POWER SYSTEM ANALYSIS Operation and Control

ABHIJIT CHAKRABART! Professor, Department of Elcdrical Engineering

Bengal Engineering and Science University Shibpur, Howrah

SUNIT A HALDER Lecturer, Department of Electrical Engineering

Jadavpur University Kolkata

Prentice-Hall of India i?oiIwlill@ ilJllMllKl@@l New Delhi-11 0001

2006 T hca. . One

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As. 450.00

POWER SYSTEM ANALYSIS: OperatIon and Control Abhij~ Chakrablortl and &.Ilta Halcr

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.

Jlen

Contents

Pre{ace ... .. .................................... ..... ... .......... ........ ................ , ........ ... ...... .. . " .............. .. .. .... .. ....... .. .. .. xiii

I . Introduction .................. " ............... " " " ............ """,,",,",, ............. ,,"",, ......... 1- 12 1.1 StruClurc of oil. Power Syslan ... ..... .. .. ............................ .......................................... ... .. ... ... I 1.2 The NC'Ccssity of Contro l o f oil. Power System ................................................................. 3

1.2,1 CnntmJ Mc1brxls ..... . ... . ........ " .. "" . " ... " " .......... . 5 1.2.2 Advanta es of Co ute r Control ._ .. , .............................. _ .............. .. ........ ... .......... 5 12.3 Trpc:s of Computer Control Syslem ... .. .. ... .. ............... ............. ........ .... ............ ...... 5

13 Power S tem R resentation ...... , ... . , .................... ... . , ...... ......... , .... , ... ... ... _ .............. , ...... .. 6 1.4 Pov.'er System at Normal Operating State .. __ _ .... _ .... _ ... .. ... _ ... .... _._, ................ ........ _ ............. 7 1.5 Operating Problems in Power Systems .... .... .. .... .. ... ................................... .............. .. ....... 8

1.5. 1 Loadabilicy of Transmiss ioR Lines ........................................................................ 8 1.52 Frequency Dynamics of Transmission Line ....................................................... 10 1.5.3 Overload and Frequency Decay Rate ......................................................... , ....... 10 1.5.4 Tra ns ient Stab ilicy Problem .................................... , ..... __ ................. ..................... 12 1

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CONTENTS

2.4 Modell ing of Generator Components ... .... .... .... .. ......... .... ... .... ...... .............. .... .... ............. 2ti 2.4.1 G

CONTeNTS

5.14 Use of Loss Formula in Economic Operation .. ..... ... .. ... ....... ....... ............ ... ...... ... ... .... ... 268 5. 14,1 Algorithm for Detennina tion of Optimal Generalion Using lAss FormuJa .. ... .. 269

5.)5 A Mtthod or Oo.:li:rmining E",onolllic Qr>

CONTENTS

7.4 Functions of Control Centres ....... ...... ........ .. ......... ............ ... ..... ............................... ...... 458 7.4.] P lanning .. .. .. .. ............ ... ... .. .. .. ... .. .... ... ....... ........ ..... ........ .. ..... ..... ... .. ............... .. ..... 458 7.4.2 Monitoring ... .......... ... ... ... ............. ..... ...... ....... ............. ... .. .. ... .... .......... ......... .. ..... . 459 7.43 Data Acquisition and System Control .... ........ ..... .. ..... ... ...... ... ........ .. ..... ........ .... 400

7.5 Sct-up ................................................................. .................................... ..... .. ....... .......... 460 7.6 locations .. ................. ... .. ...... .. .. .... ... ..... .... ..... .. ............. ..................... ................. ..... ... .... . 461 7.7 Central 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 Satelli te Communkation Channel ................ ........... ............... ...... ................. ... ... 465

7.9 Telemetry .. ..... ...... .. .. .. .. ...... .... ..... .... ........................... ... .. ...... .. .............. .. .... ... ... ........... .. .. 466 7.10 Emergency Control ....... ...... ... ... .. ........ .... ..... ... .. .... .... ........ ................. .... ..... ... ................. . 466 F_un:ises .. .... ... . _ . . _____ _ .. ___ .. _. __ ., ... , ... ... . , ....... _, ......... , ....... " .. , ....... , ... , ... , ... , ... ,' ." ... ......... , .......... 468

8. Automatic Generation Control _ ............................................................. 469-51 S 8.1 Introduction ................ .. , ................ , .. .... ...... , ... .................... ............... .. ....... ......... _ ... , .... ... 469 &.2 Types of Ahel1Ultor Eltciters .... ..... .......... ..... .. ....... ...... .. .... .... .... .. .. .. , ..... ..... .. , .... .... .... . , .... 471

8.2.1 Primitive Type Eltciters ... ... .. .... ... ... ........ .. ... .... .... ....... ... ... ........... ......................... 47] 8.2.2 Modem Eltci ters .............................................................................................. .. ... 471

8.J Exciter Modelling .......................... ... ............... ... ...... .. ...... ... ............. .... .... ... .. ..... .... , ....... .. 474 &4 Modelling of Alternator (Synchronous Gener:uor) ................. ... .. ................................. 475 &5 Statk Performance ofAVR Loop ..... ...... .... ... .. .. ....... , .. .... ..... ... .... .... .... ............. ....... .... .... 476 8.6 Dyll.lmic Performance of the AVR Loop ... ....... .. .. .. ... ........... _., .. _ ... . _ ........ .. ....... ... ........... 4n 8.7 Compe nsa liOn in A VR Loop .. ......... .... .. ........ ........ .. .......... .... .... ................. ....... .. ... ... .. .. .. 478 &8 Automatic Load Frequency Control (ALFC) ................................................................. 478 8.9 Types of Turbine Representation .. ..... .... ............ , ......... ... ........ ............ .. ... ... .. ...... ....... ... 481 8.10 Steady State Perfonnance of the Speed Governing System .... ................... ..... ......... .. .. 483 &1 1 Complete StruCTUre of Primary Al FC Loop ..... ................... .... ... ........................... ...... ... 486 8.12 RespoMes of Primary ALFC Loop ... ............................................................................. 488

8.12. 1 Steady Stalc Response ....... ......................... ... ........................... ............. ........ .... 488 8,12.2 Transient Response .......................................................... , ................ , ... ......... .. ... 489

&13 Secondary ALFC loop ...................... , ... ........ ... ... , ... ....... ... .......... ... .. .. ... .. ..... ..... .. ..... ...... 492 8. 13.1 Abou t the Controller ., ................ , ..... ... . , ................... .. ........... ........ , ..................... 492 8.13.2 Modelling of Secondary ALfC Loop ...... ........................ .. ................................. 493

8.]4 Performance of Secondary ALFC Loop . __ . ___ ... _, ,_ .. _ .. , ____ ____ __ .. _. __ _ . __ .. __ _ ....... .. ..... _ ... _ 494 &15 Elttension of ALFC Loop to Mult i-area Systems ......... ......... .. .. .... .... .. .... .. .. , ................. 495 8.16 Tic-line Power Flow Model ............................................................................................ 496 8.17 Static Response of Two-area System ... ............ .. ...... .... ... , ... ....... ......... , ... ... ... ......... , ...... . 498 8.18 Transien t Response of a Two-area System .... ........................ ... .. ....... .... .............. .. ....... 503 8.19 Application Aspects of Primary A lFC loop ........... ... ,,_ ... __ .. _. " .. ,, __ ... __ .. __ .. _ .. ,, _ ... __ ... _ .... SOl 8.20 Application Aspect of Secondary ALFC Loop ......... _ ... ... ....... " .. "._ .... _ .......... _ ... .. ........ 505

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8.21 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 J 5

9. Reactive Power Control and Voltage Stability .................................... 516-575 9.1 Introduction .. ........... ....... .......... .. ...... ..... ............ .. .. ..... .... ........ .. ............ ..... ... .... ..... ... ....... 516 92 Po,,"'!:r Flow in a Two-Bus System ..... .. .... ... ..... ................. .. ...... .. ...... ................... ..... ..... 516 93 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 Critical Re

CONTfNTS

10. Computerised Fault Analysis ............................................................... 576- 61 3 10.1 Introduction ..................... .... ........ _ .. .. ....... ...... .... ........... ... ... .. ........ .... .. .. ................ .. .... . , ... 576 IQ.2 Detenninatilln of Symmetrical Fault Curren! Using ZII", Inversion .... , ............. , ........... 5n 10J Detcnnin.ation or Fault Cumnt by Fonnulating the Impedancc Matrix Using

Network Theory ..... ..................... __ ... ........ ........ ....................... ....................... ........ ......... 578 lOA Generalised Fault Analys is Using Zo ... Building 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 uercis/!s .. , ........... , ... , ... " .. , ........ , .... , .... , ... " .. , .... , ... , ....... " ....... , ... , ... " .. . , ... " ... , ... .... .. , .. __ .. , .... , ... ,_ 599

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 bn 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

PREfAC E.

resplive lk~ns, Registrars, and Heads 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-

MJter! 1:m1 dlreibs J!c>raJ

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 syStem 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 ionally, loads are represenled in a lumped or composile fonn.

The best location of a generating station 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 power plant is frequently governed by its doseness co the energy resource and transponmion facility of Ihe fuel as well as availability ofneareslload cenlre. Ellv ironmenlal 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, non-conventional sources like wind power, 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.

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Mat Tlal Jm dIem 81

POWER S)'STtM ANALYSIS, OPERATION AND CONTROL

NeighOOuring syst~

~ SIC'P "r transfOl'1'l'lel'll 'iii!' Ci~uit breakers ~

nus Bo,

To other ~ys lcm

Tic lin~ To Other s)'litem ~---- -- ---------- ----------------------- ------ , Tran'lni", ion level I

--- -----l ;-;t::~~.:~r::.~:~--~!----------JL-_____________________ l _, '"~:"jf;::cr : Sub Iran~ nll!ion kId ~ f -- ----- ----------r--------------:

To _ Step down

CO~:l~~~ ~ _________ t ____ ~~~S:~'~C~ , : Distribution k wl I ---X-----------!--- ---

I-- - - ---J ___ $1"1' Secondary di st. (LT raJ

INTII.ODi/Cn ON

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 level. the circuits diSTribute electric power to a number of distribution centres in a cenain geographical region at a vohage level 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 level 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 only 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 out 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:gulation, power optimisation need to b1.2 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 control is based on a combination of manual inlenention, feedback loops, optimisation techn iques and load demand. The rjuirements for cont rol of frequency and power exchange can be implemented by load frequency control. This control is genemlly autonomous and each area is responsible for il.'l 0"'1'1 steady state power balance. The need for direct aCTion to control network voltage is usually done by the installation of automatic voltage regu lating equipment.

The control responsibility is basically divided according to the frequency of intervention of the physical phenomenon in~o l ved. The area level 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.

Material I Jm dire-,to ]to ai

POWER SYSTE,\i ,

The management and comrol of a power is a complex process and it requires proper interaction between many levels. Figure 1.2 i ,~~. salient clements of Ihe cOnlrol hierarchy. Manual cOnlrol is generally slower Ihan .. ~!~i~-~ . . The a~ailabi1ity of digital computers has resulted in consideration of digital ~~L . , coordinating the control parameters of

~arious le~e l s previously under La l ~ ~i' . Pr imary control at the lowest i control structure is most fundamental for ,

the proper operation of the power s)"5tem. , lowest le~e l control was analog conlrol, but small digital systems based on microcomputer

, are now being imroduced, This comrol ,

inc ludes the control of go~emor set point and , and ~oltage con!T(l1 of the station. The higher levels of control range for longer , intervals and are largely manually controlled. , However. computer control schemes ha~e been for economic {O

tN'tRODlIcrION

fl1!qllcm:y cOlllrol. Though unit commitment has been computerised in presen t-day power sys tems. syslem maintenance and system plann ing are most ly manually controlled. Ada!'t;'"

poWeR SY5T!;M A.NALYSIS: OpeRATION A.ND

When Ihe COnlro! is off-line, the computer is Ihe dala regarding the process Ihrough II human operator. The computcr is not the actual system. The duty of the computer is \0 process these data and output the results operator can ~ommend a control action.

In on-line systems. the computer i Ihe power system through suitable intcrfacing circu it ry and receives Ihe nece$Silry data any human intervention. The com puler processes the input data and outputs the result who then implemcnts the control action. This is basically the simplest on- line control and is 1 as open-loop on-line control. It is also possible to have closed-loop on-line control where computer requires no manual inlervenlion in implementing the output dedsion. The computer i I' is transmitted to the power system network through necessary interfacing network automalically.

In the in-line type comrol, Ihe operntor d". from the system and enlers them rapidly and directly into the computer through the keyboard.

The digital computer is not only the moS! I sophistiemcd. To economise the t 11 analog control 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 imp1cmcnlation of a computer as a considered before implementing the decision

Training .M

implemental;1>I\

Step 7

Pr

II'ITRODUCfION

system by means of simple s)'stems for each component resulting in single-line di~.mm. ~s shown in Fig. 1.4.

G '---/CB , Gcn bus

G~. auxiliary 100'

T,

Bus )

To the neighbouring ~y'tcm ticlines

Dir.::ct;on of pow.:r now

In 10 ",On no:

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PO~\'rR 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 operating 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 operating 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()

/,\ .'TRODUCfION

( I .4)

where Zo "" surge impedance '" If, ( an~ c being the line reactance Jnd shunt c~pac itance per unit length, respectively, p .. phase constant of the wave of proPJgation ( = m./k, tlJ being the angu lar

frequ~ncy), o elect rica l line length ofthc line in radian and 0('" (lL) being small, sin 0", O. Subst ituting equation (1.4) into equation (1.3)

V1 1 . sin '; p "" "-sm';'" (SILJ",-~

s inO Z" sin O

P sin'; [in p.u .. SIL - (1'2120)] (1.5) SIL sinO

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 single c ireu it tine assuming various I inc lengths.

" r 1 0 ~

0

" ~ , , .; -

, I '0 ~

0;

"'" 200 300

"'" 300

Line length (kill)

fiR. I.S !'ronle ofline loadabi!iw

High sourCe rNClance plays a vita l role in limiting the line loadabilit),. Loadabilit)' can be improved by reducing the reactance of overhead wires and placing series capadtor in line as well 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. Figure 1.6 represents two profiles of line loadability for t\\'o systems. one having higher source reactance and line reactance (tenned as I

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POWER SYSTEM ANALYSIS, OPERATION AND CO,VTR OL

I 2.5

'.5

, .0

05

o ".

rur robust system /'

, 200 300 ".

l.im: ICl1l;lh (km)

~' ig. 1.6 1'lOliks of iu.ldabi!ily of lobus! and wea~ s)'Sl~ms_

1.5.2 Frequency Dynamics 01 Transmission Line

When the power system struc ture is such Ihal a single line oUlage in the uansmission sysTem creales is landing condition, a enrdll l evaluat ion of rrequency bell.winllr i~ nc~-dcd \0 accretion reliability.

During nannal oper.llion. Ihe power flow on the t ie- line can be represented as

If = 1v.'11 '11 ,'" (~ _ ~I ) X .

o ( 1.6)

The ra te of change of the line rea l power flow can Illen be repre$Cntcd as

(1.7)

/:if. and 6/, being Ihe Til ' Ihe Sialic transmission capacity.

In case the slatic transmission 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 known 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

oc. dw T P~ I _=_=_x_ dfJruJ

dw I ( ) - .. - Pr-P dl J(J) ~

INTRODUCTION

( 1.8)

As. p. '" .accelerating power = P, - P,; Pr 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(, _,)

dI2H T

oc. fLL(p,_, ) dl 2 H (1.9) In a SO Hz system from equation (1 .9), the initial frequency decay rate for a s~tem toeOCQullter

II sudden load demand can then be represented by

(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.

,

,

(;,L 2 (Hz/sec)

Frequcn

POWER SYSTEM ANALYSIS: OPRATION

1.5.4 Transient Stability Problem Frequent topological changes, transmission generat ion and demand insist the power ":::;,"~:: margin. causing instability ph" ,~'m~", " Basic aspects of transient stab il ity have been the generation. popularly known as AGT ("".=,', has been found to be an effective tool in such a simulation.

V (kV)

"

';1:. 1.8 I mpro~cnknl in "ol1age profile

P (I>IW)

80

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 p1ically and it reveals thm the stable voltage state ean onl}' be maintained if the SY$tcm possesses the corresponding limiting value of the reactive power tr:msfcr capability.

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 (complete 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"

I

POWER SYSTEM ANALYSIS: OPEIUoTION AND CONTROL

contingencies so thaI the operator may take cOITC{:\ive aClion in the event of the ir occurrence. Thus, the analysis 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 (a) Nonnal (b) Emergency (c) Restorative

However. later on Iwo more slates. "Alert" and "Extremis", were added. For the sake of understanding, only 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"",}, ~IJI~ Slate

I I Fig. 1.10 State transition diagram.

I

Most of the times. the system remains 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 condition is secured. The imponam and probable contingencies to be considcred are:

Outage of a line Outage of a generating unit

Single phase or three-phase fault.

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 complex bus voliage at any instal)! from the redundant SCI of

1,1 atE

/folTRODUCTIOIII

telemetered data and brcilker status. The security 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

I POWER 5Y5TUI ANALYSIS; OPERATION AND CONTROL

maintain power balance 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 signals. 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 itation 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

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 replace the mechanical operations by electronic switches. 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:

Regulate power flow on AC l ine~ wilh a view to either avoid overloading or to minimise powcr loss

To operate the system at a safe powerangle for same power del ivered To enhance the power transfer capabil ily of the system by introducing improved dynamic

characteristics System islanding under ext reme conditions Strategies to save thc system/islands from total collapse.

Aftcr the objectives have been identified, the following stra tegies need to be decided. ( i) Planning and openllional system strategies:

System 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 phaseshifters. i r required Lowfrequency oscillation dampers, if mjuired Communication network FACTS controller with online data monitoring Computer 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, emphasis should be laid on prevention of faulls spreading into the syStem and creating gr id instabil ity. This phase can be termed as system secllrity phase.

MatE all Jm direikJ Jtorai

! POWER SYSTeM ANALYSIS, OPERATION AND CO,WROL

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 necessity of any elaborate FACTS

I controller at Ihe stage. The control actions can be derived from terminal sub-stations.

Phase II: Strong Interconnections In the second phase emphasis 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 points Phase shifter, if required Development of FACTS controlle 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 requirements but alw be

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 ~nonnal opeilting stale'? 5. 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 turbo- ,&~temator following an attempted

OYffload. 7. Write short notes on

(I) Security analysis and contingency evaluation. (U) fACT system.

Mate-rial :mI dire-it) Jtorai

,

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 certain field of bypotlteses correspollding to their use . Here. the concept o f representation of tbe physical 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. This 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 machine. 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$ the modelling. The present text will give adequate stress on an alternator modelling such that the basic building blocks for computeraided analysis of the operation of the power system can be developed at this 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 dirr axis (the rotor polar axis) Of simply the daxis ~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 lrani~nt reactance (X;). In this chapter. the salient pole synchronous machine is only considered. 115 the cylindrical 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 transient reactances X~ and X; . The equations representing this model are thus

,.,

~ = VI + IdR~ + ' . X; ~=V.+lqR~-IIIX;

(2.1)

(2.2)

Male-nal, Jm dlre-itJ Jloral.

POWER SYSTEM ANALYSIS, OPERATION AND

a-axis

,

[X. =X,+X_j X, = x,+ x ..

salient pole a.l!ema101'. sumx d stands fOl'direct Sumx I indic*, leakage quantity; V4 and II/

Fit- 1.1 Phasor diagram of steady Slate operation axis quantities and suffix q fOl' quadrature axis

= 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).

[Here. E and V represem induced and terminal .,1'," 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

MODLLlNG 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. _ Ddt5) (2 .3) dlM" ~ dl dw -=m-2tr!o '"

(2.4)

where. M "" l!-.-

ModII 2

KJ, 1M .. aIlJular momentum H .. inertia,constant 10 '" base frequency OJ .. angular frequency

p. '" turbine waft power p .. generator electrical power output D .. damping coefftcient B '" rotor angle I

The drawback of Model-O and Model-! is that the eleclrical dynamics have 00{ been considered. Model2 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 J Ef -E.) .. EI +(X ... - X~ )/~ -Ef dt r; T; (2.5)

where, r;; is the dirtd lUis IrQruitnllime COlllltllll and Elis the applied field voltage. I ... is numerk:aUy negative.

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

dE; Ed - (Xq - X;)lq - ~ -= - -=

d/ T' T'

(2.6)

Here T~ is the quadrarufZ lUis /ransient/ime cons/ont .

Malenal, Jm direikJ Jtorai

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 represents the phasor diagram of the alternator during sub-transient stlte of operation. The governing equations can be wrillen as

"

I " ,

e; = Vd + fJR. + fqX; ;=V, + f. R.- fdX;

de; _1' -(X ' - X' )f - ']fr' dtJ " qf.

,,' ..... '

,,' ,,'

I

.. , , , , ,

, , , ,

.. ' . ~ ,

"

" ,

"-. v

IR.

, , , , , , , ,

,

, E"

(2.7)

(2.8)

(2.9)

(2. 10)

E

f't" 2.3 I'hawr diagram of the sub-transient stale opel3lion of the witnt alternator. E" is the transient voltagc (daxis projeCtions arc numerically DCgativc)

In Ihe above equation. r; and T.- II/'C coru;idered 10 be subtransient dtuiI and q-axiI time cons/an/I. This model is chnractcrised by equalions (2.5) 10 (2.10 ) in addition 10 equalions (2 .3) and (2.4). Groups of synchronous machines or pariS of Ihe syslem may be represenled by a single synchronous machine mcdel. An infini te bus bar. representing a large niffsy$tcm. may simi larly be modelled as a single machine (ModelO).

MJlerl 1:m1 dlreibs Jtc>raJ

MODWNG 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

Here V, and V,," repastnt components of voltagc V in real and imaginary axis.

Imaginary axis (Network) ,

dw (M.~hinc) :

,, -

" V ... "' ................... . . . ... .

. ..' , , ,

~ . .. , ,

,,-,, - " .-

.-

.-

: ... -

, ,

-

, , ,

-

,

. ,axis (Machine) "

v,

,,-,,-

" .-.-

.-

"---~', . .... . Real axis (Netwtd.) V,

~ \

(2.1l )

.".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 machine 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 ..

[/ ) =[Y] [V] (2.13) [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...] sin 28 2 X~ Xd

(2. 14)

Material, JfT1 dlreitJ Ao~.

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 if the role of AGCJLFC (auromaric generariOl1 comrolJload frequency comroi) and ucilarion ' ... not inchvled. Just as the AVR (allrOf1UJlic volrage regula/or) achieve~ reactive power by maintaining a constant V(lltlge, the load frequency control achieves real power "'::;;' maintaining a constant frequency. Govern ... r Modelling and Trurbint: Modelling are thus '~h important in implementing AGe. 2.4.1 Governor Modelling If the load increases, the speed of the thermal unit reacts to this speed variation and to the turbine which, in turn. increases the which reinstates the increase of an adequalC Fortunately. the large thtrmol ineF/to of most of the turbine. generluor and load to of load change. the boiler pressure may be short\Cnn response of the system to the load

Many rorms of the governor other, the variation of the turbinegenerator of the turbine workin!! fluid conlTol valve ."'~ !!ovemors range between 5 and 10%. The latest electronic controller. A block diagram '''''''~ '' Fig. 2.5.

,,.

P,(.)---\ (COOlm&nded chanic ~_

in power) R .. speed regulltion of the ,""=0 KfC " i,inofthe~govemor T fC .. time conSlAntofthc: 5pd

'~n'oc"d~. slightly. The governor of Illy the entry of some more steam from the boiler

increased steam flow reduces the boiler' p,esSUfe, . and water flow 10 release the steam pressure. systems enables the load frequency performance

''';m ... of the boiler. so that, for short duration constant. The generator mainly determines the

i of which include. in some way or the as the bl5is on which the change of position

Typical speed droop characteristics for most in the turbine governor design is 10 provide an

of the speed governor syste!!! is shown in

(opmini nhtcilIIl t

Malenall JfT1 dlreitJ Ao~.

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. This is ncc(SSitaled by the large inertia of the Pf'flJtod gate, which regulates the rate of water inptJt to the turbine.

Here. tJ.x =0 KSG (liP. -..!..tJ.w) < 1+5T.iG ' 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

I S s+-T~

Mathematical maniptJlation yields,

,

which on inverse Laplace transform yields

at,(t)=oKSG(I_e- tlT",) for {;?;O

(2.17)

(2.18)

(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~ 31-."

" Fl 1.6 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

.". 2.7 Block

After passing the control valve,

s,~ chest

that introduces the delay T din ~ ,,,d,,, " function

The turbine governor block diagram has been s ....

Turbine

Tocondenscr

steam enters the turbine via the steam-chcst s) in the steam flow resu lling in the tnlnsfcr

I (2 .20) l +sTr

in Fig. 2.8.

govnning '>"'=

-- -.;. ___ Turbine --..;,

IIR

Assuming the command increment to be APe, "",

II ins isIS 10 choose a scale factor so that This gi~es the model as shown in Fig. 2.9.

IIR

( ,

F\c. 2.9 Block diagram for

K.

block diagram.

(2.21) APe. This is equivalent to picking KJr= 10.

governor modelling.

. .

. ..

" '.'

Mat" II JfT1 dlreitJ Ao~.

MODfLUNG OF POWER SYSTEM COMPONNrS

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) l+sTRJI

where, TRII is the time constant of the reheatCf having typical vltlucs in the range of 5-10 s.

~.(s) X .~ LP .(age

dP,(J)

"'"

HP .tage

S To ~ondenser Re-heater

.1&.2.10 Block diagram of reheat!.leam hlrbine.

The hydro turbine design varies with the water head (Fig. 2.11) .

.. ......... .

n.m

Water h,., Penstock:

.1" 1.11 Block diagram of hydrotwbine. The overa ll tr:lnsfer function is then

G Cl-,-:"~TL' l +sT,

where. Tp is the time it takes for the water to pass through the penstock:.

2.4.3 Modelling of Exciler

(2.n)

(2.22:1)

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.

Malenal, Jm direikJ Jtorai

POWER SYSTEM AND CONTROL

,

, , , -

,

,

. ,

, ,

, \ "

-..l ~ ,

:i!2 1;; ~l! , ~.

" .. .;: 0 ~. ~;' >'0

.' t , ~ -

-

.l1 , , ,

,~ ~ " , ~ ~

, > , d: . ~ i ~ .. ~

.!

-'5 , !, i P E ~

< ." . " - .. ...:-5 t h o 0

U~! I~WV '0 ,

, ~

:x ,

i': J!;' .-

" ~ ~

~l~ .'

'i. , '"\ \ ~ .... }

.'

~alE-nall :>rTI dirE-lID Jlo ais

MODEWNG Of POWER SYSTEM COMPONE"7S

,

J ~

" ~

-' ,

~ "

" " 0 - +

.. '

, " , .~ ., 0

~ - 0

, + ~

"" " - + -

.! , ,

0 R " " 0

I~ ," .! .' -" + ~ -

g 1

, ~ -~ -<

,

.t " " ::i - t

+ ~ , i ~ >< ,

+

Malenal, Jm direikJ Jlorai

I

I I

,

I I

POWER SYSTEM. ANALYSIS: OPEfUoTlON AND CO~OL

In !he blod: diagram of Fig. 2.13. T~. the tim~ cons/ont a/1M recrifier is very small and may be neglected. The amplifier 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 the 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,

K~ .... K, q v = = V",;

K . +K~J(,(1' when; (J is a factor associated wilh the transfer function of the synchronous generator wilen loaded.

" I

'I' K~ .. -'. I I

-

FI,.2.I4. 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.

,

, prj Sec II " IniLO

Fi,. 2.15 R.gu!atini tr:tn~(orm in a two.bus network.

The current balance equations can be wrinen as

II'" Vly ........ (V1- "V0 Y" 1]1',,- '" "Vl Y.~ '" (IIVl - VI) 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

A lso from the equa1ion (2.24). 11'" (Y ........ Y..,l v1 ... (-nY .. ) V2

MJlerl

(2.23)

(2.24) (2.25)

(2.26)

(2.27)

,"

MODElliNG OF POWER SYSTEM COMPONENTS

-,- V,

RT I,

Y.

f1&.l.l6 Equivaknt ciocuit of Fig. 2.15.

Hence, rrom equations (2.27) and (2.26).

r".,. = [(yJIt+y,.) -n Y

,-V, I,

(2.28)

In practice, RT is either a yollQ8~ mDgnilud~ eomro/ IrQfufonn~r or a phos~ ongl~ eonlrol Ira/Uform~r. In the former easc. L9 = if and in the latter casc. I n I is a conSlanl

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" V, I, y" Yl2 Y!) Y~ y" Yu V, I, )",1 Y" Y" Y. YJ! y. V,

I, y" y" >'0 YM y" Y V, (2.29) I, y" Y" f" Y. Y" Y. V, I, f" f" f" fM Y" Y. V,

Partitioning the above matri)( ,

~/'~ ~Yu J [Y"~~V' ~ [I, J = [Y~ J [YIT [V, J (2.30) .ha.

(iX]=[/l/1 /Jr [tr ]=[/./}/6 ]r (2.31)

Mate-nal, Jm dlre-itJ ]toral.

POWEll: SYSTEM ANALYSIS: OPERATION AND

Yll Yll 1 1) [Y;'21 Yn Yn . [rn

Y~I Y41 1 41 [rr.< ]= YS I YSl )'Sl [rn'

161 162 )'6.1

I, I, I,

Flc. 2.17 Si~ coils with nOOaI

)'" 116 Y:u lUI 1)s 1.16

, ,,,, 1. s 1raJ

MODElliNG OF POWr R SYSTEM COM PONrNTS

For a threephase transformer. assuming yp and y, as the selfadminances of primary and secondary coils (being equivalent to Yll ' Yl l ' YJJ""') y;' the mutual admiTtance beTWeen primary coils. y: the: mutual admittance between the: secondary coils and y: the mutual admittance between primary and secondary coils on different cores. the nodal currents in the coils may be: linked with the bmnch voltages as

I, , , v, Y, Y. Y. - y. y. Y. I, , , V, Y. Y, y- Y. -y- y. I, , , , V, y. Y. y- Y. y. - y. (2.34) = I. - y; V. -,. Y. Y

" Y.

I, Y. -y. y. y. y, y V,

-I, Y. y. - y. Y. Y. y, V, 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 Y I US is formed utilising the relation [Yau,) '" [e)T[YpIUII [el (2.35)

where Ie) i ~ the eo,,,,n:rion motrix. and I Y PlrMl 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 vanish.

TABLE 1.1: E~ments ortfllll$Cormer _mi~e ntlItriccs

Y , Y (neutral solidly grounded)

where.

Y ( r ' 5iJk neuuul

solidly glOOn.ied) Y y . ,

Yn (~If. pri. )

Y.

Y,

"" Y.

Y 0 0 0 , 0 0 0 , 2y

- Y fA ' )'a "" -y 2,

- Y - y

- y , 0 -, y 0

Y .... (ulf, s.)

Y,

Y, Y,

Y,YJ - YdJ

Y, - Y,

- Y - y (2.36) 2y 0 y

- y

Mate-rial :mI dire-it) Jtorai

POIVER SYSTEM ANALYSIS: OPCRATION

It m~y be noted that any two i ~~ '"'" b, ,,~~,m,d by two compound-linked admitunces. The current voluge relationship is by

wilerc l YpsJ '" l YlI' f and is the same as OJ,,,,'m, " and secondary buses while Y ps and Ysp represent ratio is to be included. Sect ion 2.5 is to be fo"'" , .. , 2.7 MODELLING OF THREEPHASE 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.~ .

(2.37)

. YI'I'. Yss indicate self admittances 31 primary mutual admi!laoce. In C:L

MODWNG OF POWER SYSTM COMPONENTS

i' i' i' (~ i' n K i' CR)

i' i' i' [X""I

[VJl i' y' y' y' y' i' [1',1 y' y'

'" y' y' ~;'

i' i' y' y' y' 1''' [r; ] I [I'; ] (b) Equivllrnt modelling of components.

IX 1 ,

IVJ 1 ['l'-] ['l'-] 1 [V,I

(c) Equivalent mood.

"to 2.19 Moo.:lling or ihref:.phase tr.lll$Jtliuion l i~ lUing p ITICIdtI.

2.8 MODELLING OF PAIR OF THREEPHASE 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! ... yn] ,

" [ ytl' ... ylO! ]

,

"

[ -1'''] , " [-yur ,

[12xll matrix

[yI2.,.ylO] [y:l +yo.l]

[-r,,]

[ -r" ] [ - y" ]

[ -r" ] [ - y" ]

] [Y"+y'] [-r,,] [r'" + Y~']

[l2xl2] matrix

v, , v, V, , V,

112 x l ] m;ttrix

(2.39)

\.1al[ :e-itJ Jlorai

, .

POWER SYSTEM ANALYSIS, OPERATION AND CONTROL

I Y" I S, o--- ----"' R,

'" 1 I r" l I YIlI I r,,1 1 '" ~ I YIII

1 Y.,I [Y,J '"

'" IYnl I Y.,I . !YJ

MODLUNC OF PO WER SYSTM COMPONENTS

[Y ... [ s __ ---.---' '-------,--_ R

[Y,, ] [Y,.,] 1 [ ~::l Fll- 1.21(b) (6 x 6) Compound matrix represenUlion of Fig. 2.21(a).

Thus. the mutually coopled liT\e$ afe finally represented as

[::l [XNJ' +[r .. ,] -[X"J' [~::l = [::l -[ X"J' [X.r' +[Y,,] [~:l (2.40)

[12 x I] (12 x 12] (12xl] matrix matrix 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 I stllic dlunt capacitor bank with its compound admittance representation.

t.o.d bus

" 7', "

e = -: c c I I I Fl&- 1.22 Model repuKnt:l1ion of shunt ~ap.d!

POWER SYSTEM ~N;\LYS 'S: OPERATION

2.10 MODELLING OF A SERIES The capacitive element is connected in series wi"h matrix for this system has been writlcn as

The shunt dement does 001 exist

IljX" Hen:. [r,,] =

Figure 2.23 represents Ihe modelling.

, ~. ---11 1-1 ~. R ,

2.11 MODELLING OF STATIC VAR

line and betwet.'n two buses. The ~dminallCc

(2.41)

IljX M

x" X" X"

x" x" x" R

r {V~l x" x" x" [r",) - IX .... r l

Qf ~ri~s capacitor_

(SVC) Let BsV(" be the shUn! susceptance of the SVC "'-" 10 the MVAR loading of it. It is then

" ;,~;::'::,:':i;::,~ added 10 the susceptance al the busbar. The Iota I is given by B. A reduction in the L'Ontrull ing vohage V will CJuse the '''i,oj "'''' cllhanced.

The SVC injected CUfTent inlo the bus i~ then ,,,,by I =

Here. Y=G~j8 (2.42) (2.4J)

IG may be assumed 10 be zero here. ] The MVA ()IJ IPU! of the SVC is given by

S "" VI-SII

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 equiyalent to load torque and is commonly Cllprc:ssed as

(2.47) I: i$ an e;o;poncnt and is J for fan type: of loads ~nd 2 for pump type: of loads. The electrical torque is giyen by

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 x: 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,. (2.49) , (X, + 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 To is given by

To = (2.50) and the open circuit reactance Xo is given by

Xo"' X,+ X .. (2 .51) Assuming the st.ator rc:sist.ance to be R,. the governing equations of the model are given by

Vi .. - 1:.-; .. = li .. R, + I,X' (2.52) (2.53)

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

The COffC$ponding nUl( linkages are

Vq,= R, iq, +1V1V4,+Vot.

V ... , "" R,i", ';"OJSlI'dr +.r" V" = R,i", +cUSlV ... +V"

IV II< := L,I. + 4i", If'q, :: 41" + 4io' " b = 4,ill< + 4i.it. "~" 41" + 4 i",

Neglecting stalOr Iransienl'l and assuming the rotor short-circuited

;-", :: 0 and 11'" =0 VIr .. O and V.' = 0

, (-Xo J V ... '" L, IV", x; '" X, - X~/X, (2.65 )

(2.55)

(2.56)

(2.57)

(2.58)

(2.59)

(2.60)

(2.61)

(2.62)

(2.63)

(2.64)

(2.64.1)

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 using equation (2.64.1). Us ing equation (2.65) in the final form. the resultin, equation is

[V"]_[R" -X;] [,,,] . [V;:] 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!, )id.

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 I/,I =I.O. X,=X"

I , ::c018- jsin8 And from equation (2.70).

V; = jR,( X. - X;)(cos6 - jsin 6)/( R, + jsX , J Rationalising and taking the ratio of imaginary to real parts.

V; R,cos8 -sX,sin8 V;';: R, sin6 -sX,c016

Similarly, substitution of equation (2.71) in (2.67) with V, = I. yields V; _ R;sin6-X;cos8 V; - (I -R;C056- X;sin8)

[6is the motor p.f. angle]

2.13 POWER NETWORK MODELUNG lzt I{ '" injected currenl at node i (i = 1.2 ..... II)

V, "'1"'Ile i6, (voltage at node I) The current I{ can be expressed as 3 function of the volta~s. Thus.

I{" Yu V{ + L [YijkVi -Vi )' i '" 1.2, .... n ~a(l)

a (I) designates the subset of the nodes connected 10 node i and

_ ~ '. _ 1 _I 1 ( -~.) ' Y. - i... Y~' YIJ --- Yij e

}lali) 1:;1

(2.68)

(2.69)

(2.70)

(2 .71)

(2.72)

(2.73)

(2.74)

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 .

.c we '"tllne aij . lJi '"' 0; It~ . hJi = T

Mate-nail Jm dlre-itJ Jtoral.

POWER SYSTEM ANALYSIS: OPERATION AND CONTROL

In general form. the preceding eqUlltions can be written lIS

(/J=(YIIVj.

rq Xv Yjj=G~ +jH~ andG"- " --' H :o -r.:' II ~

;oa(1)

Also. at node i.

However. V;. is compleJt conjugate of VI and hence

...

...

fl=RealJVt[ YIIVj+ L YIj(Yr-V1) ]1 1 .... (I)

Simplification yields

~ =V;l L (y~ cos jI .. (i)

8ij+8ij)-VI LV'Y4'COS .... "'il

(8ij+o/-o,) ... L P4' JO"(I)

Q,=V/ L (Yi/sin8~-Iv)-Vj L Vj Yijsin(8ij+6'r-6',)" L C2u-is .. (1) ji"(I) }llftl)

(2.7~

(2.76)

(l.TI)

(2.78)

(2.79)

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,; C05( 8y + 6', -0,,) '4 '" V;l ( Yi/ sinOiJ - ~)-V;VJYiJ sin{OiJ + 6; -6', ) QjI = VI (Yi/ sinOij -Iv)-V;VJYiJ sin{O~ + oJ - O()

(2.80)

(2.81)

(2.82)

(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

Lighting and Heating Induction Motors

MODEf.l1NG OF POWER SYSTEM COMPONEM"S

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

p

Q (p.u.)

II I (p.u.j

.' , "

..-

V(p.u.)

.

Constant ClImnt

C~, impedance

1.0 p.u. ' .... .r:.J -----------------------;.~.~.----'_ Constant

". ... C\IITeM '> . .

/ .... '" " ..

' j .'

" "

V(p.u.) (h) Current CIwaClcristicl of loads. FI"l.2.S Characleri$\ics of lo.ds.

Constant

""""

(j) Cons/ant cu,rrrlf rype

1= P- jQ '" 1'1 .6 - 9, V

Male-rial :mI dire-it) Jtorai

I , I I POWER SYSTEM ANALYSIS: OPERATION A.ND CONTROL

where. V .. IVI Lo. (J .. 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 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 V Here, the load is spe.;ified in MW and MVAR al nominal voltage. Here, J "'z and Z is assumed to be constant. Here I varies wilh variation of V. The load impedance is determined by

Z = ~ = W =1L=..!.. (inp.u.). Z P - jQ P - jQ Y

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 a~lylical concepl behind differenl models conceptS of an iwlated synchronous

generator. 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 lIS. 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 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

POWEll. SYSTEM ANALYSIS, OPR.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 ) (3.2) j o4) j .o

With no ground potential (i.e. with zero reference voltage), for a linear sy!lcm,

I , ... LYj.,JIi - LYqYJ (3.3) I_~ j_t j-' j-I

This equation, for a n-bus network, in mauix form can be represented as:

I , Y" ~, ~. V, I , Y" Y" r,. v, (H) -

'. Y., Y., y- V.

oc (3.5) [Y.,.,] is called Bus Admittance matrix and it has a well-defined structure. The clements of I Y,..]

arc importanl and hence defined below: Y Ii> the diagonal elemel1l, is called u /f admiftance or~ i, while Y r the off-diagonal e/emel1l,

is caUed mutual udmillance (or ,rullS/er admittance) between nodes i Mldj.

Obviously,

,,'"

fi, '" L>' i~ i' y~ = - Yi;

(3.6)

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 ~ n.

(ii) [r ...... ] is S)mmelrical. since JlV '"' Yft-

(iii) (n"n)-n n(n+1) Only 2 +n, i.e. 2 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 Ir mrH_ . J is the sum of the admillilllce of the branches wh ich are linked with corresponding i-th, j-th nodes including branches to ground, while each off-diagonal 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 obtaining [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;

'-" YBUS1 .. _ YBUS1 ... )")'>r,KT. YBUS"'. !>"T. - - t1' .,. . .... . rBUS., . w, a YBUS"" ,7.

Is iSNL?

; roc/ - 1. 2 ... Nl.

O+jO; fod - I.2 ... NB

No

,.

, ..

No

MJlerl 1:m1 dlreibs J1c>raJ

POWER NrnVORK MATRIX OPERATIONS

A "".:.: ....

.. ' ,

,

Sol bus

,

,

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 possible by invenioll of [r _ l by using special a lgori thms.

[z l=[Y~r'=

z" Z"

z" Z"

... Z" l~. (3.8)

Z_ In the {ZH .. ) matrix the diagonal elements we shorT circ;u/f driving poim impedances while the

off-diagonal elements are shan cirCl'ittrans/er impt'donces. [l/l>.,[ is symmetric provided [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 elements (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 adminance 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 + Yn; YU " Yl1 .. - Y, l Yll " 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 - j12) (-2+)6) (- 2+ j6) [Y"", ]= (- 2+)6) {4 -jI2) (- 2+)6) p.u.

(-2+)6) (-2+)6) (4-)12) 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. Xu. (G M)]

I ' I , 1.3, (0 . 05 , 0.15),(0,0), " , 2,3, (0.05,0 .15) , (0 ,0 )

Output 01 YBUS.FOR: YBUSO,DAT No. of buses

-3

Ybus mat.th: Ybu s ( 1. .}o'- )

-( 4. 000000, -12 . 000000 I Y"

Ybus ( 1. 2 I -

( - 2.000000, 6. 000000 I Y" Ybus ( 1 3 I

-( - 2.000000, 6.000000 I

Ybus ( '. 1 I - ( -2. 000000 , 6.000000 I Ybus( 2. 2 I

-( 4 .000000 , -12 . 000000 I

Ybus ( 2. 3 I -

( - 2.000000 , 6.000000 I Thus ( 3. 1 I

-( -2 .000000 , 6.00000 0 I

Ybu!I ( 3. 2 I -

( -2 .000000 , 6.00000 0 I Ybus ( 3 . 3 I

-( 4. 000000 , - 12. 000000 I 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. due 10 prcSC'nce of 4bw; systCtn. this bus admittance matrix will be a " 4 matrix.

,

- ~(!)~, _ =::J---=:J- 0 J ) .... 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, except bus no. 3, hence. YI~ " Y~\ .. 0; Y!-I - Y~2 " O.

Final Iy .... ,] matrix thus becomes 4 - jl2 - 2 + j6 - 2+ j6 0

[ ,-] . -2+)6 4 -)12 -2 + )6 0 -2+ )6 - 2+ )6 5- )15 -1+ )3 p.u.

0 0 - I +)3 1-)3 Execution of (Y IIt..>J software ~Iso yie lds the same result. It is Idl. for the reader as an exerc i$C. Eumple 3.3: The following dOlO rtfers 10 a $ixbu$ le"..Unl' pawt r ntlltwir.

Line no. From bus To blls R /p.u.) x (p.u.) 8/1 (p.u.) 1 1 2 0 .08 0.2 0.0!8 2 1 4 0 .05 0.25 0.0) ) 1 , 0 .1 0.25 0.03 4 2 ) 0.05 0.2 0.025 , 2 4 0.05 0.15 0.015 6 2 , 0 .15 0.2 0.02 7 2 6 0.09 0.25 0 .025 8 ) , 0.15 OJ 0.03 9 4 , 0.25 0.4 0 .035 10 , 6 O. 15 0.28 0.025

Jlen

POWER NETWORK MATRIX OPERATIONS

(u) Find [Ys...l (b) Alsofind [Y80,] when line 45 is tripped.

Solution: (u) Since this problem involves six buses, [YIJ .. i matrix will be 11 6 6 matrix. Result of

calculation 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 45 is tripped,!he system reduces to a 9line 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

-6

Ybus match: Ybus { 1 , 1 I

-I 3.872679, -11. 526170

Ybus{ 1 , 2 I -

I - 1. 724138 , 4. 3103 45 Ybu.s { 1 , 3 I

-I .000000, .000000

Ybus ( 1 , 4 I -

I - . 7692 3 1 , 3 . 84615 4 Ybu.s( 1. S I

-I -1 .379310, 3.448276

Ybus { 1 , 6 I -

I . 000000 , .000000 Ybus( 2, 1 I

-I -1 . 724138. " . 3103 45

Ybus( 2. 2 I -

I 8.575396. - 21. 654300 Ybus ( 2