Short Term Load Forecasting with Expert Fuzzy-Logic System

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Short Term Load Forecasting with Expert
Fuzzy-Logic System
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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

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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

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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

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Fuzzy- Expert System

• Typical membership functions are
• Triangular
• Trapezoid

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

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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

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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

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Fuzzy- Expert System

• The equation of the triangles rising edge is

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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

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Fuzzy- Expert System

• The outside region is described by
• The combination of the equations results in the
  triangular membership function equation

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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

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Fuzzy- Expert System

• Union of two fuzzy sets points included in both
  set A and B
• The membership function is

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Fuzzy- Expert System

• Union of two fuzzy sets points included in both
  sets A or B

mB
mA
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Fuzzy- Expert System

• Intersection of two fuzzy sets points which are
  in A or B
• The membership function is

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Fuzzy- Expert System

• Intersection of two fuzzy sets points which are
  in A and B

mB
mA
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Fuzzy- Expert System

• Sum of two fuzzy sets
• The membership function is

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Fuzzy- Expert System

• Sum of two fuzzy sets

mA
ms mA mB
mB
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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

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Load forecasting with Fuzzy- expert system

• Plot the historical data of load and temperature
  relation for selected 10 Tuesdays.

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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

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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

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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

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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.

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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

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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

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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

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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.

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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.

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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

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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

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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
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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.

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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

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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.

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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

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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)

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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

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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

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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

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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

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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)
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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

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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.

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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

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Load forecasting with Fuzzy- expert system

• Combined of Model uncertainty and Forecast
  -temperature uncertainty membership function
  (F3(DL3) .

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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

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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

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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,

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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

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Load forecasting with Fuzzy- expert system

• Membership function for Operators Heuristic Rules

Not confident
Quite confident
Confident
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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

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Load forecasting with Fuzzy- expert system

• Membership functions F3 and F4, (K 0.8) which
  has to be combined together

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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

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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

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Load forecasting with Fuzzy- expert system
Dlcorrection - 273.25MW
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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