Parametrisation of domestic load profiles

Applied Energy. Vol. 54, No. 3. pp. 199-210, 1995 Copyright c, 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0306-2619/95/$15.00+0.00 ELSEVIER

0306-2619(95)00075-S

Parametrisation of Domestic Load Profiles A. G. Riddell & K. Manson Industrial

Research Limited,175 Roydvale Avenue, Bishopdale, PO Box 20-028, Christchurch, New Zealand

ABSTRACT As part of a study to determine the factors influencing the power usage patterns of domestic consumers, electrical loads have been logged at half and quarter-hourly intervals for several months. This enables the production of hundreds of daily load pro$les, the shapes of which can be analvsed with respect to various attributes of the consumers, However, in order to carry out such an analysis, it is necessary toJind a numerical representation of a load shape. It is found that the most suitable means of doing so is to calculate Fourier series approximations to the profiles, and use the coe,fficients of the series to represent the profile shapes. Copyright 0 1996 Elsevier Science Ltd

INTRODUCTION It is the desire of every electricity supplier to have as even a load as possible imposed on their distribution system. Unfortunately, it is not usual for consumers to use power in an even manner - mid-morning and early-evening peaks appear to be universal consequences of the way most people currently organise their daily lives. However, the timing and severity of these peaks are not consistent across all consumers. Obviously, houses on night rate have considerably lower day-time peaks than those not on night rate. It is quite likely, though, that there may be several other factors, in addition to the nightrate issue, which influence any given household’s daily power consumption pattern. The number and age of children in the house, the age of the house itself, the occupants’ tolerance to discomfort, the timing of meals, the amount of time the occupants spend at home, and the household’s attitudes to saving energy (and saving money) are some of the more obvious factors. 199

200

A. G. Riddell, K. Manson

The aim of our research is to identify just which are the factors that have the greatest influence on the shape of a household’s load profile. To this end, several different sets of power-consumption data have been collected: (1) very detailed logging of individual appliances within a single house - providing separate load profiles for water heating, washing, cooking, lighting, space heating, etc.; (2) logging of the total power consumption of individual houses, not broken down on a per-appliance basis; (3) logging of the power being drawn from feeder transformers, each of which typically supplies 10-50 houses. Qualitative insight gained from this logged data has been reported previously.‘** The two significant findings that were described are as follows: (1) Most of the load profiles collected from individual houses were found to fall into less than a dozen categories of characteristic shape. This indicates that it should be possible to identify a reasonably small set of types of electricity consumer. (2) In general, the shapes found at the individual-house level were also apparent at the feeder-transformer level. The implication that may be drawn from this finding is that the people living in a given neighbourhood predominantly tend to fall into just one or two consumer types. This suggests that, if it is possible to identify the causes for the electricity-usage patterns displayed by particular individuals, then we may be able to predict the likely future development of electricity demand in whole sectors of the community. What is more, we ought also to be able to predict their likely response to any specific strategies by which suppliers may undertake to influence power-consumption patterns. Since the publication of those qualitative findings, we have been working to quantify the perceived relationships between consumer types and load-profile shapes. To obtain fuller information on consumer attitudes and lifestyles, detailed interviews were carried out with occupants of those houses which were individually logged. The strategy is then to correlate household-load-profile shapes with various consumption-influencing factors as identified in the interviews. The question then arises: ‘How does one carry out statistical analyses on load profiles?’ The answer is, of course, that to perform the analysis, it is necessary to be able to represent load profiles by sets of parameters. The analysis can then be performed on these parameters.

Parametrisation

of domestic load prqfiles

201

The prime subject matter of this investigation is a description of how we went about identifying and calculating suitable parameters for the representation of our domestic load profiles.

IDENTIFICATION

OF SUITABLE

PARAMETERS

Desirable properties

There were several criteria governing scheme.

our choice of parametrisation

(1) The parameters must constitute a representation of the whole load profile, not just a few features of the profile. At first sight, it may seem that we could pick out certain characteristic features of the profile, like the area under the curve. the number of peaks in the curve, the heights of the peaks, the times of the peaks, the difference between the lowest trough and the highest peak, etc. These features certainly provide some description of the profile, and are likely to be the sort of features that are of interest to someone who is analysing load profiles with a view to eventually attempting to flatten out peaks. However, the act of just picking out a certain set of features rather puts the cart before the horse. It is effectively saying ‘These are the important features of load profiles - all other features may as well be ignored’. It may well prove that other, more subtle, features of load profiles will, in the end. give important clues to energy-usage patterns. The opportunity to uncover these subtle features should not be cut off right at the start, hence the parameters must constitute a representation of the whole load profile. (2) The number of parameters required to give a reasonable approximation to the load profile must be as small as possible. (a) Statistical analysis is a messy business at the best of times, and particularly so when applied to social (rather than physical) phenomena. When attempting to establish statistically significant correlations between certain socio-economic factors and load-profile shapes, it would be desirable that the load profiles be represented by as few parameters as possible, lest any significant correlations that do exist become lost amid a muddle of conflicting results from different parameters.

202

A. G. Riddell, K. Manson

(b) Moreover, the smaller the number of parameters, the greater the amount of information that is embodied in any given parameter. At one, impractical, extreme, it could be possible to ‘parametrise’ the load profiles by using the 48 half-hourly load readings as the ‘parameters’ representing a daily profile. In which case, the sort of statements that could be formulated from the results of the statistical analysis would be of the form ‘Consumers aged over 45 tend use more power at 3.30 p.m. than consumers aged under 30’, which, although possibly interesting, does not embody a great deal of information. At the other extreme, if a load profile could be represented by a single parameter, it may be possible to formulate statements like ‘ High income consumers in houses less than 20 years old show very little variance in their complete daily power-usage patterns’. This latter statement contains significant information, and points immediately to an area of further investigation. (c) In reality, the parametrisation will fall somewhere between these extremes; but, if the number of parameters is small, then a clear correlation between one or more parameter values and some socio-economic indicator immediately highlights a result worthy of closer study. (3) The parameters must have similar values for similarly shaped profiles. The whole purpose of the research programme is to identify factors which correlate with load-profile shapes. If the parameter values change erratically with small changes in profile shape, then the whole statistical analysis would be impossible, as it rests on the assumption that sets of similar parameter values are synonymous with similar profile shapes. (4) It would be desirable if the parameters could have some physical interpretation in terms of recognisable features of the profile shape. Whilst we are fundamentally looking for correlations of consumer attributes with profile shapes as a whole, it could well be that strong correlations will appear between certain attributes and certain individual parameters. In such a case, it would be useful if some meaningful statement could be formulated from the perceived correlation. That is, it would be useful if the parameters in question embodied some gross feature of the load profile, like the proportion of power used at night as against during the day, or the steepness of the daytime peaks, or the evenness of the profile outside of the daytime peaks, etc.

Parametrisation of domestic load prqfiles

CHOOSING

203

FROM AMONG THE CANDIDATE PARAMETRISATIONS

Time-series parameters

The application of time-series analysis to electrical-load forecasting is an established practice,3 so it seemed a logical starting place in the search for a means of characterising load profiles. We tried a number of ARIMA models ARIMA(O,l ,l). ARIMA(l,l,l), ARIMA(0,2,2), ARIMA(1,1,2) etc. - but with inconclusive results. The main problem was that parameter values varied unpredictably with small changes in profile shape, so failing one of the prime criteria listed above as a requirement of a useable parameterisation. In addition, it proved difficult to find a model which, when estimated on the first half of a daily profile, provided a reasonable forecast of the second half of the profile. It may be that we just could not hit upon the correct model, although we certainly tried a good range of candidates. However, it is more likely the case that there is insufficient autocorrelation within a single day’s profile to make it suitable for accurate time-series modelling. Whatever the case, we were forced to reject time-series modelling as a means of parametrising load profiles. Approximation

by series expansion

A time-honoured

means of obtaining descriptive parameters from a set of experimental measurements is to fit a polynomial to the data by a leastsquares procedure. Polynomials were fitted to a number of different load profiles. We found that in order consistently to achieve a good approximation, i.e. a less than 10% r.m.s. deviation between the actual profile and the approximation, it was necessary to have a nine-term polynomial (i.e. to go to the eighth power). A typical such approximation is illustrated in Fig. 1. The polynomial expansion is certainly superior to time-series modelling as a parametrisation scheme in that: (1) the values of the parameters varied only slowly with small changes in profile shape; (2) a good approximation to the whole profile could be reconstituted from the parameters. However, it does have the drawback that nine parameters are required, whereas time-series models typically involve only four or five parameters.

A. G. Riddell, K. Manson

204

lOOr

0:oo

6:OO

12:oo

18:OO

24:00

Time

Fig. 1. Polynomial approximation to a typical load profile. The solid line is the actual load profile, and the broken line is the polynomial: r.m.s. deviation of approximation is 9.96%.

In an attempt to find a scheme that would provide a good approximation for less than nine parameters, we looked beyond a simple polynomial approximation to expansion in series of orthogonal functions. We tried approximation by series of Bessel functions, Legendre functions, associated Legendre functions, Chebyshev polynomials of types I and II, etc. The coefficients of these series are not evaluated by a least-squares fit, but are calculated from explicit formulae, typically involving an integral over some domain of the product of the relevant function and the curve being fitted. In general, these series did not provide as good an approximation as the least-squares polynomial fit with the equivalent number of terms. The best of the orthogonal function expansions, namely the Legendre series and the Chebyshev I series, were effectively indistinguishable from the polynomial fit for nine terms. After having exhausted the supply of polynomial-type sets of orthogonal functions, we looked at approximating the load profiles by Fourier series (i.e. a series of sines and cosines), again to nine terms. This time we found that for some load profiles there was a slight improvement in accuracy over the least-squares polynomial fit (see Figs 4 and 5), while for other profiles, the Fourier series provided a significantly better approximation (see Fig. 3). So, the Fourier series provided a better approximation to the load profiles than any other of the series expansions we examined. In addition, the Fourier series has the advantage that some physical significance can be attached to the coefficients of the series: (1) The constant term is proportional i.e. to the total power consumed.

to the area under the load profile,

Parametrisation

qf domestic load profiles

205

Fig. 2. Approximation, by a series of Type I Chebyshev polynomials to the same load profile as in Fig. 1. The solid line is the actual load profile, and the broken line IS the approximation: r.m.s. deviation of approximation is 9.96%.

in;;;

_x_,’

Fz

\

,”

40 I\;, <\\ “__-__ J 20 IL!

0:oo

-___ ITOO

12:oo

1El:oo

24:00

Time Fig. 3. Approximation by a nine-term Fourier series to the same load profile as in Figs I and 2. The solid line is the actual load profile. and the broken line is the approximation: r.m.s. deviation of approximation is 6.43%.

10' 0:oo

I 6:OO

1200

18:OO

24:00

Time Fig. 4. Approximation by a nine-term polynomial sohd line is the actual load profile, and the broken tion of approximation

to a rather different load profile. The line is the approximation: r.m.s. deviais 9.25%.

A. G. RiddeN, K. Manson

206

1

60,

I 1t3:oo

12:oo

6:OO

24:00

Time

Fig. 5. Approximation by a nine-term Fourier series to the same load profile as in Fig. 4. The solid line is the actual load profile, and the broken line is the approximation: r.m.s. deviation of approximation is 8.62%.

(2) The relative values of the parameters

are likely to be discernible in the general shape of the profile. If the coefficient of the leading cosine term dominates, then the profile will be predominantly cosine-like in shape, which would indicate high power consumption at night. If the leading sine term dominates, the profile will be predominantly sine-like in shape.

Hence, of the parametrisation schemes we considered, the Fourier series best fulfilled the criteria we laid down, and so this scheme was chosen for use in our investigations. CALCULATIONAL The Fourier series is

cc

DETAILS lx

S(x)=a,+Ca,cosnl+Cb,sinnx 2 II=1

n=l

where the coefficients are given by 2x a,

=

;

s

f(t)dt

0

2K

a, = L

1

f(t)

cos

nt dt

0 271

b, = i

ll

J

f(t)

0

sin nt dt

n = 1,2,3...

n=

1,2,3...

Parametrisation of domestic load profiles

207

Therefore, the calculation of the coefficients involves integration. As the ‘function’ we are approximating is not an analytical function, but a curve drawn through a series of data points, the integral has to be evaluated numerically. This is achieved by dividing the area under the graph into 500 narrow strips and using simple geometry to calculate the area of each strip. Undoubtedly, this will have introduced a small inaccuracy, but we found that doubling the number of strips to 1000 made no appreciable difference to the values of the integrals, so we conclude that the inaccuracy is not noticeable at the number of significant figures to which we are working. Note that the integrals defining the Fourier coefficients run from 0 to 27r, whilst the time axes in Figs l-5 all run from 0 to 24 h. This was easily remedied by integrating over a dummy time variable t’, given by t’ = 2lrt/24

Scale invariance

There is a significant problem with the data we are dealing with. As noted in the introduction, the data were collected from several different sources, representing anywhere from 1 to 50 houses. Hence, simply just directly comparing the Fourier coefficients obtained from one given load profile to those obtained from some other load profile will not be meaningful, since, in general, the profiles will be representing different numbers of houses. Of course, the larger the number of houses contributing to a given profile, the larger the values of the parameters describing the profile (because the actual power consumption involved will be greater). So, it is necessary to find a set of ‘scale-invariant’ parameters to examine, i.e. parameters which describe the shape of the load profile, not its size. Fortunately, just such a set of parameters is near at hand. If a load profile is scaled up from, say, 10 houses to 20 houses, then (on the assumption of equally consuming houses) the load profile function F(x) would be multiplied by 2. So, the Fourier coefficients for the new profile would be given by 2K

277 A:, =

1

lr J

2F(x) cosnx dx = 2;

2F(x) cos nx dx = 2A,

0

0

2n tin =

I

1

7r J 0

2n

2F(x) sin x dx = 2-I 2F(x) sin x dx = 2B,, lr J 0

A. G. Riddell, K. Manson

208

Hence, scaling the number of houses up by a factor of h just scales the Fourier parameters by h. Therefore, if we were to divide all the parameters by AO, then the scale factors drop out: $, _ hA, A, ‘%J-K&-=Ao

So, in order to be sure of comparing like with like, do not directly compare the Fourier coefficients of all the different transformer load profiles - divide through by the A0 values first, and then compare the results of these divisions.

RESULTS Whilst the values of the Fourier coefficients are not particularly noteworthy in themselves, it is interesting to compare the coefficients obtained from similar-looking profiles, and also to compare the coefficients obtained from dissimilar profiles. For example, the coefficients for the Fourier series graphed in Fig. 3 are presented in Table 1. The load profile in Fig. 3 was obtained by logging a feeder transformer serving several houses. The coefficients for the Fourier approximation to a similarly shaped load profile obtained by logging a single house are presented in Table 2. Obviously, these values differ greatly from those obtained for the load profile of the feeder transformer. As described in the previous section, if we are to compare like with like it is necessary to divide through by Ao, in which case we obtain the values in Table 3. Whilst by no means identical, these sets of values are certainly comparable, especially so when contrasted with the comparison between the coefficients obtained from Fig. 3 and those from Fig. 5, as seen in Table 4.

Coefficients A0 55.4007

Al -4.3065

A2 0.5217

TABLE 1 of the Fourier series graphed A3

-2.7118

A4

-0.9471

Bl

- 19.0076

in Fig. 3 B2

-18.7199

B3

B4

5.0148

1.2001

Parametrisation

TABLE Coefficients

for the Fourier

in Fig. 3, but obtained

41

PO.1361

Comparison

PlYlfilf Transformer House

Comparison

2

approximation

to a load

profile

from a single house (not a transformer A7

Al

I .8584

series

A

-0.0267

of the normalised

AI/A,,

B/

A4

-0.1438

-0.0748

of similar

B-7

to that

B1

-0.4136

-0.5331

shape

feeding several houses)

0.1248

& 0.0414

TABLE 3 Fourier coefficients obtained from the two load profiles ot similar shape, but different scale

A21‘4,

-0.0777 -0.0732

209

of domestic load profiles

0.0094 -0.0143

of the normalised

&iA/J

4?/4, -0.0489 -0.0774

Fourier

-0.0171 -0.0403

TABLE 4 coefficients load profiles

B/!Ao -0.3431 -0.2869

obtained

B2:Ao -0.3379 -0.2226

BjIAo

B,:A,

0.0905 0.0672

0.0217 0.0223

from two differently

shaped

-.___ PWfik

Fig. 3 Fig. 5

A,:& ____

-0.0777 0.2299

AJI&

A .I:IA ,I

0.0094 0.1320

-0.0489 -0.0150

UAn

B//h,

-0.0171 0.0964

-0.343 -0.088

BZ :’A o

I I

-0.3379 PO.0295

WA,,

0.0905 0.1494

&A,,

0.0217 0.0272

In this latter case, both the leading cosine terms differ in sign, and in some of the other terms, the value obtained from Fig. 5 is of the order of 10 times that obtained from Fig. 3, so different profile shapes do result in noticeably differing parameter values.

CONCLUSIONS

AND

APPLICATION

OF THIS

TECHNIQUE

We have developed a technique by which typical domestic load profiles can be characterised uniquely. Using this technique, it will be possible to place numerical values on certain properties of load profiles, and also to place unambiguously given profiles into predetermined profile categories. Future papers will describe the procedures developed to link analytically appliance use and socio-economic/lifestyle indicators to load-profile characteristics.

210

A. G. Riddell, K. Manson

ACKNOWLEDGEMENTS The authors would like to thank SouthPower Ltd and the Electricity Corporation of New Zealand for their ongoing assistance in this research.

REFERENCES 1. Gardiner, A. I., Manson, K. & Dix, G. I., Domestic energy savings using energy management. Industrial Research Limited, Christchurch, New Zealand, Report No. 23, 1993. 2. Gardiner, A. I. & Manson, K., The relationship between individual domestic appliance load profiles and aggregate distribution loading. Electrical Supply Engineers Association Conference, 1994. 3. Abu-El-Magd, M.A. & Sinha, N.K., Short-term load demand modeling and forecasting: a review. IEEE Trans. Systems Man Cybernetics, (SMC-12)3, 1982, 370-382.