Statistical Approaches to Electricity Price Forecasting

Transcription

1 Saisical Approaches o Elecriciy Price Forecasing By J. Suar McMenamin, Ph.D., Frank A. Monfore, Ph.D. Chrisine Fordham, Eric Fox, Fredrick D. Sebold Ph.D., and Mark Quan 1. Inroducion Wih he adven of compeiion, hourly elecriciy prices are being deermined by a variey of marke mechanisms, raher han cos-based engineering calculaions. As a resul, elecric uiliies, generaors, and raders face a new se of shor-erm forecasing problems. These problems are unlike hose in oher indusries, since elecriciy mus be produced a he same ime ha i is consumed. As a resul prices are deermined on an hourly basis, 24 hours a day, 7 days a week. Hisorically, price forecasing has been performed wih leas-cos opimizaion models. These models compue marginal cos based on assumpions abou sysem loads, power plan availabiliy, and fuel prices. These models do no explain price variaions relaed o marke sraegy and o buyer and seller behaviour in a marke sysem. Saisical models, which by heir naure reflec acual marke oucomes, are beer suied o shor-erm forecasing in his dynamic environmen. In his paper, a variey of modeling approaches are applied and evaluaed for forecasing elecriciy prices. Mehods include ime-series models, regression models, and arificial neural nework models. The paper discusses he naure of he price-forecasing problem and idenifies reasons why flexible approaches, such as neural nework models are well suied o his applicaion. For each approach, model esimaes and forecass are developed using hourly price daa for he PJM (Pennsylvania, New Jersey and Maryland) power pool area. The modeling approaches are compared based on accuracy for day-ahead forecasing. 2. Price Modeling Approaches In compeiive elecriciy markes, he marke-clearing price is he hourly bid of he las generaion uni o mee sysem demand. In mos sysems, all suppliers are paid he hourly marke-clearing price. In a perfecly compeiive marke, he marke-clearing price will be equal o he marginal cos of he las supplier. Bu exising elecriciy markes are far from meeing he requiremens for perfec compeiion. Reflecing he fac ha here are a limied number of suppliers and ha cusomer demand is highly inelasic wih respec o he marke price, here is significan room for exploiaion of marke power, especially in periods of high demand. There are wo approaches being used o forecas marke prices. The firs is a simulaion mehod based on models of producion cos. The second involves applicaion of saisical mehods o hisorical marke daa. The producion cos mehod involves a simulaion of plan dispach and iner-pool exchanges o mee hourly demands. These mehods ypically assume ha plans are dispached based on lowes running cos 1

2 of he nex available generaion uni, subjec o operaing and ransmission consrains. This approach requires deailed daa and assumpions abou invenory of generaion plan, including operaing capabiliies, operaing coss, and geographic locaion wih respec o ransmission faciliies. While hese models have proven o be very useful for assessing long-erm marke opions, hey are no well suied o he modeling of bidding sraegies in a marke seing. In conras, saisical mehods relae marke prices o observed facors ha are believed o impac prices. These facors can include boh demand side and supply side variables, and a variey of model specificaions and echniques are available. Since bidding sraegies are embedded in he observed marke oucomes, hese mehods will work well as long as sraegies, consrains, and marke rules remain sable or evolve slowly. In wha follows, we look a several approaches o shor-erm saisical modeling using daa for he PJM marke. Daa The dependen variable daa is he average on-peak price in he PJM marke. The on-peak period is defined o be he hours beween 8 am and 11 PM, which is a 16-hour block. Daa values are available from April 1998 hrough he presen. The PJM marke is currenly in ransiion from a power pool using leas-cos dispach o a compeiive marke based on generaor bidding. A his poin PJM prices sill reflec dispach coss more han compeior bidding. Sill hese prices pose a significan modeling challenge, and i is reasonable o believe ha mehods ha work well wih hese daa will also work well in a full bidding conex. Explanaory variables fall ino hree caegories. Firs, from a ime-series perspecive, here is he hisory of he marke price iself. In a day-ahead marke, lagged price daa ofen have high explanaory power, alhough he paern of weekdays, weekends, and seasons inroduces some ineresing modeling problems for ime-series models. The second se of explanaory facors is demand-side facors. Hourly elecriciy use reflecs he life paerns of people, mechanical sysems ineracing wih weaher, cloud cover, iming of sunrise and sunse, waer emperaures, and oher similar facors. Because sysem load can ypically be modeled and forecased wih high accuracy, we proceed here using he acual demand levels as an explanaory variable, raher han he indirec variables for weaher and calendar effecs. The hird se of explanaory facors is supply-side facors. These facors include fuel prices, generaion uni availabiliy, ransmission consrains, and in some markes, hydro flows. Also, in periods of high demand he load levels in surrounding areas can have a significan impac on local prices, reflecing he high price of impored energy and he high opporuniy cos of bidding ino he local marke. The supplyside variables included here are nuclear capaciy on-line and naural gas prices. The daa are presened below. Figure 1 shows he PJM average on-peak price. Over he hisorical period, he mean value is abou $25 per MWh, and mos observaions are beween $15 and $30. On several days, however, he price shows a significan and shor-lived spike, wih hourly values nearing $1000, bringing he average on-peak price for he day above $100 on occasion. (In he numbers shown here, he hourly price has been capped a $200 before compuing he daily average). 2

3 $120 $100 $80 $60 $40 $20 $0 Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Figure 1. PJM Average On-Peak Price ($/MWh), April 98 o May 99 Figure 2 shows he corresponding values for on-peak energy use. These daa have an average value of abou 500 billion Wahours (GWh), implying an average load of abou 31 GW during on-peak hours. The daa show a srong weakly cycle as well as a weaher-driven seasonal cycle Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Figure 2. PJM On-Peak Energy Demand (GWh), April 98 o May 99 3

4 Figure 3 shows available nuclear capaciy measured in million was (MW). The average value is abou 11.5 GW, wih a maximum value of abou 14 GW. Uni availabiliy reflecs boh planned and unplanned ouages. 14,000 12,000 10,000 8,000 6,000 4,000 2,000 0 Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Figure 3. Available Nuclear Capaciy (MW), April 98 o May 99 Finally, Figure 4 shows naural gas prices a he Henry Hub. The average price over his period was abou $2.00 per mmbu, which would ranslae o a marginal fuel cos of abou $20 per MWh a a hea rae of 10,000 Bu/KWh. 4

5 $3.00 $2.50 $2.00 $1.50 $1.00 $0.50 $0.00 Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Figure 4. Gas Prices a Henry Hub ($/mmbu), April 98 o May 99 Comparing Figures 1 and 2, i is clear ha high prices occur in periods of high demand. The relaionship is no perfec, however, as shown in Figure 5. This figure provides a scaer plo of on-peak prices versus on-peak loads, coded by ype of day. The figure shows ha all of he high load and high price days are weekdays. However, no all high-load days have high prices. For example, on days wih on-peak energy near 700 GWh, he average on-peak price ranges from $40 o more han $100. Despie his wide dispersion, he char does sugges ha he relaionship beween loads and price is nonlinear. 5

6 140 Average On-Peak Price $/MWh Weekdays Saurdays Sundays Holidays On Peak Energy (GWh) Figure 5: Scaer plo of On-Peak Price vs. Energy 4. Model Specificaions A series of models is esimaed using hese daa. We begin wih an exponenial smoohing model and an ARIMA model. These daa-driven models do no ake advanage of he demand and supply-side variables. Nex a linear regression model is esimaed. Finally, a neural nework exension of he linear model is esimaed o capure key nonlineariies and ineracions. The forecasing equaion for a one-day-ahead forecas wih an exponenial smoohing model is as follows. P ( Level + Trend ) DayMul = (1) Where P is he value of on-peak prices, is he curren period, and Level, Trend, and DayMul are variables generaed by he smoohing process. The smoohing equaions use he muliplicaive seasonal form, someimes referred o as he Hol-Winers mehod. In his applicaion, he seasonal lag is se o seven, so ha he DayMul variables poin o he same day in he previous week. Wih his modificaion, he smoohing equaions are as follows: Level Trend P = a + DayMul (2) 7 P DayMul 1 1 ( 1 a) ( Level + Trend ) 1 = b Level + (3) 7 1 ( 1 b) ( Trend ) P 7 DayMul = c ( 1 c) DayMul Level + (4) 6

7 The forecasing equaion for a one-day-ahead forecas wih a seasonal ARIMA model is as follows: P = SARIMA(p, d, q)(sp,sd,sq) (5) Where SARIMA represens a seasonal auoregressive (AR) moving average (MA) process wih auoregressive erms of order p and sp, moving average erms of order q and sq, differencing of order d, and seasonal differencing of order sd. In his applicaion, he seasonal periodiciy is seven days, implying ha seasonal lags poin o he same day in he preceding week. The linear regression model is as follows: P + = c0+ c j X j e (6) j Where X represens a se of demand and supply-side explanaory variables. The exension of he regression model o include nonlinear nodes from a neural nework model can hen be represened as: P = c + 0 N c j X j + Bn + u K j n 1 an,0 a X (7) = + n, k k k= 1 1+ e 1 The firs summaion in (7) repeas he linear regression from equaion (6). The second summaion includes a se of N nonlinear nodes from a simple feedforward arificial neural nework. The specific form uses logisic ransformaion funcions, which can capure a variey of nonlinear responses. By consrucion, he variables included in a node are muliplicaively ineracive if hey have nonzero parameers in he sum ha appears in he logisic exponen. The neural nework approach and his specific funcional form are widely used in day-ahead forecasing of sysem loads. The approach is well suied o his problem because he response of sysem load o weaher is nonlinear and because here are significan ineracions among explanaory variables. (For a complee discussion of he neural nework funcional form, see McMenamin and Monfore, 1998). As seen in Figure 5, he relaionship beween sysem load and prices also appears o be nonlinear. Furher, here may be imporan ineracions beween demand and supply variables ha help explain daily variaions in price. The neural nework equaion provides a simple way o allow nonlineariies and ineracions in he model wihou imposing resricive assumpions abou he srucure of he relaionship. 5. Esimaion Resuls The firs model esimaed is an exponenial smoohing model using a Hol-Winers muliplicaive form wih rend and seasonal elemens. As discussed above, he periodiciy of he seasonal erm is se o 7 days for his applicaion wih daily daa. The resuls are summarized below. This naïve model will serve as a reference poin. The mean absolue percen error (MAPE) is 22.6% and he mean absolue deviaion is abou $6 per MWh. These saisics reflec he accuracy of he model for purposes of day-ahead forecasing. 7

8 Exponenial Smoohing Model Summary Simple Smoohing Parameer.619 Trend Parameer Seasonal Parameer.321 Adjused Observaions 397 Deg. of Freedom for Error 394 Adjused R-Squared Sd. Error of Regression 9.76 Mean Abs. Dev. (MAD) 6.02 Mean Abs. % Err. (MAPE) 22.57% Durbin-Wason Saisic Table 1: Exponenial Smoohing Summary The second model esimaed is an ARIMA model. Afer examinaion of ime-series diagnosics and experimenaion wih various forms, he final model is a (0,1,4) (1,0,0), implying ha he daa are differenced, and a model is fi wih an MA4 and a seasonal AR1. As wih he smoohing model, he seasonal periodiciy is se o 7 days. The coefficiens and summary saisics for his model are presened in Table 2. As is eviden in Table 2, he ARIMA model provides only a modes improvemen wih respec o day-ahead accuracy, wih a MAPE of 22% and a MAD of $5.8 per MWh. Variable Coefficien SdErr T-Sa CONST SAR(1) MA(1) MA(2) MA(3) MA(4) Summary Saisics Adjused Observaions 391 Adjused R-Squared AIC BIC Sd. Error of Regression 9.55 Mean Abs. Dev. (MAD) 5.78 Mean Abs. % Err (MAPE) 22.03% Durbin-Wason Saisic Table 2: ARIMA (0, 1, 4) (1, 0, 0) Summary The regression model uses a combinaion of lagged dependen variables and explanaory variables. The hree explanaory variables are on-peak energy use (OnPeakEnergy), nuclear availabiliy (NukeAvail), and he price of naural gas (HHPrice). As shown in Table 3, his specificaion improves he MAPE value o 19.5% and reduces he MAD o abou $5 per MWh. The final specificaion inroduces wo nonlinear nodes o he linear model presened above. The firs node includes only on-peak energy as an inpu variable. This node will allow represenaion of nonlinear effecs, o he exen hese effecs are presen. The second node includes he wo supply-side variables, gas prices and nuclear availabiliy, allowing he modeling of nonlineariies and ineracions wih respec 8

9 o hese variables. The exended model is esimaed using nonlinear leas squares applied o normalized daa. Variable Coefficien SdErr T-Sa CONST Saurday WkDay Lag1OnPk Lag2OnPk Lag3OnPk Lag4OnPk Lag5OnPk Lag6OnPk Lag7OnPk HHPrice NukeAvail OnPeakEnergy Summary Saisics Adjused Observaions 392 Adjused R-Squared Durbin-Wason Saisic AIC BIC Sd. Error of Regression 8.18 Mean Abs. Dev. (MAD) 5.06 Mean Abs. % Err. (MAPE) 19.52% Table 3: Regression Model Resuls As shown in Table 4, his specificaion provides a furher improvemen in model accuracy, wih dayahead MAPE values dropping o 17% and MAD values dropping o $4.5 per MWh. The acual and prediced values are presened in Figure 6. As is eviden, he model does no fully predic he price spike values ha occurred in he summer of This is no surprising given he price dispersion ha is eviden for high load levels in he scaer plo in Figure 5. Oherwise, however, he day-ahead model racks acual oucomes and changes in price fairly well. Comparing he auoregressive erms, he coefficien on he one-day lag of price (Lag1OnPk) is abou half he level in he neural nework model as i is in he regression model. This indicaes ha he neural nework model places higher reliance on he explanaory variables and lessor reliance on he ime-series properies of he residuals. 9

10 Coefficien Value SdErr T-Sa Linear: Inercep Linear: Saurday Linear: WkDay Linear: Lag1OnPk Linear: Lag2OnPk Linear: Lag3OnPk Linear: Lag4OnPk Linear: Lag5OnPk Linear: Lag6OnPk Linear: Lag7OnPk Linear: HHPrice Linear: NukeAvail Linear: OnPeakEnergy Node1: Slope Node1: Bias Node1: OnPeakEnergy Node2: Slope Node2: Bias Node2: NukeAvail Node2: HHPrice Summary Saisics Adjused Observaions 392 Adjused R-Squared AIC BIC Sd. Error of Regression 7.23 Mean Abs. Dev. (MAD) 4.49 Mean Abs. % Err. (MAPE) 17.15% Durbin-Wason Saisic Table 4: Neural Nework Model Resuls 10

11 On-Peak Price ($/MWh) Prediced Acual 20 0 Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Figure 6: Acual and Prediced Values Neural Nework Model To undersand he role of he pars of he neural nework model, he conribuion of each componen (he linear componen and he wo nonlinear nodes) o he oal prediced value is presened in Figure 7, Figure 8, and Figure 9. The linear node appears o accoun for mos of he seasonal variaion and also capures weekly cycles. The firs nonlinear node fires only under price-spike condiions. The second nonlinear node is driven mosly by naural gas prices. This node has he greaes conribuion when gas prices are high and nuclear capaciy is low, as was he case in boh he spring of 98 and he spring of

12 Toal Prediced Value Linear Terms Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Figure 7. Conribuion of Linear Terms o Prediced Value Toal Prediced Value Node1 Conribuion Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Figure 8. Conribuion of Node 1 (OnPeakEnergy) o Prediced Value 12

13 Toal Prediced Value Node2 Conribuion Apr May Jun Jul Aug Sep Oc Nov Dec Jan Feb Mar Apr May Figure 9. Conribuion of Node 2 (Supply Terms) o Prediced Value 6. Conclusion As compeiive elecriciy markes evolve, ineres in price forecasing will increase significanly. Whereas under he regulaory compac, uiliies are responsible for pruden planning and are guaraneed a reasonable rae of reurn on invesmen decisions, in a compeiive generaion marke, profis will be deermined by he relaionship beween price and cos. Owners of generaion asses will wan he bes possible price forecass o make he bes decisions abou conracing and bidding sraegies. Reailers will wan he bes possible price forecass o develop sraegies for covering he loads of heir cusomers. And uiliies, which may be boh owners of generaion and reailers, will be ineresed in price forecasing for purposes of rading and risk managemen. As he analysis above suggess, elecriciy price forecasing is a significan challenge. Price variaion is significan on a day-o-day basis, and prices are even more volaile on an hourly basis. The analysis suggess ha more advanced modeling mehods will produce beer forecass. Moving from naïve mehods o advanced mehods, such as neural neworks reduces he day-ahead forecasing error from abou $6 per MWh o $4.5 per MWh. By developing beer daa abou supply side facors, i is reasonable o expec ha furher improvemens can be made. Because of he nonlinear and ineracive naure of he price responses, his is a good applicaion for neural nework approaches, which provide a flexible nonlinear mehod. References 1. Azoff, E. M. Neural Nework Time Series Forecasing of Financial Markes, John Wiley & Sons, Kuan, C and H Whie, Arificial Neural Neworks, An Economeric Perspecive, UCSD Discussion Paper, June McMenamin, J. S. and F. A. Monfore, Shor-erm Energy Forecasing Wih Neural Neworks, Energy Journal, Volume 19, Number 4, Whie, H, Neural-Nework Learning and Saisics, AI Exper, December,

14 Auhor Biographies J. Suar McMenamin, Ph.D. Vice Presiden a Iron, where he specializes in he fields of energy economics, saisical modeling, and sofware developmen. Over he las 20 years, he has managed numerous projecs in he areas of sysem load forecasing, price forecasing, reail load forecasing, enduse modeling, regional modeling, load shape developmen, and uiliy daa analysis. In addiion o direcing large analysis projecs, Dr. McMenamin direcs he developmen of Iron s forecasing sofware producs. He has also direced he developmen of sofware packages for exernal cliens, including he EPRI end-use models, regional economic forecasing packages, and home energy raing sofware. Dr. McMenamin received his B.A. in Mahemaics and Economics from Occidenal College and his Ph.D. in Economics from UCSD. Frank A. Monfore, Ph.D. is Vice Presiden of Forecasing a Iron, where he is a leading auhoriy in he areas of shor-erm load forecasing, load profiling, reail scheduling, end-use forecasing, and saisical and mahemaical modeling. Dr. Monfore direcs he developmen, suppor, and implemenaion of Iron s forecasing and load profiling ools. In addiion o his forecasing responsibiliies, he is a naionally recognized auhoriy in he area of indusrial end-use analysis. Dr. Monfore has co-auhored award-winning publicaions on a range of problems including he use of neural neworks for shor-erm load forecasing, long-erm end-use forecasing, and he use of nonlinear programming echniques for developmen of a leas cos gas supply planning ool. Dr. Monfore received his B.A. in Economics from he Universiy of California, Berkeley and his Ph.D. in Economics from he Universiy of California, San Diego. Chrisine Fordham is a Senior Consulan in he Forecasing Division, where she specializes in design and implemenaion of leading-edge forecasing sysems. Ms. Fordham has exensive experience wih implemening forecasing sysems using enerprise relaional daabases including Oracle 8i and 9i and Microsof SQL Server. Ms. Fordham has implemened sysems o forecas sysem loads, ransmission zone loads, marke prices, reail profiles, and uiliy or provider of las resor loads for uiliies in he Unied Saes and Canada. She has developed shor-erm gas forecasing sysems for gas companies in Europe and he Unied Saes. She has also implemened long-erm forecasing sysems using end-use models for numerous uiliies. Ms. Fordham received her B.S. in Economics from he Massachuses Insiue of Technology in Eric Fox is a Vice Presiden of Forecasing a Iron, where he direcs forecasing and energy analysis projecs and manages Iron's Norheas office. Mr. Fox has over 15 years of forecasing experience wih exensive experise in elecric and gas demand forecasing. Mr. Fox is one of Iron's primary insrucors. He provides forecas raining hrough workshops sponsored by Iron, uiliy onsie raining programs and workshops held by oher organizaions including EPRI and he Insiue of Business Forecasing. Mr. Fox has also provided esimony and direced regulaory workshops relaed o forecasing and rae design issues. Mr. Fox received his B.A. and M.A. in Economics from San Diego Sae Universiy. Frederick D. Sebold, Ph.D. Vice Presiden a Iron. Dr. Sebold has been acively engaged in he energy indusry for over weny years. His chief areas of experise are energy demand analysis, energy efficiency program planning and analysis, economerics, and cos-benefi analysis. His research background includes a wide range of consuling aciviies for sae and local governmen as well as privae indusry. Dr. Sebold has direced numerous end-use analysis projecs in he residenial and commercial secors. These projecs have encompassed boh daa collecion and analysis, and have enailed he use of a wide range of analyical echniques including building simulaions and economeric approaches like SAE and condiional demand analysis. Dr. Sebold has also direced a wide range of sudies in he area of energy efficiency. These projecs have included impac and process evaluaion of resource acquisiion programs, 14

15 marke assessmens and marke effecs analyses of marke ransformaion programs, and assessmens of demand-side managemen poenial. Dr. Sebold has also played a cenral role in he design of muli-uiliy energy efficiency programs. Dr. Sebold received his B.A. in Economics from Sain Vincen College, and his M.A. and Ph.D. in Economics from Boson College. Mark Quan is a Senior Forecas Analys wih Iron's Forecasing Division. Since joining Iron in 1997, Mr. Quan has specialized in energy forecasing and mahemaical modeling. Mr. Quan has developed and implemened several auomaed forecasing sysems o predic nex day sysem demand, load profiles, and reail consumpion for companies hroughou he Unied Saes and Canada. Mr. Quan conducs several of Iron's forecasing workshops and cusomized raining sessions. His breadh of forecasing experience in boh regulaed and deregulaed markes makes Mr. Quan ideally suied o eaching pracical soluions o difficul forecasing problems. Mr. Quan received an M.S. in Operaions Research from Sanford Universiy and a B.S. in Applied Mahemaics from he Universiy of California a Los Angeles. Iron Inc. Iron is a leading echnology provider and criical source of knowledge o he global energy and waer indusries. Nearly 3,000 uiliies worldwide rely on Iron echnology o deliver he knowledge hey require o opimize he delivery and use of energy and waer. Iron delivers value o is cliens by providing indusry-leading soluions for meer daa collecion, energy informaion managemen, demand response, load forecasing, analysis and consuling services, ransmission and disribuion sysem design and opimizaion, web-based workforce auomaion, C&I cusomer care, as well as enerprise and residenial energy managemen. To know more, sar here: Iron Inc. Corporae Headquarers 2111 Norh Moler Road Libery Lake, Washingon, U.S.A. Phone: Fax: Iron Inc. Energy Forecasing - Wes El Camino Real San Diego, California, U.S.A Phone: Fax: Iron Inc. Energy Forecasing - Eas 20 Park Plaza, Suie 910 Boson, Massachuses Phone: Fax: W-02 12/06