculation Using the motorconversion the stator and rotor voltages and fluxes in asynchronously rotating 119889-119902 reference frame are given by

119906

119904= 119877

119904119894

119904+

119889120595

119904

119889119905

+ 119895120596

1120595

119904

119906

119903= 119877

119903119894

119903+

119889120595

119903

119889119905

+ 119895120596

119904120595

119903

120595

119904= 119871

119904119894

119904+ 119871

119898119894

119903

120595

119903= 119871

119898119894

119904+ 119871

119903119894

119903

(1)

where 119906 119894 and 120595 represent voltage current and fluxrespectively 119877 is the resistance and 119871 is the inductanceSubscripts 119904 and 119903 denote the stator and rotor quantitiesrespectively 119871

119898is the mutual inductance 120596

1and 120596

119904are the

synchronous speed and the slip speed respectivelyOne DFIG is connected to the infinite grid through a

grounding transformer and a tertiary transformer as shownin Figure 1 It is assumed that at 119905 = 119905

0 an external three-

phase short-circuit fault occurs and as a result the terminalvoltage of the DFIG abruptly changes from the steady-statevoltage 119906

1199040to the remaining voltage 119906

119904119896 The solution of (1)

for 119905 gt 119905

0can be written as

120595

119904=

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904

120595

119903=

119906

119903(0)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906

119904119896

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904

+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903

(2)

where 119906

119903(0) is the prefault rotor voltage 119879

119904= (119871

119904119871

119903minus

119871

2

119898)(119871

119903119877

119904) is the stator decay time constant 119879

119903= (119871

119904119871

119903minus


Table 1 Model parameters

Generator

Rater powerMW 2 Rated stator voltageV 690Rated frequencyHz 50 119883

119898pu 35

119877

119904pu 001 119883

119904pu 01

119877

119903(pu) 001 119883

119903pu 01

Grounding transformer SMVA 25 119883

119879 6

Tertiary transformer SMVA 150 119883

119879 135

Crowbar protection Maximum rotor currentpu 2 119877crpu 01Overhead line

LGJ-24035 119903

0(Ωkm) 0131 119909

0(Ωkm) 0372

LGJ-300220 119903

0(Ωkm) 0107 119909

0(Ωkm) 0414

119871

2

119898)(119871

119904119877

119903) is the rotor decay time constant 120596

119903and 119862

1

represent the rotor electrical angular speed and the integralconstant respectively and 119860 = minus119871

119898(119871

119904119871

119903minus 119871

2

119898)

When the DFIG operates in the stator voltage vectororiented control mode 119906

119904119889= 119880

119904 119906119904119902

= 0 Thus therelationship between the output powers voltage and currentcan be written as

119875

119904= minus

3

2

(119906

119904119889119894

119904119889+ 119906

119904119902119894

119904119902) = minus

3

2

119880

119904119894

119904119889

119876

119904= minus

3

2

(119906

119904119902119894

119904119889minus 119906

119904119889119894

119904119902) =

3

2

119880

119904119894

119904119902

(3)

where119880119904= |119906

1199040| is the amplitude of the steady-state terminal

voltage

Based on (1) and (3) the prefault rotor voltage 119906119903(0) can

be described as

119906

119903119889(0) =

2

3

1

120596

1119871

119898119880

119904

(119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904

+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+

3

2

120596

119904119871

119903119880

2

119904)

119906

119903119902(0) =

2

3

1

120596

1119871

119898119880

119904

(120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904

minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus

3

2

119877

119903119880

2

119904)

(4)

where 120590 = 1 minus 119871

2

119898(119871

119904119871

119903)

From (1) the rotor current 119894119903is derived as follows

119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

120595

119904+

119871

119904

119871

119904119871

119903minus 119871

2

119898

120595

119903 (5)

According to (2) (4) and (5) the dynamic representation forrotor current of the DFIG can be written as

119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

[

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904]

+

119871

119904

119871

119904119871

119903minus 119871

2

119898

[

(23) (1120596

1119871

119898119880

119904) (119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+ (32) 120596

119904119871

119903119880

2

119904)

119895120596

119904+ 1119879

119903

+

119895 (23) (1120596

1119871

119898119880

119904) (120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus (32) 119877

119903119880

2

119904)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906sk

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903] = 119891 (119877

119904 119871

119904 119877

119903 119871

119903 119871

119898 119875

119904 119876

119904 119906

119904119896 119906

1199040 120596

1 120596

119904 120596

119903)

(6)

22 Timer Action Crowbar The crowbar circuit used in thispaper is formed of a three-phase rectifier power resistorand series IGBT switch [21] The crowbar protection signalis triggered when the magnitude of the rotor current exceedsa threshold value set for crowbar protection insertion (2 pu)

and then remains engaged for a fixed time (50ms) Whenthe crowbar is released rotor-side pulse width modula-tion (PWM) and rotor current PI control are immediatelyresumed As shown in Figure 2 the DFIGwind turbinemeetsfault ride through requirements which refer to the capability


9

8

7

6

5

4

3

2

HV_B

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

0

Transmissionline

10 10 10

9

8

7

6

5

4

3

2

0

10

9

8

7

6

5

4

3

2

0

Maintransformer

Line 1 Line 2 Line 3 Line 4 Line 5

10

9

8

7

6

5

4

3

2

1

Line 6

1 1

0 0

1 1

0

A

45km 05 km 55km 15 km 65 km 3km

35kV

35kV

220 kV

220 kV

Zeq

L(1

)L

(0)

L(2

)L

(3)L

(4)L

(5)L

(6) L

(7)L

(8)L

(9)L

(10

)

Medium-voltage bus

2MW

Point ofinterconnection

Figure 5 WPP network

of generation plant to remain connected and dynamicallystable and offer network support throughout a serious voltagedisturbance [22]

23 Terminal Voltage Dip Threshold According to (6) whenthe DFIG is connected to the infinite grid the postfault rotorcurrent is jointly determined by the generator parameters

active and reactive power outputs terminal voltage dipsteady-state terminal voltage and generator speed For theDFIG with the MPPT property there is a correspondingrelationship between the generator speed and the activepower output Thus the influence of the generator speed onthe rotor current can be included in the influence of theactive power output Once the generator parameters and themaximum rotor current for crowbar protection insertion are


Table 2 Wind data of each DFIG at 105ms 05∘ wind

DFIG vmsdotsminus1

1(0) 10501(1) 8921(2) 8381(3) 8731(4) 8181(5) 7981(6) 7921(7) 9161(8) 8591(9) 8171(10) 8042(0) 8002(1) 7992(2) 8462(3) 8252(4) 8482(5) 8462(6) 8092(7) 8002(8) 7982(9) 8462(10) 8083(0) 10503(1) 9913(2) 8573(3) 8233(4) 8143(5) 8773(6) 8113(7) 10323(8) 8853(9) 8993(10) 8664(0) 8204(1) 7994(2) 7984(3) 7974(4) 7974(5) 7974(6) 7964(7) 7964(8) 7964(9) 7964(10) 7965(0) 1050

Table 2 Continued

DFIG vmsdotsminus1

5(1) 8745(2) 8325(3) 8155(4) 8205(5) 8845(6) 8255(7) 8165(8) 8635(9) 8785(10) 8066(0) 9046(1) 9686(2) 8616(3) 8356(4) 8066(5) 9346(6) 9026(7) 8936(8) 8286(9) 8086(10) 801

fixed the terminal voltage dip is the function of only thepower outputs and the steady-state terminal voltage As aresult the terminal voltage dip threshold can be quantifiedwhen the crowbar protection is just activated Crowbarconduction occurs if the terminal voltage dip is lower thanthe terminal voltage drop threshold Otherwise the crowbarprotection is not activated

We select 01MW as the power step length within thescope of the DFIG rated power and 001 pu as the voltage steplength within the amplitude scope of steady-state terminalvoltage 097ndash107 pu [22] The terminal voltage dip changesthrough regulating the external short-circuit impedance andthen finding out the terminal voltage dip threshold whichmakes the rotor current equal to the maximum rotor currentfor crowbar protection insertion The terminal voltage dipthresholds of the DFIG whose power factor is minus095 1and 095 respectively are presented in Figure 3 The DFIGparameters are specified in Table 1

3 Dynamic Equivalence of the WPP

31 Effect of Collector Line Impedance on Terminal Voltage DipThreshold In order to reduce the power loss of a low-voltagecircuit it might be a better practice to connect multiplewind turbines to a middle-voltage bus using the daisy-chainstructure within an actual WPP and then power in-feed tothe external grid from the WPP through transmission linesIn the same feeder owing to the convergence effect of output


Table3DFIGun

itswith

crow

barp

rotectioncond

uctio

n

Groun

ding

impedance

DetailedWPP

Before

revisio

nAfte

rrevision

Rightjud

gment

Wrong

judgment

Rightjud

gment

Wrong

judgment

119885

119891=11989555Ω

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(0)1(7)ndash(10)2(0)ndash(10)3(0)

3(7)3(9)-(10)4(0)ndash(10)5(0)

6(0)ndash(10)

1(1)1(3)ndash(6)3(1)3(8)5

(9)

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(2)5(10)

119885

119891=1198957Ω

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

1(0)2(0)ndash(10)4(3)ndash(10)6(5)

6(7)6(9)-(10)

3(7)4(0)ndash(2)6(1)6(6)

6(8)

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

Non

e



DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850

current from other wind turbines the voltage delta betweena wind turbine and the middle-voltage bus has increasedsignificantly which is equivalent to increasing the electricaldistance of the wind turbine to the middle-voltage bus

Within the WPP the differences in terminal voltagesamong the wind turbines are mainly caused by the differentelectrical distances from each wind turbine to the middle-voltage bus Compared with the infinite grid connectionstructure for a single DFIG in theWPP topology the increas-ing effect of electrical distance between the wind turbine andthemiddle-voltage busmeans the line impedancemust not beignored At this time the line impedance is jointed betweenthe grounding transformer and themiddle-voltage bus Basedon (6) if the impedance is incorporated into the generatorstator side the change in generator impedance parameterswill have an impact on the rotor time constant which willaffect the relationship between the terminal voltage dip androtor current Therefore the terminal voltage dip thresholdsshould be revised

32 Terminal Voltage Dip Threshold Revision In the samefeeder owing to the convergence of output currents amongthe wind turbines the voltage drop has increased Inthis paper the virtual line impedance is introduced tocharacterize the equivalent electrical distance in the daisy-chain topological structure of the WPP


Table 5 Equivalent accuracy of crowbar conduction and model error indices

Operating conditions Clustering index Equivalent accuracy Dynamic absolute error Dynamic relative error119880pu 119875MW 119876Mvar 119880 119875 119876

Figure 6(a)Crowbar conduction 1000 00004 008 012 007 065 144

Incoming wind 773 00016 053 061 039 536 398

Speed vector of generator 576 00061 067 232 142 632 2587

Figure 6(b)Crowbar conduction 1000 00007 010 023 015 077 1603

Incoming wind 545 00046 073 178 113 639 2744


Figure 6(c)Crowbar conduction 985 00012 035 050 022 246 378

Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097


As shown in Figure 4 there are n DFIGs in one feederand the voltage drop Δ119880

119896that is between the wind turbine

number 119896 and the middle-voltage bus is

Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885

119895refers to the line impedance number 119895 119878lowast

119894is

the conjugate value of apparent power of the wind turbinenumber 119894 119880 is the voltage amplitude of the middle-voltagebus

Based on the principle of equal voltage drop the virtualline impedance 119885eq-119896 which is between the wind turbinenumber 119896 and the middle-voltage bus is

119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)

In the infinite grid connection structure for a single DFIG119885eq-119896 is jointed into the grounding transformer and themiddle-voltage bus then the terminal voltage dip thresholdis redetermined by means of the method in Section 23

33 Clustering Principle of DFIGs Based on the coherency-based method in the power system dynamic equivalencecrowbar conduction is selected as the clustering principle

for example DFIGs are divided into two groups (1) crowbarconduction during the fault (2) crowbar nonconductionduring the fault Depending on the abovementioned analysiswhether the crowbar protection is activated or not can bejudged by comparing the postfault terminal voltage dip of theDFIG with its terminal voltage dip threshold

4 Simulation Results

The developed equivalent WPP is tested by comparing itstransient response with the traditional equivalent WPPs andthe detailed WPP

41 WPP Description The network of the WPP is pre-sented in Figure 5 The WPP is composed of one 35220 kVtransformer six collector systems and 069 kV DFIGs thatare connected to the collector system through 06935 kVgrounding transformers Each collector system has 11 DFIGsand the interval between twoDFIGswithin the same collectorsystem is 400m Model parameters are given in Table 1

42 Validation of Clustering Index The measured wind dataof the 66 DFIGs are listed in Table 2 All DFIGs operatewith a unity power factor Three-phase short-circuit faultsare applied at point A as shown in Figure 4 Faults startat 01 s and last for 150ms Two case studies are performedfor different short-circuit grounding impedance (1) 119885

119891=


Table 6 The crowbar conditions of the detailed WPPs (radic stands for having a working crowbar)

DFIG Figure 6(a) Figure 6(b) Figure 6(c) Figure 7(a) Figure 7(b) Figure 7(c) Figure 81(0) radic radic radic

1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic

1(9) radic radic radic


2(0) radic radic radic radic radic radic

2(1) radic radic radic radic radic radic radic

2(2) radic radic radic radic radic









3(0) radic

3(1) radic

3(2)3(3)3(4)3(5)3(6)3(7) radic radic radic

3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued

DFIG Figure 6(a) Figure 6(b) Figure 6(c) Figure 7(a) Figure 7(b) Figure 7(c) Figure 84(10) radic radic radic radic radic radic radic

5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic

5(10)6(0) radic radic

6(1) radic radic

6(2) radic radic

6(3) radic radic




6(7) radic radic radic radic





Clustering index Equivalent accuracy Dynamic absolute error Dynamic relative error119880pu 119875MW 119876Mvar 119880 119875 119876

Crowbar conduction 985 00014 017 055 029 134 1106Incoming wind 561 00057 048 189 141 361 4182Speed vector of generator 561 00055 043 182 138 324 3986Reactive-power level 894 00061 028 122 132 244 5748

11989555Ω (2) 119885119891= 1198957Ω therefore the voltage of the 35 kV

bus decreases to 537 and 557 of its nominal value Thepostfault terminal voltage of the DFIG is compared withits terminal voltage dip threshold before revision and afterrevision respectively thus judging the DFIG whose crowbarprotection is activated Then based on the simulation resultsof the detailed WPP the correctness of the abovementionedjudgment is tested and the statistical results are listed inTable 3

As can be seen from Table 3 when 119885

119891= 11989555Ω and

119885

119891= 1198957Ω based on the terminal voltage dip threshold

before revision the numbers of wrong judgments of crowbarconduction are eight and seven respectively based on the

terminal voltage dip threshold after revision the numbersof wrong judgments of crowbar conduction are reduced totwo and zero respectively The accuracy of the judgment ofcrowbar protection conduction can be significantly improvedby using the terminal voltage dip threshold after revision

43 Transient Response The detailed WPP and the equiv-alent WPPs are modeled on the DIgSILENT PowerFactoryplatform The detailed WPP consists of the 66 DFIGscollection systems grounding transformers and a maintransformer Three-phase short-circuit faults are applied atpoint A as shown in Figure 5 Faults start at 01 s and last for150ms


0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)

Figure 6119880 119875 and119876 of theWPPs at 105ms 05∘ wind (a) power factor = minus095 (b) power factor = 1 and (c) power factor = 095The threeequivalent WPPs built based on crowbar conduction incoming wind and speed vector of generator are in red dash blue dot and magentadash dot respectively and the detailed WPP is in black solid

Compared with the simulation results of the detailedWPP error indicators are defined as follows

119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884

119891119894(119896) are the output variables of the

complete collector system for the equivalent WPPs and thedetailed WPP respectively and 119899 stands for the number ofDFIGs within the WPP

431 The Same Power Factor within WPP All DFIGs oper-ate with the same power factor and two case studies areperformed for different incoming winds (1) 105ms 05∘

(2) 85ms 1745∘ Based on wake effects the incomingwinds of each DFIG are listed in Tables 2 and 4 respectivelyThree different reactive-power levels of DFIG for examplethe maximum production (095) unity power factor andmaximum consumption (minus095) are used for the casesFigures 6 and 7 illustrate the behavior of the voltages activeand reactive powers of the complete collector system for thethree equivalent WPPs and the detailed WPP following theapplication of a three-phase short-circuit fault

The equivalent accuracy of the equivalent WPPs com-pared to the detailed WPP about the crowbar protectionconduction and error indices are listed in Table 5 Duringthe fault whether the crowbar protection is triggered or notdecides the reactive power supporting ability of the DFIG tothe power grid thereby affecting the depth of the voltage dipand the active power output of theWPPThe equivalentWPP


03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)


built by this paper is more similar to the detailedWPP due toits higher equivalent accuracy for crowbar conduction

The crowbar conditions of the detailed WPPs are listedin Table 6 It should be noted that the DFIG might lose itscapability to control the generator after crowbar conductionthus a DFIG whose generator speed is lower than thesynchronous speed operates in the motor state From faultelimination to crowbar protection resection with the gridvoltage recovery a DFIG operating in the motor state willconsume more active power causing the active power at thecomplete collector system for theWPP to appear with a largernegative value The equivalent WPP built by this paper canalso accurately reflect this situation

432 Different Power Factors among WPP Feeders Theincoming wind of the WPP is 105ms 05∘ Based on wake

effects the incomingwinds of eachDFIG are listed in Table 2The power factors of the DFIGs in feeders 1-2 3-4 and5-6 are minus095 1 and 095 respectively Three-phase short-circuit faults are applied at point A as shown in Figure 5Faults start at 01 s and last for 150ms Figure 8 shows thebehavior of the active and reactive powers at the completecollector system for the equivalent WPPs and the detailedWPP following the application of a three-phase short-circuitfault The crowbar condition of the detailed WPP is listedin Table 6 The equivalent accuracy of the equivalent WPPscompared to the detailed WPP about the crowbar protectionconduction and error indices are listed in Table 7

When the DFIGs operate with different power factorswithin theWPP the traditional equivalentmethodwill dividethe DFIGs which have the same reactive-power level intogroups But the simulation results of the detailed WPP


02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50

Figure 8119880 119875 and119876 of theWPPsThe four equivalentWPPs built based on crowbar conduction incoming wind speed vector of generatorand reactive-power level are in red dash blue dot magenta dash dot and green dash dot respectively and the detailedWPP is in black solid

indicate that the crowbar conduction of DFIGs with the samepower factor is not completely consistent

5 Conclusions

After the fault the crowbar conduction of the DFIG is relatedto the depth of the terminal voltage dipThis paper simplifiesthe judgment criterion of crowbar conduction as the terminalvoltage dip threshold which is convenient to calculate andmakes it easy to identify the crowbar conduction

Considering the influence of the connection lineimpedance within the same collector system on theterminal voltage dip threshold the virtual line impedance isintroduced to characterize the equivalent electrical distancefrom the DFIG to the medium-voltage bus The equivalentaccuracy of crowbar conduction discrimination is improvedby revising the terminal voltage dip threshold

The equivalent WPP established based on the divisionprinciple of crowbar conduction is verified showing that itcan more accurately reflect the external characteristics of theWPP

Appendix

See Tables 1 2 4 and 6

Competing Interests

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

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 51207039) the Science and TechnologyFoundation of SGCC (no NY17201200073) and the NationalEnergy Application Technology Research and EngineeringDemonstration Projects (no NY20110406-1)

References

[1] Z Yu M E Elbuluk and Y Sozer ldquoStability analysis ofmaximum power point tracking (MPPT) method in wind


power systemsrdquo IEEETransactions on Industry Applications vol49 no 3 pp 1129ndash1136 2013

[2] M Tazil V Kumar R C Bansal et al ldquoThree-phase doublyfed induction generators an overviewrdquo IET Electric PowerApplications vol 4 no 2 pp 75ndash89 2010

[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] E Muljadi and A Ellis Final Project Report WECC WindGenerator Development California Institute for Energy andEnvironment 2010

[5] M A Elizondo L Shuai Z Ning and N Samaan ldquoModelreduction validation and calibration of wind power plants fordynamic studiesrdquo in Proceedings of the IEEE Power and EnergySociety General Meeting pp 1ndash8 San Diego Calif USA July2011

[6] I Erlich F Shewarega C Feltes F Koch and J FortmannldquoDetermination of dynamic wind farm equivalents usingheuristic optimizationrdquo in Proceedings of the IEEE Power andEnergy Society General Meeting (PES rsquo12) pp 1ndash8 IEEE SanDiego Calif USA July 2012

[7] Y-Q Jin P Ju and X-P Pan ldquoAnalysis on controller aggre-gation method for equivalent modeling of DFIG-based windfarmrdquo Automation of Electric Power Systems vol 38 no 3 pp19ndash24 2014

[8] D N Hussein M Matar and R Iravani ldquoA type-4 wind powerplant equivalent model for the analysis of electromagnetictransients in power systemsrdquo IEEE Transactions on PowerSystems vol 28 no 3 pp 3096ndash3104 2013

[9] M A Chowdhury W X Shen N Hosseinzadeh and HR Pota ldquoA novel aggregated DFIG wind farm model usingmechanical torque compensating factorrdquo Energy Conversionand Management vol 67 pp 265ndash274 2013

[10] LM Fernandez C A Garcıa J R Saenz and F Jurado ldquoEquiv-alent models of wind farms by using aggregated wind turbinesand equivalentwindsrdquoEnergyConversion andManagement vol50 no 3 pp 691ndash704 2009

[11] I Zubia J X Ostolaza A Susperregui and J J UgartemendialdquoMulti-machine transient modelling of wind farms an essentialapproach to the study of fault conditions in the distributionnetworkrdquo Applied Energy vol 89 no 1 pp 421ndash429 2012

[12] M Ali I-S Ilie J V Milanovic and G Chicco ldquoWindfarm model aggregation using probabilistic clusteringrdquo IEEETransactions on Power Systems vol 28 no 1 pp 309ndash316 2013

[13] E Muljadi C P Butterfield A Ellis et al ldquoEquivalencing thecollector system of a large wind power plantrdquo in Proceedings ofthe IEEE Power Engineering Society General Meeting (PES rsquo06)pp 1ndash9 Montreal Canada June 2006

[14] Y Cheng M Sahni J Conto S-H Huang and J SchmallldquoVoltage-profile-based approach for developing collection sys-tem aggregated models for wind generation resources for gridvoltage ride-through studiesrdquo IET Renewable Power Generationvol 5 no 5 pp 332ndash346 2011

[15] Q Zhu P Han M Ding X Zhang and W Shi ldquoProba-bilistic equivalent model for wind farms based on clustering-discriminant analysisrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 34 no 28 pp 4770ndash4780 2014

[16] E Muljadi Z Mills R Foster J Conto and A Ellis ldquoFaultanalysis at a wind power plant for one year observationrdquo inProceedings of the IEEE Power Energy Society General Meetingpp 1ndash7 Pittsburgh Pa USA July 2008

[17] Y Zhang E Muljadi D Kosterev and M Singh ldquoWind powerplant model validation using synchrophasor measurements atthe point of interconnectionrdquo IEEE Transactions on SustainableEnergy vol 6 no 3 pp 984ndash992 2015

[18] A Perdana S Uski-Joutsenvuo O Carlson and B LemstromldquoComparison of an aggregated model of a wind farm consistingof fixed-speed wind turbines with field measurementrdquo WindEnergy vol 11 no 1 pp 13ndash27 2008

[19] M Singh K Faria S Santoso and E Muljadi ldquoValidation andanalysis of wind power plant models using short-circuit fieldmeasurement datardquo in Proceedings of the IEEE Power amp EnergySociety GeneralMeeting (PES rsquo09) pp 1ndash6 Calgary Canada July2009

[20] J Brochu C Larose and R Gagnon ldquoValidation of single- andmultiple-machine equivalents formodelingwind power plantsrdquoIEEE Transactions on Energy Conversion vol 26 no 2 pp 532ndash541 2011

[21] G Pannell D J Atkinson and B Zahawi ldquoMinimum-thresholdcrowbar for a fault-ride-through grid-code-compliant DFIGwind turbinerdquo IEEE Transactions on Energy Conversion vol 25no 3 pp 750ndash759 2010

[22] State Grid Corporation of China Technology Rule for Connect-ing Wind Farm Into Power Grid State Grid Corporation ofChina Beijing China 2009

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


DFIG Infinitegrid

MV HV

35kV220 kV690V35kV

Figure 1 Schematic of DFIG generation system

DFIG modelGrid code

02

04

06

08

10

Volta

ge (p

u)

0 1 2 3 4minus1

Time (s)

(a)

00

05

10

15

20

Activ

e pow

er (M

W)

0 1 2 3 4minus1

Time (s)

DFIG model(b)

minus4

minus3

minus2

minus1

0

1

2

3

Reac

tive p

ower

(Mva

r)

0 1 2 3 4minus1

Time (s)

DFIG model(c)

Figure 2 (a) PCC voltage of the DFIG model (b) PCC output active power (c) PCC output reactive power

clustering wind turbines with similar wind speed In [12]the wind direction is implemented To represent the powerloss on the collective network within the WPP a method ofcalculating the equivalent impedance is introduced in [13]

Moreover research work in [14] analyzes the impact of theline impedance on the diversity in the voltage profile of eachwind turbine and describes a voltage-profile-based approachto develop multiple-machine aggregated WPP In addition a


096 098 1 102 104 106 108 11

005

115

2055

06

065

07

075


Term

inal

vol

tage

dip

thre

shol

d (p

u)

(a)

096 098 1 102 104 106 108 11

005

115

206

065

07

075

08

Term

inal

vol

tage

dip

thre

shol

d (p

u)


(b)

096 098 1 102 104 106 108 11

005

115

2

065

07

075

08

085

Term

inal

vol

tage

dip

thre

shol

d (p

u)


(c)



Z1 ZnZnminus1Z2

WT1 WTnWTnminus1WT2








119906

119904= 119877

119904119894

119904+

119889120595

119904

119889119905

+ 119895120596

1120595

119904

119906

119903= 119877

119903119894

119903+

119889120595

119903

119889119905

+ 119895120596

119904120595

119903

120595

119904= 119871

119904119894

119904+ 119871

119898119894

119903

120595

119903= 119871

119898119894

119904+ 119871

119903119894

119903

(1)



1and 120596

119904are the







for 119905 gt 119905

0can be written as

120595

119904=

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904

120595

119903=

119906

119903(0)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906

119904119896

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904

+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903

(2)

where 119906


119904= (119871

119904119871

119903minus

119871

2

119898)(119871

119903119877


119903= (119871

119904119871

119903minus



Generator


119898pu 35

119877

119904pu 001 119883

119904pu 01

119877

119903(pu) 001 119883

119903pu 01


119879 6


119879 135


LGJ-24035 119903

0(Ωkm) 0131 119909

0(Ωkm) 0372

LGJ-300220 119903

0(Ωkm) 0107 119909

0(Ωkm) 0414

119871

2

119898)(119871

119904119877


119903and 119862

1


119898(119871

119904119871

119903minus 119871

2

119898)


119904119889= 119880

119904 119906119904119902


119875

119904= minus

3

2

(119906

119904119889119894

119904119889+ 119906

119904119902119894

119904119902) = minus

3

2

119880

119904119894

119904119889

119876

119904= minus

3

2

(119906

119904119902119894

119904119889minus 119906

119904119889119894

119904119902) =

3

2

119880

119904119894

119904119902

(3)

where119880119904= |119906


voltage


be described as

119906

119903119889(0) =

2

3

1

120596

1119871

119898119880

119904

(119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904

+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+

3

2

120596

119904119871

119903119880

2

119904)

119906

119903119902(0) =

2

3

1

120596

1119871

119898119880

119904

(120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904

minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus

3

2

119877

119903119880

2

119904)

(4)

where 120590 = 1 minus 119871

2

119898(119871

119904119871

119903)


119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

120595

119904+

119871

119904

119871

119904119871

119903minus 119871

2

119898

120595

119903 (5)


119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

[

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904]

+

119871

119904

119871

119904119871

119903minus 119871

2

119898

[

(23) (1120596

1119871

119898119880

119904) (119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+ (32) 120596

119904119871

119903119880

2

119904)

119895120596

119904+ 1119879

119903

+

119895 (23) (1120596

1119871

119898119880

119904) (120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus (32) 119877

119903119880

2

119904)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906sk

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903] = 119891 (119877

119904 119871

119904 119877

119903 119871

119903 119871

119898 119875

119904 119876

119904 119906

119904119896 119906

1199040 120596

1 120596

119904 120596

119903)

(6)




9

8

7

6

5

4

3

2

HV_B

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

0

Transmissionline

10 10 10

9

8

7

6

5

4

3

2

0

10

9

8

7

6

5

4

3

2

0

Maintransformer


10

9

8

7

6

5

4

3

2

1

Line 6

1 1

0 0

1 1

0

A

45km 05 km 55km 15 km 65 km 3km

35kV

35kV

220 kV

220 kV

Zeq

L(1

)L

(0)

L(2

)L

(3)L

(4)L

(5)L

(6) L

(7)L

(8)L

(9)L

(10

)

Medium-voltage bus

2MW








DFIG vmsdotsminus1

1(0) 10501(1) 8921(2) 8381(3) 8731(4) 8181(5) 7981(6) 7921(7) 9161(8) 8591(9) 8171(10) 8042(0) 8002(1) 7992(2) 8462(3) 8252(4) 8482(5) 8462(6) 8092(7) 8002(8) 7982(9) 8462(10) 8083(0) 10503(1) 9913(2) 8573(3) 8233(4) 8143(5) 8773(6) 8113(7) 10323(8) 8853(9) 8993(10) 8664(0) 8204(1) 7994(2) 7984(3) 7974(4) 7974(5) 7974(6) 7964(7) 7964(8) 7964(9) 7964(10) 7965(0) 1050

Table 2 Continued

DFIG vmsdotsminus1

5(1) 8745(2) 8325(3) 8155(4) 8205(5) 8845(6) 8255(7) 8165(8) 8635(9) 8785(10) 8066(0) 9046(1) 9686(2) 8616(3) 8356(4) 8066(5) 9346(6) 9026(7) 8936(8) 8286(9) 8086(10) 801






Table3DFIGun

itswith

crow

barp

rotectioncond

uctio

n

Groun

ding

impedance

DetailedWPP

Before

revisio

nAfte

rrevision

Rightjud

gment

Wrong

judgment

Rightjud

gment

Wrong

judgment

119885

119891=11989555Ω

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(0)1(7)ndash(10)2(0)ndash(10)3(0)

3(7)3(9)-(10)4(0)ndash(10)5(0)

6(0)ndash(10)

1(1)1(3)ndash(6)3(1)3(8)5

(9)

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(2)5(10)

119885

119891=1198957Ω

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

1(0)2(0)ndash(10)4(3)ndash(10)6(5)

6(7)6(9)-(10)

3(7)4(0)ndash(2)6(1)6(6)

6(8)

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

Non

e



DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850








Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










096 098 1 102 104 106 108 11

005

115

2055

06

065

07

075


Term

inal

vol

tage

dip

thre

shol

d (p

u)

(a)

096 098 1 102 104 106 108 11

005

115

206

065

07

075

08

Term

inal

vol

tage

dip

thre

shol

d (p

u)


(b)

096 098 1 102 104 106 108 11

005

115

2

065

07

075

08

085

Term

inal

vol

tage

dip

thre

shol

d (p

u)


(c)



Z1 ZnZnminus1Z2

WT1 WTnWTnminus1WT2








119906

119904= 119877

119904119894

119904+

119889120595

119904

119889119905

+ 119895120596

1120595

119904

119906

119903= 119877

119903119894

119903+

119889120595

119903

119889119905

+ 119895120596

119904120595

119903

120595

119904= 119871

119904119894

119904+ 119871

119898119894

119903

120595

119903= 119871

119898119894

119904+ 119871

119903119894

119903

(1)



1and 120596

119904are the







for 119905 gt 119905

0can be written as

120595

119904=

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904

120595

119903=

119906

119903(0)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906

119904119896

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904

+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903

(2)

where 119906


119904= (119871

119904119871

119903minus

119871

2

119898)(119871

119903119877


119903= (119871

119904119871

119903minus



Generator


119898pu 35

119877

119904pu 001 119883

119904pu 01

119877

119903(pu) 001 119883

119903pu 01


119879 6


119879 135


LGJ-24035 119903

0(Ωkm) 0131 119909

0(Ωkm) 0372

LGJ-300220 119903

0(Ωkm) 0107 119909

0(Ωkm) 0414

119871

2

119898)(119871

119904119877


119903and 119862

1


119898(119871

119904119871

119903minus 119871

2

119898)


119904119889= 119880

119904 119906119904119902


119875

119904= minus

3

2

(119906

119904119889119894

119904119889+ 119906

119904119902119894

119904119902) = minus

3

2

119880

119904119894

119904119889

119876

119904= minus

3

2

(119906

119904119902119894

119904119889minus 119906

119904119889119894

119904119902) =

3

2

119880

119904119894

119904119902

(3)

where119880119904= |119906


voltage


be described as

119906

119903119889(0) =

2

3

1

120596

1119871

119898119880

119904

(119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904

+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+

3

2

120596

119904119871

119903119880

2

119904)

119906

119903119902(0) =

2

3

1

120596

1119871

119898119880

119904

(120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904

minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus

3

2

119877

119903119880

2

119904)

(4)

where 120590 = 1 minus 119871

2

119898(119871

119904119871

119903)


119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

120595

119904+

119871

119904

119871

119904119871

119903minus 119871

2

119898

120595

119903 (5)


119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

[

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904]

+

119871

119904

119871

119904119871

119903minus 119871

2

119898

[

(23) (1120596

1119871

119898119880

119904) (119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+ (32) 120596

119904119871

119903119880

2

119904)

119895120596

119904+ 1119879

119903

+

119895 (23) (1120596

1119871

119898119880

119904) (120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus (32) 119877

119903119880

2

119904)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906sk

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903] = 119891 (119877

119904 119871

119904 119877

119903 119871

119903 119871

119898 119875

119904 119876

119904 119906

119904119896 119906

1199040 120596

1 120596

119904 120596

119903)

(6)




9

8

7

6

5

4

3

2

HV_B

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

0

Transmissionline

10 10 10

9

8

7

6

5

4

3

2

0

10

9

8

7

6

5

4

3

2

0

Maintransformer


10

9

8

7

6

5

4

3

2

1

Line 6

1 1

0 0

1 1

0

A

45km 05 km 55km 15 km 65 km 3km

35kV

35kV

220 kV

220 kV

Zeq

L(1

)L

(0)

L(2

)L

(3)L

(4)L

(5)L

(6) L

(7)L

(8)L

(9)L

(10

)

Medium-voltage bus

2MW








DFIG vmsdotsminus1

1(0) 10501(1) 8921(2) 8381(3) 8731(4) 8181(5) 7981(6) 7921(7) 9161(8) 8591(9) 8171(10) 8042(0) 8002(1) 7992(2) 8462(3) 8252(4) 8482(5) 8462(6) 8092(7) 8002(8) 7982(9) 8462(10) 8083(0) 10503(1) 9913(2) 8573(3) 8233(4) 8143(5) 8773(6) 8113(7) 10323(8) 8853(9) 8993(10) 8664(0) 8204(1) 7994(2) 7984(3) 7974(4) 7974(5) 7974(6) 7964(7) 7964(8) 7964(9) 7964(10) 7965(0) 1050

Table 2 Continued

DFIG vmsdotsminus1

5(1) 8745(2) 8325(3) 8155(4) 8205(5) 8845(6) 8255(7) 8165(8) 8635(9) 8785(10) 8066(0) 9046(1) 9686(2) 8616(3) 8356(4) 8066(5) 9346(6) 9026(7) 8936(8) 8286(9) 8086(10) 801






Table3DFIGun

itswith

crow

barp

rotectioncond

uctio

n

Groun

ding

impedance

DetailedWPP

Before

revisio

nAfte

rrevision

Rightjud

gment

Wrong

judgment

Rightjud

gment

Wrong

judgment

119885

119891=11989555Ω

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(0)1(7)ndash(10)2(0)ndash(10)3(0)

3(7)3(9)-(10)4(0)ndash(10)5(0)

6(0)ndash(10)

1(1)1(3)ndash(6)3(1)3(8)5

(9)

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(2)5(10)

119885

119891=1198957Ω

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

1(0)2(0)ndash(10)4(3)ndash(10)6(5)

6(7)6(9)-(10)

3(7)4(0)ndash(2)6(1)6(6)

6(8)

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

Non

e



DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850








Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Z1 ZnZnminus1Z2

WT1 WTnWTnminus1WT2








119906

119904= 119877

119904119894

119904+

119889120595

119904

119889119905

+ 119895120596

1120595

119904

119906

119903= 119877

119903119894

119903+

119889120595

119903

119889119905

+ 119895120596

119904120595

119903

120595

119904= 119871

119904119894

119904+ 119871

119898119894

119903

120595

119903= 119871

119898119894

119904+ 119871

119903119894

119903

(1)



1and 120596

119904are the







for 119905 gt 119905

0can be written as

120595

119904=

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904

120595

119903=

119906

119903(0)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906

119904119896

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904

+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903

(2)

where 119906


119904= (119871

119904119871

119903minus

119871

2

119898)(119871

119903119877


119903= (119871

119904119871

119903minus



Generator


119898pu 35

119877

119904pu 001 119883

119904pu 01

119877

119903(pu) 001 119883

119903pu 01


119879 6


119879 135


LGJ-24035 119903

0(Ωkm) 0131 119909

0(Ωkm) 0372

LGJ-300220 119903

0(Ωkm) 0107 119909

0(Ωkm) 0414

119871

2

119898)(119871

119904119877


119903and 119862

1


119898(119871

119904119871

119903minus 119871

2

119898)


119904119889= 119880

119904 119906119904119902


119875

119904= minus

3

2

(119906

119904119889119894

119904119889+ 119906

119904119902119894

119904119902) = minus

3

2

119880

119904119894

119904119889

119876

119904= minus

3

2

(119906

119904119902119894

119904119889minus 119906

119904119889119894

119904119902) =

3

2

119880

119904119894

119904119902

(3)

where119880119904= |119906


voltage


be described as

119906

119903119889(0) =

2

3

1

120596

1119871

119898119880

119904

(119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904

+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+

3

2

120596

119904119871

119903119880

2

119904)

119906

119903119902(0) =

2

3

1

120596

1119871

119898119880

119904

(120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904

minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus

3

2

119877

119903119880

2

119904)

(4)

where 120590 = 1 minus 119871

2

119898(119871

119904119871

119903)


119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

120595

119904+

119871

119904

119871

119904119871

119903minus 119871

2

119898

120595

119903 (5)


119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

[

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904]

+

119871

119904

119871

119904119871

119903minus 119871

2

119898

[

(23) (1120596

1119871

119898119880

119904) (119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+ (32) 120596

119904119871

119903119880

2

119904)

119895120596

119904+ 1119879

119903

+

119895 (23) (1120596

1119871

119898119880

119904) (120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus (32) 119877

119903119880

2

119904)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906sk

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903] = 119891 (119877

119904 119871

119904 119877

119903 119871

119903 119871

119898 119875

119904 119876

119904 119906

119904119896 119906

1199040 120596

1 120596

119904 120596

119903)

(6)




9

8

7

6

5

4

3

2

HV_B

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

0

Transmissionline

10 10 10

9

8

7

6

5

4

3

2

0

10

9

8

7

6

5

4

3

2

0

Maintransformer


10

9

8

7

6

5

4

3

2

1

Line 6

1 1

0 0

1 1

0

A

45km 05 km 55km 15 km 65 km 3km

35kV

35kV

220 kV

220 kV

Zeq

L(1

)L

(0)

L(2

)L

(3)L

(4)L

(5)L

(6) L

(7)L

(8)L

(9)L

(10

)

Medium-voltage bus

2MW








DFIG vmsdotsminus1

1(0) 10501(1) 8921(2) 8381(3) 8731(4) 8181(5) 7981(6) 7921(7) 9161(8) 8591(9) 8171(10) 8042(0) 8002(1) 7992(2) 8462(3) 8252(4) 8482(5) 8462(6) 8092(7) 8002(8) 7982(9) 8462(10) 8083(0) 10503(1) 9913(2) 8573(3) 8233(4) 8143(5) 8773(6) 8113(7) 10323(8) 8853(9) 8993(10) 8664(0) 8204(1) 7994(2) 7984(3) 7974(4) 7974(5) 7974(6) 7964(7) 7964(8) 7964(9) 7964(10) 7965(0) 1050

Table 2 Continued

DFIG vmsdotsminus1

5(1) 8745(2) 8325(3) 8155(4) 8205(5) 8845(6) 8255(7) 8165(8) 8635(9) 8785(10) 8066(0) 9046(1) 9686(2) 8616(3) 8356(4) 8066(5) 9346(6) 9026(7) 8936(8) 8286(9) 8086(10) 801






Table3DFIGun

itswith

crow

barp

rotectioncond

uctio

n

Groun

ding

impedance

DetailedWPP

Before

revisio

nAfte

rrevision

Rightjud

gment

Wrong

judgment

Rightjud

gment

Wrong

judgment

119885

119891=11989555Ω

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(0)1(7)ndash(10)2(0)ndash(10)3(0)

3(7)3(9)-(10)4(0)ndash(10)5(0)

6(0)ndash(10)

1(1)1(3)ndash(6)3(1)3(8)5

(9)

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(2)5(10)

119885

119891=1198957Ω

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

1(0)2(0)ndash(10)4(3)ndash(10)6(5)

6(7)6(9)-(10)

3(7)4(0)ndash(2)6(1)6(6)

6(8)

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

Non

e



DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850








Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Generator


119898pu 35

119877

119904pu 001 119883

119904pu 01

119877

119903(pu) 001 119883

119903pu 01


119879 6


119879 135


LGJ-24035 119903

0(Ωkm) 0131 119909

0(Ωkm) 0372

LGJ-300220 119903

0(Ωkm) 0107 119909

0(Ωkm) 0414

119871

2

119898)(119871

119904119877


119903and 119862

1


119898(119871

119904119871

119903minus 119871

2

119898)


119904119889= 119880

119904 119906119904119902


119875

119904= minus

3

2

(119906

119904119889119894

119904119889+ 119906

119904119902119894

119904119902) = minus

3

2

119880

119904119894

119904119889

119876

119904= minus

3

2

(119906

119904119902119894

119904119889minus 119906

119904119889119894

119904119902) =

3

2

119880

119904119894

119904119902

(3)

where119880119904= |119906


voltage


be described as

119906

119903119889(0) =

2

3

1

120596

1119871

119898119880

119904

(119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904

+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+

3

2

120596

119904119871

119903119880

2

119904)

119906

119903119902(0) =

2

3

1

120596

1119871

119898119880

119904

(120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904

minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus

3

2

119877

119903119880

2

119904)

(4)

where 120590 = 1 minus 119871

2

119898(119871

119904119871

119903)


119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

120595

119904+

119871

119904

119871

119904119871

119903minus 119871

2

119898

120595

119903 (5)


119894

119903= minus

119871

119898

119871

119904119871

119903minus 119871

2

119898

[

119906

119904119896

119895120596

1

+ (

119906

1199040

119895120596

1

minus

119906

119904119896

119895120596

1

) 119890

minus1198951205961119905

119890

minus119905119879119904]

+

119871

119904

119871

119904119871

119903minus 119871

2

119898

[

(23) (1120596

1119871

119898119880

119904) (119877

119903120596

1119871

119904119875

119904+ 119877

119904120596

119904119871

119903119875

119904+ 120596

1120596

119904119871

119904119871

119903120590119876

119904minus 119877

119904119877

119903119876

119904+ (32) 120596

119904119871

119903119880

2

119904)

119895120596

119904+ 1119879

119903

+

119895 (23) (1120596

1119871

119898119880

119904) (120596

1120596

119904119871

119904119871

119903120590119875

119904minus 119877

119904119877

119903119875

119904minus 119877

119903120596

1119871

119904119876

119904minus 119877

119904120596

119904119871

119903119876

119904minus (32) 119877

119903119880

2

119904)

119895120596

119904+ 1119879

119903

minus

119860119877

119903119906sk

119895120596

1(119895120596

119904+ 1119879

119903)

minus

119860119877

119903(119906

1199040minus 119906

119904119896)

119895120596

1(minus119895120596

119903+ 1119879

119903minus 1119879

119904)

119890

minus1198951205961119905

119890

minus119905119879119904+ 119862

1119890

minus119895120596119904119905

119890

minus119905119879119903] = 119891 (119877

119904 119871

119904 119877

119903 119871

119903 119871

119898 119875

119904 119876

119904 119906

119904119896 119906

1199040 120596

1 120596

119904 120596

119903)

(6)




9

8

7

6

5

4

3

2

HV_B

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

0

Transmissionline

10 10 10

9

8

7

6

5

4

3

2

0

10

9

8

7

6

5

4

3

2

0

Maintransformer


10

9

8

7

6

5

4

3

2

1

Line 6

1 1

0 0

1 1

0

A

45km 05 km 55km 15 km 65 km 3km

35kV

35kV

220 kV

220 kV

Zeq

L(1

)L

(0)

L(2

)L

(3)L

(4)L

(5)L

(6) L

(7)L

(8)L

(9)L

(10

)

Medium-voltage bus

2MW








DFIG vmsdotsminus1

1(0) 10501(1) 8921(2) 8381(3) 8731(4) 8181(5) 7981(6) 7921(7) 9161(8) 8591(9) 8171(10) 8042(0) 8002(1) 7992(2) 8462(3) 8252(4) 8482(5) 8462(6) 8092(7) 8002(8) 7982(9) 8462(10) 8083(0) 10503(1) 9913(2) 8573(3) 8233(4) 8143(5) 8773(6) 8113(7) 10323(8) 8853(9) 8993(10) 8664(0) 8204(1) 7994(2) 7984(3) 7974(4) 7974(5) 7974(6) 7964(7) 7964(8) 7964(9) 7964(10) 7965(0) 1050

Table 2 Continued

DFIG vmsdotsminus1

5(1) 8745(2) 8325(3) 8155(4) 8205(5) 8845(6) 8255(7) 8165(8) 8635(9) 8785(10) 8066(0) 9046(1) 9686(2) 8616(3) 8356(4) 8066(5) 9346(6) 9026(7) 8936(8) 8286(9) 8086(10) 801






Table3DFIGun

itswith

crow

barp

rotectioncond

uctio

n

Groun

ding

impedance

DetailedWPP

Before

revisio

nAfte

rrevision

Rightjud

gment

Wrong

judgment

Rightjud

gment

Wrong

judgment

119885

119891=11989555Ω

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(0)1(7)ndash(10)2(0)ndash(10)3(0)

3(7)3(9)-(10)4(0)ndash(10)5(0)

6(0)ndash(10)

1(1)1(3)ndash(6)3(1)3(8)5

(9)

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(2)5(10)

119885

119891=1198957Ω

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

1(0)2(0)ndash(10)4(3)ndash(10)6(5)

6(7)6(9)-(10)

3(7)4(0)ndash(2)6(1)6(6)

6(8)

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

Non

e



DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850








Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










9

8

7

6

5

4

3

2

HV_B

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

0

Transmissionline

10 10 10

9

8

7

6

5

4

3

2

0

10

9

8

7

6

5

4

3

2

0

Maintransformer


10

9

8

7

6

5

4

3

2

1

Line 6

1 1

0 0

1 1

0

A

45km 05 km 55km 15 km 65 km 3km

35kV

35kV

220 kV

220 kV

Zeq

L(1

)L

(0)

L(2

)L

(3)L

(4)L

(5)L

(6) L

(7)L

(8)L

(9)L

(10

)

Medium-voltage bus

2MW








DFIG vmsdotsminus1

1(0) 10501(1) 8921(2) 8381(3) 8731(4) 8181(5) 7981(6) 7921(7) 9161(8) 8591(9) 8171(10) 8042(0) 8002(1) 7992(2) 8462(3) 8252(4) 8482(5) 8462(6) 8092(7) 8002(8) 7982(9) 8462(10) 8083(0) 10503(1) 9913(2) 8573(3) 8233(4) 8143(5) 8773(6) 8113(7) 10323(8) 8853(9) 8993(10) 8664(0) 8204(1) 7994(2) 7984(3) 7974(4) 7974(5) 7974(6) 7964(7) 7964(8) 7964(9) 7964(10) 7965(0) 1050

Table 2 Continued

DFIG vmsdotsminus1

5(1) 8745(2) 8325(3) 8155(4) 8205(5) 8845(6) 8255(7) 8165(8) 8635(9) 8785(10) 8066(0) 9046(1) 9686(2) 8616(3) 8356(4) 8066(5) 9346(6) 9026(7) 8936(8) 8286(9) 8086(10) 801






Table3DFIGun

itswith

crow

barp

rotectioncond

uctio

n

Groun

ding

impedance

DetailedWPP

Before

revisio

nAfte

rrevision

Rightjud

gment

Wrong

judgment

Rightjud

gment

Wrong

judgment

119885

119891=11989555Ω

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(0)1(7)ndash(10)2(0)ndash(10)3(0)

3(7)3(9)-(10)4(0)ndash(10)5(0)

6(0)ndash(10)

1(1)1(3)ndash(6)3(1)3(8)5

(9)

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(2)5(10)

119885

119891=1198957Ω

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

1(0)2(0)ndash(10)4(3)ndash(10)6(5)

6(7)6(9)-(10)

3(7)4(0)ndash(2)6(1)6(6)

6(8)

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

Non

e



DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850








Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











DFIG vmsdotsminus1

1(0) 10501(1) 8921(2) 8381(3) 8731(4) 8181(5) 7981(6) 7921(7) 9161(8) 8591(9) 8171(10) 8042(0) 8002(1) 7992(2) 8462(3) 8252(4) 8482(5) 8462(6) 8092(7) 8002(8) 7982(9) 8462(10) 8083(0) 10503(1) 9913(2) 8573(3) 8233(4) 8143(5) 8773(6) 8113(7) 10323(8) 8853(9) 8993(10) 8664(0) 8204(1) 7994(2) 7984(3) 7974(4) 7974(5) 7974(6) 7964(7) 7964(8) 7964(9) 7964(10) 7965(0) 1050

Table 2 Continued

DFIG vmsdotsminus1

5(1) 8745(2) 8325(3) 8155(4) 8205(5) 8845(6) 8255(7) 8165(8) 8635(9) 8785(10) 8066(0) 9046(1) 9686(2) 8616(3) 8356(4) 8066(5) 9346(6) 9026(7) 8936(8) 8286(9) 8086(10) 801






Table3DFIGun

itswith

crow

barp

rotectioncond

uctio

n

Groun

ding

impedance

DetailedWPP

Before

revisio

nAfte

rrevision

Rightjud

gment

Wrong

judgment

Rightjud

gment

Wrong

judgment

119885

119891=11989555Ω

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(0)1(7)ndash(10)2(0)ndash(10)3(0)

3(7)3(9)-(10)4(0)ndash(10)5(0)

6(0)ndash(10)

1(1)1(3)ndash(6)3(1)3(8)5

(9)

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(2)5(10)

119885

119891=1198957Ω

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

1(0)2(0)ndash(10)4(3)ndash(10)6(5)

6(7)6(9)-(10)

3(7)4(0)ndash(2)6(1)6(6)

6(8)

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

Non

e



DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850








Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table3DFIGun

itswith

crow

barp

rotectioncond

uctio

n

Groun

ding

impedance

DetailedWPP

Before

revisio

nAfte

rrevision

Rightjud

gment

Wrong

judgment

Rightjud

gment

Wrong

judgment

119885

119891=11989555Ω

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(0)1(7)ndash(10)2(0)ndash(10)3(0)

3(7)3(9)-(10)4(0)ndash(10)5(0)

6(0)ndash(10)

1(1)1(3)ndash(6)3(1)3(8)5

(9)

1(0)-(1)1(3)ndash(10)2(0)ndash(10)

3(0)-(1)3(7)ndash(10)4(0)ndash(10)

5(0)5(9)6(0)ndash(10)

1(2)5(10)

119885

119891=1198957Ω

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

1(0)2(0)ndash(10)4(3)ndash(10)6(5)

6(7)6(9)-(10)

3(7)4(0)ndash(2)6(1)6(6)

6(8)

1(0)2(0)ndash(10)3(7)4(0)ndash(10)

6(1)6(5)ndash(10)

Non

e



DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850








Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











DFIG vmsdotsminus1

1(0) 6361(1) 6231(2) 7441(3) 6291(4) 6931(5) 6921(6) 7451(7) 6381(8) 6401(9) 6461(10) 6992(0) 6392(1) 7642(2) 6372(3) 6372(4) 6402(5) 6652(6) 7002(7) 6502(8) 6832(9) 8502(10) 8263(0) 6393(1) 6643(2) 7113(3) 6283(4) 7053(5) 7583(6) 6793(7) 8163(8) 6583(9) 7703(10) 6364(0) 6364(1) 6364(2) 6374(3) 6374(4) 6384(5) 6394(6) 6424(7) 6494(8) 6834(9) 8504(10) 6165(0) 6405(1) 647

Table 4 Continued

DFIG vmsdotsminus1

5(2) 7135(3) 6365(4) 6395(5) 6495(6) 6505(7) 7865(8) 6355(9) 6395(10) 6526(0) 7726(1) 6406(2) 7216(3) 7026(4) 8146(5) 6476(6) 6396(7) 6426(8) 6496(9) 6836(10) 850








Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Incoming wind 773 00016 053 061 039 536 398



Incoming wind 545 00046 073 178 113 639 2744



Incoming wind 727 00035 035 125 079 222 1374



Incoming wind 545 00044 081 155 121 7475 2131



Incoming wind 500 00066 119 235 180 4576 5390



Incoming wind 803 00006 018 020 008 350 097





Δ119880

119896=

sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

radic

3119880

(7)

where 119885


119894is



119885eq-119896 =sum

119899

119895=119896(119885

119895sum

119895

119894=1119878

lowast

119894)

119878

lowast

119896

(8)








119891=




1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1(1) radic radic

1(2) radic

1(3) radic radic

1(4) radic

1(5) radic

1(6) radic

1(7) radic radic

1(8) radic radic














3(0) radic

3(1) radic


3(8) radic

3(9) radic radic

3(10) radic radic












Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 6 Continued


5(0) radic

5(1)5(2)5(3)5(4)5(5)5(6)5(7)5(8)5(9) radic


6(1) radic radic

6(2) radic radic

6(3) radic radic














119891= 11989555Ω and

119885






0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

03

04

05

06

07

08

09

10

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(a)

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

150

100

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(b)

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 06 08 1000

Time (s)

(c)



119864

119886=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

119864

119903=

1

119899

119899

sum

119894=1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119884

119891119894(119896) minus 119884

119894(119896)

119884

119894(119896)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(9)

where 119884

119894(119896) and 119884







03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

minus5

0

5

10

15

20

25

30

Activ

e pow

er at

PCC

(MW

)

minus100

minus50

0

50

100

Reac

tive p

ower

at P

CC (M

var)

06 0802 04 1000

Time (s)02 04 1000 06 08

Time (s)02 04 06 08 1000

Time (s)

(a)

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

06 0802 04 1000

Time (s)

minus5

minus10

0

5

10

15

20

25

30Ac

tive p

ower

at P

CC (M

W)

02 04 1000 06 08

Time (s)

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

(b)

06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)06 0802 04 1000

Time (s)

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

0

5

10

15

20

25

30

35

Activ

e pow

er at

PCC

(MW

)

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

(c)








02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02

03

04

05

06

07

08

09

10

11

Volta

ge at

PCC

(pu

)

02 04 1000 06 08

Time (s)

0201

030

035

040

045

0

10

20

30

40

50

60

Activ

e pow

er at

PCC

(MW

)

02 04 06 08 1000

Time (s)

10

15

5

010

012

014

016

018

020

022

024

026

minus100

minus50

0

50

100

150

Reac

tive p

ower

at P

CC (M

var)

02 04 06 08 1000

Time (s)

012 014 016 018 020 022 024

0

10

20

30

40

50



5 Conclusions




Appendix


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Equivalent Modeling of DFIG-Based Wind Power …downloads.hindawi.com/journals/mpe/2016/8426492.pdf · 2019-07-30 · Research Article Equivalent Modeling of DFIG-Based