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Short-term Feeder Load Forecasting: An Expert System Using Fuzzy Logic
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SHORT-TERM FEEDER LOAD FORECASTING: AN EXPERT SYSTEM USING FUZZY LOGIC G. Lambert Torres*, B. Valiquette and D. Mukhedkar [)f!}(/rlllll'lll IJj Einlriwi ElIgillfnillg. Ewi" Plliylnhlliljlll' til' .\/lIlIlrhd, .\/lIlllrl'fli, Clllllltill
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Abstract. A short-term feeder load forecasting using an expert system approach is presented. This approach allows merging of traditional mathematical techniques and operator's experience. The rules of this expert system are calculated using historical data of the feeder and represented by fuzzy conditional statements. An exemple using real data coming f'9Dm power systems is presented.
Keywords. Load forecasting; expert systems; artificial intelligence; fuzzy sets theor y ; linearization techniques. INTRODUCTION
AN OVERVIEW OF LOAD FORECASTING METHODS
There are two different objE' c ti v es to study feeder load forecasting: economic ope rati o n of power plants ami economic solut.ion of the distribution systems !Halldschin and Dornemann, 1988J. The first objective is to study the global performance of the load without the need to tak e into account the individual performanc e of e a c h f .. ed e r of the system. The second objecti ve must take into account thE' characteristics of each load of the system. Th e first objective is applied to power system's operation and planning, while the second one is appli ed in distribution system 's operation and planning.
Main objective of power system forecasting is to enable in a ll time an adaptation between demand and generation. This adaptation must cons ider load and generation characteristics and possible paths in transmission or distribution networks to supply energy to consumers. Load shapes must be represented by daily or weekly cyc les . Thi s representation is affected by some speculat i ve and climatical variations. For generation cllaracteristics, variations can be divised in two different kinds: due variation (such as: dry periods, preventive maintenance) and aleatory variation (such as: forced outage). Possible paths to supply energy to consumers are found by logical techniques and heuristics searchs.
Currently, there are three possible classifications for l oad forecasting: short-term (from the next half-hour to twenty four hour ahead), medium -term (from the next day to the next year), and long-term (beyond the the next year). Criterias for each method are different. As an example the most important factors are in the short-term load forecasting: day of the week, temperature, humidity, seasonal effects and so on. In long-term l oad forecasting, the factors are : ec o nomic aspects , po1 itical aspects, industrial degree of a region and so on. Objectives for each method are also differents [Bunn and Farmer, 1985J. For example, short-term forecasts da y -to-day operation and scheduling of the power system while medium-term forecasts scheduling of fuel supplies and maintemance operations. Finally, long-term forecasts planning operations .
In this section, problem formulation and applied techniques for short-term feeder forecasting are presented. Problem Formulation The load forecasting can be di v ised in two general parts: peak l oad model and load shape model. The first one deals only with daily or weekly peak load modeling. Gupta ( 1971) and Goh, Ong and Lee ( 1986) has used this model. In l oad shape model, the load forecasting F(t) is obtained through operation of a standard load (or namely, a base load), B(t), a and deviation load, Wet) . This operation is made using additive form, Eq. (I), or multiplicative form, Eq. (2).
This paper describes an expert syste m to be used as an aid in operation and planning of distribution system. This expert system makes a short-term feeder load forecasting . It can be used for system's operation as an aid for the operator to know the load from one hour to twenty four hour ahead. It is useful when the feeders are operated closely to their maximum capacity. This system is also useful in planning because it is possible to know if the nominal capacity of each equipment of the feeders is exceeded or not, according to the feeder load prospective.
F(t) - B(t) + Wet)
(1)
We t)
(2 )
F(t) - B(t)
In most of the cases, the additive form is used. An example of multiplicative form was presented by Baker et al. ( 19 78) and has been used b y the Central Electricity Generating Board. Standard Load . This value must characterize the base load of the feeder. It is calculated using historical data. These data must be merged according to time-of-the-day, day-of-the-week and time-of-the-year (week -of-the- year or month-of-the year or season-of-the-year). The division by day allows to check the l oad values for each time of
Following an outline of load forecasting methods, a brief explanation about fuzzy logic is presented. Expert system proposed is then described together with an example using Hydro-Quebec Power System data (Canada).
-EiS
G. Lambert Tones . B. \'aliqllette and D. \Illkh edkar the day separately. The division by day of the week allows to select Saturday and Monday of any one day . It is important because the load shape of these days is different of the others days. The division by year (in weeks, months or seasons) is function of the load characteristics. Current models use division by season option, i.e., they have the value of B(t) calculated for each season, separately.
and transfer function modelling), general exponential smoothing, state space using Kalman filter, and knowledge-based approach. Each method was applied to the same database, allowing a direct comparision of results.
FUZZY LOGIC APPROACH Fuzzy Statements (Fuzz y Terms)
The standard load calculation can be divided in two parts. The first one makes an average using all common days in the same period. The holidays are included with Saturdays and Mondays. The second part investigates on the particular characteristics for each day of the week, separately. For this to be done, a simple or weighted moving average is made. Equations (3) to (5) present the standard load equations. (3)
B(t) - B, (t) + B2 (t)
B, (t) -
m
n
T;:l
bl
L(t,d,w)
(l/m.n)
(4)
n B2 (t) -
L w- l
(l/n)
L(t , d,w) - B, (t)
(5)
Where L(t,d,w) represents the load at time t for da y -of-the-week d of week w. The value of n can be changed from one week to another week and represents how many common days are in this week. The value m represents the number of the weeks in the modeled period. Other factors must be summed in Eq. (3), depending on correlation of data. For example, Abu-Hussein et al . (1981) has proposed to sum of second-order polynomials and temperature-dependent components. Deviation Load. This value the most recent variations of contain information for about some models information about used [Gupta, 1972 ) .
is used to represent the load. This value last 3 hours. But in last 24 hours can be
Autoregressive and exponential smoothing are the most common methods used to calculate the deviation of load value. The first one can be described as a sum of the last differences between actual and forecasting load, Eq . (6). Galiana, Handschin and Fiechter (1974) ha v e proposed the use of last two differences. n
R(t) -
L
A(t-i) - F(t-i)
Usuall y in engineering, exact or mathematical statements are used. These statements corresponds to exact information , such as "x- 3.0·'. "2s8s6" or " y- 3t+24". All these statements c ontain a precise information, for example: the value of x is 3.0, wi t h a grade of membership like 100% (-1) and for all other value (2.8, 2.9, 3.1, 3.2) , the grades of membership in the solution is zero . But, for a value coming from the real world (a measure, for example) , this grade of membership is not true due to different kinds of imprecision such as the tool used, the influence of observer , and so on.
(6)
i~l
In addition, mo st of t he time , information from the real wo rld are impre c i se , i . e, ine xact statements, such as "the value o f x is not big" or "the temperature i s about 4 ° " , So a theory to express correctly the grade of member s hip would be desirable . Zadeh (1965) has proposed a theory to make a rapprochement b e twe e n the preci s ion of classical mathematics and the impr ec ise information coming from real world. This theory is c a lled Fuzzy Sets Theory and works with grades of membe rship o f x in A, i.e., ~A(x), taking in the set M ~ [O,l ) . Goguen (1967) has proposed a gene ralization of this theory, where the values of the membership are taken from the set (-00 ,00 ) (or [ -1,1) for a normalized set). In Fuzzy Sets Theor y , both exact and inexact (fuzzy) statements can be manipulated and operated. This is ver y important in an expert system for load forecasting because there are many factors which are fuzzy and their characterization by an number is difficult. For example: "The temperature this afternoon will be about 4'". The temperature is a very important factor in load forecasting, but it is not easy to characterize it by an exact nume rical quantit y . In addition, the linguistic hedges (such as, about , little, big, small, medium , large) can be modified by other linguistic hedges (such as, not, v ery , very very) [Zadeh, 1973 ). For example, " value of x is not big", where big is a linguistic hedge and not, a modifier. Figure 1 shows fuzzy terms.
Exponential smoothing method uses the same differences that the autoregressive method , which are related by a function of the lead time and different smoothing constants . Applied TechniQues Gross and Galiana (1987) have divised load shape forecasting in time-of-day and dynamic models . In the first one , a set of curves is composed by day-of-week , temperature conditions , wind conditions, and so on. To calculate the load forecasting, the operator must select whi c h curves will be used. In d y namic models , the load forecasting is not only function of time-of-day but also of last information about weather conditions, actual load and random inputs. Moghram and Rahman (1989) have evaluated five different methods for short-term load forecasting: multiple linear regression , stochastic time series (autoregressi ve integrated moving-average process
4
6
'9
not big 1_
3 Fig. 1.
10 Fuzzy Terms .
x
Shurt-term Feeder Luad Furecasting Or, in discrete form:
all cases, a complete database is suitable. It allows to have feeder load forecasting with major degree of hit.
(110), (21 0 .2), (310.6), (411), (5 0.6), (610.2), (710) }
1'1(8) -
(
1'2 (x) -
(
1'2 (x) -
( (011),
I
This expert system is written in Fuzzy-Prolog [Ricards, 1986J. It uses Turbo-Prolog (from Borland International Inc.) for logical part and Fortran (from Microsoft Inc.) to yield a standard and deviation feeder load (mathematical part).
(11 0 . 1), (2 I0 . 2 ), (31 0 . 3) , (40.4), ... , (910.9), (1011) } ( 0 0),
(11 0 . 9 ), (210 . 8), (410 . 7), (4 0.6), ... , (910.1), (1010) }
Following an overview of database structure, an outline of load forecasting method used (mathematical part) is presented and the logical part is described.
As seen, not express the complement of a set. Fuzzy Conditional Statements (Fuzzy Rules) Fuzzy Set Theory is also able to manipulate and operate fuzzy conditional statements (fuzzy rules). These statements are in general expressed in form "If A then B", where A and/or B are terms having fuzzy meaning . In this paper, A is a fuzzy statement and B, a linear equation. A fuzzy conditional statement has the following general form : If Xl is 1'1' x 2 is 1'2' then P,(x 1 ,x 2 ' . . . ,x.).
t t t t
is is is is
Database must contain historical data information about the load and theirs weather parameters. Dry bulb temperature, wet bulb temperature, relative humidity, wind direction and wind speed are the most common we at her parameters. For special load, others factors (such as sun inclination, cloudiness) can also have influence on the load.
of fuzzy conditional statement set is
An example shown as: : If : If : If : If
Database
Data structure is built to discriminate for each data: its value (v), hour (h), day (d), year (y) and season (s). The general forms of the data are presented below .
Where 1', is a membership function.
Rl R2 R3 R.
-157
small l big l small 1 big l
and and and and
h h h h
is is is is
smal1 2 smal1 2 big 2 big 2
then then then then
Real Power Dry Bulb Tempearture Wet Bulb Temperature Relative Humidity Wind Direction Wind Speed
P l (t, h). P 2 (t, h). P3 (t, h). P, (t, h).
The inferred value of consequence is calculated by Eq. (7) .
(7)
P(t,h) -
Where A(l'i) represents membership of rule i.
the
minimum
grade
of
Fuzzy Relation Fuzzy relation is defined as a grade of membership for a n-tuples formed by a Cartesian product. One may write:
Sometimes when the number of fuzzy variables is large, it is suitable to merge some these variables. At this moment , the use of fuzzy relation techniques to decrease order of the problem is indispensable. For example, the above fuzzy conditional statement set can be represented by only two rules.
P (v,h,d,w,y,s). D (v,h,d,w,y,s). M (v,h,d,w,y,s). H (v,h,d,w,y,s). WD(v,h,d,w,y,s). WS(v,h,d,w,y,s) .
Real power values, temperature values, relative humidity and wind speed are stored in [MWJ, [OCJ, [%J and [km/hJ, respecti ve ly. For the wind direction values, the following convention is employed: North is 1, and each 22.5° in hourly sense is more. For the calm day 0 is used. So for example, the direction WSW is 12 and NE is 3. Twenty four hours notation is used for hour values. The day values are numbered as 1 from Monday until 7 for Sunday. Holidays receive the number 6 (Saturday). The week va lues are numbered by season and the most recent receives the number 1. This database is built to manipulate using last three years and the current year. The year values are from 1 for the current year until 4 for three years ago. The season values are as follows: winter - 1, spring - 2, summer - 3, and fall - 4. For up to two missing values (v) of same one data, they are calculated using single average between the previous and next va lues. When more than two values are missing, the data is forgotten. Special days (such as: snow storm) are stored in the database with a flag. These values are not used for the current calculations, except when this weather condition is forecasted.
RI: If (t,h) is small then P{(t,h). R2 : If (t,h) is big then Pi(t,h). Where small and big represent fuzzy relations.
Mathematical Part It is also possible to make a composition of two fuzzy relations or between a fuzzy relation and a fuzzy statement. In this paper, all composltions will be made using MAX-MIN composition. Equations (8) and (9) represent such a composition.
MAX MIN
1'01 (x,y)
- MAX MIN
1', 2 (x,y)
I'o,(x,z) -
1"2 (y, z)
(8)
(9)
In the mathematical part, standard load B(t) and residual load R(t) are calculated using database information. Standard Load. The standard load is used to represent the historical data set. Hourly load shape must be represented using weather conditions. This relation between hourly load and weather conditions must be establish directly from a mathematical function:
SOFTWARE DESIGN B(t) - f(P,D,M,H ,WD,WS) there are many As seen in the previous section, methods to calculate the load forecasting. And in
or through standard shape
for
each
one
of
the
(;. LlIllbert TOITes. 13. \'a liqul'ttl' and D. elements.
in
our
~llIkht'dkar
case.
witho ut
to
make
a
better
representation of the resiadual load. The standard shape is made by weighted average using a decrease degree of membership for each year of database. If the load evolution is not large in the last years, the decrease is little. On the contrary, if the load evolution is large, the decrease is big. Following that, the memberships ~,(year) and ~2(year) represent a large and a not large evolution of the load, respectively. -
(111), (210.5), (310 . 1), (410) )
(year) -
(111), (210.8), (310.6), (4 10.4)
~,(year) ~2
Equation (10) shows calculation for a time
standard shape of the season s.
value
>%>(t,d,w,y,s)
Sometimes
a division of the day is necessary to represent the residual load depending to the degree of correlation of data used. For special characterisitcs of the load, two kinds are considered. The first one is daily routine load. The second one repres e nts thermic inertia of the load. Daily routine load can be described as usual attitudes for specific part of the day. For example, if there is a load point at 9:00 a.m. and the load at 8:00 a.m. is greater than stal~ard load for the same ". eather condi t i ons, then part of l oad point has been already consumed. In the same way, if the load at 8:00 a.m. is l ess than standard load then, possibily the load point will be increased. This residual load form is applicable in load forecasting for up to 3 hours ahead prediction.
4
L y-l
~i (y)
(10) Ilhere >%> ( ... ) repr ese nts each one of the database elements namely , real power, dry bulb temperature, wet bulb tempearature , and so on. The values k, and k2 depend of the kind of day (working days or weekend days) , and of the specific week, year and season. The value k) depends of specific year and
Thermic inertia is to have difficulty in changing from a better to a worse temperature. For example, if the average temperature in the winter for the last day was -S ·C , and today is -20·C, the load will increase due temperature differellce plus a certain thermic inertia. If for the next day, the temperature remains at -20 ·C, this inertial component will decrease or come to nothing. In Canada for example, this factor is very important because
excessive
temperature
residual load form is mainly hour ahead load forecasting .
season.
Following these calculations for all times-of-day t, an equation of the shape must be calculated. For this to be done, Lambert Torres and Mukhedkar (1989a, 1989b) has proposed an algorithm to express a shape by fuzzy conditional statements. These statements present a linear membership function as a premise and a straight line as a consequence. Lambert Torres, Valiquette and Mukhedkar (1989) has de ve lopped an example of this algorithm for load modelling.
Lo~ical
(11) For recent load performance , infomation about the last 4 weeks are used. These data are merged in a multiple linear model through the method of the least squares . Equation (12) shows the general form for this model, which uses differences between actual and standard weather information. R, (t) - a o + + + + + +
a, a2 a) a, as a6
and Ilhere >%>. (t) stantard value of respectively .
(Da (Da (Ma (Ha (Da
(t) (t) (t) (t) (t) (101. (t)
Ds Ds M, H, D. W.
(t» (t»2 (t» (t» (t -1» (t -1»
>%>, (t ) represent actual time a element of
(12 ) and t,
In Eq. (12 ), the square difference between dry bulb temperature due the strongest correlation bet~een
tile
load
and
this
temperature
This
for
24
Part
Following
the
calculation
of knowledge base, as
shown in previous subsection, the inference engine
will be actived . The engine is used to find the load forecasting value. For this to be acheived, it must infer in the knowledge base to find the standard load value B(t), load forecasting value for recent load performance R,(t), and the correction value due to
Residual Load . The residua l load contains two kinds of information, the first one being rec e nt load performance as a function of weather conditions, and the latter one correspollds to special characteristics of the load. The residual load can be calculated by the operations between these two kinds of information, Eq. (11).
ranges.
applicable
R2 (t). Equation (13) forecasting found. B(t) +
L(t) - 0, Ilhere 0i is forecasting.
a
special
shows
as
cllaracteristics,
the
final load
02
ratio
to
apportion
the
load
Figure 2 shows a pictorial representation of the proposed expert system .
ILLuSTRATIVE EXAMPLE This example uses data coming from Hydro-Quebec Power Sys tem, Canada. Numerical data extends from April 1984 to July 1988 and conta ins information related to real power flows in lilles 1288 alld 1289, as well as weather conditions such as dry blub temperature, wet bulb temperature, relative humidity, willd speed and direction. These parameters were measured hourly. The performance shown in this section is function of the addition of these two lines because they have worked in parallel. The season used in this example was winter, and the month of forecasting was January 1988. For the standard l oad calculation. Eq . (10) is used and th" results are shown in Fig. 3.
was
considered. The increase of parcel number is possible (for example, other square differences). But, it is not recommendable because problems of multivariable grows up rapidly in complexity, and
If olle applies the algorithm to compute the model of a shape proposed by Lambe rt Torres and Mukhedkar (1989a, 1989b) for the real po~er data
above, one Catl obtaitl:
Short-term Feeder Load Forecasting
Fig. 2.
Pic~orial
Representation of the Proposed Expert System.
JO
Where: )J,(t) 1'2 (t) 1'3 (t) 1', (t)
10
ID
L-____L -_ _
~~
o
__
~
____
12
~
_ _ _ __ L_ _ _ _
16
-159
t + 0.3881 0.6190 t t + 0.4495 0.4725 t
for for for for
3 :<; t < 13 5 < t :<; 15 15 :<; t < 25 16 < t :<; 27
~.
24
20
--0.0419 - 0.1079 --0.0184 - 0.0254
Note :
Hours
(a) 01)' &JIb Temperature ('Ci
1. Outside the limits for t , the membership values are zero. 2. A 24 hour notation is used. 3. 27 is 24 + 4, i.e., 3 a.m. of the next day.
Using fuzzy concepts, I',(t) )J2(t) 1'3 (t) I',(t) -
·4
one can associate: 'very small' 'big' 'small' 'very big'
For the recent load performance, it is possible to calculate the following rules. then P, (t) - P(t) + + 2.60 (D(t)-Ds(t» then P2 (t) - P(t) then P3 (t) - P(t) - 1 . 73 ( D(t)-Ds(t»
RI: If t is 1', (l ,D( t»
10
.,2 L-----''-----'----1''-2---1''-6----:20'----'240
R2 : If t is 1'2(t,D(t» R3 : If t is 1'3 (t, D( t»
Hours
Where:
(b)
for
1.0
D(t) < Om i . (t)
Relative Humidify (%)
1') (t ,D(t»
-
82
{
(l/D.i.(t)-D,(t» for Dm,.(t)
1 - 1') (t)
for
D(t)
1'3 (t)
for
D(t) > D, (t)
11 0 0 70L-----~----~-----'-----~----~----~. o 20 24 16 12 Hours
(c)
Fig. 3. Standard Curves. a . Real Power. b. Dry Bulb Temperature. c. Relative Humidity.
1'3
(t ,D (t»
{
(D(t)-Ds(t» D(t) :<; D.(t)
for D(t) > D.(t)
0.0
f
:<;
for
:<;
0, (t)
O(t) < D.(t)
(l/D.. ox (t) - 0. (t) ) (D (t) - Ds (t) ) for D,(t) :<; D(t) :<; D•• ,(t) 1.0
for O(t) > Dm.,(t)
And, P(t), D,(t), Dmi.(t) and Dmax(t) are average load, standard dry bulb temperature, minimum dry bulb temperature observed and maximum dry bulb temperature observed , for the time t, respectively.
G. Lambert Torres, B. \'aliqll ct te alld D. \llIkh edkar In
the
same
way,
~l(t,D(t» ~2 (t ,D(t» ~) (t ,D(t»
one
can
associate:
- ' small' - 'medium' - 'big'
Real Power (MW)
170
Forecasting Load
OOO~----~~----~----~t2~--~t~6----~2~0-----724~ Hours
Fig. 4. Actual and Forecasting Load for one hour ahead forecasting. Real Power (MW)
170
150
130 --- Actual Load - - - Forecasling Load ~~
__
~
____
o
~
12
____- L_ _ _ _- L_ _ _ _ 16
also
Aperfei~oamento
The load forecasting calculation for the day January 21-22 , 1988 was obtained by two different ways for one hour ahead forecasting and twenty-four h ours ahead forecasting. In the first approach, daily routine load and thermic inertia for the second one. The maximal error found in ea c h approach was 2.43% and 4.19%, while the a verage error was 1.12% and 2.01%, respectivel y. Figure 4 and 5 represent the actual and forecasting load for the two approaches used .
ooL-__
like to thank Coordena~ao de de Pessoal de Nivel Superior (CAPES) Brazil for financial support of his doctoral program. The authors would like to express their appreciation to Mr. Paul Langlois and Mr . Jean Claude Galardo from Hydro-Quebec Power System for pro viding the load data used in this work. Thanks are also expressed to Professor Remi Tougas, Director of International Relations of Ecole Poly technique. would
20
~.
24
Hours
Fig. 5. Actual and Forecasting Load for twenty-four hour ahead forecasting.
CONCLUSIONS An expert s yste m for load forecasting using fuzzy logic has been developped in this paper. The expert system approach allows to add information about the load that in traditional methods is not eas y, as for example, dail y routine load and ther~ic inertia. The use of Fuzzy Sets Theory enables the to expert system to manipulate inexact (fuzzy) statements . The average error is 1 . 12% for one hour ahead forecasting, and 2.01% for twenty-four hour ahead forecasting . These results allows the operator to make a plan about the good way between source and consumer, without exceeding nominal capacit y of feeders and equipments. ACKOIILEDGEMENT G. Lambert Torres , who is registered doctoral student, wishes to thank Ecole Poly technique de Montreal for the facilities made available. He
REFERENCES Abu-Hussien.M.S., M.S. Kandil , M.A. Tantuary. and S . A. Farghal (1981). An accuarate model for short-term load f o re ca sting . IEEE Trans Power ADD Syst Vol.PAS-lOO, no.9, 4158-4165 . Barker.R.C .• E.D. Farmer , 11 . 0. Laing and A.D.N. March (1978). The Online Demand Validation and Predi ct i on Facilit y at the National Control Centre. OD(S)/R38/78, ~ El ect ri c ity Ge ner ati n~ Board . Bunn,D.II. and E.D. Farmer (1985) . Comparative Models for Electrical Load Forecastin~ . 1st ed., John lIiley & Sons Ltd., New York, pp.3-11 and 13-30. Galiana,F.D., E. Hand sc hin , and A. Fiechter (1974). Iden t ifi ca tion of Stochastic Electric Load Models from Phys ical Data. ~ Trans Aut Cont , Vol.AC -19 . no.6, 887-893. Goguen,J.A. (1967). L-Fuzzy Sets . J Mat Anal ~. la, 145-174. Goh,T.N., H.L. Ong , and Y.O. Lee (1986). A New Approach to Statistical Forecasting of Daily Pe ak Powe r Demand. Elec Power Syst Res , Vol.lO, no.2, 145-148. Gross,G., and Galiana,F.D. (1987). Short-term Load Forecasting. Proc IEEE , Vol.75, no.12, 1558-1573. Gupta,P.C. (1971). A Stochastic Approach to Peak Power Demand Foreca s ting in Electric Utility Systems . IEEE Trans Power App Syst, Vol.PAS-90, no.2. Gupta,P.C.. and K. Yamada (1972). Adapti ve short-term forecasting of hourly loads using weather information . IEEE lIinter Power ~, New York, 2085-2094. Handschin , E. and C. Dorne mann (1988). Bus Load Modelling and Forecasting. IEEE Trans Power ~, Vol.3, no.2, 627-633. Lambert Torres,G. , and D. Mukhedkar (1989a). On-line Fuzzy Rule Set Generator. Accepted for presentation at Advanced Information Proce ssing in Automatic Control, ~, Nancy-France, July 3-7. Lambert Torres,G., and D. Mukhedkar (1989b). Fuzzy Conditional Statements to Modelling and Control. Accepted for presentation at Third International Fuzzy Systems Association lIorld Co n~ress, Seatle - USA, August, 6-11. Lambert Torres ,G .; B. Valiquette, and D. Mukhedkar (1989). Load Modelling using Fuzzy Concepts. Submitted to IEEE PES Summer Meetin~ , Long Beach-USA , July 9-14. Moghram, I . , and S. Rahman (1989). Analysis and Evaluation of Five Short-term Load Forecasting Techniques . IEEE Hinter Meet, 89 IIM 171-0 PIIRS. Richards,B.L. (1986). Programming in Fuzzy Logic: Fuzzy-Prolog. M.Sc . Thesis , AD-A177 940. ~ Force Institute of Technolo~y, IIright-Patterson Air Force Base , Ohio. Zacteh,L.A. (1965). Fuzzy Sets. Information & ~, ~, 338-353. Zadeh,L.A. (1973). Outline of a New Approach to the Anal ysis of Complex Systems and Decision Process . IEEE Trans on Syst Man and ~ , Vol.SMC-3, no.l , 28-44.
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SHORT-TERM FEEDER LOAD FORECASTING: AN EXPERT SYSTEM USING FUZZY LOGIC G. Lambert Torres*, B. Valiquette and D. Mukhedkar [)f!}(/rlllll'lll IJj Einlriwi ElIgillfnillg. Ewi" Plliylnhlliljlll' til' .\/lIlIlrhd, .\/lIlllrl'fli, Clllllltill
"Oil
inll 'I'
IIj 11 iI.II' 1I (I' jl"lJlII c·.\IlJill Fnil' mi til' ElIgl'ltll!lrill til' ! h/fll/w. illll"/W, Bm:iI
Abstract. A short-term feeder load forecasting using an expert system approach is presented. This approach allows merging of traditional mathematical techniques and operator's experience. The rules of this expert system are calculated using historical data of the feeder and represented by fuzzy conditional statements. An exemple using real data coming f'9Dm power systems is presented.
Keywords. Load forecasting; expert systems; artificial intelligence; fuzzy sets theor y ; linearization techniques. INTRODUCTION
AN OVERVIEW OF LOAD FORECASTING METHODS
There are two different objE' c ti v es to study feeder load forecasting: economic ope rati o n of power plants ami economic solut.ion of the distribution systems !Halldschin and Dornemann, 1988J. The first objective is to study the global performance of the load without the need to tak e into account the individual performanc e of e a c h f .. ed e r of the system. The second objecti ve must take into account thE' characteristics of each load of the system. Th e first objective is applied to power system's operation and planning, while the second one is appli ed in distribution system 's operation and planning.
Main objective of power system forecasting is to enable in a ll time an adaptation between demand and generation. This adaptation must cons ider load and generation characteristics and possible paths in transmission or distribution networks to supply energy to consumers. Load shapes must be represented by daily or weekly cyc les . Thi s representation is affected by some speculat i ve and climatical variations. For generation cllaracteristics, variations can be divised in two different kinds: due variation (such as: dry periods, preventive maintenance) and aleatory variation (such as: forced outage). Possible paths to supply energy to consumers are found by logical techniques and heuristics searchs.
Currently, there are three possible classifications for l oad forecasting: short-term (from the next half-hour to twenty four hour ahead), medium -term (from the next day to the next year), and long-term (beyond the the next year). Criterias for each method are different. As an example the most important factors are in the short-term load forecasting: day of the week, temperature, humidity, seasonal effects and so on. In long-term l oad forecasting, the factors are : ec o nomic aspects , po1 itical aspects, industrial degree of a region and so on. Objectives for each method are also differents [Bunn and Farmer, 1985J. For example, short-term forecasts da y -to-day operation and scheduling of the power system while medium-term forecasts scheduling of fuel supplies and maintemance operations. Finally, long-term forecasts planning operations .
In this section, problem formulation and applied techniques for short-term feeder forecasting are presented. Problem Formulation The load forecasting can be di v ised in two general parts: peak l oad model and load shape model. The first one deals only with daily or weekly peak load modeling. Gupta ( 1971) and Goh, Ong and Lee ( 1986) has used this model. In l oad shape model, the load forecasting F(t) is obtained through operation of a standard load (or namely, a base load), B(t), a and deviation load, Wet) . This operation is made using additive form, Eq. (I), or multiplicative form, Eq. (2).
This paper describes an expert syste m to be used as an aid in operation and planning of distribution system. This expert system makes a short-term feeder load forecasting . It can be used for system's operation as an aid for the operator to know the load from one hour to twenty four hour ahead. It is useful when the feeders are operated closely to their maximum capacity. This system is also useful in planning because it is possible to know if the nominal capacity of each equipment of the feeders is exceeded or not, according to the feeder load prospective.
F(t) - B(t) + Wet)
(1)
We t)
(2 )
F(t) - B(t)
In most of the cases, the additive form is used. An example of multiplicative form was presented by Baker et al. ( 19 78) and has been used b y the Central Electricity Generating Board. Standard Load . This value must characterize the base load of the feeder. It is calculated using historical data. These data must be merged according to time-of-the-day, day-of-the-week and time-of-the-year (week -of-the- year or month-of-the year or season-of-the-year). The division by day allows to check the l oad values for each time of
Following an outline of load forecasting methods, a brief explanation about fuzzy logic is presented. Expert system proposed is then described together with an example using Hydro-Quebec Power System data (Canada).
-EiS
G. Lambert Tones . B. \'aliqllette and D. \Illkh edkar the day separately. The division by day of the week allows to select Saturday and Monday of any one day . It is important because the load shape of these days is different of the others days. The division by year (in weeks, months or seasons) is function of the load characteristics. Current models use division by season option, i.e., they have the value of B(t) calculated for each season, separately.
and transfer function modelling), general exponential smoothing, state space using Kalman filter, and knowledge-based approach. Each method was applied to the same database, allowing a direct comparision of results.
FUZZY LOGIC APPROACH Fuzzy Statements (Fuzz y Terms)
The standard load calculation can be divided in two parts. The first one makes an average using all common days in the same period. The holidays are included with Saturdays and Mondays. The second part investigates on the particular characteristics for each day of the week, separately. For this to be done, a simple or weighted moving average is made. Equations (3) to (5) present the standard load equations. (3)
B(t) - B, (t) + B2 (t)
B, (t) -
m
n
T;:l
bl
L(t,d,w)
(l/m.n)
(4)
n B2 (t) -
L w- l
(l/n)
L(t , d,w) - B, (t)
(5)
Where L(t,d,w) represents the load at time t for da y -of-the-week d of week w. The value of n can be changed from one week to another week and represents how many common days are in this week. The value m represents the number of the weeks in the modeled period. Other factors must be summed in Eq. (3), depending on correlation of data. For example, Abu-Hussein et al . (1981) has proposed to sum of second-order polynomials and temperature-dependent components. Deviation Load. This value the most recent variations of contain information for about some models information about used [Gupta, 1972 ) .
is used to represent the load. This value last 3 hours. But in last 24 hours can be
Autoregressive and exponential smoothing are the most common methods used to calculate the deviation of load value. The first one can be described as a sum of the last differences between actual and forecasting load, Eq . (6). Galiana, Handschin and Fiechter (1974) ha v e proposed the use of last two differences. n
R(t) -
L
A(t-i) - F(t-i)
Usuall y in engineering, exact or mathematical statements are used. These statements corresponds to exact information , such as "x- 3.0·'. "2s8s6" or " y- 3t+24". All these statements c ontain a precise information, for example: the value of x is 3.0, wi t h a grade of membership like 100% (-1) and for all other value (2.8, 2.9, 3.1, 3.2) , the grades of membership in the solution is zero . But, for a value coming from the real world (a measure, for example) , this grade of membership is not true due to different kinds of imprecision such as the tool used, the influence of observer , and so on.
(6)
i~l
In addition, mo st of t he time , information from the real wo rld are impre c i se , i . e, ine xact statements, such as "the value o f x is not big" or "the temperature i s about 4 ° " , So a theory to express correctly the grade of member s hip would be desirable . Zadeh (1965) has proposed a theory to make a rapprochement b e twe e n the preci s ion of classical mathematics and the impr ec ise information coming from real world. This theory is c a lled Fuzzy Sets Theory and works with grades of membe rship o f x in A, i.e., ~A(x), taking in the set M ~ [O,l ) . Goguen (1967) has proposed a gene ralization of this theory, where the values of the membership are taken from the set (-00 ,00 ) (or [ -1,1) for a normalized set). In Fuzzy Sets Theor y , both exact and inexact (fuzzy) statements can be manipulated and operated. This is ver y important in an expert system for load forecasting because there are many factors which are fuzzy and their characterization by an number is difficult. For example: "The temperature this afternoon will be about 4'". The temperature is a very important factor in load forecasting, but it is not easy to characterize it by an exact nume rical quantit y . In addition, the linguistic hedges (such as, about , little, big, small, medium , large) can be modified by other linguistic hedges (such as, not, v ery , very very) [Zadeh, 1973 ). For example, " value of x is not big", where big is a linguistic hedge and not, a modifier. Figure 1 shows fuzzy terms.
Exponential smoothing method uses the same differences that the autoregressive method , which are related by a function of the lead time and different smoothing constants . Applied TechniQues Gross and Galiana (1987) have divised load shape forecasting in time-of-day and dynamic models . In the first one , a set of curves is composed by day-of-week , temperature conditions , wind conditions, and so on. To calculate the load forecasting, the operator must select whi c h curves will be used. In d y namic models , the load forecasting is not only function of time-of-day but also of last information about weather conditions, actual load and random inputs. Moghram and Rahman (1989) have evaluated five different methods for short-term load forecasting: multiple linear regression , stochastic time series (autoregressi ve integrated moving-average process
4
6
'9
not big 1_
3 Fig. 1.
10 Fuzzy Terms .
x
Shurt-term Feeder Luad Furecasting Or, in discrete form:
all cases, a complete database is suitable. It allows to have feeder load forecasting with major degree of hit.
(110), (21 0 .2), (310.6), (411), (5 0.6), (610.2), (710) }
1'1(8) -
(
1'2 (x) -
(
1'2 (x) -
( (011),
I
This expert system is written in Fuzzy-Prolog [Ricards, 1986J. It uses Turbo-Prolog (from Borland International Inc.) for logical part and Fortran (from Microsoft Inc.) to yield a standard and deviation feeder load (mathematical part).
(11 0 . 1), (2 I0 . 2 ), (31 0 . 3) , (40.4), ... , (910.9), (1011) } ( 0 0),
(11 0 . 9 ), (210 . 8), (410 . 7), (4 0.6), ... , (910.1), (1010) }
Following an overview of database structure, an outline of load forecasting method used (mathematical part) is presented and the logical part is described.
As seen, not express the complement of a set. Fuzzy Conditional Statements (Fuzzy Rules) Fuzzy Set Theory is also able to manipulate and operate fuzzy conditional statements (fuzzy rules). These statements are in general expressed in form "If A then B", where A and/or B are terms having fuzzy meaning . In this paper, A is a fuzzy statement and B, a linear equation. A fuzzy conditional statement has the following general form : If Xl is 1'1' x 2 is 1'2' then P,(x 1 ,x 2 ' . . . ,x.).
t t t t
is is is is
Database must contain historical data information about the load and theirs weather parameters. Dry bulb temperature, wet bulb temperature, relative humidity, wind direction and wind speed are the most common we at her parameters. For special load, others factors (such as sun inclination, cloudiness) can also have influence on the load.
of fuzzy conditional statement set is
An example shown as: : If : If : If : If
Database
Data structure is built to discriminate for each data: its value (v), hour (h), day (d), year (y) and season (s). The general forms of the data are presented below .
Where 1', is a membership function.
Rl R2 R3 R.
-157
small l big l small 1 big l
and and and and
h h h h
is is is is
smal1 2 smal1 2 big 2 big 2
then then then then
Real Power Dry Bulb Tempearture Wet Bulb Temperature Relative Humidity Wind Direction Wind Speed
P l (t, h). P 2 (t, h). P3 (t, h). P, (t, h).
The inferred value of consequence is calculated by Eq. (7) .
(7)
P(t,h) -
Where A(l'i) represents membership of rule i.
the
minimum
grade
of
Fuzzy Relation Fuzzy relation is defined as a grade of membership for a n-tuples formed by a Cartesian product. One may write:
Sometimes when the number of fuzzy variables is large, it is suitable to merge some these variables. At this moment , the use of fuzzy relation techniques to decrease order of the problem is indispensable. For example, the above fuzzy conditional statement set can be represented by only two rules.
P (v,h,d,w,y,s). D (v,h,d,w,y,s). M (v,h,d,w,y,s). H (v,h,d,w,y,s). WD(v,h,d,w,y,s). WS(v,h,d,w,y,s) .
Real power values, temperature values, relative humidity and wind speed are stored in [MWJ, [OCJ, [%J and [km/hJ, respecti ve ly. For the wind direction values, the following convention is employed: North is 1, and each 22.5° in hourly sense is more. For the calm day 0 is used. So for example, the direction WSW is 12 and NE is 3. Twenty four hours notation is used for hour values. The day values are numbered as 1 from Monday until 7 for Sunday. Holidays receive the number 6 (Saturday). The week va lues are numbered by season and the most recent receives the number 1. This database is built to manipulate using last three years and the current year. The year values are from 1 for the current year until 4 for three years ago. The season values are as follows: winter - 1, spring - 2, summer - 3, and fall - 4. For up to two missing values (v) of same one data, they are calculated using single average between the previous and next va lues. When more than two values are missing, the data is forgotten. Special days (such as: snow storm) are stored in the database with a flag. These values are not used for the current calculations, except when this weather condition is forecasted.
RI: If (t,h) is small then P{(t,h). R2 : If (t,h) is big then Pi(t,h). Where small and big represent fuzzy relations.
Mathematical Part It is also possible to make a composition of two fuzzy relations or between a fuzzy relation and a fuzzy statement. In this paper, all composltions will be made using MAX-MIN composition. Equations (8) and (9) represent such a composition.
MAX MIN
1'01 (x,y)
- MAX MIN
1', 2 (x,y)
I'o,(x,z) -
1"2 (y, z)
(8)
(9)
In the mathematical part, standard load B(t) and residual load R(t) are calculated using database information. Standard Load. The standard load is used to represent the historical data set. Hourly load shape must be represented using weather conditions. This relation between hourly load and weather conditions must be establish directly from a mathematical function:
SOFTWARE DESIGN B(t) - f(P,D,M,H ,WD,WS) there are many As seen in the previous section, methods to calculate the load forecasting. And in
or through standard shape
for
each
one
of
the
(;. LlIllbert TOITes. 13. \'a liqul'ttl' and D. elements.
in
our
~llIkht'dkar
case.
witho ut
to
make
a
better
representation of the resiadual load. The standard shape is made by weighted average using a decrease degree of membership for each year of database. If the load evolution is not large in the last years, the decrease is little. On the contrary, if the load evolution is large, the decrease is big. Following that, the memberships ~,(year) and ~2(year) represent a large and a not large evolution of the load, respectively. -
(111), (210.5), (310 . 1), (410) )
(year) -
(111), (210.8), (310.6), (4 10.4)
~,(year) ~2
Equation (10) shows calculation for a time
standard shape of the season s.
value
>%>(t,d,w,y,s)
Sometimes
a division of the day is necessary to represent the residual load depending to the degree of correlation of data used. For special characterisitcs of the load, two kinds are considered. The first one is daily routine load. The second one repres e nts thermic inertia of the load. Daily routine load can be described as usual attitudes for specific part of the day. For example, if there is a load point at 9:00 a.m. and the load at 8:00 a.m. is greater than stal~ard load for the same ". eather condi t i ons, then part of l oad point has been already consumed. In the same way, if the load at 8:00 a.m. is l ess than standard load then, possibily the load point will be increased. This residual load form is applicable in load forecasting for up to 3 hours ahead prediction.
4
L y-l
~i (y)
(10) Ilhere >%> ( ... ) repr ese nts each one of the database elements namely , real power, dry bulb temperature, wet bulb tempearature , and so on. The values k, and k2 depend of the kind of day (working days or weekend days) , and of the specific week, year and season. The value k) depends of specific year and
Thermic inertia is to have difficulty in changing from a better to a worse temperature. For example, if the average temperature in the winter for the last day was -S ·C , and today is -20·C, the load will increase due temperature differellce plus a certain thermic inertia. If for the next day, the temperature remains at -20 ·C, this inertial component will decrease or come to nothing. In Canada for example, this factor is very important because
excessive
temperature
residual load form is mainly hour ahead load forecasting .
season.
Following these calculations for all times-of-day t, an equation of the shape must be calculated. For this to be done, Lambert Torres and Mukhedkar (1989a, 1989b) has proposed an algorithm to express a shape by fuzzy conditional statements. These statements present a linear membership function as a premise and a straight line as a consequence. Lambert Torres, Valiquette and Mukhedkar (1989) has de ve lopped an example of this algorithm for load modelling.
Lo~ical
(11) For recent load performance , infomation about the last 4 weeks are used. These data are merged in a multiple linear model through the method of the least squares . Equation (12) shows the general form for this model, which uses differences between actual and standard weather information. R, (t) - a o + + + + + +
a, a2 a) a, as a6
and Ilhere >%>. (t) stantard value of respectively .
(Da (Da (Ma (Ha (Da
(t) (t) (t) (t) (t) (101. (t)
Ds Ds M, H, D. W.
(t» (t»2 (t» (t» (t -1» (t -1»
>%>, (t ) represent actual time a element of
(12 ) and t,
In Eq. (12 ), the square difference between dry bulb temperature due the strongest correlation bet~een
tile
load
and
this
temperature
This
for
24
Part
Following
the
calculation
of knowledge base, as
shown in previous subsection, the inference engine
will be actived . The engine is used to find the load forecasting value. For this to be acheived, it must infer in the knowledge base to find the standard load value B(t), load forecasting value for recent load performance R,(t), and the correction value due to
Residual Load . The residua l load contains two kinds of information, the first one being rec e nt load performance as a function of weather conditions, and the latter one correspollds to special characteristics of the load. The residual load can be calculated by the operations between these two kinds of information, Eq. (11).
ranges.
applicable
R2 (t). Equation (13) forecasting found. B(t) +
L(t) - 0, Ilhere 0i is forecasting.
a
special
shows
as
cllaracteristics,
the
final load
02
ratio
to
apportion
the
load
Figure 2 shows a pictorial representation of the proposed expert system .
ILLuSTRATIVE EXAMPLE This example uses data coming from Hydro-Quebec Power Sys tem, Canada. Numerical data extends from April 1984 to July 1988 and conta ins information related to real power flows in lilles 1288 alld 1289, as well as weather conditions such as dry blub temperature, wet bulb temperature, relative humidity, willd speed and direction. These parameters were measured hourly. The performance shown in this section is function of the addition of these two lines because they have worked in parallel. The season used in this example was winter, and the month of forecasting was January 1988. For the standard l oad calculation. Eq . (10) is used and th" results are shown in Fig. 3.
was
considered. The increase of parcel number is possible (for example, other square differences). But, it is not recommendable because problems of multivariable grows up rapidly in complexity, and
If olle applies the algorithm to compute the model of a shape proposed by Lambe rt Torres and Mukhedkar (1989a, 1989b) for the real po~er data
above, one Catl obtaitl:
Short-term Feeder Load Forecasting
Fig. 2.
Pic~orial
Representation of the Proposed Expert System.
JO
Where: )J,(t) 1'2 (t) 1'3 (t) 1', (t)
10
ID
L-____L -_ _
~~
o
__
~
____
12
~
_ _ _ __ L_ _ _ _
16
-159
t + 0.3881 0.6190 t t + 0.4495 0.4725 t
for for for for
3 :<; t < 13 5 < t :<; 15 15 :<; t < 25 16 < t :<; 27
~.
24
20
--0.0419 - 0.1079 --0.0184 - 0.0254
Note :
Hours
(a) 01)' &JIb Temperature ('Ci
1. Outside the limits for t , the membership values are zero. 2. A 24 hour notation is used. 3. 27 is 24 + 4, i.e., 3 a.m. of the next day.
Using fuzzy concepts, I',(t) )J2(t) 1'3 (t) I',(t) -
·4
one can associate: 'very small' 'big' 'small' 'very big'
For the recent load performance, it is possible to calculate the following rules. then P, (t) - P(t) + + 2.60 (D(t)-Ds(t» then P2 (t) - P(t) then P3 (t) - P(t) - 1 . 73 ( D(t)-Ds(t»
RI: If t is 1', (l ,D( t»
10
.,2 L-----''-----'----1''-2---1''-6----:20'----'240
R2 : If t is 1'2(t,D(t» R3 : If t is 1'3 (t, D( t»
Hours
Where:
(b)
for
1.0
D(t) < Om i . (t)
Relative Humidify (%)
1') (t ,D(t»
-
82
{
(l/D.i.(t)-D,(t» for Dm,.(t)
1 - 1') (t)
for
D(t)
1'3 (t)
for
D(t) > D, (t)
11 0 0 70L-----~----~-----'-----~----~----~. o 20 24 16 12 Hours
(c)
Fig. 3. Standard Curves. a . Real Power. b. Dry Bulb Temperature. c. Relative Humidity.
1'3
(t ,D (t»
{
(D(t)-Ds(t» D(t) :<; D.(t)
for D(t) > D.(t)
0.0
f
:<;
for
:<;
0, (t)
O(t) < D.(t)
(l/D.. ox (t) - 0. (t) ) (D (t) - Ds (t) ) for D,(t) :<; D(t) :<; D•• ,(t) 1.0
for O(t) > Dm.,(t)
And, P(t), D,(t), Dmi.(t) and Dmax(t) are average load, standard dry bulb temperature, minimum dry bulb temperature observed and maximum dry bulb temperature observed , for the time t, respectively.
G. Lambert Torres, B. \'aliqll ct te alld D. \llIkh edkar In
the
same
way,
~l(t,D(t» ~2 (t ,D(t» ~) (t ,D(t»
one
can
associate:
- ' small' - 'medium' - 'big'
Real Power (MW)
170
Forecasting Load
OOO~----~~----~----~t2~--~t~6----~2~0-----724~ Hours
Fig. 4. Actual and Forecasting Load for one hour ahead forecasting. Real Power (MW)
170
150
130 --- Actual Load - - - Forecasling Load ~~
__
~
____
o
~
12
____- L_ _ _ _- L_ _ _ _ 16
also
Aperfei~oamento
The load forecasting calculation for the day January 21-22 , 1988 was obtained by two different ways for one hour ahead forecasting and twenty-four h ours ahead forecasting. In the first approach, daily routine load and thermic inertia for the second one. The maximal error found in ea c h approach was 2.43% and 4.19%, while the a verage error was 1.12% and 2.01%, respectivel y. Figure 4 and 5 represent the actual and forecasting load for the two approaches used .
ooL-__
like to thank Coordena~ao de de Pessoal de Nivel Superior (CAPES) Brazil for financial support of his doctoral program. The authors would like to express their appreciation to Mr. Paul Langlois and Mr . Jean Claude Galardo from Hydro-Quebec Power System for pro viding the load data used in this work. Thanks are also expressed to Professor Remi Tougas, Director of International Relations of Ecole Poly technique. would
20
~.
24
Hours
Fig. 5. Actual and Forecasting Load for twenty-four hour ahead forecasting.
CONCLUSIONS An expert s yste m for load forecasting using fuzzy logic has been developped in this paper. The expert system approach allows to add information about the load that in traditional methods is not eas y, as for example, dail y routine load and ther~ic inertia. The use of Fuzzy Sets Theory enables the to expert system to manipulate inexact (fuzzy) statements . The average error is 1 . 12% for one hour ahead forecasting, and 2.01% for twenty-four hour ahead forecasting . These results allows the operator to make a plan about the good way between source and consumer, without exceeding nominal capacit y of feeders and equipments. ACKOIILEDGEMENT G. Lambert Torres , who is registered doctoral student, wishes to thank Ecole Poly technique de Montreal for the facilities made available. He
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