1
Extended Transmission Line
Loadability Curve by Including Voltage
Stability Constrains
Jin Hao, Member, IEEE, and Wilsun Xu, Fellow, IEEE

Abstract— St. Clair curve provides a simple means for
estimating power transfer capabilities of transmission lines. It
concerns three limiting factors: thermal limit, voltage quality (or
drop) limit, and angular stability limit. This paper illustrates the
influence of voltage stability limit and presents an extended
loadability curve. Moreover, the impacts of line resistance and
shunt compensation on line loadability are investigated.
Index Terms— Transmission line loadability, St. Clair curve,
voltage stability limit, surge impedance loading.
I. INTRODUCTION
T
ransmission line loadability curve, also known as St. Clair
curve [1] has been a valuable tool for quickly estimating
the power transfer capabilities of transmission lines. Due to
its universal characteristics, i.e. applicable to all voltage
levels, St. Clair curve are generally accepted in the industry as
a convenient reference for estimating the maximum loading
limits on transmission lines.
The St. Clair curve [2],[3], presented in Fig. 1, shows the
loadability of transmission line in terms of their Surge
Impedance Loading (SIL). It is well known that the per-unit
line data normalized using SIL and Surge Impedance is
constant, i.e. independent of line construction and voltage
rating. Therefore, this curve can be used universally.
In recent years, power system voltage stability has attracted
considerable interest in industry. Therefore, it is important to
include the voltage stability limit in the line loadability curve.
This paper illustrats the influence of voltage stability limit on
line loadability. The impacts of line resistance and shunt
compensation are also investigated. Besides, the voltage drop
limit and voltage stability limit are compared for different
transmission lines.
II. EXTENDED TRANSMISSION LINE LOADABILITY CURVE
In this section, we first introduce the basic concepts of
voltage stability limit and then present the extended
transmission line loadability curve with considering the
voltage stability limit.
A. Voltage Stability Limit
A simple system is shown in Fig. 2. In the system, to
simplify the calculation, the voltage phase angle at the
receiving end is seen as reference, and the voltage magnitude
of the sending end, VS, is constant.
I
VS 
jX
P+jQ
VR 0
3
Line load limit in pu of SIL
Fig. 2. System diagram
2.5
At the receiving end,
VR I *  P  jQ
2
1.5
(1)
So
1
I
0.5
P  jQ
VR
(2)
The sending end voltage is:
0
0
160
320
480
640
Line length (km)
800
960
 P  jQ 

 PX 
QX 
VS  VR  
 jX   VR 
 j

V
V
R
R 



 VR 
Fig. 1. Transmission line loadability curve (St. Clair curve) [2]
The corresponding magnitude equation is
J. Hao and W. Xu are with the Department of Electrical and Computer
Engineering, University of Alberta, Edmonton, Alberta, T6G2V4, Canada (email: jhao@ece.ualberta.ca, wxu@ualberta.ca ).
(3)
2
2

QX   PX 
V   VR 
 

VR   VR 

2
2
S
(4)
The power delivered to the load as a function of receiving
end voltage when Q = 0 can be solved as:
P
VS2  VR2
X
VR
(5)
Since VS is constant and close to 1 per-unit, and X can not
change. VR is the only variable that can vary. So the power
will vary with VR, which is shown in Fig.3.
It can be seen from the equivalent circuit, when the line
length increases, the open circuit voltage VS-eq increases
accordingly because of the line charge. This is the well-known
Ferranti effect. This effect leads to the increase of nose point
voltage Vnose. We can further deduct from the figure that when
the length increases to a specific value, the nose point voltage
will become higher than the sending end voltage. When this
happens, it becomes impossible to operate the system as any
operating point with acceptable voltage level will be below
the nose point, which is an unstable case. The line length at
which the nose point will move above the sending end voltage
can be determined using the following condition:
Vnose  VS
P
assuming that the receiving end Q load is equal to 0, we can
establish
Pmax
VS  eq
Vnose 
Operating
point
L
Fig. 3. Transfer capability curve
The maximum power that can be transmitted is reached
when dP/dVR = 0, which can be determined as
VS2
2X
(6)
The voltage corresponding to (6) is
Vnose 
VS
(7)
2
The above limit (6) is called the Voltage Stability Limit of
power transmission and the subscript “Vstab” is used for this
consideration.
If we use the nominal PI circuit to approximate the line, the
transmission scheme and its equivalent circuit are shown in
Fig. 4.
Z=j0.0013L
P
VS 0
2

VS
1
 VS
2
2 1  (0.0013L ) / 2
(9)
Solving the above equation yields:
VR
Pmax_ Vstab 
(8)
Y=j0.0013/2*L
2 2
 588.7km
(0.0013) 2
(10)
This demonstrates that the power transfer capability is
limited by voltage stability concern when the line length is
greater than 588.7km.
B. Extended Line Loadability Curve
In this subsection, we will use the results obtained above to
estimate the power transmission capabilities of 72kV, 138kV,
240kV, 345kV and 500kV lines for different line length. The
lines used for this study are shown in Tab. I. In this table,
Zsurge stands for surge impedance of the line.
It can be seen from the table that the per-unit, per-km X and
B values of overhead lines are all equal to 0.00126pu/km
regardless of the voltage ratings.
TABLE I
TYPICAL LINE DATA AND SURGE IMPEDANCE LOADING
Voltage
R
(Ω/km)
X
(Ω /km)
B
(mΩ /km)
Zsurge
(Ω)
SIL
(MW)
72kV
138kV
240kV
345kV
500kV
Per-unit
0.3970
0.2140
0.0626
0.0370
0.0280
vary
0.4923
0.4801
0.3681
0.3670
0.3250
0.00126
3.6567
3.4321
4.4936
4.5180
5.2000
0.00126
367
374
286
285
250
1
14
51
201
418
1000
1
A) Nominal PI circuit of a transmission line
jX eq 
VS  eq
1
 Vs
1  (0.0013L) 2 / 2
j 0.0013L
1  (0.0013L)2 / 2
B) Equivalent circuit
Fig. 4. Transmission line model and its equivalent circuit
P
The line loading limitations considered here are: thermal
limitation, voltage stability limitation, voltage quality
limitation, and angular stability limitation. A voltage drop of
10% is used as the voltage quality threshold. The angular
stability limit is defined as the maximum transfer capability of
the system. It should be noticed that in the orginal St. Clair
curve [1], the load angle 440 (the corresponding stability
margin is 30%) is selected as the angular stability limit. In this
paper, in order to consist with the voltage stability limit, the
load angle 900 is defined as angular stability limit.
3
4
Voltage stability limit
3
2
Angular stability limit
Thermal limit
1
Voltage quality limit
0
200
400
600
Line Length (km)
800
1000
5
4
Voltage stability limit
3
Angular stability limit
2
1
C. Comparison of stability limits
The practical stability limits of power systems are the load
angle 440 for the angular stability limit and the margin of 5%
for the voltage stability limit. Fig. 7 and Fig. 8 show the
power transfer capability curves with these realistic
constrains. Fig.7 shows the limits when line charging is
included while Fig. 8 shows the results when the charging is
compensated to zero.
Power Transfer Limit (pu of SIL)
Power Transfer Limit (pu of SIL)
6
Thermal limit
6
5
4
2
1
400
600
Line Length (km)
Voltage stability limit
0
0
0
200
Angular stability limit
3
Voltage quality limit
0
800
1000
Fig. 6. Loadability curves of transmission line (Compensated)
As discussed early, the per-unit X and B data are the same
regardless of the line types. So each limit curve is applicable
to all lines except the curve corresponding to the thermal limit.
The thermal limit is line-dependent. A further note is that the
lines are assumed to be lossless. This assumption is not quite
accurate when dealing with low voltage lines. The impact of R
on the curves will be discussed in Section III.
The results lead to the following conclusions:


5
Fig. 5. Loadability curves of transmission line (Uncompensated)


6
0

becomes zero from the voltage stability perspective. The
cause of this phenomenon is explained in Fig.4 and (10).
The angular stability limit is the least restrictive one.
However, the limit goes below 1.0 SIL after the line
approaches about 760km.
If the line charging is compensated (Fig. 6), it becomes
possible to transfer some power over long distance
without violating the voltage stability or voltage quality
limits. However, the amount of power transferred is
below 1.0 SIL.
A short line is limited by thermal constraint. In the
figures, the thermal limit curves slope down because
when load increases, the increase of line current is not
linear.
The voltage quality limit is the most restrictive one as the
line length increases. This is because voltage instability
often occurs after the receiving end voltage drops beyond
power quality limit. However, when the line length
approaches to 500km, the voltage quality limit no longer
exists. This is due to the voltage rise effect of long lines.
If the line is not compensated, it is not possible to transfer
power without causing voltage stability problem when the
line length is over 588km. So the power transfer limit
200
400
600
Line Length (km)
800
1000
Fig. 7. Comparison of realistic stability limits (Uncompensated)
6
Power Transfer limit (pu of SIL)
Power Transfer Limit (pu of SIL)
Fig. 5 and 6 show the power transfer limits as a function of
line length. The power level is expressed in per-unit of the
SIL of the respective lines. Fig.5 shows the limits when line
charging is included while Fig. 6 shows the results when the
charging is compensated to zero.
5
4
Angular stability limit
3
2
1
Voltage stability limit
0
0
200
400
600
Line length (km)
800
1000
Fig. 8. Comparison of realistic stability limits (Compensated)
It can be observed from the figures that the angular stability
limit is less restrictive than voltage stability limit under the
above assumption. In Fig. 7, the angular stability limit reaches
1.0 SIL when the line length is about 537km. This is slightly
different from the original St. Clair curve in which the angular
stability limit approaches 1.0 SIL at 480km [1]. This
difference is due to the fact that, in the original St. Clair curve,
the combined reactance of step-up transformers and
generators as well as of receiving systems was added directly
4
to the reactance of the line, while in this paper only the line
reactance is considered.
voltage transmission lines. This subsection further investigates
the effect of shunt compensation on the power transfer
capabilities.
III. EFFECTS OF SYSTEM PARAMETERS ON LINE LOADABILITY
A. Effect of Line Resistance
Fig. 9 shows the loadability curves with only voltage
stability as concern for different transmission lines. It can be
seen that high resistance (e.g. 25kV line) will severely depress
line loadability, particularly for short lines. This effect
becomes much smaller for high voltage levels (e.g. 500kV
line).
3
Voltage rating
increase
2
R=0.000219(240kV-VQ)
2
R=0.000130(345kV-VQ)
Voltage rating increase
1
100
200
300
400
500
Line length (km )
600
700
800
Fig. 12 and Fig. 13 show the loadability curves with only
voltage stability as concern for different transmission lines. In
the figure, ‘Compensated’ denotes that the line charging is
fully compensated while ‘Uncompensated’ denotes that the
line is uncompensated. Fig. 12 shows the power transfer limits
for 25, 72, and 138 kV lines while Fig. 13 shows the power
transfer limits for 240 kV and 345 kV lines.
R=0.001082(72kV)
R=0.000572(138kV)
R=0.000219(240kV)
R=0.000130(345kV)
R=0.000113(500kV)
1
3
0
0
100
200
300
400
Line length (km)
500
600
700
Fig. 9. Voltage stability constrained power transfer limits (Uncompensated)
Fig. 10 and Fig. 11 compare voltage stability limit and
voltage quality limit for different transmission lines. In the
figure, ‘VS’ and ‘VQ’ stand for the voltage stability limit and
voltage quality limit, respectively.
3
R=0.002478(25kV-Compensated)
R=0.001082(72kV-Compensated)
R=0.000572(138kV-Compensated)
2
R=0.002478(25kV-Uncompensated)
R=0.001082(72kV-Uncompensated)
R=0.000572(138kV-Uncompensated)
1
Voltage rating increase
0
R=0.002478(25kV-VS)
R=0.001082(72kV-VS)
R=0.000572(138kV-VS)
R=0.002478(25kV-VQ)
R=0.001082(72kV-VQ)
R=0.000572(138kV-VQ)
2
Voltage rating increase
1
0
0
100
200
300
400
Line length (km )
500
600
700
Fig. 10. Voltage stability & quality limits with resistances for 25kV, 72kV,
and 138 kV lines (Uncompensated)
It can be seen that for the given voltage levels, the voltage
stability limit is higher than voltage quality limit, i.e., the
voltage quality limit is more restrictive.
B. Effect of Shunt Compensation
Compensating line charging is a common practice for high
0
100
200
300
400
Line length (km)
500
600
700
Fig. 12. Voltage stability constrained power transfer limits for 25kV, 72kV,
and 138 kV lines
3
Power transfer limit (pu in SIL)
Voltage stability & quality limit
R=0.000130(345kV-VS)
0
Power transfer limit (pu in SIL)
Voltage stability limit (pu of SIL)
R=0.0 (All voltages)
R=0.002478(25kV)
4
R=0.000219(240kV-VS)
Fig. 11. Voltage stability & quality limits with resistances for 240 kV and 345
kV lines (Uncompensated)
6
5
3
Voltage stability & quality limit
The effect of line resistance and shunt compensation are
investigated at 25, 72, 138, 240, 345, and 500kV transmission
levels under the criteria of voltage drop of 10% and voltage
stability constraint.
R=0.000219(240kV-Compensated)
R=0.000130(345kV-Compensated)
R=0.000219(240kV-Uncompensated)
2
R=0.000130(345kV-Uncompensated)
Voltage rating increase
1
0
100
200
300
400
500
Line length (km )
600
700
800
Fig. 13. Voltage stability constrained power transfer limits for 240 kV and 345
kV lines
It can be seen that for low voltage levels (e.g. 25kV), the
loadability curve of the line with shunt compensation is much
close to the curve without compensation. For high voltage
5
levels (e.g. 345kV), when the line charging is not
compensated, the line can not transfer power over certain
distance (e.g. 640km) without violating the voltage stability
constraints; however, when the line charging is compensated,
it is possible to transfer some power over longer distance.
Therefore, the effect of shunt compensation is to extend the
line length for which the loadability is constrained by voltage
stability.
Fig. 14 and Fig. 15 compare voltage stability limit and
voltage quality limit for different transmission lines which are
fully compensated. Again, ‘VS’ and ‘VQ’ stand for the
voltage stability limit and voltage quality limit, respectively.
Fig. 14 shows the power transfer limits for 25, 72, and 138 kV
lines while Fig. 15 shows the power transfer limits for 240 kV
and 345 kV lines. It can be observed that the voltage quality
limit is more restrictive.
Voltage stability & quality limit (pu)
3
R=0.002478(25kV-VS)
R=0.001082(72kV-VS)
R=0.000572(138kV-VS)
R=0.002478(25kV-VQ)
2
R=0.001082(72kV-VQ)
R=0.000572(138kV-VQ)
Voltage rating increase
1
0
0
100
200
300
400
Line length (km)
500
600
700
Fig. 14. Voltage stability & quality limits with resistances for 25kV, 72kV,
and 138 kV lines (Compensated)
Voltage stability & quality limit (pu)
3
R=0.000219(240kV-VS)
R=0.000130(345kV-VS)
R=0.000219(240kV-VQ)
2
R=0.000130(345kV-VQ)
Voltage rating increas e
1
0
100
200
300
400
500
600
700
Line length (km)
800
900
1000
Fig. 15. Voltage stability & quality limits with resistances for 240kV and
345kV lines (Compensated)
IV. CONCLUSIONS
The universal St. Clair curve provides a means of depicting
transmission line loadability as a function of its length. This
paper further investigates the influence of voltage stability
limit on the line loadability. Studies on the effects of various
limiting factors lead to the following main conclusions:
1. The voltage quality limit has dominating influence on the
loadability of short lines, while the voltage stability limit
is the main constrain for long lines.
2. Both analytical and numerical results show that, for the
uncompensated line, it is not possible to transfer power
without causing voltage stability problem when the line
length is over 588km.
3. The resistance has remarkable effect on line loadability,
especially for low voltage levels. This effect will be
decreased as voltage class increases.
4. With shunt compensation, it becomes possible to transfer
power over long distance without violating the voltage
stability or voltage quality limits.
V. REFERENCES
[1]
[2]
[3]
H. P. St. Clair, “Practical concepts in capability and performance of
transmission lines,” A.I.E.E. Transactions Part III. Power Apparatus
and Systems, Vol. 72, No.2, pp. 1152-1157, Jan. 1953.
P. Kundur, Power System Stability and Control, McGraw-Hill, 1994
J. D. Glover, M. S. Sarma, Power System Analysis and Design, (3rd ed.)
Brooks/Cole 2002.