Rotational Load Flow Method for Radial Distribution Systems

Transcription

1 Insttute Insttute of of Advanced Advanced Engneerng Engneerng and and Scence Scence Internatonal Journal of Electrcal and Computer Engneerng IJECE) Vol. 6, No. 3, June 016, pp ISSN: , DOI: /jece.v Rotatonal Load Flow Method for Radal Dstrbuton Systems Dego Isscaba * and Jorge Coelho ** * Department of Electrcal Engneerng, Federal Unversty of Technology - Parana UTFPR), Curtba-PR, Brazl ** Department of Electrcal Engneerng, Federal Unversty of Santa Catarna UFSC), Floranopols-SC, Brazl Artcle Info Artcle hstory: Receved Feb 5, 016 Revsed Mar 4, 016 Accepted Apr 5, 016 Keyword: Power engneerng Power dstrbuton systems Load flow analyss Radal networs Coordnate rotaton ABSTRACT Ths paper ntroduces a modfed edton of classcal Cespedes load flow method to radal dstrbuton system analyss. In the developed approach, a dstrbuton networ s modeled n dfferent complex reference systems and reduced to a set of connected equvalent subnetwors, each wthout resstance, whle graph topology and node voltage soluton are preserved. Actve power losses are then not dsspated n the modeled subnetwors and actve power flows can be obtaned as a consequence of radalty. Thus, the proposed method preprocesses a seres of varable transformatons concomtant to an teratve algorthm usng a forward-bacward sweep to arrve at the load flow soluton. The proposed approach has been tested usng lterature and actual dstrbuton networs, and effcency mprovements are verfed n comparson to Cespedes load flow method. Copyrght c 016 Insttute of Advanced Engneerng and Scence. All rghts reserved. Correspondng Author: Dego Isscaba Department of Electrcal Engneerng, Federal Unversty of Technology - Parana UTFPR) Av. Sete de Setembro, 3165, Sector D, Rebouas, Curtba-PR, Brazl sscaba@utfpr.edu.br 1. INTRODUCTION Load flow analyss [1, ] provdes the steady-state condton for power systems and s one of the most mportant numercal tools to system plannng and desgnng. In order to desgn solutons to power dstrbuton system real-tme operaton [3], modelng and analyss mght tae nto account unbalanced operaton and detaled features of each component connected to the networs. On the other hand, n long-term plannng stages, some hypotheses and smplfcatons can be assumed e.g. balanced operaton, constant power load) due to, for nstance, uncertantes regardng future load profles and dstrbuted generaton productons. Ths s the case n adequacy evaluatons [4, 5], hundreds of thousands of load flow analyss may be executed, whle performance ndces are estmated by modelng falure/repar cycles of components and load/generaton profles as stochastc processes. Detaled dstrbuton system modelng and analyss s a well establshed topc, covered n boos such as [6]. Smplfed/sngle-phase load flow modelng and analyss dates mostly to the 80 s and 90 s and can be dvded nto two groups: the frst group comprses Newtonan based methods adjusted to dstrbuton system analyss [7, 8] and the second group ncludes methods based on teratve forward bacward sweep processes [9, 10, 11, 1, 13, 14]. The sweep based technques are nown by ther effcency and tae advantage of the fact that dstrbuton networs usually are radally operated. Forward bacward sweep methods employ the followng general procedures: a) assumng a flat start or an approxmate soluton, current or power downstream each node s estmated n a bacward sweep from end nodes towards the substaton node); b) usng estmates from the prevous step, node voltages are updated n a forward sweep from the substaton node towards end nodes); c) these two steps are repeated up to the convergence of node voltages. Among the several varatons of sweep technques, the one proposed by Baran and Wu [11, 1] stands out by employng a set of equatons, nown as Dstflow branch equatons, that recursvely relates actve power, reactve power and voltage magntudes. Cespedes [9] steps ahead by addressng the use of a bquadratc equaton to relate node voltages as functons of downstream actve and reactve power flows and mmedate upstream node voltages. Ths paper brngs out the fact that, although Cespedes method s effcent and robust to dstrbuton system analyss, t nvolves a non-neglgble amount of unnecessary calculatons assocated to updatng actve and reactve Journal Homepage: w w w.. a a e e s s j j o u r r n a a l l.. c c o m

2 IJECE ISSN: power flows n networ branches. In fact, a fnte and relatvely small number of standardzed conductor types are utlzed n each dstrbuton feeder, composng subnetwors wth the same r+jx Ω/m and r /x rato. As consequence, gven a possble teratve load flow soluton, there s a constant relaton between actve and reactve losses at each subnetwor branch, mang the concomtant effort of updatng both, unnecessary. The proposed approach, of the Rotatonal Load Flow Method, goes beyond Cespedes formulaton by performng generalzed complex reference system rotatons and by tang advantage of radalty to mprove the effcency of sweep procedures. The approach has been tested usng academc and actual electrcal dstrbuton networs, such that effcency mprovements are verfed n comparson to Cespedes load flow method. The paper s organzed nto fve sectons as follows. Secton and 3 present mathematcal bacground and the proposed method, respectvely. Secton 4 shows numercal results for dfferent test systems and an actual system. Conclusons and fnal remars are outlned n secton 5.. MATHEMATICAL BACKGROUND Ths secton presents a bref revew of the load flow formulaton developed by Cespedes [9]. A dscusson about complex reference system rotaton, focused on dstrbuton system analyss and applcatons, s also approached..1. Cespedes Load Flow Formulaton Consder the radal dstrbuton system schematc shown n Fg. 1, dstrbuton lnes are modeled as seres mpedances Z = z φ = r + jx whle complex voltages and complex load demands are represented, respectvely, by V = v θ and S L = s L δ L = P L + jq L, = 1,..., N. Note that the ndex s utlzed to dentfy the node and the lne upstream ths node, dependng on the varables nvolved. In addton, u denotes the node upstream node. S V u Z V Z +1 V +1 V N 1 Z N V N }{{} branch N S L S L+1 S LN 1 S LN S Fgure 1. Radal dstrbuton networ schematc. The complex power flow njected at node through an upstream branch, denoted by S = s δ, can be wrtten as a functon of downstream node voltages, loads and lne losses. Ths power flow can be also nterpreted as an accumulated power at node gven by S = S L + S L + Z s v ) 1) or, separatng n real and magnary terms, P = P L + P P L + r + Q ) v, Q = Q L + Q L + x P + Q ) v ) denotes the set of nodes downstream node and the set of lnes downstream node, dependng on the varables nvolved. Once accumulated actve and reactve powers at node are nown, the voltage magntude at node can be obtaned by solvng the bquadratc equaton [9] v 4 + A v + B = 0 3) A = P r + Q x ) vu, B = P + Q ) r + x ) Smlarly, the angle at node can be obtaned through the followng equaton. 4) Rotatonal Load Flow Method for Radal Dstrbuton Systems Dego Isscaba and Jorge Coelho)

3 1346 ISSN: θ = θ 1 + tan 1 P x Q r P r + Q x + v Thus, assumng that the substaton node complex voltage V 0 = v 0 0 o s specfed and node voltage magntudes are ntated usng a flat start or an approxmate soluton, the load flow soluton can be found through teratve forward bacward sweeps as follows. In a bacward process, accumulated power flow at each node s calculated usng ), startng at end-nodes and stoppng at the frst node mmedately downstream from the substaton node. In a forward process, voltage magntudes are updated, away from the substaton node, usng 3) 4). These procedures are contnually repeated untl the convergence of voltage magntudes s reached. The bacward process s the most tme consumng part of the algorthm and nvolves the calculaton of power losses and power accumulaton through networ branches. However, a fnte and relatvely small number of standardzed conductor types are nstalled n a dstrbuton networ, each one wth a specfed r /x rato. These ratos can be mathematcally controlled through a proper reference system rotaton, as t wll be exposed n the followng subsecton... Complex Reference System Rotaton Assume an mpedance Z represented n an alternatve complex reference system rotated by ϕ, as llustrated n Fg.. ) 5) I ϕ I x Z ϕ = r ϕ +jxϕ x ϕ Z = r +jx R ϕ r r ϕ ϕ R Fgure. Impedance represented nto two dfferent complex reference systems phased by a ϕ angle. The complex value of the mpedance n the alternatve reference system can be obtaned by Z ϕ = Z e jϕ = r ϕ + jxϕ 6) r ϕ = r cos ϕ x sn ϕ, x ϕ = r sn ϕ + x cos ϕ 7) and the updated r /x rato gven by r ϕ x ϕ = r cos ϕ x sn ϕ r sn ϕ + x cos ϕ for x ϕ 0 8) Analogously, a complex apparent load n the alternatve reference system s gven by S ϕ L = S L e jϕ = P ϕ L + jq ϕ L 9) P ϕ L = P L cos ϕ Q L sn ϕ, Q ϕ L = P L sn ϕ + Q L cos ϕ 10) Once all mpedances and loads are rotated by 6) and 9), the system model s represented n the alternatve reference system. Hence, the complex power njected at node through an upstream branch s also rotated by ϕ, such that S ϕ = S e jϕ = P ϕ + jq ϕ 11) IJECE Vol. 6, No. 3, June 016:

4 IJECE ISSN: P ϕ = P cos ϕ Q sn ϕ, Q ϕ = P sn ϕ + Q cos ϕ 1) Snce the system model s represented n the alternatve reference system, the load flow voltage soluton s also rotated by ϕ, though voltage magntudes reman unaltered. In fact, usng 3) 4), the bquadratc voltage equaton can be rewrtten n the alternatve reference system as v ϕ )4 + A ϕ vϕ ) + B ϕ = 0 13) A ϕ = P ϕ rϕ + Qϕ xϕ ) vϕ u ), B ϕ = P ϕ ) + Q ϕ )) r ϕ ) + x ϕ )) 14) By substtutng 7) and 1) n 14), as well as by assumng vu ϕ = v u, we obtan that A ϕ = A and B ϕ = B. As consequence v ϕ = v, causng that voltage magntude solutons are preserved. Once the same voltage value s chosen for the substaton node n both reference systems,.e. v ϕ 0 = v 0, the condton vu ϕ = v u s always satsfed because of the order n whch the nodes are vsted n the ladder teratve procedures. Therefore, a networ represented n the alternatve reference system and other represented n the conventonal reference system present both the same load flow voltage magntudes, ndependently of the rotaton angle chosen. In addton, the load flow procedures wll present the same voltage terates n both reference systems. One mportant applcaton of reference system rotaton s that a dstrbuton networ can now be represented n a coordnate system the r /x ratos assume low fgures, mprovng the Newtonan based load flow algorthms n terms of robustness and effcency. The proposed approach goes beyond such concept by ntroducng generalzed complex reference system rotatons to tae advantage of dstrbuton networ radalty n Cespedes ladder teratve formulaton. 3. ROTATIONAL LOAD FLOW METHOD Ths secton concerns the proposed approach and s organzed for ddactc reasons nto two cases: a case comprsng networs wth a sngle conductor type and a general case wth dfferent conductor types nstalled through dstrbuton networs Sngle cable type case Consder a networ wth only one conductor type nstalled. By hypothess, complex mpedances present the same r /x rato and φ angle. As exposed prevously, r /x ratos can be controlled by a complex reference system rotaton. For the general schematc shown n Fg. 1, by convenently choosng the rotaton angle ϕ = π φ and by Krchhoff s current law we obtan V u V = Z ϕ S ϕ V ) S ϕ ) = jz 15) V Moreover, usng 1) and 11), the rotated accumulated complex power at node can be computed by S ϕ = S L e jϕ + S L e jϕ + Z e jϕ s v ) = S ϕ L + S ϕ L + jz s v ) 16) Separatng the real and magnary terms, and denotng L ϕ Q as the rotated reactve networ losses at branch, we have Q ϕ P ϕ = P ϕ L + = Q ϕ L + L ϕ Q = P ϕ L 17) Q ϕ L + L ϕ Q 18) z s v ) 19) Usng the varable transformaton, the accumulated actve power at node can be drectly obtaned summng rotated actve load demands downstream from node u. Therefore, there wll not be any actve losses found and actve Rotatonal Load Flow Method for Radal Dstrbuton Systems Dego Isscaba and Jorge Coelho)

5 1348 ISSN: power flow calculaton through networ branches becomes straghtforward. Consequently, load and networ data can be stored n a convenent coordnate reference system and the sweep algorthm can now be formulated to concern only the calculaton of reactve losses, as descrbed n Fg. 3. 1: Read node and networ data, both rotated by ϕ = π φ, usng 6) and 11); : Intate node voltages assumng v = v 0, = 1,..., N; 3: Compute accumulated actve power usng 17) at all networ nodes; 4: whle Changes n v tolerance, do 5: for all, followng the bacward drecton do 6: Calculate accumulated reactve power at networ nodes usng 18); 7: end for 8: for all, followng the forward drecton do 9: Obtan node voltage magntudes startng from the frst node usng 1) 0); 10: end for 11: end whle 1: Prnt result reports. Fgure 3. Rotatonal Load Flow algorthm Sngle conductor type case wth constant power load modelng. The bquadratc equaton becomes smplfed by removng resstances from the formulaton, as follows. v = A ϕ + A ϕ ) 4B ϕ ) 1/ )1/ 0) A ϕ = x ϕ Qϕ v u, B ϕ = P ϕ ) + Q ϕ )) x ϕ ) 1) By elmnatng all these calculatons, non-neglgble CPU tme can be saved. 3.. General case Consder a general dstrbuton networ wth lateral branches and voltage dependent load demands. Notce that ths networ can be dvded nto a set of connected subnetwors wth the same conductor type. These subnetwors can be represented n a convenent coordnate system to elmnate the lne resstances from modelng, as llustrated n Fg. 4. conductor type II conductor type III conductor type I conductor type IV Angle mars e jϕ I,II e jϕ I,III e jϕ I e jϕ I,IV Fgure 4. Rotatonal Load Flow prncple General case. By preservng the topologcal structure and load flow voltage soluton, the power wll flow through the subnetwors assumng dfferent representatons for each reference system. Hence, the load flow formulaton can be smplfed to elmnate the actve losses from modelng by mathematcally handlng the dfferences of coordnate reference systems among subnetwors. Ths accomplshment s obtaned through a generalzed complex reference system rotaton algorthm, shown n Fg. 5, n whch stored data s changed and connectons between subnetwors are assgned by angle mars. The rotaton procedures are dentcal for the case of modelng other seres components by seres mpedances. IJECE Vol. 6, No. 3, June 016:

6 IJECE ISSN: : Read node and networ data; : S ϕ = S, = 1,..., N; 3: counter 0; 4: Start a lst Λ = Λ 0 = {1} wth the frst node mmedately downstream of the substaton; 5: whle counter N do 6: for all node Λ do 7: ϕ π / φ ; 8: for all do S ϕ S ϕ ejϕ ; Z ϕ Z ϕ ejϕ ; 9: 10: S ϕ Sϕ ejϕ ; end for Z ϕ Zϕ ejϕ ; 11: counter counter + 1; 1: end for 13: Λ {m, such that m Λ, Λ}; 14: end whle Fgure 5. Generalzed complex reference system rotaton. 1: Read rotaton angles, as well as the node and networ data, both obtaned by the the algorthm shown n Fg. 5; : Intate node voltages assumng v = v 0, = 1,..., N; 3: whle Changes n v tolerance, do 4: Compute load demands usng an approprate voltage dependent model; 5: for all, followng the bacward drecton do 6: Calculate reactve losses at branch usng 19); 7: f ϕ = 0 then 8: Calculate accumulated actve and reactve powers through 17) and 18); 9: else 10: Obtan accumulated actve and reactve powers usng 17) and 18), as well as a proper coordnate rotaton through 1); 11: end f 1: end for 13: for all, followng the forward drecton do 14: Obtan node voltage magntudes startng from the frst node usng 0) 1); 15: end for 16: end whle 17: Prnt result reports. Fgure 6. Rotatonal Load Flow algorthm General case. The Rotatonal Load Flow method for radal dstrbuton networs s presented n Fg. 6. The approach utlzes a sweep algorthm and performs complex reference system rotatons of accumulated powers as a subnetwor connectons s assgned. In the forward procedure, rotatons are not requred and the basc bquadratc equaton s smplfed wth the removng of lne resstances from modelng. 4. RESULTS AND DISCUSSIONS The proposed approach was mplemented n MATLAB usng an old computer 1,66 GHz, Core Duo), and t has been extensvely tested wth dstrbuton networs obtaned from lterature and wth actual dstrbuton networs. For all smulatons, tolerance was set up n and ntal voltage magntudes were assumed 1 pu. Voltage dependent loads were modeled usng the polynomal form P L = PL 0 αp + β P v + γ P v ), QL = Q 0 L αq + β Q v + γ Q v ) ) P 0 L and Q 0 L are the rated actve and reactve loads, respectvely. The coeffcents α P, β P, γ P ) and α Q, β Q, γ Q ) were set n 0.8, 0.1, 0.1). Cespedes method and the Rotatonal Load Flow approach were appled to solve load flow problems n a 1- node networ [15], 7-node networ [15], 9-node networ [9], 3-node networ [1] and a 69-node networ [11]. Rotatonal Load Flow Method for Radal Dstrbuton Systems Dego Isscaba and Jorge Coelho)

7 1350 ISSN: Table 1 shows smulaton results acheved wth these comparatve case-studes. Table 1. Smulaton results for the rotatonal and Cespedes methods usng dstrbuton networs obtaned n the lterature Ref. N Runtme ms) Savng CM RLFM Iter. τ %) [15] [15] [1] [9] [11] CM: Cespedes method. RLFM: Rotatonal Load Flow method. τ: Rato between the number of changes on conductor types and the number of networ nodes. The comparatve studes ndcate that the proposed approach provdes superor results n term of effcency when compared to Cespedes technque. In fact, calculatons of actve losses have been elmnated n the proposed approach and sweep procedures were adapted to handle dfferences of complex reference system representatons. Node voltages are equally updated n both methods followng that the number of teratons necessary for convergence are also preserved n the proposed approach. Also, t was not found any sacrfce n soluton qualty by usng the rotaton technque. A smlar case study was devsed for an actual dstrbuton networ, shown n Fg. 7. Ths networ s composed of 36 nodes, 1337 lometers of lnes and 5 conductor types nstalled. Smulaton results for ths networ are presented n Table. Fgure 7. An actual radal structured dstrbuton networ. Table. Smulaton results for the rotatonal and Cespedes methods usng an actual dstrbuton networ N Runtme ms) Savng CM RLFM Iter. τ %) CM: Cespedes method. RLFM: Rotatonal Load Flow method. τ: Rato between the number of changes on conductor types and the number of networ nodes. Extensve smulatons ponted out runtme mprovements depend upon the rato between the number of changes n conductor types across the networ and the number of networ nodes. Ths fact was expected snce a large number of changes n conductor type mght ncrease the number of rotatons durng the bacward procedures, as formalzed n Appendx A. All outcomes, ether usng lterature or actual networs, ndcate that the proposed method provdes mproved results n comparson to Cespedes method n term of effcency. IJECE Vol. 6, No. 3, June 016:

8 IJECE ISSN: CONCLUSIONS AND FINAL REMARKS Adequacy evaluatons usually requre a large number of load flow computatons to estmate a small set of performance ndces. These evaluatons are utlzed n long-term plannng stages, smplfcatons such as balanced operaton and constant power load demands can be assumed, whle efforts are drected to accurately modelng the falure/repar cycles of networ components and load/generaton profles as stochastc processes. Amng at achevng mproved effcency n adequacy evaluatons, a modfed verson of Cespedes load flow method to radal dstrbuton systems s proposed. The approach elmnates unnecessary calculatons n Cespedes load flow method by representng dstrbuton systems n dfferent and convenently chosen coordnate reference systems. For ths accomplshment, a generalzed complex reference systems rotaton algorthm was ntroduced and the load flow formulaton was adapted to handle dfferences n coordnate reference system representatons. Result analyss obtaned from both lterature and actual networs ndcate that the proposed method provde mproved effcency n comparson to Cespedes method. Future wors wll nvestgate extensons of the proposed approach wth regard to unbalanced dstrbuton networs, dspersed generaton, and wealy-meshed dstrbuton systems. ACKNOWLEDGMENTS The authors would le to acnowledge the fnancal, techncal and human support of the CNPq, CAPES and INESC P&D Brasl. REFERENCES [1] H. Marefatjou and M. Sarv, Dstrbuted generaton allocaton to mprove steady state voltage stablty of dstrbuton networs usng mperalst compettve algorthm, Internatonal Journal of Appled Power Engneerng IJAPE), vol., no. 1, pp. 15 6, 013. [] N. Ghaffarzadeh, M. Abar, and A. Khanjanzadeh, Dstrbuted generaton allocaton to mprove steady state voltage stablty of dstrbuton networs usng mperalst compettve algorthm, Internatonal Journal of Appled Power Engneerng IJAPE), vol., no., pp , 013. [3] D. Isscaba, A. J. S. Costa, and J. L. Colombo, Real-tme montorng of ponts of common couplng n dstrbuton systems through state estmaton and geometrc tests, IEEE Transactons on Smart Grd, vol. 7, no. 1, pp. 9 18, Jan [4] D. Isscaba, J. A. P. Lopes, and M. A. Rosa, Adequacy and securty evaluaton of dstrbuton systems wth dstrbuted generaton, IEEE Transactons on Power Systems, vol. 7, no. 3, pp , Aug. 01. [5] D. Isscaba, M. A. Rosa, and J. A. P. Lopes, Dstrbuton systems performance evaluaton consderng slanded operaton, n Proceedngs of the Power System Computaton Conference, Stocholm, Sweden, 011. [6] W. H. Kerstng, Dstrbuton System Modelng and Analyss, nd ed., ser. The Electrcal Engneerng Seres, L. Grgsby, Ed. CRC Press, Jul [7] D. Radcc and A. Bose, A modfcaton to the fast decoupled power flow to networs wth hgh r/x ratos, IEEE Transactons on Power Systems, vol. 3, no., pp , May [8] F. Zhang and C. S. Cheng, A modfed newton method for radal dstrbuton system power flow analyss, IEEE Transactons on Power Systems, vol. 1, no. 1, pp , Feb [9] R. G. Cespedes, New method for the analyss of dstrbuton networs, IEEE Transactons on Power Delvery, vol. 5, no. 1, pp , Jan [10] D. Shrmohammad, H. W. Hong, A. Semlyen, and G. X. Luo, A compensaton-based power flow method for wealy meshed dstrbuton and transmsson networs, IEEE Transactons on Power Systems, vol. 3, no., pp , May [11] M. E. Baran and F. F. Wu, Optmal szng of capactor placed on a radal dstrbuton system, IEEE Transacton on Power Delvery, vol. 4, no. 1, pp , Jan [1], Networ reconfguraton n dstrbuton systems for loss reducton and load balancng, IEEE Transactons on Power Delvery, vol. 4, no., pp , Apr [13] M. H. Haque, Load flow soluton of dstrbuton systems wth voltage dependent load models, Electrc Power Systems Research, vol. 36, no. 3, pp , Mar [14], Effcent load flow method for dstrbuton systems wth radal or mesh confguraton, n IEE Proceedngs on Generaton, Transmsson and Dstrbuton, vol. 143, no. 1, Jan. 1996, pp [15] D. Das, H. S. Nag, and D. P. Kothar, Novel method for solvng radal dstrbuton networs, n IEE Proceedngs on Gen., Tran. and Dstr., vol. 141, no. 4, Jul. 1994, pp Rotatonal Load Flow Method for Radal Dstrbuton Systems Dego Isscaba and Jorge Coelho)

9 135 ISSN: APPENDIX A. RELATION BETWEEN NUMBER OF ROTATIONS AND NUMBER OF NODES The proposed approach can be compared to Cespedes method accordng to the usage of basc mathematcal operatons per teraton such as: multplcaton/dvson ), sum/subtracton ) and square root.). Although the usage of mathematcal operatons may vary accordng to actual mplementaton and the number of operatons to rotatng system data depends on the system tself, t s possble to relate the operatons per teraton of both methods n order to dentfy whether the proposed approach wll be more effcent for a gven adequacy study. In our mplementaton, Cespedes method utlzes n each teraton the calculatons lsted below. P +Q ; v Pu Pu + P + r I ; Qu Qu + Q + x I N tmes 1/ 1/ A +A 4B ) A r P + x Q ) vu ; B P + Q r + x ; v N tmes I totalzng, n terms of number of operatons, 18N ) + 11N ) + N.) 3) Smlarly, let M be the number of power flow rotatons between subnetwors, the basc mathematcal operatons per teraton of the proposed approach are the followng: P +Q LQ x v N tmes Pu Pu + P N M ) tmes Qu Qu + Q + LQ N M ) tmes A x Q vu N tmes B P,ac + Q,ac x N tmes 1/ 1/ A +A 4B ) N tmes v Pu Pu + P cos ϕ Q + LQ ) sn ϕ M tmes Qu Qu + P sn ϕ + Q + LQ ) cos ϕ M tmes totalzng, also n terms of number of operatons, 15N + 4M ) ) + 8N + 3M ) ) + N.) 4) By usng 3) and 4), we conclude that the proposed approach wll be advantageous n terms of number of operatons per teraton whether 3N 4M ) ) + 3N 3M ) ) > 0 Ths nequalty s true, for nstance, f M < 0, 75N, whch s a condton easly satsfed n actual networs. BIOGRAPHIES OF AUTHORS Dego Isscaba receved the B.S. and M.S. degrees n Electrcal Engneerng from the Federal Unversty of Santa Catarna UFSC), Santa Catarna, Brazl, n 006 and 008, respectvely. Furthermore, he receved the Ph.D. degree on Sustanable Energy Systems, under the MIT Doctoral Program, from the Faculty of Engneerng of the Unversty of Porto, Portugal. Hs research nterests nvolve smart grds, mutagent systems, dstrbuted generaton and dstrbuton systems. He s currently a full Professor at Federal Unversty of Technology Parana UTFPR), Assocate Researcher and Coordnator of the Research Area on Energy and Management of INESC P&D Brasl. Jorge Coelho receved the B.S. and M.S. degrees n electrcal engneerng from the Federal Unversty of Santa Catarna, Brazl, n 1977 and 1980, respectvely. In 1990, he receved the Ph.D. degree n electrcal engneerng from the Catholc Unversty of Ro de Janero, Brazl. He s a Professor of the Department of Electrcal Engneerng at the Federal Unversty of Santa Catarna, Brazl, snce March Hs research nterests nclude dstrbuton systems expanson and operaton plannng, power systems relablty, probablstc methods appled to power systems, and power qualty. IJECE Vol. 6, No. 3, June 016: )