Short Term Load Forecasting with Expert Fuzzy-Logic System.

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


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


Typical membership functions areTypical membership functions are• TriangularTriangular• Trapezoid Trapezoid

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0.2

0.4

0.6

0.8

11

0

F 3 L( )

5001.5 103 L

x variable

Mem

bers

hip

func

tion


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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


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


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


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


The equation of the triangle’s rising edge is:

)( 11 xxmyy

)(1

1 minmin

LxLL

ymid


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)(


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


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)(


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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0.8

11

0

F 4 L( )

F 3 L( )

F3 7 L( )

5001.5 103 L

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0.2

0.4

0.6

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11

0

F 4 L( )

F 3 L( )

5001.5 103 L

A

B

BAC


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)(


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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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


Sum of two fuzzy setsSum of two fuzzy sets The membership function is :The membership function is :

})(),({minsup)( xxx BAyxzS


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


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


• 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


• 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


• 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


• 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


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


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


Determination of the error coefficient by Fuzzy method.

Lp error coefficient has three

components:• Statistical model error• Temperature forecasting error• Operators’ heuristic rules


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


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.


• 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


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


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


• 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


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.


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


Temperature forecasting error• The error in peak load is proportional with

the error in temperature

• This suggests a triangular membership function.

ppp TgL


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


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)


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


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


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


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


Temperature forecasting error• An expert select the appropriate membership function

for the study

• The membership functions are:

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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


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


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)


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


Combined of Model uncertainty and Forecast -temperature uncertainty membership function (F3(L3) .

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11

0

F 1 L 1

F 2 L 1 2

F 3 L 1

5001500 L 1


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


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


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,


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


Membership function for Operators Heuristic Rules

Quite confidentConfident

Not confident


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


Membership functions F3 and F4, (K= 0.8)

which has to be combined together


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


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


lcorrection = - 273.25MW


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

Short Term Load Forecasting with Expert Fuzzy-Logic System.

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