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AC TRANSMISSIONAC TRANSMISSION
1539pk
Copyright © P. Kundur
This material should not be used without the author's consent
Performance Equations and Parameters Performance Equations and Parameters
of Transmission Linesof 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 and
usually neglected
� series inductance (L) due to magnetic field
surrounding the conductor
� shunt capacitance (C) due to the electric field
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� 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
� For balanced steady-state operation, the performance of
transmission lines may be analyzed in terms of single-
phase equivalents.
Fig. 6.1 Voltage and current relationship of a distributed
parameter line
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The general solution for voltage and current at a
distance x from the receiving end (see book: page 202)
is:
where
(6.8)
(6.9)
xRCRxRCRe
IZVe
IZVV γγ −−
++
=2
~~
2
~~~
xR
C
R
xR
C
R
eI
ZV
eI
ZV
I γγ −−
−+
=2
~~
2
~~
~
βαγ jzy
yzZC
+==
=
The constant ZC is called the characteristic
impedance and γγγγ is called the propagation constant.
� The constants γγγγ and ZC are complex quantities. The
real part of the propagation constant γγγγ is called the
attenuation constant α, and the imaginary part the
phase constant β.
� If losses are completely neglected,
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)resistance (pure
Number Real==C
LZC
numberImaginary== βγ j
� 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, ZC with
losses 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 surge
(6.17)
(6.18)
xIjZxVV RCR ββ sincos~~
+=
xZ
VjxII
C
RR ββ sin
~cos
~~
+=
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surge impedance is known as the natural load or surge
impedance load.
where V0 is the rated voltage
� At SIL, Equations 6.17 and 6.18 further simplify to
wattsZ
VSIL
C
2
0=
x
R
x
R
eII
eVV
γ
γ
=
=
~
~~(6.20)
(6.21)
� 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 quantity for evaluating and expressing line
performance
� Typical values of SIL for overhead lines:
nominal (kV): 230 345 500 765
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nominal (kV): 230 345 500 765
SIL (MW): 140 420 1000 2300
� Underground cables have higher shunt capacitance;
hence, ZC is 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
Typical ParametersTypical Parameters
Table 6.1 Typical overhead transmission line parameters
Note: 1. Rated frequency is assumed to be 60 Hz
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Table 6.2 Typical cable parameters
2. Bundled conductors used for all lines listed, except for the 230 kV line.
3. R, xL, and bC are 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)
Voltage Profile of a Radial Line at NoVoltage Profile of a Radial Line at No--LoadLoad
� 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),
( )( ) ( )xsinZV~
jI~
xcosV~
V~
CR
R
β=
β=
θ=
β=
cosV~
lcosV~
E~
R
RS
(6.31)
(6.32)
(6.33)
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where θ = βl. The angle θ is referred to as the
electrical length or the line angle, and is expressed
in radians.
� From Equations 6.31, 6.32, and 6.33
(6.35)
(6.36)θβ
=
θβ
=
cos
xsin
Z
EjI
cos
xcosE~
V~
C
S
S
� As an example, consider a 300 km, 500 kV line with β = 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, that affects the results of Figure 6.5 is β. Since β is practically the same for overhead lines of all voltage levels (see Table 6.1), the results are universally applicable, not just for a 500 kV line.
pu411.0I
pu081.1V
3.22
rads39.00013.0x300
S
R
=
=
=
==θo
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applicable, 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 of capacitive charging current flowing through line inductance.
� known as the "Ferranti effect".
(a) Schematic Diagram
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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
(c) Current Profile
Voltage Voltage -- Power Characteristics Power Characteristics
of a of a Radial LineRadial Line
� Corresponding to a load of PR+jQR at the receiving end, we have
� Assuming the line to be lossless, from Equation 6.17 with x = l
� Fig. 6.7 shows the relationship between VR and PR for a 300 km line with different loads and power factors.
The load is normalized by dividing P by P , the natural
*~~
R
RRR
V
jQPI
−=
−+=
*~sincos~~
R
RRCRS
V
jQPjZVE θθ
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The load is normalized by dividing PR by P0, the natural load (SIL), so that the results are applicable to overhead lines of all voltage ratings.
� From Figure 6.7 the following fundamental properties of ac transmission are evident:
a) There is an inherent maximum limit of power that can be transmitted at any load power factor. Obviously, there has to be such a limit, since, with ES constant, the only way to increase power is by lowering the load impedance. This will result in increased current, but decreased VR and large line losses. Up to a certain point the increase of current dominates the decrease of VR, thereby resulting in an increased PR. Finally, the decrease in VR is such that the trend reverses.
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Figure 6.7 Voltage-power characteristics of a 300 km
lossless radial line
Voltage Voltage -- Power Characteristics Power Characteristics
of a Radial Line of a Radial Line (cont'd)(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, and
may lead to voltage instability.
c) The load power factor has a significant influence
on VR and the maximum power that can be
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on VR and 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, VR is very sensitive to variations
in PR.
� For lines longer than 600 km (θ > 45°), VR at
natural 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
VoltageVoltage--Power Characteristic of a Line Power Characteristic of a Line
Connected to Sources at Both EndsConnected to Sources at Both Ends
� With ES and ER assumed to be equal, the following
conditions exist:
� the midpoint voltage is midway in phase between
ES and ER
� the power factor at midpoint is unity
� with PR>P0, both ends supply reactive power to the
line; with PR<P0, both ends absorb reactive power
from the line.
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� Fig. 6.8 (developed for a radial line) may be used to
analyze how Vm varies with PR.
� with the length equal to half that of the actual line,
plots of VR shown in Figure 6.8 give Vm.
Fig. 6.9 Voltage and current phase relationships with ESequal to ER, and PR less than Po
Power Transfer and Stability Power Transfer and Stability
ConsiderationsConsiderations
� Assuming a lossless line, from Equation 6.17 with
x = l, we can show that
where θ = βllll is the electrical length of line and is the
angle by which ES leads ER, i.e. the load angle.
� If ES = ER = rated voltage, then the natural load is
(6.51)δθ
sinsinC
RSR
Z
EEP =
C
RSO
Z
EEP =
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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 δ ---- PR relationship for a 400 km
line.
For comparison, the Vm - PR characteristic of the line is
shown in Fig. 6.10(b).
δθ
sinsin
OR
PP =
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Figure 6.10 PR-δ and Vm-PR characteristics of 400 km lossless
line transmitting power between two large systems
Reactive Power RequirementsReactive Power Requirements
� From Equation 6.17, with x = l and ES = ER = 1.0, we can show that
� Fig. 6.11 shows the terminal reactive power requirements of lines of different lengths as a function of PR.
� Adequate VAR sources must be available at the two ends to operate with varying load and nearly constant voltage.
( )θ
θδsin
coscos2
C
S
SR
Z
E
−=
−=
1539pkACT - 17
constant voltage.
General Comments
Analysis of transmission line performance characteristics presented above represents a highly idealized situation
� useful in developing a conceptual understanding of the phenomenon
� dynamics of the sending-end and receiving-end systems need to be considered for accurate analysis.
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Figure 6.11 Terminal reactive power as a function of power
transmitted for different line lengths
Loadability CharacteristicsLoadability 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)
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� 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
"St. Clair Curve""St. Clair Curve"
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Figure 6.13 Transmission line loadability curve
Stability Limit Calculation for Line Stability Limit Calculation for Line
LoadabilityLoadability
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Figure 6.14 Steady state stability margin calculation
Factors Influencing Transfer of Active Factors Influencing Transfer of Active
and Reactive Powerand 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
1539pkACT - 22
Figure 6.21 Power transfer between two sources
(a) Equivalent system diagram
(b) Phasor diagram
δ = load angle
Φ = power factor angle
The complex power at the receiving end is
Hence,
Similarly,
−+=
−==+=
jX
EjEEE
jX
EEEIEjQPS
RSSR
RSRRRRR
δδ sincos
~~~~~~ *
X
EEEQ
X
EEP
RRSR
RSR
2cos
sin
−=
=
δ
δ (6.79)
(6.80)
1539pkACT - 23
Similarly,
� Equations 6.79 to 6.82 describe the way in which
active and reactive power are transferred
� Let us examine the dependence of P and Q transfer
on the source voltages, by considering separately
the effects of differences in voltage magnitudes and
angles
X
EEEQ
X
EEP
RSSS
RSS
δ
δ
cos
sin
2 −=
=(6.81)
(6.82)
� From Equations 6.79 to 6.82, we have
� With ES > ER, QS and QR are positive
With ES < ER, QS and QR are negative
� As shown in Fig. 6.22,
� transmission of lagging current through an
inductive reactance causes a drop in receiving
end voltage
(a) Condition with δ = 0:
0== SR PP
( ) ( )X
EEEQ
X
EEEQ RSS
SRSR
R
−=
−= ,
1539pkACT - 24
� 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 δ = 0
( ) 2
2
XIX
EEQQ RS
RS=
−=−
(a) ES>ER(b) ER>ES
� From Equations 6.79 to 6.82, we now have
� With δ positive, PS and PR are positive, i.e., active
power flows from sending to receiving end
(b) Condition with ES = ER and δ ≠≠≠≠ 0
( )
2
2
2
2
1
cos1
sin
IX
X
EQQ
X
EPP
RS
SR
=
−=−=
==
δ
δ
1539pkACT - 25
� In each case, there is no reactive power transferred
from one end to the other; instead, each end
supplies half of Q consumed by X.
Figure 6.23 Phasor diagram with ES = ER
(b) δ < 0(a) δ > 0
� We now have
� If, in addition to X, we consider series resistance R
of the network, then
(c) General case applicable to any condition:
( ) 2
2
22 cos2
sincos
XIX
XI
X
EEEEQQ
jX
EjEEI
RSRSRS
RSS
==
−+=−
−+=
δ
δδ (6.83)
(6.84)
1539pkACT - 26
� The reactive power "absorbed" by X for all
conditions is X I 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.
2
222
2
222
R
RRloss
R
RRloss
E
QPRIRP
E
QPXIXQ
+==
+== (6.85)
(6.86)
Conclusions Regarding Transfer of Active and Conclusions Regarding Transfer of Active and
Reactive PowerReactive Power
� The active power transferred (PR) is a function of voltage magnitudes and δ. However, for satisfactory operation of the power system, the voltage magnitude at any bus cannot deviate significantly from the nominal value. Therefore, control of active power transfer is achieved primarily through variations in angle δ.
� Reactive power transfer depends mainly on voltage magnitudes. It is transmitted from the side with higher voltage magnitude to the side with lower voltage magnitude.
1539pkACT - 27
� Reactive power cannot be transmitted over long distances, since it would require a large voltage gradient to do so.
� An increase in reactive power transfer causes an increase in active as well as reactive power losses.
Although we have considered a simple system, the general
conclusions are applicable to any practical system, In fact, the basic
characteristics of ac transmission reflected in these conclusions
have a dominant effect on the way in which we operate and control
the power system.
Appendix to Section on AC TransmissionAppendix 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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