Stochastic Behaviour of the Electricity Bid Stack: from Fundamental Drivers to Power Prices

Transcription

1 Sochasic Behaviour of he Elecriciy Bid Sack: from Fundamenal Drivers o Power Prices Michael Coulon Mahemaical Insiue, Universiy of Oxford, S. Giles, Oxford OX1 3LB, UK coulon@mahs.ox.ac.uk ( ) Sam Howison Oxford-Man Insiue, Universiy of Oxford, Blue Boar Cour, 9 Alfred Sree, Oxford OX1 4EH, UK & Mahemaical Insiue, Universiy of Oxford howison@mahs.ox.ac.uk ( ) 23 Ocober 28 Absrac We develop a fundamenal model for spo elecriciy prices, based on sochasic processes for underlying facors (fuel prices, power demand and generaion capaciy availabiliy), as well as a parameric form for he bid sack funcion which maps hese price drivers o he power price. Using observed bid daa, we find high correlaions beween he movemens of bids and he corresponding fuel prices. We fi he model o he PJM and New England markes in he US, and discuss is performance, in erms of capuring key properies of simulaed price rajecories, as well as comparing implied forward prices wih observed daa. Keywords: elecriciy, bid sack, fundamenal, margin, demand, naural gas 1

2 1 Inroducion The ques o model he unusual behaviour of elecriciy prices ofen leads o a choice beween realisic, bu complicaed, models and simpler reduced-form models wih convenien pricing formulas. In paricular, due o he non-sorabiliy of elecriciy, here is a closer link beween power and is fundamenal underlying price drivers (in paricular fuel prices, load and generaing capaciy) han in oher markes. We propose a supply and demand based hybrid model, which explois hese links wih underlying facors and allows flexibiliy and realism, while sill reaining racabiliy for derivaive pricing purposes. Elecriciy spo prices ypically exhibi periodiciy (a annual, weekly and daily horizons), meanreversion, very high volailiy and sudden price spikes, all of which can be raced o supply and demand relaed causes. (see eg Burger e al (27) and Eydeland and Wolyniec (23)) Forward prices show none of his volaile behaviour, bu insead sugges he need for muli-facor models o capure he movemens of differen pars of he forward curve; his is illusraed by he Principal Componen Analysis of Koekebakker and Ollmar (25) and by discussions of volailiy erm srucure by Clewlow and Srickland (2). Furhermore, boh forwards and longer-erm spo dynamics reveal close links wih underlying drivers such as demand (or weaher paerns) and especially fuel prices. Burger e al (27) and Emery and Liu (22) discuss he apparen coinegraion, raher han he simple correlaion of power and fuel prices, as long-erm levels move ogeher. All of hese characerisics sugges he imporance of looking beyond hisoric price series o beer undersand he dynamics of power prices in relaion o fundamenal drivers. Noneheless, he desire for analyic formulas and efficien pricing echniques has led o a large lieraure on direc spo or forward price modelling. Early spo price models by Lucia and Schwarz (22), and Schwarz and Smih (2) proposed a wo-facor diffusion model o capure he differen shor and long-erm dynamics of power prices. However, he imporance of elecriciy spikes has led o he use of jump-diffusion processes by many auhors, including Carea and Figueroa (25), and Kluge (26). An affine jump diffusion framework leads o convenien formulas for derivaive prices; see, for example Deng (1999), and Culo e al (26), who also sugges regime-swiching jumps o capure he behaviour of shor-lived price spikes. Pure regime swiching models have also been sudied by De Jong and Huisman (25) and Weron e al (24), while an alernaive approach is proposed by Geman and Roncoroni (26), forcing jumps o be downwards when prices are above a cerain hreshold. While many of hese models produce useful resuls and realisic price dynamics, hey ofen face calibraion challenges due eiher o unobservable facors, choices of probabiliy measure, or o he complicaion of idenifying hisorical spikes. In addiion, hey fail o capure he imporan correlaions beween power prices and oher energy prices. The alernaive caegory of srucural, fundamenal, or supply and demand based models consiss of a wide range of work including agen-based models of marke power and bidding sraegy (e.g. Ruibal and Mazumdar (28), Supagia e al (21)), producion cos based equilibrium models (eg Bessembinder and Lemmon (22)) and high-frequency analyses of he dominan spo price drivers (eg Karakasani and Bunn (28)). Many auhors have sudied he impac on power prices of movemens in physical variables such as emperaure, rainfall paerns and oher demand-side facors (eg Huisman (28), Vevhilainen and Pyykkonen (24)), while some have focused primarily on specifying he correc shape of he elecriciy supply funcion (e.g. Kanamura and Ohashi (24)). These models all aemp o bridge he gap beween pure power price models and he complex equilibrium models used in indusry o forecas prices based on deailed marke knowledge such as specific generaors coss, schedules and consrains. Amongs he class of sochasic, economeric syle models mos useful for derivaive pricing, Eydeland and Wolyniec (23) discuss he role of hybrid models, hough hey sill rely on fairly deailed local marke knowledge and large simulaions. A he oher end of he specrum, Barlow (22) proposes a simple non-linear Ornsein-Uhlenbeck model consising of a mean-revering demand process combined wih a supply curve. A similar approach is aken by Skanze e al (2), Eydeland and Geman (1999), Villaplana (24), and Carea and Villaplana 2

3 (27), wih he inclusion of capaciy as a sochasic process, incorporaed hrough an exponeniallyshaped supply curve. Burger e al (24) consider a non-parameric supply curve, fied using cubic splines, and assumed o be a funcion of demand over capaciy. A slighly differen perspecive is gained by considering reserve margin (exra capaciy available beyond demand) o be a key driver, paricularly as a low level of margin corresponds o a period of marke srain, and consequenly a higher chance of a sudden price spike. This idea is exploied in a regime-swiching framework by Moun e al (26) and Anderson and Davison (28), a non-parameric approach by Booger and Dupon (28) and a model which incorporaes forward-looking margin informaion by Carea e al (28). While he appropriae choice of facors may vary across differen markes, he aricles above focus mainly on load (demand) and capaciy flucuaions, which are known o be he main shor-erm drivers of power prices. Moving o medium or long erm dynamics, daa suggess ha i is very imporan o incorporae fuel price risks ino he modelling framework. Pirrong and Jermakyan (25,25) propose a fairly simple and inuiive model based on he wo risk facors of demand and naural gas price. As in our work, hey sudy he PJM marke and advocae he use of hisorical generaor bid daa o esimae he ransformaion from underlying facors o power prices, suggesing a simple non-parameric approximaion o he bid curve, derived from he bid curve on he same dae of a previous year and adjusing by he raio of gas prices (or forward gas prices) on he wo daes. We propose a more realisic, parameric approach o he bid sack funcion, allowing for he overlap of bids from generaors of differen fuel ypes. Thus, our model exends heir approach by allowing muliple fuel prices as facors, as well as addressing capaciy or margin issues such as ouages. This allows us o capure he raher complex dependence srucure of power and fuel prices, as needed o price a variey of cross-commodiy spread opions and oher derivaives. The remainder of he paper is organised as follows. Secion 2 inroduces he wo Norh Eas US markes used in our analysis. Secion 3 presens our approach o modelling he bid sack funcion, which links elecriciy spo prices o underlying drivers. We specify slighly differen models for PJM and NEPOOL and demonsrae he srong fi o hisorical bid daa. Secion 4 complees our modelling framework by proposing sochasic processes for each of he underlying risk facors: fuel prices, demand or load, and available capaciy or margin. Secion 5 assesses he performance of he model hrough analysis of boh simulaed price pahs and forward prices, in comparison o observed daa. Secion 6 concludes. 2 Elecriciy Marke Daa Se We shall focus on wo US elecriciy markes: he PJM marke (Pennsylvania, New Jersey and Maryland, plus pars of nine oher Easern saes) and he NEPOOL marke (he New England region). While our mehodology can be adaped o fi many differen local characerisics, we choose hese wo US markes primarily for he availabiliy of hisorical bid daa (a a six-monh lag), as well as oher useful hisorical informaion also published on he PJM and NEPOOL websies. 1 The availabiliy of bid daa in a convenien form is sill uncommon in power markes and is cerainly cenral o he mehodology described here, bu i is possible o imagine approximaions or variaions of he model wih parameers esimaed for example from cos or hea rae daa. PJM is a large marke currenly serving over 5 million people, wih a oal capaciy which grew from 5, MW o 16, MW beween June 2 and July 27. This ime period forms our daase for parameer esimaion, and we use PJM Wes price daa hroughou. 2 The PJM marke 1 See hp:// and hp:// for bid daa. 2 Power prices in PJM are complicaed by he exisence of Locaional Marginal Prices (LMPs), corresponding o he differen regions of each marke. Each LMP is deermined by calculaing he price required o deliver one exra uni of power o ha poin on he grid. Facors such as local ransmission consrains can lead o significan variaion across regions. In all our analysis, we use he PJM Wes region, as hese prices are used o calculae he value of PJM fuures 3

4 has an ineresing mix of fuel ypes (power sources), wih a significan proporion of capaciy coming from coal, gas and oil, and nuclear. Table 1 illusraes how his fuel ype breakdown has changed slighly from year o year over he period considered. 3 Table 1: Fuel breakdown (percen) in he PJM marke (22-6), and NEPOOL (26) Fuel Type Jan 2 Jan 3 Jan 4 Jan 5 Dec 5 Dec 6 NEPOOL Nuclear Hydro Coal Gas Gas/Oil Oil Oher In conras o PJM, New England has a much smaller and younger marke wih capaciy fairly sable around 3, MW hroughou he daase, which covers he ime period March 23 o Augus 27. NEPOOL has a simpler fuel mix han PJM, having lile coal-powered generaion. Furhermore, gas and oil ogeher represen nearly half of he capaciy in he marke, while he remainder is primarily nuclear and hydro. Mos nuclear and some hydro generaors ypically make bids of zero as hey have lile flexibiliy in erms of swiching on or off in response o demand. As we shall see in Secion 3, NEPOOL can herefore be reaed in a simplified framework as a one-fuel marke, whereas PJM requires a leas wo fuel ypes o capure he dynamics realisically price ($) price ($) PJM peak NEPOOL peak Gas price (x1) 2 2 Jun- Dec- Jun-1 Dec-1 Jun-2 Dec-2 Jun-3 Dec-3 Jun-4 Dec-4 Jun-5 Dec-5 Jun-6 Dec-6 Jun-7 Jun- Dec- Jun-1 Dec-1 Jun-2 Dec-2 Jun-3 Dec-3 Jun-4 Dec-4 Jun-5 Dec-5 Jun-6 Dec-6 Jun-7 Figure 1: Daily average real-ime peak prices for PJM (lef graph); Monhly average real ime peak prices for PJM and NEPOOL, as well as monhly average gas prices, muliplied by a facor of 1 for comparison purposes (righ graph). (Noe ha he highes daily PJM price achieved in he lef graph is $637, and here are 8 daa poins beyond $2, bu only hree since 22, corresponding o he firs hree days of Augus 26, wih a maximum price of $327.) Figure 1 illusraes he dynamics of he real-ime daily average peak price (average of hours 8-23, weekdays only) for PJM. The dynamics for NEPOOL are slighly less volaile bu generally similar in appearance. Boh real-ime (RT) and day-ahead (DA) prices exis for boh markes, wih real-ime prices ypically more volaile. We shall only consider peak prices, as he modelling mehodology is conracs raded on NYMEX. 3 PJM daa has been aken from annual generaing capaciy repors (hp:// while NEPOOL daa from Eydeland and Wolyniec (23). Noe ha gas generaors ha also have oil-based generaion capabiliy are lised separaely. So if all of hese choose o use gas, he proporion of gas generaors in he marke is approximaely 26% and 32% for PJM and NEPOOL respecively. 4

5 less well suied o describing off-peak prices, and hus a daily price will refer o a daily peak average. Alhough daily price series clearly reveal occasional price spikes, he magniude and frequency of spikes is much higher a an hourly level (i.e., before averaging), as a resul of brief ouages or ransmission consrains. The price series show high correlaions wih naural gas prices, also as illusraed in Figure 1. Here we have removed he noise by considering only monhly average prices, and obain a remarkable visual correlaion wih monhly average Henry Hub gas prices. This link beween gas and power prices is exremely srong for he gas and oil-dominaed New England marke, bu also remarkably srong for PJM, as he marke clearing price in peak hours is ofen se by he bids from gas generaors. Capuring his relaionship accuraely is one of he primary advanages of our modelling approach s Feb s Aug s Mar s Sep 25 price ($) 3 price ($) quaniy (MW) quaniy (MW) Figure 2: Sample bid sacks for PJM (lef) and New England (righ), showing movemen as gas prices change. (The region $6 o $1 is no shown as he bid sack is almos verical for his range.) In Secion 3, we inroduce our model for he bid sack, which can be undersood as he map from he underlying random facors o he spo elecriciy price, and is hus he key componen of he power price model. We use daily bid daa, consising of day-ahead bids from all available generaors (close o 1 for PJM, close o 3 for NEPOOL), for calibraion. 4 These bids describe he prices a which he generaor is willing o sell varying amouns of elecriciy. Thus each generaor submis a non-decreasing sep funcion wih a maximum of 1 seps, or price and quaniy pairs. For example, a generaing uni which bids (2MW, $4), (3MW, $5), and (35MW, $8) is willing o sell is firs 2MW of power a $4, is nex 1MW a $5, and is final 5MW a $8. 5 By sacking all bids from all generaors in order from lowes o highes price, we can creae he marke bid sack, ypical examples of which are provided in Figure 2. The marke adminisraor hen deermines he hourly spo price by maching wih he oal demand for power. Noe ha some generaors submi mus-run bids, which we rea as bids of $, as hey mean ha he generaor mus sell is power no maer wha he price is. 6 As peak prices ypically say above $3 bu below $15 (for over 9% of he hours observed during he mos recen hree years) he middle secion of he bid sack is mos relevan in deermining prices, hough he righ hand side becomes imporan in he even of spikes. Figure 2 also shows ha significan movemen can occur from monh o monh, especially during imes of large gas price increases, as was he case in boh February 23 and Augus 25. We observe ha 4 Noe ha only daily observaions of he bid sack are available for PJM, whereas hourly sacks are available for NEPOOL, hough inra-day variaion is low. We creae an average bid sack for each day before performing he maximum likelihood esimaion. Generaor-specific issues such as sar-up imes and maximum run imes per day are ignored for simpliciy. Deails of he name of or ype of generaor making each bid are no revealed. 5 In PJM, generaors have he alernaive of connecing hese bid poins linearly insead of using a sep funcion. 6 In fac, his requiremen of some generaors can very occasionally produce negaive prices in elecriciy markes, bu his is only realisic during off-peak hours, which we do no model here. 5

6 during hese monhs he majoriy of NEPOOL s bid sack shifed upwards, while only he righ hand half of PJM s was affeced. Clearly power generaors adjus heir bids according o changes in heir generaion coss. We herefore expec a srong correlaion beween bid sack movemens and fuel price changes, hough he possible impac of oher facors such as he exercise of marke power or sraegic bidding should be acknowledged. In order o undersand his relaionship, i is useful o consider he bid sack as a hisogram of bids, as shown in Figure 3. We simply add up he oal amoun of capaciy in MW ha has been bid wihin each price bin. 7 This provides an alernaive perspecive on he same daa shown in Figure 2, and ineresingly reveals one main cluser of bids for NEPOOL, bu a pair of clusers for PJM separaed by a region of few bids. We expec bids o be ordered roughly by fuel ype corresponding o he meri order for each marke (Table 1). This suggess ha mos bids in he lef cluser of PJM s hisogram correspond o nuclear or coal generaors, while he righ cluser is primarily gas and oil, alhough low gas prices in paricular can cause hese clusers o merge somewha. While his clusering provides our primary moivaion for sudying he bid sack, i is also imporan o discuss he far lef and far righ of he sack. The far lef is less imporan as i consiss of zero bids (including mus-run ) or very low bids, boh corresponding primarily o nuclear power generaors. This firs 2-3% of capaciy almos never deermines he marke clearing price during peak hours. On he oher hand, he far righ of he sack ypically consiss of a scaering of bids beween abou $25 and $1 and herefore ses he price only during imes of srain on he marke. As we have seen, hese insances occur fairly frequenly during peak hours, as hey correspond o he disincive spikes visible in power prices. Our approach o modelling he enire bid sack focuses on he movemen of hese clusers of bids as fuel prices change bid amoun (MW) bid amoun (MW) bid price ($) bid price ($) Figure 3: Sample hisograms of bids for PJM (lef) and New England (righ). The quaniy of bids a zero (including mus-run bids) for NEPOOL is far beyond he scale of he graph, a 858MW. 3 The Bid Sack Model Le S represen he spo price a ime, and D [, 1] he demand a ime, assumed o be inelasic wih respec o price, as is ofen he case for elecriciy. We model demand no in erms of megawa-hours bu raher as a proporion of oal marke capaciy, simplifying noaion and allowing for growh in he marke size. For generaor i of n, le x i be quaniy supplied (again, normalised by oal capaciy) and b (i) (x i ) be he bid curve a ime, where b (i) : [, c max i /c max ] [, p max ]. 7 In order o plo he enire bid sack, we have chosen o vary he hisogram bin size for differen pars of he sack. For PJM, he bins covering he region [$,$64] have widh $4, while hose covering [$64,$32] have widh $8, and finally hose in [$32, $1] have widh $4. Similarly, for New England, he bins covering he region [$, $112] have widh $4, while hose covering [$112,$28] have widh $8, and finally hose in [$28,$1] have widh $4. 6

7 Generaor i s maximum capaciy is c max i, oal marke capaciy is c max = i cmax i, and p max is he maximum bid allowed in he marke (e.g. $1 for PJM). Then he marke clearing price which allows supply and demand o mach is given by { { } ( ) S = max b (i) (x i ), where {x 1...x n } = argmin max b (i) (x i ) : 1 i n x 1,...,x n 1 i n } n x i = D. Combining bid curves from differen generaors inuiively means sacking heir componen bids in order from lowes o highes. If we assume ha all bid curves are sricly increasing (and sep funcions can be approximaed by sricly increasing funcions), hen his corresponds o invering each bid curve, adding he inverses, and invering he sum. Leing B obs ( ) denoe he exac bid sack observed in he marke a ime, we can wrie S = B obs (D ), where B obs (x) = I 1 (x), and I (x) = n ( i=1 b (i) i=1 ) 1(x). These equaions provide a simplified descripion of an elecriciy marke s srucure, bu rarely hold in pracice. This is due o a variey of complicaions including generaor ouages, ransmission consrains, impors or expors, variaions in geographical disribuion of demand, possible demand elasiciy, and oher rebalancing effecs, especially for real-ime prices. In order o capure hese effecs ye reain racabiliy, we inroduce a process C for capaciy available a ime (again normalised wih c max ), and assume ha now he bid sack is a funcion of D /C. In oher words, any loss of supply is assumed o be equally spread hroughou he sack, an approach also aken by Burger e al (24). A fundamenal requiremen of his framework is ha demand never exceeds capaciy, so < D /C < 1. Typically we also observe < D < C < 1, hough someimes C > 1 (for example hrough impors of capaciy). The spo price is now given by 8 ( ) D S = B, for < D < 1, (1) C C where he ime dependence of he funcion B ( ) is in fac a dependence on fuel prices, as hese are he primary drivers of generaors bids. While his basic framework is only an approximaion o he complexiies of elecriciy markes, i allows us o reain he direc link o supply and demand facors, he flexibiliy o adap o differen markes, and he abiliy o price derivaive producs fairly easily. We now inroduce our model for his funcion B ( ), which we esimae direcly from available bid daa for PJM and NEPOOL. 3.1 General Case - fiing disribuions o bids As illusraed in Figure 3, a hisogram of bids provides a useful alernaive o simply observing he bid sack direcly in Figure 2, and moivaes fiing a densiy funcion o hese hisograms. Bids from generaors wih differen fuel ypes are driven by differen coss, leading o a mix of disribuions in he overall marke, wih weighs corresponding o he breakdown of fuel ypes in he marke. Wih his new approach, he spo price S = B (x) can be reinerpreed as he x-quanile of our bid disribuion. Thus we fi a funcion o he densiy of bids and hen deduce he quanile funcion (inverse cumulaive disribuion funcion), as opposed o fiing he bid sack (or quanile funcion) direcly. One advanage is he wide range of well-known disribuions ha we can es. Furhermore, we can link disribuions parameers o he underlying fuel prices in an inuiive manner. 8 We can inerpre his spo price as being eiher a day-ahead or real-ime price and eiher an hourly or daily average price, depending on wha we are ineresed in modelling. The framework of he model remains he same. In our analysis, S is he hourly peak price process. 7

8 In he mos general case, le F 1 (x),...,f N (x) (F i (x) : R [, 1]) be he proporion of bids below $x for generaors of fuel ype i = 1,...,N, wih weighs w i,...,w N, summing o uniy. Then he spo power price S solves F(S ) = N i=1 w i F i (S ) = D C, and he bid sack is simply he inverse of he cdf, F(x), of he mixure disribuion. To improve he fi in he mos relevan region of he bid sack, we may wish o runcae he domain of D /C from (, 1) o (b L, b U ) and ignore he ails of he bid disribuion. This is only appropriae if P[b L < D /C < b U ] = 1 (ypically se b L =.2 or.3 and b U =.9 or.95). We hen have N ( ) 1 D F(S ) = w i F i (S ) = b L. b U b L C i=1 The requiremen ha D /C (b L, b U ) poses problems from a modelling perspecive. Therefore, we sugges an alernaive approach of simply linearly rescaling boh D and D /C such ha for new variables D and C, we require D / C (, 1), jus as in he unruncaed case. In pracice his means ha he lowes porion of boh demand and capaciy is fixed, and changes in boh demand and capaciy only occur beyond his poin in he sack. Moreover, any drop in available capaciy is now assumed o affec only he region (b L, b U ) of he bid sack. We hen have F(S ) = N w i F i (S ) = D 1, where D = (D b L ) C b U b L and D C = 1 ( ) D b L C b i=1 U b L (2) Noe ha C C, as C represens he percenage availabiliy of capaciy in he relevan region of he sack, no in he marke in oal. We fi disribuions o he bid daa by maximum likelihood esimaion, where a bid of q megawas a price p is reaed as q separae observaions of a bid a p. Le (p j, q j ), j = 1,...M represen all he price quaniy pairs ha make up he porion [b L, b U ] of bid sack. So M j=1 q i is (b U b L ) imes he oal capaciy of he marke. Consider he general case of fiing a mix of N disribuions, wih weighs w i (where w w N = 1), and wo-parameer densiy funcions f i (x; α i, β i ), for i = 1,...,N. Then he log-likelihood funcion (for a given day) is { M N } qj L(w 1,..., w N, α 1,...,α N, β 1,...,β N ) = log w i f(p j ; α i, β i ) = j=1 i=1 [ M N ] q j log w i f(p j ; α i, β i ). We have compared resuls using he following disribuions: Gaussian, logisic, Cauchy, and Weibull. These all have appropriae humped shapes and only wo parameers, one corresponding a leas roughly o he mean, and he oher roughly o he sandard deviaion or shape. Hence we use he noaion m i, s i, i = 1...N for hese parameers. The performance of he four disribuions is fairly similar in erms of boh likelihood and capuring fuel price correlaions, hough hicker-ailed disribuions dominae for higher choices of he cuoff poin b U, where he hin-ailed Gaussian performs erraically. Ulimaely, we advocae he logisic disribuion (wih mean m i and scale parameer s i equal o 3/π sandard deviaions) as he bes choice, since i performs consisenly for boh markes and leads o he simples mahemaical expressions. For a mix of N logisic disribuions, he log-likelihood funcion is given by L(w 1,...,w N, m 1,...,m N, s 1,...,s N ) = j=1 i=1 { M N q j log j=1 i=1 w i 4s i sech 2 ( ) } pj m i. (3) 2s i 8

9 3.2 One Fuel Case (N = 1) - NEPOOL Beginning wih he simpler NEPOOL case, he bids of generaors can be spli ino bids of zero (roughly 3%) by nuclear and some hydro producers, and a cluser of bids primarily from oil and gas generaors. Removing he lowes bids and considering he close relaionship beween gas and oil prices, a single fuel model is reasonable. We herefore esimae he bid sack parameers for each hisorical dae as follows. Firsly, we ignore bids of below $1 and also he highes 5%, 1% or 15% of bids, so [b L, b U ] [.3,.95], [.3,.9] or [.3,.85]. 9 Then (for he logisic disribuion), we maximise wih respec o m 1 and s 1 he likelihood funcion (3) wih N = 1. We use a sandard numerical opimisaion scheme in MATLAB, while noing ha closed form expressions exis for ˆm 1 and ŝ 1 only in he Gaussian case. Figure 4 illusraes he MLE resuls for ˆm 1 and ŝ 1 wih b U =.9, also showing Henry Hub naural gas prices over he corresponding ime period. Apar from wo surprising spikes in bid levels in January 24 and 25, he correlaion wih gas prices is very high, and as much as 95% in he more recen daa. 1 The wo spikes could perhaps have been caused by sraegic bidding, hough no paricular informaion has been found o explain hese evens. 11 Noneheless, he resuls srongly sugges assuming a linear dependence srucure beween m 1 and s 1 and he naural gas price G : m 1 = α + α 1 G, s 1 = β + β 1 G. value ($) mean s dev Mar-3 Jun-3 Sep-3 Dec-3 Mar-4 Jun-4 Sep-4 Dec-4 Mar-5 Jun-5 Sep-5 Dec-5 Mar-6 Jun-6 Sep-6 Dec-6 Mar-7 Jun-7 price ($) Mar-3 Jun-3 Sep-3 Dec-3 Mar-4 Jun-4 Sep-4 Dec-4 Mar-5 Jun-5 Sep-5 Dec-5 Mar-6 Jun-6 Sep-6 Dec-6 Mar-7 Jun-7 ( Figure 4: NEPOOL esimaion resuls (lef graph) for he mean ˆm 1 and sandard deviaion compared wih Henry Hub naural gas prices (righ graph). π 3 ) ŝ 1, Esimaing he parameers {α, α 1, β, β 1 } by regression produces slighly differen resuls depending on he choice of disribuion, upper cuoff of bids, b U, and ime period considered. The upper secion of Table 2 shows he resuls for b U =.9, which appears o be a reasonable choice, reducing he influence of he far righ ail of he bid sack while keeping enough of he relevan region. 12 The resuls are very encouraging, showing high values of R 2, paricularly for ˆm 2, and paricularly over recen years, avoiding he wo spikes described above. While all four disribuions sudied share he useful propery of having a fairly simple explici inverse cumulaive disribuion funcion, his is paricularly rue for he logisic case. As a resul, 9 We firs remove all bids beween $994 and $1, as here are someimes large clusers of irrelevan bids a hese levels which complicae maers if hey are considered o be par of oal capaciy, c max. 1 Noe also ha correlaions are higher if he gas price series has a lag of one day wih respec o he bid sack parameers, as we would expec for day-ahead bidding. Thus we use a one day lag in our regression as well. 11 The PJM bid sack dynamics also show spikes in hese monhs, bu less dramaic ones. I is worh menioning ha no spike was observed in January 6 so here is no reason o expec his behaviour every January. 12 Tess reveal ha as we increase b U from.8 o 1, values of R 2 in he regressions remain sable before falling off sharply afer.9, in paricular for ŝ 1. For PJM, hey increase gradually unil abou.95 before falling off, paricularly for ŝ 2. 9

10 he one-fuel model for NEPOOL leads o a convenien equaion for he bid sack, and hence he spo elecriciy price. Under he assumpions inroduced above, (2) can be wrien as follows: ( S = α + α 1 G + (β + β 1 G ) log( D ) log( C D ) ). (4) Thus we obain a spo elecriciy price S which is linear in he naural gas price G, similar o he model for PJM prices by Pirrong and Jermakyan (25). However, his only occurs in he one-fuel case in our model, so for NEPOOL, bu no PJM. Noe ha he fairly simple form of (4) is very appealing, paricularly for he pricing of forwards presened in Secion Two Fuel Case (N = 2) - PJM As discussed briefly in Secion 2 and illusraed in Figure 3, he variey of fuel ypes in he PJM marke suggess he use of a leas wo disribuions o capure he behaviour of he bid sack. We choose a pair of disribuions o roughly represen he coal and gas porions of he marke, and esimae he bid sack parameers for each hisorical dae as follows. Firsly se [b L, b U ] = [.2,.85], [.2,.9] or [.2,.95], such ha we ignore he highes 5, 1 or 15% and lowes 2% of bids for each dae. The low bids in paricular correspond primarily o he nuclear generaors and hence do no move in he same manner as he neighbouring coal bids. 13 Nex, calculae fixed weighs w 1 and w 2 = 1 w 1 based on he spli of coal versus gas and oil in Table 1. These weighs change only a a few discree poins in ime o reflec marke changes. Finally (for he logisic disribuion), maximise he likelihood funcion (3) for N = 2 wih respec o {m 1, s 1, m 2, s 2 } mean s dev value ($) price ($) Jun- Dec- Jun-1 Dec-1 Jun-2 Dec-2 Jun-3 Dec-3 Jun-4 Dec-4 Jun-5 Dec-5 Jun-6 Dec-6 Jun-7 Jun- Dec- Jun-1 Dec-1 Jun-2 Dec-2 Jun-3 Dec-3 Jun-4 Dec-4 Jun-5 Dec-5 Jun-6 Dec-6 Jun-7 Figure 5: ( PJM) esimaion resuls (lef graph) for he second disribuion s mean ˆm 2 and sandard devaion π 3 ŝ 2, compared wih Henry Hub naural gas prices (righ graph). Figure 5 shows he gas disribuion s esimaed mean ˆm 2 and sandard deviaion ŝ 2 ploed agains ime in he logisic case wih b U =.95. As expeced, boh ˆm 2 and ŝ 2 show a srong correlaion wih he Henry Hub naural gas price, ploed again in Figure 5 over he corresponding ime period. The resuls are paricularly encouraging using he more recen daa, wih correlaion as high as 96%. The resuls for ˆm 1 and ŝ 1 are ploed in Figure 6, along wih he changes in Appalachian coal prices over he same period. Though no as sriking as he gas correlaion, some correlaion is visible, paricularly wih he period of significan increase during he year 24. As for New England, hese resuls for PJM sugges a linear dependence srucure of he form, m 1 = α + α 1 P, s 1 = β + β 1 P, m 2 = α + α 1 G, s 2 = β + β 1 G, 13 Tess reveal ha including hese lowes bids produces worse resuls by disoring he rends in coal parameers ˆm 1 and ŝ 1. Tess wih more han wo disribuions also do no appear o improve he resuls, and in fac end o reduce he sabiliy of he parameers ˆm i and ŝ i over ime. 1

11 value ($) mean s dev price ($) Jul-1 Nov-1 Mar-2 Jul-2 Nov-2 Mar-3 Jul-3 Nov-3 Mar-4 Jul-4 Nov-4 Mar-5 Jul-5 Nov-5 Mar-6 Jul-6 Nov-6 Mar-7 Jul-7 Jul-1 Nov-1 Mar-2 Jul-2 Nov-2 Mar-3 Jul-3 Nov-3 Mar-4 Jul-4 Nov-4 Mar-5 Jul-5 Nov-5 Mar-6 Jul-6 Nov-6 Mar-7 Jul-7 Figure 6: ( PJM ) esimaion resuls (lef graph) for he firs disribuion s mean ˆm 1 and sandard devaion π 3 ŝ 1, compared wih Appalachian coal prices (righ graph). where P and G are sochasic process for coal and gas prices respecively. Regression can again be used o esimae he parameers { α, α 1, β, β 1, α, α 1, β, β 1 }. Resuls are shown in he middle and lower secions of Table 2, and are very posiive, paricularly for recen years. The R 2 values for ˆm 2 are remarkably high, and explain he source of he high correlaion observed earlier in he monhly power and gas price series ploed in Figure 1. As before for NEPOOL, he sandard deviaion or scale parameer (here ŝ 2 ) shows a weaker relaionship wih gas prices han he mean, due in par o he mehod of bid daa runcaion a b U. Under he assumpions inroduced above, our modelling framework from (2), can be wrien for PJM in he logisic case as follows: where B 1 (x) = w 1 2 anh S = x such ha B 1 (x) = D, C ( x ( α + α 1 P ) 2( β + β 1 P ) ) + 1 w ( ) 1 x (α + α 1 G ) anh 2 2(β + β 1 G ) As his funcion is no inverible explicily, we canno wrie down he bid sack funcion, bu can easily solve for S numerically. (5) Table 2: Regression resuls for ˆm 1, ŝ 1, ˆm 2, ŝ 2 versus fuel prices, for NEPOOL (b U =.9) and PJM (b U =.95) Dae range ˆm 1 or ˆm 2 ŝ 1 or ŝ 2 iner slope R 2 iner slope R 2 NE (Gas) Mar3-Aug Mar5-Aug Jun-Jul PJM (Coal) Jun3-Jul Jun5-Jul Jun-Jul PJM (Gas) Jun3-Jul Jun5-Jul

12 3.4 Discussion The high fuel price correlaions observed in boh PJM and NEPOOL bids jusify he mehodology of he bid sack model, boh in erms of he use of acual bid daa and he represenaion of he bid sack as an inverse cdf funcion. Furhermore, he use of Gaussian, logisic, Cauchy or Weibull disribuions wih parameers linked o fuel prices can be undersood inuiively in erms of he mix of differen hea raes for differen generaing unis in he marke. For example, suppose ha for a given fuel, H is a random variable describing he variey of hea raes exising among differen generaing unis due o facors such as age and echnology. Suppose also ha consan fixed coss A exis for each generaor per MWh of power generaed. Finally suppose ha generaors make bids corresponding exacly o heir coss. Then he random variable X = A+HG describes he bid of a randomly chosen gas generaing uni given some gas price G. If H N(µ H, σ 2 H ), hen X N(A + µ HG, σ 2 H G2 ). This gives us precisely he Gaussian version of he model above, wih parameers α = A, α 1 = µ H, β = and β 1 = σ H. Wih fixed coss A > we expec α >, as we observe in our regressions for boh coal and gas. The observed mix of posiive and negaive values of β could be reproduced by leing A insead have a Gaussian disribuion correlaed wih H. 14 Of course, his argumen for hea rae disribuions would no work so convenienly mahemaically using oher disribuions, bu he inuiion sill holds. Ineresingly, he values we observe for α 1 (7.67 and 8.51 for NEPOOL and PJM respecively) also correspond closely o average PJM hea raes for gas generaors, lised by PJM as approximaely 7.3 MBu/MWh in 24 (PJM repor (25)). Similarly, α 1 =.48 maches equally well wih he coal hea hae of.378 /MWh used by Fehr and Hinz (26). This suggess he possible use of hea rae or cos daa as an alernaive o bid daa, hough i should be remembered ha sraegic bidding could also have an influence on he parameers. Ineresingly, recen work by Horacsu and Puller (27) on sraegic bidding suggess ha while marginal cos curves ofen conain prominen fla or verical secions, opimal bid curves are ypically smoohed o more closely resemble he bid sacks above. Thus, even for a marke wih a very narrow range of hea raes, sraegic bidding could resul in he fairly wide bid disribuions we observe. The resuls presened above for boh PJM and NEPOOL srongly suppor he overall framework of he bid sack model o connec elecriciy, gas and coal prices. However, here are several complicaions ha should be handled carefully when esimaing parameer values. Clearly, we need o find a compromise beween sriving for a perfec fi of he bid sack and a rough approximaion which capures he basic relaionships. The primary issues are he bes choices for b U, b L and w 1, and he corresponding impac on spo price behaviour. The value of w 1 is calculaed from he percenages in Table 1, and changes a six differen daes during he ime period, corresponding o imes of marke expansion. 15 The value of b L is much less significan han b U since he peak power price is much more likely o be se in he far righ han far lef of he sack. Tess of regressions using differen values of b U confirm he choices of.9 (NEPOOL) and.95 (PJM) as appropriae. Only a few hourly prices are se beyond hese poins, and Secion 4.4 complees our mehodology for capuring he far righ ail of he price disribuion, by compensaing for errors produced by runcaing he bid daa a b U. 4 Modelling he Price Drivers The bid sack model is supplemened by sochasic processes for he primary risk facors which drive he spo power prices: gas prices G (and coal prices P ), demand (or load) D, and capaciy available C. An advanage of he supply and demand approach is ha while choosing fairly simple processes for he underlying facors, we can sill replicae he unusual feaures of power prices hrough he 14 Of course he disribuions for H and A may change over ime for example as echnology improves, which suggess ha regressions over more recen daa migh be considered more useful. 15 We make he assume ha nuclear is grouped wih coal and hydro wih naural gas while oher is spli equally beween coal and gas, before removing b L =.2 from coal/nuclear and rescaling. This process for w 1 is hen esed by comparing wih he value of w 1 for each dae which minimises he sum of squared errors beween he cenral porion (25%-75%) of he model bid sack and he observed sack. Resuls lend suppor he mehod of choosing w 1 as described. 12

13 choice of bid sack funcion. 4.1 Fuel Prices For boh he PJM and NEPOOL markes, he imporan fuel price o model is he US naural gas price, as discussed earlier and confirmed by he resuls of he bid sack fi. Coal prices are less imporan, even for PJM, because hey are generally less volaile han gas, hey drive a flaer and lower region of he bid sack and hey are less significan in seing prices paricularly during peak hours. Neverheless, we propose a simple mehod for incorporaing some coal price informaion wihou increasing he complexiy of he model. In a more general seing, oher fuel prices and even carbon emissions prices can also be included, as is clearly necessary for European power markes oday Gas Prices We choose a sandard approach o modelling gas prices, by fiing log gas prices wih he Schwarz 2-facor model described by Schwarz (1997), 16 capuring he mean-reversion which is widely believed o exis in mos commodiy prices, as well as changes o he long erm equilibrium level of prices. Le X 1 and X 2 be he wo independen sochasic facors driving he spo gas price G, and h() be a seasonal componen. Dynamics under he risk-neural measure Q are given by dx 1 = κ(µ 1 X 1 )d + σ 1 dw, (6) dx 2 = µ 2 d + σ 2 d W, G = exp(h() + X 1 + X 2 ). Gas forward prices F G (, T) for delivery a a discree poin in ime T have value a ime given by E Q [G T ], he condiional expecaion of G T under Q. Hence we have: ( [ ]) log(f G (, T)) = log E Q e h(t)+x1 T +X2 T = h(t) + X 1 e κ(t ) + µ 1 (1 e κ(t )) + X 2 + µ 2 (T )... ( + σ2 1 1 e 2κ(T )) + 1 4κ 2 σ2 X(T ) (7) As X 1 and X2 are unobservable facors, we use he Kalman Filer o calibrae our model o available Henry Hub gas daa (see eg Schwarz (1997), Lucia and Schwarz (22) and Culo e al (26) in he elecriciy lieraure). The ransiion equaion is easily wrien in vecor form using he SDEs in (6). As liquid fuures or forward prices are widely available for he US gas marke, (7) forms he measuremen equaion, which imporanly is linear in he sae variables. NYMEX has provided us wih daily hisorical forward curves from January 2 hrough November 26. We use forward prices for mauriies of 1 monh, 3 monhs, 6 monhs, 1 year, 2 years, 3 years (and when available 4, 5 and 6 years), hus keeping 6 o 9 forward prices for each hisorical dae and reducing compuaion ime. We also assume ha he forwards maure a he middle of each monh, hough in fac here is a monh-long delivery period. The nex sep is o remove seasonaliy from he forward curves. Though he shape of he seasonal paern appears fairly consisenly hroughou he hisorical daa, he ampliude varies significanly over ime for boh he forward curve and log-forward curve. Thus, we canno easily remove he seasonaliy using he same funcion h(t) hroughou. Insead, we deseasonalise each forward curve independenly as follows. We firsly idenify he linear rend in he curve beyond he one-year mauriy poin, hus avoiding shor end deviaions. We hen le he curren monh equal he base monh, and calculae for every oher monh he average difference beween forwards wih mauriy in ha 16 This approach seems o give more reasonable parameer values han simply modelling G, paricularly for recen daa where he posiive skew in gas prices is significan. I also ensures ha gas prices remain posiive. 13

14 monh and he mos recen base monh forward. We accoun for linear rend in he process and average from he one-year poin onwards. Finally, we deseasonalise he forward curve by adding or subracing hese average monhly differences as appropriae, relaive o he mean of all monhs. As is ypical for he Kalman Filer, resuls are very sensiive o he choice of observaion noise σ 3. Hence we perform he opimisaion in wo sages, similarly o he mehod of Culo e al (26). For fixed σ 3, we implemen he filering wih respec o {κ, µ 1, σ 1, µ 2, σ 2 }, choosing iniial parameers X 1 =.7, X 2 = for Jan 2 (X 1 = 1.2, X 2 =.6 for Jan 23), and he iniial value of he condiional covariance marix of X o be.1i. Thus we find he parameer se ha maximises he oal likelihood of observing he enire hisory of forward curves. We hen vary σ 3 and rerun he filering for each new value, in order o minimise he sum of squared errors of all naural gas opion prices available on NYMEX. As hese are opions on forward gas conracs and forwards are lognormally disribued as in (7), opion prices (wih srike K, opion mauriy T 1, forward mauriy T 2, consan ineres rae r) are given in closed form by V G (, T 1, T 2 ) = e r(t1 ) [ E (F G (T 1, T 2 ) K) +] [e µ+ { ( )} { = e r(t 1 ) 12 log(k) µ σ 2 σ2 1 Φ K 1 Φ σ where Φ is he sandard Gaussian cdf, ( log(k) µ σ )}], ( ) µ = h(t 2 )+X 1 e κ(t2 ) +µ 1 1 e κ(t 2 ) +X 2 +µ 2 (T 2 )+ 1 ( ) 2 σ2 X(T 2 T 1 )+ σ2 1 1 e 2κ(T 2 T 1 ), 4κ and σ 2 = σ2 1e 2κ(T 2 T 1 ) 2κ ( 1 e 2κ(T 1 ) ) + σ 2 2(T 1 ). Table 3: Kalman Filer resuls for naural gas parameers Dae range κ µ 1 σ 1 µ 2 σ 2 σ 3 Jan - Mar Jan3 - Mar Jan - Nov Jan3 - Nov The resuls are shown in Table 3 for differen dae ranges. The parameers seem fairly sable over ime, hough he more recen daa is characerised by a slighly higher volailiy and faser speed of mean reversion for X 1 and larger negaive drif for X 2. Noe ha µ 2 < is necessary o accoun for he fac ha he gas forward curve was in backwardaion (downward sloping) in 82.5% of he observaions in he daase Coal Prices As PJM is roughly 4% fuelled by coal, changes in coal prices can have a significan effec on he level of power prices. However, as we have seen (e.g. Figure 6), coal prices move only gradually, suggesing ha volailiy is low, and herefore ha i is more imporan o capure he marke s expecaions of fuure rends in coal prices, han o capure he sochasic componen. Hence we use NYMEX Appalachian coal fuures curves, and ake he very simple approach of assuming ha he coal price a ime T in he fuure will exacly equal he fuures price F C (, T) wih mauriy T. Thus we have effecively assumed a deerminisic coal price model which is mached o he curren forward curve. While his seems arificially simple, i provides saisfacory resuls in he model wihou inroducing an addiional sochasic facor which is likely o have lile impac on power prices. 14

15 Table 4: Resuls of fiing PJM and NEPOOL demand (Aug 4 - Jul 7) Marke ˆκ Y ˆµ Y ˆσ Y PJM (RT) PJM (DA) NEPOOL (RT) NEPOOL (DA) Marke a 1 a 2 a 3 a 4 a 5 a 6 PJM (RT) PJM (DA) NEPOOL (RT) NEPOOL (DA) Demand (Load) Demand or load is easily observable in all elecriciy markes, and is ypically characerised by muliple periodiciies, a annual, weekly and inra-day levels. The lower lines in Figure 7 show average daily peak demand (real-ime) in PJM and New England respecively. As we use daily daa, we have averaged ou inra-day effecs and removed weekends and public holidays, so he primary seasonaliy remaining is annual. Peaks occur in boh summer and winer corresponding o higher air condiioning and heaing needs, wih summer peaks larger han winer peaks, paricularly for PJM. Iniially we model his deerminisic behaviour hrough a linear rend and a combinaion of wo cosine funcions wih periods one year and six monhs. However, daa shows ha a linear rend is only necessary for he early years of PJM when he marke expanded significanly in size. Thus we insead choose he hree year period Augus 24 o July 27 and se a 2 = for boh markes, as here is no significan rend. Finally, he deseasonalised process is fied by maximum likelihood esimaion using an exponenial OU process, as shocks o demand are considered o rever rapidly o he seasonal level. We work wih rescaled demand D hroughou, wih (b L, b U ) = (.2,.95) for PJM, and (b L, b U ) = (.3,.9) for NEPOOL. Modelling log( D ) ensures ha D mus remain posiive, as required. Hence we have log( D ) = f() + Y (8) f() = a 1 + a 2 + a 3 cos(2π + a 4 ) + a 5 cos(4π + a 6 ) dy = κ Y (µ Y Y )d + σ Y db where B is a Brownian Moion independen of W and W in he gas process. We assume hroughou ha fuel prices are independen of demand and capaciy, which flucuae on shorer ime scales and are driven by more local condiions; Pirrong and Jermakyan (25) also sugges his o be a reasonable assumpion, hough of course a prolonged cold spell over a large region is likely o impac boh power demand and gas prices. Table 4 liss he resuls for boh PJM and NEPOOL, and boh real-ime (RT) and day-ahead (DA) demand, wih = corresponding o June 1s 2. The resuls show ha mean-reversion raes κ Y and volailiy σ Y are higher for NEPOOL han PJM over he chosen period. Alhough we are ineresed in hourly spo prices, we fi our demand model only o daily peak average demand as inra-day movemens are fairly small and dominaed by he inra-day periodiciy. This can easily be incorporaed ino he model, for example wih hourly indicaor variables, bu has very lile effec. Inra-day demand movemens are ulimaely overshadowed by inra-day capaciy jumps, as discussed below. 4.3 Capaciy Available As explained in Secion 3, he process C (or C ) capures a variey of supply-side informaion relaing o ouages, ransmission consrains, expors, impors and oher power delivery issues. Therefore i 15

16 is no easily observable or even inuiively undersood, hough we can hink of i simply as he percenage of maximum capaciy available. Since we observe hourly hisorical prices, demand and bid sacks, we can calculae he implied capaciy available C imp which allows (1) o hold as closely as possible. 17 As B obs ( ) is non-decreasing, C imp is uniquely defined by 18 { ( ) } C imp = max c R + : B obs D S. c Jun- Dec- Jun-1 Dec-1 Jun-2 Dec-2 Jun-3 Dec-3 Jun-4 Dec-4 Jun-5 Dec-5 Jun-6 Dec-6 Jun-7 percenage of max capaciy demand implied capaciy available percenage of max capaciy Mar-3 Jun-3 Sep-3 Dec-3 Mar-4 Jun-4 Sep-4 Dec-4 Mar-5 Jun-5 Sep-5 Dec-5 Mar-6 Jun-6 Sep-6 Dec-6 Mar-7 Jun-7 demand implied capaciy available Figure 7: Daily average peak demand D (lower line) and implied capaciy available C imp (upper line), for PJM (lef) and NEPOOL (righ). (RT daa is ploed for PJM, while DA is used for NEPOOL.) The upper lines in Figure 7 show he hisorical evoluion of he daily peak average implied capaciy availabiliy C imp for boh PJM and NEPOOL. I has clear seasonaliy maching roughly wih demand seasonaliy, hus dampening he seasonaliy of prices. This is due o generaors mainenance schedules which are designed o avoid high demand periods. 19 C imp hus incorporaes boh expeced and unexpeced ouages. Using insead he runcaed bid daa and rescaled D and C involves solving for rescaled from (2) wihou breaking he condiion < D imp / C < 1. This holds as long as B obs (b L ) < S < B obs (b U ) for all hisorical daa, or equivalenly, b L < D /C imp < b U. However, for PJM, we find ha D /C imp >.95 for.41% of recen hourly daa, while for NEPOOL D /C imp >.9 for.6%. imp For hese observaions, we canno define C as above, wihou breaking he resricion D imp < C. mod However, we can always define he (rescaled) model-implied capaciy available C as he unique soluion o ( ) D S = B, C mod where B( ) is our model bid sack, as defined in (4) or (5). In general, we expec close o for he majoriy of daa bu o behave differenly in he ails. C imp C mod C imp o remain 17 Recall ha B obs is likely o be a sep funcion, alhough some generaors in PJM may bid coninuous curves making i really a combinaion of sep funcions and piecewise linear funcions. 18 Occasionally, we observe D > D /C imp, implying excess capaciy available in he marke relaive o normal maximum capaciy (C imp > 1 in Figure 7). While his could realisically be caused by impors or sligh demand elasiciy o price, i can be a modelling concern paricularly for low values of D. In some of hese cases our assumpion ha he bid sack is a funcion of demand over capaciy will lead o very high values of C imp such as 1.5 or 2. In hese cases, we adjus C imp by assuming ha he exra capaciy eners he marke only in he porion of he bid sack below where he price is se (as here would be no paricular reason for he exra capaciy o appear hroughou he irrelevan porion of he sack). Hence we se C imp = D +(exra capaciy in he righ of he sack)= D + 1 D /C imp. 19 While i migh be expeced ha bid daa should no include generaors known o be undergoing mainenance, hisorical bid daa is in fac observed prior o he incorporaion of scheduled ouages. 16

17 4.4 Margin Given our model for D in (8) above, modelling C separaely makes i difficul o saisfy our fundamenal requiremen ha demand does no exceed capaciy. One approach is o model D / C direcly, bu his prevens us from disenangling demand and supply effecs, and from using easily observable and well behaved demand daa D. Furhermore, i sill leaves us he problem of ensuring D / C (, 1). Insead we propose a sochasic process for he reserve margin M = C D, represening he amoun of exra capaciy available in he marke bu no needed o mach demand. By modelling boh D and M as sricly posiive processes, we auomaically fulfil he required condiion. Hence we define (rescaled) implied margin M imp = M imp M imp imp C D and Mmod = and (rescaled) model-implied margin Hourly hisorical daa for suggess he need for a wo-facor model for margin. The movemen of daily peak averages over weeks or monhs shows boh mean-reversion and some clear negaive correlaion wih demand D, as one would expec. Upward shocks o demand ofen lead o downward shocks o margin, hough marke mechanisms such as impors, exra capaciy reserves and ransmission facors can dampen he effec and reduce he correlaion. In addiion, inra-day hourly margin reveals quie noisy behaviour wih sudden and shor-lived jumps due o a variey of shor-erm effecs such as ouages. We are ineresed less in describing he precise iming or auocorrelaion of hese spikes han in describing heir magniude and likelihood. Therefore, we propose a simple regimeswiching model for log M consising of an OU process for he normal regime and an independen sample of a shifed exponenial random variable for he spike regime : C mod D M mod as: log( M ) = { Z OU Z SP wih probabiliy 1 p i wih probabiliy p i where he normal regime is given by Z OU ( = κ Z µz Z OU ) d + σz d B (9) and he spike regime is given by db d B = ρ d Z SP = α J, J Exp(λ i ), for seasons i = 1, 2, 3, 4. Thus, each value of Z SP (in pracice sampled hourly) is independen of previous values, and he probabiliy of being in he spike regime in any fuure hour is also independen of he curren regime. 2 Daa suggess ha he lef ail of he hourly margin disribuion for PJM is significanly hicker in he summer monhs, suggesing a higher chance of ouages. Therefore we fi seasonal spike parameers p i, λ i : i = 1, 2, 3, 4 where i = 1 corresponds o winer (Dec - Feb), i = 2 o spring (Mar - May), i = 3 o summer (Jun - Aug) and i = 4 o auumn (Sep - Nov). In order o esimae he parameers above, we firsly use daily average implied margin, which averages over inra-day spikes, o help beer idenify he behaviour of Z OU and especially is correlaion wih D. We esimae he parameers {κ Z, µ Z, σ Z, ρ} by maximum likelihood, condiioning on he observed daily value of demand D. 21 We use he same dae range as for he MLE of he demand process. We hen move o hourly daa o fi he spike regime hrough a momen maching procedure. In order o esimae he poin α beyond which he spike regime should operae, we exploi 2 Tess of hisorical daa using a more formal coninuous ime Markov Chain wih ransiion marix lead o an expeced duraion of say in he spike regime of approximaely 2 hours for PJM and 3 hours for NEPOOL. Thus a more complee regime swiching model for M leads o similar conclusions regarding he rapid speed of recovery from spikes. 21 We cap D imp / C imp a.99 o he avoid he rare cases of negaive M when D /C imp > b U and reduce he larges downwards spikes in margin. This ulimaely has lile impac as he lef ail of margin is capured in he second sage of esimaion. M imp 17

18 Table 5: Resuls of fiing PJM and NEPOOL margin (Aug 4 - Jul 7) Marke ˆκ Z ˆµ Z ˆσ Z ˆρ adjused ˆµ Z adjused ˆσ Z PJM (RT) NEPOOL (DA) Marke winer spring summer auumn ˆp i PJM (RT) ˆλ i ˆα R 2 of fi ˆp i NEPOOL (DA) ˆλ i ˆα R 2 of fi he fac ha Z OU has a Gaussian disribuion and hence a skew of zero. In conras, he hisorical imp disribuion of log M has significan negaive skew due o he hick lef ail caused by ouages. We remove as many daa poins as necessary o obain a non-negaive skew and choose he las poin removed o be our parameer esimae for α. Since he remaining hisorical disribuion (wih poins imp log M < α removed) is a more accurae represenaion of he invarian disribuion of Z OU, we re-esimae parameers for µ Z and σ Z o mach he firs wo momens of his disribuion. In oher words, we equae µ Z o he mean of he runcaed disribuion, and σ Z o is sandard deviaion muliplied by 2ˆκ Z, wih ˆκ Z as before. Essenially, we argue ha parameer esimaes ˆκ Z and ˆρ are well fied by our iniial procedure, while he iniial esimaors ˆµ Z and ˆσ Z are disored by he spikes and require adjusmen via momen maching. Therefore, { ( ( ) ) } ˆα = min α R : Skew log Mimp 1 {log( M imp ) α}, [ ( ) ] ˆµ Z = E log Mimp 1 {log( M imp ) ˆα}, ˆσ Z = ( ( ) ) 2ˆκ Z SDev log Mimp 1 {log( M imp ) ˆα}. The final sep of he parameer esimaion is o find seasonal spike regime parameers p i, and λ i, for i = 1, 2, 3, 4. Here we swich from using he lef ail of he disribuion M imp o ha of M mod. This key sep allows us o compensae for any errors made in fiing he ail of he bid sack, in paricular by no capuring he bids in he region (b U, 1). Though we may reain an arificially seep ail for he bid sack, we correc his by allowing he lef ail of he margin disribuion o be mod arificially sreched o produce he observed price spikes. Figure 8 shows log-hisograms of log M using he logisic disribuion (and as usual b U =.9 for NEPOOL and b U =.95 for PJM). The observed lineariy in he lef ail jusifies he use of an exponenial disribuion for he ouage regime. Clearly for PJM he summer monhs require a differen fi han oher seasons, hough for NEPOOL he difference beween seasons is much less. The parameers p i for each season are esimaed simply by finding he proporion of observaions below ˆα, while λ i is esimaed as he slope of an ordinary leas squares linear fi o he ail of he log-hisograms. 22 Table 5 liss all he esimaed parameers for he margin process (wih RT daa for PJM, and DA daa for NE), as well as he R 2 values for he linear fis o he ail, which are all above.9. Unlike demand, margin appears o be more volaile and spikier for PJM han for NEPOOL, wih higher values for κ Z and σ Z, as well as he spike regime probabiliies p i. Finally, PJM s values of p 3 =.174 and λ 3 =.712 confirm ha much larger and more frequen spikes occur in he summer, as also noed by Geman and Roncoroni (26). 22 Firsly, a sligh correcion is required for p i, since here exiss a posiive probabiliy q = Φ ˆα ˆµZ ˆσ Z of values below ˆα occurring in he non-spike regime. Hence, if p i is he percenage of observaions below ˆα, hen p i = p i q 1 q. Secondly, he regression o find λ i has been performed over he ranges [ˆα 3, ˆα] and [ˆα 1.5, ˆα] for PJM and NEPOOL respecively, each spli ino six equal widh probabiliy bins. 18

19 <-4 <-3.6 <-3.2 <-2.8 <-2.4 <-2 <-1.6 <-1.2 <-.8 <-.4 < <-5.6 <-5.2 <-4.8 <-4.4 <-4 <-3.6 <-3.2 <-2.8 <-2.4 <-2 <-1.6 <-1.2 <-.8 <-.4 < log of probabiliy winer spring summer auumn log of probabiliy winer spring summer auumn margin bin -9 margin bin Figure 8: Log hisograms of model implied log-margin log M for NEPOOL (lef) and PJM (righ), over he hree year period from Augus 4 o July 7. A his sage i easy o demonsrae he addiional convenience of choosing he logisic disribuion for our bid sack model in Secion 3. Firsly, for he one-fuel case of NEPOOL, our equaion for elecriciy spo prices, (4), can be rewrien as follows: S = α + α 1 G + (β + β 1 G ) (log D log M ) As S is linear in log M, for a given demand and gas price, he exponenially disribued lef ail of log-margin (from he spike regime) ranslaes o exponenially disribued price spikes as well. Moreover, for a fixed demand, wriing log M = log D S m 1 s 1, we can ranslae he margin spike hreshold α ino a price spike hreshold corresponding o a cerain number of sandard deviaions in he bid disribuion for gas. 23 Unlike ypical regime-swiching models for power prices (eg De Jong and Huisman (25), Weron e al (24)), we hus have an exponenial spike disribuion for power prices which shifs over ime as gas prices vary. For example, Figure 1 suggess ha while an hourly price of $15 could reasonably be considered a spike in early 24, i could no be in lae 25 when high gas prices caused daily average peak prices o approach hese levels. While formulas are no as simple in he wo-fuel case for PJM, he linear relaionship beween S and log M sill holds approximaely for he ail, and hence he spike regime. This follows because he influence of he coal disribuion is negligible in he far righ of he bid sack, so we can hink of using a one-fuel model as an approximaion. Then log M log D S m 2 s 2, (1) where M and D (and C below) represen a second rescaling of demand and margin from (b L, b U ) o he gas-dominaed porion of he bid sack (b L + w 1 (b U b L ), b U ). The rescaling echnique inroduced 23 Ignoring he role of demand in deermining price spikes is an approximaion, bu demand varies much less han D margin. As he bid sack is an increasing funcion of D + M, demand shocks combined wih margin shocks will clearly produce higher spikes han margin shocks alone. 19

20 in (2) also implies [ ( ) ] 1 log( M ) = log D D D / C [ ] 1 w 1 = log( D ) + log D / C 1 w 1 = log( D ) + log 1 D D + M D w D + M 1 = log( D ) + log( M ) log ((1 w 1 ) D ) w 1 M log( D ) + log( M ) log( D ) log(1 w 1 ) where he las line holds for small margin M. Thus, in he righ ail of he bid sack, using (1), log M log D + log(1 w 1 ) S m 2 s 2. The approximae linear relaionship beween S and log M again suggess ha, holding gas and demand consan, price spikes caused by margin spikes are also exponenially disribued. 5 Empirical Resuls The performance of he model can be evaluaed according o several differen crieria for each of he wo markes considered. We firsly compare he properies of simulaed elecriciy price pahs wih hose observed in he marke, as is suggesed by Geman and Roncoroni (26). Typical saisics such as mean, variance, skew and kurosis are considered, bu also oher key feaures in power markes, such as correlaion wih fuel prices and he probabiliy of spikes above cerain hreshold prices. We make comparisons of price series a various ime-scales, since saisics such as correlaion are mos relevan in erms of daily, weekly or even monhly averages. Though mos of our sae variables are mean-revering, he componen X 2 of gas prices is simply a Brownian Moion wih drif, implying ha no invarian disribuion for spo prices S exiss. Hence some saisics are likely o be unsable over ime and should be analysed wih care. Our second ool for evaluaing model performance is a comparison of model-implied forward prices and observed forward prices. In power markes, i is paricularly imporan for a model o capure he forward curve accuraely, a comparison we make for he sample daes 3 December 25, 31 March 26, and 29 Sepember 26. These form a fairly represenaive sample as hey correspond o imes of high, medium and low gas prices respecively, and hus will also provide hree differen saring poins for he simulaion analysis. Throughou his secion, we use he logisic disribuion for he bid sack model (b U =.9 for NEPOOL, b U =.95 for PJM), wih parameers given in Table 4. For he PJM gas disribuion, we use he regression resuls for {α, α 1, β, β 1 } over he enire daase June 2 - July 27. However for he coal disribuion, his leads o a consisen underesimaion of ˆm 1 and ŝ 1 during 26. This could be due in par o esimaion difficulies during periods of significan gas and coal disribuion overlap (24-5), or perhaps o a gradual change in he hea raes of coal generaors over ime. As a resul we use insead he regression resuls for { α, α 1, β, β 1 } over he mos recen period June 25 o July 27. For NEPOOL, we use he parameers {α, α 1, β, β 1 } from regression over he enire period March 23 o Augus 27. For he gas process G, we use parameers in Table 5 for January 2 - November 26, making use of he mos daa. Finally, for demand and margin, we use he parameers in Tables 6 and 7, noing ha forward prices (aken from NYMEX) are seled using real-ime prices for PJM, bu day-ahead prices for NEPOOL. 2