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Forecasting Conditional Quantiles of
Electricity Demand:
A Functional Data Approach
Franziska Schulz
Brenda López Cabrera
Ladislaus von Bortkiewicz Chair of StatisticsHumboldt�Universität zu Berlinhttp://lvb.wiwi.hu-berlin.dehttp://www.case.hu-berlin.de
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Motivation 1-1
Electricity Market
� Electricity is not economically storable
� Demand must be served (to avoid blackouts)
� Hence, demand and supply have to be balanced at every pointin time
� Load forecasting is a central and integral process in theplanning and operation of electric utilities
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Motivation 1-2
Electricity Market
� Mainly traded on day-ahead market
I Knowledge about whole load curve needed one day ahead
� Adjustments possible in intraday market up to 45 min ahead
I less liquid than day-ahead market
� Forecast errors have to be balanced out (Regelenergie)
I very expensive for system operator
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Motivation 1-3
Electricity Market
� Electricity suppliers usually enter into long-term contracts withconsumers
� They buy electricity by short-term contracts, supply sidecarries risk
� Knowlegde about future demand required to make pro�t
� Miscalculations can lead to huge losses, e.g Flexstrom
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Motivation 1-4
Electricity Market Merit-Order
Figure 1: Load Curve
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Motivation 1-5
Conditional Quantiles
� Quantiles give a picture of the whole distribution instead ofjust the mean
� Quanti�cation of how di�erent determinants e�ect variousquantiles
� Value at Risk modeling and forecasting
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Motivation 1-6
Functional Data Analysis
� Daily load curve regarded as one functional observation
� One step ahead forecast yields load forecast for the next day
� Forecast for every point in time (continuous)
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Motivation 1-7
Objectives
� Modeling conditional quantiles of electricity load
I Understand dynamics
I Find main drivers
� Day-ahead forecasting of conditional quantiles
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Outline
1. Introduction X
2. Data
3. Methodology
4. Results
5. Outlook
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Data 2-1
Transmission System Operators in Germany
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Data 2-2
Data
� Data obtained from the transmission system operator Amprion
� Quarter hourly observations of electricity demand
� Time interval from January 2011 to April 2013
� Deseasonalized using local linear regression
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Methodology 3-1
Functional Time Series
� Time series {lk , k ∈ Z}, where lk(t), t ∈ [a, b] is a randomfunction
� Temporal dependence between observations
� Under stationarity:Mean function: E{l(t)} = µ(t)Covariance function: c(s, t) = Cov{l(s), l(t)}, s, t ∈ [a, b]
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Methodology 3-2
Functional Time Series
1800
022
000
2600
0Lo
ad
2009/01/05 2009/01/06 2009/01/07 2009/01/08 2009/01/09
Figure 2: Electricity load at �ve consecutive days in January 2011
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Methodology 3-3
Conditional Quantile Curves
F−1Y |t(τ) = lτ (t)
lτ (t) = argminf ∈F
E[ρτ{Y − f (t)}],
where ρτ (u) = u{τ − I(u < 0)}
� lτ (t) is the τ -th quantile curve
� F is a collection of functions s.t. the expectation is wellde�ned
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Methodology 3-4
Estimation of Conditional Quantile Curves
Estimation using B-spline smoothing
lτ (t) = minf
n∑i=1
ρτ{Y − f (t)}+ λmaxtf ′′(t)
where
� f (t) =∑q
j=1ajBj(t), q = 20
� Bj(t) normalized B-spline basis functions� aj coe�cients of B-spline basis functions
λ is chosen to minimize the SIC suggested by Koenker et al. (1994):
SIC (λ) = log
[1
n
n∑i=1
ρτ{yi − lτ (t)}
]+
1
2pλ
log(n)
n
with pλ = # of interpolated data points
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Methodology 3-5
Conditional Quantile Curves
0 20 40 60 80
−50
00−
3000
−10
000
Time
Det
rend
ed L
oad
Figure 3: Electricity load at 20110414 together with estimates of the me-
dian and the conditional quantiles with τ = 0.1 and τ = 0.9Forecasting Conditional Quantiles of Electricity Demand
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Methodology 3-6
Functional Principal Component Analysis
� Tool to reduce dimensionality
� Yields direction of largest variability
� Express data as weighted sum of orthogonal curves
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Methodology 3-7
Mercer's Lemma
If∫ b
ac(t, t)dt <∞, then
(Cφj)(s)def=
∫ b
a
c(s, t)φj(t)dt = λjφk
c(s, t) =∞∑j=1
λjφj(s)φj(t)
where
� φj is an orthonormal sequence of eigenfunctions of C
� λj is a non-increasing sequence of eigenvalues of C
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Methodology 3-8
Karhunen-Loève Expansion
l(t) = µ(t) +∞∑j=1
αjφj , αj = 〈l, φj〉
lk(t) ≈ µ(t) +m∑j=1
αk,jφj
where 〈., .〉 denotes inner product� φj is the jth principal component
� αj is the jth principal component score withE(αj) = 0,Var(αj) = λj
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Methodology 3-9
Weak dependence
Hörmann and Kokoszka (2010) show that under weak dependenceL-4-approximable
µ =1
n
n∑k=1
lk
c(s, t) =1
n
n∑k=1
{ln(s)− µ(s)}{ln(t)− µ(t)}
(Cφ)(s) =
∫ b
a
c(s, t)φ(t)dt
� are√n-consistent estimators
� the eigenvalues and eigenfunctions of C are√n-consistent
estimators for λ and φForecasting Conditional Quantiles of Electricity Demand
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Methodology 3-10
Forecasting Conditional Quantile Curves
Truncated Karhunen-Loève Expansion:
lk+h(t) = µ(t) +
m∑j=1
αk+h,j φj
where
� lk+h(t) is the h-step forecast of the conditional quantile curve
� αk+h,j is the h-step forecast of the j-th prinipal componentscore
� m is the number of included principal components
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Methodology 3-11
Forecasting Principal Component Scores
� αk+h can be obtained using multivariate time series techniques
� possible to include external regressors, e.g. temperature
� in case of linearity: VARX(p)
αk =
p∑i=1
Φk−iαk−i + βxt + ηk
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Results 4-1
Functional Data
−3
−2
−1
01
23
Time
Det
rend
ed L
oad
00:00 06:00 12:00 18:00 24:00
Figure 4: 90% Quantile curves from 20110104 to 20121219
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Results 4-2
Functional Mean
−1.
0−
0.5
0.0
0.5
1.0
Time
µ
00:00 06:00 12:00 18:00 24:00
Figure 5: Estimate of the functional mean
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Results 4-3
Principal Components
−2
−1
01
2
Time
Prin
cipa
l Com
pone
nts
00:00 06:00 12:00 18:00 24:00
Figure 6: Estimates of the �rst four principal components. Explained vari-
ance: 94%Forecasting Conditional Quantiles of Electricity Demand
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Results 4-4
Principal Component Scores
0 100 200 300 400 500 600 700
−1.
00.
01.
0
Day
α 1
0 100 200 300 400 500 600 700
−1.
00.
0
Day
α 2
Figure 7: Estimates of the scores of the �rst and second principal compo-
nentForecasting Conditional Quantiles of Electricity Demand
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Results 4-5
Principal Component Scores
0 100 200 300 400 500 600 700
−0.
50.
51.
0
Day
α 3
0 100 200 300 400 500 600 700
−0.
50.
5
Day
α 4
Figure 8: Estimates of the scores of the third and fourth principal compo-
nentForecasting Conditional Quantiles of Electricity Demand
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Results 4-6
Forecasted Median Curves
Figure 9: Observed data together with forecasted median and forecast
provided by Amprion from 20130105 to 20130222
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Outlook 5-1
Outlook
� Improve forecasts
I Try di�erent rotations of PC
I Use better suited multivariate TS model
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References 6-1
References
Aue, A., Norinho, D.D., Hörmann, S.On the prediction of functional time series
arXiv preprint arXiv:1208.2892, 2012
Guo, M., Zhou, L., Huang, J.Z., Härdle, W. H.Functional Data Analysis of Generalized Quantile Regressions
SFB649 Discussion Paper, 2013
Hörmann, S., Kokoszka, P.Weakly dependent functional data
The Annals of Statistics, 2010
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References 6-2
References
Koenker, R., Ng, P., Portnoy, S.Quantile Smoothing Splines
Biometrika, 1994
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Forecasting Conditional Quantiles of
Electricity Demand:
A Functional Data Approach
Franziska Schulz
Brenda López Cabrera
Ladislaus von Bortkiewicz Chair of StatisticsHumboldt�Universität zu Berlinhttp://lvb.wiwi.hu-berlin.dehttp://www.case.hu-berlin.de
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Appendix 7-1
Weak dependence
De�nition (Hörmann and Kokoszka (2010))
A sequence {Xn} ∈ pH is called Lp-m-approximable is each Xn
admits the representation
Xn = f (εn, εn−1, . . .)
where the εi are iid elements taking values in a measurable spaceS , and f is a measurable function f : S∞ → H.
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Appendix 7-2
Weak dependence
Moreover, we assume that if {ε′i} is an independent copy of {εi}de�ned on the same probability space, then letting
X(m)n = f (εn, εn−1, . . . , εn−m+1, ε
′n−m, ε
′n−m−1, . . .)
we have∞∑
m=1
{E(|Xm − X(m)m |p)}1/p <∞
� Hörmann and Kokoszka (2010) show that a linear process{Xn} with Xn =
∑∞j=0
Ψj(εn−j) is L4-m-approximable if
∞∑m=0
∞∑j=m
Ψj <∞Return
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Appendix 7-3
Merit Order
Demand
Marginal
Cost
OilGasRenewable Nuclear Hard CoalBrown Coal
Price
Supply
Figure 10: Pricing according to Merit-Order
Return
Forecasting Conditional Quantiles of Electricity Demand