Research Article Transient Simulation of Wind Turbine ...downloads.hindawi.com/journals/mpe/2013/142765.pdfsystems. e lightning protection design needs to obtain the lightning transient

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 142765 9 pageshttpdxdoiorg1011552013142765

Research ArticleTransient Simulation of Wind Turbine Towers underLightning Stroke

Xiaoqing Zhang

National Active Distribution Network Technology Research Center Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Xiaoqing Zhang xqzhang2bjtueducn

Received 23 March 2013 Revised 16 July 2013 Accepted 17 July 2013

Academic Editor Ming-Hung Hsu

Copyright copy 2013 Xiaoqing Zhang 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

A simulation algorithm is proposed in this paper for lightning transient analysis of the wind turbine (WT) towers In the proposedalgorithm the tower body is first subdivided into a discrete multiconductor system A set of formulas are given to calculate theelectrical parameters of the branches in the multiconductor system By means of the electrical parameters each branch unit in themulticonductor system is replaced as a coupled 120587-type circuit and the multiconductor system is converted into a circuit modelThen the lightning transient responses can be obtained in different parts on the tower body by solving the circuit equations of theequivalent discretization network The laboratory measurement is also made by a reduced-scale tower for checking the validity ofthe proposed algorithm

1 Introduction

Global warming effect accelerates the utilization of windenergy As a clean energy source wind energy can be used togenerate electric power without emission of carbon dioxideinto the atmospheric environment In consequence of therapid growth in the utilization of wind energy for electricpower supply wind turbines (WTs) have increased constantlyin size and rated power during recent decades WTs areparticularly vulnerable to lightning strokes due to their greatheight distinctive shape and rather exposed position Thelightning stroke effect on WTs has become a major concernas the number of the installed WTs continues to increaseTherefore lightning protection of WTs is crucially importantfor the operational reliability of large wind power generationsystems The lightning protection design needs to obtain thelightning transient responses on WT towers since the towerbody is the main conducting path of lightning current Thesimulation algorithms were presented in the literature [1ndash3] in which the tower body was simply represented by atransmission line or a chain capacitance circuit Howeverthese existing algorithms are difficult to use for calculatingthe transient responses in different parts on the tower bodydue to the fact that they ignore the structural feature of thetower body To overcome the shortcoming in these existing

algorithms a novel simulation algorithm is proposed in thispaper The proposed algorithm subdivides the continuousconducting shell of the tower body into a discrete multi-conductor system and can give due consideration for thestructural feature of the tower body In the multiconduc-tor system the electrical parameters of the branches arecalculated by an efficient procedure and the branch unitsare represented by the coupled 120587-type circuits A circuitmodel constituted by a series of the coupled 120587-type circuitsis built for the multiconductor system On the basis ofthe circuit model the equivalent discretization network isfurther formed The transient calculation is performed forthe equivalent discretization network and then the lightningtransient responses can be obtained in different parts on thetower bodyThemeasurement of the transient response is alsotaken on a reduced-scale WT tower and the validity of theproposed algorithm is examined by comparing simulated andmeasured results

2 Calculation of Electrical Parameters

21 Discretization Description of the WT Tower After aWT is struck by lightning lightning current usually passesfrom the blade root to the tower and then dissipates in the

2 Mathematical Problems in Engineering

Earth surface

WT tower

Lightning current

Earth-termination system

120575

r1

h

r2

Figure 1 WT tower

ground through the earth-termination system Since mostmanufactures have used many kinds of brushes and slidingcontact systems to divert the lightning current from themainshaft only a much smaller part of lightning current flowsthrough the gearbox and the generator For theWT structurethe tower is the longest conducting path of lightning currentWhen the tower conducts lightning current serious transientpotential rise appears on the tower body and may resultin flashback to the electrical and electronic componentsinstalled inside the tower Much attention to the potentialrise has been paid in the lightning protection design ofWTs For the sake of the transient analysis of the potentialdistribution on the tower the blade and the conducting pathin the nacelle are left out of consideration Instead lightningcurrent is injected into the tower body from its top asshown in Figure 1 [4] In order to build the circuit modelof the tower the continuous conducting shell of the towerbody needs to be subdivided into a discrete multiconductorsystem formed by longitudinal and transverse branches asillustrated in Figure 2 [5] TheWT earth-termination systemis modeled as the grounding resistances (119877119892) The electricalparameters of the branches in the multiconductor systemare represented by resistances inductances and capacitancesFor simplifying the parameter calculation the segmental arcof each transverse branch is replaced by its chord and allthe branches are taken as the cylindrical conductors Theconductor radii are estimated from the average cross-sectionsof the respective branches

22 Inductance and Resistance The inductance parameterscan be determined by Neumannrsquos integral method [6 7] Thetypical branches in the multiconductor system are illustratedin Figure 3 where the presence of the ground is consideredby symmetrically installing the image branches (depictedin dotted lines) below the earth surface [7] According toNeumannrsquos integral method the mutual inductance between

Earth surface

Lightning current

Rg

Rg

Rg

Figure 2 Multiconductor system

the branches 119895 and 119896 is calculated by

119871119895119896 =

1205830

4120587

[int

119904119895

int

119904119896

119889s119895 sdot 119889s119896D119895119896

+ int

1199041015840119895

int

119904119896

119889s1015840119895sdot 119889s119896

D1015840119895119896

] (1)

The dot products of the vector differential segments become

119889s119895 sdot 119889s119896 = cos120593119889s119895119889s119896

119889s1015840119895sdot 119889s119896 = cos120593

10158401198891199041015840

119895119889119904

(2)

where 120593 is the direction angle between the branches119895 and 119896and 1205931015840 is that between the branches 1198951015840 and 119896 Substituting (2)into (1) gives

119871119895119896 =

1205830

4120587

[cos120593119873 (119895 119896) + cos1205931015840119873(1198951015840 119896)] (3)

where119873(119895 119896) and119873(1198951015840 119896) are two double line integrals

119873(119895 119896) = int

119904119895

int

119904119896

119889119904119895119889119904119896

119863119895119896

119873 (1198951015840 119896) = int

1199041015840119895

int

119904119896

1198891199041015840

119895119889119904119896

1198631015840

119895119896

(4)

Equations (3) and (4) indicate that the calculation of themutual inductance depends on the double line integrals119873(119895 119896) and 119873(1198951015840 119896) From the viewpoint of integral oper-ation the way to evaluate 119873(119895 119896) is the same as that toevaluate 119873(1198951015840 119896) For this reason the solution of 119873(119895 119896)is only presented for the typical space positions in themulticonductor system which holds for the evaluation of119873(1198951015840 119896)The integral 119873(119895 119896) is first evaluated in the case of the

coplanar branch pairs For the two coplanar longitudinal

Mathematical Problems in Engineering 3

Earth surfaceDjk

dsk

dsj

sj

sk

D998400jk

s998400j

ds998400j

(a)

Earth surface

ds998400j

s998400j

dsj

sj

Djk

dsk

sk

D998400jk

(b)

Figure 3 Branches in the multiconductor system

D4D2

D3

D1

lj

sj

120593

sk

2r0

lk

(a)

2r0

h

sj

sk

(b)

Figure 4 Coplanar branch pairs (a) longitudinal pair and (b) transverse pair

branches as shown in Figure 4(a) the integral119873(119895 119896) is givenby

119873(119895 119896) = (119897119895 + 119904119895) ln1198632 + 1198634 + 119904119896

1198632 + 1198634 minus 119904119896

minus 119897119895 ln1198631 + 1198633 + 119904119896

1198631 + 1198633 minus 119904119896

+ (119897119896 + 119904119896) ln1198633 + 1198634 + 119904119895

1198633 + 1198634 minus 119904119895

minus 119897119896 ln1198631 + 1198632 + 119904119895

1198631 + 1198632 minus 119904119895

(5)

If the two longitudinal branches are flush with each otherthe integral 119873(119895 119896) is evaluated by putting 119904119895 = 119904119896 and1198632 = 1198633 into (5) Under the flush condition the integral119873(119895 119895) is evaluated by again putting 1198631 = 1198634 = 1199030 into (5)

This can result in determining the self-inductance 119871119895119895 of thelongitudinal branch 119895 by (3)

For the two coplanar transverse branches (120593 = 0) asshown in Figure 4(b) the integral119873(119895 119896) is given by

119873(119895 119896)

= minus (119904119896 minus 119904119895) sinhminus1119904119896 minus 119904119895

2ℎ

+ (119904119895 + 119904119896) sinhminus1119904119895 + 119904119896

2ℎ

+ 2radic(

119904119896 minus 119904119895

2

)

2

+ ℎ2minus 2radic(

119904119895 + 119904119896

2

)

2

+ ℎ2

(6)


E

FD1D2

D3

D4

O

h

G

lk

sk

sj

lj 120593

(a)

E

O

F

G

h

D1

D2

D3

D4

sk

sj

lj

lk

120593

(b)

Figure 5 Noncoplanar branch pairs (a) longitudinal pair and (b) transverse pair

In (6) letting 119904119895 = 119904119896 and ℎ = 1199030 gives the integral 119873(119895 119895)Then 119873(119895 119895) can be used to further determine the self-inductance 119871119895119895 of the transverse branch119895 by (3)

The noncoplanar branch pairs are shown in Figure 5where line OG is the common perpendicular to the twobranches and line OF lying on the plane containing thebranch 119895 is parallel to the branch 119896 The integral 119873(119895 119896)for both the longitudinal branch pair (Figure 5(a)) and thetransverse branch pair (Figure 5(b)) is given in the followingcomplicated expression

119873(119895 119896)

= 2 [(119897119895 + 119904119895) tanhminus1 119904119896

1198631 + 1198632

minus 119897119895tanhminus1 119904119896

1198633 + 1198634

+ (119897119896 + 119904119896) tanhminus1

119904119895

1198631 + 1198634

minus 119897119896tanhminus1

119904119895

1198632 + 1198633

]

minus

Φℎ

sin120593

(7)

where Φ is the solid angle

Φ = tanhminus1 [ℎ2 cos120593 + (119897119895 + 119904119895) (119897119896 + 119904119896) sin

2120593

ℎ1198631 sin120593]

minus tanhminus1 [ℎ2 cos120593 + 119897119896 (119897119895 + 119904119895) sin

2120593

ℎ1198632 sin120593]

+ tanhminus1 [ℎ2 cos120593 + 119897119895119897119896sin

2120593

ℎ1198633 sin120593]


2120593

ℎ1198634 sin120593]

(8)

On the basis of the formulas given previously the selfand mutual inductances can be calculated for a branch unitwith119872 coupled branches in the multiconductor system thusforming the inductance matrix

L = 119871119895119896119872times119872 (9)

The branch resistance per unit length is approximatelycalculated by [8]

119877 =

120588

120587120590 [(1 minus exp (minus1199030120590))] [21199030 minus 120590 (1 minus exp (minus1199030120590))]

(10)

where 1199030 is the radius of the branch 120588 the material resistivityand 120590 the skin depth

120590 =

1

radic120587119891120583120574

(11)

where119891 is the maximum frequency likely to affect the systemtransient 120583 the material permeability and 120574 = 1120588 119891may beroughly evaluated by the waveform parameter of the injectedlightning current [9]

23 Capacitance The mutual potential coefficient betweenthe branches 119895 and 119896 as shown in Figure 3 can be calculatedby the average potential method [10]

119901119895119896 =

1

41205871205760119904119895119904119896

[int

119904119895

int

119904119896

119889119904119895119889119904119896

119863119895119896

minus int

1199041015840119895

int

119904119896

1198891199041015840

119895119889119904119896

1198631015840

119895119896

]

=

1

41205871205760119904119895119904119896

[119873 (119895 119896) minus 119873(1198951015840 119896)]

(12)

From (12) the self-potential coefficient 119901119895119895 of the branch 119895can also be obtained in a similar way to the self-inductance


A1

A2

A3

B1

B2

B3

1

2

3

(a)

L13

A1

A2

A3

B1

B2

B3

1

2C11

1

2C12

1

2C23

1

2C22

1

2C13

1

2C11

1

2C13

1

2C22

1

2C23

1

2C33

1

2C33

1

2C12

R1L11

R2

R3

L12

L22

L23

L33

(b)

A BR L

1

2C

1

2C

(c)

A B

C(t minus Δt) C(t minus Δt)

(t minus Δt)

C CRLRR R II

IRL

(d)

Figure 6 Coupled 120587-type circuit of three coupled branches (a) three coupled branches (b) coupled 120587-type circuit (c) simplified 120587-typecircuit expressed in matrix and (d) discretization 120587-type circuit

Impulsegenerator

Digital oscilloscope

Steel plate

90∘

u

i

Rg

Rg

Rg

Figure 7 Experimental setup

119871119895119895 Consequently the potential coefficientmatrix of a branchunit with119872 coupled branches in the multiconductor systemis formed by

P = 119901119895119896119872times119872 (13)

Calculating the inverse of P gives

Pminus1 = 119902119895119896119872times119872 (14)

From the inverse matrix Pminus1 the capacitance matrix can beobtained as [10 11]

C = 119862119895119896119872times119872 (15)

where the diagonal and off-diagonal elements are determinedas

119862119895119895 =

119872

sum

120577=1

119902119895120577 (119895 = 1 2 119872)

119862119895119896 = minus119902119895119896 (119895 119896 = 1 2 119872 119895 = 119896)

(16)


0 200 400 600 800 10000

10

20

30

40

50

60Cu

rren

t (m

A)

Time (ns)

(a)

0 200 400 600 800 1000Time (ns)

MeasuredSimulated

0

16

32

48

64

8

Volta

ge (V

)(b)

Figure 8 Measured and simulated waveforms (a) injected current waveform and (b) potential waveform at the top of the tower

3 Simulation of Lightning Transients

By means of the electrical parameters obtained previouslya branch unit with 119872 coupled branches is represented by acoupled 120587-type circuit formed by resistances inductancesand capacitances A coupled120587-type circuit representing threecoupled branches (119872 = 3) is shown in Figures 6(a) and6(b) Expression of the electrical parameters in matrix formgives a simplified circuit as shown in Figure 6(c)With a timediscretization treatment for the inductance and capacitanceelements these two kinds of circuit elements are replaced bycurrent sources and parallel equivalent resistances [12 13]Thus Figure 6(c) is further converted into a discretization 120587-type circuit as shown in Figure 6(d) where Δ119905 is the timestep size I119877119871(119905 minus Δ119905) and I119862(119905 minus Δ119905) are the current sourcevectors determined by the current and voltage values fromprevious time steps and R119877119871 and R119862 are the equivalentresistance matrices determined by the electrical parametersand the time step size The formulas for calculating I119877119871(119905 minusΔ119905) I119862(119905 minus Δ119905) R119877119871 and R119862 have been given in [13 14]For 119872 (119872 gt 3) coupled branches the discretization 120587-type circuit is similar to that in Figure 6(d) After all thebranch units in the multiconductor system are replaced bythe discretization 120587-type circuits the tower body can beconverted into an equivalent discretization network Thelightning current source is injected to the top nodes of theequivalent discretization network According to the topologyconnection mode of the equivalent resistances and currentsources the node voltage equations are formed and thensolved numerically at each time step The solving processhas been described in detail in [13 14] As a result lightningtransient responses can be obtained in different parts on thetower body

If the blade and the conducting path in the nacelle aretaken into account the former can be converted into a seriescircuit unit consisting of a few 120587-type circuits and the latterinto a parallel resistance-capacitance unit In the parallel unitthe resistance is the contact one of the brush and the slidingcontact system while the capacitance represents the mainshaft bearings [2 15] After the two circuit units are seriallyconnected to the equivalent discretization network of thetower the complete circuit model can be built for the WTHowever the two circuit units in the complete circuit modeldo not change the lightning current and transient potentialdistributions on the tower body due to their series connectionto the tower Thus their removal from the complete circuitmodel will not make a significant influence on calculation ofthe potential responses on the tower body

4 Verification of the Proposed Algorithm withLaboratory Measurement

An experimental setup was built in the laboratory space asshown in Figure 7 The reduced-scale tower is made of theiron sheet whose dimensions are taken as 1199031 = 00283m1199032 = 0067m ℎ = 2m and 120575 = 0002m (see Figure 1)The potential measurement wire and the current leadwire areorthogonal toweaken the electromagnetic induction betweenthem

The potential measurement wire is grounded that isconnected to the steel plate at a point 9m apart from thetower Five resistances of 5Ω representing the groundingresistances are connected between the tower bottom andsteel plateThe impulse current with fast wavefront is injectedto the top of the tower Under the excitation of the injected


0 20 40 60 80 1000

02

04

06

08

1

12

14Vo

ltage

(MV

)

Time (120583s)

(a)

0 20 40 60 80 100Time (120583s)

0

02

04

06

08

1

12

Volta

ge (M

V)

(b)

Figure 9 Transient potential waveforms (a) potential waveforms at the top of the tower and (b) potential waveforms at the bottom of thetower

current the transient potential response is measured at thetop of the tower The frequency bandwidth of the potentialprobe is 0sim500MHz and that of the current probe 20 kHzsim500MHz which is competent to measure the fast transientresponses The measured waveforms of the injected currentand the transient potential are shown in Figure 8 wherethe corresponding waveform simulated by the proposedalgorithm is given simultaneously for comparison It can befound that a better agreement appears betweenmeasured andsimulatedwaveformsThis agreement confirms the validity ofthe proposed algorithm

5 Application Example

This example takes into account an actual WT with a ratedpower of 25MW The dimensions of the WT tower are 1199031 =135m 1199032 = 217m ℎ = 82m and 120575 = 0025m and thegrounding resistance 119877119892 is 4Ω (see Figure 1) The param-eter of the injected lightning current is taken as 10350 120583s100 kA according to the technical specification of lightningprotection [16] By using the proposed algorithm to performtransient simulation the transient potential responses areobtained on the tower body The potential waveforms at thetop and the bottom of the tower are shown in Figure 9 andthe peak potential distribution along the height of the toweris also plotted in Figure 10 As seen in Figures 9 and 10 thetransient potential rise on the tower body is extremely seriousand may do damage to the facilities installed inside the towerunder lightning stroke

In an actual wind farm individualWT earth-terminationsystems are usually connected by the metallic armor of thepower cable running between the WTs An interconnectedgrounding system consisting of 5WTs and a substation isshown in Figure 11The distance between two successiveWTsis 380m The substation (SS) is located 390m away from

0

03

06

09

12

15

0 15 30 45 60 75 90Positional height on the WT tower (m)

Volta

ge (M

V)

Figure 10 Peak potential distribution along the height of the tower

WT5 Inner and outer radii of the power cable are 00135mand 0016m respectively The burial depth of the powercable is 1m and the grounding grid of the substation has agrounding resistance of 10Ω The power cable between twosuccessive WTs or SS and WT5 is divided into 5 segmentsEach segment is represented by a120587-type circuit [17]The peakearth potential distribution in the interconnected groundingsystem is shown in Figure 12 where the peak earth potentialat each WT or SS is expressed by the percentage of themaximum earth potential As expected the maximum earthpotential always appears at WT3 where the lightning strikesThe earth potential rise at WT2 and WT4 is more seriousthan that at WT1 and WT5 for WT2 and WT4 are adjacentto WT3 This transferred overvoltage is very harmful to theneighboring WT transformers and the power cables


WT1 WT2 WT3 WT4

WT5

SS

Figure 11 Layout of the interconnected grounding system

0

20

40

60

80

100

WT1 WT2 WT3 WT4 WT5 SS

Peak

eart

h po

tent

ial (

)

Figure 12 Earth potential distribution in the interconnectedgrounding system

In view of the results obtained above the efficient protec-tive measures against the transient potential rise on the towerbody and the transferred overvoltage in the interconnectedgrounding system need to be taken for the multimegawattWTs [18]

6 Conclusions

The lightning transient analysis has been carried out in thispaper for the WT towers The subdividing treatment of thelarge-sized continuous conducting shell allows a WT towerto be converted into a discrete multiconductor system Forcalculating the electrical parameters of the branches in themulticonductor system a set of analytical formulas have beengiven These formulas have the capability of considering theelectromagnetic coupling between the branches in differentspace positions By means of the electrical parameters thebranch units are represented by the coupled 120587-type circuitsand the equivalent discretization network is further builtfor the multiconductor system The solution of lightningtransients is then obtained from the equivalent discretizationnetwork The proposed algorithm can take into account thestructural feature of actual large-sized WT towers and givethe lightning transient responses in different parts on thetower body The laboratory measurement has been made bya reduced-scale tower A better agreement appears between

simulated and measured potential waveforms The applica-bility of the proposed algorithm to the lightning transientanalysis has also been examined by an actual example of a25 MWWT tower and its interconnected grounding system

Acknowledgments

This work was financially supported by the National NaturalScience Foundation of China under Award no 509770926and the Fundamental Research Funds for the Central Uni-versities under Award no 2012JBZ006 The author expresseshis thanks to the foundation committees

References

[1] P Sarajcev and R Goic ldquoA review of current issues in state-of-art of wind farm overvoltage protectionrdquo Energies vol 4 no 4pp 644ndash668 2011

[2] D Romero J Montany and A Candela ldquoBehavior of the wind-turbines under lightning strikes including nonlinear groundingsystemrdquo in Proceedings of the International Conference on RenewEnergy and Power Quality September 2004

[3] Z Haixiang and W Xiaorong ldquoOvervoltage analysis of windturbinesrdquo Power System Technology vol 24 no 5 pp 27ndash292006

[4] X Zhang and C Liu ldquoLightning transient modeling of windturbine towersrdquo International Review of Electrical Engineeringvol 7 no 1 pp 3505ndash3511 2012

[5] X Wang and X Zhang ldquoCalculation of electromagnetic induc-tion inside a wind turbine tower struck by lightningrdquo WindEnergy vol 13 no 7 pp 615ndash625 2010

[6] C R Paul Inductance-Loop and Partial John Wiley amp SonsNew York NY USA 2010

[7] P L Kalantrarov and L A Ceitlin Inductance CalculationEnergy Press Moscow Russia 1992

[8] M M Al-Asadi A P Duffy A J Willis K Hodge and TM Benson ldquoA simple formula for calculating the frequency-dependent resistance of a round wirerdquo Microwave and OpticalTechnology Letters vol 19 no 2 pp 84ndash87 1998

[9] M B Kostenko Analysis of Lightning Protection Reliability ofSubstations Science Press Moscow Russia 1991

[10] U Y Iosseli A S Kothanov and M G Stlyrski CapacitanceCalculation Energy Press Moscow Russia 1990

[11] X Zhang ldquoA circuit approach to the calculation of lightningtransients in cage-like multiconductor systemsrdquo InternationalJournal of Electrical Engineering Education vol 47 no 2 pp213ndash222 2010

[12] J Beiza S H Hosseinian and B Vahidi ldquoMultiphase trans-mission line modeling for voltage sag estimationrdquo ElectricalEngineering vol 92 no 3 pp 99ndash109 2010

[13] H W Dommel Electromagnetic Transients Program TheoryBook BPA Portland Ore USA 1995

[14] W Shi Overvoltage Calculation in Power Systems High Educa-tion Press Beijing China 2009

[15] F Napolitano M Paolone A Borghetti et al ldquoModels of wind-turbine main-shaft bearings for the development of specificlightning protection systemsrdquo IEEETransactions on Electromag-netic Compatibility vol 53 no 1 pp 99ndash107 2011

[16] National Standard GB 50057-2010 Design Code for LightningProtection of Structures China Planning Press Beijing China2010


[17] E Pyrgioti and V Bokogiannis ldquoLightning impulse perfor-mance of a wind generator grounding grid considering soilionizationrdquo in Proceedings of the 7th Asia-Pacific InternationalConference on Lightning (APL rsquo11) pp 103ndash107 November 2011

[18] IEC61400-24 ldquoWind Turbine Generation System-24 LightningProtectionrdquo Ed 10 IEC Geneva Switzerland 2010

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Stochastic AnalysisInternational Journal of


Earth surface

WT tower

Lightning current

Earth-termination system

120575

r1

h

r2

Figure 1 WT tower

ground through the earth-termination system Since mostmanufactures have used many kinds of brushes and slidingcontact systems to divert the lightning current from themainshaft only a much smaller part of lightning current flowsthrough the gearbox and the generator For theWT structurethe tower is the longest conducting path of lightning currentWhen the tower conducts lightning current serious transientpotential rise appears on the tower body and may resultin flashback to the electrical and electronic componentsinstalled inside the tower Much attention to the potentialrise has been paid in the lightning protection design ofWTs For the sake of the transient analysis of the potentialdistribution on the tower the blade and the conducting pathin the nacelle are left out of consideration Instead lightningcurrent is injected into the tower body from its top asshown in Figure 1 [4] In order to build the circuit modelof the tower the continuous conducting shell of the towerbody needs to be subdivided into a discrete multiconductorsystem formed by longitudinal and transverse branches asillustrated in Figure 2 [5] TheWT earth-termination systemis modeled as the grounding resistances (119877119892) The electricalparameters of the branches in the multiconductor systemare represented by resistances inductances and capacitancesFor simplifying the parameter calculation the segmental arcof each transverse branch is replaced by its chord and allthe branches are taken as the cylindrical conductors Theconductor radii are estimated from the average cross-sectionsof the respective branches

22 Inductance and Resistance The inductance parameterscan be determined by Neumannrsquos integral method [6 7] Thetypical branches in the multiconductor system are illustratedin Figure 3 where the presence of the ground is consideredby symmetrically installing the image branches (depictedin dotted lines) below the earth surface [7] According toNeumannrsquos integral method the mutual inductance between

Earth surface

Lightning current

Rg

Rg

Rg

Figure 2 Multiconductor system

the branches 119895 and 119896 is calculated by

119871119895119896 =

1205830

4120587

[int

119904119895

int

119904119896

119889s119895 sdot 119889s119896D119895119896

+ int

1199041015840119895

int

119904119896

119889s1015840119895sdot 119889s119896

D1015840119895119896

] (1)

The dot products of the vector differential segments become

119889s119895 sdot 119889s119896 = cos120593119889s119895119889s119896

119889s1015840119895sdot 119889s119896 = cos120593

10158401198891199041015840

119895119889119904

(2)

where 120593 is the direction angle between the branches119895 and 119896and 1205931015840 is that between the branches 1198951015840 and 119896 Substituting (2)into (1) gives

119871119895119896 =

1205830

4120587

[cos120593119873 (119895 119896) + cos1205931015840119873(1198951015840 119896)] (3)

where119873(119895 119896) and119873(1198951015840 119896) are two double line integrals

119873(119895 119896) = int

119904119895

int

119904119896

119889119904119895119889119904119896

119863119895119896

119873 (1198951015840 119896) = int

1199041015840119895

int

119904119896

1198891199041015840

119895119889119904119896

1198631015840

119895119896

(4)

Equations (3) and (4) indicate that the calculation of themutual inductance depends on the double line integrals119873(119895 119896) and 119873(1198951015840 119896) From the viewpoint of integral oper-ation the way to evaluate 119873(119895 119896) is the same as that toevaluate 119873(1198951015840 119896) For this reason the solution of 119873(119895 119896)is only presented for the typical space positions in themulticonductor system which holds for the evaluation of119873(1198951015840 119896)The integral 119873(119895 119896) is first evaluated in the case of the

coplanar branch pairs For the two coplanar longitudinal


Earth surfaceDjk

dsk

dsj

sj

sk

D998400jk

s998400j

ds998400j

(a)

Earth surface

ds998400j

s998400j

dsj

sj

Djk

dsk

sk

D998400jk

(b)


D4D2

D3

D1

lj

sj

120593

sk

2r0

lk

(a)

2r0

h

sj

sk

(b)



119873(119895 119896) = (119897119895 + 119904119895) ln1198632 + 1198634 + 119904119896

1198632 + 1198634 minus 119904119896

minus 119897119895 ln1198631 + 1198633 + 119904119896

1198631 + 1198633 minus 119904119896

+ (119897119896 + 119904119896) ln1198633 + 1198634 + 119904119895

1198633 + 1198634 minus 119904119895

minus 119897119896 ln1198631 + 1198632 + 119904119895

1198631 + 1198632 minus 119904119895

(5)




119873(119895 119896)


2ℎ

+ (119904119895 + 119904119896) sinhminus1119904119895 + 119904119896

2ℎ

+ 2radic(

119904119896 minus 119904119895

2

)

2

+ ℎ2minus 2radic(

119904119895 + 119904119896

2

)

2

+ ℎ2

(6)


E

FD1D2

D3

D4

O

h

G

lk

sk

sj

lj 120593

(a)

E

O

F

G

h

D1

D2

D3

D4

sk

sj

lj

lk

120593

(b)




119873(119895 119896)

= 2 [(119897119895 + 119904119895) tanhminus1 119904119896

1198631 + 1198632

minus 119897119895tanhminus1 119904119896

1198633 + 1198634

+ (119897119896 + 119904119896) tanhminus1

119904119895

1198631 + 1198634


119904119895

1198632 + 1198633

]

minus

Φℎ

sin120593

(7)



2120593

ℎ1198631 sin120593]


2120593

ℎ1198632 sin120593]


2120593

ℎ1198633 sin120593]


2120593

ℎ1198634 sin120593]

(8)


L = 119871119895119896119872times119872 (9)


119877 =

120588


(10)


120590 =

1

radic120587119891120583120574

(11)



119901119895119896 =

1

41205871205760119904119895119904119896

[int

119904119895

int

119904119896

119889119904119895119889119904119896

119863119895119896

minus int

1199041015840119895

int

119904119896

1198891199041015840

119895119889119904119896

1198631015840

119895119896

]

=

1

41205871205760119904119895119904119896

[119873 (119895 119896) minus 119873(1198951015840 119896)]

(12)



A1

A2

A3

B1

B2

B3

1

2

3

(a)

L13

A1

A2

A3

B1

B2

B3

1

2C11

1

2C12

1

2C23

1

2C22

1

2C13

1

2C11

1

2C13

1

2C22

1

2C23

1

2C33

1

2C33

1

2C12

R1L11

R2

R3

L12

L22

L23

L33

(b)

A BR L

1

2C

1

2C

(c)

A B


(t minus Δt)

C CRLRR R II

IRL

(d)


Impulsegenerator


Steel plate

90∘

u

i

Rg

Rg

Rg



P = 119901119895119896119872times119872 (13)


Pminus1 = 119902119895119896119872times119872 (14)


C = 119862119895119896119872times119872 (15)


119862119895119895 =

119872

sum

120577=1

119902119895120577 (119895 = 1 2 119872)

119862119895119896 = minus119902119895119896 (119895 119896 = 1 2 119872 119895 = 119896)

(16)


0 200 400 600 800 10000

10

20

30

40

50

60Cu

rren

t (m

A)

Time (ns)

(a)

0 200 400 600 800 1000Time (ns)

MeasuredSimulated

0

16

32

48

64

8

Volta

ge (V

)(b)









0 20 40 60 80 1000

02

04

06

08

1

12

14Vo

ltage

(MV

)

Time (120583s)

(a)

0 20 40 60 80 100Time (120583s)

0

02

04

06

08

1

12

Volta

ge (M

V)

(b)






0

03

06

09

12

15


Volta

ge (M

V)




WT1 WT2 WT3 WT4

WT5

SS


0

20

40

60

80

100


Peak

eart

h po

tent

ial (

)



6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Earth surfaceDjk

dsk

dsj

sj

sk

D998400jk

s998400j

ds998400j

(a)

Earth surface

ds998400j

s998400j

dsj

sj

Djk

dsk

sk

D998400jk

(b)


D4D2

D3

D1

lj

sj

120593

sk

2r0

lk

(a)

2r0

h

sj

sk

(b)



119873(119895 119896) = (119897119895 + 119904119895) ln1198632 + 1198634 + 119904119896

1198632 + 1198634 minus 119904119896

minus 119897119895 ln1198631 + 1198633 + 119904119896

1198631 + 1198633 minus 119904119896

+ (119897119896 + 119904119896) ln1198633 + 1198634 + 119904119895

1198633 + 1198634 minus 119904119895

minus 119897119896 ln1198631 + 1198632 + 119904119895

1198631 + 1198632 minus 119904119895

(5)




119873(119895 119896)


2ℎ

+ (119904119895 + 119904119896) sinhminus1119904119895 + 119904119896

2ℎ

+ 2radic(

119904119896 minus 119904119895

2

)

2

+ ℎ2minus 2radic(

119904119895 + 119904119896

2

)

2

+ ℎ2

(6)


E

FD1D2

D3

D4

O

h

G

lk

sk

sj

lj 120593

(a)

E

O

F

G

h

D1

D2

D3

D4

sk

sj

lj

lk

120593

(b)




119873(119895 119896)

= 2 [(119897119895 + 119904119895) tanhminus1 119904119896

1198631 + 1198632

minus 119897119895tanhminus1 119904119896

1198633 + 1198634

+ (119897119896 + 119904119896) tanhminus1

119904119895

1198631 + 1198634


119904119895

1198632 + 1198633

]

minus

Φℎ

sin120593

(7)



2120593

ℎ1198631 sin120593]


2120593

ℎ1198632 sin120593]


2120593

ℎ1198633 sin120593]


2120593

ℎ1198634 sin120593]

(8)


L = 119871119895119896119872times119872 (9)


119877 =

120588


(10)


120590 =

1

radic120587119891120583120574

(11)



119901119895119896 =

1

41205871205760119904119895119904119896

[int

119904119895

int

119904119896

119889119904119895119889119904119896

119863119895119896

minus int

1199041015840119895

int

119904119896

1198891199041015840

119895119889119904119896

1198631015840

119895119896

]

=

1

41205871205760119904119895119904119896

[119873 (119895 119896) minus 119873(1198951015840 119896)]

(12)



A1

A2

A3

B1

B2

B3

1

2

3

(a)

L13

A1

A2

A3

B1

B2

B3

1

2C11

1

2C12

1

2C23

1

2C22

1

2C13

1

2C11

1

2C13

1

2C22

1

2C23

1

2C33

1

2C33

1

2C12

R1L11

R2

R3

L12

L22

L23

L33

(b)

A BR L

1

2C

1

2C

(c)

A B


(t minus Δt)

C CRLRR R II

IRL

(d)


Impulsegenerator


Steel plate

90∘

u

i

Rg

Rg

Rg



P = 119901119895119896119872times119872 (13)


Pminus1 = 119902119895119896119872times119872 (14)


C = 119862119895119896119872times119872 (15)


119862119895119895 =

119872

sum

120577=1

119902119895120577 (119895 = 1 2 119872)

119862119895119896 = minus119902119895119896 (119895 119896 = 1 2 119872 119895 = 119896)

(16)


0 200 400 600 800 10000

10

20

30

40

50

60Cu

rren

t (m

A)

Time (ns)

(a)

0 200 400 600 800 1000Time (ns)

MeasuredSimulated

0

16

32

48

64

8

Volta

ge (V

)(b)









0 20 40 60 80 1000

02

04

06

08

1

12

14Vo

ltage

(MV

)

Time (120583s)

(a)

0 20 40 60 80 100Time (120583s)

0

02

04

06

08

1

12

Volta

ge (M

V)

(b)






0

03

06

09

12

15


Volta

ge (M

V)




WT1 WT2 WT3 WT4

WT5

SS


0

20

40

60

80

100


Peak

eart

h po

tent

ial (

)



6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










E

FD1D2

D3

D4

O

h

G

lk

sk

sj

lj 120593

(a)

E

O

F

G

h

D1

D2

D3

D4

sk

sj

lj

lk

120593

(b)




119873(119895 119896)

= 2 [(119897119895 + 119904119895) tanhminus1 119904119896

1198631 + 1198632

minus 119897119895tanhminus1 119904119896

1198633 + 1198634

+ (119897119896 + 119904119896) tanhminus1

119904119895

1198631 + 1198634


119904119895

1198632 + 1198633

]

minus

Φℎ

sin120593

(7)



2120593

ℎ1198631 sin120593]


2120593

ℎ1198632 sin120593]


2120593

ℎ1198633 sin120593]


2120593

ℎ1198634 sin120593]

(8)


L = 119871119895119896119872times119872 (9)


119877 =

120588


(10)


120590 =

1

radic120587119891120583120574

(11)



119901119895119896 =

1

41205871205760119904119895119904119896

[int

119904119895

int

119904119896

119889119904119895119889119904119896

119863119895119896

minus int

1199041015840119895

int

119904119896

1198891199041015840

119895119889119904119896

1198631015840

119895119896

]

=

1

41205871205760119904119895119904119896

[119873 (119895 119896) minus 119873(1198951015840 119896)]

(12)



A1

A2

A3

B1

B2

B3

1

2

3

(a)

L13

A1

A2

A3

B1

B2

B3

1

2C11

1

2C12

1

2C23

1

2C22

1

2C13

1

2C11

1

2C13

1

2C22

1

2C23

1

2C33

1

2C33

1

2C12

R1L11

R2

R3

L12

L22

L23

L33

(b)

A BR L

1

2C

1

2C

(c)

A B


(t minus Δt)

C CRLRR R II

IRL

(d)


Impulsegenerator


Steel plate

90∘

u

i

Rg

Rg

Rg



P = 119901119895119896119872times119872 (13)


Pminus1 = 119902119895119896119872times119872 (14)


C = 119862119895119896119872times119872 (15)


119862119895119895 =

119872

sum

120577=1

119902119895120577 (119895 = 1 2 119872)

119862119895119896 = minus119902119895119896 (119895 119896 = 1 2 119872 119895 = 119896)

(16)


0 200 400 600 800 10000

10

20

30

40

50

60Cu

rren

t (m

A)

Time (ns)

(a)

0 200 400 600 800 1000Time (ns)

MeasuredSimulated

0

16

32

48

64

8

Volta

ge (V

)(b)









0 20 40 60 80 1000

02

04

06

08

1

12

14Vo

ltage

(MV

)

Time (120583s)

(a)

0 20 40 60 80 100Time (120583s)

0

02

04

06

08

1

12

Volta

ge (M

V)

(b)






0

03

06

09

12

15


Volta

ge (M

V)




WT1 WT2 WT3 WT4

WT5

SS


0

20

40

60

80

100


Peak

eart

h po

tent

ial (

)



6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A1

A2

A3

B1

B2

B3

1

2

3

(a)

L13

A1

A2

A3

B1

B2

B3

1

2C11

1

2C12

1

2C23

1

2C22

1

2C13

1

2C11

1

2C13

1

2C22

1

2C23

1

2C33

1

2C33

1

2C12

R1L11

R2

R3

L12

L22

L23

L33

(b)

A BR L

1

2C

1

2C

(c)

A B


(t minus Δt)

C CRLRR R II

IRL

(d)


Impulsegenerator


Steel plate

90∘

u

i

Rg

Rg

Rg



P = 119901119895119896119872times119872 (13)


Pminus1 = 119902119895119896119872times119872 (14)


C = 119862119895119896119872times119872 (15)


119862119895119895 =

119872

sum

120577=1

119902119895120577 (119895 = 1 2 119872)

119862119895119896 = minus119902119895119896 (119895 119896 = 1 2 119872 119895 = 119896)

(16)


0 200 400 600 800 10000

10

20

30

40

50

60Cu

rren

t (m

A)

Time (ns)

(a)

0 200 400 600 800 1000Time (ns)

MeasuredSimulated

0

16

32

48

64

8

Volta

ge (V

)(b)









0 20 40 60 80 1000

02

04

06

08

1

12

14Vo

ltage

(MV

)

Time (120583s)

(a)

0 20 40 60 80 100Time (120583s)

0

02

04

06

08

1

12

Volta

ge (M

V)

(b)






0

03

06

09

12

15


Volta

ge (M

V)




WT1 WT2 WT3 WT4

WT5

SS


0

20

40

60

80

100


Peak

eart

h po

tent

ial (

)



6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 200 400 600 800 10000

10

20

30

40

50

60Cu

rren

t (m

A)

Time (ns)

(a)

0 200 400 600 800 1000Time (ns)

MeasuredSimulated

0

16

32

48

64

8

Volta

ge (V

)(b)









0 20 40 60 80 1000

02

04

06

08

1

12

14Vo

ltage

(MV

)

Time (120583s)

(a)

0 20 40 60 80 100Time (120583s)

0

02

04

06

08

1

12

Volta

ge (M

V)

(b)






0

03

06

09

12

15


Volta

ge (M

V)




WT1 WT2 WT3 WT4

WT5

SS


0

20

40

60

80

100


Peak

eart

h po

tent

ial (

)



6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

02

04

06

08

1

12

14Vo

ltage

(MV

)

Time (120583s)

(a)

0 20 40 60 80 100Time (120583s)

0

02

04

06

08

1

12

Volta

ge (M

V)

(b)






0

03

06

09

12

15


Volta

ge (M

V)




WT1 WT2 WT3 WT4

WT5

SS


0

20

40

60

80

100


Peak

eart

h po

tent

ial (

)



6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










WT1 WT2 WT3 WT4

WT5

SS


0

20

40

60

80

100


Peak

eart

h po

tent

ial (

)



6 Conclusions



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 Transient Simulation of Wind Turbine ...downloads.hindawi.com/journals/mpe/2013/142765.pdfsystems. e lightning protection design needs to obtain the lightning transient