SCENARIO SELECTION PROBLEM IN NORDIC POWER MARKETS. Juha Ojala. Systems Analysis Laboratory, Helsinki University of Technology

Transcription

1 SCENARIO SELECTION ROBLEM IN NORDIC OER MARKETS MAT-2.08 INDEENDENT RESEARCH ROJECT IN ALIED MATHEMATICS Juha Oala Sytem Analy Laboratory, Heln Unverty of Technology ortfolo Management and Tradng, Fortum Corporaton

2 Scenaro Selecton roblem n Nordc ower Maret TABLE OF CONTENTS INTRODUCTION OER RICE DETERMINATION NORDIC OER MARKET BALANCE OF SULY AND DEMAND MOST IMORTANT FACTORS AFFECTING RICE FORECASTING OER RICE GENERATION OF RICE SCENARIOS SCENARIO EIGHTS SCENARIO SELECTION ROBLEM MOTIVATION DECREASING THE NUMBER OF SCENARIOS BINOMIAL OTIMISATION ROBLEM ROFIT AND RISK OF A ORTFOLIO ROFIT OF A ORTFOLIO ROFIT AT RISK VaR veru ar Calculaton wth Scenaro Analy FORMULATION OF THE OTIMISATION ROBLEM MEASURES FOR THE QUALITY OF THE SOLUTION FORMULATION OF CONSTRAINTS Formaton of Scenaro Subet Calculaton of New eght FORMULATION OF THE OBJECTIVE FUNCTION ERROR ANALYSIS Error n ower rce Error n ar wth Lnear ortfolo Improvng Accurary of ar SUMMARY OF THE METHOD Intalaton Frt Optmaton roblem Second Optmaton roblem Thrd Optmaton roblem COMLEXITY ANALYSIS ILLUSTRATIVE EXAMLE

3 Scenaro Selecton roblem n Nordc ower Maret 7. MODEL FOR OER RICE SELECTING THE SUBSET OF SCENARIOS Frt Two Optmng Round Thrd Optmng Round COMARATIVE AR CALCULATIONS CONCLUSION... 3 AENDIX SUMMARY OF THE NOTATION ROOF OF THE EQUATION GRAHS AND TABLES RELATED TO THE EXAMLE REFERENCES

4 Scenaro Selecton roblem n Nordc ower Maret INTRODUCTION Snce the deregulaton of the Nordc power maret tarted from Norway n 99, the table electrcty prce have turned to extremely volatle pot maret prce. In the new envronment, producton plannng of hgh mportance. By tmng ther producton correctly, generator can tae advantage of the pea n the power prce. However, wth ncreaed proft poblte come ncreaed r. The purpoe of r management to evaluate the ze of the poble downde n proft, and ether reduce the r or prepare for t. In order to plan t producton or to etmate t r, a maret partcpant ha to have a comprehenon of how the prce are lely to develop. Becaue of the many uncertan factor, t not poble to gve an accurate forecat for longer tmer perod. A commonly ued way to expre the uncertanty n the prce forecat to ue a et of prce cenaro. Every cenaro repreent one poble tate of the power maret, and wth many varable n the maret, the et of cenaro ealy grow n hundred. From the prce cenaro, a maret partcpant can etmate the r of t portfolo. The r meaure value at r can be ued to repreent the poble lo of value of a portfolo of fnancal power contract. However, mot maret partcpant are more ntereted n the poble downde n ther proft, whch can be decrbed wth the commonly ued r meaure proft at r. th ome confdence level, t tell what the maxmum poble dfference between expected and actual proft. Both value at r and proft at r can be traghtforwardly calculated from the et of power prce cenaro. roducton plannng can be done n numerou way, rangng from heurtc method to tochatc optmaton ee, e.g. Santze and Ilc, 200. If cenaro analy ued, the producton plan done o that the expected future proft over the et of cenaro maxmzed. The complexty of th ta depend on the number and type of the producton aet, on the length of the plannng perod, and on the number of prce cenaro. However, th may be a nonlnear optmng problem wth thouand of decon varable. In order to mae the optmaton computatonally vable, t may be neceary to reduce the complexty of the problem. One way to do th to decreae the number of cenaro ued. 4

5 Scenaro Selecton roblem n Nordc ower Maret Th paper concentrate on the followng problem: From a et of cenaro, how to elect a ubet of cenaro uch that the tattcal properte of the cenaro et change a lttle a poble. Later on, the problem formulated n more detal. Th a necety nce the oluton of the problem heavly depend on the tructure of the cenaro, on the underlyng dynamc of the ytem, and on the purpoe for whch the cenaro are ued. However, I feel confdent that the method decrbed n th paper can be ued alo wth a wder cla of cenaro electon problem. The ret of th paper organed a follow: Secton two provde a hort overvew of the Nordc power maret, cenaro electon problem and the proft at r meaure. In ecton three, the problem formulated n detal, and n ecton four the propoed oluton method preented. Secton fve preent an llutratve example, and ecton x conclude the paper. 5

6 Scenaro Selecton roblem n Nordc ower Maret 2 OER RICE DETERMINATION In th ecton the determnaton of power prce dcued. Frtly the buldng bloc of the Nordc power maret and the bac of prce determnaton are preented. After that the mot mportant factor affectng power prce are dcued. 2. Nordc ower Maret By January 2003, the Nordc power maret wll be fully open to competton, currently 65% of Danh power maret tll regulated Nord ool 2002, p. 8. The power maret cont of retal maret and wholeale maret, whch can be furthermore dvded nto OTC maret over the counter, blateral maret and power exchange maret. In th paper the dcuon concentrate on the power exchange maret, becaue the power prce n the other maret cloely follow the prce n exchange maret. Fgure preent the dfferent component of the Nordc power maret. In the power exchange Nord ool, a power pot prce calculated for each hour. The mot mportant pot prce the ytem prce, whch calculated wth the aumpton that there are no tranmon contrant between dfferent countre or regon. In order to tae nto account the tranmon contrant, the Nordc countre are dvded nto dfferent prce area. If the tranmon capacty exceeded, the prce n dfferent area change o that n the new maret equlbrum the tranmon capacty uffcent. In addton to the power exchange Nord ool, tranmon ytem operator TSO 2 and maret partcpant are the ey element of power maret. Grd owner operate regulated monopole n ther area and do not tae part n the everyday competton. Maret partcpant generally fall nto the categore of generator, retaler, enduer, or trader. The role of the ey element can be roughly defned a follow ower generaton and power tranmon are completely eparated n Nordc countre. Generaton open to competton, wherea tranmon regulated becaue grd owner operate natural monopole. The term 'power maret' ued to refer to the compettve power generaton maret. 2 There one TSO n Fnland, Sweden, and Norway, and two TSO n Denmar. 6

7 Scenaro Selecton roblem n Nordc ower Maret The power exchange operate the phycal pot maret and an organed maret for fnancal product. Exchange alo act a a neutral contract counterpart to maret partcpant. The TSO man reponblte are to operate and mantan the man grd and to handle non-predctable mbalance of upply and demand durng real-tme operaton. In addton of upplyng power, generator contrbute to levellng prce dfference between the maret by operatng n dfferent wholeale maret. Retaler erve the end-uer baed on ther own producton or power purchaed n wholeale maret. All power conumer are end-uer. Larger end-uer may operate n wholeale maret, whle retaler erve maller uer. Trader trade phycal and fnancal contract 3 to proft on prce dfference and volatlty. OTC Maret Large ndutry Generator Trade & clearng repreentatve Retaler Blateral holeale Maret Fnancal Contract hycal Contract Talor made or tandarded Nord ool Fnancal Maret hycal Maret Large ndutry Generator Trade & clearng repreentatve Retaler Retal Maret Small cale ndutry Servce bune Houeh Fgure. Nordc power maret Nord ool 2002, p.3. 3 Fnancal power contract are contract whch payoff depend on the power prce, but whch do not nclude phycal tranmon of power. 7

8 Scenaro Selecton roblem n Nordc ower Maret 2.2 Balance of Supply and Demand In the Nordc power maret, a n any compettve maret, the power pot prce determned from the balance of upply and demand. That, the prce correct when the amount of power generator are wllng to upply and the amount of power conumer are wllng to buy are equal. Graphcally the maret equlbrum can be preented a the nterecton of upply and demand curve. Fgure 2 preent the etmated upply and demand n the Nordc maret n the year 2005 VTT Energy 999, p. 85. Demand practcally prce nentve becaue power uch a vtal commodty. From the nterecton of upply and demand curve n Fgure 2, the power prce n an average hydropower year can be etmated to be 8 /Mh. However, a dcued n the followng ecton, the varance of the prce large compared to the expected prce. 70 EUR/Mh Hydro & wnd Hydro varance Nuclear power Combned producton Condenng, coal & ga Condenng, ol Ga turbne Demand Th/year Fgure 2. Etmated power upply and demand n the Nordc maret n the year 2005 VTT Energy 999, p Mot Important Factor Affectng rce The mot mportant factor affectng yearly demand temperature. Anyway, at a yearly level demand relatvely table, and the mall movement of the demand curve doe not change the power prce remarably. On the upply de, the amount of 8

9 Scenaro Selecton roblem n Nordc ower Maret hydropower avalable water nflow nto reervor and the coal maret prce are the two mot mportant factor affectng power prce. From the Fgure 2 the effect of water nflow and coal prce on power prce can be etmated. If, for example, the yearly nflow would equal only 50Th ntead of the normal 80Th, the upply curve would move 30Th to left, and the new prce level would be around 50 /Mh. Change n coal maret prce change the the varable cot of power generated by coal condenng. Th hft the coal condenng phae of upply curve hgher, and the maret equlbrum prce change. For example, 20% ncreae n the varable cot of coal condenng would re the power prce 20%. Thee artfcal example do not tae account the effect of mport, whch would balance the prce omewhat. However, they demontrate the hgh volatlty of power prce. Fgure 3 llutrate htorcal daly ytem prce. In addton to the yearly varaton, ytem prce alo exhbt heavy eaonal and daly varaton. The eaonal prce varaton manly due to temperature varaton and the uneven tmely dtrbuton of water nflow. Thee ue caue prce to be low durng ummer and hgh durng wnter. The mot mportant factor caung daly varaton of prce are publc hay and temperature. ublc hay, le weeend, caue the prce to come down gnfcantly. The effect of temperature motly een n the hgh pe of power prce, whch are uually caued by extremely c weather. 9

10 Scenaro Selecton roblem n Nordc ower Maret EUR/Mh Day Fgure 3. Htorcal daly average of ytem prce 4 from..995 to Becaue the hay are nown beforehand, and the coal prce varaton relatvely low, nflow and temperature are the two mot mportant factor caung unpredctable power prce varaton. 3 FORECASTING OER RICE In th ecton the medum term few year forecatng of power prce dcued. A dcued n the lat ecton, the power prce n Nordc maret hghly dependent on water nflow and alo on temperature. Thee factor are almot random varable, and thu t not poble to gve one prce forecat. Intead, uually a et of prce cenaro 5 contructed to tae account of the randomne. 3. Generaton of rce Scenaro In many maret htorcal prce can be ued a prce cenaro. However the problem n the Nordc power maret that there not much htorcal maret prce data, and even the uefulne of extng data dubou becaue the maret ha changed contnuouly a the deregulaton ha proceeded. 4 The orgnal data ha been converted from Norwegan rone to euro wth rate NOK/EUR. 5 A cenaro repreent a poble future tate of the world. In the power maret, a cenaro could, for ntance, cont of weely prce for the next four year. The probablty of a cenaro may not be nown, but the relatve probablte of the cenaro can uually be etmated. 0

11 Scenaro Selecton roblem n Nordc ower Maret In fnancal lterature tochatc procee le the Brownan movement are often ued a the prce model ee, e.g., Luenberger 998. However, n power maret th nd of model would wate a lot of uable nformaton, nce a lot of the underlyng factor of the power prce are predctable. E.g. publc hay, coal prce n the near future, generaton capablte etc. are not random varable. Alo the charactertc of the behavour of temperature and water nflow are nown. One ueful way to generate prce cenaro to tae water nflow and temperature cenaro a the tartng pont. A noted n the lat ecton, thee are the two mot mportant factor caung unpredctable varaton n power prce. There ext a lot of htorcal data of temperature and water nflow, o the generaton of thee cenaro not a problem. The et of htorcal cenaro denoted wth S H, S H {, K } =, 3- H,, H, N S H where N ued a a ze functon, NX = number of element n the et X 6. Each htorcal cenaro could cont, for example, of nflow and temperature tme ere from dfferent part of the Nordc countre From the htorcal cenaro, pot prce cenaro are generated ung ome model of the Nordc power maret. The exact model wth whch the pot prce cenaro are generated are out of the cope of th paper. However, a good model hould tae account of the nown dynamc of the maret and the predctable factor affectng power prce. Of coure, uch model wll nevtably ental a et of parameter whoe value are not exactly nown. Example of uch parameter are the future fuel prce level, hydropower generatng polce, rate of economc growth for model wth long tme frame etc. In order to tae account of the uncertanty of the parameter, more than one pot prce cenaro can be generated from each htorcal cenaro by ung a dfferent et of parameter. Durng the ret of th paper t aumed that the model for power prce of the followng type prce t = f t, htorcal, parameter Table 3 n Appendx ummare the notaton related to the dfferent cenaro et.

12 Scenaro Selecton roblem n Nordc ower Maret That, for each pot prce cenaro the prce at any tme t a functon of the tme t, the underlyng htorcal cenaro and the et of model parameter. For each htorcal cenaro, m pot prce cenaro are generated ung m dfferent et of parameter. Totally are generated m et of NS H cenaro, S { K, }, {, m},,, N S H K, =. 3-3 The total et of pot prce cenaro denoted wth S, {, } m U S =, m N SH. 3-4 = S = K 3.2 Scenaro eght For mplcty, t aumed that each htorcal cenaro condered to be equally probable. However, the et S may not be equally probable becaue of the dfferent parameter value ued. In order to tae account of the dfferent probablte, t aumed that the cenaro are weghted to reflect ther relatve lelhood. The weght are denoted wth,. The ubcrpt ued becaue the cenaro electon proce may change the weght of cenaro. Becaue the htorcal cenaro were aumed to be equally probable, {, m}, = w, K,. 3-5 For notatonal mplcty and wthout lo of generalty, t can be aumed that the um of weght, N S = = SCENARIO SELECTION ROBLEM The prevou two ecton concentrated on the power prce determnaton and how cenaro can be ued n forecatng prce. In th ecton the problem of cenaro electon ntroduced and dfferent oluton method are preented. Fnally the oluton preented a an optmaton problem wth tll unnown obectve functon and contrant. 2

13 Scenaro Selecton roblem n Nordc ower Maret 4. Motvaton The power maret partcpant ue power prce cenaro at leat for two purpoe. Frtly, power generator ue cenaro n producton plannng. Th epecally mportant wth the lmted hydropower reource. Secondly, all maret partcpant can etmate the r of ther portfolo producton, procurement, fnancal etc. wth cenaro analy. R calculaton do not uually nvolve optmaton, thu they are computatonally eay. roducton plannng, ntead, may nclude the optmaton of hourly producton of ten of generatng aet over the next year. Th may be a nonlnear optmaton problem wth thouand of decon varable. In order to mae the optmaton computatonally vable, t may be neceary to reduce the complexty of the problem. One way to do th to decreae the number of cenaro ued. 4.2 Decreang the Number of Scenaro There are two way to decreae the number of cenaro n a cenaro et S:. Create a new, maller cenaro et, whch tattcal properte are mlar to the et S. 2. From the et S, elect a ubet of cenaro uch that the tattcal properte of the ubet are mlar to the et S. Høyland and allace 200 preent a method baed on nonlnear programmng that can be ued to generate a et of cenaro, whch atfy pecfed tattcal properte. Even though the method alo applcable for multperod problem, the queton how to elect the relevant tattcal properte. A noted n the prevou ecton, the dynamc of power maret are not mple and cannot be tated wth any mall or even large et of tattcal properte. Moreover, the large amount of dered tattcal crteron for hundred of perod would lead to a complex, nonlnear optmng problem. In the method decrbed n th paper, the ze of the cenaro et decreaed by electng a ubet of cenaro ntead of creatng a new et of cenaro. The maor problem n th approach what are the relevant tattcal crteron for the electon proce. In the preented method th queton bypaed by utlzng the tructure of 3

14 Scenaro Selecton roblem n Nordc ower Maret the cenaro et n a way that quarantee that all the tattcal properte reman relatvely unchanged. 4.3 Bnomal Optmaton roblem The problem can be defned a follow: From the orgnal et S of NS cenaro, elect a ubet G of NG cenaro uch that the eental properte of the cenaro et would be a mlar a poble. The tructure of the orgnal cenaro et the one preented n Secton 3. The oluton for th problem can be formulated a the followng bnomal optmng problem: { x K } mn f X, X =, x N S, uch that x { 0,} x X and poble addtonal contrant are fulflled. The decon varable x ha the value f the cenaro elected, otherwe x ha the value 0. The dcuon n the next ecton concentrate on the queton of what are the eental properte of the cenaro et. Baed on the analy, the optmaton problem formulated n Secton 6. 5 ROFIT AND RISK OF A ORTFOLIO In order to formulate the obectve functon and the contrant of the optmaton problem preented n lat ecton, the eental properte of the cenaro et have to be nown. Becaue the et of prce cenaro ued n producton optmaton and r calculaton, the electon hould be done uch that the reult of producton optmaton and r calculaton would not change. In th ecton the relaton between power prce and the proft of a portfolo dcued, and the wdely ued r meaure roft at R ntroduced. 5. roft of a ortfolo The portfolo n conderaton could be, for ntance, a portfolo of power generaton aet or a portfolo of purely fnancal power contract. Here two nd of portfolo are condered, one wth coal condenng plant and one wth hydropower plant. 4

15 Scenaro Selecton roblem n Nordc ower Maret roducton plannng for coal condenng plant relatvely mple. If the power prce above the varable cot of producton, the plant hould be ued n generaton 7. Of coure, one ha to tae nto account the tartng cot of the plant and poble contrant regardng the amount of fuel the plant can or ha to conume. However, n prncple, when power prce above the varable cot of producton, the proft lnearly dependent on the power prce, and the power prce level the mot mportant factor determnng the proft. The logc behnd the producton plannng of nuclear power and many conventonal producton form except from hydropower mlar to coal condenng power. Thu, except from hydropower, the prevou cae cover almot all producton n the Nordc area. For hydropower plant, the relaton between power prce and proft much more complex. There are two man reaon for that. Frtly, the water nflow lmted and unpredctable, and econdly, the capacty of reervor lmted. The lmted water nflow mean that t not poble to generate hydropower alway when power prce above the varable cot whch, n prncple, alway the cae, becaue producer would oon run out of water. Intead, t reaonable to produce hydropower only when the prce hgh. However, the lmted reervor capacty mean that ometme generator are oblged to produce even f the prce low, becaue otherwe there would not be room for any future nflow. Thu, the ey queton n plannng the hydropower producton conder tmng: when to generate power from the lmted upply of water n the reervor whle tang nto account the uncertan future nflow. To anwer that queton, producer may etmate o called margnal value of water, whch tell the monetary value of one addtonal unt of water n reervor. If the power prce hgher than the margnal value of water, power hould be generated. Of coure, dffcult part the calculaton of the margnal value of water, whch a functon of the current amount of water n reervor, predcted future nflow and power prce. Eentally, the margnal value 7 In ome tuaton a generator may not ue ome generatng aet even f the prce above the varable cot of producton. By not producng, the generator can re the power prce and thu ncreae the proft of t other generatng aet. 5

16 Scenaro Selecton roblem n Nordc ower Maret of water, and thu the producton plan, hghly dependent on cenaro path, not ut average level of power prce. The prevou dcuon howed that the average power prce not the only thng affectng proft; alo the development of the prce over tme mportant. 5.2 roft at R In th ubecton the r meaure proft at r ar preented and a method to calculate t wth cenaro analy ntroduced VaR veru ar erhap the mot ued meaure of r value at r Joron 200. Let the be the random value of an aet e.g. bond after ome elected tme perod. The abolute value at r abolute VaR wth confdence level α can then be defned by VaR α = f w dw, 5- where fw the dtrbuton of. Thu, abolute VaR the mnmum value of the aet after the elected tme perod wth confdence level α. Uually t more nformatve to calculate the relatve VaR, whch relate the mnmum poble aet value to t expected value, E, VaR relatve = E VaR abolute. 5-2 However, power generator are not very ntereted n the value of ther aet, whch are motly power plant 8. After all, they have decded to tay n the power generaton bune and wll thu own ther plant whatever ther value may be. Intead, they are more ntereted n the future proft ther bune generate. Analogouly to abolute value at r, the abolute proft at r can be defned a the mallet poble proft wthn ome tme nterval and confdence level. Let be the random proft of ome tme perod. Then the abolute proft at r wth confdence level α can be defned by ar α = f p dp, In fact, contnuou etmaton of the value of power plant mpoble, becaue n prncple there are no maret for ued power plant. Thu, there are no maret value. 6

17 Scenaro Selecton roblem n Nordc ower Maret where fp the proft dtrbuton of the elected tme perod. Once agan, t often more nformatve to relate the abolute mnmum proft to the expected proft, E. The relatve ar can be defned a ar relatve = E ar abolute. 5-4 Durng the ret of th paper, ar ued a a ynonym of relatve ar Calculaton wth Scenaro Analy Aume that the et S of n cenaro ued to repreent the poble future tate of the world. The cenaro are weghted uch that the weght um up to, and the weght of a cenaro denoted wth. Furthermore, aume that for each of the cenaro the proft of an aet can be calculated to be R. For mplcty of notaton, aume that cenaro are orted uch that R R. Then, accordng to the + defnton 5-4 and 5-3, the ar of the aet wth confdence level α can be calculated to be ar = E R S R, uch that α and > = = α 5-5 where n E R S = R. = The reult calculated wth the equaton 5-5 are drectly dependent on R. th mall number of cenaro the proft dtrbuton uually pare n the tal, and the ar value thu heavly dependent on the proft outcome n that ngle cenaro. A more robut way to calculate proft at r to meaure the downde by expected proft over the et of cenaro wth mallet proft. ar = E R S uch that = = R / 2α and = =, > 2α. 5-6 Th not n accordance wth the equaton 5-3, only a reaonable approxmaton. However, the ar value calculated wth 5-6 hould be more table than the value calculated wth 5-5. Th becaue the downde meaured by the expected proft over a et of cenaro rather than by proft outcome of one cenaro. To be more 7

18 Scenaro Selecton roblem n Nordc ower Maret exact, the method 5-6 aume that the proft dtrbuton n the tal flat. If the dtrbuton get thnner to the end, then 5-6 overetmate ar. 6 FORMULATION OF THE OTIMISATION ROBLEM The ey queton to be anwered how the goodne of the oluton meaured. How to meaure whether two et of cenaro are mlar? It clear that a pot prce cenaro can be charactered by t average prce and path form. Here the cenaro path form undertood to mean the tmely dfference of the pot prce and the average prce. Fgure 4 llutrate th dea. In the lat ecton t wa hown that thee two ue, the average prce and the prce development, are mportant factor determnng the proft of a portfolo. Value Tme Contant value dfference Fgure 4. Two cenaro wth the ame path form but wth dfferent average value. 6. Meaure for the Qualty of the Soluton In the problem n queton, each cenaro contan one pot prce tme ere wth tme pan of everal year. The cenaro are generated ung htorcal tme ere of nflow and temperature and a et of model parameter ee Secton 3. Thu, each cenaro charactered by the htorcal tme ere, the pot prce cenaro, and the et of parameter ued. In order to cut down the dmenon of the problem, the followng two aumpton are done: Aumpton. hen cenaro are ued n ar calculaton or n producton plannng, t aumed that the et of parameter behnd each cenaro not of concern. 8

19 Scenaro Selecton roblem n Nordc ower Maret Aumpton. It aumed that for every cenaro the relaton 3- can be wrtten a follow, prce t = f t,, htorcal, parameter = h t,, htorcal + g, htorcal, parameter. 6- The frt aumpton tate that after a pot prce cenaro generated, the et of parameter ued not of concern. The econd aumpton mean that the dfference n power prce between cenaro generated from the ame htorcal tme ere but wth dfferent et of parameter contant. Thu, the prce cenaro path form are the ame. Th aumpton reaonable nce the analy of ecton 2.3 howed that water nflow and temperature are the mot mportant factor affectng power prce varaton. Other factor, le coal prce, affect motly the level of the prce. After the prevou aumpton, the number of dfferent prce cenaro path form the number of htorcal cenaro, NS H. Moreover, there are m cenaro path of each form becaue of the m dfferent et of aumpton. If the electon done uch that both the average prce dtrbuton and the path form dtrbuton 9 tay cloe to the orgnal, alo all the tattcal properte tay cloe to the orgnal perhap except for the correlaton between average prce and path form. It hould be clear that thee two obectve, to eep the average prce dtrbuton and the path form dtrbuton cloe to orgnal, can be contradctory. Thu, mot probably there not a electon n whch both the average prce dtrbuton and the path form dtrbuton would be dentcal to the orgnal. Becaue of that both crteron can not be treated a contrant. Intead, one of them ha to be taen a the obectve functon and the other a a contrant. There not a way to ay whch one more mportant, the average prce dtrbuton or the path form dtrbuton. It alway depend on the purpoe for whch the cenaro et ued. However, f the path form dtrbuton contraned, t would confne the total number of cenaro elected to be a multple of NS H. Becaue of th, the dtrbuton of cenaro path form wll not be contraned. Intead, the target functon of the problem wll be formulated n uch a way that the path form dtrbuton tay cloe to orgnal. 9 The path form dtrbuton mean the number of cenaro of each path form. The Average prce dtrbuton can be condered a a htogram of the average prce. 9

20 Scenaro Selecton roblem n Nordc ower Maret 6.2 Formulaton of Contrant In order to eep the average prce dtrbuton cloe to the orgnal, the et S dvded nto ubet of cenaro wth relatvely mlar 0 average prce. From each of thee ubet, one cenaro elected to repreent the whole ubet. The weght of the elected cenaro et to be the um of the weght of the cenaro n the ubet Formaton of Scenaro Subet The dvon of the et S nto ubet can be done n multple way. One method would be to do the dvon uch that the um of weght of the cenaro n dfferent ubet would be the ame. However, aumng that the weght of cenaro wth hgh or low pot prce value are maller than the weght of cenaro wth moderate value, th method would elect more frequently cenaro wth moderate prce average. From the vewpont of r analy th not derable, becaue the tal of the dtrbuton are uually more nteretng than the centre. Another method would be to fx the number of cenaro n a ubet. Th way cenaro would be elected a frequently from the tal a from the centre of the dtrbuton. However, the way ued n th paper to ue a fxed average prce range a a crteron, uch that the prce range of each ubet for example at mot 0% of the whole range. Aumng that the tal of the proft dtrbuton are more pare than the center, th method wll elect cenaro wth low or hgh average prce more frequently than cenaro wth moderate average prce. The ubet of cenaro are denoted wth Z. Thu, n ubet Z are formed uch that wth ome fxed ε {, n} ε, Z K,, 6-2 where the average pot prce of cenaro. For mplcty of notaton, t aumed that the et S orted uch that. Addtonally t can be + demanded that the ze of one et Z may not exceed ome fxed fgure. Th may be we f otherwe the weght of one cenaro would be exceptonally hgh becaue of a large ubet. Thu, t can be demanded that {, n} N Z C K,, Th looe expreon pecfed n the followng ubecton. 20

21 Scenaro Selecton roblem n Nordc ower Maret wth ome fxed C. The neceary contrant for the decon varable can be wrtten wth x {, n} =, K, Z Calculaton of New eght hen the dtrbuton of elected cenaro retrcted le decrbed n the prevou ubecton, the new weght of the elected cenaro can be calculated wth new Z {,,n} Z =, K, 6-5 Thu, the weght of a cenaro the um of the weght of all the cenaro n the ubet from whch the cenaro wa elected. The calculaton of the new weght n th way ha two mportant mplcaton. Frtly, the total um of weght n the elected ubet of cenaro the ame than n the orgnal whch wa aumed to be. Secondly, the new weght of cenaro can be calculated before actually electng cenaro, becaue the new weght of cenaro n the ame ubet are the ame. 6.3 Formulaton of the Obectve Functon In the orgnal et of cenaro the NS H ubet of cenaro wth the ame path form have the ame um of weght, that m = = N S, {, K, N S }, H H. 6-6 Th evdent from the Equaton 3-2. A noted n the prevou ecton the um of the new weght the ame a the um of the weght. Now the natural thng to do to dtrbute that weght unformly nto the NS H ubet of cenaro wth dfferent path form. For one ubet, the weght dfference between the orgnal et of cenaro and the elected ubet can be calculated wth m = x, new, N S H, 6-7 where x, the decon varable aocated wth,. The dfference are ept cloe to zero by mnmng the followng functon 2

22 Scenaro Selecton roblem n Nordc ower Maret N S H m = = x, new,. 6-8 N S H Quadratc functon would perhap be more utable, becaue t would penale larger dfference more. However, becaue the problem may nclude hundred of bnary decon varable, t neceary to eep the target functon lnear. Functon 6-8 can be ealy turned nto a lnear target functon by ntroducng NS H new decon varable A and replacng the formula 6-8 wth N S H = A, A {, K,N S } R. 6-9 H The A are contraned uch that the value of 6-9 can not be maller than the value of 6-8, A m = x, new, N S H, and 6-0 A N S H m = x {, K, N S }, new, H. 6- On the other hand, mnmng the functon 6-9 guarantee that n the optmum t value equvalent to the value of Error Analy Baed on the analy of the frt two followng ubecton, the target functon of the optmaton problem changed uch that the maxmum error of ar wth lnear portfolo get maller Error n ower rce It relatvely mple to prove that the maxmum dfference between the expected average power prce over the orgnal et of cenaro, ES, and over the elected ubet of cenaro, EG, ε, where ε the parameter of equaton 6-2. That, E S E G ε. 6-2 The proof of th n the appendx. 22

23 Scenaro Selecton roblem n Nordc ower Maret Error n ar wth Lnear ortfolo Aume that the proft of the portfolo n conderaton lnearly dependent on the average power prce, R = a + b. 6-3 Then t poble to calculate the maxmum poble dfference between the ar calculated wth the orgnal et of cenaro and the elected ubet. By denotng the et of cenaro wth mallet proft wth S, the et of cenaro elected from th et by G, and ung the defnton 5-6, the dfference can be calculated wth ar S E R S = b E S ar G = E R G + E G + bε + bε = 2 bε. [ E R S E R S ] [ E R G E R G ] E R G E R S b E G E S ε The nequalty E S E G ε can be proved n the ame way than the nequalty 6-2. Clearly, the naccuracy of the ar value due to the naccuracy of the expected average prce over the et G and G. In the next ubecton a method developed to eep the expected average prce over the et G orgnal, thu reducng the maxmum ar error to bε Improvng Accurary of ar A noted n the prevou ubecton, the maxmum error n the ar value of lnear portfolo can be halved f the expected average prce over the et S and G are the ame. In order to eep the expected value a cloe to each other a poble, a new obectve functon ntroduced, N Z = N Z 2 N Z N Z / xnew / xnew = = =, 6-4 where n U = Z = Z, 6-5 Th can be the cae wth e.g. ome fnancal power contract or wth power generatng aet f the power prce over the varable cot of producton. 23

24 Scenaro Selecton roblem n Nordc ower Maret and n an nteger maller than n. Mnmng th obectve functon mnme the dfference n the expected value of average power prce, and thu mnme the maxmum poble error n the ar value. Th functon can be ued to fx the value of x {, K, N Z },, and then the functon 6-9 can be ued to fx the value of the ret decon varable. Of coure, becaue the value of ome decon varable are already fxed, the optmum value of functon 6-9 probably grow. Becaue N Z = N Z {, K,n} = xnew, = ee Equaton 6-5, the obectve functon 6-4 can be wrtten n a mpler form, N Z = N 2 xnew = Z. th ome portfolo, the lowet proft may be aocated wth the hghet power prce. In that nd of cae, target functon le 6-4 can be ued to fx the value of { N Z + N Z +, K, N } x, S m, where n + n U m = n + Z m = Z, 6-6 n + nm + n U h = n + nm + Z h = Z, 6-7 n = n + n + n. 6-8 m h Note that the target functon 6-4 not lnear. If needed, t can be turned nto a lnear functon wth the method ued n the ubecton 6.3. However, the quadratc target functon hould not be a problem becaue the number of decon varable relatvely mall. 6.5 Summary of the Method Table 3 n Appendx ummare the notaton related to the dfferent et of cenaro. 24

25 Scenaro Selecton roblem n Nordc ower Maret 6.5. Intalaton Set the value of parameter ε and C and dvde S nto et Z. Select the value n, n m, and n, and form the et Z, Z m, and Z h. Calculate the new weght of cenaro wth new Z {,,n} Z =, K Frt Optmaton roblem By changng decon varable x Z, mn N Z = uch that N 2 xnew = Z, 6-20 Z x {,, n } =, K Second Optmaton roblem By changng decon varable x Z h, N S N S mn xnew, 6-22 = N Z + N Z m + = N Z + N Z m + uch that 2 Z x { n + n +,, n} =, K m Thrd Optmaton roblem By changng decon varable x, and A {, K,N S } Z m, H 6-24 N S H mn A, 6-25 = 25

26 Scenaro Selecton roblem n Nordc ower Maret uch that A m = x, new, N S H, 6-26 A N S H m = x, {, K, N S }, new, H, 6-27 Z x { n +,, n } =, K m Complexty Analy In Table 2 are hown the number of decon varable and contrant n the three optmng problem. Table 2. Complexty of the optmaton problem. Frt roblem Second roblem Thrd roblem Bnary deccon varable NS NS h NS m Other decon varable 0 0 NS H Contrant n n h n m +2 NS H 26

27 Scenaro Selecton roblem n Nordc ower Maret 7 ILLUSTRATIVE EXAMLE Th ecton preent an example llutratng the ue of the method. In the frt ubecton a mple model ntroduced for generatng power prce cenaro. In the econd ubecton the method preented n prevou ecton ued to elect a ubet of cenaro. In the thrd ecton ar calculated for an artfcal power generatng aet ung both the orgnal cenaro et and the elected cenaro et n order to compare the reult. 7. Model for ower rce The pot prce dynamc modelled wth the followng tatc, lnear model, n whch the nflow and temperature are the ndependent varable. In addton, the frt level autocorrelaton n the error term ncluded n the model. S t 2 = β + β + β T + ε, ε = ρε + v, v ~ NID0, σ, 7-0 t 2 t t t t t t v where S t pot prce Eur/Mh, t water nflow Th/wee, and T t temperature Celcu at tme t, and the tme unt one wee. Becaue the redual term are not aumed to be ndependent, t not poble to ue the leat quare method to olve the coeffcent β. However, the problem can be ealy tranformed nto a problem n whch the redual term are ndependent and the leat quare method can thu be ued. By wrtng the Equaton 7- for tme perod t- and multplyng t by ρ we get the followng equaton, ρ S. 7-2 t = ρβ0 + ρβ t + ρβ2tt + ρεt By ubtractng 7-2 from 7- we get S t = β ρs 0 t = β ρ + 0 β ρ + t ρ β t t + ρ 2 t β T t + ρt β T 2 t t + v. t ρt t + ε t ρε t 7-3 Th a model whch can be olved wth leat quare method ndyc & Rubnfeld 99, p. 73. To olve the coeffcent of the model, the followng ource data wa ued. Nordpool pot prce, wee /998-52/999. Total nflow nto reervor n Norway, Sweden and Fnland, wee /998-52/

28 Scenaro Selecton roblem n Nordc ower Maret Average temperature of Olo, Stocholm and Heln, wee /998-52/999. The model wa olved wth dfferent value of ρ and the ρ aocated wth the mallet redual quare um wa elected. In lterature th called the Hldreth-Lu procedure ndyc & Rubnfeld 99, p.42. The reult were the followng ρ = 0.86, β β β σ v ρ = = 0.244, = 0.230, = β 0 = 7.69, The -value for the coeffcent β 0, β and β 2 were 0, 0.03 and 0, repectvely. The coeffcent of determnaton, R 2, wa The low value of the coeffcent mean that the model doe not explan the pot prce dynamc well. However, for the purpoe of th paper the model adequate. The fnal model thu S t = t t t t t T + ε, ε = 0.86 ε v, v ~ N0, Fgure 5 preent the ource data and the pot prce cenaro gven by the determntc part of the model. t t ee Inflow Th/wee Temperature Celu Spot rce EUR/Mh Model rce EUR/Mh Fgure 5. eely nflow, temperature, power prce and the modelled power prce for year The model 7-4 wa ued to generate 5 pot prce cenaro ung the nflow and temperature data from year 950 to In addton, for each of the pot prce cenaro, two addtonal cenaro were generated by aumng exceptonally hgh or low coal prce. Thee cenaro were mply formed by addng hgh coal prce or 28

29 Scenaro Selecton roblem n Nordc ower Maret ubtractng low coal prce two euro from the orgnal pot prce cenaro. Thu, fnally the et of pot prce cenaro cont of 53 cenaro, wth 5 dfferent cenaro path form. The graph of the nflow and temperature cenaro a well a the generated pot prce cenaro are n the Appendx. 7.2 Selectng the Subet of Scenaro The et of pot prce cenaro wth the aumpton of low, medum and hgh coal prce are denoted wth S, S 2, and S 3, repectvely. The probablte of low, medum and hgh coal prce were etmated to be 0.25, 0.5, and 0.25, repectvely. Thu, the cenaro are weghted uch that, {,,5}., = 0.25/ 5, 2, = 0.5 / 5, 3 = 0.25/ 5 K The value of parameter e and C were et to be 0.35 and 4, repectvely. The nteger n and n h were et uch that the weght of the et Z and Z h would be over 0% of the total weght. The new weght of the cenaro were calculated wth the equaton 6-9. The new weght, a well a other data related to the example, are hown n Table 4 n Appendx Frt Two Optmng Round In the frt optmng round the value for the decon varable X, {, K,22}, were fxed. The value of the target functon 6-20 wa , meanng that the dfference of expected pot prce average over the elected ubet of 0 cenaro and the orgnal et of 22 cenaro wa n practce zero. In the econd optmng round the value for the decon varable { 36,,53} X, K, were fxed. The value of the target functon 6-22 wa , thu, the dfference of expected pot prce average over the elected ubet of 8 cenaro and the orgnal et of 8 cenaro wa n practce zero Thrd Optmng Round In the lat optmng round the value for the remanng decon varable { 23,,35} X, K were fxed. The value of the target functon 6-25 wa

30 Scenaro Selecton roblem n Nordc ower Maret 7.3 Comparatve ar Calculaton The aet under conderaton 2 a peat condenng power plant wth effect 50M and varable cot of producton 4 /Mh. The fxed cot are aumed to be /wee. lant aumed to be n generaton f the weely average pot prce over the varable cot of producton. Thu, the weely proft of the plant can be calculated wth proft = max 0, prce 4EUR / Mh 50M 68h 20000EUR. The proft aocated wth dfferent pot prce cenaro are preented n Table 5 n Appendx. The proft at r wa calculated wth the Equaton 5-6 for both the orgnal et of cenaro and the elected ubet. The reult are hown n Table 3. The upde ar meaure the maxmum poble proft compared to the expected proft. In Table 5 n Appendx are hown the et of proft cenaro from whch the upde and downde proft were calculated. Table 3. The reult of ar calculaton. Orgnal Set Selected Subet Dfference Expected roft ,29 % Downde ,86 % Upde ,08 % Downde ar ,2 % Upde ar ,60 % Only the meaure of downde ha changed remarably. However, the large relatve change due to the mall abolute fgure of downde. 2 The fgure n th example are artfcal. However, the magntude are approxmately correct. 30

31 Scenaro Selecton roblem n Nordc ower Maret 8 CONCLUSION In th paper an optmaton model wa preented a a oluton for a power prce cenaro electon problem. The problem defnton wa done trctly, becaue the goal wa to olve an extng problem. Furthermore, t wa aumed that the parameter of the prce model do not affect on the cenaro path form. Th aumpton wa baed on the concluon that water nflow and temperature are the two mot mportant factor caung unpredctable varaton n power prce. Thu, thoe two factor determne the form of a prce cenaro path. All the other factor, whch are parameter of the prce model, motly affect the prce level only. Th aumpton wa done n order to cut down the dmenon of the problem. The propoed oluton ha four mportant properte. Frtly, depte of the trct tructure of the orgnal cenaro et, the number of cenaro to be elected can be freely decded by adutng parameter ε and C ee Equaton 6-2 and 6-3 for defnton. Secondly, the maxmum poble dfference between the expected average prce over the et S and G wa hown to be ε. Thrdly, for portfolo whch proft lnearly dependent on the average power prce, the maxmum dfference of ar value calculated wth the et S and G wa hown to be lnearly dependent on ε. Even though lnear aet are rare, lnearty a vable aumpton n many crcumtance. For ntance, the proft of mot generatng aet lnearly dependent on the power prce when the prce over the varable cot of producton. Fourthly, the man target functon of the problem lnear. Th mportant nce nonlnear problem wth hundred of bnomal decon varable are dffcult to olve, epecally wth commonly avalable oftware. In addton to thoe four proven properte, the dtrbuton of dfferent prce cenaro path form n the elected ubet reemble the dtrbuton n the orgnal et. Combned wth the reult that the average prce dtrbuton of the elected ubet cloe to the orgnal, th mple that all the tattcal properte of the two cenaro et are mlar. Fnally, the propoed method tranparent and can be mplemented alo wth tandard oftware. 3

32 Scenaro Selecton roblem n Nordc ower Maret AENDIX Summary of the Notaton Set Decrpton Element Indce S H Htorcal cenaro H, {,...,NS H } S Scenaro generated wth ame model, {,...,m}, {,...,NS H } parameter S Unon of S {,...,NS} Z Scenaro wth mlar average value {,...,n} Z Z m Z h Scenaro wth mall prce average Scenaro wth medum prce average Scenaro wth hgh prce average X Decon varable for S x, {,...,m}, {,...,NS H } X Unon of X x {,...,NS} G All elected cenaro Table 3. The notaton ued for cenaro et and decon varable. 32

33 Scenaro Selecton roblem n Nordc ower Maret 33 roof of the Equaton 4-2, = = = = = = = n new Z n Z new Z n n Z new Z G E S E where the cenaro elected from the et Z. Ung the equaton X-X, th can be wrtten [ ] [ ] [ ] [ ], ε ε ε = = + = + = = = = = = = = = n Z n Z n Z n Z n Z Z Z n Z Z n Z Z becaue the um of the weght wa aumed to be.

34 Scenaro Selecton roblem n Nordc ower Maret Graph and Table Related to the Example Th/ee ee Fgure 6. Total weely nflow nto reervor n Nordc countre. Year from 950 to Celu ee Fgure 7. eely average temperature of Heln, Stocholm and Norway. Year from 950 to EUR/Mh ee Fgure 8. Spot prce cenaro generated by the determntc model. 34

35 Scenaro Selecton roblem n Nordc ower Maret EUR/Mh ee Fgure 9. Spot prce cenaro generated by the tochatc model EUR/Mh ee Fgure 0. All the pot prce cenaro ued n the example EUR/Mh ee Fgure. The elected ubet of pot prce cenaro. 35

36 Scenaro Selecton roblem n Nordc ower Maret Table 4. The cenaro electon data. The border ndcate the et Z. The grey colour ndcate the et Z and Z h. The weght of cenaro are multpled by 204. Scenaro # Htorcal Scenaro # Set # Average rce Orgnal eght New eght Decon Varable 39 0, , ,6 4 50, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

37 Scenaro Selecton roblem n Nordc ower Maret Table 4 Contnue. Scenaro # Htorcal Scenaro # Set # Average rce Orgnal eght New eght Decon Varable , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,2 2 37

38 Scenaro Selecton roblem n Nordc ower Maret Table 5. ar calculaton data. Scenaro n the orgnal cenaro et that are ued n upde / downde calculaton are coloured grey. Scnearo n the elected ubet that are ued n upde / downde calculaton are mared wth heavy border. roft # Scenaro # Decon Varable Orgnal eght New eght roft

39 Scenaro Selecton roblem n Nordc ower Maret Tabe 5 contnue. roft # Scenaro # Decon Varable Orgnal eght New eght roft