A Dynamic Economic Dispatch Model Incorporating Wind Power Based on Chance Constrained Programming

Transcription

1 Energes ; do:.339/en8233 Arcle OPEN ACCESS energes ISSN A Dynamc Economc Dspach Model Incorporang Wnd Power Based on Chance Consraned Programmng Wushan Cheng and Hafeng Zhang * School of Mechancal Engneerng Shangha Unversy of Engneerng Scence Shangha 262 Chna; E-Mal: cwushan@63.com * Auhor o whom correspondence should be addressed; E-Mal: hfzhang@se.xju.edu.cn; Tel.: ; Fax: Academc Edor: Erk Gawel Receved: 3 Sepember 24 / Acceped: 8 December 24 / Publshed: 29 December 24 Absrac: In order o manan he sably and secury of he power sysem he uncerany and nermency of wnd power mus be aken no accoun n economc dspach (ED) problems. In hs paper a dynamc economc dspach (DED) model based on chance consraned programmng s presened and an mproved parcle swarm opmzaon (PSO) approach s proposed o solve he problem. Wnd power s regarded as a random varable and s ncluded n he chance consran. New formulaon of up and down spnnng reserve consrans are presened under expecaon meanng. The mproved PSO algorhm combnes a feasble regon adjusmen sraegy wh a hll clmbng search operaon based on he basc PSO. Smulaons are performed under hree dsnc es sysems wh dfferen generaors. Resuls show ha boh he proposed DED model and he mproved PSO approach are effecve. Keywords: wnd power; dynamc economc dspach; spnnng reserve; chance consran programmng; parcle swarm opmzaon. Inroducon Dynamc economc dspach (DED) whch deermnes he opmal generaon scheme o mee he predced load demand over a me horzon sasfyng he consran such as ramp-rae lms of generaors beween me nervals s crucal for power sysem operaon [ 3]. Pror o he wdespread

2 Energes use of alernae sources (solar wnd) of energy he DED problem nvolved only convenonal generaors. In recen years wnd power has experenced an explosve growh and has shown grea poenal n fuel savngs and envronmenal proecon. However s uncerany and nermency also makes challengng o fnd a proper dspach scheme for a wnd peneraed power sysem. The key ssue assocaed wh he ncorporaon of wnd power s how o deal wh s flucuaon and nermence consderng he requred relably and secury of power sysems [2]. Up o now dfferen auhors have proposed models o solve economc dspach (or un commmen) problems for wnd-peneraed power sysems [4 6]. Chen [4] proposed a mehod o ncorporae wnd generaors no he generaon schedulng problem. Specal reserve consrans are esablshed o operae he power sysem whn he requred sably margn whch s also adoped by Zhou e al. [5] and Jang e al. n [6]. These models are deermnsc and canno accuraely descrbe he effec of wnd flucuaons on sysem operaon. To effecvely and safely use wnd power researchers usually forecas he wnd speed or wnd generaon over a me horzon n advance and hen oban he sasc dsrbuon of wnd speed or wnd generaon. Based on he dsrbuon funcon and esmaed loads researchers can deermne he dspach scheme. Hezer e al. presened an ED model ncorporang wnd power wh he sochasc wnd speed characerzaon based on he Webull probably densy funcon [7]. Penaly coss for overesmaon and underesmaon of avalable wnd energy are also consdered. In [8] a modfed ED opmzaon model wh wnd power peneraon s presened and rsk-based up and down spnnng reserve consrans are presened consderng he unceranes of avalable wnd power load forecas error and also generaor ouage raes. I has been dscussed n [9 ] ha forecased wnd generaon usually follows a bea dsrbuon funcon and followng hs he auhors of [2] and [3] proposed ED models ncorporang he mpac of wnd varably. Mranda and Hang developed n [4] an ED model ncludng wnd generaors usng conceps from he fuzzy se heory and hey added a penaly cos facor for no usng he avalable wnd power capacy. I s noed clearly n references [782 4] ha he consrans of ramp raes are no aken no accoun. In [5] and [6] a scenaro-based approach s ulzed o model he uncerany of wnd power and sochasc models are proposed o solve he opmal schedulng of he generaors n wnd peneraed power sysem. Ths scenaro approach requres a large number of scenaros o ensure he qualy of soluon and usually suffers huge compuaon cos. Chance consraned programmng (CCP) s a knd of sochasc opmzaon approach [7]. I s suable for solvng opmzaon problems wh random varables eher ncluded n consrans or n he objecve funcon. CCP has been suded o solve he ransmsson plannng problem n [78] and he sochasc un commmen problem wh unceran wnd power oupu n [9]. In [7] and [8] he same formulaon of chance consran s appled o ransmsson plannng and s n he form ha he no-overload probably for ransmsson lne s requred o be more han a specfed confdence level. In [9] he chance consran s appled o descrbe polces o ensure he ulzaon of wnd power and a wo sage sochasc opmzaon s presened o solve he un commmen problem. In hs paper he chance consran s used o descrbe ha he probably ha acual wnd generaon s greaer han or equal o he scheduled wnd power s more han a gven confdence level. In a manner somewha smlar o he wnd model n [2] and [3] hs paper uses a bea dsrbuon o characerze he acual wnd generaon for each ndvdual perod. Up spnnng reserve (USR) and

3 Energes down spnnng reserve (DSR) are requred o deal wh he forecasng errors n load and he sudden flucuaons n wnd generaon [5]. In general he reserve requremen wh respec o forecas errors n load s defned o be a fxed percenage [5] (e.g. 5%) of he load demand. As o he reserve requremens caused by varaon of wnd generaon up reserve requremen (URR) nduced by he sudden fall n wnd generaon and down reserve requremen (DRR) nduced by he sudden rase n wnd generaon are boh aken no accoun n our work. The condonal expecaon when acual wnd generaon s smaller han he scheduled wnd s calculaed and URR s defned as he dfference beween he scheduled wnd power and he condonal expecaon of wnd power. Smlarly DRR s defned as he dfference beween he condonal expecaon when acual wnd generaon s greaer han or equal o he scheduled wnd power and he scheduled wnd power. Ths paper proposes a sochasc DED model based on chance consrans n wnd power peneraed sysems. New formulaons of he spnnng reserve consrans are consdered under he expeced meanng. An mproved parcle swarm opmzaon (PSO) approach s proposed o opmze he model. In order o demonsrae he effcacy of he proposed model and he PSO approach varous comparsons are performed under wo dfferen es sysems. Resuls show ha he above-menoned model s effecve and he proposed PSO approach s able o solve such model. The res of hs paper s organzed as follows: n Secon 2 he sochasc DED model s formulaed. Secon 3 nroduces he bea dsrbuon as he bass of he equvalen ransformaon of he proposed DED model and a deermnsc DED model s obaned. In Secon 4 an mproved PSO approach ncludng feasble regon adjusmen (FRA) sraegy and hll clmbng search operaon (HCSO) s appled o solve he deermnsc model. Secon 5 presens a dscusson of he numercal resuls. Fnally conclusons are drawn n Secon DED Problem Formulaon In hs secon he objecve funcon of DED problem s descrbed and he DED model based on CCP s formulaed. 2.. Objecve Funcon In wnd peneraed power sysems wnd producon s usually regarded as zero cos he DED model wh he valve pon effec usually akes he followng form [2 24]: T I mn f = [ C ( p ) + E ( p )] () cos = = where: f s he oal generaon cos over he whole me horzon; cos T s he number of perods; I s he number of hermal uns; p s he power oupu (MW) of he h un correspondng o me perod ; C( p ) s he generaon cos of he h un correspondng o me perod ; E( p ) s he valve pon loadng effec of he h un correspondng o me perod ;

4 Energes For he hermal uns he generaon cos can be approxmaed by a quadrac funcon of he power oupu whch s praccal for mos of he cases and s gven by: 2 C( p ) = ap + bp + c (2) where a b and c are cos coeffcens for he h un. E( p ) s expressed as follows: E( p ) = esn[ f( pmn p )] (3) where e and f are coeffcens relaed o valve pon effec of he h un. generaon lm of un Sysem and Un Consrans p mn s he mnmum Ths DED problem s subjeced o a varey of sysem and un consrans whch nclude power balance consrans generaon lms of uns ramp rae lms and spnnng reserve consrans. These consrans are dscussed below Power Balance Consrans Toal power generaon mus equal he load demand pd n all me perod: where pw s he scheduled wnd power of wnd farm a me Generaon Lms of Thermal Uns and Wnd Farm I p + pw = pd (4) = The oupu of each hermal un and wnd farm mus le n beween a lower and an upper bound. These consrans are represened as follows: pmn p p (5) pw w (6) where p s he mum generaon lm of hermal un and w s he nsalled capacy of wnd farm Chance Consran on Wnd Power Due o he sochasc naure of wnd power he schedule for wnd may no be realzed on a scheduled day. So we nroduce he followng chance consran: P { w pw } ρ (7) where w s a random varable represenng he wnd power generaon a me. ρ s he confdence level. Equaon (7) defnes he probably ha he scheduled wnd power can be realzed s greaer han or equal o ρ. Furhermore Equaon (7) ses a reasonable upper bound for wnd power generaon and he probably ha hs upper bound can be realzed s no less han ρ. The larger he confdence level s

5 Energes he less sochasc he wnd power s and hence he more relable he power sysem s. Especally when ρ = here s no wnd power n sysem and he DED problem s changed no a deermnsc one Ramp Rae Lms of Thermal Uns The ramp rae lms resrc he operang range of all he uns for adjusng he generaon beween wo perods. The generaon may adjus he dspach level as: Δ d T6 p p Δ u T6 (8) where p s he oupu of un a me and Δ u and Δ d are he upper and lower ramp rae lms respecvely. T6 s he operang perod.e. h Spnnng Reserve Consrans Due o he peneraon of wnd power and s nermen naure addonal reserve needs o be provded o suppor he wnd generaon varaon [5]. The USR suppors he forecas errors n load and a sudden decrease n wnd power and he DSR conrbues o he sudden rase n wnd power. Boh he USR and DSR are suppled by he rampng capacy of hermal uns and are formulaed by Equaons (9) and () respecvely: n u mn( p p Δ u T ) pr + rw (9) n mn d mn( p p Δ d T ) rw () where p r s he reserve level o suppor forecas error n demand. T s mn. mn p and p are upper and lower generaon lms of un ncludng ramp rae lms a me and mn u d p = mn( p p +Δ u ) p = ( pmn p Δ d ). r w and r w are he URFW and he DRRW o follow he sudden decrease and ncrease n wnd power a me. u d u Fgure shows he concepual llusraon of r w and r w. r w s calculaed by he dfference beween he scheduled wnd power and condonal expecaon of wnd power when he acual wnd generaon s less han scheduled and s shown n Equaon (): The condonal expecaon s obaned by he followng equaon: u rw = pw E( w w < pw ) () Ew ( w p) < = pw w pw wf ( w)dw W f ( w )dw W where f ( w ) s he probably densy funcon of wnd power a me. W d Smlarly r w s calculaed usng he wo equaons as menoned n Equaons (3) and (4): (2) d rw = E( w w pw ) pw (3)

6 Energes w pw w w Ew ( w p) = pw wf ( w)dw W f ( w)dw W (4) d r w x = Ew ( w p ) w u r w x x = Ew ( w< p ) 2 w p w x 2 u d Fgure. Concep llusraon of r w and r w. Takng a closer look a he opmzaon model () (4) conans chance consran and hus s a sochasc DED model. In he nex secon we wll dscuss he equvalen ransformaon of he sochasc model o a deermnsc one. 3. Equvalen Transformaon of he DED Model 3.. Bea Dsrbuon of Wnd Power In he real word he acual wnd generaon s a funcon of wnd speed ha randomly changes all he me [3]. In our sudy normalzed wnd power n each perod s regarded as a random varable whch follows bea dsrbuon [9 3]. We denoe he probably densy funcon (PDF) by f ( x ) as follows: f X α β x ( x) < x < ( x) = B( αβ ) (5) oherwse X where x = w / w and x []. α> β > and B( α β ) s he well-known bea funcon defned as: ( α β ) = α ( ) β d Parameers α and β can be deermned by followng equaons: where μ and B u u u (6) μ α = w α +β σ 2 αβ ( ) = w α+β α +β+ 2 ( ) ( ) σ are he mean value and sandard devaon of he predced wnd generaon a me on he schedulng day respecvely. The cumulave densy funcon (CDF) of X can be easly obaned as shown below: (7)

7 Energes < x x α β FX ( x) = u ( u) d u < x< B( αβ ) (8) x 3.2. Equvalen Transformaon Pung he aforemenoned dscusson of he bea dsrbuon we can ge he followng CDF ( FW( w ) ) of w : { } { } F w W w w X w F w w W( ) P P ( / ) = = = X (9) Then Equaons (2) and (4) are ransformed no he followng equaons as expressed n Equaons (2) and (2) respecvely: pw / w w w pw / w Ew ( w p ) Ew ( x x p / w ) w < = < = w pw / w w w = Ew ( w p ) w pw / w xf ( x )dx X f ( x )dx X x f ( x )dx X f ( x )dx X (2) (2) Boh Equaons (2) and (2) wll be solved usng numercal negraon mehods because of he complexy. In addon we rewre he Equaon (7) as follows: { } P w p = F ( z) ρ (22) w X where pw / w = z form FX () z = ρ we learn ha z s he upper quanle of FX ( x ) hen we ge p w F ( ρ ) and hus oban he followng consran of scheduled wnd power: w X pw w F X ( ) ρ (23) Gven all ha he equvalen ransformaon of he sochasc DED model s mahemacally represened as: T I 2 mn{ ap + bp + c+ esn[ f( pmn p )]} = = I p + pw = pd = pmn p p pw w FX ( ρ) Δ d T6 p p Δ u T6 (24) s.. pw / w I xf ( )d X x x mn( p p Δ u T ) pr + pw w pw / w f ( )d X x x I xf ( )d mn p / X x x w w mn( p p Δ d T) w p w f ( )d p / X x x w w

8 Energes Improved PSO Approach The problem n Equaon (24) s a hgh-dmenson and non-convex opmzaon problem so s very dffcul o fnd analycal soluons. In recen decades many salen approaches have been developed o solve such problems such as genec algorhm [22] dfferenal evoluon [22] evoluonary programmng [2526] and PSO [627 32]. PSO frs nroduced by Kennedy and Eberhar s a populaon-based opmzaon echnque and conducs s search usng a populaon of parcles [2728]. Each parcle s a canddae soluon o he problem and s moved oward he opmal pon by addng a velocy wh s poson. The poson and he velocy of he jh parcle n he D T T dmensonal search space can be expressed as Yj = [ yj yj2 yjd] and Vj = [ vj vj2 vjd] k respecvely. Each parcle has s own bes poson ( pbes j = 2 J ) correspondng o he k personal bes objecve value obaned a generaon k. The global bes parcle s denoed by gbes whch represens he bes parcle found so far a generaon k among he whole populaon. The new velocy and poson of each parcle a generaon k + are calculaed as shown below [629]: k k k k k k k k V + j =ω( k) Vj +ϕ( k) rand ( pbes j Yj ) +ϕ2( k) rand2 ( gbes Yj ) (25) j k + k k + Y j = Yj + Vj j J (26) where: J s he populaon sze; ω( k ) s he dynamc nera wegh facor and can be dynamcally se wh he followng equaon [6]: ω ( k) =ω ( ω ωmn ) k / K (27) where ω and ω mn are nal and fnal nera wegh facors and se o.9 and.4 respecvely. K s he mum eraon number. ϕ ( k) and ϕ ( k) 2 are me-varyng acceleraon coeffcens correspondng o cognve and socal behavor [6] and are se wh he followng equaons shown n Equaons (28) and (29): 2 f ϕ ( k) = ϕ + ( ϕf ϕ ) k / K (28) ϕ 2( k) = ϕ 2 + ( ϕ2 f ϕ2) k / K (29) where ϕ ϕ 2 are he nal values of ϕ ( k) and ϕ ( k) and are se o 2.5 and.5 respecvely; 2 ϕ f ϕ are he fnal values of ϕ ( k) and ϕ ( k) 2 and are se o.5 and 2.5 respecvely. I s no always effecve o solve he problem wh equaly and nequaly consrans usng basc PSO and he parcles (soluons) whch sasfy he nequaly consrans usually volae he equaly consrans. In hs secon we propose a FRA sraegy over hese parcles whch volae equaly consrans. In addon n order o mprove he qualy of he bes soluon HCSO s appled o updae he global bes parcle along wh he eraon. In subsecons 4. and 4.2 we wll separaely dscuss he FRA sraegy and he HCSO n deal. 4.. Feasble Regon Adjusmen Sraegy Le us rewre he poson of he jh parcle as he followng marx:

9 Energes Y j y y2 y y T p p2 p p T y2 y22 y2 y 2 T p2 p22 p2 p 2 T = ym ym2 ym ym T = p p2 p p T yi yi2 yi yi T pi pi2 pi pi T ym ym2 ym y M T pw pw2 pw p w T (3) where M s he number of generaors ncludng hermal uns and one wnd farm.e. M = I + and M T = D. The elemen of marx ym s he oupu of he mh generaor a me. The row vecors represen he oupu of he ndvdual generaor each hour one day. The sum of he oupu of ndvdual generaor (.e. power supply) each hour mus be equal o he I demand.e. p + pw = pd = 2 T. If he power suppled n each hour s greaer or smaller = han he demand he FRA sraegy wll be appled o he jh parcle. The FRA sraegy s descrbed as follows: I () f p + p < p = w d We use he followng equaon o adjus he oupu of he hermal uns and he wnd farm as shown n Equaon (3): p p p p ( p p p ) * = + d w p + w FX ( ρ) p pw (3) In Equaon (3) p p s he range ha he oupu of hermal un can be ncreased a me. p p p w F p p + X ( ρ) w he oal adjusable range of all generaors. he adjusable value of un. I can be easly seen ha he value of less han so he value of * refers o he fracon of he adjusable range of un compared wh p afer adjused sll sasfes p p ( p + pw pd ) s p + w F ρ p p X ( ) w p p p p d w p + w FX ( ρ) p pw * p. Smlar o Equaon (3) we use he Equaon (32) o adjus he oupu of he wnd farm as shown below: w * FX ( ρ) p w pw = pw + ( p d p pw ) (32) p + w F ( ρ) p p X w s and he oupu of wnd farm p afer adjused s sll smaller han * w w F ( ) ρ. X

10 Energes I (2) f p + p > p = w d Smlar analyss can be appled o he condon p + p > p I = w d. We use he followng Equaons (33) and (34) o adjus he oupu of he hermal uns and he wnd farm respecvely: and we can oban ha p p p p ( p p p ) mn * = mn + w d p + pw p (33) p p = p ( p + p p ) + p * w w w mn w d p pw p mn * * p and pw. (34) From above dscusson we can learn ha he FRA sraegy ensures ha he parcles afer adjused sll sasfy he generaon lm consrans Hll Clmbng Search Operaon Hll clmbng s an opmzaon echnque whch belongs o he local search famly. I s an erave algorhm ha sars wh an arbrary soluon and hen aemps o fnd a beer soluon by ncremenally changng a sngle elemen of he soluon. In our work n order o oban a beer soluon we adop HCSO o updae he global bes parcle along wh he eraon. When he fness of he global bes parcle does no change connuously for a fxed number of mes HCSO can be used as explaned n Equaons (35) and (36): ' pm = pm + mn( Δm Δn) ε (35) ' pn = pn mn( Δm Δn) ε (36) where p m and p n are generaons of hermal un m and n whch belong o he same perod and are ' ' randomly seleced from all he hermal uns. p and p are generaons of hermal uns afer hll m clmbng operaon. Δ m and Δ n are he upper ramp raes of un m and n. In hs paper we adop he upper ramp raes n smulaon. However he lower ramp raes can be used eher. mn( Δm Δn) ε s he lnear decrease sep sze of hll clmbng operaon and ε s lsed n Table 2 n Secon 5. Boh Equaons (35) and (36) have he same sep sze and he oppose drecon whch ensures ha he global bes parcle wll no volae equaly consrans afer hll clmbng operaon DED Consrans Handlng Usng PSO-HCSO The fness funcon of parcle s he generaon cos bounded wh he penaly funcons as shown n Equaon (37): 6 n cos g g= = T fness = f + λ PF (37)

11 Energes and: I PF = ( p + p p ) (38) w d = PF2 = ( pmn p ) + ( p p) (39) PF = p + p w F ρ (4) 3 ( w ) ( w X ( )) PF4 = ( Δ d T6 p + p ) + ( p p Δ u T6) (4) pw / w ( ) I xf X x dx 5 = r + w pw / w Δ u f ( ) X x dx PF ( p p w mn( p p T )) xf ( x) dx PF ( w p mn( p p T )) I p / w w X mn 6 = w Δ d ( ) f p / X x dx w w (42) (43) where λ s he penaly facor correspondng o he consrans. Here we use unform penaly facor for PF g.e. λ= e + 8; PF g s he penaly funcon. I s noed ha he soluon should no conan any penaly for he consran volaon The Procedure of Improved PSO Approach The procedures for mplemenng he PSO approach are gven as he followng seps: Sep : Inalze he parameers such as populaon sze J he mum eraon number K and he mum number of hll clmbng search operaon H. Se he sequence number of eraon K = and he number of mes ha he global bes parcle does no change connuously hcoun =. Sep 2: Creae a swarm of parcles as he nal populaon ncludng random poson and velocy. For any parcle whch volaes he equaly consrans FAR sraegy s ulzed. Evaluae he fness of parcles and oban he nal gbes and pbes j = 2 J. j Sep 3: Calculae ω ( k) ϕ ( k) and ϕ ( k) 2 and hen updae he poson and velocy of each parcle among he populaon accordng o Equaons (25) and (26). FAR sraegy s appled o any parcle whch volaes he equaly consrans. k k Sep 4: Evaluae he fness of parcles and updae pbes j j = 2 J and gbes of he k populaon. Check gbes. If hcoun reaches he fxed number of mes hll clmbng search operaon s appled unl he mum number H s reached; oherwse se hcoun =. k Sep 5: k = k + f k > K sop he algorhm and oupu he global bes soluon ( gbes ) wh he bes fness value; oherwse go back o sep Smulaon Resuls and Dscussons In order o verfy he effecveness of he proposed DED model wh wnd power hree dsnc es sysems (.e. sysem sysem 2 and sysem 3) are employed n hs paper. Sysem conans sx hermal uns and one wnd farm (WF) whch s also used n sysem 2 and 3. Sysem 2 conans 5

12 Energes hermal uns. The characerscs of he 6 and 5 hermal uns can be obaned from [27]. Sysem 3 has 26 hermal uns derved for a IEEE 24-bus sysem. The wnd farm s locaed n he Inner Mongola Auonomous Regon n Chna and conans 32 wnd urbnes. The oal nsalled capacy s 98 MW. Table gves he hourly expeced value and sandard devaon of wnd power forecas on he scheduled day and also he correspondng bea parameers whch are calculaed by Equaon (7). The parameers of mproved PSO are shown n Table 2. In he able h s he sequence number of Hll clmbng search operaon. The mproved PSO approach has been mplemened on a personal compuer wh four processors runnng a 3.2 GHz and equpped wh 4 GB of RAM memory usng Malab Table. Hourly expeced value and sandard devaon of wnd power forecas and bea parameers. Perod Expeced Value (MW) Sandard Devaon (MW) α β Perod Expeced Value (MW) Sandard Devaon (MW) α β Table 2. Parameers for mproved PSO algorhm. J K H hcoun ε h/(h+.) 5.. Comparsons Among he Three Cases of he DED Model In each es sysem we perform 5 rals usng he mproved PSO approach consderng hree cases as follows: Case (): he DED model whou consderng wnd power; Case (2): he DED model whou consderng wnd effec n he reserve consran; Case (3): he proposed DED model n hs paper. Boh casea () and (2) can be derved from case (3). The frs case can be obaned by seng he confdence level (ρ) of case (3) o and he laer s obaned by removng he URFW and DRRW n he reserve consran. Comparson resuls among hree cases menoned above are shown n Table 3.

13 Energes Table 3. Comparsons among he hree cases n he hree sysems. Cases Confdence Level Average Generaon Cos ($) Sysem Sysem 2 Sysem 3 Case () Case (2) Case (3) I s evden ha he average generaon cos n case () s he hghes among he hree sysems; hs s because ha no wnd power s ulzed n case (). Wnd power n cases (2) and (3) affords a ceran proporon of he load and hus reduces he fuel consumpon. Average generaon cos decreases gradually as he confdence level decreases for cases (2) and (3) among he hree sysems. The smaller he confdence s he lower he average generaon cos s. Takng sysem for example he average cos n cases (2) and (3) decreases by 5.2% and 4.93% respecvely compared o case () when ρ =.9. When ρ decreases o. he average generaon cos decreases by 9.% and 7.78% respecvely compared o case (). Fgures 2 4 show he average cos change rend for dfferen confdence level scenaros for cases (2) and (3) among he hree sysems. As seen n he fgures he average cos n he hree sysems ncreases gradually whle he confdence level ncreases. There s no much dfference n he average cos beween case (3) and case (2) however case (3) consders he URFW and DRRW n he reserve consrans and hs makes he sysems more relable. Average generaon cos/$ 2.66 x case 3) case 2) Confdence level Fgure 2. Average generaon cos under dfferen confdence level n sysem.

14 Energes x case 3) case 2) Average generaon cos/$ Confdence level Fgure 3. Average generaon cos under dfferen confdence level n sysem x 5 case 3) case 2) Average generaon cos/$ Confdence level Fgure 4. Average generaon cos under dfferen confdence level n sysem 3. We choose he soluons wh he mnmum generaon cos from 5 rals as he opmal soluons by varyng ρ from. o.9. The opmal soluons (power oupu durng each perod for each un and wnd farm) for sysem and 2 under ρ =.9 are shown n Fgures 5 and 6. Power oupus for perod n sysem 3 are lsed n Table 4.

15 Energes Fgure 5. Opmal generaon oupu of uns and wnd farm n sysem wh ρ =.9. Fgure 6. Generaon oupu of uns and wnd farm n sysem 2 wh ρ =.9.

16 Energes Un Index Table 4. Opmal oupu of uns and wnd farm (WF) n sysem 3 wh ρ =.9. Oupu/MW ( = ) Oupu/MW ( = 5) Oupu/MW ( = 9) Oupu/MW ( = 3) Oupu/MW ( = 7) Oupu/MW ( = 2) WF Wnd Peneraon as a Funcon of he Confdence Level Under Two Sysems ( Comparsons beween wnd peneraon ( pw % / pd ) and wnd peneraon lms w F X ( ρ ) % / p d ) under he opmal soluons n each sysem are shown n Fgures 7 9. As shown wnd peneraon n each sysem under ρ =.9 s he lowes because under ha suaon each sysem requres he hghes relably and secury. As he confdence level decreases wnd peneraon almos ncreases and does no exceed he wnd peneraon lms n he hree sysems.

17 Energes wnd peneraon lm wnd peneraon Wnd peneraon (%) Confdence level Fgure 7. Wnd peneraon and s lm under dfferen confdence levels n Sysem. Wnd peneraon (%) wnd peneraon lm wnd peneraon Confdence level Fgure 8. Wnd peneraon and s lm under dfferen confdence levels n Sysem 2. Wnd peneraon (%) wnd peneraon lm wnd peneraon Confdence level Fgure 9. Wnd peneraon and s lm under dfferen confdence levels n sysem 3.

18 Energes Dscusson abou he Opmal Reserve Allocaon Due o he sochasc and nermen naure of wnd power he USR and DSR are used o ensure he relably and secury of sysems wh wnd farms. Accordng o Inequaons (9) and () he USR DSR URFW and DRRW for each perod are calculaed based on he opmal soluons when ρ =.9 serves as an example. Fgures and show he opmal USR and DSR allocaon for 24 h n sysem respecvely. Fgures 2 and 3 gve he opmal USR and DSR allocaon for 24 h n sysem 2. I can be learned ha he USR and DSR provded by hermal uns n boh sysems can effecvely cover he sudden fall and ncrease n wnd power. The same concluson can also be drawn for sysem 3 so he correspondng fgures for USR and DSR are no dsplayed. USR USR plus reserve for load forecas error Reserve capacy/mw Perod/H Fgure. The opmal USR allocaon and URR plus reserve for load forecas error n sysem wh ρ =.9. 2 DSR DRR Reserve capacy/mw Perod/H Fgure. The opmal DSR allocaon and DRR for 24 h n sysem wh ρ =.9.

19 Energes USR URR plus reserve for load forecas error Reserve capacy/mw Perod/H Fgure 2. The opmal USR allocaon and URR plus reserve for load forecas error for 24 h n sysem 2 wh ρ = DSR DRR Reserve capacy/mw Perod/H Fgure 3. The opmal DSR allocaon and DRR for 24 h n sysem 2 wh ρ = Comparsons beween he PSO-HCSO and PSO whou HCSO In order o nvesgae he effec of he parameer H on he mproved PSO algorhm we se H as and perform he algorhm on sysem wh ρ =.9 for 5 rals. The average generaon cos and average me are lsed n Table 5. The average generaon coss decrease gradually and average me ncreases as H ncreases. Table 5. Effec of H on he mproved PSO algorhm wh ρ =.9. H Average Generaon Cos ($) Average Tme (s)

20 Energes Fnally n order o show he effecveness of HCSO PSO whou HCSO s used for comparson. Parameer K for PSO whou HCSO s se as 3. Resuls of he proposed DED model under hree confdence levels are shown n Table 6. Alhough PSO whou HCSO consume less compuaon me han PSO-HCSO leads o much hgher generaon coss. From hs pon of vew he mproved PSO algorhm s more effcen. Table 6. Comparsons beween PSO-HCSO and PSO whou HCSO among hree sysems. Approach Sysem Sysem 2 Sysem 3 Confdence Level Average Generaon Cos ($) Average Tme (s) PSO-HCSO PSO whou HCSO PSO-HCSO PSO whou HCSO Conclusons Wnd power provdes energy savngs and envronmenal proecon benefs. However he nermency and uncerany of wnd power generaon requre ha convenonal uns provde addonal reserve o ensure he sably and relably of a wnd power-peneraed power sysem. Ths paper formulaes a DED model ncorporang wnd power based on CCP. The unceran naure of wnd generaon s represened by a bea dsrbuon funcon. In order o ensure he relably of he power sysem a chance consran s ncluded and condonal expecaon s presened o calculae he up and down spnnng reserves. The proposed DED model s hen numercally solved usng an mproved PSO approach n hree dfferen es sysems. Resuls show ha he proposed model can effecvely respond o sudden wnd power falls or rases. The mproved PSO approach s f o solve he DED model. The resuls also show ha he average generaon cos and wnd peneraon are dependen on he confdence level. If he confdence level s ncreased he wnd peneraon wll be reduced whch resuls n hgher relably of he power sysem and he average generaon cos wll be ncreased. Conversely f he confdence level s decreased less relably allows more wnd power o be ncorporaed n he power sysem. The average generaon cos wll be reduced. Acknowledgmens The auhors graefully acknowledge he suppor of he Naonal Naural Scence Foundaon of Chna (63549).

21 Energes Auhor Conrbuons All he auhors conrbued o hs paper. Hafeng Zhang formulaed he DED model performed expermens and eded he manuscrp. Wushan Cheng revewed and revsed he manuscrp. Ls of Abbrevaons and Symbols Abbrevaons ED DED USR DSR URR DRR PSO HCSO PSO-HCSO PDF CDF FRA Economc dspach Dynamc economc dspach Up spnnng reserve Down spnnng reserve Up reserve requremen Down reserve requremen Parcle swarm opmzaon Hll clmbng search operaon Parcle swarm opmzaon wh hll clmbng search operaon Probably densy funcon Cumulave densy funcon Feasble regon adjusmen Symbols T Number of Perods I Number of hermal uns Index of me perod = 2 T Index of hermal un = 2 I f Toal generaon cos cos p Power oupu of hermal un a me p w Scheduled wnd power of wnd farm a me p d Load demand a me C( p ) Generaon cos of hermal un a me a b c Cos coeffcens of hermal un E ( p ) Valve pon loadng effec of hermal un a me f e Coeffcens relaed o valve pon effec of hermal un p mn p Mnmum and Maxmum generaon lms of hermal un w w ρ Acual wnd generaon a random varable Insalled capacy of wnd farm Confdence level

22 Energes Δ u d Δ Upper and lower ramp rae lms of hermal un T T6 mn and h respecvely u r URFW a me w r d w Conflcs of Ineres The auhors declare no conflc of neres. References DRRW a me α β Parameers of he bea funcon μ σ Mean value and he sandard devaon f W F PDF and CDF of acual wnd generaon a me W f X X F PDF and CDF of normalzed acual wnd generaon a me D Dmenson of he parcle k K Curren number and mum number of eraon V Y Velocy and poson of he jh parcle a generaon k k j k j ω ( k ) Dynamc nera wegh facor ϕ k () 2 Acceleraon coeffcens correspondng o cognve and socal behavor () k pbes j k gbes λ Penaly facor PF Penaly funcons g H hcoun Personal bes poson of jh parcle a generaon k Global bes poson a generaon k n he whole populaon Maxmum number of he HCSO The number of mes ha gbes does no change connuously. Ren B.Q.; Jang C.W. A revew on he economc dspach and rsk managemen consderng wnd power n power marke. Renew. Susan. Energy Rev Xa X.; Elaw A. Opmal dynamc economc dspach of generaon: A revew. Elecr. Power Sys. Res Guo C.X.; Zhan J.P.; Wu Q.H. Dynamc economc emsson dspach based on group search opmzer wh mulple producers. Elecr. Power Sys. Res Chen C.L. Opmal wnd-hermal generang un commmen. IEEE Trans. Energy Convers Zhou W.; Peng Y.; Sun H.; We Q.H. Dynamc economc dspach n wnd power negraed sysem. Proc. CSEE Jang W.; Yan Z.; Hu Z. A novel mproved parcle swarm opmzaon approach for dynamc economc dspach ncorporang wnd power. Elecr. Power Compon. Sys Hezer J.; Yu C.D.; Bhaara K. An economc dspach model ncorporang wnd power. IEEE Trans. Energy Convers

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24 Energes Gang Z.L. Parcle swarm opmzaon o solvng he economc dspach consderng he generaor consrans. IEEE Trans. Power Sys Lee T.Y. Opmal spnnng reserve for a wnd-hermal power sysem usng PSO. IEEE Trans. Power Sys Park J.B.; Lee K.S.; Shn J.R.; Lee K.Y. A parcle swarm opmzaon for economc dspach wh non-smooh cos funcons. IEEE Trans. Power Sys L X.H.; Jang C.W. Shor-erm operaon model and rsk managemen for wnd power peneraed sysem n elecrcy marke. IEEE Trans. Power Sys Pangrah B.K.; Pand V.R.; Das S. Adapve parcle swarm opmzaon approach for sac and dynamc economc load dspach. Energy Convers. Manag Park J.B.; Jeong Y.W.; Shn J.R.; Lee K.Y. An mproved parcle swarm opmzaon for nonconvex economc dspach problems. IEEE Trans. Power Sys by he auhors; lcensee MDPI Basel Swzerland. Ths arcle s an open access arcle dsrbued under he erms and condons of he Creave Commons Arbuon lcense (hp://creavecommons.org/lcenses/by/4./).