A novel analysis of Optimal Power Flow with TCPS

Transcription

1 IOSR Joural of Electrcal ad Electrocs Egeerg (IOSR-JEEE) e-issn: ,p-ISSN: , Volume 10, Issue Ver. I (Mar Apr. 015), PP S.Saar* S.Saravaaumar **, C.T.Maada***, M.Padmarasa*** *Professor, Dept of EEE,Pamalar Isttute of Techology, Chea, ** Professor, Dept of IT,Pamalar Isttute of Techology, Chea, ***Assstat Professor(G-I), Dept of EEE,Pamalar Isttute of Techology, Chea saravaaumars81@gmal.com, ssaarphd@yahoo.com, padmaras_math@yahoo.com Abstract: Optmal power flow (OPF) s oe of the most mportat algorthms avalable to utlty for geeratg least cost geerato patters a power system satsfyg trasmsso ad operatoal costrats. A wde varety of covetoal techques are avalable to obta soluto. I day to day lfe, the forecasted loads used classcal OPF algorthms are creasg wth tme ad are also ot completely free from errors. By varyg the load demads leads to overloadg of the trasmsso les ad forecasted errors cause loss of optmal system. So ths moder optmal power flow algorthms may ot be able to provde optmal solutos. Ths paper presets soluto of optmal power flow problem for 30bus system va a smple geetc algorthm aalyss. Our obectve s to mmze the fuel cost ad eep the power outputs of Alterators, Le bus voltages, le shut capactors/reactors ad trasformers tap-settg ther secure lmts. Keywords: FACTS; Geetc algorthm; Optmal power flow. I. Itroducto The Optmal Power Flow (OPF) problem was troduced by Carpeter 196 as a etwor costraed ecoomc dspatch problem. OPF formulato s amed to mmze the operatg costs whle satsfyg costrats cludg voltage lmts, geerato capablty etc. It determes the optmal settg of geerato operatg uts. It s ecessery mportace to solve ths problem as qucly ad accurately as possble [1]. There are a umber of covetoal methods le Lambda terato method, Gradet method, Newto s method etc []. However, these methods suffers from some shortages le eormous computatoal efforts ad tme cosumpto, sestveess to startg pots, frequet covergece to local optmum soluto, o applcable wth dscrete varables etc[3]. It s well ow that FACTS devces ca mprove the steady state as well as traset performace of power systems. The degree of o covexty o OPF problems are further creased wth the cluso of FACTS devces o the etwor ad the usual covetoal methods wll ot produce optmal result. Hece, t s ecessary to employ a ew, relable ad moder method. It s thus ecessary to solve OPF usg oe of the moder methods- Geetc Algorthm. GA, veted by Hollad [4] the early 1980s, s a stochastc global search method that mmcs the metaphor of atural bologcal evaluato. GAs are a attractve alteratve to other optmzato methods because they overcome the lmtatos of the covetoal methods ad are robust [5]. II. Optmal Power Flow Problem Formulato The mathematcal formulato of the OPF problems s a well ow optmzato problem. The basc formulato of ay optmzato ca be represeted as mmzg a defed obectve fucto subect to ay physcal or operatoal costrats of the system. Hece a optmzato problem cossts of obectve fucto subected to equalty costrats ad equalty costrats. The ma obectve s to mze the total potetal eergy of water stored all the reservors. The formulato must tae to accout the fact that the water stored oe reservor wll be used all ts dowstream reservors, hece, the water stored the upstream reservor s more valuable tha that stored the dowstream reservor [1]: Dr.S.Saar s a Professor Pamalar Isttute of Techology, Chea ssaarphd@yahoo.com Dr.S.Saravaaumar s a Professor Pamalar Isttute of Techology, Chea.saravaaumars81@gmal.com DOI: / Page

2 -1. Obectve fucto: f E ( x ) E ( u ) 1 f P p m a Where: : Last perod of the plag horzo. f f Ep( x ): Potetal eergy of water stored reservor at the ed of the plag horzo f. Ths eergy depeds o the amout of water x f stored the reservor, o ts effectve water head ad o the effectve water head of the dowstream reservors. Ep ( um ) : Total potetal eergy of the released water u m from reservor m, whch wll reach later the dowstream reservor after the last perod of the plag horzo. a f Sm m : The reservor mmedately precedg the reservor. S : Durato perod for the water dscharged from reservor m to reach ts drect dowstream reservor, m hours. -. Operatoal costrats: The prcpal operatoal costrats of the system are the followg [1-, 5-8]: - Hydraulc cotuty costrat: For each reservor at each tme perod, the followg costrat establshes the water balace equato: x x y u v 1 Where: x : Cotet of the reservor at perod, Mm 3. u : Turbe dscharge of hydroelectrc plat perod, Mm 3. v : Spllage of hydroelectrc plat perod, Mm 3. y : Total flows to reservor perod, Mm 3. Tag to accout hydraulc couplg, the total flow to reservors s determed as follows: S y q u m m q : Natural flows to reservor perod, Mm 3. S u m m perod : Turbe dscharge from hydroelectrc plat m, whch wll reach later the dowstream reservor at S. m - Mmum ad mum storage lmts: x x x x, x : Lower ad upper lmts o reservor storage capacty, respectvely, for reservor, Mm 3. - Mmum ad mum dscharge lmts: u u u u, u : Mmum ad mum lmts o water dscharge, respectvely, of hydroelectrc plat. - Demad-geerato balace: The total power geerated by all the hydroelectrc plats must satsfy the system load demad at each perod of the plag horzo. I mathematcal terms, ths has the followg form: P 1 Where: D DOI: / Page f

3 D : The demad for electrcal power at each perod, Mw. P : Electrc power geerated by hydro plat at perod, Mw. : Number of reservors of the system. III. Soluto method I mathematcal terms, we formulate the short term schedulg problem of a hydroelectrc plat system as follows [1]: f E ( x ) E ( u ) 1 f (1) P p m a Subect to the followg costrats: 1 x x y u v () P 1 0 x 0 D (3) x (4) u u (5) v 0 (6) Ths problem ca be solved by applyg the dscrete mum prcple as follows [1, 9-10]: Assocate the costrat () to the crtero (1) wth the dual varable.to satsfy the balace betwee electrc eergy demad ad geerato we assocate the costrat (3) to the crtero (1) wth the Lagrage multpler form: 1, ad the we defe the fucto H, called the Hamltoa fucto, whch has the followg sm H [ ( x y u u u )] m m ( P D ) The problem (1)-(6) becomes: H (7) Subect to costrats (4)-(6), wth the coverso equato of the assocate varable [9]: 1 H 1 x (8) Whe costrats (4), (5) ad (6) are actve, the optmal traectory u wll be reached whe the followg optmalty codtos for each hydroelectrc plat ad at each perod are satsfed: H 0 (9) u I order to solve these equatos, we eeds to ow the operatg lmts, whch are: - The frst oe s the tal state, whch s specfed,.e., at the tal tme, the tal cotet of all reservors are ow, thus: 0 x b (10) - The secod oe s the termal codto for the ad ot equato: f E ( ) f p x (11) f x Cosequetly, equatos () ad (8)-(11) costtute a two-pot boudary value problem whose soluto determes the optmal state ad cotrol varables. Ths problem s solved teratvely by usg the gradet method. To tae to accout possble volatos of costrats (4) ad (5) we proceed respectvely as follows: DOI: / Page

4 - If the value of some u whch satsfes the optmalty codto (8) volates the costrat (5), we ll fx them to ther boudary lmts, the others are left free. The a ew research for the optmum s made but wth oly the free varables. - To deal wth possble volatos of costrat (4), we use the augmeted Lagrage method [1,3,11], whch cossts addg a fucto R to the Hamltoa H, whch pealzes the volato of costrats (5). The the Hamltoa H becomes: H [ ( x y u )] 1 ( P D ) R The fucto R s defed as follows: (1) R r( ) (13) Where: r : Pealty weght. : Lagrage multplers, updated as follows: r ( x x, ) (14) r The operatg fucto s determed as follows: ( x x, ) r (15) IV. Model Of TCPS Fgure 1 shows the schematc dagram of a TCPS. The seres trasformer ects a voltage seres wth the system. The actve ad reactve power ected by the seres trasformer s tae from the shut coected trasformer. Here the losses the trasformers ad the coverter are eglected. Thus the et complex power (actve ad reactve power) exchage betwee the TCPS ad the system s zero. The ected complex power of the seres trasformer depeds o the complex ected voltage ad the le curret. Fgure shows the equvalet crcut of Fgure 1, where V s ad V sh represet by the voltage of the seres ad shut trasformer, respectvely. X s ad X sh represet the leaage reactace of the seres ad shut trasformers, respectvely. X s represets the leaage reactace see from the prmary sde of the seres trasformer [7] ad s gve by X s = X s + X sh (16) where s the turs rato of the shut trasformer The shut voltage source ad the assocated leaage reactace X sh ca be represeted by a shut ected curret source (I sh ) as show Fgure 3. The shut ected curret has two compoets: phase compoet (I p ) ad quadrature compoet (I q ) wth respect to the bus voltage V m. Thus I sh ca be expressed as m I I p I e (17) Sh q Fg.1.TCPS schematc dagram DOI: / Page

5 Fg..TCPS equvalet crcut Fg.3.TCPS curret ecto model. V. Severty Idex Here ths system operatg uder ormal ad cotgecy cases ca be descrbed by a real power le flow performace dex as 1 N (18) s lm PI wm P m Plm where s the real power flow ad s the rated capacty of le-m, s the expoet ad a real Plm Plm wm oegatve weghtg coeffcet whch may be used to reflect the mportace of le coected system. I ths operatg PI wll be small whe all the les are wth ther lmts ad reach a hgh value whe there are hgh levels operatg loads. Thus, t provdes a good measure of severty of the le overloads for a gve state of the power system. I solvg the OPF problem, two types of varables eed to be determed by the optmzato algorthm: geerator actve power geerato P g ad geerator bus voltages V g whch are cotuous varables ad trasformer tap settg t whch are dscrete varables. The dscrete varables are the formulato due to the dscrete ature of the trasformer tap postos. Covetoal methods are ot effcet hadlg problems wth dscrete varables. I ths paper Geetc Algorthm s appled to solve the OPF problem. Fg.4.Flowchart of GA-OPF Algorthm. DOI: / Page

6 Whle applyg GA, to solve the OPF problem, ether t ca be used to obta the optmal values of all the cotrol varables or to search the dscrete varables aloe, leavg the cotuous varables to be searched by the covetoal method. I the frst approach, the dvduals the GA populato cossts of bary strgs correspodg to all the varables ad the ftess value of each dvdual s evaluated by rug the power flow algorthm usg the cotrol varables represeted by the dvdual. The the geetc operators are appled o the GA populato. Ths process s cotued utl the covergece crtero s satsfed. Ths s pctorally represeted Fgure 4. I the secod approach, oly the trasformer tap settg whch s a dscrete quatty aloe s coded as the dvdual of the GA populato. I ths case each dvdual s evaluated by rug the optmal power flow program usg the trasformer tap settg represeted by the dvdual. Thus ths approach, the covetoal LP- based algorthm searches the cotuous varables ad GA searches the dscrete varables. Populato represetato ad talzato Ftess evaluato ad applcato of geetc operators eed to be addressed whle applyg GA for the OPF problem A GA wors o chromosomes, whch are strgs of zeros ad oes. Implemetato of a problem a GA starts from the parameter ecodg (.e., the represetato of the problem).the ecodg must be carefully desged to utlze the GA s ablty to effcetly trasfer formato betwee chromosome strgs ad the obectve fucto of problem. Each dvdual the populato represets a caddate OPF soluto. The elemets of that soluto cosst of all the cotrol varables the system. For the OPF problem uder cosderato, geerator actve power P g ad geerator termal ode voltages V ad trasformer tap settgs are the cotrol varables P P 3 P V 1 V V T 1 T T I the OPF problem uder cosderato, the obectve s to mmze the total fuel cost satsfyg the costrats. For each dvdual, the equalty costrats are satsfed by rug the Newto Raphso power flow algorthm ad the costrats o the state varables are tae to cosderato by addg a quadratc pealty fucto to the obectve fucto. Wth the troducto of pealty fucto the ew obectve fucto becomes, N l N g N l (19) M f F SP T VP QP LP Here SP, VP, QP ad LP are the pealty terms for the referece bus geerator actve power operatg lmt volato, load bus voltage lmt offece; ths reactve power geerato lmt volato ad le flow lmt volatg system respectvely. These quattes are defed by the followg equatos: K f P P (0) s Ps Ps s s m m SP K P P f P s s Ps VP QP 0 0 v s s K v V V K V V 0 q K q Q Q K Q Q otherwse f V V m f V V otherwse f Q Q m f Q Q otherwse m m (1) () DOI: / Page

7 LP K l 0 S S l l f S l S l otherwse (3) Where K s, K v, K q ad K l are the pealty factors. The success of the approach les the proper choce of these pealty parameters. Usg the above pealty fucto approach, oe has to fd a correct combato of pealty parameters K s, K v, K q ad K l. However, order to reduce the umber of pealty parameters, ofte the costrats are ormalzed ad oly oe pealty factor R s used. Sce GA mzes the ftess fucto, the mmzato obectve fucto s trasformed to a ftess fucto to be mzed as, (4) Ftess f Where s a large costat. VI. Results Ad Dscussos The frst part deals wth solvg Optmal Power Flow problem for a 30- bus test system wthout FACTS devces usg Geetc Algorthm. The cotrol parameter settgs for solvg OPF wthout FACTS devces are umber of geeratos (= 60), Populato sze (= 50), Cross over rate (= 0.6), Mutato rate (= 0.05). Table: Cotrol Parameter Settgs For Opf For A 30 Bus System No. of Populato Crossover Mutato geeratos sze rate rate We cosder real power of slac bus, reactve power of geerator bus, voltage of load bus ad power flow through the braches as the costrats. The graph showg geerato versus ftess s show Fg. 5. The mmum cost of $/hr s obtaed wth the followg values Table 1 for real powers ad voltages. Fg.5.Geerato versus ftess for a 30 bus system DOI: / Page

8 Table real power ad voltage for geerator buses P P 6.86 P 3 3. P P P 6.66 V V V V V V Table: cotrol parameter settgs cludg tcps Le No. of Populato Crossover Mutato outages geeratos sze rate rate It s see that, o the outage of some of the braches, the power flow through the braches gets volated. We cosder outage of braches 1-, 1-3 ad 3-4. I order to releve these les from overloadg, we place oe of the FACTS devces, TCPS approprate postos, whch are determed by sestvty aalyss. The problem represetato the wll have a addtoal costrat whch s the phase shftg trasformer costrat. The cotrol varables wll the be the real power of geerator ad phase agle of the TCPS[9]. The umber of cotrol varables, ther rage whch specfes the mmum lmt ad mum lmt, umber of bts etc are cluded the GA codg. Table: smulato result o cludg tcps for varous le outages Le outage P P P P P P X TCPS X TCPS X TCPS X TCPS SI Matpower solves power flow ad provdes real power of slac bus, reactve power of geerator bus, voltage of load bus ad power flow through the braches. The settg of cotrol parameters for varous le outages cludg TCPS are gve Table. The value of real power, phase agle of TCPS ad cost are gve Table 3. It s see that the le overloads were releved through the adustmet of phase agle of TCPS. VII. Cocluso The mmum cost obtaed wthout cludg FACTS devces was ear to the cost obtaed by usg Gradet method. We ca coclude, the proposed wor has gve a better global soluto. I addto, t saves computatoal tme as well as computer memory ad produce most optmal result. The ext part deals wth the securty ehacemet. I the base case, we see that the real power of the slac bus, reactve power of the geerator bus, voltage of the load bus ad power through the braches are wth the lmts. However, o le outages, some of the les get overloaded. FACTS devces are cluded order to releve les from overloadg. By proper postog of TCPS, the les get releved from overloadg ad hece the severty dex s obtaed to be zero. DOI: / Page

9 Refereces [1]. M. S. Osma, Abo-Sa M. A. ad Mousa A. A., A soluto to the optmal power flow usg geetc algorthm, Appled mathematcs ad computato, vol. 155, pp , 01. []. W. Herma Dommel ad F. Wllam Tey, Optmal Power Flow Solutos, IEEE Tras. Power apparatus ad systems, October 000. [3]. M. Youes, M. Rahl ad L. Abdelhaem- Korda, 007, Optmal Power Flow Based o Hybrd Geetc Algorthm, Joural of Iformato Scece Ad Egeerg 3, pp [4]. J. H. Hollad, Adaptato atural ad Artfcal Systems, The uversty of Mchga Press, A Arbor, USA, 01. [5]. K. Vayaumar, R. P. Kumuddev, D. Suchtra, A Hybrd Geetc Algorthm for Optmal Power Flow Icorporatg FACTS Devces, IEEE Computer Socety Washgto, DC, USA. [6]. D.E. Goldberg, Geetc Algorthms search, optmzato, ad Mache Learg, Addso Wesley Publshg Compay,013. [7]. T. S. Chug, Y. Z. L, A hybrd GA approach for OPF wth cosderato of FACTS devces, Power Egeerg Revew, IEEE, Volume 0, pp.54 57, Aug 009. [8]. Z. Mchalewcz, A survey of costrat hadlg techques evolutoary computato methods, Proceedgs of the 4th Aual Coferece o Evolutoary Programmg, pp , 01. [9]. Tare Boutr, Lda Slma, M. Belacem, A Geetc Algorthm for Solvg the Optmal Power Flow Problem, Leoardo Joural of Sceces Issue 4, Jauary-Jue 014. [10]. O. Alsac, B. Scott, Optmal Load Flow wth Steady State Securty, IEEE Trasactos, PAS-93, pp , 004. DOI: / Page