forecasting electricity demand distributions using a semiparametric additive model

Forecasting electricity demanddistributions using asemiparametric additive model

Rob J HyndmanJoint work with Shu Fan

Forecasting electricity demand distributions 1

Outline

1 The problem

2 The model

3 Long-term forecasts

4 Short term forecasts

5 Forecast density evaluation

6 Forecast quantile evaluation

7 References and R implementation

Forecasting electricity demand distributions The problem 2

The problem

We want to forecast the peak electricitydemand in a half-hour period in twenty yearstime.

We have fifteen years of half-hourly electricitydata, temperature data and some economicand demographic data.

The location is South Australia: home to themost volatile electricity demand in the world.

Sounds impossible?

The problem

Sounds impossible?

The problem

Sounds impossible?

The problem

Sounds impossible?

The problem

Sounds impossible?

South Australian demand data

Black Saturday→

The 2009 heatwave

The 2009 heatwaveAverage temperature (January−February 2009)

Date in January/February 2009

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 312 4 6 8 10 12 14 16 18 20 22 24 26 28 30 1 3 5 7 9 11 13 15 17 19 21 23 25 272 4 6 8 10 12 14 16 18 20 22 24 26 28

The 2009 heatwaveAverage temperature (January−February 2009)

Date in January/February 2009

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 312 4 6 8 10 12 14 16 18 20 22 24 26 28 30 1 3 5 7 9 11 13 15 17 19 21 23 25 272 4 6 8 10 12 14 16 18 20 22 24 26 28

The 2009 heatwave

Black Saturday→

South Australia state wide demand (summer 10/11)

Oct 10 Nov 10 Dec 10 Jan 11 Feb 11 Mar 11

South Australia state wide demand (January 2011)

Date in January

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3111 13 15 17 19 21

Demand boxplots (Sth Aust)

Temperature data (Sth Aust)

Demand densities (Sth Aust)

Industrial offset demand

Winter

Industrial offset demand

Summer

Outline

1 The problem

2 The model

Forecasting electricity demand distributions The model 13

Predictorscalendar effectsprevailing and recent weather conditionsclimate changeseconomic and demographic changeschanging technology

Modelling framework

Semi-parametric additive models withcorrelated errors.Each half-hour period modelled separately foreach season.Variables selected to provide bestout-of-sample predictions using cross-validationon each summer.

Modelling framework

Monash Electricity Forecasting Model

log(yt) = hp(t) + fp(w1,t,w2,t) +

J∑j=1

cjzj,t + nt

yt denotes per capita demand (minus offset) at time t(measured in half-hourly intervals) and p denotes thetime of day p = 1, . . . ,48;

hp(t) models all calendar effects;

fp(w1,t,w2,t) models all temperature effects where w1,t isa vector of recent temperatures at location 1 and w2,t isa vector of recent temperatures at location 2;

zj,t is a demographic or economic variable at time t

nt denotes the model error at time t.

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

hp(t) includes handle annual, weekly and daily seasonalpatterns as well as public holidays:

hp(t) = `p(t) + αt,p + βt,p + γt,p + δt,p

`p(t) is “time of summer” effect (a regression spline);

αt,p is day of week effect;

βt,p is “holiday” effect;

γt,p New Year’s Eve effect;

δt,p is millennium effect;

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

Fitted results (Summer 3pm)

0 50 100 150

Day of summer

Mon Tue Wed Thu Fri Sat Sun

Day of week

Normal Day before Holiday Day after

Holiday

Time: 3:00 pm

J∑j=1

cjzj,t + nt

fp(w1,t,w2,t) =6∑

[fk,p(xt−k) + gk,p(dt−k)

]+ qp(x+

t ) + rp(x−t ) + sp(xt)

+6∑j=1

[Fj,p(xt−48j) + Gj,p(dt−48j)

]xt is ave temp across two sites (Kent Town and AdelaideAirport) at time t;dt is the temp difference between two sites at time t;x+t is max of xt values in past 24 hours;x−t is min of xt values in past 24 hours;xt is ave temp in past seven days.

Each function is smooth & estimated using regression splines.Forecasting electricity demand distributions The model 18

J∑j=1

cjzj,t + nt

fp(w1,t,w2,t) =6∑

]+ qp(x+

t ) + rp(x−t ) + sp(xt)

+6∑j=1

J∑j=1

cjzj,t + nt

fp(w1,t,w2,t) =6∑

]+ qp(x+

t ) + rp(x−t ) + sp(xt)

+6∑j=1

J∑j=1

cjzj,t + nt

fp(w1,t,w2,t) =6∑

]+ qp(x+

t ) + rp(x−t ) + sp(xt)

+6∑j=1

J∑j=1

cjzj,t + nt

fp(w1,t,w2,t) =6∑

]+ qp(x+

t ) + rp(x−t ) + sp(xt)

+6∑j=1

J∑j=1

cjzj,t + nt

fp(w1,t,w2,t) =6∑

]+ qp(x+

t ) + rp(x−t ) + sp(xt)

+6∑j=1

J∑j=1

cjzj,t + nt

fp(w1,t,w2,t) =6∑

]+ qp(x+

t ) + rp(x−t ) + sp(xt)

+6∑j=1

Fitted results (Summer 3pm)

10 20 30 40

Temperature

10 20 30 40

Lag 1 temperature

10 20 30 40

Lag 2 temperature

10 20 30 40

Lag 3 temperature

10 20 30 40

Lag 1 day temperature

10 15 20 25 30

Last week average temp

15 25 35

Previous max temp

10 15 20 25

Previous min temp

Time: 3:00 pm

J∑j=1

cjzj,t + nt

Other variables described by linearrelationships with coefficients c1, . . . , cJ.Estimation based on annual data.

J∑j=1

cjzj,t + nt

Other variables described by linearrelationships with coefficients c1, . . . , cJ.Estimation based on annual data.

J∑j=1

cjzj,t + nt

log(yt) = log(y∗t ) + log(yi)

log(y∗t ) = hp(t) + fp(w1,t,w2,t) + et

log(yi) =

J∑j=1

cjzj,i + εi

yi is the average demand for year i where t is inyear i.y∗t is the standardized demand for time t.

Annual model

log(yi) =∑j

cjzj,i + εi

log(yi)− log(yi−1) =∑j

cj(zj,i − zj,i−1) + ε∗i

First differences modelled to avoidnon-stationary variables.Predictors: Per-capita GSP, Price, Summer CDD,Winter HDD.

Annual model

log(yi) =∑j

cjzj,i + εi

Annual model

log(yi) =∑j

cjzj,i + εi

zCDD =∑

summer

max(0, T − 18.5)

T = daily mean

Annual model

log(yi) =∑j

cjzj,i + εi

zHDD =∑

winter

max(0,18.5− T)

T = daily mean

Annual modelCooling and Heating Degree Days

1990 1995 2000 2005 2010

Cooling and Heating degree days

Annual model

Variable Coefficient Std. Error t value P value∆gsp.pc 2.02×10−6 5.05×10−6 0.38 0.711∆price −1.67×10−8 6.76×10−9 −2.46 0.026∆scdd 1.11×10−10 2.48×10−11 4.49 0.000∆whdd 2.07×10−11 3.28×10−11 0.63 0.537

GSP needed to stay in the model to allowscenario forecasting.

All other variables led to improved AICC.

Annual model

89/90 91/92 93/94 95/96 97/98 99/00 01/02 03/04 05/06 07/08 09/10

ActualFitted

Half-hourly models

Separate model for each half-hour.Same predictors used for all models.Predictors chosen by cross-validation onsummer of 2007/2008 and 2009/2010.Each model is fitted to the data twice, firstexcluding the summer of 2009/2010 and thenexcluding the summer of 2010/2011. Theaverage out-of-sample MSE is calculated fromthe omitted data for the time periods12noon–8.30pm.

Half-hourly models

Half-hourly modelsx x1 x2 x3 x4 x5 x6 x48 x96 x144 x192 x240 x288 d d1 d2 d3 d4 d5 d6 d48 d96 d144 d192 d240 d288 x+ x− x dow hol dos MSE

1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1.0372 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1.0343 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1.0314 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1.0275 • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1.0256 • • • • • • • • • • • • • • • • • • • • • • • • • • • 1.0207 • • • • • • • • • • • • • • • • • • • • • • • • • • 1.0258 • • • • • • • • • • • • • • • • • • • • • • • • • • 1.0269 • • • • • • • • • • • • • • • • • • • • • • • • • 1.035

10 • • • • • • • • • • • • • • • • • • • • • • • • 1.04411 • • • • • • • • • • • • • • • • • • • • • • • 1.05712 • • • • • • • • • • • • • • • • • • • • • • 1.07613 • • • • • • • • • • • • • • • • • • • • • 1.10214 • • • • • • • • • • • • • • • • • • • • • • • • • • 1.01815 • • • • • • • • • • • • • • • • • • • • • • • • • 1.02116 • • • • • • • • • • • • • • • • • • • • • • • • 1.03717 • • • • • • • • • • • • • • • • • • • • • • • 1.07418 • • • • • • • • • • • • • • • • • • • • • • 1.15219 • • • • • • • • • • • • • • • • • • • • • 1.18020 • • • • • • • • • • • • • • • • • • • • • • • • • 1.02121 • • • • • • • • • • • • • • • • • • • • • • • • 1.02722 • • • • • • • • • • • • • • • • • • • • • • • 1.03823 • • • • • • • • • • • • • • • • • • • • • • 1.05624 • • • • • • • • • • • • • • • • • • • • • 1.08625 • • • • • • • • • • • • • • • • • • • • 1.13526 • • • • • • • • • • • • • • • • • • • • • • • • • 1.00927 • • • • • • • • • • • • • • • • • • • • • • • • • 1.06328 • • • • • • • • • • • • • • • • • • • • • • • • • 1.02829 • • • • • • • • • • • • • • • • • • • • • • • • • 3.52330 • • • • • • • • • • • • • • • • • • • • • • • • • 2.14331 • • • • • • • • • • • • • • • • • • • • • • • • • 1.523

Half-hourly models

R−squared

Time of day

12 midnight 6:00 am 9:00 am 12 noon 3:00 pm 6:00 pm 9:00 pm3:00 am 12 midnight

Half-hourly models

South Australian demand (January 2011)

Date in January

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

ActualFitted

Temperatures (January 2011)

Date in January

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Kent TownAirport

Half-hourly models

Adjusted model

Original model

J∑j=1

cjzj,t + nt

Model allowing saturated usage

qt = hp(t) + fp(w1,t,w2,t) +

J∑j=1

cjzj,t + nt

log(yt) =

{qt if qt ≤ τ ;τ + k(qt − τ) if qt > τ .

Adjusted model

Original model

J∑j=1

cjzj,t + nt

Model allowing saturated usage

qt = hp(t) + fp(w1,t,w2,t) +

J∑j=1

cjzj,t + nt

log(yt) =

{qt if qt ≤ τ ;τ + k(qt − τ) if qt > τ .

Outline

1 The problem

2 The model

Forecasting electricity demand distributions Long-term forecasts 31

Peak demand forecasting

qt,p = hp(t) + fp(w1,t,w2,t) +

J∑j=1

cjzj,t + nt

Multiple alternative futures created:hp(t) known;

simulate future temperatures using doubleseasonal block bootstrap with variable blocks(with adjustment for climate change);

use assumed values for GSP, population andprice;

resample residuals using double seasonal blockbootstrap with variable blocks.

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

Seasonal block bootstrapping

Conventional seasonal block bootstrap

Same as block bootstrap but with whole years as theblocks to preserve seasonality.But we only have about 10–15 years of data, so there is alimited number of possible bootstrap samples.

Double seasonal block bootstrap

Suitable when there are two seasonal periods (here wehave years of 151 days and days of 48 half-hours).Divide each year into blocks of length 48m.Block 1 consists of the first m days of the year, block 2consists of the next m days, and so on.Bootstrap sample consists of a sample of blocks whereeach block may come from a different randomly selectedyear but must be at the correct time of year.

Actual temperatures

0 10 20 30 40 50 60

Bootstrap temperatures (fixed blocks)

0 10 20 30 40 50 60

Bootstrap temperatures (variable blocks)

0 10 20 30 40 50 60

Problems with the double seasonal bootstrapBoundaries between blocks can introduce largejumps. However, only at midnight.Number of values that any given time in year isstill limited to the number of years in the dataset.

Variable length double seasonal blockbootstrap

Blocks allowed to vary in length between m−∆and m + ∆ days where 0 ≤ ∆ < m.

Blocks allowed to move up to ∆ days from theiroriginal position.

Has little effect on the overall time seriespatterns provided ∆ is relatively small.

Use uniform distribution on (m−∆,m + ∆) toselect block length, and independent uniformdistribution on (−∆,∆) to select variation onstarting position for each block.

Actual temperatures

C0 10 20 30 40 50 60

Bootstrap temperatures (fixed blocks)

0 10 20 30 40 50 60

Bootstrap temperatures (variable blocks)

0 10 20 30 40 50 60

Climate change adjustmentsCSIRO estimates for 2030:

0.3◦C for 10th percentile0.9◦C for 50th percentile1.5◦C for 90th percentile

We implement these shifts linearly from 2010.

No change in the variation in temperature.

Thousands of “futures” generated using aseasonal bootstrap.

J∑j=1

cjzj,t + nt

Multiple alternative futures created:hp(t) known;simulate future temperatures using doubleseasonal block bootstrap with variableblocks (with adjustment for climate change);use assumed values for GSP, population andprice;resample residuals using double seasonal blockbootstrap with variable blocks.

Peak demand backcasting

J∑j=1

cjzj,t + nt

Multiple alternative pasts created:hp(t) known;simulate past temperatures using doubleseasonal block bootstrap with variableblocks;use actual values for GSP, population andprice;resample residuals using double seasonal blockbootstrap with variable blocks.

Peak demand backcasting

PoE (annual interpretation)

98/99 00/01 02/03 04/05 06/07 08/09 10/11

10 %50 %90 %

●●

● ●

J∑j=1

cjzj,t + nt

Multiple alternative futures created:hp(t) known;simulate future temperatures using doubleseasonal block bootstrap with variableblocks (with adjustment for climate change);use assumed values for GSP, population andprice;resample residuals using double seasonal blockbootstrap with variable blocks.

South Australia GSP

1990 1995 2000 2005 2010 2015 2020

HighBaseLow

South Australia population

1990 1995 2000 2005 2010 2015 2020

HighBaseLow

Average electricity prices

1990 1995 2000 2005 2010 2015 2020

HighBaseLow

Major industrial offset demand

1990 1995 2000 2005 2010 2015 2020

HighBaseLow

Peak demand distribution

Annual POE levels

98/99 00/01 02/03 04/05 06/07 08/09 10/11 12/13 14/15 16/17 18/19 20/21

●●

● ●

●●

1 % POE5 % POE10 % POE50 % POE90 % POEActual annual maximum

2.5 3.0 3.5 4.0 4.5 5.0 5.5

Demand (GW)

2.5 3.0 3.5 4.0 4.5 5.0 5.5

Demand (GW)

2.5 3.0 3.5 4.0 4.5 5.0 5.5

Demand (GW)

2011/20122012/20132013/20142014/20152015/20162016/20172017/20182018/20192019/20202020/2021

2.5 3.0 3.5 4.0 4.5 5.0

Quantile

2.5 3.0 3.5 4.0 4.5 5.0

0 Base

Quantile

2.5 3.0 3.5 4.0 4.5 5.0

0 High

Quantile

2011/20122012/20132013/20142014/20152015/20162016/20172017/20182018/20192019/20202020/2021

Outline

1 The problem

2 The model

Forecasting electricity demand distributions Short term forecasts 46

Short term forecasts

J∑j=1

cjzj,t + nt

Bootstrapping temperatures and residuals is okfor long-term forecasts because short-termdynamics wash out after a few weeks.But short-term forecasts need to take accountof recent temperatures and recent residualsdue to serial correlation.Short-term temperature forecasts are available.Building a separate model for nt is possible, butthere is a simpler approach.

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

Short-term forecasting model

log(yt,p) = hp(t) + fp(w1,t,w2,t) + ap(yt−1) +

J∑j=1

cjzj,t + nt

yt,p denotes per capita demand (minus offset) at time t(measured in half-hourly intervals) during period p,p = 1, . . . ,48;hp(t) models all calendar effects;fp(w1,t,w2,t) models all temperature effects where w1,t isa vector of recent temperatures at location 1 and w2,t isa vector of recent temperatures at location 2;zj,t is a demographic or economic variable at time tnt denotes the model error at time tyt = [yt, yt−1, yt−2, . . . ]ap(yt−1) models effects of recent demands.

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

J∑j=1

cjzj,t + nt

ap(yt−1) =n∑

bk,p(yt−k) +m∑j=1

Bj,p(yt−48j)

+ Qp(y+t ) + Rp(y−t ) + Sp(yt)

y+t is maximum of yt values in past 24 hours;

y−t is minimum of yt values in past 24 hours;

yt is average demand in past 7 days

bk,p, Bj,p, Qp, Rp and Sp are estimated usingcubic splines.

ap(yt−1) =n∑

Bj,p(yt−48j)

ap(yt−1) =n∑

Bj,p(yt−48j)

ap(yt−1) =n∑

Bj,p(yt−48j)

Weakest assumptions

Temperature effects are independent of day ofweek effects.

Historical demand response to temperature willcontinue into the future.

Climate change will have only a small additiveincrease in temperature levels.

Locally generated electricity (e.g., PVgeneration) is not captured in demand data.

Weakest assumptions

Outline

1 The problem

2 The model

Forecasting electricity demand distributions Forecast density evaluation 51

Forecast density evaluation

PoE (annual interpretation)

98/99 00/01 02/03 04/05 06/07 08/09 10/11

10 %50 %90 %

●●

● ●

Qt(p) = forecast quantile of yt, to be ex-ceeded with probability 1− p.

G(p) = proportion of times yt less thanQt(p) in the historical data.

If Qt(p) is an accurate forecast distribution, thenG(p) ≈ p.

Excess probability

E(p) = G(p)− pE(p) does not depend on t.

Excess probability

KS = maxp |E(p)|MAEP =

∫ 10 |E(p)|dp

Cramer-von-Mises=∫ 1

0 E2(p)dp

Excess probability

∫ 10 |E(p)|dp

0 E2(p)dp

Excess probability

∫ 10 |E(p)|dp

0 E2(p)dp

Density evaluation

0.0 0.2 0.4 0.6 0.8 1.0

Density evaluation

0.0 0.2 0.4 0.6 0.8 1.0

Density evaluation

0.0 0.2 0.4 0.6 0.8 1.0

Area = MAEP: Mean Absolute Excess Probability

Density evaluation

0.0 0.2 0.4 0.6 0.8 1.0

Probability p

Area = MAEP: Mean Absolute Excess Probability

Density evaluation

0.0 0.2 0.4 0.6 0.8 1.0

Probability p

Area = Cramer−von−Mises statistic

Probability integral transform

Ft(y) = Prob(yt ≤ y) = distribution of yt.

Ft(Qt(p)) = p.

Zt = Ft(yt) is the PIT.

If Ft(y) is correct, then Zt will follow a U(0,1)distribution.

Ft(Qt(p)) = p.

4 5 6 7 8 9

Quantile: Q(p)

4 5 6 7 8 9

Quantile: Q(p)

4 5 6 7 8 9

Quantile: Q(p)

0.0 0.2 0.4 0.6 0.8 1.0

KS (same value as before)

0.0 0.2 0.4 0.6 0.8 1.0

MAEP (same value as before)

0.0 0.2 0.4 0.6 0.8 1.0

PIT not necessary as G(p)gives same informationand more interpretable.

MAEP (same value as before)

MAEP for density evaluation

MAEP more sensitive and less variable

than KS.

MAEP more interpretable than

Cramer-von-Mises statistic.

Calculation and interpretation of MAEP

does not require a Probability Integral

Transform

than KS.

Transform

than KS.

Transform

Outline

1 The problem

2 The model

Forecasting electricity demand distributions Forecast quantile evaluation 60

Quantile evaluation

Apply density evaluation measures to tail ofdistribution only.

E(p) = G(p)− p = excess probability

Quantile evaluation measures

KS = maxp |E(p)| where p > q

MAEPq =∫ 1q |E(p)|dp

Quantile evaluation

0.0 0.2 0.4 0.6 0.8 1.0

MAEP0.9

0.0 0.2 0.4 0.6 0.8 1.0

Probability p

MAEP0.9

q must be small enough for some observationsto have occurred in the tail.If yt values independent and there are nforecast distributions, then probability of Q(q)being exceeded at least once is 1− qn.Let Xq = number of observations > Q(q). ThenXq ∼ Binomial(n,1− q).Select n to ensure probability of at least 5 tailobservations is at least 0.95.q = 0.9⇒ n > 89.q = 0.95⇒ n > 181.q = 0.99⇒ n > 913.

0.90 0.92 0.94 0.96 0.98 1.00

We need forecasts of half-hourly demand with αannual probability of exceedance.

Insufficient data to look at annual maximums(less than 15 years)

Create approximately independent weeklymaximum forecasts (21 weeks each summer)

For these weekly forecasts, q = (1− α)1/21.

For 15 years of data, n = 315.

Therefore q ≤ 0.971 and α ≥ 0.46.

Model evaluation

A relatively large number of historicaldistributions are needed to compute MAEP.

We use weekly maximum demand for MAEP toallow a larger sample size.

MAEP5 MAEP10 MAEP50 MAEP90 MAEP100

Summer ex ante 5.46 10.27 22.73 21.64 19.92Summer ex post 20.08 17.58 18.12 12.63 11.59Winter ex ante 3.00 4.35 3.68 4.48 4.26Winter ex post 3.92 12.58 11.84 10.12 9.27

Model evaluation

Outline

1 The problem

2 The model

Forecasting electricity demand distributions References and R implementation 67

References and R implementation

Main papers

å Hyndman, R.J. and Fan, S. (2010) “Density forecasting forlong-term peak electricity demand”, IEEE Transactionson Power Systems, 25(2), 1142–1153.

å Fan, S. and Hyndman, R.J. (2012) “Short-term loadforecasting based on a semi-parametric additive model”.IEEE Transactions on Power Systems, 27(1), 134–141.

R package

We have an R package that implements allmethods, but it is not publicly available forcommercial reasons.

References and R implementation

Main papers

å Hyndman, R.J. and Fan, S. (2010) “Density forecasting forlong-term peak electricity demand”, IEEE Transactionson Power Systems, 25(2), 1142–1153.

å Fan, S. and Hyndman, R.J. (2012) “Short-term loadforecasting based on a semi-parametric additive model”.IEEE Transactions on Power Systems, 27(1), 134–141.

R package

We have an R package that implements allmethods, but it is not publicly available forcommercial reasons.

forecasting electricity demand distributions using a semiparametric additive model

volatile electricity

peak electricity demand

demand gw2

south australian demand

demand gwqqqqqq q q

q qqq q q qqq q q q

demand boxplots sth

demographic data

Business

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