Multi-Step Methods Applied to Nonlinear Equations of Power Networks

Transcription

1 Electrcal ad Electroc Egeerg 03, 3(5): 8-3 DOI: 0.593/j.eee Mult-Step s Appled to olear Equatos of Power etworks Rubé llafuerte D.,*, Rubé A. llafuerte S., Jesús Meda C. 3, Edgar Meja S. 3 Departmet of Electrcal Egeerg, Faculty of Egeerg, Uversdad eracruzaa, Cudad Medoza er, 9474, Meco Departmet of electrcalegeerg, Isttuto Tecológco de Orzaba, Orzaba, er, 9430, Méco 3 Departmet of Mechacal Egeerg, Faculty of Egeerg, Uversdad eracruzaa, Cudad Medoza er, 9474, Meco Abstract The power flow calculato power etworks geerally carred out usg teratve methods such as the Jacob method, the Gauss-Sedel method, ad the ewto-raphso method. The authors preset ths paper a vestgato o the methods for solvg olear equatos proposed by other authors, ad are modfed to solve olear equatos geerated the load flow study of a power system ad s obtaed effcet teratve method based o the three steps orgal formula of ewto-raphso method ad mult-step optmal algorthms for the soluto of olear equatos. Keywords Iteratve s, olear Equatos, Electrcal etwork, Power Flow, Mult-Step Algorthms. Itroducto I electrcal egeerg there are dfferet types of studes to determe how they behave dfferet varables whch are referred to as steady state ad traset state. The electrcal eergy used a cty s mafested electrc geerators are desged to geerate ad subsequetly be set to users. The electrcal power trasmtted to the ctes receves the ame of power flow, ths s accomplshed what s kow permaet state ad volves the establshmet of several olear equatos, whch are geerated from the applcato of the Krchhoff's curret law at each of the pots or odes of terest, whch are defed by the aalyst grds. The lterature electrcal egeerg[,] cotas methods that have tradtoally bee used to solve the power flow problem, whch s to calculate the voltages at each grd substato, kowg the demad ad electrc power geerato each of these.the Jacob method: S * (k ) (k) Y Y Y j j () j j j, j For, It's a smple formulato that teratvely calculates the voltage the curret terato ad oly requres the value of the other voltages the prevous terato. The Gauss-Sedel method: * Correspodg author: rubev46@yahoo.com.m (Rubé llafuerte D.) Publshed ole at Copyrght 03 Scetfc & Academc Publshg. All Rghts Reserved Y Y * ( k ) S ( k ) ( k) ( k ) j j j j Y j j j () For, Whch order to accelerate the covergece requres voltage values of the prevous terato, ad voltage values of the curret terato. The ewto-raphso method: f( ) (3) f ( ) Matr formulatos of ths method whch evaluates the jacoba matr formed by the dervatve of the fucto at each ode geerated accordg to voltages ad agles, or fucto of the rectagular compoets of voltages, are used to solve the olear equatos whch volve the study of flow electrc power load. Based o that, there are several compaes that have bee developg commercal software fo r solvg very effcet electrc grds. I a electrcal etwork wth odes depedet set a equato for ode, has the followg form: * S * Y j j (4) j For, Where: S s the comple cojugate et power demaded at ode. Equato (4) ca also be wrtte as: * * S * Y j j ( Y Y,, Y Y ) (5) j

2 Electrcal ad Electroc Egeerg 03, 3(5): Or as a comple fucto, depedg o all the voltages. * * f (,,,, ) S Y j j (6) j The formulas (4) ad (6) are used to solve the olear equatos of a power grd wth two, twety, or more odes.. Iteratve s for Solvg olear Equatos There are the lterature a mportat umber of methods for the soluto of olear equatos, amog whch we ca meto the followg[]: Steffese s method ( f( )) f f f Secat method: ( ( ) ( ) f( ) (8) f( ) f( ) I electrcal etworks the Gauss-Sedel method s used to calculate the voltage at each ode by usg equato [,3]: * (k ) S (k) Y *(k) j j Y (9) j, j I the soluto of a olear equato, due to ther stablty ad covergece the ewto-raphso method [4] has bee wdely used f( ) (7) f ( ) (0) 3. Mult-Step Iteratve s for Solvg olear Equatos May multstep teratve methods for solvg o-lear real equatos have bee proposed, amog whch are the followg: Froze slope method[4]: y z f( ) f ( ) f(y ) y f ( ) f(z ) f ( ) z Weerako ad Ferado method [5]: f( ) f( ) f ( ) f f ( ) () () Kug-Traubmethod [6]: f( ) y 3 f ( ) f( ) z y 6 f ( ) f( ) (3) u z f ( ) 3f (y ) f( ) v u f ( ) 3f (z ) 5f ( ) 3f (y ) f( ) v f ( ) f ( ) Cordero s method [3] f( ) y 3 f ( ) 3f (y ) 3f ( ) f(y ) (4) z y 3f (y ) f ( ) f (y ) f(z ) z 3 f ( ) f (y ) Lag F. algorth ms[8] are ffth order whch the umber of teratos s reduced sgfcatly solvg olear equatos wth real umbers. f( ) y 3 f ( ) (5) f( ) H(,y ) f ( ) Where H(, y ) are as follows: 3f () f (y) H (,y) (6) f () 5f (y) 5f () f ()f (y) f (y) H (,y) (6a) 4f ()f (y) 3f () f (y) H 3(,y) (7) f ()f (y) f (y) 4f () H 4(,y) (8) f () 6f ()f (y) f (y) 5f () 3f (y) H 5(,y) (9) f () 7f (y) 6f () f ()f (y) f (y) H 6(,y) (0) f () 7f ()f (y) f (y) Jarratt s method P. [7]: f( ) y 3 f () () 3f (y ) f ( ) f( ) 3f (y ) f ( ) f ( )

3 30 Rubé llafuerte D. et al.: Mult-Step s Appled to olear Equatos of Power etworks 4. Mult-Step s for Solvg olear Equatos of Power Systems Fgure shows a power system of four odes, the values of the voltages at odes,, 3, ad 4, depeds o the geerated power ad demad each ode or substato. The voltages of each ode ca obta geeratg equato (6), ad f desred usg the method of Jacob, Gauss-Sedel method, or a matr formulato of the ewto-raphso method[]. F gure. System power of four odes To calculate the voltage of ths system, we ca establsh the followg equatos: For ode two: * (k) *(k) f (,, 3, 4) S Y j jj () For ode three: * (k) *(k) f 3(,, 3, 4) S3 Y j 3jj 3 (3) For ode four: * (k) *(k) f 4(,, 3, 4) S4 Y j 4jj 4 (4) I addto, at the four odes we have to calculate: (kk) QQ (kk) IIII (kk) YY jj (5) Yj system costats are calculated usg the seres mpedace, parallel admttace of the trasmsso le, ad are: Y j9078; Y j5.8478; Y 4 0j0 Y j ; Y 3 0j0; Y j Y 3 0j0; Y j ; Y j5.85 Y j5.8478; Y j5.85; Y j s for olear equatos metoed above are appled to solve equatos wth real umbers. The authors refer to ther covergece characterstcs, ad have show a favourable performace. However, the equatos (), (3), (4), ad (5), are comple. All prevous methods have preseted covergece problems. Because t has bee used the followg methods of several steps: yy ff( ) ff ( ) zz yy ff(yy ) ff (yy ) (6) zz ff(zz ) ff (zz ) Because Equato (6), the dervatve s chagg, s called varable slope method.a multstep method s formed wth the orgal formula of ewto-raphso, two terms of the formula of ffth order of Fag Lag, s as follows [8], Other authors have proposed methods wthout the secod dervatve [0,]. f( ) f ( ) 3f ( ) f (y ) f(y ) y f ( ) f (y ) f (y ) 7f ( ) 7f (y ) f(z ) z 6f ( ) 0f ( )f (y ) 6f (y ) y z f (z ) 5. Results (7) We vestgate the performace of fve methods that have bee modfed to determe the voltages for each ode the system show Fgure. Table lsts all methods ad teratos they eed to acheve covergece.we developed a program for each equato vsual Fortra, because ths laguage s easy to work wth comple umbers[]. Table. Iteratos for a four-ode system 4 odes Iteratos e wto-raphso 6 Cordero s method 34 Equato (6) 6 Equato (7) 8 Table shows the odal voltages values calculated for each ofthe methods. The large electrcal etworks geerally have a greatest umber of olear equatos, ad more restrctos. Ths makes the umber of teratos creases, ad causes slower covergece. The fve prevous methods have bee used to solve the olear equatos of a etwork of e odes, eght olear equatos[9]. Fgure shows the etwork of e odes.

4 Electrcal ad Electroc Egeerg 03, 3(5): Ta ble. Fal voltage values oltages ewto- Raphso Cordero s method method Equato (6) Equato (7) [Grager] Fal values j j j j j j F gure. e-odes test system [9] The results of a program developed wth equatos (6) ad (7) are show Table 3. Table 3. alues obt aed wth equatos (6) ad (7) Equato (6) (37 t erat os) Equato (7) (8 t erat os) o Mag Ag. Mag Ag Table 4 also shows the umber of teratos of each method a system of eght lear equatos (9 odes). Table 4 shows the teratos that used the ewto-raphso method, Cordero s method ad Jarrat s method the system of Fgure 6. Coclusos Based o the orgal form of the ewto-raphso method ad multstep methods, ad establshg a mult-step algorthm for solvg olear equatos geerated the steady-state study of electrcal etworks. The orgal equatos[3, 8] were take as the bass for creatg a mu lt-step method that would acheve covergece whe there are power systems wth voltage cotrolled odes. It was cosdered the establshmet of a fucto at each etwork ode that depeds o several varables, stead of creatg a system of equatos where the matr mapulato s usually very elaborate. Po wer systems so far are of moderate sze ad almost ddactc purposes; however, let you kow a fast steady-state behavour of a electrcal etwork. Its applcato s certaly possble to other felds of egeerg.we are smulatg the behavour of methods thrd ad ffth order, whch have prelmary results that wll be reported other work later. Table 4. Iteratos for a system of e odes 9 odes Iteratos e wto-raphso 98 Cordero s method 80 The soluto of olear equatos wth the classcal formula of ewto-raphso method s smple, although the umber of teratos to acheve covergece creases whe the equatos are umerous. Usg hgh-order formulas such as the Cordero, ad Goupg Lag Fag He, ad adaptg, accelerates the covergece solvg systems of olear equatos o a grd, as show Table 3. REFERECES [] J. J. Grager ad W. D. Steveso Jr, Aálss de sstemas de poteca, McGraw-Hll, 996. [] Jzhog Zhu, Optmzato of Power System Operato, Wley/IEEE, 009. [3] Cordero A., Hueso, J. L., Martíez, E. Terregosa J. R., A Modfed ewto- composto, umercal Algorthm, 55 (00), [4] Hueso J. l.,martíez E., Terregrosa, J. R., Thrd order teratve s free from secod dervatve for olear systems, Appled mathematcs ad Computato, (009),

5 3 Rubé llafuerte D. et al.: Mult-Step s Appled to olear Equatos of Power etworks [5] Weerako S., Ferado T. G. I, A varat of ewto s method wth accelerated thrd-order covergece, Appled Mathematcs Letters, 3(8) (000)87-93 [6] Kug H. T., Traub J. F., Optmal order of oe-pot ad multpot terato, Joural ACM, (974) [7] Jarratt P., Some fourth order multpot methods for solvg equatos, Mathematcs ad Computatos, 0 (966) [8] Lag F. ad Goupg He, Some modfcatos of ewto s method wth Hgher-order covergece for solvg olear equatos, Joural of Computatoal ad Appled Mathematcs 8(008) [9] P. M. Aderso & A. A. Fouad, Power System Cotrol ad stablty, d edto IEEE Press Power Egeerg Seres, Wley-Iterscece, 003. [0] Muhammad Aslam oor, Fourth-Order Iteratve Free from Secod Dervatve for Solvg olear Equatos, Appled Mathematcal Sceces, ol. 6, 0, o. 93, [] Gustavo Feradez Torres ad Jua azquez Aquo, Three ew Optmal Fourth-Order Iteratve s to Solve olear Equatos, Advaces umercal Aalyss, olume 03, (03) Artcle ID , 8 pages. [] sual Fortra 90, User Gude.