Load flow solution of the Tanzanian power network using Newton-Raphson method and MATLAB software

Transcription

1 Internatonal Journal of Energy and Power Engneerng 2014; 3(6): Publshed onlne December 05, 2014 ( do: /.epe ISSN: X (Prnt); ISSN: X (Onlne) Load flow soluton of the Tanzanan power networ usng Newton-Raphson method and MATLAB software Mashaur Adam Kusewa Electrcal Engneerng Department, Dar es Salaam Insttute of Technology (DIT), Dar es Salaam, Tanzana Emal address: To cte ths artcle: Mashaur Adam Kusewa. Load Flow Soluton of the Tanzanan Power Networ Usng Newton-Raphson Method and. Internatonal Journal of Energy and Power Engneerng. Vol. 3, No. 6, 2014, pp do: /.epe Abstract: Load flow studes are the bacbone of power system analyss and desgn. They are necessary for plannng, operaton, optmal power flow and economc schedulng and power exchange between utltes. Ths paper descrbes modellng procedure and present models of system components used n performng load flow analyss. The developed models are oned together to form a system networ representng an approxmate Tanzanan power networ model. A load flow problem s formulated usng the model and a MATLAB program developed usng Newton-Raphson algorthm s appled n solvng the problem. Smulaton results are presented and analysed. The results ndcate that the voltage magntude and voltage phase angle profles are wthn the operatng lmts of the system; t means that the selecton of system components and modellng process s approprate and accurate. The results wll form the bass of other crtcal power system studes of the networ n the future such as power system state estmaton, optmal power flow and securty constraned optmal power flow studes. Keywords: System Component Modellng, Power System, Load Flow Analyss, Newton-Raphson Method, 1. Introducton The am of load flow analyss program [1-5] s to determne the steady-state operatng condton of the power system for a gven load dstrbuton. The steady-state of a power system may be determned by fndng out of real and reactve power throughout the system networ and the voltage magntude and voltage angle at all buses of the networ. The plannng and day-to-day operaton of modern power systems call for numerous load flow analyss. Informaton obtaned from the analyss s useful n fndng component or crcut loadngs, bus voltages, real and reactve power flows, transformer tap settngs, system losses, excter voltage set ponts, and performance under emergency condtons. The load flow model also forms the bass for other types of analyss such as short crcut, angle and voltage stablty, motor startng and harmonc studes. Load flow problem s bascally nvolves the soluton of a set of non-lnear equatons for real and reactve powers at each bus. Several methods have been developed and successfully appled n solvng the problem [6-9]. Methods [8-9] are derved from the Newton-Raphson method gven n [6]. Method [6] s appled n ths paper. The problem to be solved s that of computng the steady-state load flow through dfferent transmsson components.e. computng the voltage magntude and voltage angle at all buses, real and reactve power flows and losses n the system. In ths way a smplfed modellng through whch the whole generatontransmsson-consumpton system can be rapdly smulated s adopted. The frst smplfcaton s by consderng only the electrcal varables wth angular frequency correspondng to fundamental frequency whch wll vary only slghtly from the nomnal frequency. Fast transent phenomena happenng n the system are not consdered, the tme constants n the transmsson lnes, n the transformers and generators are so low, such that are neglected. Hence, no component of the system s modelled usng dfferental equatons. Voltage, currents, power, and mpedance are expressed as complex varables n polar coordnates. The second smplfcaton s by lmtng the modellng to equvalent sngle-phase crcut, whch corresponds ether to the balanced states of a threephase networ or to postve, negatve or zero sequence components of the unbalanced states. The second assumpton s adopted n order to avod a detaled modellng.

2 278 Mashaur Adam Kusewa: Load Flow Soluton of the Tanzanan Power Networ Usng Newton-Raphson Method and In ths paper, system components from the Tanzanan power networ are used n the modellng process. Load flow analyss s mplemented under MATLAB envronment usng Newton-Raphson algorthm. The obectve of the study s to develop alternatve load flow software to PSS/E, whch s used by the Tanzana Electrc Supply Company Lmted (TANESCO) at the moment. The structure of the paper s as follows. Secton 2 presents a bref account of Tanzanan power networ status (generaton and hgh voltage transmsson). Secton 3 presents materal and method that nclude system component modellng procedures and developed models for load flow analyss s gven. Secton 4 presents overall system networ modellng. Formulaton of load flow problem usng Newton- Raphson method and ts soluton algorthm s gven n secton 5. Secton 6 presents nput data, algorthm, smulaton procedures and results. Secton 7 dscusses the obtaned results, and secton 8 concludes the paper. 2. Tanzanan Power Networ 2.1. Generaton The Tanzanan power networ comprses of hydro, thermal and gas plants [10]. The hydro system s comprsed of 6 plants wth a total nameplate of 561MW (see Table 1). The nstalled capacty of thermal generatng plants totals 453.6MW (see Table 2). The nstalled capacty of solated thermal generatng plants totals 33.80MW. Currently, the total nameplate capacty s 1, MW. Coal power generaton s between 4 to 6 MW; mport of power s about 5 and 10 MW of bul power from Uganda and 3 MW from Zamba. The demand of electrcty n Tanzana whch s a large country (950,000 square lometres) s however growng at a relatvely fast rate. Whle the annual average growth rate between 1990 and 1998 was 4.45 percent, the average load growth rate between 2003 and 2006 has been above 8 percent [11] KEY: G-Generatng plant, T-Two wndng transformer Fgure 1. One-lne dagram-tanzanan Power Networ (not to scale) Table 1. Installed hydro grd generaton capacty Plant Name Fuel Type Installed Capacty [MW] Ownershp Khans Hydro TANESCO Kdatu Hydro TANESCO Mtera Hydro TANESCO NPF Hydro TANESCO Hale Hydro TANESCO NYM Hydro TANESCO TOTAL Tanzana Electrc Supply Company Lmted (TANESCO) owns all of the hydro generatng plants n the country and some of the thermal generatng plants, although there are some ndependent power producers (IPPs) owned by prvate operators. Tables 1 and 2 show the nstalled grd connected generaton capactes for the country. The system presently conssts of an nterconnected grd and several solated systems. The model of the nterconnected grd s shown n

3 Internatonal Journal of Energy and Power Engneerng 2014; 3(6): Fgure 1. The nterconnected system conssts of hydro and thermal generatng plants provdng power to ctes, Muncpals and townshps Hgh Voltage Transmsson Networ TANESCO owns hgh voltage and low voltage transmsson and dstrbuton lnes of dfferent voltage levels scattered all over the country. The hgh voltage transmsson lnes are estmated to comprse of 2, m of system voltage 220 V; 1, m of 132 V and m of 66 V, totallng to 4, m by the end of December 2006 [12]. Hgh voltage transmsson lnes use pylons made of steel. Almost of all HV transmsson lnes are radal sngle crcut lnes. The country power system s alternatng current (AC) and the system frequency s 50 Hz. The TANESCO grd comprses of: South-West grd, North-West grd and North-East grd. South-East grd s stll under plannng stage. South-West grd mostly of 220 V connects: Ubungo- Morogoro-Kdatu-Khans-Irnga-Mufnd-Mbeya. North-West grd connects: Ubungo-Morogoro-Kdatu- Khans-Irnga-Mtera-Dodoma-Sngda-Shnyanga-Mwanza (220 V); Mwanza- Musoma (132 V)-Shnyanga- Tabora (132 V) North-East grd connects: Ubungo-Tegeta-Zanzbar (132 V); Ubungo-Chalnze- Hale-NPF-Tanga (132 V); Chalnze Mosh Arusha (132 V); NYM Mosh (66 V); Arusha- Babat-Sngda (220 V). Table 2. Installed thermal grd generaton capacty Plant Name Fuel Type Installed Capacty [MW] Ownershp Songas Natural gas Prvate Ubungo Natural gas TANESCO IPTL HFO Prvate Dodoma IDO TANESCO Mbeya IDO TANESCO Mwanza IDO TANESCO Musoma IDO TANESCO Tabora IDO TANESCO TOTAL Source: Economc Survey Report: 2007 and2009 IDO Industral Desel Ol HFO- Heavy Fuel Ol 3. Materals and Methods The data used for ths study were obtaned from TANESCO, Ubungo power staton. Computer software programmed usng MATLAB 2013 were used n conductng the smulaton Modellng of System Components A state of a power system s defned by ts topology.e. by the lst of components n operaton at the tme of analyss and by the connectons between these components. In the loadflow analyss, the system s represented n the nodal topology. The nodal topology can be defned by a graph, the buses of whch are electrcal buses and branches of whch are the transmsson system components (lnes, cables, transformers). Models of these components are presented n the followng subsectons AC synchronous Generator Two approaches are possble for modellng AC synchronous generators: representng n detal the exctaton control system or usng approxmate models. In system studes where system alternatves must be explored n detal, or when developng protecton schemes and operatng crtera to mantan the power system stablty, accurate modellng s necessary and requres detaled generator models. But n the ntal stages of a plannng study or n operatng studes such as load-flow, smplfed models may be adequate for realtme determnaton of operatng lmts and for some contngency analyss studes [13]. Thus, the AC synchronous generator s modelled as a voltage-controlled bus wth constant real power and voltage Transmsson Lnes A transmsson lne or a cable connectng two buses (Fgure 1) and s modelled by a π- crcut wth seres mpedance: and δ z = z e = r + x And a shunt admttance on the sde The π-crcut s symmetrcal, thus: (1) y = g + b (2) g g = 0 (3) = Cω b = b = (4) 2 Where C ω denotes the susceptance ω s the angular frequency correspondng to the fundamental frequency C s the capactance 3.4. Regulatng Transformer (RT) Regulatng transformers can be used to control real and reactve power flows n a crcut [14].Regulatng transformers (RT) can also be used for control of voltage magntude as well as phase angle. Thus, t s necessary to develop bus admttance equatons that can be ncluded n load flow analyss from regulatng transformers. Fgure 2 shows a detaled representaton of a practcal regulatng transformer. Fgure 2 shows currents I and I enterng the two buses, and the voltage at these buses are V and V referred to the reference bus. The complex expresson for power nto the deal transformer wth turns rato 1: t from bus and bus are,

4 280 Mashaur Adam Kusewa: Load Flow Soluton of the Tanzanan Power Networ Usng Newton-Raphson Method and respectvely Fgure 4. π-equvalent model of RT Fgure 2. Transmsson lne representaton = VI S (5) = tvi S (6) Assumng the deal transformer has no losses, the power S nto the deal transformer from bus must equal the power S out of the deal transformer on the bus sde, so from (5) and (6) S = S V I = tv I I = ti The current I can be expressed by I = ( V tv ) Y = tyv + V Y Multplyng (8) by -t and substtutng I for -t I yeld: (7) (8) I = tt YV t YV (9) Settng tt = t 2 and re- arrangng (8) and (9) nto Ybus admttance matrx form, gves I t Y = ty I 2 t Y V Y V (10) The π-equvalent model correspondng to (10) s presented n Fgure Loads It has been suggested [15] that the actual load be modelled as lnear combnaton of constant load, constant current and constant mpedance. Ths approach would requre consderable nowledge of load composton or nowledge of real and reactve power varaton wth voltage magntude. In real power system, load models are categorzed as statc models or dynamc models. Statc models normally express the characterstc of the load at any nstant of tme as algebrac functon of the bus voltage magntude and frequency at that nstant tme. In ths way the real power component P and the reactve power component Q are consdered separately. The load characterstc n terms of voltage for statc loads s represented by exponental model: a V P = P0 V (11) 0 b V Q = Q0 V (12) 0 Where P and Q are real and reactve components of the load when the load voltage magntude s V. Subscrpt 0 dentfes the values of the respectve varables at the ntal operatng condtons The parameters for statc load the exponents a and b. When these exponents are equal to 0, 1, 2 [15] the statc load model represents constant power, constant current or constant mpedance characterstc, respectvely. In case of composte load, ther parameter values depend on the aggregate characterstcs of load components. In ths study statc model type of loads s adopted. 4. System Networ Modellng Fgure 3. Detaled representaton of RT Gven a bus load and specfed voltage magntudes/power nectons at generaton buses, usually conventonal load flow analyss determnes the steady-state operatng condton of a power system based on the bus/branch networ model. Such a networ model s produced by mergng adacent substaton buses present at the actual bus-secton level topology. The basc power flow equatons are obtaned by applyng the Krchhoff s laws to the networ represented by the bus/branch model. The results can be grouped as ether nodal

5 Internatonal Journal of Energy and Power Engneerng 2014; 3(6): or branch equatons. The nodal equatons for an N-bus networ are the real and reactve power nectons at each bus, gven by [16-17]. where V, V N P = V V = 1 = 1, N N Q = V V = 1 = 1, N ( G cosδ + B sn δ ) ( G sn δ B cosδ ) δ δ δ : voltage magntudes at buses and δ, δ : bus voltage angles at buses and P : real power necton at bus Q : reactve power necton at bus G + B : entry (, ) of the nodal admttance matrx (13) (14) The branch equatons provde the real and reactve power flows through the branches of the networ, whch are respectvely, gven by [17] Q 2 ( G cos + B sn ) V G P = VV δ δ (15) = VV 2 shunt ( G sn B cosδ ) + V ( B b ) δ (16) where P : real power flow through branch Q : reactve power flow through branch Shunt b : shunt susceptance of branch - Equatons (15) and (16) can be extended to represent the power flow through tap-changng and phase-shftng transformers as gven n [15]. Power nectons gven by (13) and (14) can also be wrtten as the sum of the real power flow through the branches ncdent to bus p that s gven n [15] P = P Ω = 1, N, = 1, N, ( V, V, δ, δ ) shunt 2 Q = b V + Q Ω ( V V, δ δ ), (17) (18) where Ω : Set of buses adacent to bus (bus not ncludes) : Shunt susceptance at bus shunt b 5. Load Flow Problem Formulaton The load flow problem s formulated as a set of non-lnear algebrac equatons, normally represented by (13) and (14) and a set of nequalty relatonshp to tae nto account operatng lmts such as reactve power nectons/voltage magntudes at generaton buses. The problem solvablty s guaranteed by the classcal bus classfcatons: Slac/reference bus (V-δ), voltage-controlled/regulated buses (P-V) and load buses (P-Q) [16] and [18]. Load flow usually defnes a sngle bus.e. reference bus, whch plays a double functon: t provdes the phase reference angle, and snce the transmsson losses are unnown n advance, ths bus s used to balance generaton losses and load [15] and [18]. Consder an electrcal power system (the Tanzanan system) comprsng of n L buses, n PV generaton buses and one reference bus. The vector of state varables.e. voltage magntudes and phase angles determned by the load flow formulaton s gven by: where = ( n, ) L n PV T T [ V, δ ] δ Vector of phase angles V = Vector of voltage magntudes n L x = (19) The set of equalty equatons, whch represents the system of power flow problem, s gven by [18-19] (, δ) (, δ) P ( ) = PSche PV f x = = 0 Q QSch QV (20) where P and Q s the real and reactve vector of power msmatches, respectvely P sche and Q sche the vectors of scheduled values of real and reactve power nectons, respectvely P and Q are vectors of non-lnear equatons of real and reactve power nectons, represented n equatons (15) and (16), respectvely 5.1. Load Flow Soluton Usng Newton-Raphson Method The followng lnear system s generated when applyng Newton-Raphson method to solve eqn (20) [5], [14], [18], [19] and [20-21]: P Q = J ( x ) δ V where : the teraton counter J (x ): the problem s Jacoban matrx gven by J ( x ) J = J 1 3 J J 2 4 P P Q Q V = δ δ V (21) (22)

6 282 Mashaur Adam Kusewa: Load Flow Soluton of the Tanzanan Power Networ Usng Newton-Raphson Method and If m buses of the system are PV, m equatons nvolvng Q and V and the correspondng column of the Jacoban matrx are elmnated because for PV buses, the voltage magntudes are nown. Accordngly, there are n-1 real power constrans and (n-1-m) reactve power constrants, and the Jacoban matrx s of order (2n-2-m) x (2n-2-m). J 1 s of the order (n-1) x (n-1) J 2 s of the order (n-1) x (n-1-m) J 3 s of order the (n-1-m) x (n-1) J 4 s of the order (n-1-m) x (n-1-m) The load flow soluton can be teratvely obtaned by solvng the lnear system represented n Eqn (21). The voltage magntudes and phase angles are updated as: δ V = δ = V + δ + V (23) Untl convergence s obtaned The procedure for load flow soluton by the Newton- Raphson method s gven n flow chart of Fgure Results 6.1. Input Data Input data for load flow smulaton are gven n Tables 3 and 4. Table 3 gves the transmsson lnes of the Tanzanan Networ whle Table 4 provdes power generaton and demand of all buses n the system Smulaton Table 3. Lnedata 30-Bus Tanzana System Networ From To Impedance Half of lne Tap rato chargng settng From To Impedance Half of lne chargng Tap rato settng Bus No. Table 4. Busdata 30 -Bus Tanzana System Networ Load demand Generaton MW MVAr MW MVAr A computer program has been developed n MATLAB envronment to mplement the load flow descrbed n secton 5. MATLAB software development s based on [22]. The MATLAB software comprses of 7 fles namely: LF30.m, ths fle runs the software. Other fles nclude: LD30, BD30, Lfybus.m, Lfnewton.m, Lneflow.m and Busout.m. LD30 and BD 30 are excel fles descrbe arrangement of transmsson lne data gven n Table 3 and bus data of the system of Table 4. LD and BD fles were prepared accordng to IEEE Data Format from [23]. These MATLAB and Excel fles must be n the same current drectory n order to mplement the load flow analyss. The fles are used n sequence to calculate and dsplay load flow soluton n the MATLAB command

7 Internatonal Journal of Energy and Power Engneerng 2014; 3(6): wndow. The summary of the MATLAB software can be found n Table 5. Algorthm used n developng the MATLAB software s gven n Fgures 7a and 7b. The MATLAB computer software was test usng a PC wth CPU Pentum IV, 3.33 GHz and 0.99 GB of RAM. The effcency of the software was demonstrated by IEEE standard bus test systems IEEE14, IEEE30 and later on the 30-bus system of the Tanzanan networ. The approxmate one-lne dagram model of the Tanzanan networ shown n Fgure 1 comprses of 12 generatng plants (hydro and thermal), 6 power transformers, and 17 load centres. All power transformers of the system are assumed to be twowndng transformers. Voltage magntudes for voltage controlled buses were not set at 6.3. Computatonal Results Fle LF30 LD30 BD30 Lfybus.m Lfnewton.m Lneflow.m Busout.m VOLTAGE MAGNITUDE IN P.U. VOLTAGE ANGLE IN DEGREE Table 5. MATLAB fles for computaton of load flow Descrpton M-fle to run the software Excel-fle gvng transmsson lne parameters Excel-fle gvng bus data M-fle whch calculates bus admttance matrx M-fle whch calculates load flow usng Newton-Raphson algorthm M-fle whch calculates lne-flow and losses of the system M-fle whch prnts the output on the computer screen n tabular form VOLTAGE MAGNITUDE PROFILE-TANZANIAN NETWORK BUS NUMBER Fgure 5. Voltage magntude profle-tanzanan Networ VOLTAGE ANGLE PROFILE-TANZANIAN NETWORK BUS NUMBER Fgure 6. Voltage angle profle-tanzanan Networ Table 6. Voltage magntude and Voltage angle profles: Tanzana networ Bus No. Volt. Mag. [p.u] Volt. Angle [Deg.] Bus No. Volt. Mag. [p.u.] Volt. Angle [Deg.] Computatonal results are dvded nto A and B parts. Part A refers to results from IEEE 14 and IEEE 30 bus test systems. These results are presented to chec the convergence characterstcs of the developed MATLAB software. Part B presents result from the Tanzanan networ model; computatonal result from ths system are used to valdate the one-lne dagram model as well as valdty of nput data collected from Tanzana Electrc Supply Company Lmted (TANESCO). The computatonal results obtaned after smulaton are presented n both tabular and graphcal form. Table 6 shows the correspondng voltage magntude and voltage angle profle of the Tanzanan networ. Table 7 presents a summary of load flow results of other IEEE standard bus test systems ncludng the Tanzanan networ. The am of presentng ths summary s to mae comparson of/observe teraton counts for dfferent IEEE bus test systems, maxmum power msmatch; CPU tme elapsed tll convergence, total system loss, total MVAr system loss, and accuracy of the MATLAB software f t remans wthn predefned tolerance for all test systems. Fgures 5 shows voltage magntude profle and Fgure 6 presents voltage angle profle of the Tanzana networ. Table 7. Summary of load flow results Test System IEEE 14 IEEE 30 Number of Lnes Tanzanan Networ Transformer Tap settng FIXED FIXED FIXED Max. Power Msmatch 1.15E E E-007 No. of Iteratons Comp. Accuracy CPU tme [Seconds] Inected MVAr Total MVAr Loss Total System Loss

8 284 Mashaur Adam Kusewa: Load Flow Soluton of the Tanzanan Power Networ Usng Newton-Raphson Method and Fgure 7a. Newton-Raphson Algorthm n Flowchart Format

9 Internatonal Journal of Energy and Power Engneerng 2014; 3(6): Fgure 7b. Newton-Raphson Algorthm n Flowchart Format 7. Dscusson The followng observatons from smulaton results can be made. The load flow soluton for the Tanzanan networ has a maxmum power msmatch of about 7.17x10-7 ; and converged after 4 numbers of teratons. The total real and reactve power losses n the system durng ths partcular scenaro were MW and MVAr. The voltage magntude and voltage angle profles of the Tanzanan power networ are wthn acceptable lmts.e per unt for voltage magntude and degree for voltage angle. The power factor (pf) of the system s around (+30 0 degree) whch s the operatng value requred by TANESCO. It means that the selecton of system components and modellng procedure was successful. Also, t means that the hgh voltage transmsson system has the requred nomnal capacty to meet the current power demand. 8. Concluson Ths paper has presented an overvew of the Tanzanan power networ structure. Modellng procedure of crtcal system components for load flow analyss and ther correspondng models have been developed and presented. An approxmate model of the Tanzanan power networ model s bult from the developed models and used n load flow analyss. The load flow problem, whch nvolves n determnng voltage, and lne flow n an electrcal networ s dscussed and then formulated usng Newton-Raphson algorthm. Algorthm formulaton n form of flowchart and MATLAB software for smulaton as well as results from smulaton are developed and presented. The load flow soluton presented n ths paper forms the bass for future nvestgaton and development of MATLAB software as an alternatve to PSS/E software. In addton, the

10 286 Mashaur Adam Kusewa: Load Flow Soluton of the Tanzanan Power Networ Usng Newton-Raphson Method and results can be used n other crtcal studes such as power system state estmaton (PSSE), optmal power flow (OPF), securty constraned optmal power flow (SCOPF) etc of the Tanzanan power networ. Acnowledgement I would le to than the Tanzana Electrc Supply Company Lmted (TANESCO) for ts cooperaton and readness to supply most of the needed data and nformaton to mae ths wor possble. Ther support s gratefully acnowledged. References [1] W.F. Tnney & C.E. Hart, Power flow soluton by Newton s method, IEEE Transactons on Power Apparatus and Systems, Vol. PAS-86, November 1967: [2] A.O. Ewue, & J.F. Macqueen, Comparson of Load Flow Soluton Methods, Electrc power System Research 22 (1991): [3] L. Srvastava, S.C. Srvastava & L.P. Sngh, Fast decoupled load flow methods n rectangular coordnates, Electrcal Power and Energy Systems (1991): [4] A.E. Gule & W.D. Paterson, Electrcal Power Systems, Vol.2,( Pergamon Press, 2 nd Edton, 1977) [5] W.D. Stevenson Jr, Elements of Power System Analyss (McGraw-Hll, 4 th Edton, 1982) [6] B. Stott, Effectve startng process for Newton-Raphson load flows, IEE Proceedng, 118, No. 8, August 1971: [7] W.F. Tnney & W.L. Powel, Notes on Newton-Raphson method for soluton of AC power flow problem, IEEE Short course, Power System Plannng, 1971 [9] B. Stott & O. Alsac, Fast decoupled load flow, IEEE Transactons on Power Apparatus and Systems, Vol. PAS-83, 1974: [10] [11] [12] [13] S.S. Vadhera, Power System Analyss and Stablty (Khanna Publshers, 1 st Edton, 1981) [14] J.J.Granger & W.D. Stevenson Jr. Power System Analyss (McGraw-Hll, Inc. Sngapore, 1994) [15] J.A. Momoh, Electrc Power System Applcaton of Optmzaton (CRC Press, 2 nd Edton, 2009) [16] A. Montcell, State Estmaton n Electrc Power Systems: A Generalzed Approach. (Norwell, MA: Kluwer, 1999) [17] J. Arrlaga, C.P.Arnold & B.J. Harer, Computer Modellng of Electrcal Power Systems. (New Yor: Wley, 1983) [18] A.G. Exposto, Analss y operacon de sstemas de energa Electrca-Madrd Span (McGraw-Hll/Interamercan de Espana, 2002) [19] H.Saadat, Power System Analyss. Internatonal Edton (McGraw-Hll, Sngapore, 2 nd Edton, 2004) [20] A.R. Bergen & V.Vttal, Power System Analyss, Internatonal Edton (Pearson Prentce Hall, 2 nd Edton, 2000) [21] L. Powel, Power System Load Flow Analyss. (McGraw-Hll, New Yor, 2004) [22] F.L. Alvarado, Solvng Power flow Systems wth MATLAB mplementaton of the Power System Applcaton Data Dctonary, Proceedngs of the 32 nd Hawa Internatonal Conference on System Scence, Hawa, USA, 1999, 1-7 [23] [8] B. Stott, Fast decoupled Newton Load flow, IEEE Transactons on Power Apparatus and Systems, Vol. PAS-91, October 1972: