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 and
usually 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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
For balanced steady-state operation, the performance of
transmission lines may be analyzed in terms of singlephase equivalents.
Fig. 6.1 Voltage and current relationship of a distributed
parameter line
The general solution for voltage and current at a
distance x from the receiving end (see book: page 202)
is:
~
~
~
~
~ VR  Z C I R  x VR  Z C I R   x
V 
e 
e
2
2
~
VR
~
I 
where
ZC
e
2
ZC 
 
~
VR
~
 IR
x

ZC
~
 IR
e
(6.8)
 x
(6.9)
2
z
y
zy    j 
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The constant ZC is called the characteristic
impedance and K is called the propagation constant.

The constants K and ZC are complex quantities. The
real part of the propagation constant K is called the
attenuation constant α, and the imaginary part the
phase constant β.

If losses are completely neglected,
ZC 
L
 Real Number
C
(pure resistance )
  j   Imaginary
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
For a lossless line, Equations 6.8 and 6.9 simplify to
~
~
V  V R cos  x  jZ C I R sin  x
(6.17)
 V~

~ ~
 sin  x
I  I R cos  x  j  R

ZC 


(6.18)
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
impedance load.
2
SIL 
V0
watts
ZC
where V0 is the rated voltage

At SIL, Equations 6.17 and 6.18 further simplify to
~
~ x
V  VR e
(6.20)
~
x
I  I Re
(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 quantity for evaluating and expressing line
performance

Typical values of SIL for overhead lines:
nominal (kV): 230
SIL (MW):
140

345
420
500
1000
765
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
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Typical Parameters
Table 6.1 Typical overhead transmission line 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 bC are per-phase values.
4. SIL and charging MVA are three-phase values.
Table 6.2 Typical cable parameters
* 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
~
~
V  VR cos  x 
~
~
I  j VR Z C sin  x 



(6.31)
(6.32)
At the sending end (x = l),
~
~
E S  VR cos  l
~
 VR cos 
(6.33)
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
~
~ cos  x
V  ES
cos 
I j
E S sin  x
(6.35)
(6.36)
Z C cos 
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
As an example, consider a 300 km, 500 kV line with
β = 0.0013 rads/km, ZC = 250 ohms, and ES = 1.0 pu:
  300 x 0 . 0013  0 . 39 rads
 22 . 3

V R  1 . 081 pu
I S  0 . 411 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.

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".
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(a) Schematic Diagram
(b) Voltage Profile
(c) Current Profile
Figure 6.5 Voltage and current profiles for a 300 km lossless
line with receiving end open-circuited
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Voltage - Power Characteristics
of a Radial Line

Corresponding to a load of PR+jQR at the receiving end, we
have
P  jQ
~
IR  R ~ R
*
VR

Assuming the line to be lossless, from Equation 6.17
with x = l
 P  jQ
~
~
E S  V R cos   jZ C sin   R ~ R
*

VR






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 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
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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, and
may lead to voltage instability.
c) The load power factor has a significant influence
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
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Voltage-Power Characteristic of a Line
Connected 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.
Fig. 6.9 Voltage and current phase relationships with ES
equal to ER, and PR less than Po

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.
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Power Transfer and Stability
Considerations

Assuming a lossless line, from Equation 6.17 with
x = l, we can show that
PR 
ESER
Z C sin 
sin 
(6.51)
where θ = βl 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
PO 
ESER
ZC
and Equation 6.51 becomes
PR 
PO
sin 
sin 
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).
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Figure 6.10 PR-δ and Vm-PR characteristics of 400 km lossless
line transmitting power between two large systems
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Reactive Power Requirements

From Equation 6.17, with x = l and ES = ER = 1.0, we can
show that
Q R  Q S
E S cos   cos 
2



Z C sin 
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.
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
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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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"St. Clair Curve"
Figure 6.13 Transmission line loadability curve
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Stability Limit Calculation for Line
Loadability
Figure 6.14 Steady state stability margin calculation
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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
(a) Equivalent system diagram
δ = load angle
Φ = power factor angle
(b) Phasor diagram
Figure 6.21 Power transfer between two sources
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The complex power at the receiving end is
~
~
~
~ ~*
~  ES  ER 
S R  PR  jQ R  E R I  E R 

jX


 E cos   jE S sin   E R 
 ER  S

jX


Hence,
PR 
ESER
X
(6.79)
sin 
E S E R cos   E R
2
QR 
(6.80)
X
Similarly,
PS 
ESER
X
sin 
E S  E S E R cos 
2
QS 
(6.81)
(6.82)
X

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
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(a) Condition with δ = 0:

From Equations 6.79 to 6.82, we have
PR  PS  0
QR 

E R E S  E R 
QS 
,
X
E S E S  E R 
X
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
 transmission of leading current through an
inductive reactance causes a rise in receiving
end voltage

Reactive power "consumed" in each case is
QS  QR 
E
S
 ER 
X
2
 XI
2
(b) ER>ES
(a) ES>ER
Figure 6.22 Phasor diagrams with δ = 0
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(b) Condition with ES = ER and δ  0

From Equations 6.79 to 6.82, we now have
PR  PS 
E
2
sin 
X
Q S  Q R 

2
E
X
1
1  cos  
X I
2
2

With δ positive, PS and PR are 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 end
supplies half of Q consumed by X.
(a) δ > 0
(b) δ < 0
Figure 6.23 Phasor diagram with ES = ER
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(c) General case applicable to any condition:

We now have
I 
E S cos   jE S sin   E R
jX
E S  E R  2 E S E R cos 
2
QS  QR 

(6.83)
2
X
 XI 
(6.84)
2
 XI
2
X

If, in addition to X, we consider series resistance R
of the network, then
P R Q
2
Q loss  X I
2
Ploss  R I
2
 X
2
R
2
ER
P R Q
2
R
E
2
(6.85)
2
R
(6.86)
R

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.
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Conclusions Regarding Transfer of Active and
Reactive 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.

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.
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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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