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should trip a nmachine or sound ani alarm is not easy to answer. To me it does not appear to be a matter of maxinmum importance. At times, a machine should be tripped immediately upon field failure, and at times an alarm would be adequate. The total number of machine trips from this cause in the life of a machine in cases where the sounding of an alarm would be adequate is extremely small, too small, in fact, to

justify loading another duty on the oper-

ator. We have not had an operation of these relays to trip the machine following field failure. It is true that there is no protection against machine motoring as such. In an attempt to protect against damage in as direct a method as possible there are two thermostats which sense shell temperatures. The first one to operate sounds an alarm.

Estimating Kw Demand ror Future Loads on Rural Distri6ution Systems STANLEY J. VEST

ASSOCIATE MEMBER AIEE

SINCE 1939 REA (Rural Electrification Administration) distribution borrowers have been estimating the capacity needed for future loads on the basis of curves relating kw demand to the number of consumers and the average kw-hr (kilowatt-hour) usage. This method has proved reliable and the curves have been revised when necessary because of changing conditions. Prior to the l)resent revision the values desired were read from a family of curves or from tables made up from these curves. A method has now been developed for determining kw demand by the multiplication of two factors corresponding to the number of consumers and kw-hr usage. These factors may be read from tables or determined mathematically. This paper outlines the history of the demand curves, the techniques used in preparing the latest revision, and the manner in which the curves and factor tables are used.

History The first kw-demand curves used by REA distribution borrowers were prepared in 1939 for a maximum of 2,000 consumers and 120 kw-hr/mo/consumer (kw-hr per month per consumer). They were revised in 1945 to provide for 5,000 consumers and 500 kw-hr/mo/ consumer, and again in 1949 for 10,000 consumers and 1,000 kw-hr/mo/conPaper 57-600, recommended by the AIEEE System Engineering Committee and approved by the AIEE Technical Operations Department for presentation at the ATEE Great Lakes District Meeting, Des Moines, Iowa, April 15-17, 1957. Manuscript submitted December 20, 1956; made available for printing February 18, 1957. STANLEY J. VEST is with the Rural Electrification Administration, Washington, D. C.

652

sumer. As a result of numerous requests for demand informnation corresponding to consumptions above 1,000 kw-hr/ mo/consumer, curves were prepared in 1955, based on 1949 information, extending the previous curves to 2,000 kw-hr/ mo/consumer. These curves were used only as an interim measure until current information could be assembled and new curves prepared.

Basic Information The information used in preparing the revised curves and factor tables, except for the smaller numbers of consumers (less than 50), was taken from operating reports and power bills furnished by REA borrowers. A 255% sample was taken, representing as far as possible 25% of those in each state. One substation was selected from each system avoiding those with unusual loads, such as army installations, large industrial plants, and seasonal cottages. In several cases it was necessary to discard a sample because of unusual conditions which were not representative of a rural system, resulting in a reduction of the sample to approximately

23%.

To avoid irregularities in meter-reading times, the 4-month peak demand period (4 consecutive months) was selected for determining monthly usage and demand. The values used in preparing the curves were the average monthly usage and the average monthly demand during this 4month peak period of maximum demand. Therefore, the kw-demand data are those which may be expected for any particular monthly usage. To obtain the maximum yearly demand, the maximum monthly usage rather than the average usage

If the slhell temiiperature increases to the setting of the second thermostat the generator is tripped. Our experience with the oscillographs and highi-speed chart recorders has been such that we are installing similar types of equipmenit on subsequent machines. They have proved very valuable in trouble analysis. The instrumentation of our newer units is inuch the same as that described.

should be applied in reading the factor tables prepared from the curves. The kw-hr values used are based on kw-hr sold at the consumer's meter, making it unnecessary to correct for losses. Information available in REA for small numbers of consumers could not be used, since it was found that substations with only a few consumers were those with unusual loads and not representative of a typical rural area. The data in this range were obtained from a studv made on 42 farms by the Agricultural Experiment Station, Iowa State College, and the Farm Electrification Section, Agricultural Research Service, U. S. Department of Agriculture.,

Method of Plotting Data The data were plotted as kw-hr/mo/ kw versus consumers, the ordinate being a measure of diversity. A notation was made at each point indicating the area from which it was taken. The peak month and density were also noted on a separate sheet. An examination of the points showed no noticeable difference because of area or density. However, a spot check indicated that about three fourths of the summer peaking systems will have a lower demand than the average and one fourth higher than average. From p)ast experience and by inspection it could be seen that the plot of kw-hr/ mo/kw versus consumers would be a family of curves, each curve representing a particular value of kw-hr/mo/consumer. Knowing this to be true, one curve would have been sufficient, but to prove the point three curves were plotted. Because of the lack of sufficient points to plot specific values ot usage, curves were plotted for three ranges: 100 to 200, 201 to 400, and 401 to 600 kw-hr/mo/consumer. Fig. I shows the spread of points for 401 to (600 kw-hr/mo/consumer. A curve was drawn through these points by the method of moving averages. The original curve was carried to 10,000 con sumers but, as a matter of convenience, Fig. 1 has not been reproduced beyond

5Iest-Estimating Kw Demand for Rural Distribution Systems

AUGUST 1957

2,000 consumers since the curve continues in a straight line. In Fig. 2 this same curve has been reproduced with the other two curves, mentioned in the foregoing for comparative purposes. Note that the three curves have the same shape and level off at approximately 1,400 consumers. Any point on either curve may now be identified as a given percentage of the maximum kw-hr/mo/kw for that curve. For example, using a given number of consumers, a point which is 50%,0 below the maximum on one curve will be 30% below the maximum for any curve in the

380 _

_

___-

.-401-600 KWH MoS onsumer-C

c

-

300.. 3280 ------I

ax240. 260

¢ 22a0

L_

20

- I -

16 140

a0O4Q 0

380

-

-

-

-

600 )-

--80 IMO 1400 CONSUMERS Fig. 1. Kw-hr/mo/kw versus consumers

~~~~~~~I

-

-

-

-

_

-

_

_

_

_

_

_

_

i6

1600

lI _

_

_

!

00

_

l l 360360L - ~~~~~~~~ ------- 401-600L KWH/Mo/Consumer i I I I 34 _ 201-400 KWH/Mo/Consumer | || 0|

L I

-007320F1 300 -

-

IO

200OKWHt'Mo/Cons5umSr

280

.--

260 240 1

x22C C2200

1180 -160--.

Calculation of Kw Demand

-

140 -120 --

The following equation was derived for

calculating demand from the information

100--

available from the curves kw-hr X kw kw='

so

200

400

600

800

100

1200 14001T60 Ts0 2000

CONSUMERS

kw-hr

F'ig. 2. Kw-hrfmo/kw versus consumers

Dividing numerator and denominator by

kw gives kw =

_

340C320

family.

To find the maximum for any curve in the family, it was necessary to find some relationship between the maximum values for the three curves plotted. The three points appeared to fonn a straight line on log-log paper, but more points were needed to verify this assumption. This was accomplished by making a plot of kw-hr/mo/kw versus kw-hr/mo/consumer for all points above 1,400 consumers (Fig. 3) since the effect of the third variable, consumers, is constant in that range as shown by the curves in Fig. 2. A curve drawn through these points by the method of moving averages verified the previous assumption that the plot would form a straight line on log-log paper. This straight-line curve was also in agreement with a straight line drawn through the three points of maximum value found in Fig. 2.

. - _ - - --

40--

360

I000

kw-hr

kw-hr/kw

where kw-hr = kw-hr/mo/consumer X consumers

kw

kw-hr/mo/consumer Xconsumers maximum kw-hr/mo/kw X per cent of

maximum

kw=

-kw-hr/mo/consumer

consumers

0

5400

~300

I

200

maximum kw-hr/mo/kw per cent of

maximum

The first term has been designated as kw-hr factor or factor B and the second term as consumer factor or factor A. After determining factor A (Table I) for

AUGUST 1 957

1001DV a

1VU

I

1. 200 300

I500 l

KWH/ Mo/Consumer

1000

1 11

2000 3000

Fig. 3. Kw-hr/mo/kw versus kw-hr/mo/consumer, For 1,400 consumers or more

Vest-Estimating Kw Demand for Rural Distribution Systems

653

Table 1. Consumer Factor, Factor A No. Con- Factor sumers A

No. Con-

Factor A

sumers

No. Consumers

41..... 53.4 42 .... 54.5 43 .... 55.5 5

44

6.... 10.8

7 . 12.1 8.13.5

48

9 . 14.8 10 .. 16.1

49

...

61.4

62.4 50.... 63.5

51..... 64.7

17.4

11 .

56.7

45 .... 57.9 46 ..... 59.0 47 .6...60.2

9.49

..

....

12. 18.7 13 20.1 14 . 21.4 15 . 22.7 16. .24.0 17 25.3

52 .... 53 .... 54 .... 55 .... 56 .... 57 58 .... 59 60 ....

18 . 26.6 19.27.8 20.29.2

65.7 66.7 68.0 69.0 70.2 71.2 72.3 73.6 74.5

62 ..... 7

21 . 30.4

..80... 96;.0

4.41.9

82 ..... 98.3

31

64..... 66... 68 70 .....

43.1 34.... .45.4 35 ..... 46.6 36i .47.7 37.. 48.9 385...50.0 39 .51.2 .2 40.... 52.3 32...

153 159 163 168

155 . 160 .

173 178

170 . 175 . 180 . 185 . 190 195 . 200 .

188 193 198 203 208 213 218

81.1

85.4

84 ..... 100 86 ..... 102 88 ..... 104 90 ..... 107 92..... 109

Consumer

50.

0. 189

55 .

0.203

65. 70 . 75 . 80.

0. 237 0.254 0.2 70

90.. 95.. 100.. 110..

0.317 0333 0.348 0.379

60 .

0.220

02'86

85 ..

0.301

120 ..

0 .409

140 ..

0.468 0.497

130 ..

0.439

150.. 160 .. 170.. 180..

190..

200. 210

220. 230. 240. 250 . 260 . 270 . 280.. 290 .. 300.. 320..

340

0.525

0.554 0.583 0.641 0.669 0. 697

360

..

400

...

380..

0. 612

0.726 0.755 0.784 0.810 0.836 0.864 0.893 0. 923 0.972 1.03

.08

1.14 1.19

Kw-Hr/Mo/ Consumer

654

148

272 282

275 .

291 296 301

270 .

280 .

285 .

290 .

295 . 300 .

Factor B

420 .. 1.24 440 .. 1.29 460 .. 1.34 480 .. 1.40 500 ... 1 .45 525 .. 1.51 550 .. 1.58 575 ..... 1f64 600 .. 1.70 625 .. 1.77 650 .. l1 .83 675 .. 1.90 700 .. 1.96 725 ... 2 .02 750 2.08 775 .. 2.14 800 .. 2.20 825 ... 2.26 850 .. 2.32 875 .. 2.38 900 ... 2.44 925 .. 2.50 950 .. 2.56 975 2.62 1,000 .. 2.68 1,100 .. 2.92 3.15 1,200 1,300 .. 3.39

1,400 .. 3.62 1,500 .. 3. 84 1,600 .. 4.07 1,700 .. 4.29 1,800 .. 4.51 1,900 .. 4.73 2,000 .. 4.95

Note: The data may be plotted as a straight line log-log paper.

on

133 138 143

255

260 .

94.....111 96 ..... 113 98 ..... 115 100 ..... 117

Factor B

128

265 .

Table II. Kw-Hr Factor, Factor B Kw-Hr/Mo/

122

205 . 223 210 . 228 215 . 233 220 . 238 225 . 243 230 . 247 235 . 252 240 . 257 245 . 262 250 . 267

78.9

83.2 72 ..... 87.6 74 ..... 89.7 76..... 91.8 78 93.9

33. ....44.3

105 . 110 . 115 . 120 . 125 . 130 . 135 . 140 . 145 . 150 .

165. 183

7

22. 31.7 23 . 32.8 24 ....339 25. 34.9 26. 36.0 27 37. 2... 28 . 38.9 29. 39.5 30 .40.7

Factor A

276

287

306 310 315

No. Consumers

Factor A

310 . 325 320 . 335 330 . 344 340 . 354 350 . 364 360 . 373 370 . 383 380... 393 390 . 403 400 . 412 410 .. 420 430 . 440 . 450 . 460 . 470 . 480 . 490 500 .

422 432 442 452 462 472 481 491

510 . 520 530 . 540 . 550 ..

522 532

501

512 542

551 561

560 .... 571

570 . 582 580 592 590 . 601 600 . 612 620 .. 631

640 6.652 .. 66X;0 ,72

680 ..... 692 700 .. 713 720 . 733 740 .. 753 760 .... 772 780 793 800 . 812

problems for solution by electronic computers, the following may be used

No. Consumers

Factor A

820..... 840 .. 860.... 880.....

900....

832 853 873 891 911 931 951 972 992

920..... 940 ... 960..... 980..... 1,000 ..... 1,010

1,050.

1.100.

1, 059 1,108

1,150..... 1,157 1 ,200. 1, 207 1,250. 1,255 1, 300..... 1,304 1,350..... 1 ,353 1, 400..... 1,400 1 ,450..... 1,450 1,500..... 1 .500

1,600. 2,000.... 2,400. 2,800. 3,200.....

1 ,600 2,000

2,400 2,800 3,200 3,600O..... 3,600 4,000..... 4,000 4,400..... 4,400 4,800..... 4,800 5,200..... 5,200

5,500 . .. 5,500 6,000 ..... 6,000 6,500 ..... 6,500 7,000 ..... 7,000 7,500..... 7,500 8,000 ..... 8,000

8,500 ..... 8,500 9,000 ..... 9,000 9,500..... 9,500

10,000.....10,000

all numbers of consumers and factor B (Table II) for all values of kw-hr, the kw demand for any consumer density and usage may then be calculated by multiplying the two corresponding factors. Factor A reflects the improved diversity resulting from an increase in the number of consumers. Factor B reflects the improvement in load factor with increased usage, and is the kw demand per consumer to be expected on an average substation having maximum diversity (more than 1,400 consumers). Kw demands may be calculated for usages higher than the 2,000 kw-hr/mo/ consumer, shown in the factor B table, by plotting and extending the curve for factor B which is a straight line on log-log paper as shown in Fig. 4. This factor may also be calculated for any usage value by using the equation of the line shown in Fig. 4. factor B = 0.005925 (kw-hr/mo/con-

sumer)0°886

Since factor A is equal to the number of consumers beyond 1,400 consumers, an equation for this curve would not ordinarily be needed. In special instances, however, such as in the programming of

factor A = C[1 -0.4C+0.4( C2+40)1/uI

where C= number of consumers

This equation is not exact but closely approximates the curve for factor A. A nomogram, Figs. 5(A) and (B), has also been prepared as an added convenience. The divisions on the left-hand scale (in each figure) of the nomogram represent the log of factor A corresponding to the number of consumers, the divisions on the right-hand scale represent the log of factor B corresponding to kwhr/mo/consumer and the center scale represents log factor A +log factor B found by placing a straightedge from A to B. In this way factor A and factor B are multiplied by adding their logarithms.

Adjustment for Difference in Load Factor

The demand curves and factor tables are based on the average system, and some systems will deviate from the average be-

cause of load factor. If the load factor and diversity are expected to continue

to bear the same relationship to the average, the information may be easily adapted to the particular system. Two methods of doing this and comparisons of kw demand found by each method for 5,000 consumers are shown in the following, where the present values are 254 kw, 500 consumers, and 200 kw-hr/mo/ consumer; and the multiplying factor or shift in factor B curve required for the same area with a usage of 600 kw-hr/ mo/consumer is to be found. Method 1. For 500 consumers and 200 kw-hr/mo/consumer, the load factor from the factor tables equals consumers X kw-hr/mo/consumer kw X hours

500(200) =41.8% =-500(-00) 328(730)

The actual load factor equals consumers X kw-hr/mo/consumer kw X hours

500(200) 254(730)

Multiplying factor=41.8/54.0 =0.775 or

254/328 = 0.775

Method 2. Consumer factor (factor A) does not change but kw-hr usage factor (factor B) changes with load factor. Therefore, adjustment may be made by

Vest Estimating Kw Demand for Rural Distribution Systems

AUGUST 1 957

a

XF

6 %A

7-

3

c

0 U

.4 _

l--L

l

5

4

____,

-

.3

- -

IL

CONSUMERS

Fig. 4 (left). Kw-hr Factor, Factor B Fig. 5 (below). Nomogram for kw demand B-140 to 10,000 consumers A-5 to 140 consumers Fig. 6 (above). Kw-hr/mo/kw versus consumers, for 50 kw-hr/mo/ consumer.

.2

50

N 0. CONSUM ERS

KWH/Mo/Consumer

DEMAN D IN KW 1200

140 100 80+

.1000

-700 -500

601

t300 200

40

30

t70

20 16+ 12

30

..3000 t 2400 t 2000

1600 1200 1000

t 400 t 300

20 10

6" St

KWH/MO

/CONSUMER

600

-50

8t

400

200

100

1000

2000 30i)00

NO. CONSUMERS

DEMAND IN KW

8000

60,000 40,000 30,000 20,000

10,000 6000

5000

4000 3200

2400 2000 1600 1200 000

7

200

800

5

150

600 500

t3

2 -1.5

100

400

75

320t 380t 240 200j 1601

5a

140

100,000

KWH/MO

/CONSUMER

3000 2400 2000 1600

15,000

1200

10,000

7000

1000

5000

800

3000

600 500 400

2000 1400 1000

300 250 200

700 500

300

150 125 100 75

200 150

100 80 60 40 32

s50

26

drawing a factor B curve for the particular system parallel to the average curve. For 500 consumers

factor, the corrected demand equals 8,500 kwXO.775=6,600 kw

factor A =512

factor B curve drawn

kw kw

254

factor A

512

0.496

A straight line through the point for kwhr/mo/consumer 200 and factor B = 0. 496, drawn parallel to the average curve, is the factor B curve for this system. Kw Demand by Method 1. For 600 kw-hr/mo/consumer and 5,000 consurmers the factor tables show a demand of 8,500 kw. Applying the multiplying =

AUGUST 1957

Kw Demand by Method 2. The new as mentioned in the foregoing shows this factor to be 1.32 for 600 kw-hr/mo/consumer. For 5,000 consumers

factor A 5,000 factor A Xfactor B 5,000 X 1.32 6,600 kw =

=

compared in Fig. 6 with the

=

new

Fifty

kw-hr/mo/consumer has been used as a basis for comparison since the 1939 curve was based on this usage. From 1949 to 1955 the maximum kw-lir/mo/kw shows very little change. This indicates that the addition of new uses for electricity should not appreciably affect estimates made on the basis of present information.

Conversion Equations Those who wish to relate information in the factor tables to coincidence factor, diversity factor and average undiversified individual consumer demand may do so by use of the following coincidence factor=

diversity factoraverage

(B)

factor B=

are

curve for the same usage value.

factor A A

3.29 Xconsumers

3.29

X(consumers factor A

kw/consumer (undiversified)= 3.29 Xfactor B

where 3.29 =factor A for

one consumer

Conclusions Substation demand is related to the number of consumers and the average Demand curves and kw-hr usage. factor tables, prepared on the basis of present information, provide a convenient means for predicting future kw demands.

Comparison of Demand Curves 1939 to 1955

Reference

The cuirves prepared in 1939 and 1949 plotting kw-hr/mo/kw versus consumers

1. LOAD) CHARACTERISTICS OF SOUTHEASTERN IOWA FARMS USING ELECTRIC RANGES, Landy B. Altman, Jr., Emil H. Jebe. Research Bulletin 420, Iowa Stete College, Ames, Iowa, Jan. 1955.

Vest-Estimating Kw Demand for Rural Distribution Systems

n L

5