Lecture Notes
EE 330 Power System Analysis I
Lecture Notes
EE 330 Power System Analysis I
Chapter 4: Transmission Line Parameters
Lecturer: Dr Ibrahim Rida
Electrical Engineering Department
University of Hail
First Semester (101) – October, 2010
1. Introduction
The purpose of transmission line:
• To transfer energy from generating units to distribution system to
supply load.
• To interconnect neighboring utilities.
Parameters of transmission line:
• Resistance, due to the resistance of the conductor
• Inductance, due to effects of magnetic fields around the conductor
• Capacitance, due to effects of electric fields around the conductor
• Conductance, due to the leakage currents flowing across insulators
and the ionized pathways in the air (usually neglected due to small
compared to the line currents)
Run “lcgui” for the MATLAB program of the transmission lines.
2. Overhead Transmission Lines
Transmission lines consists of conductors, insulators, and shield wires, see
Figure 4.1. The tower is made of steel, wood, or reinforced concrete with its
own right of way. The tower is being built for single circuit, double circuit
or multi-circuit (3 to 10). Underground transmission line (which less than
1% of the total line number in the world) is done due to technical and
economic reasons.
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Lecture Notes
EE 330 Power System Analysis I
Selection of voltage level is based on the amount of power and the length of
transmission line.
Selection of conductor size (which contributed to the fixed charges of the
investment) is based on:
• R I2 losses
• Audible noise
• Radio interference level
In the US, American National Standard Institute (ANSI) set the voltage (line
to line) standard for transmission line (above 60 kV):
• High Voltage (HV): 69 kV, 115 kV, 138 kV, 161 kV, 230 kV
• Extra HV (EHV): 345 kV, 500 kV
• Ultra HV (UHV): 765 kV
Conductor materials for HV transmission line is:
• Aluminum Conductor Steel Reinforced – ACSR
• All Aluminium Conductor – AAC
• All Aluminium Alloy Conductor – AAAC
• Aluminium Conductor Alloy Reinforced - ACAR
The reason for popularity of these types is due to the low cost and high
strength-to-weight ratio compared to copper conductors. Also Aluminium is
widely available compared to Copper. See Table “acsr.m” for most
commonly used ACSR or you may run “acsrgui”.
Conductor is stranded to have flexibility. See Figure 4.2 for ACSR: center
core of steel strands and surrounded by layer of Aluminium. Size of
conductor is expressed in circular mils (cmil), i.e. 1 mil = 0.001 in, therefore
1,000,000 cmil is the area of conductor of 1 in diameter.
EHV uses more than 1 conductor/phase, which is the bundling of conductors
(2, 3, or 4 conductors), which:
• Increase the radius of conductors
• Reduces the electric field strength (reduce corona power loss, audible
noise, radio interference)
• Reduce line reactance
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Lecture Notes
EE 330 Power System Analysis I
3. Line Resistance
For a solid round conductor at a fixed temperature, the dc resistance is given
as
Rdc = ρ ℓ / A
(1)
Where: ρ = conductor resistivity, ℓ = length of the conductor, A = crosssectional area.
Rdc is affected by the frequency, spiraling and the temperature.
When AC current is flowing, the current density is greatest at the surface,
which makes Rac > Rdc. This phenomenon is called the skin effect. At 60 Hz
frequency, Rac = 102% Rdc. Since stranded conductor is spiraled, each strand
is longer than the finished conductor which causes slightly higher Rdc.
With temperature changes, the resistance will change, following:
R2 = R1 [(T + t2)/(T + t1)]
(2)
Where T = temperature constant depending on the material (T for
Aluminium is 228). Manufacturing data of the conductor can be used to
calculate the conductor resistance.
3. Inductance of a single conductor
Magnetic field around the conductor is produced due to the current, circling
around the conductor given by the right-hand rule.
For nonmagnetic material, the inductance L is the ratio of magnetic flux
linkage to the current I, given by:
L = λ/ I
Where λ = flux linkages (Wb-turns).
(3)
Consider a long round conductor with radius r, carrying a current I as shown
in Figure 4.3. The magnetic field intensity around a circle of radius x is
constant and Tangent to the circle is given by Ampere’s law as:
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Lecture Notes
EE 330 Power System Analysis I
Hx =
Ix
2πx
(4)
The inductance of conductor is the sum of contributions from flux linkages
internal and external to the conductor.
Neglecting skin effect and assuming uniform current distribution, it is shown
on page 107 of the text book, that the internal conductance Lint is a constant,
independent of radius r and given as follow:
Lint = μ0 /8π = 0.5 10-7 H/m
(5)
For flux linkage external to the conductor, i.e. when x > r, flux circle
encloses the entire current, Ix = I, See Figure 4.4. The flux density at radius
x is given by:
Bx = μ0 Hx = μ0I / (2 π x)
-7
(6)
μ0 = 4 π 10
Following the discussion on page 108, it is shown that the inductance
between two points, external to a conductor, D1 and D2, is given by:
Lext = 2 × 10− 7 ln
D2
H/m
D1
(7)
4. Inductance of single-phase line
See Figure 4.5 for 1m length of single-phase line consisting of 2 solid round
conductors of radius r1 and r2. D is the distance between them. I1 towards the
page and I2 coming out of the page, where I2 = -I1. These currents set up
magnetic field that links between the conductors.
Total Inductance of conductor 1 due to internal and external fluxes is given
by:
L1 = L1(int) + L1(ext) = 0.5 x 10-7 + 2 x 10-7 ln (
D
) H/m
r1
(10)
Or can be written as equation (4.19) or equation (4.20) for L1 and (4.21) for
L2, respectively.
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Lecture Notes
EE 330 Power System Analysis I
If r1 = r2 = r and L1 = L2 = L, the inductance per conductor per meter length
of the line is given by:
1
r'
L = 2 x10-7 ln ( ) + 2 x 10-7 ln (
D
) H/m
1
(11)
Where r’ = r e- ¼ is the self geometric mean distance (GMR) of a circle of
radius r. GMR can be considered as radius of fictitious conductor assumed to
have no internal flux with the same inductance as the actual conductor with
radius r.
The first component of equation (11) is a function of conductor radius and
the second component is a function of conductor spacing (known as
inductance spacing factor).
GMR is commonly referred to as Geometric Mean Radius and designated as
Ds. Thus, the inductance per conductor in mH/km is given by:
L = 0.2 ln (
D
) mH/km
Ds
(12)
5. Flux linkage in terms of Self and Mutual inductances
Consider one-meter length single-phase two-wire circuit as in Figure 4.6,
characterized by:
L11 and L22 = self inductance
L12 = mutual inductance
The flux linkage λ1 and λ2 is as in equation (4.24). Since I2 = -I1 we have
equation (4.25). Comparing (4.25) with (4.20) and (4.21) we have:
1
r '1
1
L22 = 2 x 10-7 ln
r '2
L11 = 2 x 10-7 ln
L12 = L21 = 2 x 10-7 ln
1
D
For a group of n conductors, we have equations (4.27), (4.28) and (4.29).
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Lecture Notes
EE 330 Power System Analysis I
6. Inductance of three phase transmission line
For symmetrical spacing 3-phase conductor as in Figure 4.7, we have
equation (4.30), (4.31) and (4.32) applies, and since λb = λc = λa, we then
have equation (4.33); that is:
L = 0.2 ln (
D
) mH/km
Ds
That is, inductance per phase for 3-phase circuit with equilateral spacing is
the same as for 1 conductor of a 1-phase circuit.
For unsymmetrical spacing as in Figure 4.8, from equation (4.34), (4.35),
(4.36) and (4.37) we obtain phase inductances (equation 4.38):
La =
Lb =
⎛ 1
1
1 ⎞
⎟
= 2 × 10 − 7 ⎜⎜ ln + a 2 ln
+ a ln
Ia
D12
D13 ⎟⎠
⎝ r'
λa
⎛
1
1
1 ⎞
⎟⎟
= 2 × 10− 7 ⎜⎜ a ln
+ ln + a 2 ln
'
Ib
D
r
D
12
23 ⎠
⎝
λb
⎛
1
1
1⎞
= 2 × 10− 7 ⎜⎜ a 2 ln
+ a ln
+ ln ⎟⎟
Ic
D13
D23
r' ⎠
⎝
o
2
o
Where a = 1∠120 and a = 1∠240 .
Lc =
λc
These equations show that the phase inductances are not equal and they
contain an imaginary term due to the mutual inductance:
For a transposed line as in Figure 4.9, we have equations (4.39), (4.40),
(4.41) and (4.42) apply. The relationship has the same form as the
inductance of 1-phase except that the D is replaced by GMD = 3 D12 D23 D13
(geometric mean distance) as the equivalent conductor spacing. The GMR
(geometric mean radius) for stranded conductors, Ds, is obtained from
manufacturing data. For solid conductors, Ds = r’ = r e- ¼. In modeling
transmission lines, lines are assumed continuously transposed, hence the
transposed line model is mainly used, allowing for a small error. In practice,
long transmission lines are artificially transposed to avoid asymmetry in line
parameters.
7. Inductance of composite conductors
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Lecture Notes
EE 330 Power System Analysis I
In practical transmission lines, stranded and bundled conductors (composite
conductors) are used. Consider Figure 4.10 where Ix flows in conductor x
(towards page) which has n strands and Iy flows in conductor y (out of page)
which has m strands. Lx is given as in equation (4.48):
Lx = 2 × 10− 7 ln
GMD
GMRs
H/m
where GMD and GMRx are as given in equation (4.49) and (4.50)
respectively.
See Example 4.1.
The GMR of bundled conductors as in Figure 4.12 can be calculated using
equation (4.50) also can be found using equations (4.51), (4.52) and (4.53).
8. Inductance of 3-phase double circuit line
A three-phase double-circuit line consists of two identical three-phase
circuits, as shown in Figure 4.13. Voltage drop due to line inductance will
be unbalanced due to geometrical differences between conductors. To
balance it, each phase conductor must be transposed within its group and
with respect to the parallel three-phase line.
The method of GMD can be used to find the inductance per phase. To do
this, use equations (4.54)- (4.57).
The inductance per phase in millihenries per kilometer will then be as in
equation (4.58), that is:
L = 0.2 ln
October, 2010
GDM
mH/km
GMRL
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Lecture Notes
EE 330 Power System Analysis I
9. Line capacitance
The potential differences between conductors of a transmission line create
capacitance across them, depending on: conductor size, spacing, the high
above the ground. By definition the capacitance is given in equation (4.59).
C = q/V
Consider a conductor with radius r carrying charge of q coulombs per meter
length, as shown in Figure 4.14. The induced electric field can be drawn as
radial flux lines. The total electric flux is numerically equal to the value of
charge on the conductor. From Gauss’ law, the electric flux density, D, in 1
m length of the conductor is given in equation (4.60), and the electric field
intensity, E, is given in equation (4.61) and (4.62). The potential difference
between D1 to D2 (voltage drop) is given in equation (4.63), that is:
V12 =
q
2πε 0
ln
D2
D1
10. Capacitance of single-phase lines
Consider one meter length of a single-phase line consisting of two long solid
round conductors, as shown in Figure 4.15. Conductor 1 carries a charge of
q1 coulombs/meter. The presence of the second conductor and ground
disturbs the field of the first conductor. Since D is much larger than r and the
height of conductor is much larger than D, the distortion effect is small.
The analysis given using equation (4.64) to equations (4.69) concludes that
the capacitance for the single-phase line is given by equation (4.70)as:
C=
0.0556
μF/km
D
ln
r
Note: ln D/r is used for C, while ln D/r’ is used for L, where r’ = GMR = Ds
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Lecture Notes
EE 330 Power System Analysis I
11. Potential difference in multiconductor configuration
Considering Figure 4.17, it is shown in equation (4.72) that the potential
difference between conductors due to the presence of conductor charges is
given by:
Vij =
1
2π ε 0
n
∑q
k =1
k
ln
Dkj
Dki
,
Where, ε0 = 8.85 x 10-12 F/m – is permittivity of free space.
When k = i, Dii is the distance between the surface of the conductor and its
center (its radius r)
12. Capacitance of 3-phase lines
Consider one-meter length of a three-phase line with three long conductors,
as shown in Figure 4.18, for a balanced 3-phase system, equation (4.73)
holds:
qa + qb + q c = 0
Neglecting the effect of ground and the shield wires and assuming the line is
transposed, equations (4.74), (4.75) and (4.76) hold for each section.
The average Vab is as in equation (4.77) or (4.78). The GMD is given as in
(4.79), so Vab as in (4.80) is given by:
Vab =
GMD
r ⎞
1 ⎛
+ qb ln
⎜ qa ln
⎟
r
GMD ⎠
2πε 0 ⎝
Similarly Vac is given as in (4.81) and Vab + Vac = 3 Van as in (4.82), (4.83)
and (4.84). From this, the capacitance/phase to neutral can be calculated as
given in equation (4.85), or as given in (4.86). It can be seen that, this is the
same form as given for single-phase capacitance. The GMD (geometric
mean distance) is the equivalent conductor spacing.
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Lecture Notes
EE 330 Power System Analysis I
13. Effect of bundling
Following the same steps as given for the inductance, the capacitance for
bundle-conductors can be found as given in equation (4.87):
C=
2πε 0
GMD
ln b
r
F/m
The effect of bundling is introduced as equivalent radius rb, is given in
(4.88) for two-sub-conductor bundle, for three-sub-conductors in (4.89) and
for four-sub-conductor bundle in (4.90).
14. Capacitance of 3-phase double-circuit lines
Phase positions a1b1c1-c2b2a2 as in Figure 4.13 is the transposed lines. The
effect of the shield wires and ground are negligible.
The per phase equivalent capacitance to neutral is calculated using (4.91), or
per-phase capacitance to neutral in μF/km is calculated using (4.92):
C=
0.0556
GMD
ln
GMRC
μF/km
The GMD is as in (4.55), similar as inductance GMD. The GMRC of each
bundle (group) of conductors is as in (4.93) and (4.94) where rb is the
geometric mean radius of bundled conductors given by (4.88) to (4.90).
15. Effect of earth on the capacitance
The presence of earth will alter the distribution of electric flux lines, which
will change the effective capacitance of the line. The earth level is an
equipotential surface, therefore the flux lines are forced to cut the surface of
the earth orthogonally.
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Lecture Notes
EE 330 Power System Analysis I
Using method of image charges by Kelvin, earth surface can be treated as a
mirror, where –q charge is located –H from surface (for q at height H from
surface), so procedure of Section 4.13 in your text book can be used to
calculate the capacitance.
The effect of earth is to increase the capacitance. However, since H is much
greater than D, the earth effect is negligible, therefore for balanced steady
state analysis, the effect of earth capacitance can be neglected, but for
unbalanced analysis of faults, the earth effect and the shield wires should be
considered. See Example 4.2 and Example 4.3.
Using MATLAB (run “lcgui”), see Example 4.4, Example 4.5 and
Example 4.6.
16. Magnetic field induction
Magnetic fields (due to currents) induce voltage in objects that have length
parallel to the line such as fences, pipelines and telephone wires. It also
affects blood composition, growth, behavior, immune system and neural
functions. See Example 4.7.
17. Electrostatic induction
Electric fields (due to voltage) are the caused of induction to vehicles,
buildings and objects (big size) as a steady state currents or spark discharges.
For human body, the current densities induces by electric fields is larger than
caused by magnetic field.
18. Corona
Corona is the partial ionization of air close to conductor surface due to
potential gradient exceeds the dielectric strength (30 kV/cm) under fair
weather (25 °C, 1 atm). Corona produces power loss, audible hissing sound,
ozone and radio interference (AM band).
Corona is the function of conductor diameter, line configuration, conductor
type, condition of conductor surface, and atmospheric conditions (air
density, humidity, wind influence). During rain or snow the corona losses is
higher.
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