Calculations of Capacitance for Transposed Bundled Conductor Transmission Lines

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1 Calculations of Capacitance for Transposed Bundled Conductor Transmission Lines

2 Multi-conductor Lines. An example with a conductor bundle r: conductor radius, d: distance between conductors of the same phase Example: V a,,ii Voltage of conductor # in phase a, in section II We calculate the voltage drop of the st conductor in phase a : +q / +q / h h -q / -q / ' 3 +q / +q / 3 h h -q / -q / ' 3 +q 3 / +q 3 / ' 3 h 3 h 3 -q 3 / -q 3 / V a,, I V q q q ln ln ln πε r d We then calculate the voltage drop of the nd conductor in phase a : a,, I q q ln ln πε r d + +

3 3 Multi-conductor Lines () +q / +q / h h -q / -q / ' 3 +q / +q / 3 h h -q / -q / ' 3 +q 3 / +q 3 / ' 3 h 3 h 3 -q 3 / -q 3 / For phase a in a transposed line with sections I, II and III, we calculate the average voltage for each section and then the average voltage drop for the whole line (all sections, phase a ) V V V V ai, aii, a, III a V V a,, I a,, I V + V + V a,, II a,, II + V a,, III a,, III V + V + V ai, aii, aiii, 3

4 Geometric Mean of Heights Power Engineering -- Egill Bundled Benedikt Hreinsson 4 Conductors We get a combination of factor with logarithms that for instance lead to roots as follows +q i / +q i / Phase # "i" h a h b a b h i 4 h a h b a b -q i / -q i / Image of phase # "i"

5 5 Geometric Mean istances to Images phase # "i" +q i / a +q i / b +q j / +q j / d c phase # "j" Similarly for a combination of distances between phases These lead to roots as follows: ij 4 a b c d ' c ' a ' b Image of phase # "j" ' d -q j / -q j / 4 ij a b c d

6 6 Effective Radius - Bundled Conductors Radius of each conductor r The effective radius of each conductor bundle R. Compare with the GMR d -q j /4 -q j /4 -q d j /3 d d d -q j /3 d -q j / -q j / -q j -q j /4 -q j /4 -q j /3 A phase with 4 conductors A phase with 3 conductors A phase with conductors 4 3 R r d d d R r d R r d A phase with conductor R r

7 7 Summary of capacitance calculations With earth s influence: C r ln πε h R ' Without earth s influence: C r πε ln R ' ' ' ' 3 3 h h h h R r d d d R r d R r d

8 8 Capacitance - Inductance Relation

9 9 Capacitance - Inductance Relation Transposing of lines allows us to form a symmetric circuit model or single phase equivalent which is identical for all phases both regarding reactance and capacitance We remember that the earth is conductive while it is not ferromagnetic Therefore 3-phase transmission lines with equi-distant conductors (located at the corner of a triangle with equal sides) will ensure a symmetric model regarding inductance while the earth will influence its capacitance. This is because the conductor closest to the ground has a different geographical relation than the other conductors - to the earth but not to the other conductors d d d

10 0 Lecture 4 Inductance Matrix Δ Δ Δ Δ u u u u j L I I I n n ω n n n n r r r L ln ln ln ln ln ln ln ln 0 π μ Inductance for a system of parallel conductors without considering the internal inductance

11 Capacitance Matrix - Beta Matrix We can now compare the previous matrices regarding both inductance and capacitance. In both cases these matrices can not exist physically, although mathematically there is no problem. This is because each element in these matrices is a logarithm of a factor which has a dimension of m!! C C n C β C C n nn β C β β n β β n nn

12 The Beta-Matrix Therefore the Beta-matrix shown here is not physically possible since each element is a logarithm of the quantity /length where the length is measured in m!! β ln r ln πε ln n ln ln r ln ln n ln r n n

13 3 Capacitance - Inductance We now consider the product of these matrices shown to the right. The result is that the product of the capacitance matrix and the inductance matrix is constant for a system of thin conductors E L μ/π C πε L C με E E c 0 is the unit matrix 0

14 4 Inductive Reactances (ohm/km) The inductance for a power line lies in the range of ohm/km

15 5 ACSR Table ata Code words GMR Inductance and Capacitance

16 Typical values of overhead Power Engineering - Egill line Benedikt Hreinsson 6 characteristics at 50 Hz Typically for voltages below 60 kv line charging may be ignored. For extra high voltages (400 kv+) line charging must be carefully analyzed

17 Typical values for underground Power Engineering - Egill Benedikt cable Hreinsson 7 characteristics at 50 Hz For underground cables SIL exceeds the thermal rating which means that underground cable connections are always net producers of reactive power

18 8 Capacitances (nf/km)

19 9 Additional Transmission topics Ground wires: Transmission lines are usually protected from lightning strikes with a ground wire. This topmost wire (or wires) helps to attenuate the transient voltages/currents that arise during a lighting strike. The ground wire is typically grounded at each pole. Corona discharge: ue to high electric fields around lines, the air molecules become ionized. This causes a crackling sound and may cause the line to glow!

20 0 Resistance of transmission lines and transmission real losses

21 Factor Influencing Line Resistance R dc ρ A Skin effect (0-5%) Temperature Conductor winding (Spiraling effect ) (0-5%) Because ac current tends to flow towards the surface of a conductor, the resistance of a line at 60 Hz is slightly higher than at dc. Resistivity and hence line resistance increase as conductor temperature increases (changes is about 8% between 5 C and 50 C)

22 Real losses an example Line resistance per length, is given by ρ R where ρ is the resistivity A Resistivity of Copper.68 0 Ω-m Resistivity of Aluminum.65 0 Ω-m Example: What is the resistance in Ω / mile of a " diameter solid aluminum wire (at dc)? ( ) -8 A π r R Ω-m m Ω A mile mile m -8

23 3 Effective Resistance (ohm/km)

24 4 Circuit models for short transmission lines Transmission Capacity

25 One phase equivalent model 5 for a short line R+jX i j C/ C/

26 One phase equivalent model 6 for a short line () Equivalent circuit of a short line.

27 One phase equivalent model 7 for a long line Sending end Receiving end Equivalent circuit for a long transmission line.

28 One phase equivalent model 8 for a long line (3) Equivalent circuit for a long transmission line. Sending end Receiving end Series L draws reactive power Q L wli decreases V along line Line charging C generates reactive power Q C wcv increases V along line

29 Voltage balance along the line 9 Q L << Q C Light load voltage increases along line Q L >> Q C Heavy load voltage decreases V(x) Light V Heavy What happens if: Q L Q C? x

30 30 Transmission capacity definitions Thermal limits: With V being constant, I is the limiting factor (I max ) Steady State Stability Limits: P 3 V Icosφ P 3 V I cosφ max max Natural Loading Surge impedance Loading or (SIL) Z c P V V sinδ X L C S SIL P max V V X V V ( P ) * SIL Zc L C

31 3 Surge Impedance Loading (SIL) SIL is reached, when the generated reactive power equals the consumed power in the high voltage line. SIL is not maximum loading but a characteristic loading consumed L Q X I ωl I Q Q generated generated ωli Z c V V ωc V X c ωc Q ωcv consumed X L ω C C V I L Z c C

32 3 Surge Impedance Loading Surge Impedance: Also called characteristic impedance. this is the impedance with which you can insert a surge the sending end of the line and not get any reflection back at the receiving end. X is the reactance of the line (in Ohm/km or in Ohm) B is the succeptance of the line (in Siemens/km or in Siemens) Z c Sending end i C/ R+jX C/ Surge impedance Receiving end X L X ω ( ohm) C C B j

33 Ractive power ballance of Power a Engineering transmission - Egill Benedikt Hreinsson 33 line For light loading the line produces more Mvars than it consumes For heavy loading the line is a net consumer of reactive power

ECE 325 Electric Energy System Components 5 Transmission Lines. Instructor: Kai Sun Fall 2015

ECE 325 Electric Energy System Components 5 Transmission Lines Instructor: Kai Sun Fall 2015 1 Content (Materials are from Chapter 25) Overview of power lines Equivalent circuit of a line Voltage regulation