Short Term LdL oad orecasting - IIT Kanpur 2015 Training... · sec, min, hours, days, months and...
Transcript of Short Term LdL oad orecasting - IIT Kanpur 2015 Training... · sec, min, hours, days, months and...
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: [email protected]
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