Short Term Load Forecasting Using Weekday Load Models and Bias Models

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SHORT TERM LOAD FORECASTING USING WEEKDA Y LOAD MODELS AND BIAS MODELS M. Nakamura

Abstract. This paper proposes one - day - ahead load fore casting using daily updated weekday load models and weekly updated bias models for e veryday - of - the-week loads. The load characteristics are examined first for actual data from Kyushu Electric Power Company and weather stations in Kyushu throughout 1982. Then , according to properties of the loads, the algorithm of the load forecasting is derived. Based on actual data, the accuracy of the proposed load forecasting is found to be very high, the standard deviation of the error of the load forecast being about 3%. Keywords.

Least square approximation; load forecasting; modelling; optimal filtering;

parameter estimation; power system cont rol; weather factor .

This paper proposes one - day - ahead load forecasting using daily updated weekday load models and weekly updated bias models for everyday-of - the - week loads . The load characteristics are e xamined first for actual data from Kyushu Ele c tri c Power Company and weather stations in Kyushu throughout 1982 . Then , according to p r operties of the loads, t he algo rithm of the l oad forecasting is derived. Based on actual data , the accuracy of the proposed load forecasting is found to be very high , the standard deviation of the error of the load forecast being about 3% .

1. INTRODUCTION There

has

been

much recent research (Load Fore -

casting Working Group , 1980) on load f orecasting which is one of the most important tasks for powe r companies where power is generated to follow fluctuating load demands. Load forecasting is classified into three classes; l ong term , short term and instantaneous load forecasting. A long term load fore cast determines the required future expantion of generation ,

bution of term) load scheduling generated

transmission and distri -

electric ene rg y . One - day - ahead (short forecasting is necessa r y for optimum of generating units, and then power is by the load frequency control based on

instantaneous load forecasts.

'3 :E

Aam, Skarstein and Gagnat (1981) repo r ted a one day - ahead load forecasting method in which Kalman filter (kalman, 1960) techniques were used for updating the state of load models and the model parameters

were

estimated

with

the

4000

'"

... Cl

maximum

ex: 0

likelihood method. Nakamura , Tahara and Hatazaki (1981) also proposed a one - day - ahead load forecast ing method for weekday loads based on a comb ination o f peak and minimum load models and a load ratio model with updating by the Kalman f i l ter.

.>
... Q

--.J

Fig . 1.

Daily load curves for a week (The shaded area represents the bias; a difference between the load and a smoothed weekday load) .

6000 4000 2000

0

0

-'

0

HOUR j o 12 0 12 0 12 0 12 0 12 0 12 0 12 0 I I DAY k TUE WED THU FRI SAT SUN MaN

8000

:;: ::;:

c:x:

8000

A.M .

0 kO DAY 0

20

40

80

60

100

120

140

160

180

1""1,,,,1"1,1111111"11'1,,1'11 1111 11111111111 111 1111 11111'111111111111 111111111111111111111!111111 1111 11111''111111111'1'1I1 111 '111 111'11 1' 1,111'11''1 ,1 11 ,,11' 11!1 '111 111 11111111

MONTH Fig. 2.

5

1

2

3

4

5

6

7

8

9

10

11

12

Hourly weekday loads at 0 a.m. , 6 a.m. and 12 p . m. from Kyushu Electric Power Company throughout 1982.

2097

2098

M. Nakamur a

a3 (kO)

2. LOAD CHARACTERIST I CS AND LOAD FORECASTING SCHEME

• • • aj(kO)

t ion , Actual data fro m Kyushu Electr i c Powe r Company a nd weathe r s t ations in Kyus h u throug h out 1982 a r e used to analyse the load characte ri s ti cs. Da i ly load cu rves , depicted i n Fi g . 1, are c l assified into fi ve pa t te r ns; those f o r weekda ys (e x c l ud i ng Monday s) , Saturda y s , Sunda y s , Mond ay s and spec i al days (holida ys and other days of un usua l load s ) . One - day- ahead load forecasting is ou tli ned b e l o w accor d i ng to patte r n .

the

load

] T) is an n- dimens i onal regresT repr esents t r ansposi vj(kO) is scalar noise. S i nce t he

Slon coef f i c l en t vector,

characte r is t ics

fo r each

and

r eg r ession coef f icient vector changes r andomly ,

a

tr ans i t i on equation f o r Xj (k O) i s const r ucted as (2 )

whe r e Wj (k O) i s an n- dirnensiona l noise v ecto r with mean z e r o.

In sect i on 3 .1, using the exponential least squa r es method , a weekday load constructed

weighted model is

which gives estimates for the regres -

sion coeff i c i ent vector Xj(k O) in (1) and (2) . 2.1 Weekday Load Characteristics Load curves for the weekday pattern are similar to each other and there is litt l e daily va r iation for 24 hourly loads tj (k O) at the j - th hour of the k O- th day. Figure 2 illustrates t h e hourly weekday loads at 0 a . m. , 6 a . m. and 1 2 p.m. fo r a one yea r pe r iod . The l oad changes a r e mainly caused by weathe r

factors

such

as

the re is an o bvi o us

the we2 kday fa c tors.

l o ads

< j(k O)

correlat i on

and

Characteristics To extract a difference between weekday loads othe r loads , biases are defined as

between

the load

change

b~ (k 2 )=i j (kD) _
( 3)

b5 (k 3 )=i j (k D) - t j (k 3 ) where bj(k 1 ) is Saturday bias at the the kl _ th day in Saturday pattern , bias and bj(k 3 ) Monday bias. For the l atest value of a smoothed updated by the equation

j - th hour o f b}(k:) Sunday (3) , £ j(k O) i s weekday l o ad

For certain parts of the correlation i n Fig. 3, the load ~j (k O) is expressed by a linear regres sion of l oad change factors as

(4)

where t h e i ndex AO is in the range O< AD ~l . App r op r i at e selection of the value of AD is discussed i n section 4.2 . Figure 4 illustrates the biases for AO=0.4 at noon dur i ng a one year period.

tj (k O)=a3 (kO)+a} (k O) £j (kO - l) +a5 (k O) tj (k O)+ • • ' +Vj (k O) =H j (k O)x j (k O)+vj (k O)

(1) In section 3 . 2 ,

wher e Hj (k O) (= [ 1 £ j (k O- l) tj (k O) •• • ]) is an n - dimensional measurement vecto r , Xj (k 0 ) (= [aj (k 0 )

3: :E

8000

'n

6000

Cl

5000

":'~~f~{~~~':':"

"

5000

6000

LOAD

7000

i j (kD- l )

8000

'n

6000

Cl

5000

:E

.: ..

la

20

30

tj ( k O)

[ ·C)

-'

:·: ~·.:>~r!: ;':: .:'.~ . . ..

5000

'

..

8000

..

..I<:

..

...

'n

6000

Cl

5000

c:(

0

20

40

HUM IDITY Fig. 3.

60 h j (k 0)

80

100

.....J

40

50

60

[ %]

Correlation between the weekday loads tj (k O) and load change factors 'j (k O- l) , tj (k G) , h j (k O), dj(k O) ) at noon fo r the year 1982.

70

with

smoothing

.:~. .

7000

0

'n

0

TEMPERATURE :::

8000

... 6000 --l

...

--l

7000

c:( C)

exponential

8000

[MW]

..I<:

Cl

an

7000

c:( C)

0 --l

0

by

..I<:

...

3: :E

bias models are constructed

updating

equation.

0

..I<:

c:(

weekly

3: :E

.

".'

7000

0

and

bj(k1) =iJ(kO) - tj(k 1 )

temperature,

humidity and discomfort index. Figure 3 shows the re l ation between the loads £j(k O) and l oad change factors ; the load for the previous day £ j(k D- l), the current temp erature tj (k O), the cu r rent humidity hj (k O) and the current discomfort index dj (k O) at noon during the year. As seen in Fig . 3 ,

2.2 Satu r day , Sunday and Monday Load

80

2099

Short Te rm Load Forecasting

2 .3 Load Forecasting Scheme

load model updating, the bias models updating and the load fore cast ing . Both model updatings depend

For the weekday pattern , a one- day - ahead load forecast i j (k+llk) is made on the basis of th: daily updated weekday l oad model (the estimate x of the regression coefficient vector) and the load change factors f or the next day ; for examp le, the current load £j(k) , the fore cas~ by weather stations ~f the tempera ture tj (k+llk) , the humidity h j (k+l lk) and the discomfo r t index d j (k+llk).

on patte rn q ,

and the load forecasting depends on

pat t ern r. Introdu cing identity transition equations in the model upda ti ngs , k is asigned to days for days kO , kl , k 2 , k 3 , k" in each patte rn . To make the algorithm unif orm in different patterns , estimates o f the weekday biases are formally introduced (5 )

For Saturday , Sunday and Monday patterns, a pseudo-weekday load ~;(k) is calcu lated from th e current load tj (~) and the we ek ly updated b,as mod el (the estimate b of the bias). Using £* (k) , forecast ~j(k+llkj is a pseudo- weekday load obtained using the same method as for the weekday pattern . A one -day-ahead ~~ad forecast i j (k+llk) is generated by correc ting lj (k+l l k) by once again using the bias model.

fo r a 11 j and k. Since the algorithms at each hour during a 24 hour

pe ri od are the same , only the algorithm at j - th hour is shown in this section . A weekday load

WEATHER For special days , a one - day - ahead load forecast i J (k+llk) is made using other techniques; such as i ntuition based on many years experience .

~j

The outline of the load forecasting Fig. 5 .

is

ONE-DAY-AHEAD

LOAD

I NFORMA TI ON

LOAD FORECAST

(k l

given in

3. ALGORITHM OF LOAD FORECASTING A number i s given to each pattern: the weekda y pattern is 0 , the Satu r day pattern 1, the Sunday pattern 2 , the Monday pa tt ern 3 and the special day pattern 4. The algor i thm is shown for obtaining a one - day-ahead l oa d f orecast ij (k+l lk) in patte rn r (the pattern for t he next day) , when the current time is the j-th hour of the k - th day in pattern q (the pattern for the cu rrent day). The algorithm i nvolves three parts ; the weekday

Fig . 5 .

Outline of the load forecasting .

~

::;:

2000 .... ~

·,..,j -n

.0

1000

= =

0

U)

(J)

c:t:

tO~-;(~~

-1000

o:l

~

1000 -1000

DAY

ki

III1

MONTH Fig . 4.

I 5

1

I11

2

1111111111111111111111111111111111

I

3

I

4

1

5

I

6

I

r

7

Saturday bias, Sunday bias and Monday bias at the year 1982.

8

° a.m . ,

I

9

10

11111111

I

11

I

12

6 a . m. and 12 p .m . for

2100

M. Nakamur a

model whose load change factors are
updated on weekdays. Then, e x cep t for special days, either the bias model s or the smoothed we e kday load is updated everyday . A ;;riori information at the j-th hour of the k - th day in pattern q (the current day) is bJ\ (klk - ll, b'2 (k k - l) , b'3j (k k - lI and ~j (k - l) . j

I

3.1 Weekday Load Mode l Updating The weekday load model which gives estimates for the reg r ession coefficient vector in (11 and (21 is updated eve ryday (excluding special daysl by the exponential weighted least squares method. ~ ;riori inf ormation at the j - th hour of the k-th day in pattern q (the current dayl is an n-dimensional vector i j(klk- l), an nxn positive definite matrix Cj(klk- ll and the scalar values £j(k- ll , b'j(k l k- lI. In the case of weekday, Saturday, Sunday and Monday patterns f or the current day (q=0,1,2 or 31 , the current information on the current load 'j(kl and the discomfort index dj (kl is obtain-:d .. USlng the current load, £j (kl and an:z ;;r1-or,. estimate of the bias bj(klk - ll from the bias model, the pseudo- weekday load is defined as:

I

The estimates of the biases by an exponential smoothing equation are summarized as follows

I

(61

As seen from (51 , £* (kl equals £ j (kl for q=O. Using £;(k- ll and dj(~1 the measurement vector is cons tructed as:

j

(k I

b 2 (k +l lkl

The estimate i j(k+llkl for the regression coef ficient vector in (11 and (21 , given the pseudo- weekday load £ ;(kl, is generated by the e xpone ntial weight ed least squares method: Xj (k+llkl={I - Kj (klHj (kl }Xj (klk-ll

(91

Kj (kl=C j (klk-lIHj(kI T

/{ Hj(kICj(k l k - lIHj(k)T+ ~l.

(101

The symbol I is an identity matrix . The pa rameter ~ is an arbitrary positive scalar constant and does not affe ct the estimate I j(k+ llk). The index \ is in the range O<\~l and dlrectly effects the estimate Xj (k+llkl. The value of the index A is appropriately se lected to minimize the load forecasting error in section 4.1 . In the case of the specia l day pattern for the c urrent day (q=41 , the weekday load mod el , not being updated, is formally expressed by identity transition equations: Xj (k+llkl=x j(klk- ll ,

(11)

I

(121

C j (k + 1 k I =C j (k k - 1 I .

2;

load,

£*J' (k-ll

I

b 3 (k+llk) J

b3 (k k - lI

~j

b3(klk - l)

~j (k - lI - £j (k)

(k - lI - £ j (k)

(131 where

~

is a 4x4 diagonal matrix

diag ( '. 0 1 dlag (

I,

(for weekday,

I,

(for Saturday ,

q = lI

I,

(for Sunday ,

q=2)

\3 I ,

(for Monday,

q=31

\1 . '

diag (

/,"-

diag (

diag ( 1 1 1 1 I,

q=OI

(for special day , q=4) . (141

~

\.? are in the range O
,\

' .

,

~oad ~ore (;astin 2 ' and b1(k+l l k) , b3(k +l l kl, bj(k+llkl and ~j(kl are sto r ed for ,; ij'Ll'i i nfo rm.J.tion for the bias model updating for the next day .

(81

Cj (k+l lkl=f{I - Kj(kIHj(kl}Cj(klk - ll,

For the pseudo - weekday (kl =£; (k- ll .

tj (k - l) - £j (k) + (1 - ,\ )

3 . 3 Load Forecasting

+Kj (kl £j (kl ,

I

= :~

J

Tih? indices

(71

I

b3 (k k - lI

bj(k+l l kl

,\=

q (kl=£j (kl+bj(klk-lI .

-

is

used;

In all cases , Ij(k+l l k) and tj(k) are used for the load forecasti ng , and Xj (k+llk), Cj (k+llk) and ~j(kl are stor ed for a ;;riori informati on for the weekday load model updating for the next day.

The load forecasting depends on pattern r for the next day. In the case of weekday , Saturday , Sunday and Monday patterns for the next day (r=0 , 1 , 2 or 31, the load forecast is made on the basis of the weekday load mod e l, the bias models and the forecast by weather stations; the discom f ort index dj (k+llkl for the next day. Using the pseudo:weekday load Zj(k) and the weather for ecas t dj (k+llkl the following matrix is co nstructed :

(15) Using this matrix and the estimate i j (k+llkl from the weekday load model, the pseudo -weekday load for ecast can be obtained by the equation :

Th e one - day-ahead load fore cast is obtained by i ;(k+llk) and the estimate bj(k+llkl as: ij(k +l lk)=i.j(k +l lk) - bj(k+lik). As seen from (51, r=O.

ij( k+l ikl equals

(17) 2;(k+l lk) for

In the case of the special day pattern for the next day (r=4 1 , a one - day-ahead load forecast tjP(k+l l k) is made:

i

i

Zj (k+l kl =i.j P (k +l kl .

(18)

3 . 2 Bias Model Updating The bias models are updated by an exponential smoothing equati on . The Saturday bias model is updated on Saturdays, the Sunday bias model on Sundays, the Monday bias model on Mondays. The smoo thed weekday load in each bias model is

A flow chart of th e load forecasting given in Fig. 6.

4. EVALUATION OF THE LOAD FORECASTING SCHEME

algor i thm is

2101

Short Term Load Forecasting

Using actual data from Kyushu Electri c Power Company and weather stations in Kyushu throughout 1982, the one - day - ahead load f orecast ing scheme is evaluated. The evaluation is performed in three parts; evaluation of the weekday load mode ls, the bias models, and the load forecasts.

from

February

to

Oecember,

since

those months

were not affected very much by the a information (1 10). In the evaluation,

x·

t~ere

assumed that

pri or i it is

is no weather fore c ast error by

weather stations; tj(k O+l l k O)=tj(k O+l), hj(k O+l l k O) =hj(k O+l) and d j (k O+llk O)=dj(k O+l), Eleven weekday load models are evaluated and their

4.1 Evaluation of the Weekday Load Models

measurement matrices are as follows:

H . (k O) = [ J Hj(k O)=[

(1)

Using on l y weekday data, the weekday load models are evaluated by the standard deviations of the load forecasting error

(L) (T)

(19)

o ~= J

o s=

0 j=

0 =

IjYO( oj) 2/24

(21)

(LH)

J

(LD)

II

n O (o s) 2/ nO s J s=2 s

Table 1

A PRIORI INFORMATION l (klk-l) C l (klk-l) 'i(k-l)

b,lklk-l)

(,(k-l)

b~(klk-ll bjlk!k-ll

-, I I I I I I I

YES

CURRENT INFORMATION ' l (k) d

l

(k)

WEEKDAY LOAD t'ODEL UPDATING "l(k) -11 'i(k-l) dl(k)1

' j (k) -'l

(k) +;;1 (k I k-l)

;j (k+ljk) "Xj (kilt-I) Cl(k+llk) -C l (kl k-l )

9

..

Ij

(It-l)

b;Ck+1Ik) "bjlklk-ll bjCk+llk) -bjlklk-ll bj(k+llk) -bjlklk-ll

ij

(k)

from

the

e leven

mode ls f o r

weekday load the standard

update d

by

is

(4)

deviation:

12 24 [ n g . (25) s=2 The bias models,

for AO =0 .4 updated by (13) ,

are

Standard De viations of th e Load Forecasting Errors from Eleven Mod e l s for A=0.85

(tor q-lI (tor q-2 )

tjllt)

~ OS

0j

0 3 6 9 12 15 lB 21

140 104 96 275 379 397 304 25B

130 108 95 225 276 336 26B 229

Ul 98 90 185 24B 239 234 164

'" 0

(k.likl -ijP (k+llk)

1) Ck+llkl -t; ("+11 kl-bj (k+ll kl

Flow c hart of the load forecasting algorithm (Identity transition equations are abbreviated).

~

'"z 0

..'" VI

0

H

0

L T

L H

L 0

L T

LL 00

H

163 204 139 253 195 333 326 237 133 114 197

>

ij

T

163 205 134 229 200 350 395 465 187 109 245

z

LOAD FORECAST

L

2 3 4 5 6 7 8 9 10 11 12

..!8 LOAD FORECAST

1

142 171 115 199 139 192 210 235 211 118 182

~

~i (k+ llk)-~, (k+1Ik ); , (k+llk)

erro rs

The smoothed evaluated by

TABLE 1

(for q-l l

J.

4.2 Evaluation o f the Bias Models

i

Fig. 6.

d (k 0 ) d (k O-ll j j £ j (k O- 2 )

The standard deviations of the forecasting errors from the LD model for different values of the index A are shown in Table 2. Th e pertinent valu e of A, which minimizes 0 , is found to be 0 . 83.

b, lk+llkl bj (k+llk) hi (k+llkl

H;lk ... llk).[l (jlk) d ;(k +llkl)

(k 0 ) h j (k 0 )

A=0. 85 . As se e n fr o m 0 in Table 1 , the LD mo d e l is found to be appropriate f o r the weekday l o ad model.

(for q-Ol

STORE ;'j(k+llkl Cj( k+llk)

(2 4 )

(k O)

shows the standard deviations of th e l o ad

forecasting

BIAS MODELS UPDATING i, (It)

(k O)

tj (k 0 ) tj (k O- l) h j (k O) h j (k O-l)

(23)

where ng denotes the number of the weekdays in the s - th month. The s tand ard deviations 0 j and 0 are calculated by the load forecasting erro rs

(k 0 )

£ j (k O- 2)

(22)

12 n O(o s)2 / 24s~2 n O s s J

j~O

i

~ j (k O- l)

(LLTTHH) Hj(k O)=[ 1 £ j(k O- l)

23

s=2

Hj(k O)=[

(LT) (20)

s=2

(LLDD)

(D)

In,} ( n . (k O) }2/ n O k° s

IV IV

(LTH)

(H)

The standard deviations are as follows

t j (k O-l)

Hj(k O)=[ tj (k 0 ) J H (k O)=[ 1 h (k O) j j d (k O) H (k O) = [ j J Hj(k O)=[ £ j (k O- l ) t j Hj (k O)=[ £ j (k O-l) h j Hj( k O)=[ £ j (k O- l) d j Hj(k O)=[ £j (k O-l) t j

LL 'IT HH

170 194 129 227 201 345 353 539 185 108 251

140 168 118 195 131 162 224 173 199 119 183

147 177 117 221 136 164 205 206 149 U6 175

177 188 131 251 200 335 277 287 143 U3 210

144 173 120 215 126 157 226 160 146 114 175

174 156 118 208 125 155 221 194 155 U4 167

156 191 127 254 134 166 236 176 16. 126 175

192 168 134 240 138 160 240 237 lBl 128 168

146 Ul 101 290 3B4 394 303 264

Ul 98 91 183 217 21B 210 170

113 105 91 178 20B 216 224 167

137 113 97 238 276 319 266 240

112 105 92 179 193 210 20B 171

U7 UO 94 IBB 202 199 200 176

120 US 98 199 210 225 219 lBO

127 131 108 217 222 220 213 200

266 222 17B 271 167 167 222 162 164 1 76 lB2

2102

M. Nakamura

evaluated by the standard deviation of the form .

12 n~ 23

.

s=2 kl j=O

J

/

L

£1=

.

.

..

2

12

L (bl(kl) _t;l(kljkl_l)} /24 L

L.

.

n~,

s=2

J

i=1,2,3

(26)

where n§ denotes the number of days on pattern i in the s-th month. Table 3 shows the standard deviations for differ ent values of the indices ,i. The pertinent values,

which

minimize

the standard deviations,

are found to be; '0=0 . 4, ,1=0.6 , ,2=0.6 and ,3=0.9.

(1) According to properties of l oad curves, 24 hourly weekday load models wer e const ructed individually during a 24 hour pe ri od. (2) The weekday load models, whi ch give estimates for the regression coe ffi c ients of weather and other facto rs, were updated everyday (excluding special days) by the exponential weighted least squares method. (3) By using the bias models, the load fore casts for Saturday, Sunday and Monday patterns were also obtained by the same method used as for the weekday pattern. (4) The values of the indices " ,0 , ,1 , ,2 , ,3 were appropriately selected to minimize the forecasting errors.

4.3 Evaluation of the Load Forecasts ACKNOWLEDGMENT Using data from everyday in the week, the one-day ahead load forecast is evaluated. For the proposed load forecasting, the LD model with , =0.83 is used as the weekday load model and bias models with '0=0 .4, ,1 =0.6, ,2=0.6 and ,3=0.9 are adopted . The A columns in Table 4 show the standard deviations of the load forecast i ng errors for all days , weekdays , Saturdays, Sundays and Mondays. The standard deviations of two other schemes are also shown in Table 4. One is an o scheme (a load forecast for the next day is simply the actual load for the current day) and the other is a PM scheme (Nakamura , Tahara and Hatazaki, 1981) . As seen in Table 4, the accuracy of the proposed load forecasting is found to be very high , the standard deviation of the error of the load forecast being about 3\.

5. CONCLUSIONS One - day- ahead load forecasting based on a combination of the weekday load model and the bias models was proposed. Based on actual data , the accuracy of the proposed load forecasting was found to be very high, the standard deviation of the error of the forecast being about 3\. Features of the load forecasting were summarized as follows:

The author is grateful to Mr. W. J. Herlofsky of Saga University for revising the English in this paper, to Mr. S. Tahara and Mr . T. Yoshida of Saga University for their help in the FACOM computation and Mr. H. Hatazaki of Kyushu Electric Power Company for providing the actual data .

REFERENCES Aam, S., Skarstein, <1>. and Gagnat, L. (1981). Implementation of a load prediction program system for the Norwegian power pool. Proceedings of t he 8th I FAC Congress on Control Science and Technology for the Progress of Society. ~ , 3085 - 3090 Kalman, R. E. (1960). A new approach to linear filtering

and

prediction problems .

filter using atmospheric

~ z

;::

OS

~ > c c a:

'"

"'z" '"

.

VI

the standard deviat ions expressed as

Il

12 0

TABLE 3

141 164 125 342 214 317 315 219 143 1 20 194

14 2 166 124 27 5 175 255 273 1 78 162 146 208

14 3 169 122 235 14 3 176 228 149 165 12 3 190

14 4 173 120 215 126 157 226 160 149 ll4 175

14 5 175 121 2ll 1 23 157 230 162 141 11 2 173

146 179 123 207 122 159 2 37 165 137 112 173

152 194 134 202 131 172 264 175 136 12 1 180

162 215 148 206 147 188 297 188 145 138 187

178 238 166 216 168 208 3 31 206 161 171 198

20 1 263 187 232 196 232 369 232 195 24 1 21 7

236 291 213 265 237 264 4 22 276 281 367 252

222 197 170 162 162 163 173 189 209 237 286

Standard Deviations £i for Different Values of the Index

,1

a percentage)

~ E' E'

1.0 0.9 0 . 8 0.7 0 . 6 0.5 0. 4 0.3 0.2 0. 1 472 281 860 267

308 215 298 230

24 3 207 239 23 0

221 203 223 234

213 201 22 1 24 0

210 202 225 24 9

209 204 2 32 259

210 207 242 271

2ll 212 2 54 285

215 219 268 302

WEEKDAY

ALL DAY

O'

15 ;: ~

~

Q

~ ~ ~

0j

0

I"

MON

SUN

0

A

0

P"

A

0

A

0

A

0

4 5 6 7 8 9 10 11 12

142 169 128 145 113 161 228 184 161 107 148

178 205 156 204 227 J2J 365 294 168 123 20)

135 112 115 157 116 146 211 168 108 109 125

171 222 167 236 202 307 321 222 134 125 193

145 259 149 206 140 206 230 225 131 145 197

121 19) 189 130 98 205

188 170 194 177 198

189 220 100 100

491 ))3 276 125 231

149 149 110 117 130 176 320 234 2)4 76 212

18) 149 102 155 234 233 378 262 112 9) 181

176 146 116 127 92 146 17) 178 159 124 183

184 211 10) 165 329 217 ,97 456 153 118 230

0 3 6 9 12 15 18 21

118 102 93 182 194 207 190 IS)

154 121 111 241 311 339 283 226

107 98 94 167 177 176 196 141

134 116 99 219 274 327 268 220

128 109 101 210 234 255 246 195

119 185 101 116 10) 113 165 332 190 408 258 397 215 JJ5 210 264

129 110 82 252 261 261 172 154

174 138 112

141 105 90 170 18) 198 156 118

169 125 127 283 381 381 300 207

145 219 190 2.8 4 . 0 3.6

170 285 J.3 5.2

187 206 4.1 4.4

)

!

SAT

A

,

ii!

E' El

1153 - 1163

Standard Deviations of the Load Forecasting Errors by Three Schemes (The numbers in the bottom line show

1. 0 .95 .90 .85 .83 .80 .70 .60 .50 . 4 0 .30

2 3 4 5 6 7 8 9 10

0

lQ ,

Standard Deviations of the Load Forecasting Errors from the LD Model for Different Values of the Index, TABLE 4

~

and noise variances.

ll,

Int. J. System Sci.,

i"

temperature informa -

tion. Proceedings of the 8th IFAC Congress on Control Science and Technology for the Progress of Society. ~ , 3067-3072 Nakamura , M. (1982). Relationship between steady state Kalman filt er gain

TABLE 2

Trans .

ASME Ser. D (USA) , 35 -4 5 Load Forecasting Working Group (1980) . Load forecast bibliography. IEEE Trans. Power Apparatus & Syst. (USA) , PAS-99, l' 53-58 Nakamura , M., Tahara, S. and Hatazaki, H. (1981). One-day - ahead l oad forecasting by Kalman

156 23 4 3 . 1 4.3

~34

~76

;'5]

247 272 265 222

149 258 ).0 4.7