Short Term Load Forecasting with Expert Fuzzy-Logic System

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 = - 750 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) andmB(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 10-12 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= 300.006 and h= 871.587

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. 1 1 F3 (D L3) 0.8 ( ) F D L D L1 1 1 D L2 0.6 ( ) F D L , - 2 2 1 0.4 ( ) F D L D L3 3 1 0.2 0 0 1500 1250 1000 750 500 250 0 250 500 - 1500 D L 500 1 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.

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 = 0 + 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 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,