Techref_overheadlineconstants

DIgSILENT PowerFactory Technical Reference Documentation

Overhead Line Constants TypGeo, TypTow

DIgSILENT GmbH Heinrich-Hertz-Str. 9 72810 - Gomaringen Germany T: +49 7072 9168 00 F: +49 7072 9168 88 http://www.digsilent.de [email protected] r994

Copyright ©2011, DIgSILENT GmbH. Copyright of this document belongs to DIgSILENT GmbH. No part of this document may be reproduced, copied, or transmitted in any form, by any means electronic or mechanical, without the prior written permission of DIgSILENT GmbH. Overhead Line Constants (TypGeo, TypTow)

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Contents

Contents 1 Introduction

3

2 Definition in Terms of Geometrical Data

3

3 Conductor Type

4

3.1 Geometrical Mean Radius (GMR) of a Conductor . . . . . . . . . . . . . . . . . .

4

3.1.1 GMR of solid conductors . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3.1.2 GMR of tubular conductors . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3.1.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.2 Bundle Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

4 Calculation of Overhead Line Constants

8

4.1 Series Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.1.1 Internal impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.1.2 Geometrical impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.1.3 Earth correction term . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

4.1.4 Series impedance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

4.2 Shunt Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

4.3 Output Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

5 Definition in Terms of Electrical Data

16

A Parameter Definitions

17

B References

20

List of Figures

21

List of Tables

22

Overhead Line Constants (TypGeo, TypTow)

2

2

1

Definition in Terms of Geometrical Data

Introduction

This document describes the calculation of the electrical parameters of an overhead line system from its configuration characteristics like tower geometry, conductor types, number, phasing and grounding condition of its circuits, etc. The calculation function is available for lines having a tower type (TypTow) or a tower geometry type (TypGeo). The line parameters calculation function, or so-called line constants, supports overhead lines systems with any number of parallel circuits of the same or different nominal voltage, 3-ph, 2-ph and single phase, with or without earth wires and neutral conductors and different types of transpositions. The calculation accounts for the skin effect in the conductors and for the frequency dependency of the earth return path. The calculation function can be used in a stand-alone mode, in which case PowerFactory prints the calculation results (impedance and admittance matrices) to the output window, or it can be automatically called by the line (ElmLne) or line coupling (ElmTow) elements when associated to a tower type (TypTow) or a tower geometry type (TypGeo). In the last case, the parameters calculation function will automatically return the resulting impedance and admittance matrices of the overhead line system to the simulation model. Finally, the tower type (TypTow) does also support the definition of the transmission system in terms of its electrical parameters, so that the user has the option to enter the impedance and admittance matrices either in natural or in sequence components. This is especially useful when the user has to define an unbalanced system (eg. untransposed line) with multiple circuits not supported by the line type (TypLne).

2

Definition in Terms of Geometrical Data

The overhead line system is defined in terms of geometrical data, i.e. the physical dimensions of the towers (geometrical data) and the conductor data. The model consists in the conductor types (TypCon) and the tower geometries (TypTow or TypGeo). The input parameters of the conductor type are listed in Table A.1 and discussed in 3. The geometry of the tower is entered in either a tower type (TypTow) or a tower geometry type (TypGeo). In the tower type (TypTow) the user associates to the geometry of the tower the corresponding conductor types of each circuit and therefore the tower type (TypTower ) contains all data of the overhead line transmission system as required for the calculation of the electrical parameters. The tower geometry type (TypGeo) instead, does not contain a reference to the conductor type, so that the definition is not complete. The conductor types are added later on in the line (ElmLne) or coupling (ElmTow) elements. For that reason the tower geometry type (TypGeo) is more flexible, and should be therefore the preferred option, when combining the same tower geometry with different conductor types. This is quite often the case in distribution systems. The input parameters of the tower type (TypTow) are shown in Table A.3 and those of the tower geometry type (TypGeo) in Table A.2. The calculation of the electrical parameters is discussed in 4.

Overhead Line Constants (TypGeo, TypTow)

3

he input parametes of the tower type (TypTow) are shown in Table 4 and those of the tower geometry type TypGeo) in Table 3. The calculation of the electrical parameters is discussed in 4. 3

Conductor Type

3 Conductor type 3 Conductor Type

he geometrical and characteristics of the conductor are defined in a conductor typetype (TypCon). The Theelectrical geometrical and electrical characteristics of the conductor are defined in a conductor user and has the optionconductors, between solid and and tubular conductors, and betweenand single ser has the option(TypCon). betweenThe solide tubular between single conductors a bundle of subconductors and a bundle of sub-conductors (number of Sub-conductors >1) in which case he onductors (number of Sub-condcutors >1) in which case he further specifies bundle spacing. further specifies bundle spacing.

Figure 3.1: Conductor type dialog

Figure 1: Conductor type dialog.

The parameters of the sub-conductors are used to calculate the internal impedance. These parameters are the DC resistance, the diameter (or radius) and the internal inductance. The he parameters ofinternal the sub-conductors arealso used to calculate the internal impedance. These parameters are the impedance can be defined in terms of relative permeability or geometrical mean GMR(or as explained in 3.1. DC resistance, the radius diameter radius) and the internal inductance. The internal impedance can be also defined

n terms of relativeIfpermeability geometrical radius of GMR as explained in 3.1. the ”Skin effect”orflag is asserted, mean the calculation the internal impedance will account for the skin effect (Bessel functions) as described in section 4.

v e r h e a d L i n e S3.1 y s t eGeometrical ms Mean Radius (GMR) of a Conductor

Page 5 of 22

Geometrical Mean Radius (GMR) is usually provided in manufactures data sheets. This is the data the user is encouraged to give as input for the conductor type definition. If not available, Power Factory can also calculate the GMR at power frequency starting from the conductor diameter (or radius) or the relative permeability; however, under the assumption of an uniform current distribution over its cross section. In other words, the influence of elementary wires is not taken into account [1, 2, 3].

Overhead Line Constants (TypGeo, TypTow)

4

Geometrical Mean Radius (GMR) is usually provided in manufactures´ datasheets. This is the data the user is encouraged to give as input for the conductor type definition. If not available, Power Factory can also calculate the GMR at power frequency starting from the conductor 3 Conductor Type diameter (or radius) or the relative permeability; however, under the assumption of an uniform current distribution over its cross section. In other words, the influence of elementary wires is not taken into account. 3.1.1

GMR of solid conductors

3.1.1 GMR of solid conductors

Consider a solid conductor of nonmagnetic material and radius r as depicted in Figure 3.2. Consider a solid conductor of nonmagnetic material and radius r as depicted in Figure 2.

Figure 3.2: Solid conductor

Figure 2: Solid conductor From theFrom magnetic field theory, autothe inductance of the conductor can be can calculated as in (1), the magnetic fieldthe theory, auto inductance of the conductor be calculated as in (1),   m æ1 1 ö µ00 µm 00 L = + · ln Lself self = 8π + 2π × ln ç · r× r ÷ 8p 2p è2 2 ø

(1)

(1)

where the first term of the sum is the internal inductance associated with the magnetic flux where theinside first term of the sum the the internal inductance associated the magnetic flux inside thetoconductor the conductor (2)isand second one represents the with external inductance associated thesecond externalone fluxrepresents (3). (2) and the the external inductance associated to the external flux (3). Linternal =

µm0

(2)

Lint ernal = 8π0 8p

Lexternal =

Lexternal

µ0 m0 · ln 2π



1 æ2 ·1r

(2)



ö = × ln ç ÷ 2p è 2×r ø

(3)

(3)

In means of the Geometrical Mean Radius (GMR) of the conductor, (1) can be rewritten as (4)

(1) In means of the Geometrical Mean Radius (GMR) of the conductor,   can be rewritten as (4) 1 µ0 · ln Lself = 2π 2 · GM R

(4)

m0 1 æ ö × ln ç ÷ deduced for the GMRof a solid Therefore, between (1) and (4) the following can be 2pexpression è 2 × GMR ø Lself =

(4)

conductor under the above mentioned assumptions:

Therefore, between (1) and (4) the following expression can be1 deduced for the GMR of a solid conductor under GM R = r · e− 4 (5) the above mentioned assumptions:

3.1.2

GMR of tubular conductors

GMR = r × e

-

1 4

(5)

A similar procedure can be used to calculate the GMR of a tubular conductor. A uniform mag-

O v e r h e anetic d L i ndistribution e S y s t e m of s the current over the conductor section is here to be assumed as inP the age 6 of 22

preceding case as well.

Overhead Line Constants (TypGeo, TypTow)

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3.1.2 GMR of tubular conductors Conductor A similar 3procedure canType be used to calculate the GMR of a tubular conductor. A uniform magnetic distribution of the current over the conductor section is here to be assumed as in the preceding case as well.

Figure 3.3: Tubular conductor

Figure 3: Tubular conductor. For the tubular conductor depicted in Figure 3.3, the auto inductance is calculated as in (6):

For the tubular conductor depicted in Figure 3, the auto inductance is calculated as in (6): " #   µ0 q4 r 3q 2 − r2 µ0 1 Lself = · + · ln · ln − 2π é (r2 −4 q 2 )2 q 4 · (r2 2− q 2 )2 ù 2π 2·r

(6)

m q r 3q - r ú m0 æ 1 ö Lself = 0 × ê ln × + × ln ç ÷ 2 2 2 2 p tubular 2p ê ( r 2the q 4 × ( r and rø - qthus è 2 ×conductor: Again, the first term represents ) ú for2the - qinternal ) inductance, ë

û

" # r q4 3q 2 − r2 µ0 Again, the first term represents the internal inductance, and thus for the tubular conductor: · Lint = 2 · ln q − 4 · (r 2 − q 2 ) 2π (r2 − q 2 ) 4 é = ×ê 2p êë r"2 - q 2

(

2

)

2

× ln

q

-

(7)

ù ú 4 × r 2 - q#2 úû

q of a tubularr conductor: 3q - r Between (4) and (6) results (8)mfor 0 the GMR Lint

(6)

(

2

)

q4 r 3q 2 − r2 − · ln GM R = r · exp 2 2 2 4 (r − q ) (r2 − q 2 ) q Between (4) and (6) results (8) for the GMR of a tubular conductor:

(7) (8)

It should be emphasized here again that (5) and (8) assume a uniform distribution of the current é and over the cross section of the conductor wires 3q 2therefore - r 2 the elementary q4 r ù are not taken into GMR = r × exp ê 2 ln × ú account. 2 2

(

êë 4 r - q

) (r

2

- q2

)

q úû

(8)

3.1.3 Numerical example It should be emphasized here again that (5) and (8) assume a uniform distribution of the current over the cross section of the conductor and therefore the elementary wires are not taken into account. For an Al/St Conductor, 120/20 mm2 , Radius 7,75 mm and q/r= 0,226, the following values can be verified in PowerFactory :

3.1.3 Numerical example

• From Eq. (7), Linternal2 =0,045479 mH/km For an Al/St Conductor, 120/20 mm , Radius 7,75 mm and q/r= 0,226, the following values can be verified in Power factory: • From Eq. (6), Lself = 0,878862 mH/km

-

• From Eq. (8), GMR = 6,17369 mm From (7), Linternal=0,045479 mH/km

- From Eq. (6), Lself= 0,878862 mH/km 3.2 Bundle Conductor - From Eq. (8), are GMRfrequently = 6,17369used mm in high voltage transmission lines to reduce corona Bundle conductors losses and fulfill electromagnetic radiation requirements. Two or more subconductors per phase are held together by spacers conforming a symmetrical bundle conductor.

Overhead Line Constants (TypGeo, TypTow) Overhead Line Systems

6 Page 7 of 22

3

Conductor Type

3.2 Bundle conductor Bundle conductors are frequently used in high voltage transmission lines to reduce corona losses and fulfill

For electromagnetic the calculation of electrical parameters, PowerFactory replaces the bundle subconductors radiation requirements. Two or more subconductors per phase are held together by spacers withconforming a conductor of equivalent radius. Subsequent calculations of line parameters (internal a symmetrical bundle conductor. impedance and geometrical impedance due to external flux) are then carried out considering the calculation of electrical parameters, Factory replaces the bundle subconductors with a conductor of thisFor equivalent single conductor as if Power it were located in the middle of the bundle. equivalent radius. Subsequent calculations of line parameters (internal impedance and geometrical impedance to external is flux) are then out considering equivalent single conductor as if it were located in the Thisdue approach based oncarried the following two this assumptions: middle of the bundle.

• The bundle is symmetrical This approach is based on the following two assumptions: • The current distribution among the subconductors within a bundle is uniform § The bundle is symmetrical §

The current distribution among the subconductors within a bundle is uniform

Under these assumptions, the radius of the equivalent conductor will be: Under these assumptions, the radius of the equivalent conductor will be:

√ n rB = n · r · Rn−1 n rB = n × r × R n-1

(9) (9)

where r is the radius of an individual subconductor, n the number of subconductors and R the where r is the radius of an individual subconductor, n the number of subconductors and R the radius of the radius of the bundle (calculated from the bundle spacing a as depicted in Figure 3.4). bundle (calculated from the bundle spacing a as depicted in Figure 4 ).

2r

R a

Figure 3.4: Symmetrical bundle of radius R with n subconductors. Bundle spacing a Figure 4: Symmetrical bundle of radius R with n subconductors. Bundle spacing a

The relationship between the bundle spacing a and angle

a is given by:

The relationship between the bundle spacing a and angle α is given by:

æp ö a = 2 R × sinç  ÷  π a = 2R · sinè n ø n The equivalent GMR results:

The equivalent GMR results:

GMRB = n n × GMR × R n-1 √ n GM RB = n · GM R · Rn−1

where GMR is the geometrical mean radius of individual sub-conductor in bundle.

(10)

(10)

where GMR is the geometrical mean radius of individual sub-conductor in bundle. A different method for calculating line parameters of bundle conductors consists in computing Overhead Line Systems Page 8 of 22 the parameters for each sub-conductor as if it were represented as an individual conductor. Since all sub-conductors within a bundle have the same voltage, the order of the geometrical matrices is then reduce to matrices of equivalent phase conductors. Even though nonsymmetrical bundle conductor can also be considered in this case, a uniform current distribution among sub-conductors within the bundle is still to be assumed. Differences between both methods seem however not to be relevant. PowerFactory supports the first approach described above. For calculating the internal impedance of the bundle, the internal impedance of one sub-conductor must be divided by the number of sub-conductors n. Overhead Line Constants (TypGeo, TypTow)

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4

Calculation of Overhead Line Constants

4

Calculation of Overhead Line Constants

4.1

Series Impedance

The line impedance consists of three components: 0 0 1. The internal impedance ZInt = RInt (ω)+jω·L0 Int (ω), which accounts for the voltage drop due to conductor resistance and the magnetic field inside the conductor itself. Because of skin effect, both the internal reactance and resistance are frequency dependent. 0 2. Geometrical impedance ZG , being the impedance of an ideal conductor without any magnetic field inside and an ideally conducting ground. The geometrical inductance is not frequency dependent. 0 3. Earth correction term ZE = Rt0 (ω) + jω · L0 t (ω), a frequency dependent term considering finite earth conductivity and proximity effects. It depends on the earth resistivity and the line geometry (different coefficients for self and mutual impedances).

4.1.1

Internal impedance

Without considering skin effect If skin effect is not considered (skin effect flag on the conductor type) following formulae are used for the calculation of the internal impedance:

L0Int

=

0 0 ZInt = RInt + jωL0Int

(11)

0 RInt = RDC

(12)

µ0 8π

    

µ0 2π

·

h

q4 (r 2 −q 2 )2

· ln rq −

solide conductor 3q 2 −r 2 4·(r 2 −q 2 )

i

(13) tubular conductor

where RDC is the DC resistance of the conductor in Ω/km and r and q the outside and inside radius of the tubular conductor respectively.

Considering skin effect If skin effect is considered, the internal impedance of the conductor becomes a function of the frequency and it is calculated using the complex Bessel functions:

ZInt

 p  J ς · √−j
1 0 = · RDC · ς · −j · √
2 J1 ς · −j

(14)

where Overhead Line Constants (TypGeo, TypTow)

8

4

Calculation of Overhead Line Constants

r ς =r·

Jn (x) =

∞ X k=0

ω · µr · µ0 =r·m ρ

k  x n+2k (−1) · k! · (n + k)! 2

(15)

(16)

The parameter r m=

ω · µr · µ0 ρ

in (15) is the reciprocal (absolute value) of the complex depth of penetration p. Eq. (15) can also be expressed in terms of the DC resistance of the conductor as follows: r ς=

ω · µr · µ0 RDC · π

(17)

The relative permeability µr accounts for the conductor geometry (µr = 1 for solid conductor and µr ≤ 1 for tubular conductors), with µr 6= 1 if the conductor material is magnetic.

Temperature coefficient If the temperature dependency of line/cables option is enabled in the load flow calculation, the resistivity of the conducting layers is adjusted by the following equation ρT = ρ20◦ C · [1 + α (20 − T )] where α is the temperature coefficient of resistance. The resistivities and temperature coefficient of common metals are given in Table 4.1 for reference. Table 4.1: Resistivities and temperature coefficient of resistance Material Aluminum Copper, hard drawn Copper, annealed Brass Iron Silver Steel

4.1.2

Resistivity at 20 ◦ C [µΩ.cm] 2.83 1.77 1.72 6.4-8.4 10 1.59 12-88

Temperature coefficient at 20◦ C [1/◦ C] 0.0039 0.00382 0.00393 0.0020 0.0050 0.0038 0.001-0.005

Geometrical impedance 0 ZGii = jω

µ0 Nii 2π

(18)

0 ZGik = jω

µ0 Nik 2π

(19)

The frequency independent coefficients N can be calculated as follows: Overhead Line Constants (TypGeo, TypTow)

9

Z 'Gik = jw

m0 N ik 2p

(19)

4 The Calculation Overheadcoefficients Line Constants frequencyof independent N can be calculated as follows:

N ii = ln

2hi Nii = ln ri

N ik d=ikln

Nik = ln with

h,d

and

r

d0ik

2hi ri

(20)

d ik d 'ik

(21)

(20)

(21)

as defined in Figure 5.

ij r asi defined in Figure 4.1. with hi , diij and i

ri i

rk

dik

k hi hk

d

' ik

k i

'

'

Figure 4.1: Calculation of the geometrical coefficients. Conductor profiles between towers. Figure 5: Calculation of the geometrical coefficients. Conductor profiles between towers.

In case of bundle conductors, the radius ri in (15) is to be replaced by the equivalent radius as calculated (9). conductors, the radius ri in (15) is to be replaced by the equivalent radius as calculated in (9). In case ofinbundle hi is the average height above ground of conductor i. If the conductor profile can be described hi parabola is the average height aboveaccurate ground of for conductor If the conductor profilethen can be a parabola as a (which is quite spans i.below 500 meters), thedescribed averageasheight above ground is: (which is quite accurate for spans below 500 meters), then the average height above ground is: 2 1 hi = haverage = · hmidspan + · htower 3 3 2 1 hi =athaverage = and × hmidspan htower as shown in (6). where hmidspan is the conductor height midspan htower+ at ×tower

3

4.1.3

3

Earth correction term

Overhead Line Systems

Page 12 of 22

The earth correction term and hence the impedance of the earth return path is calculated in PowerFactory according to the Carson’ series given by: 0 ZE =ω

µ0 (P + jQ) π

(22)

The coefficients P and Q are highly frequency dependent and are calculated as follows:

Overhead Line Constants (TypGeo, TypTow)

10

4

Calculation of Overhead Line Constants

π P = − 8   − b1 x cos ϑ + b2 (c2 − ln x) x2 cos 2ϑ + x2 ϑ sin 2ϑ + b3 x3 cos 3ϑ − d4 x4 cos 4ϑ   − b5 x5 cos 5ϑ + b6 (c6 − ln x) x6 cos 6ϑ + x6 ϑ sin 6ϑ + b7 x7 cos 7ϑ − d8 x8 cos 8ϑ . . .

(23)

1 Q = (k − ln x) x6 + 2   + b1 x cos ϑ − d2 x2 cos 2ϑ + b3 x3 cos 3ϑ − b4 (c4 − ln x) x4 cos 4ϑ + x4 ϑ sin 4ϑ   + b5 x5 cos 5ϑ − d6 x6 cos 6ϑ + b7 x7 cos 7ϑ − b8 (c8 − ln x) x8 cos 8ϑ + x8 ϑ sin 8ϑ . . .

(24)

where: √ x = 2hi µ · κ · ω and ϑ = 0 for the self impedance √ x = dij µ · κ · ω for the mutual impedance bi = |bi−2 |

sign i · (i + 2)

sign = +1 for I = 1,2,3,4. . . . sign = +1 for I = 5,6,7,8 . . . and alternating after 4 terms ci = ci−2 + di =

1 1 + i i+2

π · bi 4

1 + ln 2 − C = 0, 61593 2 ! n X 1 C = lim − ln n = 0, 577215 n→∞ ν ν=1 k=

Following initial values are being used: √ b1 = c2 =

2 1 b2 = 6 16

5 − C + ln 2 = 1, 3659315 4

Overhead Line Constants (TypGeo, TypTow)

11

b1 =

1 2 b2 = 16 6

c2 =

5 4 Calculation of Overhead Line Constants - C + ln 2 = 1,3659315 4 4.1.4

Series impedance matrix

4.1.4 Series impedance matrix

4.2:(left) Definition the self(right) (left) and mutual (right) impedance of a line. Figure 6: Definition ofFigure the self and of mutual impedance of a line.

According to Figure 6 the self and mutual impedances can hence be expressed as follows: According to Figure 4.2 the self and mutual impedances can hence be expressed as follows:

Z 'ii = Z 'Gii + Z ' Eii + Z ' Lii

Z 'ij = Z 'Gij + Z ' Eij

0 0 0 Zii0 = ZGii + ZEii + ZLii

0 Zij

=

0 ZGij

+

0 ZEij

(25) (25) (26) (26)

'ij defines mutual impedance, it still has a real component due to It should be noticed thatbe even though 0 It should noticed thatwhen evenZthough whena Z ij defines a mutual impedance, it still has a real component to the resistance of the earth return path. the resistance of the earth due return path. For an overhead line with multiple conductors the voltage drop along the line can be written For an overhead with conductors thewhich voltage dropthe along the line can be written in terms oftoan in line terms of multiple an impedance matrix, relates voltage drop along every conductor the currents across them: impedance matrix, which relates the voltage drop along every conductor to the currents across them: ∆U = Z × I

DU = ZThe ´ I dimension of the matrix depends on the total number of conductors in the line and therefore corresponds to the number of phase conductors plus the number of earth wires. Such a matrix is called the ”natural” impedance matrix of the line. The dimension of the matrix depends on the total number of conductors in the line and therefore corresponds to   wires. Such  a matrix   is called the “natural” impedance the number of phase conductors plus the number ofearth ∆UE ZEE ZEP I = × E matrix of the line. ∆UP ZP E ZP P IP Overhead Line Systems

Page 14 of 22

For using the impedance matrix in a three phase model, it must be reduced to the number of phase conductors. Therefore, an ideal grounding is considered leading to the assumption that there is no voltage drop across the earth conductors. The reduced matrix is hence:   −1 ∆UP = ZP P − ZP E ZEE ZEP IP = Zred IP If a detailed modelling of earth wires is required, earth wires must be entered as phase conductors.

Overhead Line Constants (TypGeo, TypTow)

12

4

Calculation of Overhead Line Constants

The symmetrical impedance matrix is calculated as follows: Z012 = S · Zred · T 

1 where T =  1 1

1 a2 a

 1 ◦ a , S = T −1 and a = ej120 2 a

Only for perfectly transposed lines the resulting symmetrical matrix is diagonal, with all elements outside the main diagonal equal to zero, and hence there is no coupling between sequence modes. In this case, the resulting matrix looks like:   Zs + 2Zm 0 0  0 Zs − Zm 0 Z012 =  0 0 Zs − Zm where Zs is the self impedance and Zm the mutual impedance of the perfectly transposed line.

4.2

Shunt Capacitance

The natural potential coefficient matrix P relates the voltage to the charge of each conductor. The dimension of this ”natural” matrix corresponds to the number of phase conductors + the number of earth wires. U =P ×Q Analogous to the impedance matrix, also the matrix of potential coefficients is reduced to the number of phase conductors, under the assumption that the voltage at the earth wires is zero. It follows:       UE PEE PEP QE = × UP PP E PP P QP UE = 0   −1 UP = PP P − PP E PEE PEP · QP and hence the reduced coefficient matrix is   −1 Pred = PP P − PP E PEE PEP The matrix of capacitance coefficients can be obtained now by inverting the potential coefficient matrix: −1 Cred = Pred Using the same transformation matrix T and S as for the impedance, we can now calculate the symmetrical admittance matrix as follows: C012 = S · CRST · T As in the impedance case, the resulting symmetrical matrix Z012 is diagonal only for perfectly transposed lines, with all elements outside the main diagonal equal to zero. Hence there is no coupling between sequence modes. In this case, the resulting admittance matrix looks like: 

C012

Cs + 2 · Cm 0 = 0

0 Cs − Cm 0

 0  0 Cs − Cm

where Cs is the self capacitance and Cm the coupling capacitance of the perfectly transposed line. Overhead Line Constants (TypGeo, TypTow)

13

4

Calculation of Overhead Line Constants

4.3

Output Results

The line constants calculation function in stand-alone mode can be started from the Calculate button on the edit dialog of the tower type TypTow. Then PowerFactory prints the resulting impedance and admittance matrices to the output windows. The natural impedance matrix corresponds to the system of physical conductors including the earth wires. The size of the natural impedance matrix results: Size (Zn) = NEW +

Nc X

(Nph )j

j=1

where NEW is the total number of earth wires, Nc the line circuits and Nph the number of phases of the corresponding line circuit. Rows and columns of the natural impedance matrix proceed in the sequence ground wire 1, 2, . . . NEW followed by phases A, B C for line circuits 1, 2 . . . .Nc . The reduced impedance matrix represents the system of equivalent phase conductors after reduction of the earth wires. Rows and columns proceed in the same sequence as before but without the earth wires. The symmetrical components matrix proceed in the sequence 0, 1, 2 for line circuits 1, 2, . . . Nc . The size of the reduced impedance matrix ZABC is equal to the size of the symmetrical components matrix Z012 : Size (ZABC ) = Size (Z012 ) =

Nc X

(Nph )j

j=1

It follows for reference an extract of the output window for a 132 kV, 2 x 3-phase circuits, 1 x earth wire, circuit-wise transposed. First matrix corresponds to the natural impedance matrix (7 x 7), the second and third one to the reduced impedance matrix in natural components and symmetrical components respectively.

Overhead Line Constants (TypGeo, TypTow)

14

Size ( Z ABC ) = Size ( Z 012 ) =

Nc

å( N ) ph

j =1

4

j

It follows for an extract of the output window for a 132 kV, 2 x 3-phase circuits, 1 x earth wire, circuitCalculation ofreference Overhead Line Constants wise transposed. First matrix corresponds to the natural impedance matrix (7 x 7), the second and third one to the reduced impedance matrix in natural components and symmetrical components respectively.

DIgSI/info - Natural Impedance Matrix (R+jX) [ohm/km] DIgSI/info - Earth conductors first, followed by phase conductors in same order as the input. DIgSI/info - Rows follow R,X, R,X... in [ohm/km]

Phase conductors, in same order as the input 1.93180e-001

4.48600e-002

4.52553e-002

4.55627e-002

4.48600e-002

4.52553e-002

4.55627e-002

6.87017e-001

2.64516e-001

2.32909e-001

2.19884e-001

2.64516e-001

2.32909e-001

2.19884e-001

4.48600e-002

1.13864e-001

4.58758e-002

4.61923e-002

4.54536e-002

4.58589e-002

4.61788e-002

2.64516e-001

6.63646e-001

2.91102e-001

2.60068e-001

2.70790e-001

2.47761e-001

2.40699e-001

4.52553e-002

4.58758e-002

1.14699e-001

4.66221e-002

4.58589e-002

4.62714e-002

4.66000e-002

2.32909e-001

2.91102e-001

6.62649e-001

3.08999e-001

2.47761e-001

2.42940e-001

2.46523e-001

4.55627e-002

4.61923e-002

4.66221e-002

1.15352e-001

4.61788e-002

4.66000e-002

4.69345e-002

2.19884e-001

2.60068e-001

3.08999e-001

6.61891e-001

2.40699e-001

2.46523e-001

2.59351e-001

4.48600e-002

4.54536e-002

4.58589e-002

4.61788e-002

1.13864e-001

4.58758e-002

4.61923e-002

2.64516e-001

2.70790e-001

2.47761e-001

2.40699e-001

6.63646e-001

2.91102e-001

2.60068e-001

4.52553e-002

4.58589e-002

4.62714e-002

4.66000e-002

4.58758e-002

1.14699e-001

4.66221e-002

2.32909e-001

2.47761e-001

2.42940e-001

2.46523e-001

2.91102e-001

6.62649e-001

3.08999e-001

4.55627e-002

4.61788e-002

4.66000e-002

4.69345e-002

4.61923e-002

4.66221e-002

1.15352e-001

2.19884e-001

2.40699e-001

2.46523e-001

2.59351e-001

2.60068e-001

3.08999e-001

6.61891e-001

Earth wire

Circuit 2

Circuit 1

DIgSI/info - Reduced Impedance Matrix (R+jX) [ohm/km] DIgSI/info - Circuits (phases A,B,C...) follow in same order as the input. DIgSI/info - Rows follow R,X, R,X... in [ohm/km]

1.06521e-001

3.78915e-002

3.78915e-002

3.81026e-002

3.78741e-002

3.78741e-002

5.79697e-001

2.04396e-001

2.04396e-001

1.74662e-001

1.62667e-001

1.62667e-001

3.78915e-002

1.06521e-001

3.78915e-002

3.78741e-002

3.81026e-002

3.78741e-002

2.04396e-001

5.79697e-001

2.04396e-001

1.62667e-001

1.74662e-001

1.62667e-001

3.78915e-002

3.78915e-002

1.06521e-001

3.78741e-002

3.78741e-002

3.81026e-002

2.04396e-001 2.04396e-001 Overhead Line Systems

5.79697e-001

1.62667e-001

1.62667e-001

1.74662e-001 Page 17 of 22

3.81026e-002

3.78741e-002

3.78741e-002

1.06521e-001

3.78915e-002

3.78915e-002

1.74662e-001

1.62667e-001

1.62667e-001

5.79697e-001

2.04396e-001

2.04396e-001

3.78741e-002

3.81026e-002

3.78741e-002

3.78915e-002

1.06521e-001

3.78915e-002

1.62667e-001

1.74662e-001

1.62667e-001

2.04396e-001

5.79697e-001

2.04396e-001

3.78741e-002

3.78741e-002

3.81026e-002

3.78915e-002

3.78915e-002

1.06521e-001

1.62667e-001

1.62667e-001

1.74662e-001

2.04396e-001

2.04396e-001

5.79697e-001

DIgSI/info - Symmetrical Impedance Matrix (R+jX) [ohm/km] DIgSI/info - Circuits (seq. 0,1,2...) follow in same order as the input. DIgSI/info - Rows follow R,X, R,X... in [ohm/km]

Overhead Line Constants (TypGeo, TypTow)

15

1.82304e-001

1.38778e-017

2.77556e-017

1.13851e-001

1.86483e-017

0.00000e+000

9.88490e-001

0.00000e+000

0.00000e+000

4.99996e-001

4.33681e-018

9.54098e-018

5

1.74662e-001

1.62667e-001

1.62667e-001

5.79697e-001

2.04396e-001

2.04396e-001

3.78741e-002

3.81026e-002

3.78741e-002

3.78915e-002

1.06521e-001

3.78915e-002

1.62667e-001

1.74662e-001

1.62667e-001

2.04396e-001

5.79697e-001

2.04396e-001

3.78741e-002 3.81026e-002 Definition in Terms3.78741e-002 of Electrical Data

3.78915e-002

3.78915e-002

1.06521e-001

2.04396e-001

2.04396e-001

5.79697e-001

1.62667e-001

1.62667e-001

1.74662e-001

DIgSI/info - Symmetrical Impedance Matrix (R+jX) [ohm/km] DIgSI/info - Circuits (seq. 0,1,2...) follow in same order as the input. DIgSI/info - Rows follow R,X, R,X... in [ohm/km]

1.82304e-001

1.38778e-017

2.77556e-017

1.13851e-001

1.86483e-017

0.00000e+000

9.88490e-001

0.00000e+000

0.00000e+000

4.99996e-001

4.33681e-018

9.54098e-018

5.55112e-017

6.86296e-002

0.00000e+000

0.00000e+000

2.28545e-004

-1.82146e-017

0.00000e+000

3.75301e-001

5.55112e-017

0.00000e+000

1.19951e-002

5.63785e-018

5.55112e-017

0.00000e+000

6.86296e-002

0.00000e+000

1.60462e-017

2.28545e-004

0.00000e+000

2.77556e-017

3.75301e-001

0.00000e+000

4.33681e-019

1.19951e-002

1.13851e-001

1.86483e-017

0.00000e+000

1.82304e-001

1.38778e-017

2.77556e-017

4.99996e-001

4.33681e-018

9.54098e-018

9.88490e-001

0.00000e+000

0.00000e+000

0.00000e+000

2.28545e-004

-1.82146e-017

5.55112e-017

6.86296e-002

0.00000e+000

0.00000e+000

1.19951e-002

5.63785e-018

0.00000e+000

3.75301e-001

5.55112e-017

0.00000e+000

1.60462e-017

2.28545e-004

5.55112e-017

0.00000e+000

6.86296e-002

0.00000e+000

4.33681e-019

1.19951e-002

0.00000e+000

2.77556e-017

3.75301e-001

The output admittance matrices follow the same structure.

The output admittance matrices follow the same structure.

5

Definition in Terms of Electrical Data

The overhead line system can be alternatively defined in terms of electrical data. In that case, only the tower type (TypTow) can be used. In the tower type set the input parameter i mode to 1 (electrical parameters, see Table A.3). On the load flow page you are able now to enter the impedance and admittance matrices in ohm/km. Note that you can choose between phase/symmetrical components, reactance/inductance and O v e r h e a d L i n e S y s tby e mclicking s susceptance/capacitance . Use the arrows swap between the impedance P a g e 1 8 o f 2 2 and admittance matrices. On the basic data page, note that you still need to specify the number of circuits and the number of phases per circuit as they define the size of the Z/Y matrix. Besides you will be prompted to select a conductor type for each circuit as it defines the nominal voltage of the circuit (see parameter uline in TypCon, Table A.1). It should also be noted that in case of electrical parameters the user just enters the reduced matrices, i.e. after elimination of the earth wires. In case that only the non-reduced natural matrices were available, the user shall define single-phase circuits for the earth wires and then ground these circuits externally in the network model.

Overhead Line Constants (TypGeo, TypTow)

16

A

A

Parameter Definitions

Parameter Definitions Table A.1: Input parameters of the conductor type (TypCon) Name loc name uline sline ncsub

Description Name Nominal Voltage Nominal Current Number of Subconductors

Unit

Range

Default Symbol

kV kA

x>0 x>0 x>0 and x<100

6 1 1

dsubc iModel

Bundle Spacing Conductor model: iModel = 0 : solide conductor iModel = 1 : tubular conductor; in this case inner diameter is compulsitory (Sub-)Conductor: DCResistance (20◦ C) (Sub-)Conductor: GMR (Equivalent Radius) (Sub-)Conductor: Internal Inductance (Sub-)Conductor: Relative Permeability (Sub-)Conductor: Outer Diameter (Sub-)Conductor: Outer Radius (Sub-)Conductor: Inner Diameter (Sub-)Conductor: Inner Radius Consider skin effect (iskin=1) Neglect skin effect (iskin=0) (Sub-)Conductor: Max. operational temperature (Sub-)Conductor: DCResistance at max. operational temperature (Sub-)Conductor: Temperature Coefficient (Sub-)Conductor: Conductor Material (Sub-)Conductor: Max. End Temperature (Sub-)Conductor: Rated ShortTime (1s) Current

m

rpha erpha Lint my r diaco radco diatub radtub iskin tmax rpha tmax

alpha mlei rtemp Ithr

Overhead Line Constants (TypGeo, TypTow)

x=0 or x=1

0.1 0

Ohm/km

x>0

0.05

mm

x>0

11.682

mH/km

x>0

0.05

x>0

1

mm

x>0

30

mm mm

x>0 x>0

15 16

mm

x>0 x=0 or x=1 x>=0

8 1

Ohm/km

x>=0

0.05

1/K

x>=0

0

◦

C

80

Al ◦

C

x>0

80

kA

x>=0

0

17

A

Parameter Definitions

Table A.2: Input parameters of the tower geometry type (TypGeo) Name loc name nlear nlcir xy e xy c

Description Name Number of Earth Wires Number of Line Circuits Coordinates Earth Wires Coordinates Phase Circuits

Unit

Range

Default

x>=0 x>0

1 1 0 3

m m

Symbol

Table A.3: Input parameters of the tower type (TypTow) Name loc name frnom nlear nlcir iTransMode

Description Name Nominal Frequency Number of Earth Wires Number of Line Circuits Transposition

i mode

Input Mode: =0 : geometrical parameters. Coordinates of earth and phase conductors required (parameters “xy e” and “xy c”) = 1 : electrical parameters: R, X, G and B matrices required (parameters R c, X c, G c and B c or their equivalent matrices in sequence components) Earth Conductivity Earth Resistivity Conductor types of earth wires Conductor types of line circuits Num. of Phases per line circuit Assert this flag to enable transposition of the corresponding circuit in case if iTransMode=circuit-wise. For iTransMode other than circuitwise this option is read only. Coordinate of Earth Conductors Coordinate of Line Circuits : Matrix of Resistances R ij (natural components) : Matrix of Reactances X ij (natural components) : Matrix of Inductances L ij (natural components)

gearth rearth pcond e pcond c nphas cktrto

xy e xy c Rc Xc Lc

Overhead Line Constants (TypGeo, TypTow)

Unit

Range

Hz x>=0 x>=1 None Circuit-wise Symmetrical perfect x=0 or x=1

uS/cm Ohmm TypCon

x>0 x>0

Default Symbol 50 1 1 none

0

100 100

TypCon 3 0

m

0

m Ohm/km

0 0

Ohm/km

0

H/km

0

18

A

Parameter Definitions

R c0 R c1 X c0 X c1 L c0 L c1 Gc Bc Cc G c0 G c1 B c0 B c1 C c0 C c1 pStoch

: Matrix of 0-SequenceResistances R ij 0 : Matrix of 1-SequenceResistances R ij 1 : Matrix of 0-SequenceReactances X ij 0 : Matrix of 1-SequenceReactances X ij 1 : Matrix of 0-SequenceInductances L ij 0 : Matrix of 1-SequenceInductances L ij 1 : Matrix of Conductances G ij (natural components) : Matrix of Susceptances B ij (natural components) : Matrix of Capacitances C ij (natural components) : Matrix of 0-SequenceConductances G ij 0 : Matrix of 1-SequenceConductances G ij 1 : Matrix of 0-SequenceSusceptances B ij 0 : Matrix of 1-SequenceSusceptances B ij 1 : Matrix of 0-SequenceCapacitances C ij 0 : Matrix of 1-SequenceCapacitances C ij 1 Stochastic model (StoTyplne)

Overhead Line Constants (TypGeo, TypTow)

Ohm/km

0

Ohm/km

0

Ohm/km

0

Ohm/km

0

H/km

0

H/km

0

uS/km

0

uS/km

0

uF/km

0

uS/km

0

uS/km

0

uS/km

0

uS/km

0

uF/km

0

uF/km

0

19

B

B

References

References

[1] B. Oswald D. Oeding. Elektrische Kraftwerke und Netze. Springer Verlag, 6 edition, 2004. [2] H. Dommel. EMTP Theory Book. Microtran Power System Analysis Corporation, 1 edition, 1996. [3] B. Oswald. Netzberechnung 2: Berechnung transienter Elektroenergieversorgungs-netzen. VDE-Verlag, 1 edition, 1996.

Overhead Line Constants (TypGeo, TypTow)

Vorgnge

in

20

List of Figures

List of Figures 3.1 Conductor type dialog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

3.2 Solid conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3.3 Tubular conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.4 Symmetrical bundle of radius R with n subconductors. Bundle spacing a . . . . .

7

4.1 Calculation of the geometrical coefficients. Conductor profiles between towers. .

10

4.2 Definition of the self (left) and mutual (right) impedance of a line. . . . . . . . . .

12

Overhead Line Constants (TypGeo, TypTow)

21

List of Tables

List of Tables 4.1 Resistivities and temperature coefficient of resistance . . . . . . . . . . . . . . .

9

A.1 Input parameters of the conductor type (TypCon) . . . . . . . . . . . . . . . . . .

17

A.2 Input parameters of the tower geometry type (TypGeo) . . . . . . . . . . . . . . .

18

A.3 Input parameters of the tower type (TypTow) . . . . . . . . . . . . . . . . . . . . .

18

Overhead Line Constants (TypGeo, TypTow)

22

List of Tables

Overhead Line Constants (TypGeo, TypTow)

23