An Efficient Procedure for Solving Radial Distribution Networks through the Backward/Forward Method

Transcription

1 Proceedngs of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetc Compatblty, Corfu, Greece, August 23-25, 2005 (pp ) An Effcent Procedure for Solvng Radal Dstrbuton Networks through the Backward/Forward Method A. AUGUGLIARO, L. DUSONCHET, S. FAUZZA, M. G. IPPOLITO, E. RIA SANSEERINO Dpartmento d Ingegnera Elettrca, Elettronca e delle Telecomuncazon Unverstà degl Stud d Palermo ale delle Scenze, Palermo ITALY Abstract: - In the paper, after havng presented the general backward/forward methodology for radal systems analyss, a new b/f procedure showng some nterestng features that mprove ts performance n terms of convergence speed and calculaton effort s presented. The features that fundamentally are responsble for such mprovements concern the man steps of the b/f procedure. The startng voltage profle soluton s dfferent from the flat profle and s sutably modfed. In the backward phase and startng from the second teraton, the branch currents varatons due to the loads changes are evaluated; the latter varatons are calculated on the bass of the dfference of nodal voltages at the begnnng and at the end of each teraton. Fnally, the adopted convergence crteron s based on the entty of the dfference between each load node current n two subsequent teratons. The results of the applcatons of the proposed methodology to a set of networks taken from the lterature on the topc are reported. In ths way, the performance of the proposed methodology has been evaluated n terms of computatonal effcency. The results of other tests have evdenced the nfluence of each feature of the modfed b/f methodology over ts performances. Key-words: - Backward/forward method, Load flow analyss, Power dstrbuton. 1 Introducton For the analyss of dstrbuton systems, characterzed by a hgh R/X rato and a radal structure, n recent tmes t has been developed the backward/forward method. Such method proceeds through the followng steps: a) calculaton of the currents requred by the loads and the lnes shunt admttances, on the bass of the calculated or fxed values of nodal voltages; b) evaluaton of the current (or power) flows n the branches composng the electrcal system, startng from the termnal branches and gong up to the source node (backward sweep); c) nodes voltages calculaton, startng from the source node and proceedng to the termnal ones (forward sweep); d) verfcaton of a convergence crteron; f t s satsfed the process stops, otherwse t restarts from step a). The backward/forward method shows, as compared to the Newton methods, a hgh probablty to converge, a lmted requred computatonal effort and an easy mplementaton; the only nconvenence, especally for heavly loaded systems, s the ncreased number of teratons to attan a satsfyng soluton. The measures to lmt such nconvenence manly are of two knds: 1. reducton of the number of operatons through the adopton of smplfed network models; 2. dentfcaton of startng solutons that are closer to the fnal one. Obvously, n the frst case t s necessary to dentfy the entty of the errors caused by the ntroduced approxmatons, n the second case t s requred to evaluate the reducton of the number of teratons and of the CPU tme as compared to ts ncrease due to the evaluaton of the startng soluton. The load flow problem n dstrbuton systems has been wdely dealt wth n the lterature snce the eghtes, n relaton wth the development of those problems that are connected to dstrbuton automaton. Intally only radal systems have been consdered, then also weakly meshed systems have been studed; when there are meshes n the system, t s turned nto radal, by means of some cuts and by the ntroducton, n the cuts sectons, of the so-called breakpont currents calculated usng the mult-port compensaton method. The basc backward/forward method s presented n [1] where a power flow method s developed for weakly meshed dstrbuton and transmsson networks; the radal network s solved by the drect applcaton of Krchhoff's voltage and current laws. In [2] a smplfed method for the load flow analyss of radal dstrbuton systems based on the use of voltage

2 Proceedngs of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetc Compatblty, Corfu, Greece, August 23-25, 2005 (pp ) magntudes and neglectng the voltage phase angle of each bus s presented. A three-phase power flow, based on the methodology developed n [1], s presented n [3] to solve dstrbuton networks showng meshes, dstrbuted generaton, unbalanced loads, voltage regulators and shunt capactor banks. In [4] the load-flow soluton s carred out through the teratve calculaton of the bus voltage magntudes expressed n terms of the real and reactve powers flowng n the branches keepng nto account even the losses; the convergence crteron s based on the dfference at each branch between the real and reactve power evaluated n two followng teratons. The same methodology s agan consdered n [5] where the convergence crteron s modfed as t takes nto account the real and reactve powers, flowng from the substaton, n two subsequent teratons. For dstrbuton systems havng both radal and meshed topology, Haque ([6]) has developed an teratve soluton method usng the bus voltage magntudes and phases equatons; the meshes are opened addng some dummy buses; the power flows njected at these buses are evaluated by means of mpedance matrces of reduced order. The branch by branch computatonal approach s agan consdered n [7] to solve both radal and meshed systems n normal and faulty condtons; the load models here consdered are wth constant admttance, wth constant power and wth load proportonal to the voltage magntude. In [8] three sets of recursve load flow equatons for dfferent load models are compared. In [9] the dstrbuton network s solved teratvely consderng as state varables the bus voltages. The method developed by Haque n [6] s agan consdered and studed by the same Author ([10]) n order to keep nto account shunt elements and more than one supply node. In [11] three load flow soluton methodologes for radal systems are descrbed and compared. An effcent method for radal networks soluton s proposed n [12]. It s based on an teratve algorthm wth some specal measures to ncrease the convergence speed; the bus voltages are consdered as state varables accordng to approaches that are common n lterature. It uses a smple matrx representaton for the network topology and branch current flows management. The soluton method developed n [13] s based on the teratve backward/forward process, whch s appled to a set of partal networks obtaned by the orgnal radal system. As compared to classcal backward/forward sweep methods, n the proposed method the state varables calculaton s performed startng from the termnal nodes and movng to the source node; ths allows to verfy the convergence of the process performng a test on the source node voltage only. In [14] the methodology developed by Baran and Wu ([15]) s smplfed usng the real and reactve powers and the bus voltage magntudes. Some measures and procedures to mprove the effcency of the classcal backward/forward method are presented n [16] and [17]. In ths paper, after the presentaton of the state of the art on the soluton of radal dstrbuton networks wth the b/f technque, the same methodology s descrbed. In partcular t s examned consderng four man steps: ntalzaton of the state varables; backward sweep; forward sweep; dentfcaton of a convergence crteron. For each of these, except for the forward sweep, a new procedure s defned. In partcular, a methodology for the dentfcaton of a startng soluton that s as close as possble to the desred one s proposed; the backward phase s modfed for the evaluaton of the branch currents and a new convergence crteron s proposed. Fnally, the results of the applcaton of the proposed methodology to a set of systems already studed by other Authors are reported. 2 The B/F Method for Radal Systems Analyss The man features of the system are the followng. The dstrbuton system has radal topology and a sngle feedng pont wth constant voltage; the branches are three-phase; the loads are three-phase and symmetrcal and can be modelled as constant power snks; the capactve admttances of the branches are neglgble. The radal topology allows to know the drecton of the power flows n the branches and therefore to dentfy, for each branch, the sendng node through whch the power s njected n the branch and the endng bus through whch the power s transferred to the buses downstream the branch. The b/f method can be dvded nto the followng steps: 1. ntalzaton of the bus voltages; 2. backward sweep: evaluaton of the branch currents, startng from the termnal branches and gong up to the source node; 3. forward sweep: calculaton of the bus voltages startng from those mmedately downstream the source node and arrvng at the termnal nodes; 4. verfcaton of a convergence crteron. 2.1 Intalzaton of the bus voltages The flat profle of the voltages,. e. all bus voltages equal to the source node voltage, s the startng profle adopted

3 Proceedngs of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetc Compatblty, Corfu, Greece, August 23-25, 2005 (pp ) n most of the paper about load flow n dstrbuton systems. Such assumpton has the man advantage of a lmted computatonal effort and does not prevent from convergng to a fnal soluton, because of the robustness of the b/f analyss method. It s clear that, anyway, supplyng a startng soluton closer to the fnal one can reduce the number of teratons and the computatonal tme. Consder the sngle branch system, suppled at one end by a constant voltage o source, that supples, at the other end, a constant power load (A=P+jQ), fg R+jX Fg. 1. Sngle branch system. It s known that such system can be drectly solved wthout usng any teratve method; ndeed the complex voltage at the load bus s gven by: ( R + jx )( P + jq) = o (1) * where R and X are the resstance and the reactance of the branch and * s the complex conjugate of the voltage ; expressng the voltages 0 and as summaton of real and magnary parts: o = o (2) A = P + jq radal system, to dentfy, for each of ts nodes, an equvalent at a sngle branch that allows an approxmate evaluaton of the voltage at that node. The approxmaton s due to the fact that losses and voltage drops (for load current calculaton) are neglected. Consder the system n fg. 2 a), made of two branches connected n seres, suppled by node 0 at constant voltage o ; the loads at the two nodes are: A 1 =P 1 + jq 1 and A 2 =P 2 + jq 2. In the am of evaluatng the voltage 1 the sngle branch equvalent s depcted n fg. 2 b), where the load at bus 1 s the summaton of the loads A 1 and A 2, and the mpedance of branch 1 of the equvalent network remans the same as that of the orgnal system. For the evaluaton of the voltage at node 2, the equvalent network, fg. 2 c), shows n node 2, n the same way as before, the summaton of the loads A 1 and A 2, whereas the sngle branch mpedance between the source node 0 and node 2 s gven by the weghted sum of the mpedances of the branches that, n the orgnal network, connect node 0 and node 2; for each of these branches the weght s gven by the rato between the power flowng n the branch (whch, neglectng the losses, s the total power requred by the loads downstream the branch) and the power flowng n the branch connected to the source node (whch, neglectng the losses, s the total power requred by the networks loads). a) 0 Z1 1 Z2 2 A1=P1+jQ1 A2=P2+jQ2 = r + j (3) from (1) t can be obtaned: ( RQ XP) = (4) r o [ + ( RP )] o + o = XQ (5) 2 b) c) 0 Zeq,1 1 A1+A2 0 Zeq,2 2 A1+A2 In ths way, under the hypothess of constant voltage at the sendng bus of the branch, and once the real and reactve powers are known at the endng bus of the branch, usng (4) and (5) the complex voltage components can be calculated for the endng node. In the partcular case of the network n fg. 1, the voltage calculated n ths way s the exact one; t s anyway possble, for a generc Fg. 2. Radal network wth two branches connected n seres, a); network equvalent for the approxmate evaluaton of the voltage at node 1, b), and at node 2, c). As a result, for the evaluaton of the voltage at the two ends, the equvalent mpedances to be consdered n a sngle branch system are:

4 Proceedngs of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetc Compatblty, Corfu, Greece, August 23-25, 2005 (pp ) 1 ) Z eq, 1 = Z1 (6) 2 ) Z = Z + Z A 2 2 eq,2 1 (7) A1 + A2 The evaluaton of the voltages 1 and 2 through the equvalent network gves rse to errors; ndeed the losses n the branches downstream the branch connected to the source node are not consdered n the evaluaton of the power snk at the end of the sngle branch equvalent. For a generc system, wth laterals and sub-laterals, the approxmated evaluaton of the bus voltages can be executed by means of the followng procedure: 1. evaluaton of the apparent complex powers, A b,, crculatng on each branch and due to the loads; 2. for the generc node j, dentfcaton of the set {B j } of the branches connectng t to the source node; 3. calculaton of the equvalent mpedance: Z eq, Z Ab, j = Z1 + (8) A b,1 where the summaton s extended to the branches that belong to the set {B j }; 4. calculaton of the real and magnary components of the voltage j through (4) and (5) where: R= Real [Z eq,,j ] (9) X= Imag [Z eq,j, ] (10) P= Real [A b,1 ] (11) Q= Imag [A b,1 ] (12) where A b,1 s the complex apparent power requred by all the network loads and flowng, not consderng the losses, on branch 1 connected to the source node 0. The bus voltages calculated n ths way can be used as startng soluton of the teratve backward/forward process. 2.2 Backward sweep In ths phase, startng from the termnal branches and gong to the source node, the current flows n each branch are evaluated startng from the load demand. The bus voltages, calculated as startng soluton or attaned at the end of the precedng teraton, are useful to evaluate the currents requred at the bus loads. At the begnnng of the frst teraton the load current, I, at node s gven by: (1) A I = (13) (0)* where A s the complex apparent power at node and (0)* s the complex conjugate voltage at the same node used as startng soluton. At the end of the frst teraton, the voltage at node, (1), dffers from the startng voltage value, (0), thus the complex power requred by the load, evaluated wth the new voltage value and wth the ntal value of the current (13), s not equal to the fxed value. Therefore, at the end of the frst teraton, and n general, at the end of the k-th teraton, a dfference between the calculated and fxed values of power requred by the loads comes up. In the methodology here proposed, for each load, the current due to the dfference between the calculated and fxed values of power requred by the loads s evaluated at the end of each teraton. Such current, n what follows, wll be ndcated as load current dfference. Ths value s summed to the load current at the bus; n ths way, the load current at the begnnng of each teraton s the same as the one n the precedng teraton ncreased, from the second teraton, of the load current dfference. In general terms, the load complex power at node, at the end of teraton k, s gven by: A A = (14) ( k 1) * * I = ( k 1)* and dffers from the requred power at node, A, of the followng quantty: A = A A (15) The load current dfference assocated to ths power dfference s gven by: A ( A A ) 1 1 I = = = A ( ) * * (16) * ( k 1)* Thus, the load current at node, at the begnnng of the subsequent teraton, s gven by: I ( k + 1) = I + I (17) The load current dfference n (16) approaches zero, as the bus voltage dfference n two subsequent teratons gets smaller, thus when the overall voltage profle converges to the fnal value.

5 Proceedngs of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetc Compatblty, Corfu, Greece, August 23-25, 2005 (pp ) 2.3 Forward sweep In ths phase, for each branch, the voltage at the endng bus EB, s calculated on the bass of the known values of the sendng node, SB, voltage and of the current, I b,, flowng n the branch. The need to know the sendng node voltage makes t mandatory to proceed from the source node towards the endng branches. For each branch then, f n the backward sweep the currents calculaton was performed, the followng equaton can be used: EB = ZI (18) SB b,1 2.4 Convergence crteron Each teraton termnates wth the calculaton of the bus voltages; to decde whether to contnue the teratve process or not, the attaned results are compared to those obtaned n the precedng teraton; ths s done n order to evaluate the errors and, on the bass of a prefxed convergence factor, ε, to verfy whether the errors are larger or smaller than ths factor; f these are greater than ths factor, another teraton s performed calculatng the currents dfference and assumng as bus voltages the last calculated values; otherwse, f they are smaller, the teratve process stops and the requred results are prnted out. The easest convergence crteron conssts n the comparson between the voltages n two subsequent teratons; the error s then evaluated as: ( k 1) ε (19) If, for all the nodes, (19) s verfed, the convergence has been reached. It s possble to reduce the overall number of operatons to execute, and thus the calculaton tme, by verfyng the crteron not for the set of all nodes, but smply node by node. If for the generc node, (19) s verfed, the voltage at the -th node s consdered constant on the subsequent teratons; n ths way, n the backward phase, the calculaton of the load current dfference at node can be avoded as well as, n the forward phase, the calculaton of the voltage at node. Another convergence crteron, smlar to the latter that can be adopted, conssts n the comparson for each node of the calculated and fxed values of power requred by the loads at the end of each teraton. The load current dfference s a feature that s representatve of the error n terms of power snce t reachng zero ndcates the equvalence between the calculated and fxed values of power requred by the loads, or also ndcates the equvalence at each node of the voltage n two subsequent teratons. Thus the nequalty to be verfed s the followng: ( k ) I ε (20) In the same way as before, the convergence crteron expressed by (20) can be verfed, at each teraton, for all the load nodes of the network (global convergence crteron) or, node by node, not consderng n the followng evaluatons the nodes for whch the crteron has been already verfed (local convergence crteron). In fg. 3 the flow-chart of the proposed algorthm n shown. Data acquston (lnes and loads) Calculaton of the complex apparent power n the branches, not consderng the power losses Calculaton of the equvalent mpedance between every node and the source bus (eqn.8) Calculaton of the approxmate voltages at every node (eqns.4,5,3) NO IT = 1 Calculaton of the load currents at nodes (eqn.13) Branches currents calculaton Bus voltages calculaton (eqn.18) Modfcaton of the loads currents due to the dfference currents IT=IT+1 Loads dfference currents calculaton (eqn.16) erfcaton of the convergence crteron YES Prnt results Fg. 3. Flow-chart of the proposed algorthm.

6 Proceedngs of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetc Compatblty, Corfu, Greece, August 23-25, 2005 (pp ) 3 Applcatons In order to evaluate the performance of the proposed methodology, the relevant algorthm has been mplemented n FORTRAN 90 and the program has run on a manframe IBM S/ /225. The applcatons concerned the soluton of some networks wth 12, 15, 28, 33, 69 and 85 nodes; the data about the lnes and the loads of these networks are respectvely reported n [5], [4], [5], [18], [15], [4]. These networks have been used by some of the cted Authors to test dfferent b/f radal networks soluton methodologes. In partcular the 85 nodes test systems has been studed by Das et al. ([5]) wth other several Indan rural dstrbuton networks. In Table 1 for each of the consdered systems are reported: col. 1 the number of network buses, NB; col. 2 the reference n whch the system was analyzed, [..]; col. 3 and 5 the number of teratons, IT, and the CPU tme, n s, attaned n the ref. reported n col. 2; col. 4 and 6 the number of teratons, IT, and the CPU tme, n s, attaned wth the procedure here presented; col. 7 the maxmum percent error n the voltage magntudes evaluated comparng the startng and the fnal soluton, err x 0 %(= max{( o - f ) 100/ f }); col. 8 the maxmum percent error n the voltage magntudes evaluated, comparng the end of the frst teraton and the fnal soluton, err x 1 %(= max{( 1 - f ) 100/ f }). In all cases, the proposed procedure reaches the fnal soluton wth a number of teratons that s smaller than that of other methodologes. The convergence factor has been fxed to for all the consdered cases; ths value s the same as that fxed by other Authors to test the developed b/f methodologes. In order to verfy the nfluence of the modfcatons concernng the startng soluton and the convergence crteron, the same networks have been solved usng the proposed procedure, also consderng a flat voltage profle and the global convergence crteron. In Table 2 the number of teratons, the CPU tme and the maxmum errors attaned consderng the global convergence crteron are shown; n Tables 3 and 4 are nstead reported the same features consderng, as startng soluton, the flat voltage profle and adoptng ether the local convergence crteron (Table 3) or the global convergence crteron (Table 4). In these Tables err x j %(= max{( j - f ) 100/ f }) s the maxmum percent error n the voltage magntudes evaluated comparng the end of the j-th teraton and the fnal soluton. From the analyss of the results, the followng consderatons can be deduced: - the ntalzaton of the bus voltages through the approxmated soluton gves rse to a reducton of the number of teratons and of the CPU tme; - the local convergence crteron does not sensbly modfy the number of teratons (only n one case a reducton can be observed); t allows a lmted reducton of the CPU tme when the number of teratons s greater than 2; - the course of the maxmum errors shows that choosng an approxmated voltage profle as startng soluton ncreases the effcency and provokes a reducton of the number of teratons. 4 Conclusons The b/f method specfcally developed for radal dstrbuton systems, even f t s conceptually qute easy, shows, n ts mplementaton, a set of alternatves n each of the steps nto whch t can be dvded. In ths paper some modfcatons concernng a sutable ntalzaton of the bus voltages, the backward phase and the convergence crteron are presented. The results of some applcatons carred out on some test systems show the effcency of these modfcatons n terms of reducton of the number of teratons and of CPU tme. Improvng the performance of the b/f methodology mples also an mprovement for those methods orented to the soluton of weakly meshed networks and/or wth P nodes (nodes njectng only real power, produced by means of renewable sources, cogeneraton plant, etc., and requrng voltage regulaton systems). These networks are usually solved reducng them to radal systems, by means of cuts n real and fcttous meshes, the latter beng assocated to the P nodes, and by means of the ntroducton of compensaton currents. References: [1] D. Shrmohammad, H.W. Hong, A. Semlyen, G.X. Luo, A compensaton-based power flow method for weakly meshed dstrbuton and transmsson networks, IEEE Trans. Power Systems, ol.3, No.2, May 1988, pp [2] G. R. Cespedes, New method for the analyss of dstrbuton networks, IEEE Trans. Power Delvery, ol.5, No.1, January 1990, pp [3] C.S. Cheng, D. Shrmohammad, A three-phase power flow method for real-tme dstrbuton

7 Proceedngs of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetc Compatblty, Corfu, Greece, August 23-25, 2005 (pp ) system analyss, IEEE Transactons on Power Systems, ol. 10, No. 2, May 1995, pp [4] D. Das, D. P. Kothar, A. Kalam, Smple and effcent method for load flow soluton of radal dstrbuton networks, Electrc Power & Energy Systems, ol. 17, No.5, 1995, pp [5] D. Das, H. S. Nag, D. P. Kothar, Novel method for solvng radal dstrbuton networks, IEE Proc. - Gener. Transm. Dstrb., ol. 141, No. 4, July 1994, pp [6] M. H. Haque, Effcent load flow method for dstrbuton systems wth radal or mesh confguraton, IEE Proc. - Gener. Transm. Dstrb., ol. 143, No.1, January 1996, pp [7] D. Rajcc, R. Talesk, Two novel methods for radal and weakly meshed network analyss, Electrc Power Systems Research, ol.48, No. 2, 1998, pp [8] M. Haque, Load flow soluton of dstrbuton systems wth voltage dependent load models, Electrc Power Systems Research, ol.36, No. 3, 1996, pp [9] S. Ghos, D. Das, Method for load-flow soluton of radal dstrbuton networks, IEE Proc. - Gener. Transm. Dstrb., ol. 146, No. 6, November 1999, pp [10] M. H. Haque, A general load flow method for dstrbuton systems, Electrc Power Systems Research, ol. 54, No. 1, 2000, pp [11] J. Nanda, M. S. Srnvas, M. Sharma, S. S. Dey, L. L. La, New fndngs on radal dstrbuton system load flow algorthms, Proc. IEEE Power Engneerng Socety Wnter Meetng, 2000, ol. 2, pp [12] A. Auguglaro, L. Dusonchet, M.G. Ippolto, E. Rva Sanseverno, An effcent teratve method for load-flow soluton n radal dstrbuton networks, Proc. IEEE Porto Power Tech Conference, Porto (Portugal), September 10-13, [13] A. Auguglaro, L. Dusonchet, S. Mangone, E. Rva Sanseverno, An alternatve forward/backward method for radal networks soluton, Proc. 2 nd IASTED Internatonal Conference on Power and Energy Systems (Europes 2002), Crete (Greece), June 25-28, [14] S. F. Mekhamer, S. A. Solman, M. A. Moustafa, M. E. El-Hawary, Load flow soluton of radal dstrbuton feeders: a new contrbuton, Electrc Power & Energy Systems, ol.24, No. 9, 2002, pp [15] M. E. Baran, F. F. Wu, Optmal szng of capactors placed on a radal dstrbuton system, IEEE Transactons on Power Delvery, ol. 4, No.1, January 1989, pp [16] A. Auguglaro, L. Dusonchet, S. Favuzza, M. G. Ippolto, E. Rva Sanseverno, Smple measures to mprove the performances of the backward/forward method for radal dstrbuton network analyss, Proc. 5 th IASTED Internatonal Conference on Power and Energy Systems (Europes 2005), Benalmadena (Span), June 15-17, [17] A. Auguglaro, L. Dusonchet, S. Favuzza, M. G. Ippolto, E. Rva Sanseverno, Some Improvements n Solvng Radal Dstrbutons Networks Through the Backward/Forward Method, Proc. IEEE St. Petersburg Power Tech Conference, St. Petersburg (Russa), June 27-30, [18] M. E. Baran, F. F. Wu, Network reconfguraton n dstrbuton systems for loss reducton and load balancng, IEEE Transactons on Power Delvery, ol. 4, No.2, Aprl 1989, pp Table 1 - Performance of the proposed methodology for some networks studed n the lterature IT CPU [s] err x [%] nodes Ref. [ ] proposed [ ] proposed err x 0 err x ; ; ; ;

8 Proceedngs of the 5th WSEAS Int. Conf. on Power Systems and Electromagnetc Compatblty, Corfu, Greece, August 23-25, 2005 (pp ) Table 2 - Performance of the proposed methodology for the networks of Table 1 wth global convergence crteron Table 3 - Performance of the proposed methodology for the networks of Table 1 for flat voltage profle and local convergence crteron err x [%] err x [%] nodes IT CPU [s] err x 0 err x Nodes IT CPU [s] err x 1 err x 2 err x Table 4 - Performance of the proposed methodology for the networks of Table 1 for flat voltage profle and global convergence crteron err x [%] nodes IT CPU [s] err x 1 err x 2 err x 3 err x