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Abstract This paper deals with the characterization of the

probability distributions of the aggregated residential load. A

detailed statistical study has been performed on a set of data

referred to single-house extra-urban customers, in order to assess

the time evolution of the average value and standard deviation of

the aggregated load and its possible representation with some

typical probability distributions. The results have shown that the

aggregated residential load data can be satisfactorily represented

by using a Gamma probability distribution with parameters

variable in function of time and number of customers.

Index Terms Residential aggregated load, Probability

distributions, Goodness-of-fit, Monte Carlo simulations.

I. INTRODUCTION

he recent evolution of the electricity systems towards

time-dependent tariff rates and integration of distributed

generation is increasing the importance of assessing the

time evolution of the electricity consumption. Taking into

account the effects of the aggregation of residential loads is

now essential for studying the time evolution of the load in the

distribution system feeders. In fact, the electricity

consumption of the single residential customer is too variable

in time to allow for obtaining a sound estimate of its

individual load pattern [1]. The residential load aggregation

can be obtained by either working directly at the distribution

system level (if the results of measurements carried out on

several feeders are available) [2], or resorting to a bottom-up

approach in which the aggregated load patterns of single-

house customers are computed on the basis of information

obtained from real case investigations on customer behaviour,

lifestyle, and usage of the appliances [3]. In particular, it is

important to assess not only the average value of the

aggregated load, but also how its probability distribution

varies during the day and in function of the number of the

residential customers. Previous studies have shown that the

time evolution of the averagepower, normalized with respect

to the total contract power of the customers, has a predictable

behaviour, especially when the number of customers is

relatively high (e.g., over 100) [4]. Yet, when the number of

customers is low, the possible variations of the load power at

any given time instant are significantly high and strongly

depend on the number of customers and on the randomness in

the customer composition and lifestyle [5].

This paper presents the results of a study aimed at

characterizing the time evolution of the probability

distributions of the aggregated residential load when the

E. Carpaneto and G. Chicco are with the Dipartimento di IngegneriaElettrica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino,Italy (e-mail [email protected], [email protected]).

number of residential customers varies. Starting from the

statistical characterization of the aggregated load patterns of

single-house customers carried out in [4] by using the bottom-

up approach, the aggregated load power has been assessed in

function of the number of residential customers by using

Monte Carlo simulations. Then, a goodness-of-fit analysis has

been extensively performed with several probability

distributions (Normal, Log-Normal, Gamma, Gumbel,

Inverse-normal, Beta, Exponential, Rayleigh and Weibull) inorder to assess which probability distribution fits the

distribution of the load power at each time instant most

satisfactorily. The results obtained allow for establishing a

sound basis to be used within a comprehensive probabilistic

evaluation of the residential load or to be integrated into more

general procedures of analysis where detailed knowledge of

the variability of the residential load patterns is required [6].

Running Monte Carlo simulations is essential to obtain the

customer data for variable numbers of customers. In fact, field

measurements [2,7-9] would be possible only on a pre-

selected set with fixed number of customers. In addition, it is

sometimes difficult to gather only the data of the residential

customers, without superposition of other loads (e.g. buildingservices). This difficulty emerges in particular in urban areas,

where the residential load and the general services of the

buildings are supplied by the same feeder.

Section II of the paper deals with the formation of the data

set for extra-urban customers. Section III illustrates the

characteristics of the statistical tests used. Section IV provides

the numerical results and their discussion. Section V contains

the concluding remarks.

II. FORMATION OF THE DATA SET

The analysis has been structured on the basis of the results

obtained for extra-urban customers in a previous study [4]

carried out by using a comprehensive approach including twophases. In the first phase, a direct investigation of the

customers electricity usage has been performed for a real set

of single-house extra-urban customers. The results were

related to the presence at home of the family members and to

the detailed usage of the appliances, and were processed and

validated on the subset of customers who gave acceptable

information. In the second phase, an overall Monte Carlo

simulation was conducted to form the data set for the

successive statistical study. Different types of days (working

days and weekend days) and periods (summer and winter)

were considered. For space limitations, only the results

obtained for winter working days are presented here.

T

Probability distributions of the

aggregated residential load

Enrico Carpaneto and Gianfranco Chicco

9th International Conference on Probabilistic Methods Applied to Power SystemsKTH, Stockholm, Sweden June 11-15, 2006

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III. PROBABILITY DENSITY FUNCTIONS FOR THE STATISTICAL

TESTS

Various probability distributions have been used for the

goodness-of-fit statistical tests [10], including two one-

parameter distributions (Exponential and Rayleigh), five two-

parameter distributions (Gamma, Gumbel, Weibull, Normal,Log-normal and Inverse Normal), and the three-parameter

Beta distribution (whose third parameter has been set to the

maximum value of the sample data). Table I contains some

details on the probability distributions tested, with the

corresponding expressions of the Probability Density Function

(PDF) and Cumulative Distribution Function (CDF).

The Chi-square, Kolmogorov-Smirnov (KS) and

geometrical adaptation statistical tests have been used for

investigating the goodness-of-fit of the various probability

distributions as a function of the load powerP.

TABLE I

PROBABILITY DISTRIBUTIONS USED FOR THE GOODNESS-OF-FIT TESTSname

PDF

f(P)

CDF

F(P)parameter

limits

Beta 1

11

)()(

)()(+

+ba

ba

cba

PcPba

( )

( ) dxxx

dxxx

ba

cPba

1

0

11

0

11

1

1

= Beta incomplete(P/c,a,b)

0 P c

a > 0

b > 0

Exponential

b

P

eb

1

bP

e1 P 0

b > 0

Gamma( )

b

P

a

a

eab

P 1 ( )

P

b

x

a

a

dxeab

x

0

1

= Gamma incomplete(P/b)

P 0

a > 0

b > 0

Gumbel

b

aP

eb

aP

eeb

1

b

aP

ee1 P

Inverse

Normal

( )

3

2

2

2

bP

eabP

aP

( ) ( )

++

bP

aPerfe

bP

aPerf b

a2

P 0

a > 0

b > 0

Lognormal

( )

2

2

2

2

ln

Pa

e abP

+

2

ln1

2

1

a

bPerf P 0

a> 0

Normal

( )

2

2

2

2

P

e

21

2

1

Perf P

> 0

Rayleigh

2

2

2

bP

eb

P

2

1

bP

e P 0

b > 0

Weibull

a

b

P

a

a

eb

Pa

1

a

b

P

e

1

P 0

a > 0

b > 0

A. Chi-square test

The parametric chi-square test has been used, adopting the

same data sample for the statistical test. The Yates correction

[11] has been introduced in order to better estimate the

significance level. The results of the test depend on the pre-

specified number and structure (uniform or non-uniform) of

the classes, and on the level of significance%. If the value of% is specified in advance, the observed value is compared

to the critical range of values (min, max), depending on the

number of degrees of freedom, so that the test is successful if

min, unsuccessful if > max, whereas for min< maxthe result is undefined. Alternatively, the maximum level of

significancemax% corresponding to = minsets the limits of

acceptance of the chi-square test results.

B. Kolmogorov-Smirnov (KS) test

The error of the KS test is given by the maximum mismatch

between the Empirical CDF (ECDF) obtained by the set of

data under analysis and the CDF of the probability distribution

under test. The error is compared to a critical value critand

the test is successful if crit. If the CDF under test is fullyspecified by assigning all its parameters, the result of the test

is independent of the distribution, and the critical values gencrit

are found in specific tables in function of the level of

significance (see [11] p.797 and Table 1 of [12]). However, if

the CDF parameters are estimated from the data, these critical

values are no longer valid and must be determined by

simulation or by specific tables. Specific tables have been

found for the Exponential distribution (p.798 of [11] and

Table 1 of [13]) and for the Normal distribution (p.799 of [11]

and Table 1 of [14]). For the other cases in which the

distribution parameters are extracted from the data, the critical

values corresponding to a generic distribution can be seen

only as upper bounds of the actual critical values. The

assessment of the critical values is then performed by using a

Monte Carlo simulation. At first, a set of m= 1 , ,Mvalues

is specified, at which the CDF under test is calculated. Then, a

specified numberKof Monte Carlo simulations is performed.

At each simulation, a vector of length H is filled with H

random values extracted from the CDF under test. The

extraction is carried out by using Hrandom extractions in the

interval (0,1) from a uniform probability distribution, that are

considered as the values of the CDF under test, whose

corresponding values are computed from the abscissa of the

CDF under test. Then, the simulated empirical CDF is built at

the Mpredefined locations, and the KS error is computed as

the absolute value of the maximum difference between the

points of the simulated empirical CDF and of the CDF under

test referred to the same values m= 1, ,M. TheKerrors of

the KS test are then used to build the related CDF, and the

critical value is evaluated for a given level of significance.

The Monte Carlo simulations performed in this paper assume

M= 1000,K= 5000 andHset to the sample set size.

C. Geometrical adaptation tests.

The assessment of the fitness of the Empirical CDF (ECDF) to

a reference CDF has been performed by using two graphical

representations, with suitable functions of the generic CDF

valueF. The first representation transforms each valueFinto

( )FaE = 1ln (1)

and plots aEversus the power P, so that an exponential CDF

would be represented by a straight line in the (P,aE) plane.

The second representation transforms each valueFinto

( )( )FaW =

1lnln (2)


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and plots aW versus ln(P), so that a Weibull CDF would be

represented by a straight line in the (ln(P),aW) plane. In

addition, plotting aWversusPwould represent a Gumbel CDF

as a straight line in the (P,aW) plane.

IV. NUMERICAL TESTS AND RESULTSA. Tests for a 100-customer case

A first set of tests has been performed on the data obtained

from the Monte Carlo simulation at the same time instant. An

example is presented for hour 12:00 of a winter working day

with N = 100 customers. The number of Monte Carlo

simulations carried out to obtain the ECDF for this situation is

100. The characteristics of the complete data sample include

the minimum value 23.69 kW, maximum value 51.69 kW,

average value 39.21 kW, standard deviation 5.77 kW (14.7%),

and skewness -0.0004. The results of the KS test with level of

significance 5% are shown in Table II. The critical values of

the KS test have been computed by running 5000 Monte Carlo

simulations for each probability distribution (other thanExponential and Normal), resulting in values less restrictive

than gencrit = 0.1360. In particular, in this case the tests are not

accepted only for the Exponential and Rayleigh probability

distributions, whereas all the other distributions exhibit

acceptable goodness-of-fit. Fig. 1 reports the various CDFs.

More details are reported for the Gamma CDF with average

value and standard deviation equal to the ones of the data

sample (shape factor a= 46.2 and scale factor b= 848.5 W

according to Table I). Fig. 2 shows the details of the KS test.

Fig. 3 reports the results of the chi-square test with 7 degrees

of freedom (maximum acceptable error 14.07). Appling the

Yates correction, the observed error (6.85) has been

acceptably low and the test has been passed with a maximum

level of significance 44.5%, and with a non-excessive

adaptation (the critical value being 2.17).

Fig. 1. ECDF of the load at hour 12:00 for N = 100 and CDFs of various

probability distributions with the same average value and standard deviation.

Fig. 4 shows the results of the geometrical adaptation tests.

The acceptability of the Normal distribution is also confirmed

by the high value of the shape factor of the Gamma

distribution. As indicated in Fig. 5, for the Gamma CDF the

KS observed error during the day never exceeds the 5%

acceptance threshold.

TABLE II

KSTEST ERRORS FOR THE POWER AT HOUR 12:00WITH 100CUSTOMERS

(LEVEL OF SIGNIFICANCE 5%)

CDF KS test error critical value result

Beta 0.0832 0.1255 accepted

Exponential 0.5101 0.1060 rejected

Gamma 0.0800 0.1313 acceptedGumbel 0.1160 0.1314 accepted

Inverse-Normal 0.0659 0.1225 accepted

Log-Normal 0.0887 0.1230 accepted

Normal 0.0653 0.0886 accepted

Rayleigh 0.3434 0.1318 rejected

Weibull 0.0855 0.1306 accepted

Fig. 2. KS test for the Gamma CDF.

Fig. 3. Results of the chi-square test for the Gamma CDF forN= 100.

Fig. 4. Geometrical adaptation tests for the Gamma CDF.


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0

0.1

0.2

0.3

0 240 480 720 960 1200 1440

time (min)

KSobserv

ederror

KS 5% acceptance threshold

Fig. 5. Results of the KS 5% test with the Gamma CDF forN= 100.

B. Tests for a 10- customer case

The results of the same tests indicated in the previous

subsection are shown here on the data obtained from 100

Monte Carlo simulations referred to hour 12:00 of a winterworking day withN= 10 customers. The characteristics of the

complete data sample include the minimum value 1.508 kW,

maximum value 11.011 kW, average value 3.805 kW,

standard deviation 1.618 kW (42.5 %), and skewness 0.1109.

Table III shows the results of the KS test with level of

significance 5%. The Log-Normal and Inverse Normal

distributions exhibit the better goodness-of-fit. In this case, the

Normal probability distribution no longer fits the data. The

Gamma CDF has shape factor 5.53 (much lower than in the

previous case) and scale factor 688 W.

TABLE III

KSTEST ERRORS FOR THE POWER AT HOUR 12:00WITH 10CUSTOMERS

(LEVEL OF SIGNIFICANCE 5%)CDF KS test error critical value Result

Beta 0.1134 0.1263 accepted

Exponential 0.3721 0.1060 rejected

Gamma 0.1007 0.1325 accepted

Gumbel 0.2108 0.1334 rejected

Inverse-Normal 0.0851 0.1280 accepted

Log-Normal 0.0889 0.1263 accepted

Normal 0.1469 0.0886 rejected

Rayleigh 0.1588 0.1346 rejected

Weibull 0.1208 0.1322 accepted

0.0

1.0

2.0

3.0

4.0

5.0

0 240 480 720 960 1200 1440

time(min)

KSe

rrorratio

Exponential

Rayleigh

Normal

Gumbel

Fig. 6. Observed error to critical value ratio of the KS test with significance

level 5% for a winter working day withN= 10.

The results obtained for hour 12:00 can be generalized by

considering the results of the KS test with the various CDFs

for the 1440 minutes of the day. Fig. 6 shows that the

Exponential and Rayleigh distributions do not fit the ECDF

satisfactorily. A zoom into a specific time interval (from hour

11:40 to hour 12:20, Fig. 7) allows for identifying the Log-Normal, Inverse Normal and Gamma CDFs as the ones

exhibiting the best values of goodness-of-fit, compared to the

corresponding thresholds. However, as indicated for the

Gamma CDF in Fig. 8, in some hours of the day the KS

observed error could exceed the 5% acceptance threshold.

0.0

0.5

1.0

1.5

2.0

700 710 720 730 740

time (min)

KSe

rrorratio

Normal

Weibull

GammaInverse Normal

Log-Normal

Beta (max)

Gumbel

Fig. 7. Observed errors of the KS test with significance level 5% from hour

11:40 to hour 12:20 forN= 10.

0

0.1

0.2

0.3

0 240 480 720 960 1200 1440

time (min)

KSobserv

ederror

KS 5% acceptance threshold

Fig. 8. Results of the KS 5% test with the Gamma CDF forN= 10.

V. EXTENDED TESTS AND RESULTS

A. Extended tests for variable numbers of customers

The same set of tests specified in the previous subsections

have been carried out for a different number of customers(from 10 to 300) for all the 1440 minutes of the day, and for

the 4 types of days considered (working day and weekend day

in the summer and winter seasons). Each customer has a

contract power of 3 kW. Some significant results are

summarized in the sequel.

B. Time evolution of the aggregated load patterns and

standard deviations (winter working days)

A first result can be achieved by comparing the time evolution

of the aggregated load power for different numbers of

customers. Fig. 9, Fig. 10 and Fig. 11 show the load patterns

for a winter weekday with N = 10, N = 100 and N = 300,

respectively. The internal filled band represents the regions


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(-,+), where is the average value and is the standard

deviation of the data concerning each minute. The upper and

lower lines represent the maximum and minimum values

obtained in the Monte Carlo simulation. It is evident how

whenNincreases there is a reduction in the range of variation

of the aggregated load power, as well as a trend to obtainmore symmetrical probability distributions. In particular, Fig.

12 shows how the uncertainty of aggregated load (represented

by the standard deviation in per cent of the average value)

depends on the number of aggregated customers and varies

during the day.

C. Evaluations at specific hours

Further evaluations have been carried out by comparing the

evolution of the load in function of the number of customers.

A first case is presented in Fig. 13, considering the CDF of the

load at hour 12:00. When the number of customers increases,

the CDFs move from left to right, but the standard deviation

does not increase in the same way as the increase of theaverage value. This fact is well highlighted by the

representation of the specific power (W/customers) shown in

Fig. 14, where it is clear that whenNvaries the average value

of the specific power remains within a narrow range, whereas

the standard deviation varies considerably. This fact is

important to establish a reference value of specific power that

can be used to make good estimates of the consumption of the

group of customers tested. Extending the calculation to all the

time instants allows for observing that the specific load power

profile shown in Fig. 15 remains very similar.

0

5

10

15

20

0 240 480 720 960 1200 1440

time (min)

power[kW]

Fig. 9. Aggregated load patterns forN= 10.

0

20

40

60

80

100

120

0 240 480 720 960 1200 1440

time (min)

power[kW]


0

50

100

150

200

250

300

350

0 240 480 720 960 1200 1440

time (min)

power

[kW]


0

5

10

15

20

25

30

35

40

0 240 480 720 960 1200 1440

time (min)

standarddeviation

(%ofaveragev

alue)

N = 20

N = 40

N = 80 N = 150 N = 300

Fig. 12. Time evolution of the standard deviation of the load power in per cent

of the average value.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60

load [kW]

CDF

N = 1020 30

40

50 60 7080

90 100

Fig. 13. CDF of the load at hour 12:00 forN= 10 to 100.

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

specific power (W/customer)

CDF

N = 20

N = 300

N = 150N = 80

N = 40

Fig. 14. CDFs of the aggregated specific load power at 16:00 for

different numbers of customers.


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0

200

400

600

800

1000

0 240 480 720 960 1200 1440

time (min)

specificpower(W/customer)

Fig. 15. Specific load power profiles (winter working day).

In order to assess the most suitable probability distribution, a

comparison has been made by taking into account as

parameter the ratio between the observed error of the KS test

and the KS error threshold for the corresponding probabilitydistribution. For each number of customers, the probability

distributions for which this ratio is the lowest at the various

time instants have been identified. The results are summarized

in Table IV, showing the percentage of winning time instant

for the various probability distributions. From this point of

view, the Gamma distribution emerges as the most promising

one for the various numbers of customers. Only the Inverse

Normal distribution could be a viable alternative for a low

number of customers.

TABLE IV

PERCENTAGE OF WINNING TIME INSTANTS FOR THE VARIOUS PROBABILITY

DISTRIBUTIONS

probability distributions and percentages of winning timeinstants

N

Gamma

Exponential

Log-Normal

Normal

Inverse

Normal

Beta

Weibull

Rayleigh

Gumbel

10 43.1 0 2.9 0 46.0 2.2 4.8 1.0 0

20 58.9 0 4.7 0 32.4 1.7 2.2 0 0.1

30 74.9 0 7.1 0 14.5 1.7 1.8 0 0

40 70.5 0 11.5 0 8.7 3.2 5.8 0 0.3

50 86.0 0 7.8 0 3.2 1.5 1.5 0 0

60 81.5 0 12.5 0 3.5 1.6 0.9 0 0

70 80.8 0 11.3 0 3.1 2.7 2.0 0 0.1

80 76.0 0 13.5 0 5.5 2.8 2.1 0 0.1

90 80.3 0 11.1 0 5.1 2.1 1.4 0 0

100 80.1 0 12.1 0 5.1 1.9 0.8 0 0

VI. CONCLUDING REMARKS

The results presented in this paper form a useful basis for

preparing a meaningful set of probabilistic data concerning the

time-evolution of the aggregated residential load, to be used

into any tool for probabilistic simulation (e.g., based on

analytical or Monte Carlo simulations). The results shown are

specifically referred to groups of residential customers in

extra-urban areas. It has to be stressed that these values cannot

be generally applied to all residential customers, regardless of

their location. In fact, for urban areas generally lower values

of specific power are expected for the same number of

customers, due to the reduced size of the houses, reduced

number of persons per house, and different types of activity.The whole analysis could be repeated with a different data set,

with initial parameters concerning the composition of the

residential customer set (e.g., including new appliances,

customer preferences, or customer willingness to participate to

tariff-driven programs), to perform scenario studies and

assessing the effects of the penetration of new technologies on

the time evolution of the aggregated residential consumption.

Examples are the assessment of the distributed generation

impact on residential districts, distribution system load

forecasting, and simulation of unbalanced loads.

The Gamma probability distribution has clearly emerged as

the one with the best goodness-of-fit. This result is

particularly interesting, since the simple relationships betweenthe Gamma parameters and the average value and standard

deviation, as well as the existence and easy formulation of its

characteristic function, make the Gamma distribution

particularly flexible and powerful for many applications.

VII. REFERENCES

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customer behavior,IEEE Trans. on Power Apparatus and Systems, Vol.

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[2] I.C.Schick, P.B.Usoro, M.F.Ruane and J.A.Hausman, Residential end-

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on Power Systems, Vol.3, No.3, August 1988, pp.986-991.

[3] A.Capasso, W.Grattieri, R.Lamedica and A.Prudenzi, A bottom-up

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[8] K.Rosen and A.Meier, Energy use of televisions and videocassette

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Available: http://eetd.lbl.gov/EA/Reports[9] B.Lebot, A.Meier and A.Anglade, Global implications of standby power

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[10] R.B.DAgostino and M.A.Stephens (ed.), Goodness-of-fit techniques,

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[11] K.S.Trivedi, Probability and Statistics With Reliability, Queuing and

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