Complex Dynamics Analysis for Cournot Game with Bounded Rationality in Power Market

Transcription

1 J. Elecromagnec Analyss & Applcaons 9 : 48- Publshed Onlne March 9 n ScRes ( Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke Hongmng Yang Yongx Zhang College of Elecrcal and Informaon Engneerng Changsha Unversy of Scence and echnology Chna. Emal: yhm58@homal.com Receved January h 9; revsed February h 9; acceped February 3 rd 9. ABSRAC In order o accuraely smulae he game behavors of he marke parcpans wh bounded raonaly a new dynamc Courno game model of power marke consderng he consrans of ransmsson nework s proposed n hs paper. he model s represened by a dscree dfferenal equaons embedded wh he maxmzaon problem of he socal benef of marke. he ash equlbrum and s sably n a duopoly game are quanavely analyzed. I s found ha here are dfferen ash equlbrums wh dfferen marke parameers correspondng o dfferen operang condons of power nework.e. congeson and non-congeson and even n some cases here s no ash equlbrum a all. he marke dynamc behavors are numercally smulaed n whch he perodc or chaoc behavors are focused when he marke parameers are beyond he sably regon of ash equlbrum. Keywords: Chaos Dynamc Model ash Equlbrum Power Marke. Inroducon Some foundaon ndusres such as elecrc power avaon elecommuncaon ralroad ec. are radonally hough of havng naural monopoly characerscs. Wh he developmen of echnology economy and socey n recen years hese ndusres have been undergong a marke reformaon de of deregulaon and compeon n order o reduce he cos and prce of monopoly ndusry and promoe he enhancemen of socal economy benef. All hese ndusres have he naural monopoly nework wh he complex nheren physcal propery by whch he marke parcpans can provde commody servces. he complex monopoly nework causes he reformaon and operaon of marke o be more complcaed and dffcul han ha of general commody marke especally for he reformaon of elecrc power ndusry. In he process of reformaon and operaon of marke how o effecvely maser and supervse he dynamc marke behavors s an mporan research opc especally for he power marke whose reformaon s carred ou n s nfancy sage. akng an exreme example of Calforna power marke he negleced sudy of he dynamc marke behavors led o a severe suaon causng he elecrc power wholesale prce o rse sharply and hus affecng he power supply o a lo of cusomers. hs happened n less han hree years of marke operaon whch has made a grea mpac on he economy of Calforna and even he USA []. he sysem of marke economy s essenally a dynamc sysem whch s mahemacally represened by he dfferenal or dfference equaons. In he dynamc heory of economcs here are a lo of dfferenal or dfference dynamc models such as he classcal cobweb model descrbng he varaon of he supply and demand he Courno dynamc model reflecng he olgopoly marke he Haavelmo model descrbng he economc growh problem and so on [3]. Based on hese models he analyss and conrol of he sable perodc and chaoc dynamc evoluon of he marke economy sysem are nvesgaed and a seres of resuls have been yelded [45]. However he complex nheren physcal propery of nework and he parculary of marke ransacon n he marke wh he monopoly nework are no aken no accoun n hese models and mehods. herefore n vew of he characerscs of power marke he research on he dynamc evoluon of power marke s carred ou by some scholars. he research on he dynamcs of power marke was frs launched by F. L. Alvarado e al. va a se of one-order dfferenal equaons of power generaons and consumpons. hs work provdes nsghs o he condons for he evolvng process convergng o he marke equlbrum.e. he sably condon of power marke [67]. Wh he same dynamc model a seres of suffcen condons are gven o deermne he sably of power marke n Reference [8]. Reference [9] esablshes he dfference equaons by akng he elecrcy prce as a varable and analyzes he sably condon needed for he elecrcy prce convergng o he equlbrum. Alhough he resuls acheved are neresng hese models are esablshed based on a perfec compeve model. I Copyrgh 9 ScRes

2 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke 49 neglecs he game behavors of generaon companes as well as her mpacs on he elecrcy prce and hus can no raonally descrbe he acual power marke. In order o accuraely smulae he game behavors of he marke parcpans he olgopolsc game models n economcs are furher nroduced o research he dynamcs of power marke. References [] adop he Courno model o esablsh he dfferenal equaons of he dynamc power marke. hen he marke equlbrum of he generaon quanes s calculaed under he gven demand funcon and he varyng curve of elecrcy prce convergng o he equlbrum s numercally smulaed. In References [3] he dynamc dfferenal or dfference equaons are esablshed based on he perfec compeve model he Sackelberg game model and he Courno game model respecvely. However he consrans of power nework and her mpacs on he elecrcy prce are no aken no accoun nor are he sably analyss of he marke equlbrum nvolved. In Reference [4] he evoluonary game s nroduced o esablsh he dynamc evoluonary dfferenal equaons by akng he generaon bds as varables. However he consrans of power nework are no aken no accoun oo. Consequenly no only he raonal game behavors of marke parcpans bu also he nheren physcal properes of power nework need o be consdered n he dynamc modelng of power marke. In addon due o he complex dynamc characerscs of he acual power marke n some cases here exss no marke equlbrum a all or even f here s mgh le n he non-sably regon of he marke equlbrum. I s sgnfcan for he marke operaors o sudy he dynamc behavors of he power marke assocaed wh hese cases. herefore he am of hs paper s o make a horough sudy concernng he dynamc Courno game behavors of he power marke wh bounded raonaly under he consderaon of he power nework consrans. he followng aspecs are focused: A new dynamc Courno game model of power marke represened by he dfference equaons embedded wh he maxmzaon problem s proposed. he remarkable characersc of he model s wofold: adops a dynamc adjusmen where he lm pon s he ash equlbrum of power marke; and he sysem of dscree dfference equaons embedded wh he maxmzaon problem consders he consrans of power nework. he exsence and sably of ash equlbrum for a duopoly game are quanavely analyzed wh dfferen marke parameers under dfferen operang condons of power nework; 3 he dynamc behavors of power marke especally he perodc and chaoc dynamc behavors when he marke parameers are beyond he sably regon of equlbrum are numercally smulaed.. Dynamc Courno Game Model of Power Marke wh Bounded Raonaly Consderng ework Consrans. Dynamc Courno Game Model wh Bounded Raonaly Power marke s dfferen from general compeve commody marke n whch he producon of power energy needs very hgh cos and echnology and here are fne elecrc power producers. hs naure of elecrc power ndusry mples ha power marke does no have he characersc of perfec compeve marke bu should belong o an olgopolsc marke. In economcs several knds of game models have been proposed o smulae he olgopolsc behavors of marke parcpans. he Courno game model s mos commonly used whch smulaes he compeon of oupu quanes beween he olgopolss [5]. Recenly he sac Courno models are appled o analyze he ash equlbrum of power marke [67]. In hs case he game of marke parcpans s done based on a fully raonaly. Each parcpan has complee marke nformaon (ncludng he compeors prof funcons when he makes hs opmal producon decson. If here s a ash equlbrum n he marke he olgopolss can move sragh (n one sho o he ash equlbrum. he process s ndependen of he nal condon and does no relae o any dynamc adjusmen of power marke. However n he acual power marke he marke parcpans are no fully raonal and unable o know he compeors producon decson and prof funcons. hey are unable o reach he equlbrum condon a once. In fac each parcpan s bounded raonal and can only decde he producon sraegy accordng o hs expeced margnal prof a each perod. For each marke parcpan he evaluaon of hs own margnal prof s more accurae han he predcon of he compeors oupus [89]. herefore he marke parcpans play a Courno game wh bounded raonaly n a dynamc adjusmen process descrbed as follows. In he marke operaon a generaon producer decdes he opmal producon sraegy accordng o s own generaon cos and marke nformaon n order o oban maxmum prof. he opmal decson problem can be mahemacally wren as q ( q max π = Pq C ( where C ( q s he generaon cos of generaon company a node ; P s he elecrcy prce a node whch s decded by he Independen Sysem Operaor (ISO. By applyng he margnal prof funcon of a generaon company he opmal generaon quanes can be obaned. π P C = P + q q q q ( q = ( Copyrgh 9 ScRes

3 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke Due o he lack of global nformaon of power marke each generaon company adjuss s supply quanes for obanng more benef accordng o he local esmae of s own margnal prof. he mahemacal model of he adjusmen mechansm of he generaon quanes.e. he dynamc Courno game model wh bounded raonaly s π q ( + = F ( q ( = q ( + α ( q ( q ( ( q ( q ( P C = q ( + α ( q ( P + q ( q ( = L (3 ( where q ( ( + q are he generaon quanes of he generaon company a node a me and + ; α s a posve funcon whch gves he exen of ( ( q he producon varaon of he h generaon company α s assumed followng a gven prof sgnal. If ( q ( o be a lnear funcon hen ( q ( =αq α can be obaned where he posve consan α s called he speed of adjusmen. From (3 can be seen ha n order o cause he generaon company o oban a more economcal prof n he power marke f s margnal prof s greaer han he generaon company wll ncrease q n he nex me; oherwse he generaon company wll decrease q n he nex me.. ISO Opmzaon Model In he power marke he decson behavors of marke parcpans should be checked by he ISO o sasfy he nheren physcal characerscs of power nework and ensure he secury of power sysem operaon. In he cenralzed marke clearng on he premse ha he supply quanes of he generaon companes are known (whch can be deermned by he dynamc Courno game model of he marke parcpans n (3 he ISO allocaes he marke demand o maxmze he oal marke benef wh sasfyng he power nework consrans such as he power balance consran and he lne flow consrans. he mahemacal model based on he DC power flow can be expressed as follows: ( d B ( d max e B + (4 s.. H e d + d = e q + ( q d K where s he oal number of nodes ( where node s assumed o be he slack node L s he oal number of lnes; dq R are he nodal demand and generaon power vecors excludng he slack node d q are he demand and generaon power a he slack node ; L ( H R denoes he ransfer admance marx ha q represens he sensvy of he nodal power njecon o lne power flow; e R s a vecor of all ones; L K R s he vecor of maxmum power flow on he ransmsson lne; ( B d R s he vecor of he nodal benef of consumer excludng he slack node B ( d s he benef of consumer a he slack node and assumed as B( d = a d.5b d ( = L (5 where a b are he lnear and quadrac coeffcens of he consumer benef funcon. he Lagrange funcon for he opmzaon problem n (4 can be se up (n he consrans of lne power flow only he equaly consrans are aken no accoun: L = e µ B( d + B ( d λ( e d + d e q q ( H& &&( q d K&&& where λ µ are he Lagrange mulplers for he power balance consran and he lne flow consrans; H &&&&& K & are he marces HK excludng he erms correspondng o he non-congeson lnes. By L d = L d = he funcon relaonshp beween he elecrcy prce and he generaon quanes can be obaned as follows: When he congeson occurs n he power nework P B = d B P = d ( d λ a b ( q e H K &&& = = + &&& ( d λe H&&& µ ae b( q H&&& = = K&&& When he congeson does no occur n he power nework b P = λ = a ( e q + q (8 b P = λe = ae ( e q + q e where P R s he nodal prce vecor excludng he slack node P s he nodal prce a he slack node. From (7 and (8 can be concluded ha when here s no congeson n he power nework all nodal prces are dencal; whle durng congeson he nodal prces are dfferen and relaed o he congeson condons of power nework. Wh he change of congesed lnes he marces H &&&&& K & s vared and hen he funcon relaonshp beween he nodal prce and generaon quanes s changed. A furher analyss s performed wh an example of smple power marke as shown n Fgure. here are wo zonal markes conneced by a ransmsson lne wh capacy k. he elecrcy prces of he wo zonal markes are P P wh he demand quanes beng d d and he generaon quanes as q q. (6 (7 Copyrgh 9 ScRes

4 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke 5 Zonal Marke P /($/MWh q -k q ransmsson lne Fgure. Srucure of power marke Lne s no congesed q +k Zonal Marke For smplcy he benef of consumers s dencal n hese wo zonal markes. In he calculaon suppose node s he slack node he posve drecon of lne power flow denoed by he arrow n Fgure. By esablshng he opmzaon model n (6 he funcon relaon beween he zonal prces and he generaon quanes are deduced from (7 and (8. When he ransmsson lne s no congesed he zonal prces are b P = P = a ( q + q (9 In hs case he power flow on he ransmsson lne sasfes: ( q q k q d = k.e. k q q k. When he ransmsson lne s congesed and s power flow s k he zonal prces are ( q k P = a b( q k P = a b + ( Smlarly when he lne power flow s prces are k ( q + k P = a b( q k he zonal P = a b ( herefore under he consderaon of power nework consrans he prce funcon of power marke exhbs he followng pecewse form: ( q + k ( q + q ( q k a b q q < k b P = a k q q k ( a b q q > k ( q k ( q + q ( q + k a b q q < k b P = a k q q k (3 a b q q > k Fgure shows he pecewse connuous curve of elecrcy prce funcon n he zonal marke. q /(MWh Fgure. Curve of elecrcy prce of zonal marke.3 Dynamc Model of Power Marke he dynamc model of power marke s an negraon of he dynamc Courno game model wh bounded raonaly.e. he dscree dfference equaons n (3 and he maxmzaon model of marke benef consderng he power nework consrans.e. he opmzaon model n (4. herefore he dynamc model of power marke s represened by he dscree dfference equaons embedded wh he opmzaon problem. Compared wh he exsng dynamc models he remarkable characerscs of he proposed one are he marke parcpans need no have global marke nformaon such as he marke demand and he compeors cos. hey decde her generaon quanes by esmang her own margnal prof. hs decson process reflecs he acual suaon of he economc sysem o a ceran exen ndcang some feasble and raonal feaures. If he dynamc sysem s fnally able o converge o he equlbrum condon.e. π q ( = each generaon company reaches s own maxmum prof and s unable o mprove he prof only by changng s own generaon sraeges. In hs suaon he marke reaches he condon of ash equlbrum. π s he margnal prof funcon. From 3 q ( (3 can be observed ha f π ( he generaon company wll ncrease q > q n he nex me; oherwse he generaon company wll decrease q. 4 he sysem of dscree dfference equaons embedded wh he opmzaon problem consders he mpac of he power nework consrans on he behavors of he marke parcpans. I can ndcae ha he dynamc model of power marke s more complex han ha of general commody marke. For a duopoly Courno game as shown n Fgure he dynamc Courno model of power marke wh bounded raonaly consderng he power nework consrans s q q ( + = q( + αq( [ a c bk bq ( ] ( + = q ( + α q ( [ a c + bk bq ( ] f q ( q ( < k q q (4 ( + = q( + αq( [ a c bq (.5bq ( ] ( + = q ( + α q ( [ a c.5bq ( bq ( ] f k q ( q ( k q q (5 ( + = q( + αq( [ a c + bk bq ( ] ( + = q ( + α q ( [ a c bk bq ( ] f q ( q ( k (6 > 3. ash Equlbrum and Local Sably of Power Marke Copyrgh 9 ScRes

5 5 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke 3. ash Equlbrum of Power Marke Defnon : A ash equlbrum for ( s a vecor ( q q q q L such ha for each parcpan q = L gven all oher parcpans oupu q q maxmzes he h parcpan s prof ha s q arg max π ( q q. In he dynamc model (3 f q ( + = q ( he marke arrves a a fxed pon. I s called he equlbrum pon n economcs where he fxed pon q ( = s he boundary equlbrum pon. I s easy o verfy ha he nonzero fxed pon s he ash equlbrum pon. For he duopoly dynamc game n he smple power marke as shown n Fgure represened by (4 (5 (6 he equlbrum pons of he marke are analyzed under he dfferen operang condons of power nework.e. congeson or non-congeson. In he model suppose he generaon cos funcon s n lnear form.e. ( q cq C( q c C = = q (7 where c c are he margnal generaon coss. If k q q k he ransmsson lne s no congesed. By solvng he fxed pons n (5 we can have a mos 4 equlbrum pons: where a c a c q = ( q = q = b b q = ( a + c c ( a + c c 3b 3b q q q are he boundary equlbrums and s he ash equlbrum. Due o he sasfacon of he condons -k q -q k q q only he equlbrum pons q q and q q are effecve f <a-c bk <a-c bk a+c -c a+c -c -bk<c -c bk. If q -q <-k or q -q <k he ransmsson lne s congesed. By solvng he fxed pons n (4 we can have a mos 4 equlbrum pons: a c bk a c + bk q = ( q = q = b b q a c bk a c + bk = b b Due o he sasfacon of he condons q -q <-k q q only he equlbrum pons q and q are effecve f a-c >bk c -c >bk. By solvng he fxed pons n (6 we can have a mos 4 equlbrum pons: a c + bk a c bk q = ( q = q = b b a c + bk a c bk q = b b Due o he sasfacon of he condons q -q >k q q only he equlbrum pons q and q are effecve f a-c >bk c -c >bk. From he above analyss s found ha here are dfferen ash equlbrums n he power marke under dfferen operaonal condons of power nework such as congeson and non-congeson whle n some cases here s no ash equlbrum a all f he marke parameers sasfy bk < c c < bk. 3. Local Sably of ash Equlbrum he local sably of equlbrum pon s suded based on he complex plane of he egenvalues of he Jacoban marx of he mappng q ( + = F( q( Defnon : For a dynamc sysem x ( + = F( x(. ( x R wh a fxed pon q f all he egenvalues of he Jacoban marx F(q s less han n modulus here exss an open neghbourhood I of q. When x I such ha lm x( = q here q s called he local sable fxed pon []. If k q -q k he ransmsson lne s no congesed he Jacoban marx F(q s denoed as + α ( a c bq.5bq F( q =.5bq.5bq + α ( a c.5bq bq when he marke les n he ash equlbrum pon ( a + c c ( a + c c q = he egenvalue 3b 3b equaon of he Jacoban marx F( q s: λ ( α bq α bq λ 3 + α bq α bq + αα b q q = 4 he sably condon of ash equlbrum pon s λ < λ < and hus he marke parameers should sasfy: 4α bq + 4α bq 6 < 3α α b q q 4α bq 4α bq < (8 when he marke equlbrum pon s a c q = b he wo egenvalues of he Jacoban marx F( q are λ = α( a c < Copyrgh 9 ScRes

6 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke 53 λ = + α( a + c c > hus he boundary equlbrum pon q s unsable. Smlarly s easy o prove ha he boundary equlbrum pon q s unsable oo. If q q <-k he ransmsson lne s congesed he Jacoban marx F(q s denoed as + α( a c bk 4bq F( q = + α( a c + bk 4bq when he marke les n he ash equlbrum pon a c bk a c + bk q = he sably condon of b b ash equlbrum pon s: α < α < (9 bq bq when he marke equlbrum pon s a c + bk q = one of he egenvalues of he Jaco- b ban marx F( q s greaer han. hus he boundary equlbrum pon q s unsable. If q -q >k s easy o prove smlarly ha he boundary equlbrum pon q s unsable and whle he ash equlbrum pon q s sable f α < α <. bq bq herefore n he dynamc Courno game wheher he marke can fnally converge o a ceran ash equlbrum pon s decded by he marke parameers and he lne flow lms.e. When he dfference beween he margnal cos of generaon companes s less han bk.e. bk <c -c <bk (he oher marke parameers sasfy a + c c a + c c and he marke parameers sasfy he condon n (8 he generaon quanes of generaon companes do no grealy dffer n dfferen zonal markes. hus he ransmsson lne can no be congesed. In hs suaon f he generaon quanes fall nsde he sably regon of ash equlbrum he marke wll be able o gradually converge o he ash equlbrum pon q = ( a + c c ( a + c c ( 3b 3b When he dfference beween he margnal cos of generaon companes s greaer han bk.e. c c > bk (he oher marke parameers sasfy a-c >bk or a-c >bk and he marke parameers sasfy he condon n (9 he generaon quanes of generaon companes grealy dffer n dfferen zonal markes. hus he ransmsson lne s congesed. In hs suaon f he generaon quanes fall nsde he sably regon of ash equlbrum he marke wll be able o gradually converge o he ash equlbrum pon q + = ( a c bk a c bk b b or q + = ( a c bk a c bk ( b b 3.3 Effec of Marke Parameers on Sably he equaon n (8 gves he sably condon of ash equlbrum f he lne s no congesed. Fgure 3 shows he correspondng sably regon of ash equlbrum pon n he plane of he adjusmen speeds ( α α whch s bounded by he poron of hyperbola.e. where: 3α α b q q 8α bq 8α bq + 6 = A = 3 + a c c A 3 = a + c c ( α α nsde he sably regon he ash equlbrum s sable. From Fgure 3 he ncremen of he adjusmen speeds wll reduce he sably margn when he oher parameers are fxed. If he adjusmen speeds go beyond he sably regon he ash equlbrum pon loses s sably hrough a perod-doublng bfurcaon. If he parameer a he maxmum elecrcy prce of elecrc power s ncreased and he oher parameers α α c c are fxed he sably regon becomes smaller as can be easly deduced from (. Oherwse f he parameer a s reduced he sably of ash equlbrum can be renforced. If he oher parameers are fxed an ncremen of he margnal generaon cos c causes a dsplacemen of he pon A o he rgh and of A downwards. Insead an ncremen of he margnal generaon cos c causes a dsplacemen of he pon A o he lef and of A upwards. In boh cases he effec on he sably of ash equlbrum pon depends on he poson of he pon α For he values ( A c=c=a=b=.5 unsably regon A α Fgure 3. Sably regon of ash equlbrum under noncongeson Copyrgh 9 ScRes

7 54 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke α B Sably Regon Unsably Regon Fgure 4. Sably regon of ash equlbrum under congeson B ( α. In fac f he pon ( α α α α s above he dagonal α= α.e. α < α an ncremen of c can desablze he ash equlbrum pon whereas an ncrease of c renforces s sably. he suaon s reversed f α > α. he equaon n (9 gves he sably condon of ash equlbrum f he lne s congesed. Fgure 4 shows he correspondng sably regon of ash equlbrum pon n he plane of he adjusmen speeds ( α α. where: f or f B = B a c bk c c bk > B = + a c bk B = a c + bk = a c bk c c bk (3 > From Fgure 4 he ncremen of he adjusmen speeds ( α α and he maxmum prce a can cause a loss of sably of ash equlbrum and whle he ncremen of he margnal cos c and c can renforce s sably. herefore he power marke can be kep n he sable equlbrum condon by he followng measures n he acual operaon. he plenful compeon s nroduced o reduce he dfference beween he generaon margnal cos of generaon companes n he power marke; and he raonal power nework plannng can mprove he ransfer capacy of lnes n order o keep he marke n he sable equlbrum. he varaon exen of he generaon quanes s no oo large; and he smooh operaon of he generaor has mporan effec no only on he sably of power sysem bu on he sably of power marke. 3 he maxmum prce of marke s no oo hgh; and he resrcon of he maxmum value of elecrcy prce can renforce he sably of power marke. 4. umercal Smulaon of Dynamc Marke Behavors he dynamc behavors of power marke are demonsraed wh an example of wo-node power marke as shown n Fgure. he evolvng characerscs of marke behavors are analyzed when he parameers le n dfferen ranges by usng he bfurcaon dagram phase dagram Lyapunov exponen and fracal dmenson. In he erave process of he numercal smulaon he benef of consumers s dencal and assumed wh a = $ / MWh and b =. 5 $ / MWh ; he maxmum producon oupus of he wo generaon companes boh are MWh; he flow lms of he lne s MW. 4. Case : Dfference beween Margnal Cos of Generaon Companes s Less han bk Frsly he dynamc behavors of power marke are numercally smulaed when he dfference beween he margnal cos of he wo generaon companes s less han bk.e. bk < c c < bk. he generaon margnal coss are aken as c = $ MWh c = $ MWh. If dfferen values are seleced smlar resuls can be obaned. In hs case a ash equlbrum pon s obaned. By ( he correspondng generaon quanes of he wo zones are (66.67MWh 6.67MWh. Le α =.3 / $ he adjusmen speed of he MWh generaon quanes of generaon company s changed. Fgure 5 shows he bfurcaon dagram of he sable soluons of he generaon quanes and elecrcy prce wh α. When he adjusmen speed of generaon company s changed smlar resuls can be obaned. If he adjusmen speed of he generaon quanes of generaon company sasfes α <. he marke le n he sably regon of ash equlbrum. he generaon quanes wll gradually converge o he unque sable soluon.e. he ash equlbrum pon (66.67MWh 6.67MWh. In hs case he power flow on he lne s MW ha s he lne s no congesed. hus he elecrcy prce of he wo zones s dencal boh beng 36.67$/MWh. Fgure 6 shows he evolvng curve of he marke convergng o he ash equlbrum f α =.. Wh he ncremen of he adjusmen speed α when α >. he marke wll go beyond he sably regon of ash equlbrum and hus loses sably. If. < α <.75 he dynamc evoluon of he generaon quanes and elecrcy prce wll converge o he wo perodc pons and he wo-perod varaon s exhbed. Sequenally wh he ncremen of α he more complex dynamc behavors are exhbed such as four perods egh perods sxeen perods ec. Fgure 7 shows he perodc evolvng curve of he marke f α =. 6. Wh he connuous ncremen of he adjusmen speed α when α >. 75 he marke converges o many Copyrgh 9 ScRes

8 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke nfne pons nsde he bounded range and he seemngly random chaoc varaon s exhbed. When α s n he neghborhood of.8 he sable soluons of he marke le whn a smaller range. In hs case he power flow on he lne s less han MW ha s he lne s no congesed. hus he elecrcy prce of zonal marke and s dencal. Fgure 8 shows he chaoc evolvng curve of he generaon quanes and elecrcy prce f α =.8 ; and he correspondng chaoc aracors as shown n Fgure α α α α Fgure 5. Bfurcaon dagram of sable soluons of power marke wh α f dfference beween margnal cos of generaon companes s less han bk q q P P q 7 P q 5 P Fgure 6. Dynamc marke behavors convergng o ash equlbrum f α =. q 8 P q P Fgure 7. Perodc dynamc marke behavors f α =.6 Copyrgh 9 ScRes

9 56 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke However when he adjusmen speed α les n he oher chaos area he sable soluons of he marke fall whn a greaer range. In hs case s found ha he lne somemes s congesed o cause dfferen elecrcy prce n he zonal marke and ; he chaoc aracors of marke nclude no only he nvarable manfold under non-congeson condon (as shown n Fgure 9 bu he nvarable manfold under congeson condon. Fgure shows he chaoc aracors of he generaon quanes and elecrcy prce f α =. her maxmum Lyapunov exponens and fracal dmensons beng.34 and. respecvely. q q P P Fgure 8. Chaoc dynamc marke behavors f α = q q P( P( Fgure 9. Chaoc aracors of generaon quanes and elecrcy prce f α = q P(+ 7 8 q P( Fgure. Chaoc aracors of generaon quanes and elecrcy prce f α =. Copyrgh 9 ScRes

10 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke Case : Dfference beween Margnal Cos of Generaon Companes s Greaer han bk he dynamc behavors of he power marke are numercally smulaed when he dfference beween he margnal cos of he generaon companes s greaer han bk.e. c c bk. he generaon margnal coss are aken as > c = $ MWh c 5$ MWh =. If dfferen values are seleced smlar resuls can be obaned. In hs case a ash equlbrum pon s obaned. By ( he correspondng generaon quanes of he wo zones are (5MWh 7MWh. Le α =.MWh / $ he adjusmen speed of he generaon quanes of generaon company s changed. Fgure shows he bfurcaon dagram of he sable soluons of he generaon quanes and elecrcy prce wh α. When he adjusmen speed of generaon company s changed smlar resuls can be obaned. If he adjusmen speed of he generaon quanes of generaon company sasfes α <.8 he marke le n he sably regon of ash equlbrum. he generaon quanes wll gradually converge o he unque sable soluon.e. he ash equlbrum pon (5MWh 7MWh. In hs case he power flow on he lne s MW ha s he lne s congesed. hus he elecrcy prce of he wo zonal markes s no dencal beng 4.5$/MWh and $/MWh respecvely. Wh he ncremen of he adjusmen speed α when α >.8 he marke wll go beyond he sably regon of ash equlbrum and hus loses sably. he dynamc marke behavors exhb he perodc and chaoc varaon. he consran of ransmsson lne change he roue of perod-doublng bfurcaon o chaos exhbng nermency. Fgure shows he chaoc evolvng behavors f α =. 3. he correspondng chaoc aracors are shown n Fgure 3 her maxmum Lyapunov exponens and fracal dmensons beng.53 and.. q α P α q P α α Fgure. Bfurcaon dagram of sable soluons of power marke wh α =. f dfference beween margnal cos of generaon companes s greaer han bk q P q Fgure. Chaoc dynamc marke behavors f α =.3 P Copyrgh 9 ScRes

11 58 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke q 8 P( q P( Fgure 3. Chaoc aracors of generaon quanes and elecrcy prce f α = Case 3: Dfference beween Margnal Cos of Generaon Companes Les n [bk bk] he dynamc behavors of he power marke are numercally smulaed when he dfference beween he margnal bk bk.e. cos of he generaon companes les n [ ] bk < c c bk. he generaon margnal coss are < aken as c = $ MWh c 5$ MWh =. Smlar resuls can be obaned for oher seleced values. By he analyss of Secon 3. here s no ash equlbrum pon n hs case ha s no maer how large he adjusmen speeds are he marke canno converge o a sable ash equlbrum a all. Fgure 4 shows he bfurcaon dagram of he sable soluons of he generaon quanes and elecrcy prce wh α (where α =.MWh / $. From Fgure 4 s found ha he dynamc marke behavors exhb he perodc and chaoc varaon; and he chaoc and perodc wndows appear n urn. Fgure 5 shows he chaoc aracors of he generaon quanes and elecrcy prce f α =. 5 her maxmum Lyapunov exponens and fracal dmensons beng.3 and.8 respecvely. Wheher he ransmsson lnes s congesed or no f he marke parcpans wh bounded raonaly connuously adjus her producon sraeges he marke wll fnally converge o he ash equlbrum under he sasfacon of s sably condon. Sequenally a sae ha he marke parcpans smulaneously maxmze her respecve prof s acheved. In he complex dynamc power marke he equlbrum condon s shor-erm and emporary. In he equlbrum condon many unceran facors such as he adjusmen speeds and margnal cos of generaon companes he maxmum elecrcy prce of marke are changng he operang condon of marke and pushng owards chaos. he appearance of marke chaos s very sensve o he marke parameers. he change of parameers can lead o a grea dfference beween he long-erm evolvng rajecores of he dynamc marke. Once he marke eners he chaoc condon wll be unpredcable n whch he generaon companes are unable o effecvely deermne he adjusmen of oupu quanes n he long erm. q q α α P P Fgure 4. Bfurcaon dagram of sable soluons of power marke wh α f dfference beween margnal cos of generaon companes les n [bkbk] α α Copyrgh 9 ScRes

12 Complex Dynamcs Analyss for Courno Game wh Bounded Raonaly n Power Marke 59 q 7 P( q P( Fgure 5. Chaoc aracors of generaon quanes and elecrcy prce f α =.5 However when he marke les n he chaoc condon s sll possble o effecvely predc he shor-erm dynamcs and change he chaoc marke aracors o conrol he chaos. herefore n he case he generaon companes wh bounded raonaly should connuously survey her own surroundngs and adjus her operaon objecves. he marke managers should mely modfy he operaon rules n order o change he chaoc marke aracors and adap he varaon of he marke envronmen. 5. Conclusons hs paper proposes he dynamc Courno game model wh bounded raonaly consderng he power nework consrans.e. he dfference equaons embedded wh he opmzaon problem. By usng he heory of nonlnear dscree dynamc sysem he ash equlbrum and s sably for a duopoly marke are quanavely analyzed. I s found ha he power marke has dfferen ash equlbrums wh dfferen marke parameers correspondng o dfferen operang condons.e. congeson and non-congeson whle n some cases has no ash equlbrum a all. he effec of marke parameers s nvesgaed on he sably of ash equlbrum. I s also revealed ha he smooh adjusmen of he generaon quanes and he resrcon of he maxmum value of elecrcy prce can renforce he sably of he power marke. In he dynamc evoluon he marke exhbs a varey of dynamc behavors.e. convergng o he ash equlbrum perod and chaos. Based on he above work here are he followng ssues need o be explaned and dscussed. (a For descrpve smplcy he generaon margnal cos s assumed o be a lnear form n hs paper. If s a quadrac funcon he ash equlbrums of marke and her sably condons may be smlarly obaned as well as he perodc and even chaoc dynamc behavors when he marke go beyond he sably regon. (b In he dynamc Courno game he generaon quanes are regarded as he decson varables of generaon companes whch may be sold o he users hrough he conrac ransacon also o he Power Exchange hrough he Pool ransacon. So long as he relaonshp beween he demand and elecrcy prce s dencal n he wo ransacon models he smlar resuls such as he same marke equlbrum pons and dynamc behavors can be obaned. 6. Acknowledgemens hs work s suppored by he aonal aural Scence Foundaon of Chna (o.73 and he Program for ew Cenury Excellen alens n Unversy of Chna. REFERECES [] W. W. Hogan Calforna marke desgn breakhrough Harvard Unversy Cambrdge. [] D. X. Zhang and Z. H. Chen onlnear dynamc economcs-bfurcaon and chaos Ocean Unversy Press of Qngdao Qngdao 995. [3] H.. Agza A. S. Hegaz and A. A.Elsadany he dynamcs of Bowley s model wh bounded raonaly Chaos Solons and Fracals pp [4]. A. Hamdy I. B. Gan and K. Mchael Mulsably n a dynamc Courno game wh hree olgopolss Mahemacs and Compuers n Smulaon 5 pp [5] R. Herdrk and S. Arco Conrol of he rple chaoc aracor n a Courno ropoly model Chaos Solons and Fracals 9 pp [6] F. L. Alvarado he sably of power sysem markes IEEE ransacons on Power Sysems pp [7] F. L. Alvarado J. Meng and C. L. DeMarco Sably analyss of nerconneced power sysems coupled wh marke dynamcs IEEE ransacons on Power Sysems 4 pp [8] Z. H. Yang Y. F. Lu and Y. ang Analyss of power marke sably Proceedngs of he Chnese Socey for Elecrcal Engneerng pp [9] Y. D. ang J. J. Wu and Y. Zou he research on he sably of power marke Auomaon of Elecrc Power Sysems 4 pp. -6. [] A. Maorano Y. H. Song M. rovao Dynamcs of non-collusve olgopolsc elecrcy markes IEEE Copyrgh 9 ScRes

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