Load Models

EE 455

Introduction to Energy Distribution Systems

Lecture Notes # 11

Prof. S. S. Venkata

Copy Right 2001 S. S. Venkata. No part of the notes may bereproduced in any form without the consent of the author.

Load Modeling

Under steady-state, complex power S at any location in a distribution

system is a function of voltage, V

Model I

It is usual to represent both P & Q as a polynomial function of |V|. That

is

P = ao + a1|V| + a2|V|2 + a-1|V|-1 + (1)

Q = bo + b1|V| +b2|V|2 + (2)

Such a representation is valid for individual type of loads or aggregate

(composite) type of loads.

In equations (1) & (2)

(i) if only ao and bo are only non-zero, then with all other a si and b si

being zero,

(ii) if only a1 and b1 are non-zero,

(iii) if only a2 and b2 are non-zero,

S VI P jQ f V= = + =* ( )

P aQ b CONSTANT POWER MODEL

=

=

0

0

P a VQ b V CONSTANT CURRENT MODEL

=

=

1

1

| || |

P a VQ b V

CONSTANT IMPEDANCE MODEL=

=

2

2

22

| || |

Example [1]

(a) Air-conditioning load demand (per unit values)

P = 2.97 4.00 |V| + 2.02 |V|2

Q = 12.90 26.8 |V| + 14.90 |V|2

(b) Flourescent lighting

P = 2.18 + 0.286 |V| - 1.45 |V|-1

Q = 6.31 15.60 |V| + 10.3 |V|2

(c) Induction motor

P = 0.720 + 0.109 |V| + 0.172 |V|-1

Q = 2.80 + 1.63 |V| - 7.6 |V|2 + 4.89 |V|3

These models are usually arrived at by conducting a series of tests and

performing suitable regression analyses:

Model II

Composite loads which are assumed to be mixtures of the types

discussed above can be represented as

P P VVn n

k

=

| || | , k varies between 0 to 3 (3)

Q Q VVn n

l

=

| || | , l varies between 0 to 3 (4)

Examples [2]

(a) If k = 1, l = 0, it implies the load is a constant current type, with

power factor unity.

(b) If k = l = 2, then the load is constant impedance type.

(c) If k = 2.5 and k = 2.7, then it represents an aluminum reduction

plant. This model is simpler and can be determined empirically knowing

|Vn| = initial or base value of voltage

Pn = initial or base value of real power

Qn = initial or base value of reactive power

Note: In all these models, one should be aware of the range of |V| for

which the equations (1) to (4) are valid.

Model III

Loads are seldom modeled to include frequency effects. If loads (or

demands) are sensitive to frequency effects, then these effects should be

included: In this case

P f V P VVn n n

k

( , | | ) | || |=

(5)

Q f V Q VVn n n

l

( ,| | ) | || |=

(6)

where, = 2( )f

n nf= 2( )

f base frequencyn =

, = constant exponents.

With this type of model, it is often a practice to determine four sensitivity

coefficients defined below:

dPdf

dPd V

dQdf

and dQd V

, | | , , | |

Knowing these coefficients, the new values for P and Q can be

determined from f and |V|. All the changes are assumed to be

small: Thus

PQ

dP df dp d VdQ df dQ d V

fV

=

( / ) ( / | | )( / ) ( / | | ) | |

Then: P = Pn + P

Q = Qn + Q

for f = fn + f

and |V| = |Vn| + |Vn|

The following tables show the results of a data survey conducted by EPRI

[3].

Model IV [4]

This is particularly suitable for modeling aggregate loads at a node

knowing the demand profiles for a day, a season or a year.

Assuming Gaussian Distribution

P P kp n p= + (7)

Q Q kp n p= + (8)

where, Pp = Power having an excess probability by p %

kp = Coefficients related to p

= Standard Deviation

The power values in excess probability around 10 % are used in volt-drop

calculations; The mean values (50% excess probability) are used for loss

calculations. Smaller values of (p%) are used for overload and

emergency conditions determination.

References

[1] R. B. Adler, C. C. Mosher, Steady-state Power Characteristics for

Power Loads , Stability of Large Electric Power Systems, IEEE Press,

1974, (This book has a series of papers on modeling.) pp. 147-153.

[2] M. H. Kent et-al., Dynamic Modeling of Loads in the Stability

Studies, ibid, pp. 139-146.

[3] EPRI Project Report, Long Term Power System Dynamics , Vol. I,

EPRI RP. 90-7-0, June 1974.

[4] E. Lakervi & E. J. Holmes, Electricity Distribution network Design

(Book) , Peter Peregrinus Ltd., 1989.

Demand: Demand is the load averaged over a specified interval of time and is

expressed in kilowatts, kilovoltamperes, amperes or other suitable units applicable to the

load commodity. Demand interval is the period over which the load is averaged.

Figure 1 shows how demand varies with time and demand interval at it occurs.

This is a daily demand curve. Note, the demand interval could be 15, 30, or 60 minutes.

Maximum demand is the greatest of all demands which have occurred during the specified

period of time. It must state demand interval and period such as daily, monthly, or annual.

Load Duration Curve: Figure 2 shows the demand of a particular value cumulatively

occurring during the interval versus duration. It is a monotonically decreasing function as

to be naturally expected and is derived from Figure 1. Diversified or coincident demand is

the demand of a composite group of loads. The maximum diversified demand or

Maximum Coincident Demand, Dm (henceforth called maximum demand) is usually the

value of which is of most importance for related loads. Non-coincident demand is the sum

of the demands of a group of loads with no restrictions on the interval to which each

demand is applicable. Here also the maximum non-coincident demand, DM is of interest

for unrelated loads.

Demand factor, FD, is the ratio of the maximum demand of a system to the total

connected load, LC (or sum of continuous ratings) of the systems.

Utilization factor is the ratio of maximum demand (DM) of a system to the rated

capacity of the system, which is the smaller of thermal or voltage drop capacity.

Load factor, FLD is the ratio of average load, Davg over a designated period of time

to the peak load (or maximum demand) occurring during the period. Load factor varies

between 0 and 1.

Diversity factor, FDV is the ratio of the sum of individual maximum demands of

various subdivisions of a system to the maximum demand of the whole system. It is

greater than or equal to 1. This means it is the ratio of maximum non-coincident demand

to maximum coincident demand.

Coincidence factor, FC 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 times.

Coincidence factor is the reciprocal of the diversity factor.

Load diversity, LD is the difference between the sum of the peaks of two or more

individual loads and the peak of the combined load.

(1) Diversity factor = FDV =

D D DD D

Dm m mNmi m

i

N

m cF1 2

1 1+ + + ==

=

D D D

D Dm

Dm m mNmi

i

N

m cF1 2

1 1+ + + ==

=

(22)

where N = Number of loads connected

(2) Coincidence factor = FC = 1FDV1

FDV(23)

(3) Load diversity = LD = DM - Dm (24)

The maximum demand of the group of (1+2+3+. . . . +N) loads can also be expressed as:

Dm = c1D1 + c2D2 + c3D3 + . . . . . . .+ cNDN (25)

where cN = the contributing factor of the Nth load which is defined as the contribution

of the nth load to the group maximum demand. It is equivalent to hourly

variation factor listed in Table 2.

With the above definition, the coincidence factor can be expressed as

Fc = 1 1 2 21 2

c D c D c DD D D

N N

N

+ + +

+ + +

L L

L L(26)

If D1 = D2 = . . . . . . = DN then

Fc = D c c c

NDc c c

NN N1 1 2

1

1 2( )+ + +=

+ + +L L (True for ranges) (27)

If c1 = c2 = . . . . . . = cN then

Fc = c D D D

D D DcN

N

1 1 2

1 21

( )+ + ++ + +

=

L

L(True for air conditioners) (28)

Loss Factor: Ratio of average power loss to peak load power loss during a specified

period of time.

Referring to Figure 3

x = peak load of duration t

y = minimum load of duration (T - t)

Average load = xt y T tT

+ ( ) (a)

Load factor = FLD = xt y T tTx+ ( ) (b) (29)

If y = 0, FLD = tT

Peak loss = x2R for duration t (a)

Minimum loss = y2R for duration (T-t) where R is a constant (b)

Average loss = x Rt y R T tT

2 2+ ( ) (c) (30)

Loss factor = FLS =Rx t y R T t

RTxtT

yx

T tT

2 2

2

2+

= +

( ) (d)

If y = 0 and x 0, FLS = FLD

If T tT

1 0. tT

0 and y x 0 FLS (FLD)2

This means loss factor varies between FLD and (FLD)2

Figure 4 shows load and loss factors as functions of peak load duration for load cycle of

Figure 2.

Figure 5 has curves of FLS as a function of FLD.

Empirical formula: FLS = 0.3 FLD + 0.7 F2LD for 30-minute monthly demand curve. In

general, this can be expressed as:

FLS = (1-A)FLD + AF2LD

The loss factor is primarily used in loss evaluation studies.

Power factor: Power factor of a load = Fp

= cos ( - ) = cos (33)= Active Power, P Apparent Power,

rS

In equation (33), = arc tan QP

. Here again this is true only for pure sinusoidal qualities.

The above definition is not generally applicable to a distributed load or a group of

individual loads which are continually changing. In this case, such as light load or peak

load in which case the loads are expressed in demand. The concept of group power factor

is a complex subject. It may be more desirable to determine the average power factor,

rather than the power factor at a particular load condition. Such is the case in considering

industrial and commercial services, the contracts of which might contain power factor

clauses.

Balanced load: A balanced polyphase load is that which draws symmetrical currents when

symmetrical voltages are applied to it.

Balanced voltages = symmetrical polyphase voltages.

Voltage unbalance = maximum deviation from the average base phase voltage/average phase voltage. (34)

or V2 = negative-sequence voltage V1 positive-sequence voltage (35)

This could easily be achieved more easily with the aid of computers.

The second definition is widely used. We will get more into it when we discuss the

subject of symmetrical components and overcurrent protection.

Load Distribution and Load Density

2 methods of representing distributed load

a) uniform distribution

b) loads are all same fraction of distribution transformer rating

2 methods of load density representation

a) linear density - kVA/1000

b) area density - MVA/mi2

Determining Load Characteristics

1) Measurements

2) Statistics (using a few spot measurements)

The greatest interest is in residential loads as opposed to commercial or industrial.

Methods of Making Field Measurements

Instruments include ammeter (rms & thermal lag)

wattmeter

demand meter

recording - disk chart, strip chart, paper tape, magnetic

tape.

Means of measurement: spot check, maximum demand test, recording demand.

Estimating Load Characteristics

1) Diversified Demand Method (covered before)

2) Relationship between maximum demand and KWH of compensation.

The second method uses factors to estimate maximum demand for each consumer.

The factors are:

0.0039 for 0-500 kWh

0.0035 for 500-1500 kWh

0.0030 for > 1500 kWh

Coincidence or Diversity Factor

FcN = Fc +1

FN

c (51)

FcN = coincidence factor for N loads

Fc = coincidence factor for loads

FcN =DD

N

1(52)

DN = maximum diversified demand for load

D1 = average individual maximum demand

Fc = nFn

cn

11

(53)

DN = D1FcN (54)

Motor Starting Characteristics

Motor starting causes voltage dips. Starting currents are roughly 6 to 8 times

corresponding to full-load components.

EXAMPLE: DISTRIBUTION TRANSFORMER SERVING RESIDENTIAL LOADS:

(REF: Electric Power Distribution Engineering by Gonen)

Let s consider the concept of Maximum Diversified Demand (Coincident Demand) for

residential loads. The objective is to estimate distribution transformer loads serving

residential areas. Figure 7 shows typical maximum diversified demands for various loads.

Table 1 shows hourly variation factors for these load types on an hourly basis.

Let s also assume that a given distribution transformer is serving 6 houses.

Usually such transformers serve 4 to 14 houses depending upon the type of heating and

transformer sizes. Further, let us consider that each house has a clothes dryer, a range, a

refrigerator, lighting and miscellaneous loads. We want to estimate the 30-minute

maximum diversified demand as seen by the transformer. We are also given that the

maximum coincident demand seen by each of these load types for each house is (see

Figure 7)

Dryer 1.6 kW distribution transformer

Range: 0.8 kW

Refrigerator: 0.0666 kW 6 similar houses

Lighting: 0.61 kW

Hence, the 30-min. maximum coincident demand

= 6(1.6 + 0.8 + 0.66 + 0.61) = 18.5 kW

(2) Suppose the primary feeder has 150 DT s serving 900 houses. Again from the above

curve,

1.2 kW/house dryer

0.53 kW/ house range

Pav, max = 0.52 kW/house refrigerator

0.044 kW/house lighting, etc.

hence, 30-min max diversified demand

= (1.2 + 0.53 + 0.52 + 0.044) x 900 = 2064.6 kW

(3) Hourly variation factor of a particular load

= demand (of a particular load) coincident with group demand max diversified demand (of the particular type of load)

At 4pm,

clothes dryer Dryer (0.38 x 1.6 x 6) kW

12am 0.031 am 0.022 am 0.003 am 0.004 am 0.005 am 0.006 am 0.007 am 0.008 am 0.089 am 0.20

Dryer 0.38 x 1.6 x 6 kW

Range 0.24 x 0.8 x 6 kW

Refrigerator 0.90 x 0.066 x 6 kW

lighting 0.32 x 0.61 x 6 kW

10 am 0.6511 am 1.0012 noon 0.981 pm 0.702 pm 0.653 pm 0.634 pm 0.38 Table 1Hourly variation factors

Refer to (1), 6 house case 6.344 kW = total hourly diversified demand

(contribution of the particular type of load to group max demand)

Rate Structures

1. Flat Demand Rate: Does not require meters. Used for street lighting.

2. Straight-Line Meter Method: Single Price based on kWh consumed. Most

residential customers are charged on this basis.

3. Blocked-Meter Rate: Graduated structure, in which higher the consumption lower

the rate applied. Not suitable for conservation.

4. Demand Rate: Recognizes load factor and involves a two-part tariff, one based on

maximum (monthly) demand and the second one based on actual energy

consumed.

5. Seasonal Rate: Summer versus Winter Rate. Here in the Northwest we pay

higher rate for winter and lower rate for summer. (Why?)

6. Time-Of-Day (TOD) Rate: Recognizes on-peak versus off-peak duration

consumption. The philosophy used here is similar to that for seasonal rates.

Load Growth

Pn = P0(1 + g)n (load forecasting)

Pn: load forecast at nth year

g: annual growth rate

n: # of years

P0: initial load

Pn

P0

n

0 1 2 3

5. Rate Structure

Sample:

Min change (first 20 kWh) $2.25/mo.

Next 80 kWh $0.0355/kWh




Consumption in excess of 800 kWh $0.0220/kWh

Ex: Consumption 2200kWh

2.25 + 80 x 0.0355 + 100 x 0.0321 + 200 x 0.0296 + 400 x 0.0265 + 1400 x 0.022

= 55.62

Environmental surcharges:

County tax, fuel cost adjustment, state sales tax

Total: $ 85.42

*rate schedule may be seasonal.

Northwest higher rate in Winter higher cost due to power import from California.

Fuel Cost Adjustment (factor FCAF)

FCAF = A x B106

x C x 11 D

$/kWh BtukWh

$Btu

A: weighted average Btu/kWh

B: amount by which average cost of fuel per million Btu exceeds or is

less than $1/million Btu.

C: ratio of total net thermal generation to total net generation

KWH x 11 D

: total generation to meet the given demand

loss factor

KWH x 11

D

C : total thermal generated to meet the given

demand

Documents

Load Models