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AC TRANSMISSION
Copyright P. Kundur
This material should not be used without the author's consent
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Performance Equations and Parameters
of Transmission Lines
A transmission line is characterized by four
parameters:
series resistance (R) due to conductor resistivity
shunt conductance (G) due to currents along
insulator strings and corona; effect is small andusually neglected
series inductance (L) due to magnetic field
surrounding the conductor
shunt capacitance (C) due to the electric field
between the conductors
These are distributed parameters.
The parameters and hence the characteristics of
cables differ significantly from those of overhead
lines because the conductors in a cable are
much closer to each other surrounded by metallic bodies such as shields,
lead or aluminum sheets, and steel pipes
separated by insulating material such as
impregnated paper, oil, or inert gas
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The constant ZCis called the character is t icimpedanceand is called the propagat ion constant.
The constants and ZCare complex quantities. The
real part of the propagation constant is called the
attenuation constant , and the imaginary part the
phase constant .
If losses are completely neglected,
)resistance(pure
NumberReal
C
LZC
numberImaginary j
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For a lossless line, Equations 6.8 and 6.9 simplify to
When dealing with lightening/switching surges, HV
lines are assumed to be lossless. Hence, ZCwithlosses neglected is commonly referred to as the surge
impedance.
The power delivered by a line when terminated by its
surge impedance is known as the natural load or su rge
impedance load.
where V0 is the rated voltage
At SIL, Equations 6.17 and 6.18 further simplify to
(6.17)
(6.18)
xIjZxVVRCR
sincos~~
xZ
VjxII
C
RR sin
~cos
~~
wattsZ
VSIL
C
2
0
x
R
x
R
eII
eVV
~
~~
(6.20)
(6.21)
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Hence, for a lossless line at SIL,
V and I have constant amplitude along the line
V and I are in phase throughout the length of the line
The line neither generates nor absorbs VARS
As we will see later, the SIL serves as a convenient
reference q uant i tyfor evaluating and expressing line
performance
Typical values of SIL for overhead lines:
nominal (kV): 230 345 500 765
SIL (MW): 140 420 1000 2300
Underground cables have higher shunt capacitance;
hence, ZCis much smaller and SIL is much higher than
those for overhead lines.
for example, the SIL of a 230 kV cable is about
1400 MW
generate VARs at all loads
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Typical Parameters
Table 6.1 Typical overhead transmission line parameters
Table 6.2 Typical cable parameters
Note: 1. Rated frequency is assumed to be 60 Hz
2. Bundled conductors used for all lines listed, except for the 230 kV line.
3. R, xL, and bCare per-phase values.
4. SIL and charging MVA are three-phase values.
* direct buried paper insulated lead covered (PILC) and high pressure pipe
type (PIPE)
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Voltage Profile of a Radial Line at No-Load
With receiving end open, IR= 0. Assuming a
lossless line from Equations 6.17 and 6.18, we have
At the sending end (x = l),
where = l. The angle is referred to as the
electr ic al lengthor the l ine angle, and is expressed
in radians.
From Equations 6.31, 6.32, and 6.33
xsinZV~jI~xcosV
~V~
CR
R
cosV~
lcosV~
E~
R
RS
(6.31)
(6.32)
(6.33)
(6.35)
(6.36)
cos
xsin
Z
EjI
cos
xcosE~
V~
C
S
S
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As an example, consider a 300 km, 500 kV line with = 0.0013 rads/km, ZC= 250 ohms, and ES= 1.0 pu:
Base current is equal to that corresponding to SIL.
Voltage and current profiles are shown in Figure 6.5.
The only line parameter, other than line length, thataffects the results of Figure 6.5 is. Sinceispractically the same for overhead lines of all voltagelevels (see Table 6.1),the results are universallyapplicable, not just for a 500 kV line.
The receiving end voltage for different line lengths:
- for l= 300 km, VR= 1.081 pu- for l= 600 km, VR= 1.407 pu- for l= 1200 km, VR= infinity
Rise in voltage at the receiving end is because ofcapacitive charging current flowing through line
inductance.
known as the "Ferranti effect".
pu411.0I
pu081.1V
3.22
rads39.00013.0x300
S
R
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Figure 6.5 Voltage and current profiles for a 300 km lossless
line with receiving end open-circuited
(b) Voltage Profile
(a) Schematic Diagram
(c) Current Profile
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Voltage - Power Characteristics
of a Radial Line
Corresponding to a load of PR+jQRat the receiving end, wehave
Assuming the line to be lossless, from Equation 6.17with x = l
Fig. 6.7 shows the relationship between VRand PRfor a300 km line with different loads and power factors.
The load is normalized by dividing PRbyP0, the naturalload (SIL), so that the results are applicable to overhead
lines of all voltage ratings.
From Figure 6.7 the following fundamental properties of actransmission are evident:
a) There is an inherent maximum limit of power that can betransmitted at any load power factor. Obviously, therehas to be such a limit, since, with ESconstant, the onlyway to increase power is by lowering the loadimpedance. This will result in increased current, butdecreased VRand large line losses. Up to a certain pointthe increase of current dominates the decrease of VR,thereby resulting in an increased PR. Finally, thedecrease in VR is such that the trend reverses.
*~~
R
RRR
V
jQPI
*~sincos~~
R
RRCRS
V
jQPjZVE
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Figure 6.7 Voltage-power characteristics of a 300 km
lossless radial line
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Voltage - Power Characteristics
of a Radial Line (cont'd)
b) Any value of power below the maximum can be
transmitted at two different values of VR. The
normal operation is at the upper value, within
narrow limits around 1.0 pu. At the lower voltage,
the current is higher and may exceed thermal
limits. The feasibility of operation at the lower
voltage also depends on load characteristics, andmay lead to voltage instability.
c) The load power factor has a significant influence
on VRand the maximum power that can be
transmitted. This means that the receiving end
voltage can be regulated by the addition of shunt
capacitive compensation.
Fig. 6.8 depicts the effect of line length:
For longer lines, VRis very sensitive to variations
in PR.
For lines longer than 600 km ( > 45),VRatnatural load is the lower of the two values which
satisfies Equation 6.46. Such operation is likely
to be unstable.
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Figure 6.8 Relationship between receiving end voltage,
line length, and load of a lossless radial line
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Voltage-Power Characteristic of a Line
Connected to Sources at Both Ends
With ESand ERassumed to be equal, the following
conditions exist:
the midpoint voltage is midway in phase between
ESand ER
the power factor at midpoint is unity
with PR>P0, both ends supply reactive power to theline; with PR
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Power Transfer and Stability
Considerations
Assuming a lossless line, from Equation 6.17 with
x = l,we can show that
where = lis the electrical length of line and is the
angle by which ESleads ER, i.e. the load angle.
If ES= ER=rated voltage, then the natural load is
and Equation 6.51 becomes
The above is valid for synchronous as well as
asynchronous load at the receiving end.
Fig. 6.10(a) shows the PRrelationship for a 400 km
line.
For comparison, the Vm- PRcharacteristic of the line is
shown in Fig. 6.10(b).
(6.51)sin
sinC
RSR
Z
EEP
C
RSO
Z
EEP
sin
sinO
R
PP
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Figure 6.10 PR- and Vm-PRcharacteristics of 400 km lossless
line transmitting power between two large systems
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Reactive Power Requirements
From Equation 6.17, with x = land ES= ER=1.0, we canshow that
Fig. 6.11 shows the terminal reactive powerrequirements of lines of different lengths as a functionof PR.
Adequate VAR sources must be available at the twoends to operate with varying load and nearlyconstant voltage.
General Comments
Analysis of transmission line performancecharacteristics presented above represents a highlyidealized situation
useful in developing a conceptual understanding ofthe phenomenon
dynamics of the sending-end and receiving-endsystems need to be considered for accurateanalysis.
sin
coscos2
C
S
SR
Z
E
QQ
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Figure 6.11 Terminal reactive power as a function of power
transmitted for different line lengths
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Loadability Characteristics
The concept of "line loadability" was introduced by
H.P. St. Clair in 1953
Fig. 6.13 shows the universal loadability curve for
overhead uncompensated lines applicable to all
voltage ratings
Three factors influence power transfer limits:
thermal limit (annealing and increased sag)
voltage drop limit (maximum 5% drop)
steady-state stability limit (steady-state stability
margin of 30% as shown in Fig. 6.14)
The "St. Clair Curve" provides a simple means of
visualizing power transfer capabilities of transmission
lines.
useful for developing conceptual guides to
preliminary planning of transmission systems
must be used with some caution
Large complex systems require detailed assessment
of their performance and consideration of additional
factors
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Figure 6.13 Transmission line loadability curve
"St. Clair Curve"
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Figure 6.14 Steady state stability margin calculation
Stability Limit Calculation for Line
Loadability
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Factors Influencing Transfer of Active
and Reactive Power
Consider two sources connected by an inductive
reactance as shown in Figure 6.21.
representation of two sections of a power system
interconnected by a transmission system
a purely inductive reactance is considered
because impedances of transmission elements
are predominately inductive
effects of shunt capacitances do not appear
explicitly
Figure 6.21 Power transfer between two sources
(a) Equivalent system diagram
(b) Phasor diagram
= load angle
= power factor angle
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The complex power at the receiving end is
Hence,
Similarly,
Equations 6.79 to 6.82 describe the way in which
active and reactive power are transferred
Let us examine the dependence of Pand Qtransfer
on the source voltages, by considering separately
the effects of differences in voltage magnitudes and
angles
jX
EjEEE
jX
EEEIEjQPS
RSSR
RSRRRRR
sincos
~~~~~~
*
X
EEEQ
X
EEP
RRSR
RS
R
2cos
sin
(6.79)
(6.80)
X
EEEQ
XEEP
RSSS
RSS
cos
sin
2
(6.81)
(6.82)
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From Equations 6.79 to 6.82, we have
With ES> ER, QSand QRare positive
With ES< ER, QSand QRare negative
As shown in Fig. 6.22,
transmission of lagging current through an
inductive reactance causes a drop in receiving
end voltage
transmission of leading current through an
inductive reactance causes a rise in receiving
end voltage
Reactive power "consumed" in each case is
Figure 6.22 Phasor diagrams with = 0
(a) Condition with = 0:
0 SR PP
X
EEEQ
X
EEEQ RSSS
RSRR
,
22
XIX
EEQQ RSRS
(a) ES>ER (b) ER>ES
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From Equations 6.79 to 6.82, we now have
With positive, PSand PRare positive, i.e., active
power flows from sending to receiving end
In each case, there is no reactive power transferred
from one end to the other; instead, each endsupplies half of Qconsumed by X.
(b) Condition with ES= ERand 0
Figure 6.23 Phasor diagram with ES= ER
2
2
2
2
1
cos1
sin
IX
X
EQQ
X
EPP
RS
SR
(b) < 0(a) > 0
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We now have
If, in addition to X, we consider series resistance R
of the network, then
The reactive power "absorbed" by Xfor all
conditions is XI 2. This leads to the concept of
"reactive power loss", a companion term to active
power loss.
An increase in reactive power transmitted increases
active as well as reactive power losses. This has an
impact on efficiency and voltage regulation.
(c) General case applicable to any condition:
22
22 cos2
sincos
XI
X
XI
X
EEEEQQ
jX
EjEEI
RSRSRS
RSS
2
22
2
2
22
2
R
RRloss
R
RRloss
E
QPRIRP
E
QPXIXQ
(6.83)
(6.84)
(6.85)
(6.86)
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Appendix to Section on AC Transmission
1. Copy of Section 6.4 from the book Power System
Stability and Control
provides background information related to
power flow analysis techniques
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