Optimal Bidding Strategies for Generation Companies in a DayAhead Electricity Market with Risk Management Taken into Account


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1 Amercan J. of Engneerng and Appled Scences (): 86, 009 ISSN Scence Publcatons Optmal Bddng Strateges for Generaton Companes n a DayAhead Electrcty Market wth Rsk Management Taken nto Account Azm Saleh, Takao Tsu and Tsutomu Oyama Department of Electrcal and Computer Engneerng, Yokohama Natonal Unversty, 795 Tokwada, Hodogayaku, Yokohama , Japan Abstract: Problem statement: In a compettve electrcty market wth lmted number of producers, Generaton Companes (Gencos) s facng an olgopoly market rather than a perfect competton. Under olgopoly market envronment, each Genco may ncrease ts own proft through a favorable bddng strategy. The obectve of a Genco s to maxmze ts proft and mnmze the assocated rsk. In order to acheve ths goal, t s necessary and mportant for the Genco to make optmal bddng strateges wth rsk management before bddng nto spot market to get an expected hgh proft, snce spot prces are substantally volatle. Ths study propose a method to buld optmal bddng strateges n a dayahead electrcty market wth ncomplete nformaton and consderng both rsk management and unt commtment. Approach: The proposed methodology employs the Monte Carlo smulaton for modelng a rsk management and a strategc behavor of rval. A probablty densty functon (pdf), Value at Rsk (VaR) and Monte Carlo smulaton used to buld optmal bddng strateges for a Genco. Results: The result of the proposed method shows that a Genco can buld optmal bddng strateges to maxmze expected total proft consderng unt commtment and rsk management. The Genco controls the rsk by settng the confdence level. If the Genco ncrease the confdence level, the expected total VaR of proft decrease. Conclusons/Recommendatons: The proposed method for buldng optmal bddng strateges n a dayahead electrcty market to maxmze expected total proft consderng unt commtment and rsk management s helpful for a Genco to make a decson to submt bddng to the Independent System Operator (ISO). Key words: Electrcty market, bddng strateges, day ahead electrcty market, rsk management, unt commtment, monte carlo INTRODUCTION Deregulaton and reformng n the electrcty market have created a compettve open market envronment. Under the deregulated envronment, the Genco operate own generatng resources ndependently accordng ther ndvdual profts. The obectve of a Genco s to max ts proft and mn the assocated rsk. In order to acheve ths goal, ts necessary and mportant for the Genco to make optmal bddng strateges wth rsk management before bddng nto a spot market to get an expected hgh proft, snce spot prces are substantally volatle. The dayahead electrcty market s cleared on an hourly bass. The supply curve s bult up for each hour consderng the sellng bds ordered by ncreasng prces and also a demand curve s bult up consderng the buyng bds ordered by decreasng prces. The ntersecton of the supply curves and demand curves determnes the sellng and buyng bds that are accepted. The hourly market prce s the prce of the last accepted sellng bd. Ths process results n a unform prce for every hour. A Unt commtment becomes responsble for each Genco and dffcult for small Genco have one generaton or small generaton capacty. In order to buld optmal bddng strateges wth rsk management, a Genco should consder a unt commtment wth constrans n tme perods (for example mnmum up and down tme, startup and shutdown cost) for possbltes to get dscontnuous dspatch that could reduce expected total profts. In perfectly compettve electrcty market, the optmal bddng strateges for a Genco s to bd ts margnal operaton cost. However, the emergng electrcty markets are not perfectly compettve due to Correspondng Author: Azm Saleh, Department of Electrcal and Computer Engneerng, Yokohama Natonal Unversty, 795 Tokwada, Hodogayaku, Yokohama , Japan Tel: , Fax: , Emal: 8
2 Am. J. Engg. & Appled Sc., (): 86, 009 specal features, such as large nvestment sze and economy of scale n the generaton sector and therefore more akn to olgopoly. In an olgopoly electrcty market, Genco could exercse strategc bddng to maxmze own proft. The problem of how to develop optmal bddng strateges for compettve Genco n the electrcty market was addressed for the frst tme []. Many strategc bddng models have been reported n recent years, whch can be grouped nto three man categores as follows: () estmatng the Market Clearng Prce (MCP): Bd prce s slghtly lower than the estmated prce [,3]. () game theores: base ether on beneft matrx or mperfectly compettve game model [4,6]. () estmatng the bddng behavors of rvals: usually based on probablstc method [7,8]. All strategc bddng models are wthout assocated rsks taken nto account. The frst category s smple n prncple. Based on the estmaton of the MCP, the Genco determne ts bddng strateges by offerng n a prce cheaper than the MCP. Ths method s based on an mplct assumpton that the own bd wll not nfluence the MCP. The game theory s appled n the second category. The bddng strateges have to be represented as dscrete quanttes such as bddng hgh, bddng medum and bddng low n order to make problem smple. However, n realstc stuaton, bddng strateges can be contnues quanttes. The probablstc method s appled n the thrd category. Based on hstorcal data, the Genco can estmate bddng strateges of rvals. The Genco determne ts bddng strateges based on the estmaton of bddng of rvals. Ths method s assumed to be more practcal n electrcty market. Therefore t s used n ths paper to buld optmal bddng strateges. Ths study proposes a method to buld optmal bddng strateges n a dayahead electrcty market wth ncomplete nformaton and consderng both rsk management and unt commtment. A probablty densty functon (pdf) for bddng behavors of rvals s consdered to model the uncertanty n the rval behavor. The proposed methodology employs the monte carlo smulaton for solvng stochastc optmzaton. The rsk management n uncertanty of rval behavor s analyzed usng Value at Rsk (VaR). The numercal test result of a smulated electrcty market wth four Genco used to demonstrate the essental features of the developed model and method. MATERIALS AND METHODS Market Structure: In the poolbased electrcty market, every Genco submts a bddng prce functon 9 to the Independent System Operator (ISO) for every hour of the plannng horzon. The ISO uses the bddng prce functon and forecastng demand to determne the Market Clearng Prce (MCP) and hourly generaton outputs by maxmzng the total surplus of generators and consumers. Assume that each Genco s requred to submt lnear bddng prce functon α + βp, where P s the generaton output and α and β are bddng coeffcent of bddng prce functon. The ISO determne the MCP and the hourly generaton outputs each Genco usng formulaton as: n α n Q0 = β = β () R = + P = R α β Subect to: () N P = Q (3) = P,mn P P,max (4) Where: R = The margnal clearng prce Q = The pool load forecasted P = The generaton output of the th Genco When the soluton set from () volates the generaton outputs lmt (4), t modfed based on these constrants. When P s larger than upper lmt constrants P,max, P should be set to P,max. When P s smaller than lower lmt constrants P,mn, P should be set to zero. The proft functon of th Genco s defned by the dfference between the total revenue and the total producton cost as: π = R.P C(P ) (5) Subect to: (4) Where: π = The proft of th Genco C (.) = The producton cost functon of the th Genco The producton cost functon of the th Genco s assumng a quadratc functon as:
3 Am. J. Engg. & Appled Sc., (): 86, 009 C(P) = a + b.p+ c.p (6) where, a, b and c s the coeffcents of producton cost. The problem of buldng optmal bddng strateges for the th Genco s determned α and β so as to max proft as: max π( α, β,p ) = ( α +β.p )P C (P ) (7) Subect to: (4) To solve Eq. 7, the th Genco need data of bddng coeffcents of rvals. Because the bddng coeffcents of rvals are confdental, the th Genco could be estmated based on hstorcal bddng data. The problem of th Genco to estmatng the bddng coeffcents. Estmatng opponents unknown nformaton: Generally, the Gencos do not have access to complete nformaton of ther opponent, so t s necessary for a Genco to estmate opponents' unknown nformaton. It s assumed that the past data of bddng coeffcents are avalable. The th Genco can determne mean and standard devatons of bddng coeffcents based on hstorcal data. Suppose that the data of bddng coeffcents are normally random varables wth the followng probablty densty functon (pdf) as: Fg. : The pdf of two varable α and β Ths can be expressed n the compressed form as: ( α, β ) N, ( ) ( ) ( α) α ( α) ( β) t t µ σ ρ. σ. σ ( β) ( ) ( ) ( ) α β β µ ρσ.. σ σ ( ) (0) Where: ρ = The correlaton coeffcent between α and ( α) ( β) β µ, µ = The mean values σ, σ = The standard devatons ( α) ( β) pdf (x ) (x µ ) πσ =.exp. σ Where: µ I = The mean values σ I = The standard devatons (8) The data of bddng coeffcents have two values α and β (the ntercept and slope) of bddng prce functon, respectvely. The pdf functon wth two varables that represent the ont dstrbuton of α (t) and (t) β ( =,,, n, ; t =,,,4) can be formulated as: (t) (t) (t) pdf ( α, β ) = ( ) ( ) πσ α σ β ρ (t) ( α) α µ xexp ( α) ( ) ρ σ (t) ( α) (t) ( β) (t) ( β) ρ ( α µ )( β µ ) β µ + ( α) ( β) ( β) σ σ σ (9) 0 The correlaton coeffcent s a number among  and. If there s no relaton of two varables, the correlaton coeffcent s 0. The perfect relatons of two varables, the correlaton coeffcent s or . The two varables that represent the ont dstrbuton of bddng coeffcents can be vsualzed n Fg.. Based on estmaton of bddng coeffcents, the th Genco can determne α (t) and β (t) (t =,,, 4) so as to maxmze proft usng Eq. 8. The optmal bddng problem became a stochastc problem. Optmal bddng strateges n a dayahead electrcty market: The problem of developng bddng strateges for th Genco n a dayahead energy market can be formulated as maxmzaton of total proft durng 4 h as: 4 (t) (t) t t () t= max Ω= π ST ( u ) u Subect to:,f Ton > Tup ut = 0,f Toff > Tdown 0or,otherwse ()
4 Where: ST = The startup cost u t = The status of the th Genco (: operaton, 0: down) T on = On tme duraton of the th Genco T off = Off tme duraton of the th Genco T up = Mnmum up tme T down = Mnmum down tme When an th Genco s n operaton, t cannot shut down before a mn up tme perod s met. On the other hand, when an th Genco s n shut down, t cannot start up agan before a mn down tme have passed. If mn up tme and mn down tme perod have passed, the status of the th Genco can set to or 0 as to max total proft. To solve Eq. drectly s dffcult and should be solved separately as follow: Step : Developng the bddng strateges and the status (ut) of the th Genco for each hour of the schedule day, separately, usng Eq.. If the th Genco gets dspatch from ISO, set value of ut to, otherwse, set to 0 (zero). Step : Checkng the status of the th Genco wth the unt commtment constrant. If the unt commtment constrant s satsfed, then these strateges are optmal for a dayahead market and the th Genco should reman n operaton for the whole day and the procedure s completed here. Otherwse, go ahead to step 3 to update status for th Genco. Step 3: Determnng the status for th Genco to satsfy the unt commtment constrant usng dynamc programmng to maxmze total proft durng 4 h. If th Genco should be n operaton because of the commtment constrant (due to constrants of mnmum up tme), update bddng coeffcents and reduce bddng offer so that th Genco can obtan dspatch from ISO. The dynamc programmng represents a mult stage decson problem as a sequence of sngle decson problems. The advantage of dynamc programmng s ts ablty to mantan soluton feasblty, unlke prorty lst method. Dynamc programmng bulds and evaluates the complete decson tree to optmze the problem. The two possble states for th Genco (u t = 0 or ) problem can be solved usng forward dynamc programmng algorthm to run forward n tme from ntal hour to the fnal hour. The ntal condtons are easly specfed and the computatons can go forward n tme as long as requred. Am. J. Engg. & Appled Sc., (): 86, 009 The proposed soluton method: In order to solve the problem of buldng optmal bddng strateges for th Genco n a dayahead electrcty market, frst buldng optmal bddng strateges for each hour of the schedule usng Eq. 7. Secondly s determnng the unt status of th Genco that satsfed unt commtment constrants usng forward dynamc programmng algorthm. The th Genco s estmatng the bddng coeffcents based on pdf to model the uncertanty n the rval behavor. In ths paper a Monte Carlo method s consdered to deal wth the aforementoned uncertantes. The Monte Carlo method s a numercal smulaton procedure appled to problem nvolvng random varables wth kwon or assumed probablty dstrbutons. It conssts n repeatng a determnstc smulaton process, usng n each smulaton a partcular set of values for the random varables that generated accordng to the correspondng probablty dstrbutons. The basc procedure n Monte Carlo method s frst s generatng random samplngs. The random samplngs of two bddng coeffcents from each rval accordng to ther probablty densty functon (pdf). Secondly s solvng the optmzaton problem n Eq. 7 wth all the bddng coeffcents from rval partcpants as fxed constant numbers n Monte Carlo smulaton teraton. Fnally s calculaton the statstc parameters such as the expectaton value and standard devaton of the proft. In order to solve the maxmzng proft problem of the th Genco, the two coeffcents α (t) and β (t), cannot be selected ndependently. The th Genco can fx one and then determne the other by usng an optmzaton procedure. In ths work, the lagrangan relaxaton method s used for ths purpose. It s assumed that the th Genco fxes bddng coeffcents of α (t) and determne the bddng coeffcents of β (t). After all number of smulatons s done, the expectaton value of β (t) s adopted as the optmal bddng strateges of th Genco. Optmzaton based lagrangan relaxaton method: The problem of buldng the optmal bddng strateges for th Genco as Eq. 7 could be expressed mnmzng producton cost as: mn π( α, β,p) = C(P) ( α +β.p)p (3) Subect to: (3) α Denote u = Q0 + and v = β. Eq. and β could be expressed as:
5 Am. J. Engg. & Appled Sc., (): 86, 009 α+ uβ R = vβ+ (4) P u αv = (5) v β+ Equaton 3 could be solved through the generalzed Lagrange multpler method as: L( α, β,p, λ ) = C (P ) ( α +β.p )P λ(u α v P (vβ + )) (6) The optmal soluton of Eq. 6 wth applyng Kuhn = Tucker condtons, could be obtaned wth assume α = b are: β = ( ) cv + b u cu v vb u Subect to: P,mn P P,max (7) When P s less than P,mn, update β usng the formula as: u bv P,mn v.p,mn β = (8) When P s greater than P,max, update β usng formula (9) wth replace P,mn wth P,max. Rsk consderaton: To ths pont the problem of buldng the optmal bddng strateges wthout consderng rsk. However, ths concept s an mportant subect from Genco's vewpont. Method for handlng the rsk s ntroduced. Value at Rsk (VaR) s an estmate that shows how much a portfolo could lose due to market movements at a partcular tme horzon and for a gven probablty of occurrence. The gven probablty s called a confdence level, whch represents the level of certanty of VaR. In ths study, the VaR s defned as the expected mnmum proft of a portfolo over a target wthn a gven confdence nterval. The confdence level depends on the extent of the Genco's rskaverson. Normally, a Genco wth moderate rskaverson adopts 95% confdence level; a more rskaverse Genco may requre 99% confdence level and a less rskaverse Genco could use 9.5% confdence level. Fg. : Cumulatve dstrbuton functon of profts Monte Carlo smulaton s used to calculate the VaR based on cumulatve dstrbuton functon (cdf) of proft durng the smulaton process. For example, Fg. show the cfd of the 000 smulatons. The proft correspondng wth (0095% = 5%) s VaR, whch s $,5x0 5. Accordng to Fg., there s a 95% probablty that the actual proft wll be larger than $,5x0 5. Expected proft correspondng wth (0050% = 50%) or wthout consderng the rsk s $,30x0 5 whch s a 50% probablty that the actual proft wll be larger than $,30x0 5. The Value at Rsk (VaR) s calculated based on expected proft obtaned by Monte Carlo method. The calculaton of VaR s a threestep process: Step : Aggregate the expected proft to form a dstrbuton of expected proft. Step : Create cumulatve dstrbuton functon of expected proft from dstrbuton of expected proft. Step 3: Determne expected VaR of proft based on confdence level. The buldng optmal bddng strateges n a dayahead electrcty market wth rsk management taken nto account s presented n flowchart n Fg. 3. Two stages to buldng optmal bddng strateges: Stage : Determnng the optmal bddng strateges and expected proft each hour. Stage : Determnng the status for th Genco to satsfy the unt commtment constrants. In stage, creatng the samplng of bddng coeffcents rval every hour before determnng optmal bddng parameter and expected proft each teraton (samplng). After the teraton fnshed, creatng the cumulatve probablty dstrbuton (cdf) of expected proft. Fnally, a determnng the optmal bddng parameter and expected VaR of proft based on cdf and confdence level.
6 Am. J. Engg. & Appled Sc., (): 86, 009 START Set confdence level Hour = Create the samplng of bddng coeff cents rvals usng eq. (0) Iteraton = Table : Generaton data Genco a b c Pmn (MW) Pmax (MW) Table : Estmatons of the rvals ( α) ( β) Genco µ µ σ ( α) ( β) σ ρ.b w t x.c b c 0..b w t x.c b c b w t x.0c b c 0. wt = + 0.3(Q Q mn)/(qmax Q mn). Fnd Optmal Bddng Parameter. Calculate the Expected Proft Last teraton? Yes. Create cd f of Expect ed Proft. Determne Bddng Parameter 3. Determne Expected VaR of Pro ft No No Power (MWh) Tme (h) 9 3 Last hour? Yes Unt commtment constrants satsfed? Yes END No Stage Determne commtment status Fg. 3: Flow chart of the proposed method Stage In stage, check the unt commtment constrants. If satsfed, the calculaton process s fnsh. If dd not satsfed, a determnng commtment status usng dynamc programmng. RESULTS The four Gencos s used to llustrate the results of applcaton of the proposed method for buldng optmal bddng strateges n a dayahead electrcty market wth consderng rsk management. 3 Fg. 4: Hourly loads The producton cost functon of the th Genco s C (P ) = a +b.p +c.p. The coeffcents of producton cost functon and output lmts of all Gencos are shown n Table. Hourly loads n a dayahead electrcty market that ISO broadcast as shown n Fg. 4. In the followng case studes, we suppose that the fourth Genco n our subect research estmated ont normal dstrbuton for the two bddng coeffcents α (t) and β (t) of the rvals. The estmated parameters n the ont normal dstrbuton for the rvals' as descrbed n Eq. 5 are shown n Table. When suffcent bddng data from past bddng hstores s avalable, these parameters can be estmated usng stochastc methods. Here, Q mn and Q max are the mnmum and maxmum loads n the 4 h of a dayahead electrcty market, respectvely and n ths smulaton Q mn = 335MW and Q max = 755MW. Hence, wt takes a value rangng from .3 and s a lnearly ncreasng functon of the system loads. Optmal bddng wthout consderng the rsk: In frst case, the fourth Genco buldng optmal bddng strateges n a dayahead electrcty market wthout consderng the rsk usng the proposed method.
7 Table 3: Optmal β and expected proft wthout consderng unt commtment constrant Expected Expected T β Proft ($) T β Proft ($) Table 4: Optmal β and expected proft wth consderng unt commtment constrant Expected Expected T β Proft ($) T β Proft ($) Expected total proft ($).3 In ths case, the confdence level s settng wth 0.5 or 50%. The smulaton results of bddng coeffcents and expected hourly proft s shown n Table 3. From Table 3, the fourth Genco gettng to dscontnue dspatch and unt commtment constrant dd not satsfy. The fourth Genco determned the commtment status usng dynamc programmng to make unt commtment constrant dd not satsfy wth assume ntal condton for fourth Genco s off. Fnal result from the commtment status s shown n Table 4. The expected proft dstrbuton and cumulatve dstrbuton of expected proft durng Monte Carlo smulaton process at h and 6 are shown n Fg. 5 and 6, respectvely. The margnal clearng prce durng 4 h s shown n Fg. 7. Optmal bddng wth consder the rsk: In the second case, Gencos consder the rsk for bddng optmal bddng strateges n a dayahead electrcty market wth settng confdence level above 50%. In ths smulaton, the fourth Genco settng value of a confdence level s 95% and 99%, respectvely. The smulaton results of bddng coeffcents and expected hourly proft s shown n Table 5. Am. J. Engg. & Appled Sc., (): 86, Count Expexted Proft ($) Cumulatve probablty Fg. 5: Dstrbuton and cumulatve dstrbuton of expected hourly proft at h Count Expexted Proft ($) Fg. 6: Dstrbuton and cumulatve dstrbuton of expected hourly proft at 6 h Fg. 7: The expected hourly margnal clearng prce From Table 5, the fourth Genco gettng to dscontnue dspatch and unt commtment constrant dd not satsfy. The fourth Genco determned the commtment status usng dynamc programmng to make unt commtment constrant dd not satsfy wth assume ntal condton for fourth Genco s off. Fnal result from the commtment status s shown n Table 6. Cumulatve Probablty
8 Table 5: Optmal β and expected var of proft wthout consderng unt commtment constrant Wthout Rsk VaR = 95% Var = 99% Expected Expected Expected T β Proft ($) β Proft ($) β Proft ($) Table 6: Optmal β and expected var of proft wth consderng unt commtment constrant Wthout rsk VaR = 95% Var = 99% Expected Expected Expected T β Proft ($) β Proft ($) β Proft ($) Total proft The margnal clearng prce durng 4 h wth a dfference confdence level s shown n Fg. 8. Am. J. Engg. & Appled Sc., (): 86, Fg. 8: The expected hourly margnal clearng prce wth dfference confdence level DISCUSSION In Table, fourth Genco make estmaton for two bddng coeffcents α (t) and β (t) of the rvals. For the frst rvals, the mean value of α (t) and β (t) are 0% above b and wt tmes of.xc, respectvely. Also, the standard devatons of α (t) and β (t) are xb and xc, respectvely. The correlaton coeffcent among α (t) and β (t) s 0.. The correlaton s negatve because when rvals decde to ncrease one of ts two bddng coeffcents, t wll decrease rather than ncrease the other coeffcent. For example, n h the mean value of α (t) and β (t) are 0.34 and wtx0.088 or 0.03, respectvely. The standard devatons of α (t) and β (t) are and The range value of α (t) and β (t) are [ ; 0.695] and [0.08 ; 0.046] Optmal bddng wthout consderng the rsk: In Table 4, the fourth Genco dd not submt offer durng untl 7 h and durng untl 4 h. The fourth Genco stll submt offer n h. to satsfy unt commtment constrant wth negatve expected proft s $ The total expected proft of fourth Genco s $.3. In Fg. 5, the probablty for fourth Genco gettng proft $8.4 or more s 50% and the probablty gettng negatve proft s about 5%. It means, the fourth Genco have been rsk to gettng proft negatve n h. In Fg. 6, the probablty for fourth Genco gettng hourly proft $7.7 or more s 50% and the probablty gettng negatve proft s 0%. It means, the fourth Genco no rsk to gettng negatve profts n 6 h. Optmal bddng wth consder the rsk: In Table 6, the fourth Genco wth confdence level 95% and 99% dd not submt offer durng hour untl hour 8 and
9 Am. J. Engg. & Appled Sc., (): 86, 009 durng untl 4 h. The fourth Genco stll submt offer n h to satsfy unt commtment constrant wth negatve expected VaR of proft are $ and  $80.466, respectvely. The total expected VaR of proft are $ and $470.06, respectvely. In ths case, ncreasng a confdence level only decreasng the expected proft. CONCLUSION In ths research, bddng decsons problem n dayahead electrcty market consderng rsk management and unt commtment s formulated from the Genco vewpont. A stochastc optmzaton model s bult for ths purpose usng Monte Carlo smulaton and Lagrangan Relaxaton and effcent for solvng ths problem. Smulaton results of a numercal example have demonstrated the effect of a confdence level to the decson submts offer and expected total VaR of proft. In ths case study only shown effect of confdence level to the expected total VaR of proft. If the Genco ncreased the confdence level, the expected total VaR of proft wll decreased. REFERENCES. Davd, A.K., 993. Compettve bddng n electrcty supply. IEE Proceedng of the Generaton Transmsson Dstrbuton., Sep. 993, IEEE Xplore, USA., 40: ber= Shangyou, H., 000. A study of basc bddng strategy n clearng prcng auctons. IEEE Trans. Power Syst., 5: DOI: 0.09/ Ferrero, R.W. and J.F. Rvera, 00. Prcetaker bddng strategy under prce uncertanty, IEEE Trans. Power Syst., 7: DOI: 0.09/TPWRS Ferrero, R.W. and J.F. Rvera, 998. Applcaton of games wth ncomplete nformaton for prcng electrcty n deregulated power pools. IEEE Trans. Power Syst., 3: org/xpl/freeabs_all.sp?arnumber= Gan, D., J. Wang and D.V. Bourcer, 005. An aucton game model for poolbaed electrcty markets. Elect. Power Energ. Syst., 7: DOI: 0.06/.epes Kang, D.J., B.H. Km and D. Hur, 007. Supler bddng strategy based on noncooperatve game theory concepts n sngle aucton power pools. Elect. Power Energ. Syst., 77: DOI: 0.06/.epsr Fushuan, W. and A.K. Davd, 00. Optmal bddng strateges and modelng of mperfect nformaton among compettve generators. IEEE Trans. Power Syst., 6: 5. DOI: 0.09/ Shreshta, G.B. and L.K.S. Goel, 00. Strategc bddng for mnmum power output n the compettve power market. IEEE Trans. Power Syst., 6: DOI: 0.09/
benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
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