eee471_ch56662

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Chapter 5: Line Model and Performance

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This chapter deals with the representation and performance of

transmission lines under normal operating conditions.

Transmission lines can be represented by an equivalent circuit with

appropriate circuit parameters on a «per-phase» basis.

The terminal voltages are represented as «line-to-neutral»

The currents are represented as «phase current»

The three-phase system is reduced to an equivalent single-phase

system

The aim of transmission line is to transmit power from one end

to another over a distance with high efficiency and low voltage

regulation

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The transmission lines are categorized into three types

1) Short transmission line– the line length is up to 80 km.

2) Medium transmission line– the line length is between 80km to 160 km.

3) Long transmission line – the line length is more than 160 km.

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Short Line Model

The short line model is suitable for the lines up to 80 km.

The short line model is suitable for the voltage level up to 69 kV.

The line is modeled using resistance R and inductive reactance X.

Due to smaller length and lower voltage the capacitance of the line

can be ignored.

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Short Line Model

Z: per-phase impedance of the short line

r: per-phase resistance per unit length

L: per-phase inductance per unit length

l: length of the line

R: per-phase resistance of the line

X: per-phase inductive reactance of the line

Receiving-end current

Sending-end voltage

Sending-end current

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Two-port network model of short transmission line

The ABCD parameters are given as in matrix form:

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Voltage Regulation:

Voltage regulation of the line may be defined as the percentage change in

voltage at the receiving-end of the line (expresed as percent of full-load

voltage) in going from no-load to full-load.

Voltage regulation is a measure of voltage drop of the line and depends on the power factor of

the load.

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Total Line Loss and Efficiency:

The total line loss is

total line losssending-end

side complex

power

receiving-end side

complex power

Transmission line

efficiency

Total real power at the

receiving-end

Total real power at the

sending-end

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f=60 Hz

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Medium Line Model

The medium line model is suitable for the lines between 80-250 km.

The transmission voltage level is generally bigger than 69 kV.

Since the distance and voltage are increased, shunt capacitance

should be taken into account.

Series resistance R and inductive reactance X of the line are still

used.

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Nominal π-Model:

Half of the shunt capacitance may be considered to be lumped at each end of the line.

g represents the leakage current over the insulators and corona effects

Y: total shunt admittance of the line

g: shunt conductance per-unit length, generally

taken as zero under normal operating conditions

C: Line-to-neutral capacitance per unit length

l: length of the line

Z: Total series impedance of the line Nominal π-model for medium length line

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Derivations of equations to obtain ABCD parameters:

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Receiving-end side voltage and current can be found

in terms of sending-end side voltage and current:

Since

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Long Line Model

For short and medium length lines, the models are obtained using lumped

parameters.

For lines longer than 250 km, the distributed parameters should be

considered for accurate solutions.

Expressions for voltage and current at any point on the line are derived.

Based on these equations an equivalent π-model is obtained.

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Long line with distributed parameters

z: series impedance per phase per unit length

y: shunt admittance per phase per unit length

z = r + jwL

y = g + jwC

Δx = small segment

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KVL

KCL

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2

Combining 1 and 2 with differentiating

Let

2nd order DE is obtained

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The solution of the differential equation is

Propagation constant

(complex number)

Real Part

(attenuation constant)

Imaginary Part

(phase constant)

From (1)

or

Characteristic impedance

of the line

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To find constant A1 and A2 we need boundary conditions

V(x) = VR and I(x) = IR at x=0

From (3) and (4)

The general expressions for voltage and

current along a long transmission line

By rearranging (3) and (4)

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By rearranging (5)


By recognizing the hyperbolic functions

sinh and cosh

At x = l we reach sending-end side:

V(x=l) = VS and I(x=l) = IS

Rewriting the equations in ABCD form

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Equivalent π model for long line

The parameters of the equivalent π model

Note that:

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When line losses are neglected (g=r=0), the characteristic impedance becomes purely resistive:

Also known as «surge impedance»

For a lossles line the following equations can be derived:

Velocity of propagation

Wavelength

When the internal flux

of the conductor is neglected

sm /7.637,795,299

Speed of light (3x108 m/s)

km5000

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Surge Impedance Loading

Surge Impedance Loading (SIL) is the case when the line is terminated with an

impedance equal to the characteristic impedance at the receiving-end side.

For a lossless line ZC is a real number so there is no reactive power in the line.

The reactive power consumption in the line by series inductive reactance (reactive

losses) are exactly offset by reactive power supplied by shunt capacitance.

SIL for typical transmission lines varies from approximately 150 MW for 230-kV lines

to about 2000 MW for 765-kV lines.

Shunt capacitive compensation may be required to increase voltage at certain

buses for the cases where line loading is bigger than SIL.

Shunt inductive compensation may be required to decrease voltgae at certain buses

for the cases where line loading is smaller than SIL.

= since

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Thermal Limit of Transmission Lines

Thermal limit of a transmission line is defined in terms of the maximum current carrying

capacity (ampacity).

The excess amount of current flowing on the line produces heat leading to undesirable

results such as

annealing loss

gradual loss of mechanical strength of the conductor caused by temperature extremes

increase sag and decreased clearance to ground due to conductor expansion at higher

temperatures

So the transmission line can be utilized best only if it is loaded up to its thermal limit

which cannot be done normally without line compensation.

Thermal loading limit

of a line

Obtained from manufacturer’s data

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Maximum Power Transfer and Angle Stability

Power-angle curve of a transmission line

Stable region unstable regionFor a lossless line three-phase real power transfer

from sending-end to receiving-end side

practice

SILpu pu

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Voltage Stability

Voltage stability is defined as “the ability of a power system to maintain steady acceptable

voltages at all buses in the system under normal operating conditions and after being subjected

to a disturbance” (Kundur, 1994).

A typical voltage-power characteristics (Kundur, 1994)

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A typical reactive power – voltage curves (Kundur, 1994)

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Transmission Line Loadability Curve (Kundur, 1994)

Transmission Line Loadability Curve

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Transmission Line Loadability Characteristics

Loadability characteristics of transmission lines (Zhang et al., 2006)

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SILpu pu

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Transmission Line Compensation

Normally transmission lines are not loaded at their surge impedance, the ideal

case in which neither reactive power is produced nor consumed in the line.

On long transmission lines, light loads less than SIL result in a voltage rise at

the receiving-end.

On long transmission lines, heavy loads greater than SIL result in a voltage dip

at the receiving-end.

Shunt reactors are widely used to reduce high voltages under light load or

open line conditions.

Shunt capacitors, static var compensators, and synchronous condensers

(very old technique) are used to boost voltage, increase power transfer and

improve system stability.

Today transmission line compensation is done using modern power electronic

based high power converters called «Flexible AC Transmission Systems»

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Shunt Reactors

Shunt reactors are conventional solutions to compensate for the undesirable voltage

effects associated with line capacitance.

Shunt reactors are used to control voltage during low-load period.

Shunt reactors are usually unswitched.

Photo: http://kiran111.hubpages.com/hub/electrical-substation

Shunt reactor banks

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The amount of shunt reactive power required on a transmission line to maintain the receiving-end

voltage at a specified value can be obtained as follow:

Reactance of shunt reactor

connected at the receiving-end

Since

Solving for XLsh

If VS=VR is required

The amount of shunt reactive power

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Shunt Capacitive Compensation

Photo: http://kiran111.hubpages.com/hub/electrical-substation

Shunt capacitor banks

Shunt capacitors are used for

compensating reactive power of lagging power factor load

improving power factor

voltage control during heavy lagging power factor loads

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Single-Line Diagram of the Thyristor controlled SVC (Static Var Compensator)

Figure: http://www.mathworks.com/help/releases/R2013b/physmod/sps/powersys/ug/pe_applications6a.gif

Operate

as a variable

shunt reactor

Operate

as a variable

shunt capacitor

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Single-Line Diagram of the STATCOM (Static Synchronous Compensator)

STATCOM configuration:(a) single-line diagram (b) operating modes

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Series Capacitive Compensation

Series capacitors are connected in series with the line (at sending/receiving-end or mid-point)

To reduce the series reactance between the load and the supply point

To increase power transfer

To improve transient and steady-state stability of the power system

Series capacitors are provided with bypass circuit breaker and protective spark–gaps

Studies show that the addition of series capacitors on EHV lines can more than double

the transient stability load limit of long lines at a fraction of the cost of a new transmission line

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Series Capacitors

Photo:http://1.bp.blogspot.com/-VuW2DoFs420/Tg3ocZTd0yI/AAAAAAAAAMY/pZyF63u7JJY/s1600/Series-Capacitors.jpg

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With the series capacitor switched, the power transfer over the line for a lossless line becomes:

Series and shunt compensation

XCser Series capacitor reactance

XCser / X’ Percentage compensation (25-70 % in practice)

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There are two drawbacks of series capactive compensation:

Protection devices are required to protect capacitors and bypass the high current produced

when a short circuits occurs in the power system.

Series capacitors can lead to subsynchronous resonance due to the resonant circuit that

can oscillate at a frequency below the normal synchronous frequency (50/60 Hz) when stilmulated

by a disturbance. Subsynchronous resonance can damage turbine-generator.

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End of Chapter 5

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