Probabilistic forecasting of wholesale electricity prices

Transcription

Probabilistic forecasting
of wholesale electricity prices
Rafal Weron
Department of Operations Research
Wroclaw University of Technology (PWr), Poland
http://kbo.pwr.edu.pl/pracownik/weron
Rafal Weron (PWr)
Forecasting electricity prices
1.11.2015, CAS, Beijing
1 / 41
Introduction: What and how are we forecasting?
The vocabulary
Smart grids (smart meters, appliances, houses, ... cities)
Prosumers = producing consumers
Load = consumption (≈ demand) + losses
Non-storability
Power grid/network
Interconnector
Power exchange,
power pool
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Power markets in Europe
Nord Pool (DK, EST,
FIN, NOR, SWE)
N2EX (UK)
APX-ENDEX
(NL)
PolPX (PL)
OTE (CZ)
Belpex (BE)
OKTE (SK)
EPEX Spot
(AT,CH, DE, FR)
OPCOM (RO)
OMIE (ES, PT)
HUPX (HU)
EXAA (AT)
GME (IT)
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Borzen (SLO)
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... in North America and Australia
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GEFCom2014: electricity price track
Electricity prices and loads (GEFCom2014)
Data cont.
Seasonality, floor reversion and price spikes
400
350
Zonal price
300
250
200
150
100
50
0
Jan 01, 2011
Jan 01, 2012
Jan 01, 2013
Jul 04, 2013
Dec 17, 2013
4
x 10
System load
3.5
Zonal load
Forecasted Load
3
2.5
2
1.5
1
0.5
Jan 01, 2011
K. Maciejowska, J. Nowotarski
Rafal Weron (PWr)
Jan 01, 2012
Jan 01, 2013
Price forecasting: a hybrid model
Jul 04, 2013
Dec 17, 2013
London, 11.09.2015
7 / 31
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GEFCom2014: electricity price track
Electricity
loads (GEFCom2014)
Data cont.:prices
price vs.
vs load
Non-linear, time-varying dependence
K. Maciejowska, J. Nowotarski
Rafal Weron (PWr)
Price forecasting: a hybrid model
London, 11.09.2015
9 / 31
6 / 41
Supply stack and price formation
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The electricity ‘spot’ price
Day d – 2
Bidding for
day d – 1
Day d – 1
Bidding for
day d
24 hours (48 half-hours)
of day d – 1
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Day d
24 hours (48 half-hours)
of day d
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Prices for different load periods
Strongly correlated but seem to follow different data generating processes (DGPs)
6.5
Load period 6 (2:30−3:00)
Load period 36 (17:30−18:00)
6
5.5
Log−price
5
4.5
4
3.5
3
2.5
2
1.5
06−Nov−2008
27−Mar−2010
15−Aug−2011
02−Jan−2013
Time
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A commodity ... but a very special one
Not storable (economically)
Time consuming shut-down/start-up procedures for some
technologies
Extreme price changes → spikes
Possible negative prices
Pronounced daily and weekly cycles, annual seasonality
Mean (floor) reversion
Highly volatile
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Forecasting horizons
Short-term
From a few minutes up to a few days ahead
Of prime importance in day-to-day market operations
Medium-term
From a few days to a few months ahead
Balance sheet calculations, risk management, derivatives pricing
Inflow of ‘finance solutions’
Long-term
Lead times measured in months, quarters or even in years
Investment profitability analysis and planning
Beyond the scope of this review
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A taxonomy of (price) modeling approaches
(Weron, 2014, Int. J. Forecasting)
Electricity price
models
Multi-agent
Fundamental
Reduced-form
Statistical
Computational
intelligence
CournotNash
framework
Parameter
rich
fundamental
Jumpdiffusions
Similar-day,
exponential
smoothing
Feed-forward
neural
networks
Supply
function
equilibrium
Parsimonious
structural
Markov
regimeswitching
Regression
models
Recurrent
neural
networks
Strategic
productioncost
AR, ARX-type
Fuzzy neural
networks
Agent-based
Threshold AR
Support
vector
machines
GARCH-type
Hybrid
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Combining point forecasts
Individual forecasts
Point forecast averaging: The idea
f1
f2
…
Weights
estimation
fC
Combined
forecast
fN
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Forecast combinations, forecast/model averaging
The idea goes back to the 1960s to the seminal papers of
Bates and Granger (1969) and Crane and Crotty (1967)
In electricity markets:
Electricity demand or transmission congestion forecasting
(Bunn, 1985a; Bunn and Farmer, 1985; Løland et al., 2012;
Smith, 1989; Taylor, 2010; Taylor and Majithia, 2000)
Only recently applied in the context of electricity price
forecasting (EPF): Bordignon et al. (2013), Nowotarski et al.
(2014) and Raviv et al. (2013)
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Case study I: Combining price forecasts
150
→ Individual forecasts (weeks 1−34)
120
90
Combined forecasts (weeks 5−34)
60
30
0
8.8.2012
WMAEi−min(WMAEi)
NP Price [EUR/MWh]
(Weron, 2014, Int.J.Forecasting)
5.6.2013
Hours [8.8.2012−31.12.2013]
31.12.2013
3
Individual models
Simple
CLS
LAD
2
1
0
5
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10
15
20
25
Weeks [5.6.2013−31.12.2013]
30
34
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Case study I: Combining price forecasts
Summary statistics for 6 individual and 3 averaging methods: WMAE is the mean value of
WMAE for a given model (with standard deviation in parentheses), # best is the number of
weeks a given averaging method performs best in terms of WMAE, and finally m.d.f.b. is the
mean deviation from the best model in each week. The out-of-sample test period covers 30
weeks (5.6.2013–31.12.2013).
WMAE
# best
m.d.f.b.
Individual models
SNAR
MRJD
4.77
4.98
AR
5.03
TAR
5.07
(3.40)
(3.53)
(3.26)
1
1.01
3
1.05
4
0.75
Rafal Weron (PWr)
Forecast combinations
Simple
CLS
LAD
4.47
4.29
4.92
NAR
4.88
FM
5.36
(3.17)
(1.62)
(3.17)
(2.87)
(1.88)
(2.41)
1
0.96
2
0.86
4
1.34
8
0.45
6
0.27
1
0.89
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In the ‘AI world’ ...
Committee machines, ensemble averaging:
Guo and Luh (2004) combine a RBF network (23 inputs and six
clusters) and a MLP (55 inputs and eight hidden neurons) to
compute daily average on-peak electricity price for New England
Forecast combinations and committee machines seem to evolve
independently, with researchers from both groups not being
aware of the parallel developments !
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Beyond point forecasts
Reviews and competitions
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Beyond point forecasts
Interval forecast averaging
P
For point forecasts: fc = M
i=1 wi fi
(e.g. a linear regression model)
For interval forecasts the above formula does not hold
A linear combination of α-th quantiles is not the α-th quantile
of a linear combination of random variables
qcα
6=
M
X
wi qiα
i=1
→ Need for development of new approaches
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Quantile Regression Averaging (QRA)
Quantile Regression Averaging
Individual point forecasts
(Nowotarski & Weron, 2015, Computational Statistics)
f1
f2
…
Quantile
regression
Combined
interval
forecast
(2 quantiles)
fN
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fC
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Quantile Regression Averaging cont.
The averaging problem is given by:
Qp (q|pbt ) = pbt wq
Qp (q|·) is the conditional q-th quantile of the electricity spot
price distribution,
pbt are the regressors (explanatory variables)
wq is a vector of parameters for a given q-th quantile
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Quantile Regression Averaging cont.
The weights are estimated by minimizing:


X
X
min 
q|pt − pbt wt | +
(1 − q)|pt − pbt wt | =
wt
{t:pt ≥b
p t wt }
{t:pt <b
p t wt }
"
min
wt
1.4
q=50%
q=25%
q=5%
1.2
1
#
X
(q − 1pt <bpt wt )(pt − pbt wt )
t
0.8
0.6
0.4
0.2
0
−2
−1
0
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1
2
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Quantile regression
300
Linear regression
250
200
Y
150
100
50
0
−50
10
15
20
25
30
X
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Quantile regression
300
Linear regression
Quantile regression, α=0.95, α=0.05
250
200
Y
150
100
50
Interval forecast
0
−50
10
15
20
25
30
X
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Case study II: Combining individual price forecasts
(Nowotarski & Weron, 2015, Computational Statistics)
Six individual point forecast models:
Autoregression (AR)
Threshold AR (TAR)
Semi-parametric AR (SNAR)
Mean-reverting jump diffusion (MRJD)
Non-linear AR neural network (NAR)
Factor model (FM)
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The data
NP Price [EUR/MWh]
150
120
Forecast
(weeks 1−26)
90
60
30
0
Aug 08, 2012
Jul 03, 2013
Hours [Aug 08, 2012 − Dec 31, 2013]
Dec 31, 2013
Seven months for calibration of individual models
Four weeks for calibration of quantile regression
26 weeks for evaluation of interval forecasts
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Evaluation of forecasts
50% and 90% two-sided day-ahead prediction intervals
Two benchmark models: AR and SNAR
Christoffersen’s (1998) test for unconditional and conditional
coverage
(
1 pt ∈ [Lt|t−1 , Ut|t−1 ]
The focus on the sequence: It =
0 pt 6∈ [Lt|t−1 , Ut|t−1 ]
Conditional Coverage test
(UC + independece)
Asymptotically χ2 (2)
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Unconditional Coverage test
Asymptotically χ2 (1)
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Results: Unconditional coverage
PI
50%
90%
AR
SNAR
Unconditional coverage
77.50
61.93
97.53
96.41
QRA
49.77
89.33
Mean width (STD of interval width)
50% 4.55 (1.34) 2.76 (0.61) 2.23 (0.81)
90% 11.14 (3.31) 9.33 (2.45) 6.78 (2.20)
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Results: Christoffersen’s test
Conditional coverage LR
Unconditional coverage LR
20
20
15
15
AR
10
5
0
10
5
1
6
12
18
0
24
20
15
15
10
SNAR 10
5
5
0
1
6
12
18
0
24
20
6
12
18
24
1
6
12
18
24
1
6
12
18
24
20
15
15
QRA
10
5
0
1
20
10
5
1
6
12
Hour
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18
24
0
50% PI
90% PI
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GEFCom2014 Price Track: 1st and 2nd place for QRA!
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Case study III: Combining sister load forecasts
(Liu, Nowotarski, Hong & Weron, 2015, IEEE Transactions on Smart Grid)
Variable selection may be difficult in load forecasting
Sister models – constructed by different subsets of variables
with overlapping components
Here: 2 or 3 years for calibration and 4 ways of partitioning
training and validation periods
p̂t = β0 + β1 Mt + β2 Wt + β3 Ht + β4 Wt Ht + f (Tt ) +
X
X
+
f (T̃t,d ) +
f (Tt−lag ),
d
lag
Sister forecasts are generated from sister models
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The data
(from the load forecasting track of GEFCom2014)
2 or 3 years for calibration of sister (individual) models
1 year for validation of sister (individual) models (variable selection)
1 year for validation of probabilistic forecasts (best models selection)
1 year for testing probabilistic forecasts
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Benchmarks
Two naive benchmarks
Scenario generation from historical weather data, no recency
effect (Vanilla)
Quantiles interpolated from 8 individual forecasts (Direct)
Benchmarks from individual models
8 individual models (Ind) with residuals’ distribution
Best Individual (BI) individual model according to MAE
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Pinball loss function for 99 percentiles (as in GEFCom2014)
(
(1 − q)(p̂tq − pt ), pt < p̂tq
Pinballt =
q(pt − p̂tq ),
pt ≥ p̂tq
Winkler score for central (1 − α) × 100%, α = 0.5, 0.9,
two-sided day-ahead PI:


for pt ∈ [Lt|t−1 , Ut|t−1 ],
δ t
2
Wt = δt + α (Lt|t−1 − pt ) for pt < Lt|t−1 ,


δt + α2 (pt − Ut|t−1 ) for pt > Ut|t−1 ,
where δt = Ut|t−1 − Lt|t−1 is the PI width
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Results: Test period
Model class
QRA(8,183)
Ind(1,91)
BI(-,365)
Direct
Vanilla
Pinball Winkler (50%) Winkler (90%)
2.85
25.04
55.85
3.22
26.35
56.38
3.00
26.38
57.17
3.19
26.62
94.27
8.00
70.51
150.0
Sister forecasts easy to generate
No need for independent expert forecasts
Simple way to leverage from point to probabilistic forecasts
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Factor Quantile Regression Averaging (FQRA)
Extension: A large number of predictors
Individual point forecasts
(Maciejowska, Nowotarski & Weron, 2015, Int.J.Forecasting)
f1
F1
f2
PCA
…
fN
Rafal Weron (PWr)
…
Quantile
regression
FK
K factors extracted
from a panel of
point forecasts,
K<N
fC
Combined
interval
forecast
(2 quantiles)
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Case study IV
(Maciejowska, Nowotarski & Weron, 2015, Int.J.Forecasting)
APX Price [GBP/MWh]
350
300 Start of calibration period
250 (individual models)
Start of QR
calibration period
Start of PI
validation
200
150
100
50
0
Jul 01, 2010
Jul 01 2011
Jan 01, 2012
Hours [Jul 01, 2010 − Dec 31, 2012]
Dec 31, 2012
32 individual forecasting models
One year for calibration of individual models
Half a year for calibration of quantile regression
One year for evaluation of interval forecasts
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50% and 90% two-sided day-ahead prediction intervals
Three methods: QRA, FQRA and ARX (benchmark)
Christoffersen’s (1998) test for unconditional and conditional
coverage
Winkler score:


for pt ∈ [Lt|t−1 , Ut|t−1 ],
δt
2
Wt = δt + α (Lt|t−1 − pt ) for pt < Lt|t−1 ,


δt + α2 (pt − Ut|t−1 ) for pt > Ut|t−1 ,
where δt = Ut|t−1 − Lt|t−1 is the interval’s width
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Results: Christoffersen’s test
CC
ARX
FQRA
20
20
15
15
15
10
10
10
5
5
5
0
UC
QRA
20
12
24
36
48
0
12
24
36
48
0
20
20
20
15
15
15
10
10
10
5
5
5
0
12
24
36
Load period (h)
48
0
12
24
36
Load period (h)
48
0
12
36
48
12
24
36
Load period (h)
48
50% PI
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24
90% PI
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Relative Winkler
score, 90% PI
Relative Winkler
score, 50% PI
Results: Winkler score
25%
20%
15%
10%
5%
0%
−5%
1 − WQRA
/WARX
h
h
6
12
18
6
12
18
24
1 − WFQRA
/WARX
h
h
30
36
42
48
30
36
42
48
25%
20%
15%
10%
5%
0%
−5%
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24
Load period (h)
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The end
Take-home message(s)
Combining point forecasts is a robust technique, generally
improving the performance
The new trend is probabilistic forecasting
Combining interval (or density) forecasts is more tricky than
combining point forecasts
QRA is a simple way to leverage from point to probabilistic
forecasts
QRA is potentially useful for VaR calculations
Forecast evaluation is a critical issue
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The end
Evaluating probabilistic forecasts
For interval forecasts
The pinball function, as in GEFCom2014
The interval or Winkler score, see e.g. Maciejowska et al. (2015)
For density forecasts
The Continuous Ranked Probability Score (CRPS), see e.g.
Gneiting and Raftery (2007)
Statistical tests
The conditional coverage test of Christoffersen (1998); for
extensions and alternatives see Berkowitz et al. (2011)
The Berkowitz (2001) approach to the evaluation of density
forecasts (→ VaR backtesting)
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Probabilistic forecasting of wholesale electricity prices

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