3𝑟𝑑 Year Electric Power
REPORT
Transmission Lines
‫ أحمد محمد حسن عالم‬/ ‫االسم‬
Sec./ 2
B.N/ 29
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List of contents
• Introduction
• Transmission Line Parameters
I. Resistance
II. Inductance
III. capacitance
• Performance of Transmission Lines
• Conclusion
• References
List of figures
I.
II.
Skin effect
Two Parallel Wires with Single–Phase Circuit
III.
Single-Phase Line double circuit
IV.
Unbalanced Three–Phase Line
V.
VI.
VII.
VIII.
IX.
Single Phase Two-wire Line
conductors and a neutral point between them
Phase Overhead Line with equal spacing
Phase Overhead Line with unequal spacing
Single Phase Short Transmission Lines
X.
Medium Transmission Lines Nominal π Method
XI.
Medium Transmission Lines Nominal T Method
XII.
Long Transmission Lines
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Introduction
Transmission lines is a very important part of electrical system as it is used to carry
large amount of generated power at very high voltage from one station to another.
we are going to study the transmission line parameters and its performance.
1. Transmission Line Parameters
Transmission lines have the electrical properties resistance, inductance and
capacitance.
1. Series resistance: due to resistivity of conductors material.
2. Series inductance: due to magnetic field produced by ac current.
3. Shunt capacitance: due to electric field from potential difference between
conductors.
1.1. Resistance of overhead transmission line
DC resistance is calculated by
Where:
ρ is conductor resistivity at a given temperature (Ω.m)
L is conductor length (m)
A is cross-section area (m2)
If AC current is flowing, rather than DC current, the conductor effective resistance
is higher due to skin effect.
Skin Effect
• The alternating current concentrating near the surface of a conductor (see
figure 1) is known as skin effect.
• The effective area of cross-section of the conductor is reduced.
• The resistance of the conductor is increased when carrying an ac current.
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Figure 1 Skin effect
Temperature Effect
As temperature rises, the conductor resistance increases linearly, over normal
operating temperatures, according to the following equation:
1.2. Inductance of overhead transmission line
When an alternating current flows in lines, a changing magnetic flux is produced.
The changing flux links the lines, and the lines possesses inductance.
It depends on
• Size of conductor (r).
• Distance between conductors (D).
• Material of conductors (𝜀)
1. Inductance of Two Parallel Wires with Single–Phase Circuit
𝐷𝑚
𝐿 = 2 ∗ 10−7 𝐿𝑛( )
𝐷𝑠
Dm= D
DS=re-0.25
Figure 2 Two Parallel Wires with Single–Phase Circuit
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2. Inductance of Single-Phase Line double circuit
𝐿 = 2 ∗ 10−7 𝐿𝑛(
Where:
𝐷𝑚 = 2∗2√𝐷
𝐷𝑚
𝐷𝑠
)
𝑎𝑥 𝐷𝑎𝑦 𝐷 𝑏𝑥 𝐷𝑏𝑦
2
𝐷𝑠 = 2 √𝑟𝑎 𝐷𝑎𝑏 𝑟𝑏 𝐷𝑏𝑎
Figure 3 Single-Phase Line double circuit
3. Inductance of a Three–Phase Line
a) Balanced Three–Phase Line
𝐷
𝐿 = 2 ∗ 10−7 𝐿𝑛( )
𝐷𝑠
DS=re-0.25
Figure 4 Balanced Three–Phase
Line
b) Unbalanced Three–Phase Line
𝐷𝑚
𝐿 = 2 ∗ 10−7 𝐿𝑛( )
𝐷𝑠
𝐷𝑚 = 3√𝐷𝐴𝐵 𝐷𝐵𝐶 𝐷𝐶𝐴
DS=re-0.25
Figure 5 Unbalanced Three–Phase Line
1.3. Capacitance of Overhead transmission lines
If we have two conductors separated from each other by an insulator and have a
potential difference, there will be capacitance between them. In transmission lines
conductors there is potential difference between any two conductors and they are
separated by air so they will exhibit capacitance effect.
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We need to calculate the capacitance to consider it while design of transmission
lines.
A. Capacitance of a Single Phase Two-wire Line
1. capacitance between two wires
A
is equal to
B
d
Figure 6 Single Phase Two-wire Line
2. capacitance between one of the conductors and a neutral point between them
is equal to
A
N
B
Figure 7 conductors and a neutral point between them
B. Capacitance of a 3-Phase Overhead Line
1. Equal spacing
Capacitance between conductor and neutral is equal
to
equal to
𝐶𝑝ℎ =
2𝜋𝜀0
𝐷
𝑟
ln( )
2. Unequal spacing
Capacitance between conductor and neutral is
equal to
𝐶𝑝ℎ =
Figure 8 3-Phase Overhead Line
with equal spacing
2𝜋𝜀0
𝐷
ln( 𝑚 )
𝐷𝑆
where
𝐷𝑚 = 3√𝐷𝐴𝐵 𝐷𝐵𝐶 𝐷𝐶𝐴
𝐷𝑆 = 𝑟
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Figure 9 3-Phase Overhead Line with
unequal spacing
2. Performance of Transmission Lines
To study the Performance of Transmission Lines we will study some terms such voltage drop,
line losses and efficiency of transmission which will indicate how successfully is the design and
operation of a transmission line. So first we will represent the transmission line and study it
mathematically.
Classification of Overhead Transmission Lines
1. Short transmission lines:
• The length of an overhead transmission line is up to about 80 km.
• The capacitance effects are small and is neglected.
Figure 10 Single Phase Short Transmission Lines
2. Medium transmission lines:
• The length of an overhead transmission line is between 80 km and 240 km.
• The capacitance effects are considered.
Figure 12 Medium Transmission Lines Nominal π
Method
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Figure 11Medium Transmission Lines Nominal T
Method
3. Long transmission lines:
• The length of an overhead transmission line is more than 240 km.
• The line constants are considered uniformly distributed over the whole
length of the line.
Figure 13 Long Transmission Lines
,For mathematical analysis we will use Two Port Network
Where:
• Vr, Ir are the receiving end voltage, current.
• Vs, Is are the sending end voltage, current.
• To calculate values of A, B, C and D for all cases:
Table 1
Long
A
cosh 𝜃
B
Zsinh 𝜃
𝜃
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Medium π
Medium T
Short
1
Z
Z
C
Ysinh 𝜃
𝜃
D
cosh 𝜃
•
•
•
•
•
Y
0 (Because y=0 and
Is=Ir )
1
R: Total resistance of the line
X: Inductive reactance of the line
Z: Total line impedance
B: Capacitive susceptance
1
Y: Admittance Y= = 𝐽𝐵
𝑋𝐶
• 𝜃 = √𝑍𝑌
Now we will define voltage regulation, line losses and efficiency of transmission.
1. voltage regulation
The difference in voltage at the receiving end of a transmission line and the
sending end and is expressed as a percentage of the receiving end voltage.
2. Line losses
𝑆𝐿𝑂𝑆𝑆𝐸𝑆 = 𝑆𝑆𝐸𝑁𝐷𝐼𝑁𝐺 − 𝑆𝑅𝐸𝐶𝐸𝐼𝑉𝐼𝑁𝐺
3. Transmission efficiency
The ratio of receiving end power to the sending end power of a transmission line
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Conclusion
As transmission lines is a very important part of electrical system we have
illustrated
1. The transmission line parameters (their meanings and how to calculate them):
• Resistance
• Inductance
• Capacitance
2. The transmission line performance by viewing short, medium and long
representation of it and defining voltage regulation, line losses and efficiency of
transmission.
References
I.
II.
III.
Principles of Power Systems By V.K Mehta 4th edition .
Power System Analysis - Hadi Saadat.
Lectures notes.
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