A long-term risk management tool for electricity markets using swarm intelligence

Transcription

1 A long-term rsk management tool for electrcty markets usng swarm ntellgence F. Azevedo, Z.A. Vale, P.B. Moura Olvera, H.M. Khodr abstract Ths paper addresses the optmal nvolvement n dervatves electrcty markets of a power producer to hedge aganst the pool prce volatlty. To acheve ths am, a swarm ntellgence meta-heurstc optmzaton technque for long-term rsk management tool s proposed. Ths tool nvestgates the long-term opportuntes for rsk hedgng avalable for electrc power producers through the use of contracts physcal (spot and forward contracts) and fnancal (optons contracts) settlement. The producer rsk preference s formulated as a utlty functon (U) expressng the trade-off between the expectaton and the varance of the return. Varance of return and the expectaton are based on a forecasted scenaro nterval determned by a long-term prce range forecastng model. Ths model also makes use of partcle swarm optmzaton (PSO) to fnd the best parameters allow to acheve better forecastng results. On the other hand, the prce estmaton depends on load forecastng. Ths work also presents a regressve long-term load forecast model that make use of PSO to fnd the best parameters as well as n prce estmaton. The PSO technque performance has been evaluated by comparson a Genetc Algorthm (GA) based approach. A case study s presented and the results are dscussed takng nto account the real prce and load hstorcal data from manland Spansh electrcty market demonstratng the effectveness of the methodology handlng ths type of problems. Fnally, conclusons are dully drawn. Keywords: Electrcty markets Load forecast Optmzaton Partcle swarm optmzaton Portfolo Prce forecast Rsk management 1. Introducton Long-term contractual decsons are the bass of an effcent rsk management. On a vertcal ntegrated electrcty market, prce varatons were often mnmal and heavly controlled by regulators. In ths structure, electrcty prce evoluton s drectly dependent on the government s socal and ndustral polcy. The prce forecastng was manly focused on the underlyng costs (namely, fuel prces and technologcal nnovaton among others). Any prce forecastng made on ths bass was tended to be over the long-term. Wth electrcty markets re-regulaton process, aforementoned features have been changed dramatcally. Thus, ownershp on ths actvty sector becomes prvate rather than publc or a mxture of both. Moreover, pools or power exchanges have been ntroduced for wholesale tradng. Prce forecast on re-regulated electrcty markets s a hard task due to the hgh pool prce volatlty. Charge characterstcs (seasonalty, mean-reverson and load stochastc growth) and producer s characterstcs (technology, generaton avalablty, fuel prces, techncal restrctons, mport/export, etc.) are at the orgn of hgh prce volatlty n electrcty markets. Several technques have been used for short-term prce forecast n electrcty markets. In [1], artfcal ntellgent tools were proposed to forecast spot prces, namely, a combnaton of neural networks and fuzzy logc. Indeed, neural networks have now an extensve use n load and n prce forecast [2 6]. Fuzzy technques mxed neural networks are used to predct possble prces range [7,8]. Stochastc processes are also used to analyze tme seres as ARIMA processes [9]; a class of stochastc processes was used to predct next-day electrcty prces n manland Spansh and n Calforna markets. In [10], two forecastng tools based on dynamc regresson and transfer functon models are presented. However, for the market agents who want to maxmze ther profts and smultaneously to practce the hedge aganst the market prce volatlty, the use of forward, futures and optons contracts become a constant n developed electrcty markets. Those types of contracts have a maturty that goes from 1 year to several years n the future, turnng more dffcult the decson process related to contracts establshment f they are not supported a robust prce forecast methodology. Due to long delvery perods of the contracts descrbed above that make more sense to forecast the market prce mean value

2 for each month. Moreover, contractual postons should be contnuously revew (say once a month) or each tme the agent needs to consder the ones already locked. After revew references, we assure that ths problem has not been treated deeply n the scentfc lterature. It s not a good practce n rsk management to take contractual decsons based exclusvely on a sngle forecasted value. In Secton 3 s presented a dfferent approach for long-term prce forecast. The proposed methodology, based on regresson model, ams to fnd a maxmum and a mnmum monthly Market Clearng Prce (MCP) average for a programmng perod, a desred confdence level. Ths method makes use of statstcal nformaton extracted from manland Spansh market hstorcal data. Due to the complexty of the problem, the parameters are obtaned usng the meta-heurstc PSO [11 14]. Ths model could also be appled to forecast electrcty market prce for more than 1 year. The long-term prce forecast makes use of the monthly load average forecast for the same perod. For ths reason, the authors have also developed a long-term monthly load forecast usng PSO. For both models, long-term prce and load forecast, PSO performance has been evaluated by comparng t a GA. To fnd best portfolo for a market agent and partcularly for producers, whch allow to hedge aganst market prce volatlty and smultaneously ncreasng ther profts. Ref. [15] proposes solutons for electrcty producers n the fnancal rsk management feld for electrc energy contract evaluaton usng effcent fronter as a tool to dentfy the preferred contract portfolo. In [16] a decson-support system based on stochastc smulaton, optmzaton and mult-crtera analyss, s appled to electrcty retaler. A statstcal study of drect and cross hedgng strateges usng futures contracts n an electrcty market s presented n [17,18]. A framework to obtan the optmal bddng strategy of a thermal prce-taker producer on a pool-based electrc energy market s presented n [19]. The optmal nvolvement n a futures electrcty market of a power producer to hedge aganst the rsk of pool prce volatlty usng condtonal Value-at-Rsk as rsk measure s presented n [20]. A rsk-constraned stochastc programmng framework to decde whch forward contracts the retaler should sgn and at prce t must sell electrcty and ts expected proft s maxmzed at a gven rsk level has been proposed n [21]. A technque based on stochastc programmng to optmally solve the electrcty procurement problem faced by large consumer s presented n [22]. Ref. [23] analyzes the mpact of the degree of unavalablty of the generatng unt on ts forward contractng decsons. In ths work, long-term rsk management tool makes use of a long-term prce range forecast has been developed and dscussed. The proposed long-term rsk management tool ams to fnd the best portfolo n functon of the rsk averson factor () of the producer, whch maxmzes the expected return and, smultaneously, allows hedgng aganst market prce volatlty. To acheve ths, the decson-support system maxmzes a mean-varance utlty functon (U) of the total return (). In ths methodology, a portfolo model based on utlty functons nstead of opton prcng models [24,25] has been used, because the fnancal markets on electrcty markets are ncomplete (hedgng nstruments unavalable). Uncertantes assocated to generators avalablty, fuel prces, techncal restrctons and weather condtons, turn dffcult, f not mpossble, to fnd a replcatng portfolo, whch perfectly matches the future spot market payoffs. The power market exercse by some agents s also a source of uncertanty. Moreover, several markets around the world are stll on ther recent stage, a small number of fnancal tools for an effcent rsk management. Another ssue n power markets s that electrcty cannot be stored for later use. Consequently, the strategy of buyng the asset today to offset part of future losses does not apply. The closest strategy s to buy a forward or futures contracts. Therefore, the delvery prce of these mentoned contracts should be equal to the expected spot market prce for the delvery perod, whch not always happens. Consequently, the electrcty markets are not complete (.e., any desred fnancal hedges are not avalable at a prce), so rsk atttudes and mean-varance fronters are stll relevant. PSO and GA algorthm performance are evaluated to show PSO s a very successful meta-heurstc technque for solvng ths problem n partcular. The paper s organzed as follows: Secton 2 presents a long-term load forecastng model, followed by a case study applyng aforementoned model. Secton 3 presents a long-term prce forecastng method, followed by the applcaton of ths method to a case study. Secton 4 shows the problem formulaton of the rsk management. Fnally, Secton 5 draws the relevant conclusons. 2. Long-term load forecast The proposed method s based on regresson models. The man goal of ths methodology s to fnd the regresson parameters that mnmze the absolute error to consdered load hstorcal data for monthly tme nterval for 1-year perod. Load pattern s not complex as the revealed by the market prce. To fnd the best regresson parameters, load hstorcal data of the prevous 2 years has been used General descrpton The optmzaton problem that allows to fnd the best regresson parameters for the monthly load average s gven by (1) and (2): 12 Mn C Ĉ =1 (1) Subjected to : Ĉ 0 Ĉ = ω 1, C 1,j + ω 2, C 1 + ω 3, C 2 (2) 2.2. Penalty functon To solve the optmzaton problem, PSO has been used to fnd the best soluton. To satsfy constrant (2) for each perod, the penalzaton functon (3) has been added to the optmzaton problem: { 0 f Ĉ p f = 0 (3) e 100 a2 1 otherwse where a = Ĉ (4) 2.3. PSO and GA parameters Tables 1 and 2 present the parameters of PSO and GA algorthm, respectvely. Besdes the optmum parameters of PSO method, beng also dependent on the ftness functon. Expermentatons show that the number of evaluatons used cannot compromse the results and allow to acheve the best soluton Case study Ths test case uses a real load hstorcal data ( ) that has been extracted from the manland Spansh market has been used to forecast the monthly load average for the year Fg. 1

3 Table 1 PSO parameters. partcles teratons evaluatons Cogntve acceleraton Socal acceleraton Intal nerta weght Fnal nerta weght Maxmum velocty (V max) 20 20, , Table 2 GA parameters. Populaton sze generatons evaluatons Crossover rate Mutaton rate , Table 4 GA results. ω 1, ω 2, ω 3, January February March Aprl May June July August September October November December Table 5 PSO and GA ftness functon comparson. Fg. 1. Monthly load average for manland Spansh market from 2001 to shows the mentoned monthly load average hstorcal evoluton curve. To fnd the best parameters, t was used the meta-heurstc PSO has been used and compared a GA from performance pont of vew. The stoppng crteron s the maxmum number of evaluatons, fxed n 400,000 evaluatons for both algorthms. To acheve the convergence PSO are necessary 20 partcles and 20,000 teratons for PSO and a populaton sze of 50 ndvduals and 8000 generatons for GA. Results are presented n Tables 3 5, respectvely. It can be seen from Table 5 that, for ths partcular problem, PSO s faster than GA (less tme) and fnds better solutons (lower ftness value). These smulatons have obtaned by a computer Pentum 4, 3.2 GHz processor and 1 GB of RAM. The monthly load average forecast for the year 2007 s presented n Fg. 2. Algorthm Ftness functon value Tme (s) PSO GA Long-term prce range forecast The proposed model ams to forecast the maxmum and mnmum monthly market prce average a desred confdence level, by usng load and prce statstcal hstorcal nformaton data. The followng methodology s deployed: (1) For each month of the hstorcal data, the monthly average load s dvded on fxed and equal wndows of sze L. Then, for each wndow L and for the desred confdence level, statstcal nformaton s extracted from the correspondent monthly average market prce namely, the sgnfcance nterval corre- Table 3 SO results. ω 1, ω 2, ω 3, January February March Aprl May June July August September October November December Fg. 2. Monthly load average forecast for the year 2007 on manland Spansh market.

4 spondng to maxmum and mnmum prce s between the nterval (50 + /2) and (50 /2), respectvely. (2) For each hstorcal monthly market prce average, we fnd the correspondent maxmum and mnmum monthly market prce average from the wndow L that t belongs. Based on ths nformaton, for each month of the year a regresson s determned for the maxmum and mnmum monthly market prce average the objectve to mnmze the absolute error consderng annual hstorcal data. The regresson ndependent varables used are: the monthly market prce average of the prevous month and the monthly load average Case study The optmzaton problem allow to fnd the best regresson parameters for the maxmum monthly market prce average confdence level, s gven by (5) and (6): Mn 12 N =1 j=1 Subjected to : ˆP max, 0 ˆP max, P max, ˆP max, = 1, C + 2, P 1,j (6) The optmzaton problem allow to fnd the best regresson parameters for the mnmum monthly market prce average confdence level, s formulated by (7) and (8): Mn 12 N =1 j=1 Subjected to : ˆP mn, 0 ˆP mn, P mn, ˆP mn, = ˇ1, C + ˇ2, P 1,j (8) After fndng the best regresson parameters for the maxmum and mnmum monthly market prce average, t s necessary to forecast the load for the months whch s ntended to forecast the monthly market prce average range. To forecast monthly market prce average range for the month, (6) and (8) should be changed to (9) and (10), respectvely: ˆP max, ˆP mn, = 1, Ĉ + 2, P 1 (9) = ˇ1, Ĉ + ˇ2, P 1 (10) Due to the hgh value of the monthly load average, the monthly market prce average, a scale factor ı s appled to the monthly load average to avod hgh values of 1, and ˇ1, to facltate the convergence of the optmzaton problems. (5) (7) Based on the aforementoned dea, Eqs. (9) and (10) should be changed to (11) and (12) as follows: ˆP max = 1, Ĉ ı + 2, P 1 (11) ˆP mn = ˇ1, Ĉ ı + ˇ2, P 1 (12) Ths method can also be used to forecast the monthly market prce average range for several programmng perods ahead usng the followng strategy: the average of monthly market prce average range forecast for the programmng perod p wll act as the prevous monthly market prce average for the programmng perod p +1. To solve the optmzaton problems the PSO algorthm was used and ts performance compared the performance of a GA Penalty functons To fnd the best soluton of problems (5) and (7) PSO and GA algorthm were used. In order to satsfy the constrant of the optmzaton problem (5), the penalty functon gven by (13) and (14) was used: { 0 f ˆP max, 0 p = (13) e 100 b2 1 otherwse b = ˆP max, (14) To satsfy the constrant of the optmzaton problem (7) the penalty functon gven by (15) and (16) s used: { 0 f ˆP mn, 0 p 2 = (15) e 100 c2 1 otherwse c = ˆP mn, (16) 3.3. PSO and GA parameters The PSO and GA parameters used to fnd the best soluton are presented n Tables 6 and 7, respectvely. Besdes the optmum parameters of PSO method, beng also dependent on the ftness functon. Expermentatons show that the number of evaluatons used cannot compromse the results and allow to acheve the best soluton Case study To test our model, n ths secton the monthly market prce average s forecasted for the year 2007 n manland Spansh market a confdence level = 95%. It s assumed that t = 0 corresponds to 31 December of the year The hstorcal data used was the monthly market prce and load average from 2001 to 2006 presented n Fgs. 1 and 3, respectvely. Table 6 PSO parameters. partcles teratons evaluatons Cogntve acceleraton Socal acceleraton Intal nerta weght Fnal nerta weght Maxmum velocty (V max) 20 20, , Table 7 GA parameters. Populaton sze generatons evaluatons Crossover rate Mutaton rate ,

5 5 Table 10 Standard devaton of the regresson parameters usng PSO algorthm. 1, 2, ˇ1, ˇ2, January February March Aprl May June July August September October November December Table 11 Standard devaton of the regresson parameters usng GA. Fg. 3. Monthly market prce average for manland Spansh market from 2001 to Table 8 Average of the regresson parameters usng PSO algorthm. 1, 2, ˇ1, ˇ2, January February March Aprl May June July August September October November December The proposed model needs the monthly forecasted load for the year The results of PSO algorthm have been used and presented n Fg. 2 because t acheve better results, namely lower Mean Absolute Percentage Error (MAPE). PSO performance s compared the one acheved by GA. The stoppng crteron s the maxmum number of evaluatons, fxed n 400,000 evaluatons for both algorthms. To acheve the convergence PSO are necessary 20 partcles and 20,000 teratons for PSO and a populaton sze of 50 ndvduals and 8000 generatons for GA. Due to random ntalzaton, the trajectory for each run s dfferent; so 10 runs are used to fnd the average and the standard devaton of the results. The scale factor ı appled the monthly load average was set equal to Results are presented n Tables 8 and 9, respectvely. The regresson parameters standard devaton for the monthly market prce range forecast usng the PSO and a GA s presented n Tables 10 and 11, respectvely. Table 9 Average of the regresson parameters usng GA. 1, 2, ˇ1, ˇ2, January February March Aprl May June July August September October November December , 2, ˇ1, ˇ2, January February March Aprl May June July August September October November December The mean and the standard devaton of the ftness functons for the 10 runs are presented n Table 12. Table 12 also ncludes the mean tme necessary to reach the optmal soluton for PSO and GA. Analyzng Tables 10 and 11, t can be verfed that, for ths partcular problem, PSO solutons are more robust (smaller standard devaton values) than the ones attaned the GA. It can be verfed from Table 12 that, for ths partcular problem, PSO s faster than GA (smaller mean tme), fnds better solutons (smaller mean ftness value) and s more robust (smaller standard devaton values). These smulatons were on the same computer. The monthly market prce range forecast for the year 2007 n manland Spansh market confdence level = 95%, usng the PSO and GA are presented n Fgs. 4 and 5, respectvely. In these fgures we verfy results PSO are better for ths partcular problem when compared GA results, because the sum of the MAPE for the maxmum and mnmum monthly market prce average forecast s lower for PSO. Also from the same fgures t s possble to verfy the monthly market prce average range forecast nclude n 91.67% the real monthly market prce average. Comparng Fgs. 4 and 5, we verfy that for the December the monthly market prce average range forecast does not nclude the real monthly market prce average. Ths occur because from Fg. 3, the excepton of the year 2002, the monthly market prce average varaton between November and December was not so hgh when compared the same perod of the year Indeed, we use a regresson model, objectve to mnmze absolute error for the hstorcal data. Ths s not possble to capture such pattern. Table 12 PSO and GA ftness functon comparson. Algorthm Mean Std. Dev. Mean tme (s) PSO (max) GA (max) PSO (mn) GA (mn)

6 has a long poston) obtaned by the producer s dependent of the perod and scenaro j and s gven by (17): r ss = MCP e ss (17) 4.2. Forward contracts Fg. 4. Monthly market prce range forecast for the year 2007 n manland Spansh market, confdence level = 95%, usng the PSO algorthm. The strategy of buyng the asset today to offset part of future losses does not apply. The closer strategy s to buy a forward or futures contracts. Forward contracts are blateral agreements n whch two parts agree mutually on the characterstcs (quantty, prce, pont of delvery and date/tme). The payment s made only on a future date, elmnatng the rsk assocated to prce varaton. The producer revenue from a short forward poston on forward contracts s gven by (18). In ths rsk management tool t was consdered that the delvery perod for forward contracts s the same of all perod n analyss: r sf = k sf e sf (18) The delvery prce s fxed on forward contracts. So, ts revenue s only dependent on the delvery prce and quantty establshed n the contract. To avod producers take advantage of any arbtrage opportunty, ths long-term rsk management tool does not allow long postons on forward contracts Optons contracts Fg. 5. Monthly market prce range forecast for the year 2007 n manland Spansh market, confdence level = 95%, usng the GA. Maybe these results wll be better than the obtaned results, f we have other type of nformaton, lke temperature for example. 4. Long-term rsk management Fndng the optmal portfolo of contracts for a deregulated electrcty market agent s a hard task due to the characterstcs of the product electrcty and to the non-lnear characterstcs of the avalable contracts. Ths paper presents a new methodology for long-term rsk management tool, takng nto account, large perod prce forecast and contractual dversfcaton that are the key ssue for an effcent rsk management. To acheve ths, t s assumed that producers can make use of contracts physcal settlement (spot and forward contracts) and contracts fnancal settlement (optons contracts). It has been solved by a combnaton of PSO and regresson algorthms Spot contracts Electrcty market desgn s manly dependent of fnancal, poltcal and deology reasons. Therefore, deregulated electrcty markets desgn vares from country to another. However, spot market becomes the core of the man deregulated electrcty markets around the world. Producers make extensve use of ths market to sell ther energy on an hour or half-hour bass. The revenue from the short poston (who sells has a short poston and who buys On optons contracts the buyer have the rght, but not the oblgaton, to exercse the contract. Based on that, there are four postons on optons contracts: short call, long call, short put and long put. However, rsk management tool only allows the producers to establsh short call and long put postons. These postons are smlar to the postons of the producer can establsh to sell the produced energy physcal settlement. If producer s allowed to establsh four postons types, quanttes to practce the hedge would be almost nfnte f a fnancal lmt s not establshed. It was assumed they are fnancal type and European-style optons (European-style optons can only be exercsed at the begnnng of the delvery date whle Amercan-style optons can be exercsed at any tme untl the delvery date). The payoff for the short call poston s gven by (19): P sc = esc [mn(k sc MCP, 0) + p sc ] (19) From Eq. (19) t s clear that the buyer only exercses the call opton f the Market Clearng Prce s hgher than the delvery prce. Also we can see the seller (producer) payoff s postve only f the Market Clearng Prce at the expraton date s lower than the call opton exercse prce plus the premum. For the long put poston, opton buyer (producer) wll exercse t f the MCP s lower than the exercse prce. The payoff for long put poston s gven by (20): P lp = elp [max(k lp MCP, 0) p lp ] (20) From Eq. (20) t can be seem the long put poston payoff s postve only f the MCP s hgher than the exercse prce Mathematcal optmzaton problem To fnd optmal energy quanttes establshng on each contract type, t was developed an optmzaton problem based on a meanvarance of the return. Ths formulaton allows fndng the optmal energy quanttes that maxmzes the profts and smultaneously practces the hedge aganst the MCP volatlty n functon of the producer rsk averson factor.

7 The mathematcal formulaton s stated as followng: Maxmze U() = E() Var() (21) Subjected to: e mn e cs + e cf e max (22) e cs,e cf,e cc,e lp 0 (23) where E() = E( max ) + E( mn ) (24) and 2 2 Var() = cov ( max, mn ) (25) =1 j=1 max = [ max 1,..., max T ] (26) and mn = [ mn 1,..., mn T ] (27) The mean-varance formulaton resemble closely the Valueat-Rsk (VaR) formulaton and have as man advantage to be computatonally more effcent for a gven rsk averson factor. Moreover, VaR formulaton needs hgher order of nformaton about the jont probablty dstrbuton of the payoffs and s hghly senstve to the hgh mpact of low probablty events, whch create fat tals n payoff dstrbuton. In ths formulaton we assumed the rsk averson factor s equal for the whole perod n analyss. The return for each perod, expressed n Eur, s a functon of the consdered mnmum or maxmum prce forecast scenaro j for that perod, and s equal to the sum of all revenues and optons payoffs mnus the costs of producton. Mathematcally, the return s = r ss + rsf + P sc + Plp C (28) C = C(e ss + e sf ) (29) Optons contracts have fnancal settlement; the total producton cost s only dependent on the energy that the producer wll sell on spot market, and on forward contracts, meanng that s only dependent of the energy establshed on contracts physcal settlement Penalty functons Due to optmzaton problem complexty, PSO was used to fnd the optmal soluton and results were compared GA results. To satsfy constrant (22) for each perod, penalty functon (30) and (31) s added to (21): { 0 f e emn and e e max p f 1 = (30) e 100 d2 1 otherwse where d = mn[ e e mn, e e max ] (31) To satsfy constrant (23) for each perod, penalty functon (32) and (33) s added to (21): { 0 f e ss,sf,sc,lp 0 p f 2 = (32) e 100 e2 1 otherwse where e = e ss,sf,sc,lp (33) 4.6. PSO and GA parameters The parameters of PSO and GA, used fndng the best soluton are presented n Tables 13 and 14, respectvely. Besdes these parameters beng dependent on the ftness functon, expermentatons show that the number of evaluatons used does not compromse the results and allow achevng the optmal soluton Producer characterstcs It was assumed that producer cost functon s equal for the entre perod n analyss (1 year) and s gven by Eq. (34): C(P g ) = P g Pg 2 (34) where P g s n MW, C n Eur/h, P max g = 200 MW and Pg mn = 5 MW. The cost of sales (lke taxes, market commssons and others) s not addressed. Moreover, there s just as much rsk n the cost of sales as there s n the generaton of revenue Contracts characterstcs Optons contracts characterstcs delvery perod for the year 2007 are presented n Table 15. It was assumed that forward contracts delvery perod for the year 2007 are traded at a prce equal to 40 Eur/MWh Case study In ths case a producer ams (n December 2006) to fnd the optmal contracts portfolo for the entre year of However, although to be beyond the purpose of ths work, the producer must adjust ts contractual postons contnuously (say once a month) and whenever, he needs to reconsder hs contractual postons already establshed n forward and other contracts, before adjustng the portfolo. Although PSO acheve better results on monthly prce range average forecast, Fg. 4 present the results for the year An evaluaton of PSO and GA performance for ths partcular problem has been carred out. The algorthms stoppng crteron was the maxmum number of evaluatons (fxed n 400,000 evaluatons). Wth 20 partcles n the PSO 20,000 teratons were performed. For GA a populaton sze of 50 ndvduals and 8000 generatons was used. Due to random ntalzaton, the trajectory for each run s dfferent; so, we used 10 runs to calculate the average and the standard devaton of the results. Due to the problem complexty, the entre perod was dvded n sub-perods of one month of duraton allowng to reduce the number of varables and, consequently, turnng the optmzaton problem lghter. In Tables 16 and 17 results are presented for the average quanttes, n MWh, for each contractual poston and rsk averson factor usng PSO and GA, respectvely. The standard devaton of the results usng PSO and GA s presented n Tables 18 and 19, respectvely. Comparng the standard devaton for each soluton (Tables 18 and 19), we conclude that PSO s more robust than the GA. The mean and the standard devaton of the ftness functons for the 10 runs and for each rsk averson factor are presented n Table 20. Table 20 also ncludes the mean tme necessary to reach the optmal soluton for PSO and GA. It can be verfed from Table 20 that, for ths partcular problem, PSO s faster than GA (smaller mean tme), fnds better solutons (smaller mean ftness value) and s more robust (smaller standard

8 Table 13 PSO parameters. partcles teratons evaluatons Cogntve acceleraton Socal acceleraton Intal nerta weght Fnal nerta weght Maxmum velocty (V max) 20 20, , Table 14 GA parameters. Populaton sze generatons evaluatons Crossover rate Mutaton rate , devaton). These smulatons were made on an ASUS L5GX laptop, P4 3.2 GHz processor and 1 GB of memory. Because PSO acheve better results n ths partcular problem, n Fgs. 6 and 7 s presented ts results for the expected return and the assocated rsk for each month, as functon of the rsk averson factor, respectvely. From Fgs. 6 and 7 we conclude that, for the same rsk averson factor, the bgger the expected return the bgger the rsk (standard devaton of the return) that the producer s exposed to. Analyzng Fgs. 7 and 8 we verfy that the rsk (standard devaton of the return) s nversely proportonal to the rsk averson factor, and so s the energy that the producer wll sell n the spot market. Fg. 8. Optmal energy quanttes that producer should sell n spot market n functon of rsk averson factor. Ths happens because the lower the rsk averson factor the most ndfferent the producer wll be to the rsk and therefore he wll have more rsky atttudes and sell more energy on the spot market, as t can be seen n Fg. 8. Table 15 Optons contracts characterstcs. Exercse prce (Eur/MWh) Premum (Eur/MWh) Short call Long put Fg. 6. Producer expected return n functon of rsk averson factor. Table 16 Average quantty, n MWh, to establsh by contractual poston and rsk averson factor usng PSO. Postons Average quantty (MWh) =0 =1 =2 =3 Short spot Short forward Short call Long put Table 17 Average quantty, n MWh, to establsh by contractual poston and rsk averson factor usng GA. Postons Average quantty (MWh) =0 =1 =2 =3 Fg. 7. Rsk n functon of rsk averson factor. Short spot Short forward Short call Long put

9 Table 18 Standard devaton quantty, n MWh, to establsh by contractual poston and rsk averson factor usng PSO. Postons Std. Dev. (MWh) =0 =1 =2 =3 Short spot Short forward Short call Long put Table 19 Standard devaton quantty, n MWh, to establsh by contractual poston and rsk averson factor usng GA. Postons Std. Dev. (MWh) =0 =1 =2 =3 Short spot Short forward Short call Long put Table 20 PSO and GA ftness functon comparson. Algorthm Mean Std. Dev. Mean tme (s) PSO ( = 0) GA ( = 0) PSO ( = 1) GA ( = 1) PSO ( = 2) GA ( = 2) PSO ( = 3) GA ( = 3) Conclusons In ths paper an effectve long-term rsk management tool has been proposed that allows producers to maxmze ther expected return whle practcng the hedge aganst spot prce volatlty. Moreover, to do ths, three models have been elaborated n ths work. One s for rsk management that depends on the prce forecastng, whch also depends on the load forecastng. The man advantage of the prce forecastng model s the fact that does not make any statstcal dstrbuton functon for market prce assumpton. For solvng the model of load and prce forecast, PSO technque has been used to fnd the best regresson parameters. The methodology has been appled to a case study to carry out load and prce forecastng. Results demonstrate that ths methodology can effectvely support handle the decson-support for long-term rsk management n power systems. Acknowledgments The authors would lke to acknowledge the sponsor and fnancal support to FCT, FEDER, POCTI, POSI, POCI and POSC for ther support to R&D Projects and GECAD Unt. Appendx A. Lst of symbols The authors would lke to acknowledge the sponsor and fnancal support to FCT, FEDER, POCTI, POSI, POCI and POSC for ther support to R&D Projects and GECAD Unt. C Ĉ real monthly load average for month, year j monthly load average forecast for month, year j ω 1, parameter assocated to the monthly load average of the month 1 ω 2, parameter assocated to the monthly load average of the month, year j 1 ω 3, parameter assocated to the monthly load average of the month, year j 2 L wndow sze to extract statstcal nformaton from hstorcal prce data desred confdence level for the monthly prce range forecast P max, real maxmum monthly market prce average for month, year j confdence level P mn, real mnmum monthly market prce average for month, year j confdence level ˆP max, maxmum monthly market prce average forecast for month, year j confdence level ˆP mn, mnmum monthly market prce average forecast for month, year j confdence level P 1,j monthly market prce average of the month 1, year j 1, parameter assocated to the load average forecast for the month to forecast the maxmum market prce average for the same month 2, parameter assocated to the monthly market prce average of month 1 to forecast the maxmum market prce for month ˇ1, parameter assocated to the load average forecast for the month to forecast the mnmum market prce average for the same month ˇ2, parameter assocated to the monthly market prce average of month 1 to forecast the mnmum market prce month N number of hstorcal years used ı represents a scale factor to be appled to the monthly load average forecast r ss revenue of the short poston obtaned by the producer n the spot market for perod, scenaro j (Eur) MCP Market Clearng Prce for perod, scenaro j (Eur/MWh) e ss energy amount that the producer decdes to sell n the spot market for perod (MWh) r sf revenue of the short poston obtaned by the producer n forward contracts (Eur) k sf delvery prce of the forward contract (Eur/MWh) e sf energy amount that the producer decdes to sell n forward contracts (MWh) P sc payoff of the short call poston, for the perod, scenaro j (Eur) p sc premum of the call opton (Eur/MWh) k sc delvery prce of the call opton (Eur/MWh) e sc energy assocated to the short call poston obtaned by the producer (MWh) P lp payoff of the long put poston, for perod, scenaro j (Eur) p lp premum of the put opton (Eur/MWh) k lp delvery prce of the put opton (Eur/MWh) e lp energy assocated to the long put poston obtaned by the producer (MWh) producer return for the entre perod n analyss (Eur) E() expected value of the return based on the forecasted prce nterval for the entre perod n analyss (Eur) Var() varance of the return based on the forecasted prce nterval for the entre perod n analyss (Eur) cov ( max, mn ) element (, j) of the covarance matrx of the returns for all perods based on maxmum and mnmum prce forecast (Eur) C(P g ) cost to produce P g (Eur/h)

10 P g Pg max Pg mn max mn T e mn e max References power energy of producer (MW) maxmum electrcal power of producer (MW) mnmum electrcal power of producer (MW) perod return based on the maxmum prce forecast (Eur) perod return based on the mnmum prce forecast (Eur) number of the consdered perods for the entre perod n analyss producer rsk averson factor mnmum energy that the producer can produce (MWh) maxmum energy that the producer can produce (MWh) [1] C.P. Rodrguez, G.J. Anders, Energy prce forecastng n the Ontaro compettve power system market, IEEE Transactons on Power Systems 19 (February(1)) (2004) [2] H. Chen, C. Cañzares, A. Sngh, ANN-based short-term load forecastng n electrcty markets, n: Power Engneerng Socety Wnter Meetng, vol. 2, IEEE, February, 2001, pp [3] B.R. Szkuta, L.A. Sanabra, T.S. Dllon, Electrcty prce short-term forecastng usng artfcal neural networks, IEEE Transactons on Power Systems 14 (August(3)) (1999) [4] F. Azevedo, Z.A. Vale, Short-term prce forecast from rsk management pont of vew, n: 13th Internatonal Conference on Intellgent Systems Applcaton to Power Systems ISAP 2005, Arlngton USA, November, 2005, pp [5] F. Azevedo, Z.A. Vale, Forecastng electrcty prces hstorcal statstcal nformaton usng neural networks and clusterng technques, n: PSCE Power Systems Conference and Exposton, Georga, Atlanta, USA, October, 2006, pp [6] T. Nmura, Forecastng technques for deregulated electrcty market prces extended survey, n: PSCE Power Systems Conference and Exposton, Georga, Atlanta, USA, October, 2006, pp [7] T. Nmura, H. Ko, K. Ozawa, A day-ahead electrcty prce predcton based on a fuzzy-neuro autoregressve model n a deregulated electrcty market, n: Proceedngs of the 2002 Internatonal Jont Conference on Neural Networks, vol. 2, May, 2002, pp [8] L. Hongje, W. Xufeng, Z. Wecun, X. Guohua, Market Clearng Prce forecastng based on dynamc fuzzy system, n: Proceedngs of the 2002 Internatonal Conference on Power System Technology, vol. 2, October, 2002, pp [9] J. Contreras, R. Espínola, F.J. Nogales, A.J. Conejo, ARIMA models to predct nextday electrcty prces, IEEE Transactons on Power Systems 18 (February(3)) (2003) [10] F.J. Nogales, J. Contreras, A.J. Conejo, R. Espínola, Forecastng next-day electrcty prces by tme seres models, IEEE Transactons on Power Systems 17 (May(2)) (2002) [11] J. Kennedy, R. Eberhart, Partcle swarm optmzaton, n: IEEE Internatonal Conference on Neural Networks, Perth, Australa, 1995, pp [12] Y. Sh, R. Eberhart, Parameter selecton n partcle swarm optmzaton, n: Proceedngs of the Seventh Annual Conference on Evolutonary Programmng, 1998, pp [13] J. Holland, Genetc Algorthms, Scentfc Amercan, July de 1992, pp [14] M. Mtchell, An Introducton to Genetc Algorthms, MIT Press, [15] R. Bjorgan, C. Lu, J. Lawarrée, Fnancal rsk management on a compettve electrcty market, IEEE Transactons on Power Systems 14 (November(4)) (1999) [16] S. Makkonen, R. Lahdelma, A. Asell, A. Joknen, Mult-crtera decson support n the lberalzed energy market, Journal of Mult-Crtera Decson Analyss 12 (2004) [17] E. Tanlapco, J. Lawarrée, C. Lu, Hedgng futures contracts n a deregulated electrcty ndustry, IEEE Transactons on Power Systems 17 (August(3)) (2002). [18] R. Bjorgan, H. Song, C. Lu, R. Dahlgren, Prcng flexble electrcty contracts, IEEE Transactons on Power Systems 15 (May(2)) (2000) [19] J. Conejo, F.J. Nogales, J.M. Arroyo, Prce-taker bddng strategy under prce uncertanty, IEEE Transactons on Power Systems 17 (November(4)) (2002) [20] J. Conejo, R. Garca-Bertrand, M. Carron, A. Caballero, A. Andres, Optmal nvolvement n futures markets of a power producer, IEEE Transactons on Power Systems 23 (May(2)) (2008) [21] M. Carron, J. Conejo, M. Arroyo, Forward contractng and sellng prce determnaton for a retaler, IEEE Transactons on Power Systems 22 (November(4)) (2007) [22] M. Carron, B. Phpott, J. Conejo, M. Arroyo, A stochastc programmng approach to electrc energy procurement for large consumers, IEEE Transactons on Power Systems 22 (May(2)) (2007) [23] S. Pneda, J. Conejo, M. Carron, Impact of unt falure on forward contractng, IEEE Transactons on Power Systems 23 (November(4)) (2008) [24] F. Black, M. Scholes, The prcng of optons and corporate labltes, Journal of Poltcal Economy 81 (June(81)) (1973) [25] R.C. Merton, Theory of ratonal opton prcng, Bell Journal of Economcs and Management Scence 4 (4) (1973) Flpe Azevedo receved hs dploma n 1999, the M.Sc. n 2003, both n Electrcal Engneerng from Porto Unversty, Portugal, and the Ph.D. degree n 2008 on Electrcal Engneerng at the Unversty of Trás-os-Montes e Alto Douro (UTAD), Vla Real, Portugal. He s currently an Assstant of Power Systems at the Insttute of Engneerng Polytechnc of Porto (ISEP/IPP), Portugal, and a researcher at the Knowledge Engneerng and Decson-Support Research Group (GECAD). Hs man research nterests are Economcs appled to Energy Markets, Decson-Support System technques appled to Economcs Engneerng problems and Artfcal Intellgence (A.I.) applcatons. Zta A. Vale s a Coordnator Professor of Power Systems at the Insttute of Engneerng Polytechnc Insttute of Porto (ISEP/IPP), Portugal. She receved her dploma n Electrcal Engneerng n 1986 and her Ph.D. n 1993, both from Unversty of Porto. Her man research nterests concern Artfcal Intellgence (A.I.) applcatons to Power System operaton and control, Electrcty Markets and Dstrbuted Generaton. She s nvolved n several R&D projects concernng the applcaton of A.I. and Decson-Support technques to Engneerng problems. P.B. Moura Olvera s an Assstant Professor of the Engneerng Department Systems at the Unversty of Trás-os-Montes e Alto Douro (UTAD), Portugal. He receved hs dploma n Electrcal Engneerng n 1991, from the UTAD Unversty, M.Sc. n Industral Control Systems n1994 and Ph.D. n Control Engneerng n 1998, both from Salford Unversty, UK. Hs man research nterests concern Computatonal Intellgence applcatons to automaton and control. Hussen M. Khodr receved hs Ph.D., M.Sc. and Engneer degrees n Electrcal Engneerng from the José Antono Echeverría Hgher Polytechnc Insttute (ISPJAE) n 1997 and 1993, respectvely. He s a Former Assocate Professor of Electrcal Engneerng at Unversdad Smón Bolívar, Venezuela. He also was a Researcher at INESC Porto, Portugal. Presently, he s a Researcher at GECAD Knowledge Engneerng and Decson-Support Research Group Polytechnc Insttute of Porto (ISEP/IPP). He has partcpated n a number of projects performed for the local ndustres. Hs current research actvtes are concentrated on plannng, operaton, and economcs of electrcal dstrbuton and ndustral power systems, power qualty, groundng systems and optmzaton.