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Non-parametric short-term load forecasting
D. Asber a, S. Lefebvre a,*, J. Asber b, M. Saad b, C. Desbiens
c
a IREQ, Institut de Recherche dHydro-Quebec, 1800 Boul. Lionel
Boulet, Varennes, Que., Canada J3X 1S1b Ecole de Technologie
Superieure 1100 Notre-Dame Ouest, Montreal, Que., Canada H3C1K3
c Hydro-Quebec Distribution 680 Sherbrooke Ouest, Montreal,
Que., Canada H3C4T8
Received 22 March 2005; received in revised form 30 May 2006;
accepted 5 September 2006
Abstract
Load forecasting is an important problem in the operation and
planning of electrical power generation, as well as in transmission
anddistribution networks. This paper is interested by short-term
load forecasting. It deals with the development of a reliable and
efficientKernel regression model to forecast the load in the Hydro
Quebec distribution network.
A set of past load history comprising of weather information and
load consumption is used. A non-parametric model serves to
estab-lish a relationship among past, current and future
temperatures and the system loads. The paper proposes a class of
flexible conditionalprobability models and techniques for
classification and regression problems. A group of regression
models is used, each one focusing onconsumer classes characterising
specific load behaviour. Each forecasting process has the
information of the past 300 h and yields esti-mated loads for next
120 h. Numerical investigations show that the suggested technique
is an efficient way of computing forecast statistics. 2007 Elsevier
Ltd. All rights reserved.
Keywords: Distribution network; Forecasting; Time series;
Regression; Non-parametric
1. Introduction
In constructing a load forecasting model, a mathemati-cal
relation is established between the measured load andvarious
factors of influence. The model contains severalcoefficients, with
values to be determined, that quantifythe magnitudes of each
influence. The coefficient valuesare chosen such that the overall
error between model esti-mates and actual measured loads is
minimized. The modelis considered valid if tests conducted with
numerous histor-
ical data sets result in small overall errors. Improvementsto a
deficient model could involve the use of a differentmathematical
relationship or of data that are more refined.
Irrespective of the forecasting techniques, weatherparameters
are the key factors of Hydro-Quebec short-term load forecasts:
temperature and humidity are the mostcommonly used load predictors.
Thermostat-based modelshave thus been developed (explanatory
models). The time
factors include the day of the week, and the hour of theday
because there are important differences in load betweenweekdays and
weekends. Furthermore, the load on differ-ent weekdays can behave
differently. For example, Mon-days and Fridays being adjacent to
weekends, they oftenhave structurally different loads than Tuesday
throughThursday. This is particularly true during summer. Holi-days
are more difficult to forecast than non-holidaysbecause of their
relative infrequent occurrence. For cus-tomer classes with similar
load patterns, standard load
curves, namely diagrams of loads as a function of time,can be
obtained through load research studies based onmodelling individual
customer demands within a specificinterval.
For short-term load forecasting (STLF) several factorsmust be
considered such as time factors, weather data,and possible
customers classes. Several forecasting meth-ods have been developed
using parameter regression ortime series [1,2]. These technologies,
despite some limita-tions, are widely used in the industry. Neural
networks[36], fuzzy techniques [7] have also been applied, and
there
0142-0615/$ - see front matter 2007 Elsevier Ltd. All rights
reserved.
doi:10.1016/j.ijepes.2006.09.007
* Corresponding author.E-mail address: [email protected]
(S. Lefebvre).
www.elsevier.com/locate/ijepes
Electrical Power and Energy Systems 29 (2007) 630635
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are numerous publications in scientific journals. Accordingto
Hippert et al. [8], although these technologies seem toprovide
valid load forecasts, most investigators have usedseemingly
misspecified models that have been incompletelytested. More
research on the behavior large neural net-works is needed before
definite conclusions are drawn on
the suitability of these approaches. To alleviate these
prob-lems, a heuristic approach in [9] has been rejuvenated in[10]
that proposes using abductive networks. The techniqueis reported to
offer the advantages of simplified and moreautomated model
synthesis. It also provides analyticalinputoutput models that
automatically select influentialinputs. Nonetheless, tuning is
still required to improveforecasting accuracy through, for example,
the inclusionof hourly temperature data and the development of
dedi-cated seasonal models. Charytoniuk et al. [11] presentanother
approach to short-term load forecasting. It is anon-parametric
regression. The main advantage of thenon-parametric approach is
that it is data driven and elim-
inates a need for the statistical analysis aimed at selecting
amultivariate distribution fitting the data. This also
assuresportability of the proposed method. It can be used in
anyutility, regardless of the type of its load distribution.
This paper builds on non-parametric regression andcompares its
outcome with the classical approaches. In Sec-tion 2, the paper
first presents a simple time series basedforecasting model and a
thermostat-based regressionmodel. In Section 3, non-parametric
algorithms are usedfor STLF for commercial loads. In Section 4, the
probabil-ity forecast functions of the aggregated load are
developed.In Section 5, results from different models are presented
for
residential and commercial loads. For residential loads,
thepaper compares a thermostat-based model and a non-para-metric
model. For commercial loads, times series estimatesand
non-parametric estimates are compared since thermo-stat-based
regression is not representative of these loads.
2. Time series and regression models
2.1. Basic time series model
Time series models use previous or historical values, ofthe data
as input to the model, thus
yt %Ffxt;xt dt; :::;xt ndt;yt dt;yt 2dt; :::;yt ndtg 1
where y(t), y(t-dt), y(t-2dt) represent the load at t, t-dt
andt-2dt, dt being the time period considered. Here dt is 1 h,xt
represents a vector of other factors such as the day,the hour or
the temperature.
2.2. A thermostat-based regression model for heating loads
In this model, system load follows closely home
heatingrequirements as controlled by thermostats actions. The
ran-
dom heating cycles of numerous homes is not represented,
thus the model cannot reproduce the combined load overtime.
However, with a single equation, it can reproducethe average load
over the considered time period.
The dynamic model for the temperature of an averagehouse having
a heater regulated by a thermostat is of theform
dxdt
Ax BM CQT Hx
2
where T is the room temperature as measured by the ther-mostat;
x is the thermostat state; M is a vector of possiblynon-linear
function of meteorological inputs such as out-side air temperature,
solar intensity and wind velocity; Qis the heat flow of space
heaters; A, B, C, H are constantmatrices whose values depend on
house construction.
The thermostat reduces the thermal characteristics (andthe
effects of weather and lifestyle) to two variables: on-duration and
off-duration. On-duration d1 is the time inminutes required for the
room temperature Tto raise fromthe lower set point (Ts D) to the
upper set point Ts whenthe heater is on. The off-duration dois the
time in minutesrequired for T to decrease from Ts to (Ts D) when
theheater is off. The mean value of the thermostat status
undernormal conditions is
b d1d1 do 3
In this model, we substitute in the matrix H the value
ofbinstead of using directly the temperature forecast.
3. Non-parametric models
In load forecasting, optimal algorithms often require
theknowledge of underlying densities of measurements and/ornoise.
As these densities are usually unknown, assumptionsare frequently
made that compromise the algorithms per-formance. A common approach
to this problem is datadensity estimation. If a particular density
form is assumedor known, then parametric estimation is used. If
nothing isassumed about the density shape, non-parametric
estima-tion is the choice.
The kernel density estimator, also commonly referred toas the
Parzen window estimator [12] is non-parametric. It isparticularly
attractive when no a priori information isavailable to guide the
choice of density with which to fitthe data, for example the number
of variables affecting aforecast. A comprehensive review of
non-parametric esti-mation is presented in [13]. In time series,
the Parzen win-dow is a weighted moving average transformation used
tosmooth measurements.
3.1. Parzen window
In the Parzen window approach, a hypercube cell offixed width is
used to investigate a region Rn. The region
volume is
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Vn hdn 4where hn is the length of the edge of Rn . Define, for
exam-ple, the function
uu 1 jujj 612j 1; :::; d
0 otherwise
5
u((x xi)/hn) is equal to unity if xi falls within the hyper-cube
of volume Vn centered at x and is equal to zero other-wise. The
number of samples (or independent observations)in this hypercube is
n, let
Kn Xni1
ux xi
hn
1
n
Xni1
1
Vnu
x xihn
6
The probability estimates Pn(x) are
Pnx 1n
Xni1
1
hnu
x xihn
7
The definitions for u may vary from uniform, triangular,Gaussian
and others since the choice in Eq. (5) is not un-ique. This
function must satisfy two conditions, namelyRjudu 1 and j(.) is
symmetric. By considering a
Gaussian distribution u, Eq. (7) becomes an average ofnormal
densities centered on the samples xi. The probabil-ity density
function PDF is
^fkx 1nh
Xni1
Kxi xh
8
The scalar function j(.) or u(.) is called Parzen windowfunction
or kernel function. The Gaussian kernel is used
in rest of the paper:
uu 1ffiffiffiffiffiffiffi2P
p
expu2=2 and hn h1ffiffiffin
p n 1 1 9
In practice, the kernel function chosen is not nearly
asimportant as the kernel size h.
3.2. Simple Kernel regression
Regression estimation aims at finding a relationshipbetween a
dependent variable and a set of independentvariables. Kernel
regressions are used when we are unwill-
ing to impose a parametric form on the regression equationand
there is lot of data.Let the scalars yi be the outputs and xi the
data inputs.
Regression equations are specified as
yi mxi ei; 10where E[ei] = Cov [m(xi),ei] = 0 and m(.) is a
possibly non-linear function. The term ei is random with mean zero
andvariance r2. It defines the variation of yi around its mean,m(xi
) . The mean can be expressed as a function of theprobability
density f
m
xi
E
Yi
jxi
x
Ry fx;ydy
Rfx;ydy Ry fx;ydy
fx 11
The kernel smoothed density estimator is assumed to be
acombination of the Gaussian distribution as a function of nand the
kernel bandwidth h. The general form for this typeof estimator
follows
m
xi
Pn
i1Khxi xyiPni1Khxi x where Khxt x exp xtx
2
2h2
h ihffiffiffiffiffiffiffi2PpKu 1ffiffiffiffiffiffiffi
2Pp exp u
2
2
12
The term KkxixPni1Kxix
is the weight given to observation i.
The denominator makes the weights sum equals to unity.As an
example, the weight function could give equal weightto the kvalues
ofxi that are closest to x and zero weight toall other
observations. The N(0, h2) PDF is commonlyused. The choice of h
allows us to easily vary the relativeweights of different
observations. This weighting function
is positive so all observations get a positive weight.
Theweights are largest for observations near x and then tapersoff
in a bell-shaped way. A low value of h means that theweights taper
off fast; the weight function is then a normalPDF with a low
variance.
3.3. Practical regression
In practice we have to estimate mx at a finite number ofpoints
x. The load forecast yt depends on many variables(xt, zt, lt, . .
., ht) and the general formulation of a local aver-aging estimator
uses the multivariate Kernel regression:
mx Pni1Khxxi xKhzzi zKhlli lyiPni1Khxxi xKhzzi zKhlli l
13
where
Khxi x 1hffiffiffiffiffiffiffi
2Pp exp xi x
2
2h2i
" #14
Optimization estimators, on the other hand, are more ame-nable
to incorporating additional structure.
As a prelude to our later discussion, consider the follow-ing
estimator. Given
^yi
mxi;y
ih; :::; li
e
15
Then we solve:
minh
1
T
Xi
yih ^yi2 16
3.4. Confidence intervals and bands
Define the confidence intervals D(k) of the estimatedload Pt as
follows:
D L; U; L the band lower and U the upper band
of the estimator
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Dk fpavr Cstdpe;pavr Cstdpegwhere pavr is the mean value for
estimated load,C 1 x1a2x1a2 is the (1 a/2) quintile of the
standardGaussian distribution with the probability equal top = 1
a.
4. Smoothing parameters for feeder load profile forecasting
Electric load aggregation is the process by which
largecommercial, residential and industrial loads are combinedto
form an aggregated load with homogeneous and non-homogeneous
loads.
The main objective of this section is to obtain the prob-ability
forecast functions of the aggregated load. Themethod consists of
two steps. First, smoothing techniquesbased on kernel estimates are
applied to derive non-para-metric estimators. The aggregated load
is the sum of theindividual category forecasts. Then, parameter
smoothing
is applied to form the aggregated load. Parameter smooth-ing is
modelled as being unknown but bounded in ampli-tude by a closed
convex set.
At any given instant, the aggregated smoothing param-eter hiagr
is equal to the sum of the category parameters.
hu agrt XCi1
kihuit 17
Subject to:XCi1
ki 1 18
where u = 1. . ., m, with m the total number of
parametersincluded in the load model. Here the parameters
areweather and time for a weekday load. This added con-straint
requires that the aggregated parameters lie withinthe smallest
convex set containing the points formed by
the parameters category loads. The problem is then castin the
form of a constrained optimization problem:
minh
1
T
Xt
yth byt2subject to X
C
i1ki
1
ki P 0
19
5. Illustration and comparison of methods
This section is a scoping study for the methodologybased upon a
data set from customers in the Montrealregion of the Hydro-Quebec
network. In this study, theelectricity consumption of five load
categories sampled at60 min intervals was recorded over a month
period. The
0 5 10 15 20 25
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
Time in hours
Loadinp.u.
Forecasting for the residential load
non parametric
method
with measure
thermstat
method
Fig. 1. Load forecast over a 24 h period for a residential
load.
0 5 10 15 20 25-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time in hours
ErreurRelative
thermostat
method
non parametric
load
Relative error over a 24 hours period for a residential load
Fig. 2. Forecasting error for the residential load.
0 5 10 15 20 250.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Time in hours
Loadinp.u.
series
method
measured
load
non
parametric
load
Load forecast over a 24 hours period for a commercial load
Fig. 3. Forecast over a 24 h period for a commercial load.
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recorded historical data are used to compute the parame-ters of
each forecasting technique in the paper. The histor-ical load, the
hour and the corresponding temperature areused for the modeling
procedure.
Fig. 1 shows the forecasts for a residential load and
therecorded values. The non-parametric approach ensures
forecasts very close to the measured load over a 24 h periodof
time. The thermostat-based method does not yield thesame accuracy;
the forecasting error with these two meth-ods is illustrated in
Fig. 2. The non-parametric techniquehas an average error of 0.01
p.u. while the forecasting errorusing the thermostat approach
varies between 0.04 and0.05 p.u. From Fig. 2, we compute MAPE =
1.5% for thenon-parametric model, and MAPE = 2.2% with the
ther-mostat model.
The second data set represents a commercial load (officeload).
Fig. 3 illustrates the forecasted load using long pro-
ven time series [14] and non-parametric techniques.
Bothtechniques ensure satisfactory results for very
short-termforecasts up to about 7 h ahead. However, for larger
peri-ods, the non-parametric approach has a better
trackingperformance than the time series approach. The averageerror
is illustrated in Fig. 4. From this figure we compute
MAPE = 0.87% for the non-parametric model, andMAPE = 3.04% with
time series. Time series models maybe made more accurate, but this
requires large amountsof historical good quality data.
Finally, Fig. 5 shows the normalized error between theaggregated
commercial load and the sum of the individualcategory forecasts. In
this case, MAPE = 2.69%, higherthan without
decompositionaggregation.
6. Conclusion
The proposed non-parametric methods for forecastingproposed
exhibit a very good performance with respect to
the well-known thermostat-based techniques. These non-parametric
algorithms give results that are more consis-tent. Their
performance is also very competitive evenwith linear time series.
They are automatic proceduresthat do not need of any prior
information. In short,the method searches a collection of
historical observa-tions for records similar to the current
conditions anduses these to estimate the future state of the
system.The method is simple, has a sound theoretical basisand
provides the best forecast in the sense of a minimumexpected
squared error. It requires few parameters thatcan be easily
calculated from historical data by applying
the cross validation technique.There are drawbacks to
non-parametric estimation.
Whereas parametric models compress all training data intoa set
of equations through the process of parameter
fitting,non-parametric regression retains the data and
searchesthrough them for past similar cases each time a forecastis
made. If the number of variables is very large, then theresult is
inefficiency of the non-parametric methods. Thiswas not a problem
for the load forecasting problem, butmay become one in very
short-term estimation.
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0 5 10 15 20 25-0.05
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