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

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1 Fakultät IV Department Mathematik Probabilistic of Medium-Term Electricity Demand: A Comparison of Time Series Kevin Berk and Alfred Müller SPA 2015, Oxford July 2015

2 Load forecasting Probabilistic forecasting of electricity demand is an important task for all active market participants! pricing of electricity contracts for end customers basis for risk management and hedging strategies of suppliers and vendors optimization of electricity procurement (generation) and consumption evaluation of the fair market price of electricity Kevin Berk and Alfred Müller Electricity load forecasting 2

3 Literature approaches vary from AI-based like neural networks to parametric approaches like exponential smoothing or time series analysis (Weron (2006), Alfares and Nazeeruddin (2002)) typically considered: load forecasting models for households or for the total system load (Paatero and Lund (2005), Dordonnat et. al (2008)) lack of information about uncertainty (point forecasts) 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. Kevin Berk and Alfred Müller Electricity load forecasting 3

4 Our aim: load forecasting for companies depending on business sector medium-term, i.e. year-ahead probabilistic forecasts total system load as an exogenous factor SARIMA model for stochastic load component challenge: consumption patterns can vary significantly among different sectors Which model is the best one? Kevin Berk and Alfred Müller Electricity load forecasting 4

5 Outline Kevin Berk and Alfred Müller Electricity load forecasting 5

6 metering of electricity demand hourly load data for a few years the cumulated load of a set of households or the total system load show a quite homogeneous behavior (diversification) in contrast, the load of a single industry customer is more heterogeneous reason: stochastic events like machine failures have a greater impact on the individual demand profile Kevin Berk and Alfred Müller Electricity load forecasting 6

7 Some examples Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Metal Processing Industry Metal Forming Technologies Mo Tu We Th Fr Sa Su Connection Technology Mo Tu We Th Fr Sa Su Automotive Parts Supplier Mo Tu We Th Fr Sa Su Retail Mo Tu We Th Fr Sa Su Service Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Electric Steel Mill Paper Mill Kevin Berk and Alfred Müller Electricity load forecasting 7

8 Fundamental model equation Let C t denote the electricity consumption at time t, our model equation has the following form: log C t = D t + u(g t ) + R t Here, D t is a deterministic seasonal pattern, u(gt ) is a function of the residual grid load G t R t is a residual time series. and Kevin Berk and Alfred Müller Electricity load forecasting 8

9 A similar-day approach for D t The deterministic seasonal pattern D t is given by 2πt D t =β 0 + β 1 sin{ } + β 2πt 2 cos{ } + 16 β j ϑ j 2 (t) + 40 j=3 j=17 β j ϱ j 16 (t), where the β j are determined via OLS (using outlier-cleaned historical data). The dummies ϑ j, j = 1,..., 14 are described in table 1 and ϱ 1,..., ϱ 24 denote the hours. Kevin Berk and Alfred Müller Electricity load forecasting 9

10 A similar-day approach for D t ϑ 1 ϑ 2 ϑ 3 ϑ 4 ϑ 5 ϑ 6 ϑ 7 ϑ 8 ϑ 9 ϑ 10 ϑ 11 ϑ 12 ϑ 13 ϑ 14 Table : Overview of dummy variables for D41sincos. Mondays (if not a public holiday or a bridge day) Fridays (if not a public holiday or a bridge day) Tuesdays, Wednesdays and Thursdays (if not a p. hol.) Saturdays (if not a public holiday) Sundays (if not a public holiday) Winter holiday period Summer holiday period Public holidays Bridge day (Monday before a public holiday) Bridge day (Friday after a public holiday) January 1st December 24th December 25th and 26th New Year s Eve Kevin Berk and Alfred Müller Electricity load forecasting 10

11 G t as an exogenous factor denotes the residual grid load, i.e. the total system load minus its deterministic part G t maps the stochastic fluctuations of the market correlation with a customer s electricity demand can be positive or negative the function u(gt ) 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 Kevin Berk and Alfred Müller Electricity load forecasting 11

12 The residual time series R t covers remaining autocorrelations and seasonalities often modeled by a Gaussian white noise process (iid normal distributed) general assumption of independence and light tails is wrong choose a (seasonal) ARIMA model for R t number and choice of parameters depend on business sector R t = log C t D t u(g t ) Jan Kevin FebBerkMar and Alfred AprMüller May Jun Electricity Jul loadaug forecasting Sep Oct Nov Dec 12

13 Model choice for R t BIC suggests a SARIMA(p, 0, q) (P, 0, Q) 24 model 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. innovations are not normally distributed Figure : normal distribution fit for SARIMA model innovations Kevin Berk and Alfred Müller Electricity load forecasting 13

14 Model choice for R t Best fit for innovations given by NIG-distribution 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. Kevin Berk and Alfred Müller Electricity load forecasting 14

15 Medium-term forecasts year ahead forecast scenarios Monte-Carlo approach for retail pricing (see Burger and Müller (2012)) trading strategy evaluation investment decisions (thermal power station, photovoltaic system) peak load management strategic planning Kevin Berk and Alfred Müller Electricity load forecasting 15

16 Real load vs one-year forecast scenarios Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Kevin Berk and Alfred Müller Electricity load forecasting 16

17 Real load vs one-year forecast scenarios Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Kevin Berk and Alfred Müller Electricity load forecasting 16

18 Real load vs expected consumption / forecast scenarios Mo We Fr Su Tu Th Sa Kevin Berk and Alfred Müller Electricity load forecasting 17

19 Real load vs expected consumption / forecast scenarios Mo We Fr Su Tu Th Sa Kevin Berk and Alfred Müller Electricity load forecasting 17

20 Real load vs expected consumption / forecast scenarios Mo We Fr Su Tu Th Sa Mo We Fr Su Tu Th Sa Kevin Berk and Alfred Müller Electricity load forecasting 17

21 different variants of the model: different number of parameters for seasonal patterns monthly dummies, sin-cos, area-preserving splines of order 4 innovations normal, t or NIG Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Figure : Regression coefficients of monthly dummy variables and fitted spline of order four. Kevin Berk and Alfred Müller Electricity load forecasting 18

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) = = 1 0 (F (z) F(z)) 2 dz QS α (F 1 (α), x)dα with QS α (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). Kevin Berk and Alfred Müller Electricity load forecasting 19

23 Data of four companies 2008 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 Regime Kevin Berk and Alfred Müller 2010 Electricity load forecasting

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 (10) 100% (10) 100% D31sincos-Normal (9) % 9.01 (9) % D35-SARIMA-Normal (8) % 8.95 (8) % D35-SARIMA-NIG (7) % 8.89 (5) -37.6% D35spline-SARIMA-NIG (6) % 8.93 (7) % D41sincos-SARIMA-NIG (3) % 8.83 (3) % D41sincos-GL-SARIMA-NIG (1) % 8.72 (1) -38.8% D51-SARIMA-Normal (5) % 8.89 (6) % D51-SARIMA-NIG (4) % 8.86 (4) % D51spline-GL-SARIMA-NIG (2) -27% 8.73 (2) % Kevin Berk and Alfred Müller Electricity load forecasting 21

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 (10) 100% (10) 100% D31sincos-Normal (6) -6.53% (3) % D35-SARIMA-Normal (9) -4.53% (2) % D35-SARIMA-NIG (8) -5.02% (7) % D35spline-SARIMA-NIG (7) -5.36% (6) % D41sincos-SARIMA-NIG (1) % (5) % D41sincos-GL-SARIMA-NIG (2) % (8) -38.3% D51-SARIMA-Normal (3) % (1) % D51-SARIMA-NIG (4) % (4) % D51spline-GL-SARIMA-NIG (5) % (9) % Kevin Berk and Alfred Müller Electricity load forecasting 22

26 Evaluation of probabilistic forecasts Figure : Rank histograms of the empirical PITs for the best ranked models. Top row: customer I and II. Bottom row: customer III and IV. Kevin Berk and Alfred Müller Electricity load forecasting 23

27 switching load profiles Jan'10 Jan'11 Jan'12 Jan'13 Figure : Load profile with stochastic switches in regime (Electric wire manufacturing). two-regime time series different behavior in each regime switches are random but seem to depend on time summer and winter holiday periods are special Kevin Berk and Alfred Müller Electricity load forecasting 24

28 Markov-switching model Idea: Inhomogeneous markov-switching model ARMA (R t ) plus seasonal component (D t ) in upper regime iid rv (Z t ) in lower regime markov-chain with time-varying transition probabilities Let C t be the electricity consumption and S t {0, 1} be the state of regime at time t. Then C t = (D t + R t )S t + Z t (1 S t ). S t can not be observed directly, we rather have to infer its operation through the observed behavior of C t. Kevin Berk and Alfred Müller Electricity load forecasting 25

29 Markov-chain S t The two-point stochastic process S t is described by the transition matrix P(S t = s t S t 1 = s t 1, z t 1 ; β 0 ) [ ] q(zt 1, β = 0 ) 1 q(z t 1, β 0 ) 1 p(z t 1, β 1 ) p(z t 1, β 1 ) where z t is a vector of economic-indicator variables, β = (β 0, β 1 ) is a set of parameters governing the transition probabilities and p, q : R 2k [0, 1] are piecewise continuously differentiable functions (see Filardo (1994)). We use a logistic functional form. Kevin Berk and Alfred Müller Electricity load forecasting 26

30 Parameter estimation MS26-D14-ARMA model 2 13 = 26 parameters for the transition probabilities 14 parameters for D t, i.e. seasonality in upper regime ARMA-parameters for R t two parameters for Z t 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(S t = 0 {C t } T t=1, {z t } T t=1, θ) >.99) Kevin Berk and Alfred Müller Electricity load forecasting 27

31 Simulation results Jan' Jan' Jan'12 Jan' Jan'12 Jan'13 Figure : Test-period time series (top), D51-Normal forecast (middle) and MS26-D14-ARMA forecast (bottom). Kevin Berk and Alfred Müller Electricity load forecasting 28

32 CRPS results conventional models detrending routine performs badly as a result, innovation variance is highly overestimated markov-switching approach improves forecasting performance significantly Table : CRPS of different models for regime switching customer Model CRPS pct. Benchmark (8) 100% D (9) % D51-Normal (3) % D51-SARIMA-Normal (4) % D51-SARIMA-Students (6) -7.71% D51-SARIMA-NIG (2) % D51spline-SARIMA-Students (7) -4.6% D51spline-SARIMA-NIG (5) % MS26-D14-ARMA 8.72 (1) -41% Kevin Berk and Alfred Müller Electricity load forecasting 29

33 modeling of electricity demand via time series analysis CRPS for comparison of probabilistic forecasts and observations more parameters better performance sin/cos approach for seasonality superior to dummies/area-preserving splines inhomogeneous markov-switching model for regime-switching load profiles significant improvement of forecast quality Current research other methods of evaluating probabilistic forecasts improve computation effort for regime model estimation empirical work on load data analysis Kevin Berk and Alfred Müller Electricity load forecasting 30

34 Literature I R. Weron. Modeling and Electricity Loads and Prices. Chichester: Wiley, K. Berk. Modeling and Electricity demand: A Risk Management Perspective. Springer, K. Berk and A. Müller. Probabilistic of Medium-Term Electricity Demand: A Comparison of Time Series. 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, Kevin Berk and Alfred Müller Electricity load forecasting 31

35 Literature II M. Burger, B. Klar, A. Müller, G. Schindlmayr. A spot market model for pricing derivatives in electricity markets. Quantitative Finance, 4: , A.J. Filardo. Business-Cycle Phases and Their Transitional Dynamics. Journal of Business & Economic Statistics, 12: , Kevin Berk and Alfred Müller Electricity load forecasting 32

36 Thank you for your attention! Kevin Berk and Alfred Müller Electricity load forecasting 33

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