1. Short Term Load Forecasting with Expert Fuzzy-Logic System

2. 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

3. 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

4. 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

5. Fuzzy- Expert System Typical membership functions are Triangular
Trapezoid
Membership function
x variable

6. Fuzzy- Expert System
Membership function
DLmid
DLmin
DLmax
x variable

7. Fuzzy- Expert System
A fuzzy set A in X is defined to be a set of ordered pairs
Example: Figure before shows that x = belongs a value of A = 0.62

8. 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

9. 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

10. Fuzzy- Expert System
The equation of the triangle’s rising edge is:

11. 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

12. Fuzzy- Expert System The outside region is described by
The combination of the equations results in the triangular membership function equation

13. 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

14. Fuzzy- Expert System
Union of two fuzzy sets: points included in both set A and B
The membership function is :

15. Fuzzy- Expert System
Union of two fuzzy sets: points included in both sets A or B mB mA

16. Fuzzy- Expert System
Intersection of two fuzzy sets: points which are in A or B
The membership function is :

17. Fuzzy- Expert System
Intersection of two fuzzy sets: points which are in A and B mB mA

18. Fuzzy- Expert System Sum of two fuzzy sets
The membership function is :

19. Fuzzy- Expert System
Sum of two fuzzy sets: mA
ms = mA + mB mB

20. 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 Tuesdays

21. Load forecasting with Fuzzy- expert system
Plot the historical data of load and temperature relation for selected 10 Tuesdays.

22. 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

23. 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= and h=

24. 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:

25. 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.

26. 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:

27. 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

28. 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

29. 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.

30. 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.

31. 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:

32. 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:

33. 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

34. 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.

35. 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

36. 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.

37. 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

38. 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)

39. 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

40. 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

41. 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

42. 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

43. Load forecasting with Fuzzy- expert system
Temperature forecasting error
An expert select the appropriate membership function for the study
The membership functions are: ML L C H MH
Membership function
Load ( MW)

44. 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

45. 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.
1500
1250
1000
750
500
250
0.2
0.4
0.6
0.8 1 F D L ( ) 2 - , 3
D L1
D L2
D L3
F3 (D L3)

46. 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 = g d
The substitution of this values in the general equation gives the membership function

47. Load forecasting with Fuzzy- expert system
Combined of Model uncertainty and Forecast -temperature uncertainty membership function (F3(DL3) .

48. 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

49. 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

50. 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 MW
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,

51. 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

52. Load forecasting with Fuzzy- expert system
Membership function for Operators Heuristic Rules
Not confident
Quite confident
Confident

53. 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

54. Load forecasting with Fuzzy- expert system
Membership functions F3 and F4, (K= 0.8) which has to be combined together

55. 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:

56. 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

57. Load forecasting with Fuzzy- expert system
Dlcorrection = MW

58. 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: