COST OF CONDUCTORS FOR ELECTRIC-POWER TRANSMISSION.
Electrical transmission of energy involves problems quite distinct from
its development. A great water-power, or a location where fuel is cheap,
may offer opportunity to generate electrical energy at an exceptionally
low cost. This energy may be used so close to the point of its
development that the cost of transmission is too small for separate
consideration.
An example of conditions where the important problems of transmission
are absent exists in the numerous factories grouped about the great
water-power plants at Niagara and drawing electrical energy from it. In
such a case energy flows directly from the dynamos, driven by
water-power, to the lamps, motors, chemical vats, and electric heaters
of consumers through the medium, perhaps, of local transformers. Here
the costs and losses of transmitting or distributing equipments are
minor matters, compared with the development of the energy.
If, now, energy from the water-power is to be transmitted over a
distance of many miles, a new set of costs is to be met. In the first
place, it will be necessary to raise the voltage of the transmitted
energy much above the pressure at the dynamos in order to save in the
weight and cost of conductors for the transmission line. This increase
of voltage requires transformers with capacity equal to the maximum rate
at which energy is to be delivered to the line. These transformers will
add to the cost of the energy that they deliver in two ways: by the
absorption of some energy to form heat, and by the sum of annual
interest, maintenance, and depreciation charges on the price paid for
them. Other additions to the cost of energy delivered by the
transmission line must be made to cover the annual interest,
maintenance, and depreciation charges on the amount of the line
investment, and to pay for the energy changed to heat in the line.
Near the points where the energy is to be used, the transmission line
must end in transformers to reduce the voltage to a safe figure for
local distribution. This second set of transformers will further add to
the cost of the delivered energy in the same ways as the former set.
From these facts it is evident that, to warrant an electrical
transmission, the value of energy at the point of distribution should at
least equal the value at the generating plant plus the cost of the
transmission. Knowing the cost of energy at one end of the transmission
line and its value at the other, the difference between these two
represents the maximum cost at which the transmission will pay.
Three main factors are concerned in the cost of electric power
transmission, namely, the transformers, the pole line, and the wire or
conductors. These factors enter into the cost of transmitted energy in
very different degrees, according to the circumstances of each case. The
maximum and average rates of energy transmission, the total voltage, the
percentage of line loss, and the length of the line mainly determine the
relative importance of the transformers, pole line, and conductors in
the total cost of delivered energy.
First cost of transformers varies directly with the maximum rate of
transmission, and is nearly independent of the voltage, the length of
the transmission, and the percentage of line loss. A pole line changes
in first cost with the length of the transmission, but is nearly
independent of the other factors. Line conductors, for a fixed maximum
percentage of loss, vary in first cost directly with the square of the
length of the transmission and with the rate of the transmission; but
their first cost decreases as the percentage of line loss increases and
as the square of the voltage of transmission increases.
If a given amount of power is to be transmitted, at a certain percentage
of loss in the line and at a fixed voltage, over distances of 50, 100,
and 200 miles, respectively, the foregoing principles lead to the
following conclusions: The capacity of transformers, being fixed by the
rate of transmission, will be the same for either distance, and their
cost is therefore constant. Transformer losses, interest, depreciation,
and repairs are also constant. The cost of pole line, depending on its
length, will be twice as great at 100 and four times as great at 200 as
at 50 miles. Interest, depreciation, and repairs will also go up
directly with the length of the pole lines.
Line conductors will cost four times as much for the 100- as for the
50-mile transmission, because their weight will be four times as great,
and the annual interest and depreciation will go up at the same rate.
For the transmission of 200 miles the cost of line conductors and their
weight will be sixteen times as great as the cost at 50 miles. It
follows that interest, depreciation, and maintenance will be increased
sixteen times with the 200-mile transmission over what they were at 50
miles, if voltage and line loss are constant.
A concrete example of the cost of electric power transmission over a
given distance will illustrate the practical application of these
principles. Let the problem be to deliver electrical energy in a city
distant 100 miles from the generating plant. Transformers with
approximately twice the capacity corresponding to the maximum rate of
transmission must be provided, because one set is required at the
generating and another at the delivery station. The cost of these
transformers will be approximately $7.50 per horse-power for any large
capacity.
Reliability is of the utmost importance in a great power transmission,
and this requires a pole line of the most substantial construction. Such
a line in a locality where wooden poles can be had at a moderate price
will cost, with conductors in position, about $700 per mile, exclusive
of the cost of the conductors themselves or of the right of way but
including the cost of erecting the conductors. The 100 miles of pole
line in the present case should, therefore, be set down at a cost of
$70,000.
A large delivery of power must be made to warrant the construction of so
long and expensive a line, and 10,000 horse-power may be taken as the
maximum rate of delivery. On the basis of two horse-power of transformer
capacity for each horse-power of the maximum delivery rate, transformers
with a capacity of 20,000 horse-power are necessary for the present
transmission. At $7.50 per horse-power capacity, the first cost of these
transformers is $150,000.
Before the weight and cost of line conductors can be determined, the
voltage at which the transmission shall be carried out and the
percentage of the energy to be lost in the conductors at periods of
maximum load must be decided on. The voltage to be used is a matter of
engineering judgment, based in large part on experience, and cannot be
determined by calculation. In a transmission of 100 miles the cost of
conductors is certain to be a very heavy item, and, as this cost
decreases as the square of the voltage goes up, it is desirable to push
the voltage as high as the requirements for reliable service permit.
A transmission line 142 miles long, from the mountains to Oakland, Cal.,
has been in constant and successful use for several years with 40,000
volts pressure. This line passes through wet as well as dry climate. It
seems safer to conclude, therefore, that 40,000 volts may be used in
most places with good results.
Having decided on the amount of power and the voltage and length of the
transmission, the required weight of conductors will vary inversely as
the percentage of energy lost as heat in the line. The best percentage
of loss depends on the number of factors, some of which, such as the
cost of energy at the generating plant, are peculiar to each case.
As a provisional figure, based in part on the practice elsewhere, the
loss on the line here considered may be taken at 10 per cent. when
transmitting the full load of 10,000 horse-power. If the line is
constructed on this basis the percentage of loss will be proportionately
less for any smaller load. Thus, when the line is transmitting only
5,000 horse-power, the loss will amount to 5 per cent. During the
greater portion of each day the demand for power is certain to be less
than the maximum figure, so that a maximum loss of 10 per cent will
correspond to an average loss on all the power delivered to the line of
probably less than 7 per cent.
In order to deliver 10,000 horse-power by the transformers at a
receiving station from a generating plant 100 miles distant where the
pressure is 40,000 volts, the copper conductors must have a weight of
about 1,500,000 pounds, if the loss of energy in them is 10 per cent of
the energy delivered to the line. Taking these conductors at a medium
price of 15 cents per pound, their cost amounts to $225,000.
The combined cost of the transformers, pole line, and line conductors,
as now estimated, amounts to $445,000. No account is taken of the
right-of-way for the pole line, because in many cases this would cost
nothing, the public roads being used for the purpose; in other cases the
cost might vary greatly with local conditions.
The efficiency of the transmission is measured by the ratio of the
energy delivered by the transformers at the receiving station for local
distribution to the energy delivered by the generating plant to the
transformers that supply energy to the line for transmission. If worked
at full capacity the large transformers here considered would have an
efficiency of nearly 98 per cent; but as they must work, to some extent,
on partial loads, the actual efficiency will hardly exceed 96 per cent.
The efficiency of the line conductors rises on partial loads, and may be
safely taken at 93 per cent for all of the energy transmitted, though it
is only 90 per cent on the maximum load. The combined efficiencies of
the two sets of transformers and the line give the efficiency of the
transmission, which equals the product of 0.96 × 0.93 × 0.96, or almost
exactly 85.7 per cent. In other words, the transformers at the
water-power station absorb 1.17 times as much energy as the transformers
at the receiving station deliver to distribution lines in the place of
use.
Interest, maintenance, and depreciation of this complete transmission
system are sufficiently provided for by an allowance of 15 per cent
yearly on its entire first cost. As the total first cost of the
transmission system was found to be $445,000, the annual expense of
interest, depreciation, and repairs at 15 per cent of this sum amounts
to $66,750.
In order to find the bearings of this annual charge on the cost of power
transmission the total amount of energy transmitted annually must be
determined. The 10,000 horse-power delivered by the system at the
sub-station is simply the maximum rate at which energy may be supplied,
and the element of time must be introduced in order to compute the
amount of transmitted energy. If the system could be kept at work during
twenty-four hours a day at full capacity, the delivered energy would be
represented by the product of the numbers which stand for the capacity
and for the total number of hours yearly.
Unfortunately, however, the demands for electric light and power vary
through a wide range in the course of each twenty-four hours, and the
period of maximum demand extends over only a small part of each day. The
problem is, therefore, to find what relation the average load that may
be had during the twenty-four hours bears to the capacity required to
carry this maximum load. As the answer to this question depends on the
power requirements of various classes of consumers, it can be obtained
only by experience. It has been found that some electric stations,
working twenty-four hours daily on mixed loads of lamps and stationary
motors, can deliver energy to an amount represented by the necessary
maximum capacity during about 3,000 hours per year. Applying this rule
to the present case, the transformers at the sub-station, if loaded to
their maximum capacity of 10,000 horse-power by the heaviest demands of
consumers, may be expected to deliver energy to the amount of 3,000 ×
10,000 = 30,000,000 horse-power hours yearly.
The total cost of operation for this transmission system was found above
to be $66,750 per annum, exclusive of the cost of energy at the
generating plant. This sum, divided by 30,000,000, shows the cost of
energy transmission to be 0.222 cent per horse-power hour, exclusive of
the first cost of the energy. To obtain the total cost of transmission,
the figures just given must be increased by the value of the energy lost
in transformers and in the line conductors. In order to find this value,
the cost of energy at the generating plant must be known.
The cost of electrical energy at the switchboard in a water-power
station is subject to wide variations, owing to the different
investments necessary in the hydraulic work per unit of power developed.
With large powers, such as are here considered, a horse-power hour of
electrical energy may be developed for materially less than 0.5 cent in
some plants. As the average efficiency of the present transmission has
been found to be 85.7 per cent of the energy delivered by the
generators, it is evident that 1.17 horse-power hours must be drawn from
the generators for every horse-power hour supplied by the transformers
at the sub-station for distribution. In other words, 0.17 horse-power
hour is wasted for each horse-power hour delivered.
The cost of 0.17 of a horse-power hour, or say not more than 0.5 × 0.17
= 0.085 cent, must thus be added to the figures for transmission cost
already found, that is, 0.222 cent per horse-power hour, to obtain the
total cost of transmission. The sum of these two items of cost amounts
to 0.307 cent per horse-power hour, as the entire transmission expense.
It may now be asked how the cost of transmission just found will
increase if the distance be extended. As an illustration, assume the
length of the transmission to be 150 instead of 100 miles. Let the
amount of energy delivered by the sub-station, the loss in line
conductors, and the energy drawn from the generating plant remain the
same as before. Evidently the cost of the pole line will be increased 50
per cent, that is, from $70,000 to $105,000. Transformers, having the
same capacity, will not be changed from the previous estimate of
$150,000. If the voltage of the transmission remain constant, as well as
the line loss at maximum load, the weight and cost of copper conductors
must increase with the square of the distances of transmission. For 150
miles the weight of copper will thus be 2.25 times the weight required
for the 100-mile transmission.
Instead of an increase in the weight of conductors a higher voltage may
be adopted. The transformers for the two great transmission systems that
extend over a distance of about 150 miles, from the Sierra Nevada
Mountains to San Francisco Bay, in California, are designed to deliver
energy to the line at either 40,000 or 60,000 volts, as desired. Though
the regular operation at first was at the lower pressure, the voltage
has been raised to 60,000.
The lower valleys of the Sacramento and the San Joaquin rivers, which
are crossed by these California systems, as well as the shores of San
Francisco Bay, have as much annual precipitation and as moist an
atmosphere as most parts of the United States and Canada. Therefore
there seems to be no good reason to prevent the use of 60,000 volts
elsewhere.
The distance over which energy may be transmitted at a given rate, with
a fixed percentage of loss and a constant weight of copper, goes up
directly with the voltage employed. This rule follows because, while the
weight of conductors to transmit energy at a given rate, with a certain
percentage of loss and constant voltage, increases as the square of the
distance, the weight of conductors decreases as the square of the
voltage when all the other factors are constant.
Applying these principles to the 150-mile transmission, it is evident
that an increase of the voltage to 60,000 will allow the weight of
conductors to remain exactly where it was for the transmission of 100
miles, the rate of working and the line loss being equal for the two
cases.
The only additional item of expense in the 150-mile transmission, on the
basis of 60,000 volts, is the $35,000 for pole line. Allowing 15 per
cent on the $35,000 to cover interest, depreciation, and maintenance, as
before, makes a total yearly increase in the costs of transmission of
$5,250 over that found for the transmission of 100 miles. This last sum
amounts to 0.0175 cent per horse-power hour of the delivered energy.
The cost of transmission is thus raised to 0.307 + 0.0175 = 0.324 cent
per horse-power hour of delivered energy on the 150-mile system with
60,000 volts.
Existing transmission lines not only illustrate the relations of the
factors named above to the cost and weight of conductors, but also show
marked variations of practice, corresponding to the opinions of
different engineers. In order to bring out the facts on these points,
the data of a number of transmission lines are here presented. On these
lines the distance of transmission varies between 5 and 142 miles, the
voltage from 5,000 to 50,000, and the maximum rate of work from a few
hundred to some thousands of horse-power. For each transmission the
single length and total weight of conductors, the voltage, and the
capacity of the generating equipment that supplies the line is recorded.
From these data the volts per mile of line, weight and cost of
conductors per kilowatt capacity of generating equipment, and the weight
of conductors per mile for each kilowatt of capacity in the generating
equipment are calculated. In each case the length of line given is the
distance from the generating to the receiving station. The capacity
given for generating equipment in each case is that of the main dynamos,
where their entire output goes to the transmission line in question, but
where the dynamos supply energy for other purposes also, the rating of
the transformers that feed only the particular transmission line is
given as the capacity of generating equipment.
DISTANCE AND VOLTAGE OF ELECTRICAL TRANSMISSION.
+------------------------------------+-----------+--------+---------+
| |Distance in| Volts. |Volts per|
| | Miles. | | Mile. |
+------------------------------------+-----------+--------+---------+
|Colgate to Oakland, Cal. | 142 | 60,000 | 422 |
|Cañon Ferry to Butte, Mont. | 65 | 50,000 | 769 |
|Santa Ana River to Los Angeles | 83 | 33,000 | 397 |
|Ogden to Salt Lake City, Utah | 36.5 | 16,000 | 438 |
|Madrid to Bland, N. M. | 32 | 20,000 | 625 |
|Welland Canal to Hamilton, Can. | { 35 | 22,500 | 643 |
| | { 37 | | |
|San Gabriel Cañon to Los Angeles | 23 | 16,000 | 695 |
|Cañon City to Cripple Creek, Colo. | 23.5 | 20,000 | 851 |
|Apple River to St. Paul, Minn. | 25 | 25,000 | 1,000 |
|Yadkin River to Salem, N. C. | 14.5 | 12,000 | 827 |
|Into Victor, Colo. | 8 | 12,600 | 1,575 |
|Montmorency Falls to Quebec | 7 | 5,500 | 785 |
|Farmington River to Hartford | 11 | 10,000 | 909 |
|Sewall’s Falls to R.R. shops Concord| 5.5 | 10,000 | 1,818 |
|Wilbraham to Ludlow Mills | 4.5 | 11,500 | 2,555 |
|To Dales, Ore. | 27 | 22,000 | 814 |
+------------------------------------+-----------+--------+---------+
The transmission systems here considered have been selected because it
was possible to obtain the desired data as to each, and it may be
presumed that they fairly illustrate present practice. It may be noted
at once that in general the line voltage is increased with the length of
the transmission. Thus, the transmission for the Ludlow Mills over a
distance of 4.5 miles is carried out at 11,500 volts. On the other hand,
the transmission between Cañon Ferry and Butte, a distance of 65 miles,
employs 50,000 volts and represents recent practice. The system from
Colgate to Oakland, a distance of 142 miles, the longest here
considered, now has 60,000 volts on its lines. In spite of the general
resort to high pressures with greater distances of transmission, the
rise in voltage has not kept pace with the increasing length of line.
For the Wilbraham-Ludlow transmission the total pressure amounts to
2,555 volts per mile, while the line from Colgate to Oakland with 31.5
times the length of the former operates at an average of only 422 volts
per mile. Of the fifteen transmissions considered, six are over
distances of less than 15 miles, and for four of the six the voltage is
more than 900 per mile. Eight transmissions range from 23 to 83 miles in
length, with voltages that average between 1,000 volts per mile at 25
miles and only 397 per mile on the 83-mile line. The volts per mile are
6 times as great in the Ludlow as in the Oakland transmission.
CAPACITY OF GENERATING STATIONS AND WEIGHT OF CONDUCTORS.
+---------------------------------+-----------+------------+------------+
| | Kilowatt |Total Weight| Pounds of |
| |Capacity at| of | Conductors |
| Location of Transmission. |Generators.|Conductors. |per Kilowatt|
| | | | Capacity. |
+---------------------------------+-----------+------------+------------+
|Wilbraham to Ludlow | 4,600 | 17,820 | 3.7[A] |
|Sewall’s Falls to railroad shops | 50 | 6,914 | 15 |
|Into Victor, Colo. | 1,600 | 15,960 | 10 |
|To Dales, Ore. | 1,000 | 33,939 | 34 |
|Apple River to St. Paul | 3,000 | 159,600 | 53 |
|Farmington River to Hartford | 1,500 | 54,054 | 36 |
|Cañon City to Cripple Creek | 1,500 | 59,079 | 39 |
|Yadkin River to Salem | 1,500 | 58,073 | 39 |
|Montmorency Falls to Quebec | 2,400 | 189,056 | 79 |
|Cañon Ferry to Butte | 5,700 | 658,320 | 115 |
|San Gabriel Cañon to Los Angeles | 1,200 | 73,002 | 61 |
|Welland Canal to Hamilton | 6,000 | 376,494 | 63 |
|Madrid to Bland, N. M. | 600 | 127,680 | 212 |
|Ogden to Salt Lake City | 2,250 | 292,365 | 129 |
|Santa Ana River to Los Angeles | 2,250 | 664,830 | 295 |
|Colgate to Oakland | 11,250 | {906,954 | 81 |
| | | {446,627 | 40[A] |
+---------------------------------+-----------+------------+------------+
[A] Aluminum.
These wide variations in the volts per mile on transmission lines and in
length of lines lead to different weights of conductors per kilowatt of
generator capacity. All other factors remaining constant, the weight of
conductors per kilowatt of generator capacity would be the same whatever
the length of the transmission, provided that the volts per mile were
uniform for all cases. One important factor, the percentage of loss for
which the line conductors are designed at full load, is sure to vary in
different cases, and lead to corresponding variations in the weights of
conductors per kilowatt of generator capacity. In conductors of equal
length one pound of aluminum has nearly the same electrical resistance
as two pounds of copper, and this ratio must be allowed for when copper
and aluminum lines are compared.
From the table it may be seen that the weight of conductors per kilowatt
of generator capacity for the transmission from Santa Ana River is 29.5
times as great as the like weight for the line into Victor. But the
volts per mile are four times as great on the Victor as they are on the
Santa Ana River line. The extreme range of the cases presented is that
between the Ludlow plant, with the equivalent of 7.4 pounds, and the
Santa Ana River system with 295 pounds of copper conductors per kilowatt
of generator capacity. Three transmissions with 1,575 to 2,555 volts
per mile have the equivalent of 7.4 to 15 pounds of copper each, per
kilowatt of generator capacity.
WEIGHT AND COST OF CONDUCTORS.
+--------------------------------+----------+-----------+
| |Pounds per|Dollars per|
| | Kilowatt | Generator |
| | Mile. | Kilowatt. |
+--------------------------------+----------+-----------+
|Wilbraham to Ludlow | 0.86[A] | 1.11 |
|Sewall’s Falls to railroad shops| 2.7 | 2.25 |
|Into Victor, Colo. | 0.9 | 1.50 |
|To Dales, Ore. | 1.2 | 5.10 |
|Apple River to St. Paul | 2.1 | 7.95 |
|Farmington River to Hartford | 3.2 | 10.80 |
|Cañon City to Cripple Creek | 1.6 | 5.85 |
|Yadkin River to Salem | 2.6 | 5.85 |
|Montmorency Falls to Quebec | 11.2 | 11.85 |
|Cañon Ferry to Butte | 1.7 | 17.25 |
|San Gabriel Cañon to Los Angeles| 2.6 | 9.85 |
|Welland Canal to Hamilton | 1.7 | 9.45 |
|Madrid to Bland, N. M. | 6.6 | 31.80 |
|Ogden to Salt Lake City | 3.5 | 19.35 |
|Santa Ana River to Los Angeles | 3.5 | 44.25 |
|Colgate to Oakland | { .56 | 24.15 |
| | { .27[A] | |
+--------------------------------+----------+-----------+
[A] Aluminum.
Of the seven transmissions using between 36 and 79 pounds of copper for
each kilowatt of generator capacity, four have voltages ranging from 827
to 1,000 per mile, and on only one is the pressure as low as 643 volts
per mile. Five transmission lines vary between 115 and 295 pounds of
copper, or its equivalent, per kilowatt of generator capacity, and their
voltages per mile are as high as 769 in one case and down to 281 in
another. Allowing for some variations in the percentages of loss in
transmission lines at full load, the fifteen plants plainly illustrate
the advantage of a high voltage per mile, as to the weight of
conductors. This advantage is especially clear if the differences due to
the lengths of the transmissions are eliminated by dividing the weight
of conductors per kilowatt of generator capacity in each case by the
length of the transmission in miles. This division gives the weight of
conductors per kilowatt of generators for each mile of the line, which
may be called the weight per kilowatt mile. For the Ludlow transmission
this weight is only 0.86 pound of aluminum, the equivalent of 1.72
pounds of copper, while the like weight for the line into Quebec is 11.2
pounds of copper, or 6.5 times that for the former line. But the
voltage per mile on the Ludlow is 3.2 times as great as the like voltage
on the Quebec line.
The weight of conductor per kilowatt mile in the Victor line is only 0.9
pound, and the like weight for the line between Madrid and Bland is 6.6
pounds, or 7.3 times as great. On the Victor line the voltage per mile
is 2.5 times as great as the voltage for each mile of the Bland line.
Comparing systems with nearly equal voltages per mile, it appears in
most cases that only such difference exists in their pounds of
conductors per kilowatt mile as may readily be accounted for by designs
for various percentages of loss at full load. Though the transmission
line into Butte is nearly twice as long as the one entering Hamilton,
the weight of conductors for each is 1.7 pounds per kilowatt mile. The
line from Santa Ana River is more than twice as long as the one entering
Salt Lake City, but its voltage per mile is only nine per cent less, and
there are 3.5 pounds of copper in each line per kilowatt mile.
The final, practical questions as to conductors in electrical
transmission relate to their cost per kilowatt of maximum working
capacity, and per kilowatt hour of delivered energy. If the cost of
conductors per kilowatt of generator capacity is greater than that of
all the remaining equipment, it is doubtful whether the transmission
will pay. If fixed charges on the conductors more than offset the
difference in the cost of energy per kilowatt hour at the points of
development and delivery, it is certain that the generating plant should
be located where the power is wanted. The great cost of conductors is
often put forward as a most serious impediment to long-distance
transmission, and the examples here cited will indicate the weight of
this argument. In order to find the approximate cost of conductors per
kilowatt of generator capacity for each of the transmission lines here
considered, the price of bare copper wire is taken at 15 cents, and the
price of bare aluminum wire at 30 cents per pound. In each case the
weight of copper or aluminum conductor per kilowatt of generator
capacity is used to determine their costs per kilowatt of this capacity
at the prices just named. This process when carried out for the 15
transmission lines shows that their cost of conductors per kilowatt of
generator capacity varies between $1.11 for the 4.5 mile line into
Ludlow and $44.25 for the line of 83 miles from the Santa Ana River. It
should be noted that the former of these lines operates at 2,555 and the
latter at 397 volts per mile. The line into Madrid shows an investment
in conductors of $31.80 per kilowatt of generator capacity with 625
volts per mile. That a long transmission does not necessarily require a
large investment in conductors per kilowatt of generator capacity is
shown by the line 65 miles long into Butte, for which the cost is
$17.25 per kilowatt, with 769 volts per mile. For the transmission to
St. Paul, a distance of 25 miles, at 1,000 volts per mile, the cost of
conductors is $7.95 per kilowatt of generator capacity. The seven-mile
line into Quebec shows an investment of $11.85 per kilowatt of generator
capacity.