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8/12/2019 Ch1 Current Volt Relationship TrasCh1 Current Volt Relationship Trasmission LineCh1 Current Volt Relationship Trasmission LineCh1 Current Volt Relationship Trasmission LineCh1 Current Volt Relationship Trasmission LineCh1 Current Vol

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POWER SYSTEM ANALYSIS

AND SIMULATION

150902

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SYALLBUS OVER VIEW


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Book References

Elements of Power Systems Analysis : W. D. Stevenson Jr., 4thEdition, McGraw Hill International.

Power system analysis and design by B R Gupta Electrical Power systems: C. L . Wadhwa, 5th Edition, New Age

International Publishers

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CURRENT AND VOLTAGE

ELATIONS ON ATRASMISSION LIN

Prepared by:SUMIT K RATHOREGCET

Website : www.sumitrathor.co.inEmail id : [email protected]

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Transmission Line


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LINESGenerators and loads are connected together through transmissionlines 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 toindividual consumers.

The power capability of a transmission line is proportional to thesquare of the voltage on the line . Therefore, very high voltage levelsare used to transmit power over long distances. Once the powerreaches the area where it will be used, it is stepped down to a lower

voltages in distribution substations, and then delivered to customersthrough distribution lines.


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Preliminaries

Dual 345 kV transmission line

Distribution line with no ground wire.


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PreliminariesAn overhead transmission line usually consists of three conductors

or bundles of conductors containing the three phases of the powersystem. 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.


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Preliminaries

In overhead transmission lines, the conductors are suspended from a poleor a tower via insulators.


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PreliminariesIn addition to phase conductors, a transmission line usually includes one or two

steel wires called ground (shield) wires. These wires are electrically connectedto the tower and to the ground, and, therefore, are at ground potential.

In large transmission lines,these wires are located abovethe phase conductors,

shielding them from lightning.


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Preliminaries

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


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PreliminariesCable lines are designed to be placed underground or

under water. The conductors are insulated from oneanother and surrounded by protective sheath. Cablelines are usually more expensive and harder tomaintain. They also have capacitance problem notsuitable for long distance.


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Resistance

The DC resistance of a conductor is given by

DC

l R

A

Where l is the length of conductor; A cross-sectional area, is the resistivity ofthe conductor. Therefore, the DC resistance per meter of the conductor is

(9.9.1)

(9.9.1

)

DC m

r A

The resistivity of a conductor is a fundamental property of the material that theconductor 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 resistivityof copper.


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ResistanceWe notice that silver and copper would be among the best conductors. However,

aluminum, being much cheaper and lighter, is used to make most of thetransmission line conductors. Conductors made out of aluminum should have bigger diameter than copper conductors to offset the higher resistivity of thematerial and, therefore, support the necessary currents.

AC resistance of a conductor is always higher than its DC resistance due to theskin effect forcing more current flow near the outer surface of the conductor. Thehigher 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 commonfrequencies (50 and 60 Hz). Therefore, the resistance can be determined fromsuch tables.


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Inductance and inductivereactance

The series inductance of a transmission line consists of two components: internaland external inductances, which are due the magnetic flux inside and outside theconductor respectively. The inductance of a transmission line is defined as thenumber of flux linkages [Wb-turns] produced per ampere of current flowingthrough the line:

L I

Internal inductance:Consider a conductor of radius r carrying a current I . At a

distance x from the center of this conductor, the magneticfield intensity H x can be found from Amperes law:

x x H dl I (9.12.2)


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Inductance of a transmission line

A two-conductorbundle

A four-conductorbundle


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Capacitance and capacitivereactance

Since 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:

q CV (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 differentin each phase. This time-varying voltage causes the changes in charges stored onconductors. Changing charges produce a changing current, which will increasethe current through the transmission line and affect the power factor and voltagedrop of the line. This changing current will flow in a transmission line even if it isopen circuited.


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Does open circuit current flow

possible??


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Capacitance of a single phasetwo-wire transmission line

lnabc D

r

Thus:(9.29.1)

Which is the capacitance per unit length of a single-phase two-wire transmissionline.

The potential difference between each conductor and the ground (or neutral) isone half of the potential difference between the two conductors. Therefore, the

capacitance to ground of this single-phase transmission line will be

2

lnn an bnc c c D

r

(9.29.2)


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Shunt capacitive admittance

The shunt capacitive admittance of a transmission line depends on both thecapacitance of the line and the frequency of the power system. Denoting thecapacitance per unit length as c , the shunt admittance per unit length will be

2C y j c j fc

The total shunt capacitive admittance therefore is

2C C Y y d j fcd

where d is the length of the line. The corresponding capacitive reactance is thereciprocal to the admittance:

1 1

2C C

Z jY fcd

(9.31.1)

(9.31.2)

(9.31.3)


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Inductive reactance of a line

The series inductive reactance of a transmission line depends on both theinductance of the line and the frequency of the power system. Denoting theinductance per unit length as l , the inductive reactance per unit length will be

2 I x j l j fl

where f is the power system frequency. Therefore, the total series inductivereactance of a transmission line can be found as

I I X x d

(9.22.1)

(9.22.2)

where d is the length of the line.


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2-port networks and ABCDmodels

A transmission line can be represented by a 2- port network a network that can be isolatedfrom 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 endvoltages and currents as

S R R

S R R

V AV BI

I CV DI

Here constants A and D are dimensionless, a constant B has units of , and aconstant C is measured in siemens. These constants are sometimes referred to asgeneralized circuit constants, or ABCD constants.


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2-port networks and ABCDmodels

The ABCD constants can be physically interpreted. Constant A represents the effectof 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 endcurrent. Naturally, both constants A and D are dimensionless.

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

Transmission lines are 2-port linear networks, and they are often represented byABCD models. For the short transmission line model, I S = I R = I , and the ABCD

constants are 1

0

1

A B Z

C

D

(9.51.1)


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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 itharder to model. Such parameters may be approximated by many smalldiscrete resistors, capacitors, and inductors.

However, this approach is not very practical, since it would require to solvefor voltages and currents at all nodes along the line. We could also solve theexact differential equations for a line but this is also not very practical forlarge power systems with many lines.


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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)

R i f T i i


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Representation of Transmissionline models

Fortunately, certain simplifications can be used

Overhead transmission lines shorter than 80 km (50 miles) can be modeled as aseries resistance and inductance, since the shunt capacitance can be neglectedover short distances.

The inductive reactance at 60 Hz for overheadlines is typically much larger than the resistanceof the line.For medium-length lines (80-240 km), shuntcapacitance should be taken into account.However, it can be modeled by two capacitors ofa half of the line capacitance each.

Lines longer than 240 km (150 miles) are long transmission lines and are to bediscussed later.

R i f T i i


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General equation relating voltage and current on a transmission line recognizethe fact that all four of the parameters of transmission line discussed so far areuniformly distributed along the line Lumped parameter gives good accuracyfor 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 seriesinductance L for the total length of lineA medium length line can be represented sufficiently well by R and L aslumped parameter with half the capacitance to neutral of the line lumped ateach end of the equivalent circuit. Shunt conductance is usually neglected inoverhead power transmission lines when calculating voltage and current

R i f T i i


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

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

l = length of transmission line

Z= zl total impedance per phase

Y= yl total admittance per phase


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Transmission line models

The total series resistance, series reactance, and shunt admittance ofa transmission line can be calculated as

R rd

X xd

Y yd

(9.37.1)

(9.37.2)

(9.37.3)

where r , x, and y are resistance, reactance, and shunt admittance perunit length and d is the length of the transmission line. The values ofr , x, and y can be computed from the line geometry or found in thereference tables for the specific transmission line.


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Short transmission lineThe per-phase equivalent circuit of a short line

V S and V R are the sending and receiving endvoltages; I S and I R are the sending and receivingend currents. Assumption of no line admittanceleads to

S R I I

We can relate voltages through the Kirchhoffs voltage law

S R R LV V ZI V RI jX I

R S LV V RI jX I

which is very similar to the equation derived for a synchronous generator.

(9.38.1)

(9.38.2)

(9.38.3)++


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Why conductance is neglected inshort lines?


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Short Transmission line

The effect of load variation of the power factor of the load onthe voltage regulation of line is most easily understood for shortline.

Voltage regulation of line is rise in voltage at receiving endexpressed in percent of full load voltage when full load at aspecified power factor is removed while the sending end voltageis held constant

Sh t t i i li h


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Short transmission line: phasordiagram

AC 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 V S and magnitude of theline current, the received voltage is lower for laggingloads and higher for leading loads.


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Short Transmission line

At no load condition V S=V RNLIn various load conditions with various pf receiving end voltagevaries as well sending end voltage requirement also required tochange.Smaller sending end voltage is required to maintain the givenreceiving end voltage when the receiving end current leads thevoltageThe 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 eachcaseThe regulation is greatest for lagging power factor and least oreven negative for leading power factors


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Medium-length transmission lineConsidering medium-length lines (50 to 150mile-long), the shunt admittance must beincluded in calculations. However, the totaladmittance is usually modeled ( model) astwo capacitors of equal values (eachcorresponding to a half of total admittance)

placed at the sending and receiving ends.

The current through the receiving end capacitor can be found as

2

2C R

Y I V

And the current through the series impedance elements is

2 ser R RY

I V I

(9.52.1)

(9.52.2)


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Medium-length transmission line

From the Kirchhoffs voltage law, the sending end voltage is

2 12S ser R C R R R R

YZ V ZI V Z I I V V ZI

(9.53.1)

The source current will be

1 1 2 1 1

2 2 4 2S C ser C C R S R R R RY Y ZY ZY I I I I I I V V I Y V I

(9.53.2)

Therefore, the ABCD constants of a medium-length transmission line are

1

2

14

12

ZY A

B Z

ZY C Y

ZY D

(9.53.3)If the shunt capacitance of the lineis ignored, the ABCD constantsare the constants for a shorttransmission line.


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Medium Transmission lineABCD constants are sometimes called the generalized constants ofthe transmission lineIn general they are complex numbersA and D are dimensionless and equal each other if the line is samewhen viewed from either end

The dimension of B and C are ohms and mhosThe constants apply to any linear, passive and bilateral four terminalnetwork having tow pairs of terminalSuch a network is called a two port networkBy letting IR=0 A=V s/VR at no load

Similarly B is the ratio V s/IR when the receiving end is short circuitedConstant A is useful in computing regulationIf V RFL is receiving end voltage at full load for sending end voltage ofVS becomes %regulation = V s/A-V RFL

VRFL


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REACTIVE POWER COMPENSATION OF LINE

The desired reactance of the capacitor bank can be determined bycompensating for a specific amount of the total inductive reactance of thelineThis leads to the term compensation factor which is defined by Xc/X L 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 thecapacitor bank by its ABCD constants

in long distance power transmission if generators are located far from loadsseries compensation is requiredLower voltage drop in line with series compensation is an additionaladvantageSeries capacitors are also useful in balancing the voltage drop of tow parallel

lines


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THE LONG TRANSMISSION LINE: SOLUTION OF THEDIFFERENTIAL EQUATIONS

For 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 isdistributed 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


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THE LONG TRANSMISSI ON LINE:INTERPRETATION OF THE EQUATIONS

Both y and Zc are complex quantities. The real part of the propagation constant y iscalled the attenuation constant alpha and is measured in nepers per unit length . Thequadrature part of gama is called the phase constant beta and is measured in radians perunit length . Thus,


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THE LONG TRANSMISSION LINE:HYPERBOLIC FORM OF THE EQUATIONS


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THE EQUIVALENT CIRCUIT


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THE EQUIVALENT CIRCUITOF A LONG LINE

z


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Power circle diagram of transmission line


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Performance of transmission line can be studied either by analyticalmethod 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 andreactive power as horizontal and vertical coordinates respectively



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Receving 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 forinductive load.

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

S R R

S R R

V AV BI

I CV DI


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Procedure


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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 receivingend voltage remains constant.

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


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Procedure


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Procedure

Because of sending and receiving end voltage is constant and with parameter Aand 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


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MVAr


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Cr

O

Radius

MW


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MVAr


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Cr

Radius

r

P

MW

MVAr


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Cr

O

Radius

rPr

Qr

P

MW


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Power circle diagram of line

The operating point P on circle is loacted by amount of real power delivered tothe 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 equipmentmaximum receiving end power etc can also be obtained from power circlediagram


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REACTIVE POWER COMPENSATION OF LINE

SHUNT COMPENSATIONSERIES COMPENSATION

Q i


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Questions

Difference between nominal pi and nominal T circuits. Difference between series and shunt compensation. What is ferranti effect What are ABCD parameters What is lossless line What is surge impedance What is wavelength Surge impedance loading SLACK BUS?

INFINE BUS PV BUS? PQ BUS?

67

B k R f


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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

68


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Thank YouDownload available on

www.sumitrathor.co.in Email: [email protected]
http://www.sumitrathor.co.in/http://www.sumitrathor.co.in/

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