forecasting conditional quantiles of electricity demand: a...

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

http://lvb.wiwi.hu-berlin.de

http://www.case.hu-berlin.de

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

Forecasting Conditional Quantiles of Electricity Demand

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


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


Motivation 1-4

Electricity Market Merit-Order

Figure 1: Load Curve


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


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)


Motivation 1-7

Objectives

� Modeling conditional quantiles of electricity load

I Understand dynamics

I Find main drivers

� Day-ahead forecasting of conditional quantiles


Outline

1. Introduction X

2. Data

3. Methodology

4. Results

5. Outlook


Data 2-1

Transmission System Operators in Germany


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


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]


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


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


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


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

Methodology 3-6

Functional Principal Component Analysis

� Tool to reduce dimensionality

� Yields direction of largest variability

� Express data as weighted sum of orthogonal curves


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


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


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

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


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


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


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


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

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

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

Results 4-6

Forecasted Median Curves

Figure 9: Observed data together with forecasted median and forecast

provided by Amprion from 20130105 to 20130222


Outlook 5-1

Outlook

� Improve forecasts

I Try di�erent rotations of PC

I Use better suited multivariate TS model


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


References 6-2

References

Koenker, R., Ng, P., Portnoy, S.Quantile Smoothing Splines

Biometrika, 1994


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

http://lvb.wiwi.hu-berlin.de

http://www.case.hu-berlin.de

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.


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


Appendix 7-3

Merit Order

Demand

Marginal

Cost

OilGasRenewable Nuclear Hard CoalBrown Coal

Price

Supply

Figure 10: Pricing according to Merit-Order

Return


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