Importance of the long-term seasonal component in day-ahead electricity price forecasting: Regression vs. neural network models

Transcription

1 Importance of the long-term seasonal component in day-ahead electricity price forecasting: Regression vs. neural network models Rafa l Weron Department of Operations Research Wroc law University of Science and Technology, Poland Based on a working paper with Grzegorz Marcjasz and Bartosz Uniejewski, available from RePEc: Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 1 / 38

2 Introduction Electricity markets and prices Markets for electricity in Europe Nord Pool (DK, EST, FIN, NOR, SWE) N2EX (UK) Belpex (BE) EPEX Spot (AT,CH, DE, FR) APX-ENDEX (NL) PolPX (PL) OTE (CZ) OKTE (SK) OPCOM (RO) OMIE (ES, PT) GME (IT) HUPX (HU) EXAA (AT) Borzen (SLO) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 2 / 38

3 Introduction Electricity markets and prices... in North America and Australia Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 3 / 38

4 Introduction Electricity markets and prices Electricity price time series Seasonality, mean-reversion and price spikes Daily POLPX spot price [PLN/MWh] Days [ ] Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 4 / 38

5 Introduction Electricity markets and prices The electricity spot (day-ahead) price Day D Day D + 1 Day D + 2 Bidding for day D + 1 Bidding for day D hours of day D hours of day D + 2 Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 5 / 38

6 Introduction Electricity markets and prices Supply and demand, renewables and negative prices Source: Ziel & Steinert (2016) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 6 / 38

7 Introduction Electricity markets and prices Prices for different load periods Strongly correlated but seem to follow different data generating processes (DGPs) Load period 6 (2:30 3:00) Load period 36 (17:30 18:00) Log price Nov Mar Aug Jan 2013 Time Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 7 / 38

8 Introduction Electricity markets and prices First read on electricity price forecasting (EPF) R.Hyndman: this paper alone is responsible for 0.7 of the current IF 2Y =2.642 ;-) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 8 / 38

9 Introduction Motivation A look into the future of EPF EPF directions in the next decade (according to Weron, 2014, IJF): 1 Modeling and forecasting the trend-seasonal components 2 Beyond point forecasts probabilistic forecasts 3 Combining forecasts 4 Multivariate factor models 5 Guidelines for evaluating forecasts Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 9 / 38

10 Introduction Motivation Role of the long-term seasonal component (LTSC) for short-term EPF Significant prediction accuracy gains possible for linear regression models (Nowotarski & Weron, 2016, ENEECO): Unknown effects for non-linear (e.g., ANN) models Is this phenomenon more general? Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 10 / 38

11 Agenda Introduction Electricity markets and prices Motivation Trend-seasonal components Wavelets The Hodrick-Prescott (HP) filter Case study Datasets and LTSCs ARX and SCARX models ANNs in EPF Committee machines of (SC)ANN networks Results and conclusions Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 11 / 38

12 Trend-seasonal components Wavelets Wavelets Decomposition of a signal 1000 Original signal Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 12 / 38

13 Trend-seasonal components Wavelets Wavelets Decomposition of a signal 1000 Original signal 1000 Approximation 1 level S Details 1 level D Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 12 / 38

14 Trend-seasonal components Wavelets Wavelets Decomposition of a signal 1000 Original signal 1000 Approximation 1 level S Approximation 7 level S Details 1 level D Details 7 level D Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 12 / 38

15 Trend-seasonal components Wavelets Sample fits to Nord Pool data 4.5 log(price) LTSC S 5 LTSC S Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 13 / 38

16 Trend-seasonal components HP-filter The Hodrick-Prescott (1980, 1997) filter A simple alternative to wavelets Originally proposed for decomposing GDP into a long-term growth component and a cyclical component Returns a smoothed series τ t for a noisy input series y t : { T } T 1 [ ] 2 min (y t τ t ) 2 + λ (τ t+1 τ t ) (τ t τ t 1 ), τ t t=1 t=2 Punish for: deviating from the original series roughness of the smoothed series Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 14 / 38

17 Trend-seasonal components HP-filter Sample fits to EEX and PJM data (Weron & Zator, 2015, ENEECO) Price [EUR/MWh] EEX spot price HP, λ =5x10 4 HP, λ =5x10 5 HP, λ =5x10 7 Price [USD/MWh] Days ( ) PJM spot price HP, λ = 5x10 4 HP, λ = 5x10 5 HP, λ = 5x Days ( ) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 15 / 38

18 Agenda Introduction Electricity markets and prices Motivation Trend-seasonal components Wavelets The Hodrick-Prescott (HP) filter Case study Datasets and LTSCs ARX and SCARX models ANNs in EPF Committee machines of (SC)ANN networks Results and conclusions Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 16 / 38

19 Datasets and LTSCs Datasets: GEFCom 2014 LMP [USD/MWh] GEFCom2014 data ( , hour , hour 24) Initial calibration period Test period System load [GWh] Datasets are the same as in Nowotarski & Weron (2016, ENEECO) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 17 / 38

20 Datasets and LTSCs Datasets: Nord Pool Price [EUR/MWh] Nord Pool data ( , hour , hour 24) Initial calibration period Test period Consumption [GWh] Datasets are the same as in Nowotarski & Weron (2016, ENEECO) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 18 / 38

21 Datasets and LTSCs Long-Term Seasonal Components (LTSCs) Like in Nowotarski & Weron (2016, ENEECO), we consider 18 LTSCs from two categories: Wavelet filters S 5, S 6,..., S 14, ranging from daily smoothing (S hours) up to biannual (S hours) Models with wavelet filters are denoted by suffixes -S J HP-filters with λ = 10 8, , 10 9,..., , also ranging from daily up to biannual smoothing Models with HP filters are denoted by suffixes -HP λ Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 19 / 38

22 ARX and SCARX models Benchmark: The ARX model For the log-price, i.e., p d,h = log(p d,h ), the model is given by: p d,h = β h,1 p d 1,h + β h,2 p d 2,h + β h,3 p }{{ d 7,h + β } h,4 p }{{ d 1,min } autoregressive effects non-linear effect + β h,5 z }{{} t + 3 β h,i+5d i +ε d,h (1) i=1 }{{} load weekday dummies p d 1,min is yesterday s minimum hourly price z t is the logarithm of system load/consumption Dummy variables D 1, D 2 and D 3 refer to Monday, Saturday and Sunday, respectively Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 20 / 38

23 ARX and SCARX models The SCAR modeling framework (Nowotarski & Weron, 2016, ENEECO) The Seasonal Component AutoRegressive (SCAR) modeling framework consists of the following steps: 1 (a) Decompose the series in the calibration window into the LTSC T d,h and the stochastic component q d,h (b) Decompose the exogenous series in the calibration window using the same type of LTSC as for prices 2 Calibrate the ARX model to q t and compute forecasts for the 24 hours of the next day (24 separate series) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 21 / 38

24 ARX and SCARX models The SCAR modeling framework cont log(price) LTSC LTSC forecast Add stochastic component forecasts ˆq d+1,h to persistent forecasts ˆT d+1,h of the LTSC to yield log-price forecasts ˆp d+1,h 4 Convert them into price forecasts of the SCARX model, i.e., ˆP d+1,h = exp (ˆp d+1,h ) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 22 / 38

25 ARX and SCARX models Sample LTSC and stochastic component forecasts log(price) LTSC S Stoch. comp. Forecast Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 23 / 38

26 ANNs in EPF ANNs in other EPF studies Variety of ANN implementations, as well as considered inputs, making it impossible to compare with commonly used methods based on linear regression Several studies that acknowledge the need of removing seasonal components from time series for neural network models: Andrawis et al. (2011) Zhang and Qi (2005) Keles et al. (2016), the only one in the context of EPF Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 24 / 38

27 ANNs in EPF ANN: Based on Matlab s NARXnet x(t) 5 y(t) 1 0 W W Hidden b 5 W b Output 1 y(t+1) 1 One hidden layer with 5 neurons and sigmoid activation functions Inputs identical as in the ARX model Trained using Matlab s trainlm function, utilizing the Levenberg-Marquardt algorithm for supervised learning Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 25 / 38

28 ANNs in EPF Seasonal Component ANN (SCANN) The SCANN modeling framework is a generalization of the ANN model, analogous to the SCAR framework for the ARX model: 1 (a) Decompose the series in the calibration window into the LTSC T d,h and the stochastic component q d,h (b) Decompose the exogenous series in the calibration window using the same type of LTSC as for prices 2 Calibrate the ANN model to q t and compute forecasts for the 24 hours of the next day (24 separate series) 3 Add stochastic component forecasts ˆq d+1,h to persistent forecasts ˆT d+1,h of the LTSC to yield log-price forecasts ˆp d+1,h 4 Convert them into price forecasts of the SCANN model, i.e., ˆP d+1,h = exp (ˆp d+1,h ) Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 26 / 38

29 ANNs in EPF Number of hidden neurons Average WMAE (in %) GEFCom: ANN 5 (SC)ANN 5 (SC)ARX #hidden neurons GEFCom: SCANN 5 -HP 1e #hidden neurons NP: ANN #hidden neurons NP: SCANN 5 -S #hidden neurons There is no universally optimal number, but the errors are smallest for 4 to 6 neurons in the hidden layer Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 27 / 38

30 Committee machines of (SC)ANN networks Committee machines of (SC)ANN networks Every forecast yields slightly different results two model categories are considered: ANN 1 the expected result for a single ANN network, an average of error scores across separate runs ANN 5 a forecast average of 5 runs (hour-by-hour) with identical parameters, a so-called committee machine Analogously: SCANN 1 the expected result for a single SCANN network SCANN 5 a committee machine of 5 SCANNs Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 28 / 38

31 Committee machines of (SC)ANN networks Committee machines of (SC)ANN networks Real price Single run Forecast Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 29 / 38

32 Committee machines of (SC)ANN networks Sample gains from using committee machines Average WMAE (in %) GEFCom: ANN n (SC)ANN n (SC)ARX n GEFCom: SCANN n -HP 1e n NP: ANN n n NP: SCANN n -S n Forecast errors roughly scale as a power-law function of the number of networks in a committee machine We should use as large committee machines as we can... Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 30 / 38

33 Committee machines of (SC)ANN networks Sample gains cont.... however, the time needed may be substantial, e.g., for generating forecasts for the next 24 hours: Model ARX SCARX-HP 10 8 SCARX-S 9 ANN 1 ANN 5 Time 8.6ms 13.5ms 37.3ms 7.6s 38.2s SCANN times are omitted here, because LTSC computation is negligible compared to training the ANN Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 31 / 38

34 Results Weekly-weighted Mean Absolute Error (WMAE) Following Conejo et al. (2005), Weron & Misiorek (2008) and Nowotarski et al. (2014), among others, we use: WMAE w = 1 P 168 MAE w = P 168 where P 168 = 1 Sun d=mon h=1 P d,h Sun 24 d=mon h=1 WMAE = 1 w max WMAE w w max w=1 where w max = 103 for GEFCom and 104 for Nord Pool P d,h ˆP d,h Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 32 / 38

35 Results Average WMAE for GEFCom2014 Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 33 / 38

36 Results Average WMAE for Nord Pool Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 34 / 38

37 Results Aggregate results of SCANN performance GEFCom2014 Average WMAE (in %) SCANN 5 with Step 1(b) SCANN 5 w/o Step 1(b) ANN 5 ( ARX) Best SCARX with Step 1(b) S 5 S 6 S 7 S 8 S 9 S 10 S 11 S 12 S 13 S 14 HP 1e8 HP 5e8 HP 1e9 HP 5e9 HP 1e10 HP 5e10 HP 1e11 HP 5e11 Average WMAE (in %) Nord Pool S 5 S 6 S 7 S 8 S 9 S 10 S 11 S 12 S 13 S 14 HP 1e8 HP 5e8 HP 1e9 HP 5e9 HP 1e10 HP 5e10 HP 1e11 HP 5e11 Note: Step 1(b) is important (green vs. yellow)! Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 35 / 38

38 Results The Diebold-Mariano test (1995) We define the error function as 24 L(ε d ) = ε d 1 = P d,h ˆP d,h h=1 For each pair of models we compute the loss differential D d = L(ε model X d ) L(ε model Y d ) Hypothesis H 0 :E(D d ) 0, model X outperforms model Y Reversed hypothesis H R 0 :E(D d ) 0, model Y outperforms model X Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 36 / 38

39 Results Diebold-Mariano test results Naive ANN 1 1 : over 24h, GEFCom2014 SCANN 1 -S 9 SCANN -HP 1 1e9 ARX SCARX-S 9 SCARX-HP 1e9 ANN 5 SCANN -S 5 9 SCANN 5 -HP 1e Naive ANN 1 1 : over 24h, Nord Pool SCANN 1 -S 10 SCANN -HP 1 5e9 ARX SCARX-S 9 SCARX-HP 1e8 ANN 5 SCANN -S 5 10 SCANN 5 -HP 5e Naive ANN 1 SCANN 1 -S 9 SCANN 1 -HP 1e9 ARX SCARX-S 9 SCARX-HP 1e9 ANN 5 SCANN -S 5 9 SCANN 5 -HP 1e9 Naive ANN 1 SCANN 1 -S 10 SCANN -HP 1 5e9 ARX SCARX-S 9 SCARX-HP 1e8 ANN 5 SCANN -S 5 10 SCANN 5 -HP 5e9 Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 37 / 38

40 Conclusions Conclusions Using Seasonal Component ANN (SCANN) models can yield statistically significant improvement over the ANN benchmark SCANN 5 returns % lower WMAE than ANN 5 The accuracy gains from using LTSC are greater in ANN models than in regression models SCARX models yield only a % improvement in WMAE vs. the ARX benchmark Forecast averaging is crucial in outperforming the SCARX model SCANN 5 yields % lower WMAE Rafa l Weron (Wroc law, PL) Seasonal Component EPF models , Uni Bolzano 38 / 38