DECOMPOSITION ALGORITHM FOR OPTIMAL SECURITY-CONSTRAINED POWER SCHEDULING

DECOMPOSITION ALGORITHM FOR OPTIMAL SECURITY-CONSTRAINED POWER SCHEDULING Jorge Martínez-Crespo Julo Usaola José L. Fernández Unversdad Carlos III de Madrd Unversdad Carlos III de Madrd Red Elétra de Espana (REE) Leganés, Span Leganés, Span Madrd, Span jorgemar@ng.u3m.es jusaola@ng.u3m.es joselfdez@ree.es Abstrat - Ths paper presents an approah to determne the optmal daly generaton shedulng n a ompettve envronment takng nto aount dspath onstrants (power reserve, ramp rate lmts, real and reatve power output lmts), network and power system seurty onstrants (real and reatve power flow equatons, bus voltages and transmsson flow lmts n pre and post-ontngeny states). The proposed approah onsders Benders deomposton for solvng the seurty-onstraned optmal generaton shedulng problem. It has been developed usng market algorthm appled to a Sngle System Operator Model whh onsders the daly market and the tehnal onstrants resoluton proess as a sngle one. The objetve funton mnmzes the energy purhase ost subjet to prevously defned onstrants, n a market where all avalable generatng unts must make energy bds arranged n dfferent bloks wth ther own pre. As a result of the optmzaton proess, the on/off status of the generatng unts and voltage ontrol deves, the atve and reatve power output of eah generatng unt and the tap transformer value are provded. Mathematally the power shedulng s defned as a nonlnear mxed-nteger optmzaton problem wth lnear objetve funton, bnary deson varables (on/off status of the generatng unts, reators and apators), ontnuous varables for the operaton proesses, tme ouplngs (ramp rate lmts and mnmum up/down tme) and non-lnear onstrants as omplete flow load equatons. The model has been tested n dfferent ases of the IEEE 24-bus Relablty Test System and an adapted IEEE 118-bus Test System. It has been programmed n GAMS mathematal modellng language, usng CONOPT and CPLEX solvers for non-lnear and lnear mxed-nteger programmng problems, respetvely. Keywords - Power system regulaton, eletrty markets, non-lnear mxed-nteger optmzaton problem, Benders deomposton, preventve seurty analyss 1 INTRODUCTION THIS paper presents a Benders deomposton approah [1] to defne the optmal seurty-onstraned daly unt ommtment n a ompettve envronment. Eletrty markets an be organzed n dfferent ways. The analyzed model s based on the struture of ertan eletrty markets [2, 3] that solve jontly the generaton dspath problem and the network onstrants. A sngle entty, System Operator (SO), determnes the daly generaton shedulng mnmzng the energy purhase ost and onsderng the seure operaton of the power system. The market partpants submt hourly energy multblok pre bds. The dspather (SO) problem s to selet the heapest offers from the set of the supply-sde offers to math the real power demand (demand-sde bds are not onsdered n ths study). The soluton sets the aepted offers, the ommtted energy and the hourly pre pad to every ommtted generatng unt. The ommtted unts an be pad at ther offer pres or at the spot pres, whh are dretly provded by the soluton. The seure power shedulng nvolves two lassal problems n the eletr systems operaton: the unt ommtment and the seurty-onstraned optmal power flow. In order to defne a preventve seure power shedulng, dspath onstrants (generaton lmts, power reserve, ramp rate lmts, mnmum up and down tme) as well as network (omplete load flow equatons) and seurty onstrants (bus voltages and transmsson flow lmts n pre and post ontngeny states) have been nluded n the model. Ths preventve seurty rteron also norporates the ommtment of voltage ontrol deves, suh as reators or apators. Therefore, n ths work, the daly power shedulng s a mxed-nteger non-lnear optmzaton problem that nludes: lnear objetve funton based on hourly energy mult-blok pre bds, bnary deson varables (off-lne or on-lne generatng unts, reators and apators), ontnuous varables for the operaton proesses (real and reatve power, transformer taps, voltage magntude and phase angle), tme ouplngs (ramp rate lmts, mnmum up and down tme), non-lnear network onstrants (omplete power flow equatons, transmsson flow lmts, reatve power njeted by voltage ontrol deves). non-lnear seurty onstrants (post-ontngeny real and reatve power flow equatons, transmsson flow lmts,... ). Ths optmzaton problem has been solved usng Generalzed Benders deomposton [4, 5]. Ths partton algorthm s a deomposton tehnque n two-levels, master and slave. The master problem solves the unt ommtment problem whereas the slave problem deals wth the seurty-onstraned optmal power flow (SCOPF). Ths Benders method allows to broah the presene of bnary 15th PSCC, Lege, 22-26 August 2005 Sesson 18, Paper 1, Page 1

varables n a non-lnear model. Besdes, the tme ouplngs are arranged so that t an be treated n a optmal way by the Benders algorthm. Ths method mproves presently avalable approahes as t supples an optmal 24-hour shedulng onsderng a prese model of the transmsson network and a omplete seurty analyss, provdng an optmal and preventve real power dspath wth voltages ontrol deves. The model has been tested n the IEEE 24-bus Relablty Test System [6] and an adapted IEEE 118-bus System [7]. The exeuton tmes and teratons number are provded and some results for the IEEE 24-bus System are reported. The model has been programmed n GAMS [8], usng CONOPT and CPLEX solvers [9] for non-lnear and lnear mxednteger programmng problems, respetvely. Ths paper s organzed n the followng way. The notaton used throughout the artle s provded n Seton 2. Seton 3 expounds the model formulaton. The struture of the Benders deomposton algorthm and ts applaton to the optmal seure power shedulng are explaned n Seton 4. Seton 5 presents the test systems and the results and fnally, Seton 6 states the onlusons. 2 NOMENCLATURE The notaton used throughout the paper s: Varables P t, real power output of unt at perod t. P g t,j real power njeted by all the generatng unts onneted at the bus j at perod t. P t,,b real power of blok b offered by unt at perod t. rt t,nj transformer tap for transformer (nj) at perod t. Q reatve power output of generatng unt, reator or apator k at perod t. Q g t,j reatve power njeted by the generatng unts, reators and apators onneted at the bus j at perod t. u deson varable (0/1) that represents the ommtment state of unt, reatane or apator x t, k at perod t. number of hours that generatng unt has been on (+) or off (-) at the end of hour t. V t,n bus voltage at bus n durng perod t. V t,n bus voltage magntude at bus n durng perod t. δ t,n phase angle at bus n durng perod t. G t,jn B t,jn G t,jn B t,jn λ m α t real term of the element j,n n bus admttane matrx. magnary term of the element j,n n bus admttane matrx. real term of the element j,n n bus admttane matrx for the post-ontngeny state. magnary term of the element j,n n bus admttane matrx for the post-ontngeny state. dual varable suppled by the slave subproblem n eah teraton m, whh s assoated to the deson of onnetng of unt k at perod t. underestmaton of the operaton osts omputed n the slave subproblem at perod t. Parameters p t,,b pre offered by unt at hour t for blok b. Pt,,b max maxmum real power offered by the generatng unt for the blok b at perod t. P max maxmum real power output of generatng unt. P mn Pt, max mnmum real power output of generatng unt. maxmum real power output of generatng unt at perod t. mnmum real power output of generatng unt at perod t. maxmum reatve power output of unt. P mn t, Q max Q mn mnmum reatve power output of unt. Pt,n d real load demand at bus n durng perod t. Q d t,j reatve load demand at bus n durng perod t. UR ramp-up and start-up rate lmt of unt. DR ramp-down and shut-down rate lmt of unt. UT mnmum up tme of unt. UT mnmum down tme of unt. state (on/off) of the unt k at perod t for teraton m. Pt, m real power of the unt at perod t for teraton m. R t reserve requrement durng perod t. U m V mn V max V mn V max Rt mn Rt max Bj sh S max jn S max jn y jn y jn ε mnmum voltage at any node n and any perod t for n state. maxmum voltage at any node n and any perod t for n state. mnmum voltage at any node n and any perod t for any (n-1) state. maxmum voltage at any node n and any perod t for any (n-1) state. jn mnmum transformer tap at any perod t. jn maxmum transformer tap at any perod t. suseptane of the apator or reator k onneted at bus n. maxmum transmsson apaty (MVA) at lne (jn) for n state. maxmum transmsson apaty (MVA) at lne (jn) for any (n-1) state. lne seres admttane. hargng admttane. ost tolerane. Sets G set of ndexes of all generatng unts. GR set of ndexes of all generatng unts, reators and apators. RC set of ndexes of all reators and apators. B set of ndexes of energy sale bloks. N set of ndexes of all buses. N set of all load buses. C set of seleted ontngenes. T set of ndexes of all perods n hours. Φ set of all system branhes and transformers. Φ n subset of all system branhes onneted at bus n. Φ RT n subset of all transformers onneted at bus n. Ψ n subset of all generatng unts at bus n. M set of teraton ndexes. 15th PSCC, Lege, 22-26 August 2005 Sesson 18, Paper 1, Page 2

It should be noted that the supersrpt n a state varable (Vt,n, Q t,j, δ t,n) represents the value of that varable n the post-ontngeny state. The varable Vt,n s only defned for N buses. 3 PROBLEM FORMULATION After the lberalzaton of the eletrty ndustry, the unt ommtment s solved as a market problem based on offer pres, nstead of the ost-based mnmzaton of the lassal model. In the analyzed model, the OS operates the power system n a entralzed way. Its am s to mnmze the energy purhase ost takng nto aount the bds submtted by the generatng agents nto the market, the hourly demand and the dfferent onstrants of the power system. In ths market, all avalable generatng unts must submt energy bds arranged n dfferent power-prze bloks for eah hour. Mathematally the model an be stated as follows: Mnmze Pt,,b p t,,b P t,,b (1) t T G b B The objetve funton nludes the energy bds P t,,b dvded n bloks as well as ther bd pres, p t,,b. It should be noted that the model assumes, by smplty, that the mnmum power of thermal plant, P mn, s always offered as the frst blok, whh s onsdered as ndvsble energy blok through the deson bnary varable u t,. The aeptane of ths frst blok nvolves the generatng unt start-up. Wth ths onsderaton, the objetve funton (1) ould be formulated as follows: Mnmze ut,;p t,,b t T G p t,,1 u t, P mn + t T G b B b>1 p t,,b P t,,b (2) The njeted power to or drawn from the system s lmted by a set of onstrants, whh an be lassfed n three dfferent groups: Dspath onstrants: (3)-(10), Network onstrants: (11)-(17), Seurty onstrants: (18)-(25). The network onstrants nlude a omplete AC model. The n-1 seurty onstrants are addtonal equalty and nequalty onstrants assoated wth those preseleted prevalng outages. These outages produe a volaton of the seurty lmts and they are haraterzed by a new set of nodal power flow equatons and transmsson system operatng lmts (n-1 state equatons), n whh the ontrol varables (generator real power, generator voltage magntude and transformer taps) are kept n equal value that n the base-ase (n state). The exepton would be the generatng unt outage, n whh the lost generaton wll be suppled by the rest of the ommtted generatng unts (all of them or a subset, smulatng the aton of the P-f regulaton) aordng to the equaton (22). In the ase of lne or transformer outage, t s verfed that G t,jn =G t,jn and Bt,jn = B t,jn exept for the terms of the bus admttane matrx related to the lost element. Voltage magntudes are norporated as they are a rtal fator n some real power systems. By smplty, for (n-1)-state onstrants, t has been onsdered n the formulaton all voltage magntudes as Vt,j. Ths s only true n PQ buses. The prevously mentoned onstrants are the followng ones: a) Energy bloks lmts: 0 P t,,b P max t,,b t T, G, b B/{1} (3) b) Relaton between energy bloks and real power output: P t, = u t, P mn + b B b>1 ) Real power output lmts: u t, P mn P t,,b t T, G (4) P t, u t, P max t T, G (5) d) Reatve power output lmts: u t, Q mn e) Ramp rate lmts: Q t, u t, Q max t T, G (6) DR P t, P t 1, UR t T, G (7) f) Mnmum startng up tmes: [x t 1, UT ] [u t 1, u t, ] 0 t T, UG (8) g) Mnmum startng down tmes: [x t 1, +DT ] [u t, u t 1, ] 0 t T, UG (9) h) System operatng reserve requrements: u t, P max Pt,n d + R t t T G u Q max k Q d t,n t T ) n-state real power flow equatons: V t,j V t,n (G t,jn os(δ t,j δ t,n )+B t,jn sn(δ t,j δ t,n ))= = P g t,j P d t,j t T, j N : P g t,j = (10) Ψ j P t, (11) j) n-state reatve power flow equatons: V t,j V t,n (G t,jn sn(δ t,j δ t,n ) B t,jn os(δ t,j δ t,n ))= = Q g t,j Qd t,j t T, j N : Q g t,j = k Ψ j Q (12) k) n-state transmsson apaty lmts of lnes: ( ( ) ) V t,j [(V t,j V t,n ) y jn ] 1 +V t,j V t,j 2 y jn Sjn max t T, j, n N : (jn) Φ j (13) 15th PSCC, Lege, 22-26 August 2005 Sesson 18, Paper 1, Page 3

l) Transformer tap lmts: Rt mn jn rt t,jn Rt max jn t T, j, n N :(jn) Φ RT j (14) m) Pre-ontngeny reatve power njeted by shunt reators or apators: Q =u B sh k,j V 2 t,j n) n-state bus voltage magntude lmts: t T, k RC, j N:B sh k,j 0 (15) V mn V t,n V max t T, n N (16) o) n-state bus angle lmts: - π δ t,n π t T, n N/{ns} δ t,ns = 0 ns: swng bus (17) p) (n-1)-states real power flow equatons: V t,n( G t,jn os ( δt,j δ t,n) +B t,jn sn ( δt,j δ t,n)) = t,jv =P g t,j P d t,j C, t T, j N :P g t,j = Ψ j P t, (18) q) (n-1)-states reatve power flow equatons: V t,n( G t,jn sn ( δt,j δ t,n) B t,jn os ( δt,j δ t,n)) = t,jv =Q g t,j Qd t,j C, t T, j N :Q g t,j = Ψ j Q t, (19) r) (n-1)-states transmsson apaty lmts of lnes: [( ( ( ) V t,j Vt,j V t,n) yjn ] ) +V 1 t,j Vt,j 2 y jn Sjn max C, t T, j, n N : (jn) Φ j (20) s) Post-ontngeny reatve power njeted by shunt reators or apators: Q =u B sh k,j (V t,j) 2 C, t T, k RC, j N :B sh k,j 0 (21) t) Real power output of unts after generatng unt outage: P max P =P + k P (Pk max P ) P t, k G:P >0 k P t, = 0 G: P t, > 0 (22) u) (n-1)-states bus voltage magntude lmts: V mn V t,n V max C, t T, n N (23) v) (n-1)-states reatve power output lmts: u t, Q mn Q t, u t, Q max C, t T, G (24) w) (n-1)-states bus angle lmts: - π δ t,n π C, t T, n N/{ns} δ t,ns = 0 ns: swng bus (25) 4 BENDERS DECOMPOSITION The short-term power shedulng addressed n ths paper s a mxed-nteger non-lnear optmzaton problem whh nludes lnear objetve funton, bnary deson varables, ontnuous varables for operaton proesses, tme ouplngs and lnear and non-lnear onstrants. The dffultes assoated to the resoluton of nonlnear optmzaton problems wth bnary and/or nteger varables fore to make use of parttonng tehnques. The Benders partton algorthm s a deomposton tehnque n two-levels, master and slave. The master level represents the deson problem, unt ommtment, whereas the slave level deals wth the operaton problem: SCOPF. The Benders algorthm defnes an teratve proedure between both levels. The master problem s formulated as a lnear mxed-nteger problem whh determnes the ommtted generatng unts, reators and apators. Ths shedule s transferred to the slave subproblem nonlnear optmzaton problem whh alulates the operatng ost and the dual values assoated to the shedulng deson prevously taken by the master problem. Ths new nformaton s suppled to the master problem through the Benders uts n order to mprove the new deson of the master problem. The slave level solves the operaton problem by means of a seurty onstraned AC optmal power flow. In daly shedulng, the slave problem an be deomposed n 24 subproblems (one per hour), whh are sequentally solved. Ths s possble beause of the ramp rate lmts, tme ouplng onstrants, are formulated as power lmts [10] that are updated after solvng eah hourly slave subproblem so that produton lmts for eah generatng unt are ompled wthn the followng hourly slave subproblem. Therefore, ths method allows to broah the nononvexty assoated wth the bnary varables and to dvde the global problem nto two smaller problems easer to solve. Besdes, the tme ouplngs an be treated n an optmal way. The algorthm optmzes jontly the 24- hour problem (unt ommtment wth SCOPF) provdng better results than those obtaned n ase of solvng the onstrants separately hour by hour, as t has usually been solved. The proedure followed n ths paper nludes the steps llustrated n the flowhart of the Fgure 1. 4.1 Master Problem The master level solves the deson problem, a mxednteger lnear problem, whose soluton provdes the on/off state of the generatng unts and voltage ontrol deves gven by the bnary varables u. The master problem ontans any onstrant wth bnary varables n ts formulaton. At the same manner, the objetve funton s made of those terms of the funton (2) that nlude bnary varables. The objetve funton s defned as follows: Mnmze ut,, α t t T G p t,,1 u t, P mn + t Tα t (26) 15th PSCC, Lege, 22-26 August 2005 Sesson 18, Paper 1, Page 4

Mnmze Pt,,b p t,,b P t,,b t T (28) G b B b>1 subjet to the rest of onstrants that have not nluded n the master problem, n addton to: u = U m : λ m+1 t T, k GR (29) To onsder the onstrant (29) n the slave problem, t s neessary to defne temporally the nteger varable u as a ontnuous varable to obtan ts senstvty or dual value λ m+1. Eah slave problem determnes the values of the operaton varables (P t,, Q, V t,n, r t,nj, δ t,n, Q, V t,n, δ t,n ) at eah perod for n and n-1 states. Fgure 1: Flowhart of the deomposton proedure The objetve funton onssts of two terms, the frst one represents the start-up ost of the unt at ts mnmum power and the seond one s a lower estmaton of the operaton osts omputed n the slave problem. Therefore, the optmzaton varables are αt and u. Ths funton s subjet to the onstrants (8), (9), (10) and the Benders lnear uts whh are formulated as: α t α t ( U m 1 )+ ( λ m 1 u U m 1 ) t T, m M(27) The key ssue n Benders deomposton s loated at these Benders lnear uts. Both levels, master and slave, are oupled by the Benders uts that are updated at eah teraton for all operaton problems. These uts nludes the term α t (U m 1 ), whh represents the ost of the slave subproblem orrespondng to the perod t for the generaton shedulng establshed by the master problem n the prevous teraton, and a seond term related to the senstvty assoated to the unt ommtment provded by the master problem n the prevous teraton. 24 new uts are added to the master problem at eah teraton. On the other hand, the mnmum and maxmum startng up tmes onstrants (8) and (9) are lnearzed to be nluded n the master problem [10]. 4.2 Slave Problem The slave level solves the operaton problem by means of a seurty onstraned optmal power flow. The slave problem s deomposed n hourly subproblems, whh are sequentally solved. Eah hourly slave subproblem solves the system operaton problem, a SCOPF, mnmzng the ost of the produton bds (exept for the frst blok bd) submtted to the market by eah generatng unt ommtted by the master problem at eah hour. Therefore, the objetve funton for eah hourly subproblem s: 4.3 Benders onvergene rteron The teratve Benders deomposton proedure stops when the value of the objetve funton omputed n the master problem reahes the same value than the startup osts (frst bd blok) plus operatng osts omputed through the slave problem, exept for a small tolerane ost ε. Atually, as the start-up osts are the same n both problems, slave and master, they an be omtted n the onvergene rteron (CC), onsderng only the operatng osts of both problems. The fnal onvergene rteron s establshed as the equaton (30) shows. ( ) ) (α t U m αt t T CC = ) ε m M (30) α t (U m t T 4.4 Slave problem feasblty Feasblty uts have been added to the master problem to enfore the feasblty of the hourly slave subproblems. However, sne voltage ontrol s a loal problem, there ould be some ases where the reatve power onstrant (32) does not guarantee the problem feasblty. It would fore to add fttous soures at some PV buses or at buses wth voltage ontrol deves and to nlude them wth a hgh penalty fator n the objetve funton of the slave problem so that t s optmzed the global ost mnmzng the ost of the nfeasbltes as well. Nevertheless, these uts mprove the problem onvergene, redung the number of teratons. The uts are formulated as: u t, Pt, max Pt,n d + R t t T G G u t, P mn t, P d t,n t T (31) u Q max k Q d t,n t T u Q mn k Q d t,n t T (32) 15th PSCC, Lege, 22-26 August 2005 Sesson 18, Paper 1, Page 5

5 TEST SYSTEMS AND RESULTS The test systems are the IEEE 24-bus Relablty Test System and an adapted IEEE 118-bus Test System wth standard osts for the generatng unts. Only results for the IEEE 24-bus system are shown. The test for the IEEE 118-bus system seeks to ompare the behavour of the proposed approah n a larger power system wth an mportant number of voltage ontrol deves (12 apators and 2 reators) and to ompare the exeuton tmes. The IEEE 24-bus Test System nludes 32 unts: nulear, oal, ol and hydro plants, dstrbuted all over the generatng buses and rangng from 12 to 400 MW. The total generaton apaty amounts to 3405 MW. The peak load s 2850 MW and ours n hours T18 and T19. The mnmum load s 1682 MW and takes plae n hours T4 and T5. A 98% power fator s appled to all load levels. The transmsson network ontans 24 load/generatng buses onneted by 38 lnes or transformers at two voltages, 138 and 230 kv. The swng bus s N13. The power system has voltage ontrol deves at buses N14 (synhronous ondenser) and N6 (reator). The transformers taps are modelled as ontnuous varables. The voltage lmts n n-state are 0.95 and 1.05. In a post-fault state the lmts are set to 0.93 and 1.11. The flow lmts are provded n the referene [6]. The seleted prevalng ontngenes are shown n Table 1. Hours (t) Lnes/transformers (jn) Unts () t= T1,T2,T4, (N7-N8), (N8-N9), G9, G10 or G11 T5,T22,T24 (N8-N10) at bus N7 t = T3, T6 (N7-N8), (N8-N9) G9, G10 or G11 t = T23 (N7-N8), (N8-N9), G9, G10 or G11 (N8-N10), (N3-N24), (N9-N11) t = T7 (N7-N8), (N8-N9), G9, G10 or G11 (N8-N10), (N3-N24), (N9-N11), (N11-N14), (N12-N23), (N15-N24) t= [T8,T21] (N7-N8) Table 1: Seleted ontngenes at eah perod (IEEE 24-Bus System) The generators bd pres have been taken aordng to the margnal osts of eah energy blok. The number of energy bloks s 5 and the bd pre rses wth the blok number. The model manages n the master problem 816 bnary varables u, 624 nteger varables x onverted n 1248 new bnary varables as a result of the lnearzaton of equatons (8) and (9), and 24 ontnuous varables α t. On the other hand, eah hourly slave problem nludes from 357 varables and 644 onstrants on, dependng on the number of ontngenes. Eah ontngeny adds 227 new onstrants and 77 new varables. The master problem s solved usng CPLEX under GAMS, whereas the slave subproblem s solved usng CONOPT. The per unt ost tolerane ε s fxed to 1e-3. Avalable soluton data nlude: system total ost. on/off status of every generatng unt, reator and apator per hour. atve power output of every plant per hour. voltage magntude of every bus per hour. reatve power output of every plant and voltage ontrol deves per hour. power flow of every lne per perod. spot pre per bus and perod. Table 2 shows the evoluton of the Benders teratve proess. The dfferene between operatng osts of the slave and master problems s progressvely dereasng wth the number of teratons. The operatng osts are very hgh n the frst teratons due to the ommtment of fttous generaton soures n some buses. The master total ost reports on the total ost of unt ommtment wth onstrants resoluton. Iter. Master Slave Master CC Total Cost Operatng Cost Operatng Cost 1 200275.38 0.00 12754747.16 1.000 2 212400.71 4826.19 5623158.21 0.999 3 505353.91 253161.62 815471.30 0.690 4 537442.34 292004.22 399780.87 0.270 5 564275.97 318100.91 359046.42 0.114 6 588114.92 336825.69 614480.42 0.452 7 590622.53 341848.27 348124.03 0.018 8 595639.04 348235.17 348513.89 0.001 Table 2: Evoluton of the onvergene of osts ($) wth the teratons Fgure 2 shows the hourly maxmum pre obtaned wthout onstrants (Case1) and the hourly maxmum pre wth onstrants soluton (Case2). It hghlghts the ndene of onstrants soluton n the daly market. Obvously, ths last maxmum pre s determned by some generatng unt ommtted to solve onstrants. The generatng unts loated at buses N7 and N13, whh are more expensve, determne the fnal pre at eah perod. It may be dedued that these unts have a prvleged loaton n the network to solve onstrants. It stands to reason that the hgher dfferene n pres between both ases ours n the valley load perods, as t s neessary to onnet new generators to fulfll the network and seurty onstrants. Fgure 2: System maxmum pre wth or wthout onstrants Spot pres per bus at perods T18 (peak load) and T2 (valley load) are shown n the Fgure 3. Fgure 4 detals the tme evoluton of the spot pre n the buses N2 (PV bus) and N19 (PQ bus). Spot pres per bus and hour show, respetvely, ther spae dstrbuton (maxmum spot pres at rtal buses N7 and N8) and the orrelaton between the tme evoluton of total generaton and the tme evoluton 15th PSCC, Lege, 22-26 August 2005 Sesson 18, Paper 1, Page 6

of spot pres spot pres per hour follow approxmately the load urve. Fgure 3: Tme evoluton of the spot pres (perods T2 y T18) Fgure 4: Spot pres per bus (buses N2 and N19) The number of teratons to reah the onvergene and the total CPU tme requred arryng out the study ases on an INTEL P-IV (3.06 GHz.) are shown n Table 3. The method has been appled to real systems but t s neessary to mprove ts performane n the future to redue the omputaton tme. In large-sale power system, the exeuton tme s hghly dependent on the preson requred to the lnear mxed-nteger optmzer, the number of ontngenes analyzed and the omputaton effeny of non-lnear optmzer. Computaton tme Iteratons IEEE 24-Bus System 322 8 IEEE 118-Bus System 2h. 01 21 18 Table 3: Computaton and number of teratons n test systems 6 CONCLUSIONS The Generalzed Benders deomposton method s used to solve a mult-perod dspath problem wth seurty onstrants n a wholesale market model. Ths model solves the daly market and onstrants soluton proess n a sngle stage. The method shows good onvergene propertes for the developed applaton. The proposed algorthm mproves presently avalable approahes n the followng respets: a daly unt ommtment s onsdered smultaneously wth a AC prese model of the transmsson network nludng load flow equatons, lne apaty lmts and voltage lmts n both pre and post-ontngeny states; branhes overload and bus voltages problems are jontly analyzed; and the model provdes an optmal and preventve real power shedule wth voltages ontrol deves ommtment. Fnally, the algorthm optmzes jontly the 24-hour problem (unt ommtment wth SCOPF) provdng better results than those obtaned solvng eah hourly problem separately. A small-sale ase study based on the IEEE 24-bus s analyzed and the omputaton tme and number of teratons are ompared wth the IEEE 118-bus System. The hourly maxmum pre allows to know the last generatng unt ommtted to solve onstrants n eah perod. Spot pres per bus and hour show, respetvely, ther spae dstrbuton and the orrelaton between the tme evoluton of total generaton and the tme evoluton of spot pres. Tme omputaton problems an be deteted n largesale power system. Ths exeuton tme s hghly dependent on the preson requred to the lnear mxed-nteger optmzer, the number of ontngenes analyzed and the omputaton effeny of non-lnear optmzer. Parallel omputaton and/or reatve reserve by areas ould be appled to mprove the tme omputaton soluton. Referenes [1] J. F. Benders, Parttonng Proedures for Solvng Mxed-Varables Programmng Problems, Numershe Mathematk, vol. 4, pp. 238-252, 1962 [2] H. Rudnk, R. Varela and W. Hogan, Evaluaton of Alternatves for Power System Coordnaton and Poolng n a Compettve Envronment, IEEE Transatons on Power Systems, vol. 12, no. 2, pp. 605-613, May 1997 [3] R. A. Drom, Operatng Agreement of PJM Interonneton, L.L.C., FERC, Aprl 2002 [4] H. Ma and S. M. Shahdehpour, Unt ommtment wth transmsson seurty and voltage onstrants, IEEE Transatons on Power Systems, vol. 14, no. 2, pp. 757-764, May 1999 [5] N. Algual and A. J. Conejo, Multperod optmal power flow usng Benders deomposton, IEEE Transatons on Power Systems, vol. 15, no. 1, pp. 196-201, February 2000 [6] The IEEE Relablty Test System-1996, IEEE Transatons on Power Systems, vol. 14, no. 3, pp. 1010-1020, August 1999 [7] IEEE 118-bus System. Avalable at: http://www.edu.es [8] A. Brooke, D. Kendrk, A. Meeraus and R. Raman, Release 2.50 GAMS A Users Gude, Washngton, DC: GAMS Development Corporaton, 1998 [9] GAMS, GAMS-The Solver Manuals, Washngton, DC: GAMS Development Corporaton, Marh 2001. [10] J. M. Arroyo and A. J. Conejo, Optmal Response of a thermal unt to an eletrty spot market, IEEE Transatons on Power Systems, vol. 15, no. 3, pp. 1098-1104, August 2000 15th PSCC, Lege, 22-26 August 2005 Sesson 18, Paper 1, Page 7