2 Load Characteristics
2.1 basIc deFInItIons
Demand: “The demand of an installation or system is the load at the receiving terminals averaged over a specified interval of time” [1]. Here, the load may be given in kilowatts, kilovars, kilovoltamperes, kiloamperes, or amperes.
Demand interval: It is the period over which the load is averaged. This selected ∆t period may be 15 min, 30 min, 1 h, or even longer. Of course, there may be situations where the 15 and 30 min demands are identical.
The demand statement should express the demand interval ∆t used to measure it. Figure 2.1 shows a daily demand variation curve, or load curve, as a function of demand intervals. Note that the selection of both ∆t and total time t is arbitrary. The load is expressed in per unit (pu) of peak load of the system. For example, the maximum of 15-min demands is 0.940 pu, and the maximum of 1-h demands is 0.884, whereas the average daily demand of the system is 0.254. The data given by the curve of Figure 2.1 can also be expressed as shown in Figure 2.2. Here, the time is given in per unit of the total time. The curve is constructed by selecting the maximum peak points and connecting them by a curve. This curve is called the load duration curve. The load duration curves can be daily, weekly, monthly, or annual. For example, if the curve is a plot of all the 8760 hourly loads during the year, it is called an annual load duration curve. In that case, the curve shows the individual hourly loads during the year, but not in the order that they occurred, and the number of hours in the year that load exceeded the value is also shown.
The hour-to-hour load on a system changes over a wide range. For example, the daytime peak load is typically double the minimum load during the night. Usually, the annual peak load is, due to seasonal variations, about three times the annual minimum.
To calculate the average demand, the area under the curve has to be determined. This can easily be achieved by a computer program.
Maximum demand: “The maximum demand of an installation or system is the greatest of all demands which have occurred during the specified period of time” [1]. The maximum demand statement should also express the demand interval used to measure it. For example, the specific demand might be the maximum of all demands such as daily, weekly, monthly, or annual.
example 2.1
Assume that the loading data given in Table 2.1 belongs to one of the primary feeders of the No Light & No Power (NL&NP) Company and that they are for a typical winter day. Develop the ideal-ized daily load curve for the given hypothetical primary feeder.
solution
The solution is self-explanatory, as shown in Figure 2.3.
Diversified demand (or coincident demand): It is the demand of the composite group, as a whole, of somewhat unrelated loads over a specified period of time. Here, the maximum diversified demand has an importance. It is the maximum sum of the contributions of the individual demands to the diversified demand over a specific time interval.
For example, “if the test locations can, in the aggregate, be considered statistically representa-tive of the residential customers as a whole, a load curve for the entire residential class of custom-ers can be prepared. If this same technique is used for other classes of customcustom-ers, similar load
1.0
0.9 0.8 0.7 0.6
0.5 0.4
Load (pu peak)
0.3
0.2 0.1
0.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 pu time
FIgure 2.2 A load duration curve.
0.9801.0
0.884
Load (pu peak)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
12 AM 20.0 4 6 8 10 12 N 2 4 6 8 10 12 PM
Average demand = 0.254 Maximum 15 min demand Maximum 30 min demand Maximum 1 h demand
Δt = 1 h 0.940
Time (h) FIgure 2.1 A daily demand variation curve.
Utilization factor: It is “the ratio of the maximum demand of a system to the rated capacity of the system” [1]. Therefore, the utilization factor (Fu) is
Fu Maximum demand
Rated system capacity (2.2)
The utilization factor can also be found for a part of the system. The rated system capacity may be selected to be the smaller of thermal- or voltage-drop capacity [2].
Plant factor: It is the ratio of the total actual energy produced or served over a designated period of time to the energy that would have been produced or served if the plant (or unit) had operated continuously at maximum rating. It is also known as the capacity factor or the use factor. Therefore,
Plant factor Actual energy produced or served Maximum plant rati
= ×T
nng×T (2.3)
It is mostly used in generation studies. For example,
Annual plant factor Actualannual energy generation Maximum plant
= rrating (2.4)
or
Annual plant factor Actualannual energy generation Maximum plant
100 Street lighting load (kW)
800 800 800 800 700
1000 1200
1000
Commercial load (kW) FIgure 2.3 The daily load curve for Example 2.1.
100Urban residential load 80 60 40 20 Percent of peak load 0 12
12481248 PMAM ++=
++++ 0 PMAM
100Rural residential load 80 60 40 20
Per cen t o f p eak load
1212481248 Losses in transmission and distribution
0 PMAM
100Rural commercial load 80 60 40 20 Percent of peak load
1212481248 PMAM
0100 80 60 40 20
Commercial load
Per cen t o f p eak load
1212481248 100Industrial load 80 60 40 20 Percent of peak load 0 12
124812480
100Miscellaneous load 80 60 40 20 Percent of peak load
12124812480
100System load 80 60 40 20
Per cen t o f p eak load
1212481248 PMAMPMAMPMAM FIgure 2.4 Development of aggregate load curves for winter peak period. Miscellaneous load includes street lighting and sales to other agencies. Dashed curve shown on system load diagram is actual system generation sent out. Solid curve is based on group load study data. (From Sarikas, R.H. and Thacker, H.B., AIEE Trans., 31(pt. III), 564, August 1957. Used by permission.)
Load factor: It is “the ratio of the average load over a designated period of time to the peak load occurring on that period” [1]. Therefore, the load factor FLD is o average load:
FLD Average load Peak load
(2.6)
or
F T
T
T
LD Average load Peak load Unitsserved Peak load
×
×
= × (2.7)
where T is the time, in days, weeks, months, or years. The longer the period T, the smaller the resultant factor. The reason for this is that for the same maximum demand, the energy consumption covers a larger time period and results in a smaller average load. Here, when time T is selected to be in days, weeks, months, or years, use it in 24, 168, 730, or 8760 h, respectively. It is less than or equal to 1.0.
Therefore,
Annual load factor Total annual energy Annual peak load 8760
= × (2.8)
Diversity factor: It is “the ratio of the sum of the individual maximum demands of the various subdivisions of a system to the maximum demand of the whole system” [1]. Therefore, the diversity factor (FD) is
FD Sum of individual maximum demands
Coincident maximum demand (2.9)
or
F D D D D
D D n
g
= 1+ 2+ 3+ + (2.10)
or
F D
D D
i i n
g
=
∑
=1 (2.11)where
Di is the maximum demand of load i, disregarding time of occurrence
D D
n
g= n
=coincident maximum demand of group of loads1 2 3+ + + +
The diversity factor can be equal to or greater than 1.0.
From Equation 2.1,
DF Maximum demand
Total connected demand
=
or
Maximum demand Total connected demand DF= × (2.12) Substituting Equation 2.12 into 2.11, the diversity factor can also be given as
FD D
TCDi is the total connected demand of group, or class, i load DFi is the demand factor of group, or class, i load
Coincidence factor: It is “the ratio of the maximum coincident total demand of a group of consumers to the sum of the maximum power demands of individual consumers comprising the group both taken at the same point of supply for the same time” [1]. Therefore, the coincidence factor (Fc) is
Fc = Coincident maximum demand
Sum of individual maximum demands (2.14)
or
Thus, the coincidence factor is the reciprocal of diversity factor, that is,
Fc FD
= 1 (2.16)
These ideas on the diversity and coincidence are the basis for the theory and practice of north-to-south and east-to-west interconnections among the power pools in this country. For example, in the United States during winter, energy comes from south to north, and during summer, just the opposite occurs. Also, east-to-west interconnections help to improve the energy dispatch by means of sunset or sunrise adjustments, that is, the setting of clocks 1 h late or early.
Load diversity: It is “the difference between the sum of the peaks of two or more individual loads and the peak of the combined load” [1]. Therefore, the load diversity (LD) is
LD Di D
Contribution factor: Manning [2] defines ci as “the contribution factor of the ith load to the group maximum demand.” It is given in per unit of the individual maximum demand of the ith load.
Therefore,
Dg c1×D1+ ×c2 D2+ ×c3 D3+ + × cn Dn (2.18) Substituting Equation 2.18 into 2.15,
F c D c D c D c D
That is, the coincidence factor is equal to the average contribution factor.
Case 2: c1 = c2 = c3 = … = cn = c. Hence, from Equation 2.20,
That is, the coincidence factor is equal to the contribution factor.
Loss factor: It is “the ratio of the average power loss to the peak-load power loss during a specified period of time” [1]. Therefore, the loss factor (FLS) is
FLS Average power loss Power lossat peak load
(2.25)
Equation 2.25 is applicable for the copper losses of the system but not for the iron losses.
example 2.2
Assume that the annual peak load of a primary feeder is 2000 kW, at which the power loss, that is, total copper, or
∑
I R2 loss, is 80 kW per three phase. Assuming an annual loss factor of 0.15, determinea. The average annual power loss
b. The total annual energy loss due to the copper losses of the feeder circuits solution
a. From Equation 2.25,
Average power loss = power loss at peak load × FLS
= 80 kW × 0.15
= 12 kW b. The total annual energy loss is
TAELCu = average power loss × 8760 h/year
= 12 × 8760 = 105,120 kWh
example 2.3
There are six residential customers connected to a distribution transformer (DT), as shown in Figure 2.5. Notice the code in the customer account number, for example, 4276. The first figure, 4, stands for feeder F4; the second figure, 2, indicates the lateral number connected to the F4 feeder; the third figure, 7, is for the DT on that lateral; and finally the last figure, 6, is for the house number connected to that DT.
Assume that the connected load is 9 kW per house and that the DF and diversity factor for the group of six houses, either from the NL&NP Company’s records or from the relevant handbooks, have been decided as 0.65 and 1.10, respectively. Determine the diversified demand of the group of six houses on the DT DT427.
1
Distribution transformer DT427 Lateral L41
F1 F2 F3
Lateral L42 Feeder F4
Customer 4276
2 3 4 5 6
FIgure 2.5 Illustration of load connected to a distribution transformer.
solution
From Equation 2.13, the diversified demand of the group on the DT is
Dg F
Assume that feeder 4 of Example 2.3 has a system peak of 3000 kVA per phase and a copper loss of 0.5% at the system peak. Determine the following:
a. The copper loss of the feeder in kilowatts per phase
b. The total copper losses of the feeder in kilowatts per three phase
solution
a. The copper loss of the feeder in kilowatts per phase is
I R2 0 15 kW per phase
×
×
=
=
b. The total copper losses of the feeder in kilowatts per three phase is
3I R 2 3 15
= 45 kW per three phase
×
example 2.5
Assume that there are two primary feeders supplied by one of the three transformers located at the NL&NP’s Riverside distribution substation, as shown in Figure 2.6. One of the feeders supplies an industrial load that occurs primarily between 8 AM and 11 PM, with a peak of 2000 kW at 5 PM.
The other one feeds residential loads that occur mainly between 6 AM and 12 PM, with a peak of 2000 kW at 9 PM, as shown in Figure 2.7. Determine the following:
a. The diversity factor of the load connected to transformer T3 b. The load diversity of the load connected to transformer T3 c. The coincidence factor of the load connected to transformer T3
solution
a. From Equation 2.11, the diversity factor of the load is
F D
D D
i i
g
=
= + =
∑
=1 22000 2000 3000 1 33.
b. From Equation 2.17, the load diversity of the load is
LD
kW
= −
= − =
∑
= D Di g i12
4000 3000 1000 Transformer
T3
Subtransmission Riverside distribution
substation
Primary feeders Reserved for
future loads
Residential Industrial load
load
FIgure 2.6 NL&NP’s riverside distribution substation.
4000
3000
2000
Load (kW)
1000
12 AM 20 4 6 8 10 12 N
Residential load peak Industrial
load peak System load peak
2 4 6 8 10 12
Time (h) FIgure 2.7 Daily load curves of a substation transformer.
c. From Equation 3.16, the coincidence factor of the load is
Fc FD
=
=
≅ 1
1 1 33 0 752
. .
example 2.6
Use the data given in Example 2.1 for the NL&NP’s load curve. Note that the peak occurs at 4 PM.
Determine the following:
a. The class contribution factors for each of the three load classes b. The diversity factor for the primary feeder
c. The diversified maximum demand of the load group d. The coincidence factor of the load group
solution
a. The class contribution factor is
ci ≅ Class demand at time of system (i.e., group) peak Class noncoinciddent maximum demand
For street lighting, residential, and commercial loads,
cstreet kW
= 0 kW =
100 0
cresidential kW
= 600 kW = 1000 0 6.
ccommercial kW
=1200kW = 1200 1 0.
b. From Equation 2.11, the diversity factor is
F D
D D
i i n
g
=
∑
=1and from Equation 2.18,
Dg c D1× 1+c2×D2+c3×D3+ +cn×Dn
Substituting Equation 2.18 into 2.11,
Therefore, the diversity factor for the primary feeder is
F D Therefore, from Equation 2.13, the diversity factor is
FD D
where the maximum demand, from Equation 2.12, is
Maximum demand Total connected demand DF= × Substituting Equation 2.12 into 2.13,
F D
Therefore, the diversified maximum demand of the load group is
D D
d. The coincidence factor of the load group, from Equation 2.15, is
F D
c gD
i i
= n
∑
=1or, from Equation 2.16,
Fc FD
=
=
= 1
1 1 278 0 7825
. .