POWER SYSTEM ANALYSIS AND SIMULATION150902*

SYALLBUS OVER VIEW

SYALLBUS OVER VIEWCurrent and Voltage Relations on a Transmission LineSystem Modeling:Symmetrical Three-Phase Faults:Symmetrical Components:Unsymmetrical Faults:Corona:Neutral Grounding:Transients in Power Systems:

Book References Elements of Power Systems Analysis : W. D. Stevenson Jr., 4th Edition, McGraw Hill International.Power system analysis and design by B R GuptaElectrical Power systems: C. L . Wadhwa, 5th Edition, New Age International Publishers*

CURRENT AND VOLTAGE RELATIONS ON ATRASMISSION LINE Prepared by:SUMIT K RATHOREGCET Website : www.sumitrathor.co.inEmail id : sumitrathor@gcet.ac.in*

Transmission Line

LINESGenerators and loads are connected together through transmission lines transporting electric power from one place to another. Transmission line must, therefore, take power from generators, transmit it to location where it will be used, and then distribute it to individual consumers.

The power capability of a transmission line is proportional to the square of the voltage on the line. Therefore, very high voltage levels are used to transmit power over long distances. Once the power reaches the area where it will be used, it is stepped down to a lower voltages in distribution substations, and then delivered to customers through distribution lines.

PreliminariesDual 345 kV transmission lineDistribution line with no ground wire.

PreliminariesAn overhead transmission line usually consists of three conductors or bundles of conductors containing the three phases of the power system. The conductors are usually aluminum cable steel reinforced (ACSR), which are steel core (for strength) and aluminum wires (having low resistance) wrapped around the core.

PreliminariesIn overhead transmission lines, the conductors are suspended from a pole or a tower via insulators.

PreliminariesIn addition to phase conductors, a transmission line usually includes one or two steel wires called ground (shield) wires. These wires are electrically connected to the tower and to the ground, and, therefore, are at ground potential.In large transmission lines, these wires are located above the phase conductors, shielding them from lightning.

PreliminariesThere two types of transmission lines: overhead lines and buried cables.

PreliminariesCable lines are designed to be placed underground or under water. The conductors are insulated from one another and surrounded by protective sheath. Cable lines are usually more expensive and harder to maintain. They also have capacitance problem not suitable for long distance.

PreliminariesTransmission lines are characterized by a resistance, inductance, capacitance, and conductance per unit length. These values determine the power-carrying capacity of the transmission line and the voltage drop across it at full load.

Resistance The DC resistance of a conductor is given byWhere l is the length of conductor; A cross-sectional area, is the resistivity of the conductor. Therefore, the DC resistance per meter of the conductor is(9.9.1)(9.9.1)The resistivity of a conductor is a fundamental property of the material that the conductor is made from. It varies with both type and temperature of the material. At the same temperature, the resistivity of aluminum is higher than the resistivity of copper.

Resistance We notice that silver and copper would be among the best conductors. However, aluminum, being much cheaper and lighter, is used to make most of the transmission line conductors. Conductors made out of aluminum should have bigger diameter than copper conductors to offset the higher resistivity of the material and, therefore, support the necessary currents.AC resistance of a conductor is always higher than its DC resistance due to the skin effect forcing more current flow near the outer surface of the conductor. The higher the frequency of current, the more noticeable skin effect would be.At frequencies of our interest (50-60 Hz), however, skin effect is not very strong.

Wire manufacturers usually supply tables of resistance per unit length at common frequencies (50 and 60 Hz). Therefore, the resistance can be determined from such tables.

*Inductance and inductive reactanceThe series inductance of a transmission line consists of two components: internal and external inductances, which are due the magnetic flux inside and outside the conductor respectively. The inductance of a transmission line is defined as the number of flux linkages [Wb-turns] produced per ampere of current flowing through the line:Internal inductance:Consider a conductor of radius r carrying a current I. At a distance x from the center of this conductor, the magnetic field intensity Hx can be found from Amperes law:(9.12.2)

Inductance of a transmission lineA two-conductor bundleA four-conductor bundle

Capacitance and capacitive reactanceSince a voltage V is applied to a pair of conductors separated by a dielectric (air), charges of equal magnitude but opposite sign will accumulate on the conductors:(9.23.1)Where C is the capacitance between the pair of conductors.In AC power systems, a transmission line carries a time-varying voltage different in each phase. This time-varying voltage causes the changes in charges stored on conductors. Changing charges produce a changing current, which will increase the current through the transmission line and affect the power factor and voltage drop of the line. This changing current will flow in a transmission line even if it is open circuited.

Does open circuit current flow possible??

Capacitance of a single phase two-wire transmission lineThus:(9.29.1)Which is the capacitance per unit length of a single-phase two-wire transmission line.

The potential difference between each conductor and the ground (or neutral) is one half of the potential difference between the two conductors. Therefore, the capacitance to ground of this single-phase transmission line will be(9.29.2)

Shunt capacitive admittanceThe shunt capacitive admittance of a transmission line depends on both the capacitance of the line and the frequency of the power system. Denoting the capacitance per unit length as c, the shunt admittance per unit length will beThe total shunt capacitive admittance therefore iswhere d is the length of the line. The corresponding capacitive reactance is the reciprocal to the admittance:(9.31.1)(9.31.2)(9.31.3)

Inductive reactance of a lineThe series inductive reactance of a transmission line depends on both the inductance of the line and the frequency of the power system. Denoting the inductance per unit length as l, the inductive reactance per unit length will bewhere f is the power system frequency. Therefore, the total series inductive reactance of a transmission line can be found as(9.22.1)(9.22.2)where d is the length of the line.

2-port networks and ABCD modelsA transmission line can be represented by a 2-port network a network that can be isolated from the outside world by two connections (ports) as shown.If the network is linear, an elementary circuits theorem (analogous to Thevenins theorem) establishes the relationship between the sending and receiving end voltages and currents asHere constants A and D are dimensionless, a constant B has units of , and a constant C is measured in siemens. These constants are sometimes referred to as generalized circuit constants, or ABCD constants.

2-port networks and ABCD modelsThe ABCD constants can be physically interpreted. Constant A represents the effect of a change in the receiving end voltage on the sending end voltage; and constant D models the effect of a change in the receiving end current on the sending end current. Naturally, both constants A and D are dimensionless.

The constant B represents the effect of a change in the receiving end current on the sending end voltage. The constant C denotes the effect of a change in the receiving end voltage on the sending end current.

Transmission lines are 2-port linear networks, and they are often represented by ABCD models. For the short transmission line model, IS = IR = I, and the ABCD constants are(9.51.1)

Transmission line modelsUnlike the electric machines studied so far, transmission lines are characterized by their distributed parameters: distributed resistance, inductance, and capacitance.The distributed series and shunt elements of the transmission line make it harder to model. Such parameters may be approximated by many small discrete resistors, capacitors, and inductors.However, this approach is not very practical, since it would require to solve for voltages and currents at all nodes along the line. We could also solve the exact differential equations for a line but this is also not very practical for large power systems with many lines.

CLASSIFICATION OF LINEBased on distance of transmission of power lines are classified as

Short lines(80km or 50mile)Medium line(80 to 240km (50 to 150mile))Long lines( more than 240km or 150mile)

Representation of Transmission line modelsFortunately, certain simplifications can be used

Overhead transmission lines shorter than 80 km (50 miles) can be modeled as a series resistance and inductance, since the shunt capacitance can be neglected over short distances.The inductive reactance at 60 Hz for overhead lines is typically much larger than the resistance of the line.For medium-length lines (80-240 km), shunt capacitance should be taken into account. However, it can be modeled by two capacitors of a half of the line capacitance each.Lines longer than 240 km (150 miles) are long transmission lines and are to be discussed later.

Representation of Transmission line modelsGeneral equation relating voltage and current on a transmission line recognize the fact that all four of the parameters of transmission line discussed so far are uniformly distributed along the line Lumped parameter gives good accuracy for short lines and for lines of medium length. If an overhead line is classified as short shunt capacitance is ignored with little loss of accuracy and we need to consider only series resistance R and series inductance L for the total length of lineA medium length line can be represented sufficiently well by R and L as lumped parameter with half the capacitance to neutral of the line lumped at each end of the equivalent circuit. Shunt conductance is usually neglected in overhead power transmission lines when calculating voltage and current

Representation of Transmission line modelsNormally transmission lines are operated with balance three phase loads. Although the lines are not spaced equilaterally and not transposed the resulting dissymmetry is slight and the phases are considered to be balancedIn order the distinguish between total series impedance of line and the series impedance per unit length of line the following nomenclature is adopted

z = series impedance per unit length per phase y = series admittance per unit length per phase

l = length of transmission line

Z= zl total impedance per phase

Y= yl total admittance per phase

Transmission line modelsThe total series resistance, series reactance, and shunt admittance of a transmission line can be calculated as(9.37.1)(9.37.2)(9.37.3)where r, x, and y are resistance, reactance, and shunt admittance per unit length and d is the length of the transmission line. The values of r, x, and y can be computed from the line geometry or found in the reference tables for the specific transmission line.

Short transmission lineThe per-phase equivalent circuit of a short lineVS and VR are the sending and receiving end voltages; IS and IR are the sending and receiving end currents. Assumption of no line admittance leads to We can relate voltages through the Kirchhoffs voltage lawwhich is very similar to the equation derived for a synchronous generator.(9.38.1)(9.38.2)(9.38.3)++

Why conductance is neglected in short lines?

Short Transmission line The effect of load variation of the power factor of the load on the voltage regulation of line is most easily understood for short line.

Voltage regulation of line is rise in voltage at receiving end expressed in percent of full load voltage when full load at a specified power factor is removed while the sending end voltage is held constant

Short transmission line: phasor diagramAC voltages are usually expressed as phasors.Load with lagging power factor.Load with unity power factor.Load with leading power factor.For a given source voltage VS and magnitude of the line current, the received voltage is lower for lagging loads and higher for leading loads.

Short Transmission line At no load condition VS=VRNLIn various load conditions with various pf receiving end voltage varies as well sending end voltage requirement also required to change. Smaller sending end voltage is required to maintain the given receiving end voltage when the receiving end current leads the voltageThe voltage drop is same in series impedance of line in all cases because because of different power factor the voltage drop is added to the receiving end voltage at different angle in each caseThe regulation is greatest for lagging power factor and least or even negative for leading power factors

Medium-length transmission lineConsidering medium-length lines (50 to 150 mile-long), the shunt admittance must be included in calculations. However, the total admittance is usually modeled ( model) as two capacitors of equal values (each corresponding to a half of total admittance) placed at the sending and receiving ends.The current through the receiving end capacitor can be found asAnd the current through the series impedance elements is(9.52.1)(9.52.2)

Medium-length transmission lineFrom the Kirchhoffs voltage law, the sending end voltage is(9.53.1)The source current will be(9.53.2)Therefore, the ABCD constants of a medium-length transmission line are(9.53.3)If the shunt capacitance of the line is ignored, the ABCD constants are the constants for a short transmission line.

Medium Transmission line ABCD constants are sometimes called the generalized constants of the transmission lineIn general they are complex numbersA and D are dimensionless and equal each other if the line is same when viewed from either endThe dimension of B and C are ohms and mhosThe constants apply to any linear, passive and bilateral four terminal network having tow pairs of terminalSuch a network is called a two port networkBy letting IR=0 A=Vs/VR at no loadSimilarly B is the ratio Vs/IR when the receiving end is short circuited Constant A is useful in computing regulationIf VRFL is receiving end voltage at full load for sending end voltage of VS becomes %regulation = Vs/A-VRFL VRFL

One-line (single-line) diagramsAlmost all modern power systems are three-phase systems with the phases of equal amplitude and shifted by 120. Since phases are similar, it is customary to sketch power systems in a simple form with a single line representing all three phases of the real system.Combined with a standard set of symbols for electrical components, such one-line diagrams provide a compact way to represent information.*@Sumit K Rathore

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One-line (single-line) diagrams

If a power system containing two synchronous machines, two loads, two busses, two transformers, and a transmission line to connect busses together.All devices are protected by oil circuit breakers (OCBs). We notice that the diagram indicates the type of connection for each machine and transformer, and also the points in the system connected to the ground.The ground connections are important since they affect the current flowing in nonsymmetrical faults. These connection can be direct or through a resistor or inductor (they help reducing the fault current that flows in unsymmetrical faults, while having no impact on the steady-state operation of the system since the current through them will be zero). Machine ratings, impedances, and/or consumed (or supplied) powers are usually included in the diagrams.*@Sumit K Rathore

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One line representation*@Sumit K Rathore

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One line representation*@Sumit K Rathore

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REACTIVE POWER COMPENSATION OF LINEThe performance of medium and long line can be improved by reactive power compensation by shunt and series typeSeries compensation consist of capacitor bank places in series with each phase conductor of the lineShunt compensation refers to the placement of inductor from each line to neutral to reduce partially or completely the shunt susceptance of a high voltage line which is which is particularly important at light loads when the voltage at receiving end may otherwise become very highSeries compensation reduces the series impedance of line which is the principal cause of voltage drop and the most important factor in determining the maximum power which the line can transmitIn order to understand the effect of series impedance Z on maximum power transmission out of ABCD constant B is useful for understanding power transmission in short line B=Z

REACTIVE POWER COMPENSATION OF LINEThe desired reactance of the capacitor bank can be determined by compensating for a specific amount of the total inductive reactance of the lineThis leads to the term compensation factor which is defined by Xc/XL Physical location of capacitor bank is important for power system studies

This can be easily accomplished by determining ABCD constant of the portions of line on each side of the capacitor bank and by representing the capacitor bank by its ABCD constantsin long distance power transmission if generators are located far from loads series compensation is requiredLower voltage drop in line with series compensation is an additional advantageSeries capacitors are also useful in balancing the voltage drop of tow parallel lines

Per-phase, per-unit equivalent circuitsAs we have learned, the easiest way to analyze a balanced three-phase circuit is by a per-phase equivalent circuit with all connections converted in their equivalent Y connections. The solution obtained can be extended to three phases knowing that the voltages and currents in other two phases would be the same except for the 120 phase shift.An advantage of per-unit representation is that circuits containing transformers can be easily analyzed.Real power systems are convenient to analyze using their per-phase (since the system is three-phase) per-unit (since there are many transformers) equivalent circuits. The per-phase base voltage, current, apparent power, and impedance are*@Sumit K Rathore

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Advantage of per unit systemManufacturers usually specify the impedance of a piece of apparatus in percent of per unit on the base of the nameplate ratingThe per unit impedance of machine of the same type and widely different rating usually lie within a narrow range although the ohmic values differ materially for machines of different ratings. For this reason when the impedance is not known definitely it is generally possible to select from the tabulated average values which is reasonably correctWhen impedance in ohms is specified in an equivalent circuit each impedance must be referred to the same circuit by multiplying it by the square of the ratio of the rated voltage of the two sides of transformer connecting the reference circuit and the circuit containing the impedance. The per unit impedance once expressed on the proper base it is same referred to either side of any transformer The way in which transformer are connected in three phase circuits does not affect the per unit impedance of the equivalent circuit although the transformer connection does determine the relation between the voltage bases on the two sides of the transformer.*@Sumit K Rathore

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Per unit system*@Sumit K Rathore

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Per unit system*@Sumit K Rathore

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PER UNIT VALUEFor three phase to per phase per unit values

kVA1 = kVA3 /3

kV1 = kV3 /3*@Sumit K Rathore

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Changing the base of per unit quantitiesSometimes the base impedance of component of system is expressed on a base other than selected one as base for the part of the system in which component is locatedSince all impedance in any one part of system must be expressed on same impedance base when making computations so it is necessary to have means of converting per unit impedance from one base to another.*@Sumit K Rathore

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Changing the base of per unit quantities

Which shows per unit impedance is directly proportional base kVA and inversely proportional to kV. So change from old impedance to new one is given by *@Sumit K Rathore

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Changing the base of per unit quantitiesFor example you have generator whose base kV and KVA or MVA changes to new one

E.g: the reactance of geneator designated X given by 0.25pu based in generator name plate of 18kV and 500MVA. The base for calculations is 20kV, 100MVA. Find X new base?*@Sumit K Rathore

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Long transmission lineFor long lines, it is not accurate enough to approximate the shunt admittance by two constant capacitors at either end of the line. Instead, both the shunt capacitance and the series impedance must be treated as distributed quantities; the voltages and currents on the line should be found by solving differential equations of the line.However, it is possible to model a long transmission line as a model with a modified series impedance Z and a modified shunt admittance Y and to perform calculations on that model using ABCD constants. The modified values of series impedance and shunt admittance are:(9.54.1)(9.54.2)

Long transmission lineHere Z is the series impedance of the line; Y is the shunt admittance of the line; d is the length of the line; is the propagation constant of the line:(9.55.1)where y is the shunt admittance per kilometer and z is the series impedance per km.As d gets small, the ratios approach 1.0 and the model becomes a medium-length line model. The ABCD constants for a long transmission line are(9.55.2)

THE LONG TRANSMISSION LINE: SOLUTION OF THE DIFFERENTIAL EQUATIONSFor line length of more than exact 50Hz and 240km or 150mi is considered to be the long length line and the parameter of the line is distributed along the line.

Figure shows the line with one phase and neutral and the parameter is distributed along the line

Lumped parameters are not shown because we are ready to consider thesolution of the line with the impedance and admittance uniformly distributed

THE LONG TRANSMISSI ON LINE:INTERPRETATION OF THE EQUATIONSBoth y and Zc are complex quantities. The real part of the propagation constant y is called the attenuation constant alpha and is measured in nepers per unit length . The quadrature part of gama is called the phase constant beta and is measured in radians per unit length . Thus,

THE LONG TRANSMISSION LINE:HYPERBOLIC FORM OF THE EQUATIONSz

THE EQUIVALENT CIRCUITOF A LONG LINEz

Power circle diagram of transmission linePerformance of transmission line can be studied either by analytical method or graphical method

Analytical method is cumbersome and laborious

Graphical method analyzing by means of circle diagram approve to be convenient method

Receiving end circle diagram is drawn with receiving end real and reactive power as horizontal and vertical coordinates respectively

Power circle diagram of transmission lineReceving end circle diagram is drawn from voltage phasor diagram.

Phasor drawn with Vr as reference and Ir lags behind Vr for lagging power load or for inductive load.

Then phasors can be converted to power phasor by multiplying the entire voltage phasor by current phasor = Vr/B

Power circle diagram of transmission line

Procedure Take x axis as active power reference and Y- axis as reactive power reference.

This is receiving end power circle diagram and it is assumed that sending and receiving end voltage remains constant.

So first draw a line with angle of in anticlock wise direction because of minus sign. This line having constant magnitude and constant phase angle.

Reference line for deltaMWMVAr

Procedure Because of sending and receiving end voltage is constant and with parameter A and B, Cr will be centre of circle.

The radius of the receiving end circle is also remains constant but only

changes angle by with angle of in positive direction

CrOMWMVAr

CrORadius MWMVAr

CrRadius rPMWMVAr

CrRadius rPMWMVAr

CrORadius rPrQrPMWMVAr

Power circle diagram of lineThe operating point P on circle is loacted by amount of real power delivered to the load i.e Pr the corresponding value of Qr can be read from circle diagram.

The power angle is the angle between reference line shown and phasor CrP

Many other useful information e.g capacity of compensation equipment maximum receiving end power etc can also be obtained from power circle diagram

REACTIVE POWER COMPENSATION OF LINESHUNT COMPENSATIONSERIES COMPENSATION

Questions Difference between nominal pi and nominal T circuits.Difference between series and shunt compensation.What is ferranti effectWhat are ABCD parametersWhat is lossless lineWhat is surge impedanceWhat is wavelength Surge impedance loading SLACK BUS?INFINE BUSPV BUS?PQ BUS?

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Book References POWER SYSTEM ANALYSIS BY JOHN J GRAINGER & WILLIAM D STEVENSONS JR.POWER SYSTEM ANALYSIS AND DESIGN BY B R GUPTA*

Web References Z*

Thank You

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