An electricity price model with consideration to load and gas price effects

http:/www.zju.educn/jzus;E-mail:jzus@zju.edu.cnISSN 1009-3095 Joumal of Zhejiang University SCIENCE V 4 No 6 P 666-671 Nov.-Dec., 2003An electricity price model with considerationto load and gas price effectsHUANG Min- xiang黄民翔 )i TAO Xiao-hu(陶小虎)han Zhen-xang韩祯祥)College of Electrical Engineering, Zhejiang University, Hangzhou 310027, ChinaTE-mailhuangmx@zju.edu.cnReceived May 10 2002; revision accepted Apr 19 22003Abstract: Some characteristics of the electricity load and prices are studied and the relationship betweenlectricity prices and gas( fuel prices is analyzed in this paper. Because electricity prices are strongly depen-dent on load and gas prices the authors constructed a model for electricity prices based on the effects of thesetwo factors i and used the Geometric Mean Reversion Brownian Motion( GMRBM)model to describe the electricity load process and a Geometric Brownian Motion( GBM ) model to describe the gas prices deduced theprice stochastic process model based on the above load model and gas price model. This paper also presentsmethods for parameters estimation and proposes some methods to solve the modelKey words: Electricity market, Stochastic process, Electricity price GasDocument code: ACLC number: TM73 F123INTRODUCTIONparameters models based on the general BlackScholes model and only considered the effects ofIn an electricity market the electricity pric- historical prices and used historical price data toes have important impact on such diverse issues construct the electricity price process model in aas asset pricing contracting planning, and method like that for constructing the stock marchoice of operation policies for generation and ket price model. Because thee non -stotransmission of electricity. Analysis of the his- transmission limitation of electricty the electrictorical electricity prices suggested that it is very ity market displays different behaviors from thatdifficult to model the behaviour of electricity of the commodities market. There are many facprices because they are highly volatile. From the tors which significantly affect the electricity pricecurve of the historical price we can see that there behaviour such as the load behaviour differentare many price"spikes"( price behaviour defined time periods( hour, day week and year ) difas suddenly upward or downward movement ) ferent regions, the gas price behaviour and theConsidering that the number of price spikes in generation and transmission capacitiesne year is not negligible we must take them inThe electricity market deregulation processesto consideration when we model the price behav- had been carried out for a short time only andiours. Deng( 2000) described three types of due to shortage of historical electricty pricesmean-reversion jump-diffusion models for model- there are few papers reporting on resulting energy commodity spot prices with jumps and ysis on the electricity price behaviours based onspikes Barz and Johnson( 1999) described the affecting factors and the relationship betweenbrownian motion, mean reversion, geometric thebrownian motion and geometric mean reversion iour中国煤化工cars, generation andmodels and also detailedly analysed the behav-tranCNMHGprature, etc.Alleniour of prices in different regions and time. Mon-( 1998) dealt with correlations among electricitte-Carlo simulation is used for electricity deriva- prices and time load and temperature. But thetive pricing in the jump-diffusionels. models( Brownian MotionMost of the electricity price models reported and AR( 1 process )are very simple and thein the literattre are only one-parameter or few- analysis can only be taken as an introductionAn electricity price model with consideration to load and gas price effects667Kreuzberg( 1999) took the generation ,transmision, efficiencies of power plants and startup Table 1 The correlation matrix CM) in PJM in 1999costs, into consideration, but did not analyze theTimecorrelations between them and the electricityLoadprices i he took these factors into the model di0.68rectly. In fact some parameters can be ignoredo small. This pa-0.440.231.O0per' s analysis results showed thatthe gas price and load behaviour is very imporGenerally speaking the load has the followtant. Considering the difficulty of including all ing characteristicsthe factors in thctricity price mod1. Strongly time-dependent( time mean re-took the most important factors, load and gas version and seal). Load changes accordiprice behaviour, as the variables to construct the to different hours everyday and everyday thermodel in the Part Ill of this paper. Simulation are two load peaks one at about 9: 00 AM toresults demonstraded that this model is feasible. 13 00 PM, the other from 16 00 PM to 19 :00PM. In one year most load curves have also twoCHARACTERISTICS OF THE PRICE AND LOAD peaks one in summer and the other in winteBEHAVIOURThe height of the load peaks in different areasare different. For example in the USa the peakin summer is higher than the one in winter butBecause of its non-storablility and limitationsIn Germanythe opposite is trueof transmia2. Region dependent. In different areasspecial commodity. The limitations on the possi- even at the same time, the electricity loads arebilities of transmitting electricity across time and different. It is caused by the different consumpspace strongly affect the behaviour of electricity tion structure at each region. The characteristicsspot and derivative prices as compared to other of electricity load seasonality are also differentcommodities especially the behaviorof the For example, in California or PJM the peak inprice"spikes". The non-storability of electricity summer is higher than that in winter, but in Gerrequires that electricity must be used at the same many the opposite is truetime it is generated. Limitations in transmissiongly temperature dependent. Allencapacity and transmission losses are important (1998)described in detail the relationship beconsiderations for deciding if transmission among tween temperature and electricity load for PJMome zones is economical. These limitations where in summer the temperature is very highmake electricity prices depend highly on the re- so that the electricity demand will also be verygions, which means that the prices depend on high because a great part of the electricity Isthe local level of supply and demand weather, consumed to turn on fans or other cooling electriand the fuel or gas pricescal equipment. In winter when it is very coldhe following sections will show the price the electricity demand will also increase, be-behaviours in different time period regions, cause much electricity will be used for heatingtemperature, load and gas price behaviour. The And the relationships between temperature andorrelation between electricity prices and load electricity load are also greatly different in eachgas( fuel prices will also be describedregionCharacteristics of load behaviourFig. I shows the price and load behaviouur inFrom the following anaysis we can see thatelectricity prices are strongly load-and time -deorn中国煤化工199. HthatCN Saviours are periodItthe factors of the behaviour of the load and time and strongly time-dependent. Fig. 2 shows theinto the electricity price model. The correlationbehaviour of on peak price and on peak loadfactor between load and time, 0. 44, isfrom 1999 to 2000 in CalPX. It demonstrates thegh.T搬 that load iseasonal effects of the log price and log loaddependent( Table 1668HUANG Minxiang TAO Xiaohu et al20040000z3000081020000Fig 1 Price and load of the lst and 2nd day of April and August CalPX in 1999(a)lst day i (b)2nd day州h邮啊州制1101201301401501601701(h)01201301401501601701(h)Fig 2 Relatioship between log price and log load 1999-2000 in CalPX(a) Log on-peak price 1999-2000 in CalPX b) Log on-peak load 1999-2000 in CalPX2. Characteristics of electricity price3. Correlation of price with load and gad fuel ) priceThe most important characteristics of thelectricity price areIt is well known that price will go up with in-1)High load-dependencecrease of electricity demand and down with2 )Mean Reversion and Seasonal depen- crease of electricity demand. The relationshipdencebetween price and demand for electricity is the3)High volatility and occasional price same as that of some other commodities.Fig.1show the relation between electricityFig 2 shows mean reversion and seasonalprices and loadIf there are lots of gas genefects. Price"spikes"can be defined as abrupt up tors and most of them are marginal units,thenor down movements of electricity prices. Price the behaviour of gas prices will also affect elecikes"can be regarded as the result of the non- tricity prices significantly because the cost ofstorable nature of electricity. But price"spikes" electricity produced by gas-powered generatorselectricity markets. For ex- depends on gas prices. Then it is reasonable andample the price curves in EEX( European EnerImpgy Exchange are relatively flat it means that中国煤化工 e effects of load andthere are no price"spikes". In the CalPx thereCNMHGCIICIty price modelare such price " spikes". So we should constructMODELSdifferent electricity price models for each market. The model used in this paper is based onEEX witheconsideration of price"spikesPrices for most common commodities, suchas copper are mostly modeled using aAn electricity price model with consideration to load and gas price effects669stochastic process in which the price at a future terms. Here we express O t ) as a sum of sinetime is a random function of the current price. a terms to reflect seasonal, predictable variationscommon price process model is Geometric in electricity load this assumption was first prop-Brownian Motion( sometimes called Wiener Pro- osed by Craig Pirrong in 1998)cess)which is a continuous-time process as de-In order to simplify the problem, we canscribed belowtake all parameters as constantdP=u, P,dt+oP, dwLet a ti)=Inqi and (( t )=xrexp(t)+re, P,, spot price at time (;Ar, drift of 3 si( x4t+as)+x6sir( x,t+x8)price i o volatility of spot price ; dw,rankelihood method to es-dom brownian motionthe following expres-Another price process model is the mean-re- sionsverting process. Whereas a random walk can dea ti)1-x21)+x3sin(x4l12+x5)+viatfar from the starting point and not return fora very long time ,a mean-reverting process will sIr( xit;+x8)+Ei,Ei-NO 02)have a tendency to return to a mean value over the equation becomes, Ei-MO02)Tscribed in discrete time as followsat:)MxJexp-x2t: )+x3sir( x4t; +xs)+xsi x7t i+x8)+02)dp,=ku, t))P, dt+ oP, dw, where a ti), A(ti) are independent each othere lwhere, u,=a, +0/2, h is the mean reversion Then the combination function isfactor. y( t) is the equilibrium process forPt)Lxt.Here we represent the electricity load andthe electricity price with time meaersion (xiexp-x2t)+x3sir( x4t:+xs)+model. And for the fuel or gas price we adopt x6sir(x,t; x&))the geometric brownian motion model, becausethis model is widely used in the energy except To obtain the maximum of the above is equivaelectricity )market by many researcherslent to solving the followingThe electricity load modelhe first state variable is a demand variableMin:[ d(xuexiThe load stochastic process can be decribed as x] sir( x4l;+x5 )+x6sir x74+a8 ))I=dqn=从(t)Dai dt oq, dwa (1)The parameters xHere we define d qu t )=k, (t)-Inqrx& can be obAssume that y, Inq, then rewrite Eq. 1 )as tained by solving the above equationFig 3 shows the recorded log load curve inby using the estimated parameters. ve urve indy,=k 0 (t)-y t +odwx (2)1999 in CalPX and the simulated cuwheyI the log of the electricity load-- Predicted basisk the mean reversion factor8 t): the equilibrium process for y.中国煤化工o: the volatility of the log electricity load 8CNMHGdwx the stochastic variable for log electricity31619112115118l211241271301331361y, reverts to a time-varying mean 0 t).InT(day)Fig. 21gsee that the historical log loadI方据 function of the sum of siFig 3 Comparation of original and simulated670HUANG Minxiang TAO Xiaohu et al2. The gas fuel price Modelo.qad,aPa(gnt)m习P3Papqr gThe geometric brownian motion model is fre- Q g, xg,,2+0.5 -Qiu( g, aquently used to model the price behaviour in normal commodities markets( including fuel and gas d22+0522P2x( g,t X22*3yosaPoPe the characteristic of time mean reversion ;the half-life is u qu t b(gr t dudt +ogx, t ddudt+always about a few months much longer than a PaPhours). The risk-neutralised process for the q,ag,go t Xudw+0532P103du'+the half-life of electricity load about a few aprice of the marginal fuel or the gas isg2872a-P)+aa2y人)dg1=x(614)+可(gdm(3)ag dtdu+52gix gr ,t o( g,, xtdw (6)gr: the gas or fuel price at time tdrift of the gas pricas dt->0, dt2 and dt. can be ignored com-pared with dt. And we know that: dw= dtthe volatility of the gas pricedw: standard Brownian motiondu2= dt and dwdt= dt. 5, dudt= dtl. 5 andAssume that z,= Ing, thenassume dudu= Ppg dt, then the above equationbecomesdz=x(zt划t+a(zt知aTo simplify the problem, we can assume that drag qu gr nt Xit+pa Pa+3. The0.522dt+0.5( g, t XItthe forward price is P( q1 18 ,t), which is a ag,agogo gr ,t og dt+opIn any forward contract it is assumed that apapqro,du+function of t ,q, and g, i then the forward price a Psatisfies the following PDE Partial Differential agg,o,(g,,t xw(7)EquatiqUthen we can assume thatdP=sdea PaP dq, dgudP=y( qu ,gt ,t )dPdPqdgqro0.52(d)+0.5(dg(4)(g1,t(8)Combining eqs. ( 1) and(4) into one yieldsthaPa Pdp=oPgl g, t+o, du ]+aPg4)=a0994)+Bx8n)[xg)t+og〃1aPPa-P(g1t)g+0grog2q0[Aqr tt +o, du lg[xg,,ttapOcgi ,t)(9)o(g,t )du]+0.5(q[( q, , t )dtdu 1)+0.5[x g, Xlt中国煤化工aPCNMHG2+(g1,t)])g(5)aPaP(10)written in another form, Eq (5)becomesdP=Dg 73sgE. Xlt+ag, 8x g,, t Xlt+ followg Price P( q,,g, ,t)can be written asaPForwardAn electricity price model with consideration to load and gas price effects671pq,,gt,t)=(mo mi g, )exp( n( t)+ni, er is for gas( fuel price processn2 97Among many methods that can be used tosolve the model pDe )are the Monte CarloWheremethod and finite diffence mesh method Thegr gas or fuel price at time texplicit methods include binomial trees providedqr the electricity load at time Lby Cox et al. 1979), and trinomial trees( HullAgain , using the maximum-likelihood meth- and White, 1994). Although the explicit meth-od we can estimate the parameters mo mI, ods are not so accurate compared with finite dif-nI, n2, no( t). Here no( t ) can exhibit sea- ference methods they are very simple to implesonal effectsment and useful for futher research on options orRewrite the Eq (8)as followfuture s electricity pricesdP=g u ,g, t Xt+(ni+2n2 qr )q,o, Pdu(11)SUMMARYmgo pduThis paper first analyses some characteristicsg qr ,g, t)=(ni +2n2 qr )qu qr ,t )P+of the electricity load and prices then it demig g,t )P+mi(nI +2n2 9, o,o gr ,t) scribes the relation between electricity price andoP+0.x2n2+(n1+2n29))Py22+load and gas( fuel )prices. It is obvious that0Ca2p(12) electricity prices are strongly dependent on loadand the variance iselectricity price based on the effects of these two02(9,,,,t)=(( n 2n2q1 )qo, P)+ factors. In this paper we also present methods forparameters estimation, propose some methods for(m1g , P)+X n +2n2 q: )q, a, Pm1,0g PPg solving the model(13)Define X=L( P), the electricity price process Referencecan be described asAllen e. 1998. Price- Based Commitment Decisions in thelectricity Market. Ph. D dissertion, MIT.dX= pnew qr g,,t X+(nI +2n2, )no, du+ Barz, G., 1999. Stochastic Financial Models for Elem1 gandu(14)Derivatives. Ph D Dissertation Standford UniversityBarz, G. and Johnson, B., 1999. Modeling ElectricityPrices. Working Paper, Stanford UrCox ,J C., Ross ,S.A. and Rubinstein M., 1979. Op-中t)=(n+2nqr )q gr,)+on pricing: A simplified approach. Journal of Finanmigxg,d)+m (nI +2n2 q b, o g,cial Economics, x October)g+0.52n2+(m1+2m29))Deng,SJ., 2000. Stochastic Models of Energy CommodityPrices and their Applications0. 5migiog(15)Jumps and Spikes, WeWorkable Energy Regulation( POWER ) University ofand the new variance is1)=((n1+2n2q)q)+Hull, J. and White, A., 1994. Numerical procedures formplemnting term structure models I: single factor mod-(m1g8)+xm1+2m2q1o,m129(16)els. Journal of Derivaties,X 1): 7-16Kreuzberg,M., 1999. Forecasting Spot Prices for the EuroThere are stochastic motions in the electricitypean Power Market. EWI Working Paper 98/2, Instituteprice model i one is for load process and the oth-onomics, University of Cologne中国煤化工CNMHG