POWER SYSTEM ANALYSIS Operation and Control
ABHIJIT CHAKRABART! Professo r, Department of Elcdrical Engineering Bengal Engineering and Science University Shibpur, Howrah
SUNITA HALDER Lecturer, Department of Elect rical Engineering Jada vpu r University Kolkata
Prentice-Hall of India i?oiIwlill@ ilJllMllKl@@l New Delhi-11 0001
2006
T hca. .
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POWER SYSTEM ANALYSIS: OperatIon and Control Abhij~ Chakrablortl
and &.Ilta Halc»r
o 2006 by p,entic..... of india PrIvate Llmlted, New DeIhl ..... ~ _lid. No PIIrt oIlhis book may be 'iiIj)OoduI:ed In any lam, by mimeogIaph or any oIhItr~. ",Ithoul pennIssion r. writing from the publisher. IS8N~I-203-2777-2
The export rights of Ihls book ala vested solely with the publl$her. Second Pflntlng
SII'N"'W, 2001
PuIlIishod by Asoka K. Gt.:>sh, Prentice-lid 01 India Privaltl Umi\otd, M-97, CornIo.VlI Circus, New Dalhi· l10001 and Printed by J.y Pmt PKk PM. limited, New o.flI.l10015.
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Contents Pre{ace ..... .................................... ..... ............. ........ ................ ,........ ......... ..." ................ ................. .... xiii
I.
Introduction .................. " ............... " " " ............ """,,",,",, ............. ,,"",, ......... 1- 12 1.1 1.2
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StruClurc o f oil. Power Syslan ............ ............................ ..................................................... I The NC'Ccssity of Contro l o f oil. Power System ................................................................. 3 1.2,1 Cn ntmJ M c1brxls ..... . ... . ........ " .. "" . " ... " " .......... . 5
1.2.2
Advanta es of Co
ute r Control ._.. ,.............................. _........................... .......... 5
12.3
Trpc:s of Computer Control Syslem ..... ..... ................. ............. ........ .... .................. 5
Power S tem R resentation ...... ,.... ,....................... .,...... ......... ,.... ,... ...... _.............. ,........ 6
1.4 1.5
Pov.'er System at Normal Operatin g State .. __ _.... _.... _........ _....... _._,........................ _............. 7 Operating Prob lems in Power Systems ........ ..................................................................... 8 1.5. 1 Loadabilicy of Transmiss io R Lines ........................................................................ 8 1.52 Frequency Dynamics of Tra nsmission Line ....................................................... 10 1.5.3 Overload and Frequency Decay Rate ......................................................... ,....... 10 1.5.4 Tra ns ient Stab ilicy Problem .................................... ,..... __ ...................................... 12 1 <; 5 power Osc illa tions ]3 1.5.6 Reac tive PO\\'er Limita tions and Voltage Contro l Problems ........... .................... 13 1.6 S«uricy AlllIl ys is and Contingency Evaluation ................................................. ,........... 13 12 Automatic Control IS 1.7. 1 Automatic Load Frequency Control (ALFC) ......... ................. ............................ 15 1.7.2 Automatic Voltage Control (AVC) ......... .......................... " ....... ............................ 16 1.7.3 Co ntrol Components in Power System ............................................... ................ 16 1.8 Use of Computers and Futun: Trends ..... .................... ,......................................... ........ 18 Exercises .... ...... .. ..... .. .. .. ..... ... . ..... .. .. ... ...... ..... ........... . ... ... ..... 19
2. Modellin or Power S tern Com onents ................................................ 20-46 2 J
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lnlroduc lion. ...... . .... .... . . .. . . .. ... . ...... ;n Modelling o f Synchronous Generator (Altemalor) ........................................................ 21 Modelling o f a Synchronous Generator in I Netwo rk .................................................. 2S
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Modell ing of Gene rator Components ........................................... ................................... 2ti 2.4.1 G
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Power Network l\'(atr ix 0 eratlons .................. __ .................... _ ............ 47- 119 lnt rodoction to [ r& n) Fo rrnu l~t ion ..... ........ .. .. ............ .... .... 47 Nooal Method fo r Deve lop ment o f [ Y" ,,) ... .. .................. ,.. ... ...... ...... ..... ........... .. ...... n Modification of {YlIgsj Due to Inclusion of Regu lating Tra nsformer between Two Buses 57 3.4 Form-uion of [Y_l wi th Tr~nsfnnno::r Pre""nt in the line ............................................ (D 3.5 Development of [ Y&wI Using Singular Transformati on .................... .............................. 14 3.6 Development of [Y1I..J Matrix Using Coefficient Matrix ................................................ KZ 3.6.1 Steps of Algorithm to Develop [YIIm ] When chere is no Mutual Coupli ng between Branches (Using Coefficient Matrix) .................................... 84 3.7 Formulation of Complete [ Ys.J for a General Network ................................................. 96 38 Modifica tion of [ Y_ l for Branch AddilionIDeletion ............................................. ....... 106 3.&1 Qe"elopntent nf [Ya..J by Step by Step [yJ Array Formation ........................ 107 3.9 [Ya. l Forma tion with Consideration of Mutually Coupled linc:s Us ing Step by Step [y ] Formation ...................................................................... 110 3.10 Computational Aspe<:ts of ( YIIuoJ FormaTion .................................................................. II I 3. 11 [Zp..l Build ing by Step by Step Method ...... ................................................................ 11 1 3.11.1 ,\ddit;on of a Branch (or Link ) Zb from a New Bus to the Reference Bus (Type_I Modification) (Fig. 3.18) ................................................................ 111 3.11.2 Additi on of a Branch (or Link) Zb from a Ne w Bus to an Old Bus (Typc-2 ModificaTion) (Fig. 3. 19) ..................... ................................................... 112 3. 11J Addition of a Branch (or Link) Zb from an Old Bus 10 the Reference Bus (Type-3 Modification) (Fig. 3.20) ....................................... .. .................. 113 3.1104 Add ition of a Branch (or Link) Zb between Two Old Buses (Type-4 Modification) (Fig. 3.21) ........................................................................ 114 3.!l.5 Add ition of Two Branches (Z. aDd Zt ) with Mutual Impedam:e (Z.) betwccn Four Buses (Type-S Modificati on) (Fig. 3.22) .. __ ................................ 11 6 Exercises .................. ............ " ............................. .......... ......... ........ ............ .......... ........... ......... III 11 12 33
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5.14 Use of Loss Formula in Economi c Operation ............................................................... 268 5. 14,1 Algorithm for De tenni na tion of Optimal Generalion Using lAss FormuJa ....... 269 5.)5 A Mtthod or Oo.:li:rmi ning E",onolllic Qr>
6.
Computrr.Aided Economic Load Dispatcb and Optimal Power Flow ••••••..••••.•.••.•..•....•.........••..... _.......••••••.•....•. _............•.• _....•.•••.......••••••. 313-455 Introduc tion ........................................................................ .............. ............................... 313 Economic Load Dispatc h by NeW\ozr Raphson Melhod .................. ............................ 313 6.3 Economi c ~d Dispatch by Appro:timatc Ne""10 zr Raphson Method ........................ 323 6.4 Economic Load Dispatch Using Eo ct Loss Formula ................................................... 329 6.4.1 Formation of E)(aet Loss F onnula .................................................. ..................... 329 6.42 Economic Load Dispatch .................................................................................... 333 Economic Load Dispatch Using Loss Fomtula which is a Func tion of Real and Reactive Po ....'Cr .............................................. _.................................................. __ ............_346 6.5.1 Derivatioo of Re al 3nd Reactive Po,,"'Cr Govcmcd Loss Formu la .................... J46 6.52 Economic Load Dispatc h Using Loss Formula ( Func tion of Rcal and Reactive Power) ............................................. ......... ............................................. 348 6.6 E~ onomie Load Dispalch ror Rcal and ReaClive Powcr B31ance ................................. 363 6.7 Optima l Power Flow Using C l a ssic~ 1 Met hods ....................... ... ,.................................. 380 6.8 Modem Approach to Optimal Power Flow Solution .................................................... 392 6.8.! Ne ....10n·Raphson (N-R) Method ........................................................................ 392 6.8.2 Fast De coupled Method .......................................................... ........................... 426 6.9 Gradient Mcthod ................................................................................. ............................ 436 Exercises ........... " .. __ ,... ,.. __ .... ,...... ,... ,... ,.... ... ,... ,.... .. __ ..... ___ .___ ..... ' ... __.___.__ __ ,___,__ ._____ ___ _... ___... ' . 444 6. 1 6.2
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Power System CODtrol CeDtres .................................. _..................._..... 456-468 7. 1 7.2 73
Introd uction ......................... ........................ .................................................................... 456 Aim of Control Centres ....................................................................... ...... _..................... 456 Planning Objective ... ................................. __ .. __ ... __..__ ................__..........................._......... 458
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Functions of Control Centres ......................................................................................... 458 7.4.] P lanning ............................................................................................................... 458 7.4.2 Monitoring ... ................................................................................................ ........ 459 7.43 Da ta Acquisition and System Control ................................................... ............ 400 7.5 Sct-up ............................................................................................................................. 460 7.6 loca tio ns ........................ ................................................................... .............................. 461 7.7 Centra l Facililie$ .............................................................................................................. 461 7.7. 1 Civil Facilities ......... ........................................................ ..................................... 461 7.7.2 Facil ities in Control Room ....................................................................... ........... 462 7.8 Communication ................................................................................................................ 463 7.8.] Power Line Carrier Comrmnication (PLCC) ........................................................ 463 7.8.2 Leased Te lephone Lines ...... ............................................................................... 464 7.83 Mierowave Channel ................................................. ............................................ 464 7.8.4 Fibre Optic Communication ........... ..................................................................... 465 7.8.5 Satellite Communkation Channel ................................................................. ...... 465 7.9 Telemetry .. ................... .................................................................................................... 466 7.10 Emergency Control .......................................................................................................... 466
F_un:ises .. .... ... ._. ._____ _..___ .. _.__ ., ... ,... .... ,....... _, ......... ,....... " .. ,....... ,... ,... ,... ,... ,' ." ... ......... ,.......... 468
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Automatic Generation Control _............................................................. 469-51 S 8.1 &.2
8.J &4 &5 8.6
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8.9 8.10 &1 1
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Introduction .................. ,................ ,............ ,........................................................ _... ,....... 469 Types of Ahel1Ultor Eltciters ......... .................................................. ,............ ,.... ......... ,.... 471 8.2.1 Primitive Type Eltciters ........................................................................................ 47] 8.2.2 Modem Eltci ters ................................................................................................... 471 Exciter Modelling ................................................................................................... ,......... 474 Modelling of Alternator (Synchronous Gener:u or) ....................................................... 475 Statk Performance ofAVR Loop ............................. ,........... ........................................... 476 Dyll.lmic Performance of the AVR Loop .............................. _., .. _...._........ ....................... 4n Compe nsa liOn in AVR Loop ........................................................................................... 478 Automatic Load Frequency Control (ALFC) ................................................................. 478 Types o f Turbine Represen tation ....................... ,.......................................................... 481 Steady State Perfonnance of the Speed Governing Sys tem ....................... .................. 483 Complete StruCTUre of Primary Al FC Loop ............................ ....................................... 486 RespoMes of Primary ALFC Loop ................................................................................ 488 8.12. 1 Steady Stalc Response ........................................................................... ............ 488 8,12.2 Trans ient Response .......................................................... ,................ ,................. 489 Secondary ALFC loop ...................... ,.............. ... ,... .................... ................................... 492 8. 13.1 Abou t the Controller .,................ ,......... ,........................................ ,..................... 492 8.13.2 Modelling of Secondary ALfC Loop ................................................................. 493 Performance of Secondary ALFC Loop . __.___... _, ,_.. _.. , ____ ____ __.. _. __ _. __..__ _.............. _... _494 Elttension of ALFC Loop to Mu lt i-area Systems ........................................ ,................. 495 Tic-line Power Flow Model ............................................................................................ 496 Static Response of Two- area System ........................... ... ,... ................ ,.................. ,....... 498 Transien t Response of a Two-area System ................................................................... 503 Application Aspects of Primary A lFC loop .............. ,,_ ... __.. _. " .. ,, __... __ .. __.. _.. ,, _... __ ... _.... SOl Application Aspect of Secondary ALFC Loop ......... _............. " .. "._ .... _.......... _............. 505 Malenal,
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CONTENTS
Interfacing of AGe with Economic Dispatch ................................................................ S06 822 Application of Optimal Control CO~epIS in ALFC ...................................................... 'YJ1 8.23 Fundamenta! Aspects of Optimal Linear Regulator (OLR) Design ...... ........................ 510 813.1 Significance of Q and R in the State RcgulalOT Problem ..... ............................ 511 Exercises .... ,................... ,....... ,.... _........ '. ___ ._____ _.___. _, ___ _,__ __ . __.. __ __ ___,,", ..... ,_.. ,... ,' ........ ,........ ,..... J J5 8.21
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Reactive Power Control and Voltage Stability .................................... 516-575 9.1 92 93
Introduction ...................................... .......................... ..................................................... 516 Po,,"'!:r Flow in a Two-Bus System ....... .......................................................................... 516 Vollage Regulation in a Transmission System and lis Relatioo with Reactil'e PO,,"'!:t ............................................................................................................................ 518 9.4 Reactive Power and Voltage Collapse ............................................................................ 522 9.5 Changes in Power System Contributing to Vo ltage Collapse ...................................... S22 9.6 Concept of Stability of Transmission System ............................................................... 5.22 9.7 Definition and Classification of Voltage Stability .......................................................... ill 9.8 Mechanism of Voltage Collapse ........ ......... ............................. ....................................... 525 9.9 Analytical Concept of Voltage Stability for a Two-bus System ................................... 526 9.\0 E ~pression for Critica l Re<:eiving End Voltage and Critica l Power Angle at Voltage Stability Limit for a T\I,'o-Bus Power System ................................................................. 531 9.11 Relatioo of Voltage Stability and Rotor Angle Stability ............. .................................. 532 9.12 Factors Affecting Voltage Stability .................................................................. ............... 532 9.12.1 Reactive Power Capability of Synchronous Generator ..................................... 533 9.12.2 Automatic Voltage Control of Synchronous Generator ... .................................. S34 9.13 Voltage Stability of Non-linear Power System .................................. ............................. 535 9.13.1 Static and Dynamic Analysis ........................., ......................................... .......... 535 9.13.2 Stabi lity of Non-linear System ............................................................................ 536 9.133 Bifun:ation.'.nalysis .......................................... .................................................. 536 9.14 Computation of Voltage Collapse Point ......................................................................... 537 9.\4. 1 Minimum Singular Value Method ....................................................................... 537 9.14.2 Point of Col!apse Method ........................................................... ....................... 538 9.14.3 Opt imisation Method ........................................................................................... 539 9.14.4 Continualion Load Flow Method ....................................................................... 540 9.14.5 Comparison of Computation Methods ............................................................... 541 9.1 5 Role of TransfOllller on Voltage Control of a Power System ....................................... 542 9.15.1 Method of Voltage Control by Tap-changing Transformers ............................. 542 9.152 Effect of On-load Tap Changer Transformer on Voltage Stability ...... ............. 545 9.\6 Reactive Compensation Methods for Heavily Loaded and Voltage Stressed Power Systems to Enhance Voltage Stability ................................................................ 548 9.16.1 Line Series Compensation ................................................................................... 548 9.162 Shun! Compensa tion ........................................................................................... 549 9.163 Static VAR Compensators ............... .............................................. ...................... 550 9.16.4 Synchronous Condenser at the Load Bus ........................................................ 553 9.1 7 De termination of Vohage Stability Using Sensitivity Indicator .................................... 564 9.18 A Voltage Security Indicator (VSI) Combining Fast De<:ouplcd Load Flow (FDLF) and Ne"10n-Raphson Load Flow Methods ................................. .............. ................... 9.19 Determination of Voltage Stability by g.V Modal An~lysi s ......................................... 573 Exercises ........ ............. ... ................ ............ ............................. ....' in T .. "l:[j ' r ....... 74 to
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10. Computerised Fault Analysis ............................................................... 576- 61 3 10.1 Introduction ................................. _.. ...................................... ....................................... ,... 576 IQ.2 Detenninatilln of Symmetrical Fa ult Curren! Using ZII", Inversion .... ,............. ,........... 5n 10J Detcnnin.ation or Fault Cumnt by Fonnulating the Impedancc Matrix Using Network Theory .......................... __ .................................................................................. 578 lOA Generalised Fault Ana lys is Using Zo ... Buil ding Algor ithm .......................................... 519 10.4.1 Sequcnce Network Modelling .......................................................... ......... ........ .. 579 10.42 Three_phase Balanccd Fault ........................................................................ ,...... 5!Kl 1O.4J Single Line to Ground Fault ............................................................................... 5&l 10.4.4 Line 10 Line Fault ............................................................................................. ... 5M3 10.4.5 Double Line to Ground Fault .... " ....................................................................... S8S 10.5 Detcrmination of Line Current During Fault Condition ................................................ 587 10.6 Utility of Fault Studics .................................... .... ............ ................................. __... __ ....... S88 10.7 Flowchart for Short Circuit Studics (Fig. 10.9) ........................... " .................. ,............ .. 588
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Appendix A
Unit Commitment ................................................................. 615-620
Appendix B
Applications of Computer Methods .................................... 621-M9
Bibliography .......................................................................................................... 641 In de-x .................................." .............................. .. ................................. .........• 643--645
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Preface The fundamental aim or this Ic)(t is to present a number of engineering and economic matters in power system planning operation and control in a comprehensive way. The topics substantiated by a number
of illustrations and computer progralllS describe analytical methods of power system and their
operation and control. To understand the text, some acquaintance with the basic concepts in power system as well as advanced calculus methods is needed. The chapters have b« n methodically arranged, starting with the basic aspects of power engineering problems. In each chapter, the relevant methods have been dealt with the help of suitable computer-based examples. In a few se<:tions, while dealing with operalKmal problems, optimiution methods have been preferred as they can be u5Cd without extensive mathematical proofs and arc useful in solving practical problems.
The text begins with an introductory discussion on common operating problems and basic lISpects of powe r system operation, including structures of power system, power system representation. and representation of power system clements. Different co nventional models arc brie fly described and analytical trealmCnts are presented to show the modeling concepts of power apparatus like Synchr01lO11$ generator, transformer, transmission lines, motors, etc. Matrix operational methods applicable to powe r network also get proper anention. Exhaustive analytical treaunents arc presented for the conventional load flow methods. A ll the conventional methods of optimization are explained with the help o f suitable examples. Some practical and applicational aspects of basic philosophy of ALFC also form part of the discussion. Fundamental aspects of reactive power control and voltage problems in transmission network followed by modem developme nts in this field including advanced treatments have been detailed. Compu terized methods for the analysis of faulted power system have been furnished as well. The text is self-contained and thorough . It is intended for a one·semester course for postgraduate students as well as II one-year course for senior undergraduate students in electrical engineering. Practising engineers and researchers will also find the book suitable for their use. The authors acknowledge the constant encouragement they received from the respected Vice·Chancellors Prof. N.R. Banerje& of Bengal Engineering and &ience University. Shibpur and Prof. A.N. Basu of Jadavpur University for th is project. They also express their gratitude 10 the xiii
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PREfAC E.
resp«live lk~ns, Registrars, and Head s of the lkpartments ofbolh these univers ities for offering all facililies in course of preparation of Ihe manuscript. The authors cordially invite any constructive criticism of or comment about the book.
Abllljit Chakrabuti Sunil' Hallla-
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Introduction 1.1
STRUCTURE OF A POWER SYSTEM
EleclTicily is the only fonn of energy used in Ihe industrial. domestic, comrnen; ial, and lransponalion sectors. II is a coveted fonn of energy. since it can be gener-l1ed in bulk and transmitted economically over long distances. Electric power sySte m deals with the generation. transmission and distribution of
electric enc:rgy associaTed with the unique feature of control of the flow or demand of energy al desired nodes throughout the power network. Figure 1.1 represents the fundamental structure of a power
network where generators produce electric energy. transfonners transfonn this energy into one voltage level from another voltage level and transmission lines wheel Ihe power from the generating stations to the load centres for the final distribution ofelccrrical energy to different loads. Tie-lines interconnect one system with the neighbouring electric system belonging to the same grid. The circuit breakers isolate a faulty pan of the network (the fault being sensed by the relays) while slatid rotary compensators may be used for voltage control at load or remote buses, Convent ion ally, loads are represenled in a lumped or composile fonn. The best location of a generating st ation being at a place very c lose to electrical load centre (i.e .. the region where the major energy demand exists). the practical location of the primary conventional energy sources does not necessarily coincide with the urban centres, The locat ion of a powe r plant is frequently governed by its doseness co the energy resource and transponmion facility of Ihe fuel as well as availability ofneareslload cenlre. Ellv iron menlal aspects arc also key factors in detennining the site of the plant. Mostly. a generating plant consists of generating units comple!e wilh necessary accessories. Control elements like different valves. e;.;citers. regulators etc .. also step up transfonners. and instrument transformers a long wilh breakers arc intended in the stat ion swi tchyard for the transmission of power and protection of Ihe system. Sources of inpul to the geueruting system arc conventional1y fossil fllels (e.g .. coa!. oil and gas). hydrosource and nuclear fuel. However, nonconventional sources like wind powe r, solar energy. tidal power. geothemlal power etc. arc a lso being used for .Ita"d~alone systems. An electric J)Qw~r system. e~en a small one, usually constitutes an eieetric network of ~ast complex,ty, The diversity of the system magnitude being great. ther~ is no general role rega rding the structure of the system that applies to any power system. However, any J)Qwer system could be categorised by a combination of generation. transmission and distribution networks. After generation. trallsmission plays D vi tal role in transponing power from the generating station 10 load centres. 1
Mat Tlal Jm dIem
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• POWER S)'STtM ANALYSIS, OPERATION AND CONTROL
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Transmission of power is usually done at HV / EHV / UHV range due to the kno",1'1 fact that it reduct'S the power loss in the line as well as improves stability. The common transmission voltages acros.s the globe are 3J kVI66 kVlIJ4 kV/ 1J2 kVII38 kVJI61 kV1220 kV1230 kV/345 kV/400 kV/500 kV in the HV and EHV ranges and 765 kV/800 kVl llOO kV/ 1500 kV in the UIIV ranges in most partS of the world whi le the generat ion voltages have commonly been 6 kV/ 11 kV/ 12.47 kV/I3.2 kV/ 13.& kV/ 15 kVI16 kV122 kV (all an: line-to· line voltage). In sub-transmis.sion leve l. the circuits diSTribute electric power to a number of distribution centres in a cenain geographical region at a vohage leve l that typically varies between 1J kV and 138 kV, the most common grades being 3J kV/66 kVl flO kV/ 120 kV/ 132 kV. The sub-transmission circuits may also receive electric power directly from any generator bus. Larger customers are mostly served by sub-transmission leve l circuits. In small power systems. the sub-transmission level may coincide with the distribution level. The distribution le,·el consists of the d istribution circuits in the o"eral l region of distribution. The larger consumers, i.e. high tension (H.T.) have been termed as primary distributors while low tension (L. T.) consumers an: the secondary distributors. The consumers consuming energy between 3 kVand 23 kV an: H.T. consumers while the consumers in the category of I 10 V-4001440 V lie in the class of secondary or L.T. consumer'S. The increasing demand on the electrical energy has led not on ly to diversificat ion of the generation. transmission and distribution network but also raised the points of proper utilisation and reliability of the e lectric pow ... r. This, in tum. has necessitated the pooling of larger number of powcr systems into a common grid and consequently insisting for proper scheduling of generat ion and demand. It also turned ou t that the incorporation of a large number of systems into a common grid makes the operation of the entire system very sensitive to the operaling conditions. Thus in addition to the study of po....er system opera/ion. the knowledge of p{)lver system ron/rol is very much required in order to run the system economically and to maintain a continuous balance between generation and varying load demand. In one way, the problems of dyn3m ic and transient stability. steady state stability, voltage and frequenc), n:gu lation , power optimisation need to b
1.2 THE NECESSITY OF CONTROL OF A POWER SYSTEM Present-day power systems operating as interconnected grid networks have sevcral advantages. First lransfer of power between areas is made feasible. enabling the advantage of each generation to be exploited and resulting ill improved compensation ofl03d fluctuations with reduced running costs. A reduction in the spare capacity of each of the interconnected system is also possible as a result of mUlUal auiSlancc between the areas. Power system control is vel)' much rcquiTl'd to maintain demand. whi le the system frequency. voltage level and security au maintained. Overall system contro l is based on a combination of manual inlen·ention, feedback loops, optimisation tech n iques and load demand. The r«juirement s for cont rol of frequency and power exchange can be implemen ted by load frequency control. This contro l is genemlly autonomous and each area is responsible for il.'l 0"'1'1 steady state power ba lance. The need for direct aCTion to control network voltage is usually done by the installation of automatic voltage regu lating equipment. The contro l responsibility is basically divided according to the frequency of intervention o f the physical phenomenon in~o l ved. The area leve l decision may also inc lude system voltage control with an optimal schedu ling of reaCTive power flow. Distribulion of reactive powcr generally docs not affect the system operating cost significantly, bUI an optimum aUocation may be important for maintaining steady slate system stabil ity and vollage levels.
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POWER SYSTE,\i
,
The management and comrol of a power interaction between many levels. Figure 1.2 i Manual cOnlrol is generally slower Ihan..·~!~i~-~ . resulted in consideration of digital ~~L . , ~arious le~e l s previously under La l ~ ~i' Pr imary control at the lowest i, the proper operation of the power s)"5tem. , small digital systems based on microcomputer inc ludes the control of go~emor set point and The higher levels of control range for longer , Howeve r. computer control schemes ha~e been
,
,
is a complex process and it requires proper ,~~. salient clements of Ihe cOnlrol hierarchy. . The a~ailabi1ity of digital computers has coordinating the control parameters of
.
control structure is most fundamental for lowest le~e l control was analog conlrol, but are now being imroduced , This comrol and ~oltage con!T(l1 of the station. intervals and are largely manually controlled. for economic {O
,
,
Power plant
.
e;t;t~~l
Manua! com",!
•• •• ••• •• ••
contro l
•
,
,
lood fn:q.
I Lower Ofdcr
~ont ro!
,
Digital
load fn:q. CQflt.oJ
I Manual
cont rol
", ,~
,
•
Digital control
" ,onom;e
0
,,,
"
,
, ", ,
, ,
M:muall
digital control
C
Manual
System
control
", • •
•• •
•
Mauua'
syst,.n+
control
• f ill. 1.1
,
conlrol SlruCtllt'C .
Material,
JfT1
dlreitJ
,to
~.
tN'tRODlIcrION
fl1!qllcm:y cOlllrol. Though unit commitment has been computerised in presen t-day power sys tems. syslem maintenance and system pl ann ing are most ly manually controlled. Ada!'t;'"<' control. the newe r control conce pt , is being imple mented to the sect ion in the power network fo r impleme nti ng most desired operat ion.
1.2.1
Control Methods
On ce the need for an efficient control of a power system is understood. thc next q uest ion that arises is. "How to control?" In old t imes, the po wer system \\"as mosl ly cOfltrolled by ma nu al in tervention. But Ihe man ual control be ing slow and dependent 0 11 the reaction of ind ividua Is, a large number of nHlnua l contro ls we re replaced by electron ic or analog co ntrols wi th the pass;.gc of t ime. In the lateu devc lopme nts, digi tal controls we re incorporated replacing some key analog cont rols and made Ihe overnll conlrol more relia bl e, faster and adaptive. Computer is th e key cleme nt in Ihe digital contro l of th e po wer system. It enables pr!X:ess ing of large num!J.c,r of differe nt data and lakes C9re of tht' con stra ints invo lved in ttle operation of t he power system. The computer repl aces the convcnt iona l hardware contro l cirtuitry in the conlrol loop and offers the most feasible control ompu t, taki ng inlo account the com plexity and variety of decisions that have to be laken in view of d li deU! operation o f a modem inle rconn ccll-d po .... er s)'stem.
Advantages of Computer Control
1.2.2
Th e majO!'
a dv~mages
of computer contro l of a power system arc as fo ll ows:
•
Highest speed of operation and fastest control action, maximum accuracy and high reli ab;1 il)"
•
O pt imal opera tion and control
•
Fast network state s.:a nning and monitoring
•
&ope of implement ing adaptil'e con trol
•
Low ma intenance and ope rating
co~t
1.2.3 Types of Computer Control System Basically, the dut ies of Ihe compute r system to control the power system operation arc of two types: (a)
Supervisory
(b)
Direct
In supervisory type, the computer generates an output to change the set point of tile cootro! ler. In th is case, the computer is JUSt the decision·making tool while the controller is the WO,kllOfSC in the control s)'stcln. The contro ller could !J.c, an analog or a digil3! t)'pe. In direct control, the compute r it.selfacts as controller and executes the decision laken by itself in order to cO ll tro! Ihe process. Depending 01, the Ucsign of the system, the computer conl ro l can be
(i)
Off·line
(ii) (ii i)
O!'-line tn·) ine
Mate-rial
:mI
dire-it)
Jtorai
poWeR SY5T!;M A.NALYSIS: OpeRATION A.ND
When Ihe COnlro! is off-line, the computer is human operator. The computcr is not \0 process these data and output the results In on-line systems. the computer i intcrfacing circu it ry and receives Ihe nece$Silry data processes the input data and outputs the result This is basically the simplest on- lin e control and is possible to have closed-loop on-line control where implementing the output dedsion. The computer i I' through necessary interfacing network automalically. In the in-line type comrol, Ihe operntor and directly into the compute r through the keyboard. The digital computer is not on ly the moS! I sophistiemcd. To economise the t 11 analog contro l equipment can still be used. This too. Analog conlTollers can be used to i also. Before applying direct computer control. it is gather data and provide track up for analog i storage and control of power system elements in successful imp1c mcnlation of a computer as a considered before implementing the decision
Ihe dala regarding the process Ihrough II the actual system. The duty of the computer is operator can ~ommend a control action. Ihe power system through suitable any human intervention. The com puler who then implemcnts the control action. 1 as open-loop on-line control. It is also computer requires no manual inlervenlion in is transmitted to the power system network
d". from the system and enlers them rapidly tool in power syS!em control but also th e most in the lower levels of control. some the purpose of instrumentation and metering system control aspects for training purposes
.,.
possible to use cheapcr digital instruments to Digital means may also be adopted for data sector. Careful planning is needed for the Figure 1.3 shows the steps that are to be I Fe~sibility
Pr
study of
definition
comp. oonl.
Step· I
St"f'l·J
ScIC(:(ion
Simul.lron
"'
~"""
obje-ctivc
Step·6
Training
.M implemental;1>I\
Step· 7 Fil. IJ
1.3
Sleps of planning
"~ I '."P"'" conlrol.
POWER SYSTEM
As a complete diagram of a practical power system ITansmission and d ist ribution) is tOO compl icated, it
"",",ii.
a ll the three phases (generation, a normal practice to represent a power
Mat, '
"0
II'ITRODUCfION
system by means of simp le s)'ste ms for each component resulting in si ngle-line Fig. 1.4.
di~£.mm. ~s
To the nei ghbouri ng ~y'tcm tic·lines Bus-2
Gcn.bu .•
shown in
~
T,
,
•
G '---/CB
Gcn bus
G~.
auxiliary
100'
Fi&- 1.4
Bus· )
G,
Dir.::ct;on of pow.:r now
In 10 ",On no:<: lion
Iie·1;IIeI, "b· I", nsmi~,;on !)'stcm fnr distribUlion S;"8k· lin~ "'p~mal ;on
LQad
(au~ iliary)
of a simple tw..... bus sysl( ln.
Any particular component mayor may not be shown in the diagram depend ing on the in format ion required in a system swdy. e.g. drcllit break ers need not be shoWl1 in a load now analysis diagram but are 10 be shoWl1 for a protection study. Different generator ~nd transfonncr con nections an: indicated by proper symbols. Equivalent circuits of power circuit components can be represented in the d iagrams.
1.4
POWER SYSTEM AT NORMAL OPERATING STATE
A power system operates in a normal st3l( if the following condit ions an: sat isfied: (a) Then: is a perfect balance between powe r generation and demand; consequently, the load flow equations an: satisfied. (b) The frequency./. is constant throughout the system. (c) The bus voltage magnitude IV.! is with in the presc ribed limit, i.e. IVJ",," !i IV~ ~ IV'!,,,,,, (This is requi red a s all the p<>wcr c'Iuipmcm and
apparulu~cs
( 1.1 ) arc s upposed 10 be
opera t~
at a specified voltage.) (d) No power system component is to be overloaded. However, the load is mOSily a cons! ant l ~ \'arying parameter and in order to !:leet th is slow change of the load demand, the nonna l operating state drifts with time (the load is mostly met by optimal gClleratiOfl schedu ling), Change in frcquenc~ causes change in the speed oflhe drives in the consumer's plant. Funher, it is necessary to ma int ain network frequency constant so that the powe r stat ions run satisfactorily in paraJlel, the vari ous motors operati ng on the system run at the desired speed and other de vices function proper ly. However, the most im ponant reason for keeping frequency of the electrica l system constant is that its constancy indicates power ba lance of the total system.
Mate-n II
Jtn
dlre-ibs
Jtoral.
PO~\'r R
SYST(M /lNAI.YSIS: OP(RATION AND CONTROL
O~erlOJding
of any power system component results in higher temperature of operalion and the component is likely 10 bi: d~Ul aged. System stab ili ty, given by Ihe mimI/Will power that can be transmined. also ind icates the power system operatin g at normal stale. Th is .m ·ady $llIIe ••,abili,y fimil (also known a~ .wllic I,,,,,-,mis.fi(m C"pocily) is given by
( 1.2) In atl altempl to lransmit mote power than this limit. synchronism is lost and the transmission system collapses. For short lines (less than 100 km). the thermal limit capability fi~es the loading of line whereas for medium or long line. the stat ic transmission capacity becomes the limiting facto!". Vo/wge .l labilily is another operati ng par.\mcter that needs to be considered.
1.5
OPERATING PROBLEMS IN POWER SYSTEMS
An insight in to the operat ion of any electric power system reveals 1hat frequent)' and vollage arc the prime ~lId main indications of proper system operation. Any di sturbance in the s>"stem operation causes varia! ion in Ihese two parameters separately or jointly and in cases of ~cvere system disturballces, Ihe freqnent), atallor ~oltagc varimions may be abnormally high indicating the I()<;s of .\ystem stability. Frequency variation being tile ,ause of real power mistualch, voltage is the sole indicator of Ihe rcactivc powcr imbalances illlhc syslem. Common operating problems tllat are inh·crelH in EH V power lines hn~c been ch ss ificd ""d bricny d~scrilx-d belQw, Major "r~a S "rstudy in Ih~ rclevant M"3 con sist of loadability. frequcncy d}'1lamics, transient swbilil}'. power line oscillations and I·oltage stab ilit}" problem. in addition 10 th e conventional steady state and tran sient stme power stability.
1.5.1
Loadability of Transmission Lines
II is defined as the oplimtlm power transfer capability of an EJ/I' lille IIIlder a specified sel uf opf!rating crilf!rill. In an EHV power system, the power tran sfer capabilit), of a transmission line is substantially aflected by nodal power injections and topological changes. II has been generally 3ccrpted thm the "mitt! slre"gth, i.e. capabi lity of a tran smi$Sion line , is substan tially affected by noda l power injection s and topological ~hange.<;. It has been generally accepted thaI noda l strength. i.e. the short drcrlil Cllp'lbilil)' (S.C.C.) of the system, is the inverse of the posilive seq uence equiva lenl impedance in per un it and it intlic~tes the robrmn"~·.f of the power network concerned. Thi s itnfl"'danee, also consisting of the source reac tance. is USUJlly dicI 31ed by the series rcactance of thc line whcn analysed in a loss- less frJme. Thc mOSI convcn tional fonn of,,·prcscnting the loadabi lity being in tenns of srtrge inrped(llrc~ Irwding (SIl.). where SfL ~ (V1rL.;,J in 1'.11., the basic express ion of power transfer is gh'en by /'-
,
(1.3)
(assllming equal sending and receiving end vollag.e, the power an·g le being Jand transfer re~clance be in~ X. X - XL..f being the reactance per unil length and L, the length of the line). Also.
X = xl. = rof L : ro,[fJh·: {f .tu-!k .L
V-;
,
V-:;
~ale"nall :>rTI
dlrelID
JIO ais
/,\ .'TRODUCfION
where
Zo "" surge impedance '"
If,(an~
( I .4)
c being the li ne reactance Jnd shunt
c~pacitance per unit
length, respectively, p .. phase constan t of the wave of proPJgation ( = m./k, tlJ being the angu lar frequ~ncy), oelect rica l line length ofthc line in radian and 0('" (lL ) being small, sin 0", O. Subst ituting equa tion (1.4) into eq uation (1.3)
p ""
V1
1 . sin '; " - sm ';'" (SIL J",-~ s inO Z" sin O
P
sin';
SIL
sinO
[in p.u .. SIL - (1'2120)]
(1.5)
Equation ( !.5) ind icates that the power transfer c~pabi I ity can be repre~cnled in tcnns of SIL. Figure 1.5 represents the loadabil ity of a typical EHV sing le c ireu it tine assuming variou s I inc lengths.
r
~
0
,"
~
,•-
" 10
,.;
I '0
~
0;
"'"
200
Line length (kill)
fiR. I.S
300
•
"'"
300
!'ronle ofline loadabi!iw
Hi gh sourCe rN Clance plays a vita l role in limiting the line loa dab ilit),. Loadabilit)' can be improved by reduci ng the reactance of over head wires and placi ng series capadtor in line as we ll as relaxing the vo ltage drop constraint. Reduction of line power loss improves line loadability also. This can be achieved by utilising low resistance conductors. parallel wires in the transmission system as well as by placing shunt capacity at the load end. Loadability is severel), impaired b)' the application of shunt reactors in the Jines up to 500 kV, This impainnent is not much for UHV lines. Fi gure 1.6 represents two profiles of line loadability for t\\'o systems. one having higher source reactance and line reactance (tenned as I<'ellk system) and the other for lower so urc e reactance and line reac tance (lermed as robust system ).
Mate-nail Jm dlre-itJ
ltoral.
,, • POWER SYSTEM ANALYSIS, OPERATION AND CO,VTR OL
I
,,
•
2. 5
,,
rur robust system
I
I
/'
'.5
I
•
I
, .0
05
i
o
i,
,
~' ig.
".
,
200
300
l.im: ICl1l;lh (km)
".
1.6 1'lOliks of iu.ldabi!ily of lobus! and
•
wea~ s)'Sl~ms_
•
,•
1.5.2
I
When the power system struc ture is suc h Ihal a si ngle line oUlage in the uansmission sysTem creales is lan ding condition, a enrdll l evaluat ion of rrequency bell.w inllr i~ nc~-dcd \0 accretion reliability. During nannal oper.llion. Ihe power flow on the t ie- line can be represented as
, •
II I
Frequency Dynamics 01 Transmission Line
If• = 1v.'11'11 X ,'"(~. _~I ) o
( 1.6)
The ra te of change of the line rea l power flow can Illen be repre$Cntcd as
I
(1.7)
I
! !
I I I •
/:if. and 6/, being Ihe
Til ' Ihe Sialic tran smission
capacity. In case the slatic tran smi ssion capacity oflhe line is low. higher frequency deviations are to be tolerated for signific~nt contribution of power flow through the link_
1.5.3 Overload and Frequency Oecay Rate II is well kn own that if there is any mismatch be-Iwttn the mcrhanica l power input and the e lectr ica l power ou tput lIos~ being negJe<:ted), there will be a tcndcncy of speed change in the generator rotor. As the machine consists of hc~vy rotational mass. the speed change c~nnOI be insta ntan eous and is governed by the following equJ ti on .
T - Jdw
I
,"
Jto ais
INTRODUCTION
dw T P~ I _=_=_x_
oc.
dfJruJ
-dw .. - I ( Pr-P ) dl
J(J)
( 1.8)
~
As. p. '" .accelerating power = P, - P,; P r being the turbine po"'er output and PI' the electric.al power output. Also,
M = Joo
then,
dw
-
I.e. .
=
where M = (2H/w,,.), w,,., being the synchronous speed of alternator rotor. Thus.
d(211f) = 2Jrf(, _,) T • dI2H
fLL(p,_, ) dl 2 H •
oc. II
(1.9)
In a SO Hz system from equation (1 .9), the initial frequency decay rate for a sudden load demand can then be represented by
s~tem
toeOCQullter
(UO)
The initial frequency decay r.lte for diffCR:nt types of loads for a typical radial system for varying attempted overloads has been graphically reprucnted in Fig. 1.1.
,
Z"'''
of
Frequcn<::y
mi~ro
dependent __ I""d
'~d
,
(;,L
2
Frequency indcpatcknlload
(Hz/sec)
Ancmplcd overload (p.u.) FIg.1.7
Init ial frequency r.lIe for varying attempted o,·er\oads of static and frequc:ncy..(\epf:ndtnl h...Js.
Mat,
POWER SYSTEM ANALYSIS: OP£RATION
1.5.4
Transient Stability Problem
imbalances between the rea l and reactive power Frequent topological changes, transmission to I\Ifl the systcm near the transient stabil ity ge nerat ion and demand insist the powe r ":::;,"~:: causing the loss of transient stabil ity. margin. causing instability ph" ,~'m~", " Basic aspects of transient stab il ity have been a number OfIC"15. The reccnt lrend 10 adjust ,"" "" ,,",',;p . du ring lransicnl stabililY problem the generation. popu larly known as AGT has been found to be an effective too l in transient stab: lity. Figures 1.8 and 1.9 ill ustrate such a simulation.
("".=,',
V (kV )
---__
--- - --' \vilh ,\(;T
WililOttl AGT
" •';1:. 1.8
I mpro~cnk·nl
in "ol1age profile
10 MiT in Iflmsicnt slability pro;>blcm .
P (I>IW)
,With "GT
-- ------ -
80<)
"JO
.,
ol--~--~Ti me (""c)
,
Wi~",'"
fig. 1.9
Improve ment ,,,'" ''
AGT
stabilily due 10 AGT.
Jlen
1:m1 dlreibs
Jtc>raJ
1.'1T/I.OOLle r/ON
1.5.5 Power Oscillations Sustained law frequmcy oscil lat ions lIave been reported in po,",er supply systems. A comprellensive literature review reveals tha t the amount of p
1.5.6 Reactive Power Limitations and Voltage Control Problems In an EHV p
'c power availability may be restricted. leading to the system voltase collapse in case the line is reactivc power constrained. A sudden increase in reactive power demand in a reacti ve power constrained line is generally due to the cont ingcncy in transmission network (e.g .. tile tripp ing ofa heavily loaded EHV line causing an in~rt'asc of the load bu rden of the ~djacent line(s) for maintenance of th e constant system load). The addition~1 reactive demand caused by the d isturbance is generally compensated by the system reac tive re serve. if ava il ab le. allowing the systcm to sell Ie down at a reduced leve l 0'transmission Voltage. On the other hand. whcre the reactive re serve cannot cope with the sudden ri<;e ofrt'aclivc demand. 5)"!;tcm vollage instab ili ty results. This collapse may occu r even though the real power requirements o fthc system are met and the frequency is stabi lised. In case of a cont ingency in a transmission system. the seri"$ rl'ClCfive lau (Q,t) increases. When the remain ing healthy line(s) loading surpasses the SIL (surgc imperlance loading). the ris ing Tilte of series reactive losses is s Ubst~nlial showing a steep increase: in Ihe rate of scries reactive loss aga inst SIL load ing at its hi gher magn itudcs. This. in Turn , depresses the system voltage in the lines. As the SIL is directly proponionalto the $Gua re of the sySlem I"Ohage (SIL - 1' 11Z<)). it starts to drop as the vo ltage dccreascs causing further series reactive loss. In addit ion to thi s cneet. reacth'c charging capability. being propon ional to thc square of the transmission volmse. decreases with decaying \'o ltagc causing funher deTerioration of the system voltage Sla bl e stale. In extreme cases. allthcse elTccts may add to crea te high magnitude of th~ line react ive loss for ea~h c.~ tr.l un it o r rise in rea l lo~d . This cnonnous rise in demand of reactive powcr invi tcs severe volt,,!:c control prob lem s. and in case of weak systcm.s. there ma y be sponL1ncous volt:lgc in stability. In a react ive power constrained system. tlte condition of voltage instability is always governed by the lim itation of reactive power ava ilabil ity. The magn itude of the Iimit iLL g Vl\ lue of the reacti I'e pown at any operat ing conditiOl1 can be determined anal>1ic ally and it reveals thm the stable voltage state ean onl}' be maintained if the SY$tcm possesses th e corresponding limiting value of the reactive power tr:msfcr capab ility.
1.6
SECURITY ANALYSIS AND CONTINGENCY EVALUATION
Under nonnal operat ing cond itions 3 power system may face a cont ingency cond it ion such as oLltage (comp lete or partial) of a gener.lting ~n it or of a line, a sudden increase or decrease of the power demand on the system. A system operdtor has to ana lyse tile effcct of suc ll hig.llly probable
MatE
Jtor"
POWER SYSTEM ANALYSIS: OPEIUoTION AND CONTROL
contingenc ies so thaI the operator may take cOITC{:\ive aClion in the event of the ir occurrence. Thus,
the ana lysis of some of the most probable ronlingencies helps in enhancing system seeurity. The security assessment and its enhancement fonn an importanT part of planning and operation of power systems that arc continuously expanding.
The main operating Slates of a power system may be classified as
I
(a) Nonnal (b)
Emergency
(c) Restorative However. later on Iwo more slates. "Alert" and "Extremis", were added. For the sa ke of understanding, on ly the three-Slate trans ition diayam of Fig. 1. 10 will be considered here as this diagram provides a good conceptual picture oflhe overall computer control requ irements of a po""cr system.
Normal Slale
ReSIOmli,-c
Emcr!:e"",}, Slate
~ IJI~
I
I
Fig. 1.10
State transition diagram.
Most of the times. the system rema ins in the nonnal state as stated earlier. In this state, the load now equations are sat isfied and voltage constancy is maintained. with all operating (or inequality) constraints being satisfied. When these constraints are not satisfied. the system is said to be in a lcn state. Contingency evaluation is. therefore. required to find out if the prevailing nonna l operating cond ition is secured. The imponam and probab le contingencies to be considcred are: • Outage of a line • Outage of a generating unit • Single phase or three-phase fault.
I
The modem powcr system conlro l centres (or load dispatch centres) are the places for secur ity monitoring , In these centres. on-line identification of the actual operating condition is undcnaken util ising a computer-based technique. known as stale estima/ion , The state estimat ion gives the load dispatcher the best (stim31e of 1he comp lex bus voliage at any instal)! from the redundant SCI of
1,1 atE
/folTRODUCTIOIII
telemetered data and brcilker status. The sec urity anal)'1i5, with the help of the state estimator. then finds out the impact of the contingencies using somc fast load flow method such as Fill! Decoupled Load Flow (FOI..F). In this way, the real time data obtained at the energy control (entres are examined by the security analyser to find out the security of the system. If the sys tem ;s found to be insc(ure, then the system engineer dctemlines the preventive (ontrols \0 be applied to brillg the system back into the se<:ure zone. This may require geneT"lltion Tesdeduling andfoT a change in the interchange schedu le. This nonna lly would deviate the system from the most e<:Ollomic operation, but is quite justified and is very much desirable. In case the emc-rgency occurs due to cascading events or contingencies, the corrective emergellcy comrols such as optimum load shedding, the network rearrangement, starting up of some quick-stilrl units are 10 be applied to bring the system back into a secured state. Contingency such as outage ofa line. generator or loss of a transformer would reduce the securil)' leve!. The operating prob lems as indicated earlier may also lead the system to a state having lesser , secur ity. This state is now the alert state where the system remains stab le alld the operating constraints are satisfied, but an abnormal voltage and frequency condition may arise. This type of state can be tolerated for some time. Preventive controls (for example. start up of standby units andfor compensators) may bring the system from this state to th e nonnal sta te. However, when Ihe system is in alert stale, some additional contingency may take p lace such as fu rther loss of unit or line. The (ontingencies in the distribution or sub-transm ission levels may also lead the system to another state with lesse r securil)'. This is an emergency state and emergency comrols must be imp lemented to $ave the system from vulnerdble collapse. There may be undue voltage depression andfor overloadillgs of lines during emergen(y state. If the emergency controls fail. lhen the over loaded line must be tripped and the system faces the risk of total shutdovo"ll (the extreme state). Load sheddi ng and intentiona l voltage degradation are the two most effective means of imp lementing emergency control in order to $ave the power system. The restora tive stale invo lves reschedul ing of active and reactive power, re-synchronisation and gradual load pick-up. The system now returns either to a new normal state or to the previous normal state. In order to avoid the damage to the costly (omponents of the power network. as a first line of defence, protective devices are used allhe appropriate pbces in the system. FUlKtions such as re lay ing and voltage contro l are carried out within milliseconds and executed I(l(:a lly throughout the system and no cCIlualised de<:ision-milking process is involved. Typically. a re lay detects the fau lt and initiates a circuit breaker tr ipping to remove the unhealthy part of the network or faulted components from the rest of the system. Another importanl objL""Ctive of lhe emergency control is to perfonn automatic reorganisation of components. The re-closing of a line must be fast enough. The fast application of emergency controls saves the system from the loss of synchronisalion and subsequent island ing.
1.7
AUTOMATIC CONTROL
The necessity of conlrol of a power system being highlighted, it is imperative to mention that Ihe con trol measures are most effc(tive once the automat ic devices are the control elements. All10matic load frequency control (ALFC) and automatic voltage control (AVC) are the tll"O most important aspects that can be im plemcnt;:d to ensure proper system operat ion.
1.7.1
Automatic Load Frequency Control (ALFC)
In th is contro l circuit. there are two feedback loops, primary and secondary. The purpose of both these loops is to achieve re~1 p<>wer balance or load tracking in the system. ALFC loops are designed to MatE
Jtor"
I POWER 5Y5TUI ANALYSIS; OPERATION AND CONTROL
maintain power ba lan ce by an appropriate adjusunent of the wrbine torque. By means orllle primary
loop, a relatively fast but course frequency control is achieved. The secolldary ALFC loop works in a slow reset mode to eliminate the remaining small frequency errors. This loop also controls Ihe power
interchange between pool members. While the primary loop response is over seconds, the secondary line adjustments may take about minutes and will stop only after achieving zero frequency error. [t may be nOled that since the whole group of generators within II given area move coherently, the frequency dynamics is slow. thus charact~ising them all wilh the same lI/(frequcncy crror). In the case of interconnected power systems, tie-lines are erected \0 interconnect the neighbouring areas. Muhi-area d)l1amic is imponant to ~ di$Cussed. All the power commands can ~ executed in unison among all the g~nerators under control. The secondary AI..FC loops in a multi-area system contain control si gnals. now referred as area conlrol errors (ACE). wllich , in addition to frequency error I!/. a lso contain tile errors in the lie-line powers. These corn:epls have ~en discussed in Cllaptcr 8.
1.7.2
Automatic Voltage Control (AVC)
In tlles.c control systems, bus voltage is measured utilising a potential transformer and is compared to a reference after being rectified and filtered. Tile resulting error voltage. after amplification. serves as input to an cxc it ation control systcm whcre output direct ly feeds the generator field. A drop ill thc tenninal voltage caus.cs a boost in the field current This increases the reactive power output of the machine. thus tending to offset the init ial voltage drop. The AVR loop maintains reactivc power balance ofa generator by maintaining a COI\5tant voltage ltvel. Besides generator buses. shunt capac ilon are: used to kcy buses to ensure: an overal l S
1,7,3 Control Components In Power System AC Power System is controlled primarily by mechanica l me ans such as circuit breakers. OLTC of tran sfonners and isolators. The sol id state devices and communications system are used for monitoring, da1a-logging and protec1ion. Thus, the primary control 011 power side suffers from the following disadvantages: • Comparatively slow operation • Controls are of On/Off nature and are not regu latory • Contro ls cannot be used frequently as the controlmcchanism has a tendency to .... c~r-Oul. Apart from the abovc. some time the present-day power system sutTers from lack of a decision suppon systcm for optima! and reliable power system operation. Sophistkated computer systems and faster and reliable communkation network betwee n regional and state load dispatch centres (RLDCs and SLDCs) should form the backbone oflhe decision suppon system. The data from remotc substat ion s need to be gathered at RLDC~SLDCs for the analysis of grid operating conditions. The analysis can then be used to gcncrote control commands for transmis sion 10 SLDCs and remote substations. tn a number of cases. due to the mccllanica! contro l mechan ism supplemented by lack of proper load di spaTch tool s. the present-day power system is inflex ible and stiff. This makes Ihe system
INTRODlICTION
manager. a helpless speclalor of various grid problems such as overloading of transmission elements, poor VAR management etc. Lack of controlmcasures to deal with emergent operating conditions often leads to grid disturbances Bnd blackouts. However. with the availabilily oflhyristor valves for power applications. it has become possible to repl ace the mechanical operations by electronic sw itches. Though the ONIOFF operation can still be performed by mechanical closing/opening of circuit breaker. it is now possible to change the basic characteristic of tlte network by electronic devices to achieve the requisite flexibility. The availabilily of faster control is a necessity but not sufficient for making the AC system flexible . One should first address the objectives to be achieved by the FACTS (Flexible AC Transmiss ion System). Some of the objectives can be as follows: l ine~
wilh a view to either avoid overloading or to minimise
•
Regulate power flow on AC powcr loss
•
To operate the system at a safe power·angle for same power del ivered
•
To enhance th e power transfer capabil ily of the system by introdu cing improved dynamic characteristics
•
System island ing under ext reme conditions
•
Strategies to save thc system/islands from total collapse.
Aftcr the objectives have been iden tified, the following stra tegies need to be decided. ( i)
Planning and openllional system strategies: •
Syste m analysis and planning
• •
Loss optimisation System security
(ii) FACTS controllers stratcgy (ii i)
Inter-uti lity communication strategy.
The details of the FACTS project for a region can be worked out based on the following: •
Installation of s.eries capacitors on ceruin sect iooslJines
•
Installation of statk VAR compensator.f (SVCs) at strategic loca tions
•
Insta 11ation of phase·shifters. i r required
•
Low·frequency oscillation dampers, if mjuired
•
Commu nication network
•
FACTS controller with on·line data monitoring
•
Com puter software for grid analysis.
The above items not only require huge investments but also coordination among the various utilities. A systematic approach is to be adopted and the investments are to be phased out over a period of time. The following phases arc important aspects in FACT planning. Phase I:
System Security
In the first phase, emphas is shou ld be laid on prevention of faulls spreading into the syStem and creating gr id instabil ity. This phase can be termed as system secllrity ph ase.
MatE all
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! POWER SYSTeM ANALYSIS, OPERATION AND CO,WROL
I
In tllis phase a few pi lo! project(s} can be taken up for the installation of switched series capacitors in ceNin selected locations, There shall no! be any nec essity of any elaborate FACTS controller at Ihe stage. The control actions can be derived from terminal sub-stations. Phase II: Strong Interconnections
In the second phase em phasis should be on strong inter-utility interconnections free from 101'.'frequency oscillations. This shal! invo lve: •
Extension of switched series capacitors 10 many other sections
•
Installation ofSVCs at grid po ints
• Phase shifter, if required •
Development of FACTS co ntrolle r
•
Communication means .
Phase III: Optimal Operation
I
Optimal grid operation can be the walch-ward orthe third and last phase. In this, loss optimisation can be carried out through the FACTS controller. Many other advanced control means can be used for optimal system operation. viz _ phase-shifters. SSR (sub-synchronous rtso!lllnce) dampers. dynamic loads, etc. The evolution of FACTS has to be progressive with time, not only because orthe huge resource requireme nts but alw be<:ause of Ih e fast development on technologica l fron1. FACTS i. likely to become more and more economically viable with the passage of time.
1.8 USE OF COMPUTERS AND FUTURE TRENDS The use of digital computers for solving a load flow problem was first made in 1956. Subsequen tly, other studies such as fault level anal ysis, transient stab ility analysis, economic load dispatch etc. were carried out off-line using digital compUlers. In more recent times, however, it has been realised that digital computeT can also be greatly helpful as on-line monitorin g and controlling agent5 for the modem large-scale integrated power systems. As mentioned earl ier, a compelling reason for th is is the critical dependencc of the rel iable operation oflhe large in tereonnected power system on a proper selling of the operati ng conditions. Yet another reason is the dramati c improvement in the capabili ties of the computing system assoc iated with a steep fal! in their costs. Due to Ihe avai la bility of modem. fast and cost-effective computers. it has been possible to achieve a greatn fl e;o;ibility. accuracy, speed and economy in real time control and mon itoring of power systems. Mi cro, mini and mainframe contputers are increasingly being ~d for both off- line computation and on-line monitorin g and contro l of power systems. bolh inside and ou tside energy comro! centres. The add it ion of system security function has initiated a significant change in Ihe scope of control ce ntres. This addition in volves major changes in real time data requ irements and the sophistication in data and in foml3tion processing. The functions can be implemented in a completely automatic manner using $UpervifOry control and daro aequiJ"ilion (SCADA ) systems. It invo lves data colle<:t ion (the da ta involved arc active and reactive powers flowing th ro ugh the tines and trans follTl ers. ,'o liage and frequencies al various bus bars. status of breakers ami sw itch es etc.) and display of the desired data as well as data processing for netwon:: state estimat ion. automatic generation control etc. The output commands are used 10 alter generation, open or close cireuit breakers. switch on or offreacth·e powe r
Jten
INrRODUCTION control elements and so on. In addition to automatic genemtion control (AGe) and automalic voltage control (AVC), the olller denigrated works of the computer' control are economic dispatch, security monitorins;, security analysis, off-line short circuit calculations and state estimation.
EXERCISES I.
Draw a block diagram of a Hierarchical Control StnK:tun:.
2. What are the advantages of computer control in power system'! What are the types of computer control'! 3,
Draw the single-line diagram of a two-bus power system. What is the usual range of transmission voltage in India?
4 . What are the 'staleS' in a power system? What do you mean by 5.
opeilting stale'?
What do you mean by 'loadability' of transmission line? Derive an expression for il.
6. Find the expression for the frequency decay rate of a OYffload. 7.
~nonnal
turbo- ,&~temator
following an attempted
Write short notes on (I) Security analysis and contingency evaluation.
(U) fACT system.
Mate-rial
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, ·•
Modelling of Power System Components
•
2.1
INTRODUCTION
In order to implement computer COli/rot of a power system, il is imperative 10 gam I clear understanding of the representalion of the power system components. Component modelling thus becomes very important Studies of electrical energy systems are based on the simulation of actual phenomena using models behaving exactly in lbe identical way as the elements in the physical system. In research, it is necessary to have models pel ",illing precise and detailed simulation. The different parameters must be accessib le and the models are required 10 follow the physical prIX'S' as closely and as faithfully as possible. Then it is required to solve mathematical equations governing these phenomena. Modelling of active elements, c.g. generator, transformer etc. is relatively difficult while that of passive clements. e.g. transmission line, relay. inductive VAR compensator tIl: , is easier. Passive circuit elements are mostly II'IOdelled by their parameters in the equivalent circuits wbile the active power system components are modelled by their operation in steady, transient.1Id sub-transient state, Tbe models used in tbe power system give precise results in a certai n field of bypotlteses correspolldi ng to their use . Here. the concept o f representation of tbe physica l reality of the phenomena disappears and onl y the relationsbip between data and results e~ists. Their limited use leads to s impler models tban the preceding ones and necessitates fewer data processing requirements. Thi s means that they can be more easi ly integrated into large simulation packages. In these models the process representation is based 011 the fundamental physical laws. lbougb the model is simplified, its method of representation takes into aC(:ount tbe principle of non-linearity inherent in the physical phenomena involved. The models can be structured in modules to simplify subsequent upgrading and correction of tbe network. To a greater or lesser e~lent, tbe system variables requ ire time in order to respond to any change in their operatio n. Modelling sbould take care of tbe cbange and system equ;uions arc to be written to designate tbe state of the operation o f the element. However. writing of these equations obviously requires assumptions and bence no clear definitive model e~ists for most of the active elements. Proper model is to be selected by the programmer that suits the requirements of the problem.
"
Mal
I
"
MODEl1JNG OF POWER SYSTEM COMPONENTS
The modelling of a synchronous generator needs utmost care as it is the heaTl of the power system. It may be observed that its modelling is the most difficult task due to its "stiffness" to the changes in the operating conditions external to the mach ine. On the other h3nd. there is transmission network that respooos almost immediately to the configurational change and loading alteration. Thc time constants associated with the network are insignificant in comparison to those of the synchronous machine. The rotary swing funher complicatC$ th e modelling. The present text will give adequate stress on an alternator modelling such that the basic building blocks for computer·aided analysis of the operation of the power system can be developed at thi s stage.
2.2 MODELLING OF SYNCHRONOUS GENERATOR (ALTERNATOR) In modelling of the synchronous generator. the mOSt appropriate frarne: of reference is one that is attached to the rotor. This frame rotates at the same speed of the rO(or. The major axis of this frame is known as dir«r axis (the rotor polar axis) Of si mply the d·axis ~r.J the second axis is 90° (elec!.) Ipan from this polar axis and is known as the q,tadratu~ axis (the inler polar axis ) Of the q-uis . In this text. the synchronous generator has been modelled in five different modes. Each mode is associated with some assumptions aoo the programmer is 10 select the panicular model depending on the requirements of the following assumptions: (i) Thc rotor speed of the alternator does not vary more than the prescribed limit (ii) Rotational power loss due to windage and friction are neglected (iii) Mechanical power input is constant.
Mod.! '0' From the basic eonce~ on electrical machines. it is well known that a group of synchronous m:IChines or a pin of the power system may be represented by • Jingle eq"imlenf synchronous mDchint'. Similarly. an infinite bllJ. representing a pan of the system having 1:ero impedance and infinite rotational inertia. may be similarly modelled using the operating stlue equations while the: machine voltage is assumed to be constant behind d-axiJ lran£i~nt reactance (X;). In this chapter. the salient pole synchronous machine is only considered. 115 the cy lindrical rmor machine model may be regarded lIS II special case of a salient machine model with Xd = X~; X~ and X~ are the direct axis and quadrature axis synchronous reactana:s. tespectively. To model II !iJIIlient pole generator in transient state, two transient voltagcs Ire to be assumed (E:: and~) representing the flu" linkage in the rotor wiooing. The transient operation is assoc iated with addition of transient reactance and voltage to the Sleady sl3te model (Fig. 2.1). The phasor diagram of the transient condition in the machine has been sllo ..... n in Fig. 2.2. where the induced voltage E h:ts been ronsidered the sum of the two vol tages EI3nd E. unlike to that in the sleady state model when E = E, and Ed = O. The: transient voltage in this model can be sho ..... n to e"i5t behind tlte The equations representing this model are thus transient reactances X~ and
X; .
,.,
~ = VI + IdR~ + ' . X;
(2.1)
~=V.+lqR~-IIIX;
(2.2)
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POWER SYSTEM ANALYSIS, OPERATION AND
,
a-axis
X. [ Fit- 1.1 Phasor diagram of steady Slate operation axis quantities and suffix q fOl' quadrature axis
=
=X,+X_j
X, = x,+ x.. •
salient pole a.l!ema101'. sumx d stands fOl'direct Sumx I indic*, leakage quantity; V4 and II/
I negative.
d-axis
f. )0' •• 2..1
Phasor
dial""" of \1\(0 uanm~':i;':;":' ~.:;:'=;~ .':;::'.Wicnt a1tematOl'. E' is the traJ15ient voltage et liT II negative).
.,1',"
[Here. E and V represem induced and terminal uansient reactance of the salient pole alternator. axis and quadtature axis components of the
while I is the machine currenl and X' is the d and q are used to designate the direct ~. Vd and 1.1 are numerically negative .]
1,1 atE
MOD£LLlNG OF POWEI! SYSTEM COMPONENTS
Mode! 1
Here. the model of the machine has been assumed to have the magnitude of COII5tanl voltage behind the d-axis transient reactance only; q-axis transient flullO linkage has been assumed to be so small that it has been neglected. However. the mechanical system equations have been considered in thislllOdel. Hence, the modelling has been done utilising the equalions (2. 1) and (2.2) in addition 10 rotor swing equations given by equations (2.]) and (2.4).
daJ=....!..(p' _ P. _ D dt5 ) dlM"
. '"
~
(2 .3)
dl
dw
-=m-2tr!o
where. M ""
(2.4)
'"
l!-.-
KJ, 1M .. aIlJular momentum
p. '" turbine waft power p • .. generator electrical power output
H .. inertia,constant 10 '" base frequency OJ .. angular frequency
D .. damping coefftcient B '" rotor angle I
ModII 2 The drawback of Model-O and Model-! is that the eleclrical dynamics have 00{ been considered. Model·2 includes the machine operation with time Yarying equations assuming d-uis transient effects only. The equations represenling this model are given by equations (2.1), (2.2), (2.3) and (2 .4) in addition to equation (2.S) that represents the governing differential equation to allow the rotor flux linkage to change with time. From the phasor diagram of FiS. 2.2.
dEf dt
r;;
where, negative.
J E -E.) .. EI +(X ... - X~ )/~ -Ef f
r;
(2.5)
T;
is the dirtd lUis IrQruitnllime COlllltllll and Elis the applied field voltage. I ... is numerk:aUy
Modi! 3 In this model. tbe InInsient effects in both the d and q-ues have been oolUidered. The soverning equations are represented by equations (2.1) to (2.6). Equation (2.5) considered the flux linkage changing with time for the q-axis while equation (2.6) describes the same for the d-axis. From phasor diagram of Fig. 2.2. equalion (2.6) can be formed as
Ed - (Xq - X;)lq - ~ -= - -= d/ T' T'
dE;
•
(2.6)
•
Here T~ is the quadrarufZ lUis /ransient/ime cons/ont .
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POWER SYSTEM ANALYSIS: OPERATION AND CONTROL
Mod.14
Sub ·/ransien/ nate or operation has not yet been considered in any of the models discussed 110 rar. Due to the presence of a damfN' winding . sub-transient state or operation needs attention. Similar to the transient modelling. in this case also. two sub-transient new voltages (Ej and E:;) have been assumed. Figure 2.3 represe nts the phasor diagram of the alternator during sub-transient stlte of operation. The governing equations can be wrillen as
e; = Vd + fJR. + f qX;
(2.7)
£;=V, + f. R. - fdX;
(2.8) (2.9)
de; _1£' -(X ' - X' )f - £']fr' dtJ " qf.
(2. 10)
E
,, ,, ,
,,
.
,
,, ,, ,
..
"
..... ,,' '
,,' ,,'
,
, E"
,, ,,
"
' .~
" ,
,
"-. v IR.
I I
,
"
f't" 2.3 I'hawr diagram of the sub-transient stale opel3lion of the witnt alternator. E" is the transient voltagc (d·axis projeCtions arc numerically DCgativc)
r;
In Ihe above equation. and T.- II/'C coru;idered 10 be sub·transient d·tuiI and q-axiI time cons/an/I. This model is chnractcrised by equalions (2.5) 10 (2.1 0 ) in addition 10 equalions (2 .3) and (2.4). Groups of synchrono us mach ines or pariS of Ihe syslem may be represenled by a sing le synchronous machine mcdel. An infini te bus bar. representing a large niffsy$tcm. may simi larl y be modelled as a single machine (Model·O).
MJlerl
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Jtc>raJ
MOD£WNG OF POWER SYSTeM COMPONENTS
2.3
MODELLING OF A SYNCHRONOUS GENERATOR ' IN A NETWORK
The synchronous machine equations have been framed wilh a reference roIIling wilh its own rotor. The rcal and imaginary components of the vollages in a network reference frame (Fig. 2.4) can thus be formed as
(2.1l )
Here V, and V,," repastnt components of voltagc V in real and imaginary axis. Imaginary axis (Network)
,
d·w
(M.~hinc)
:
. V .., "'................... . ... . ..' ,, ." ,, - ,,-
"...
,
~
..: ..., ,- ,
. .. ,
- v," "---~', . .... .
. ,·axis (Machine)
, ,,
,, -
,,- ,,-
Real axis (Netwtd.)
V,
." ..-
~\
•.".1.4
Co-~lation
between alternation and network frame of ~ferellCe.
It may be noted here thai the two reference frames and the relationship between components of the reference frames (equation 2.11) are commonly discussed in the lilerature. II may also be oolCd that a given phasor V has been distributed inlo two very different forms of components depending on the angle lj of the mach ine reference frame. II may be observed that the vector V can also be represented in the form of equation (2.12). (2.12)
where V, and Vd are purely real quantiti .... Assuming the positive scqu~nce volt~ges and CUrTen" with the ampJirude and phases, the general relation between tllest variables may be wrinen for the network
.
(2.13)
[/ ) =[Y] [V]
[Tn case of representation of the variables of the machine, the expressed quantities in d-q reference frame must be convened inlO a common referellCe frame by axis transformations.] The power equations for I salient pole altemaror can be modelled by anyone of the models. lbc power equations in the steady stote and transient state are given by sin 8 +
rt(...!.... -..2...] 2
X~
Xd
(2. 14)
sin 28
Material,
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POWER SYSTEM ANALYSIS, OPERATION
....
P.
,
.
=~sin.5 x'
-....!....] sin 2.5 [....!... X' X',
,
(2.15)
when . .5 '" LE - LV = LE:' - LV.
2.4
MODELLING OF GENERATOR
The modelling of the generator remaiM comrolJload frequency comroi) and ucilarion volrage regula/or) achieve~ reactive power frequency control achieves real power "'::;;' Modelling and Trurbint: Modelling are thus
2.4.1
if the role of AGCJLFC (auromaric generariOl1 ' ... not inchvled. Just as the AVR (allrOf1UJlic by maintaining a constant V(lltlge, the load maintaining a constant frequency. Govern ...r '~h important in implementing AGe.
Governor Modelling
If the load increases, the speed of the '~n'oc"d~. slightly. The governor of Illy thermal unit reacts to this speed variation and the entry of some more steam from the boiler to the turbine which, in turn. increases the increased steam flow reduces the boiler' p,esSUfe, . and water flow 10 release the steam pressure. which reinstates the increase of an adequalC Fortunately. the large thtrmol ineF/to of most systems enables the load frequency performance of the turbine. generluor and load to ''';m ... of the boiler. so that, for short duration of load change. the boiler pressure may be constant. The generator mainly determines the short·\Cnn response of the system to the load Many rorms of the governor i of which include. in some way or the other, the variation of the turbine·generator as the bl5is on which the change of position of the turbine workin!! fluid conlTol valve ."'~ Typical speed droop characteristics for most !!ovemors range between 5 and 10%. The latest in the turbine governor design is 10 provide an of the speed governor syste!!! is shown in electronic controller. A block diagram '''''''~ '' Fig. 2.5.
,,. P,(.)---\ (COOlm&nded chanic in power)
R
~_
(opmini nhtcilIIl
t
.. speed regulltion of the ,""=0
KfC " i,inofthe~govemor TfC .. time conSlAntofthc: 5p«d
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MODELUNC Of POWER SYSTEM COMPONENTS
The speed governing system of hydroturbine is more complicated. An ~dditionaJ feedbac::k loop provides temporary droop compensation to pr-.:vellt instability. Th is is ncc(SSitaled by the large inertia of the Pf'flJtod gate, which regulates the rate of water inptJt to the turbine. tJ.x
Here.
=0
<
KSG 1+5T.iG
(liP.' -..!..tJ.w) R
(2.16)
Equation (2.16) plays an important role in modelling the governor operation. Let us consider a simple e.lample. Alisuming an increment llP, =0 1.0 at t =0 0, for a speed governing system under test (i.e. operating on open loop resulting tJ.w:: 0), tke iocremcnt in sturn valve opening at, is obtained from equation (2.16) using the Laplace transform of &P,:
( K 59 ). using the Laplace transform of llP, s l+sTSG =
K.iG TSG
(2.17)
I S s+T~
Mathematical maniptJlation yields,
,
(2.18)
which on inverse Laplace transform yields at,(t)=oKSG(I_e- tlT",) for {;?;O
(2 .19)
The response curve has been ploned in Fig. 2.6. Thus. the governor action has been modelled utilising the concept of transfer functions.
----------------------
""1 T~
Fl • • 1.6
31-."
"
Speed governor response curve.
2.4.2 Turbine Modelling Turbine dynamics are of prime importance as they also affect the overall response of the generating plant to load changes. The actual dynamics of course greatly depends on the type of turbine used . A non-reheat type of steam wrbine has been shown in Fig. 2.7. MatE
·,to
]\0
POWER SYSTEM ANALYSIS: OPERATION
s,~
Turbine
chest
Tocondenscr .". 2.7 Block steam enters the turbine via the steam-chcst s) in the steam flow resu lling in the tnlnsfcr
After passing the control valve, that introduces the delay din ~ function
T
,,,d,,, "
I
(2 .20)
l +sTr
The turbine governor block diagram has been
in Fig. 2.8.
s....
govnning - - -.;._ _ _ Turbine - - . . ; ,
•
'>"'=
IIR
K.
block diagram. Assuming the command increment to be APe, "", (2.21)
II ins isIS 10 choose a scale factor so that This gi~es the model as shown in Fig. 2.9.
APe. This is equivalent to picking KJr= 10.
IIR
(
F\c.
2.9
.....
,
Block diagram for
'.'" governor modelling. Mat" II
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MODfLUNG OF POWER SYSTEM COMPON£NrS
This model can a lso be modified to account for re heat cycle steam IUrbine (Fig. 2. 10). This is more efficient and is used fOf modcro-day large scu;. The overall transfer function of the reheat type unit is given by G = AI( (s) = I +O.5.sTRH
l1x.(.s)
(2.n)
l+sTRJI
where, TRII is the time constant of the reheatCf having typical vltlucs in the range of 5-10 s.
~.(s)
.~
X
HP .tage
"'"
S .1&.2.10
LP .(age
To
dP,(J)
~ondenser
Re-heater
Block diagram of reheat!.leam hlrbine.
The hydro turbine design varies with the water head (Fig. 2.11).
.. .......... n.m Penstock: Water·
h,.,
.1"
1.11
Block diagram of hydrotwbine.
The overa ll tr:lnsfer function is then G•
Cl-,-:"~TL'
(2.22:1)
l +sT,
where . Tp is the time it takes for the water to pass through the penstock:.
2.4.3
Modelling of Exciler
Figure 2.12 represents the conventional e:o;citltion system of an alternator while Figure 2.1 J its block: diagram with respective transfer functions.
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POWER SYSTEM
AND CONTROL
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MODEWNG Of POWER SYSTEM COMPONE"7S
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+
0
., .~
• 0
-, ~
·
0
,
+
~
""
,-
- +" -
I~
.'
,"" .!•• + ~ -
,
0
R
•
.!
•• " "0 g
•
1 ~
,•
~ <
"-
.t •• +
•
" ::i
t
~
,
i ~>< , •
+
Malenal,
Jm
direikJ
Jlorai
I POWER SYSTEM. ANALYSIS: OPEfUoTlON AND CO~OL
I• I
In !he blod: diagram of Fig. 2. 13. T~. the tim~ cons/ont a/1M recrifier is very small and may be neglected. The amplifi er gain K"""" is usually high (between 2S and 400). Amplifier time constant (T,t.o,;» is in the range of 0.02-0.4 sec. A stabiliser has also been dIown to stabilise the gain of th e exciter. K". the stabiliser gain. is 0.02 10 0.1 while srabiliser time COllSlanl T. is in the range 0(0.35
10 2.5 sec . Some simplifications lead 10 a simplified block diagram as shown in Fig. 2.14. Here,
v= when;
(J
K~ .... K, q
=
K•.• +K~J(,(1'
is a factor associated wilh the transfer function of the synchronous generator wilen loaded.
"
-I
K~ ..
'. I
-
FI,.2.I4.
,
V",;
I
'I'
Slmplifted block diagram.
2.5 MODELLING OF REGULATING TRANSFORMERS (Rn Let the ""su/ol;"8 transformer (Fig. 2. 1.5) be placed in a two-bus ,ystem wi lh a complu
(r(lIlsfommlion ratio " = I ~L8 The primary voltage and curren t wi l1then be (IIVl ) and (lin*'). respectively.
(2.23)
, ,
prj
Sec
IniLO
II "
Fi,. 2.15
R.gu!atini
tr:tn~(orm«
in a two.bus network.
The current balance equations can be wrinen as (2.24)
I I'" Vly........ ( V1 - "V0 Y"
1]1', ,- '"
"Vl Y.~ '" (IIVl - VI)
(2.25)
Y..
IThe equivalen1 circui1 has been shown in Fig. 2.16.1 Equ;llion (2.25) can be rewrillen as
Ii = -II -VIY...... nn" (Y..... ... Y.. ) V1 (2.26)
A lso from the equa1ion (2.24).
I
I
11'" (Y........
Y..,l v1 ... (-nY..)
(2.27)
V2 MJlerl
,"
MODElliNG OF POWER SYSTEM COMPONENTS
,-V,
- , - V,
RT
I,
Equivaknt ciocuit of Fig. 2.15.
f1&.l.l6 Hence,
I,
Y.
rrom equations (2.27) and (2.26). r".,.
=
[(yJIt+y,.)
(2.28)
-n ·Y
•
In practice, RT is either a yollQ8~ mDgnilud~ eomro/ IrQfufonn~r or a phos~ Ira/Uform~r. In the former easc. L9 = if and in the latter casc. I n I is a conSlanl
ongl~
eonlrol
2,6 THREE-PHASE MODELLING In a three-phase network., the three n{)des are mostly associated together in their interconnections. This network. is then termed IS a rompalllw n~fWOrlr and the admitt.ances are represented by compound adn!i/1anct!. Laws ~nd equations that arc valid for ordinary networks are al so valid for compound network. by simply replacing single quantities by appropriate matrices. Figure 2.17 represents si~ mutually coupled single admitlances. The node currents can be link.ed by admittance matri)( to the branch vollages as follows:
I,
Y"
Y"
Y"
y"
Y"
y"
I, y" Yl2 Y!) Y~ y" Yu I, )",1 Y" Y" Y. YJ! y. I, • y" y" >'0 YM y" Y« I, y" Y" f" Y. Y" Y. I, f" f" f" fM Y" Y.
V, V, V, V, V, V,
(2.29)
Partitioning the above matri)( ,
.ha.
~/'~ J [Y"~~V' ~ [I, J = ~Yu [Y~ J [YIT [V, J
(2.30)
(iX]=[/l/1 /Jr [t r ]=[/./}/6 ]r
(2.31)
Mate-nal, Jm dlre-itJ
]toral.
· POWEll: SYSTEM ANALYSIS: OPERATION AND
[Y;
[rr.< ]=
Yll
Yll
1 1)
)'"
116
>'21
Yn
Yn . [rn
Y:u
lUI
1)s
1.16
Y~I
Y41
1 41
YS I
YSl
)'Sl
161
162
)'6.1
I,
I,
•
[rn'
, ,,,,
1. s
1<0(1
Y,..
)'ss
)'56
)'6.1'
)'66
I,
(2,32)
I,
v.) Flc. 2.17
Si~
coils with nOOaI
Thus. the si~ coi ls can be represenled by ind ividual adminanccs. This has been shown in Fig"
t
coib (X and Y) consisting of thm:
IS.
[ ~d f
Fia. 2.18 Equi "aknt :uJminancc coils and Hence. equation (2.29) is finally represented
,0",1'' '" '0",00", foon a~
(2.33)
~.
since
[YnHr MJlerl
1:m1 dlreibs
Jtc>raJ
MODElliNG OF POWr R SYSTEM COM PON rNTS
For a three·phase transformer. assuming yp and y, as the self·adminances of primary and secondary coils (being equivalent to Yll ' Yll ' YJJ""') y;' the mutual admiTtance beTWeen primary coils. the: mutual adm ittance between the: secondary coils and y: the mutual admittance between primary and secondary coils on different cores. the nodal currents in the coil s may be: linked with the bmnch voltages as
y:
I, I, I, I. I, I,
Y, , Y. , =
y.
-,. Y.• Y.•
, Y. Y, , Y. Y.•
, Y. ,
- y.
Y.• Y.•
y-
-y.
yY•• y.•
y.•
- y.
y.•
Y.•
v,
-y-
y.•
V,
y.
-
- y.
V,
y;
V.
y,
-
y•
V,
Y.
y,
V,
,
Y.
"
y.• Y.•
•
(2.34)
TI,~ prinl~d ''Qlu~s
ort! rffectiv~ly t ro for ,il rt!e single phose units. If transformer connections are 10 be: incorporated. the YI US is formed utilising the relati on
(2.35)
[Yau,) '" [e)T[YpIUII [el
where Ie) i ~ the eo,,,,n:rion motrix. and I YPlrMl is the primitive mlltrix. Table 2. 1 represents the [Y,US) matrix for common transformer connections assuming three individual uni ts 50 that primed values vani sh.
TABLE 1.1:
E~ments
Yn
Y ,
Y
ortfllll$Cormer
(~If.
_mi~e
ntlItriccs
Y.... (ulf, s«.)
pri. )
Y.
Y,
Y,
Y,
Y,
""
Y,YJ
- YdJ
Y,
- Y,
(neutral solidly grounded)
Y ( r ' 5iJk neuuul soli dly glOOn.ied)
. Y
y
,
Y.
where.
fA ' )'a ""
Y
0
0 0
0
0 0
,
,
2y - Y -y 2, - Y• - y -y
,
0
-,
y
0
-Y -y 2y
(2.36)
0
y -y
Mate-rial
:mI
dire-it)
Jtorai
POIVER SYSTEM ANALYSIS: OPCRATION
It m~y be noted that any two i admitunces. The current voluge relationship is
~~
'"'" b, ,,~·~,m,d by two compound-linked by
(2.37) wilerc l YpsJ '" l YlI' f and is the same as OJ,,,,'m, " and secondary buses while Yps and Ysp represent ratio is to be included. Section 2.5 is to be
fo"'",..,
. YI'I'. Yss indicate self admittances 31 primary mutual admi!laoce. In C:L<;e the off-nominal IlIp in conjunction wi th the abo~e modelling.
2.7 MODELLING OF THREE·PHASE
CIRCUIT TRANSMISSION LINE
f igure 2. 19 represents the lumped line with suffix (S) as sending end and (RJ as se ri es rcaclllnce ,,·hUc Y is the shum admi!larK.~ .
si ngle circuit three-phase transm iss ion ,",;,~,'p"l I. 2. J lhe phm. es. X is tbe
-\ xSRr '
:]
(2.38)
{6xl)matrill
(6xl)m:urill
These primiti'·c matrice s in Fig. 2. 19(a) Figs. 2.19(bl and 2. I9(e) utilising the te.:hniquc s arc described in a generalised manner as urxlcr.
be rcpresentcd by equivalent matrices in described earlicr. The CUTTent voltage rel.Uion5
~
"•
) I,'
r
H
~
r " ~
Y~H
r
n
~
Material,
JfT1
dlreitJ
;\O~.
MOD£WNG OF POWER SYST£M COMPONENTS
i' i' i' i' K· i' i' i' i' n
(~
CR)
[X""I
[VJl
i' y' y' y' y' i' i' y'
'"
[r; ]
y' y' i' y' y' ~;' y' y' 1''
[1',1
I
[I'; ]
(b) Equivllrnt modelling of components.
IX, 1
IVJ
1
['l'-] (c)
"to 2.19
['l'-]
1[V,I
Equivalent mood.
Moo.:lling or ihref:.phase tr.lll$Jtliuion
l i~
lUing p ITICIdtI.
2.8 MODELLING OF PAIR OF THREE·PHASE MUTUALLY COUPLED TRANSMISSION LINES Figure 2.20 represents lhe equivalent of each of the two mutually coupled tiroe~ ut ili sing 11" model. Here each :tdmiuance matrix element is a [3 >I 31 ma trix; the curre nts :tnd vo ltages are related by the following re tation;
Is
,
", "
",
,
[yJ! . . y n] [yI2.,.ylO] [ ytl' ... ylO! ] [y:l +yo.l] [ -1'''] [-yur
[-r,,] [-r,,]
[ -r" ]
[ -r" ]
[ - y" ]
[ - y" ]
] [Y"+y'] [r'"
+ Y~']
v,,
v,
V, ,
(2.39)
V,
[12xll
[l2xl2]
112 x l ]
matrix
matrix
m;ttrix \.1al[
:e-itJ
Jlorai
POWER SYSTEM ANALYSIS, OPERATION AND CONTROL
I Y" I
S, o - - - - - - - " '
R,
I r" l
'" 1
I r,,1
I YIlI I YIII
[Y,J
1Y.,I
,.
1'"
~
'" . !YJ
s,
'" I Y.,I R,
IYnl
[Y.J
1v"
•
Hg. 2.20 R~P"''''n1al ioo "f I..·" mUlu.:1l1y Cl)Upletl li"n.
Figures 2.21(0) and 2.2I(b) represent the compound admittance form of the matrix represe"tation ~Ilown in equation (2.39) corresponding to Fig. 2.20.
" ". "
S
Y"
f,r
•
Y" y'
"
Y" Y"
[y.1•."
"
Y'1
hI!
R
Y"
Y" Y,(
Y. Y.
[r..,1606)
MJler!
1:m1 dlreibs
.
Jtc'·ilJ
MOD£ LUNC OF PO WER SYST£M COMPONENTS
[Y...[ s __- - - . - - - ' ' - - - - - - - , - - _ R
•
[Y,, ]
•
Fll- 1.21(b)
[Y,.,]
1[~::l
(6 x 6) Compound matrix represenUlion of Fig. 2.21(a).
Thus. the mutually coopled liT\e$ afe finally represented as
[::l [::l
[XNJ' +[r..,] -[X"J'
[~::l
-[ X"J' [X.r' +[Y ,,]
[~:l
=
[12 x I]
(12 x 12] matrix
matrix
(2.40)
(12xl] matrix
2.9 MODELUNG OF A SHUNT CAPACITOR/INDUCTOR For effective reactive power and bus voltage control. shunt capac itors andlor reaCIOI"$ are frequently used. Figu.re 2.22 represents representation.
I
stllic dlunt capacitor bank with its compound admittance
t.o.d bus
"
7', " c
e
=-:
I I I
c
Fl&- 1.22 Model repuKnt:l1ion of shunt
~ap.d!
As there is no coupling between the components of each phase. the
Y matrix only contain s the
dwgofl(ll elements. In a similar way, the modelling of I shunt reactor can be done.
MatE
I direikJ
Jtorai
•
POWER SYSTEM ~N;\LYS 'S: OPERATION
2.10
MODELLING OF A SERIES line and betwet.'n two buses. The
The capacitive element is connected in series wi"h matrix for this system has been writlcn as
~dminallCc
(2.41 )
The shunt dement does 001 exist IljX "
[r,,]
Hen:.
=
IljX M Figure 2.23 represents Ihe mode lling.
, ~.---111 1-~.
x" X" x" x" x" x"
,
R
[r",) Qf
2.11
~ri~s
Here.
x" x"
r{V~l
R
- IX....r l
capacitor_
MODELLING OF STATIC VAR
Let BsV(" be the shUn! susceptance of the SVC "'-" added 10 the susceptance al the busbar. The Iota I L'Ontrull ing vohage V will CJuse the '''i,oj "'''' cllhanced. The SVC injected CUfTent inlo the bus i~ then
X"
(SVC) 10 the MVAR loading of it. It is then is give n by B. A reduction in the
";,~;::'::,:':i;::,~ ,,,,by
I =
(2.42)
Y=G~j8
(2.4J)
IG may be assumed 10 be zero here. ] The MVA ()IJ IPU! of the SVC is given by S "" VI-SII<.'
Q = II~' IE
and
2.12
MODELLING OF AN
The slip of an inducti()
MOTOR
c~prcsscd - II
/I
1'"'
(2.44)
0
in "rr",
it s rotor speed and synchronous speed
'x i OO
3$
(2.45)
,\, Mate-ria
Jtore
MODf.U ING OF POW!://. SYSTI:M COMPON!:NTS
The equation of motion for the shaft power is given by
d8
dt " (T.. - T, )/2H
(2.46)
where H is tile illCflia constant, T... the mechanical torque and T, the electrical torque. Howeyer. the mechanicallOl"quc: is equiya lent to load torque and is commonly Cllprc:ssed as (2.47) ~nd
I: i$ an e;o;poncnt and is J for fan type: of loads giyen by
2 for pump type: of loads. The electrical torque is
r.. :. Real [EJ-V2nIo
(2.48)
where E is the ai r gap yolt.age, f the st.ator current .input and fo the base frequency. The transient reactance has been defined as the apparrnt reactance seen through the equivalent circuit when the rOlor is held locked and the s lip is unity. Thus. from Fig. 2.24, the equiyalent ciKuit during transient operation. we obtain
x:
X' " X + ,
X,X,. (X,
(2.49)
+ X",)
x,
x.
x' ""
1'1 .. 2.24
Equi~a1e!!l
cil"l;uil of indoction
/IIOIOf
durin, transient Stale of operation.
The transient model of the induction motor has been assumed by a Thevein equivalent circuit of a voltage E' behind the: transient reactance J( while the transient time constant T o is given by
To
(2.50)
=
and the open circuit reactance Xo is given by (2 .51)
Xo"' X,+ X ..
Assuming the st.ator rc:sist.ance to be R,. the governing equations of the model are given by (2.52) Vi ..
- 1:-. ;..
(2.53)
= li .. R, + I,X'
Here. the reactances are assumed to be unaffected by the ralor position and the model is analysed in the real (no) and imaginary{im) un for the I1Ctwork.
Material
I
Jm dire-,to
]to ai
POWER SYSTU ! ANALYSIS, OPERATION AND CONTROL
The system model is described as
X'] [V. -e:.] X' Vi"-E'...
(2.54)
The rotor reactance does not vary much with the variation of rotor resistance with slip. provided the salUration effect is neglected. Transicm reactancc X' varics with rotor reoJClance only and hence is a lmost constant ~t any slip. ~ induction machine can also be modelled in terms of d-q axis as follows: the (p.u) vollage equations for a s inglc rotor winding induction motor in d-q coordinate are given by (2.55)
Vq,= R, iq, +1V1V4,+Vot.
+.r"
(2.57)
V" = R,i", +cUSlV... +V"
(2.58)
V..., "" R,i", ';"OJSlI'dr
The COffC$ponding
nUl(
(2.56)
linkages are :=
L,I. + 4i",
(2.59)
If'q, ::
4 1" + 4io'
(2.60)
= 4,ill< + 4i.it.
(2.61)
IVII<
" b
"~" 41" + 4 i",
(2.62)
Neglecting stalOr Iransienl'l and assuming the rotor short-circuited ;-", :: 0 and
11'" =0
(2.63) (2.64)
VIr .. O and V.' = 0
, (-Xo J
V... '"
L,
(2.64.1)
IV",
x; '" X, - X~/X, (2.65 ) SulistilUling equation (2.63) in (2.55). tYII< and liI"are eliminated. ,.",and lV... are .1150 eliminated by substitution of equations (2.59) aBd (2.60) in (2.55) and (2.56); i", and i.,.. are then el iminated using rearranged equations (2.61) and (2.62). IV... and ".,.. are el iminated usi ng equation (2.64.1). Us ing equation (2.65) in the final form. the resultin, equation is
-X;] [,,,].[V;:] [V"]_[R" V" X, R, '.. Vq.,
(2.66)
(2.67)
~ale-rlall :>rTI
dlfl'IID
Jlo ais
MODEU1NC OF POWER SYSTrM COMPONENTS
The state equations C3 n be developed by substituting (he value of Voir and V., from equations (2.64) in (2.57) and (2.58). Substitution for io/rand i., is done from equations (2.61) and (1.61). In its new form. IV.... and " .. are replaced by V; and V~ using equation (2.64a). In its final ronn. (he derivative of V~ and V; are taken to give:
V;
=( ~ )v~ +$WV; -(L.X .. R)'; )itp
v; = -SlI)V; -(R,J4)V; +(L",X",R,JI!,
(2.68) )id.
(2.69)
Expressing equations (2.68) and (2.69) in phasor fe>rm,
v; '" [- ~: - js }VV; + j(R.tX, )(X, - X;}al/, At steady state
v;=O.
Assuming
(2.70)
I/,I =I.O. X,=X" I , ::c018- jsin8
(2.71)
V; = jR,( X. - X;)(cos6 - jsin 6 )/( R, + jsX , J
(2.72)
And from equation (2.70).
Rationalising and taking the ratio of imaginary to real parts.
V;
V;';:
R,cos8 -sX,sin8 R, sin6 -sX,c016
(2.73)
Similarly, substitution of equation (2.71) in (2.67) with V, = I. yields
V; _
R;sin6-X;cos8
V; - (I -R;C056- X;sin8)
(2.74)
[6is the motor p.f. angle]
2.13 POWER NETWORK MODELUNG I{ '" injected currenl at node i (i = 1.2 ..... II) i6 V, , (voltage at node I)
lzt
"'1"'Ile
The current I{ can be expressed as 3 function of the
volta~s.
Thus.
L [YijkVi -Vi )' i '" 1.2, .... n
I{" Yu V{ +
~a(l)
a (I) designates the subset of the nodes connected 10 node i _ ~ Y. - i... }lali)
'.
Y~'
and
_ 1 _I 1 ( -~.) ' YIJ - - - Yij e 1:;1
• It. line or cable connetIini two buses i :and j can be: modelled by • "pr' equivalent circuit tuvillj series impedance .too and shlmt admiuance YiI' where !4f .. Tij + ~ and y~ =,~ + jh" Since the ~p, circuit of !he line is symcfricaJ .
we '"tllne
aij .
lJi '"' 0;
It~ . hJi =
.c
T
Mate-nail Jm dlre-itJ
Jtoral.
POWER SYSTEM ANALYSIS: OPERATION AND CONTROL
In general form. the preceding eqUlltions can be written lIS (2.7~
(/J=(YIIVj. rq Yjj=G~ +jH~ andG"- " -- ' H
r.:'
II
Xv ~
:o -
;oa(1)
Also. at node i. However. V;. is compleJt conjugate of VI and hence
fl=RealJVt[
1
...
YIIVj+
L YIj(Yr-V1 ) ]1 ....
(2.76)
(I)
(l.TI) Simplification yields
~ =V;l
...
L
(y~ cos 8ij+8ij)-VI LV'Y4'COS (8ij+o/-o,) ...
jI .. (i)
Q,=V/
.... "'il
L (Yi/sin8~-Iv)-Vj L is .. (1)
L
Vj Yijsin(8ij+6'r-6',)"
ji"(I)
L P4' JO"(I)
(2.78)
C2u-
(2.79)
}llftl)
whue P" and Q4' dellOle the active and'reactive powen through the line connecting the ith andjth
"""".
Obviously. Pi! = v/
(Yif cosOij + ,~)- VjVjYi/ C05(OIj + 6; -6'J)
PjI = V/~ (Y j/ C050V+ B~ )-V;V/Yr,;
(2.80)
C05( 8y + 6', -0,,)
(2.81)
'4 '" V;l ( Yi/ sinOiJ - ~)- V;VJYiJ sin{OiJ + 6; -6', )
(2.82)
QjI = VI (Yi/ sinOij -Iv)- V;VJYiJ sin{O~ + oJ- O()
(2.83)
(Nol~; Conventionally Bli = 0 and hi) = ~I{ . It may be noted that power flow equation Itave been dealt in deLlil in Chapter 4 where we replaced the notItion ofsusceptance H by B.)
2.14 MODELLING OF LOAD Load drawn by the consumers is the toughest parameter to be assumed scientifically. The magnitude of load. in fact. changes continually so thai lite load foreca.ning problem is truly a statistical one. The loads are generally composed of Jlen
•
MODEf.l1NG OF POWER SYSTEM COMPONEM"S
Lighting and Heating Induction Motors Synchronous MOiors
ClS-1S'l>
The loads are mostly of composite character and it is prudent 10 represent them by P-V or Q-V characlC:ristics (Fig. 2.2S(I)(b)). Broadly. the loads are classified 1.5 Constant ClImnt
p
Q (p.u.)
, .'
.."
V(p.u.)
. C~,
•
impedance •
1.0 p.u.
.r:.J
'....
-----------------------;.~.~.- ---'_ Constant
/
'" .
II I (p.u.j
.'
"
"
".
..•.
....
C\IITeM '> .
"
..
' j """"
Constant
V(p.u.)
(h) Current CIwaClcristicl of loads.
FI"l.2.S Characleri$\ics of lo.ds. (j)
Cons/ant cu,rrrlf rype
1= P- jQ '"
V·
1'1 £.6 - 9, Male-rial
•
:mI
dire-it)
Jtorai
I, I
I
POWER SYSTEM ANALYSIS: OPERATION A.ND CONTROL
IVI
tan - I Q. (} is fhe power factor angle. II is k.nown as constant p current representation as the currenl remains constant. Fluorescent lamp belongs to this where. V ..
Lo.
(J ..
type of load. (il)
Corman! f'OY"t!f type
This load is specifIed by its MW and MVAR ratings and is assumed 10 be constant. Thb
type of representation is used in load flow study. Induction motors belong to these types of loads. (iiI)
COIlslant inl,ndanu type Here, the load is spe.;ified in MW and MVAR al nominal voltage. Here, J assumed to be constant. Here I varies wilh variation of V. The load impedance is determined by
Z = ~Z = PW· =1L=..!. - jQ P - jQ Y
V
"'z and Z is
(i np.u.).
Healers , domeSlic loads and incandescent lamps are conlWlt impedance loads.
EXERCISES I.
What is 'modelling of electrical components' and wily iI is required?
2. Explain the generator.
a~lylical
concepl behind differenl models conceptS of an iwlated synchronous
J, Ho w would you analytically model a regulating transformer in power network?
4. Explain the concept of 'three-phase modelling' . 5.
Analytically model the fo llowing: (il a three-phase single circuit transmission line, (ii) a pair of three-phase mutually coupled transmission lines,
(iii) a shunt capacitor,
(iv ) a series capacitor. 6.
What is SVC? How would you model it?
7.
How would you model an induction mocor in d--q reference frame?
8. Develop power flow equation in a power network.
~ate-rlall :>rTI
dlfl'IIo
Jto ais
Operations 3.1
INTRODUCTION TO [YBus:I FORMULATION
The large, inlerCOT\llecled AC power system (network) consists of numerous power stations, transmission lines, transformers. shunt reactors andlor capacitors and distribution networks through which loads arc supplied. All this leads to a high voltage, largely interconnected AC power transmission system and the assessment of the steady state behaviour of all the components o f the network acting together system requires computer-based large-scale system analysis or~ network model. In computer-based power system analysis, the network model takes on the form of BWI Admittance Motrix [Y,.,J.IY/IOr) is often used in solving loadjlow (or complex po-wer flow) problems. Its widespread application in power system compuwions is due to its simpliciry in data preparation
lIS.
and the ease with which it
tall
be formed and modified for any network change (e.g. addition or
tripping of line etc.). iY. .] matrix is highly SptnU and facilitates minimum (:omputer storage as well as redtK:es computtr operation time. There arc different methods of formulation of [YB ..J matrix and a few of them are reviewed here which are easily amenable to computer programming and easy to ,,",p.
3.2
NODAL METHOD FOR DEVELOPMENT OF [YSIIS]
In this method of [Y",, ) formation , the variables include the camp/a load vol/ages being treated as node vol/ages {the referen<:e is the 'ground' for designating the magn itudes of bus voltages and for voltage angles, the referetK:e is one of the bus (or node) voltages which are usually fixed at a datum value (say, zero». The node Cl/I?"en/ being the other variable , it is the nel current injected inlo the network al a given node (trom a source and/or load external to the network). When the curnnt enters the network from a node, the s ign of the (:lIJTent is assumed 10 be po$itive, while for the curnntleaving the network, Ihe sign is negative; the nel nodal (:urrenl being the algebraic sum of these node (:urnnts. In the nodal melhod it is usual to use branch admittances rather than branch impedances. For an isolated branch Y" (F ig. 3. 1), the node voltage being VI and VI It the buses i andj, respectively, CLUTenl flowing from node i to node j is given by
Matanal
)Ill
d ram
},
POWEll. SYSTEM ANALYSIS, OP£R.ATION AND CO.VTROL
(3. 1)
,. •
o reference node
Fia:.3. 1 Nodal rdationship bet\ocen node voltages and branch currents. In a complex network the nodes being numbered O. 1.2, ... , n. whell: node 0 indicates the reference node, by Kirchoff's cunent law, the injected cunent I; being cqualto the sum of aU currents leaving node i; thus, we can write
•
•
I, - LI,. LY,(V, - Vi ) j o4)
(3.2)
j .o
With no ground potential (i.e. with zero reference voltage), for a linear sy!lcm,
•
•
(3.3)
I, ... LYj.,JIi - LYqYJ I _~
j_t
j-'
j-I
This equation, for a n-bus network, in mauix form can be represented as: I, I,
··•
'.
-
Y" Y" Y.,
~,
Y"
·••
Y.,
•••
~.
r,.
V,
v,
• •
·
·••
y-
V.
(H)
(3.5)
oc• •
[Y.,.,] is called Bus Admittance matrix and it has a well-defined structure. The clements of I Y,..] arc importanl and hence defined below: YIi> the diagonal elemel1l, is called u /f admiftance or~ i, while Yr the off-diagonal e/emel1l, is caUed mutual udmillance (or ,rullS/er admittance) between nodes i Mldj. Obviously,
,,'"
•
fi, '"
L>' i~
i·'
(3.6)
y~ = - Yi;
M alerial I
JfT1
d rell)
Ao
~.
POWER NeTWORK MATRIX OPERATIONS
The pro~r1ies of the
Ir.....) matrix are as follows:
(i)
I r...... 1 is a square ",atrl.r: of order n
(ii)
[r...... ] is S)mmelrical. since JlV '"' Yft-
(iii) Only
(n"n)-n 2
+n, i.e.
n(n+1) 2
~
n.
terms Bre to be stored for a n-bus power system.
The e lemenl5 of ry&"] matrix are complex numbers; rY~ ... J matrix itself is thus complex. Each diagonal e lement IrmrH_ •.• J is the sum of the admillilllce of the branches wh ic h are linked with corresponding i-th, j-th nodes including branches to ground, while each offdiagonal element Yf is negative of the bl1lI1ch admittance between nodes i andj. In order to illustrate this pro~rty, lei us assume a two-bus system (Fig. 3.2) where a transmission linc: is represented by series admittance y.. and shunt admittance y*
(j) . - - -
---{~Y·:::'::]--I--·
••,
In this case the diagonal elements of IY""']
~
given as
(3.7)
(VI)
Y, {i"';1 - 0 if I_th
bus andJ-th buses arc
not connected.
In actual systems lots of interconnections do not exist bel\\.·een a number of buses and hence the [Y_ l matrix becomes highly IJparlJe (containing number of zero clements in the matrix). This saves tremendous computer storage and memory requirements. The flowchan for obtain ing [rs""l by nodal method is shown in Fig. 3.3.
Mate-nal, Jm dlre-itl
Jtoral.
POWER SYSTEM ANALYSIS, OPERATION AND
~
,,. , ,.
...
1
Ii
f""" bus .., .•
Compute yr...., NT,-
In;t;ali"". YBU$l .... 0 ~ jO;
; roc/ - 1. 2•...• Nl.
O+ jO; fod - I.2• ...• NB
No
,.
'-"
YBUS1 .. _ YBUS1 ... )")'>r,KT.
YBUS"'. !>"T. - - t1'•.,.. ..... rBUS.,•.w, a YBUS"" ,7.
Is iSNL?
,.. No
MJlerl
1:m1 dlreibs
J1c>raJ
POWER NrnVORK MATRIX OPERATIONS
"".:.:
A
,
'
..
....
,
Sol bus
SoIli... count. I _ I
"
I. t - NF,?
'"
c:omp.... YD, -
"a
lIiSNL1
I"D, r .
re,
10 I - NT, ?
..
"
'"
' ,
..
i+-i+1
No ~+-A:+
'"
Is A: '!: NB?
1
I
No
Sci bus rount, .t .. 1
Compu:c rBUS.. - rBUS I .. + rBUSl .. + rD,
k+-I+II
b.t'!:NB?
'"
No [ r ... 1matrix delTlCTlts. rBUS, . rOl" i .. 1. 2• .... NB; fOI) " 1. 2• ...• NB
.., AI.3.3 FloII/chart of [y....1foml:llion by nodal
~thod.
MJlerl
1:m1 dlreibs
Jtc>raJ
POWI:R SYSTEM ANALYSIS, OPI:RATION AND CONrROL
Formation of [ZsuJ from [Ys..J In this context, it may be noted here tllat formation of bus impedance malrix !llrw] is possib le by invenioll of [r_ l by using special algori thms.
z" z" ... Z" Z" [z••• l=[Y~r'= •
Z" l~.
(3.8)
· · •
Z_ • •
In the {ZH.. ) matrix the diagonal elements we shorT circ;u/f driving poim impedances wh ile the off-diagonal elements are shan cirCl'ittrans/er impt'donces. [l/l>.,[ is symmetric prov ided [Y/I>..J is symmetric, which is very much usual in power network structure. Ho~ver, [Zs:...l is not splll""se like IY~.,1 and is a full matrix COfItain ing non-~ero e lements (tero clements in IY_' become non-zero elements in the corresponding [ZlIo,l). Example 3.1: A three-bus system is shown in Fig. EJ.I. Each line h(JS a series impeJanct 0/(0.05 + jO.15) p.u. while the shum admillance is neglected. Find I YJt.,].
Q) ...,
=----i='=:":J--.:J!
Q)
•
-'---'- (j) Fij:.O. 1 A thrtt-bus three-line power Sl'stem.
,
Solution : Given:
:11 "' :l.J ~ Ill'" (0.05+jO.lS)p.u.
(series adminan ce of each line) I
'" (o.o5+jO.15)
.
=(2 -J6)
p.u.
Since the given problem is a three-bus system hence [ Y..... J matTix would be a 3 x 3 matTix. I'll
I'l l
1'13
[y..... ] = >11
Yn
Yll
Y"
Yn
Y))
where.
,
Material,
JfT1
dlreitJ
Ao~.
POWER NETWORK MATRlX OP[JU.TIONS
Since,
Y12
..
Yll
"
Yll + Yn; YU " Yl1 .. - Y, l Yll + Y:I1; Yll .. Yll .. - Y:IJ
Yl1 .. .I':ll" Yll" (2 - j6) p.ll .
• ••
Yll " 2 - j6+2-j6 - (4 - jI2)p.u. Y11 " y11 .. (- 2 + j6) p.u.
Yll - 2 - j6+2 - j6-(4- jI2)p.u. Yl ] " Yll - (- 2 + j6) p.u. Y] ) " (4 - j12) p.u.
•
Y,, " Yl1 " (- 2 + j6) p.u.
(4 -
[Y"", ]=
•
(-2+)6) (- 2+ j6) (- 2+)6) {4 - jI2) (- 2+)6) p.u. (-2+)6) (-2+)6) (4-)12) j12)
Identical result is obtained by executing the [Y_I software following the flowchart presented in the text. The input and output of the result are shown below.
Execution 01 the computer program YBUS.FOR lor Example 3.1 Line data: ZBUSO.DAT 3, 3 [No. of lines, No. of buses] 1. 2 , (0 . 05, ,15) , (0,0) (From bus, To bus, (R. I ' 1.3, (0 . 05 , 0.15),(0,0), " 2,3, (0.05,0 .1 5) , (0 ,0 )
°
•
Xu. (G M)] I
•
,
,
•
Output 01 YBUS.FOR: YBUSO,DAT No.
of buses
Ybus mat.th: Ybu s ( 1. Ybus ( 1. Ybus ( 1• Ybu s ( Ybus( 2.
'.
Ybus ( Thus (
2.
Ybu!I ( Ybus (
3.
3. 3.
-
.}o'- )
2 I 3 I 1 I
2 I 3
I 1 I
2 I 3 I
3
--
---
(
4. 000000, -12 . 000000
(
- 2.000000,
6. 000000
(
- 2.000000,
6.000000
(
-2. 0 00000 ,
6.000000
(
4 .000000 , -12 . 000000
I I I I I I
(
- 2.000000 ,
6.000000
(
-2 .000000 ,
6.00000 0 I
(
-2 .000000 ,
(
6.00000 0 I 4. 000000 , - 12. 000000 I
Y" Y"
Y"
EUlilple 3.2: In Example J. J. Jor the Jame three-blU ~Iem (Fig. Ell) lei a nrw bw (bw no. 4) be added with btu no. J Ihrrwgh a trQnJmiJJian line aJ p.lI. :: (- 0.1 + jO.3). Ohtain IY//luI.
Malenal'
Jm {]
reiID
Jtorai
POWER SYSTEM ANALYSIS: OPERATION AND CONTROL
Solution: Let lhe bus no . 4 be added 10 bus no. 3 through a transmiss ion line of (0. 1 + jO.3) p.u., i.c. }'ll " 1/(0.1 + jO.3)" (I - j3) p.u. (F ig. £3.2]. Sincc the ne w clemen!~ is added with bus 3, entries of Y )} will changc and new entries of Y:\oI and Yu will appear in the new bus admittance malri1t. Obviously. du e 10 prcSC'nce of 4·bw; systCtn. this bus admittance matrix will be a " • 4 matrix.
=..
,
=::J---=:J-
~(!)~,_
-
J )....
0
11:.14
fig. [J.2 A M'" bus added to three-bus S)·stem.
YJJ .. YJJ ,oIdl + ( I - )3) '" (5 - jl 5) p.L1. YJoI .. Yo " - .V)., .. ( - I +)3) p.L1. Y",", m(I _ ) 3) p.u. Since there is no conneclion of bus 4 with any other bus, exce pt bus no. 3, hence. YI~ " Y~\
Final
Iy....,]
.. 0;
Y!-I -
Y~2 " O.
matrix thus becomes
[ ,-] .
4 - jl2
- 2 + j6
- 2+ j6
0
-2+)6
4 -)12
-2 + )6
-2+ )6 - 2+ )6
5- )15
0 -1+ )3
- I +)3
1-)3
0 Execution of (YIIt..>J software Eumple 3.3:
~Iso
0
p.u.
yie lds the same result. It is Idl. for the reader as an exerc i$C.
The following dOlO rtfers 10 a $ix·bu$ le"..Unl' pawt r ntlltwir.
Line no.
From bus
To blls
R /p.u.)
x (p.u.)
8/1 (p.u.)
1 2
2 4
6 7
1 1 1 2 2 2 2
0.2 0.25 0.25 0.2 0.15 0.2 0.25
8
)
0 .08 0 .05 0.1 0.05 0.05 0 . 15 0.09 0.15 0.25 O. 15
0.0!8 0.0) 0.03 0.025 0.015 0.02 0 .025 0.03 0 .035 0.025
)
, 4
9
10
, 4
, , ,, )
4
6
6
OJ 0.4 0.28 Jlen
POWER NETWORK MATRIX OPERATIONS
(u) Find [Ys...l (b) Alsofind [Y80,] when line 4·5 is tripped. Solution: (u) Since this problem involves six buses, [YIJ .. i matrix will be 11 6 • 6 matrix. Result of calcu lation or [Y,..,] using the developed software is shown below: Here, the linelbranch admittance being ealculateif lint, diagonal clements (Y,,) of bus no. I are first obtBined followed by the calculation of off-diagonal clements oftlle same bus (Y~). The same scheme being exC'Cu\ed for each of the buses, the final [Y"",I array is oollIined. (b) For the steond case, when line 4·5 is tripped,!he system reduces to a 9·line system wi!h 6 buses. With this input, new [Y_ I is obllIined.
Execution of the computer program YBUS.FOR for Example 3.3 Line data for Example 3.3a: ZBUS1A.DAT 10, 6 1No. of lines, No. of buses] 1 , 2, (0.08 , 0,20) , (O.O , O.OlS ) 1 , 4, (0 . OS , 0.25), (O.O , O. OlO) 1 ,5, (0.10 , 0.25), (O.O , O. OlO) 2 , 3 , (0.05 , 0.20), (0.0 , 0.02S) 2 ,4, (0 . 05 , 0.15), (0.0 , 0 . 0 15) 2,5, (0 . 15 , 0,20) , (0.0,0.020) 2,6, (0 .09,0.25), (0.0,0.02S) 3,5, (0.15,0.30), (0 .0,0.030) 4, 5 , (0 . 25 , 0. 40), (0.0 , 0 . 035) 5,6, (0.I5 , 0.28), (0.0 ,0. 025)
[From bus, To bus. (R. XLJ. (G 811»
,, ,, ,, ,, ,
,,, ,
,, ,, ,, ,, ,,, ,, ,, ,, ,
Output of YBUS.FOR for Example 3.3a: YBUS1A,DAT No . o f
buses
Ybus match: Yb us { Ybus{ Ybu.s { Ybus ( Ybu.s( Yb us { Ybus( Ybus( Ybus ( Ybus { Ybus ( Ybus( Ybus{ Ybus (
-
6
1, 1,
1
1,
3 I 4 I
1, 1. 1,
2, 2. 2. 2. 2. 2, 3, 3,
I 2 I
S 6 1
2 3 4 S 6 1
I I I I I I I I I
2 I
----
----
I 3.872679, -11. 526170 I Y" 4. 3103 45 I Y , I - 1. 7 24138 , I
I I I I I I I I ( ( (
.000000 I .000000, - . 7692 3 1 , 3 . 84615 4 I -1 .379310, 3.448276 I . 000000 , .000000 I -1 . 724138. " . 3103 45 I 8.575396. - 21. 654300 I -1.176 471, 4. 70588 2 I - 2.000000 , 6.000000 I -2. 400000, 3 . 200000 I ) . 541076 I -1.27 4788. . 000000 , . 000000 I 4 . 705882 I -1 . 176471,
,, , ,, , ,
MJlerl
I
J{l1
dlreibs
JIc>raJ
PO WER SYSTI;M A.NALYSIS: OPERATION A.ND COHTROL
Ybus ( Ybus~ Ybull~
,
,
,,
Ybus ( Ybull{ 'tbus { 'tb us( Ybus! '{bus! 'tbus { 'tbus{ '{bull ( 'tbus { 'tb us( 'tbus ( 'tbus( Ybull ( Ybus( 'tbus( Ybus ( 'tbus( Ybus (
3, 3, 3, 3,
" " "
"" "5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6,
3 I 4 I 5 I 6 I 1 I 2 I 3 I I 5 I 6 I 1 I 2 I 3 I I 5 I 6 I 1 I I 3 I I 5 I 6 I
---
-, ---, --, -, ---
( ( (
( ( ( ( ( ( (
I (
I ( (
( (
( ( ( ( (
2 . 509804, -;.317549 ( . 000000 , .000000 I -1 . 333333 , 2 . 666661 I . 000000 I . 000000 , - . 169231, 3 . 8 46 154 I - 2 . 000000, 6 . 000000 I . 000000 , . 000000 I 3.89282 6, -11 .563910 I - 1.123595 , 1.191153 I .000000, . 000000 I -1.319310, 3 .448 216 I - 2.400000, 3.200000 I -1 . 333333, 2 .666667 I -1.123595, 1. 791153 I 1 . 722860 , -13.747120 I - 1.486620, 2 . 775025 I . 000000 , . 000000 I -1.274788, 3.541 01 6 I . 000000 , .000000 I .000000 , .000000 I - 1.4 S66 20 , 2.775025 I 2.761408, -6.266101 I
,, ,
y~
Line data for Example 3.3b: ZBUS1B.DAT
9, 6 1, 2, 1.4 , 1,5 , 2 ,), 2 ,4, 2,5 , 2 , 6, 3, 5, 5 , 6,
,
(0 . 08,0 . 20) , (0.05 , 0 . 25) , (0 .1 0 , 0.25). (0 . 05 , O. 20) , (0 . 05 , 0 . 15), (0.IS , 0.201 , (0 . 09 , 0 . 25 ) , (0 . 15,0 . 30) , (0 . 15,0 . 28 ) ,
{O . O, O. Ol S} (0.0,0.030) (0.0 , 0.030 )(0 . O, 0 . 02S) (0 . 0 , 0 . 015) (0.0 , 0.020) (0 . 0,0 . 025) (0 . 0 , 0 . 030) (0.0,0.025 )
Outpul of YBUS.FOR for Example 3.3b: YBUS1B.OAT No.of buses
'tbus matrix Ybus ( 'tbus { 'tbus ( Ybus! Ybus ( Ybus (
1, 1, 1,
1, 1,
,, '
-
6
, , -1
I I 3 I I 5 I 6 I
•
•
I ) . 812619 , -11. 526770 ( -1. 724138 , 4. 3103 45 ( .000000 , . 000000 ( -.769231, 3.8H154 ( -1. 379310, 3.448276 ( . 000000, . 000000
I I I I I I
Y" Y" ,
Materia! I :xn Olrf'llos
Jlorals
/'OwrR NOWORJ( MATRIX OPHU.TIONS Ybus ( Ybus { Ybus( Ybu!l ( 'tbus ( ,{busl '{bus ( '{bus { Ybus ( '{bus ( '{bus { Yb us! '{bus ( Ybus ( 'tbus ( Ybu s ( rbus ( rbus ( Ybus! rbus( rbus ! rbus{ 'tbus ( Ybus ( rbus ( rbus ! 'tbus( rbus! Ybus~
'{bus (
2, 2,
2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4,
5, 5, 5,
5, 5, 5, 6, 6, 6, 6, 6, 6,
1 ) 2 I 3 ) 4 ) 5 ) 6 ) 1 ) 2 ) 3 I 4 ) 5 ) 6 ) 1 ) 2 ) ) ) 4 ) 5 ) 6 I 1 ) 2 ) 3 ) 4 ) 5 ) 6 ) 1 ) 2) ) ) 4 ) 5 ) 6 )
.. .. .. .. .. .. .. .. .. .. .. '"' >0;
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
( - 1.72 41 38 , 4.310345 ) ( 8 . 575396 , - 21.654300 ) (-1.176 471, 4. 705882) ( - 2.000000 , 6.000000 ) (-2.400000 , 3 . 200000) ( - 1.274788 , 3 . 54101 6 ) ( .000000 , .000000) ( -1. 17 64 1 1. 4 . 705882) ( 2.50980 4, -7.317549 ) ( . 000000 , . 000000) ( - 1.)33333 , 2.666661) ( .000000 , .000000 ) -. 769231 , ).846154) (-2 .000000, 6.000000) ( .000000 , .000000) ( 2.769231. -9.8 01153 ) ( . 000000 , . 000000) ( . 000000 , . 000000) (- 1. 319310, 3. 448 216) (- 2.4 00000 , 3 . 200000) ( -1. 333)33, 2.666661 ) ( . 000000, . 000000) ( 6.599264, - 11.984910 ) ( -1.4 86620, 2 . 775025) ( .000000, .000000) ( - 1 . 274789, 3.5 4 1016) ( .000000, .000000) ( .000000 , .000000) (-1.486620, 2.775025) ( 2.761408, -6.266101 ) (
,
, r.
,
3.3 MODIFICATION OF [Yeus1 DUE TO INCLUSION OF REGULATING TRANSFORMER BETWEEN TWO BUSES In th is section we will discuss about the formation of admittance ma/rix between two buses with inclusion of a two-winding transformer between them , We take into account the p.u. transfonner adminanccs (the series and shunt using nmode l) which are actually reciprocal of the p .u. impedance of the transformer that has the complex transformation ratio a:l. The nequivalent line may be shown to the left as well 35to the right ofthc tr.ansformer. We present here both of these representat ions for beueT understanding. We assume the regulating transformer to be situated at the receiving end of the line when the nmodel is included at the left side of the transformer (Fig. 3.4). while we assume the regulating transformer to be situated at the sending end of the line when the ,T model is included at the right side of the transformer (Fig. 3.5). Since the regulating transformers arc usually provided at the: end(s) of the line. we approach a realistic situation by assuming the transformer to be placed at the ends of the line (e ither sending or receiving) and nearest to the buses.
Mate-nail Jm dlre-itJ •
Jtoral.
POWER SYSTEM I\NALYSI5: OPERATION AND CONTROL
Case A: When the regulating transformer is present between two buses and is placed, s/ the receiving end: Let us assume the regu lat ing lransfonner is having a complu transformation ralio of a ( .. I a I L a). Fi gure 3.4 represents Ihe voltages and currents al the line and bus side for Ihe transfonncr being included in the line. , ,
PrimlU)' vol4lge V,...
_ a";(mv.
)
j
,
, ,
I,
I
),
) '.
,,' (o; I)
1
, Equi,-aknl ~i",uit for a line ,ontai ning regulmi ng transformer bctwecn \WO buses and placai al the rcce;>' ing end. [S, and 5, ~ the inj ected comp lex powers al the srnding end and recei\';ng end ~s. =pccti'-c1y. while I', and I~ are respective btlS ,'oluges. IaI > 1. Le., Ioi" I + 1&>11.
~il-
3."
The IrMsfonner is assumed 10 be neMeT to the j-tll bus (re<:eiving end bus) and has ,omplc)( ofJ-nominaltap ralio a : I, wh ich CorTeSpondS 10 Vprt: V.... ; also, lot > 1. Assuming the transformer 10
be loss-less.
.'"
....L ~ a
,,
~'
.. IaILa
(i.e_. Vprl) = aVj
(3.9)
Also. iRput power being equal to outpUt power,
I Ii
V;'
Ij
• Vj
/ :_
Of
- = 11 = -
,
[ 1; is the secoRdary current of transfontterJ
[ ' _ ...J.... [,-
(3. 10)
".
Ne"t we consider the CUTTeR! ba laRCe a! two buses by the follow iRg two equations:
,
(3. 11)
,,' , I
1,1 atE
POWER NrTWOII.K MATRIX OP£JU.T10NS
I ) = (-aoYvl \.j+ooo(Yo+YvlV,
• ••
(3.12)
Let us flOW rewrite equations (3.11) and (3.12) in pair (orm u (allows:
li-(YO +Y~ ) \.j+(- oyv)V}
,,"
11 = (-aoyu) \.j+OO·( Yo+Y~ )V,
In maui)'; form these two equations can be reptcsc:nled as
["]- ['., "v I,
-o'Y
(3.13)
It may be noted that a is complu and In is not symmetric. If a is a rtal quantify. i.e. a .. (KVU(KV)...,. then
-" 1
(3. 14)
al(yo+y~) The mam)(
rn then bc:comes 5ymnwlriC.
Cese B: When the f9gu/ating transform9f" is Pl"fJssnt between two buses end is pllJCfKJ at the sending end (Fig. 3.5) Primary voltage:
Secondary voltage
v,... .. v,
Y_ - ~' - "v, y.
•
v,
J
-
I,
,. (1 : ,,)
Fit. 3.5
a.rlvaienl cirevit fCit I line coruinin& reJUlating transCo""tr bd~ two busts and placed II the se~ end. rial> I. i.e.. lor[ .. 1 + jQa[J.
Material,
JfT1
dlreitJ
Ao~.
POWER SYSTEM ANA.LYSIS: OPERATION AND CONTROL
Y'
:...L '"' Q " I II 1La, i.e. Vi " QV;
Hen!,
V.
(3 . IS)
•
Abo (Power bting equal at the transformer input and output, while transformer]
I;
is the secondary eUlTent of
I," .. ..!..
",
, (3.16)
'"At bus i we can write ,
o[
1/ '" a Ii '" a al'/yo +(aJi - Vi )y~] 0
I, '"
'" ..•
aa'yoV;+aa'y~~-aoYuYj
..
I, = Qao(yo+Yv ) l'/-a'y{iVj
(3.17)
AI$O at bus j we can write,
...
(3.11)
In matri)( form equations (3. 17) and (3.11) can bt realTllnged as ,
..
..
[',]-["o(r,+r,) -ayv Ii
c,' y ] Yo+ Y:
,
[~l
, (3.19)
..
H=
-,' r,] IYI · ["'~:r:r.) Ylj Yo +
,
It may be notw that a ~ng rtal,
[Yj .. [a
l
.. ,, ..
.
(3.20)
'
.
(~:~YU) -'Y, ]
(321)
)'0 + )'1/
and the [y] matri)( b«omcs ~mmelrir:al. [In practical cases, the regulating trlll\$fOfll1Cf is designed for either vol/age magnitude or phase angle r:ontrol. In the former case a " 0 and 101 can bt changed in discrete steps of 6101. In the laUer case, 111\ is constant and a is chan,ed in discKle steps of 6a.)
3.4 FORMATION OF [Ysus] WITH TRANSFORMER PRESENT IN THE LINE ..
,
Once the modelling oflhe branch with the transformer installed between thesc two buses is done, we proceed to modify [Y"".j . We are now in a position to draw the equivalent ,y-circuit of the line 1,1 atE
POW(R N(7WOI!J( MATRIX OPERATIONS
rransfonner system connected between bus i and j . Figure 3.6(a) rcpracnts the tr equivalent circuit when the transformer is placed at the receiving end. while Fig. 3.6(b) represents the 11" equivalent circuil when the transformer is placed al the sending end. II may ~ noted here that Y" and Y...., or Y
,
,
• ,
r"; ,
, ,
,
,
,
,
, ,
,
,
,
roO, .. (Yo'" )'~) ... (It
a)'v) " )'o· (I -
,
,
"
t )y~ ;
-,
., ,
': "
",
,
alY¥
("'I >
Equivaletild int it for the uansfonncr pI~it;rtbi'=eivilli md ,
'.
,
.. ,\
,,,
,
r.... a)'~; r....-, " a'(y. "')',1'" (--11)'). .. a' )'• .,. a(a fi" 3.6(a)
j
,
'" ,
,
,
I).
j
" r...,
--
. ~.
r..,
-
' 'T' .. .. . . ..
"
..
,.
r.. .. ")'f ; J'~, .. a' (y. -f- )', ) -+- f-V" ,,!.)'•.-+- <1(,,- 1W¥ ;· rIl," ()., -+ ),.-J ~ (- ,,)'¥) .. -", . (1 - "V',
.,
A" J.6(b)
..
11:
,
,
,.
.'-
Equivalent circuit for the transformer placed' at the sending end
..
.. -- ._. -
,
(lal > 1).
..
"
Then !he (r... 1matrix. can be modified with inclusion oflhc InnsfOilllcr 't eith« end with revised form of 11$ self (diagonal) and transfer (off-diaSOnal) demenl$ as shown below; .'. ~ I" .. : ' , ".(II) Fer the taJe whtn tht Irllll$fonntr if at tht rtcti\'ing end fid;
r~_l = Y;o'+ .•• + Yo +( I-a lYli +-ayti + ... +
'ii
(3 .22)
'" >';0+ ' ··+>'0 +)',+····)'.. Mate-rial
Itore
I
,, ,•
POWEll. SYSITM ANALYSIS, OPEItIIT/ON AND CONTROL y~
I•
,
., y.,
Yji\- )
•
i·
_I
1\-1
., -aYij
= YJO+···+alYo+a(a-I)Yi/+ayV +· ·· +Y",
, ,
)110+"'+ 0 "a+a )lv+ ···+Y,. For Ihe case whell tM tr(l1Ufomur is at t~ sendi1l& end side =:
(h)
(3.23)
(3.24)
y",_) = >';0 +"'+a1yo +a(a-I)Yi/ +ayU + "' +}'/><
• I
= riO + ...+alyo +a1yv
,•
+... + y..
(3.25)
YiJ1- ) = YJ\-t - -aYiJ
i
Yj/
I- I
=:
(326)
YjO+" ' +Yo+(I - a)Yi/+ayU+ " ' +Yjo (3.27)
'" J'jo+· ··+Yo+Yij +···+Y...,
I,
En.pIe 3.4: A thne· lnu syllem is shown ill Fig. £J.3(a). Assume all idtal trans/orme, to In XOIIMcud IHtwUl1 b/lSt!s 2 and 1 ill uriltS wilh a liM rractaflU jO.5 p . lI. if off-tlf)mini11 tap ratlD be j;W2, jinci [y....J.
I
\"C--{::"':'}-~~
, •
•
,, I
flc. E3.J(a)
A
tIne-bus three ·line po.... er l)'item.
, I
, ••
1
, •
• I • •
, •
To ""
R (ill p.... )
X (in p.... )
2
0.15 O.IS
0'
2
1 1
3
0.05 0.05
3
2
3
o
Off-nomiNIl tap ratio ofrnuuform.,
1:1.02
Soll'dotI: Given:
Zi l -
zll - (O.OS T jO.(5) p.L1.
..•
Yll-
~-
series admittance of line •
I .
O.OS+ }0.15
... (2- j6) p.LI. \,1atE
I"ID
Jlo ais
POWER Nf.TWORJ( MA.TRIX OPERATIONS
Since the given problem is. three-bus system. henee [Y"",,) matrix will be a 3 " 3 matrix. :. [Y"",,] matrix before considering ~ line with transfOlll1er (as explained earlier) is given by [see Fig. E3.3(b»).
CD -r-
=---C·,,:::J------= -
0
."
-'---(j) f1&.
£3.J(b)
The system befon: considering 1M: line with ttansfot"Kl.
(4 - j12) (-2+ j6) (- 2+ j6)
[r,.] _
(- 2+j6) (- 2 + j6)
(2 - j6) (0+ jO)
(0+ jO)
p.L1.
(2-j6)
This is the case of regLllating transfonner placed at the sending tIld,
(VifYz
~
1.02). i.e.
a - 1.02 [see Fig. EJ.3(e)].
"model orlM: line with
r-", Ll.J(r)
·•. •
ttansfo""er.
I
Yll ..
Y....
jO.S .. - j2 p.u. QYll ..
1.02>« -j2) '" -j2.04 p.u.
Material
I
Jm dire-,to
]to ai
POWER SYSTeM ANALYSIS: OPERATION AND CONTROL .
...
ylA<, "a ( ~-I ) yu [.: Yo · 0+ )0]
r;.,.,".".',
= 1.02)«(1.02 - 1))«(- )2) = - )0.0408 p.ll.
.. Y.oy ! ·
0.-a)Y1J "" (1-1.02)
For the tnmsfonner in line 2-3:
Y,.,:> '" Y""n...
'
x( -J2)..- )0.04 p.ll. . - ,
+(-)2.04)+(-)0.0408)
= (2- )6)- j2.OS08 '" 2- )"8.0808 p.u. Y,.." ~ yllou,.... +(-)2.04)+(jO.04)
'" (2- j6)- j2 = (2 -
•·•
}"g) p.u.
(4 - J12) (- 2+ ) 6) (':2+ j 6) [r,.I= (-2 +j') (2 - j 8.0S08) j2.04 )2.04 (2 -j8 ) (-2+j')
p.u.
Execution of the computer program '(BUS.FOR .for Example 3.4 line data: ZBUS11.DAT
••
3,3, 1 1,2 , (0.05,0.15) , (O , O) 1. 3, (o . 05, 0 . 15) , to, 0) 2,3, (0 , 0.5i', (1. 02;Q ) ,
[No. of lines, No. ofbuscs, No. oftransfonnersJ [From bus, To bus, (R. XJ, (4 812)] [From bus, To bus, (R. XL ), (G 812)] [From~s, To bus, (R."XU, Re<:tangular fohn of off· nominal tap ratio oftr.msformerJ
Output of TMYBUS.FOR: YBUS11.DAT
The o"'p"r/llrt)ished the [r,... l mo/rlx/irsr considering I/O trans/()I"mer and then considering the said fraM/armer.
No . of buses •
3
Before cons ide r ing line transforner Ybus m3.trix Ybus { I, 1 { • { 4 . 00000 ,-1 2 . 00000 1, Ybus { 2 { • ( -2,00000, 6 . 00000 (, ( -2 . 00000 , Ybus { 3 ) 6. 60000 6 . 00000 Ybus ( 2, 1 ) • ( - 2 , 00000 , Ybus ( 2, 2 ) • ( 2. 00000, - 6.00000 Ybus ( 2, 3 ) • ( . 00000 , .00000 ( ) • { -2,00000, Ybus { 3, 6 .00000 3, , 00000, ' ,00000 Ybus ( 2 ) • i Ybus ( 3, 3 ) • ( 2.00000, -6.00000
..
[
{
Y"
{
r"
) )
•
)
) )
) )
, Y"
,
{
•,,
I
Jlen
1:m1 dlreibs
JloraJ
POWER HETWOAA MATlUX OPEM.7lOHS
'n
For Transformer line - J (.0.0000908+00. - 2.040000.) series admittance of lin.e shunt admittance of from bus (0.000000£+00,-4.079996£-02) sliunt admittance of bus (0.0000008+0.0,3.9999968-02) . Ybus matrix after considering line transformer
,, .
'
'0
¥bus I ¥bus I ¥bus I
1.
Ybus
Ybus Ybus Ybus Ybus
Ybus
1 2
I I
1.
J
2, 2, 2,
1 2 3 1 2
I I I I I I I
"
I I I I I I
J. J, J,
Ex·mpSe 3..5:
J
• • • • • • • • •
,- '
,
4.00000. -12.00000 I Y" {-2.00000. 6.00009 I 'Y'l (-2.00000, 6.00000 I ( -2.00000, 6.00000 I I 2.00000, - 8.08080 I .00000, 2.04000 I I ( -2.00000, 6.00000 I .00000, 2.04000 I I I 2.00000, -8.00000 I Y" I
... ,
,
,
,
Afive·bus SYJIDR iJ J!wt.·11 ill Fig. £3.4. Awun"'l111 idMllrurufo,.mu 10 be conMCtM bnwun biluJ J" mvl2 uri",J with 0 lin ... of mJClonc... jO.4 p..... If I~ off-nomilllll rap ittrio be I: 1.05. find [Y,,;.,l using camp"u,. program.
in
,
(j)
,
.•
Line no. 6
" a>
,
,
""
Line 00. I
Line 00.2 00. )
Line no.
~
(l) Line no. 4 Fll- E3A Th~
lim!! do/a for
Ih~
A five-bus power system.
given Iylum is shown below (n~gltcl shllnt charging
LiM no.
From bus
To,,"
R (ill p.Il.)
X (ill p.Il.)
I 2
I 2 2
4 J
,
0.30 0.28 0.18
,
J 4
,
0.10 0.10 0.075 0.15
6
I
2
J 4
4
O.IS
0
~Jftcr)
Ojf·_iIllIl lap ratiD of rrwufonMr
0.35 0.40 0.4
1:1.05
Matf, ,
Jtorai
1 SYSTEM ANM.rSIS, OPEIVITION AND COHrROL
,
Solution:
Execution of the computer prognlm yaUS,FOA for Example 3.5 TM computer program !ol/awing the j1uwchart fiunuhed in 1M lUI for deurmining If..,.".] /(11' 1M power nefWfJrl; having trans/or"," i.J urculed wilh the/of/owing /in' dOlo (flIP"')
i
Line data: ZBUS2A.OAT
,,
6,5,1 [tfo. of lines, No. of buses, No. oflrart5formersJ 1,4,10.10,0.30), (D . O, 0.0) [From bus, To bus, (R. I I 2 , 3 , (0 . 10,0.29) , (O.O,O .OJ
XJ. (G I
&'1)1 ,
2,5, (0.075,0.18), (0.0 ,0.0)
3,4, (0.15,0.35), (0 . 0,0.0)
,
4, 5, (O.lS,O. 40), (0 . 0 , 0.0)
I
1 , 2 , (O.O , O.4), (1.05,0.0)
I
,
I
[From bus, To bus, Rectangular form of off-nominal tap
ratio oftransronner]
,
1M owput fornished 1M [r".,] malrix fUJI considering no uaruformer and then coruiliuing the said transformer.
Output of YBUS,FOR: ,,(BUS2A.DAT
,
'0,
of b uses
S e fore
-
5
con"ldering
line
tr .. n,,(ormec
Ybus matrix
, ,
,
Ybus Ybus Ybus Ybus Ybus Ybus \'bu, Ybus YbU3 Ybus Ybus YbU3 Ybus Ybus Ybus Ybus Ybus Ybus YbU3 Ybus
I I I I I I I I I I I I I I I I I I I I
-" , -1,
1
)
1,
2
)
)
)
1,
)
1,
5
)
2, 2, 2, 2, 2,
1
)
2
)
)
)
<
)
5 1
)
2
)
)
)
<
)
5
)
1
)
), ), ), ), ) ,
4,
<, <, <, <,
,
)
)
)
)
<
)
5
)
--------
I 1 . 00000, .00000, I .00000, I ( -1. 00000, .00000, I . 00000 , I I .3 .. 103 61, ( - 1.13122 , .00000, I ( - 1 . 97239 , . 00000, I \ -1 . 13122, ) 2.16570, ( - 1.03 44 8, .00000 , I ( -1.00000, . 00000 , I { -1.03448, I 2 . 85640, I -. 82192 ,
- 3.00000 .00000 .00000 3.00000 .00000 . 00000 -7 . 90115 3.16742 .00000 4.73373 . 00000 3.16742 -5.58121 2.41379 .00000 3.00000 . 00000 2.41379 - 7. 60557 2.19178
) )
Y" Y"
) )
)
)
) ) )
) ) ) ) )
) ) ) )
) )
,, ~ale-rlall :>rTI
dlfl'IID
Jlo ais
POWER NETWORK MATRIX OPERATIONS 'ibus Ybus Ybus Ybus Ybus
( ( ( { (
5,
5, 5, 5, 5,
1 ) .. 2 ) ..
J ) .. 4 ) .. 5 ) ..
For Tran l!lformer
in
( .00000, ( -1. 97239, ( . 00000, ( -.82192, ( 2.79430, line no.
.00000 4.733 73 .00000 2.19176 -6 .92551
) ) ) ) I
, ,, Y"
6
(0.000000E+00,-2.625003) I!Ihunt admittance of tram bus (0.000000£+00,-1.312526£-01) shunt admittance of to bus (0.000000£+00,1.250026£-01) l!Ieries admittance of line
Ybus m
1 ) ..
1.
2, 2,
2 ) 3 ) 4 ) 5 ) 1 ) 2) 3 )
.. .. .. .. .. .. ..
2,
4
..
2,
5 ) ..
3, 3,
1 2 3 4 5 1 2 J
( {
YbU5
(
YbU5
(
"
Ybus
(
1.
YbU5
(
2,
YbU5
(
Ybus Ybus
( (
YbU5 Ybus
( (
Ybus Ybus
( {
Ybu s Ybus
( {
¥bus
(
Ybu s
(
Ybu s Ybus Ybus
( { {
Ybus
{
Ybus Ybu s
( {
Ybus
{
5, 5, 5,
Ybu s
(
5,
E. . . ple 3.6:
Ihm
1,
Ybus Ybus
conn~Cff!d
"
3, 3,
., 3,
" "" "5,
)
) ) ) ) ) ) ) )
.. .. .. .. .. .. .. ..
4 J .. 5 ) .. 1 ) .. 2 ) 3 ) ..
4 J .. 5 ) ..
line transformer
( 1.00000, -5.75626 ( .00000, 2.62500 ( .00000 .00000, (-1.00000, 3.00000 ( . 00000 , .00000 ( .00000, 2.62500 ( 3.10361, - 10. 40 115 ( -1.13122, 3.16742 ( .00000, .00000 ( -1. 97239, 4.7337 3 ( .00000, .00000 ( -1.13122, 3.16742 ( 2.16570, -5 . 58121 (-1.03448, 2.41379 ( .00000, .00000 (-1.00000 , 3.00000 ( . 00000, .00000 (- 1.03448, 2.41379 ( 2.85640, -7.60557 ( -. 82192, 2.1 9178 ( .00000, . 00000 4.7337J (-1. 97239, ( .00000, .0000 0 ( -.82192, 2.19178 ( 2.79430, -6.92551
) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
Y" Y"
, ,, , ,,
, ,,, ,,, ,, r"
For Example J . j, conslihr lhe line charging ofjO.OJ (811) p.lI. In 011 the IifIU except In so-ie. with luuu/OI7Iff!l'. Find _ (Y"",} matrix.
Male-rial
:mI
dire-it)
Jlorai
! I
POWER SYSTEM ANALYSIS: OPEJlATION AND CONTROL
,
SoILillon:
, I
!,
i !•
, I, •
Execution 01 the computer program YBUS.FOR for Example 3.11
LIM dati: ZeUS2B.DAT 6, 5 , 1 [No. of lines, No. of buses, No. of tr'llmformers] 1,4, (0.10,0.30), (0.0,0.030) 1From bus, To bus, (R. XI)' (a 812») I " I 2,3, (0 .10,0.28), (0.0,0 . 030) 2,5, (0.075,0.18) , (O.D,O.OJ)
,
,
I
.
3,4, (O .1 5 , 0.35 ) , (0.0 , O. 030) (From bus, To bus, (R,
I
XJ. (a
4. 5, (O .15, 0. 40 ), (0.0, O. 030) (From bus, To bus, (R. XI).
I, 2, (0.0 , O. 4) , (1.05 , 0 . 0)
,
I
'
,
(a
I
8I2)J
iiI2)]
[From bus, To bus, Rectangu lar form of off-nominal tap r.tlio ortnmsformerJ
Output of YBUS.FOR: YBUS28.DAT
I
No .
,
Before considering line transformer
of buses "
5
Yhus matrix •
..
I 1.00000, I .00000, ( .00000, ( -1. 00000 , ( . 00000 , . 00000, I 3 . 10361. ( - 1.131 22 . ( . 00000 , (- 1. 97239, ( . 00000 , ( -1.13122, (2.16570 ,
) .. ) .. ) ..
(- 1.03448, ( . 00000 , (-I.OOOOO,
I,
'tbus
(
Yhus Ybu s
( (
Ybus
(
I, 1. 1. I,
4 I .. 5 I ..
rhus
(
2,
1 I
Ybus
(
2 I •
Ybu"
(
'tbus
(
'tbu s 'tbus Ybus Ybus Ybus
( ( ( ( (
Ybus
(
Yb us
(
Ybu s Ybus Ybus Ybus
( ( ( (
2, 2, 2, 2, 3, 3, 3. 3, 3, 4, 4,
•
Ybus
(
I
Ybus Ybus Ybus
( ( (
Ybus
(
, •
,, ,,• •
;
I
i, ,• , ••
I I• I
,, •
I ,•
Ybus
1 I • 2 I •
(
!
4,
4, 4,
5, 5, 5,
5, 5,
3
) ..
·,
3 I .. 4 ) .. 5 I •
1 I •
2 I • 3) 4 5 I
2 ) .. 3 ) ..
( .00000 , ( -1. 03 448, 4 ) .. ( 2. 85640, 5 ) .. ( - .9 2192, 1 ) .. ( . 00000 ,
2
-2.97000 . 00000 .00000 3.00000 . 00000 .00000 - 7 .8411 5 3.167 42 . 00000 4.73373 . 00000 3 .1 6742 -5.52121 2 .413 7 9 . 00000 3.00000 .0 0000
) ) )
Y" Y"
)
) ) ) )
) ) ) ) ) ) ) ) )
2.41379 J
- 7 . 51557 ) 2 . 19179 ) .00000 )
) ..
(-1.97239 .
4. 73 373 )
3 ) .. 4 ) .. 5) ..
( .00000, ( - .8219 2, ( 2. 79 430.
. 00000 ) 2. 19178 ) - 6.86551 )
Y" Jlen
1:m1 dlreibs
J!c>raJ
POWf //. NfTWORX MATRIX OPflVoTIONS
-
Fo, Transformer io line 00 , 6 (0.0000OOE+OO,-2.625003) series admittance of line shunt ad'TIittance o f from bus (O.OOOOOOE.OO,-1.312528E-01) shunt admittance of to b" (0.OOOOOOE +00,1.250026E-Ol) Ybus matrix after considering line transformer Ybus { { 1.00000, -5. '1 2626 I Y" 1 I 1. 1, { Yb us { 2 I . 00000, 2.62500 I r" { .00000, Ybus { 3 I .00000 I Y" 1. Ybus { 1, 4 I ( -1. 00000 , 3.00000 I rbus { 1, { .00000 I S I . 00000, 2, { 'tbus { 1 I .00000 , 2.62500 I { 3.10361, -10.34115 I YbU5 { 2, 2 I 2, 3 . 16742 I 3 I ( - 1.131 22, Ybus { { ,00000 , 2, .00000 I Ybus { I ( - 1,97239, 4 . 73373 I Yb us { 2, 5 I 3, { ,00000 , Ybus { 1 I . 00000 I 3, 3.16 7~2 'tbus { 2 I ( -1.13122, I { 2.16570, - 5 . 52121 I 3, Ybus { 3 I ,, rbus I ( -1.03448, 2.41379 I 3, 4 I , Ybus { 3, { . 00000 , 5 I . 00000 I 4, Ybus { 1 I (-1 . 00000 , 3.00000 I , { { YbU 5 4, 2 I . 00000, .0 0000 I 2.41379 I 4, 3 I ( - 1.03448, Ybus { , { { Ybus , 4 I 2.85640, - 7 . 5 1557 I , { -.82192, 2.191 78 I rbus { 5 I { r bus { 5, 1 I . 00000 , .00000 I rbus { 5, 2 I ( - 1.97239, 4.7337 3 I 5, { .00 000 I Ybus { 3 { .00000, { -.82192, Ybus { 5, 4 I 2: 19178 I , { 2.7943 0 , - 6.86 55 1 I rbus { 5, 5 I ~, ,
-------------
,
..
Eumple ),7:
,
Oinuin (Z/Iou"] mutrix lor Ihe J)'$I.,m Je.I'crib..d in uamplt! 3.6.
Solurlo n: (Z"", ] matrix is obtained by inverr ing the [Ylho ' ] matrix obtained in Example 3.6. Execution ot the computer program for " Example 3.7
m~trlx
Inyerelon MINV.FOR tor
"
,
"
Input data: YBUS2B.DAT ([Y,g..J matrIx) Output of MINV.FOR: ZBUS2C.DAT No,
0'
buses
-
5
zbu!!< lI'IlItri>: i .
Malenal,
Jm
direikJ
Jtorai
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POWER NEn,'ORK MATRIX OP£RATIONS
, T 1.-1 + I
l'J
-L
'"
lsISNBRN?
,. CQ!I1p11te -, (y] .. [z] ; [AI .. Transpose of[A];lvAj" [y][A)
,
and [Y.., ) "' [4 [y) [A]
L f'Ic. ) ,9
Display and/or OIore [A ]. [z], [y], [yA ] and [Y...l mamCd
/
'"'
flo ....chan fOl' de~e lopmr:nL o f I Y,,", I nwrix using singul ar Lran,fomution.
Example 3.10:
In a plinian of n po ....er syJfem n ef><'ort (Fig_ EJ.5(a), two brundles 1-2 und 2·) are mutunlly coupled Ihrough ~ .. (= jQ.2 p.u.). Find the bus admittance matrix using singular /fan sfo f1tIQ t iOll.
(j)-
I
• ~11l'
Solution:
.:3.5(11)
•
Mutual ly coup!ctl ""tworh.
Th e orientcd/connected graph of the system is shown in Fig. EJ.5(b).
(j) 2
• " . J.5(b)
•
Oriemedloonnl'Cled gl'llp/1 of Ihe system mo .... n in Fig. E3.S\al· Jlen
'eibs
JIc>raJ
POWER SYSTEM ANALYSIS, OPERATION AND CONTROL
(Since there is a mutual coupling between branches I and 2, and for both branches the dots are lowards bus 2, bus 2 is taken as from node for both of these twO branches.) Reduced incidence matrix is given by nodes --t I' 2 3
[..ll - br.tnchcs
.1.
I[-I 1 0]
2
0
1 - I
However, primilive impedance matrix is given by
1
2
Iz] _ I [jO.3
2 jO.2 i.e.
jO.2] jO.3
the primilivc admittance matrix becomes,
[ y ] '" [;:]
1 2 -j6 j4] '" 2 j4 -j6
_I I[
[- j6 [y][..l ]" j4
•·•
jT'
-j6
1 0 1
-~]
"[!;4
- j6+ j4 - j4] j4- j6 j6
[ j6 - j2 - j4] " - j 4 -j2 j6 -I
Now
(Arl '"
1
0 1
0 -I -I •·•
Irs... ] '" [Ar]
[yJ[A] "
1
0
-j2 -j4] [ j6 1 -j4 -j2 j6
0 -I
(-j6) (j2) (j4) • (j6- j4) (-j2-j2) (-j4+j6) (j4) (j2) (-j6)
•
-j6 j2 j2 -j4 j4 P
j4 j2
-%
Execution of the computer program ATYA.EOR Ipr ElIaMple 3.10 Input required lor computer aimulatlo.. : No. of buses - 3; No. of branches - 2 For braoch-I : Branch no ... I: branch hwpedance - jO.3 p.u.; From node'"' 2; 10 node " 1 [since the dOl is lowards node 2J Materia! I :xn olrf'llos
Jlorals
poweR NeTWORK MA.TRIX opeRATIONS
The branch is mutually coupled. (BralK'h·l is mutUllJ1y coupled with branch·2.)
The mUIUlII impedance "'Ill p.u. For braoch.2 : Branch no. = 2; branch impedance = jJ.3 p.u.: From node 2 ; 10 node 3 [since the dol is lowards node 2J Since the mutual coupling bc\"'eeo branches 2 and I has already been coll'lidered earlier, helK'e there kIlO more mutual coupling with branch·2.
=
=
Output Or ATVA.FOR: AYBUS,DAT Reduced incidence matrix
""
1. 1. 1.
""
2. 2. 2.
"
AI
1 I 2 I 3 I 1 I 2 I 3 I
-•
• ( -1.000000 , • I 1,000000, • I . 000000 , • I . 000000,
•
.000000 .000000 . 000000 .000000 .000000 .000000
I I I I I I
A" A"
.300000 ' 200000 .200000 .300000
I I I I
'" '" '" '"
.000000, .000000 , .000000, .000000,
- 6.000000 4 . 000000 4.000000 - 6 . 000000
I I I I
.000000, .000000, .000000, .000000, .000000, .000000,
6.000000 - 2.000000 -4. 000000 - 4 .000000 - 2 . 000000 6 . 000000
I yAIi I )"A'l I . I I I
I 1.000000, ( -1.0 00000,
•
,, , ,
A"
Primitive I , I matrix , , , ,
I I I I
1 I 2 I 1 I 2 I
1• 1.
2. 2.
Primitive ,I
"" "
I 2 I 1 I 2 I
2. 2.
(y * A)
yAI yA< yAI ,A I ,AI yA,
I,J 1
1. 1•
• • • •
.000000, .000000, . 000000, .000000,
I I I I
matrix
• I
•
I I I
• •
,,, ,,,
YJ2 ,..~
mat rix
,
1.
1
I I 3 I 1 I 2 I 3 I
1•
1.
2. 2. 2.
• • • •
• •
I I I I I I
,, ,
,A"
'(bus matrix -(t r a nspose of A) *y*A Ybus ( Ybus ( Ybus (
1.
1. 1.
1 I 2 I 3 I
• I .000000, -6.000000 I
• •
I .0 00000, I .000000,
2 .000000 I 4 .000000 I
Y" Y" ,
,
~ate-rlall :>rTI
dlre-'to
Jto ais
POWER SYSTEM AN .... LYSIS; OPERATION AND COM'1l.0L
'tbus' Thu s (
rhus(
2, 2, 2,
'thus ( 'tbus (
)
'thus (
),
),
,
1 I 2 I ) I I I 2 I ) I
•
{ .000000 ,
• • • •
( . 000000.
•
( . 000000. ( . 000000. ( .000000,
(.000000,
2. 000000 I ~4 . 000000 I 2 . 000000 I 4. 000000 I 2.000000 I -6.000000 I
, ,
, y"
3.6 DEVELOPMENT OF [Y8usl MATRIX USING COEFFICIENT MATRIX
I, ~
I"
/ 1,
,• ----{==;==l-------.' "
J
•
o reference nCldc=
~ig.
3. 10
Nodal currents injected into a single brand'! two·node circuit
Let liS consider Fig. 3. 10 using Cltrnnt injtlcti{H,
,,·~clors.
I; alld Ij: following the sign ~'()nvention
outlined above, we ca n write
(3.32) I
I
I
by arranging the respective ROda] CUrTent equation s in the vector form . Equation (3.32) indicates the di"CliQII of l ij from i tOj-th node with +1 and -I entries being designa1ed as rows i andj. Assumi ng vo luge drop:!Cross"i1j to be V~ and being dirult:d low"mj I.,. the node vo luge governing eq uotioo V'i'" VI- V) ca n be represented sim il arly in the vector form as
Vij
_ 1+1
-!I[~:J
(3.33)
Howe"cr, from the baliic koowledge of circuit theory, Y,j Vij '" It}
or.
Y~
[+1
-I][~:] = l ij
(3.34)
Pre-mult iplying bolt! sides of equation (3.34) by the co lumn of equation (3.32), we find (3.35)
i.e.
"',
(3 .36)
lyllvl· I'1
(3.31) Material
Ao
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POWER SYSTlM ANALYS IS: OPERATION AND CONTROL
D
C
j
F
N. Is*d - I?
Solve tbe equation
.,
Ya
[AI',] -[B2] [QDC), forl .. I. 2..... {M- I) DispllY.
(.J
Update bus volla~ magniTUde.
vt' - V: + AI',' . for j .. 2. 3, ...• M E
•
.h . .. t+
Final blls voltagc magnitlKk.
-'
j ..
forl·I,2, .... N ;
...-,
(ii,) No .. of iterltiolls
N. lsi - I" ?
Ya
•
/
1,2. .... N;
load antic .t;',
"0
i1
1'/. fOT
Ya
Dilplay. 'ltcnrion/
incomplete for i " I""
•
I.M- N ? No
Compute and display reactive power for PV buses. Q, .. QC.AL, - QD, ro,.; .. (M + 1). 1M " 2). .... N
•
L
•
S"'" f1&. 4..8
Flowchart to cllculatc bus YOlta&eS by FDLF method for the syllCm....;th Q-limit
• PV ""'" Execution of FDLF algorithm (with Q.llmlt at PV bu.) for Exampla 4.9
• •
LINE DATA· GZBUSI.DAT Ybus Matm· GYBUSI.DAT
For III of the above data, refer Example 4.1 (O-S method). WAD FLOW DATA· FDLF3.DAT
1 I
3 , 4 I no. of PQ bwos_ taUl no. of buses] 1 .0 6,0 .0,0 . 0, 0 _0 [V, 'O),I);(O)J'"Qd 1.0,0 .0 -0 _6.-0_3 1.0,0.0, -0. 7,-0 . 5 1.02', O. 0.1 . 2. 0 _0, 0.1,1
lV,°.a.°.P•. Q"
Q..... Q... ]
This dala fil e "i$ s.ame as GLDA.OI.OAT; only orientation of data is different to m l tch with algorithm. \01 alE
Jlora.
COAtPU:X POIV[R FLOWS
OUTPUT OF FDLFQ.FOR - FDVOLTJ.DAT [Final bus voltages]
No . of Iterations rcqd.
k ·
5
FI NAL BUS VOLTAGES ARE
Bus-code
VOLTAGE
------------
------------
,
1 . 0 60000 . 85 1554 . 903501
l
3
• 4.12
1.0 128 98
LOAQ ANGLE
-----------. 000000 -. 1560.69 - .031l31 .169335
DC LOAD FLOW
We rewrite first the de<:oupled power flow equations (equations (4.77) and (4.18».
Let U$ further simplify the load flow technique by dropping the Q-V iteration. It results in a completely linear, non-iterative power flow method. Assuming all bus voltage magnilUdes to be unity, the real power deeoupled equation becomes
1""1; [."]1 061 The "OC" power flow can then be calculated as
Pit = ; . (0; - OJ); P;
,
Howe~r.
this DC load flow method is good only for the calculation of MW flows in lines bUI does nOI give any indication ofMVAR or MVA flows. Figure 4.9 represents the f10wchan to calculate bus voltages by OC load flow method.
Dfn'elop OC wad Fla .... /or the prob/~mfurnished in Example 4.3 in order 10 obtain
ElImple 4.10: bin vol/agel .
Solution: If.... J Matrix NYBUSL OAT LOAD FWW DATA - NLfIA.DAT All or the above data given in Example-4.3 (N-R method). OUTPUT OF DC_ FOR -DCVOLTLDAT No.
of I te rations re qd.
k. -
2
Material,
JfT1
olreitJ
Ao~.
POW(R SYSTrM ANA LYSIS: OPERATION
Read . Tornl no. and [ r ••• 1 rrIiIl rix for ; " I. 2. ___ .
Sep,,,"~
«:al and
G,, " Re.1( 1'1. ). B,, "' - 1111
(Y I ~).
for ,
1.2 . .... N : for j .. 1, 2, ...• N
r
Convert r .... J m.ttrix inlo polar for; .. 1. 2•...• N: forj - I.2 .....
91~ - 9. _[MI. 91 1 is in d~=
,
, R=l.
,.'"''''~''. v·· ,.
\ •
No Inilialisc slack bus voltage V: .. V,';tl,' - J.'. for; - I .....
• .,Sle for
Form and store [BII malrix SI ,•. ,,,,. ,, " So' for i '" 2. J ..."
Mate-rial
:mI
dire-it)
Jlorai
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COIIIPl.EX POWER rLOWS
TABLE P4.2 G~na(Jli(Jn
BIIS
". I
2 3 4 2
Wad
.
MW
MYAR
-
-
-
-
0 0 0
0 0 0
50
20
40
10
20
0
MW
MVAR
(a) In Eumplc 4 ,1 in the text, the lillC and bus data remaining the same, compute bU$ voltage
using G-S mctllod for Qu.... of bus-2 as - 0.1 :s;
Q !:S;
0.2.
(b) Repeat 2((J) but for nat start and assum e I Vl l = 1.0 p.u. Apply Q-S method and check the results using N-R metllod.
1
Figure P4.2 represents a five-bus power system feeding constant power loads.The line data and load schedule are shown in Tables P4.3 and P4.4 (lillC data has been furnished on 100 MVA basis).
•
Q)
Q)
....d
TABLE P4.3 Un~
"
F~,
b"
T,
line
im~dance
Shunt susceplance (812)
bus
(P.II.,
(p.II.)
I
I
2
(0.01 -I- jO.OS)
2
I
5
(0.10 -I- jO.5)
jO.02 jO,(115
3
I
4
(O.IS -I- jO.SS)
jO,(.Y15
4
2
3
(0,05 -I- j O.3)
jO.02
5
3
4
(0,08 + )0.5)
jO.02
6
4
5
(0.02 -I- jO.I S)
jO.OI
M ate-ri
"'
u,
e-itJ
ltoral.
POWER SYST£M ANALYSIS: OPERATION
B"
MW
MW
MVAR
2
50
15
5
3
o
4
o o
20 25
10 10
3Q
15
110.
I
5
4.
Assume slack bus vo ltage ~nd oompare the resuhs.
G.S, N-R alld FDLF methods of load now study
In Fig. P4.2, assume a transformer
buses 2 and 4 having p.u. reactance of jO.3. If
V~: V~ =l.02 : l.(i.e.
V;':V; '"
1.02:1]
(a) Find bus vo ltages and line nows using
(b) Repeat Ihe same e~en:ise for
V~
: V. ::
1.02, [i.e. VI :V/ '" 1.02: 1 ]
S. Repel! problem no 5, but with FDLF 6.
In E~ample 4.5 in the text, the lim: and lood now metllod and compare the result.
t being the same, compute bus voltage using de
Mate-rial
:mI
dire-it)
Jtorai
COM PUX POWI:R FLOWS
1. RESULT OF EXERCISE I ELNVOLTI.DAT By G-S Method No.
of iterations required k -
8
fINAL BUS VOLTAGES ARE
---_
no. B"" .. ..... 1 2 3 4
Voltage in RECTANGULAR form
.. .. _-----------_ ..... _-----_ .... ,
( 1 . 010000 I .970951 I . 976413 I .998218
.000000) - . 090640) -.086518 ) - .042165)
,
, •
Voltage in POLAR f orm Bus no .
.. -_ .........
Voltage magnitude
.._---
1
1.010000 . 975173 .980239 .9991 08
.000000 - . 093082 - . 088377 -.042215
VOLTAGE
LOAD ANGLE
1.010000 .975016 .980207 .999 126
.000000 -.092805 -.088241 -.0421]9
2 3 4
Delta
By N-R Method Ite r ation reqd. , k -
3
Bus-code
------------
-----------1 2 3 4
2. RES ULT OF EXERCIS E 2(A) EGVOLT2.DAT No.
of iterations require d k -
12
fINAL BUS VOLTAGES ARE
B",
no.
--_ .. _-1
2 3 4
Voltage
RECTAN GULAR form
'" .. _-_.......... _-.._------------_ ( 1. 060000
I I
,
.912 647 .890217 .832520
, , , ,
.000000) . 178907) - . 03 14 88) -.132646)
Male-nai, Jm dlre-itJ
Jloral.
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,
•
POWER SYSTEM ANALYSIS: OPERAnON AND CONTROL
4. RESULT OF EXERCISE 4 ELNVOLT4.DAT When V2;V4 = 1.02 :1 Iteration reqd . ,Ie"
J VOLTAGE
Sus -code
-----------1.000000
1
.997329
-.004191
,
.968431 .91)0)2 .957131
-.066869 -.073258 -.090680
5 Base H\TA "
100
••••••••••••
• •
-----.000000 -------
2 3
• •
LOAD ANGLE
LlNEFLO'tiS
• • • •••••••••
Bus-code
MW
MVAR
-----------1 2
------------
-- ---2.017580 ----- ---
-.018221
2
1
.018246
1
5
- 3.874346 3.970911
5 1
,
•
,
•
1
1
•
- 1.971873 - 5.677482 .401535 -5 . 195072 . 001012 -3 . 213329 -,79 0563
2
3
3
,
- 3.473356 3.56256 1 - 1.661327 1 . 684468 -.336513
3
. 336990
,
- . 213457 .214491
-1.204706
2
22 . 770200 - 22.770200
15.789290 -13.5635 10
, ,
2
3
5
•
, , 5
2
• •
- 1.702069 -2.0702 12 -·~f7087
LINE LOSS
.210446 SLACK BOS
PCYtJER
MI~
.
-21.881640
HV'AR
(TN P.U.)
(-7 . 365923E-02,
-1.289013E~Ol)
Male-rial
:mI
dire-it)
Jlorai
COMPLE X POW ER
When V2:Y4
rwws
= 1:1.02
Iteration reqd. ,k =
3
Bus-code
VOLTAGE
2
3
•5 Base KVA '"'
LINEFLCMS
Bus-code
1
1 -
5
5 1
1 •
•
1
2
3
3
2
3
•
•
3
• 5
5
• •
2
•
-.090 425
••••••••••••
MW
2 -
-.004436 -.0 66669 -.07231)
100
.............. ..
2
-----------.000000
1.000000 .999884 .96 4 0 4 3 .957433 .9 4 5352
1
1
LOAD ANGLE
2
MVAR
-. 022050 .022070 -3.683256 3.790366 -3.128547
-2.005066 - 1.994568 -6 . 279184 1.009407 -6 . 016432
3.230792 -1.609983 1.635066 -.209361 .209946 - .222462 .2233 48 22.076280 -22.076280
.849839 - 3.430361 - .5784)2 -2 .219798 - 1.475970 -1.125187 -. 691824 8.516070 -6.83 6023 •
LINE LOSS
.............
. 235928 MW SLACK BUS
POWER
(IN
.
-22.277530 HVAR
P. U. l
(-6.83385 4 E-02,-1.430068E -O l)
MJlerl
1:m1 dlreibs
Jtc>raJ
POW£R SySTUvI .... N.... LySIS: OP£IUTION .... ND CONTROL
S. RESULT OF EXERCISE S ELFVOLT4.DAT When V2;V4 No.
= 1.02;1
of Iterations reqd.
k ...
3
fI NAL BUS VOLTAG ES ARE
Bus-code
VOLTAGE
------------
------------
2 3
, 5 100 *.*.* •••••••
LI N£FLOWS ••••••••••••
Bus-code 1
2
2
1
1
5
5 1
,
2
1 3
------------ . 018055 .018080 - 3.881931 3.984016 - 3. 483 368 3.572141 -1 . 664394 1. 681 4 61 - . 337932 .338 42 8 - .215099 .216125 22 . 786250 -22.786250
1
, ,, 3 5 5 - , 3 3
2
2
4
4
-
MVAR
MW
------------
,-
. 000000 -.004167 - . 066876 -.0732 46 -.090134
1.000000 .997285 .968125 .913594 .958492
1
Base MVA -
LOAD ANGLE
2
------------ 2.01111 4 -1.911503 - 5 .6410 13 .363822 -5 .164634 - .030578 -3. 20330 4 -. 801113 -1 . 690057 -2 . 0 85668 - 1 . 203204 - .671090 15.58358 0 -1).374190
LINE LOSS
-....
- "'-- ~ -
.20 9472 MW SLACK BUS
POWE R (IN
-21 . 906130 MVAR
P . U.)
(-1 .38 935 4E- 02 ,-1. 2823 ~ 2E - Ol)
Malenal,
Jm
direikJ
Jlorai
• When V2:V4
COMPLEX POIYtR FLOWS
= 1:1.02
No. of Iterations reqd. It -
3
FINAL BUS VOLTAGES ARE Bus-code
VOLTAGE
-----------1
1.000000
.000000
1.000085 .964861 .958663 .946940
-.004469 - . 066750 - .012410 -.090585
2 3
• 5
LOAD ANGLE
Base MVA .. 100
•••••••••••• LINErLOWS •••••••••••• Bus- code
HW
------------
------------
1 2 1 5 1
2 1 5 1
-.022417 .022437 -3.71 3629 3.819432 -3.156833 3.257828
2 3 3
3 2
-1.616288
• •- 1
• •
-
-.213809 . 214354
5
-.224699 .225573
••
5 2
------------
1 . 641201
•3
, •
MVAA
- 2.004093
-1. 99fi34 6
-6.203759 .933006 -5.952059 . 78 4158 -3.413584 -.59814 5 -2.203544 - 1.499638 -1.122858
-.699430
22.,130110
8.169057
-22.130110
-6.499916
LINE LOSS
'"'--------
• .233150 MW
SLACK BUS
POWER
-22.307)50 MVAR
(I N P . U . )
(-6 . 692879£- 02 ,-1.41 5991 £-01)
MJlerl
1:m1 dlreibs
Jtc>raJ
-, I
!
!
, POWER SysrEM ANALYSIS, OPERATION AND CONTROL
i
6. RESULT OF EXERCISE 6 EDCVOLT.DAT
:,•
No.
I
i I
of Iterations
Bus - code
2
VOLTAGE
toM ANGLE
-----------1 . 04 0000
-----------.000000
1 . 000000 1.000000
2 )
I
k -
FINAL SUS VOLTAGES ARE
-----------1
I
reqd.
•S
1.0 00000 1 . 000000 1 . 000000
6
-.2015 15 -. 0 65728
-.170765 -.n8810
-.238048
•
•
I
i I I
•
•
,• , •, ! •
•
I
Mate-rial
:mI
dire-it)
Jtorai
Economic Operation of Energy Generating Systems 5.1 INTRODUCTION Eronomir: operatiO/'l and pluMing of electric energy generating systems have always ~n given proper attention in the eleetric power system industry. A saving in the eost of generation represents a significant reduction in the operating cost (including the fuel cost) and hence this area has warranted a great deal of attention from operating and planning engineers. The original problem of economic dispatch ofthennal poWeT genera.ting systems used to be so lved by numerous methods. However. with the development ofmatbematical tools and advance C(lmputational methods, the economic SCMduUng of generators has become more a<:cUl1lte and tan be apptled even in complex networks. Thermal scheduling being ofprimc importance, hydrothermal coordination schedJIling has emerged as another aspect of economic scheduling. This chapter aims 10 provide the basic analytical techniques in order to detenn ine the tconomic operation schedule of the conventional energy generating plants along with illustrations and discussions. Since the fuel prices are changing globally, the prices indicated here, for illustration purposes, are indicative only and not the true market price. The following text can thus be considered a basic progress report using conventional methods of si mulations in the area of economic OJlCnltion (since the field of simulatiori and analysis is still undergoing rapid changes, only the basic building blocks are furnished here). Since the basic purpose of economic opcr.ation of power system is to reduce fuel cost for the operation of power system. economic operation is achieved when the generatOr$ in the system share load to minimifc overall generation cost. The main economic factor in the power system operation is the cost of genenlting real power. In any power system, th is cost has got two components, viz., (i) the fcxed COSI being determined by the capital investment. interest charged on the money borrowed. tax paid. labour charge, salary given to staff and any other expenses that continue irTespective of the load on the power system, and (ii) the ~ariablc COSI, a fune1ion of loading on generating units, losses. daily load requirement and pur<:hase or sale of power.
225 Mat lIal Jm dIem
81
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ECONOMIC OPERATION OF ENERGY GENERATING SYSTUfS
.
water flow COStS are due to capac ity of the storage. ~gricu1tura l requirement and cost of running the plant during dry season. Also, artifici~ 1 storage requirement imposes COSt on water input \0 lurbi ne as well as the cost to control the water output from the turbine due to agricultural needs.
5.4
INCREMENTAL FUEL RATE (IFR) CURVES
The input.()ulput curves, obtained from the operating data of power ;IIcremcltlal fu el rate (lFR or IR) curve from the relation
~lation,
can be ulilised 10 get Ihe
IFR = incrementaUlI change in input
incrementatal change in output
Thus, by calculating the sha pe of the input-<>utput '!,lIves at various points o f operation. the profile of IFR can be obtained. Figures 5.3(a) and (b) give the IFR profiles for typical thermal and hydro power slations.
,------i
• • •• • • •,
If' (Kca~MWhr)
• • ,•
,
• ,•
,,
•
Min Output (MWj
fi,. S.J(a) Typical IFR curvcs for thermal pou.er slation.
-
Min Outpul(MW)
Fig. S..3(b) Typtcal rFR curves for hydro P'O"'er station
Malenal,
Jm
direikJ
Jtorai
;
POWER SYSTEM ANALYSIS: OPE.RATION AND CONTROL
•
5.5 ·INCREMENT AL FUEL COST (IFe) CURVE This curve can be obtained from IFe curve by multiplying the 'lFe' by the ctlst of fuel per Kcal. As in a poWer station, fuel cost govems the actual lotal cost. Hence, IFe is very significant in tcOfIOmic loading of tile generation unit. The lFe CUl'Ve$ will be similar to the IFe ch.aracteristic in configuralion. 11 is obvious thai the slopes of the input-output curve and inercmental fuel me curve do not change fot differe nt fuels or for changes in the cost of me same fueL This lime a multiplying factor may be used so that the actual cost is a realistic one. The unit of IFe (or simply the IC) is unit afcost! MWht.
5.6 CONSTRAINTS IN ECONOMIC OPERATION OF POWER SYSTEM 5.6.1
Primary Constraints
These constrainlS arise out orilte necessity for the system to balance the load demand and generation. They arc also called equolity constTQilllS. If P; and Q, arc the $Chcduled electrical generation, P-. and Q_, are Ihe respeclive load demands. it is obvious that the following equations musl be satisfied althe load bus (Fig.. S.4). P,
I P,-.--~
I
p-
Rcal power position at load bus.
FIc. 5.4
0/-0_, -~ = N/ = 0 where M, and N, represent the power neighbouring syslem given by
~siduals
al bU$-i and
(S.2) ~
and Q, Ihe power flow 10 the
" ~YjY(i cos (Oij - 0iJ) L ,.,
(s.J)
" eli " L~''JYvsin(oiJ-Ou)
(S.4)
~
=
,.,
5.6.2
Secondary Constraints
These constraints arise due to physical and operational limitations ofrcspeclive units and components and are known as ineqliaUry cons/rain/so Power incqliality cons/raill/s are applicable for proper operation; for each generator we should have a minimum and maximum permissible output and the unit production should be constrained to ensure that
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ECONOMIC OPERATION
or ENERGY GENERATrNG SYST£MS
P,,-.5: P; .5: p"-' i '" I. 2•...• Np 'Ls~s~,
~d
1=1,2, ...• N Q
1'1, and NQ being the total number of real and reactive sources in the system.
if ct
In addition to the ir inequality constraints. another tonstraint + S (Sr.-l must be ~tisfied, where s~ denotes the tomplex power tapadty of the generating unit without any overloading.
5.6.3
Dynamic Constraints
These constraints arise where fast changes In generation are required for picking up the increasing load demand. Here.
dp,(t)
<
d,
=",,:'("')'1 s dl
dp,(t) dt
Similarly, for reactive power constraint,
-
< 1:d"Q,~(',,)
J,
••
"'
5.6.4 Spare Capacity Constraints In order to account for the EltOiS in load predictioo, any sudden and fast change in load demands and the inadvetlentloss ofsclleduled generation, spare capacity constraints are frequently ulilised. In this constraint, the total genention available at any time should be in excess of the total anticipaled load demand and any system loss by an amouot not less than a specified minimum spare capacity PSI's. N
P,, 2:
"" ~ P, + j- I .
.
PSI'S + P_
For groups of generators. when all planlS are not equally operatiooally suitable fot takiog up additional load, this constraint is then given by
P,,
.
2: '~ " P, + Psp(; + P_
where p.<;f'(J is the spare capacity generation for the specified generator(s).
5.6.5 Thermal Constraints of Transmission Lines These constrainlS arise when power inja:tion (+S"-) or power drawa] (- S,,--) is allowed such that
where
(t,)~
represents the number ofbranchcs and SIr the branch power transfer in MVA.
Mate-rial
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dire-it)
Jtorai
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£CONOMIC OP£RATION OF ENrRGY C£NfRATINC SYSTEM S
Eumple 5.4:
The fuel cost characteristics of fWO generators are oblined as under
f~, (~) '" 1000 + 50~ + O.OI?" 2 unit of rostlhr
F.:, (ll, ) If the
,
'" 2500 + 4Sp" + O.OOS?,,· unit of costlhr
tOlOlload supplied i.l 1000 AOY. find optimal load division /)er..·un the r..·o getrerators.
Solution: are given by
Utilising the: fuel cost characteristics. incremental fuel dF (P. J
, " <,
~
dF (P. ) <,
dF.
.,
(OS\5 for
the two generators
'" 50 + 0.02P.
'
'" 4S+0.01P.
I,
"
Normally. for any specific loading. it may be obst ..... td thatlFe , > fFC1. HenC1:. it implies that generator-2 carriC"5 more load than generator-I . However. for C(;onomic dispatch. IFC , must be equal 10 fFC1 . Implementing this along with the power balance equation, !FC, " A "SO +O.02P~ {FC! '" A '" 45 + 0.01?"
?', +?', = lOOO Solulion of these equal ions resu Its. for optimal loading.
P" '"' 166.33 MW (approx); and .
Ps." 833.67
MW (approxJ
;. - 53.33 unil ofcostfMWhr.
Eumple 5.5: Determine Ihe I'CQnomic operation schedlile for delivering a IOtalload of 1000 ,wI¥.
three thermal unilS when
1'"", '" 600
1-0 curve: Unil-B:
MW.I'.. ,n" 150 MW • H .4 '" 500+7~. "0.00151'; I'n ... ~ 500 MW.l'min '" 125 MW
1-0 curve; Unil-C:
N tI '" 300 + 7.88"8 + o.OO2Pi Pm", '" 300 MW. Pm;. " 75 MW
Uml-.4:
th~
,
1-0 Cll rvc: fie '" 80 .. 7.9')~. + O.OO5Pc . Fuel COS/$: Unit A: 1. 1 un;' of pl"icelMbfll Un it B: 1.0 unit of pricd Mbtu Unit C: 1.0 unit of priceiMbtu. Find the values of PA. 1',," Pcfor opt;mal operolion.
Solution:
F~(r.;)
rs (P
'"
tI ) '"
Il~"l.!
If tI
"
,
.. 550 + 7.7P,. +0.OOI6Sr;
1.0 '" }OO + 7.88 P, + O.D02P; Jlen
1:m1 dlreibs
J!c>raJ
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II I
• •
POWER SYSTEM ANALYSIS, OPERATION AND CONTROL
i
.,
,• ••
I
(5. 12) In generalised form. equation ( S.12) can be Written as
AP+B
I
(5.13)
••
! ,
I
whm
•
Further generalisation for N number
I
:
or units gives
,
(5.14(a»
I, I•
(S.l4(b))
i
i
(S.15)
I ,
I•
(S.16)
I
I , • •
I• I
I
Individua l generation Ph PI' . .. can be calculated from the tOl11Jl'lC)n value of J. obtained in equation (5. 16). Here a lso, if the limiting raling of any unit is violated, it is con~'entional to keep fixed the unit generating at lhal limit and e-conomic ope ration is obtained for the remaining unit re-ca1culating A and B for the other units and sening the net economic dispatch value of generation to be equal to the tobi plant load minus the limiting VlI luc of generating poW1:1 whose genmlling limit is made fi~ed. The resulting value of A. then governs the ecotl(lmic of the Icst of the units. [lImpJe 5.8:
There
(Iff!
1""0 turbo-gt",:ratort feedillS a load bus wjlh the following incrf!mt!ntai
charaCfl'rislics:
dF,(P, ) ..- 3+0.0 15P. (0)
j
Fint!th~
"'•
economic sc.hedul..
.
"d
dF,(~)
if tomlloot! is
'" 2+0.0 18Pl
/[1)
160 MW. Assume no generator limits. \.1 alE
Jtore
EOONOAtlC OPERATION OF fNE RGY GEN£IVoTlNC SYST£MS
(b)
Re~al
(a) willi Ille followillg gener(J1i()l1 IimilS ,"OIlJidered: Unil-I: p .... = 100 MW. p ... = 2Q MW: Unil·2: Pm... c tOO MW, p ... = 10 MW
(a) For economic operation,
Solution;
dfiUU _ dF. (Pz ) dfl
dP1
3 + 0 .0 15P I '" 2 + 0 .018P z •·•
(I)
0.0 15P I - 0.018P1 = - I and PI + Pz = 160 {since lotalload = 160 MW I Solving for (I) and (2), PI'" 56.97 MW;
(2)
(Ans. of (a»
PI = 103.03 MW
Uere ). = 3.855. (b) If genera tor limit is considered. PI • ••
-
= 100 MW
-.
PI " 100 MW IP! is fixed al its upper limit p,
....
P I = (160 - 100) MW '" 60 MW
1
d~(fjl
.. dF.(P.) I:. A, =3.854 and .4.: =2+0.018x I00=3.8OJ dfl dP1 Therefore. the load scheduling is not perfectly econom ic .
Hm:.
Eump)e 5,9:
The increment futl cos/! for a plant Iioving IhemlOl /lni(J are as under:
dF.(P.) I I ('" d~
A. ) <: 0 .0050fl + 6 .0
dF2<1l) (= ,t,)
'",
unit of costIMWhr
= O.00750f!, +5.0 uni t of costIM Whr
-
If !he lUwl load varies from 100 to 500 MW in Ihree sreps of 200 MW varia/ious. fiud IIle incremtntalfuel ron and rhe incrmumfutl cost of eC'OlUJmic operating schedules of rile plam. "'SSUnl t [or "nil-J: p ..... = 330 MW. P.." also
:=
75 MW,
for IInit·2: Pm... = 330 MW • Prrit! = 75 MW.
Solution: At 100 MW total 1000d. lei us fi~1 5CC lhe IC of each unil
(A,
3nd
lz ) in the plant
).1 = 0.005 x 100 + 6 = 6.5 unit of cos tlMWhr ).2
= 0.0075 x 100 + 5 = 5.75 unil of costlMWhr
The IC of unil·2 is comparatively lesser and hence il is economic 10 load unit-2 at li ght load period and also it is economical 10 supply maximum load from unit-2 till its inc remental cost approache!llhal of unit-I. We can find that value from lhe following equltion: 0.0075P2 + 5 '" 6.5
:. p. ..
2
1.5 .. 200 MW 0.0075
Mate-nail Jm dlre-itJ
Jtoral.
POWER SYSTEM ANALYSIS, OPERATION
Thus, Up to a loading of 200 MW for operation criterion is govemed by unit-2. It ~onomic loading),
and 75 MW of loading of unit-I the economic also be seen that for 200 MW loading (to have
.>A[t A ).,.6 i. 1
Let us now take the system load as 200 MW.
! 0.0075
a,
·, 1
.. 0.003
= 5.6 Then A ... AP + B '" 0.003 - 200 + 5.6 .. 6.2 ,"it ~
>
"d
='1
a,
= 160 MW
a,
For a loading of 300 MW, ..I - AP + B
'l '"
,r
~
6.5 (wh
, - /!, ,. tOO MWand a,
p .. 300 MW);
_ J.. -P; ..
a,
Exactly in a similar way for a system load ... 7.1 [ .: p .. 500 MW]
~ .. A-PI .. a,
...
220 MWand
200 MW.
MW, A-O.OO3,S-5.6and..t-AP+B
•
, - (J,•
a,
... 280 MW
The reader is encooraged to obsnve Table 5.t .
TABLE 5.1: Opel"olion
1 2 3
, 4
6
7
»0.
"
TOio/load ( MW )
100 100 200 200 200 300 '00
0
100 100 200 160 200 280
6.'
0
0
5.75
6.'
5.75
0
6.5 6.2
6.2 6.' 7.1
Material
6' 7.1
I
Jm dire-,to
,to ai
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• (CONOMIC OPeRATION OF ENERGY CENERATING SYSITMS
For N number of plants the coordination equations are given as
(5 .20)
while the constraint equalion is given by
, L (1:) - P,.",d - P, '" 0 .,
(5.21)
If the numerical value of [he partial derivative of the line 1055 wilh respect to each generator output is Known, the generator output
(op,/op,)
power may be adjusted to satisfy the following
equalion
dfi(P,)
I
dfl
\ _ 01}
=:
A, i == L2 •...,N
all The optimum economy is thus achieved when the product of the incremental fuel cost times the penalty factor is the same for all planes, i.e.
(5.22)
A stands here for the incremental cost or tile received power in unit of currencylMWhr and hence during economic operation of plants with losscs being considered,
,= ,
--cc:,,:,:,,:m::'"'",=':"'e':'c': "c''occ:
1 - Incremental transmission loss
Example 5.10: A two-bus system, wilhoUl generator limits. hQ5 been considered (shown in Fig. £5.1) where p_ • .. 400 MW. p_• .. 100 MW and
(IFC),4 .. O.OO6P,.
Fi "" O.OOOS(P...
-IOOf
+ 4.0 unit or "ostIMWhr
{IFC)II '" 0.OO7P,. +4.0 unit orcostIMWhr Find optimal generation/or each plant and the power loss in Ihe hne.
~ale-nall :>rTI
dire-lID
Jlo ais
POWER SYSTEM ANALYSIS; OPEfVoTlON AND CONTROL
~
~
I'
I', ,
-
Lil\l'
"
,
I ~.
/'~.
fl&. ES.1 A two-bus system. Solution:
As f} .. O.0008{P.I- 100t MW.
...
'" O.1)()16{P,. -100) '" O.OOI6P,. -0.16 I
of}
(Penalty factor or PF) -
1--
aP. For generator- I, assuming it to be a slack bus.
(PFI =
~d
(PF)z =
I
I
= 1-0 = I dP' lop. I
I
1- af}
ap,.
I
= 1-0.OOI6P•• +0.16 = J.J6 -0.00 16P••
For optimal dispatch,
=
(PF) dF: l
,.
dP
(I )
I.C .
...
I ( ) 1.16-0.0016P,. O.OO7P, . +4 '" A
p•., + p•• = O.OOO8(P,.
(2)
-Ioof +500
(l)
Solving for (I), (2) and 0), we gel
.,
P
"' 227.7MW, P,, - 1l7.6S, .... ... S.694and f} " 4.SS MW \.1al[
:mI
dire-it)
Jtorai
ECONOMIC OPERATION OF ENERGY GENERATING SYSTEMS
Eumple 5.11: Find the incremental transmission lones for a twa-area power system, where the bus voltages ore teptflXed and the line power flaw if afonc/ion offine ongle. Power Ion if afunction of generation of area B only. Solution:
It is evident thJI P'=f(l~. )
This also suggests that the incremental transmission loss for grid A will be zero and the inco".mlllentallTansmission loss of the line will be governed by the grid B only. Thus,
(In),j '" (ITL) •
0
= of} ap
••
Economic operation being dictated by the criterion
A"" (IFCL '" (IFC), 1- (fTL)A I - (ITL), in this case, for e<:onomic operation, =
(lFC), I - (m).
Enmplc 5.12: T1re losses in the IirHJJ slwwn in Fig. &5.1 ore proportional to the sqllare of the power flow. Barh units are loaded at 250 MW. Dlle to fronsmlljion f~ 12.5 MW ofpower is [oslo Where should the extra 12.5 MW be generated far eCOflomic op«a/ion? Attempt a reschedllling /0 minimiu the tronsminiOll [OSS . Line
P, SOOMW (to m;ei, e j
A Iwo-bm power gencnl1ing system f«ding I load.
Fie:. E5.1
Given:
~ Line loss .. 0 .000211 W .
fl
(Min .. 70 MW ( Min .. 70 MW Max .. 400 MW' Pz Max .. 400 MW
(md Solution: Due to transmission loss, the load received is 487.5 MW instead of 500 MW. Utilisation of Lagrangian multiplier results L '" Also.
p, ,.
fi(fl) +F2 (l))+,l. (500 + P, - fl-l))
(I)
12.5 MW (which is equivalent to 0.OOO2if") M- erial I :xn Olrf'ltos
Jlorals
I
POW!R SYSTrM ANA LYSIS, OP[RIITION AND CONTROL
For economic operation. (2)
(3)
;
fl
. Iw
+Pl ,. (500+12.5) MW
(4)
Howc"cr, from !he given dala,
dF;
,
un ,. 7 + O.OO4P. I' ,
dP.
ap,
ap,
'" O.OOO411, "" 0 (If] a~ Thus from (2) and (3) we can write,
i
! I !,
7+0.004fl-).(I - O.000411) = 0
(S)
- ..t .. 0
(6)
7+0.004~
Solving equatio1\$ (4), (5) aQd (6).
PI'" 179 MW, P1 " 327.5 MW and Power production cost
Fr ,
'"
,
(F~ )
P, ,. O.ooo2if ... 6.4 MW
is given by
F; (1;)+Fl (~)
= 400 + 7)( 179 + 0.002)( 179 1 + 400 + 7)( 327.5 + 0.002)( 327.51 .. 4623. 15 unit of oostlhr. Next, we suppose thai the minimum loss is desired. LeI the generation of unit 00-2 be increased to 400 MW (muimum possible) as, apparently. this is the Iros1 feasible solution to reduce transmission loss by adjusting generation;..unil-2 being connected 10 the load bus. Thus, unit 00-1 is to give 100
MW
fl + 1';
"'
= 500 + O.OOO2fr
O.OOO2fll -
fl
fl
+ 100 .. 0
[,: P1 ... 400 MWj
= 102 MW
i.e. the loss is only 2 MW utilising unit·2 \0 gencnte maxiJmm possible generation of 400 MW and allowing the remote unit-I to genClate 100 MW. Power production cost (r., ) is then given by F<, .. 400 + 1 " 102'" 0.002 ~ 1021 + 400 + 1 " 400 + 0.002 ~ 400: '" 4655 unit or costlhr
I I
Comparing the cost of F, and F, • it lIYy be minimising the transmission [0$5 'only. '
obscrv~d
that economy is not achieved by
\.1al£ ..
.1 ...
_
Jlora.
eCONOMIC OpeRATION OF ENERCY CtNERATlNG SYSTeMS
Eumple 5.13:
The fuel cosl cun'e ofrwo generators are given as under
F;.(P,)
,
=:
800 + 45P" + O.OIP;..
If the tOlalload supplied be 700 MW,jind the aplimal dispUlch with and wilhaul romidering the generator limits where the limits h([Ve been upre.sst d as
s 200 MW and 50 MW S
50 MW :S p, ..
p, . S 600 MW
Compare Ihe syslem incremtnlo{ cost with and withoul genuator limits being camidered.
Solution:
The incremental ,osts are
(/Fe) 4 '"
4 5 + O.02P,. and (IFC)II '" 43 + O.006P,.
To have eronomie openllion, equation,
I'll.'
{lFC),4 .. (lFC), .. A;
considering this along with the ,onstraint
can write .l. = 45 + O.02P,.
(I)
as well as
.l. .. 43 + O.OO6Pl •
(2)
while
P, .. +PI . =700 MW
(3)
Solving these three equations.
A - 46.7. P ,,84.6 MWand P '" 615.4 MW "
J.
In the above illustration, generator limil$ have not been included. If these lim it are included, it may be s.een that generator B has violated the limit. Fixing it at the uppermost Umil, let p,• .. 600 MW, Obviously, p,. '" 100 MW and then
P,. + pz• "
700 MW.
This time, '/',4 '" 45 + 0.02" 100 '" 47 and All '" 43 + 0.006" 600 = 46.6 Hence, it is obs.erved that '/',4 .,. AS' Le. economi, operation is not stti' tly maintained in this pani,ular condition. locremental ,ost of unit A is now marginally higher than thaI of unit B. However, since the difference of AA and '!s is not much hen,e the system operation is still justified.
5.12 A SIMPLE COMPUTER APPROACH TO SOLVE TRANSMISSION LOSS PROBLEM We have just developed the 'fiterion of economic operation of thennal plants with network losses considered. In the following section a straightforward 'algorithm has been presented in order to obtain the generation schedule with transmission losscs considerc-d. Step I:
~t
starting values of generation Pt. Pl .... that satisfies the constraint equation
LP, - P"-, , , 0 . Sup 2:
cR cR
Calculate the incremental losses ~,~, .. . as \'it'll as the total loss
, ,
Mate-n II Jm dlre-itJ
Jtoral.
roWER SYSTtM ANALYSIS: OPERATION AND
I. Tolal "':;:.~ 2. Tolal Ie
3. P,o.a~/I,.r,.ror;
,
,
. contribl1tion 10 the 1 .. 1.2.3.....
1055 for <:very
Compule 1.
,
iJP/rm
_..• Nand 2. P/ws
....
'0 No
Yn
No
1
, I.
3.
,
,
om," geno:ra,ion , .fori .. 1.2.3 ..... N Il~. line loss etc.
). from the: equalions gi"en by I.
:*
J:1: ~t.l (I~ )and 2. L." 1 ..
Fie. 5.' FlolO.'chan 10 r"v,j economic generation
.h.... i + 1
''''''''""o"d~~ ImIsmission line ~"
Material, Jm dlreo,to
•
]to
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ECONOMIC OPERATION OF ENERGY GENERATING SYSTEMS
Output or computer program WLOSS.FOR art er uecution: SOU.DAT k
lambda
,
••••••
••• 2 3
,
.000000 8.5023 76 8.497055 8.457865
••••
"
" .....
" ......
......
350.000000 395.835500 390.629200 391.270900
250.000000 257.884100 255.684100 256.188800
150.000000 10".730500 111.907600 110.767100
8 . 450000 8.220816 8.22661 4 8.226754
Iteration completed l ambda
.,
k
•
Ploss
,
• 8.497865
Net gen (MW) • 758 . 226800 (MW ) • 8 .22 6754. Powe r tolerance • .000052
Net
load
( MW )
•
750 . 000000
Bus no
......
Economic generation (HW )
................................ .
1 2 3
391.270900 256.188800 110.767100
generator no.
.....................
• ••
2
8 . 365084 8 .2 37132 8.307671
,
3 Check,
lambda .. penalty factor*cifc
................... ,
generato r
2 3
5.13
no .
( for each gen)
LAMBDA.
. .........
8.498086 9.498395 8.495883
THE TRANSMISSION LOSS FORMULA
The Iran..miss/on /oss/ormula was first introduced in early 1950s in order to obtain a practkal method for the cal culation of transmiss ion ross (includ ing incremenlaltransmission loss) in economic p lant scheduling. However. this makes the calculation somewhat more d ifficult though it is well accepted in th e power uti lity industry. We first furnish the stcps of derivation of the transmi ssion loss formula. Let two generators G t and G1 be connected to buses I and 2 injecting currents I, and 11 into the network of lines and load buses 3 and 4 (Fig . .5.10). Let n be the neutral bus wh~~ the net load com:nt is I~ . Let ' } and ,~ be the load current inj ections at buses 3 and 4 such th at the net load current M' eri
,,105
J\Om.5
POWER SYST£\.l ANALYSIS, OPEIUTION AND CONTROL
1f) .. /)+/~
Let us
00'" assume that each load
(5.23)
is a constant fraction of total load such that (5.24)
I) '" KJ/f> and I, = K,lv
abo.
(5.25)
rv
rv
G,
, I,
G~
T.
,
, ~,
- -l
I,
I,
I,
•
~. , ,~
Z ...
1
I, Fie- 5.10 Schematic rtprescntalion of bus structure for
Let us now choose the neutral bus as the
(VJ = (ZJ[/] . hence
der;~alion
of loss formulae.
reference bus. Since in a generalised network
for the gi~en network,
V" v~.
v"
-
V"
Zll
Z" ZJ 1Z (I
Z"
Z" Zl~ Z" Z" Z" Z,! Z"
Z" Z" Z" Z"
I, I, I, I,
Expanding for the fi rst now, V I < '"
(5.26)
21/ I +ZI111 +21l / J +ZI. I.
Substituting for I) and I. from equation (5.24) we can write, (5.27)
where
V
1° '" _..'.h. •
Z II
We neXI write equation (5.27) as (S.27(a»
Mate-rial
:mI
dire-it)
Jtorai
SYSTEMS
E.CONOMIC
m,
whm
- Z"
=
K ) = I)
+ K~:14
(5.2 8)
- Zll
m, •
K ):I)
+ K411 4
Substitution of (5 .27(3) in (5.2 4) yields
IJ
,,,.
K)"'I/I - KJ"'l/z -
'" -
"J"'I/!
(S.29(a») (S.29(b)
Let us now relate current II. 12 and I! with th e bus curnnt through the connecfion matrix utilising equat ions (S.29{a)) and (S .29{b» when I] and h Il:main invariant.
I, I, I, I,
I
=
I - K J lIl]
-"]m)
- "Jml
- K 4ml
- K~"'l
- K 4 "'1
I, I, I'•
=c
I, I, I'
(5.30)
•
However, we know ITom the concepts of power invariant tronsjorm(l{ion that load bus power St.• in terms oftransfonned (new) current, can be expressed as
St
,
'"
1""",Z BOll ~i
.""'"
(5.31)
Z~ ..... ) = CrZ"""C ·
But,
(5.32)
where, Z8 .. = (R_ + jX"",). Thus, equation (5.31) can be wrinen for real load power as
PL _ 1;""CrR8",C·/~
(S.33)
which. the following equation (5.30) becomes
I,
PL '" [II '2
1~l! CrR"., Cl I)
• (5. 34)
I'
•
Let us now assume that at each generator bus. the react ive power is a fmction '.,' ofrea l power, I.e.
F;+jQ I = (I+ j-JI)F;
,,'
P1+jQl = (I + j-Jl ) PI
,,' (5.35(a»
POWER SYSTEM ANALYSIS, OPEIVITION AND COI'ITROL
(S.35(b))
we can now
\\TiIC
[from equations
and (S.3S(b))J
(S.35(~»
a, (S.lS(c»
Thus. equation (S.lS(c» can be written as
"•
This finally gives,
a,
-
erR
a,
•
C.
a,
~
• (.D 5(d»
P, "•
1
P,
(5.36)
1
B. l •
In the symmetrical network.
B~l'
thus, (5.37(a»
:
=
:
z
L L 1}BijP + ~)jol} + Boo j
I~ I
(S.37(b»
'.1
jo'
In general fonn, equation (S .37(b» may be ..... riucn as
I} = pTsp+ pTSo + Boo
(5.38)
Equation (5.38) is the genera li sed tron$miuion loss formula. In a system of N sources, ,
I} "'-
H
,
L ~)~Billj + LSin?' +800 /_1
jol
(5 .39)
1.1
where B \cnns arc called I()ll cCMjficienlJ or ~coejJici(!nrs (N" N matrix) and is always symmetrical] Figure 5.11 sho ..... s the flowchart 10 find out loss coefficients (i.e. [B] matrix) and transmission line [055.
I
Mate-rial
:mI
dire-it)
Jtorai
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POWER 5Y.sTEM ANALYSIS: OPEIVoTION AND CONTROL
1
1
1
1
1
1
8.465701>< 10-1 [ - j1.294148><10-IO 1 2.832449>< 10[ - j .093726 >< 10-6
o
Now.
f. .. •
_VI ~I I
'"
2 2 (2.832 459 >< 102.9 10364 >< 10[ +j2.811132x10 ..... . +j7.088398><10-6 2 1.991409><10( + j l.644594 >< 10.....
2 1.34881 1><10[ - j4.S9S40S >< 10-9
-1+jO 0.04149603 - j6.S50091
'" (- 9.671 51 51< 10.... - jO. 1526635) p. u.
a l '"
I _ jJI 1 - j (0.06467313/0.140189) .. .. 1- jO.4S93621 " I
,
.. 0.9916434 - jI.260])3
..•
raj -
(1- jO.4S93621)
o
o
o
(0.99 16434 - jl.260133)
o
o
o
(- 0.9611 5 I 5 >< 10-' - jO.1 526635)
-
(0.1025208 ) - jl.941 570 >< 10-'1
2 ( 4.548110>< 10) _ j2.385856 >< 10- 1
) (1.951863>< 10-) - j4.331206 >< lO-l
2 ) ( 4.548 111><102 - j2.385857 >< 10-
C·061192><10-1 ) -jl.875613>< 10-t
(3.836992>< 10-) ) j3.007SJ2 >< 10-1
(1.957855>< 10-) ) -j4.319 1><10-)
(3.836998>< 10-) ) - j3.001539 >< 10-)
(3.1 43685><10..... ) 10 - jl .071456 >< 10-
B-coefficicllIS matrix i$ givcn by
0.1025208 '" 4.548111><10- 2 1.957855" 10-)
4.548110>< 10-1 0.1067192
1.951863>< 10-) 3.836992" 10-1
3.836998>< 10-)
3.143685>< 10....
•
Mate-rial
:mI
dire-it)
Jtorai
[There is a slight discrepancy ill 6th decimal point. This is due to lIumerical method adopted in matrix inyersion. ] : . Line loss
.. [0.14{)789 0.5
I)
0.1025208
4 .548110" 10- 1
1.957863" 10-1
0.140789
4.548111" 10-1
0.1067192
3.836992 x 10- 3
0.5
1.957855xl0- 3
3.836998 x 10-1
3.143685><10'"
I
0.0391322 .. [0. 140789 0.'
I)
0.0635998
.. 0.03981780 p.u.
2.SOSS 12 x 10-1 The computer method (flowclwt sho~n in Fig. 5.11), when applied to the Example 5.1 S. yields the following results:
Execution of computer algorithm of N-R Method of load flow given in 5.11 for Example 5.15 Input data to tOmputc IY..... I matrix (LINE DATA): lIMTl.DAT 5 , 4 , 0 [No. of lillCS, No. of buses. No. of trAnsfllrmcl"lll 1,2, 1,4, 2,3, 2,4, 3,4,
( 0 .1 0, 0.30) , (0.0, 0.030) [from bus, To bus. (R. i i (0 .15,0.40 ), (0 . 0,0 . 030) (0.10,0.28), (0.0,0.030) (0.075,0.18), (0 . 0,0 . 03 ) (0'.15,0.35), (0.0,0.030)
Xu, (G
,, ,,
i
,, , , ,
BI1)1 ,
,, ,
, ,
If..... ] matrix: YBUS1.DAT (here busn 2 and 4 arc illterchanged for power flow calculatioll)
•
(1.821918,-5.161781 ) (0.0000008+00,0 , 0000008+00) (-8.219178E-O l,2.191 781) ( - 1. 034483,2 .413 793) (O . OOOOOOE+OO,O . OOOOOOE+OO) (2.165705,-5.551214) (- 9.9999998 - 01,3 . 000000) { - 1.131222,3.1674211
Y\I
( -8.2191788-01,2.191781) (-9,9999998-01,3 , 000000) (3 . 828787, -9.294302 I (-1 . 972386,4.7337271 1-1 . 034483,2 . 413793 ) (-1.131222,3.167421) (-1 . 972386,4.733727) ( 4 .103608, - 10.856150)
r ,!
y.
Load flow data: LDFL01.DAT 3,4 INo. of PQ buses, Total 110. or buses1 1.0,0 . 0,0 , 0,0.0 [VIO, ~Q, P" QII Q l , O,0.O, - 0.25,-0.~ [V/. 6z , P~ 'OlI 1.0,0.0, - 0 . 35, - 0 .45 [V1a, 6,g, P,. QJI 1.0 , 0.0, 0 . 5 ,0.0 [V~ , o~, P~ Q~J •
••
Materia!, :xn Olrf"tos
Jlorals
I•
POWER SYSTtM ANALYSIS, OPERATION AND CONTROL
Flnll bus voltlees .fter Io.d flow (ll<'ul,lIolI: VOLTI.DAT
VOLTAGE
Bus~code
LOAD ANGLE
------.-.-.
~- .-- .-----
----- ------
I
1. 000000
.000000
2
1.000000
-.0066 49 -.05 1878 -.03 0806
, ,
.868215 .929816
REACTIVE POWER AT VOLTAGE-CONTROLLED BUSES bus
Reactive power
no.
---.------.----.626756
2
,
Line nows: POWERl.DAT .. ".""".,,",, ....
Bus-code
-----------
MW
-------- ---
MVAR
-----------
I ·
2
1. 996863
-3.658250
2-
,
- 1.992442
-2 . 328488
12.082040
10.125560
I
- 11.6046 50
- 14.446210
2-
J
34.582370
J.
2
27 . 44186 0 - 25.276370
I
I ·
,.
, ,. ,. , 2-
2
,
LINEFLOWS ................. .
,. Base
-33 .780400
24.528440 -23. 44986 0 -9 .619760
-28.926790
10.049380
9.515548
J
25.921710 -13.368160
• 100
MVA
. -.... "' ........
LINE LOSS
4 .155501 MW SLACK. 8US
POWBR
, -16.3G3090 MVAR
(IN P . U . )
(1 .4 07890£-01,6. 4 67)13E-02)
Execution of computer elgorlthm given In Fig. 5.11 for Example 5.15 Input of computer program to compute 18] coefficients (BCOEFF.FOR) :
\.1al[
11 dire-it)
Jtorai
[CONOMIC OPERATION OF ENERGY GENERATING SYSTEMS
YBUS1.DAT. PROB4.DAT
(Y..,lmatrix: YBUS1.DAT (B.sa 1 and 4 are DOt interchanged) (1.821918, -5.161781)
(-9. 999999E-01, 3.000000) Yu
r"
(0.000000£+00,0.0000008+00)
(-8.2191788-01,2.191781)
r"
(-9.999999E-01,3.000000) (-1.131222,3.167421)
(4 .103608, - 10.856150) (- 1.972386,4.733727)
(0.0000008+00,0.0000008+00) (2.165705,-5.551214)
r"
(-1 .131222,3.167421 )
(-8.2191788-01,2.191781)
(-1.034483,2.413793) ( - 1.972386,4.733727)
(-1.034483,2.413793)
(3.828787, -9.29430 2 )
BUJ data: PROB4.DAT 2,4 (total number of generator buses, IOlal number of buses] 1,0, 1. 407890E- 01,6. 467313E-02( 1"I
I,
1,-.006649,0.5, .626756
[Do]
.868215,-.051878,-0.35,-0.45
[Do ]
.929816,-.030806,-0.25,-0.3
[Do]
~,
PI, ,Q.,
J
Output of COmpuler proaram BCOEFF.FOR aher necullon: SOLA.DAT Tot.al
no.
of buses
4
[Zbusj
(4 .1496038-02,-6 .550091) (-1.93 46118 -02,-6.719252) ( - 5.952761£-03,-6.685503) ( - 5.9352768-03,-6.683190 ) (-1.93 45868 -02,-6 .7 19252) (4.159922E-02, - 6.559778) , (-8.698829E-03, -6.690867) (-8.783918E-03,-6.686722)
(-5 . 9529068-03,-6.685503) (-8 . 698707£-03,-6.690867) (1.4903028-02,-6 . 627306) (-6.860666£-03,-6 . 682828) (-5.935172£-03,-6.683189) (-8 .783548£-03, -6 .6 86722) (-6.860934£-03, -6.682828) (1 .86 32438-02, -6.621376)
[Rbus1
4 . 149603E-02 - 5.9527618-03 - 1.934586£-02 -8.6988291;:-03
- 5.9529068 -03 1.4903028-02 -5.9351728-03 -6.8609341i>03
-1.93 4 611£-02 -5 .935276£-03 " . 159922£-02 -8.7839188-03
-8.698707£- 03 -6.860666£-03 - 8.7835488-03 1.863243£-02
MatOor
, .n,
u
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POWER SYSTL\f ANALYSIS,
AND CONTROL
0.0096 , 6 [A].i1l [ 4 .567152£ -03, - 0.0392 1 6£-03,0.2309641>-03 - 0 . 03 9216£- 03. 4. . .; 59630&- 03,0. 1543251>- 03 [S]
matri~
0.2309641>-03,0.1543251>-03,0 . 09 46231>-03
Or
Output computer proiram BMTHD2.FOR a fter execution (Selecting initial value of A by convcntionalmtlhod); SOLSI.DAT
ITL
generator no.
,
2.647640£-02 3.326676E-02
I
PENALTY
FACTOR
generator no.
,
1.027196 l .03 4414
I
He
generator no.
, I
1 1.727960 11.646140
lambda _penalty factor* IFC
, I
k
•••
(for each gen )
12 . 0469:10 12.046920
lamMa
---_ ....
PI
(p.u.)
.. .................
F2(p.u.)
Ploss(p.u.)
. .............. . ....................
,
11.659490
2 .3 45633
3.374579
. 077515
11.6694 90
2.359492
3.383537
.076067
I
12.035500
2.864267
3.710361
. 1 00597
•
12.0 46 920
2.879941
3.720546
.101348
I
Iteration compl e ted at k .. 4 lambda .. 12.046920 Mate-rial
:mI
dire-it)
Jtorai
ECONOMIC OPERATION OF ENERGY GENERATING SYSTEMS
_
Net gen (MW) 10.134830
_ 660.048900
Power tolerance
..
a 650.000000
Loss (MW)
.000860
no
Bus
Net load (MW)
Economic generation (MW)
_ __ ._ _
_ ___ a ... _. ______ • __ _ .......
1 :2
287.994100 372.054800
Output or c:omputfr" pror;ram BMTHD2.FOR .ntr tltcution (Seltcting inlti.1 v.lut or A by ntw propoxd JQtthod): SOL52.DAT
ITL
, ,
ITL
generator no.
2.648463E -02 3.327396E-02
PENALTY FACTOR
p,
,
generator no.
,
1.027205 1.034.419
,
!FC 11.728510 11.646710
IFC
generator no .
,
lambda _ penalty factor·cifc
, ,
(for each gen)
12.047580 12.047580
k
lambda
...... _....
,
"_. _. -. .....
14.135000 14.145000 12.085390 12 .047580
5.670753 5.683769 2.932"103 2.880847
---, , •
(p.u . )
Iteration completed at k ...
............. - ........... -.....P2(p.u.)
P1ose(p.u.)
5.548358 5.5!>6965 3."154781 3.721137
.286119 .287220 .103902 .101392
4
lambda .. 12.047580
Material,
JfT1
dlreitJ
Ao~.
POWER SYSTEM ANAl.YSIS: OPEMTION AND CONTROL
..
Net gen (Mit ) _ 660 . 198400 10.139190
Net load (Mit) .. 650 . 000000
Loss (MIt)
Power tolerance .. - .000592
-_ ..... ,,
Sus
Economic generation (MIt)
no
........................ _.288.084700 372 .113600
5.15 A METHOD OF DETERMINING ECONOMIC OPERATION CRITERION USING TRANSMISSION LOSS FORMULA We have derived earlier (equation (5.38» that total IJ'arumission loss in the system is given by
I} = pTSP+B;P+iloo while the incrementalloSJ at the generator bus i = (81}/8Pr). The incremental loss is thc variation in IOSKS during an incremental variation in the generation output. Let us select a generator bus thaI can be used as a reference bus and which always adjusts any change in the generation of any other bus in the system. Nexl, we asswne thaI the generalion bus-i is changed by an incremental amount t:.P, such IMI
' \-
I
=IL _)
+~
Asswning the net load leyello be constant, to compensate for the increase in t:.P~ the reference bus is to drop an equiyalent amount ofgencration t:.P..q andthcn P-, = P..J +M'-, [obYiously, . . . "'(-1 "'I-I ~, for mcrcase of ~, t:.P..q IS to be negatlYe]. Howeyer, there would be a now change in the lines of the system following the two-generation adjustments. This, in tum, would cause change in losses and this makes
t:.P", .. ~ Le.
l>P.
=I- ~
M1
This may be written lIS D = I _
where D represents (-t:.P-.t
aPr ap,
/61'i)
the ratio of the negative change in the reference bus power to the
change in M I' However, total cost function being given by r.F,(P,), the change in the cost function for a change in generation tlP, is
,• ,•
Mate-rial
:mI
dlre-itl
Jtorai
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ECONO.\!1C OPERATION OF ENERC Y GENERATINC SYSTEMS
D
y" Isp..... , ...
P,,- P.:. - :>t, "! y"
S1<,", 00. of .{-i1<,nltion. =rd. fOf r-tb inter.·al , .1:_ " - ~
L
0;",1"
pl"
Compute. A,J. -
,.vw'~7
pt, ,J.'"
I.{; -
Y
,J.:~
rs ~ ;. I?
-r
112
No
y"
Is 10tal gen=tion grealer lhan (load "'105.$)?
y"
)'., ., - tJ. _.. _, .. -'1,
No
I' t - t_ ?
.to., , .. .t',-
!
A).
1: ; ..... .1: .. r
r ..... r .o. I
B
<
--l
C.ompulc W,(Ph', )
No
No
A
•
GUMS)" .. 0,./'..... ,- (J,.
No 1511 - 20')
,,-,
COmpUle IOIaI walerreqd.
11'..... "
No
E
rW,
y"
"N
.'7X1I
Io W_ - W • ..,
7
I
s .·) ?
0;.,1.,
y"
r ."~w "o" ,~
final hydro-thermal S<:ItroUl in ! hour b ..... i
5.""
,
Material,
JfT1
dlreitJ
Ao
~.
I,
POWER SYSTEM ANALYSIS; OPERATION AND CO,VTROL
I,
I ! ,,
E
.1
No
Is] > ! ?
I,
Yes
,
Ar- P/- .t'11 2 No
y~
"W_ > H'_?
I ,, ,
";" - rl -
.,r' _ y'+ l!.r
A,,
,
I
~
I
~
I
,
I
,
init ial value of t..\
J~j'"
I
No
y~
[s] - 20'/
tll·5.16
RG,"char1 (0 OOIain shon ·ltnn hydro-lllcrmaJ schedule " 'ilh transmission 10051 cOIlsi.rred
,
~>. ,.,
Also.
'"
(5.76)
T...,
Application of Lagrang ian teo:.:hnique IlOW yie lds
I I
I I
!
I
L •
r [oo,FI P" l' A,I p~, • P" - P, - p~ l] + ,.[roo,W,lf, )-w.. 1 ", I
(5.77)
•• 1
where )'T is the inc remental fuel COSI of thermal plant and r~ is the constan t already ddined in the ana lytical derivation of short term scheduling (Sec. 5.22). For any specific interva l T'" f. the coordina tion eq uations arc then given by
I!
'
dF( P,.. ) + 1 oli.
y~ ",
,
dP.
dlV,
'l
,
oP.
(p, )
,
dP.
""J
"
(5.78)
op" + A,
,
all '" A,
The compuler nowchart for sol"ing such a schedulin g problc:m is shown in Fig. 5.16. MatE
Jtorai
ECONOMIC OPERllTlON OF ENERCY C£NERllTlN(; SYSTEMS
Example 5.18: (a) A load of800 MW is /0 be suppliedfrom a generating system comprising a hydro plant and on equn'alem Sleam plan/. The plunt characleriJlics are
Hydro plum: IV ( '" fl_ + a_P.) '" (10" 10' + 6" IIY P_) mllh r Volumt afwaler available - 32 " 10· ml. Thermal plant (tqui,·alent): F~ (=
a,.P.i + fl,"P,~ + rll,) ~ (0 .OO2p,~ +8P,. + 12(0) unit of costI'MW
Make a hydro-thermal 5chedule per hour ba.liJfor 12.hour ptriod. !1f!Jsum e inilial valut of rIO be 0.16 ~ 10-1 unit of costlrn! and trol1$mission loss I} as O.0000045~ , (b) Makt a hydro-thermal 5chtdule per hour iJmis for Ihe 50me plant for one day. M'here Ihe load has the folfowing schedule: 12 mldnighl - 12 nOOf/: 740 MW 12 'loan - 12 midnight: 900 MW Tile hy dro plam rtsen'oir limil is 60 _ 10~ ml orer Ihe entire 24 IIr p eriod. (c) Repeat fh t same problem (Example 5. 18(0» for 0 load sched"le as f olfnws: 6 a.m. - 12 nOOf/: 740 MW 12 nOOf/'- 6 pm: 900 MW 6 pm - 12 midnight: 830 MW 12 midnighl - 6 am : 640 MW Reservoir copocify limil i5 56 " 10" ml. SoluUon :
The generalised
COSI
funcl ion characlerislic for th ermal plant being given by
F(P,.)
= a,.p,~ +fl,~P,.
+r,.
we can wrile
dF(' )
= --7.;~'~ dl',.
2aIII I',- + {J,,.
(I)
For short term hyd ro-thermal schedu le, the coordination equat ion for Ihe Ihennal p lanl with transmiss ion loss nOt considered is given by equation 5.7 1
A
= ",
(2)
Using (I) in (2) ). = " , [2a,.p,~ + P,~
I
Pili '" -"~-'-'~""~'"'"
..
2alll P'.
However. the transmission loss
(Il )
be ing eonside~d ,
F(p,~) = a,~P,i + {J,.P,~ + rl/o and l'ihe ~ C,~
(3)
Il " C",P,;
represents the loss coefficient. Material,
JfT1
dlreitJ
Ao~.
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COMPIJT[R·AIOEn
WAO DISPATCH AND OPTIMAL POWER FWW
~~~~~~~~~~~
,,'
(633c)
Equations (6.3h-6.31c) can be wrinen in a matrix
.'L ap" ap,.
.'L
rorm IS
.'L
ap,,aQ,, aPI, dA.
•'L
"L aa.,oP" oar,oa" .'L "L oAaP,, ',IDQ,,
"L oa,OA. ,
"L aA'
'L Dr,
•
tJ', •
'Q,,
=
'L
(6.343)
dQ, . •
""
'L -aA
[for i = I, 2, 3, .. ., NG and for It:: l. 2. 3.. . .. NGJ f'.
H1',0,
H",
tJ',
HQ"I',
H0,0,
H(J,~
'Q,
H",
H",
Ha
""
1/ f'.
"
~.
•
I ,, •
(6 .34b)
I" I,
In this method, Ihe active power loss in U'lInsmission lines is calculated using equation (6 ,3Ia). From equation (6.3Ia ). wc can wrile
, :: 2a;.P; + L [(all +0,, ) ~ +(b11- b",) Qd ,.,
...,
(6.358)
;
olj
oQ,•
= 2n;iQ; +
L ,,.., [ (aji + all) Q" + (blj -bji) I't]
(6.3Sb)
WhCTe 1', Qnd Q. arc aClive and reaclive power inj ections (or calculated ;!Clive and reactive power) al bus·1t respectively and i is varying from 1 to NG. Th~ fi rst ord~r d~ri ...rti.·~s "'q .. i~ for cqllations (6.JJa-6.JJc) arc gi>'cn ~Iow:
:: 2a,.p +R.+A.[dP, I I, 1', iJp. - I)
" = 2a;P,, +Pi+A.
, 2ail lj+~](
.
Mate-rial
:mI
(6,36a)
dire-it)
Jtorai
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POWER SYSTEM
Compule initial ",:II i.e . for;" 1. 2.3.. ". NG
Sel inili:>l . I. ~ '" 0 forall genrraton.
2,
0,'" 1.0 and oJ:,. 0 for 1 fori ., (NG. I).
I kla:I buSC1 as zero.
Sci inil i:ll real
~ .. 0 ond
ct., .. O. for
2), .... N
Compule the
B
Compute the ele mellIi of Hcssian
[If). using equalions (M3~H6.63e)
Solve eql1;)1ioo (6.64) I
l.
""".••, . roc •
for 1111
2. ~ ' '' I
3.
, . M'''''
, .!A.'•.. ' 5.
•
3. _.. N).
for all buse$ (i.e. ; .. 1
3. .... N).
for a1110itd buse$
(NG. 1~ (NG -+- 2). " .. N).
~l','''\ for al l load buses (Le. i
(NG . 1). (NG -+- 2) •.". N).
Malenal,
Jm
direikJ
Jtorai
COMPlITER·IIIDW· ECONOMIC WilD DISPATCH lIND OPTIlM.L POWER
rww
C
Upd:de ttwo. rontrol YlriabI<"$ I. p
fQf all &
2·(j{lp.,) -15'''' +t.5'~" ,-1 I '
for all buses excludi ng slack bus (i.e. j
J. A' Y'!> =A'''' tM' '''
for al l buses (I.e. i_ I. 2. 3•...• N),
I,
"
••
I.
).(1,0."
"
'
" .
. '
_ A("" t M , ... ,
, 1'1,11,0")1 ~
..
~
I. 2. J ..... NG). "
2. 3, ...• Nj,
for all load buso:s (I.e. i .. (NO .. 1). (NO .. 2). .... N}.
= tv.'Y'I" 610. ...'" for alllOlld buses (I.e.
. .
UJI .. [~ r (t.P.:)lt r {~)l tI: {~)lt . .,
j ..
. .,
, ..
Reactive po .... er generations (/.....1 = ~
j " (NO
.
. ']
I: (t..l.!)l .. r
' ' ' ' ' ' '. ,
t
.. 11. (NG .. 2). .... N)
(6IVofl'
j • .., . ,
0-,. I.e. for I = 1.2. J •..., NO
kp ..... kpt (
"
Yo.
Is tp > KPMAX? No
<
D
luoJ!i t: ?
,.....,
Yo.
•
Male-rial
:mI
dire-it)
Jlorai
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,
• •
• (-I) 6.616864
(- 3.275675)
(- 3.275675)
6.583783
• • • •
-0.545946
•
• • • •
•
• •
0545946
•
623.033385
(- L(I9m,) (- 1252.235527)
• • • •
1.06486
(-0.551352)
(6A)
(-0.545946)
1.064864
6.349189
•
6.349189
The JacobiOll malrix [J 1of the same problem for !he first ilerafion is given below:
••
623.033386
-623.033386
- 1252.235527
1252.235527
--0.163460
[Jj "" -
• •
-
0.163460
".en
•.en
0.233783 - 208.706080
-{):233783 208.706080
-{I.Ol7m
o.oln97
(6.8)
Therefore. updated values or control variables for the nell' iteration are:
.
~ =
if, +fJ>:, .. 0.38 - 0.004175,. 0.375825
P;, "' ~. +lJ.~... O.17+0.()()46Q5=O.17460:S ~ - ~+"'6f "' O-O.008374" - O.008374
6::
= ~ + lJ.~ = 0 - 0.039138 = - 0.039138
A~, = A;, ,I
'" =
,0 A.p
+t....l.;,
= 1902-0.l66992 => 190.1033003
AlO
, + ......p , ",0+ 189,7511 ""'189.7571
..I.'t, = -'"v. 1° +M,Q, =0-5.8422= - 5.8422
Material
I
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POWER SYSTEM ANALYSIS: OPERATION AND COIfl"ROL
Execution 01 computer algorithm given In Fig. 6.7 lor Ex.mple 6.7
UNE DATA orthe51sle... : NZPROBl.DAT Given in EumpJe 6.3
[v_I of the 1)'51.....: NVPROBl.DAT Given in Eumple 6.3
Bus data aad aeouator ruel cosl data (laplll data of eGllllpllkr prop-am, "ellllld NEWOPT.FOR b', 71 d OQ !be 110.. J ,neh.. ba J1'Ic. "7): NPROBl.DAT Given in E:u.mple 6.3
OutPlit or computer PfOVUll NEWOPT.FOR after "tclll:loa: NSOLl.DAT Iteration cot\IPleted at k .. 7 Tolerance = .004345 Final Economic Generation
,,
i
Po
1 2 3
.374900 .176972 .000000
i
Plolld
Qload
-----.068270 .131912 .000000
v
.200000 .100000 .250000
Deltll
-----.000000
-----.000000 .150000 .100000
• ---
.174960 .076912 -.250000
.068270 -.018088 -.099999
Qhm
-----189.996000
------
1.020000 -.008040 1.010000 190.618300 2 3 .996019 - .039119 192.725300 All the powers and voltages are in p.u. 1
o
.876832
Angles are in radian. All lambdas are in unit of cost/p.u.power.hr I
Total cost of generation Bus no.
Cost
(unit of cost/hr)
--------1
118.418500
2
72.794550
Total cost is 191.213000 unit of cost/hr.
•,
Malenal,
Jm
direikJ
Jtorai
COMPtlT[R·AIDED ECONOMJC WAD DISPATCH /\NO OPTIMAL POWeR fLOW
Exam. . 6.8: CoruUkr a 1u·b,u, Iel'tn ·/ille JX>lWr SY11nn as lOOwn in Fig. E6.8. The sysum Iuu Ihru gtneratars. Find 0111 the oplitnQl power Jlow solulion Illing N-R method. Tht fuel COSI cMracteristics of the thru genera/ors art 41 under;
F~. (P,. ) .. (lOP,.1 + 175P,. + 75 unit of oosll1'lr F.,(P,,) .. 110P"1+ I80P,, +50 unitofcostlhr
l Fr, (P,.) :: I60P,. + !!OP•• +80 unit
or costlhr
WMrt powtr gtfleralioru aTe p.w. with 100 MVA bast. The lint dota and bus dolO of 1M SY1tem art given below:
Line dmo of tM system: Li~
no.
Ftvm bus
To ....
Bf2
Li.M illlpMalK"e (p.w.)
(p.w.)
Off·nominol lap ratio of transformer
I
I
2
(0.01 + fJ3)
}l
2
2
3
pOI
3
2
,
(0.03 + /J.2)
(0.03 +fJ.2)
PDI
4
,
3
4
(0..01 + fJ.2)
pOI
3
6
}l
6
I
,
(0.01 + /J.2) (0.0 +/)2)
1.01
7
I
6
(0.0+/)1)
102
Male-n II
Jill
U
""'I<)
Jloral.
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COMPUTER-AIDED [CONOMIC WAD DISPATCH AND OI'TIA!AL POWER FWI\'
..... lIere P", P" and p" system 3.fe given be low.
arc in p.u. willt 100 MVA base. The line data and bus dMn of the
~
~
ICD
I
Lood
""d
';;J
L(j)
I
load
Load
Fig. P6.1
A three-bus foor-line power system.
Line data of the system
Line /w.
From blls
To blls
Une impfflmra (P.ll.)
812 (p.u. )
1
1
2
(0.04 -+ jO.3)
jO.OI
2
2
3
(O.OJ -+ jO.2)
jO.OI
3
3
4
(0.02 -+ jO.2)
jO.OI
4
1
4
(O.tu -+ jO.2)
jllO!
The po ..... er and bus Brn
/1<>.
Bru
voh~ge I)'~
of the system: V(p.u.)
p.(P.u.)
Q. (P.II.)
P,,(p.u.)
Q,,(P.u.}
?
?
0.2
1
SLx'
1.04 Lo"
2
PV
l lDl(
,
3
PV
/ 11l21
4
PQ
.,
0.25
?
, ,
0 0.25
0. 15
0.<15
0
0
025
01
Find oot economic generation schedu le for rea! and rcacth'e po""er b~lance. 4. Consider the 5i~ -bus. sevcn-line power system as shown in Fig. E6.8. Find out the optimal power now SO l\llion using conventional method. S.
Consider a three-bus. four-li ne power system as shown in Fig. P6.1. Find oot the optimal power now so lution using conventional method, N- R method. fast decoup led method and gradient method.
Mate-rial
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POWER SYSTEM ANALYSIS: OP[RJlTlON AND COIflROL
Result of exerel.. 1 Geaeralor fuel cost dati and IB) of the IJAUD (Iaput uti or computer provam, .em Ed ELDNR.FOR baud on the Clowd.art cI- i.Q ..... U): NELDPl.DAT 3,1.65 [No. of generator bw;es, tcu.lload of the system) 50,200,100 Ia. fJ, y for generatOf-l] 90,120,150 Ia.Ay forgenentor-2] 70,170,50 Ia.Ay forgenentor·3j 0.0008 IBlly2] 0.0006 lB,fll 0.0005 18)01'21
0. 00 0002
I
I,
0.006 0.0002 0.0002
IBool 0.0002 0 . 003
- 0.0001
[B II • 8 ,2. Bul [8110 8Z!. Bn]
-0.0001
0.004
[811 , Bn. B ll ]
0.0002
Output ohomputer PlVli", ELDNR.FOR Ilfler ua:alioa: NIlELDSl.DAT Iteration completed at k .. :2
Tolerance La.mbdto.
~
(uni t
Epsilon =
. 000026 of
c08 t/ p.u.power.hr.)
•
245.455300
.000100
Final Economic Generation
, I I
I
Bus no.
--------1
" ---------.436569
2
.689602
3
.529719
(p.u. )
Translllission line loss .. S.8902J8E- OJ
All results are in p.u. Lambda
is in unit of cost/p.u . power . hr.
Total cost of generation
I I I
I
I
Bus no.
Cost (unit of cost/hr)
1
---------------------196.844
2 3
275.552 159.694
Total cost is 6 3 2.089700 unit of cost/hr. Malenal,
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•
COMPUTER-AlOrD ECOHOMJC WAD DISPATCH AHD OPTIMAL POWER FlOW
Qtet,'tJI oIq "'I-
$
,
pot....
=
£LDNRA,F()R lli"a de 'oIIoH': NAELDSl.DAT
Iteration cOJ'!i)leted at k .. 4 Power tolerance
. 000000
H
!unit of
cost/p.u.
,
Lambda tolerance
power·hr)
•
245,455300
H
.000015
••
Epsilon
•
. 000100
Final EconOll'lic: Generation
1'0 (p.u.)
8ue: no.
---------
----------
1
. f.36570
2
.689602
3
.529719
TranamiBBion line 10BB "' 5.890239E-03 All resul tB are in p. u. Lambda is in unit of cost /p. u.power·hr.
Total coat of generation
BUB no.
Cost (Unit of coat/hr)
1
196,844
2
275.552
3
159.694
Total cost is 632.089700 unit of COBt/hr.
R..uM of ....-elM 2 Oalput 01 COTp"'" pl'IJITIIIII KI ».FOR .tler uecuUoll: ELDSl.DAT Iteration completed at kp ., 5 Iteration cQfI'IPleted at k • 3 Tolerance in P = .000000 Tolerance = ,000086 Epsil .. . 000100 Lambda (unit of cost/p.u .power. hr) .. 27 0.3 449 00 POIo.-er 10.. • . 002776
Malenal,
Jm
direikJ
Jtorai
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Constraints in economic operation. 228 bus vol!:Ij!;e. no dynamic. 229 primary. 228 s«ondary. 228 spau c~ity. 229
Adaptivc ronlrol. 5 AltemalOrncilCn. 47 1 Aperiodic lr.lIIS~nI component. 576 Atea coolrolle
Aspects of hydro Khcduling. 282
thermal. 229
Automatic
Conlnll cenlte. 456 aimof, 456 fUllCIion or. 458 Control
gcneralioncon11Ol.469 load frequency control. 15.469 voluae control. 15 vollqc rel"llIor. 469
emc ISCIX:y.466
loops. 469 responsibility. J
BifUR:alion. 536 Bnoshlcssc~i\a.
~~
4n
Bus .chnillanu matrix. 47 formation of. 57 modirocalion. 106 pioculurc orfOlTlla\ion. 96
Damper windin,. 24 Oua lICquisilion and system ronU'Ol. 458 Decouple
Crntnl facilities. 461 Civil racililin.. 461 Cauical methods. 380 Coeff\C~nt matrix. 82 Communkltion techniques. 46J Compu;son of comp'J!alion methods. 54 1 Compenwion, S48 Complete SUUClure. 486 Component mode ll ing. 20 active clements. 20 pusi"e clements. 20 Computcr conU'Ol syslCm. 5 di=t. 5
Distri~tion
IcYcl. 3
Double line faull. 576
bouble line to Ji'OIlfId flult. 576 Dynamic performance. 4n
Ecooornic alloc:llioo. 24S load dispatclt, 323 operation, 225 scheduling. 225 E1ecuic power syslCm. I Em:,&cncy ronU'Ol. 466
inti"". 5
omine. S ooIine. 5 sUpICl"lisory. 5 COlI«pt of stability. 522
'"
aulomatic. 466 manual. 466
...
Exxtloss ronnula. 329 Exciter rocld windin8. 469
IND[X
,
FACTS, IJ optilD&l QPeralion,
OpIimai controller (OlR) design, 5 10 power !low (OPF), 313 Optimum
" "
$IrOn, interconnections, system security. IJ Fa.\( de(OI.Ipled Ia.d flow, 195,
I I i •
,
,
,,
,
..
method, 426 Fcrranti effect, 4'9 Fibre opcic channe l~, 463
Galin &id.ll (G-S) """hod. GoVCmof modellillJ. 26 Gradi~nt method. 436
OJ'
.'
.scheduJi11i.
1oad~ ~U
,
.,
QuadratUR: axis. 21
Marginal load ~,,"atiOf\S. 436 Method of deicnninin, KOOOmic operation. 218
Microwal'c cllallnels.. 463 Mimic board, 462 Modem c~citers, 472 Monitorin, of the intclWllllCCteU system, 4'8, 459
NcwWf\-Raph'lOO "",!hod, 152 application proadu~ of, 156 N",", method, 47 S1~n,lh.
8 Node incidence nwrix. 75
'.
.
Pri...., AlFe loop. 480
Induction motor. 41 inlma.cin" S06
dispalm ~~ntr~, 461 no ... or:rrations, 130 stabi lity. 523 Loadabilily, 9 lon,-term $Chedulin,. 458
..""
f'Iwc: angle control. 60 Pilox,main exciter system, 412 Plant schedulin,. 4'9 Pow<:r line carrier communication (PlCC). 463 Power system control. 3
Hopfbifurcation, '36
"""
.~
281
control. 4 Pumped stor:age plant. J01
kascd tdephone li nel. 46J
I
.,
Real and reacti~e power balance, 363 RCi\llatinllr2nsfonnen, 32 Rotor ansJe stability. 532
Saddle node, '36 Satellite C()ITIlnunication. 463
SCR controlled static ncitu. 412 SttondaJy AlFC loop, 418
,.,.,
capacitor. 40 compc:Tmltion. Sot8 R:acli~e loss. 13 Short circuit r.... lI. 576 Shon-tmnschedulin,.458 Sh~
capaci tors. 39. 552 compcn
Static dynamic analysis. 535
performance. 476 On·load tap changclS, 545 OpenuiolW cl!ar.octeristics of hydel power plant. 226 thermal plants. 226
VAK cornpen.ator. 40. 'SO Steody Stale model. 22 vol~e lI.ability. 525 \o1al[ al
:mI
dlrl'itJ
Jlorai
IND£X
SLeam turbine
Trll/1,minion lou
non-reheat. 481 R'heat. 481 S1Cp5 of a1gorilhm to develop IY..). 84 Sub-cran,mission \eve]. 3 Surge impedance loading. 8 Syl'l\JlletricaJ fault eunen!.. S77 Synchronous condenser
fennula. 25S problem, 251 Turbine modelling. 26 R'pttscntation. 4.S I
application of. 556 modelling of. 553 Opl'ration of. 555 Synchronou.gaoUalor. 21. 469
TCR comperu.ator, S5 I Telemetry. 466
.-,. '" digital . 466
Three phase bClIlIIICtd fault. 576 Tie-line power flow model . 496 TTanJient sttbi]ity, 12 voltage stabil ity. 52S
Unit commitment. 459
\bILage collapse, S22 Vollagecollapse point. 537 continuation Iood flow, 537 optimiJafion, 537 point of collapse. 537 silljl;ularvalucpoinl, 537, \'oltage I'I!'glIlation, S 18 VoILagestability. ~16 definition and clas,iflCaliOfl, 523 faclOTSaffecting. S32 VSAT IVery Small Aperture Terminal). 466
MJlerl
1:m1 dlreibs
JIc>raJ
THE AUTHOkS
Rs.450.00
