Sh T L d F iShort‐Term Load Forecasting

Dr SN Singh, ProfessorDepartment of Electrical Engineering

Indian Institute of Technology KanpurEmail: snsingh@iitk ac inEmail: snsingh@iitk.ac.in

Basic Definition of Forecasting

Forecasting is a problem of determining the future values of a time series from current and past valuestime series from current and past values.

Past measurements Forecasted values

• one step ahead• two step ahead• Multiple step ahead

Time sampling can be in  sec, min, hours, days, months and years

Short term forecast Medium term forecast Long term forecast

Role of Forecasting in Electric Power System

Before Deregulation of Electric Power SystemOnly Load Demand forecasting is carried

‐ Economic operation and Unit commitment  

‐ Maintenance and planning  of power system   

After Deregulation of Electric Power System

MonopolisticMotivation: Creating competition 

Deregulated Market

among the suppliers Leaving choices to the 

buyers 

Role of Forecasting in Electric Power System….Contd.

Electricity Market Operation

ISO’Energy

D h d

GENCO’s/Suppliers

ForecastingLoad

ISO’s

Market Forecast• Load

Energy, Ancillary Services, and 

Transmission

Day ahead‐ Load‐ Price‐Wind Power Bids

S h d l

• Load• Price

BidsHour ahead

Real Time

Markets

Bidding strategies/Risk Management

Bidding strategies/Risk Management

SchedulesMarket Operation• SCUC• A S Auction• Cong. Mgmt.

T P i i

Schedules

• Trans. Pricing

Market Bidding Process

Day aheadDay ahead bidding for day 2

Hour‐aheadHour ahead bidding for day 2

Day 1 Day 2

Market

Real time

Market clearing for day 2

Day‐ahead Forecast (24+6hrs)

Important Tools 

1. Load Forecast2. Price Forecast3. Operating Reserve Margin Forecast

System Operator Point of View:l bl

3. Operating Reserve Margin Forecast4. Wind Forecast

Planning Problems:Due to uncertainty, unlike conventional generators, wind power generation cannot be included into ELD and UC problems.

Operational:Frequency control, Voltage control, Power Quality, Ancillary services provisionprovision.

Wind power producer point of view:

Bidding in day ahead adjustment and settling Electricity Markets toBidding in day ahead, adjustment and settling Electricity Markets to maximize profits/minimize their imbalance costs.

Factors Influencing the Forecast  Variable

Time Factor• Hour in a day• Hour in a day• Day of the Week• Holiday

Load Demand

Type of Customer• Domestic loads• Commercial loads• Industrial loads

Weather Parameters• Temperature• Humidity• Sky cover• Sun shine• Wind Speed & 

Direction

Effect of Time Factorx 104

3

3.2x 10

2.6

2.8

man

d (M

W)

2.2

2.4

Load

Dem

0 24 48 72 96 120 144 168 192 216 240 264 288 312 3361.8

2

Time sample

First Week Second Week

Time sample

Dependency on Weather Parameter:Temperature

5x 104

2

3

4

5

ad D

eman

d (M

W)

4

4.5

5x 10

4

0 1000 2000 3000 4000 5000 6000 7000 8000 90001

Loa

403

3.5

Load

Dem

and

(MW

)

0

20

Tem

p (o C

)

-10 -5 0 5 10 15 20 25 30 35 401.5

2

2.5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-20

Time sample

Temp (oC)

Factors Influencing Electricity Market Price

Load Demand

Network Congestion

ElectricityMarket Clearing Price

ReserveMarging Margin

Fuel Prices 

Available Hydro Generation

Price vs. Load Demand

250

300

8000

9000

W)

150

200

CP

($/M

Wh)

4000

5000

6000

7000

load

dem

and

(MW

3000 4000 5000 6000 7000 8000 90000

50

100MC

0 1000 2000 3000 4000 5000 60003000

time samples (hrs)

300) 3000 4000 5000 6000 7000 8000 9000load demand (MW)

200

300

arin

g pr

ice

($/M

Wh

100

0 1000 2000 3000 4000 5000 60000

100

time samples (hrs)

Mar

ket C

lea

2400 2450 2500 2550 2600 265020

40

60

80

MC

P ($

/MW

h)

2400 2450 2500 2550 2600 2650time samples (hrs)

Factors Influencing Wind Power Generation

Wind SpeedWind Speed

Wind

Wind Power

Wind Direction

Wind TurbineWind TurbineLayout

Terrain 

30

m/s

)Wind speed and wind power time series

10

20

Win

d Spe

ed (m

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

W

200

100

150

200

Pow

er (M

W)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 54

0

50

Win

d P

Timesample x 104Time sample

Wind Speed vs. Wind Power scatter plot

160

180

120

140

160

W)

80

100

nd P

ower

(MW

)

40

60Win

0 5 10 15 20 250

20

Wind Speed (m/s)

Forecasting Approaches

Linear Regression Models : The forecast value is linearly dependent on the past historical values of the time series. (AR, ARMA, ARIMA, GARCH, etc.)

AR(p):  Autoregressive model of order p

ARMA(p,q): Autoregressive Moving AverageARMA(p,q): Autoregressive Moving  Average model of order (p,q).

MA(q): Moving Average model of order qMA(q):  Moving Average model of order qGeneralized AutoRegressive Conditional Heteroskedasticity (GARCH)

The general process for a GARCH model involves three tsteps.

• The first is to estimate a best-fitting autoregressive model; g g• secondly, compute autocorrelations of the error term and• lastly, test for significance.

GARCH models are used by financial professionals in several arenas including trading, investing, hedging and d lidealing.

Forecasting Approaches (Contd..)

ARIMA: Autoregressive Integrated Moving Average Model

Fractional‐ARIMA:

This type of models are employed whenARIMA is a generalized form of ARMA model. It is applied when time series has some non‐stationary behavior (do not vary about a fixed mean)

This type of models are employed when time series exhibits long memory.

F‐ARIMA model is a special case of ARIMA(p d q) processvary about a fixed mean).  ARIMA(p,d,q) process.

Where parameter d assumes fractionally continuous values in the range (‐0.5, 0.5).

Correlation Analysis

Finding the appropriate values of p and q  can be facilitated by Partial Auto Correlation Functions (PACF) and Auto‐Correlation Functions (ACF).

If Mean of a Time series is given  :

and Variance is given by :

Then auto‐correlation at lag ‘k’ is given by :

Example in Matlab :  AR(1)

AR(1) ith 0 8

e=randn(100,1);

x=zeros(100,1);

x(1)=e(1);5

10AR(1): xt=xt-1+e, with =0.8

(1)x(1) e(1);

alpha=0.8;

for i=2:100

x(i)=alpha*x(i‐1)+e(i); 0 20 40 60 80 100-5

0AR

(

1end

subplot(3,1,1)

plot(x);

title('AR(1) {t} \alpha{ {t0

0.5

1

AC

F

title('AR(1): x_{t}=\alpha{x_{t‐1}+e}, with \alpha{=0.8}');

subplot(3,1,2);

autocorr(x,25,[],2);

0 5 10 15 20 25-0.5

Lag1

subplot(3,1,3);

parcorr(x,25,[],2);0

0.5

PA

CF

0 5 10 15 20 25-0.5

Lag

Example in Matlab :  AR(2)AR(2): x = x + x +e with 1=1 5 =-0 75

e=randn(100,1);

x=zeros(100,1);

x(1)=e(1); x(2)=e(2); 0

5

10AR(2): xt=1xt-1+2xt-2+e, with 1=1.5, 2=-0.75

R(2

)

x(1)=e(1); x(2)=e(2);

alpha1=1.5; alpha2 = ‐0.75; 

for i=3:100, x(i)=alpha1*x(i‐1)+alpha2*x(i‐2)+e(i); end 0 10 20 30 40 50 60 70 80 90 100

-10

-5

AR

1subplot(3,1,1)

plot(x);

title('AR(2): x_{t}=\alpha_{1}x_{t‐1}+\alpha {2}x {t‐2}+e with 0

0.5

1

AC

F1}+\alpha_{2}x_{t 2}+e, with \alpha{1}=1.5, \alpha_{2}=‐0.75');

subplot(3,1,2);

autocorr(x,25,[],2);

0 5 10 15 20 25-0.5

Lag1

subplot(3,1,3);

parcorr(x,25,[],2);

-0.5

0

0.5

PA

CF

0 5 10 15 20 25-1

Lag

Example in Matlab :  MA(1)

e=randn(101,1); 2

4MA(1): xt=xt-1+e, with =0.8

theta=‐0.8;

x=e(2:101,1)‐theta*e(1:100,1);

subplot(3,1,1)

plot(x);0 10 20 30 40 50 60 70 80 90 100

-4

-2

0

MA

(1)

plot(x);

title('MA(1): x_{t}=\theta{x_{t‐1}+e}, with \theta{=0.8}');

subplot(3,1,2);0

0.5

1

AC

F

autocorr(x,25,[],2);

subplot(3,1,3);

parcorr(x,25,[],2);0 5 10 15 20 25

-0.5

0

Lag1

0

0.5

PA

CF

0 5 10 15 20 25-0.5

Lag

Example in Matlab :  MA(2)

e=randn(102,1); 0

5MA(2): xt=et-1et-1-2et-2

)

theta1=‐1.5; theta2=‐0.8; 

x=e(3:102,1) ‐ theta1*e(2:101,1) ‐theta2*e(1:100,1);

subplot(3,1,1);0 10 20 30 40 50 60 70 80 90 100

-10

-5MA

(2

subplot(3,1,1);

plot(x);

title('MA(2): x_{t}=e_{t}‐\theta_{1}e_{t‐1}‐\theta_{2}e_{t‐2} );

0

0.5

1

AC

Fsubplot(3,1,2);

autocorr(x(20:end),20,[],2);

subplot(3,1,3);

parcorr(x(20:end) 20 [] 2);

0 2 4 6 8 10 12 14 16 18 20-0.5

Lag1

parcorr(x(20:end),20,[],2);

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

PA

CF

Lag

Example in Matlab :  ARMA(2,1)

5

10

)

clear all;close all;

0 50 100 150 200 250 300-10

-5

0

AR

MA

(2,1close all;

e=2*randn(300,1);x=zeros(300,1);x(1)=e(1); x(2)=e(2);l h 1 0 9 l h 2 0 3 0 50 100 150 200 250 300

0.5

1

F

alpha1=0.9; alpha2 = ‐0.3; theta1= 0.5;for i=3:300x(i)=alpha1*x(i‐1)+

0 5 10 15 20 25 30-0.5

0

Lag

AC

F( ) p ( )alpha2*x(i‐2)+e(i)‐theta1*e(i‐1) ; endsubplot(3 1 1) Lag

0.5

1

PA

CF

subplot(3,1,1)plot(x);subplot(3,1,2);autocorr(x(25:end),30,[],2);

0 5 10 15 20 25 30-0.5

0

Lag

P

subplot(3,1,3);parcorr(x(25:end),30,[],2);

AR(p) MA(q) ARMA(p,q)

ACF and PACF for casual Time series models

ACF Tails off Cuts off after lag q Tails off

PACF Cuts off after lag p Tails off Tails offPACF Cuts off after lag p Tails off Tails off

Estimation of Model Parameters

After choosing p and q the model can be fitted by linear  least squares regression to find the model parameters which minimize the error terms.

State-Space Models

Consider a univariate AR(p) process:

This could be written in state-space form as,

state equation

observation equation

Limitations of Linear Regression Models

As they are linear models, they cannot capture the non‐linear y , y prelation between the independent and dependent variable.

The forecasting error increases rapidly with the increase in look‐ahead time.

The model parameters have to be updated very frequently.

Forecasting Approaches        …..contd

Non‐Linear Regression models:

Artificial Neural Networks (ANN): are well established in function approximation, many variants of NNs are employed in the field of forecasting problem. Like FFNN, RNN, RBF, WNN.

NetworkParameters

‐+

Parameters

Back‐Propagation  Algorithm,  Evolutionary based Optimization methods like GA, PSO are also applied for network trainingPSO are also applied for network training.

Input variables are selected using ACF and PACF.

Other Methods..Other Methods..

‐ Fuzzy LogicFuzzy Logic

‐ Adaptive Neuro‐Fuzzy Inference System (ANFIS)

D t Mi i t h i lik l t i d S t‐ Data Mining techniques like clustering and Support Vector Machines (SVM) based classification and Regression models

‐ Wavelet pre‐filtering based ANN and Fuzzy models.

Benchmark Models

Measure of Errors

Then,

AWNN : Architecture

Continuous Wavelet Transforms 

• A wavelet is a small wave which grow and decays essentially in a limited time period.

• Should satisfy two basic properties

and

Training Algorithm

Case study: Load Forecasting

2 5

3

3.5x 104

and

(MW

)

0 100 200 300 400 500 600 700 800 900 10001.5

2

2.5

Load

Dem

a

0.5

1

F

0 24 48 72 96 120 144 168 192-0.5

0

Lag

AC

F

g

0

0.5

1

PA

CF

0 24 48 72 96 120 144 168 192-1

-0.5

Lag

P

Cross correlation between Load and Temp.

3

4

5x 104

(MW

)

0 1000 2000 3000 4000 5000 6000 7000 80001

2

3

Load

(

40

10

20

30

40

mp

(0 C)

0 1000 2000 3000 4000 5000 6000 7000 8000-10

0Tem

0.8

0

0.2

0.4

0.6

XC

F

-100 -80 -60 -40 -20 0 20 40 60 80 100-0.2

0

Lag

Input lag hours selected for Load Forecasting

Load Terms

Temp. Terms

Temp. time series

Load time series

Epochs:  (input, output) pairs 

AWNN

Hour ahead load forecast4

3

3.2

3.4x 104

2.6

2.8

3

man

d (M

W)

2

2.2

2.4

Load

Dem

0 24 48 72 96 120 144 168 192 2161.8

2

Hour

=  0.6905 

24 h h d F

Forecasthour

MAPE

24‐hours ahead Forecast

3 5x 104

12345

0.95511.51911.96642.25922.4768

3

3.5

(MW

)

ThuWedTueMonSunSat Fri Sat Sun 678910

2.57792.66392.68832.65562 6816

2.5

ad D

eman

d ( 10

11121314

2.68162.75652.81462.86402.8843

0 24 48 72 96 120 144 168 192 2161.5

2Loa

1516171819

2.89492.91132.91902.90962 8514

time samples1920212223

2.85142.82902.80392.76182.7373

24 2.7447

Average 2.5886

Case study: Price Forecasting

250

300

100

150

200

250

CP

($/M

Wh)

1

0 1000 2000 3000 4000 5000 60000

50

100

time samples (hrs)

MC

0

0.5

AC

F

time samples (hrs)

80

1000 50 100 150 200 250 300

-0.5

Lag

40

60

80

MC

P ($

/MW

h)

2400 2450 2500 2550 2600 265020

time samples (hrs)

150

200

250

300

CP ($

/MW

h)

0 1000 2000 3000 4000 5000 60000

50

100

time samples (hrs)

MCP

10000

Price Terms

4000

6000

8000

10000

dem

and

(MW

)

Load 

0 1000 2000 3000 4000 5000 60002000

4000

time samples (hrs)

load

1

Terms

-0.5

0

0.5

XC

F

-200 -150 -100 -50 0 50 100 150 200-1

Lag

Price Forecasting Results

150

200

)

Forecasthour

MAPE

1 8.1130 

1‐hr ahead Price Forecast  (MAPE = 8.4355)

50

100

MC

P ($

/MW

h) 234567

10.6029 11.9206 12.7485 13.2603 13.8258 14 0576

0 24 48 72 96 120 144 168 1800

time samples (hrs)

789101112

14.0576 14.2544 14.2845 14.2799 14.2896 14.2922 

h h d

150

200

Wh)

1314151617

14.2737 14.3819 14.5195 14.4968 14.5853 

24‐hr ahead Price Forecast

50

100

MC

P ($

/MW 18

1920212223

14.7405 14.9456 15.1473 15.3574 15.5360 15 7343

0 24 48 72 96 120 144 168 192 2160

time samples (hrs)

2324

15.7343 15.9714

Average 13.9841

Wind Speed Time Series

15

20

25

peed

0

5

10

win

d sp

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

time samples

15

20

eed

5

10

win

d sp

e

3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 40000

time samples

Schematic Block Diagram for Wind Speed Forecasting

Multiresolution Analysis of Wind Speed Time SeriesTime Series

05

10S

7

5

-50D

7

-505

D6

5

-505

D5

-505

D4

-505

D3

-505

D2

5

-505

D1

01020

d S

erie

s

0 1000 2000 3000 4000 5000 6000 7000 80000

time (hours)

win

d

ACF’s of Decomposed Wind Speed Time Series

-1

0

1S

7

1

-1

0D7

0

1

D6

-1

-1

0

1

D5

1

-1

0D4

0

1

D3

-1

-1

0

1

D2

1

0 100 200 300 400 500 600-1

0

1

Lag

D1

Network Architecture and Input Lag Hours

D d I L h N k A hiDecomposedSignal

Input Lag‐hours Network Architecture

AWNN FFNN

S7 1‐14,157‐159,285‐287 20‐2‐1 20‐3‐1S7 1 14,157 159,285 287 20 2 1 20 3 1

D7 1‐12,76‐83,167‐169 19‐2‐1 19‐3‐1

D6 1‐10,41‐44,84‐86 17‐2‐1 17‐3‐1

D5 1 6 21 23 44 47 13 2 1 13 3 1D5 1‐6,21‐23,44‐47 13‐2‐1 13‐3‐1

D4 1‐3,11‐13,23‐25,48,72 11‐2‐1 11‐3‐1

D3 1,2,5,6,12,60,72 7‐2‐1 7‐3‐1

D2 3,6,9,15 4‐2‐1 4‐3‐1

D1 1,2,5,22 4‐2‐1 4‐3‐1

Wind Power Forecast

Wind 

Wind Farm

dSpeed 

highly stochastic  random  non‐stationary.

putrated speed

 spe

ed

Power outp

Wind

Wind P

Manufacturer curve

Cut‐in speed

Wind Power Forecasting: Time Horizon

Very Short‐Term Forecasting : Up to 2‐3 h in steps of 10min. or 15min.‐Turbine control 

‐Real time participation in electricity market‐Real time participation in electricity market.

Short‐Term Forecasting : Up to 24h in steps of 1h.‐ Hour ahead biddingHour ahead bidding‐ Intraday market‐ Day‐ahead market‐ Unit commitment and Economic dispatch

ll‐ Ancillary services management‐ Day‐ahead reserve setting

Medium Term Forecasting ( With NWP inputs): up to 72h in steps of 1h.g ( p ) p p• In addition to the above mentioned benefits 

• Maintenance planning of wind farms

• Wind farm and storage device CoordinationWind farm and storage device Coordination

• Congestion management

• Maintenance planning of network lines. 50

Wind Power Forecasting: Approaches

NWP  Wind Speed at WPforecasts Hub heightPhysical 

Model

WP Forecast1)

Manufacturer curve

NWP forecasts

Statistical Model WP  Forecast

2) Wind speed

Wind power

Available historical measurements. ARX,  ARMAX, NN, Fuzzy, ANIF

Wind power

A Two stage approach for Wind Power Forecast

Wind speed forecasts

Historical measurements of AWNN

FFNN

forecasts

WP  ForecastWind speed

measurements of wind speed. 

AWNN

Wind power

Autocorrelation Analysis of Wind Series

1

latio

n

Sample Autocorrelation Function1

atio

n

Sample Autocorrelation Function

0

0.5

ple

Aut

ocor

re

0

0.5

ple

Aut

ocor

rela

0 20 40 60 80 100 120 140 160 180 200Lag

Sam

1n

Sample Autocorrelation Function0 20 40 60 80 100 120 140 160 180 200

-0.5

Lag

Sam

p

1n

Sample Autocorrelation Function

0

0.5

Aut

ocor

rela

tio

0

0.5

Aut

ocor

rela

tio

0 20 40 60 80 100 120 140 160 180 200-0.5

0

Lag

Sam

ple

0 20 40 60 80 100 120 140 160 180 200-0.5

0

Lag

Sam

ple

A

ag

Hour Ahead Forecast of Wind Speed

12

8

10

peed

6

8

win

d sp

AWNNFFNNActual

1 5 9 13 17 21 244

hours

Actual

30‐hours ahead Wind Speed Forecast

14

16

18 actualforecast by AWNNforecast by FFNN

10

12

14

ed (m

/s)

forecast by FFNN

4

6

8

win

d sp

ee

0 30 60 90 120 150 180 210 240 270 3000

2

4

time (hours)

Comparative Performance

3.5

4

4.5

MAE of AWNN

MAE f FFNN

2

2.5

3

erro

rs

MAE of FFNN

MAE of NR

MAE of PER

RMSE of AWNN

0 5

1

1.5

2

RMSE of FFNN

RMSE of NR

RMSE of PER

0 5 10 15 20 25 300

0.5

look-ahead time (hours)

Percentage Improvement

80

90 MAE over PERRMSE over PERMAE NR

70

prov

emen

t

MAE over NRRMSE over NR

50

60

rcen

tage

imp

40

per

0 5 10 15 20 25 3030

look-ahead time (hours)

Wind speed to Wind Power Transformation

Wi d F t dWind speed

Forecastedwind speed

Wind

FFNN 

Wind power wind power

Forecast

FFNN Inputs:  wind speed {0, 1, 2} lag hours and from wind power series {1 2 3 4 5 6} lag hourswind power series {1, 2, 3, 4, 5, 6} lag hours.

Error Distributions and Forecasting Ability

40

60

80

e of

erro

rs(%

)

-100 -50 0 50 1000

20

Error(% of P )

Occ

uren

ce

90

100

mar

gin(

%)

7.5%12.5%

Error(% of Pinst)

30%)

70

80

hin

the

erro

r

1‐hr ahead forecast error distributions

10

20

30

ce o

f erro

rs(%

50

60 o

f tim

es w

it

-100 -50 0 50 1000

10

Error(% of Pinst)

Occ

uren

0 5 10 15 20 25 3040

look-ahead time (hours)

%

inst

30th ‐hr ahead forecast error distributions