Journal of Intelligent Learning Systems and Applications, 2012, 4, 285-290
http://dx.doi.org/10.4236/jilsa.2012.44030 Published Online November 2012 (http://www.SciRP.org/journal/jilsa)
285
Weighted Time-Variant Slide Fuzzy Time-Series Models
for Short-Term Load Forecasting
Xiaojuan Liu1,2, Enjian Bai1, Jian’an Fang1
1College of Information Science & Technology, Donghua University, Shanghai, China; 2Department of Mathematics and Physics,
Shanghai University of Electric Power, Shanghai, China.
Received May 29th, 2012; revised June 26th, 2012; accepted July 3rd, 2012
ABSTRACT
Short-term load forecast plays an important role in the day-to-day operation and scheduling of generating units. Season
and temperature are the most important factors that affect the load change, but random factors such as big sport events
or popular TV shows can change demand consumption in particular hours, which will lead to sudden load changes. A
weighted time-variant slide fuzzy time-series model (WTVS) for short-term load forecasting is proposed to improve
forecasting accuracy. The WTVS model is divided into three parts, including the data preprocessing, the trend training
and the load forecasting. In the data preprocessing phase, the impact of random factors will be weakened by smoothing
the historical data. In the trend training and load forecasting phase, the seasonal factor and the weighted historical data
are introduced into the Time-variant Slide Fuzzy Time-series Models (TVS) for short-term load forecasting. The WTVS
model is tested on the load of the National Electric Power Company in Jordan. Results show that the proposed WTVS
model achieves a significant improvement in load forecasting accuracy as compared to TVS models.
Keywords: Load Forecasting; Fuzzy Time-Series; Weighted; Slide
1. Introduction
Load forecast has been a research topic for many decades
and the accuracy of load forecast is crucial to electricity
power industry due to its direct influence on generating
planning. Short-term load forecast means the forecast
time lead is in the range of hours to a few days ahead,
which plays an important role in the day-to-day operation
and scheduling of generating units. There are many fac-
tors that affect the load changes, such as calendar, wea-
ther, economical and random factors. For short-term load
forecast, weather and random factors are the most impor-
tant factors. Season and temperature are have the most
influence to the load due to the fact that changes in tem-
perature results in direct changes in energy consumption
by heating and cooling appliances. Random factors such
as big sport events or popular TV shows can change de-
mand consumption in particular hours, which will lead to
sudden load changes. A number of load forecasting mod-
els have been presented in the last decades. These models
can be divided into traditional approaches [1] and the ar-
tificial intelligence methods [2]. The former include re-
gression models, time series models et al, and the latter
provided many new tools for the forecasting of short-
term load such as neural networks [3-5], fuzzy logic [6,7],
support vector machines [8], expert systems [9], hybrid
method [10,11] et al. In recent years, many researchers
have used fuzzy time series models to handle load fore-
casting problems [12-15]. Liu et al. proposed a Time-
variant Slide Fuzzy Time-series Model (TVS) for short-
term load forecasting [13], the TVS model only uses his-
torical data to predict the load changes. Taking into ac-
count the affect of season, temperature, and random fac-
tors, a Weighted Time-variant Slide Fuzzy Time-series
Forecasting Model (WTVS) is presented. The WTVS
model is divided into three parts, including the data pre-
processing, the trend training and the load forecasting. In
the data preprocessing stage, the impact of random fac-
tors will be weakened by smoothing the history data. In
the trend training and load forecasting stage, the seasonal
factor and the weight of history data are introduced into
the TVS model. The WTVS model is tested on the load
of the National Electric Power Company in Jordan. Re-
sults show that the WTVS model achieves a significant
improvement in load forecasting accuracy as compared
to TVS models.
2. Time-Variant Fuzzy Time-Series
A fuzzy set
A
defined in the universe of discourse


12 , n
Uu u,,ucan be represented as



 
11AA2 2Ann
A
fuu fu ufu u,where A
f
is

Weighted Time-Variant Slide Fuzzy Time-Series Models for Short-Term Load Forecasting
286
the membership function of the fuzzy set
A
,


:0,fU1

AiA,
f
u denotes the degree of member-
ship of be longing to the fuzzy set
i
u
A
,



0, 1
Ai
fu



,
and .
1in

1,fti
Definition 1. Let Yt be the uni-
verse of discourse and also a subset of. It is assumed
that iis defined on and

,0,t1,2,
R

Yt

2,


F
t is
the collection of

i
f
t, therefore,

F
t is called a
fuzzy time series on .

Yt

Definition 2. It is assumed that

F
t is a fuzzy time
series and
 


11Rtt,Ft Ft , where


Rt,1t

is a fuzzy relation and × is an operator which is caused
by . The relationship between

1Ft


F
t

and
can be denoted by

1Ft

1
F
tFt when
is the first-order fuzzy time-
series model of
 
Ft Ft
 
1

1,Rtt

F
t

.
Definition 3. Let

F
t be a fuzzy time series. For
any t,

1

F
tFt and

F
t have only finite ele-
ments and therefore

F
t

is a time-invariant fuzzy time
series; otherwise, it is a time-variant fuzzy time series.
Definition 4. If
F
t

2, ,
is caused by

1,
 
,
F
tFtFtn



the fuzzy relationship is
represented by



1,2, ,,
F
tFtFtn tF

it is the nth order fuzzy time-series model.
Definition 5. It is supposed that
F
tis caused by
 


01,Ft Ft2, ,Ftmm, simultaneously
and the relations are time variant. The


F
t
11

is a time-
variant fuzzy time series and the relation can be ex-
pressed as , where
is a time parameter affecting the forecast
 
Ft

1
w
R

t

,tFt w

F
t, which is
the analysis window of time-variant models.
3. WTVS Model
This study aims to improve short-term load forecasting
using an adaptive algorithm to adjust the analysis win-
dow automatically in the training phase of weighted his-
torical data and heuristic rules for forecasting in the test-
ing phase. The WTVS model includes the following
steps: 1) Preprocessing historical data, 2) defining and
partitioning the universe of discourse, 3) defining fuzzy
sets and fuzzifying time series, 4) establishing fuzzy re-
lationships and 5) forecasting and defuzzifying fore-
casting results. These steps consist of three parts: pre-
processing phase, training phase and testing phase. The
preprocessing phase is used to eliminate the impact of
random factors by smoothing the historical data. The
training phase is used for data learning. Two values are
computed in each round based on the selected analysis
window sizes and the value with higher prediction accu-
racy is determined as the forecasting value. In this proc-
ess, a sequence of the analysis windows is obtained. The
selection of analysis window is determined by the fol-
lowing adaptive algorithm (Algorithm 1). The testing
phase is used for forecasting accuracy test. Two values
are computed by Algorithm 3 for every testing data based
on the selected analysis window sizes of testing phase.
Taking into account the affect of seasonal factor, a heu-
ristic method is proposed to select the analysis window
sizes of testing phase and determine the forecasting value
based on the sequence of analysis window obtained in
training phase. The structure of the WTVS model is pre-
sented in Figure 1.
In the following, details of each step is described.
Step 1. Preprocessing the historical load. Random
factors may cause sudden load changes. We will smooth
these sudden load changes by the following method
when the absolute difference value of the load is higher
than a threshold. The threshold is defined as
1
2
Threhold 31
n
ii
i
F
F
n



 

Suppose that 1Threshold
ii
FF

 , we will substi-
tute t
F
by1Threshold
t
F

.
Step 2. Fuzzified the revised historical load.
(1) Define the universe of discourse


min max
,ULL
,
m
u
and separate it into intervals 12
uu , m,,


min n
1,
i
uL iil
mi
lL

 



, wherelis the interval leng-
History data
Preprocessing
Fuzzfied
Fuzzy time-series
Fuzzfied time-series groups
Adaptive
algorithm
Forecasting load
training forecasting
Time and
season factor
Preprocessing
Phase
Training
Phase
Forecasting
Phase
Figure 1. WTVS model.

Weighted Time-Variant Slide Fuzzy Time-Series Models for Short-Term Load Forecasting 287
th, the midpoint ofis .
i i
(2) Define the fuzzy sets
u m
i
A
and fuzzify the data.
 


12
12
1, 2,,
ii i
AA Am
im
fu fufu
A
im
uu u
 
Step 3. Establishing fuzzy relationships of time
and and group the fuzzy time-series.
t
1t
In the training phase, the fuzzy relationship is sup-
posed to be ij
A
A. In the testing phase, the fuzzy
relationship is supposed to be . #
i
A
Generally, the trend of load in summer and winter is
shown in Tables 1 and 2, respectively. For example, in
summer, we can conclude that from 1 to 6 o’clock, the
load have the downward trend, while from 7 to 12
o’clock, the load have the upward trend. These trends can
be used to revise the forecast in the forecasting phase.
Step 4. Forecasting and defuzzifying forecasting re-
sults.
In the training phase, each round calculates values For
1 and For 2 and compares the two values to actual value
Act with the better one as the forecasting load. The ana-
lysis window is determined by Algorithm 1. The compu-
tations of For 1 and For 2 are carried out by Algorithm 2.
In the testing phase, the forecasting load is determined by
Algorithm 3.
Algorithm 1. (slide analysis window)
(1) ;; 1, whereand1i1
i
S2
i
Si S1i S

are the
sizes of the initial window. Flag . 1n
(2) If the prediction accuracy computed by 1i S

is
higher than that of i, then slide the analysis window
forward and the size of the analysis window plus 1, and
flag . Otherwise, slide the analysis window
backward and the size of the analysis window minus 1,
and flag .
S
1nn
nn1
(3) Repeat step (2) until the end of the training data.
Algorithm 2. (training phase) Suppose that the fuzzy
relationship of time and is j k1ki
A
A, and the
analysis window size is . Let n
j
M
A


be the middle
value of interval
j
u.
(1) Select two initial window sizes 11S

and
. Let .
22S1n
1For (2) If, n1
j
M
A

 . If , 2n
Table 1. Load trend in summer.
Time 1 - 6 7 - 12 13 - 20 21 - 24
Trend ↓ ↑ ↓ ↓
Table 2. Load trend in winter.
Time 1 - 6 7 - 12 13 - 16 17 - 18 19 - 24
Trend ↓ ↑ ↓ ↑ ↓

2
1
0
For 2
22
1
1
n
ttiti
ij
F
iFFM
nnn
s




 

 A


 
 








 ,
where 1111
,,, ,1,2,3,4
8642



 



and


F
t is the actual load at time . t
If

2
1
0
22
1
n
tti
i
DFiFFu
nnn





tij








then


, and 1
s
s

. Else

, and
s
s

.
(3) Compared with the actual load, if the training ac-
curacy of theis higher than, then
For 2
2
For 1
predict ForF

. Else .
predict
(4) Slide the analysis window until the end of the
whole training data.
ForF1
Algorithm 3. (forecasting phase ) Suppose that the
fuzzy relationship of timekand is, and the
analysis window size is n. let
1k#
i
A
j
M
A

 
1
be the middle
value of intervalu.
j
(1) Select two initial window sizes and
1
S22S

.
Let 1n

.
(2) If 1n

,


For 3i
M
A. If, 2n

2
1
0
22
1
For 4
n
tt
i
FiFF
nnn
s




 iti



 

 









,
where 11 11 ,,, ,1,2,3,4
8642







and


F
tis the actual load at time. t
If

2
1
0
22
1
n
tti
i
DFiFFu
nnn





tij








,
Then


, and 1
s
s

. Else

, and
s
s

.
(3) Consider the trend of load change and the sequence
of flags which obtained in training phase, there are the
following heuristic rules:
n
1) If 1tt nn

, and the actual load at timetis bigger
than that of time 1t

, then the load at time 1t

has
the trend of increasing. At the same time pay attention to
the trend in Tables 1 and 2, the forecast value is max


For 4
nn
For 3.
2) If 1tt


,and the actual load at time is smaller
than that of time
t
1t

, then the load at time 1t

has
the trend of decreasing. At the same time pay attention to
the trend in Tables 1 and 2, the forecast value is min


For4For 3.
3) Else, the forecast value is the arithmetic average of
For 3 and For 4.

Weighted Time-Variant Slide Fuzzy Time-Series Models for Short-Term Load Forecasting
288
(4) Slide the window until the end of the whole fore-
casting data.
the purpose of the comparisons of the predictive accu-
racy, we use the mean absolute percentage error (MAPE)
as the index of forecasting accuracy. MAPE can be de-
fined as
4. Experiments and Analysis
The load of the National Electric Power Company in
Jordan [12] is chosen for model validation. An empirical
analysis is conducted to validate the performance of
WTVS model by comparing the forecasted load with that
of TVS model [13]. Considering the time and season
factors, we choose the data from 1 to 24 in each day as
our research data. The data are divided into two parts: the
training data (from 1 to 20) and the forecasting data
(from 21 to 24). Table 3 lists the load in 23/5 and 29/6
and the corresponding revised load by preprocessing. For
1
1
MAPE 100%
nkk
kk
tm
nt




where ,k represent the actual and forecasting values
of thedata, respectively. Andnis the number of data.
Table 4 compares load forecasting results among WTVS
and VTS model in the forecasting phase with the same
number of intervals. The MAPE results show WTVS
model outperforms TVS model.
k
t
thkm
Table 5 shows the different forecasting accuracy in
Table 3. Load preprocessing.
23/5 Actual load 1tt
F
F
 23/5 Revised load 29/6 Actual load 1tt
F
F
 29/6 Revised load
1 1176 1176 1285 1285
2 1129 −47 1129 1210 −75 1210
3 1095 −34 1095 1170 −40 1170
4 1098 3 1098 1130 −40 1130
5 1093 −5 1093 1145 15 1145
6 1080 −13 1080 1080 −65 1080
7 1195 115 1195 1140 60 1140
8 1327 132 1327 1360 220 1350
9 1509 182 1509 1485 125 1485
10 1567 58 1567 1548 63 1548
11 1614 47 1614 1612 64 1612
12 1640 26 1640 1640 28 1640
13 1610 −30 1610 1631 −9 1631
14 1600 −10 1600 1600 −31 1600
15 1591 −9 1591 1605 5 1605
16 1547 −44 1547 1565 −40 1565
17 1528 −19 1528 1486 −79 1486
18 1482 −46 1482 1420 −66 1420
19 1418 −64 1418 1380 −80 1380
20 1700 282 1628 1535 155 1535
21 1633 −67 1633 1615 80 1615
22 1515 −188 1515 1520 −95 1520
23 1417 −98 1417 1475 −45 1475
24 1293 −124 1293 1370 −105 1370
Average 68.4 67.2
Threshold 210 210

Weighted Time-Variant Slide Fuzzy Time-Series Models for Short-Term Load Forecasting 289
Table 4. Comparison between WTVS and TVS model in
forecasting phase.
23/5 Actual
load WTVS TVS[3] 29/6
Actual
load WTVS TVS[3]
21 1633 1620 1723 21 1615 15471519
22 1515 1625 1607 22 1520 16201627
23 1417 1497 1538 23 1475 15291525
24 1293 1419 1433 24 1370 14601475
MAPE 5.8 7.74 5.2 6.0
Table 5. Comparison under different numbers of intervals
in forecasting phase.
Number of intervals
Time
4 5 8 10 16
23/5 6.94 7.23 7.74 8.27 8.18
29/6 4.8 7.28 4.6 5.87 5.43
the forecasting phase under different numbers of inter-
vals. It is shown that the forecast accuracy is influenced
by the length of intervals.
5. Conclusions
In this paper a weighted time-variant slide fuzzy time-
series model for short-term load forecasting is proposed.
The proposed model is tested for forecasting efficacy on
the load of the National Electric Power Company in Jor-
dan. Some of the heuristic knowledge generated by the
WTVS model in the training phase is used to forecast
unknown future values. The experimental results show
that the WTVS model is more accurate than TVS model.
The advantages of the WTVS model are as follows.
1) External factors were considered in WTVS model.
In the data preprocessing phase, the impact of random
factors is weakened by smoothing the historical data. In
the trend training and load forecasting phase, the sea-
sonal factor was introduced into TVS model.
2) The most recent data from the prediction load has
the greater impact. The weighted historical data are con-
sidered in WTVS model.
6. Acknowledgements
The authors would like to thank the anonymous review-
ers and the work was supported by Shanghai Municipal
Natural Science Fund under grant 10ZR1401400 and The
Fundamental Research Funds for the Central Universities
under grant 11D10417 and 11D10402.
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