Probabilistic Forecasting of Medium-Term Electricity Demand: A Comparison of Time Series Models

(1)

Fakultät IV

Department Mathematik

Probabilistic Forecasting of Medium-Term

Electricity Demand: A Comparison of Time

Series Models

Kevin Berk and Alfred Müller

SPA 2015, Oxford

(2)

Motivation Load profiles Methodology Forecasting Comparing Models Regime-Switching Conclusion

Load forecasting

Probabilistic forecasting of electricity demand is an important task for all active market participants!

I pricing of electricity contracts for end customers

I basis for risk management and hedging strategies of suppliers and vendors

I optimization of electricity procurement (generation) and consumption

I evaluation of the fair market price of electricity

(3)

Literature

I approaches vary from AI-based like neural networks to parametric approaches like exponential smoothing or time series analysis (Weron (2006), Alfares and Nazeeruddin (2002))

I typically considered: load forecasting models for households or for the total system load (Paatero and Lund (2005), Dordonnat et. al (2008))

I lack of information about uncertainty (point forecasts)

I usual forecast horizon is hours to days ((very) short term)

Suitable studies of medium term probabilistic models for industrial end customers seem to be rare so far.

(4)

Our aim:

I load forecasting for companies depending on business sector

I medium-term, i.e. year-aheadprobabilistic forecasts

I total system load as an exogenous factor

I SARIMA model for stochastic load component

I challenge: consumption patterns can vary significantly among different sectors

Which model is the best one?

(5)

Outline

1. Motivation 2. Load profiles 3. Methodology 4. Forecasting 5. Comparing Models 6. Regime-Switching 7. Conclusion

(6)

Load profiles

I metering of electricity demand

I hourly load data for a few years

I the cumulated load of a set of households or the total system load show a quite homogeneous behavior (diversification)

I in contrast, the load of a single industry customer is more heterogeneous

I reason: stochastic events like machine failures have a greater impact on the individual demand profile

(7)

Some examples

Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Retail Service

Electric Steel Mill Paper Mill

Automotive Parts Supplier Connection Technology

Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su

Metal Processing Industry Metal Forming Technologies

(8)

Fundamental model equation

Let Ct denote the electricity consumption at time t, our model

equation has the following form:

log Ct=Dt+u(Gt∗) +Rt

Here,

I Dt is adeterministic seasonal pattern, I u(G∗

t)is a function of theresidual grid load G∗t and I Rt is aresidual time series.

(9)

A similar-day approach for D

t

The deterministic seasonal pattern Dtis given by

Dt =β0+ β1sin{ 2πt 365.25 · 24} + β2cos{ 2πt 365.25 · 24} + 16 X j=3 βjϑj−2(t) + 40 X j=17 βj%j−16(t),

where the βj are determined via OLS (using outlier-cleaned

historical data). The dummies ϑj , j = 1, . . . , 14 are described in

table 1 and %1, . . . , %24denote the hours.

(10)

A similar-day approach for D

t

Table : Overview of dummy variables for D41sincos.

ϑ1 Mondays (if not a public holiday or a bridge day)

ϑ2 Fridays (if not a public holiday or a bridge day)

ϑ3 Tuesdays, Wednesdays and Thursdays (if not a p. hol.)

ϑ4 Saturdays (if not a public holiday)

ϑ5 Sundays (if not a public holiday)

ϑ6 Winter holiday period

ϑ7 Summer holiday period

ϑ8 Public holidays

ϑ9 Bridge day (Monday before a public holiday)

ϑ10 Bridge day (Friday after a public holiday)

ϑ11 January 1st

ϑ12 December 24th

ϑ13 December 25th and 26th

ϑ14 New Year’s Eve

(11)

G

∗_t

as an exogenous factor

I G∗_t denotes the residual grid load, i.e. the total system load minus its deterministic part

I maps the stochastic fluctuations of the market

I correlation with a customer’s electricity demand can be positive or negative

I the function u(G_t∗)could be any suitable function, we found a linear approach sufficient

Residual grid load vs. residual customer consumption (peak hours): Corr=23.16% Robust regression

(12)

The residual time series R

t

I covers remaining autocorrelations and seasonalities

I often modeled by a Gaussian white noise process (iid normal distributed)

I general assumption of independence and light tails is wrong

I choose a (seasonal) ARIMA model for Rt

I number and choice of parameters depend on business sector

Rt =log Ct− Dt− u(G∗t)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec −1 −0.5 0 0.5 1 1.5

Figure : Exemplary residual time series Rt

(13)

Model choice for R

t

I BIC suggests a SARIMA(p, 0, q) × (P, 0, Q)24model with

(p, 0, q) × (P, 0, Q) = =     

(3, 0, 3) × (0, 0, 2) for retail customers,

(4, 0, 4) × (1, 0, 1) for two shift operating customers, (2, 0, 0) × (2, 0, 2) for three shift operating customers.

I innovations are not normally distributed

−1.50 −1 −0.5 0 0.5 1 1.5 200 400 600 800 1000

Figure : normal distribution fit for SARIMA model innovations

(14)

Model choice for R

t

Best fit for innovations given by NIG-distribution.

−1.5 −1 −0.5 0 0.5 1 1.5 10−12 10−10 10−8 10−6 10−4 10−2 100

Figure : Fit of a normal distribution (dotted) and a normal inverse Gaussian

distribution (dashed) to the kernel density of the empirical innovations (solid) on a semi-logarithmic scale.

(15)

Medium-term forecasts

I year ahead forecast scenarios

I Monte-Carlo approach for retail pricing (see Burger and Müller (2012))

I trading strategy evaluation

I investment decisions (thermal power station, photovoltaic system)

I peak load management

I strategic planning

(16)

Real load vs

one-year forecast scenarios

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

0 100 200 300 400

(17)

Real load vs

one-year forecast scenarios

0 100 200 300 400

(18)

Real load vs

expected consumption

/

forecast

scenarios

Mo We Fr Su Tu Th Sa 50 100 150 200 Mo We Fr Su Tu Th Sa 50 100 150 200

(19)

Real load vs

expected consumption

/

forecast

scenarios

(20)

Real load vs

expected consumption

/

forecast

scenarios

(21)

Comparing Models

Comparing different variants of the model:

I different number of parameters for seasonal patterns

I monthly dummies, sin-cos, area-preserving splines of order 4

I innovations normal, t or NIG

Figure : Regression coefficients of monthly dummy variables and fitted spline of

order four.

(22)

Evaluation of probabilistic forecasts

For a given observation ˜x let ˜F (z) = 1_[z≥˜_{x ]}be the empirical cdf. We use the continuous ranked probability score (CPRS), which is defined for a probabilistic forecast with cdf F as

CRPS(F , ˜x ) = Z ∞ −∞ (F (z) − ˜F (z))2dz = Z 1 0 QSα(F−1(α), ˜x )d α with QSα(q, ˜x ) = 2(1[˜x <q]− α)(q − ˜x )

denoting the quantile score.

We estimate F via Monte-Carlo simulation and compute the CRPS for every point of time t and add up over t (of one year).

(23)

Data of four companies

Mo Tu We Th Fr Sa Su 2008 2009 2010 2011 Mo Tu We Th Fr Sa Su 2009 2010 2011 Mo Tu We Th Fr Sa Su 2009 2010 2011 Mo Tu We Th Fr Sa Su 2009 2010 2011 2012

Figure : Weekly (left) and yearly (right) electricity demand structure of the four

customers in our database. Customers I-IV from top to bottom.

(24)

Evaluation of probabilistic forecasts

Table : CRPS of different models for customer I to II. Absolute value and

percentage deviation from benchmark. Rank in brackets, best values in bold.

Model CRPS pct. CRPS pct. customer I customer II Benchmark 28.59 (10) 100% 14.24 (10) 100% D31sincos-Normal 23.43 (9) -18.03% 9.01 (9) -36.77% D35-SARIMA-Normal 23.26 (8) -18.62% 8.95 (8) -37.13% D35-SARIMA-NIG 22.84 (7) -20.12% 8.89 (5) -37.6% D35spline-SARIMA-NIG 22.78 (6) -20.33% 8.93 (7) -37.33% D41sincos-SARIMA-NIG 21.29 (3) -25.53% 8.83 (3) -38.03% D41sincos-GL-SARIMA-NIG 20.57 (1) -28.03% 8.72 (1) -38.8% D51-SARIMA-Normal 21.96 (5) -23.19% 8.89 (6) -37.56% D51-SARIMA-NIG 21.57 (4) -24.55% 8.86 (4) -37.78% D51spline-GL-SARIMA-NIG 20.87 (2) -27% 8.73 (2) -38.68%

(25)

Evaluation of probabilistic forecasts

Table : CRPS of different models for customer III and IV. Absolute value and

percentage deviation from benchmark. Rank in brackets, best values in bold.

Model CRPS pct. CRPS pct. customer III customer IV Benchmark 888.78 (10) 100% 45.33 (10) 100% D31sincos-Normal 830.72 (6) -6.53% 27.46 (3) -39.43% D35-SARIMA-Normal 848.54 (9) -4.53% 27.44 (2) -39.47% D35-SARIMA-NIG 844.13 (8) -5.02% 27.59 (7) -39.14% D35spline-SARIMA-NIG 841.13 (7) -5.36% 27.56 (6) -39.21% D41sincos-SARIMA-NIG 771.95 (1) -13.14% 27.51 (5) -39.31% D41sincos-GL-SARIMA-NIG 774.24 (2) -12.89% 27.97 (8) -38.3% D51-SARIMA-Normal 792.51 (3) -10.83% 27.39 (1) -39.58% D51-SARIMA-NIG 792.53 (4) -10.83% 27.49 (4) -39.37% D51spline-GL-SARIMA-NIG 795.26 (5) -10.52% 28.14 (9) -37.91%

(26)

Evaluation of probabilistic forecasts

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300 350 400 450 500 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700 800 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 1200 1400

Figure : Rank histograms of the empirical PITs for the best ranked models. Top

row: customer I and II. Bottom row: customer III and IV.

(27)

Regime-switching load profiles

Jan'10 Jan'11 Jan'12 Jan'13

Figure : Load profile with stochastic switches in regime (Electric wire

manufacturing).

I two-regime time series

I different behavior in each regime

I switches are random but seem to depend on time

I summer and winter holiday periods are special

(28)

Markov-switching model

I Idea: Inhomogeneous markov-switching model

I ARMA (Rt) plus seasonal component (Dt) in upper regime I iid rv (Zt) in lower regime

I markov-chain with time-varying transition probabilities

Let Ct be the electricity consumption and St ∈ {0, 1} be the state

of regime at time t. Then

Ct= (Dt+Rt)St+Zt(1 − St) .

St can not be observed directly, we rather have to infer its

operation through the observed behavior of Ct.

(29)

Markov-chain S

t

The two-point stochastic process St is described by the transition

matrix P(St =st|St−1=st−1,zt−1; β0) = q(zt−1, β0) 1 − q(zt−1, β0) 1 − p(zt−1, β1) p(zt−1, β1)

where zt is a vector of economic-indicator variables, β = (β0, β1)

is a set of parameters governing the transition probabilities and p, q : R2k → [0, 1] are piecewise continuously differentiable functions (see Filardo (1994)). We use a logistic functional form.

(30)

Parameter estimation

MS26-D14-ARMA model

I 2 · 13 = 26 parameters for the transition probabilities

I 14 parameters for Dt, i.e. seasonality in upper regime I ARMA-parameters for Rt

I two parameters for Zt

The parameters are estimated via differential evolution using the first two years of hourly data.

Jan'10 Jan'11

Figure : Timeseries with detected regimes (smoothed marginal state probabilities

P(St=0 | {Ct}Tt=1, {zt}

T

t=1, θ) > .99)

(31)

Simulation results

Jan'120 Jan'13 10 20 30 40 50 60 70 80 90 Jan'120 Jan'13 20 40 60 80 100 120 Jan'120 Jan'13 10 20 30 40 50 60 70 80

Figure : Test-period time series (top), D51-Normal forecast (middle) and

MS26-D14-ARMA forecast (bottom).

(32)

CRPS results

I conventional models’ detrending routine performs badly

I as a result, innovation variance is highly overestimated

I markov-switching approach improves forecasting performance significantly

Table : CRPS of different models for regime switching customer

Model CRPS pct. Benchmark 14.78 (8) 100% D31 20.49 (9) +38.63% D51-Normal 12.76 (3) -13.67% D51-SARIMA-Normal 12.82 (4) -13.26% D51-SARIMA-Students 13.64 (6) -7.71% D51-SARIMA-NIG 12.73 (2) -13.87% D51spline-SARIMA-Students 14.10 (7) -4.6% D51spline-SARIMA-NIG 13.21 (5) -10.62% MS26-D14-ARMA 8.72 (1) -41%

(33)

Conclusion

I modeling of electricity demand via time series analysis

I CRPS for comparison of probabilistic forecasts and observations

I more parameters 6= better performance

I sin/cos approach for seasonality superior to dummies/area-preserving splines

I inhomogeneous markov-switching model for regime-switching load profiles

I significant improvement of forecast quality Current research

I other methods of evaluating probabilistic forecasts

I improve computation effort for regime model estimation

I empirical work on load data analysis

(34)

Literature I

R. Weron.

Modeling and Forecasting Electricity Loads and Prices.

Chichester: Wiley, 2006.

K. Berk.

Modeling and Forecasting Electricity demand: A Risk Management Perspective.

Springer, 2015.

K. Berk and A. Müller.

Probabilistic Forecasting of Medium-Term Electricity Demand: A Comparison of Time Series Models.

The Journal of Energy Markets, accepted.

M. Burger and J. Müller.

Risk-adequate pricing of retail power contracts.

The Journal of Energy Markets, 4:53-75, 2012.

(35)

Literature II

M. Burger, B. Klar, A. Müller, G. Schindlmayr.

A spot market model for pricing derivatives in electricity markets.

Quantitative Finance, 4:109-122, 2004.

A.J. Filardo.

Business-Cycle Phases and Their Transitional Dynamics.

Journal of Business & Economic Statistics, 12:299-308, 1994.

(36)

Probabilistic Forecasting of Medium-Term Electricity Demand: A Comparison of Time Series Models