Transmission Lines
Content
EE 340 – Transmission Lines
Spring 2013
Physical Characteristics – Overhead lines
• An overhead transmission line usually consists of three conductors or
bundles of conductors containing the three phases of the power
system.
• In overhead transmission lines, the bare conductors are suspended
from a pole or a tower via insulators. The conductors are usually
made of aluminum cable steel reinforced (ACSR).
Physical Characteristics – underground cables
• Cable lines are designed to be placed underground or under
water. The conductors are insulated from one another and
surrounded by protective sheath.
• Cable lines are more expensive and harder to maintain.
They also have capacitance problem – not suitable for long
distance.
Electrical Characteristics
• Transmission lines are characterized by a series resistance,
inductance, and shunt capacitance per unit length.
• These values determine the power-carrying capacity of the
transmission line and the voltage drop across it at full load.
• The DC resistance of a conductor is expressed in terms of
resistively, length and cross sectional area as follows:
Cable resistance
• The resistively increases linearly with temperature over normal
range of temperatures.
• If the resistively at one temperature and material temperature
constant are known, the resistively at another temperature can
be found by
Cable Resistance
• AC resistance of a conductor is always higher than its DC
resistance due to the skin effect forcing more current flow near
the outer surface of the conductor. The higher the frequency of
current, the more noticeable skin effect would be.
• Wire manufacturers usually supply tables of resistance per unit
length at common frequencies (50 and 60 Hz). Therefore, the
resistance can be determined from such tables.
Line inductance
Remarks on line inductance
• The greater the spacing between the phases of a
transmission line, the greater the inductance of the line.
– Since the phases of a high-voltage overhead transmission line must be
spaced further apart to ensure proper insulation, a high-voltage line will
have a higher inductance than a low-voltage line.
– Since the spacing between lines in buried cables is very small, series
inductance of cables is much smaller than the inductance of overhead lines
• The greater the radius of the conductors in a transmission
line, the lower the inductance of the line. In practical
transmission lines, instead of using heavy and inflexible
conductors of large radii, two and more conductors are bundled
together to approximate a large diameter conductor, and reduce
corona loss.
Per-Phase Inductance of 3-phase transmission line
Shunt capacitance
• Since a voltage V is applied to a pair of conductors separated by
a dielectric (air), charges q of equal magnitude but opposite
sign will accumulate on the conductors. Capacitance C between
the two conductors is defined by
q
C
V
• The capacitance of a single-phase transmission line is given by
(see derivation in the book): (ε = 8.85 x 10-12 F/m)
Capacitance of 3-phase transmission line
Remarks on line capacitance
1. The greater the spacing between the phases of a
transmission line, the lower the capacitance of the line.
–
–
Since the phases of a high-voltage overhead transmission line must be
spaced further apart to ensure proper insulation, a high-voltage line will
have a lower capacitance than a low-voltage line.
Since the spacing between lines in buried cables is very small, shunt
capacitance of cables is much larger than the capacitance of overhead
lines.
2. The greater the radius of the conductors in a transmission
line, the higher the capacitance of the line. Therefore,
bundling increases the capacitance.
Short line model
• Overhead transmission lines shorter than 50 miles can be
modeled as a series resistance and inductance, since the
shunt capacitance can be neglected over short distances.
• The total series resistance and series reactance can be
calculated as
• where r, x are resistance and reactance per unit length and d
is the length of the transmission line.
Short line model
• Two-port network model:
• The equation is similar to that of a synchronous generator and
transformer (w/o shunt impedance)
Short line
Voltage Regulation:
1.
2.
3.
If unity-PF (resistive) loads are added at the end of a
line, the voltage at the end of the transmission line
decreases slightly – small positive VR.
If leading (capacitive) loads are added at the end of a
line, the voltage at the end of the transmission line
increases – negative VR.
If lagging (inductive) loads are added at the end of a
line, the voltage at the end of the transmission line
decreases significantly – large positive VR.
Power flow and efficiency
• Input powers
• Output powers
• Efficiency
Short line – simplified
• If the resistance of the line is ignored, then
• Therefore, for fixed voltages, the power flow through a
transmission line depends on the angle between the input and
output voltages.
• Maximum power flow occurs when δ = 90o.
• Notes:
– The maximum power handling capability of a transmission line is a
function of the square of its voltage.
– The maximum power handling capability of a transmission line is
inversely proportional to its series reactance (some very long lines
include series capacitors to reduce the total series reactance).
– The angle δ controls the power flow through the line. Hence, it is
possible to control power flow by placing a phase-shifting transformer.
Example
• A line with reactance X and negligible resistance supplies a
pure resistive load from a fixed source VS. Determine the
maximum power transfer, and the load voltage VR at which
this occurs. (Hint: recall the maximum power transfer theorem
from your basic circuits course)
• Ans: Pmax
VS2
,
2X
VR
VS
2
Line Characteristics
• To prevents excessive voltage variations in a power system, the
ratio of the magnitude of the receiving end voltage to the
magnitude of the ending end voltage is generally within
0.95 ≤ VS/VR ≤ 1.05
• The angle δ in a transmission line should typically be ≤ 30o to
ensure that the power flow in the transmission line is well below
the static stability limit.
• Any of these limits can be more or less important in different
circumstances.
– In short lines, where series reactance X is relatively small, the
resistive heating usually limits the power that the line can supply.
– In longer lines operating at lagging power factors, the voltage drop
across the line is usually the limiting factor.
– In longer lines operating at leading power factors, the maximum
angle δ can be the limiting f actor.
Medium Line (50-150 mi)
• the shunt admittance must be included in calculations. However, the
total admittance is usually modeled (π model) as two capacitors of
equal values (each corresponding to a half of total admittance)
placed at the sending and receiving ends.
• The total series resistance and series reactance are calculated as
before. Similarly, the total shunt admittance is given by
• where y is the shunt admittance per unit length and d is the length
of the transmission line.
Medium Line
• Two-port network:
Determining the Line Circuit Parameters
• Note: Typically, |R +jXL| << |1/Y|
• Under Open-Circuit Test (phase quantities):
VS AVR
I S CVR YVS
• Under Short-Circuit Test:
P 0, Q YVS2
VS BI R ZI R , I S DI R
P RI R2 , Q X L I R2 (Y / 2)VS2 ,
Long Lines ( > 150 mi)
• For long lines, both the shunt capacitance and the series
impedance must be treated as distributed quantities. The
voltages and currents on the line are found by solving differential
equations of the line.
• However, it is possible to model a long transmission line as a π
model with a modified series impedance Z’ and a modified shunt
admittance Y’ and to perform calculations on that model using
ABCD constants. These modified values are
where the propagation constant is defined by
Long line series and shunt compensation
• Shunt reactors are used to compensate the line shunt
capacitance under light load or no load to regulate voltage.
• Series capacitors are often used to compensate the line
inductive reactance in order to transfer more power.
Practice Problems
• 9.11 – 9.17
Spring 2013
Physical Characteristics – Overhead lines
• An overhead transmission line usually consists of three conductors or
bundles of conductors containing the three phases of the power
system.
• In overhead transmission lines, the bare conductors are suspended
from a pole or a tower via insulators. The conductors are usually
made of aluminum cable steel reinforced (ACSR).
Physical Characteristics – underground cables
• Cable lines are designed to be placed underground or under
water. The conductors are insulated from one another and
surrounded by protective sheath.
• Cable lines are more expensive and harder to maintain.
They also have capacitance problem – not suitable for long
distance.
Electrical Characteristics
• Transmission lines are characterized by a series resistance,
inductance, and shunt capacitance per unit length.
• These values determine the power-carrying capacity of the
transmission line and the voltage drop across it at full load.
• The DC resistance of a conductor is expressed in terms of
resistively, length and cross sectional area as follows:
Cable resistance
• The resistively increases linearly with temperature over normal
range of temperatures.
• If the resistively at one temperature and material temperature
constant are known, the resistively at another temperature can
be found by
Cable Resistance
• AC resistance of a conductor is always higher than its DC
resistance due to the skin effect forcing more current flow near
the outer surface of the conductor. The higher the frequency of
current, the more noticeable skin effect would be.
• Wire manufacturers usually supply tables of resistance per unit
length at common frequencies (50 and 60 Hz). Therefore, the
resistance can be determined from such tables.
Line inductance
Remarks on line inductance
• The greater the spacing between the phases of a
transmission line, the greater the inductance of the line.
– Since the phases of a high-voltage overhead transmission line must be
spaced further apart to ensure proper insulation, a high-voltage line will
have a higher inductance than a low-voltage line.
– Since the spacing between lines in buried cables is very small, series
inductance of cables is much smaller than the inductance of overhead lines
• The greater the radius of the conductors in a transmission
line, the lower the inductance of the line. In practical
transmission lines, instead of using heavy and inflexible
conductors of large radii, two and more conductors are bundled
together to approximate a large diameter conductor, and reduce
corona loss.
Per-Phase Inductance of 3-phase transmission line
Shunt capacitance
• Since a voltage V is applied to a pair of conductors separated by
a dielectric (air), charges q of equal magnitude but opposite
sign will accumulate on the conductors. Capacitance C between
the two conductors is defined by
q
C
V
• The capacitance of a single-phase transmission line is given by
(see derivation in the book): (ε = 8.85 x 10-12 F/m)
Capacitance of 3-phase transmission line
Remarks on line capacitance
1. The greater the spacing between the phases of a
transmission line, the lower the capacitance of the line.
–
–
Since the phases of a high-voltage overhead transmission line must be
spaced further apart to ensure proper insulation, a high-voltage line will
have a lower capacitance than a low-voltage line.
Since the spacing between lines in buried cables is very small, shunt
capacitance of cables is much larger than the capacitance of overhead
lines.
2. The greater the radius of the conductors in a transmission
line, the higher the capacitance of the line. Therefore,
bundling increases the capacitance.
Short line model
• Overhead transmission lines shorter than 50 miles can be
modeled as a series resistance and inductance, since the
shunt capacitance can be neglected over short distances.
• The total series resistance and series reactance can be
calculated as
• where r, x are resistance and reactance per unit length and d
is the length of the transmission line.
Short line model
• Two-port network model:
• The equation is similar to that of a synchronous generator and
transformer (w/o shunt impedance)
Short line
Voltage Regulation:
1.
2.
3.
If unity-PF (resistive) loads are added at the end of a
line, the voltage at the end of the transmission line
decreases slightly – small positive VR.
If leading (capacitive) loads are added at the end of a
line, the voltage at the end of the transmission line
increases – negative VR.
If lagging (inductive) loads are added at the end of a
line, the voltage at the end of the transmission line
decreases significantly – large positive VR.
Power flow and efficiency
• Input powers
• Output powers
• Efficiency
Short line – simplified
• If the resistance of the line is ignored, then
• Therefore, for fixed voltages, the power flow through a
transmission line depends on the angle between the input and
output voltages.
• Maximum power flow occurs when δ = 90o.
• Notes:
– The maximum power handling capability of a transmission line is a
function of the square of its voltage.
– The maximum power handling capability of a transmission line is
inversely proportional to its series reactance (some very long lines
include series capacitors to reduce the total series reactance).
– The angle δ controls the power flow through the line. Hence, it is
possible to control power flow by placing a phase-shifting transformer.
Example
• A line with reactance X and negligible resistance supplies a
pure resistive load from a fixed source VS. Determine the
maximum power transfer, and the load voltage VR at which
this occurs. (Hint: recall the maximum power transfer theorem
from your basic circuits course)
• Ans: Pmax
VS2
,
2X
VR
VS
2
Line Characteristics
• To prevents excessive voltage variations in a power system, the
ratio of the magnitude of the receiving end voltage to the
magnitude of the ending end voltage is generally within
0.95 ≤ VS/VR ≤ 1.05
• The angle δ in a transmission line should typically be ≤ 30o to
ensure that the power flow in the transmission line is well below
the static stability limit.
• Any of these limits can be more or less important in different
circumstances.
– In short lines, where series reactance X is relatively small, the
resistive heating usually limits the power that the line can supply.
– In longer lines operating at lagging power factors, the voltage drop
across the line is usually the limiting factor.
– In longer lines operating at leading power factors, the maximum
angle δ can be the limiting f actor.
Medium Line (50-150 mi)
• the shunt admittance must be included in calculations. However, the
total admittance is usually modeled (π model) as two capacitors of
equal values (each corresponding to a half of total admittance)
placed at the sending and receiving ends.
• The total series resistance and series reactance are calculated as
before. Similarly, the total shunt admittance is given by
• where y is the shunt admittance per unit length and d is the length
of the transmission line.
Medium Line
• Two-port network:
Determining the Line Circuit Parameters
• Note: Typically, |R +jXL| << |1/Y|
• Under Open-Circuit Test (phase quantities):
VS AVR
I S CVR YVS
• Under Short-Circuit Test:
P 0, Q YVS2
VS BI R ZI R , I S DI R
P RI R2 , Q X L I R2 (Y / 2)VS2 ,
Long Lines ( > 150 mi)
• For long lines, both the shunt capacitance and the series
impedance must be treated as distributed quantities. The
voltages and currents on the line are found by solving differential
equations of the line.
• However, it is possible to model a long transmission line as a π
model with a modified series impedance Z’ and a modified shunt
admittance Y’ and to perform calculations on that model using
ABCD constants. These modified values are
where the propagation constant is defined by
Long line series and shunt compensation
• Shunt reactors are used to compensate the line shunt
capacitance under light load or no load to regulate voltage.
• Series capacitors are often used to compensate the line
inductive reactance in order to transfer more power.
Practice Problems
• 9.11 – 9.17
