Probabilistic Electric Load Forecasting and R Implementation

Shu FanJoint work with Rob J Hyndman

Monash University

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Outline

• 1 The problem• 2 The model• 3 Forecasts• 4 R implementation: MEFM package• 5 References

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The problem

• We want to forecast the peak electricity demand in a half-hour period in twenty years time.

• We have fifteen years of half-hourly electricity data, temperature data and some economic and demographic data.

• The location: regions in Australian National Electricity Market (NEM).

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South Australian demand data4

South Australian demand data5

Outline

• 1 The problem• 2 The model• 3 Forecasts• 4 R implementation: MEFM package• 5 References

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Predictors

• calendar effects• prevailing and recent weather conditions• climate changes• economic and demographic changes• changing technology

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Modelling framework

• Semi-parametric additive models with correlated errors.

• Each half-hour period modelled separately for each season.

• Variables selected to provide best out-of-sample predictions using cross-validation on each summer.

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Monash Electricity Forecasting Model

log 𝑦𝑦𝑡𝑡 = ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑡𝑡 + 𝑛𝑛𝑡𝑡

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

• ℎ𝑝𝑝 𝑡𝑡 models all calendar effects;

• 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡,𝜔𝜔2,𝑡𝑡 models all temperature effects where 𝜔𝜔1,𝑡𝑡 is a vector of recent temperatures at location 1 and 𝜔𝜔2,𝑡𝑡 is a vector of recent temperatures at location 2;

• 𝑍𝑍𝑗𝑗,𝑡𝑡 is a demographic or economic variable at time t• 𝑛𝑛𝑡𝑡 denotes the model error at time t.

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Monash Electricity Forecasting Model

log 𝑦𝑦𝑡𝑡 = ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑡𝑡 + 𝑛𝑛𝑡𝑡

• ℎ𝑝𝑝 𝑡𝑡 includes handle annual, weekly and daily seasonal patterns as well as public holidays:

ℎ𝑝𝑝 𝑡𝑡 = 𝑙𝑙𝑝𝑝 𝑡𝑡 + 𝛼𝛼𝑡𝑡,𝑝𝑝 + 𝛽𝛽𝑡𝑡,𝑝𝑝 + 𝛾𝛾𝑡𝑡,𝑝𝑝 + 𝛿𝛿𝑡𝑡,𝑝𝑝

• 𝑙𝑙𝑝𝑝 𝑡𝑡 is “time of summer” effect (a regression spline);• 𝛼𝛼𝑡𝑡,𝑝𝑝 is day of week effect;• 𝛽𝛽𝑡𝑡,𝑝𝑝 is “holiday” effect;• 𝛾𝛾𝑡𝑡,𝑝𝑝 New Year’s Eve effect;• 𝛿𝛿𝑡𝑡,𝑝𝑝 is millennium effect;

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Fitted results (Summer 3pm)11

Monash Electricity Forecasting Model

log 𝑦𝑦𝑡𝑡 = ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑡𝑡 + 𝑛𝑛𝑡𝑡

𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡,𝜔𝜔2,𝑡𝑡 = ∑𝑘𝑘=06 𝑓𝑓𝑘𝑘,𝑝𝑝 𝑥𝑥𝑡𝑡−𝑘𝑘 + 𝑔𝑔𝑘𝑘,𝑝𝑝(𝑑𝑑𝑡𝑡−𝑘𝑘) + 𝑞𝑞𝑝𝑝(𝑥𝑥𝑡𝑡+

)+𝑟𝑟𝑝𝑝(𝑥𝑥𝑡𝑡−

)+ 𝑠𝑠𝑝𝑝 �𝑥𝑥𝑡𝑡 + ∑𝑗𝑗=16 𝐹𝐹𝑗𝑗,𝑝𝑝 𝑥𝑥𝑡𝑡−48𝑗𝑗 + 𝐺𝐺𝑗𝑗,𝑝𝑝(𝑑𝑑𝑡𝑡−48𝑗𝑗)

• 𝑥𝑥𝑡𝑡 is ave temp across two sites (Kent Town and Adelaide Airport) at time t;• 𝑑𝑑𝑡𝑡 is the temp difference between two sites at time t;

• 𝑥𝑥𝑡𝑡+

is max of 𝑥𝑥𝑡𝑡 values in past 24 hours;

• 𝑥𝑥𝑡𝑡−

is min of 𝑥𝑥𝑡𝑡 values in past 24 hours;• �𝑥𝑥𝑡𝑡is average temp in past seven days.Each function is smooth & estimated using regression splines.

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Fitted results (Summer 3pm)13

Monash Electricity Forecasting Model

log 𝑦𝑦𝑡𝑡 = ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑡𝑡 + 𝑛𝑛𝑡𝑡

• Other variables described by linear relationships with coefficients 𝐶𝐶1,…., 𝐶𝐶𝑗𝑗 .

• Estimation based on annual data.

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Split model

log 𝑦𝑦𝑡𝑡 = ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑡𝑡 + 𝑛𝑛𝑡𝑡

log 𝑦𝑦𝑡𝑡 = log 𝑦𝑦𝑡𝑡∗

+ log �𝑦𝑦𝑖𝑖log 𝑦𝑦𝑡𝑡

∗= ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + 𝑒𝑒𝑡𝑡

log �𝑦𝑦𝑖𝑖 = �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑖𝑖 + 𝜖𝜖𝑖𝑖

• �𝑦𝑦𝑖𝑖 is the average demand for year i where t is in year 𝑖𝑖.

• 𝑦𝑦𝑡𝑡∗

is the standardized demand for time t.

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Split model16

Annual model

log �𝑦𝑦𝑖𝑖 = �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑖𝑖 + 𝜖𝜖𝑖𝑖

log �𝑦𝑦𝑖𝑖 − log 𝑦𝑦𝑖𝑖−1 −= �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗(𝑍𝑍𝑗𝑗,𝑖𝑖 − 𝑍𝑍𝑗𝑗,𝑖𝑖−1) + 𝜖𝜖𝑖𝑖

• First differences modelled to avoid non-stationary variables.• Predictors: Per-capita GSP, Price, Summer CDD, Winter HDD.Variable selection• GSP needed to stay in the model to allow scenario forecasting.• All other variables led to improved AICC.

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Annual model18

Annual model19

Half-hourly models

log 𝑦𝑦𝑡𝑡 = log 𝑦𝑦𝑡𝑡∗

+ log �𝑦𝑦𝑖𝑖log 𝑦𝑦𝑡𝑡

∗= ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + 𝑒𝑒𝑡𝑡

• Separate model for each half-hour.• Same predictors used for all models.• Predictors chosen by cross-validation on last 2 summer.• Each model is fitted to the data twice, first excluding the last

summer and then excluding the previous summer. Average out-of-sample MSE calculated from omitted data.

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Half-hourly models21

Outline

• 1 The problem• 2 The model• 3 Forecasts• 4 R implementation: MEFM package• 5 References

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Peak demand forecasting

𝑞𝑞𝑡𝑡,𝑝𝑝 = ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑡𝑡 + 𝑛𝑛𝑡𝑡

Multiple alternative futures created:• ℎ𝑝𝑝 𝑡𝑡 known;• simulate future temperatures using double seasonal block

bootstrap with variable blocks (with adjustment for climate change);

• use assumed values for GSP, population and price;• resample residuals using double seasonal block bootstrap

with variable blocks.

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Peak demand backcasting

𝑞𝑞𝑡𝑡,𝑝𝑝 = ℎ𝑝𝑝 𝑡𝑡 + 𝑓𝑓𝑝𝑝 𝜔𝜔1,𝑡𝑡 ,𝜔𝜔2,𝑡𝑡 + �𝑗𝑗=1

𝐽𝐽

𝐶𝐶𝑗𝑗𝑍𝑍𝑗𝑗,𝑡𝑡 + 𝑛𝑛𝑡𝑡

Multiple alternative pasts created:• ℎ𝑝𝑝 𝑡𝑡 known;• simulate past temperatures using double seasonal block

bootstrap with variable blocks (with adjustment for climate change);

• use actual values for GSP, population and price;• resample residuals using double seasonal block bootstrap

with variable blocks.

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Estimated historical quantiles25

Peak demand forecasting26

Peak demand forecasting27

Outline

• 1 The problem• 2 The model• 3 Forecasts• 4 R implementation: MEFM package• 5 References

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MEFM package for R

• Available on github:

install.packages("devtools")library(devtools)install_github("robjhyndman/MEFM-package")

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MEFM package for R

Package contents:• seasondays The number of days in a season• sa.econ Historical demographic & economic data for

South Australia• sa Historical data for model estimation• maketemps Create lagged temperature variables• demand_model Estimate the electricity demand models• simulate_ddemand Temperature and demand simulation• simulate_demand Simulate the electricity demand for the

next season

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MEFM package for R

Usagelibrary(MEFM)

• # Number of days in each "season"seasondays

• # Historical economic datasa.econ

• # Historical temperature and calendar datahead(sa); tail(sa); dim(sa)

• # create lagged temperature variablessalags <- maketemps(sa,2,48)dim(salags)head(salags)

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MEFM package for R• # formula for annual modelformula.a <- as.formula(anndemand ~ gsp + ddays + resiprice)• # formulas for half-hourly model

# These can be different for each half-hourformula.hh <- list()for(i in 1:48) {formula.hh[[i]] <- as.formula(log(ddemand) ~ ns(temp, df=2)

+ day + holiday+ ns(timeofyear, df=9) + ns(avetemp, df=3)+ ns(dtemp, df=3) + ns(lastmin, df=3)+ ns(prevtemp1, df=2) + ns(prevtemp2, df=2)+ ns(prevtemp3, df=2) + ns(prevtemp4, df=2)+ ns(day1temp, df=2) + ns(day2temp, df=2)+ ns(day3temp, df=2) + ns(prevdtemp1, df=3)+ ns(prevdtemp2, df=3) + ns(prevdtemp3, df=3) + ns(day1dtemp, df=3))}

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MEFM package for R• # Fit all modelssa.model <- demand_model(salags, sa.econ, formula.hh, formula.a)• # Summary of annual modelsummary(sa.model$a)• # Summary of half-hourly model at 4pmsummary(sa.model$hh[[33]])• # Simulate future normalized half-hourly datasimdemand <- simulate_ddemand(sa.model, sa, simyears=50)• # economic forecasts, to be given by userafcast <- data.frame(pop=1694, gsp=22573, resiprice=34.65,ddays=642)• # Simulate half-hourly datademand <- simulate_demand(simdemand, afcast)

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MEFM package for R

• Results plottingplot(ts(demand$demand[,sample(1:50, 4)], freq=48, start=0),

xlab="Days", main="Simulated demand futures")plot(demand$annmax, main="Simulated seasonal maximums", ylab="GW")plot(demand$annmax, main="Simulated seasonal maximums", ylab="GW")boxplot(demand$annmax, main="Simulated seasonal maximums",

xlab="GW", horizontal=TRUE)rug(demand$annmax)plot(density(demand$annmax, bw="SJ"), xlab="Demand (GW)",

main="Density of seasonal maximum demand")rug(demand$annmax)

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References

• Hyndman, R.J. and Fan, S. (2010) “Density forecasting for long-term peak electricity demand”, IEEE Transactions on Power systems, 25(2), 1142–1153.

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

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• Ben Taieb, S. & Hyndman, R.J. (2013) “A gradient boosting approach to the Kaggle load forecasting competition”, International Journal of Forecasting, 29(4).

• Hyndman, R.J., & Fan, S. (2014). “Monash Electricity Forecasting Model”. Technical paper. robjhyndman.com/working-papers/mefm/

• Fan, S., & Hyndman, R.J. (2014). “MEFM: An R package implementing the Monash Electricity Forecasting Model.” github.com/robjhyndman/MEFM-package

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