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Real-Time Ampacity Model for Overhead Lines
Article in IEEE Transactions on Power Apparatus and Systems · August 1983
DOI: 10.1109/TPAS.1983.318152 · Source: IEEE Xplore
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IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No. 7,
July 1983
2289
REAL-TIME AMPACITY MODEL FOR OVERHEAD LINES
W. Z. Black
School of Mechanical Engineering
Georgia Institute of Technology
Atlanta, Georgia
ABSTRACT
A thermal model is formulated that has the capability of predicting
the temperature of overhead transmission lines. The thermal model is
formulated into a computer program that can calculate both steadystate and transient temperatures of a conductor. The transient portion
of the program allows the simulation of "real-time" line operation and
provides a method to predict instantaneous conductor temperatures.
In addition to predicting the temperature and ultimate sag of the line,
the program can be used to predict the operating conditions of lines
in the design stage. The program is also useful for designing new
transmission spans, and for predicting contingency ratings during
emergency conditions.
The model includes energy generated in the conductor, incident solar
energy, emitted radiant energy, convection from the surface of the
conductorand energy stored within the conductor. Both free and forced
convection modes are considered. The radiation model includes contributions due to direct and diffuse solar energy. The resulting differential
equation for the conductor temperature is solved by the computer using
a numerical technique.
The program is capable of predicting the real-time line temperature
for all conductor designs and any variation in weather conditions and
l-ne current. Coupled with a sag program, this type of information will
permit the determination of instantaneous ground clearances of
overhead conductors.
Results are presented for the time constant of several conductors
subject to step changes in current. The influence of weather conditions
on conductor temperature are examined. The wind velocity and direction are shown to be most influential in determining the conductor
temperature. The results of the thermal model have been verified by
temperatures measured on a full-scale outdoor test line.
INTRODUCTION
The safe and efficient transmission of electricity is of major importance to the electrical power industry, and high voltage overhead power
lines are an integral part of the transmission network. Overhead
transmission lines operate at high voltages and have no electrical
insulation, so they must meet certain legal ground clearances to insure
safe operation. A major factor in determining the sag, and consequently
the ground clearance of the line, is the calculation of the conductor
temperature; therefore it is imperative to be able to predict the
temperature of a conductor at any given time. Compounding the problem of maintaining adequate ground clearances is the fact that most
utilities experience peak demand periods during the summer. Therefore
the period of greatest heat generation within the conductor occurs at
aetime when the ambient temperature is not conducive to removing heat
Wm the surface of the line.
The thermal model for determining the steady state ampacity of an
o-yerhead line has been reported in several papers [1-7], but the paper
by House and Tuttle [8] is the most widely referenced paper on the
subject. If all lines operated at constant current and all weather conditions remained independent of time, then the application of a steady
state ampacity model would be perfectly adequate and line sags could
83 WM 145-0 A paper recommended and approved by
the IEEE Transmission and Distribution Committee
of the IEEE Power Engineering Society for presentation at the IEEE/PES 1983 Winter Meeting, New
York, New York, January 30-February 4, 1983.
Manuscript submitted September 1, 1982; made
available for printing November 19, 1982.
W. R. Byrd
Exxon Corporation
Pensacola, Florida
be predicted with reasonable accuracy once the conductor temperature
was determined. Unfortunately, however, the line current is a variable
and the weather conditions are rarely constant for more than a few
minutes. Air temperatures are continually varying and the wind velocity
and direction vary in a seemingly random pattern. All of these factors
provide for transient conductor temperatures and reduce the accuracy
of a steady state ampacity program. Furthermore during emergency
conditions when line currents may vary significantly over relatively
short durations of time, a real-time ampacity program is necessary to
predict the transient temperature of the conductor.
While a real-time ampacity program is a valuable tool for a utility
engineer, the task of calculating a real-time temperature is a formidable
one and few models have been successfully formulated for the solution of the transient problem. One model [9] uses a single-step linear
integration of the House and Tuttle method. The conductor current was
assumed to experience a step change from a steady-state value to an
overload value for a specific time after which the current was reduced
to the standard 24 hour emergency loading value. Wind was assumed
constant and perpendicular to the conductor for the duration of the
transient. All heat transfer and storage terms were considered constant
overeach time interval. The simplifying assumptions used in this model
make it unsuitable for simulating a practical real-time operation.
Another model [10] to calculate real-time conductor temperatures
"linearizes" convection, radiation and heat generation, and combines
these terms into a constant. This linearization is rationalized by the
assumption of a constant conductor current and constant weather
conditions over the period of interest, an assumption which is accurate
only for short intervals of time. All papers reporting transient ampacity
models thus far have had to resort to simplifying assumptions of this
type that severely limit their accuracy and restrict their usefulness as
real-time rating schemes.
This paper formulates a thermal model which can be used to determine the relationship between the real-time conductor temperature and
the line current. The model is very general in nature. It can be applied
to a wide variety of line designs including aluminum and ACSR conductors. The model takes into consideration any change in weather
condition which may affect the conductor temperature and it is capable
of considering any variation in line current. The thermal model is formulated into a computer program that can be used to predict the real-time
conductor temperature and ultimately the instantaneous line sag. The
accuracy of the thermal model is verified by measurements made on
a full-size outdoor test facility. The measured conductor temperatures
are reported in a companion paper [11] and they are shown to be within
100C of the temperatures predicted by the thermal model. This accuracy
corresponds to within approximately 15 percent of the temperature rise
of the conductor above the temperature of the ambient air.
THERMAL MODEL
The thermal model starts with a basic energy balance on a representative segment of the conductor. The model considers convection and
radiation from the surface of the conductor, energy generation inside
the conductor due to 12R heating and storage of energy within the conductor due to its thermal capacitance. All of these components are
subject to time dependent variables such as wind speed and direction,
ambient temperature, and line current, so the solution is transient in
nature.
The strands of the conductor are assumed to be in good thermal
contact so that the temperature of all strands is identical. Therefore,
the model is unable to predict the conductor temperature when the
aluminum strands expand to such an extent that they are no longer in
contact with the steel core. Under these conditions there can be significant temperature differences between the strands.
An energy balance on a unit length of conductor results in a governing
equation which can be solved for the conductor temperature, T, as a
function of time, t, the mass of the line, m, specific heat of the line, cp,
0018-9510/83/0700-2289$01.00 © 1983 IEEE
2290
and the various contributions to the heat input to the line.
P
dT
dt
gen
sun
-
Qrad Qconv
(1)
This equation is identical to the steady state energy balance on a conductor except that the term on the left-hand side of the equation has
been inserted to include energy stored in the conductor during periods
of transient operation.
The symbols mc in Eq. 1 represent the average mass-specific heat
product of the composite conductor on a per unit length basis. The
symbol Qgen represents the rate of heat generation per unit length due
to current in the line. This term is a function of both time and conductor temperature. The current is a function of time and the conductor
resistance is a function of temperature. The term Qsun isthe rateof both
direct and diffuse solar energy absorbed per unit length of conductor.
This term is a function of time due to the variation of solar energy
incident on the conductor during the day. The term Qrad is the emitted
radiation from a unit length of conductor. This term is a function of the
conductor and environment temperatures. Finally the symbol Qconv
represents the rate of heat removed from the surface of the conductor
to the ambient air by the convection mode. This term is a function of
the conductor temperature and the instantaneous weather conditions
which are functions of time.
For an ACSR conductor the product mc should be calculated as
mcp = mScPS + mAcP
(2)
where the subscript S refers to the properties of the steel core and A
refers to properties of the aluminum strands.
The generation term in Eq. 1 is calculated from
Qgen
=
12(t) RAC(T)
The AC resistance of the conductor is assumed to be a linear function
of the conductor temperature and accounts for the skin effect and line
reactance.
The sun's energy which is absorbed per unit length of conductor
is attributed to two distinct sources. The first is energy directly
(Qsun)
incident on the conductor and the second is due to solar energy which
first reflects from the surroundings before striking the line. The total
rate at which solar energy is absorbed by a unit length of conductor is
then
Qsun
D[Is
(4)
Qdiff(t)]
where D is the conductor diameter, as is the solar absorptivity of the
line. The direct incident solar flux (Qdir) and diffuse incident solar flux
(Qdiff) are functions of date, time of day, latitude and longitude of the line,
=
Qdir(t) + 'rr a
orientation of the line and amount of cloud cover. For the purposes of
formulating a computer program to calculate both of these terms, it was
found [12] that the standard solar flux equations given in Ref. 13 were
satisfactory in estimating the total amount of solar energy corrected
for atmospheric absorption that is incident on the line.
The conductor will emit radiant energy from its surface to the surroundings and this heat loss per unit length of conductor is given by
the term Qrad in Eq. 1. Since the conductor has a relatively low
temperature, the predominate portion of the emitted radiant energy is
in the infrared wavelength range. Therefore the correct line radiative
property to be used in calculating the emitted energy is the infrared
emissivity ( e ). Assuming the portion of the surroundings that has aview
of the line has the same temperature as the ambient air, T., the net
radiant energy exchange between the conductor and the surroundings
per unit line length is
Qrad = (ID 7r a- [T4 -T4(t)]
(5)
where a- is the Stefan-Bolzmann constant an-d T is the absolute
temperature of the conductor.
The convection term, Qc
in Eq. 1 must account forfree convection
when the wind velocity is zero and for forced convection effects when
wind exists. The heat removed from the surface of the conductor per
unit length by convection to the ambient air in terms of the convective
heat transfer coefficent, h, is
Qconv
=
-
D h(t)
[T -
Too (t)]
(6)
The convective heat transfer coefficient is a complex function of conductor temperature, air temperature, wind velocity and wind direction.
For still air conditions the convective heat transfer coefficient is a
function of the Prandtl number and Grashof number and for forced
convection the Reynolds number replaces the Grashof number as the
significant dimensionless group. The free convection correlation used
in the model was Eq. 19 in Ref. 14 and the forced convection correlation
used was Eq. 4 in Ref. 14 which considers airflow normal to the axis of
the conductor. Since the wind direction is seldom truly perpendicular
to the conductor, the correlation was corrected for an arbitrary wind
angle of incidence by using Eq. 13 in Ref, 14.
Substitution of the various heat transfer terms into the basic energy
balance equation results in
dT
mc dt =12(t)Ac(T) + DItS Qdir(t) + 7r a' Qdff(t)J
-
D rr
r
[T4- T4 (t)] -
n
Dh(t)fT
-
T,co(t)J
(7)
This equation is a first order, ordinary, non-linear differential equation
which must be solved for the real-time conductor temperature. Since
Eq. 7 is non-linear, it is not reasonable to expect a closed-form analytical
solution for the conductor temperature as a function of time. However,
standard numerical techniques such as a Runge-Kutta method [15] can
be used to provide a value for the conductor temperature at discrete
time intervals.
A first order differential equation in time requires a single initial
condition. Assuming steady conductor conditions exist prior to the
transient operation, the initial 6ondition for Eq. 7 is
T
=
T0(0)
(8)
where To is the steady state conductor temperature. A value forT can
be determined by setting the left-hand side of Eq. 7 equal to zero and
solving the resulting algebriac equation with a numerical technique
such as Newton-Raphson 1151 for a single value for TO'
The complexity of the model and the necessity of repetitive application of numerical techniques suggests the use of a digital computer to
solve for the real-time conductor temperature. A computer program was
written to calculate the conductor temperature for any variation in
weather conditions and conductor current. Details of the program are
reported in Ref. 12.
The program has been used to simulate the therma-l response of many
different conductors under practically all loading and weather conditions. The program results have proven to be strictly convergent in all
cases and the predicted temperatures are stab'le at al-l times. The program has been started numerous times in the middle of a transient
situation and the temperatures converge within less than ten minutes
to the actual conductor temperatures.
RESULTS
The program, based on the thermal model described in the previous
section, was initially verified by comparing the computed steady conductor temperatures with those published by the Aluminimum Association
[16]. After verification of the steady state segment of the program, a
number of hypothetical cases were used to evaluate the sensitivity of the
conductor temperature to operating and weather conditions.
The influence of incident solar energy and changes in ambient
temperature on the line temperature are shown in Fig. 1 for a Linnet conductor (2617, 336 kcmil ACSR) located in the Atlanta area during the summer. The curves show that the presence of the sun changes the conductor temperature by about 200C. This result suggests that when- all
other factors are held constant, the nighttime temperatures of a conductor will be approximately 200C less than summer, daytime temperatures. Furthermore, the curves in Fig. I are inclined at a 450 angle
indicating that an increase in ambient temperature of 100C will cause
an increase in conductor temperature of approximately 100C. This
result implies that it is a fairly simple matter to modify conductor temperatures calculated on the basis of one air temperature so that they
will apply to the case of a different ambient air temperature. A similar
method of correcting conductor temperatures for different ambient
temperatures is outlined in Ref. 17.
The dependence of the steady state conductor temperature on wind
direction and wind velocity is shown in Fig. 2. The influence of the wind
conditions on the conductor temperature is much more significant than
the effect of the sun. In fact, a study [12] of the comparitive roles that
120
Linnet Conductor
2 ft/s Cross-wind
x
//
E. = 0.76
//
June 9, 1 P.M.
100
- Atlanta Area
Cu~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Z
0
I-~~~~~~~~~~~~~~~~~I
0
60
0
C.)
40
'20
0
-20
20
Ambient Temperature OC
40
Fig. 1. Effect of Ambient Air Temperature on Conductor Temperature
120
-
\
2291
radiation and convection play on establishing the conductor temperature has shown that for a 2 ft./sec. cross-wind, convection typically
transfers between 5 and 10 times more heat from the surface of the
conductorthan radiation. For typical operating conditions, the incident
solar energy represents approximately 10 to 25 percent of the total
heat input to the conductor. The influence of radiation decreases as the
temperature of the conductor increases.
The curves in Fig. 2 show that conductor temperature is greatly
influenced by wind direction, but this dependence disappears as the
wind velocity is reduced to zero. During gusting wind conditions or situations in which the wind direction changes rapidly, the conductor
temperature can be expected to reflect these changes. The actual
response of the conductor temperature will be delayed from the variations in the weather conditions due to the thermal capacitance of the
conductor. Larger conductor sizes possess larger thermal capacitance
and their response to rapid changes in weather conditions or line current will be less amplified and more delayed than for smaller conductors. It should also be evident that the assumption of a 2 ft./sec. crosswind used by many utilities is not necessarily conservative when rating
conductors. A wind direction along the axis is just as probable as a
cross-wind and the variation of conductor temperature with wind direction for the case shown in Fig. 2 is as great as 350C for a 2 ft./sec. wind
and 550C for a 5 ft./sec. wind.
Radiation plays a secondary role in establishing the conductor temperature, while convection is the most influential factor which dictates
line ampacity. Therefore characterizing the convective mode is the
single most important task in formulating a real-time ampacity model.
As a result, those weather parameters such as wind velocity and direction which determine the convection heat transfer rate must be accurately measured at the conductor location if the program is expected
to return accurate results. However, because radiation is of lesser
importance, the incident solar energy does not need to be measured,
but simply can be approximated by using existing programs without any
significant loss in accuracy of a real-time ampacity calculation.
The values of the radiative properties as and ae influence the conductor temperature because they affect the amount of absorbed solar
energy and the amount of radiant energy emitted from the surface of
the conductor. A detailed study [l12] of these two properties has shown
that during daylight hours the difference between the emissivity and
absorptivity is more important in determining the conductor tempera-
0 fts Wind
90 -
85 -
C) 800
S0-
E
0
0
=
75 -
0
0
70
Parallel
Flow
-
Wind Incidence Angle degrees
Fig. 2. Influence of Wind Speed and Direction on Conductor
Temperature
0.1
0.2
0.3
0.4
0.5
Conductor Infrared Emissivity
I
0.6
0.7
0.8
el
Fig. 3. Effect of Radiative Properties on Conductor Temperature
0.9
2292
ture than the actual value of either property. That is, conductor temperatures calculated for e = 0.1 and as = 0.3 ( as - E = 0.2) are
practically the same as the temperature when the properties are E = 0.8
and a = 1.0( as - el = 0.2).ThistrendisshowninFig.3whichplots
the temperature of a Linnet conductor for different values of infrared
emissivity and solar absorptivity for solar conditions that exist on a
typical summer day. The results show that the magnitude of both as
and e have a strong influence on the conductor temperature with the
temperature decreasing for increasing values of e and decreasing
values of as. However the dashed curves plotted for a constant difpractically horizontal indicating only a
ference between a sand e
slight influence on the conductor temperature.
The magnitude of the infrared emissivity is important in determining
the conductor temperature during the nighttime when the absorbed
solar radiation is zero. The effect of e on the conductor temperature
during the night can be determined by observing the trend of the a s = 0
curve in Fig. 3. As the infrared emissivity decreases, the rate at which
heat is emitted from the conductor decreases and the conductor
temperature increases.
Most utilities assume the emissivity and absorptivity of the conductor
are equal. This is a common assumption used to simplify radiation
calculations. The basis for this assumption is the gray body approximation which assumes that the properties of the conductor are indepen-
160t
dent of wavelength. Unfortunately the distribution of incoming solar
radiation and outgoing emitted energy occurs over a very broad wavelength range and the gray body approximation is not satisfactory for
the case of an overhead conductor. Fortunately, the emissivity and
absorptivity have been sliown [18] to differ by about 0.2 regardless of
the age of the conductor, so the selection of the magnitude of the two
properties is not too critical as long as they always differ by 0.2. If the
absorptivity and emissivity are assumed to be equal, the thermal model
will be in error and the use of a high absorptivity in an attempt to determine a conservative line rating will actually result in an ampacity that
is too high.
Even though the real value of a transient ampacity program resides
in its ability to predict the conductor temperature under realistic or
actual operating conditions, considerable knowledge can be gained by
predicting the real-time response of conductors subject to hypothetical
changes in operating conditions. Figure 4 shows the response of a
Linnet conductor to a simple step change in current. The time constant
of the conductor is the time required for the conductor to reach 63 percent of its ultimate steady state temperature rise when subject to a step
,
500 Percent Current Overload
900 Amps
I
Time
A.M.
P.M.
Fig. 6. Typical Daily Response of Conductor Temperature
Fig. 4. Response to Step Change in Current
100
Linnet Conductor
Nighttime Rating
250C Air
Temperature
2 ft/s Wind
le = 0.76
Nighttime Ratings
0
a
250C Air
Temperature
80
E 750 -
E
0
= 600-
E
I-. 0
6a
I
0
0t
0~
40)
20
-
Curve
Conductor
1
2
3
4
5
6
7
8
Linnet
Linnet
Linnet
Linnet
Grackle
Grackle
Grackle
Grackle
From
Amps
495
495
495
495
1108
1108
1108
1108
Cret
To
Amps
743
743
743
743
1662
1662
1662
1662
Wind
Time
Witts)
CT
Constant
(Minutes)
5
5
2
2
5
5
2
2
0.76
0.23
0.76
0.23
0.76
0.23
0.76
0.23
5.5
6.5
7.5
11.0
12.5
15.5
18.0
25.5
10
20
40
Time Minutes
Fig. 5. Conductor Time Constants
60
20
30
40
Time minutes
Fig. 7. Response of a Linnet Conductor to 150 Amp/Minute
Ramp Current Change
50
2293
change in current. The time constant for a Linnet conductor is approximately 8 minutes, regardless of the percent increase in current.
Time constants for other conditions are given in Fig. 5 for two conductor sizes assuming a 50 percent step increase in normal ampacity.
Time constants are strongly dependent on prevailing weather conditions and the size of the conductor. The time constant of the conductor
increases for conditions which reduce heat transfer from the surface
of the conductor, such as low wind velocity and low values for surface
emissivity. The time constant increases as the conductor size increases,
because larger conductors have a greater thermal capacitance. For
example, a Grackle conductor (1192 kcmil) has a time constant up to
25 minutes for a 2 ft./sec. cross-wind and e = 0.23.
The curve in Fig. 6 shows the transient response of a Linnet conductor to a constant current and a hypothetical variation in daily weather
conditions. The ambient air temperature is assumed to vary sinusoidally
from 15 to 300C with the minimum and maximum temperatures occurring at 5 a.m. and 5 p.m., respectively. The dashed line shows the predicted conductor temperature without considering any incident solar
energy and the solid line during the day includes the sun's energy. The
difference between the two daytime curves shows the influence of the
sun on the temperature of the conductor.
The transient response of a Linnet conductor to a ramp change in
current from 600 amps to 750 amps over a period of 1.0 minute is shown
in Fig. 7 where it is compared to the response predicted by a steady state
model. It is apparent from the differences between the two curves that
a steady state analysis leads to temperatures that are far in excess of
the temperatures that are predicted by a real-time analysis. The error
is due to the significant thermal capacitance of the conductor which
is ignored in the steady state analysis. Notice that the conductor has
not reached its steady value even 25 minutes after the application of
the ramp change in current.
CONCLUSIONS
A thermal model has been formulated and an associated computer
has been written which can calculate the real-time temperature
of overhead conductors. The computer program will prove to be a
valuable tool for an engineer who is responsible for designing and safely
operating overhead lines. The program will aid in the design stages
when new conductor sizes are being selected and in the operating
stages when conductors are called upon to carry contingency or peak
loads. The program results show that the convective mode of heat
transfer is more important than the radiation mode in determining the
conductor temperature. The wind velocity and direction are the two
most important factors which specify the convection from the surface
of the conductor. Therefore, a knowledge of the wind velocity and direction at the line location is essential if the computer program is expected
to accurately predict the conductor temperature.
Since incident solarenergy plays a secondary role in establishing the
conductor temperature, it is not necessary to measure solar heating at
the conductor. Solar energy incident on the conductor can simply be
estimated by using established programs. Errors caused by estimating
the solar flux have been shown to be insignificant when calculating the
line ampacity. As a result of this conclusion, a weather station used to
provide input data to the real-time ampacity program needs only to
measure ambient air temperature, wind velocity and wind direction. This
eliminates the necessity of measuring solar flux, a measurement which
is expensive to achieve.
Time constants resulting from step-changes in line current for typical
overhead conductors range between approximately 5 and 30 minutes.
The time constant is insensitive to the magnitude of the current
overload and it decreases for those conditions which increase heat
transfer from the surface of the conductor. Due to the thermal capacitance of the conductor, the actual response of a conductor can be
significantly different from the temperature predicted by a steady state
ampacity model.
program
REFERENCES
[1] Schurig, 0. R. and Frick, C. W., "Heating and Current-carrying
Capacity of Bare Conductors for Outdoor Service," General Electric Review, Vol. 33, No. 3, pp. 141-157, March, 1930.
[2] Enos, H. A., "Current Carrying Capacity of Overhead Conductors,"
Electric World, pp. 60-62, May, 1943.
[3] Waghorne, J. H. and Ogorodnikov, V. E., "Current Carrying Capacity
of ACSR Conductors," AIEE Trans., Vol. 70, pp. 1159-62, 1951.
[4] Beers, G. M., Gilligam, S. R., Lis, H.W. and Schamberger, J. M.,
"Transmission Conductor Ratings," IEEE Trans., Vol. PAS-82, No.
10, pp. 767-75, Oct., 1963.
[5] Davidson, Glenn A., Donoho, T. E., Hakun, G., Hofmann, P. W.,
Bethke, T. E., Landrieu, P. R. H., McElhaney, R. T., "Thermal Ratings
for Bare Overhead Conductors," IEEE Trans., Vol. PAS-88, No. 3,
pp. 200-205, March, 1969.
[6] Koral, D. 0. and Billington, Roy, "Determination of Transmission
Line Ampacities by Probability and Numerical Methods," IEEE
Trans., Vol. PAS-89, No. 7, pp. 1485-92, Sept./Oct., 1970.
[7] Davis, Murray W., "A NewThermal Rating Approach: The Real Time
Thermal Rating System For Strategic Overhead Conductor
Transmission Lines - Part I, General Description and Justification of the Real Time Thermal Rating System," IEEE Trans., Vol.
PAS-96, No. 3, pp. 803-809, May/June, 1977.
[8] House, H. E. and Tuttle, P. D., "Current-Carrying Capacity of ACSR,"
AIEE Trans., pp. 1-9, February, 1958.
[9] Davidson, G. A., Donoho, T. E., Landrieu, P. R. H., McElhaney, R.
T., and Saeger, J. H., "Short-Time Thermal Ratings for Bare
Overhead Conductors," IEEE Trans., Vol. PAS-88, No. 3, pp. 194-199,
March, 1969.
[10] Wong, T. Y., Findlay, J. A. , and McMurtrie, A. N., "An On-Line
Method forTransmission Ampacity Evaluation," IEEE Trans., Vol.
PAS-101, No. 2, February, 1982.
[11] Bush, R. A., Black, W. Z., Champion, T. C. and Byrd, W. R., "Experimental Verification of a Real-Time Program forthe Determination
of Temperature and Sag of Overhead Lines," to be published in
IEEE-PAS, Paper No. TP&C-11-WPM-83.
[12] Byrd, W. R., "Transient Thermal Model of Overhead Electric Lines,"
M.S. Thesis, Georgia Institute of Technology, School of Mechanical
Engineering, June 1982. (Available through Price Gilbert Memorial
Library, Georgia Institute of Technology, Atlanta, GA.)
[13] Duffie, J. A., and Beckman, W. A., SolarEnergy ThermalProcesses,
A Wiley-Interscience Publication, 1974, John Wiley and Sons, New
York, NY.
[14] Davis, M. W., "A New Thermal Rating Approach: The Real Time
Thermal Rating System for Strategic Overhead Conductor
Transmission Lines, Part II: Steady-State Thermal Rating Program," IEEE Trans., Vol. PAS-96, pp. 810-825, May/June, 1977.
[15] Carnahan, B., Luther, H. A., and Wilkes, J. O., Applied Numerical
Methods, 1969 John Wiley and Sons, New York, NY.
[16] Simpson, Tracy W. and Greenfield, Eugene W., Aluminum Electrical
Conductor Handbook, Sept., 1971, Published by the Aluminum
Association, New York, NY.
[17] Hazen, Earl, "Extra-High Voltage Single and Twin Bundle Conductors," AIEE Trans. Part ll-B PAS, pp. 1425-1434, Dec., 1959.
[18] House, H. E., Rigdon, W. S., Grosh, R. J., and Cottingham, W. B.,
"Emissivity of Weathered Conductors After Service in Rural and
Industrial Environments," AIEE Trans., Vol. PAS-81, pp. 891-896,
February, 1963.
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