1 Stochastc Real-Tme Schedulng of Wnd-thermal Generaton Unts n an Electrc Utlty Alreza Soroud, Member, IEEE, Abbas Rabee, Member, IEEE, and Andrew Keane, Senor Member, IEEE Abstract The objectve of dynamc economc dspatch (DED) problem s to fnd the optmal dspatch of generaton unts n a gven operaton horzon to supply a pre-specfed demand, whle satsfyng a set of constrants. In ths paper, an effcent method based on Optmalty Condton Decomposton (OCD) technque s proposed to solve the DED problem n real-tme envronment whle consderng wnd power generaton and pool market. The uncertantes of wnd power generaton as well as the electrcty prces are also taken nto account. The above uncertantes are handled usng scenaro based approach. To llustrate the effectveness of the proposed approach, t s appled on 40, 54 thermal generaton unts, and a large-scale practcal system wth 391 thermal generaton unts. The obtaned results substantate the applcablty of the proposed method for solvng the real-tme DED problem wth uncertan wnd power generaton. Index Terms Dynamc economc dspatch, Optmalty condton decomposton, Real-tme, Scenaro based uncertanty modelng, Wnd power generaton. Pw avl s,t I,t v c n/v c out Pw f t λ f t D f t P max/mn Pp max/mn t π s P,t Pp s,t wp s,t w s λ s Pw s,t NOMENCLATURE Avalable wnd power generaton capacty at scenaro s and tme t (MW) Auxlary bnary parameter denotes on/off status of generaton unt n tme t Cut-n/out speed of wnd turbne (m/sec) Forecasted value of wnd power generaton (as a percent of Pr w ) Forecasted value of electrcty prce Forecasted value of demand Maxmum/mnmum lmt of power generaton of th thermal unt Maxmum/mnmum Power purchased from pool market n tme t (MW) Probablty of scenaro s Power produced by thermal unt n tme t (MW) Power purchased from pool market at scenaro s and tme t (MW) Percent of wnd farm capacty avalable at scenaro s and tme t Percent of wnd farm capacty avalable at scenaro s Percent of peak electrcty prce at scenaro s Power produced by wnd farm at scenaro s and tme t (MW) A. Rabee s wth the Department of Electrcal Engneerng, Faculty of Engneerng, Unversty of Zanjan, Zanjan 45371-38111, Iran, (e-mal: rabee@znu.ac.r) Alreza Soroud and Andrew Keane are wth the School of Electrcal, Electronc and Communcatons Engneerng, Unversty College Dubln, (emal: alreza.soroud@ucd.e, andrew.keane@ucd.e) The work of A. Soroud was conducted n the Electrcty Research Centre, Unversty College Dubln, Ireland, whch s supported by the Commsson for Energy Regulaton, Bord Gás Energy, Bord na Móna Energy, Cylon Controls, ErGrd, Electrc Ireland, EPRI, ESB Internatonal, ESB Networks, Gaelectrc, Intel, SSE Renewables, UTRC and Vrdan Power & Energy. A. Soroud s funded through Scence Foundaton Ireland (SFI) SEES Cluster under grant number SFI/09/SRC/E1780. A. Keane s supported by the SFI Charles Parsons Energy Research Awards SFI/06/CP/E005. Pt D Ppeak D D f t Power demand n tme t (MW) Peak power demand (MW) Percent of forecasted value of demand to the peak demand Λ Peak electrcty prce of pool market UR /DR Ramp-up/down lmt of power generaton of th thermal unt (MW/h) Pr w Rated power of wnd farm (MW) v r Rated speed of wnd turbne (m/sec) s Scenaro s t Tme nterval t NS Total number of scenaros T Total number of tme ntervals Thermal generaton unt TC Total costs ($) OF Total beneft ($) λ P s,t The prce of purchased energy from pool market n tme t and scenaro s($/mwh) λ D s,t The prce of energy sold to consumers n tme t and scenaro s ($/MWh) I. INTRODUCTION ECONOMIC load dspatch (ELD) s a non-lnear constraned optmzaton problem whch plays an mportant role n the economc operaton of power systems [1] [4]. Dynamc economc dspatch (DED), whch s an extenson of ELD for a gven operaton horzon, takes nto account the connecton of dfferent operatng tmes by consderng ramp-rate constrants of thermal generaton unts. In recent decade, several economc and envronmental reasons motvate ncreasng the share of renewable technologes n electrcty generaton [5]. However the nherent uncertantes assocated wth the operaton these energy resources and the dynamc constrants lke ramp-rate lmts make the DED problem more sophstcated. On the other hand, the recent trends toward the smart grds and also the mportance of ntegraton of renewable energy sources (regardng envronmental concerns) [6], have ncreased the need for realtme dspatchng methodologes. A real-tme dspatch method makes dspatchng decsons quckly, and s not responsble for extracton of commtment decsons and wll not consder start-up costs n any of ts dspatchng or prcng decsons n the studed horzon [7]. Thus, a powerful tool s needed for handlng the uncertantes of renewable energy resources [8] [10] along wth the techncal and economcal constrants of thermal unts. The motvaton of ths study s to provde such a tool. In other words, n ths paper the real-tme DED problem s formulated by consderng uncertan wnd power generaton (as an mportant and most popular renewable energy resource), and t s solved by utlzng OCD approach n real-tme envronment. Besdes, uncertantes n pool market prces are consdered n the proposed DED model, n order to make t more realstc and practcal.
2 A. Lterature revew The OCD technque s a powerful theoretcal and algorthmc approach for addressng contnuous NLP optmzaton problems, as well as the problems whch requre explotaton of ther nherent mathematcal structure va decomposton prncples [11]. OCD s based on relaxng the complcatng constrants [12]. Complcatng constrants are those that f relaxed, the resultng problem decomposes nto several smpler problems. The successful applcatons of the OCD technque have been reported n varous research felds such as: Mult-area optmal power flow [11] OPF for overlappng areas n power systems [13] Coordnated voltage control of large mult-area power systems [14] Optmal ntegraton of ntermttent energy sources [15] Predctve control for coordnaton n mult-carrer Energy Systems [16] Decomposed stochastc model predctve control for optmal dspatch of storage and generaton [17] Integrated water and power modelng framework for renewable energy ntegraton [18] In recent years, many approaches have been proposed for consderng the mpact of wnd power generaton on ELD problem. In [19], a method s proposed whch estmates the avalable wnd power and then solves the ELD problem. In [20], a tme seres of observed and predcted 15-mn average wnd speeds at foreseen onshore and offshore wnd farms locatons s proposed. A heurstc method (.e bacteral foragng algorthm) s proposed n [21] for solvng the ELD problem. In [22], [23], the use of battery storage s consdered for makng the wnd turbne dspatchable. In [24], a new method s ntroduced for generatng correlated wnd power values and explans how to apply the method when evaluatng economc dspatch. II. OPTIMALITY CONDITION DECOMPOSITION Consder an optmzaton problem wth the specfed decomposable structure, whch conssts of N blocks of varables as follows. max X f(x 1,,X n) = N f n(x n) (1) n=1 h n(x 1,,X n) 0, n = 1,,N (2) g n(x n) 0, n = 1,,N (3) Where X n = [ ] x n1,,x nφn are the varables for each block n n whch the orgnal problem (1 )-(3) decomposes. φ n s the cardnalty of n-th block of varables. Constrants (2) and (3) represent both equalty and nequalty constrants of the problem. In the above optmzaton problem, (2) are the complcatng constrants,.e. by relaxaton of these constrants, the overall optmzaton problem wll be decomposed to several (here N ndependent sub-problems). By nvestgaton of the frst-order KKT optmalty condtons for the aforementoned problem, whch s descrbed n detal n [25], the orgnal problem could be decomposed to N ndependent sub-problems as follows. N max f n(x n)+ λ T kh k ( X n) (4) k=1,k n h n( X n) 0 (5) g n(x n) 0 (6) Where X n = [ X1,, X n 1,X n, X n+1,, X N ]. It s worth to menton that the varables wth above them, are known for n-th sub-problem. Also, λ k s the obtaned value for Lagrange multplers of k-th complcatng constrants, whch s obtaned n k-th sub-problem. Fg.1 llustrates basc functonalty of the OCD approach. f ( X,..., X 1 N ) f1( X 1)... fn ( X N ) B. Contrbuton In ths paper, a powerful stochastc real-tme DED model s proposed for an electrc utlty to determne ts optmal strategy n supplyng the demand of ts customers. The thermal unts, wnd power generaton and pool market are taken nto account as the energy procurement resources. The uncertantes n wnd power generaton as well as pool market prces are consdered and modeled by scenaro based approach. the resultant model s solved usng the OCD approach, n real-tme envronment. g1( X 1)... gn ( X N ) N T * ( ) + λk k ( ) f1 X1 h X1 k = 2 * 1( 1 ) h1 ( X1,..., X N ) hn ( X1,..., X N ) h X... g 1( X 1 ) N 1 T * ( ) + λ ( ) fn X N k hk X N k = 1 * hn ( X N ) g ( X ) N N C. Paper organzaton Ths paper s set out as follows: secton II provdes a general descrpton of OCD algorthm. Secton III deals wth uncertanty modelng n the proposed real-tme DED. Secton IV presents DED problem formulaton. Applcaton of OCD on the DED problem s presented n Secton V. Smulaton results are presented n secton VI and fnally, Secton VII concludes the paper. Fg. 1. Decomposton by OCD technque III. SCENARIO BASED UNCERTAINTY MODELING The assumptons and techncal constrants are descrbed as follows:
3 A. Assumptons The electrc utlty s pad a fxed prce for each MWh whch sells to the customers. The electrc utlty has three optons for supplyng the demand of ts customers namely: pool market, thermal generatng unts and fnally the wnd turbne power generaton. The electrc utlty s assumed to be the owner of the thermal and wnd generatng unts. The wnd generaton and electrcty prce n pool market are assumed to be uncertan parameters. The proposed tool s run every 15-mnutes and t consders ntervals (the length of each nterval s 15 mnutes). Ths rollng wndow starts at t = 0 and moves toward the end of the operatng horzon t = T as shown n Fg.2. Fg. 3. The uncertan parameter (Pu) 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 t1 t2 t3................... T 1 T Tme (t) Scattered realzaton of the uncertan parameter over the tme... Prce or Wnd Scenaros s 1 s NS 1 s NS Expected Prce (Forecasted) Fg. 2. The rollng wndow concept of proposed method Fg. 4. t t 1 2 tt 1 Scenaro modelng around the forecasted value t T Future Horzon B. Uncertanty modelng of wnd turbne power generaton and electrcty prces The generaton power of a wnd turbne depends on ts nput source of energy. The varaton of wnd speed s a key factor for determnaton of wnd turbne s output. The prce of energy n the pool market s determned based on the competton between the market players. The value of prce n each hour s an uncertan parameter. The hstorc data of wnd speed n the regon and electrcty prces can be used to probablstcally model the uncertantes of wnd speed [26]. In ths paper, t s assumed that the forecasted values of wnd power generaton (Pw f t ) and electrcty prces (λ f t ) are avalable, as depcted n Fg. 3.To defne wnd power generaton scenaros, one can consder beta dstrbuton for wnd power [27] or webull dstrbuton for wnd speed besde wnd turbne cure [28]. The latter s consdered n ths paper whch s descrbed wth more detals n Appendx A. The realzaton of wnd power (Pw s,t ) and electrcty prce (λ P s,t) are modeled usng scenaros around the forecasted value, as shown conceptually n Fg. 4. It s assumed that the actual wnd power generaton and electrcty prce s normally dstrbuted around the correspondng forecasted value µ = Pw f t or µ = λ f t. In ths work, 7 scenaros are consdered for modelng the uncertan parameter (µ,µ ± 3σ) as depcted n Fg. 5. The probablty of fallng nto each scenaro s ndcated on the correspondng area, as shown n Fg. 5. It s assumed that for the employed normal dstrbuton, σ = 0.01µ. It s also assumed that the varaton electrcty prce values are ndependent wth the varatons of wnd speeds. The scenaros of wnd power generatons and prce values are combned and a unque set of scenaros (.e. 49 scenaros) s constructed. Each scenaro contanst avalable wnd power generaton andt prce values as follows ( t = t 1,...,t T and s = s 1,...,s NS ). λ P s,t = λ s λ f t Λ (7) wp s,t = w s Pw f t (8) Pw avl s,t = mn(wp s,t P w r,p w r ) (9) P D t = D f t P D peak (10) If expected value of wnd speed s n the nterval [v r,v c out], consderng the wnd turbne cure (see Appendx A), wnd generators produce ther maxmum value as long as wnd speed s n the above nterval. Hence, there s no scenaro wth wnd generaton greater than ts forecasted value, whch s a usual case. Ths fact s reflected n (9). IV. REAL-TIME DED PROBLEM FORMULATION The objectve functon of the proposed tool s to fnd the optmal dynamc schedule of the generatng unts to maxmze the total benefts, whch s formulated as follows: A. Total cost of energy procurement The producton cost of thermal unts s defned as: C (P,t) = a P 2,t +b P,t +c (11) where a, b and c are the fuel cost coeffcents of the th unt. The total cost pad by the electrc utlty s calculated as follows: TC = s,t ( ) π spp s,tλ P s,t + (I,tC (P,t)) (12),t
4 Fg. 5. µ 3δ µ 2δ µ δ µ µ + δ µ + 2δ µ + 3δ Normal varaton of uncertan parameter around the forecasted value problem s decomposed nto several smpler problems [25]. In the DED problem, ramp rate constrants (.e. (15)) are the complcatng constrants [29]. Snce the P D t and λ D s,t are known parameters and are gven data of the problem then maxmzng the OF (18) would be the same as mnmzng the total costs. By relaxng the ramp rate constrants, correspondng consecutve m and m + 1-th subproblems (.e. for t = t m and t = t m+1 ) of the DED n k th teraton of the OCD are as follows: For nterval t = t m (.e. m th sub-problem): In each tme step and scenaro the power balance constrants should be satsfed as follows: B. Constrants 1) Real power balance (I,tP,t)+Pw s,t +Pp s,t = Pt D (13) 2) Generaton lmts of thermal unts: P mn P,t P max (14) 3) Ramp up and ramp down constrants: The output power change rate of the thermal unt should be below the prespecfed lmts called ramp rates. Ths s to avod damagng the boler and combuston equpment. These lmts are stated as follows: P,t 1 DR P,t P,t 1 +UR (15) 4) Wnd power generaton lmts: 0 Pw s,t Pw avl s,t (16) 5) Lmts on power exchange wth pool market: Pp mn t Pp t,s Pp max t (17) C. Objectve functon (total beneft): In ths paper, the objectve functon (OF) s defned as the total money receved from the energy consumers mnus the total cost pad for operatng the thermal unts as follows: OF = s,t π sλ D s,tp D t TC (18) The OF (.e total beneft) should be maxmzed, subject to the above equalty and nequalty constrants. V. APPLICATION OF OCD ON THE REAL-TIME DED In large-scale power systems, the dmenson of the DED problem s very large and hence the soluton-tme s very crtcal for real-tme mplementaton. To speed up the soluton process, the DED problem could be decomposed to a few smpler sub-problems wth lower dmensons, and hence wth less computatonal burden. Ths s accomplshed due the multperod structure of the DED problem. In ths paper, the OCD approach s utlzed for ths am. OCD s based on relaxng the complcatng constrants of the orgnal NLP problem. Complcatng constrants are constrants that f relaxed, the resultng mn TC (k) t m = s + I,tm C,tm (P (k) )+ π spp (k) s,t m λ P s,t m (19) φ (k) C,tm (P (k) ) = a (P (k) ) 2 +b (P (k) )+c (20) φ (k) = µ UR,(k 1) + µ DR,(k 1) (P (k) Subject to: P (k) ( P (k 1) P (k) P (k 1) DR ) UR ) (21) P (k) I,tm +Pw s,tm +Pp s,tm = P D t m (22) (k 1) P 1 UR (23) P (k 1) 1 P (k) DR (24) P mn P (k) P max (25) 0 Pw s,tm Pw avl s,t m (26) Pp mn t m Pp (k) s,t m Pp max t m (27) For nterval t = t m+1 (.e. m+1-th sub-problem): mn TC (k) t m+1 = s π spp (k) s,t m+1 λ P s,t m+1 + (28) I,tm+1 C,tm+1 (P (k) )+ φ (k) C,tm+1 (P (k) ) = a (P (k) ) 2 +b (P (k) )+c (29) φ (k) = µ UR,(k 1) +2 ( P (k 1) +2 P (k) UR )+ (30) µ DR,(k 1) +2 (P (k) (k 1) P +2 DR ) Subject to: I,tm+1 P (k) +Pw s,tm+1 +Pp s,tm+1 = Pt D m+1 (31) P (k) P (k 1) P mn P (k 1) UR (32) P (k) DR (33) P (k) P max (34) 0 Pw s,tm+1 Pw avl s,t m+1 (35) Pp mn t m+1 Pp (k) s,t m+1 Pp max t m+1 (36) where, µ UR,(k 1) and µ DR,(k 1) are Lagrange multplers correspondng to complcatng constrans (15) of m + 1-th subproblem at prevous teraton (.e. teraton k 1). The dashed (k 1) parameters (lke P ) are the obtaned values of the correspondng varables at the prevous teraton (.e. teraton k 1) of the OCD. It s evdently observed that utlzaton of the OCD leads to ndependent sub-problems wth much less dmenson than the orgnal DED problem, whch can be solved quckly n
5 parallel manner. In other words, the sub-problem fort = t m only contans the varables (.e. generaton schedules) of that nterval, and the varables of neghbour ntervals are treated as some constant parameters both n objectve functon and constrants of that nterval. The OCD steps are descrbed as follows: Step. 1. Intalzaton (k = 1): In ths step, all varables and Lagrange multplers of complcatng constrans (15) are ntalzed. In ths paper, the ntal values for varables are chosen by ndependently solvng the relaxed subproblem (RSPs), wth zero ntal values for Lagrange multplers of complcatng constrants ( µ UR,(0),t and ), and neglectng constrants (15). Therefore, µ DR,(0),t t P (0),t s known. Step. 2. Independently solvng of the RSPs n teraton k: In ths phase, there are T RSPs to be solved ndependently, by parallel computaton ablty and the optmal values for all varables are obtaned, along wth the Lagrange multplers of complcatng constrants (15)..e µ UR,(k),t, µ DR,(k),t P (k),t and ;,t are determned. Step. 3. Stoppng crteron. The algorthm stops f varables (or the value of OF) do not change sgnfcantly n two consecutve teratons [25]. Otherwse, go to Step 2. The flowchart of the OCD algorthm s depcted n Fg. 6. Wnd Scenaros The OCD to decompose the DED to relaxed subproblems (RSP) Complcated DED problem RSP for T 1 RSP for T 2... Pool Prce Scenaros RSP for T n electrcty prce and wnd power combned scenaros are gven n Table.I. TABLE I THE ELECTRICITY PRICE AND WIND POWER SCENARIOS Scenaro w s λ s π s Scenaro w s λ s π s s 1 1.03 1.03 0 s 26 1 0.99 0.0924 s 2 1.03 1.02 0.0004 s 27 1 0.98 0.0233 s 3 1.03 1.01 0.0015 s 28 1 0.97 0.0023 s 4 1.03 1 0.0023 s 29 0.99 1.03 0.0015 s 5 1.03 0.99 0.0015 s 30 0.99 1.02 0.0148 s 6 1.03 0.98 0.0004 s 31 0.99 1.01 0.0586 s 7 1.03 0.97 0 s 32 0.99 1 0.0924 s 8 1.02 1.03 0.0004 s 33 0.99 0.99 0.0586 s 9 1.02 1.02 0.0037 s 34 0.99 0.98 0.0148 s 10 1.02 1.01 0.0148 s 35 0.99 0.97 0.0015 s 11 1.02 1 0.0233 s 36 0.98 1.03 0.0004 s 12 1.02 0.99 0.0148 s 37 0.98 1.02 0.0037 s 13 1.02 0.98 0.0037 s 38 0.98 1.01 0.0148 s 14 1.02 0.97 0.0004 s 39 0.98 1 0.0233 s 15 1.01 1.03 0.0015 s 40 0.98 0.99 0.0148 s 16 1.01 1.02 0.0148 s 41 0.98 0.98 0.0037 s 17 1.01 1.01 0.0586 s 42 0.98 0.97 0.0004 s 18 1.01 1 0.0924 s 43 0.97 1.03 0 s 19 1.01 0.99 0.0586 s 44 0.97 1.02 0.0004 s 20 1.01 0.98 0.0148 s 45 0.97 1.01 0.0015 s 21 1.01 0.97 0.0015 s 46 0.97 1 0.0023 s 22 1 1.03 0.0023 s 47 0.97 0.99 0.0015 s 23 1 1.02 0.0233 s 48 0.97 0.98 0.0004 s 24 1 1.01 0.0924 s 49 0.97 0.97 0 s 25 1 1 0.1459 The forecasted values of wnd power generaton, electrcty prce and demand values n percent of the correspondng peak values are gven n Table II. TABLE II THE FORECASTED VALUES OF ELECTRICITY PRICE AND WIND POWER AND DEMAND VALUES (%) OF PEAK VALUE t Pw f t λ f t D f t t 1 100.00 77.79 63.41 t 2 96.88 75.24 62.12 t 3 93.75 72.70 61.07 t 4 91.41 71.92 60.08 t 5 89.06 71.13 59.07 t 6 96.48 71.13 58.80 t 7 95.31 71.13 58.26 t 8 82.42 71.13 57.78 t 9 80.86 71.13 57.45 t 10 83.20 71.09 57.36 t 11 81.64 71.05 56.91 t 12 83.20 70.66 56.73 t 13 82.03 70.27 56.82 Fg. 6. No Solve the above RSPs By Parallel Computaton Stoppng Crteron Satsfed? The flowchart of the OCD algorthm Yes VI. CASE STUDIES The proposed model for the DED and the aforementoned OCD algorthm s mplemented n General Algebrac Modelng System (GAMS) envronment and solved by SNOPT solver. The End The length of ths movng wndow s assumed to be = 13 tme steps. Each tme step s 15 mnutes. In ths way, at t = 0 the wndow expands to the begnnng of t = 3h. The proposed approach s mplemented on 40-unts and 54-unts test systems, along wth a large-scale 391-unts system. For smplcty the prce of sellng electrcty to the consumers s assumed to be constant durng the entre horzon and equal to $15/M W h, $27/MWh and $23/MWh for the above systems, respectvely. A. Case-I: 40-unts system In ths case, there are 40 thermal unts. The techncal data of these unts s avalable n [30]. The wnd capacty s assumed to be 1800 MW. Also, the peak value of electrcty prce Λ and demand are assumed to be 12.75 $/MWh and 12000 MW, respectvely. Besdes, n ths case the maxmum/mnmum lmts on power exchange wth pool market s ±1200 MW. Wthout usng the OCD approach, total CPU tme s obtaned 2.944 seconds. n ths case the OCD algorthm s converged after 11 teratons, and the total CPU tme s equal to 1.060 seconds. Ths means 63.99% reducton n the CPU tme. Ths
6 sgnfcant reducton n the CPU tme, s substantal from the real-tme mplementaton vewpont of the proposed approach. The optmal schedule of thermal generaton unts n the studed horzon s gven n Table IV. Also, the expected value of purchased power from pool market (n each tme step) n ths case are shown n Fg. 7. It s observed from ths fgure that n some hours the electrc utlty purchases power from pool market, whereas n some others t sells energy to the pool market. The values of OF and CPU tme for the teratons of OCD algorthm n ths case are gven n Table III. TABLE III THE VALUES OF OF AND CPU TIME FOR THE OCD ALGORITHM IN CASE-I Iteraton Total beneft ($) CPU-tme (s) 1 295170.635 0.047 2 295090.969 0.078 3 295044.194 0.078 4 295040.244 0.078 5 295129.705 0.078 6 295136.803 0.078 7 295122.144 0.078 8 295129.523 0.078 9 295096.667 0.202 10 295130.516 0.078 11 295131.260 0.187 OCD s total tme 1.060 TABLE IV OPTIMAL POWER GENERATION SCHEDULE OF THERMAL UNITS IN CASE-I (MW) Generator t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 P1 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 40.0 P2 70.2 62.2 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 P3 92.2 80.0 80.0 80.0 80.0 80.0 80.0 80.0 80.0 80.0 80.0 80.0 80.0 P4 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 P5 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 P6 81.6 68.0 68.0 68.0 68.0 68.0 68.0 68.0 68.0 68.0 68.0 68.0 68.0 P7 241.4 216.4 191.4 166.4 145.3 145.3 145.3 145.3 145.3 144.6 143.8 136.9 130.0 P8 288.7 263.7 238.7 220.6 210.4 210.4 210.4 210.4 210.4 209.9 209.3 204.3 199.3 P9 286.1 261.1 236.1 224.0 215.3 215.3 215.3 215.3 215.3 214.8 214.4 210.1 205.8 P10 130.0 130.0 130.0 130.0 130.0 130.0 130.0 130.0 130.0 130.0 130.0 130.0 130.0 P11 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 P12 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 94.0 P13 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 P14 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 P15 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 P16 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 P17 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 125.0 P18 287.0 259.5 232.0 220.0 220.0 220.0 220.0 220.0 220.0 220.0 220.0 220.0 220.0 P19 289.3 261.8 234.3 220.0 220.0 220.0 220.0 220.0 220.0 220.0 220.0 220.0 220.0 P20 297.0 269.5 242.0 242.0 242.0 242.0 242.0 242.0 242.0 242.0 242.0 242.0 242.0 P21 297.0 269.5 242.0 242.0 242.0 242.0 242.0 242.0 242.0 242.0 242.0 242.0 242.0 P22 520.3 492.8 465.3 437.8 410.3 409.1 409.1 409.1 409.1 408.2 407.4 399.1 390.8 P23 520.3 492.8 465.3 437.8 410.3 409.1 409.1 409.1 409.1 408.2 407.4 399.1 390.8 P24 537.9 510.4 482.9 455.4 427.9 424.0 424.0 424.0 424.0 423.1 422.2 413.5 404.8 P25 537.9 510.4 482.9 455.4 427.9 424.0 424.0 424.0 424.0 423.1 422.2 413.5 404.8 P26 470.3 442.8 415.3 387.8 360.3 354.9 354.9 354.9 354.9 354.0 353.1 344.2 335.3 P27 470.3 442.8 415.3 387.8 360.3 354.9 354.9 354.9 354.9 354.0 353.1 344.2 335.3 P28 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 P29 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 P30 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 P31 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 P32 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 P33 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 P34 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 P35 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 P36 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 18.0 P37 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 P38 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 P39 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 P40 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 B. Case-II: 54-unts system In ths case, there are 54 thermal generatng unts. The data of these unts and the load profle of the system are gven n [30]. In ths case, total wnd power generaton capacty s assumed to be 900 MW. The peak value of electrcty prce Λ and demand are assumed to be 18.75 $/MWh and 6000 MW, respectvely. Also, for ths case the lmts on the power exchange wth pool market are ±500 MW. Table V shows the varatons of total benefts versus teratons of the OCD algorthm for ths system. As t s observed from ths table, the OCD s converged after 7 teratons n ths Expected purchased power from pool market Pp s,t (MW) 100 0 100 200 300 400 500 600 700 1 2 3 4 5 6 7 8 9 10 11 12 13 Tme (t) Fg. 7. The expected value of purchased power from pool market (n each tme step) n Case-I (+ for purchase/- for buy) case. The overall CPU tme for ths problem, usng parallel computaton ablty s 0.368 seconds, whch s qute low. If one solves the above model wthout usng the OCD algorthm, the CPU-tme would be 3.263 seconds. Ths means that OCD reduces the computaton tme about 88.72%. Also, the expected values of the purchased power from pool market (n each tme step) n ths case are shown n Fg. 8. TABLE V THE VALUES OF OF AND CPU TIME FOR THE OCD ALGORITHM IN CASE-II Expected purchased power from pool market Pp s,t (MW) 80 60 40 20 0 20 40 Iteraton Total beneft ($) CPU-tme (s) 1 695647.427 0.036 2 695617.371 0.065 3 695595.139 0.040 4 695605.825 0.050 5 695653.832 0.068 6 695631.706 0.069 7 695624.885 0.038 OCD s total tme 0.368 1 2 3 4 5 6 7 8 9 10 11 12 13 Tme (t) Fg. 8. The expected value of purchased power from pool market (n each tme step) n Case-II (+ for purchase/- for buy) C. Case-III: Practcal large-scale case study In ths case, a real-lfe power system s studed to nvestgate the applcablty of the proposed approach on the large-scale power systems. there are 391 thermal unts avalable n ths
7 system. The techncal data of unts s gven n [30]. The peak demand and nomnal wnd power generaton capacty are 45000 MW and 8000 MW, respectvely. The peak of electrcty prce Λ s assumed to be 10.50 $/MWh. Also, the lmts on the power exchange wth pool market are ±5000 MW n ths case. The optmal total beneft and CPU tme obtaned wthout usng the OCD algorthm are $1627872.913 and 75.751 seconds, respectvely. On the other hand, f OCD s used t would converge n 14 teratons and the total CPU tme s 3.186 seconds. Ths means that utlzaton of the OCD algorthm reduces the computaton tme about 95.794 %. Ths huge reducton n the CPU tme justfes the applcablty of the proposed algorthm for real-tme mplementaton of the formulated DED problem, especally n the case of large-scale power systems. The expected values of purchased power from pool market of Case-III (n each tme step) are shown n Fg.9. TABLE VI THE CPU TIME FOR OCD ALGORITHM IN LARGE-SCALE CASE STUDY WITH 391 UNITS Expected purchased power from pool market Pp s,t (MW) 1500 1000 500 0 500 1000 1500 Iteraton Total beneft ($) CPU-tme (s) 1 1627878.834 0.187 2 1627480.918 0.110 3 1627353.704 0.281 4 1626822.455 0.203 5 1626197.460 0.141 6 1625555.368 0.141 7 1625232.771 0.327 8 1624699.933 0.218 9 1624660.484 0.687 10 1624845.930 0.188 11 1624662.113 0.250 12 1624432.050 0.187 13 1624289.197 0.141 14 1624288.957 0.125 OCD s total tme 3.186 1 2 3 4 5 6 7 8 9 10 11 12 13 Tme (t) Fg. 9. The expected values of purchased power from pool market (n each tme step) n practcal Case (+ for purchase/- for buy) VII. CONCLUSION Ths paper presents a probablstc real-tme methodology to fnd the optmal schedule of thermal generaton unts at the presence of wnd power generaton. The uncertanty of wnd power generaton and electrcty prce are modeled by scenaro based technque. In order to make the proposed approach applcable n case of real-tme operaton of power systems, optmalty condton decomposton s utlzed. It s demonstrated that mplementng the proposed OCD technque along wth parallel computaton ablty, reduces computatonal burden of the DED problem and hence, facltates ts applcaton n realtme envronment. The proposed approach s nvestgated on varous test systems. Numercal results show the applcablty and usefulness of the OCD for solvng the real-tme DED problem, especally n the case of large-scale power systems. Also, t s observed form the numercal studes that by ncreasng the dmenson of the system, more reducton n CPU tme s obtaned, whch s very mportant from the vew pont of real-tme operaton of large-scale power systems. Future work wll be focused on comparng the performance of the proposed method n comparson to the other exstng methods lke as Meta heurstc methods (Partcle Swarm Optmzaton [31] and Honey Bee Matng Optmzaton (HBMO) [32]). The uncertanty of wnd power generaton as well as the demand uncertanty are consdered usng scenaro approach. However usng some rsk measures lke CVaR can enhance the proposed model. The senstvty analyss can also be used to check the robustness of the results of the model n the presence of uncertanty [33]. APPENDIX A WIND POWER GENERATION UNCERTAINTY MODELING It s assumed that the probablty densty functon (PDF) of wnd speed follows the Raylegh behavor (whch s a subset of Webul dstrbuton) and s known for the wnd ste as follows [26], [34]. PDF(v) = ( v v c2) exp[ ( ) 2 ] (37) 2c The occurrence probablty of scenaro s and the correspondng wnd speed v s s calculated as follows: π s = vf,s ( v v v,s c2) exp[ ( ) 2 ]dv (38) 2c v s = v f,s +v,s (39) 2 where v,s,v f,s defne the ntal and fnal values of wnd speed s nterval n scenaro s, respectvely. The characterstc curve of a wnd turbne [28], s depcted n Fg. 10. Thus, n each pont n the future predcton horzon, Fg. 10. Generated power of wnd turbne P w (V) n (MW) P w r v c n v r Wnd speed (v) n (m/s) The power curve of a wnd turbne the forecasted output power of the wnd turbne for the above wnd profle (as a percent of ts rated power, Pr w ), s determned usng ts characterstcs as follows: 0 f v s vn c or v s vout c Pw f v = s vn c v r v np w c r f vn c v s v r (40) 1 f v r v s vout c v c out
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Alreza Soroud (M 14) Receved the B.Sc. and M.Sc. degrees from Sharf Unversty of Technology, Tehran, Iran, n 2002 and 2004, respectvely, both n electrcal engneerng. and the jont Ph.D. degree from Sharf Unversty of Technology and the Grenoble Insttute of Technology (Grenoble-INP), Grenoble, France, n 2011. He s the wnner of the ENRE Young Researcher Prze at the INFORMS 2013. He s currently a senor researcher wth the School of Electrcal, Electronc, and Mechancal Engneerng, Unversty College Dubln wth research nterests n uncertanty modelng and optmzaton technques appled to Smart grds, power system plannng and operaton. Abbas Rabee (S 09, M 14) Receved the B.Sc. degree n electrcal engneerng from Iran Unversty of Scence and Technology, Tehran, Iran, n 2006, and the M.Sc. and Ph.D. degrees n electrcal power engneerng from Sharf Unversty of Technology (SUT), Tehran, n 2008 and 2013, respectvely. Currently, he s an Assstant Professor at the Department of Electrcal and Computer Engneerng, Unversty of Zanjan, Zanjan, Iran. Hs research nterests nclude operaton and securty of power systems, renewable energes, and optmzaton methods. Andrew Keane (S 04, M 07, SM 14) receved the B.E. and Ph.D. degrees n electrcal engneerng from Unversty College Dubln, Dubln, Ireland, n 2003 and 2007, respectvely. He s currently a lecturer wth the School of Electrcal, Electronc, and Mechancal Engneerng, Unversty College Dubln wth research nterests n power systems plannng and operaton, dstrbuted energy resources, and dstrbuton networks.