I J Nagrath is Adjunct Professor, BITS Pilani and retired as professor of. Electrical Modern Power System. Analysis. Third Edition. D P Kothari. Vice Chancellor. VIT University. Vellore energy of water. The energy is obtained almost free of nrnning cost . shutting down of the reactor with all its consequences. Because of. Ebooks:Power systems books. Modern power systems analysis 3rd Edition by D P Kothari & I J Nagrath pdf Free Download:Click Link 'Download here'. Chapter 6 gives power network modelling and load flow analysis, while 1. quite serious and may required shutting down of the reactor with all its consequences. A nuclear power plant is totally free of air pollution. transfers it to a heat exchanger to Nagrath,I.J. and D.P. Kothari, Electric Machines,Tata McGraw-Hill.
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In simple language, book providesa modernintroduction power the to system operation,controland analysis. Key Features of the Third. New appendices on: Real time computercontrol of power systems. The book is very comprehensive, well organised,up-to-dateand above all lucid and easy to follow for self-study. He has uiro t. Earlier lflaz-s:
Urban and industrial wastesare used for variousenergy applications including power generationwhich was around 17 Mw in The in thesehave already'been kinds are concerned, problerns relatedto largelrydroplantshavealsobeendwelledupon in Section 1. Therefore,we shall now focus our attentionon fossil fuel plant including plants.
Pollutants know no geographical boundary, as result the pollution issue has become a nightmarish problem and strong national and international pressuregroups have sprung up and they are having a definite has awareness Governmental of impacton the development energyresources.
Lengthy, time consuming level, PIL public interestlitigation and demonstraprocedures governrnent at tive protestshave delayedseveralprojects in severalcountries. This has led of projectsand redevelopment existing sites. But to favouring of small-size with the increasinggap in electric dernandand production,our country has to move forward fbr severallarge thermal, hydro and nuclear power projects. It has to be devektpnrenlwittr uppntpriata technolog- ' particularly assuredthat no irreversible damageis caused to environment which wouid affect the living conditions of the future generations.
Irreversible in by like ozonelayer holesand global warmingcaused increase CO2 damages up. We shall treat here only pollutrorras causedby thermalplants using coal as feedstock. Certain issues concerning this have already been highlighted in Section 1. The fossil fuel based generatingplants fonn the backbone of power generation in our country and also giobally as other options like nuclear and even hydro have even stronger hazardsassociatedwith them.
Also it should be understood that pollution in large cities like Delhi is caused more by vehicrtlar traffic and their emission. In Delhi of course Inderprastha and Badarpur power stationscontributetheir share in certain areas.
Problematic pollutants in emission of coal-basedgeneratingplants are. On the other hand CO2 has been identified developingcountries. Ifydrocarbons During the oxidation process in cornbustioncharnbercertain light weight hydrocarbon may be formed.
Tire compounds are a major source of photochemical reaction that adds to depleti,rnof ozone layer. Particulates fIY ash.
Certain hydrocarbons o Particulates Though the account that follows will be general, it needs to be mentioned here that Indian coal has comparatively low sulphur content but a very high ash content which in some coals may be as high as 53Vo. A brief account of various pollutants, their likely impact and methods of abatements presentedas follows.
Most of the sulphur present in the fossil fuel is oxidized to SO2 in the combustion chamberbefore being emittedby the chimney.
In atmosphere it gets further oxidized to HrSOo and metallic sulphateswhich are the major sourceof concern as these can causeacid rain, impaired visibility, damageto buildings and vegetation. Sulphate concenffations of 9 LElm3 of air aggravate asthma,lung and heart disease.
Sulphur emissioncan be controlledby: Removing sulphurfrom the coal by gasificationor floatationprocesses. It has been noticed that the byproduct sulphur could off-set the cost of sulphur recovery plant. Dust content is particularly high in the Indian coal. Particulatescome out of the stack in the form of fly ash. It comprisesfine particles of carbon, ash and other inert materials. In high concentrations,these cause poor visibility and respiratory diseases. Concentration of pollutants can be reducedby dispersal over a wider area by use of high stacks.
Precipitators can be used to remove particles as the flue gasesrise up the stack. If in the stack a vertical wire is strung in the middle and charged to a high negative potential, it emits electrons. These electrons are captured by the gas molecules therebybecomingnegative ions. These ions accelerate towards the walls, get neutralized on hitting the'walls and the particles drop down the walls.
Precipitatorshave high efficiency up to 99Vofor large particles, but they have poor performancefor particles of size less than is 0. The efficiency of precipitators high with reasonable in flue gasesbut drops for'low sulphurcontentcoals;99Vofor sulphur content 37o sulphur and 83Vofor 0. Fabric filters in form of bag lnuses have also been employed and are located before the flue gases enter the stack.
Thermal Pollution. Of theseNOz, nitrogenoxides,is a majorconcernas a pollutant. At ievels of parts per million NO, can cause acutebronchitis and pneumonia.
These can also be removedfrom the combustionproducts by absorptionprocessby certain solventsgoing on to the stock. Steam fronr low-pressureturbine has to be liquefied in a condenser and reduced to lowest possible temperatureto maximize the thermodynamic practicallyachievableis about efficiency. It meansthat60Vo the heat in steamat This is achievedby following two methods' cooling tubes of seaor river 1. Once through circulation through condenser of water where available.
This raises the temperature water in these two sources and threatenssea and river life around in sea and downstream in river.
ThesE,are serious environmental objections and many times cannot be overruled ard also there may be legislation againstit. Cooling tov,ers Cool water is circulatedrottnd the condensertube to remove heat from the exhaust steam in order to condenseit.
Some of the water evaporates providingcooling. The power system engineerof the first decadeof the twenty-first century has abreastof the recent scientific advancesand the latest techniques. On the planning side, he or she has to make decisions on how much electricity to generate-where, when, and by using what fuel. He has to be involved in and transmission. He constructiontasksof greatmagnitudeboth in generation has to solve the problemsof planning and coordinatedoperationof a vast and complex power network, so as to achieve a high degree of economy and reliability.
In a country like India, he has to additionally face the perennial problem of power shortages and to evolve strategies energyconservation for and load management. He has to know the principlesof economicload despatchand load frequency control. All these problems are dealt with in the next few chapters after some basic concepts in the theory of transmission lines are discussed. The solutions to these problems and the enormouscontribution problems of to made by digital cornputers solve the planning and operational power systemsis also investigated.
Closed cooling towers where condenr;ate flows through tubcs anclair is blown in thesetubesavoidsthe humidity problembut at a very high cost. In India only v,et towers are being used. Electromagnetic Radiation from Overhead Lines.
Biological effects of electromagnetic radiation from power lines and even cables in close proximity of buildings have recently attractedattentionand have also causedsomeconcern. Power frequency 50 or 60 Hz and even their harmonics are not considered harmful. Investigations carried out in certain advanced countries have so far proved inconclusive.
The electrical and electronics engineers, while being aware of this controversy, must know that many other environmentalagentsare moving around that can causefar greater harm to human health than does electromagnetic radiation. Visual and Audible Impacts These environmentalproblems are causedby the following factors.
Right of way acquiresland underneath. Not a seriousproblernin India at present. Could be a problem in future. Lines converging a large substation at mar the beauty of the lanclscape around. Underground cablesas alternativeare too expensivea proposition except in congestecl city areas. Phenomenon corona a sort of electric dischargearound the high of tension line producesa hissingnoise which is aucliblewhen habitation is in close proximity.
At the to'wers great attention must be paid to tightness of joints, avoidance of sharp edges and use of earth screen shielding to lirnit audible noise to acceptable levels. To reduce this uoise to tolerable level foundations and vibration filters have to be designed properly and simulation studiescarried out.
The worker nlust be given regularmedical examinations and sound medical advice. AC analysers network analysers florv and stability studieswhereasDC were preferredfor short-circuitstudies. In s many analogue devices were developed to control the on-line Ii'equencyand tie-line controt. Power system studiesby computers gave greater flexibility, accuracy,speedand economy. Till s,there was a widespreaduse of computersin systemanalysis.
With the entry of microprocessors the arena,now, besidesmain frame compLlters, mini, micro and in personalcomputersare all increasinglybeing used to carry out various power systern studies and solve power system problems for off-line and on-line applications.
Off-line applications include research, routine evaluation of system performanceand data assimilationand retrieval. It is mainly usedfor planning and arralysing some new aspects of the system. On-line and real time applications include data-logging and the monitoring of the system state. Microprocessors ancl computersinstalledin generating stations control various local processes such as startingup of a generator from the cold state,etc.
Table 1. The transmission loss in on a national basiswas It should be possibleto achieve considerable savingby leducingthis lossto by the end of the Tenth Five Year Plan by r-rsing well known ways and nreans and by adooting sound commercial practices. Further, evcry attempt should be made to improve system load factors by flattening the load curve by giving proper tariff incentives and taking other administrativem.
In future it is likely to be 7I7o. By i,5. Assuming a very modest averageannualenergy growth of 5Vo,India's electrical energy requirementin the year will be enormouslyhigh.
A difficult and challengingtask of planning,engineeringand constructingnew power stationsis imrninentto rneetthis situation. The governnlent has bLrilt. Manv more super thermal plants would be built in future.
Intensive work must be conductedon boiler furnaces to burn coal with high ash content. Hydro power will continue to remain cheaper than the other types for the next decade. As mentioned earlier, India has so far developed only around l87o of its estimatedtotal hydro potentialof MW. The utilization of in this perennialsource of energy would involve massiveinvestments dams, system.
The Central Electricity Authorchannelsand generation-transrnission ity, the Planning Commissionand the Ministry of Power are coordinating to work out a perspectiveplan to develop all hydroelectric sourcesby the end of this century to be executed by the National Hydro Power Corporation NHPC.
NTPC has also startedrecently developmentof hydro plants. Nuclear energy assumesspecialsignificancein energy planning in India.
According to the Atomic to to will increase Energy Commission,India's nuclearpower generation to MW by year Everything seems be set for a take off in nuclearpowel' production using the country's thorium reservesin breederreactors.
To can clemancl be met and depletingfbssil fuel resources that the coal productionwill have meet the energyrequirement,it is expected lts cotttpltrcdto q orc than. A number of kV lines are operating successfullysince s as a This was the firsi stepin working towards nationalgrid.
It is There is a need in future expecredrhat by the year 2Ol, ckt krn of kV lines and Also lines may be sericsand ckt kni gf kV lines would be in operation.
There shunt compensated is a needfor constructingHVDC High Voltage DC links in the country since more power at the samevoltageand require DC lines can carry considerably fewer conductors. At the time of writing, the whole energy sce is so clouded with future.
However, certain trends that will decide the future developments of electric power industry are clear. Generally,unit size will go further up from MW. A higher voltage kV will come eventually at the transmissionlevel. There is little chance for six-phasetransmission becomingpopular though there are few suchlines in USA.
More of HVDC lines will do-. As populhtion has already touched the million mark in India, we may see a trend to go toward undergroundtransmissionin urban areas. Public sector investment in power has increasedfrom Rs million in the First Plan to Rs million in the SevenrhPlan Shortfall in the Sixth Plan has been around 26Vo.
There have been serious power shortagesand generationand availability of power in turn have lagged too much from the industrial, agricultural and domestic requiremeni.
Huge amounts of funds of the order of Rs. Otherwise achieving a rarget of billion units of electric power will remain an utopian dream. Power grid is planning creation of transmissionhighways to conserve Right-of-way.
Strong national grid is being developedin phasedmanner. In 20Ol the interregional capacitywas MW. It is Lxpecredthat by 2OlI, it will be Mw. Improved o and M Operation and Maintenance technologieswhich are being used tgday are hotline maintenance,emergencyrestoration system, thermovision scanning, etc.
Because of power shortages,many of the industries, particularly powerintensive ones,have installed their own captive power plants. Consortiumof industrial. Import should be liberalized to support this activity.
Apart from being lower efficiency plants they use diesel which should be conservedfor transportationsector. It may also be pointed out that this book will also help in training and preparingthe large number of professionals trained in computeraided power system operationand control that would be required to handle v. New Delhi. Eilgerd, O. Sullivan, R. Krotzki, B. McMillan, J. I L Bennet, D. Power Generationand Environmentalchange, M. Steinberg, M. Systems, Power System Planning and Operations: Future Problemsand ResearchNeeds.
Twidell, J. N, spon, E. Mahalanabis, A. Kothari and S. RobertNoyes Ed. Kothari, D. Theory and practice, S. Sharma Eds , Energy Engineering. Chand, Kothari, D,P. Kiuwer, i Philipson,L and H. Papers Kusko, A. Fink, L. Talukdar, s. I, Morgen, M. Talukdar, 'Electric Power Load Management: IEEE, Feb. L, vol. Sachdev, M. Spom, P. Kothari D. P, and J. NPC Cong.
PowerLine, vol. United Nations. A General Study; Bhatti and D. IIT Saxena,Anshu, D.
Power System Analysis Books
Wb-T ' of the circuit in weber-turns ", ,ti" flux linkages 2. These are of course in phase. Therefore,from Eq. The conceptof mutual inductance requiredwhile consideringthe coupling is betweenparallel lines and the influence of power lines on telephonelines.
Transmissionlines are composedof parallel contluctorswhich, for all practical purposes, can be considered infinitely long. Let us first developexpressions as for flux linkages of a long isolatedcurrent-canying cylindricat conductor with return path lying at infinity. This systemforms a single-turncircuit, flux linking which is in the form of circular lines concentricto the conductor. The total flux can be divided into two parts,that which is internal to the conductor and the flux externalto the conductor.
Such a division is helpful as the internal flux progressivelylinks a smalleramountof current as we proceedinwards towards the centreof the conductor,while the external flux alwayslinks the total current inside the conductor. Flux Linkages due to Internal Flux.
Figure 2. Consider now an infinitesimal tubular elementof thicknessdy and length one metre. The effect of non-uniform current density is consideredlater in this chapter while treating resistance. Let the external point be at distanceD from the centre of the conductor. Flux linkages of the conductor due to external flux from the surface of the conductor up to the externalpoint is obtainedfrom Eq.
Here r' can be regarded as the radius of a fictitious conductor with no internal inductancebut the same total inductanceas the actual conductor. To start with, let us considerthe flux linkages of the circuit caused by current in conductor 1 only. We make three observationsin regard to these flux linkages: External flux from 11to D - , links all the current It in conductor 1.
For calculating the total inductance due to current in conductor 1, a simplifying assumption will now be made. Based on the above assumption,the flux linkages of the circuit causedby current in conductor 1 as per Eq. Considera simple two-wire line composed solid round conductors of carrying currents1, and 1, as shown in Fig. Similarly, the inductanceof the circuit due to current in conductor 2 is ' D. This is so because relative permeabilityof earth the is about the sameas that of air and its electricalconductivitv is relativelv small.
So far we have considered transmission lines consisting of single solid cylindrical conductors for forward and return paths. Stranded conductors are composed of strands of wire, electrically in parallel, with alternate layers spiralled in opposite direction to prevent unwinding.
The total number of strands M in concentrically stranded cables with total annular space filled with strands of rD uniform diameter is given by 2. The overall diameter D of a strandedconductoris. As shown in Fig. I istances of these an expression for the total flux linkages of the ith conductor of the group consideringflux up to the point P only. Aluminium is now the most commonlyemployedconductormaterial.
It has the advdntages being cheaperand lighter than copper though with less of conductivity and tensile strength. Low density and low conductivity result in larger overall conductordiameter,which offers anotherincidental advantagein high voltage lines. Increased diameter results in reduced electrical stress at conductor surfacefor a given voltage so that the line is coronafree.
The low tensile strengthof aluminium conductors made up by providing central is strandsof high tensilestrengthsteel. Sucha conductoris known as alurninium conductor steelreinforced ACSR and is most commonly used in overhead transmission lines.
The flux linkages of conductor i due to current in conductor 7 1rlf"r to Eq. From F,q. Theseare provided with paper or hessian between various layers of strandsso as to increasethe overall conductordiameter in an attemptto reduce electrical stressat conductor surfaceand prevent corona. The most effective per corona-free EHV linesis to provide several conductors way of constructing phase in suitable geometrical configuration.
These are known as bundled conductors and are a common practice now for EHV lines. Also for the sake of symmetry, denoting ,. We are now ready to study the inductanceof transmissionlines composedof compositeconductors. Future 2. Though the inductance each filament will of be somewhat different theirresistances be equalif conductordiameters will are chosento be unttorm , it is sufficiently accuratefo assum.
Note the similarity of the above relation with Eq. In Eq. The diameterof eachstrand 1. Solution The conductivity of steelbeing much poorer than that of aluminium and the internal inductance of steel strandsbeing p-times that of aluminium strands,the current conductedby the central strandsof steel can be assumedto be zero.
Thus, all strands are of the same diameter, say d. For the arrangement of strandsas given in Fig. A conductor is composed of seven identical copper strands, each having a radius r, as shownin Fig. Frnd the self GMD of the conductor. The arrangement conductorsof a single-phase of transmissionline is shown in Fig. Find the inductanceof each side of the line and that of the complete line. The basic equations So far we have considered only single-phase of to can,however,be easilyadapled the calculation the inductance developed line lines.
AIso voltages will be induced in adjacent communication lines even when line currents are balanced. This problem is tackled by exchanging the positions of the conductors at regular intervais aiong the line such that each conductor occupies the original position of every other conductor over an equal distance.
Such an exchange of conductor positions is called transposition. A complete transposition cycle is shown in Fig.
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This alrangement causes each conductor to have the same average inductance over the transposition cycle. Over the length of one transposition cycle, the total flux linkages and hence net voltage induced in a nearby telephone line is zero. This is the same relation as Eq. For all practical purposesthe and the inductancc an untransposecl can of line dissynrnrctry bc neglectcd can line.
Show that over the length of one transpositioncycle of a power line, the total line are z,ero, flux linkagesof a nearby telephone ftrr balanced three-phase currents. Thus thereis need to avoid the presence suchharmonic currentson power of Iine from considerations the performance nearbytelephone of of lines. It has been shown above that voltage inducedin a telephoneline running parallel to a power line is reduced to zero if the power line is transposed and provided it carries balanced currents.
It was also shown that ptwer line transpositionis ineffective in reducing the inducedtelephoneline uoitug. Power line transpositionaparrfrom being ineffectiveintroducesmechanical and insulation problems. It is, therefore,easierto eliminate induced voltages by transposing telephone the line instead.
In fact, the readercan easily verify that even when the power line currents are unbalanced or when t-h"y cgntain harmonics,the voltage induced over complete transpositioncycle called a barrel of a telephone line is zero. Someinducedvoltagewill alwaysbe present on a telephoneline running parallel to a power line becausein actu4l practice transposition never completelysymmetrical.
Therefore,when the lines is run parallel over a considerable length, it is a good practiceto transpose both power and telephone lines. Such cancellation doesnot takeplacewith harmonr. Consequently,these frequencies, if present. The wires in order are carrying currents 1o, Iu and I, and the fourth wire, which is a neutral, carries ,". The currents are:.
Also find the voltage induced in the neutral wire. Substitutingthe valuesof D,,n,Dg, and D,. A single-phase Hz power line is supported a horizontal cross-ann. A telephone line is supported symmetricallybelow the power line as shown in Fig. Find the mutual inductancebetweenthe two circuits and the voltageinducedper kilometre in the telephoneline if the current in the power line is A.
Assume the telephone Iine current to be zero. Solution Flux linkaees of conductor Z,. Lines of and Resistance Transmission lnductanee will lead to eachother. The readercan try other configurations verity that these equivalentequilateralspacingis to low D.. It is commonpracticeto build double-circuit three-phase lines so as to increase transmission reliability at somewhatenhancedcost. From the point of view of rrowertransferfrom one end of the line to the other see Sec.
Therefore,the individual conductorsof a phase should be kept as far apart as possible for high self GMD , while the distance between phasesbe kept as low as permissible for low mutual GMD. It may be noted here that conductors a and a' in parallel compose phase a and sirnilarly b and b'compose phase b and, and c'compose phase c. Lines of and Resisiance Transmission lnductance Equation 2. The GMD method, though applied above to a particular configuration of a double circuit, is valid for any configuration as long as the circuits are electricallyparallel.
While the GMD method is valid. Lr AAr? Further, because of increased self GMD- line inductance is reduced considerablywith the incidental advantageof increasedtransmissioncapacity of the line. It is economical transmit large chunksof power 10 over long dista'ces by employing EHV lines. However, the line voltagesthat can be usedare severely limited by the phenomenon corona. Critical line voltage for formation of corona can be raised considerably by the use of bundled conductors-a group of two or more conductorup".
Reichman  has shownthat the spacingof conductors in a bundle affects vortagegradient and the optimum spacing is of the order of The bundle usually comprises two, three or four conductors arrangedin configurationsillustratedin Fig' 2. The current will not divide equally amongthe conductors of the bundle unlessconductors within the bundle are ruly transposed. The GMD method is still fairly accurate for all practical purposes. Find the inductive reactancein ohms per kilometer at 50 Hz of a three-phase per phaseas shown in Fig.
Equation 2. This is '4rvo higher than the colrespondingvalue liuiritc out' Though the contribution of line resistanceto series line impedance can be neglectedin most cases'it is the main sourceof line power loss. Thus while considering transmission line economy,the presence line resistance of must be considered.
Ohmic or DC resistanceis given by the formula nl R'n - ' ' " o h l n s A. The distribution of current throughout the cross-sectionof a conductor is uniform only when DC is passing through it. This effect becornes Inore pronounced frequencyis increased. It causeslarger power loss for a given rms AC than the loss when the Sairr vaiueof DC is flowing ihroughthe conciuctor. Consequently, effective the conductor resistance more fbr AC then fbr DC.
A qualitative is explanation of the phenomenon as follows. The flux linking the filaments progressively decreasesas we move towards the outer filaments fbr the simple reasonthat the flux inside a filament does not link it.
The inductive reactance the inraginaryfilaments therefore of decreases outwards with the result that the outer filaments conduct more AC than the inner filaments filaments being parallel. With the increase of frequency the non-uniformity of inductive reactanceof the filaments becomes more pronounced,so also the non-uniformity of current distribution. For large solid conductors the skin effect is quite significant even at 50 Hz.
The analytical study of skin effect requires the use of Bessel's functions and is beyond the scope of this book. Apart fronl the skin effect, non-uniformity of current distribution is also causedby proximity eJJ'ect.
Each line conductorcan be divided into sectionsof equalcross-sectionat area say three sections. Pairs aat, bbt and, can form threeloops in parallel. Thus the density of AC flowing through the au' of the conductors is the least and is at conductors highest the inneredges at the outer edges cc'. This type of non-uniform AC current distribution Decomes more pronounceo as me olstance Detween conouctors ls reouceo.
LlKe skin effect, the non-uniformity of current distribution caused by proximity effect. For normal spacing of overhead lines, this effect is always of a negligible order. However, for whereconductors locatedcloseto eachother,proximity are underground cables etfect causesan appreciable increasein effective conductorresistance.
If the mutual reactancebetween them is Xp, what is the effective reactance between the two ends of the line? A telephoneline is also supported a horizontalcross-arm the samehorizontalplane as the on in power line.
The condttctors the telephrlncline are of solid copper of spaced 0. The distance between the nearest conductors the two lines is 20 m. Find the mutual inductance of between the circuitsand the voltageper kilometreinducedin the teiephone line for A current flowing over the power line. The power line carriesbalanced current of A per phase.
Find the mutual inductancebetweenthe circuits and calculatethe 50 Hz voltageinducedin the telephone line ptsrkm. The cable consistingof 12 equal strandsaround a nonconducting diameterof each strandis 0. Assume full transposition. What is the of 2. The line reactance must not exceed Find the maximum permissiblespacing. Each circuit remains on its own side of the tower. Let the self GMD of a single conductorbe 1 cm. Conductors a and at and other corresponding phase conductors are connected in parallel.
Find the reactance per phaseof the system. Waddicor, H. Nagrath, I. New York, Gross, C. Weedy, B. Kimbark, E. Paper I l. The radius of each conductor is 1. The conducrors are to be fully transposed. Find the spacingbetween adjacentconductorssuch that the new line has the sameinductanceas the original line.
The capacitance togetherwith conductance forms the shunt admittanceof a transmission line. As mentioned earlierthe conductance the result of leakage is over the surface of insulatorsand is of negligible order. When an alternating voltageis appliedto the line, the line capacitance drawsa leading sinusoidal current called the charging current which is drawn even when the line is open circuited at the far end.
The line capacitance being proportionalto its length,the chargingcurrentis negligible lines lessthan km long. For longerlines for the capacitance becomes increasingly importantand has to be accounted for. Since equipo',entral,Vrris obtained simply by integrating along PyP, t. By symmetry,the equipotentialsurfaces will be concentriccylinders,while the lines of electrostaticstresswill be radial.
The spacingcommonly imply that the charge on Further, these assumptions meets theseassumptions. The potentialdifferencebetweenany two conductors the group can then be of obtainedby adding the contributionsof the individual chargedconductors: Each term in Eq. Expressionson similar lines could be written for voltage drop between any two conductorsof the group. If the chargesvary sinusoidally, do the voltages this is the casefor AC so transmissionline , the expressionof Eq.
Dlr is large and ground is lar away. As shown in Figs. The voltageacross as is the lines dividesequallybetweenthe capacitances suchthat the neutralpoint n is at the groundpotential.
The capacitance eachline to neutralis then given of by. The assumptions inherent the abovederivation in are: If non-uniforrnityof chargedistributionis takeninto account,then C,. The use of the radius of the circumscribingcircle for a strandedconductorcauses insignificanterror.
With balanced three-phase voltages applied to the line, it follows from the phasor diagrarn ol'I,rig. Adding Eqs. As per Eq. Use of these relationships Eq. If the voltage drop along the line is neglected, Vno is the same in each transposition cycle. Thrce more equations can be written equating to zero the summation of all line charges in each section of the transposition cycle.
From these nine independent equations, it is possible to determine the nine unknoWn charges. The rigorous solution though possible is too involved.
In calculatingthe capacitance transrnission of lines,the presence earthwas of ignored, so far. The effect of earth on capacitance can be conveniently taken into account by the method of images. Method of Images The electric field of transmission line conductorsmust conforrn to the presence of the earth below.
The earthfor this purposemay be assumedto be a perfectty. If a conducting sheetof infinite dimensionsis placed at the zero potential plane, the electric field remains undisturbed. Further, if the conductor carrying charge -q is now removed, the electric field above the conducting sheet stays intact, while that below it vanishes. Using these well of we known resultsin reverse, may equivalentlyreplacethe presence ground below a chargedconductorby a fictitious conductor having equal and opposite chargeand locatedasfar below the surfaceof ground as the overheadconductor above it-such a fictitious conductor is the mirror image of the overhead of conductor.
This method of creatingthe same electric field as in the presence by images originally suggested Lord Kelvin. The equationfor the voltage drop Vo6as determined by the b, two chargedconductorsa and, and their images a'and b' canbe written as follows:. Proceedingon the lines of Sec.
If the fairly occurate assumption constant of chargeper unit length of the conductor throughoutthe transmissioncycle is made,the average value of Voufor the three sectionsof the cycle is given by. Comparing Eqs. If the conductors are high above earttr comparedto the distancesamong them, the effect of earth on the capacitance of three-phase lines can be neglected.
Calculate the capacitanceto neutrallkm of a single-phaseline composed of No. Solution 1 Neglecting the presenceof earth tEq. The presenceof earth increases the capacitanceby approximately 3 partsin Examp 3. The conductorsare No. The voltage of the line is kV. Find the capacitance to neutral and the charging current per kilometre of line.
The six conductorsof a double-circuitthree-phase line having an overall radius of 0. Find the capacitive reactanceto neutral and charging current per kilometre per conductor at k V, 5 0 H z. A comparison of various expressions for inductance and capacitance of transmission lines [e. This fact suggeststhat the method of GDM would be applicable the calculations capacitance well providedit is modified in for as by using the outer conductor radius for finding D,, the self geometric mean distance.
A bundled conductorline is shown in Fig. The results obtained-with theseurru'rnptions are fairly accurate for usual spacings. Thus if the charge on phase a is qo, the conductorsa and a'have a chargeof qolz each; similarly the charge is equally divided for phasesb and c.
Take the voltage 50 of phase a as referencephasor. All conductors have the sameradii. Also find the charging current of phase a. Neglect the effect of grouno. The diameterof each conductoris 2. The line is transffio and carries balanced load' Find the capacitanceper phaseto neutral of the line.
Consideringthe line to be transposed proceeding and in the usual manner,the f inal r e s u l t w i l l b e. The radius of eachconductoris 0. At a certain instant the chargeson the centre conductor and on one of the outside conductors are identical and voltage drop betweenthese identically chargedconductorsis v.
Paper insulation separatingthe conductor from the concentric lead sheath has a thickness of 2. The thickness of the lead sheath is 2 mm. Find the capacitive reactance per kilometre between the inner conductor and the lead sheath.
Assume that the line is fully transposedand carriesbalanced load.
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Calculatethe maximum potentialdifferencepermissible betweenthe conductors, the if electric lield strengthbetweenthem is not to exceed25 kY lcm, r being 0. Starr, A. A complete diagram of a power system representing all the three phases becomes complicatedfor a systemof practicalsize, so much so that it may too no longer convey the information it is intended to convey.
It is much more practicalto represent power systemby meansof simple symbbls for each a componentresultingin what is called a one-linediagram. Per unit system leads to great simplification of three-phasenetworks involvingtransformers. The synchronous machinemodel in steadystateis presented this in chapter. The transient model of the machine will be presentedin Chapter 9. Parton, J. Stevens, R. The solution of a three-phase network under balancedconditions is easily carriedout by solving the single-phase networkcorresponding the ref'erence to phase.
Thus the neutral impedance Zn does not affect network behaviour. Equation 4. If the transformeris YIA connectedas in-Fig. Since both phasevoltage and line current shift through the samephaseangle from star to delta side, the transformerper phaseimpedanceand power flow are preserved the single-phase in equivalent. In most analytical studies,we are merely interestedin the magnitude of voltages and currents so that the single-phaseequivalentof Fig.
Wherever proper phaseangles of currents and voltages are needed, correction can be easily applier after obtaining the solution through a singlephasetransformerequivalent.
It may be noted here that irrespective of the type of connection, the transformation ratio of the single-phaseequivalent of a three-phasetransformer is the same as the line-to-line transformationratio. DIAGRAM A one-linediagramof a power system shows the main connectionsand showrrdependingon the information required in a system study, e.
Power system networks are represented one-line diagrams using by suitable symbols fbr generators,motors, transformersand loads. It is a convenient practical way of network representationrather than drawing the actual three-phasediagram which may indeed be quite cumbersome and confusing for a practical size power network. Generator and transformer connections-star, delta,and neutralgrounding are indicatedby symbols drawn by the side of the representation these elements.
Circuit breakers are of represented rectangularblocks. Figure 4. The reactancedata of the elementsare given below the diagram. For the systemof conditionscan be easily transtbrmer diagramis drarvnin Frg. Single-phase Fie. The transmission not involving rotatingmachines to Chapter5. Loads are assumed be passive and are representedby resistanceand inductive reactancein series.
Neutral do grounding impedances not appearin the diagram as balancedconditions are assumed. Three voltagelevels 6. I 1 ancl33 kV are presentin this system.
Generators specified three-phase are in MVA,line-to-line voltage and per phase reactance equivalent star. Transformers specified three-phase are in MVA, line-to-line transformation ratio,and per phase equivalent star impedance one side. Loads are on specifiedin three-phase MW,line-to-line voltageand powerfactor. Per cent value is not convenient for use as the factor of has to be carried in computations. For a power system,practical choice of base values are: Tar cpr esent sasingle- phaset r ansf gr m lr int er m sof pr im ara' yand andan ideal transformer of ratio 1: From Fig.
Any other impedanceon either side of a transformeris converted to pu value just like Zo or Zr.
From a one-line diagramof a power systemwe can directly draw the impedance diagramby following the stepsgiven below: Choosean appropriate 2. Consider the system to be divided into a number of sections by the Choose an appropriatekV base in one of the sections.
Calculate kV bases of other sectionsin the ratio of transformation. Calculateper unit valuesof voltagesand impedances eqch sectionand connectthem up as per the topologyof the one-linediagram. The result per unit impedancediagram.
I4 can be represented the simple equivalent circuit by of Fig. Z pu can be determined directlyfrom the equivalentimpedance primary on or secondaryside of a transformerby using the appropriaie impedancl base. On primary side: Obtain the per unit impedance reactance 4. We shall make some further simplifying assumptions. Line capacitance a series reactanceonly. We shall assumethat the impedancediagram is meant for short circuit studies. Current drawn by static loadsunder short circuit conditions can be neglected.
Loads A and B are thereforeignored. Choosea common three-phase MVA base of 30 and a voltagebaseof 33 kV line-to-lineon the transmission line. Then the voltagebasein the circuit of generator1 is 11 kV line-to-line and 2 that in the circuits of generators and 3 is 6.
Transmission line: Transform T,: I iliffij IExample 4. I, we now calculatethe pu values of the reactances transfonners of and generatorsas per relation 4. Transformer Z: Transformet Tr:. Generator 3: The reactancc data of gencrators and transtbrmers usually specified are in pu or per cent values, based on equiprnentratings rather than in actual ohmic valuesasgiven in Exampl 4.
This is a convenient choice of sign of 0 in power systems where loads have mostly lagging power factors. With a baseMVA of The three-phase complex power fed into load is given by.
With the direction of current indicated in Fig. The synchronousmachineis the most important elementof a power system. It converts mechanical power into electrical form and feeds it into the power network or, in the caseof a motor, it draws electrical power from the network and converts it into the mechanicalform.
The machine excitation which is controllable determinesthe flow of VARs into or out of the machine. Books on electrical machines may be consulted for a detailed account of the synchronousmachine.
We shall presenthere a simplified circuit model of the undertransient which with suitable machine modifications wherever necessary conditions will be adoptedthroughout this book. The stator has a balanced three-phasewinding-aat, bbt and cct. The winding shown is a concentratedone, while the winding in an actual machineis distributed across the stator periphery.
The rotor shown is a cylindrical" one round rotor or nonsalient pole rotor with rotor winding excited by the DC source. As the rotor rotates, three-phase emfs are produced in stator winding. Since the machine is a balancedone and balanced loading will be considered, can be modelled on it per phase basis for the referencephase a.
Obviously to be unsaturated. High-speedturbo-generators have cylindrical rotors and Iow sppedhydro-generators have salient pole rotors. This flux, called armature reaction flux, is therefore stationarywith respectto field flux Qy.
It intuitively fbllows that Qois in phase with phase c current 1o which causesit. Since the magnetic circuit has been.
Phasor diagram under loaded balanced conditions showing fluxes, currents and voltagesas phasorsis drawn in Fig. NEffi t The circuit of Fig. L6 can be easilymodifiedto include the effect of armature leakage reactance resistance and these series are effects to give the complete circuitmodelof the synchronous generator in Fig. Becauseof the assumed linearity of the magneticcircuit, voltage phasorE, Eo and v, are proportionalto flux phasors dr, doand d, respectively;furthei, voltage phasors lag 90' behind flux phasori.
It therefore easily follows from Fig. This model of the synchronousmachinecan be further modified to account for the effect of magnetic saturation where the principle of super-position does not hold. Fig- 4. Therefore,in the place of the circuit model of Fig.
I7, the simplified circuit model of Fig. The corresponding phasor diagram is given in Fig. The fieici induceciemi Ey ieacisthe terminal -condition voltage by the torque load angle d This, in fact, is the for acrive power to flow out of the generator.
The magnitudeof power delivered depends upon sin d In the motoring operationof a synchronous machine,the current 1,,reverses as shown in Fig. It may be noted that by V, now leads lby d, This in fact is the condition for power to flow into motor terminals. This is discussed detail in by is mainly controllecl meansof its regulated Section5.
For bus generatorconnectedto an inJinite bus as shown in Fig' 4'22' As infinite independent remainconstant voltageand frequency whose a means largesystem machine and the bus' and of the power exchangebetweenthe synchronous machine.
As the machineexcitation is varied, armaturecurrent keep angle g, t. In the overexcited case,Io leads Vu i. Io lags V, r. This problem the at stability will be discussed length in Chapter From the above discussionwe can draw the general conclusion that a synchronousmachine generating or motoring while operating at constant power suppliespositive reactive power into the bus bar or draws negative r c q e t t v cI v.
While Figs 4. As lEtl sin dremainsconstant, the tip of phasor Ermoves along a line parallel to y, as excitation is varied. The direction of phasor1ois always 90o lagging jI"X, andits magnitudeis obtained Figurc 4. For excitation lower than this value the generatff becomesunstable.
Considernow the power deliveredby a synchronousgeneratorto an infinite bus. V1 Fig. Similar phasor diagrams can be drawn for synchronous motor as well for constant input power or constant load if copper and iron losses are neglected and mechanical ioss is combined with load. Schweit, J. Forty per cent of those displaced by dams are tribal peoples.
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