A Semismooth Inexact Newton-type Method for the Solution of Optimal Power Flow Problem

Transcription

1 Mauscrpt receved May, 7; revsed Aug. 5, 7 A Semsmooth Iexact Newto-type Method for the Soluto of Optmal Power Flow Problem XUE LI, YUZENG LI, SHAOHUA ZHANG Key Laboratory of Power Stato Automato echology Automato Departmet Shagha Uversty Zhabe Dstrct, Shagha 7 CHINA lxue_988@63.com, yzl@mal.shu.edu.c, eeshzha@6.com Abstract: - he paper presets a semsmooth exact Newto-type method for solvg optmal power flow (OPF) problem. By troducg the olear complemetarty problem (NCP) fucto, the Karush-Kuh- ucer (KK) codtos of OPF model are trasformed equvaletly to a set of semsmooth olear algebrac equatos. he the set of semsmooth equatos ca be solved by a mproved exact Leveberg- Marquardt (L-M) algorthm based o the subdfferetal. I the algorthm, the postve deftveess of the teratve coeffcet matrx s ehaced by usg the L-M parameter, whle a reformed omootoe le search s used to eforce global covergece of the algorthm. Fally, the feasblty of the proposed method for solvg the odfferetable problem s verfed o Kojma-Shdo problem, ad the effectveess of the proposed method s demostrated o the IEEE test systems. Key-Words: - exact Leveberg-Marquardt algorthm; olear complemetarty problem; optmal power flow; power system; subdfferetal Itroducto he optmal power flow (OPF) problem has bee commoly used as a effcet tool the power system plag ad operatg []-[3]. Over the years, researchers have examed varous algorthmc techques that see to speed up the OPF computato. Most of the wor doe was captured the 98s ad 99s [4]-[7], a tme whe several optmzato techques, such as reduced gradet techque, quadratc programmg ad Newto methods etc., emerged as the leadg olear programmg (NLP) algorthms for solvg the OPF problem. However, the NLP algorthms are less robust ad ofte experece coverget problems. I recet years, the algorthms based o the teror pot method (IPM), especally the prmal-dual IPM, have bee appled extesvely [8]-[]. he prmaldual IPM has may attractve features, whereas t suffers the drawbac of the requred postvty of slac varables ad ther correspodg dual varables at every terato. More recetly, the olear complemetarty method (NCM) s appled to solve the OPF problem []-[4]. he method, by troducg the olear complemetarty problem (NCP) fuctos, ca mae the Karush-Kuh-ucer (KK) codtos of the OPF model trasform to a set of olear algebrac equatos. he NCM has three appealg features: () ease of hadlg equalty costrats by the NCP fuctos, () ot ecessarly detfyg the bdg costrats ad () the teratos are ot requred to stay the postve orthat. A damped Newto-type method ad a decoupled semsmooth Newto method were proposed to solve the trasformed olear algebrac equatos, respectvely [3]-[4]. However, whe teratve equatos are ll-codtoed, the covergece of two methods ca ot be guarateed. I ths paper, the NCM s used to cope wth the complemetarty codtos of the KK system. By usg a NCP fucto, the KK system of the OPF problem s trasformed equvaletly to a set of the semsmooth olear algebrac equatos. he the semsmooth equatos ca be solved by a mproved exact Leveberg-Marquardt (L-M) algorthm based o the subdfferetal. he proposed method has the followg features: () t ca solve the semsmooth systems formulated by NCP by usg the subdfferetal. () t avods the emergece of the ll-codtog equatos by usg the well-codtoed augmeted coeffcet matrx. () t employs a reformed omootoe le search to speed up the procedure of obtag a seres of OPF solutos. 59

2 Cosequetly, the proposed method has good covergece performace ad computato accuracy, as has bee verfed by the computatoal results. he rest of ths paper s orgazed as follows: Mathematcal foudato of the proposed method s descrbed Secto. he OPF model s formulated ad the NCM s used to trasform the KK codtos to the semsmooth equatos Secto 3. A mproved exact L-M algorthm s proposed for the soluto of the semsmooth equatos Secto 4. I Secto 5, the feasblty of the proposed method for solvg the odfferetable problem s verfed o the Kojma-Shdo problem. he, the effectveess of the proposed method s aalyzed o the IEEE test systems. Cocluso s gve Secto 6. Problem Formulato. B-Subdfferetal Let G: R R be a locally Lpschtza fucto. Hece, G s dfferetable almost everywhere [5]. If we dcate by D G the set where G s dfferetable, we ca defe the B-subdfferetal of G at x [6] as BGx () = lm Gx ( ) () x x x DG Note that the geeralzed Jacoba of Clare C Gx ( ) s just the covex hull of B Gx ( ) [5]. CGx ( ) = cov( BGx ( )) () where cov( A) deotes the covex hull of the set A.. Semsmooth fucto Let G: R R be a locally Lpschtza fucto at x R. We say that G s semsmooth at x [7], f lm { Hv' } (3) H G( x+ tv') exsts for all v R. B v' v, t 3 Problem Formulato he OPF problem ca be show as the followg olear programmg problem m cx ( ) s.. t gx= ( ) (4) hx ( ) where x R s the vector of system varables, cx ( ): R R s the objectve fucto, m g( x): R R represets the olear power flow equatos ad hx ( ): R R p represets several equpmet ad system equalty costrats. he explct OPF model the paper s expressed subsequetly. I ths paper, the objectve fucto of the OPF problem s cosdered as the fuel cost mmzato. m ( apg + apg + a) (5) SG where SG s the set of power geerato; P G s the actve geerato output; a, a ad a are the geerato cost coeffcets. he equalty ad equalty costrats that clude flows, real geerato, reactve geerato, trasformer taps, voltage ad curret of a N-bus power system are cosdered as follows. PG PD P (, e f,) t = Q Q Q (, e f,) t =, =,, N, slac G D m max G G G m max G G G P P P, SG Q Q Q, S G (6 ) m max tj tj tj, (, j) S m max ( V ) ( e + f ) ( V ), =,, N m Ij ( Ij ) ax, (, j) SL where P D : real power load at load bus ; Q D : reactve power load at load bus ; Q G : reactve geerato of geerator ; P : real power jecto at bus ; Q : reactve power jecto at bus ; e : real part of odal voltage at bus ; f : magary part of odal voltage at bus ; t j : trasformer rato of trasformer brach j; V : voltage at bus ; I j : curret at le j; S : set of trasformer braches; S L : set of trasmsso le L. he superscrpt m ad max stad for the lower ad upper bouds of a costrat, respectvely. he Lagrage fucto L() for (4) ca be wrtte as follows: L( w) = c( x) λ g( x) μ h( x) (7) where λ ad μ are the vectors of Lagrage multplers about the equalty costrats ad 6

3 equalty costrats, respectvely. w= [ x, λ, μ ] s a vector of prmal ad dual varables. he KK frst-order codtos of optmalty for (4) ca be wrtte as the followg equatos ad complemetarty codtos: xlw ( ) = (8) gx ( ) = (9) hx ( ), μ, μh =,( =,, p) () where xlw ( ) = cx ( ) J λ K μ, c( x) s the gradet of prmal objectve fucto at the pot x, J ad K are the Jacoba matrces of g( x ) ad hx ( ) about x, respectvely. o deal wth a set of complemetarty codtos (), the NCM s employed by meas of a fucto ϕ : R R, called the NCP fucto, whch s troduced as follows: ϕ ( ab, ) = a + b a b () he NCP fucto () satsfes the basc property: ϕ ( ab, ) = a, b, ab= () Usg (), the complemetarty codtos () ca be expressed as the followg set of semsmooth olear equatos: ϕ ϕ( μ, h( x)) =, =,, p (3) Next, usg (3), equatos (8)-() ca be equvaletly reformulated as the olear system: x Lw ( ) Fw ( ) = gx ( ) = (4) φ( x, μ) where φ( x, μ) = ( ϕ,, ϕ ) p s a semsmooth system, ad the osmooth pots exst f : μ = h ( x) =. { } Step ) Italzato: Set =, ρ >, q>, α (,.5), ε, ad choose a startg pot w. Step ) Stoppg crtero: he atural mert fucto s defed as: ψ ( w) = F( w) F( w) (5) If ψ ( w ) ε, stop; otherwse, go to step ). Step ) Search drecto calculato: Select a B-subdfferetal elemet H BFw ( ) ad the fd a soluto Δw of the system (( H ) H + σ I) Δ w = ( H ) F( w ) + r (6) It ca be smplfed as ( W + σ I) Δ w Z = + r (7) where σ s the L-M parameter ad r s the resdul vector. Set Δ w = ψ ( w ) f the followg codto (8) s ot satsfed ψ( w ) Δw ρ Δ w (8) Step 3) Armjo le search: Fd the smallest {,,, such that } ψ( w + Δw ) ψ( w + α ψ( w ) Δw (9) Step 4) Varables update: he varable w + s updated as follows: + w : w = + Δ w : = + () the go to step ). ) q 4 he Soluto Algorthm A mproved exact L-M algorthm s proposed to solve (4). he prmal exact L-M algorthm s based o the recetly developed theory for solvg semsmooth systems formulated by NCP. Due to the semsmooth fucto (4), the oto of subdfferetal s troduced ths algorthm to determe the search drecto. Oly the approxmate soluto of a lear system s requred at every terato of the algorthm, whch redered the algorthm qute applcable to the large-scale cases. he computatoal procedure of the prmal algorthm ca be summarzed the followg steps. Note that the symbol dcates the Eucldea vector orm or ts assocated matrx orm. 4. Specfyg a B-subdfferetal elemet he matrx H (6) s ay elemet amog B- subdfferetal of F at w [7]. Let β : = { : μ = = h( x) } be the dex set. he, the matrx H s defed by ˆ H J K H = J () ( φ ) K φ ( ( )) ( ) ( ) where ˆ cx J λ K μ H =, x x x φ ( x, μ ) = dag ( A ( x, μ )), φ ( x, μ ) = dag ( B ( x, μ )), φ ad φ are p p dagoal matrces whose th dagoal elemets are gve, respectvely, by 6

4 h ( x), f β A ( x, μ) = ( μ, h( x)) ξ, f β μ, f β B ( x, μ) = ( μ, h( x)) η, f β ( =,, p ) () where ξ ad η whch are the subdfferetal parameters satsfy ( ξ, η) R ad ( ξ, η ) ( =,, p). 4. Improvemet of postve deftveess of the coeffcet Matrx W Note that the equato (7) s always solvable. I fact, If σ =, ( W + σ I) reduces to W, whch s guarateed to be oly postve semdefte. I ths case, provded that H s o-sgular, the equatos (4) s equvalet to the geeralzed Newto equato H Δ w = F( w ) ad s solvable; whe H s sgular, the equatos (4) becomes llcodtoed. If σ >, the ( W + σ I) s always postve defte ad surely solvable. herefore, the postve deftveess of the matrx ( W +σ I) (4) ca be mproved by adjustg the value of L-M parameter σ. 4.3 Improved omootoe le search he Armjo le search (9) s used to eforce global covergece of the algorthm. However, the le search ca lead to very small step szes, tur ths ca brg very slow covergece ad eve umercal falure of the algorthm. o crcumvet the problem, a geeralzato of omootoe Armjo le search s proposed to substtute the le search (9) by the followg expresso. Fd the smallest {,,, } such that ψ( w + Δw ) Ω+ α ψ( w ) Δ w j Ω= max ψ ( w ) (3) j s( ) where s( τ ) m[ s( τ ) +,], τ, s () =. If s( τ ) =, the above omootoe le search reduces to the Armjo le search. he omootoe le search has proved very useful, allowg a cosderable savg both the umber of le search ad the umber of fucto evaluatos [8]. 5 Numercal Examples 5. Feasblty for solvg the odfferetable problem o verfy the feasblty of the proposed approach for the soluto of the odfferetable problem, the Kojma-Shdo problem, whch s the stadard test problem of NCP, s used to llustrate the process. he fucto of the Kojma-Shdo problem s defed by 3x + xx + x + x3 + 3x4 6 x + x+ x + x3 + x4 F( x) = (4) 3x + xx + x + x3 + 9x4 9 x + 3x + x3 + 3x4 3 Accordg to the troduced NCM, the NCP s formulated as x, F ( x), xf ( x ) = (=,,3,4) (5) he assocated NCP has two solutos: x = (.47,,,.5), Fx ( ) = (, 3.47,, ) ; x = (,, 3, ), Fx ( ) = (,3,,4). he soluto x s a degeerate soluto because of x 3 = F 3 ( x ) =. Equatos (5) are trasformed equvaletly to the semsmooth equatos as. ϕ( x, F( x)) ϕ( x, F( x)) = (6) ϕ( x3, F3( x)) ϕ( x4, F4( x)) he equatos are solved by the mproved exact L-M algorthm. I order to demostrate the performace of the proposed method, the comparsos wth the successve quadratc programmg (SQP) algorthm for the osmooth equatos [9] s used. he results are show able. Method Proposed method able Comparso of results Startg Iterato Soluto Pot (,,,) 6 x SQP (,,,) 7 x Proposed method (,,,) 6 x SQP (,,,) 7 x F vectors at the soluto F( x ) F( x ) F ( x ) F ( x ) he solutos obtaed from the two methods are same whe the startg pot are (,,,) ad (,,,), respectvely. However, the proposed method has less computatoal expese tha the SQP algorthm. 6

5 It s worth potg out that the proposed method ca solve the semsmooth equatos (6) whe the ( ) ( ) odfferetal case,.e., x = F( x ) = ( s the terato umber) occurs. he case s show Fg.. he teratve process s gve whe the startg pot s (,,,). Values of x ad F - - x 3 x 4-3 F 3 F Iterato Fg. Iterato process () () From Fg., the expresso x4 = F4 ( x ) = ad (5) (5) x3 = F3 ( x ) = are satsfed, respectvely, amely, () () (5) (5) ( x4, F4 ( x )) ad ( x3, F3 ( x )) are the osmooth pots. Uder the crcumstaces the covetoal soluto method ca ot be used to solve the (6) sce the osmooth pots exst. However, the proposed method ca do t. It ca be see from Fg. that the terato process ca pursue at the osmooth pots. he results from Fg. also dcate that the proposed method ca solve the odfferetable problem. 5. Numercal examples o IEEE test systems A computer program was mplemeted MALAB to solve the OPF problem. he proposed method was tested o three systems,.e., the IEEE 9, 3, 8-bus test systems. he parameters ρ, q, α, ξ ad η 8 were gve as,., 4,.5,.5, respectvely. he values of L-M parameter σ ad Lagrage multplers λ ad μ are show able. Nx, Nl ad Nu deote the umber of the system orgal varables, the umber of Lagrage multplers for the equalty costrats ad equalty costrats, respectvely. he covergece crtero 6 s ε. ε ams at ay below ad s the teratve umber. () ψ( w ) ψ( w ) ; () r < (./( + )) ψ ( w ) ; () the maxmum value of odal ubalace power. I the tests, the cotrbutg covergece crtero, whch has bee verfed by the computer program, s show as = ( w ) ( w ) (7) ε ψ ψ able Iformato of test systems ad tal values est Nx Nl Nu σ λ μ Systems IEEE IEEE IEEE Covergece ad performace he SQP algorthm MAPOWER program exploted by Corell Uversty s used to compare wth the proposed method o the same test systems. he tests are coducted o the Itel CPU.6 GHz wth Gbytes RAM, rug the Mscrosoft Wdows server 3 operatg system. able 3 lsts the performace comparso of two methods computg the formulated OPF. able 3 Comparso of computatoal performaces Iteratve Numbers Executo me(s) est Systems Proposed Proposed SQP Method Method SQP IEEE IEEE IEEE It ca be see from able 3 that the proposed method has less both the teratve umbers ad executo tme tha the SQP algorthm. Wth the system scale creasg, the teratve umbers of the proposed method do ot vary too much whereas that of the SQP algorthm creases drastcally. Furthermore, the SQP algorthm tested o the IEEE 8-bus system fals to solve the OPF problem whch provdes the quadratc cost fucto however the proposed method ca do t. he results from able 3 also dcate that the proposed method shows better performaces. he covergece characterstcs for the IEEE 3- bus system are show Fg.. he results from two le search techques,.e., the Armjo le search (9) ad the mproved omootoe le search (3), are compared Fg.. he former fals to coverge eve whe the teratve umber s large tha oe hudred, whch s caused by the small step szes the process of the Armjo le search. 63

6 However, the latter ca brg the satsfyg results wth the eleve teratos. he results from Fg. llustrate that the mproved omootoe le search s superor to the Armjo le search. the teratve process presets to be coverget effcetly the case of. σ =..8 Crtero.8.6 Crtero Nomootoe Armjo Iterato Fg. Comparso of covergece characterstc wth the Armjo ad omootoe le search for IEEE 3-bus system (5 teratos are gve due to the lmted place) he covergece characterstcs for the IEEE 8-bus system are show Fg.3. Wth the system scale expadg, umercal ll-codtog frequetly occurs owg to the o-postve deftveess of the coeffcet matrx. (7). hus, there are more ucertates covergece for the IEEE 8-bus system tha IEEE 9 ad 3-bus systems. However, the proposed method, the case ca be mproved by the L-M parameter σ (7). I able, the value of parameter σ s. for IEEE 8-bus system, whch ca mprove the postve deftveess of the matrx W. It ca be show from Fg.3 that the teratve process taes o the oscllato ad expereces the coverget problem the case of σ =. eve whe the teratve umber s large tha oe hudred, whereas. σ=. σ= Iterato Fg.3 Comparso of covergece characterstc wth σ =. ad σ =. for IEEE 8-bus system (5 teratos are gve due to the lmted place) 5.. Accuracy he Lagrage multplers of the equalty costrats of OPF problem correspod to the shadow prces of odal power jectos ad have the same ecoomcal sgfcace wth the spot prces. Fg.4 shows the results of the spot prces. It s obtaed by the proposed method ad the SQP algorthm for IEEE 3-bus system. he results of two methods agree so well, whch shows the effectveess of the proposed method Some detals Numercal Smulatos () Equatos (7) are solved by the Gaussa elmato method, whch ca esure the covergece of the proposed algorthm. I addto, factorzato techques are used the process of Gaussa elmato ad sparsty techques are employed to store the odal admttace matrx, 64

7 5 Proposed Method SQP 4 Spot prce($/mwh) Bus o. whch saves the computatoal tme ad mproves the computatoal effcecy. () A rearraged varables sequece about the prmal ad dual varables, smlar to the varables sequece of Newto OPF, s used. After rearragemet, a dagoal sub-matrx of the matrx H (), whch each dagoal bloc s costructed by (4 4) small bloc elemets, s obtaed ad has smlar frame wth the odal admttace matrx. hus, the dagoal sub-matrx ca be stored by the sparsty techques. he rearragemet sequece ca reduce the fll- elemets ad save the memory. Usg the above volved techques, the proposed method ca be exteded to the real sze power etwors. 6 Cocluso hs paper proposed a semsmooth exact Newtotype method for the soluto of the OPF problem. After troducg the NCP fucto, the KK codtos of OPF model are trasformed equvaletly to a set of semsmooth olear algebrac equatos. he the set of semsmooth equatos ca be solved by a mproved exact Leveberg-Marquardt (L-M) algorthm based o the subdfferetal. Numercal studes showed that the proposed method s relable ad better tha some exstg oes. I spte of the fact that the osmooth cases were ot ecoutered o the IEEE test systems, the subdfferetal dea preseted ths paper ca be exteded to cope wth other odfferetal problems the electrcty maret, for example the odfferetal pecewse cost fucto, the real eergy, reactve eergy ad voltage support traded dscrete bds ad offers deregulated marets. Refereces: Fg.4 Comparso of actve spot prces [] H. A. Shayafar, A. Kazem, ad J. Aqhae, Effectve Locato of Facts Devces for Optmal Power Flow Deregulated Systems, WSEAS ra. Crcuts ad Systems, Vol.4, No., 5, pp [] C. Wag, C. Jag, A New Arthmetc of Optmal Power Flow Problem Amg at Iequalty Costrats, WSEAS ra. Crcuts ad Systems, Vol.4, No.8, 5, pp [3] A. Kazem, H. A. Shayafar, et al, A New Method for Optmal Reactve Power Prcg Cosderg Power Losses ad Voltage Profle, WSEAS ra. Crcuts ad Systems, Vol.4, No.9, 5, pp [4] H. W. Dommel, ad W. F. ey, Optmal Power Flow Solutos, IEEE ras. Power App. Syst., Vol.PAS-87, No., 968, pp [5] D. I. Su, et al, Optmal Power Flow by Newto Approach, IEEE ras. Power App. Syst., Vol.PAS-3, No., 984, pp [6] R. C. Burchett, H. H. Happ, D. R. Verath, Quadratcally Coverget Optmal Power Flow, IEEE ras. Power App. Syst., Vol.PAS-3, No., 984, pp [7] J. L. Carpeter, Optmal Power Flows: uses, methods ad developmets. Proc. IFAC Cof., 985, pp. -. [8] R. A. Jabr, A. H. Cooc, ad B. J. Cory, A Prmal-dual Iteror Pot Method for Optmal Power Flow Dspatchg, IEEE ras. o Power Syst., Vol.7, No.3,, pp [9] I. M. Nejdaw, K. A. Clemets, et al, Nolear Optmal Power Flow wth Itertemporal Costrats. IEEE Power Egeerg Revew, Vol., No.5,, pp

8 [] H Wag, C. E. Murllo-Sachez, et al, O Computatoal Issues of Maret-based Optmal Power Flow, IEEE ras. o Power Syst., Vol., No.3, 7, pp [] K. C. Almeda, R. Salgado, Optmal Power Flow Solutos uder Varable Load Codtos, IEEE ras. o Power Syst., Vol.5, No.4,, pp. 4-. [] H. Yag, Y. Zhag, ad X. og. Smooth Nolear Complemetarty Fucto Based Dyamc Modellg of Power Maret. Electrcal ad Electrocs Egeerg. 6 3 rd Iteratoal Coferece. 6. [3] G. L. orres, V. H. Qutaa, Optmal Power Flow by a Nolear Complemetarty Method, IEEE ras. Power Syst., Vol.5, No.3,, pp [4] X. og, Y. Zhag, F. Wu. A Decoupled Semsmooth Newto Method for Optmal Power Flow. Power Egeerg Socety Geeral Meetg, 6, pp. -6. [5] F. H. Clare, Optmzato ad osmooth aalyss, Wley, New Yor, 983. [6] L. Q, A Covergece Aalyss of Some Algorthms for Solvg Nosmooth Equatos, Mathematcs of Operatos research, Vol.8, No., 993, pp [7]. De Luca, F. Facce, C. Kazow, A Semsmooth Equato Approach to the Soluto of Nolear Complemetarty Problems, Mathematcal Programmg, Vol.75, No.3, 996, pp [8] L. Grppo, F. Lamparello, S. Lucd. A Nomootoe Le Search echque for Newto s Method, SIAM Joural o Numercal Aalyss, Vol.3, No.4, 986, pp [9] J. S. Pag ad S. A. Gabrel, NE/SQP: A Robust Algorthm for the Nolear Complemetarty Problem, Mathematcal Programmg, Vol.6, No.-3, 993, pp