3-Characteristics of Elements

Copyright © Siemens AG 2008. All rights reserved.

Sector Energy PTI NCTheodor Connor

Characteristics of Elements

01.2008For internal use only. / Copyright © Siemens AG 2008. All rights reserved.

PTD SE PTITh. Connor

Content

Introduction

Theoretical background

Determination of impedances

Determination of capacitances

Calculation of line impedances

Measurement of line impedances

Transformer characteristics



Importance of element characteristics

Characteristics of network elements are necessary for:

Network planning

Network calculation (e.g. load flow, short circuit)

Design of equipment

Protection setting (especially distance protection)

Analysis of disturbances



Line characteristics

Self impedance

Mutual impedance

Symmetrical components



Electric parameters of a conductor

Conductor

Resistance

Reactance

Capacitance



Resistance

Conductor

R in Ohm:

• Material of conductor

• Cross section of conductor



Reactance of a loop

Conductors

X in Ohm:

• Distance between conductor

• Diameter of conductor

Current



Capacitance

Conductor

Infinite ground plane

Voltage

C in Farad:

• Height above ground

• Diameter of conductor



Conductor with ground return

Conductor

Infinite ground plane

Current

Current



Self impedance of line-earth loop

[ ] [ ]extintextintS XXjRRZ +++=

ω⋅μ⋅= 0ext 81R

n1LX iint ⋅⋅ω=

rDlnfX E

0ext ⋅⋅μ=

qRint

ρ=



Input data

abds

t

ED

2r



Data

πμ⋅μ

⋅= r0i 8

1L

n

1n

t

n180sin2

snrr

−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

°⋅

⋅=

Internal Inductance

Equivalent depth of return path of earth current

Equivalent radius of bundle conductor

μ0 = 1,26 ⋅10-3 H/km μ r = 1Li = 0,05 mH/km

r t = 16 mm s = 400 mm n = 2r = 81 mm

ρ E = 100 Ωm f = 50 HzD E = 930 m

f660D E

Eρ

=



Mutual impedance between line-earth loops

MMM XjRZ ⋅+=

ω⋅μ⋅= 0M 81R

ab

E0M d

DlnfX ⋅⋅⋅μ=



Positive sequence impedanceIR R

S

T

UR

ZS R ZM RS

ZM RT

IS

IT

RTMTRSMSRSRR ZIZIZIU ⋅+⋅+⋅=

[ ]TRMSTMRSMM ZZZZ ++=31

( ) MTSSRR ZIIZIU ⋅++⋅=

STMTSSSRSMRS ZIZIZIU ⋅+⋅+⋅=

TSTSTMSRTMRT ZIZIZIU ⋅+⋅+⋅=

[ ]TSTSRSS ZZZZ ++=31



Positive sequence impedance

MS1 ZZZ −=

( )MSRR ZZIU −⋅=

0III TSR =++for positive sequence:


RTS III −=+

1ZIU RR ⋅=



Positive sequence impedance

Mextint1 RRRR −+=

MS1 ZZZ −=

111 XjRZ ⋅+=

ω⋅μ−ω⋅μ+ρ

= 001 81

81

qR

Mextint1 XXXX −+=

m00i1 d

Delnfr

Delnfn1LX ⋅⋅μ−⋅⋅μ+⋅ω=

rdlnf

n1

81f2X m

0r0

1 ⋅⋅μ+⋅πμ⋅μ

⋅⋅⋅π⋅=

qR1

ρ= ⎟

⎠⎞

⎜⎝⎛ +

⋅μ

μ⋅=r

dlnn4

fX mr01

3R,TT,SS,Rm dddd ⋅⋅=



Positive sequence impedance Example 1 overhead line – Given data

Conductor: 150 mm² Al

Cross-Section: q = 150 mm²

Diameter: d = 15.8 mm

Resistivity Al: ρAl = 29 Ω·mm²/km

Permeability Al: μr = 1

Frequency: f = 50 Hz

Soil resistivity: ρE = 100 Ω·m

Magnetic constant: μ0 = 4 · π · 10-4 H/km

=0.00126 Ω·s/km

Positive Sequence Impedance ??



Positive sequence impedance Example 1 overhead line – Result

⎟⎠⎞

⎜⎝⎛ +

⋅μ

μ⋅=r

dlnn4

fX mr01

qR1

ρ=

km19.0R 1

Ω=

⎟⎠⎞

⎜⎝⎛ +

⋅⋅

⋅⋅Ω⋅

=m0079.0

m0.5ln14

1kms

s 0.0012650X1

kmmm150mm29R 2

2

1 ⋅⋅Ω

=


m5m8m4m4d 3m =⋅⋅=

km 42.0X 1Ω

=



Positive Sequence Impedance Example 2 Overhead line – Given Data









=0.00126 Ω·s/km




Positive Sequence Impedance Example 2 Overhead line – Result

⎟⎠⎞

⎜⎝⎛ +

⋅μ

μ⋅=r

dlnn4

fX mr01

qR1

ρ=

km41.0R 1

Ω=

⎟⎠⎞

⎜⎝⎛ +

⋅⋅

⋅⋅Ω⋅

=m00525.0

m76.1ln14

1kms

s 0.0012650X1

kmmm70mm29R 2

2

1 ⋅⋅Ω

=


m76.1m8.2m4.1m4.1d 3m =⋅⋅=

km 38.0X 1Ω

=



Zero sequence impedanceIR R

S

T

UR

ZS R ZM RS

ZM RT

IS

IT

RTMTRSMSRSRR ZIZIZIU ⋅+⋅+⋅=


[ ]TRMSTMRSMM ZZZZ ++=31

[ ]TSTSRSS ZZZZ ++=31

STMTSSSRSMRS ZIZIZIU ⋅+⋅+⋅=

TSTSTMSRTMRT ZIZIZIU ⋅+⋅+⋅=



Zero sequence impedance


MS0 Z2ZZ ⋅+=

( )MSRR ZZIU ⋅+⋅= 2

TSR III ==for zero sequence:

0ZIU RR ⋅=




Mextint0 R2RRR ⋅++=

MS0 Z2ZZ ⋅+=

000 XjRZ ⋅+=

ω⋅μ+ω⋅μ+ρ

= 000 82

81

qR

Mextint0 X2XXX ⋅++=

m00i0 d

Delnf2r

Delnfn1LX ⋅⋅μ⋅+⋅⋅μ+⋅ω=

2m

3

0r

00 drDelnf

n4fX

⋅⋅⋅μ+

⋅μ

μ⋅=

ω⋅μ+ρ

= 00 83

qR ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

+⋅μ

μ⋅=2m

3r

00 drDe

lnn4

fX




Zero sequence impedance Example 1 overhead line – Given data









=0.00126 Ω·s/km

Zero Sequence Impedance ??



Zero sequence impedance Example 1 overhead line – Result

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

+⋅μ

μ⋅= 2m

3r

00 drDeln

n4fX

ω⋅μ+ρ

= 00 83

qR

km34.0R 0

Ω=

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

+⋅⋅

⋅Ω⋅= 2

3

0 50079.0930ln

41

kmss0.0012650X

skms

kmmmmmR

⋅⋅⋅⋅⋅Ω

⋅+⋅

⋅Ω=

π25000126.083

15029

2

2

0

km 41.1X 0Ω

=

m930f

660De E =ρ

=



Zero Sequence Impedance Example 2 Overhead line – Given Data









=0.00126 Ω·s/km




Zero Sequence Impedance Example 2 Overhead line – Result

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

+⋅μ

μ⋅= 2m

3r

00 drDeln

n4fX

ω⋅μ+ρ

= 00 83

qR

km56.0R 0

Ω=

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

+⋅⋅

⋅Ω⋅= 2

3

0 76.100525.0930ln

41

kmss0.0012650X

skm250s00126.0

83

kmmm70mm29R 2

2

0 ⋅π⋅⋅⋅⋅Ω

⋅+⋅

⋅Ω=

km 57.1X 0Ω

=

m930f

660De E =ρ

=



Further influences

Earthed conductors (overhead line earth wire; cable shield)

Height above ground; line sag

Length of conductor twist

Skin effect

Proximity effect

Conductor temperature



Double circuit – Zero sequence mutual impedance

Z 0 MZero sequence mutual impedance



Influence of zero sequence mutual impedance

1pol

1polX X

1pol

1pol

( )MZZZ 0021 +⋅=

0ZZ =

( )MZZZ 002 −⋅=

GZZ 0=



Capacitance of lines

Self Capacitance;Line-Earth Capacitance CE

Mutual Capacitance;Line-Line Capacitance CM

h

2 r

+

-

+

+

- -

UCQ ⋅=

rhC r ⋅⋅⋅⋅⋅=

2ln

12 0εεπ



Positive Sequence Capacitance

IRR

S

T

UR

CS

CM

CMUS

UT

( ) ( ) MTRMSRSRR CUUCUUCUQ ⋅−+⋅−+⋅=

[ ]TTSSRRS CCC31C ++=

[ ]TRSTRSM CCC31C ++=

( ) ( ) MTSMSRR CUUC2CUQ ⋅+−⋅+⋅=



Positive Sequence Capacitance

MS1 C3CC ⋅+=

( )MSRR C3CUQ ⋅+⋅=

0UUU TSR =++for positive sequence:

( ) ( ) MTSMSRR CUUC2CUQ ⋅+−⋅+⋅=

MSR

R C3CUQ

⋅+=

TSR UUU −−=



Zero Sequence Capacitance

IRR

S

T

UR

CS

CM

CMUS

UT

( ) ( ) MTRMSRSRR CUUCUUCUQ ⋅−+⋅−+⋅=

S0 CC =

SRR CUQ ⋅=

[ ]TTSSRRS CCC31C ++=

[ ]TRSTRSM CCC31C ++=

TSR UUU ==for positive sequence:

SR

R CUQ

=



Positive Sequence Capacitance:

Positive and Zero Sequence Capacitances of an Overhead Line

hRhT

hS

dRS

dTR

dST

RT

S

rdln

2C 0r1

ε⋅ε⋅π⋅=

Zero Sequence Capacitance:

3 2

0r0

drh2ln3

2C

⋅

⋅⋅

ε⋅ε⋅π⋅=

εr - relative dielectric constant (in air: εr = 1)

ε0 - dielectric constant (8.854 nF/km)

d - conductor distance

h - conductor height

s - line sag

r - radius of conductor

3TRSTRS dddd ⋅⋅=

s7.0hhhh 3TSR ⋅−⋅⋅=



Positive and Zero Sequence Capacitance Example Overhead line – Given Data




Max. line sag: s = 4 m

Dielectric constant: ε0 = 8.854 nF/km

Positive Sequence Capacitance ??

Zero Sequence Capacitance ??



Positive and Zero Sequence Capacitance Example Overhead line – Result

kmnF57.9C 1 =

kmnF33.4C 0 =

m00525.0m76.1ln

km/nF854.812

rdln

2C 0r1

⋅⋅π⋅=

ε⋅ε⋅π⋅=

3TRSTRS dddd ⋅⋅=

s7.0hhhh 3TSR ⋅−⋅⋅=

m76.1m8.2m4.1m4.1d 3 =⋅⋅=

m2.9m47.0m12h =⋅−=

( )3 23 2

0r0

m76.1m00525.0

m2.92ln3

km/nF854.812

drh2ln3

2C

⋅

⋅⋅

⋅⋅π⋅=

⋅

⋅⋅

ε⋅ε⋅π⋅=



Typical line parameters

R1 X1 C1 R0 X0 C0Ohm/km Ohm/km nF/km Ohm/km Ohm/km nF/km

400 kV overhead line 0,03 0,25 14 0,33 1,44 6,5

110 kV overhead line 0,07 0,41 10 0,35 1,65 4,7cable 0,04 0,11 400 0,5 0,28 400

20 kV overhead line 0,31 0,4 10 0,40 1,50 5,0cable 0,20 0,13 300 0,50 0,30 300



Calculation of line impedances



Calculation of Overhead Line Constants

Calculation of electrical parameters of overhead lines with

With several three phase systemsdifferent voltage levelsany number of earth wires (or counterpoise electrodes)

The three phase systems to be calculated may also run on different parallel overhead lines.



Calculation of Line Constants of Cables

Calculation of electrical parameters of cables with

with concentric screens or metal sheaths or armouring additional common sheaths around a three-phase system any number of parallel earthing conductors

Several three phase systems have to be considered in one calculation.



Typical Input Data

Conductor material

Name

Spec. resistance

Rel. permeability

Conductor types

Name

Cross-section

Material

Conductors

Earthing

Name

Conductor type

Außendurchmesser

Innendurchmesser

Bündel-Anordnung

Bündel-Anzahl

Bündel-Abstand

X-coordinate

Y-coordinate

Sag

Outer diameter

Inner diameter

Bundle arrangement

No. of sub-cond.

Sub-cond. distance

Twist factor

Rel. dialect. constant

Cross-section reinf.

Material reinforcm.

Systems

System Number

Number of phases

Rated voltage

Max. voltage

Voltage angle

Phase conductors only



Presentation of Results

Example110 kV overhead line

KENNWERTE IN SYMMETRISCHEN KOMPONENTEN======================================

Selbstimpedanzen¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Mitsystem R´(ohm/km) X´(ohm/km)

C´(nF/km)

System 1 0,063525 0,266984 13,537891System 2 0,063525 0,266984 13,537891

Koppelimpedanzen¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Mitsystem

R´(ohm/km) X´(ohm/km) C´(nF/km)

System 1 - System 2 0,002292 0,000552 0,035956

Example110 kV overhead line

KENNWERTE IN SYMMETRISCHEN KOMPONENTEN======================================

Selbstimpedanzen¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Mitsystem R´(ohm/km) X´(ohm/km)

C´(nF/km)

System 1 0,063525 0,266984 13,537891System 2 0,063525 0,266984 13,537891

Koppelimpedanzen¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Mitsystem

R´(ohm/km) X´(ohm/km) C´(nF/km)

System 1 - System 2 0,002292 0,000552 0,035956

ReportReport

110 kV overhead line (double system, Donau configuration)

Figure (Tower Outline)

PTD SE NC1-ue111824.09.01

Bild 1Leika 3.2C:\Program Files\Siemens SINCAL\Leika\Examples\ohl110-donau.lei

Example

20,200 m

1R

1T

2R

2T

24,000 m

1S

2S

32,000 m

E1

0,000 m E1 -3,300 m1T-5,200 m1S-7,100 m1R

3,300 m2R5,200 m 2S7,100 m 2T

1R

1S

1T 2R

2S

2T

E1

Maßstab: 1:200

DiagramDiagram



Impedance CalculationSelf and Mutual Impedances of Line-Earth-Loops

extextintintS XjRXjRZ ⋅++⋅+=Self Impedance:

Mutual Impedance:

MMM XjRZ ⋅+=

Calculation with series expression acc. to Carson for earth return

Internal ImpedanceCalculation of DC resistanceCalculation with Bessel function for Skin effect consideration

External ImpedanceCalculation with series expression acc. to Carson for earth return consideration

consideration



Impedance Calculation Conductor Types

Aluminium conductor steel reinforced

(ACSR, Al/St, AL1/ST1A)

Aluminium alloy conductor

(AAAC, Ald, AL2 – AL5)

Galvanized steel conductor (only for earth wires)

(St, ST1A)

Aluminium clad steel conductor (only for earth wires)

(ACS, St-Al-um, A20SA)



Impedance CalculationDC Resistance

Av_R ρ⋅=

rf

rfAA1v_R

ρ+

ρ

⋅=

Reinforced conductors(e.g. ACSR)

Conductors (not reinforced)

R_ - DC Resistance

v - Twist factor

ρ - resistivity of conductor material

A - Cross-section of conductor

ρ - resistivity of reinforcement material

A - Cross-section of reinforcement



Impedance Calculation Conductor Twist

Conductor twist results inhigher length of outer wires (red wire) of the conductorincrease of resistance

Increase of DC-resistance acc. to EN 50182:

7wires 1 layer 1.11 %19 wires 2 layers 1.68 %37 wires 3 layers 2.03 %61 wires 4 layers 2.36 %127 wires 6 layers 2.75 %



Impedance CalculationInfluence of Conductor Temperature

( )[ ]C20T*1RR C20,DCT,DC °−α+⋅=

RDC,T resistance at conductor temperature TRDC,20 DC resistance at 20°Cα temperature coefficient of conductor materialTC conductor temperatureρT resistivity at conductor temperature Tρ20 resistivity at 20°C

( )[ ]C20T*1 C20T °−α+⋅ρ=ρ

Conductor material Temperature coefficient αAluminium 0.0040 1/°CAluminium alloy 0.0036 1/°C

Copper 0.0039 1/°C



Impedance Calculation Internal self impedance

without Skin effectCurrent density:

Calculation:

with Skin effectCurrent density:

Calculation:with Bessel functions

r0f41X

AR μ⋅μ⋅⋅=

ρ= −− −− <> XXRR ~~

J [A/mm²] J [A/mm²]



Impedance Calculation External self impedance of Line-Earth-Loop

Basic approach:

Concentrated earth return path in depth De

Exact calculation:

Series expression by Carson to consider earth current distribution and conductor height above ground



Impedance Calculation Elimination of earthed conductors

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

E

T

S

R

EETESERE

ETTTSTRT

ESTSSSRS

ERTRSRRR

E

T

S

R

IIII

ZZZZZZZZZZZZZZZZ

UUUU

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

E

T

S

R

EETESERE

ETTTSTRT

ESTSSSRS

ERTRSRRR

E

T

S

R

UUUU

YYYYYYYYYYYYYYYY

IIII

Matrix Inversion

0

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

T

S

R

TTTSTR

STSSSR

RTRSRR

T

S

R

UUU

'Y'Y'Y'Y'Y'Y'Y'Y'Y

III

Elimination of one equation

UE = 0



UL1

UL2UL3

U1L1

U1L2

U2 L1

U2 L3

U0 L1

Positive sequence U1 = (UL1+ a UL2+ a² UL3)

Negative sequence U2 = (UL1+ a² UL2+ a UL3)

Zero sequence U0 = (UL1 + UL2 + UL3)

13

13

13

Symmetrical components



Calculation of Impedances Overview

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

EETESERE

ETTTSTRT

ESTSSSRS

ERTRSRRR

ZZZZZZZZZZZZZZZZ Calculation of Self- and Coupling impedances

Bessel function to consider the Skin effect Series expansions according to Carson and Dommel to consider the earth return path

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

TTSTRT

TSSSRS

TRSRRR

'Z'Z'Z'Z'Z'Z'Z'Z'Z

Elimination of conductors earthed at both sides(overhead line earth wires, Cable sheaths)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

221202

211101

201000

ZZZZZZZZZ

Balancing the impedance matrix



Basic formulae:

Partial capacitances

CRRCTT

CSS

CRS

CTR

CTT

UR

UT

US

( )( ) ( ) TTTTSSTTRRTT

STTSSSSSRRSS

RTTRRSSRRRRR

CUCUUCUUQC)UU(CUCUUQC)UU(C)UU(CUQ

⋅+⋅−+⋅−=

⋅−+⋅+⋅−=

⋅−+⋅−+⋅=

CRR, CSS, CTT – partial line-earth capacitance

CRS, CST, CTR – partial line-line capacitance

CUQ ⋅=



Capacitance Coefficients

( )( ) ( ) TTTTSSTTRRTT

STTSSSSSRRSS

RTTRRSSRRRRR

CUCUUCUUQC)UU(CUCUUQC)UU(C)UU(CUQ

⋅+⋅−+⋅−=

⋅−+⋅+⋅−=

⋅−+⋅−+⋅=

Partial Capacitances C (Equation)

( )( )

( ) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++−−−++−−−++

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

T

S

R

TTTSTRTSTR

STSTSSSRSR

RTRSRTRSRR

T

S

R

UUU

CCCCCCCCCCCCCCC

QQQ

Partial Capacitances C (Matrix)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

T

S

R

TTTSTR

STSSSR

RTRSRR

T

S

R

UUU

KKKKKKKKK

QQQ

Capacitance Coefficients K



Potential Coefficients and Capacitive Impedances

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

T

S

R

TTTSTR

STSSSR

RTRSRR

T

S

R

UUU

KKKKKKKKK

QQQ

Capacitance Coefficients K

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

T

S

R

TTTSTR

STSSSR

RTRSRR

T

S

R

QQQ

PPPPPPPPP

UUU

Potential Coefficients P

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

T

S

R

TTTSTR

STSSSR

RTRSRR

T

S

R

III

ZZZZZZZZZ

UUU

Capacitive Impedances Z

Matrix Inversion

ω−=

PjZ

QjI ⋅ω⋅=



Calculation of Potential coefficients

i

iii r

h2ln2

1P ⋅⋅

ε⋅π⋅=

ik

ikik d

Dln2

1P ⋅ε⋅π⋅

=

Potential coefficientof conductor i

Potential coefficient betweenconductors i and k

ε – dielectric constant

(8,854 nF/km)

ri – Radius of conductor i



Calculation of Capacitances Overview

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

TTSTRT

TSSSRS

TRSRRR

'Z'Z'Z'Z'Z'Z'Z'Z'Z

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

EETESERE

ETTTSTRT

ESTSSSRS

ERTRSRRR

ZZZZZZZZZZZZZZZZ

Capacitive Impedances

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

221202

211101

201000

ZZZZZZZZZ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

EETESERE

ETTTSTRT

ESTSSSRS

ERTRSRRR

KKKKKKKKKKKKKKKK

Capacitive Coefficients

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

EETESERE

ETTTSTRT

ESTSSSRS

ERTRSRRR

CCCCCCCCCCCCCCCC

Partial Capacitances

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

TTSTRT

TSSSRS

TRSRRR

'K'K'K'K'K'K'K'K'K

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

221202

211101

201000

KKKKKKKKK

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

TTSTRT

TSSSRS

TRSRRR

'C'C'C'C'C'C'C'C'C

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

221202

211101

201000

CCCCCCCCC

Inverting

Inverting

Inverting

Elimination of earthed conductors

Balancing



Other Characteristic ValuesSurge Impedance and Natural Load

ZW Surge impedance (complex)R1 Resistance in the positive phase-sequence systemX1 Reactance in the positive phase-sequence systemω Angular frequency (2·π·f)C1 Capacitance in the positive phase-sequence systemSnat Natural load

W

2N

nat ZUS =

1

11W Cj

XjRZ⋅ω⋅⋅+

=



Other Characteristic ValuesCharging Power

QC Charging powerUN Rated voltageω Angular frequency (2·π·f)C1 Capacitance in the positive phase-sequence system

12NC CUQ ⋅ω⋅=

100,00 %0,00 °

-0,00 kW51,70 kVAr



Other Characteristic ValuesReduction Factor – Calculation

E,S

EL,C

0

EE Z

ZI3

Ir −=⋅

=

rE Reduction factor IE Earth current (current flowing through earth)3·I0 Zero sequence current during a single phase faultZC,L-E Coupling impedance between phase conductors and

earthed conductorsZS,E Self impedance of earthed conductors

Reduction factor of earthed conductors

e.g. overhead line earth wires, metallic cable sheaths

Definition and Calculation acc. to IEC 60909-3



Other Characteristic ValuesReduction Factor – Importance

A low reduction factor results inlow earth, touch and step voltageslow inductive interference in case of earth faultslow magnetic fields in case of earth faults

IK1 · (1-rE)

IGRG

IK1

IK1 · rE

3·I0 = IK1



Measurement of line impedances



Measurement of Zero Sequence ImpedancePower Cable

IMeascurrent source

A

V

IMeas

U

power cable

remote earth gridlocal earth grid

V

ground potential rise

A ISheath

A IEarth

WP



Measurement of Zero Sequence ImpedanceOverhead Line

current source I

U

overhead line


W

P

V

A

earth wire



Measurement of Line ImpedancesTest Equipment

Diesel-generator set 200kVADiesel-generator set 200kVA

Clamp-Ampere-MeterClamp-Ampere-Meter

Active-Power-MeterActive-Power-Meter

Volt-MeterVolt-Meter



Measurement of Line ImpedancesSafety rules

1. Switch off

2. Lock against reclosure

3. Verify that system is de-energized

4. Earthing and short circuiting

5. Cover of nearby live parts

1. Be aware of trapped charges

2. Be aware of induced or influenced voltages

3. Have a clear predefined switching procedure

4. Have a back up switch off strategy

5. Stop work in case of confusion

6. Stop work in case of thunder storm (which may be at the remote end!)



Double Line Measurement of Zero Sequence Impedance

current source I

U

overhead line – system 1


W

P

V

A

earth wire




Double Line Measurement of Zero Sequence Coupling Impedance

current sourceI

U



WP

V

A

earth wire




Double Line Measurement of Double Line Zero Sequence Imped.

current source I

U



W

P

V

A

earth wire




Double Line Measurement of Shortened Zero Sequence Imped.

current source I

U



W

P

V

A

earth wire




Measurement of Line ImpedancesConnection of test leads

Connection to current sourceConnection to current source

Short circuiting of test line for zero sequence measurement

Short circuiting of test line for zero sequence measurement



Zero Sequence Impedance of 220 kV Double CircuitComparison of Measured and Calculated Values

17.050 m

1R2R

23.850 m

1S2S

30.650 m

1T2T

40.007 m

E1

0.000 m E1-5.400 m2T-5.800 m2R-7.100 m2S

5.400 m 1T5.800 m 1R7.100 m 1S

1R

1S

1T

2R

2S

2T

E1

Conductors:phase wire: Al/St 591/82earth wire: OPGW

57 mm² steel41 mm² ACS

measured calculated__________________________________

R0‘ 0.23 Ω/km 0.20 Ω/kmX0‘ 1.01 Ω/km 1.02 Ω/kmR0M‘ 0.16 Ω/km 0.15 Ω/kmX0M‘ 0.48 Ω/km 0.49 Ω/km


R0‘ 0.23 Ω/km 0.20 Ω/kmX0‘ 1.01 Ω/km 1.02 Ω/kmR0M‘ 0.16 Ω/km 0.15 Ω/kmX0M‘ 0.48 Ω/km 0.49 Ω/km



Impedance of 220 kV Overhead LineComparison of Measured and Calculated Values

Conductors:phase wire: Al/St 591/82

bundle of 2

earth wire: OPGW57 mm² steel41 mm² ACS

length: 26 km


R1 1.57 Ω 1.46 ΩX1 7.66 Ω 7.80 ΩR0 3.63 Ω 3.42 ΩX0 23.4 Ω 22.0 Ω


R1 1.57 Ω 1.46 ΩX1 7.66 Ω 7.80 ΩR0 3.63 Ω 3.42 ΩX0 23.4 Ω 22.0 Ω



Measurement of zero sequence impedance



Measurement of phase - earth loop

IMeascurrent source

A

V

IMeas

U

power cable


A ISheath

A IEarth

WP



Measurement of positive sequence impedance



Measurement of zero sequence impedance Variation of results for 20 kV cable NAEKBA

0,0

0,5

1,0

1,5

2,0

2,5

3,0

25 35 50 70 95 120 150 185 240

Cross-section in mm²

Res

ista

nce

R0

/ Rea

ctan

ce X

0in

/k

mΩ

measured R0measured X0



Measurement of zero sequence impedance Results for 10 kV cable NAKBA 3x240

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

Measuring current in A

Impe

danc

ein

Ωmeasured R0

measured X0



Cable zero sequence impedanceCurrent dependence due to steel reinforcement

r

B

B = µ0 · H

B = µr · µ0 · H

H

µ

H

B

r2IH⋅π

=

HB 0r ⋅μ⋅μ=

∫=

⋅⋅ω

=De

0rLE drB

IX



Calculation examples



Example1 : 10 kV Overhead LineInput Data

Conductors:

phase wire: Al 70

max line sag: 4 m

spec. soil resistivity: 50 Ωm

frequency: 50 Hz



Example 1: 10 kV Overhead LineInfluence of an Earth Wire

Conductors:

earth wire: ACS 70

max line sag: 4 m

Influence on



Positive Sequence Capacitance ??

Zero Sequence Capacitance ??



Example 1: 10 kV Overhead LineInfluence of an Earth Wire - Results

without earth wire with earth wire

_______________________________________________________________________

positive sequence resistance R1 0.414 Ω/km 0.414 Ω/km

positive sequence reactance X1 0.381 Ω/km 0.381 Ω/km

zero sequence resistance R0 0.559 Ω/km 0.631 Ω/km

zero sequence reactance X0 1.566 Ω/km 1.419 Ω/km

positive sequence capacitance C1 9.640 nF/km 9.641 nF/km

zero sequence capacitance C0 4.332 nF/km 4.999 nF/km



Example 2: 110 kV Double CircuitInput Data

Conductors:

phase wire: Al/St 230/30

2er bundle

earth wire: Al/St 95/55

max. sag: 15 m

spec. soil resistivity: 50 Ωm

frequency : 50 Hz



Example 2: 110 kV Double CircuitResult – Impedances in symmetrical components

all other impedances ZXX < 0.020 Ω/km + j 0.020 Ω/km

zero sequence impedance Z00(I,I) = Z00(II,II) = 0.223 Ω/km + j 0.970 Ω/km

positive sequence impedance Z11(I,I) = Z11(II,II) = 0.063 Ω/km + j 0.266 Ω/km

negative sequence impedance Z22(I,I) = Z22(II,II) = 0.063 Ω/km + j 0.266 Ω/km

zero sequence coupling imp. Z00(I,II) = Z00(II,I) = 0.165 Ω/km + j 0.537 Ω/km

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

)II,II(Z)II,II(Z)II,II(Z)I,II(Z)I,II(Z)I,II(Z)II,II(Z)II,II(Z)II,II(Z)I,II(Z)I,II(Z)I,II(Z)II,II(Z)II,II(Z)II,II(Z)I,II(Z)I,II(Z)I,II(Z)II,I(Z)II,I(Z)II,I(Z)I,I(Z)I,I(Z)I,I(Z)II,I(Z)II,I(Z)II,I(Z)I,I(Z)I,I(Z)I,I(Z)II,I(Z)II,I(Z)II,I(Z)I,I(Z)I,I(Z)I,I(Z

222120222120

121110121110

020100020100

222120222120

121110121110

020100020100



Example 2: 110 kV Double CircuitInfluence of zero sequence coupling on s.c. results (I)

0.00 kA 3.42 kA

0.00 kA 3.42 kA

0.00 kA 4.18 kA

0.00 kA 4.18 kA

6.83 kA21.68 kA

8.36 kA21.68 kA

Calculation of single phase short circuit currents

Without consideration of zero sequence coupling impedance

With consideration of zero sequence coupling impedance

wrong



Example 2: 110 kV Double CircuitInfluence of zero sequence coupling on s.c. results (II)

0.00 kA 5.18 kA

4.28 kA 0.00 kA

0.00 kA 5.18 kA

2.94 kA 0.00 kA

5.18 kA21.68 kA

5.18 kA21.68 kA

Calculation of single phase short circuit currents

Without consideration of zero sequence coupling impedance

With consideration of zero sequence coupling impedance

wrong



Example 3: 110 kV CableInput Data

Conductor:

Cross-Section: 630 mm²

Material: Copper

Outer Diameter: 31 mm

Insulation:

Material: XLPE

Metallic sheath:

Cross-Section: 50 mm²

Material: Copper

Inner Diameter: 69 mm

Outer Diameter: 73 mm

Cable Diameter: 84 mm

Arrangement: trefoil

spec. Soil Resistivity: 50 Ωm

Frequency: 50 Hz



Example 3: 110 kV CableResults

ArrangementSheath Bonding

R1‘[Ω/km]

X1‘[Ω/km]

R0‘[Ω/km]

X0‘[Ω/km]

0.368 0.134

0.138

1.673

0.349

0.365

0.179

0.192

r

0.039 0.120 0.196

0.070 0.175 0.212

0.031 0.190 1.000

0.035 0.183 0.189



Example 3: 110 kV CableCross Bonding



Example 4: Secondary Cable parallel to 110 kV CableInput Data

110 kV cable:

see Example 4

Cable sheath earthed at both sides.

secondary cable:

Cross-Section: 2.5 mm²

Material: Copper

Cable Diameter: 10 mmspec. Soil Resistivity: 50 ΩmFrequency: 50 Hz

500

mm



Example 4: Secondary Cable parallel to 110 kV CableInduced Voltage in Case of Single Phase Fault Current

l⋅⋅⋅⋅= 1ksec110Cind IrrZE

Eind - induced longitudinal voltageZc = 0.475 Ω / km coupling impedance between

110 kV and secondary cable(before elimination of earthed conductors)

r110 = 0.196 reduction factor of 110 kV cable sheathrsec = 1.000 reduction factor of secondary cable sheath

(no metallic sheath)Ik1 = 10 kA single phase fault currentl = 1300 m length of parallel section

Eind = 1210 V



Example 4: Secondary Cable parallel to 110 kV CableInduced Voltage in Case of Single Phase Fault Current

l⋅⋅= 1k'Cind IZE

Eind - induced longitudinal voltageZ’c = 0.094 Ω / km coupling impedance between

110 kV and secondary cable(after elimination of earthed conductors)

Ik1 = 10 kA single phase fault currentl = 1300 m length of parallel section

Eind = 1225 V



Example 4: Secondary Cable parallel to 110 kV CableInduced Voltage in Case of Zero Sequence Current

l⋅⋅⋅= 000ind I3ZE

Eind - induced longitudinal voltageZ00 = 0.092 Ω / km coupling impedance between

zero-sequence system of 110 kV cable and zero-sequence system of secondary cable

3 · I0 = 10 kA single phase fault currentl = 1300 m length of parallel section

Eind = 1196 V



Example 4: Secondary Cable parallel to 110 kV CableInduced Voltage in Case of Load Current

l⋅⋅⋅= Load10ind IZ3E

Eind - induced longitudinal voltage

Z10 = 0.0038 Ω / km coupling impedance between

positive-sequence system of 110 kV cable and

zero-sequence system of secondary cable

ILoad = 600 A maximum load current

l = 1300 m length of parallel section

Eind = 9 V



Transformer characteristics



Transformer types

Two winding

Two winding with delta winding

Three winding

Autotransformer



Loading capacity of transformer neutral

YNy, YynFive-limb core:

No capability

Three-limb core:Neutral rating: 25 % of rated current for1.5 hoursNeutral rating: 20 % of rated current for 3 hours

YNy(d), Yyn(d) (Delta-winding of 1/3 of rated capacity)Neutral rating: 100 % of rated current

Dyn, YNdNeutral rating: 100 % of rated current



Neutral treatment at both transformer neutrals



Neutral treatment at both transformer neutralsRecommendation

High zero sequence system voltage at secondary side duringearth fault at high voltage site

impermissible (detailed investigation required)

Small zero sequence system voltage at secondary side duringearth fault at high voltage site

possible

Zero sequence system voltage at secondary side during earthfault at high voltage site

not recommended (detailed investigation required)

Negligible zero sequence system voltage at secondary sideduring earth fault at high voltage site

possible without restrictions

3-Characteristics of Elements

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