short term load forecasting with expert fuzzy-logic system
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Short Term Load Forecasting with Expert
Fuzzy-Logic System
Load forecasting with Fuzzy- expert system
Several paper propose the use of fuzzy system for short term load forecasting
Presently most application of the fuzzy method for load forecasting is experimental
For the demonstration of the method a Fuzzy Expert System is selected that forecasts the daily peak load
Fuzzy- Expert System
X is set contains data or objects. • Example: Forecast Temperature valuesExample: Forecast Temperature values
A is a set contains data or objects• Example : Maximum Load data
x is an individual value within the X x is an individual value within the X data setdata set
x) the membership function that connects the two sets together
Fuzzy- Expert System
The membership function The membership function x) x)
• Determines the degree that x belongs to A
• Its value varies between 0 and 1
• The high value of x) means that it is very likely that x is in A
• Membership function is selected by trial and error
Fuzzy- Expert System
Typical membership functions areTypical membership functions are• TriangularTriangular• Trapezoid Trapezoid
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0
F 3 L( )
5001.5 103 L
x variable
Mem
bers
hip
func
tion
Fuzzy- Expert System
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11
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F 3 L( )
5001.5 103 L
x variable
Mem
bers
hip
func
tion
Lmin Lmax
Lmid
Fuzzy- Expert System
A fuzzy set A in X is defined to be a A fuzzy set A in X is defined to be a set of ordered pairsset of ordered pairs
Example: Figure before shows that x = - 750 belongs a value of A = 0.62
}])(,[{ XxxxA
Fuzzy- Expert System
Triangular membership function Triangular membership function equationequation
Triangular membership function is defined by• Lmax or Lmin value when function value is 0
• Lmaid value when function value is 1
Between Lmax and Lmin the triangle gives the
function value
Outside this region the function value is 0
Fuzzy- Expert System
The coordinates of the triangle are:• x1 = Lmin and y1 = 0 or (x1) = 0
• x2 = Lmid and y1 = 1 or (x2) = 1
The slope of the membership function between x1 = Lmin and x2 = Lmid is
min12
12 1
LLxx
yym
mid
Fuzzy- Expert System
The equation of the triangle’s rising edge is:
)( 11 xxmyy
)(1
1 minmin
LxLL
ymid
Fuzzy- Expert System
The complete triangle can be described by taking the absolute value:
This equation is valid between Lmin and Lmid
Outside this region the (x) = 0
midLL
Lxx
min
min )(1)(
Fuzzy- Expert System
The outside region is described by
The combination of the equations results in the triangular membership function equation
0,)(
1,)(min
minmin
midmidmid LL
LxLLLxifx
midmid LLLx min
Fuzzy- Expert System
Combination of two fuzzy setsCombination of two fuzzy sets• A and B are two fuzzy sets with A and B are two fuzzy sets with
membership function of membership function of x) x) andand x) x)
• The two fuzzy set is combined together– Union – Intersection– sum
• The aim is to determine the combined membership function
BAC BAC
BAS
Fuzzy- Expert SystemFuzzy- Expert System
Union of two fuzzy sets: points Union of two fuzzy sets: points included in both set A and Bincluded in both set A and B
The membership function is :The membership function is :
Xxxxx BABA })(),({max)(
Fuzzy- Expert SystemFuzzy- Expert System
Union of two fuzzy sets: points Union of two fuzzy sets: points included in both sets A or Bincluded in both sets A or B
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11
0
F 4 L( )
F 3 L( )
F3 7 L( )
5001.5 103 L
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0.4
0.6
0.8
11
0
F 4 L( )
F 3 L( )
5001.5 103 L
A
B
BAC
Fuzzy- Expert SystemFuzzy- Expert System
Intersection of two fuzzy sets: Intersection of two fuzzy sets: points which are in A or Bpoints which are in A or B
The membership function is :The membership function is :
Xxxxx BABA })(),({min)(
Fuzzy- Expert SystemFuzzy- Expert System
Intersection of two fuzzy sets: Intersection of two fuzzy sets: points which are in A and Bpoints which are in A and B
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0.4
0.6
0.8
11
0
F 4 L( )
F 3 L( )
F2 6 L( )
5001.5 103 L
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11
0
F 4 L( )
F 3 L( )
5001.5 103 L
A
BAD
Fuzzy- Expert SystemFuzzy- Expert System
Sum of two fuzzy setsSum of two fuzzy sets The membership function is :The membership function is :
})(),({minsup)( xxx BAyxzS
Fuzzy- Expert SystemFuzzy- Expert System
Sum of two fuzzy sets:Sum of two fuzzy sets:
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F 1 L( )
F 2 L( )
F 3 L( )
5001.5 103 L
s = A +
B
Load forecasting with Fuzzy- expert system
Steps of the proposed peak and through load forecasting method• Identification of the day (Monday, Tuesday, etc.).
Let say we select Tuesday.
• Forecast maximum and minimum temperature for the upcoming Tuesday
• Listing the max. temperature and peak load for the Listing the max. temperature and peak load for the last 10-12 Tuesdayslast 10-12 Tuesdays
Load forecasting with Fuzzy- expert system
• Plot the historical data of load and temperature relation for selected 10 Tuesdays.
30 31 32 33 34 351 10
4
1.05 104
1.1 104
1.15 104
Temperature
Pea
k L
oad
11140.786
10171.767
Li
Loadi
3530 T i
Load forecasting with Fuzzy- expert system
• The data is fitted by a linear regression curveThe data is fitted by a linear regression curve
• The actual data points are spread over the regression curve
• The regression curve is calculated using one of the calculation software (MATLAB or MATCAD)
• As an example
– MATCAD using the slope and intercept function
– MATLAB use
• to determine regression curve equation
Load forecasting with Fuzzy- expert system
• The result of the linear regression analysis isThe result of the linear regression analysis is : :
• LLp p is the peak load, is the peak load,
• Tp is the forecast maximum daily temperature,
• g and h are constants calculated by the least-square based regression analyses.
• For the data presented previously g= 300.006 and h= 871.587
pppp hTgL
Load forecasting with Fuzzy- expert system
• This equation is used for peak load forecasting::
• As an example if the forecast temperature is Tp= 35C
• The expected or forecast peak load is:
587.871006.300 ppppp ThTgL
MWLp 797.371,11587.87135006.300
Load forecasting with Fuzzy- expert system
The figure shows that the actual data points are spread over the regression curve.
The regression model forecast with a statistical error.
30 31 32 33 34 351 10
4
1.05 104
1.1 104
1.15 104
Temperature
Pea
k L
oad
11140.786
10171.767
Li
Loadi
3530 T i
Load forecasting with Fuzzy- expert system
In addition to the statistical error, the uncertainty of temperature forecast and unexpected events can produce forecasting error.
The regression model can be improved by adding an error term to the equation
The error coefficient is determined by Fuzzy method. The modified equation is:
ppppp LhTgL
Load forecasting with Fuzzy- expert system
Determination of the error coefficient by Fuzzy method.
Lp error coefficient has three
components:• Statistical model error• Temperature forecasting error• Operators’ heuristic rules
Load forecasting with Fuzzy- expert system
Statistical model error• The data is fitted by a linear regression curveThe data is fitted by a linear regression curve
• The actual data points are spread over the regression curve
• The statistical error is defined as the difference between the each sample point and the regression line
• This statistical error will be described by the fuzzy method
Load forecasting with Fuzzy- expert system
Statistical model error• Different membership function is used for each day
of the week (Monday, Tuesday etc.)
• The membership function for the statistical error is determined by an expert using trial and error.
• A triangular membership function is selected.
• The membership function is 1, when the load is 0 and decreases to 0 at a load of 2.
Load forecasting with Fuzzy- expert system
• is calculated from the historical data with the following equation:
– Lpi is the peak load
– Tpi is the maximum temperature
– n is the number of points for the selected day
• = 450 MW in our example shown before.
n
i
ppippi
n
hTgL
1
2
Load forecasting with Fuzzy- expert system
The data of the triangular membership F1(L1) function is:• L1_min = - 450MW, L1_mid = 0 MW
The substitution of these values in the general
equation gives:
0,
2
11,2)( 1
111
LLifLF
Load forecasting with Fuzzy- expert system
The data of the triangular membership F1(L1) function is:• L1_min = - 450MW, L1_mid = 0 MW
The substitution of these values in the general
equation gives:
0,)(
1,)(_1min_1
min_1_1min_1_11
midmidmid LL
LLLLLLifLF
Load forecasting with Fuzzy- expert system
• The membership function is shown below if = 450MW and L = -1500MW..500MW
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F 1 L( )
5001.5 103 L
L1_min = - 450MW
L1_mid = 0 MW
L1_max = 450MW
Load forecasting with Fuzzy- expert system
Temperature forecasting error• The forecast temperature is compared with
the actual temperature using statistical data (e.g 2 years)
• The average error and its standard deviation is calculated for this data.
• As an example the error is less than 4 degree in our selected example.
Load forecasting with Fuzzy- expert system
Temperature forecasting error produces error in the peak load forecast
The error for peak load is calculated by the derivation of the load-temperature equation
ppppp hTgL
pp
p gdT
dL ppp TgL
Load forecasting with Fuzzy- expert system
Temperature forecasting error• The error in peak load is proportional with
the error in temperature
• This suggests a triangular membership function.
ppp TgL
Load forecasting with Fuzzy- expert system
Temperature forecasting error• A fuzzy expert system can be developed using
the method applied for the statistical model• A more accurate fuzzy expert system can be
obtained by dividing the region into intervals• A membership function will be developed for
each interval• The intervals are defined by experts using the
following criterion's
Load forecasting with Fuzzy- expert system
Temperature forecasting error• The intervals for the temperature forecasting
error are defined as follows:– The temperature can be much lower than the
forecast value. (ML)
– The temperature can be lower than the forecast value. (L)
– The temperature can be close to the forecast value. (C)
Load forecasting with Fuzzy- expert system
Temperature forecasting error– The temperature can be higher than the forecast
value. (H)
– The temperature can be much higher than the forecast value. (MH)
• A membership function is assigned to each interval.
• d = -4 for ML, d = -2 for L, d=0 for C, d = 1 for H and d = 2 for MH
Load forecasting with Fuzzy- expert system
Temperature forecasting error• The membership functions are determined by
expert using the trial and error technique
• A triangular membership function with the following coordinates are selected:
– Lmin = 2 gp+ d g and Lmid = d gp
• These values are substituted in the general membership function
Load forecasting with Fuzzy- expert system
Temperature forecasting error• The membership function for change in peak
load due to the error in temperature forecasting is :
• Where: d and gp are a constants defined earlier
0,
2
11,2)(
2
222p
p
pp g
dgLgdgLifLF
Load forecasting with Fuzzy- expert system
Temperature forecasting error• The membership function for change in peak
load due to the error in temperature forecasting is :
• Where: d and gp are a constants defined earlier
0,)(
1,)(_1min_1
min_1_1min_1_11
midmidmid LL
LLLLLLifLF
Load forecasting with Fuzzy- expert system
Temperature forecasting error• An expert select the appropriate membership function
for the study
• The membership functions are:
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0
F 2 L 2 4
F 2 L 2 2
F 2 L 2 0
F 2 L 2 2
F 2 L 2 4
15001500 L 2
Mem
bers
hip
func
tion
Load ( MW)
ML MHHL C
Load forecasting with Fuzzy- expert system
Combination of Model uncertainty with Forecast -temperature uncertainty.• The peak load should be updated by an
amount :
• The membership function for L3
213 LLL
})(),({minsup)( 221133 213LFLFLF LLL
Load forecasting with Fuzzy- expert system
The analytical method to calculate the combined membership function F3(L3) is based on:
Every value of the membership function value has to be updated using:
The method is illustrated in the figure below.213 LLL
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F 1 L 1
F 2 L 1 2
F 3 L 1
5001500 L 1
L1L2
L3
F3 (L3)
Load forecasting with Fuzzy- expert system
The combined membership function will be a triangle with the following coordinates:• L3_min= L1_min + L2_min = + (2gp + d gp)
• L3_mid= L1_mid + L2_mid = + g d
The substitution of this values in the general equation gives the membership function
0,)(
1,)(_3min_3
min_3_3min_3_33
midmidmid LL
LLLLLLifLF
Load forecasting with Fuzzy- expert system
Combined of Model uncertainty and Forecast -temperature uncertainty membership function (F3(L3) .
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F 1 L 1
F 2 L 1 2
F 3 L 1
5001500 L 1
Load forecasting with Fuzzy- expert system
Operators Heuristic Rules The experienced operator can update the forecast by
considering the effect of unforeseeable events or suggest modification based of intuition.
The operator experience can be included in the fuzzy expert system
The operator recommended change has to be limited to a reasonable value.
The limit depend on the local circumstances and determined by discussion with the staff
Load forecasting with Fuzzy- expert system
Operators Heuristic Rules The operator asked :
• How much load change he/she recommends. (X MW)
• What is his confidence level– Quite confident, use factor K = 0.8
– Confident, use factor K= 1
– Not confident, use factor K = 1/0.8 = 1.25
Triangular membership function is selected
Load forecasting with Fuzzy- expert system
Operators Heuristic Rules Triangular membership function parameters determined
through discussion with operators.
Historically the operator prediction error is in the range of 200-300MW
The selected data are:• L4_mid = X selected value for the example is X = -250MW
• L4_min = K X+X selected value for the example is K = 0.8,
Load forecasting with Fuzzy- expert system
Operators Heuristic Rules The substitution of this values in the general
equation gives the membership function
The membership function for the operators heuristic rule is shown the next slide
0,)(
1,)(_4min_4
min_4_4min_4_44
midmidmid LL
LLLLLLifLF
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11
0
Fa 4 L 0.8( )
Fa 4 L 1( )
Fa 4 L1
0.8
250750 L
Load forecasting with Fuzzy- expert system
Membership function for Operators Heuristic Rules
Quite confidentConfident
Not confident
Load forecasting with Fuzzy- expert system
The prediction of the Lp error coefficient requires the combination of the membership function of• Operators Heuristic Rules (F4(L4) with the
• Combined of Model uncertainty and Forecast -temperature uncertainty membership function (F3(L3)
The next slide shows the two function
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Fa 4 L( )
Fa 3 L( )
15001500 L
Load forecasting with Fuzzy- expert system
Membership functions F3 and F4, (K= 0.8)
which has to be combined together
Load forecasting with Fuzzy- expert system
The error coefficient is determined by combination of combined Model & Temperature error and Operators Heuristic Rule.
The and relation suggests that the intersection of two fuzzy sets, which are points in F3 and F4
The membership function in case of the intersection is:
})(),({min)( 435 LFLFLF
Load forecasting with Fuzzy- expert system
The membership function can be calculated by the following equation:
The combined membership function is presented on the next slide.
The maximum of the membership function gives the error coefficient Lp
})(),(),()({)( 43435 LFLFLFLFifLF
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Fa 4 L 0.8( )
Fa 3 L( )
F 5 L( )
6001400 L
Load forecasting with Fuzzy- expert system
lcorrection = - 273.25MW
Load forecasting with Fuzzy- expert system
The error coefficient Lp is determined by the presented fuzzy expert system method
This coefficient has to be added to the load forecast obtained by the liner regression method
The corrected load forecast is:
MW
LhTgL ppppp
045.645,11248.273797.371,11