modern-power-systems-analysis-d-p-kothari-i-j-nagrath-

I-,-r6rsHs 1 . A c t u a ll o a d ECONOMETRTC MODELS 2. 15 mts.predictionerror If theloadforecastsarerequiredfor planningpurposesi,t is necessartyo select the lead time to lie in thrro aJeulyears-In sucLcases- the load demandshould be decomposedin a manner that reflects the dependence 3 of the load on the various segmentsof the economy of the concernedregion. For 200 example, the total load demand y(ft) may be decomposedas EI M (16.2Oa) cl OI y(k)=D\",y,(k)+ e(k) EI 16= i:l oO Il 1 8 0 12.=1 where aiare the regressioncoefficients,y,(k) arethe choseneconomicvariables E 9l and e(k) representsthe error of modelling. A relatively simple procedure is to rewrite the model equation in the tamiliar vector notation o(D I o lL I cl y(k)=h'(k)x+ e(k) (16.20b) ol where h'(k)= [yr(ft)yz(k)... yu&)] and x = far a2... ayl. 4 E'1C i3 14 15 16 E The regressioncoefficients may then be estimated using the convenient least Fig. 16.3 squaresalgorithm. The load forecastsare then possible through the simple nfL relation 17 18 Comment i(k + 1)= i'(k) it(k +Ilk) \\ (16.21) Application of Kalman filtering and prediction techniquesis often hamperedby where i (k) is the estimate of the coefficient vector basedon the data available the non-availability of the required state variable model of the concernedload data' For the few casesdiscussedin this section,a part of the model has been till the ftth samplingpoint ana fr6 + Uk) is the one-step-aheadprediction of the obtainable from physical considerations.The part that has not been available include the state and output noise variancesand the data for the initiai state regressionvector h(k). estimate and the conesponding covariance. In a general load forecasting situation, none of the model parametersmay be available to start with and it 16.8 REACTTVE LOAD FORECAST would be necessaryto make use of systemidentification techniquesin order to obtain the required statemodel. It has been shown that the Gauss-Markov Reactive loads are not easy to forecast as compared to active loads, since model described by Eq. (16.18) in over-parameterisedfrom the model reactive loads are made up of not only reactivecomponentsof loads,but also identification point of view in thescnsethat the datafory,(k) dr: not permit the of transmissionand distribution networksand compensatingVAR devicessuch estimation of ali the parametersof this model. It has been shown in Ref. [12] as SVC, FACTs etc. Therefore, past datamay not yield the correct forecast as that a suitable model that is identifiable andis equivalentto the Gauss-Markov reactive load varies with variations in network configuration during varying model for state estimationpurposesis the innovation model useclfor estimation operating conditions. Use of active load forecast with power factor prediction of the stochasticcomponent. may result in somewhat satisfactory results.Of course, here also, only very on'line Techniques f,or Non-stationanl Load predietion recent past data (few minutes/hours) may be used, thus assuming steady-state Most practical load data behaveas non-stationaryand it is therefore imponant network configuration. Forecasted reactive loads are adapted with current to consider the questionof adaptingthe techniquescliscussecslo f,arto the non- reactive requirementsof network including var compensationdevices. Such stationarysituation.Ref. [2] hasdiscussedthe threemodels for this purposeviz. rf -or-ecas- tsare neeo| e|diaor secunty anaiysls, voitageireactivepower scheduling (i) ARIMA Models, (ii) Time varying model and (iii) Non-dynamic models. etc. If control action is insufficient, sffucturalmodifications have to be carried out, i.e., new generatingunits, new lines or new var compensatingdevices normally have to be installed.

mitul ModrrnPo*\"r Syrt\"r An\"turi. li':l;{lrJ. sututtilARy ** and integrationwith conventionatlime-seriesmethodsin orderto provide a Load forecastingis the basic stepin power systemplanning. A reasonablyself- morepreciseforecasting. containedaccountof the various techniquesfor the load prediction of a modern REVIEWQUESTIONS predictionproblems.Applications of time series,Gauss-Markovand innovation modelsin setting up a suitabledynamic model for the stochasticpart of the load L6.7 Which method of load forecasting would you suggestfor long term and datahave been discussed.The time seriesmodel identification problem hasalso why? been dealt with through the least squaresestimation techniquesdeveloped in Chapter 14. 16.2 Which methodof load forecastingwould you suggestfor very short term and why? In an interconnectedpower system,load forecastsare usually neededat all the important load buses.A great deal of attention has in recent years been 16.3 What purposedoes medium term forecastingserve? given to the question of setting up the demand models for the individual 16.4 How is the forecaster'sknowledgeand intuition consideredsuperiorto any appliances and their impact on the aggregated demand. It may often be necessaryto make useof non-linearforms of load modelsand the questionof load forecasting method? Should a forecaster intervene to modify a identification of the non-linear models of different forms is an important issue. forecast,when, why and how? 16.5 Why and what are the non-stationarycomponentsof load changesduring Finally, a point may be made that no particular method or approach will very short, short, medium and long terms? work for all utilities. All methods are strung on a common thread, and that is the judgement of the forecaster. In no way the material presented here is REFTRNTCES exhaustive. The intent has been to introduce some ideas currentlv used in forecastingsystemload requirements. Books Future Trends 1. NagrathI.J. andD.P. Kothari, Power SystemEngineering,Tata McGraw-Hill, New Delhi, 1994. Forecastingelectricity loads had reacheda comfortable stateof performancein the yearspreceding the recent waves of industry restructuring.As discussedin 2. Mahalanabis,A.K., D.P. Kothari and S.I. Ahson, ComputerAided Power System this chapter adaptive time-series techniques based on ARIMA, Kalman Analysis and Control, Tata McGraw-Hill, New Delhi, 1988. Filtering, or spectral methods are sufficiently accurate in the short term for operational purposes, achieving errors of l-2%o. However, the arrival of 3. Sullivan,R.L., Power SystemPlanning, McGraw-Hill Book Co., New York, 1977. competitive markets has been associated with the expectation of greater 4. Pabla,A.5., ElectricolPower SystemPs lanning,Macmillan lndia Ltd., New Delhi, consumerparticipation. Overall we can identify the following trends. 1998. (i) Forecast errors have significant implications for profits, market shares, 5. Pabla,4.5., ElectricPower Distribution,4th Edn, Tata McGraw-Hill, New Delhi. and ultimately shareholdervalue. 1997. (ii) Day ahead, weather-based,forecasting is becoming the rnost crucial 6. Wang, X and J.R. McDonald (Eds), Modern Power SystemPlanning, McGraw- activity in a deregulated market. Hill, Singapore,1994. (iii) Information is'becoming commercially sensitive and increasingly trade secret. Papers (iv) Distributed, embeddedand dispersedgenerationrhay increase. 7. Bunn, D.W., \"ForecastingLoads andPrieesin CompetitivePowerMark-ets\"P- rac- of the IEEE, Vol. 88, No. 2, Feb 2000, pp 163-169. A recentpaper [7] takesa selectivelook at someof the forecastingissueswhich are now associatecwlith decision-makingin a competitivemarket.Forecasting 8. Dash,P.K. et. al., \"Fuz,zyNeural Network and Fuzzy Expert Systemfor Load loads and prices in the wholesale markets are mutually intertwined activities. Forecasting\"P, roc. IEE, Yol. 143, No. l, 1996, 106-114. Models based on simulated artificial agents may eventually become as importanton supply side as artificial neural networks have already become 9. RamanathanR, . ef. a/., \"Short-run Forecastsof Electricity Loads and Peaks\". lnt J, ForecasringV, ol. 13, 1997,pp 161-174.

Mod\"rnpo*\", syrt\"r An\"ly.i. 77 @ 10. Mohammed,A. et. al., \"Short-term Load Demand Modelling and ForecastingA Review,\" IEEE Trans. SMC, Vol. SMC-12, No. 3, lggl, pp 370_3g2. ll. Tyoda, J. et. al., \"An Application of State Estimation to Short-term Load Forecasting\",IEEE Trans., Vol. pAS-89, 1970,pp l67g-16gg. 12. Mehra,R.K., \"On-line Identificationof to Kalmann Filtering\", IEEE Trans. Vol. AC-16, lg7l, pp lZ_21. T7.T INTRODUCTION Voltage control and stability problems are very much familiar to the electric utility industry but are now receiving specialattention by every power system analyst and researcher.With growing size alongwith economic and environ- mental pressures, the possible threat of voltage instability is becoming increasingly pronouncedin power system networks. In recent years, voltagi instability has been responsible for severalmajor network collap$s in New York, France,Florida, Belgium, Swedenand Japan [4, 5]. Researchworkers, R and D organizations and utilities throughout the world, are busy in understanding, analyzing and developing newer and newer strategiesto cope up with the menaceof voltage instability/collapse. Voltage stability* covers a wide range of phenomena.Because of this, voltage stability meansdifferent things to different engineers.Voltage stability is sometimesalso called load stability. The terms voltage instability and voltage collapse are often used interchangeably.The voltage instability is a dynamic process wherein contrast to rotor angle (synchronous) stability, voltage dynamics mainly involves loads and the means for voltage control. Voltage collapseis also definedas a processby which voltageinstabilityleadsro very low voltageprofile in a significant part of the system.Voltage instability limit is not directly correlated to the network maximum power transfer limit. A CIGRE Task Force [25] has proposedthe following definitionsfor voltage stability. Small-disturban-ce voltage stability A power systemat a given operating stateis small-disturbancevoltage stable if, following any small disturbance,voltagesnearloadsdo not changeor remain *The problemof voltagestability hasalreadybeenbriefly rackledin Ch. 13. Here it is againdiscussedin greaterdetailsby devotinga full chapter.

lEq!tl#* l,rc'tr close to the pre-disturbancevalues. The concept of small-disturbancevoltage consideredM. any of the indicesusedto assesvsoltagestabilityare relatedto stability is related to steady-statestabiiity and can be analysed using small- NR load flow study.Detailsof staticand dynamicvoltagestabilitywill be signal (linearised)model of the system. considerefdurtherin Section17.5. Voltage Stability Some Counter Measures to a certain disturbance,the voltagesnear loads approachthe post-disturbance n counter measuresto avord voltage r lty are: equilibrium values. (i) generator terminal voltage increase (only limited control possible) The concept of voltage stability is related to transient stability of a power system. The analysis of voltage stability normally requiressimulation of the (ii) increase of generator transformer tap system modelled by non-linear differential-algebraicequations. (iii) reactive power injection at appropriatelocations Voltage Collapse (iv) load-end OLTC blocking Following voltage instability, a power system undergoesvoltage collapse if the post-disturbanceequilibrium voltagesnear loads are below acceptablelimits. (v) strategicload shedding (on occurrenceof undervoltage) Voltage collapse may be total (blackout) or partial. Counter measures to prevent voltage collapse will be taken up in Voltage security is the ability of a system, not only to operatestably, but Section17.6. also to remain stable following credible contingenciesor load increases. T7,3 REACTIVE POWER FLOW AND VOLTAGE COLLAPSE Although voltage stability involves dynamics, power flow based static analysis methods often servethe purposeof quick and approximateanalysis. Certain situations in power system cause problems in reactive power flow which lead to systemvoltage collapse.Someof the situationsthat can occur are T7.2 COMPARISON OF ANGTE AND VOLTAGE STABILITY listed and explained below. The problern of rotor angle (synchronous)stability (coveredin Ch. 12) is well (1) Long Transmission Lines.' In power systems,long lines with voltage understood and documented t3l. However, with power system becoming uncontrolled busesat the receiving ends create major voltage problems overstressedon accourrt of economic and resource constraint on addition of during light load or heavy load conditions. generation, transfofiners,transmissionlines and allied equipment, the voltage instability has become a seriousproblem. Therefore, voltage stability studies (ii) Radial TransmissionLines: In a power system,most of the parallel EHV have attracted the attention of researchersand planners worldwide and is an networks are composedof radial transmissionlines. Any loss of an EHV active area of research. line in the network causesan enhancementin system reactance.Under certain conditions the increasein reactive power delivered by the line(s) Real power is related to rotor angle instability. Similarly reactive power is to the load for a given drop in voltage, is lessthan the increasein reactive central to voltage instability analyses.Deficit or excessreactivepower leads to power required by the load for the samevoltage drop. In such a casea voltage instability either locally or globally and any increasein loadings may small increasein load causesthe systemto reach a voltage unstable state. lead to voltage collapse. (iii) .Sftortageof Local Reactive Power: There may occur a disorganised Voltage Stability Studies combination of outage and maintenanceschedulethat may causelocalised reactivepower shortageleading to voltage control problems.Any attempt The voltage stability can be studied either on static (slow time frame) or to import reactive power through long EHV lines will not be successful. dynamic (over long time) considerations.Depending on the nature of distur- Under this condition, the bulk system can suffer a considerable voltage bance and system/subsystemdynamicsvoltage stability may be regardeda slow drop. or fast phenomenon. I7.4 MATHEMATICAL FORMULATION OF VOLTAGE Static Voltage Analysis crrrr,lltE l.? Trnl't littD/^t T Erilt Load flow analysis reveals as to how system equilibrium values (such as rit .l l.l.EDllrr I I r.ClLr.Cr!.GrlVl voltage and power flow) vary as various system parametersand controls are changed.Power flow is a staticanalysistool wherein dynamicsis not explicitly The slower forms of voltage instability are normally analysedas steady state problems using power flow sirnulation as the primary study method. \"Snap- shots\" in time following an outage or during load build up are simulated. Besides these post-disturbancepower flows, two other power flow based

tmweothmoedtshaoredosgftievenussteeadd;pyv-sctautrlevoeasdaanbdilviteylicmuirtvsewsh.(siceheaarlseoresleact.e1d3to.6v)oThhaegsse V I 's-ls{ stability. conventional analysis. load flow programs can be used for approxlimate 0.8pf -'<- Locus of V66 and Pr\", lag Noseof the curve P-V. curvesare useful for conceptualanalysis of voltage stability and for 0.9 pf lead The model that will be employedhere to judge voltage stability is basedon a single line performance. The voltage performun\"\" of this simile system is qualitatively similar to that of a practical system with many voltage sources, Ioads and the network of transmissionlines. Considerthe radial two bus systemof Fig. 17.1.This is the samediagrarn Fig.17.2 PV curvesfor variouspowerfactors as that of Fig. 5.26 exceptthat symbols are simplified. Here Eis 75 and yis vn and E and v are magnitudes with E leading v by d, Line angle\"p: tunli As in the case of single line systerrr,,r, a general power system, voltage XlR and lzl = X. instability occurs above certain bus loading and certain Q injections. This condition is indicated by the singularity of the Jacobianof Load Flow equations and level of voltage instability is assessedby the minimum singular value. Certainresults that are of significancefor voltage stability are as under. o Voltage stability limit is reached when (r7.2) Fig. 17.1 In termsof P and e,the systemloaclenclvoltage can be expressedas [l]. where S = complex power at load bus L, - z z xz- z : -- _i 'il rrz o x - E ) ' - 4 x 2 G 2 +-o1\\-l: e, ,7_ <. t\\) Yrt= load bus admittance ', V = load bus voltage It is seenfrom Eq. (17.1)that Vis a double-valuedfunction(i.e.ithas two Nearerthe magnitud'ae in Eq. (17.2) to unity, lesserthe stability margin. solutions) of P for a particular pf which determines e in terms of p. The pV o The loading limit of transmissionline can.be determinedfrom curves for various values of pf are plotted in Fig. 17.2. For each value of pf, lsl= v\"3lx\"; (r7.3) the higher voltage solution indicates stable zvoonl tea.g- feh ec acshea,nwgheiol ev etrhoec cluorws eart X\".i is the critical systemreactancebeyond which voltage stability is lost. It can voltage lies in the unstable voltage operation points for variJus pfs is drawn be expressedas v\".t (critical) and Pro*. The locus of v.r,-p^u* in dotted line in the figure. Any attemptto iniiease the load abov\" causes Fz (r7.4) a reversal of voltage and load. Reducing voltage causesan increa\"s-i*ng current X \" n = *2 P ( - t a n Q + s e c Q ) to be drawn by the load. In turn the larger reactive line drop causesthe voltage We have so far consideredhow the PV characteristicswith constant load to dip further. This being unstableoperationcausesthe systemto suffer voltage power factor affect the voltage stability of a system. A more meaningful collapse.This is also brought out by the fact that in upperpart of the curve charrcteristic for certain aspectsof voltage stability is the QV characteristic, ff. 0 a n d i^nrr flrl rrav lr^v.w' ,wor- ^p^a*n L /..^.,+^Ll^ -p-d, -tL- \\) dP lOad means which brings out the sensitivity and variation of bus voltage with respectto dV \\u'Dr.rurc g,,V, > U (feduclng reactivepower injections (+ve or -ve). reducing voltage and vice-versa).It may be noted here that the t-ytpire\" of load Consideronce again the simple radial systemof Fig. 17.1.For p flow it is assumedin Fig' 17.2 is constantimpedance.In practical syste-. type of voltage sufficiently accurateto assumeX > R i.e. 0 = 90\".It then follows that loads aremixed or predomirrantlyconstantpower type such that system o = EY.o,a-11 (17.s) degraclationisinore and voltagelnstability occursmuch prior to the theoretical power limit. XX

or a - EV cos6+ QX = 0 (r7.6) Using ihe decoupiing principle 1.\". dP = 0, we set dV V\" +=.\"'r[#.+] Taking derivativewrt V gives d Q = E c o s6 - 2 V (17.7) dVX The QV characteristic on normalized basrs(etf**. VIE) for various values of P/P^ are plotted in Fig. 17.3.The system is voltage stablein the region o r I s c = c odsl o O + 1 1 where dQldv is positive, while = 0 which may also be termed the voltage stability limit is reachedat d,eldV LdV X J as the critical operating point. or Ers6= E cos5l !9 + 2Y] o LdV X J ,D max Voltagestabilityis achievedwhen E cos, (# . > Ers, (shortcircuitMVA of powersource) +) Pr\", isthemaxlmum 1.0 powertransferat upf (17.10) P I P ^ \" , =9 . 5 0.75 (,l.U. \\ -d z cntenon Unstable. . operation dV voltage instability occurs when the system Z is such that -d= o o ov r - = uM (r7.1r) dZ dV VIE Application of this criterion gives value of z\";. \\ for 0 0.2 0.6 0.8 1.0 (iii) Ratio of source to load reactance is very important dnd voltage Fig- 17.3 QV characteristicfsor the systemof Fig. 17.1tor stability variousvaluesof plp^^r. Xsource (17.r2) a o2 The limiting value of the reactive power transfer at the limiting stageof voltage stability is given by X load a indicatesthe off-nominal tap ratio of the OLTC transformerat the load end. Q,n=^ t-\"o'26 (17.8) T7.5 VOLTAGE STABILITY ANALYSIS The inferencesdrawn from the simple radial system qualitatively apply to a The voltage stability analysis for a given system state involves examining practicalsize system.Other factorsthat contributeto systemvoltagecollapse following two aspects. are:strengthof transmissionsystem,power transferlevels, load chaiacteristics, generatorreactivepower limits and characteristicsof reactive power compen- (i) Proximity to voltage instabitity: Distance to instability may be measured satingdevices. in terms of physical quantities,such as load level, rea power flow through a critical interface, and reactive power reserye.posiible contingencies Other Criteria of Voltage Stability suchas a line outage,loss of a generatingunit or.a reactiu. po*\"rio*.. mustbe given dueconsideration. (i) 34 ..it.rion: (E'=generatovr oltage; V=load voltage). Using this crite- (ii) iuiechanismof voitage instabiiity: How and why does voltage instabitity take place? What are the main factors leading to instability?'ulhat voltage-weakareas?What arethe most effectiveways to improv\" are the rion, the voltage stability limit is reachedwhen stability? \"of,ug, c ods{ # . # } + s i nd f f i- } = o (r7.e)

.ffiiffi| . ModernPowerSystemAnalysis The static analysis techniquespermit examination of a wide range of system Static AnalYsis conditions and can descriUethe nature of the problem and give the main The staticapproachcaptures.snapshotsof systemconditionsat variousdme contributing factors. Dynamic analysis is useful for detailed study of specific frames along the time-domain trajectory. At eachof thesetime frames' X in voltage coliapsesituations,coordination of protectionand controls, and testing of remedial rneasures.Dynamic simulations further tell us whether and how the ii h?lrJdil. frameT. hus,theoverausystemequationrsedPceto purely steady-stateequilibrium point wr Modelling Requirements of various Power system components \"acrmine'ul?fogprov'doeJreamrb\"stle6)aradat;tiafioisnecfnneiaqfllifeflturyiec;osa;trmei;atsgditoula.hon\"raeasrasvyid.ulesMlbotueeubwlmtseihrei-enovwsndg.iS'dshdpeb.eetapv.cse.iisYearsedltptdit:oesfe.nccdTh,tVtihnev'1QeietqsetuaiestreneTmsdmnluavsreslin:sttiinhio:ebvgoiidytddsy-ec:st:asno*gutt:iimc1icfvyha,ep:naa:utasrs\"eletttysaai'isngsb.ivvsoeillhfniPptapvyava-oqrbltneunebldenraevtt(tieeooaQndrl Loads p r o b l e m st 1 3 - 1 5 1 . Load modelling is very critical in voltage stability analysis. Detailed Proximity to InstabilitY subiransmissionsystemrepresentationin a voltage-weakarea may be required' pitlo;r^oc\"x;orimuniviitoeynrgteo.Rsmefiansl.ls-td1tei6sp-tsu1ur8bn1atdinl,ictsheceuvssoysltssatpgeeemciibnaesltcteaocbmhilenitsyiquunisesstdafeobtrleedromertienthreemdliobnayindignfcltohrweeapsfoainiinlgst This may includeiransformer ULTC action, reactivepower compensation'and of voltage collapse and proximity to volage instability' voltage regulators The Continuation Power-flow Analysis It is essentialt.o consider the voltage and frequency dependenceof loads' Induction motors should also be modelled' Generators and their excitation controls It is necessary to consider the droop chatacteristics of the AVR, load compensation,SVSs (static var system), AGC, protection and controls should also be modelledappropriatelyl4l. Dynamic AnalYsis TcrcoseoohovmnneeluvdraJtceiiiatononicomtnoiwnooebsenfsialatlnlh-nlocetimoasoaadnarpd-dtrfr-toilihftlolbioeowlwbenesmepcatdraolbogbmayboitlleriretiesatymhsfllo*ilfn,iropmmgoruriiusntol.alsutahriybtTaihlnhetaguetvhlpoteehpaccevoedoonlirnonlt*tvagiadnedgucr-egaoflletosoninto*wdac\"nibtteeiipoplqirpntouoysowba.lrilettmeTiioromh-intfni.slssossaAwoaot!floltophaathweanreertasastlPythutisn-ehltVigye's 'lhe generalstructurc of the systcm moclel for voltage stability analysisis similar to that for transientstability analysis.overall system equationsmay be cxprefihcdas *=f(X,n (r7.r3) and a set of algebraicequations (r7.r4) *fi\"t:lltinuation-method of power-flow analysisis ,obrrrt and flexible and I (X, V) = YxV with convergencedifficulties' However' suming.Hence the better approachis to with a setof known initial conditions (Xo' Ve)' flow irethod (NR/FDLF) and continua- where X = systemstatevector :ase,LF is solved using a conventional nnHssef.oNreroasrfumtecarc,lelytsh,setihveceolcyniontnicntriuenaautsaioitnniognlmomaeedthtlheoodvdeliisss Y = but voltage vector I exactly at and past the critical point' 1= currentinjeition vector Ilv = rletwork node admittance matrix' Voltage StabilitY with HVDC Links for extremelylong distance -tl:1r:t.:d Equations(17.13)and(17'14)can be s High voltagedirect current* (HVDC).litt oi tt e numerical integration methods analysisnrethoclcslescribeidn Ch' 6' T ffansmissiclnanclftrrasynchronousinterconnections.AnHVDClinkcanbe ,ninutes. As the special models repl ieading to voltage collapsehave bee *For dctailcd accountof HVDC, the reader maY refer to [3]' differential equationsis considerably models.Stiffnessis alsocalled synchrc

f00 | ModernpowerSystemAnatysis ' VoltaseStability I __{-tr*g either a back-to-back rectifier/inverter link or can include long distance dc monitoringand analysisto identifypotentialvoltagestabilityproblems transmission. Multi-terfninal HVDC links are also feasible. and appropriateremedialmeasureas re extremelyhelpful. The technology has come to such a level that HVDC terminals can be T7.7 STATE-OF-THE.ART, FUTURE TRENDS AND connected even at voltage-weakpoints in power systems.HVDC CHALLENGES present unfavourable \"load\" characterisficsfo fhe nnrr/rr links mav converter consumesreactivepower equal to 50-60vo of the dc power. The presentday transmissionnetworks are getting more and more stresseddue HVDC-related voltagecontrol (voltage stability and fundamentalfrequency temporary over voltages)may be studied using a transient stability program. to economic and environmental constraints.The trend is to operatethe existing Transient stability is ofteninterrelatedwith voltage stability. Ref. .onJia.r, networksoptimally close to their loadability limit. This consequentlymeansthat this problem in greaterdetail. t2i the system operationis also near voltage stability limit (nosepoint) and there is increasedpossibility of voltage instability and even collapse. 17.6 PREVENTION OF VOLTAGE COLLAPSE Off-line and on-line techniquesof determiningstate of voltage stability and when it entersthe unstablestate,provide the tools for systemplanning and real (i) Application of reactivepower-compensatingdevices. time control. Energy managementsystem(EMS) provide a variety of measured Adequate stability margins should be ensured by proper selection of and computer processeddata. This is helpful to system operatorsin taking compensationschemesin terms of their size,ratingr-undlocations. (ii) control of network voltage and generator reactive output critical decisions inter alia reactive power managementand control. In this Several utilities in rhe world such as EDF (France), ENEL (Italy) are developing specialschemesfor control of network voltages and reactive regardautornationand specializedsoftwarerelieve the operatorof good part of power. the burden of system managementbut it does add to the complexity of the systemoperation. Voltage stability analysis and techniques have been pushed forward by several researchersand several of these are in commercial use as outlined in (iii) Coordination of protections/controls this chapter.As it is still hot topic, considerableresearcheffort is beingdevoted Adequatecoordinationshould be ensuredbetweenequipmentprotections/ to it. , controls basedon dynamic simulation studies.Tripping of equipment to avoid an overloadedcondition should be the last alternative. Controlled Pw et al. l8l consideredan exponentialtype voltage dependentload model system s€parationand adaptive or irrtelligent control could also be used. and a new index calledcondition numberfor staticvoltagestabilityprediction. (iv) Control of transfurmer tap chan.gers Eigenvalue analyseshas been used to find critical group of busesresponsible T'apchangerscanbe controlled, either locally or centrally, so as to reduce for voltage collapse. Some researchers[26] have also investigatedaspectsof the risk of voltagecollapse. Microprocessor-basedOLTC controls offer bifurcations(local, Hopf, global) and chaosand their implicationson power almost unlimitedflexibility for implementingULTC control strategiesso as to take advantageof the load characteristics. systemvoltage stability. FACTS devicescan be effectively usedfor controlling (v) Under voltage load shedding the occurrenceof dynamic bifurcations and chaos by proper choice of error signaland controllergains. Tokyo Electric Power Co. has developed a pP-based controller for coordinated control of capacitor bank switching and network transformertap For unplanned or extreme situations, it may be necessary to use ctranging.HVDC power control is usedto improve stability. undervoltageload-sheddingschemes.This is similar to under ir\"q1r\"rr.y load shedding,which is a common practiceto deal with extremesituations- More systematicapproach is still required for optimal siting and sizing of resulting from generationdeficiency. FACTS devices.The availability of FACTS controllersallow operationclose to the thermal limit of the lines without jeopardizing security. The reactivepower Strategicload sheddingprovides cheapestway of preventing widespread compensationclose to the load centresand at the critical busesis essentialfor overcoming voltage instability. Better and probabilistic load modelling [11] V O l t a S g C O_-l_l a_n_sr_e L n a r l s h e r l d i n o uc vnrl r, avr-rorovJ oorl rr\\n_,r,rrl,lr\\.t Lu^g ur ^g^s:rB-l^rt^r)u s^u^ -- -- should be tried. It will be worthwhile developingtechniquesand models for as t() differentiate berween faults, transient voltage dips unJ lo* voltage conditions leadingto voltage collapse. studyof non-lineardynamicsof large sizesystemsT. his may requireexploring (vi) Operators' role new methods to obtain network equivalentssuitable for the voltage stability Operatorsmust be able to recognise voltage stabiiity-related symptoms analysis. AI is another approach to centralized reactive power and voltage and take requiredremedialactions to prevenrvoltage collapse.On-line control. An expert system [9] could assistoperatorsin applying C-banksso that

a00.l,*l ModernPo I .. t AQ = reactivepower variation (i.e. the size of the compensator) generatorsoperate near upf. The design of suitable protective measuresin the Srr.= system short circuit capacity eventof voltage instability is necessary. Then AV = trg So far, computed PV curvesarethe most widely usedmethod of estimating d,. voltage security, providing MW margin type indices. Post-disturbanceMW or AQ= AVSsk MVAr margins should be translated to predisturbanceoperating limits that =1(0.05x5000) operators can monitor. Both control centre and power.plant operators should be = + 250MVAR The capacityof the staticvAR compensatoirs +250MVAR. trainedin the basicsof voltage stability. For operatortraining simulator [10] a REFERNECES real-timedynamic model of the power systemthat interfaceswith EMS controls Books suchas AGC is of great help. 1. Chakrabarti,A., D.P. Kothari and A.K. Mukhopadhyay,Perfurmance,Operation Voltage stability is likely to challengeutility plannersand operatorsfor the and Control of EHV Power Transmission Systerts,Wheeler Publishing, New Delhi, 1995. foreseable future. As load grows and as new transmission and load area 2. Taylor,c.w., Power system voltage stability,McGraw-Hill, New york, 1994. generationbecome increasingly difficult to build, rnore and more utilities will 3. Nagrath, I.J. and D.P. Kothari, Power SystemEngineering, Tata McGraw-Hill, facethe voltage stability'challenge.Fortunately,many creativeresearchersand New Delhi, 1994. 4. Kunclur,P., Powcr Sy,stemStubilityatul Control,McGraw-Hill, New york, 1994. plannersare working on new analysismethods and an innovative solutionsto 5. Padiyar,K.R., Power System Dynamics: Stability and Control, John Wilev. the voltage stability challenge. Singapore,1996. 6. Cutsem,T Van and CostasVournas, Voltage Stabitityof Electric power Systems, A load bus is composedof induction motor where the nominal reactive power is I pu. The shunt compensationis K,n. Find the reactivepower sensitivityat Kluwer Int. Series,1998. the bus wrt changein voltage. Solution Papers Qrcot= Qno V2 [given] ' 7. Concordia,C (Ed.), \"special Issue on Voltage Stability and Collapse,', Int. J. of Electrical Power and Energy systems,voi. i5, no. 4, August 1993. Qcomp=- Krt V2 [-ve signdenotesinductive reactivepowerinjection.l 8 . Pai, M.A. and M.G.O, Grady,\"VoltageCollpaseAnalysis with ReactiveGeneration and voltage Dependentcons,traints\", J. of Elect Machines and power systems, .'. Here, Qn\"t= Qnua t Qcomp V o l . 1 7 ,N o . 6 , 1 9 8 9 ,p p 3 7 9 - 3 9 0 . Qn r= v' - Krn vz lQoo^ = 1.0givenl 9. cIGRE/ Task Forcc 38-06-0l,\"Expcrt SystemsApplied to voltage and var C o n t r o l / / ,1 9 9 1 . dQn\"t= 2v-2v Ki.'.,, dv 1 0 . \"operator Training simulator\", EpRI Final Report EL-7244, May 1991,prepared by EMPROS SystemsInternational. Sensitivity increasesor decreaseswith Krn as well as the magnitude of the voltage.Say at V - 1.0 pu, Krn = 0.8 l l . Xu, W and Y Mansour,\"Voltage StabilityusingGenericDynamicLoad Models\", IEEE Trans. on Power Systems,Vol 9, No l, Feb 1994,pp 479493. dQn t - 2 - 1 . 6 = 0 . 4 p u . dv Example:,17.2 Find the capacity of a static VAR compensatorto be installed at a bus with x 5Vovoltage fluctuation.The shortcircuit capacityis 5000 MVA. Solution For the switching of static shunt compensator,

ModernPo AppnNDrx A t In this appendix, our aim is to presentdefinitions and elementaryoperationsof IZ. VerTnat,l.lt., L.IJ. Arya ano u.r. I\\.oma.n,. .'t Y? oil- tage Jd - laDt ! rl ! t- rtyDF nLn- a- n- - , sE- - - - ^ ulenurL - - y vectors and matrices necessaryfor power system analysis. ReactivePower Loss Minimization\", JIE (I), Vol. 76, May 1995, pp.4449. VECTORS 13. IEEE, special Publication90 TH 0358-2 PWR, \"Voltage Stability of Power A vectorx is definedasan orderedsetof numbers(realor complex)i,.e. Systems:Concepts,Analytical Tools, and Industry Experience\" 1990. 14. FlataboN, ., R. Ogncdaland T. Carlsen,\"VoltagcStabilityCondition,in a Power , (A-1) Tiansmisiion System calculated by Sensilivity Methods\", IEEE Tians. VoI. xp ...t xn areknown asthe componentsof the vectorr. Thus the vector x is PWRS-5,No. 5, Nov 1990,pp 12-86-93. a n-dimensionalcolumn vector.Sometimestransposedform is found to be more 1 5 . Gao,8., G.K. Morison and P. Kundur,\"Voltagc StabilityEvaluationUsing Modal convenient and is written as the row vector. Analysis\", IEEE Trans. Vol. PWRS-7, No. 4, Nov. 1992, pp 1529-1542. T6, Cutsem, T. Van, \"A Method to Compute Rbactive Power Margins wrt Voltage rT A fx1, x2,..., xrf (A-2) Collapse\",IEEE Trans.,Vol. PWRS-6,No. 2, Feb 1991, pp 145-156. 1 7 . Ajjarapu, V. and C. Christy, \"The continuationPower Flow: A Tool for Steady Some Special Vectors StateVoltageStabilityAnalysis\",IEEE PICA Conf.Proc., May 1991,pp 304-311. 1 8 .Ldf, A-P, T. Sined, G. Andersonand D.J. Hill, \"Fast Calculationof a Voltage The null vector 0 is one whose each componentis zero, i.e. Stability Index\", IEEE Trans., Vol. PWRS-7, No. 1, Feb 1992,pp 54-64. 19. Arya, L.D., S.C. Chaubeand D.P. Itothari, \"Line Outage Ranking based on EstimatedLower Bound on Minimum Eigen Value of Load Flow Jacobian\", JIE (1),Vol. 79, Dec 1998,pp 126-129. '20. Bijwe, P.R,, S.M. Kelapure,D,P. Kothari and K.K. Saxena,\"OscillatoryStability Limit Enhancemenbt y Adaptive Control Rescheduling\",Int J. of Electrical Power and Energy SystemsV, ol. 21, No. 7, 1999,pp 507-514. 2t, Arya, L.D., S,C, Chaubeand D.P. Kothari, \"Linc Switchingfor Alleviating Overloadsunder Line OutageCondition taking Bus Voltage Limits into Account\", Int J. of Electric Power and Energy System,YoL 22, No. 3, 2000, pp 213-221. . ) Bijwc, P.R., D.P. Kothari and S. Kclapure,\"An Efficient Approach to Voltage Sccurity Analysis and Enhancement\",Int J. of EP and,ES., Vol. 22, No. 7, Oct. 2000, pp 483486. 2 3 . Arya, L.D., S.C., Chaubeand D.P. Kothari, \"Reactive Power Optimization using Static Stability Index (VSD\", Int J. of Electric Power Componentsand Systems, Vol. 29, No. 7, July 2001,pp 615-628. 24. Arya, L.D., S.C. Chaubeand D.P. Kothari, \"Line Outage Ranking for Voitage Limit Violations witti Conective ReschedulingAvoiding Masking\", Int. J. of EP and ES,Vol. 23, No. 8, Nov. 2001, pp 837-846. 25. CIGRE Task Foice 38-02-10, \"Cigre Technical Brochure: Modelling of Voltage CollapseIncludihgDynamicPhenomena\"E, lecta, No. 147,April 1993,pp' 7l-77. 2 6 . Mark J. Laufenbergand M.A. Pai, \"Hopf bifurcation control in power systemwith static var compensators\",Electrical Power and Energy Systems,Vol. 19, No. 5, 1997,pp 339-347. The sum vector i has each of its componentsequaito unity,i.e.

- . flTY' t I A The unit 'tector e 0x.=0 the.rest of the componentbare zero,i.e. The multiplication of two vectors x and y of samedimensions results in a very important product known as inner or j9g!g! pfodqcij.e. 0 *tv AD\",y,Aytx (A_3) 0 i:l (A-4) A €k: Also, it is interesting to note that xTx= lx 12 1 kth component 0 cos d' 4 l\"xtlYl y l , (A-5) 0 wtere Q is angle between vectors, lxl and lyl are the geometric lengths of Some Fundamental Vector Operations vectorsx and y, respectively.Two non-zero vectors are said to be orthJgonral, Two vectors x and y areknown as equalif, and only if, .yk= !*for k = r,2, ..., n. Thenwe sav if x=y *ty= o (A-6) The product of a vector by a scalar is carried out by multiplying each component of the vector by that scalar,i.e. MATRICES Definitions '' Matrix '\\n m x rnea(ol nt umm' bne) rmsca, otrmixpisleaxnnuomrdbeerersd,fruenctcatniognusloarraorrpaeyroaftoerlseT.mheenmtsawthriixch maybe (A-7) If a vector y is to be addedto or subtractedfrom anothervectorx of the same is a rectangular array of mn elements. dimension, then eachcomponentof the resulting vector will consistof addition or subtraction of the correspondingcomponentsof the vectors x and vr,i.e. 7.t.hoc,ot!l\"unmonte. sThtheem(ai,tir)itxh element,i.e. the elementlocatedin the ith row and the mxn. A hasm rows anon coiumnsand is said to be of order T h e f n l l n u r i nYcv r r r 6 ynr vr nPnvAr !raf ivoDo 4 r v 4 y^ I*-^, u U^<- -r+U:I^t^rlL^Ul ^u r c v erL^: t 0 rr La-r g e -D- r' a :' - When m = rt, i.e. the number of rows is equalto that of columns, thematrix x*!=y+x is said to be a square matrix of order n. x+(v+'z)=(x+y)+z An m x 1 matrix, i.e. a matrix having only one column is called a column ar @zx)- (afiz)x vector. An I x n matix, i.e. a matrix having only one row is called a (ar+ e)x- dF+ a2x vector. rore *Sometimes . ,1r\\!r inner product is also representedby the following alterr ative forms x . y, (x, y),(x,y). 's.t

frffi-H ModernPoweSr vstemAnalvsis T Diagronal matrix (A-e) A diagonal matrix is a squarematrix whoseelements off the main diagonal are a l l z e r o s( a i j = 0 f o r i + j ) . det(A)= tAt= 2' 11 3, 2l l -r -t 1l-l I 2l , l-1 3l + 4l*| 1 2l =2(8)+('6)+(-5)=5 (A-10) NUII matrix Transpose of a matrix If all the elements of the square matrix are zero, the matrix is a nuII or zero 'l.h-.etransposeof matrixA denoted by At is the matrix formed by interchanglng matrix. the rows and columns of A. Note that (Ar)r = A Symmetrtc matrix (A-8) A squarematrix is symmetric, if it is equal to its transpose,i.e. AT=A Unit (identity). matrix Notice that the matrix A of Eq. (A-9) is a symmeffic matrix. A unit matrix / is a diagonal matrix with all diagonal elementsequal to unity. If a unit matrix is multiplied by a constant(),), the resulting matrix is a diagonal Minor matrix with all diagonal elementsequal to 2. This matrix is known as a scalar maffix. The mino, Mij of an n x n rnatrix is the determinant of (n - l) x (n - 1) matrix formed by deleting the ith row and the 7th column of the n x n matrix. Cofactor The cofactor AU of element a,, of the matrix A is defined as aU = Gl)'*t M,i Adjoint matrix ll:Lll;l The adjoint matrix of a squarematrix A is found by replacing each element au of matrix A by its cofactor A,, and then transposing. =3x3scalarmatrix For example, if A is given by Eq. (A-9), then l3 l-t lz Ita o J A = l-1 l1 L- ^J l2 -l =4x4unitmatrix lr l2 l-1 l2 Determinant of a matrix I3 l-1 For each square matrix, there exists a determinant which is formed by taking the determinantof the elementsof the matrix.

MotJellr Powor S AppendixA I ,6tl I- _;l6 - \\ l 15 -1-l (A-11) C=A+B=l' I 7 -5 )l L2 2) Addition andsubtractionaredefinedonly for matricesof the sameorder. The fbllowinglawsholdlbr addition: (+ fhe cammatatLveln+y:A + B -B + ,{ : A squarematrix is called singular, if its associateddeterminant is zero, and (ii) The associativelaw: A +-(B + C) = (A + B) + C non-singular,if its associateddeterminantis non-zero. Further ELEMENTARY MATR.IX OPERATIONS (A tB)r= Ar + Br Eguality of matrices Matrix Multiplication Two matrices A(m x n) and B(m x n) are said to be equal, if the only if The product of two matrices A x B is defined if A has the samenumber of ai= bij fori = 7,2, ..., ffi, .i = 1,2, ..., fl columns as the number of rows in B. The natrices are then said to be Then we write conform.able.Ifamatrix A is of order mx n and B is an n x q matrix, the product C = AB will be an m x q matrix. The elementc,,of theproduct is given A=B bv Multiplication of a matrix by a scalar A matrix is multipliedby a scalar a if all the mn elementsare multiplied by cii = r ,l , a. i.e. )_rai*0*i (A-13) ( A -1 2 ) k:l Thus the elements cu are obtained by multiplying the elementsof the ith row ofA with the correspondingelementsof theTth column of B and then summing theseelementproductp. For example Addition (or suhtraction) of matrices 1\"\" arzllbn b,rl_['' r cnl Lau azz)Lbn brrJ Lr^ czrl where To add(or subtract)two matrices of the sameolder (samenumber of rows, and ctt= attbt,+ arrb^ samenumber of columns),simply add (or subtract)the correspondingelements ctz= anbn + arrb2 nf the frxrn rnefr;cce i e rrrhen frxrn rnqfri/-ec A enA Il nf lhe cqrnc o,rAcr qrc added,a new matrix C results such that czl= uzlbt,+ arrb21 C-A+ B; czz= azt bt z+ cr r b2 whose rythelernentequals If the product AB is defined, the product BA may or may not be defined. Even if BA is defined, the resulting products of AB andB, A arenot, in general, cij= aij * bij equal. Thus, it is important to note that in generalmatrix multiplicationis not c o m m u t a t i v ei,. e . l- - - -,-E- _x- _a_m- i pr 1l eI AB+BA r I l:r The associariveand distributive larr-shold for matrix mulriplicarion (wtren the 'aryprrtixiateerpernirnys.aredef;tv.4)- re. I Asscrica'rv &rrt': 4{B [ = A(BC) = -tBC IA A' -=-iil- ?z O-r f) -11 Distibutive law:. A(B + C) - AB + AC -rl\"-Lo 3.1 then,

tffiWl ModerPno I,j].qffil*l I l.,i95I3n or n D c i ii = 1 , 2 , . . . ,m \"*t= i:l [1 -1 3] Using the rules of matrix multiplication defined above. Eqs (A-14) canle written in the compact notation as Lo rl z Ax=c (A-1s) where Find AB and BA. A and B areconformable(A has two columns and B has two rows), thus we have '4\"8 -=|l -L11 -1 ; 39l|l t B\" ^A =-rl-l-,-1 40, 't 1 ) 2; 4 l ; A matrix remainsunaffected,if a null matrix, defined by Eq. (A-8) is added to it, i.e. A+0=A If a null matrix is multiplied to another matrix A, the result is a null matrix It is clear that the vector:mntrix Eq. (A-15) is a useful shorthand A0= 0A = 0 representationof the set of linear algebraic equations(A-14). Also Matrix Inversion A-A=0 Note that equationAB = 0 does not mean that either A or B necessarilyhas Division does not exist as such in matrix algebra. However, if A is a square non-singularmatrix, its inverse (A-l) is defined by the relation to lre a null matrix, e.g. ,l[ 3 ol=fo 0l A_IA_AA_I_I (A-16) [r 0 JL - l 0l Lo 0J The conventional method for obtaining an inverse is to use the following relation L0 Multiplication of any matrix by a unit matrix resultsin the original maffix, A-; _ adjA det A l.e. (A-17) AI-IA=A The transposeof the product of two matrices is the product of their It is easyto prove that the inverseis unique transposesin reverse order, i.e. The following are the important properties charactenzing the inverse: (AB)-r = 3-t4-l 6Dr = BrAr (A-tf = (Ar)-r (A-18) The conceptof matrix multiplication assistsin the solution of simultaneous linear algebraicequations.Consider such a Setof equations 14-r1-t= 4 atft + anxz+ ... + abJn= ct aLlxt+ azzxz+... + azrtr= c2 : ... + A-rJn = C^ (A-14) I f A is given by Eq. (A-9), then from Eqs. (A-10), (A-1I), (A-17), we get Q^lXl * C^*z+

ffd-ffi.:| ModernpowerSystemRnatysis A--+1:if : i _;l:fi:,i:;_il It may be noted that ihe derivative of a scaiar function with respect to a vector deAt t 5j L-; -; l.J of dimension n is a vector of the samedimension. L-r -s The derivative of a vector function (A-22) with respectto a vector variable r is defined as SCALAR AI\\rD VE CTOFFUNCTI O]NIS 0*, 0*, o*n ?fz A scalarfunctionof n scalarvariablesis definedas o f a }fz 0fz (A-zs) y ! f\\r, x2, ..., xn) (A-1e) 0x 0*, 0*, 0rn It can be written as a scalarfunction of a vector variable x, i.e. (A-20) af- af; .. af; y - f(x) where x is an n-dimensionvector, 0*, 0*, o*n (A-26) In general,a scalarfunction could be a function of several vector variables,e.g. Considernow a scalarfunctiondefinedas t - ff@, u, p) y-f(x,u,p) (A-2r) where x, u andp arevectorsof various dimensions. (A-27) 6-2g) A vector function is defined as = 21fi(x, u, p) + Lzfz(*,u, p) + ... + 2*f*(4, u, ) Let us fincf j{ ^ . Accordingro Eq. (A-24),we can write a In general, a vector function is a function of severalvector variables, e.g. f (x, u, p) (A-2e) y-f(x,u,p) (A-23) LJm\\rrUrP)J DERTVATNTES OF SCALAR AND \\ZECTOR FUNCTIONS Let us now find -d. s A^ccoi rd. ingto Eq. (A-24), we can write ox A derivafit'e of a scalar funcritrn 1A-lt)) s,ith re-sper.rtr) u \\.eL-R)rr.ariable -r is defind as -af 0r, (A-24) la\"l af LarJ, At 0x 0*, oJ orn -

fijfilftf.f ModernPower'systemAnatysts I }ft 0f, af^ AppnNDrx B 0*, )r 0*, 0*, af^ Ofi }fz 0*, ^2 0r, 0r, VJI UJ2 o*n 0*n 0r, (A-30) REFERNCEES We can represent,as we saw in Chapter5, a three-phasetransmissionline* by a circuit with two input terminals (sending-end,where power enters) and two l, Shiplcy, R,8,, Intoduction to Matric:r:sand Power Sy,rtemsW, iley, New York, output terminals (receiving-end,where power exits). This two-terminal pair r976. circuit is passive (since it does not contain any electric energy sources),linear. (impedancesof its elements are independentof the amount of current flowing 2. Hadley, G., Linear Algebra, Addison-WesleyPub.Co. Inc., Reading,Mass., 1961. through them), and bilateral (impedances being independent of direction of 3. Bellman, R., Introduction to Matrix Analysis, McGraw-Hill Book Co.. New York, current flowing). It can be shown that sucha two-terminal pair network can be representedby an equivalent T- or zr-network. 1960. Considerthe unsymmetrical T-network of Fig. B-1, which is equivalenrto the general two-terminal pair network. s Z1 22 lp Y Vp Vg Flg. B-1 UnsymmetricTa-lcircuietquivalentot a generatlwo-terminal nair \"n- e- -t' w- \" _ork r-\" For Fig. B-1, the following circuit equationscanbe written (B-1) 1s- I^+ Y(Vo+I^Zr) or Is= YV*+(l+ YZr)I* * : V*+ I^Zr+ IrZ, .+ V^ + I^2, + ZrWn+ I^2,,+ I^YZ.Z, iqY *A transformer is similarly represented by a circuit with two input and two ouput terminals.

tlCtf I Modernporelsyslern inal,sis or V5= (1 + YZr) V* + (2, + Z\" + yZrZr)Io (B-2) (B-3) Equations(B-1) and(B-2)canbe simplifiedby letting A-l + YZ, B= 2.,+Zr+ yZrZ, C=Y D=l+yZ, using these,Eqs. (B-1) and (B-2) can be written in matrix form as Fig. B-3 Unsymmetricazlr-circuit Llyrrl,.=JLlAt Bllv*1 ,t-0, A seriesimpedanceoften representsshorttransmissionlines and transform- ers. The ABCD constants for such a circuit (as shown in Fig. B-4) can olj^l immediately be determinedby inspection of Eqs. (B-1) and (B-2), as follows: This equationis the sameas Eq. (5.1) and is valid for any linear, passiveand A=1 bilateral two-terminal pair network. The constantsA, B, C andD arecalled the generalizedcircuit constantsor the ABCD constantsof the network, and they B=Z (B-7) can be calculated for any such two-terminal pair network. C=0 It may be noted that ABCD constantsof a two-terminal pair network are complex numbers in general,and always satisfy the following relationship AD-BC=1 (B-s) D=l Also, for any symmetrical network the constantsA andD, areequal. From Eq. (B-4) it is clear thatA andD are dimensionless,while B has the dimensionsof impedance(ohms) and c has the dimensionsof admittance(mhos). The ABCD constantsare extensivelyused in power system analysis.A generaltwo-terminal pair network is often representedas in Fig. B_2. Fig. B-4 Seriesimpedance Another simple circuit of Fig. B-5 consistingof simple shunt admittancecan be shown to possessthe following ABCD constants A=7 Fig. B-2 of a two-rerminaprair networkusing B=0 /El_a\\ :#:::representation C=Y \\s-u,, ABCD CONSTANTS FOR VARIOUS SIMPLE NETWORKS D=I We havealreadyobtained theABCD constants.ofan unsymmetricalT-network. The ABCD constantsof unsymmetricala-network shown in obtainedin a similar manner and are given below: Fi-g. B-3 may be A=l+ YrZ B = Z, Fig. B-5 Shuntadmittance r C=Yr+Yr+ZYryz (B-6) It may be noted that whenever ABCD constantsare computed, it should be checked that the relation AD-BC = 1 is satisfied.For examole, using Eq. D=I+ YrZ (B-8) we get AD-BC=1x1-0xY=1

:fflfuii'l Modernpower:Systemnnatysis ,, Append8B t, tr= (ArBr+ ArBr)/(8+, Br) II ABCDconstantsof a circuitaregiven,its equivalentI- or a-circuitcan B = B r B z l ( B t +B r ) l,;621+ be determinedby solvingEq. (B-3) or (8-6) respectivelyf,or the valuesof C= (Cr* Cr)+ (Ar- Ar) (Dz- D)l(Br * Ur) I seriesandshuntbranchesF. or the equivalentn-circuitof Fig. B-3, we liuu. (B-13) (B-e) ,r=# ABCD CONSTANTS OF NETWORKS IN SERIES AND FANEr,r,Ur, Whenever a power system consists of series and parallel combinations of networks, whoseABCD constantsare known, the overall ABCD constantsfor the system may be determined to analyze the overall clperationof the system. Fig. B-7 Networksin parallel Measurement of ABCD Gonstants Fig. 8-6 Networksin series The generuIized circuit constants rnay be computed for a transmission line which is being designedfrom a knowledge of the systemimpedance/admittance Considerthe two networks in series,as shown in Fig. 8-6. This combination parameters using expressionssuch as those clevelopedabove. If\\the line is already built, the generalizedcircuit constantscan be measuredby making a can be reducedto a single equivalent network as follows: few ordinary testson the line. Using Eq. (B-4), theseconstantscan easily be shown to be ratiosof eithervoltage or currentat the sending-endto voltageor For the first neL[twuroJ,r,kl-_. w1|eo.h,,al'eo\",,)'ll[tu, current ai the receiving-endof the network with the receiving-end openor short- circuited. When the network is a transformer, generator, or circuit having ,l ( B -1 0 ) lumped parameters,voltage and current measurementsat both ends of the line ) can be made, and the phase angles between the sending and receiving-end quantities can be found out. Thus rhe ABCD constantscan be determineC. For the secondnetwork, we can write It is possible, also, to measurethe rnagnitudesof the required voltages and I y*I I A, Brll vo] (B-11) currents sinrultaneouslyat both ends of a transmissionline, but there is no I t=t D r|J|l r *|l simple method to find the difference in phaseanglebetweenthe quantitiesat the two ends of the line. Phasedifference is necessarybecausetheABCD constants lrr) l_c, are complex. By measuringtwo impedancesat eachend of a transmissionline, however',the generalizedcircuit constantscan be computed. From Eqs.(B-10)and (B-11), we can write The following impedancesare to be measured: [u'l_- lo' u'11ou, 'l[u^] L+I 1., n,ll.c, or)j^l = lf A , A , + B t C . , A t B z+ q D 2 l [ y * I 7 -- rovor- r' lu. il-r-S-vr:\\or - . l iu* u^ oP' lv^ -\\^r(^rrrlvs .w. ' ir+lLrl -r^9^L^ 9: . r, :v- l-ttE-trlru^ - l u^ -P^ -(,il-ull^':u- ^u-lteo. : - - r lcrAz + D1C2 CrB' -t DrD, Jlt o l \"so If two networksare connectedin parallel as shown in Fig. B-7, the ABCD constantsof thecombinednetworkca-nbe found out similarly with somesimple Zss= sending-endimpedancewith receiving-endshortcircuited manipulations of matrix algebra.The results are presentedbelow: zno = receiving-endimpedancewith sending-endcpen-circuiteci Zns = receiving-endimpedancewith sending-endshort-circuited The impedancesmeasuredfrom the sending:endcan be determinedin terms of the ABCD constantsas follows:

From Eq. @- ), with 1R= 0, ( B -1 4 ) Z s o =V s l l s =A l C AppENDIX C andwith VR= 0, When the impedancesare measuredfrom the receiving-end,the direction of current flow is reversedand hencethe signs of all current terms in Eq. (5.25). We can thereforerewrite this equation as Vn= DVs + BIt (B-16) In= CVs+ AIt From Eq. (8-16), with Is = 0, ( B -1 7 ) Z n o = V R | I R =D / C and when Vs = 0, Zns=VRllR= BIA ( B -1 8 ) We know that the nodal maffix Yu.r, and its associatedJacobian are very Solving Eqs.(B-14),(B-15), (B-17) and(B-18) we canobtainthevaluesof 'ssppaarrssiety,wohf ethreeassetmhaetirricinevsemrsaey matrices are full. For large power systems the the ABCD constantsin terms of the measuredimpedancesas follows. be as high as 98Voand must be exploited. Apart AD-BC 1 [usingEq. (B-5)] fiom reducing storageand time of computation, sparsity utilization limits the AC AC round-off computational errors. In fact, straight-foru'ard application cf the iterative proceduretor systemstudieslike load flow is not possiblefor large IC1 systemsunless the sparsity of the Jacobianis dealt with effectivet). AC, A A2 o=(ffi)''' ( B -1 e ) GAUSS ELIMINATION One of the recent techniquesof solving a set of linear algebraic equations, By substitutingthis value of A in Eqs. (B-14), (B-18) and substituting the called triangular factorizatiott, replaces the use of matrix inverse which is value of C so obtainedin Eq. (B-17), we get highly inefficient for large sparsesystems.In triangularization the elements of .u.h ,o* below the main diagonal are madezero and the diagonal element of ( z \\- ' 1 l / 2 each row is normalized as soon as the processingof that row is completed- It S,l o - z )I , is possible to proceedcolumnwise but it is computationally inefficient and is Dr >=-L. 7R z * n r (B-20) theref,re not used.After triangularizationthe solution is easily obtained by hack sub.stitutionT. he techniqueis illustratedin the examplebelow' (B-21) C _ (Zss(Zno- Zo))''' (B-22) p = Z^o (zso(z^o- zo))t'' Consider the linear vector-maffix equation t;ll' L3 REFERNECE ,1l'l[1't ,I ,=I l . Cotton,H. and H, Barber,The Transmi,ssionand Distribution of Electrical Energy, 3rd edn., B.I. Publishers,New Delhi. 1970.

Procedure 1. Divide row 1 by the self-elementof the row, in this case2. Consider the following systemof linear equations: (Z)xt + Q)x2 + (3)x, = $ 2. Eliminate the element (2, 1) by multiplying the modified row 1, by element (2, I) and subtract it from row 2. (2)xr+(3)xz+(4)xt=) 3' Divide the modified row 2 by its self-element (f); and stop. (c-2) Following this procedure,we get the upper triangular equation as ( 3 ) x r+ ( 4 ) x r + ( 7 ) 4 = 1 4 l[ r z+ li [l ' 't -=tlt +o _l + l For computer solution, maximum efficiency is attained when elimination is l o ? : r l l _l - l , r carried out by rows rather than the more tamiliar column order. The successive L t )Lxzl Lt J reduced setsof equationsare as follows: Upon back substituting,that is first solvingfor x2 andthen for 11,we get (l)xr+ (t)*z+ (*)rt = * x 2 = -o- 7-:+- - 1 (Z)xt+ (3)x2+ (4)xt = ) (c-3) 2 j (3)xr+ (4)xz+ (7)x3= 14 - - I _ I Y - I _ 11_3\\- 5 ( 1 ) . r+, ( i l r r + ( | ) x t - z (2)xz+ (1)x, = l Check (c-4) (3)xr+(4)x2+(7)4=14 b r + x z =2 ( * ) - t r = t ( 1 ) x , +( t r ) r r +( * ) \" , - 3 3x,+ 5rz-3(+)- s(f) = o Thus,we havedemonstratetdheuseof thebasicGausseliminationandback ( 1 ) x 2(+* ) r , - + (c-s) substitutionprocedurefor a simplesystemb, ut the sameprocedureappliesto any generalsystemof linearalgebraicequationsi,.e. (3)xr+(4)x2+Q)xt=14 Ax=b (c-1) ( l ) x r+ ( t ) r r + ( J ) x , = 3 (l)x2+(*)\"r' = +, i An added advantageof row processing(elimination of row elementsbelow (c-6) the main diagonal and normalization of the self-element) is that it is easily (t),r+(*)r,-s amenableto the useof low storagecompactstorageschemes-avoiding storage of zero elements. GAUSS ELIMINATION USING TABLE OF FACTORS (1)x,+ (t)rr+ (*)\", = J Where repeatedsolutionof vector-matrixPq. (C-1j with constant Abut varying (t)xr+(t)*t - + (c_'7\\ values of vector D is required, it is computationally advantageousto split the (*)rt = + matrix A into triangular f'actor (termed as 'Table of factors' or 'LU decomposition') using the Gauss elimination technique. If the matrix A is (1)x+, (tr)*r+(*)r, _ J sparse,so is the tableof factorswhich can be compactly storedtherebynot only reducing core storagerequirements,but also the computational effort. Gauss (1)x,+ (I)*t 3 (c-8) elimination using the table of factors is illustrated in the following example. 2 x3= I These steps are referred to as 'elimination' operations. The solution r may be immediately determinedby 'back substitution' operationusing Eq. (C-8).

Thesolutionfor a new setof valuesfor b canbe easilyobtainedby using lt= lt- fn)z a tableof ftahcettoarbspleroefpfaarcetdboyrsaFcaasrebfeulleoxwafmorinthaetieoxnoaf meqpsle.i1nch-la) ntod.(c-g). canwrite we - 5- ( ; X ; ) : i ft, ftz fn rl3 (c-e) and for heading3, 3 fu fzz fzt ').ra lt = fszlz L =(+x+=)I Ln Tl T l In fact, operation(C-10) representsrow normalization and (C-11) represents elimination and back substitution procedures. 32 Optimal Ordering The row elements of F below the diagonal are the multipliers of the In power systemstudies,the matrix A is quite sparseso that the number of non- normalizedrows requiredfor the elimination of the row element, e.g.fzz = *, zero operationsand non-zero storagerequired in Gauss elimination is very the multiplier of normalized row 2 l&q. (c-6)l to eliminate the element(3,2), sensitive to the sequencein which the rows are processed.The row sequence that leadsto the leastnumber of non-zerooperationsis not, in general,the same i'e- (])x2. The diagonal elementsof F are the mulripliers neededto normalize as the one which yields least storage requirement. It is believed that the absoluteoptimum sequenceof orderingthe rows of a large network matrix (this the rows after the row elimination has beencompleted,e.g..fzz= ! , thefactor is equivalent to renumbering of buses)is too complicated and time consuming by which tow 2 of Eq. (C-4) must be multiplied to normalize the row. The to be of any practical value. Therefore,some simple yet effective schemeshave elementsof F above the diagonal can be immecliately written down by been evolved to achieve near optimal ordering with respectto both the criteria. inspection of Eq. (C-8). These are neededfor the back substitution process. Some of the schemesof near optimal ordering the sparsematrices,which are fully symmetrical or at least symmetric in the pattern of non-zero off-diagonal In rapidly solvingEq. (C-1) by useof the tableof factorsF, succesiivesteps terms, are describedbelow [4]. appearas columns (left to right) in Table C.l below: Scheme I Table C.l Number the matrix rows in the order of the fewest non-zerotermsin each row. h l, I 2,1 2,2 3,I 3,2 a a 2,3 1,3 1,2 x If more than one unnumbered row has the same number of non-zero terms. number thesein anv order. 6 3 3 3 3 33 a 5/2 I J Scheme 2 9 9 3 3t2 3/2 3t2 3t) 3I I Number the rows in the order of the fewest non-zerotermsin a row at each step of elimination. This schemerequires updating the count of non-zeroterms after t4 T4 14 t4 5 5t4 1 1I 1 each step. The headingrow (i, j) of Table C-1 representsthe successiveelimination and Scheme 3 back substitutionsteps.Thus, Number the rows in order of the fewestnon-zero off-diagonalterms generated 7,7 representsnormalization of row 1 in the renrainingrows at eachstepof elimination.This schemealsoinvolves an updatingprocedure. 2,L representselimination of element(2, l) I he chotceot schemets a trade-otf betweenspeedof executionand the 2,2 representsnonnalization of row 2 number of times the result is to be used.For Newton's method of load flow solution, scheme2 seemsto be the best. The efficiency of scheme3 is not 3, L; 3,2 representelimination of elements(3, 1) an,J(3,2) respectively sufficiently establishedto offset the increasedtime required for its execution. 3. 3 representsnormalizationof row 3 2,3 representselimination of element(2,3) by back substitution r, 3; 7, 2 reprelentelimination of elements(1, 3) and (r, z) respectivelyby back substitution. The solution vector at any stageof developmentis denotedby Ut lz y3Jr= y The modification of solution vector fiom column to column (left to right) is carried out for the heading (i, j) as per the operationsdefined below: !i= fiiji ttj=i (c-10) !i= !r- fi/i lt i + i (c-11) Thus for heading3, z

Mffi# Modernpo t- Scheme I is useful for problems requiring only a single solution with nq iteration. Compact Storage Schemes AppBNDrx I) The usefulnessof the Newton's method depends largely upon conserving computer storage ucmg the nu oI non-zerocomputatlons.'l'o ettect theseideason the computer, elimination of lower triangle elements is carried out a row at a time using the concept of compact working row. The non-zero modified upper triangle elements and mismatches are stored in a compact and convenientway. Back substitution progressesbackwardsthrough the compact upper triangle table. A properly programmed compact storageschemeresultsin considerablesaving of computer time during matrix operations. Naturally, there are as many compact working rows and upper triangle storageschemesas there are programmers.One possible schemefor a general matrix stores the non-zero elements of successiverows in a linear array. The column location of thesenon-zero elementsand the location where the next row Expressiontso be usedin evaluatingtheelementsof theJacobianmatrix of a powersystemarederivedbelow: starts (row index) is stored separately.The details of this and various other FromEq. (6.25b) schemesare given in [2]. REFERNECES P,- jiQ,=t', fr*rr (D-l) k:r l . Singh,L.P., Advanced Power SystemAnalysis and Dynamics, 2nd edn., Wiley Eastern,New Delhi, 1986. A\\ Agarwal, S.K., 'Optimal Power Flow Studies', Ph.D. Thesi.r,B.I.T.S.r Pilani, = lvil exp(- i6,)L lY,/ exp (i?il lVll exp (7dn) r970. k:r 3 . Tinney, W.F. and J.W. Walker, \"Direct Solutionsof SparseNetwork Equationsby Differentiating partially with respect to 6* (m * i) Optintally OrderedTriangular Factorizations\",Proc. IEEE, Nov. 1967,.55: 1801. +0-6^ i-+a6^ =Tvil exp(-r4) (Yi^l expQ0,^)tv^texp(j5^)) 4. Tinney, W.F. and C.E. Hart, \"Power Flow Solution by Newton's Method\", IEEE Trans.,Nov. 1967,No. ll, PAS-86: 1449. -- j(ei - jf) (a^ + jb^) (D-2) Y,^= G,* + jB,^ where Vi= €i+ jfi ' (a^ * jb*) = (c* * jBi) @^+ jf^) Although the polar form of the NR method is beingused,rectangular complexarithmeticis employedfor numericael valuationasit is faster. FromEq. (D-2), we canwrite #=(aJi-b^e)=Hi^ - ( a ^ e i +b , f , ) =J i ^ #=

tvrOtlcItI rOWgf lr' q-\"\" For the case of m = i. we have I f.lr.$ *0 6 , -i* = - jtvitexp(- j6,)i ty,nei xp(j4) tV1e,txpQfi) = l%l exp(- j6,)ilY,pl exp(i0*) lvplexp(7d') \" k:l 06, k:l + jlV,l exp (- j6 ) (tytil expQ?,,)tV;et xp (id)) + lV,lzlY,,el xp(7d,,) = - j(Pi - jQ) + jlV,l'(G,, + jB,,) (D-3) = \\Ft- lQ) + it* (D-s) From Eq. (D.3), we can write It followsfromEq.(D-5)that = - Qli- Biliv;12= H\" oPi- \\v'l = Pt * GiilViP = Nii # ' av,l ao. = Pi- G ' i t v i l 2= J' , e t _ lvil = Qi - Biilvil, = L t avil ;t Now differentiate Eq. (D-1) partially with respectto lV^l (m r i). We have The aboveresults are'summattzedbelow: a4 \" 0Q, = lv'il exLp (- j6) (lYi*l exp(j0,^) expQd^)) Case 7 m* i -- alv^l alv) Multiplyingby lV^l on bothsides, Hi^= Li*= aJ,- b*e, Ni*=- Ji*= d*€i+ b\"ft (D-6) aaPt tvI^ t v AO. f f i w ^ t Yi*= Gt^ + jBt* ^t-i - lvil exp(- j6) lY,^l exp(j1i)'tV*l exp(i6^) Vi= ei+ jfi (D-7) = (ei- jf,) (a^ + jb*) (D-8) It follows from Eq. (D-4) that (D-4) (a* * jb^) = (Gi^ + jBi^) @* + jk) Case 2 m= i !a! rl vr *^tl= a ^ e , +b , f i = N i ^ H,=- Qi- Biilvilz Nii= P,+ G,,lV,lz a-!l,v?.:,ltv^l= a,,f-i b^e=, L,^ Jii - Pi- Giilvr Now for the caseof m = i. we have Lii= Qr- Biilvilz aPt - \"j-7Pl .v-i l ex, .pe. jilf tyaet xp(j0*)tvetexp(jQ,) REFERNCEES alvil o:, Tinney,W.F. and C.E. Hart \"PowerFlow Solutionby Newton'sMethod\", IEEE Trans.,Nov 1967,No. 11, PAS-86:1449. + lV,lexp (- j6) lyi expQ0,,)exp (/4) Van Ness,J.E., \"Iteration Methodsfor Digital Load Flow Studies\", Trans.AIEE, Multiplyingby lV,l on bothsides Aug 1959,78A: 583. a- -nt l v', l - \"i A\" \" o, ' l v , l a l v i l awil

AppBNDrx E If z, violares a limit, it can either be upper or lower limit and not both simultaneously.Thus, either inequality constraint (E-3) or (E-4) is active at a time, that is, either ei.^^ of oi.minexists, but never both. Equation (E-5) can be written as AL 0x Ax oL= 0f +' \\raqu)l' \\*o-0 (E-e) du 0u In Eq. (E-9), di= Qi.^u* tf Ui- ui,*o ) 0 ai= - Ai,-io if ,r, ,t, - ui ) 0 The Kuhn-Tucker theorem makes it possible to solve the general non-linear aL,. (E-10) #=g(x,u,p)=o OA programming problem with several variables wherein the variables are also It is evident that a computed from Eq. (E-9) at any feasible solution, with constrained to satisfy certain equality and inequality constraints. ) from Eq. (E-8) is identical with negativegradient, i.e. We can state the minimization problem with inequality constraints for the control variables as d = - = negativeof gradientwith respectto u (E-11) K nln / (x' u) (E-l) At the optimum, a must also satisfy the exclusion equations (E-7), which subject to equality constraints state that ,\\ g(x,u,P)=o (E-2) di= 0 ff ,i, ,rt, < ui < ui, ,n\"* and to the inequality constraints di= di, -* 2 0 lf u, = ui, ,iru* u-u^u1 0 (E-3) Qi=- Q,rnin S 0 lf ui= 4i, *in u^1-- u < 0 (E-4) which can be rewritten in ternts of the gradient using Eq. (E-11) as follows: The Kuhn-Tuckertheorem[] givesthenecessarcyonditionsfor theminimum, o x= o if ur, ,n;o< ui < ui, ^ assumingconvexityfor the functions(E-1)-(E-4),as 0u, A.C = 0 (gradientwith respect to u, x, )) (E-5) where .C is the Lagrangian formed as of,.o tf u,= ui,^* (E-12) -C=f(x, u)+ )Tg(x,u,p)+ of*u*{, -u^u*)+ oT*,n1r-in-z) 0u, and (E-6) of,ro tf u, - ili, ^in out (E-7) Equations (E-7) are known as exclusion equations. REFERNECE The multipliers a,rr* and @,rri,?, re the dual variables associated with the 1 . Kuhn, H.W. and A.W. Tucker, \"Nonlinear Programming\", Proceedings of the upper and lower limits on control variables. They are auxiliary variables similar Second.BerkeleySymposium on Mathematical Statisticsand Probability, Univer- to the Lagrangian multipliers ) for the equality constraintscase. sity of California Press,Berkeley, 1951.

AppnNDrx F In developedcountriesthe focus is shifting in the power sectorfrom the creation Fig. F.l Realtime monitoringand controllingof an electricpowersystem. of additional capacity to better capacity utilization through more effective SCADA refers to a system that enablesan electricity utitity ro remotely management and efficient technology. This applies equally to developing monitor, coordinate,control and operatetransmissionand distribution compo- countrieswhere this focus will resultin reduction in needfor capacityaddition. nents,equipmentand devicesin a real-time mode from a remote location with acquisitionof datafor analysisand planningfrom one con[ol location.Thus, Immediate and near future priorities now are better plant management, the purpose of SCADA is to allow operatorsto observeand control the power higher availability, improved load management,reduced transmissionlosses, system. The specific tasks of SCADA are: revamps of distribution system, improved billing and collection, energy efficiency, energy audit and energy management.All this would enable an o Data acquisition,which providesmeasurementasnd statusinformationto electric power systemto generate,transmit and distribute electric energy at the operators. lowest possibleeconomicand ecologicalcost. . Trending plots and measurementson selectedtime scales. These objectives can only be met by use of information technology (IT) . Supervisorycontrol, which enablesoperatorsto remotely co:rtrcl devices enabled services in power systems management and control. Emphasis is such as circuit breakersand relays. therefore, being laid on computer control and information transmission and exchange. Capabilityof SCADA systemis to allow operatorsro control circuit breakers ancldisconnectswitchesand changetransformertapsand phase-shiftepr osition The operations involved in power systemsrequire geographically dispersed remotely. It also allows operatorsto monitor the generationand high-voltage and functionally complex monitoring and control system. The monitory and transmission systemsand to take action to ccrrect overloads or out-of-limit supervisory control that is constantlydeveloping and undergoing improvement voltages. It monitors all status points such as switchgear position (open or closed), substationloads and voltages, capacitorbanks, tie-line flows and in its control capabilityis schematicallypresentedin Fig. F.1 which is easily interchange schedules.It detects through telemetry the failures and errors in seento be distributed in nature. bilateral communication links between the digital computer and the remote equipment.The most critical functions, mentionedabove,are scannedevery few Starting from the top, control systemfunctions seconds.Other noncritical operations, such as the recording of the load, foiecastingof load, unit start-upsand shut-downsare carried out on an hourlv EMS Energy Management System- It exercisesoverall control over the basis. total system. Most low-priority programs (thoserun lessfrequently) may be executedon scADA Supervisory control and Data Acquisition system - It covers demand by the operatorfor study purposesor to initialize thepower system.An generationancitransmissionsystem. operator may also change the digital computercode in the execution if a parameterchangesin the system. For example,the MWmin capability of a DAC Distribution Automation and Control System - It overseesthe generatingunit may changeif one of its throttlevaluesis temporarilyremoved distribution systemincludingconnectedloads. for maintenance,so the unit's share of regulating power must accordingly be decreasedby the code.The computer softwarecompilers and datahandlersare Automation, monitoring andreal-time control have alwaysbeen a designedto be versatile and readily acceptoperatorinputs. part of scADA system. with enhancedemphasis on IT in power systems,scADA has beenreceiving a lot of attention lately.

DAC is a lower level versionof SCADA applicablein distributionsystem I (including loads), which of course draws power from the transmission/ An energycontrolcentremanagesthesetasksandprovidesoptimaloperation subtransmissionlevels. Obviously then there is no clear cut demarcation of the system.A typicalcontrolcentrecanperformthe following functions: betweenDAC and SCADA. (i) Short,mediumandlong-termloadforecastin(gLF) (ii) Systemplanning(SP) In a distribution network, computerisationcan help manage load, maintain quality,detectheftandtamperingandthusreducesystemlossesC. ornputeri- (iii) Unit commitmen(tUC) andmaintenancsecheduling(MS) sation alsohelps in centralisationof datacollection. At a central load dispatch 'centre, data such as culTent, voltage, power factor and breaker status are (v) State estimation(SE) telemeteredand displayed.This gives the operatoran overall view of the entire (vi) Economic dispatch(ED) distribution network. This enableshim to have effective control on the entire (vii) Load frequencycontrol (LFC) network and issue instructions for optimising flow in the event of feeder overload or voltage deviation. This is carried out through switching inlout of The above monitoring and control functions are performed in the hierarchical shunt capacitors,synchronouscondensersand load management.This would order classified accordingto time scales.The functions perfotmed in the control help in achievingbetter voltage profile, loss reduction, improved reliability, centre are based on the availability of a large information base and require quick detectionof fault and restorationof service. extensive software for data acquisition and processing. At a systems level, SCADA can provide status and measurementsfor At the generation level, the philosophy of 'distributed conffol' has distribution feedersat the substation.Dictribution automationequipmentcan dramatically reducedthe cabling cost within a plant and has the potential of replacing traditional control rooms with distributedCRT/keyboard stations. monitor selectionalisingdeviceslike switches,intemrpters and fuses.It can also Data acquisition systems provide a supporting role to the application operate switches for circuit reconfuration, control voltage, read customers' software in a control centre. The data acquisition system(DAS) collects raw meters, implement time-of-day pricing and switch customer equipment to data from selectedpoints in the power system and converts these data into manage load. This equipment significantly improves the functionality of engineering units. The data are checked for limit violations and statuschanges distribution control centres. and are sent to the data base for processing by the application software. The SCADA can be used extensively fbr compilation of extensive data and real-time data baseprovidesstructuredinformation so that applicationprograms managementof distribution systems.Pilferagepoints too can be zeroedin on, needing the information have direct and efficient accessto it. , as the flow of power can be closely scrutinised.Here again, trippings due to The Man-Machine interface provides a link between the operator and the human efforscan be avoided.Modern meteringsystemsusing electronicmeters, software/lrardwareused to control/monitor the power system. The interface automatic meter readers(AMRs), remote meteripg and spot billing can go a generally is a colour graphic display system.The control processorsinterface long way in helping electric utility. Thesesystbmscan bring in additional with the control interfaceof the display system.The DAS and Man-Machine revenuesand also reducethe time lag betweenbilling and collection. interface support the following functions: Distribution automationthrough SCADA systemsdirectly leadsto increased (i) Load/GenerationDispatching reliability of power for consumersand lower operatingcostsfor the utility. It (ii) Display and CRT control resultsin fbrecastingaccurarteclenrandand supply management,tasterrestora- (i i i ) Dat t BaseM aint cnancc tion of power in case of a tailure and ahernative routing of power in an (iv) AlarrnHandling emergency, (v) Supervisorycontrol A k t ' yl' c i l tu rrclll ' th c s t$r y s tc l l l si s l h c ru r l ol ccl ontnrll i rci l i tyl l rl t l l l ow s l nstcr (vi I l) r ogr ar nnr inlbgnct ions executionof decisions.Manual errors and oversightsare eliminated.Besideson (vii) DaLalogging line andrcal-timeinlbnnution, the systenrprovidesperiodicreportsthathelp in (viii) Eventlogging the analysisof performanceof the power system. Distributi'onautomation (ix) Real-timeNetwork Analysis combinesdistributionnetwork monitoring functionswith geographicalmapping, With the introduction of higher size generating units, the monitoring fr^i.l-uf er! ll u^ ^L- r ;. al ^ L- ^ tuil, a- - il lu su un, r1.() t1mprove a v a i l at D| t !t ', . trt . ' aISo lntegrates load rrvryt zurr rt vr irruavrrnr raon f < ' rhroq rvrwa frnia rurph ui -r y^ \\or.yrvrvor- *plr^4^r r+L.D (^rlr-D^( J . I- ^-l^- a^:-.--^-.^ rL- -r--^ 6vuw Ill uluttt tu lrrrpl'uvc ule Ixant lt y management,load despatchand intelligent metering. performance, now all the utilities have installed DAS in their generating units Data Acquisition Systems and Man-Machine Interface of sizes 200 MW and above.The DAS in a thermal power station collects the following- inputs from various locations in the plant and converts thern into The use of computers nowadays encompassesall phases of power system engineering'units. operation:planning, forecasting,scheduling,security assessmenta, nd control.

l,@E+l ModernPo@is I | -^^ Analog Inputs a aDJt, \" (i) Pressuresf,lows, electrical parameters,etc. power system engineerswho are adopting the low-cost and relatively powerful (ii) Analog input of 0-10 V DC computing devicesin implementing their distributedDAS andcontrol systems. (iii) Thermocoupleinputs Computer control brings in powerful algorithms with the following advan- (iv) RTD input paurly ururzauon rn generailon, (u) savings in energy Digital Inpurs: and so in raw materialsdue to increasedoperationalefficiency, (iii) flexibiliiy and modifiability, (iv) reduction in human drudgery, (v) improved operator (i) Contractoutputs effectiveness. (ii) Valve position, pressureand limit switches Intelligent databaseprocessorswill becomemore cornmonin power systems All these processinputs are brought from the field through cables to the since the search, retrieval and updating activity can be speededup. New terminals. The computer processesthe information and ,uppli\", to the Man- functional concepts from the field of Artificial Intelligence (AI) will be Machine interfaceto perform the following functions. integrated with power system monitoring, automatic restoration of power networks, and real-time control. (i) Display on CRT screen (ii) Graphic disptay of plant sub-systems Personalcomputers'(PCs)are being usedin a wide rangeof power system (iii) Datalogging operationsincludingpower stationcontrol, loaclmanagementS, CAOe systems, protection, operator training, maintenance functions, administrative data (iv) Alarm generation processing,generatorexcitation control and control of distribution networks. IT enabledsystemsthus not only monitor and control the grid, but also improve (v) Eventlogging operational efficiencies and play a key part in maintaining the security of the power system. (vi) Trending of analoguevariables (vii) Performancecalculation (viii) Generationof control signals Some of the above functions are briefly discussedas follows. RFFERNECES I. Power Line Maga7ine,yol.7, No. l, October2002,pp 65_71. 2. A.K' MahalanabisD, .P. Kothari and S.I. Ahson, ComputerAided power System Analysis and Control, TMH, New Delhi, 199g. 3. IEEE Tutorial course, Fundamentals of supervisory control system, l9gr. 4. IEEE Tutorial course, Energy control centre Design, 19g3. The DAS softwarecontainsprogramsto calculateperiodically the efficiency of various equipmentlike boiler, turbine, generator, condenser,fans, heaters, etc.

F;;ff$: AppBNDrx G You canstartMATLAB by doubleclicking on MATLAB icon on your Desktop of your computeror by clicking on StartMenu followed by 'programs' and MATLAB has been developed by MathWorks Inc. It is a powerful software ciicking appropriateprogram group such as 'MATLAB Release12,. you then package used for high performance scientific numerical computation, data will analysis and visualization.MATLAB standsfor MATrix LABoratory. The visualizea screenshown in Fis. G.1. combination of analysis capabilities, flexibility, reliability and powerful graphics makes MATLAB the main software package for power system Click on this to change engineers.This is becauseunlike other programming languageswhere you have the cunent directory to declarernatricesand operateon them with their indices, MATLAB provides matrix as one of the basic elements.It provides basic operations,as we will see Commandprompt later,like addition, subtraction,multiplication by use of simple mathematical operators. Also, we need not declare the type and size of any variable in Simulink browser advance.It is dynamically decideddependingon what value we assignto it. But MATLAB is casesensitiveand so we have to be careful about the caseof Flg.G.l variables while using them in our prograrns. The command prompt (characterisedby the symbol >>)'is the one after MATLAB gives an interactiveenvironmentwith hundredsof reliableand which you type the commands.The main menu containsthe submenussuch as accuratebuilrin functions.Thesefunctions help in providing the solutions to a Eile, Edit, Help, etc. If you want to start a new program file in MATLAB variety of mathematical problems including matrix algebra, linear systems, (denotedby .m extension)one can click on File followed by new and select the differentialequationso, ptimization,non-linearsystemsand manyother typesof desiredM'file. This will open up a MATLAB File Editor/Debugger window scientific and technicalcomputations.The most important featureof MATLAB where you can enter your program and save it for later use.You can run this is its programming capability, which supportsboth types of programming- program by typing its name in front of command prompt. Let us now learn some object oriented and structuredprogrammingand is very easyto learn and use b a s i cc o m m a n d s . and allows user developedfunctions. It facilitates accessto FORTRAN and C codes by means of externalinterfaces.There are severaloptional toolboxes for Matrix Initialization simulating specializeepl roblems of rJifferentareasanrder-tensionsto link up NIATLAB and other programs. SIMULINK is a program build on top of A matrix can be initialized by typing its name followed by = sign and an MATLAB environment,which along with its specializedproducts,enhancesthe opening squarebracket after which the user suppliesthe valuesand closes the power of MATLAB for scientific simulationsand visualizations. square brackets. Each element is separatedfrom the other by one or more spacesor tabs.Each row of matrix is separatedfrom the other by pressing Enter For a detailed description of commands,capabilities, MATLAB functions key at the end of each row or by giving semicolon at its and many other useful features, the reader is referred to MATLAB lJser's potiowing Guide/lvlanual. examplesillustrate this. \"na. 7 2 -9 -3 2 -51 The above operation can also be achievedby typing

64?,..1 ModernPo@sis t em*, t This storesdeterminant of matrix A in H. VaIues If we do not give a semicolon at the end of closing square brackets, obtainseigenvaluesof rnatrixA andstorestlremin K. MATLAB displaysthe value of matricesin the commandwinCow. If you do G.2 SPECIAL MATRICES AND PRE.DEFINED VARIABLES not want MATLAB to disptay the results, just type semicolon(;) at the end AND SOME USEFUL OPERATORS MATLAB has some preinitialised variables. These variablescan directly be of the statement. That is why you rvill find that in our programmes,whenever usedin programs. pi we want to displayvalue of variablessayvoltages,we havejust typedthe name This gives the value of n of the variablewithout semicolonat the end, so that user will seethe values convertsdegreesd rnto radians and storesit in variable r. inf during the program.After the programis run with no valuesof the variables You can specify a variable to have value as oo. being displayed,if the user wants to seethe valuesof any of thesevariablesin iandj These arepredefined variableswhose value is equitl to sqrt (- 1). This is used the program he can simply type the variable narneand seethe values. to define complex numbers. One can specify complex mafficesas well. Common Matrix Operations The abovestatementdefines complex power S. A word of caution here is that if we use i andj variablesas loop counters First let us declaresome matrices then we cannot use them for defining complex numbers.Hence you may find Addition i that in someof the programswe haveused i I and.i1 as loop variablesinstead of i and 7. Adds matricesA and B and storesthem in matrix C I eps Subtraction This variabie is preinitiiized to 2-s2. 't Identity Matrix Subtractsmatrix B from matrix A and storesthe result.in D t To generatean identity matrix and store it in variabte K give the following Multiplication command I Multiplies two conformablematricesA and B and storesthe result in E Inverse This calculatesthe inverse of matrix A by calculating the co-factors and stores the resultin F. Transpose Single quote ( / ) operator is used to obtain transposeof matrix. In case the matrix elernentis complex, it storesthe conjugateof the element while taking the transpose,e.g. or >>Gt=[1 3;45;63]' also storesthe transposeof matrix given in squarebrackets in matrix Gl In case one does not want to take conjugate of elements while taking transposeof complex matrix, one should use . 'operator instead of ' operator.

#ffif Modernpo*er SystemAnalysis [-ait+ So K after this becomes \" (.. oPERATOR) K=[1 0 0 This operation unlike complete matrix multiplication, multiplies element of one 010 matdx WitlreOnrespond'rnegi;ementof other matrixlaving sameindex. However 0 0 1l in latter caseboth the matricesmust have equal dimensions. Zeros Matrix We have used this operatorin calculating complex powers,at the buses. Say V = [0.845+ j*0.307 0.921+ j*0.248 0.966 + 7*0.410] generateas 3 x 2 matrix whoseall elementsare zeroands, toresthemin L. And I = [0.0654- j*0.432 0.876 - j*0.289 0.543 + j*0.210]' Ones Matrix Then the complex power S is calculated as generateas 3 x 2 matrixwhoseall elementsare oneandstoresthem in M. Here, conj is a built-in t'unctionwhich gives complex conjugateof its argument. : (Colon) operator So S is obtainedas [- 0.0774+ 0.385Li 0.7404+ 0.4852t 0.6106+ 0.0198i] Note here,that if the result is complex MATLAB automaticallyassignsi in the This is an important operator which can extract a submatrix from a given result without any multiplication sign(*). But while giving the input ascomplex matrix. Matrix A is given as below number we have to use multiplication (*) along with i or 7. A= [1 5 7 8 G.4 COMMON BUILT.IN FUNCTIONS ? 6 910 5 43 1 sum e 3 r 2l sum(A) gives the sum or total of all the elementsof a given matrix A. This commandextractsall columns of matrix A correspondingto row Z and min stores them in B. SoBbecomes1269l,}lt This function can be used in two forms Now try this command. (a) For comparing two scalarquantities The abovecofirmandextractsall the rows correspondingto column 3 of matrix if either a or b is complex number, then its absolutevalue is taken for A and storesthem in matrix C. comparison So C becomes (b) For finding minimum amongsta matrix or ilray e.g.if A = [6 -3;2 -5] l7 >> min (A) results in - 5 9 abs 3 If applied to'a scalar, it gives the absolute positive value of that element 1l (magnitude).For example, >>x=3+j*4 Now try this command abs(x)gives 5 If applied to a matrix, it results in a matrix storing absolutevalues of all the This command extracts a submatrix such that it contains all the elements elementSof a given matrix. correspondingto row number 2 to 4 and column number 1 to 3. SoD=[2 6 9 543 9 3 1l

t -'__ I 041/ G,5 CONTROL STRUCTURES IF Statement comesto an end when k reachesor exceedsthe final value c. For example, for i - 1:1:10, The generalfrom of theIF statemenits IF expression a(i) - 1 end statements This initializes every elementof a to 1. If increment is of 1,asin this case,then the incrementpart may as well be omitted and the aboveloop could be written E L S Ee x p r e s s i o n AS statements f o ri - 1 : 1 0 , ELSEIF a(i) = 1 statements end END While Loop ' f a Expres s io ni s a l o g i c a l e x p re s s i o nr e sultin g i n an answer,true,(l) or lse'(0). The logical expressioncan co n sistof (i) an expressioncontainingrelational operatorstabulatedalong with their This loop repeatsa block of statementstill the condition given in the loop is true m e a n i n g si n T a b l eG . l . while expression Table G.1 statements end Relational Operator Meaning For example, j -1 Greater than ' Greater than or Equal to while i <= 10 Less than a(i) = 1 Less than or Equal to i = i + 1; Equal to Not equalto end This loop makes first ten elementsof array a equal to l. (ii) or many logicalexpr essioncsombincd with Logi caloperato rs .V ar i ous logical operatorsand their meanings are given in Table G.2. break statement Table G.2 This staternenatllowsoneto exit prerr'aturelyfrom a for or while loop. Logical Operator Meaning G.6 HOW TO RUN THE PROGRAMS GIVEN IN THIS APPENDIX? & AND OR NOT 1. Copy theseprogramsinto the work subfolderunderMATLAB foider. FOR Loops 2. Justtype the nameof the programwithout'.m' extensionand theprogram will run. This repeatsa block of statementsprecleterminednumberof times. The most common{&m of FOR loop usedis 3. If you wanLto copy them in some other folder say c:\\power,then after fork=a:btc, copyingthosefiles in c:\\power.changethe work folderto e:\\power.you cando this by clicking on toolbarcontainingthreeperiods... which is on statements the right side to the Current Directory on the top rilht corner. end 4. You can seeor edit theseprogramsby going through Fite - open menu and opening the appropriatefile. However do not savethose programs, where k is the loop variable which is initialised to value of initial variable a. unless you are sure that you want the changesyou have made to these If the final value (i'e. c) is not reached,the state mentsin the body for the loop orisinal files.

ffiffi| ModernPowersystemAnalysis 5. Youcanseewhicharethevariableaslreadydefinedby typingwhosin front of command prompt. That is why you will normally find a clear command at the beginningof our programs.This clearsall the variables defined so far from the memory, so that thosevariablesdo not interfere G.7 SIMULINK BASICS flH Dbcret i U ccrlrnn Fqdin t l$hr SIMULINK is a software packagedevelopedby MathWorks Inc. which is one i\".Sloircrctc Ma[' of the most widely usedsoftwarein academiaand industry for modeling and Nmlle simulatingdynamical systemsI.t can be usedfor modelinglinearand nonlinear '. S furtmsaroUos systems,either in continuoustime frame or sampledtime frame or even a hybrid of the two. It provides a very easydrag-drop type Graphical user interface to Unu build the models in block diagram form. It has many built-in block-library componentsthat you canuseto modelcomplex systemsI.f thesebuilt-in models , H nmr'oer are not enough for you, SIMULINK allows you to have userdefined blocks as i .fi sgn*aryrtan well. However, in this short appendix, we will try to cover some of the very common blocks that one comesacrosswhile simulatinga system.You can try iHs*' to construct the models given in the examples. :.fl sar:er How to Start? f,l crrtrd9y*entooDox You can start SIMULINK by simply clicking the simulink icon in the tools bar or by typing Simulink in front of the MATLAB command prompt >>. This 1 Sign* t SJ,s{emt opens up SIMULINK'hbrary browser, which should look similar to the one :l Si*r shown in Fig. G2. There may be other tool boxes dependingupon the license you have,The plus sign thatyou seein the right half of the window indicates Fig. G.2 that there are more blocks availableunder the icon clicking on the (+) sign will expand the library. Now for building up a new model click on.File and select 5 . Similarly click on continuous library icon. You can now see various New Model. A blank model window is opened.Now all you have to do is to built-in blocks such as derivative, integrator, transfer-function, state- select the block in the SIMULINK library browser and drop it on your model spaceetc. Selectintegrator block and drag-dropit in your model window. window. Then connect them togetherand run the simulation. That is all. 6. Now click on sinks and drag-drop scopeblock into your model. This is An Example one of the most common blocks used for displaying the values of the blocks. Let us try to simulate a simple model where we take a sinusoidal input, integrate it and observethe output. The steps are outlined as below. 7' . Now join output of sine-wavesourceto input of Integratorbloc(. This can be done in two ways. Either you click the left button and drag mousefrom 1. Click on the Sourcesin the SIMULINK library browserwindow. output of sine-wave source to input of Integrator block and leave left 2. You are able to see various sourcesthat SIMULINK provides. Scroll button or otherwise click on right button and drag the mouse to form connection from input of integratorblock to output of sine,wavesource. down and you will seea Sine Wave sourcesicon. 3. Click on this sourcesicon and without releasing the mouse button drag 8. Now in the main menu, click on Simulation and click Start. The simulation runs and stopsafter the time specifiedby giving readyprompt and drop it in your model window which is currently namedas 'untitled'. at bottom left'corner. 4. If you double-click on this source, you will be able to see Block 9. Now double-click on scope to seethe output. Is something wrong? The parametersfor sinewave which includes amplitude,frequency,phase,etc. result is a sine wave of magnitude 2. Is there something wrong with Let not changetheseparametersright now. So click on cancel to go back. SIMULINK software? No, we have in fact forgotten to specify the integration constAnt! Integrationof sin ?is - cos d+ C. At 0=0, C - - 1. If we do not specify any initial condition for output of the integrator,simulink assumesit to be 0 and calculatesthe constant. So it calculates

ffi ModernPo - cos 0+ C = 0 at r = 0 givinl C = 1. Sotheequationfor outputbecomes ,*rr-rr r^t - cos 0 + l. Thusnaturally,it startsfrom 0 at t = 0 andreachesits peak in SourcesblockunderSimulink.Wehavemadeuseof thisblockin valueof 2 at 0= n, i.e.3.I4. stability studiesto provide constantmechanicalinput. 10. To rectifv this error. double click on Inte block and in the initial conditionsenter- 1 whichshouldbetheoutputof theblockatt = 0. Now generatea sine wave of any amplitude,frequency and phase. run the simulationagainandseefor yourselfthat the resultis correct. The reader is encouraged to work out, the examples given in this Appendix to gain greater insight into the software. Some Commonly used Blocks 1. Integrator We have alreadydescribedits use in the aboveexample. G.8 SCRIPT AND FUNCTION FILES 2. Transferfunction Using this block you can simulate a transferfunction Types of m-files of the form Z(s) = N(s)/D(s), where N(s) and D(s) are polynomials in s. You can double click the block and enter the coefficients of s in numerator There are two types of m-files used in Matlab programming: and denominator of the expressionin ascendingorder of s which are to be enclosedinside squarebrackets and separatedby.a space. (i) Script m-file - This file needs no input parametersand doesnot return any values as output parameters. It is just a set of Matlab statements 3. Sum You can find thisblock underMath in Simulink block.By default, which is storedin a file so that one can executethis set by just typing the it hastwo inputs with both plus signs.You can modify it to haverequired file namein front of the command prompt, eg. prograrnmesin G2 to Gl8 number of inputs to be surnmed up by specifying a string of + or are script files. - dependingupon the inputs. So if there are3 inputs you cangive the list of signs as + - - . This will denoteone positive input and two inputs with (ii) Function m-files - This file acceptsinput agrumentsand return values as - signs which are often used to simuiate negative feedback. output parameters. It can work with variables which belong to the workspace as well as with the variables which are local to the functions. 4. Gain This block is alsofound underMath in Simdlink block.It is used These are useful for making your own function for a\\ particular to simulate static eain. It can even have fractional values to act as application, eg. PolarTorect.m in Gl is a function m-file. attenuator. The basic structureof function m-file is given below: 5. Switch This block is availableunder Non-linear block in Simulink. It (a) Function definition line - This is the first line of a function. It specifies has3 inputs with the top irrputbeing numbered 1.When the input number function name, number and order of input variables. 2 equals or exceedsthe thresholdvalue specified in the propertiesof this block, it allows input number 1 to pass through, else it allows input Its syntax is- function [output variables]= function-name (list of input number 3 to pass through. variables) (b) First line of help - Whenever help is requestedon this funciton or look 6. Mux and Demux Theseblocks areavailableunderSignals and systems for is executedMATLAB displays this line. block in Simulink. The Mux block combinesits inputs into a singleoutput Its syntax is - Vofunction-name help and is mostly used to form a vector out of input scalarquantities.Demux (c) Help text - Whenever help is requestedon the function-name help text block does the reversething. It splits the vector quantity into multiple is displayed by Matlab in addition to first help line. scalaroutputs. Its syntax is - Vofunction-name (input variables) (d) Body of the function - This consistsof codesacting on the set of input 7. Scope We have alreadydescribedits use. However; if you are plotting variables to producethe output variables. a large number of points, click on properties toolbar and select Data History tab. Then uncheckthe Limit datapoints box so that all points are Tlccr nqn fhrrs rlcwelnn hic/hcr nrrrn rtrntrrqrnc end fi'rnnfinnc qnA qAA fham fn fha plotted. Also when the rvaveformdoesnot appearsmooth,in the general LarvrII tab of propertiestoolbar, selectsampletime insteadof decimationin the existing library of functions and blocks. uv ulv sample time box and enter a suitable value like le-3 (10-3). The scope then usesthe value at sample-timeinterval to plot. G.9 SOME SAMPLE EXAMPLES SOLVED BY MATLAB 8. Clock This block is used to supply time as a source input and is In this section 18 solved examples of this book are solved again using availablein Sources block under Simulink. MATLAB/SIMULINK to encourage the readerto solve more power system problems using MATLAB.

ffil uodernpowerSystefnnalysis ' AppendiGx ffi Ex G.I T- % varying line lengths from 10 to 300 kmin steps of 10 km % T h i s f u n c t i o n c o n v e r t s p o 1a r t o r e c t a n g u l a rc o o r d i n a t e s % a n d c o m p a r et h e ma s a f u n c t i o n o f l i n e l e n g t h s . YoThe argumentt o t h i s f u n c t i o n i s 1 ) M a g n i t u d e& ? ) A n g l e i s . i n d e g r e e s % This function has been used i % T h e m e t h o d sa r e a p p e n dxi % l) Short Line approximatiom f u n c t i o n r e c t = p o l a r T o r e c(ta , b ) r e c t = a * c o s( b * p i / 1 S 0 ) + j * a * s ni ( b * p i/ l g 0 ) ; N o m i n a l - P Im e t h o d % 3 ) E x a c t t r a n s m i s s i o nl i n e e q u a t i o n s % 4 ) A p p r o x i m a t i o no f e x a c t e q u a t i o n s i =1. function is used in solutionof many of the examplessolvedlater. for I=10:10:300, Ex G.2 (Example S.O) % Short Iine approximation V s _ s h o r t l 'ni e( i ) = Y q +( 2 * 1) * I R Is shortline(i)=1P % Thls illustrates the Ferranti effect s p f _ s h o r t li n e( i ) = c o s( a n g 1e ( v s _ s h o r t li n e( i ) - a n g le( I s _ s h o rtline( i ) ) ) ) % It simulatesthe effect by varying the length of line from S p o w e r _ s h o r t ln' ie( i ) = r e a l ( V s s h o r t l i n e( j ) * conj( Is_shortl i n e( i ) ) ) % z e r o ( r e c e i v i n g e n d ) t o 5 0 0 0 k mi n s t e p s o f 1 0 k m % andplots the sendingend voltage phasor % N o mnia l P I m e t h o d % T h i s c o r r e s p o n d st o F i g . 5 . 1 3 a n d d a t a f r o m E x a m p l e5 . 6 cl ear 4 = 1 +( y * 1) * ( z * 1 )/ Z D=A V R = 2 2 0 e 3 / s q(r3t ) ; a 1p h a = 0 . 1 6 3 e - 3 ; B=z*l C = y * t* ( l + ( y * 1) * ( z * 1 )/ a ) b e t a = 1. 0 6 8 e - 3 ; V s _ n o mnia lp i ( i ) = A * V R + BR* I I =5000; I s_nomn'ia'pl i (i ) =C*VR+DR*l Spf_nominalpi(i)=cos(ang1e(Vs_nominalp'i(i)-angte(Is_nominalpi(i)))) k=1; Spower_nomnai 'lpi ( i ) =real (Vs_nomnialpi ( i ) *conj ( I s_norpniaI pN.i) ) ) for i=0:10:1, % E x a c t t r a n s m j s s i o nL i n e E q u a t j o n s Lc=sqrt(z/y) VS=(VR/z)*exp(a'lpha*i)*exp(j*beta*j)+(VR/Z)*exp(-alpha*i)*exp(j*beta*j); g a r n m a = s q(ryt* z ) X ( k )= r e a l( V S ); Y(k)=imag(VS); V s _ e x a c t ( i ) = c o s h ( g a m m)a**Vl R + Z c * s i n h( g a n r m a)**1I R I s _ e x a c t( i ) = ( I l l c ) * s i n h ( g a m m a *)1 + c o s h ( g a m m )a**IlR ' k=k+1; S p f _ e x a c (t i ) = c o s( a n g 1e ( V s _ e x a c(ti ) - a n g 1e( I s _ e x a c t( j ) ) ) ) Spower_exac(ti ) =real (Vs_eiact(i ) *conj (Is_exlct (i ) ) ) end p 1o t ( X ,Y ) Ex. G.3 (Example 8.7) % A p p r o x i m a t i o no f a b o v ee x a c t e q u a t i o n s % T h i s P r o g r a mi l l u s t r a t e s t h e u s e o f d i f f e r e n t l i n e m o d e l sa s 4 = 1 + ( y *)l * ( z * t ) / Z % i n E x a m p ' l5e. 7 D=A cl ear g = ( z * 1) * ( t + ( y * l ) * ( z * t ) / 6 ) f=50 C =( y * l ) * ( 1 + ( y * l) * ( z * 1 )/ 6 ) | =300 Vs_aPPro(xi ) =A*VR+B*IR 7=4Q+j*lZ5 I s _ a p p r o x( i ) = [ * ! P + P *1P Y =I e - 3 S p f _ a p p r o(xi ) = c o s( a n g 1e ( V s _ a p p r o(xI ) - a n gI e( I s _ a p p r o(xi ) ) ) ) PR=50e6/3 S p o w e r _ a p p n o x) (=ir e a 1( V s _ a p p r o(x' i) * c o n j ( I s _ a p p r o x ( i) ) ) YR=?20e3(s/ qrt (3) ) point(j)=i pf1oad=0.8 I R = P R / ( V R * pofa1d ) i=i+L z=7/1 end y = Yl 1 % T h e r e a d e r c a n u n c o m m e natn y o f t h e f o u r p l o t s t a t e m e n t sg i v e n b e l o w % N o ww e c a l c u l a t e t h e s e n d i n g - e n dv o l t a g e , s e n d . i n g - e ncdu r r e n t % by removingthe percenlagesign against that statement % a n d s e n d i n g - e npdf . b y f o l l o w i n g m e t h o d sf o r v a r i o u s l e n g t h s o F I i n e % f o r e x . i n t h e p l o t s t a t e m e n t u n c o m m e n t ebde l o w % i t p l o t s t h e s e n d i n ge n d v o ' l t a g e sf o r s h o r t l i n e m o d e li n r e d

__ff-f-i % b y n o m i n a lp i - m o d e 1i n g r e e n , b y exact p a r a m e t e r sm o d e r i n black bu ses=ma x(ma x ( y( 2d,a: )t)a, m a x( y d a t a( 3 ,: % andby approx. pi model in blue A= z e r o (se ' le m etns , b us e s) ) ) ) plot ( p o'i n t a, b s( V s _ s h o r til n e ), ' r' , p o in t , a b s( V s _ no mnia . l p)i , , ' , p o in t ,a b s . . . (Vs_ e xa c t ) ,' b ' , Dp o in t ,a b s( V s _ a p p r o, x, k), ) , p o in t , a b s . . . f o r i = 1 : e le m e n t s , p o in t , a b s . . . %ydata(i,?) gives the ,from' bus no. % f h e e n t r y c o r r o s p o n d i n gt o c o l u m n c o r r o s p o n d i n gt o t h i s b u s Ex G.4 (Eiample 6,2) % i n A m a t r i x i s m a d e1 i f t h i s i s n o t g r o u n db u i % T h i s p r o g r a mf o r m s y B U Sb y S i n g u l a r T r a n s f o r m a t i o n i f y d a t a ( i , 2 1- = g A(i,ydata(i,Z;;=1. end %ydata(i,3) givesthe 'to' bus no. % T h e e n t r y c o r r o s p o n d i n gt o c o l u m nc o r r e s p o n d i n gt o t h i s b u s % i n A m a t r i x i s m a d e1 i f t h i s i s n o t g r o u n db u s & T h e d a t a f o r t h i s p r o g r a mi s a p r i m i t i v e a d m i t t a n c em a t r i x y i f y d a t a( ' i, 3 ; - = 9 % whichis to be given in the foilowing format and stored in ydata. A(i,ydata(i,3;;=-i % G r o u n di s g i v e n a s b u s n o 0 . end % I f t h e e l e m e n ti s n o t m u t u a l ] y c o u p l e dw i t h any other element, % t h e ( ' t h e e n t r y c o r r e s p o n dni g t o 4 t h a n d 5 t h coi umnof ydata end % has'Io be zero % YBUS=A'*yprimitive*A % e l e m e nnt o I c o n n e c t e dI y (self) lMutually I v(mutual) t c o u p l e dt o l ?% | F r o mI T o I Ex G.5 (data same as Example 6.2) |B u s n oIB u s nIo % ydata=[ I t 2 L/(0.05+;*9.15; 0 0 % T h i s p r o g r a mf o r m s y B U Sb y , a d d i n g o n e e r e m e n ta t a t . i m e 2 I 3 t / ( 0 . 1 + j * 0 . 3) 0 % T h e d a t a f o r t h i s p r o g r a mi s a p r i m i t i v e a d m i t t a n c em a t r i x y 3 2 3 t/ (0.1s+j*0.4s) 0 0 % which is to be given in the following formatand stored in ydata 4 ? 4 1/(0.10+j*9.39; 0 % G r o u n di s g i v e n a s b u s n o . 0 5 3 4 1 /( O. 0 5 +*j 0 .1 5 ) 0 0 % I f t h e e l e m e n ti s n o t m u t u a l l y c o u p l e dw i t h a n y o t h e r e l e m e n t , % t h e n t h e e n t r y c o r r e s p o n d i n gt o 4 t h a n d 5 t h c o i u m no f y d a t a 0 % has to be zero ol; % formprimitive y matrix fromthis dataandi n i t i a l i z e it to zero. % T h e d a t a m u s t b e a r r a n g e di n a s c e n d i n go r d e r o f e l e m e n tn o . % to start with % e l e m e n t s = m(ayxd a t a( : , 1 )) % elemennt o I connectedI y ( s e lf ) l M u t u al yl I y ( m u t u a) l y p r i m it i v e = z e r o s( e 1e m e n t s, e 1e m e n t )s yo l F r o m l T oI Icoupel d t o I % P r o c e s sy d a t a m a t r . i x r o w w j s et o f o r m y p r i m it i v e % IB u s nIoe u s nIo for i=1:elements, t5 yp rim it i v e( i , . i )= y da t a( i so 'if th e e l e me n ti s m , 4) ydata=[1 1 2 L /( O. O s +*j0 .1 5 ) 0 % A l u tuall y c o u p le d w it h 2 0 ated i n 5th colum any othe r element 3 0 e o ( w h o s ee l e m e n tn o i s i n d i c the i th row i no f y d a t a above) 4 1 3 r/(o.i+j*0.3) 0 0 o in s m a d ee q ual to y(m 5 2 3 t/(0.r5+j*0.45) 0 0 % t h e c o r r e s p o n d i n gc o l u m nn u t u a ). l 2 4 t / ( o. 1 9 +*30. 3 0 ) 0 oj; if (ydata(i,S)-= g ; j i s i h e e l e m e n tn o w i t h w h i c hj t h v? , Ar r1/ l\\ lvn. s 3 nr .c.';:\"+U^ . I C ,1f r \\ j = y d a t a( j , 5 ) e l e m e n ti s mutuallycoupled U Y m u t u a=i; ' d a t a ( i , 6 ) % F o l 1 o w j n gs t a t e m e n tg i v e s m a x i m u mn o . o f e l e m e n t sb y c a l c u l a t i n g t h e % m a x i m u mo u t o f a l l e l e m e n t s( d e n o t e db y : ) o f f i r s t c o l u m no f y a a t a yprimitive(i,j) = ymutual e l e m e n t s = m a( yxd a t a( : , l ) ) \\ end % t h i s g i v e s n o . o f b u s e sw h i c h i s n o t h i n g b u t t h e m a x i m u me n t r y % o u t o f ? n d a n d 3 r d c o l u m no f y d a t a w h i c h i s ' f r o m ' a n d , t o ' a o l r r n , end b u s e s = m (amx a x( y d a t a( z, Z ) ) , m a x( y d a t a( : , 3 )) ) ; % F o r m B u s ' i n c i d e n c em a t r i x A f r o m y d a t a

ffi,W uooernPoweSr ystemnnavsis nppencoix Hffi Y B U S = z e(rbouss e sb,u s e s;) Y B U S ( k=, l Y) B U S ( k-, l y) s m ( 2 , 2 ) ; - Y B U S ( l , =k )Y B U S ( k , l ) ; for row=1:elements, Y B U(Si 1, k ) = Y B U(Si 1 ,k ) + y s m ( 1 , 2 ) ; % i f y d a t a ( r o w , 5 )i s z e r o t h a t m e a n st h e c o r r e s p o n d i n ge l e m e n t Y B U( Sk ,i 1 ) = Y B U( iS1 ,k ); % i s n o t m u t u a l l y c o u p l e dw i t h a n y o t h e r e l e m e n t YBUS(II,I) = YBtll(j!,L) + v s m ( 1 . 2 ) : j f y d a t a ( r o w , 5 )= = 0 Y B U(Sl , j 1 ) = Y B U(Sj 1 , I ) ; i 1 =ydata(row,2); Y B U S ( i 1 , .=| ) Y B U S ( i 1 , 1-) y s m ( 1 , 2 ) ; j1 =ydata(row,3); Y B U(S. |, i 1 ) = Y B U(Sj 1 , . |) ; Y B U S ( j 1 , k=) Y B U S ( j 1 , k-) y s m ( 1 , 2 ) ; if i1 -= 0 & jl -= 0 Y B U S ( k , j l )= Y B U S ( j 1 , ;k ) Y B U S ( i 1 , ' i 1=) Y B U S ( j 1 , j 1+) y d a t a ( r o w , 4 ) ; Y B U S ( i 1 , j l )= Y B U S ( i 1 , j 1-) y d a t a ( r o w , 4 ) ; end Y B U S ( j 1 , i 1=) Y B U S ( i 1 , j l ) ; i f i 1 = = 0 & j 1 - = 0 & k - = 0& l - = 0 Y B U S ( j 1 , j 1=) Y B U S ( j 1 , j 1+) y d a t a ( r o w , 4 ) ; Y B U S ( j 1 , j 1= )Y B U S ( i 1 , i 1+ )y s m ( 1 , 1 ) ; end Y B U S ( k ,=k )Y B U S ( k ,+k )y s m ( 2 , 2 ) ; if il ==0 & jl -=Q Y B U S,(l )l = Y B U S ( 1 ,+. | )y s n ( ? , 2 ) ; Y B U S ( k ,=l ) Y B U S ( k ,-l )y s m ( 2 , 2 ) ; Y B U S ( i 1 , j 1=) Y B U S ( i 1 , i 1+) y d a t a ( r o w , 4 ) ; Y B U S,(kl) = Y B U S ( k) ,; l Y B U S ( j 1 ,=. | )Y B U S ( i 1 ,+1 )y s m ( 1 , 2 ) ; end Y B U S ( . l , j=1 )Y B U S ( i l ,;l ) if il -= 0 & jl ==Q Y B U S ( j 1 ,=k )Y B U S ( i 1 , -k )y s m ( 1 , ;2 ) Y B U S ( k , j =1 )Y B U S ( i 1 ;, k ) Y B U S ( j 1 , j 1=) Y B U S ( j 1 , j 1+) y d a t a ( r o w , 4 ) ; end i f i 1 - = 0& j l = = Q& k - = 0& l - = 0 end Y B U S (ji11), = Y B U S ( i 1 , '+i 1y)s m ( 1 , 1 ) ; Y B U S ( k , k=)Y B U S ( k , +k )y s n ( 2 , 2 ) ; end Y B U S ( l , l=) Y B U S ( 1 ,+' l )y s n ( z , 2 ) ; e o i f y d a t a ( r o w , 5 )i s N O Tz e r o t h a t m e a n st h e c o r r e s p o n d i n ge l e m e n t Y B U S ( k ,=l ) Y B U S ( k ,-l )y s m ( 2 , 2 ) ; Y B U S ( l , k=)Y B U S ( k , l ) ; % i s m u t u a l l yc o u p l e dw i t h e l e m e n tg i v e n i n y d a t a ( r o w , 5 ) Y B U S ( i 1 ,=k )Y B U S ( i 1 ,+k )y s m ( 1 , 2 ) ; Y B U S ( k1,)i= Y B U S ( ' i l;, k ) i f y d a t a ( r o w , 5 )- = g Y B U S ( i 1 ,=. | )Y B U S ( i 1 , -1 )y s m ( 1 , 2 ) ; i 1 =ydata(row,2); Y B U(Sl , ' i1 ) = Y B U(Si 1 , .)| ; j 1 = y d a t a ( r o w , 3; ) end % m u t u a l w i t hg i v e s t h e e l e m e n tn o w i t h w h i c h t h e c u r r e n t e l e m e n t j f i 1 - = 0& j 1 - = Q& k = = 0& l - = 0 Y B U( Si 1 ,i 1 ) = Y B U( Si 1 ,i 1 ) + y s m ( 1 , l;. ) % i s m u t u a l l y c o u p l e dw j t h k a n d 1 g l v e t h e b u s n o s b e t w e e n Y B U S ( j 1 , i 1= )Y B U S ( i 1 , i 1+ )y s m ( 1 , 1 ) ; Y B U S ( l , l=) Y B U S ( 1 ,+. |y) s m ( 2 , 2 ) ; % w h i c ht h e m u t u a l l y c o u p l e de l e m e n ti s c o n n e c t e d Y B U(Si 1 ,j 1 ) = Y B U( Si 1 , i 1 ) - y s m ( 1 , l;. ) m u t u a*l . i 1 1 = y d a t(ai 1 , 5 ); Y B U S ( j 1 , j 1= )Y B U S ( i 1 , i;1 ) ; q = y d a t(am u t u awl it h , 2 ) ; Y B U S ( j 1 ,=1 )Y B U S ( j 1 ,+1 )y s m ( I , 2 ) ; 1= y d a t a( m u t u awl it h , 3 ) ; Y B U S ( . l , j=1 )Y B U S ( i 1;, 1 ) Y B U S ( i 1 ,=. | )Y B U S ( i 1 ,-. | y) s m ( 1 , 2 ) ; zsI=I/ydata(row,4); Y B U S ( l , j l=) Y B U S ( i 1 , 1 ) ; zs?=L/ydata(mutuawl 'ith,4) ; end zm=l/ydata(row,6); if i1-=0 & jl-=Q & k-=0& l==0 7 5 6 =[ z s1 z m Y B U S ( i 1 , i 1=) Y B U S ( i 1 , i 1+) y s m ( 1 , 1 ) ; Y B U S ( j 1 , j l )= Y B U S ( j 1 , i 1+) y s m ( 1 , 1 ) ; zn zs?l; Y B U S ( k , k=) Y B U S ( k , k+) y s m ( 2 , 2 ) ; Y B U S ( i 1 , j 1=) Y B U S ( i 1 , j 1-) y s m ( 1 , 1 ) ; ysm=n'i v(zsm); % F o l ' l o w i n gi f b l o c k g i v e s t h e n e c e s s a r ym o d i f i c a t i o n s i n Y B U S % w h e nn o n eo f t h e b u s e s i s r e f e r e n c e ( g r o u n d ) b u s . if il -= 0 & jl -= 0 & k -=0 & l-=0 Y B U S ( ' i 1 , i 1=) Y B U Si(,ii 1 ) + y s m ( 1 , 1 ) ; Y B U S ( j 1 , j l )= Y B U S ( j 1 , j l )+ y s m ( 1 , 1 ) ; Y B U S (k ), = Y B U S ( k ,) + y s m ( 2 , 2 ) ; Y B U S ( l , l )= Y B U S ,(ll) + y s n ( Z , 2 ) ; Y B U SI(,ij 1 ) = Y B U S1( i, j 1 ) - y s m ( 1 , 1 ) ; Y B U S ( j 1 , i 1=) Y B U S 1( i, j 1 );

W MooernpowersystemA_nalysis Appendixc lFitlilE4lq. ffi. Y B U S ( j 1 , i=1 )y B U S ( i t , j t ) ; t ypechang(eid) =1t else NilFffi Y B U S ( i 1 ,=k )Y B U S ( i 1 ,+k )y s m ( 1 , 2 ) ; T- Y B U(Sk ,i 1 ) = Y B U(Si 1 ,k ); t ype( i ) =2; Y B U S ( i 1 ,=k )y B U S ( i 1 , k_)y s m ( 1 , 2 ) ; Y B U S ( k , i=l ) Y B U S ( j 1 ;, k ) end end end YBUS Ex G.6 sumyv=0; for k=L:n, % T h i s ' i s p r o g r a mf o r g a u s ss i e d e rL o a df r o w . T h ed a t a i s f r o m E x a m p l e6 . 5 clear if(i -= k) s u m y v = s u m y v(+i ,Yk )* V( k ) ; n=4 end V = [ 1 . 0 41 . 0 41 t ] end v ( i ) =( 1 / Y( i , i ) ) * ( ( p ( i ) - j * Q ( i ) ) / c o n j ( v ( i ) ) - s u m y v;) Y=[3-i*9 -2+i*6 -l+5*3 0 if type(i;==2 & typechanged(i)-=1, -2+i*6 3 6 6 6 _ j * 1 1 _ 0 . 6 6 6j *+2 _t+3*3 -1+;*3 - 0 . 6 6 6j *+2 3 . 6 6 6j -* 1 1 _Z+i*6 V ( i ) = p e 1a r T o r e c t ( V m a g f i x e d)(, ia n g 1e ( V ( i) ) * 1 g 0 / p i) ; - 1+ 3 ' * 3 - 2 + t r * 6 end 0 3-j\"91 end d i f f = m a x( a b s( a b s( U( Z z n )) - a b s( V p r e v( Z :n ) ) ) ) ; t y p e = o n(ens, 1 ) noofiter=noofiter+l; end type changed = z e( nr o,1s) Q li m it m a x = z e r(ons,1i V Q li m it m in = z e r o(sn ,l ) Ex. G.7 (Example 6.6) V m agxf ied= z e ro(sn, 1) % Programf o r l o a d f l o w b y N e w t o n - R a p h s oMne t h o d . tYpe(2)=2 clear; Q li m it m ax( 2 )= 1. g % n s t a n d s f o r n u m b e ro f b u s e s Q 1i m it m in( 2 )= 9 .2 n=3; V m agfxied( 2=) 1 .0 4 % Y , v o l t a g e sa t t h o s eb u s e sa r e i n i t j a l j s e d d i f f = 1 0 ;n o o f it e r = 1 v = [ 1 . 0 41 . 0 1 . 0 4 ] ; Vprev=V; %Y is YBus w h i l e ( d i f f > O . 0 0 0 0| 1n o o f i t e r = = 1 ) , a b s( V ) Y=[ 5 . 8 8 2 2 8 - j ' t 2 3 . 5 0 5- 21.49 4 2 7 + j * L t . t 6 t 6 - Z. 9 4 2+7j * 7 I. t 6 76 -2.942+7j*Lt .7676 5.88228_j*2.3505L4 - 2. 942+7j* I I . 7676 abs( V pr ev ) - ?. 9 4 2+7j * L l . 76 76 - 2 . 9 4 2+7j * t t . 76 t 6 5.88228-j*23.5051a1; %pause % B u st y p e sa r e j n i t i a l j s e d i n t y p e a r r a y t o c o d e1 w h i c hs t a n d sf o r PQ Vprev=V; %bus. P = [ i n f0 . 5 - 1 0 . 3 ] ; %code2 stands for PV bus Q = [ i n f0 0 . 5 - 0 . 1 ] ; type=ones(n,1); $ = [ i n f + j * i n f( 0 . 5 - j * 0 . 2 )( - 1 . 0 + for i=Zin, j*0.5) (0.1-1*g.1)]; % w h e nQ l ' i m j t s a r e e x c e e d e df o r a p V b u s B u s t y p e i s c h a n g e dt o p Q % t e m p o r a r1i y if t ype(i) = = 2l t y p e c h a inig#ieldll)l(i,.;=Qi=m1,t, mini()i) L a n a ' l' eo m\" ,oLn f, , e iI nvfr #u. rJ^po ashLr nrasn 'rl v s u .ri )a -J^t+r L .L^ u 1r ir -l r c d s g r r 5 D u s s t a t u s \",lli lil;:il;l# , % i s t e m p o r a r i l y c h a n g e df r o m p Q t o p V . O t h e r w i s e i t i s z e r o Q ( j) = Q il m it m in( i ) ; t y p e c h a n g e d = z e r o( ns, 1 ) ; else % s i n c e m a xa n d m i n Q l ' i m i t s a r e c h e c k e do n l y f o r p v b u s e s , % max& min Q limlts for other types of busescan be set to any values. Q('i)=Qlimitmax(i); % here we have set themto zeros for convenience end Q li m it m a x = z e r o(sn , 1 ) ; t Y P e( i ) = 1 ; Q 1i m it m in = z e r o (sn , l ) ; V m a g fxi e d = z e r o(sn , 1 ) ;

W ModernPowersystemAnalysis I t I t o t o t a l n o . o f e q u a t i o n s( e q c o u n t ) for ceq=1:eqcount, pause u p d a t e =ni v( J a c o b )* m is m a t c h;' for ccol=1:eqcount, noofeq=L; vsJ\\eey/tsJrvvv r\\uJJvevruuJ\\vwwrllr, b m =mi a g( Y( a s s o e q b u(sc e q ) ,a s s o c obl u s( c c o :)l ) * V( a ss o c o lb u s( c c o l) ) ) ; if type(i;==1 e i = r e a l ( V( a s s o e q b u(sc e q )) ) ; n e w c hni a n g V = u p d a(tneo o f e q ); f i = i m a g( V( a s s o e q b u( sc e q )) ) ; n e w a n g V = a neg( V1( i ) ) + n e w c hnia n g V; i f a s s o e q v a r ( c e=q=) ' P\" ' & a s s o c ovl a i ( c c o 1) = = ' d ' , n e w c hni m a g V = u p d a t e ( n o o f e Q + l ) *(aVb( is) ) ; newmagV=a(bVs('i) ) +newcjhnmagV; i f a s s o e q b u(sc e q )- = a s s o c obl u s( c c o l) , V( j ) = p o la r T o r e c t( n e w m a gnVe, w a n g V * 1 8 0 / p;i ) H = a m * f-ib m * e i; n o o f e q = n o o f e q +;l else else n e w c hni a n g V = u p d a(tneo o f e q ); 1=1- Q( a ss o e q b u(sc e q )) i m a g( Y(a s s o e q b u(sc e q ) ,a s s o c obl u s( c e q ) ). . . n e w a n g V = a neg(1V( ' i) ) + n e w c tnr ia n g V; * a b s( V( a s s o e q b u(sc e q )) ) ^ 2 ) ; V ( i ) = p o la r T o r e c t( a b s( V ( i) ) , n e w a n g V * l 8 O /)p;i n o o f e q = n o o f e q +;l end J a c o b( c e q ,c c o l) = H end end end if a sso e qvar ( c e=q=) ' P' &a s s o c ovl a r ( c c o )1= = ' V ', % A l I t h e f o l I o w in g v a r i a b le s / a r r a y s a r e c l e a r e d f r o m ' if a ssoe q b u(sc e q )- =as s o c obl u s( c c o l) , % m e m o r y .T h i s i s b e c a u s et h e ' i r d i m e n s i o n sm a y c h a n g ed u e t o N=am*ei+bm*fi; % b u s s w i t c h e da n d o n c e u p d a t e sa r e c a l c u l a t e d , t h e v a r i a b l e s 9rare of no use as they are being reformulated at the el se N=P(assoeqbu(cseq)) +real(Y(assoeqbu(cseq), assocobl us(ceq)) . . . % end of each iterat'ion \\ * a b s( V( a s s o e q b u(sc e q )) ) ^ 2 ) ; c l e a r m i s m a t c hJ a c o b u p d a t e a s s o e q v a ra s s o e q b u sa s s o c o l v a fa s s o c o l b u s ; end J a c o b( c e q ,c c o l) = 1 1 d i f f = m 'ni ( a b s( a b s( V( 2 : n ) ) - a b s( V p r e v( 2 :n ) ) ) ) ; end n o o fi t e r = n o o f it e r + 1 ; end i f a s s o e q v a r ( c e q ) = = &' Qa' s s o c o l v a r ( c c o 1 ) = = ' d ' , Ex. G.8 (Table 7.1) i f a s s o e q b u(sc e q )- = a s s o c obl u s( c c o l) , % M A T L A PB r o g r a mf o r o p t ' i m u ml o a d i n g o f g e n e r a t o r s J=am*ei+bm*fi; % T h e d a t a a r e f r o m E x a m p l e7 . 1 % r t f i n d s l a m d ab y t h e a l g o r i t h m g i v e n o n t h e s a m ep d 9 € , o n c e t h e d e m a n d else rkis spec'ified % W eh a v e t a k e n t h e d e m a n da s 2 3 1 . 2 5 M Wc o r r o s p o n d i n gt o t h e l a s t b u t o n e J = P( a ss o e q b u(sc e q) ) r e a l( Y( a s s o e q b u( cse q) , a s s o c obl u s( c e q )) % r o w o f T a b l e 7 . 1 a n d c a l c u l a t e dl a m d aa n d t h e l o a d s h a r l n g * a b s( V( a s s o e q b u(sc e q )) ) ^ 2 ) ; % n is no of generators end n=2 J a c o b ( c e q , c c)o=l J % Pd stands for load demand. end % alpha and beta arrays denotealphabeta coeffjcients % for given generators. i f a s s o e q v a r ( c e q ) = =&' Qa' s s o c o l v a r ( c c o 1 ) = = ' V ' , 1f a s s o e q b u( sc e q )- - a s s o c o bl u s( c c o l) , Pd=?31.25 L = a m * f-i b m * e i' a l p h a =[ 0 . 2 0 else 0.251 L = Q ( a s s o e q b( uc es q )) - . . . b e t a =[ 4 0 irmf fr fnu/ vV \\/ r' c\\ .cu^ Jo J^ vh s, ,q. vt /u^Jo\\^l \\s Y / r q J J v u v' el p. ^u^J^\\ 'L1s Y /hr ,/,,. / ^ ^ ^ \\ \\ * . h . / \\ ,q w J \\ r \\ q 1J .J.v. E^ ^V^Uh U, ,J- \\/ ^L ^g^r \\{\\,\\,' \\ D \\ . l l Ll, 301 end % i n i t i a l g u e s sf o r l a m d a J a c o b( c e q ,c c o l) = L I amda=20 en0 end en0 % N e wU p d a t ev e c t o r i s c a l c u l a t e d f r o m I n v e r s e o f t h e J a c o b i a n

ffil Moderpno Hffi I amdapreva=ml da m 1 n\" l n f i cost=0; % t o l e r a n c ei s e p sa n di n c r e m e nitn l a m d ai s d e l t a l a m d a for j=0:n, eps=1 f o r i = 0 :n , de'tlalam d a = 0 .2 5 m u m| ' rm tt s o t e a c h g e n e r a t in g u n m'tn'tmum % a r es t o r e d i n a r r a y sP g m i na n d P g m a x . for I=0:n, unit = [0 0 0 0]; % I n r e a l l i f e l a r g es c a l e p r o b l e m s ,w e c a n f i r s t i n i t i a l s e the Pgmax % H e r e w e e l i m n a t e s t r a i g h t a w a yt h o s e c o m b i n a t i o n sw h i c h % a r r a yt o i n f u s i n gf o r I o o p a n d % d o n t m a k eu p t h e % n M Wd e m a n do r s u c h c o m b i n a t i o n sw h e r em a x i m u mg e n e r a t i o n % P g n i na r r a y t o z e r o u s i n g P g m i n = z e r o s ( n , 1c) o m m a n d % on lndividual % L a t e r w e c a n c h a n g et h e l ' i m i t s i n d i v ' i d u a l l y % g e n e r a t i o ni s e x c e e d i n gt h e m a x i m u cma p a c i t y o f a n y o f t h e % generators P g m a x = [ 1 2 I52 5 ] i f ( ' i + j + k + ' l ) = = n& i c P g m a x ( 1&) i < P g m a x ( 2&) k c P g m a x ( 3&)I < . . . P g m a(x4 ) P g m i n = 1 2Z00 ) if i-=0 P g - - 1 O 0 * o (nne,s1 ) unit(1,1)=i; w h il e a b s( s u m ( P S- e) d ; ' g t t % F t n d o u t t h e c o s t o f g e n e r a t i n gt h e s e u n i t s a n d % qdd 'it up to total cost for i=1:n, c o s t = c o s t + 05.* a lp h a( I ) * i * i + b e t a( I ) * i ; end P s ( i) =( 1a m d a - b e t(ai ) ) / a l p h a( i ) ; i f P g ( i) > P s m a x)( i i f 3-=g u n it ( L , 2 ) = i ; P g ( i) = P s m a x)(;i c o s t = c o s t + 0 . 5 * a l p h a ( 2 ) * j * j + b e t a ( 2 ) * i; \\ end end if Ps(i).Pgmln(i) i f k-=0 u n it ( 1 , 3 )= k ; Pg(i)=Pgmin(i); c o s t = c o s t + 0 . 5 * aplh r ( 3 ) * k * k + b e t a ( 3 ) * k; end end end i f l-=0 i f ( s u m ( P g ) - P)d. 0 unit(1,4)=l; l a m d a p r e va=ml d a ; c o s t = c o s t + 0 . 5 * ap1h a ( a ) * l* l + b e t a ( 4 ) * l; l a m d a =al m d a + d et al 1a m d a ; el se end I a m d a p r e vI =a m d ;a % If the total cost i s coming out to be I ess than l a m d a =alm d a - d etla l a m d a ; % m i n i m u mo f t h e c o s t i n end % p r e v i o u s c o m b i n a t ' i o ntsh e n m a k em i n e q u a l t o c o s t a n d end % c u n i t ( s t a n d f o r c o m m i t t e du n i t s ) e q u a l t o u n ' i t s d i s p ( ' T h e f i n a l v a l u e o f L a m d ai s ' ) % c o m m i t t e di n t h i s i t e r a t i o n 1a m d a p r e v t ( d e n o t e db Y v a r i a b l e u n i t s ) d i s p ( ' T h ed i s t r i b u t i o n o f I o a d s h a r e d b y t w o u nj t s ' i s ' ) if cost < min Pg Ex. G.9 (Table 7.2) cun'it- un'iti % M A T LAPBro g ra mfo r o p ti mu mu n i t c o mmi ttmenbty B rute Forcemethod mi n=nnc.f : f t T h ed a t a f o r t h l s p r o g r a mc o r r e s p o n dtso T a b l e7 . 2 rrr . | | ev v v t cI ear; else % a l p h aa n db e t aa r r a y sd e n o t ea l p h ab e t ac o e f f i c i e n t sf o r g i v e n g e n e r a t o r s a l p h a = 1 0 . 71 7. 6 02 . 0 02 . 5 0 1 ' ; c o st = 0 ; b e t a = 1 2 32. 56 . 5 3 0 . 03 2 . 0 1 ' ; P g m i n =1[ 1I 1 ] ' ; end Pgmax=ll1?2 12 I2l':' end end n=9 end % n denotestotal MWto be commttied

,f,f6.'l M o d e r nP o @ E end for i=1:n, si gma=(Bi , : ) *Pg-B(i , i ) *Pg(i ) ; end P g( i ) =( 1 - ( b e t a( i ) / l a m d a-)( 2 * si s n a ) )/ ( a l p h a( i ) / l a m d a + z *(Bi , i ) ) ; , d i s p ( ' c u n i t d i s p l a y t h e n o o f c o m m j t t e du n i t s o n e a c ho f t h e f o u r g e n e r a - I tors') d i s p ( ' I f c u n i t f o r a p a r t i c u ) a r g e n e r a t o r i s 0 j t m e a n st h e u n i t i s n o t if Pg(i)'Pgmax(j) comm Pg(i ) =Pgma(xi ) ; d i s p ( ' T h e t o t a l n o o f u n i t s t o b e c o m mt ti e d a r e ' ) cunit end if Pg(i)<Pgmin(i) Ex G.lO (Ex. 7.A) Ps(i)=Psmin(i); clear end % M A T L A BP r o g r a mf o r o p t i m u ml o a d i n g o f g e n e r a t o r s % T h i s p r o g r a mf i n d s t h e o p t i m a l l o a d i n g o f g e n e r a t o r si n c l u d i n g end % penalty factors PL=Pg'*B*Pg; % I t i m p l e m e n t st h e a l g o r i t h m g i v e n j u s t b e f o r e E x a m p l e7 . 4 . % T h e d a t a f o r t h i s p r o g r a ma r e t a k e n f r o m E x a m p l e7 . 4 i f ( s u m ( P g ) - P d - P)L< 0 % H e r e w e g i v e d e m a n dP d a n d a l p h a , b e t a a n d B - c o e f f i c i e n t s t l , { ec a l c u l a t e l o a d s h a r e d b y e a c h g e n e r a t o r 1a m d a p r e v =alm d a ; % n is no of generators n=2 I amda=al mda+detal l amda; ? Pd stands for I oad demand % alpha and beta arrays denotealpha beta coefficjents for given else % generators I amdaprev=a1mda; P d = 2 3.70 4 ; a l p h 6 =[ 0 . 0 2 0 I amda=al mda-detlal amda; 0.0a1; end b e t a =[ 1 6 noofiter=noofi ter+1 ; 2 o li ' I i n i t i a l g u e sfsor I amda Pg; Iamda=20; end I amdaprevI=amda; d i s p( ' T h e n o o f i t e r a t i o n s r e q u i r e d a r e ' ) % t o l e r a n c e i s e p s a n d i n c r e m e n ti n l a m d aj s d e l t a l a m d a eps=1; noofiter 'l amda s ' ) deltal amda=O.25; d i s p( ' T h e f i n a l v a l u e o f i % t h e m i n i m u ma n d m a x i m u ml i m i t s o f e a c h g e n e r a t i n gu n i t % a r e s t o r e d i n a r r a y s P g m i na n d P g m a x . 1a m d a p r e v % I n a c t u a l l a r g e s c a l e p r o b l e m s ,w e c a n f i r s t i n i t i a l i s e t h e P g m a xa r r a y d i s p( ' T h e o p t ' i m a 1I o a dni g o f g e n e r a t o r si n c lu d in g p e n atly f a c t o , r s' i s ') % to inf using for loop Pg % a n d P g m i na r r a y t o z e r o u s i n g P g m i n = z e r o s ( n , 1c)o m m a n d d i s p( ' T h e I o s s e sa r e ' ) % L a t e r w e s h o u l d c a n c h a n g et h e l i m i t s i n d i v ' i d u a 1 1 y P q m a x = f 2 0 02 0 0 . l : PL P g m i n = [ 00 ] ; B = [ 0 . 0 0 10 Ex. G.II 0 0l; % Iri this examplethe parametersfor all the blocks for the systemin Pg=zeros(n,1); % Fis. 8.8 are initialized This programhas to be run prior to the noofi ter=0; s ' i m ualt i o n b o t h f o r F i g s . G . 3 a n d G . 4 . PL=0; T s g =. 4 Pg=zeros(n,1); Tt=0.5 Tps=20 Kps=100 R=3 K sg1=g K t= 0I. K i= 0 . 0 9

SS1 Moo\"rnpo*b,syrt\"r Rn\"trrri, . lffiw ffiflI Conetant Ksg Kps .s+1 Tps.s+ 1 I curren t busniondicat esm axim unmo. of busesaddedunt il now currentbusno=0 TraneferFcn TraneferFcn 1 Fig. G.3 Firstorderapproximationfor loadfrequencycontrol for count=1:elements, [ r o w sc o 1s ]= s j z e ( z b u s ) f rom=zpmr i ar y( count2, ) to= z prmi ar y( count , 3) valu e=zprmi ar(ycount4, ) % n e w b uvsa r i a b l ei n d i c a t e st h e m a x i m u mo f t h e t w o b u s e s % newbusbus mayor maynot a l r e a d y b e a p a r t o f e x i s t i n g z b u s Ex. G.12 newbus=m( far oxmt,o) The system of Fig. 8.10 is simulated using simulink as was done in % r e f v a r i a b l e i n d i c a t e s t h e m i n i m u mo f t h e t w o b u s e s E x am pleG.l l . % & not necessarjly the reference bus Ksg % ref bus must alwaysexist in the existing zbus Tsg.s+ 1 r e f = m in ( f r o m , t o ) TransferFcn TransferFcnl Kps % Modjfication of typel Tps.s+ 1 % A n e w e l e m e n t i s a d d e df r o m new b u s t o r e f e r e n c eb u s I TransferFcn2 i f n e w b u s> c u r r e n t b u s n o& r e f ==0 z b u s = [ z b u sz e r o s ( r o w s1, ) z e r o s ( 1 , c o l s )v a l u e l c ur r e n t b u s n o = n e w bsu \\ continue end % Modification of type? Flg.G.4 ProportionpallusIntegralloadfrequenccyontrol % A newelement is addedfrom newbus to old bus other than reference bus Ex. G.13 (Example 9.8) i f n e w b u s> c u r r e n t b u s n o& r e f - = 0 % P r o g r a mf o r b u i ] d i r r g o f Z b u s b y a d d i t ' i o n o f b r a n c ho r l . i n k z b u s = [ z b u sz b u s (: , r e f ) % Z p r i m a r y = [ e l e m e n t n of r o m t o v a l u e % z b u s( r e f , : ) v a l u e + z b u(sr e f , r e f ) ] % c ur r e n t b u s n o = n e w bsu yo continue % --l end % H e r ec a r e s h o u l db e t a k e nt h a t t o b e g i nw i t h a n e l e m e n ti s a d d e dt o % r e f e r e n c ea n d b o t h f r o ma n dt o n o d e ss h o u l dn o t be new nodes % M o d fi i c a i t o n o f t y p e 3 clear zpri mary=[ % A n e w e l e m e n t i s a d d e db e t w e e na n o l d b u s a n d r e f e r e n c e b u s i f n e w b u s< = c u r r e n t b u s n o& r e f = = O 1 I 0 0.25 2 2 I 0.1 z b u s = z b u s - 1 / ( z b(unse w b u sn,e w b u s ) + vuael )* z b u s( : , n e w b u s*)z b u s( n e w b u s: ,) 3 3 1 0.1 continue 4 2 0 0.25 s 2 3 0.11 end [ el em ent sco 1u mn s=]s i z e (z p ri m a ry ) % M o d ' i f i c a t j o no f t y p e 4 t T o b e g i nw i t h z b u sm a t r i xi s a n u l l m a t r i x % A new element is addedbetwentwo old buses i f newbus<= currentbusno & ref -=0 z b u s = z b u-s L / ( v a l u e + z b u(sf r o m ,f r o m ) + z b u(st o , t o ) - 2*zbus(from,to))*((zbus(:,from)-zbus(:,to))*((zbus(from,;|-zbuS(tor:)))) continue end end

Wl uodginPoweSr ystemRnatysis j=j+1 Ex. G.14 (Example 12,10) end t P r o g r a mf o r t r a n s ' i e n ts t a b i ' l i t y o f s i n g l e m a c h i n ec o n n e c t etdo l n f i n i t e a x i s ( [ 00 . 6 0 1 6 0 ] ) % b u s t h i s p r o g r a ms i m u l a t e sE x a m p l1e2 . 1 0u s j n gp o i n t b y p o i n t m e t h o d clear Ex. G.15 t-0 tf=0 Here the earlier Example G.l4 is solved again using SIMULINK. tfi nal=0.5 Before running simulation shown in Fig. G.5 integratert has to be initialized t c = O .1 2 5 to prefault value of 4 i.e. 4. Thit can be doneby double-clicking on integrater t s t e P = 9 .9 5 1 block and changing the initial value from 0 to 6s(in radius). Also double click l4=?.5?/( 180*50) the switch block and changethe threshold value from 0 to the fault clearing time j=2 (in sec.). d e 1t a = 2 1 . . 6 4 * p/i1 8 0 Ex. G.16 (Ex. 12.11) d d e lt a = 0 t i m e( 1 )= 9 v\" Th'tsprogramsimual tes transient stab'1i i ty of mutlimachni e systems ang(1)=2I.64 % Thedata i s f r om Exam p'1le2. 11 Pm=0.9 clear alI Pnaxbf=2.44 formatlong Pmaxdf=0.88 % S t e p1 I n ' i t i a l i s a t i o nw i t h l o a d f l o w a n dm a c h ' i ndea t a Pmaxaf=2.00 f = 5 0 ; t s t e p = 9 . 961=. [ 1 29 J ,; P g n e t t e r m =3 .[ 2 52 . 1 0 ] '; w h iI e t < t f i n a l , Q g n e t t e r m =0[. 6 9 8 60 . 3 1 1 0 ];' ' i f ( t = = t f) , X g = [ 0 . 0 607. 1 0 ] ' ; P a mn' iu s = 0 . 9 - P m a x b f *ns(i d e lt a ) % Note the .use of .' operator here P a p lu s = O . 9 - P m a x d f * s('di ne lt a ) % This does a transposewjthout taking the conjugateof each element Paav=(Pamn'ius+Papu1s) /2 y g = [ p o ' l a r T o r e c t ( 1 . 0 83 ., 2 3 5 ) p o l a r f o r e c t ( 1 . 0 2 ,7 . 1 6 ) ]. ' ' Pa=Paav % n is no of generatorsother than sl ack bus end i f (t==tc), n=2i P a mni u s = 0 . 9 - P m a x d f *ns(id e lt a ) % S t e p2 P a p lu s = 0 .9 - P m a x a f *jsn( d e lt a ) V 0 c o n i = c o r(iV 0 ) ; Paav=(Pamnj us+Paplus)/2 I g 0 = c o n j( ( P g n e t t e m + j * Q g n e t t e r m.)/ U O ); Pa=Paav E d a s h 0 = V 0\"+(jX g .* I g 0 ) ; end P g 0 = r e a('El d a s h 0 . * c o n( jI g 0 )) ; x 1 r = a n g le ( E d a s h 0;) if(t>tf & t<tc), % lnit'ial isation of state vector P a = P m - P m a x d fn*(sdi e lt a ) Pg r=Pg0; % Pg_ro1usl.=PgO; end xZ r=[0 0],; i f(t>tc), xtdot r=[0 0] ,; x2dot r=[o o] ,; P a = P m - P m a x a fn*(sdi e lt a ) end xldotrplrtl= [0 0 J' i x 2 d o t r p lr t l = J0 0J' i t, Pa d d e lt a = c i d etla + ( t s t e p * t s t e p * p a / M ) d e lt a - ( d e lt a * 1 8 0 / p j+ d d etla ) * p i / 1 8 0 d e 'tl a d e g = d et1a * 1 8 0 / P ' i t=t+tstep pause t i m e ( i) = 1 ang(i)=deltadeg

S S t e p3 % H e r ei n t h i s e x a m p lwe e h a v en o t r e a l l y c a l c u l a t e dY B u sB u t - s n e % canwrite seParateProgram. yB usdf[= 0 -0.0681+i*5.1661 1f986-j*35-6301 0 0 -j*11.236 0.1362-*j6. 2737if - 0 . 0 6 8 l + j * 51. 6 6 1 0 o Y B uspf[= -0.2214+j*7.6289 - 0 . 0 9 0 l +*j6 . 0 9 7 5 0. 5+j*7. 7898 f I .3932-j*13.8731 0 0 - 0 . 2 ? I + + j * 7. 6 2 8 9 0 . 1 5 9 1 = j * 6 . 1 1 6; 8 1 d] - 0 . 0 9 0 l +*i6 . 0 9 7 5 .o % S t e p4 E % S e t t h e v a ] u e sf o r i n i t i a l t i m e t ( o c c u r a n coef f a u l t ) a n d c % t c i s t i m e a t w h i c hf a u l t i s c l e a r e d .g t = 0 ;t c = 0 . 0 8t;f i n a l=1. 0 ; o r=1; P E dash_r =Eda;sEhd0ash_rupsl1=Edash; 0 E whilet < tfinal, o() %Step5 ComputGe eneratorPowersusing appropriateYBus % t h e Y B u sc h o s e ni n t h e f o l l o w i n gs t e p i s s e t a c c o r d i n gt o t h e c u r r e n t oc =(\\' c %time . c o) o) o i f t < = t c Y B u s = Y B u sedlfs;e Y B u s = Y B u s p f ; e n d E % N otet hat her ewe obt ain t he cur r ent siniect ed at gener at orbus c .E % by mu lt ip'lyingt he cor r espond'inr ogw of t he Ybuswjt h t he ve\\ t or of at, (ou % v o l t a g e sb e h i n dt h e t r a n s i e n t r e a c t a n c e sT.h i s s h o u l da l s o j n c l u d e IU EL % slack bus voltage % a n d h e n c e t h e e n t r Y 1 a P P e a r sj n t h e b u s v o l t a g e v e c t o r i n a d d i t i o n t o .s gc Lo oo % generator bus vo'ltages eo, - 1 , (I' I=YBus(2:m+1,:)*[1 L oO ' o /C - o) E d a s hr l ; o oL c P g _ r = r e a 1( E d a s h _ r . * c o n(iI ) ) ; ts %Step6 computexldot-r and xZdot-r =F xldot_r=x2_r; for k=1:m, _(oU x Z d o t - r (k , 1) =( p i * f / H ( k ) ) * ( P g O( k )- P g - r ( k )) ; o end c % S t e p7 C o m p u t ef i r s t s t a t e e s t i m a t e s f o r t = t ( r + 1 ) .9 x 1_rp1usI =xl_r+xldot-r*tsteP ; o xZ_rpl us1=x2_r+x2dot_r*tsteP; c (g % S t e p8 C o m p u t ef i r s t e s t i m a t e s o f E ' - r + 1 E d a s hr p l u s 1 = a b s ( E d a s h. *0()c o s( x 1 - r p 1u s l ) + i * s i n ( x l - r p l u s l ) ) ; L % S t e p9 C o m p u t eP g f o r t = t ( r + 1 ) F I=YBus(2:m+1,:)*[1 Edash_rpu'ls1l ; u? (5 P g _ r p 'ul s 1 = r e a l( E d a s h _ r pu1s 1 . * c o n i( I ) ) ; 6 % S t e p1 0 C o m p u t eS t a t e - d e n i v a t i v e s ' a t t = t ( r + 1 ) IL

W Mod\"rnpo*\",.sy.t.r An\"ryri, Iffi xldot_rplrtl=J0 J,i 'Fl x Z dot _r prlrl = 1 0 0 J, i for k=1:m, 88 x l d o t _ r p ul s l( k ,1 )= x 2 _ r pu'sl l( k,l) ; x 2 d o t r. p l u s l( k ;l ) = p i* f / H( k )* (pq g( % s t e p1 1 c o m p u taev e r a g ev a r u e so f s t a t e d e r i v a t i v e s xI dotav_r=(xldot_r+x1dot_rpu'sl l)/ ?.0; x2d o t av _r(=x 2 d o t_ r+ x Z d o t_urps1l )/2 .0; %st e pL2 C o mp u tfej n a l State estt'ma t efsb r t=t(r+1) xl_r p l us1= x l _ r+ x l d o ta v _ r* ts t;e p x 2_r plus lix Z r+ x 2 d o ta v _ r* ts te; p % s t e p1 3 C o m p u ft ien a l e s t i m a t Edash_r ups11 = a b(sE d a s h.0*)( c o s( ef o r Eda s ha t t =t(r+ 1) 1u s l )) ; Print StateVector xl_r p lu s1 )+ j* si n( x t _ rp % S t e p1 4 x ? _ r = x ? _ r pu ls l ; x l _ r = x l _ r pu1s l ; E N o tt oE E das h _ r= E d a s hu_sIrp; ' l o o %S t ep15 o sa o o 3o E> Il E E i l $ n 6J .tt -8 .= II f o oll time(r)=1; 6 U' ct for k=l:m, o a n g( r , k )=( x t _ r ( k*) 1 8 0/)p i ; c end .9 t=t+tstep; cU,' r-r+1; 6 F end .o= plot(time,ang) .cct, Ex. G. 17 (Example IZ,II) G' % E x a m p7l e2 . 1 1i E s o l v e du s i n g S I M U L I N K % T h ec o d eg i v e nb e l o ws h o u l db e r u n p r i o r t o E clear alI s i m u ' l a t i osnh o winn Fig.G.6. = g l o b a 1n r y y r q (J dt IL g l o b a l P m f H Ey n g g global r t d d tr % c o n v e rs j ofna c to r ra d /d egree 91obal Y bf Yd f y a f f=50; ngg=2; r=5; nbus=r; rtd=180/pi; d t r = p i/ 1 8 0 i 9o Gen. Ra Xd' H gendata=[ 00 inf 1 0 0.067 12.00 0 0.100 9.00 l; 2 3

iltl ModerPnowesrvstemAnatvsts - AppendiGx ydf'I 0 -0.0681+J*5.1661 IE 5.798 6 -*J3 5 .6 3 0 1 -j*11.236 0 E2 0 0 0 . 1 3 6 2 - j * 6 . 2 7 3i 7 1 - 0 . 0 6 8 1 +* 5i .1 6 6 1 Y af =[ -0.2 2 I4 + j * 7.6?89 - 0 . 0 9 0 1 +* j6. 0 9 75 Complex 0.5+j*7.7898 0 1.3932-j*13.8731 0 Yaf(2,1) - 0 . 2 2 L 4 + j * 7. 6 2 8 9 0 . 1 5 9 1 - j * 6 . 1 1 6; 8 1 - 0 . 0 9 0 1 +* 6j . 0 9 7 5 f c t = i n p u t ( ' f a u l t c l e a r i n gt i m e f c t = ' ) ; Product21 &Damnpgi factors d a m p 2 =0O; . dam p 3 =. 00; %Ini t i a l g e n e ra to rA n ge' ls d2-0,3377*rtdl d3=0.31845*rtd; %Initial Powers Pn2=3.25; Pm3=. 210; %G en e ra toInr te rn a l V o l ta g e s E l . =1. 0 ; E 2 =.10 3 ; E3=1.02; % M a c h i nIen e r t i a C o n s t a n t s ; H 2 = g e n daa( 2t , 4 )t 1 1 3 = g e n d(a3t,a4 ); & M a cnhel X d '; 1 6 6 1 = g e n d(a2t a, 3 ); 1 6 6 1 = g e n d (a3t a, 3 ); Note : For the simulationfor multimachine stability the'two summation boxes Flg. G.7 MultimachinetransientStability: Subsystem1 sum 2 & sum 3 give the net acceleating powers Po2 and Por.T}iregains of the gain blocks Gl, G2 G3 and G4 are set equal to pi*fAlZ, dampZ,pi*f/Fl3 and damp3, respectively. The accelerating power P\" is then integrated twice for each mahcine to give the rotor angles 4 ^d 4. Th\" initial conditions for integrator biocks integrater i, integrator 2,integrator 3 anciintegrator 4 are set to 0, d2/r+d, 0 and d3lrtd, respectively.The gain blocks G5 and G6 convert the angles $ and d, into degreesand hencetheir gains are set to rtd. The electrical power P\"2ts calculatedby using two subsystems1 and 2.The detailed diagram for subsystem 1 is shown in Fig. G.7. Subsystem I gives two outputs (i) complex voltage Etl6 and (ii) current of generator 12 which is equal to

vodernpowesr ysterRnalysis ffi E/-61 *Yar(2,1)+ Erlq*Yq{z,z) + ErlErYar(2,3).Theswitchesareusedto switch betweenprefault andpostfault conditionsfor eachmachineand their thresholdvaluesare adjustedto fct, i.e. the fault clearingtime. clear stabi 'li ty rs to Problems % T h i s Program fi nds the reduced matrix for s t u d ie s % w h i c h e l i m n a t e s t h e I o a dI b u s e s a n d r e t a i n s o n l y t h e g e n e r a t o rb u s e s % lIgll IYll YL? . Yln Yln+l YLn+Z. Y1n+ml IV1 CHAPTER 2 . YZn YZn+l YZn+Z. Yln+ml IV2 % lIszl iIY21. Y?r %l.l tl I vol.l I 2.1 Lirt = -1x ro-x7 -,lt-+n,e-|,,,)+arlr?] 2 *Vlr,: Y ol l g n l = l y n l YnZ . Ynn Ynn+1Ynn+2Ynn+m Vn IYn+12 Yn+Lm V n+ 1 2.2 0.658ohmlkm % l r L ll V n+ 1 yo l r L 2l I %l.l I 2.3L= F loR w^ I 2r r vol.l I 2.4 260.3Vlkm % lILml Yn+mn+m Vn*m IYn+m1 Yfull= _j*5 0 7*5 2.5 Hne= - 3.,:nId, AT/mz(directedupwards) -j*7.5 j*2.5 0 j*5 j*?.5 -j*12.51 [ r o w c o l u m n s ] = s j z e( Y f u l l ) (Xt-Xn)(X2-Xn) n=2 2.6X= \\ * x2-zxn Y A A = Y f u l (l 1 : n , 1 : n ) YAB;Yfull(1:n, n+1;columns) 2.7 0.00067m}J/krlt,0.0314V/r<Tl Y B A = Y f u1l ( n + 1 :r o w s ,n + L :c o lu m n s ) 2.8 0.W44/-t4C, ml{lkm, 0.553/.l40 Vlkm Y B B = Y f u1l ( n + 1 :r o w s ,n + 1: c o 1u m s ) % T h i s g i v e s t h e r e d u c e dm a t r i x 2.9 0.346ohmtkm 2.rc 1.48m Y r e d u c e d = Y A A - Y AnBv*(iY B B*)Y B A 2.ll 0.191ohmlkm/phas2e.12 0.455mlllkm/phase 2 . 1 32 . 3 8m 2.14 (i) 0.557 dtt2Att4 (ii) 0.633d2t3A16 er) 0.746d3t4Ar/8 Chapter 3 3.1 nru^_ 2 rk lVl(ln(r/ 2D)130\"_!n(DlLr)l_30o) -t',tm, 2h(D/r)ln(r/2nffi Io = 2rfqo/90\" A 3 . 2 0.0204p,Flkrrr 3.3 0.0096pFlkm 3.5 3.08x iO-scoulomb/km 3\"4 O OlO? ttFl1r'm rLnv unvau' lL. -l^4rl 3.7 8.54x 103ohmslkrn /r^,rutr 3.9 71.24kv 3.6 5,53x 10{ mhoslkm 3.8 8.72x l0-3 1fi/xm Chapter 4 4.1 12kV

W ModrrnPo*\"r. syrt\"t An\"lv.i. IE Chapter 5 4 5.1 (a) 992.75kW (b) No solutionpossible (b) 5 . 2 A t - 0 . 9 1 I . 5 o ,B ' = 2 3 9 . 9 1 6 6 . 3C\" ,' = 0 . 0 01 1 0 2 . 6 ,D t = 0 . 8 5 1 I . 9 6 j0.3049 j0.1694 (b) 165.44ky,0.2441-28.3\"kA, 0.808lagging,56.49MW j0.1948 j0.3134 (c) 7O.8Vo (d) 28.L5Vo 0.807_ js.6s 0.645_ j4,517 5.4 (a) 273.5MVA (b) l,IT4 A (c) 467.7MVA 0.968- j6.776 0.968_ j6.776 5.5 133.92kV, 23.12MW 5.6 202.2kV 0.880_ j6.160 5.7 At x = 0, irr = 0.3L4cosQ,s-t21.7\"), irz=0.117 cos (utt+ 109\"), (c) At x = 200km, it = 0.327cos(c..r-/ 9.3o),irz = 0.t12 cos(,*t+ 96.6) tz} 5.8 135.817.8okV, 0.138115.6okA, 0.99 leading,55.66 MW, 89.8Vo, 373.L1- t.5o,3,338km, 1,66,900km/sec 5.g v - 12g.3172.60Y,' - 0.000511g9.5\". 2 5.10 7.12\",pfr = 0.7 laggingp, fz= 0.74 \\agging 5.11 47.56MVAR lagging 5.I2 10.97kV, 0.98leading-, 0.27Vo,86.2Vo 5.13 51.16kV, 38.87MVAR leading4, 0 MW 6.5 Pn = - 0.598pu, PB = 0.2 pu, pzs= 0.796pu 5.I4 238.5kV, P,+ jQ, = 53 - jI0, pf - 0.983leading 5\"15 17.39MVAR leading3, .54MW Qn= = 0.036pu, (a) Pn 0.58pu, P* Chapter 6 6.6 Qzr Qn= Qn= 0.004 pu,ezz=ezz = 0.064pu =- = 0.214p u, pzl = 0.792pu 6.1 For thisnetworktreeis shownin Fig.6.3a;Ais givenby Eq. (6.17).The Qn = - 0.165pu, ezt _ 0.243pu, en = 0.204pu matrix is not unique.It dependsuponthe orientationof the elements. Qy = - 0.188pv, Qzt = 0.479pu, en= - 0.321pu (b) Pn = - 0.333Pu, Pzz=0.664pu, Pr: = - 0.333pu 6.2 V) = 0.9721-8.15\" 6.3 V) = 1.26l:74.66\" ' o n = Q z r= 0 ' 0 1 1 p uQ, B = Q z r= 0 . 0 1 1p u ,e z t = e n = g . g 4 4 p u 6.a@) 6.7 ( a )(i) f-iro.rors j5.o5o5 js I l--ito s/s3\" i5 I ;s.osos -j10 js -ilo | I tul ls tj s8 7\" j s I j5 _j10 js L js J L _j l0j t I UUU \\f .hr\\,/,i \\\\ i,, iD 12:- n <vn.niJ- .V. U pu, tns13= Un .AL^UA Z pu, pzZ = 0.794 pu 2 0 I 0 Qr Qn= 0.087pu, Qzt = - 0.014pu Qn= QzF 0.004 pu, Qzz= Qsz= 0.064pu 3 00I 0 \\//u::\\ , , fI) 1 2 : - tu\\ ..ooEo J p u , 40 00 I --:- Pn = 0.287pu, Pzt= 0.711pu 0 1 -l )0 00 1 Qn = 0.047 Pu,Qn= 0.008 ptr, Qzt = 0.051pu 6 -l 6.8 Vrt = I.025- y0.095= l.AZ9/.- 5.3. pu 7 l-1 0 0 I 0-l 0 1 9 - l0 l 0

4 Mooernpowerslrstemnnatysii RL.,-iiiis;E: Chapter 7 lit+i(Atlldt I-- 7.1 Rs 22.5/hr 8.1 Load on 123MW, Load on G2= 277 MW, 50.77H).. 2 7.2 (a) Pcr = 140.9MW, 1 5 9 . 1M W S4VlIl$ = 7 . 3 (i) Gen A will sharemore load than Gen B 8.2 Af (t) = - 0.029 - 0.04ea.58c' os (1.254t + 137.g\") (ii) Gen A andGen B will shareload of po each 8.3 1/(50K) sec (iii) Gen B will sharemore load thanGen A. 9.4 aPtie. | - 7.4 Pcr = 148MW, Pez=, L42.9MW, Pct = 109.1MW 7.5 (dcldPd=0.175Pc+23 _ ll R2) an (K iz+D0 / K p,1* K ipl +l / Rt)+ (K, r 7 . 6 (a) Pq = 138.89MW, Pcz = 150MW, Po = 269.6MW 0) Pcr = 310.8MW, Pcz = 55.4MW 8.5 APt i. , r ( s)= - t O OOf 'zt t t 0. 9t L 8 0 s 51 (c) For part(a): Cr = Rs 6,465.141hr Forpart (b): Cr = Rs 7,708.15/hr System is found to be unstable. 7 . 7 Bn = 0.03387pu or 0.03387x ITa MW-l Chapter 9 Brz = 9.6073x 10-5pu or 9.6073x 10-7MW-t Bzz= 0.02370pu or 0.02370x 10-2MW-r. 9 . r it = 3.14 sin (314t - 66.) + 2.97e sor,;\"* = 5A 7 . 8 EconomicallyoptimumUC 9.2 (a) 81\" (b) - 9' Load (MW) Unit Number 9.3 (1)Ie = 2.386kA, .IB- 1.75kA (ii) IA = 4.373 kA, IB = |.ZS ta 1234 9.4 8.87kA, 4.93kA 9.5 26.96kA 1111 9.6 6.97kA r110 0-4 20 1100 9.7 (a)0.9277kA (b) 1.312kA (c) 1.4843kA (d) 1.0205kv, 53.03MVA 4-8 l4 1110 (e) 0.1959kA 8-r2 6 1000 t2-16 l4 1100 9 . 8 8 . 3 1 9k A 9.9 2.39pu r6-20 4 20-24 10 9 . 1 0 132.1,47.9; 136.9,45.6 gJt 0.6pu 9 , 1 2{ - - j8,006pu, 1{: = - 74.004pu Optimal andsecureUC Chapter 10 Period Unit Number 10.1 (i) 1.7321270(\"ii) zl0\" (iii) 1.7321150(.iv) tl2r0\" lO.2 IA= jl.l6 pu, Vtn = 1.171109.5p\"u 04 t234 4-8 lvl B C -- vn . zO <i r2z _. /- o4 .J< . +A o- p- -u- , YV' C B= U . 9 9 5 2 _ I l 3 . l o p u . 8-12 1111 193 Vor= 1l997.2.V3_l-13.3V.3A\" ,VI,oVzo=zJIg2.203.ZlIlSl5V1.Alo,1Vo,oV=\"o0=A21.61110.63V\" I2_16 1 ' 11 0 10.4 lor = l6-20 1 10 0. 20-24 1 1 , 10 10.5 1,,=27-87,/3ff A ^tA,Z- - '-t ? L. J/1-- AAo.loA r n a 1100 ^r a-'.ZJ Il, lA0 = U A rt00 Iobr= 16.1A, Iab2= 7.5/_-75\"A, Iob1=l.,S/lS e, 10.6Io = 1 6 . 1 6+ j 1 . 3 3 5A , Iu= _ 9 . 2 4_ \" 1 \\ O . A Oe , Ic - - 6.93+ j9.32A, lVN,l= lVos=l 40.75y I l 7.9 Total operatingcost (both units in service for24 hrs) = Rs 1,47,960 t0.7 1,500.2w Total operating cost (unit tr put off in light load period) = Rs 1,45,g40

ffi4 ModernPoweSr ystemAnal'sis Chapter 11 Index 11.1 - i6.56 kA, lV6,l- 12.83kV, lvobl= 6.61 kV, ly\"rl = 6'61 kV = I = -2./1 (b) V*= Vo, = 0'816pu, 116=l llrl = 5'69 pu ABCD constants 61,7 B-coefficients 261.267 11.3 (i) - i6.2.5(ii) - 4.33 (iii) 6'01 (iv) - /5 pu for varioussimplenetworks 618 Bad data detection 547 In order of decreasingmagnitudeof line currentsthe faults can be listed in powerflow equations lS9 Bad data AS measurementof 621 detection 547 (a) LG (b) LLG (c) 3-Phase(d) LL of networks in seriesand parallel 620 identification 547 Il.4 0.1936ohm,0.581ohm,- 4.33pu,j5 pu AC calculatingboard lB4 suppression 548 11.5(a)3.51pu(b)Vt,=I.|91_159.5\"po,V,=I,681|29.8\"pt1 Accelerationfactor 207 treatment 546 Acceleratingpower 462 Baseload 3 (c) 0.726Pu Acid rain 16 Basevalue 99 1 1 . 6I u = - I r = - 2 . 8 8 7 P u Adjoint matrix 609 BharatHeavy Electricals ||.7 (a) Iv = _ 5.79+ j5.01kA, 18= 5.79+ j5.01kA, ,IG= j10.02kA Admittancematrix (seeBus admittance Lrd (BHEL) 14,31 (b) /n - - IY = - 6'111kA, Ic= 0 matrix) Boiling water reactor 19 Advancedgasreactor(AGR) 19 Breakingresistorsfor 11.8 In, = 0 Io^ = - 73'51Pu Algorithm improving stabiliry 499 Ius= - 72.08pu Ib^= - jl.Z Piu for building the Zru, 355 Branch 190 I,^ = - jl.Z Pu for load flow solution Breakers (see Circuit breakers) IrB= 72.08pu by GS method 205 Brown,H. E. 368 1 1 . 9 5 , 2 6 6A by NR method 214 Bundledconductors, \\ 11.10j2.0 pt by FDLF 223 capacitivereactance 92 11.11 / = - j6.732Pt, Io(A)= - i4.779pu, for optimum generation inductivereactance 68 /a(A) = - i0.255 po, 1\"(A) - - j0'255 prt scheduling 262 Bus l|lz 0.42ptt,- j9'256Pu 11.13 -ill.l52 pu,-i2.478pu, -Jl -239pn for optimal hydrothermal generator 198 '1.14 4.737Pu,1 Pu r 1.15 f, = - i12.547pu, Ifrr(b)= - i0.0962pu scheduling 280 Ioad 198 for optimal loadingof generators PQ 198 on a bus 246 PV 198 for optimal load flow solution 273 reference 198 for shortcircuit current slack computation 343 Bus admittancematrix 188 Chapter 12 for short circuit studies 349 formulationof 187.189 for staticstateestimationof Bus impedancematrix l2.I 4.19MJA4VA,0.0547MJ-sec/eledceg power systems 540 building algorithm 355 12.2 4.315MJA4VA 12.3 40'4 MJiTVIVA for transientstability, analysis for unsymmetricalfault analysis 416 of largesystem 4gS 12.4 140.1MW, 130.63MW, 175.67MW ri nr r Dc rJr ar r* r^ r+t-lls^t^tlr u c u lf a^ .t.uf ir l . ---!---' -Er Alternator (seaSynchronousmachines) Aluminum conductorsteelreinforced al|alysls JJ I 12.64 = 58' Bus incidencematrix I92 r2.5 72.54MW (ACSR) conductors 52 Capacitance !2.7 L27.3MW AAr v*ca a aL^r- r+ l- l^l1l ( I ^-^- / a ^F\\ 4A f, u^ ^ 4rf ^ - - u1 ^ u- i -l-i- luontoy me-mt oooI modrlred geometric mean distances 9l Ellul \\f\\\\-.8, JU+ 12.8 53'. We need to know the inertia constant M to determine /.' Armature leakage reactance 331 12.9 The system is stable 12.1070.54\"0, .1725sec Armaturereaction 109, 110,330 effect of earth on 83 l2.Il The systemis unstable t2.t2 63.36\" Attenuationconstant 143 effectof non-uniform 12.14 The systemis stable Augmentedcost function 279 chargedistribution 79 12.13 The system is stable Automatic generationconirol (seeLoad effect of strandedconductors 79 frequencycontrol) line-to-line 78 12.15The system is unstablefor both three pole and single pole switching Automaticvoltagccontroi 318 line-to-neutral 79

tndex of parallel circuit three-phase for hydro-thermal scheduling lines 88 . problems 277 of three-phaselines: with Contingency equilateral spacing 80 analysis 51 with unsymmetricalspacing 81 rankins 520 a two-wire line (seealso Reactance) 78 screening 520 Digiral LF controllers 322 the medium length line Direct axis reactance llg CentralElectricity Authority (CEA) Zg selection 515 Disturbanceparameters 271, 272 nominal_zr representation l3g Charpcteristic (surge) impedance l4l ' directmethod 515 Distribution factors nominal-Trepresentation 137 short transmission line l}g qf .linesand cables 145 indirect method 515 generation_shift SZ0 synchronousgenerator llg Chaiging current 76, 180 Iine-outage 520 Error amplifier 3lg Circle diagrams 167 Contingencyanalysis Diversity factor 4 Circuit breakers 327, 329 Dommel, H. W. 270 by dc model 520 Doubling effect 3Zg autoreclose 460 Double line-to-ground(LLG) fault 404 rated intemrpting capacity of 344 by distribution factors 521 Driving point (self) admittance lg7 Error signat 319 ratedmomentarycurrentof 344 Dual variables 279,632 selectionof 344 modelling for 5I4 Dynamic programming Estimation methods Circulating currents 389 Cogeneration 15 Contingencyevaluation applied to unit commitment 251 least squares 532 Coherentgroup 291,303 Compact storagescheme 189,628 (seecontingencyanalysis) weighted least-squares 533 Comparisonof angle and voltage Control Estimationof periodic components 5gl stability 592 Exact coordination equation 26I Compensation area 291,303,306 series 779,498, ,Sg by transformers 23I Expectation 532 shunt 179,498,562 Compensator integral 304 External systemequivalencing 545 combinedTCR and TSC .564 isochronous 305 Facrors affecting srability 496 shunt 562 FACTS controllers 569 of voltage profile 230 Economic dispatch(seealso optimum Failure rate 255 static synchronousseries(SSCC) 571 generationscheduling) 306 Complex power 105 of WAfiS andVARS along a Fast breeder reactor 22 Compositeconductors Electricity Board Zgl Fast decoupled load flow 223 transmission line 230 Elements 189 Fasr valving 49,9 capacitance 9l inductance 54 optimal 310 Energy conservation 3l Faults Computational flow chart for load flow Energy control centre 637 parameters 199,272 solution using FDLF 227 proportioiral pft\"s-fuBral 303 GS method 20S NR method 224 suboptimal 318 !MS.(Energymanagement system) 634 balanced(seesymmetricalfault Conductors Energy sources analysis) Control areaconcept 303 ACSR 52 conventional 13 calculationusing Zsus 351,417 Bundled 52,68 Control strategy 303 expandedACSR 52 hydroelectricpower generation l7 openconductor 414 iypes 5i Control variables 199, 632 nuclearpower stations lg unbalanced thermalpower stations 13 Conductance 45 Control vector 199 biofuels 28 series type 397 Constraints shunttype 397 Controller biomass 28 Ferranrieffect 150 equality 272, 632 gas turbines 16 Field erciretion l0g inequality 632 interline power flow (IpFC) 571 geotbermal power plants 24 Field rriading 32g interphasepower (IPC) 572 magnetohydrodynamic (MHDI Flat voltagestarr 205,209,21g on control variables 274 Flexible AC transmissionsystems on dependentvariables 274 unified power flow (UPFC) 571 generation 23 (F-ACTS) 566 Converters 567 OTEC 27 Fluidized-bedboiler 15 Flux linkages Ccxrrdinatisn equations 2fr renewable Zs Corona 52,68 solar energy 26 wave energy 27 Cost function 251,270 \\i ,-tu^ el l-(^ i^( ' 1nn L>V Covariancematrix 535 Critical clearing angleand time 467 wind power stations 25 external 49 Current distribution factors 265 Energy storage Zg Current limiting reactors 346 internal 46 fuel cells 29 DAC (Distribution Automation and of an isolatedcurrentcarrying Control System) 634 hydrogen Zg conductor 46 Damper winding 438 pumpedstorageplanf lg ,rFolyrebcaalsl stlpnegemdegtohvoedronloorg Zgz DC network analyser 39 secondarybatteries 2g y 577 Data acquisition systems(DAS) 636 Equal area criterion 461 Fortescue,C.L. 370,3t6 Equal incremental fuel cost criterion 246 Frequencybias 309 Fuel cost ofgenerators 243