Applcato of Modfed Subgradet Algor Based o Feasble Values to Securty Costraed Ecooc Dspatch roble w rohbted Operato Zoes Salh Fadıl, Burak Urazel, Eskşehr Osagaz Uversty, Faculty of Egeerg, Departet of Electrcal Egeerg, Esksehr, Turkey sfadl@ogu.edu.tr, burazel@ogu.edu.tr Abstract A securty costraed ecooc dspatch proble w prohbted operato zoes for a lossy electrc power syste s forulated. A teratve soluto eod at s based o odfed subgradet algor operatg o feasble values s eployed to solve t. Bus voltage agtudes ad phase agles, off-oal tap settgs ad susceptace values of svar systes are take as depedet (decso) varables e soluto algor. Sce load flow equatos are added to e odel as equalty costrats, actual power syste loss s used soluto of e optzato odel. The proposed techque s tested o IEEE 30-bus test systes. The u total cost rates ad e soluto tes obtaed fro F-MSG algor ad fro e oer techques are copared, ad e outperforace of e F- MSG algor w respect to e oer eods each test syste s deostrated.. Itroducto Ecooc dspatch proble power systes s a costraed o-lear optzato proble. The soluto of t gves e u of total actve power geerato cost rate so at all equalty ad equalty costrats of e proble are satsfed. I e lterature, ay eods have bee developed ad appled to solve ecooc dspatch proble w prohbted operato zoes. Soe of ese eods use e quatuspred evolutoary algor [], e hybrd partcle swar optzato techque [], partcle swar optzato techque [3], dfferetal haroy search algor [4], e θ- SO algor [5] The odfed subgradet algor operatg o feasble values (F-MSG) s a deterstc soluto eod, whch uses deterstc equatos at oe pot to produce e ext soluto pot beg closer to e optu soluto e soluto space. I e proposed dspatch tecque based o F-MSG eod [6], e bus voltage agtudes ad phase agles, e off oal tap settgs ad e susceptace values of svar systes are take as depedet varables. Sce all e costrats ad cost fucto ca be expressed ters of ose depedet varables, e trassso le capacty costrats, bus voltage agtude costrats ad svar systes susceptace value costrats are hadled togeer e sae odel easly. The load flow equatos are serted to e odel as equalty costrats; erefore, e actual syste loss s added to e soluto process autoatcally. I e F-MSG algor, e upper boud for e cost fucto value s specfed advace ad e algor tres to fd a soluto where e cost fucto s less a or equal to e upper boud ad all costrats are satsfed. If t fds t (feasble total cost), e upper boud s decreased a certa aout, oerwse (feasble total cost) e upper boud s creased a certa aout. The aout of decrease or crease o e upper boud for e ext terato depeds o f ay feasble or feasble total cost value was obtaed e prevous teratos. Ths process cotues utl absolute value of e chage e upper boud s less a a predefed tolerace value. F-MSG algor has already appled to o-covex ecooc dspatch proble [7]. Furerore, power dspatch proble cludg lted eergy supply eral uts [8] ad ocovex puped-storage hydraulc ut schedulg proble [9] were solved va F-MSG eod. However oe of e aeatcal odels [7-9] cosder e prohbted operato zoes. To our kowledge, e proposed algor has ot bee appled to e proble cosdered s paper so far.. roble Forulato I s secto, a olear prograg odel s preseted for e ecooc power dspatch proble cosdered s paper. M = F( G) Subject to G p = 0 G Load, j j B q = 0, =,,, G Load, j j B ( ) ( ) pz { pz pz }, G G G G G G () () (3), (4) G G G a pl pl, l L (5) U U U, =,,,, ref, vc (6) a a a, tap (7) b b b, (8) svar svar svar svar ote at actve geerato of e ut,, should satsfy G oe of e equalty show equato (3). I oer words, G
should ot be cotaed by ay of e utually dsjot prohbted zoe sets G ( pz, pz ), =,, pz. The eags of e sybols used s paper are gve e lst of sybols secto... Deterato of Le Flows ad ower Geeratos I order to express e total cost rate fucto ters of depedet varables of our optzato odel, le flows should be wrtte ters of bus voltage agtudes ad phase agles, off-oal tap settgs, susceptace values of svar systes (see equatos () ad ()). The followg equatos gve e actve ad reactve power flows over e le beg coected betwee buses ad j [0]. gj pj = U g sh a j jcos( δ δ j) b js( δ δ j) ( ) p = U g g j j j sh j j jcos( δ j δ) b js( δ j δ) bj qj = U b sh a j js( δ δ j) b jcos( δ δ j) ( ) q = U b b j j j sh j js( δ j δ) b js( δ j δ) (9) (0) () () I e equatos above, U s e voltage agtude of bus, δ s e phase agle of bus, r j jxj s e seres pedace of e le betwee buses ad j, gj jbj s e seres adttace of e le betwee buses ad j where gj jbj = /( rj jxj), gsh jbsh = gsh j( bcap bsvar ) s e su of e half le chargg adttace ad exteral shut susceptace (svar syste) f ay, ad a s e off-oal tap settg w tap settg faclty at bus. p j ad q j are e actve ad reactve power flows gog fro bus to j at bus border, respectvely. p j ad q j are e actve ad reactve power flows gog fro bus to j at bus j border, respectvely. W e help of equatos (9)-(), fro equato (), e actve ad reactve power geeratos of e ut (coected to bus ) ca be calculated by e followg expressos: = p G Load j j B = q G Load j j B The total loss of e etwork ca be calculated as follows: (3) (4) ploss j = p j pj (5) p LOSS = (6) j, j j The cost rate fucto of e ut s take as F = b c d (7) ( G) G G, G where b, c, d are costat coeffcets. The total cost rate s e detered as: = F( G) ( R/ h) G (8).. Covertg Iequalty Costrats to Equalty Costrats Sce e F-MSG algor requres at all costrats should be expressed equalty costrat for, e equalty costrats e optzato odel should be coverted to correspodg equalty costrats. The followg eod s used for s purpose sce t does ot add ay extra depedet varable (lke e slack varable approach) to e optzato odel. It s erefore e soluto te of e cosdered dspatch proble s reduced furer []. The double sded equalty x x x ca be wrtte as e followg two equaltes: h ( x ) = ( x x ) 0, h ( x ) = ( x x ) 0 (9) The we ca rewrte e above equaltes as a sgle equalty costrat for as follows: { [ { } { }]} eq h ( x) = 0, 0, ( x x ) 0, ( x x ) = 0 (0) If x x x, t s obvous at ( x x ) 0, ( x x) 0 ad { 0, ( )} 0, { 0, x ( )} 0 x = x x =. So, e equalty costrats (9) ca be represeted by e correspodg sgle equalty costrat (0). I s paper, e double sded equalty costrats gve equatos (3)-(8) are coverted to e correspodg sgle equalty costrats s aer. By usg e sae logc at s explaed e above, e uo of two sded equaltes show equato (3) ca be coverted to e correspodg sgle equalty costrat at s gve e followg equato. { G G } { G pz } { } { } { pz } { } pz G G G 0, ( ) 0, ( ), eq h ( G) = 0, ( pz G) 0, ( G pz ), = 0, 0, ( ) 0, ( ) G () It should be oted at whe G takes a feasble value, all quattes sde e square brackets equato () becoe postve ubers ad erefore e equalty costrat s ot satsfed. I e opposte case, oce G takes a feasble value,
oe of e quattes cotaed by e square brackets becoes zero, so e equalty costrat s satsfed s case. 3. The Modfed Subgradet Algor Based o Feasble Values The olear optzato proble descrbed by equatos ()-(8) ca be represeted e stadard for gve below: M F T ( x) hx ( ) = 0 Subject to x K () where x = U, U,, U, δ, δ,, δ, a, a, a, bsvar, bsvar,, b tap svarsvar s e depedet varable vector cosstg of e voltage agtudes ad phase agles of e buses (except e referece bus), tap settgs of e off-oal tap rato trasforers ad susceptace values of e svar systes e etwork. ( x ) s e objectve fucto at s gve equato (8), ad hx ( ) = h( ), h( ),, h ( ) x x x () s e equalty costrat vector. It cludes all e orgal equalty costrats, whch are gve (), ad e equalty costrats whch are obtaed fro covertg all e equalty costrats gve (3)-(8) to e correspodg equalty costrats va e eod gve Secto.. K s a suffcetly large copact set cotag e potetal values of x. Rego K s bouded by e upper ad e lower lts of e voltage agtudes of e buses ad e upper ad e lower lts of e tap settgs of e off oal tap rato trasforers, ad e upper ad e lower lts of e susceptace values of svar systes whch are gve equatos (6)-(8). ote at e voltage agtude ad phase agle of e referece bus, ( U ref, δ ref ), ad voltage agtudes of e voltage cotrolled buses are ot cluded to x sce ey are ot depedet varables ad rea costat durg e soluto process. I solvg e costraed optzato proble gve by equato (), e frst step s to covert t to ucostraed oe by costructg e dual proble. Ths ca be doe usg varous LaGrage fuctos []. LaGrage fucto ust guaratee at e optal soluto of e dual proble be equal to at of e pral costraed proble. Oerwse, ere wll be a dfferece betwee e optal values of ese probles, oer words, a dualty gap wll occur. Classcal LaGrage fucto guaratees e zero dualty gaps for e covex probles. However, f e objectve fucto or soe of e costrats are ot covex, e e classcal LaGrage fucto caot guaratee s. Therefore, for e o-covex probles, sutably selected augeted LaGrage fuctos should be used. Cosderg e o-covex ature of our proble, we for e dual proble usg e followg sharp augeted LaGrage fucto: L( xu,, c) = ( x) c hx ( ) uhx, ( ) = ( x) c [ h( )] [ h( )] h ( ) x x x ( uh ( x) uh ( x) u h ( )) x / (3) where u, u,, u R ad c 0 are LaGrage ultplers (dual varables). The dual fucto assocated w e costraed proble s defed as H ( u, c) = M L( x, u, c). x K The, e dual proble s gve by (4) Max H ( u, c) (5) ( u, c) R R For e gve dual proble, e codtos of ( guarateeg zero dualty gaps are prove [3]. 3.. The F-MSG Algor Italzato Step: Select arbtrary actve ad reactve power geeratos, tap settgs ad susceptace values of e svar systes for all subtervals. The, perfor AC power flow calculatos w e correspodg selected actve ad reactve power geerato values all subtervals to obta e tal values for e voltage agtudes ad phase agles of e buses all subtervals. Calculate e tal total cost F T. Step ) Choose postve ubers ε, ε, Δ ad M (upper boud for ). Set =, p = 0, q = 0, ad H =. Step ) Choose ( u, c ) R R ad () > 0 ad set =, u = u, c = c, Step 3) Gve ( u, c ) satsfacto proble (CS) Fd a soluto x K such at F( x ) c h( x ) u, h( x ) H, solve e followg costrat (6) If a soluto to (6) does ot exst or ( ) > M, e go to Step 6; oerwse, f a soluto x exsts e check wheer hx ( ) = 0. If hx ( ) = 0 (or f hx ( ) ε ) e go to step5, oerwse go to step 4. Step 4). Update dual varables as u = u α s h ( x ) (7) c = c ( α) s hx ( ) (8) where s s a postve step sze paraeter defed as ( H L( x, u, c )) λα 0 < s = (9) α ( α) hx ( ) where α ad λ are costat paraeters w α > 0 ad 0< λ<. Step sze s correspodg to e dual varables ( u, c) should also satsfy e followg property: ( hx u ) s ( ) c > ( ). (30) Set =, update ( ) such a way at ( ) as, ad go to step 3. Step 5) If p = 0, t eas at ay feasble total cost rate value has ot bee chose yet, e set Δ = Δ, oerwse set
Δ = (/ ) Δ. If Δ < ε, e stop, x s a approxate optal pral soluto, ad ( u, c) s a approxate dual H = F( x ), H Δ, soluto; oerwse set { } q = q, =, ad go to step. Step 6) If q = 0, t eas at ay feasble cost rate value has ot bee chose yet, e set Δ = Δ ; oerwse, set Δ = (/ ) Δ. If Δ < ε e stop, ad s case, e last calculated feasble x s a approxate optal pral soluto, ad ( u, c) s a approxate dual soluto; oerwse, set H = H Δ, p = p, = ad go to step-. I s algor, steps 3 ad 4 ca be cosdered as e er loop, ad steps, 5 ad 6 ca be cosdered as e outer loop. We call ay outer loop, whch a feasble cost rate value s geerated by e algor, as a feasble state, f. The followg proble s solved by usg GAMS solver: Mze f = 0 L( xu,, c) H 0 (3) Subject to x K where f s a fcttous objectve fucto whch s detcally zero, or ca be take as ay costat value [6]. The way of updatg e dual varables ( u, c) step 4 wll force e soluto Step 3 to coverge to e feasble soluto (see Theores [6]). 4. uercal Exaple The proposed dspatch techque was tested o IEEE 30-bus test syste. lease refer to referece [] for ecessary data for e test syste such as actve ad reactve geerato lts of e geerators, prohbted operato zoes for all geerators, actve power trassso capacty lt for all trassso les, actve ad reactve load schedules for e test syste, coeffcets of fuel cost rate fuctos. The paraeters, 5 explaed secto 3., ad 3., are chose as ε = 5 0, = 0.05, Δ = 00, M = 500, u = [0,0,...0,0], c = 5000, ε ( 3) ( ) =. Bus s take as e referece bus ad ts coplex voltage s take as.05 0 pu. The upper ad lower lts of e bus voltage agtudes for all buses are take as U = 0.95, U =.05 pu,. Ital bus voltage agtudes ad phase agles each subterval are calculated by carryg out a load-flow soluto w e selected tal geerato values, whch are gve Table. o ore load flow calculato s carred out e subsequet stages of e soluto process. The sulato progra was coded MATLAB ad GAMS was used to solve CS gve by equato (6). I e followg two cases, we solved e dspatch proble by usg e F-MSG algor ad copared e foud results w e oes obtaed va dfferet eods such as sulated aealg (SA), shuffle frog leapg (SFLA), partcal swar optzato (SO) ad hybrd shuffle frog leapg ad sulated aealg (Hybrd SFLA-SA) eods (please see referece []). Table. Selected tal geerato values IITIAL GEERATIOS (MW) G G G 5 G G3 LOSS 03.40 60.00 45.00 5.00 30.00 35.00 5.00 885.5 4.. Case : rohbted Operato Zoes are ot Cosdered To show at e proposed eod satsfes e prohbted operato zoe costrats, frst we solved e dspatch proble w e assupto at e prohbted operato zoes do ot exst. Therefore, we dd ot cosder e prohbted operato zoe lts equato (3) ad we appled e F-MSG algor to e dspatch proble w e calculated tal bus voltage agtudes ad phase agles. The soluto-pot actve ad reactve power geeratos for e curret case are gve Table. 4.. Case : rohbted Operato Zoes are Cosdered I s case, e prohbted operato zoe lts equato (3) are added to e dspatch proble ad t s solved by eas of e F-MSG algor by usg e sae tal bus voltage agtudes ad phase agles. The soluto-pot actve ad reactve power geeratos for e curret case are gve Table 3. 5. Dscusso ad Cocluso I s paper, we propose a soluto to securty costraed power dspatch proble w prohbted operato zoes by usg e F-MSG algor for a lossy power syste area. The dspatch techque s tested o e IEEE 30 bus test syste. As see fro Table ad Table 3, e proposed techque provdes e lowest total cost ad e shortest soluto te values, aog e results obtaed fro e techques gve referece []. We are curretly perforg research o applcato of e F-MSG algor to ecooc power dspatch probles cludg prohbted operato zoes w o-covex total cost curve. Table. Coparso of e results at are obtaed by e oer eods w oes foud va e F-MSG for case. METHODS F-MSG Hybrd SFLA-SA SFLA SA G MW 76.0 7.78 8.6 9.5 G MW 48.84 48.0 5. 48.40 G 5 MW.54 3.94.8 9.55 MW.70 3.85 5.59.6 G MW.3.60 0.00 0.00 G3 MW.00.00.8.00 LOSS ( MW ) 9.8 9.79 0.46 0.68 ( R/ h ) 80.79 a 80.97 a 803.67 (805.6) b (804.9) b 804.0 ST* (sec).73 8.93 9. 98.74 * Soluto Te a The values take fro ref []. b The values, whch are calculated by usg e respectve geerato values show Table ad cost rate fuctos gve [].
Table 3.. Coparso of e results at are obtaed by e oer eods w oes foud va e F-MSG for case. SOLUTIO METHODS F-MSG Hybrd SFLA-SA SO SA G MW 78.3 8.45 74.6 8.70 G MW 45.00 45.00 57.0 57.9 G 5 MW.8.53 3.54 7. MW 3.6.08 4.6 6.5 G MW 3.7.98.00 0.00 G3 MW.00.00 4.5.00 LOSS ( MW ) 0.05 0.64.09.85 ( R/ h ) 806.43 c 808.7 c 804.74 805.8 (809.75) d (80.74) d ST(sec) 5..9 4.94 3.86 c The values take fro ref []. d The values, whch are calculated by usg e respectve geerato values show Table 3 ad cost rate fuctos gve []. 6. Lst of Sybols R : a fcttous oetary ut : uber of buses e etwork. G,: set cotas all buses to whch a geerator s coected.,: set cotas all buses to whch a reactve power source s coected. : set at cotas all buses drectly coected to bus. B tap, L : sets at cotas all tap chagg trasforers ad les e etwork, respectvely. p : actve power flow o le l, (pu or MW). l G, G : actve/reactve power geeratos of e ut, respectvely, (pu or MW, MVar). Load, Load : actve/reactve loads of e bus, respectvely, (pu or MW, MVar)., ;, G G G G : lower/upper actve/reactve geerato lts of e geerato ut, respectvely, (pu or MW, MVar). pz, pz : lower ad upper lts of e prohbted zoe for e ut s actve power geerato, respectvely. p l : u actve trassso capacty of trassso le l, (pu or MW). pz : uber of prohbted zoes for geeratg ut. rohbted zoes are ubered such a way at pz( ) < pz, =,3,..., pz., VAR : uber of equalty costrats ad depedet varables, respectvely svar : uber of statc var systes e etwork. tap : ubers of off oal tap rato trasforers e etwork. x : depedet varable vector obtaed at e terato of e er loop of e u, c of e outer loop terato. : dual varables calculated at e terato of e outer loop. terato of e er loop s : postve step sze paraeter calculated at e er loop. F : total cost value whch wll be checked e T Δ : decreet/creet o F value, at e ed of terato, accordg to wheer F s feasble or ot, (R). ε, ε : tolerace values for h( x ) ad Δ, respectvely. 7. Refereces terato of e outer loop, (R). outer loop [] eto, JXV., Berert, DLdA., Coelho, LdS., "Iproved quatuspred evolutoary algor w dversty forato appled to ecooc dspatch proble w prohbted operatg zoes", Eergy Coverso ad Maageet, vol.5, pp.8 4, 0. [] ka, T., ara, MR., Azzpaah-Abarghooee, R., "A ew hybrd algor for optal power flow cosderg prohbted zoes ad valve pot effect", Eergy Coverso Ad Maageet, vol.58, pp.97 06, 0. [3] Mohaad-Ivatloo, B., Rabee, A., Soroud, A., Ehsa, M., "Iterato SO w te varyg accelerato coeffcets for solvg o-covex ecooc dspatch probles", Electrcal ower ad Eergy Systes, vol.4, pp.508 56, 0. [4] Lg Wag., Lg-po L., "A effectve dfferetal haroy search algor for e solvg o-covex ecooc load dspatch probles", Electrcal ower ad Eergy Systes, vol.44, pp.83 843, 03. [5] Hosseezhad, V., Babae, E., "Ecooc load dspatch usg θ- SO", Electrcal ower ad Eergy Systes, vol.49, pp.60 69, 03. [6] Kasbeyl, R., Ustu O., Rubov, AM., The odfed subgradet algor based o feasble values, Optzato, vol. 58, (5), pp. 535-56, 009. [7] Fadıl, S., Yazıcı, A., Urazel, B., A soluto to securty costraed o-covex ecooc dspatch proble by odfed subgradet algor based o feasble values, Iteratoal Joural of Electrcal ower ad Eergy Systes, vol.43, pp.849 858, 0. [8] Fadıl, S., Yazıcı, A., Urazel, B., A Securty-costraed Ecooc ower Dspatch Techque Usg Modfed Subgradet Algor Based o Feasble Values ad seudo Scalg Factor for a ower Syste Area Icludg Lted Eergy Supply Theral Uts, Electrc ower Copoets ad Systes, vol.39, pp.748 768, 0. [9] Fadıl, S., Urazel, B., Soluto to Securty-costraed ocovex uped-storage Hydraulc Ut Schedulg roble by Modfed Subgradet Algor Based o Feasble Values ad seudo Water rce, Electrc ower Copoets ad Systes, vol.4(), pp. 35, 03. [0] Jegaeesa R, or M, Role MF. ewto-raphso power flow soluto eployg systeatcally costructed Jacoba atrx. d IEEE Iteratoal Coferece o ower ad Eergy;008:80-6. [] Burachk, RS., Gasov., Isaylova, A., Kaya CY., O a odfed subgradet algor for dual probles va sharp augeted lagraga J. Of Global Optzato, vol. 34, pp. 55-78, 006. []Rubov, AM., Gasov, R., The olear ad augeted lagragas for o-covex optzato probles w a sgle costrat Appled ad Coputatoal Maeatcs, vol, p.4-58, 00. [3] Gasov, R., Augeted lagraga dualty ad odfferetable optzato eods o-covex prograg, Joural of Global Optzato, vol. 4, pp. 87 04, 00.