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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 values
- A is a set contains data or objects
- Example Maximum Load data
- x is an individual value within the X data set
- mA(x) the membership function that connects the

two sets together

Fuzzy- Expert System

- The membership function mA(x)
- Determines the degree that x belongs to A
- Its value varies between 0 and 1
- The high value of mA(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 are
- Triangular
- Trapezoid

Membership function

x variable

Fuzzy- Expert System

Membership function

DLmid

DLmin

DLmax

x variable

Fuzzy- Expert System

- A fuzzy set A in X is defined to be a set of

ordered pairs - Example Figure before shows that x - 750

belongs a value of A 0.62

Fuzzy- Expert System

- Triangular membership function equation
- Triangular membership function is defined by
- DLmax or DLmin value when function value is 0
- DLmaid value when function value is 1
- Between DLmax and DLmin the triangle gives the

function value - Outside this region the function value is 0

Fuzzy- Expert System

- The coordinates of the triangle are
- x1 DLmin and y1 0 or m(x1) 0
- x2 DLmid and y1 1 or m(x2) 1
- The slope of the membership function between x1

DLmin and x2 DLmid is

Fuzzy- Expert System

- The equation of the triangles rising edge is

Fuzzy- Expert System

- The complete triangle can be described by taking

the absolute value - This equation is valid between DLmin and DLmid
- Outside this region the m(x) 0

Fuzzy- Expert System

- The outside region is described by
- The combination of the equations results in the

triangular membership function equation

Fuzzy- Expert System

- Combination of two fuzzy sets
- A and B are two fuzzy sets with membership

function of mA(x) and mB(x) - The two fuzzy set is combined together
- Union
- Intersection
- sum
- The aim is to determine the combined membership

function

Fuzzy- Expert System

- Union of two fuzzy sets points included in both

set A and B - The membership function is

Fuzzy- Expert System

- Union of two fuzzy sets points included in both

sets A or B

mB

mA

Fuzzy- Expert System

- Intersection of two fuzzy sets points which are

in A or B - The membership function is

Fuzzy- Expert System

- Intersection of two fuzzy sets points which are

in A and B

mB

mA

Fuzzy- Expert System

- Sum of two fuzzy sets
- The membership function is

Fuzzy- Expert System

- Sum of two fuzzy sets

mA

ms mA mB

mB

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 last 10-12 Tuesdays

Load forecasting with Fuzzy- expert system

- Plot the historical data of load and temperature

relation for selected 10 Tuesdays.

Load forecasting with Fuzzy- expert system

- The 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 is
- Lp 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

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

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.

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

Load forecasting with Fuzzy- expert system

- Determination of the error coefficient e by Fuzzy

method. - DLp 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 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 2s.

Load forecasting with Fuzzy- expert system

- s 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
- s 450 MW in our example shown before.

Load forecasting with Fuzzy- expert system

- The data of the triangular membership F1(DL1)

function is - DL1_min - 450MW, DL1_mid 0 MW
- The substitution of these values in the general

equation gives

Load forecasting with Fuzzy- expert system

- The data of the triangular membership F1(DL1)

function is - DL1_min - 450MW, DL1_mid 0 MW
- The substitution of these values in the general

equation gives

Load forecasting with Fuzzy- expert system

- The membership function is shown below if s

450MW and DL -1500MW..500MW

DL1_min - 450MW

DL1_mid 0 MW

DL1_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

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.

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, d0 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 - DLmin 2 gp d g and DLmid 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

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

Load forecasting with Fuzzy- expert system

- Temperature forecasting error
- An expert select the appropriate membership

function for the study - The membership functions are

ML

MH

H

L

C

Membership function

Load ( MW)

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 DL3

Load forecasting with Fuzzy- expert system

- The analytical method to calculate the combined

membership function F3(DL3) is based on - Every value of the membership function value has

to be updated using - The method is illustrated in the figure below.

Load forecasting with Fuzzy- expert system

- The combined membership function will be a

triangle with the following coordinates - DL3_min DL1_min DL2_min s (2gp d gp)
- DL3_mid DL1_mid DL2_mid 0 g d
- The substitution of this values in the general

equation gives the membership function

Load forecasting with Fuzzy- expert system

- Combined of Model uncertainty and Forecast

-temperature uncertainty membership function

(F3(DL3) .

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 XX 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

Load forecasting with Fuzzy- expert system

- Membership function for Operators Heuristic Rules

Not confident

Quite confident

Confident

Load forecasting with Fuzzy- expert system

- The prediction of the DLp error coefficient

requires the combination of the membership

function of - Operators Heuristic Rules (F4(DL4) with the
- Combined of Model uncertainty and Forecast

-temperature uncertainty membership function

(F3(DL3) - The next slide shows the two function

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

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 DLp

Load forecasting with Fuzzy- expert system

Dlcorrection - 273.25MW

Load forecasting with Fuzzy- expert system

- The error coefficient DLp 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