By
Prof. C. Radhakrishna
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MODELING OF TRANSMISSION LINES
Electrical Characteristics
Natural or Surge Impedance Loading
Classification of line length Performance Requirements of Power Transmission Lines
Terminal V, I Relations
Transmission Matrix
Lumped‐Circuit Equivalent
Simplified Models
MODELING OF TRANSFORMERS
Representation of Two‐Winding Transformers Standard equivalent circuit
Equivalent π circuit representation
Consideration of three‐phase transformer connections
Representation of Three‐Winding Transformers Phase‐Shifting Transformers
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Modeling of Transmission Lines
For the same rating, cables are 10 to 15 times more expensive than overhead lines
Electrical Characteristics
Overhead lines characterized by four parameters: series resistance R due to
the conductor resistivity, shunt conductance G due to leakage
currents between the phases and ground, series inductance L
due to magnetic field surrounding the conductors, and shunt
capacitance C due to the electric field between conductors.
Underground cables
have the same basic parameters as overhead lines
values of the parameters and hence the characteristic of cables differ significantly from those of overhead lines 10/17/2010 12:03 AM
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Natural or Surge Impedance Loading
• Characteristic impedance Zc with losses neglected is
commonly referred to as the surge impedance. It is equal to
√(L/C) and has the dimension of a pure resistance.
• The propagation constant, γ, is defined as,
γ = √(yz) = α + jβ
The power delivered by a transmission line when it is
terminated by its surge impedance is known as the
natural load or surge impedance load (SIL):
V 02
SIL 
ZC
W
where V0 is the rated voltage of the line. If V0 is the line‐
to‐neutral voltage, SIL given by the above equation is the
per‐phase value; if V0 is the line‐to‐line value, then SIL is
the three‐phase value.
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Natural or Surge Impedance Loading contd…..
At SIL, transmission lines (lossless) exhibit the following special characteristics: • V and have constant amplitude along the line. I
• V and are in phase throughout the length of the I
line. • The phase angle between the sending end and receiving
end voltages (currents) is equal to βl.
 At the natural load, the reactive power generated by
C is equal to the reactive power absorbed by L, for each
incremental length of the line.
 No reactive power is absorbed or generated at either
end of the line, and the voltage and current profiles are
flat.
 This is an optimum condition with respect to control
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of voltage and reactive power.
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Classification of line length a) Short lines: lines shorter than about 80 km (50 mi). They
have negligible shunt capacitance, and may be
represented by their series impedance.
b) Medium‐length lines: lines with lengths in the range of
80 km to about 200 km (125 mi). They may be
represented by the nominal π equivalent circuit.
c) Long lines: lines longer than about 200 km. For such
lines the distributed effects of the parameters are
significant. They need to be represented by the
equivalent π circuit. Alternatively, they may be
represented by cascaded sections of shorter lengths,
with each section represented by a nominal π
equivalent.
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Performance Requirements of Power Transmission Lines
Voltage regulation, thermal limits, and system stability are
the factors that determine the power transmission
capability of power lines.
Terminal V, I Relations
z = r + jωι = series impedance per meter
y = g + jωc = shunt admittance per meter to neutral
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V1 = V2 cosh γl + Zc I2 sinh γl
I1 = I2 cosh γl + (V2 / Zc) sinh γl
where γ  √ (yz) is called the propagation constant.
and Z  z / y
is called the characteristic impedance of the line.
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c
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Transmission Matrix
V1 = AV2 + BI2
I2 = CV2 + DI2
where A = cosh γl
C = (l / Zc) sinh γl
B = Zc sinh γl
D = cosh γl
Lumped‐Circuit Equivalent
The A, B, C and D parameters for the circuit are A  1
Z Y 
2
Z Y  

C  Y  1 

4


B  Z
D  1
Z Y 
2
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Simplified Models
Experience indicates the following classification of lines to be reasonable. Long line ( l > 150 mi, approximately): Use the π‐equivalent circuit model with Z  and ( / 2 ) ; Y
Medium‐length line (50 < l < 150 mi, approximately): Use the circuit model with Z and (Y/2) instead of Z  and ( Y  / 2), where Z = zl and Y = yl. This is called the nominal π‐
equivalent circuit. Short line (l < 50 mi, approximately): same as the medium‐
length line except that we neglect Y/2. 10/17/2010 12:03 AM
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Modeling of Transformers
 In addition to voltage transformation, transformers are often used
for control of voltage and reactive power flow.
 Practically all transformers used for bulk power transmission and
many distribution transformers have taps in one or more windings for
changing the turns ratio.
 Changing the ratio of transformation is required to compensate for
variations in system voltages.
 Two types of tap‐changing facilities are provided: off‐load tap
changing and under load tap changing (ULTC).
 The ULTC is used when the changes in ratio need to be frequent.
 The taps normally allow the ratio to vary in the range of ±10% to
±15%.
 When the voltage transformation ratio is small, autotransformers
are normally used.
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Representation of Two‐Winding Transformers Zp = Rp + jXp ; Zs = Rs + jXs
Rp, Rs = primary and secondary winding resistances
Xp, Xs = primary and secondary winding leakage reactances
np, ns = number of turns of primary and secondry winding
Xmp = magnetizing reactance referred to the primary side
Fig. 3 Basic equivalent circuit of a two‐winding transformer
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Standard equivalent circuit: Figure 4 Standard equivalent circuit for a transformer
 If the actual turns ratio is equal to np0 / ns0, then =1.0, and the ideal
transformer vanishes. When the actual turns ratio is not equal to the
nominal turns ratio, represents the off‐nominal ratio (ONR).
 The equivalent circuit of Figure 4 can be used to represent a
transformer with a fixed (or off‐load) tap on one side and an under‐load
tap changer (ULTC) on the other side.
 The off‐nominal turns ratio is assigned to the side with ULTC and
has a value corresponding to the fixed‐tap position of the other side.
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Equivalent π circuit representation
(a) General π network
(b) Equivalent π circuit
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Consideration of three‐phase transformer connections
 In establishing the ONR, the nominal turns ratio
(np0/ns0) is taken to be equal to the ratio of line‐to‐line base
voltages on both sides of the transformer irrespective of the
winding connections (Y‐Y, ∆‐∆, or Y‐∆).
 For a Y‐∆ connected transformer, this in addition
accounts for the factor √3 due to the winding connection.
 In the case of a Y‐∆ connected transformer, a 30o phase
shift is introduced between line‐to‐line voltages on the two
sides of the transformer.
 It is usually not necessary to take this phase shift into
consideration in system studies.
 The single‐phase equivalent circuit of a Y‐∆
transformer does not account for the phase shift.
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Representation of Three‐Winding Transformers Figure 6 Equivalent circuit of a three‐winding transformer
The three windings of the transformer may have different MVA
ratings.
Zps = leakage impedance measured in primary with secondary
shorted and tertiary open
Zpt = leakage impedance measured in primary with tertiary shorted
and secondary open
Zst = leakage impedance measured in secondary with tertiary shorted
and primary open
Zp = ½ (Zps + Zpt – Zst)
Zs = ½ (Zps + Zst ‐ Zpt)
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Zt = ½ (Zpt + Zst – Zps)
Phase‐Shifting Transformers
It consists of an admittance in series with an ideal
transformer having a complex turns ratio, n n
Figure 7 Phase‐shifting transformer representation
 Ye
 ip   a s2  b s2
 
 is    Ye
 a  jb
s
 s
 Ye 
a s  jb s   v p 
 
 v
Ye   s 

 The admittance matrix in the above equation is not symmetrical.
 Therefore, a π equivalent circuit is not possible.
 If the turns ratio is real (i.e., and bs = 0), the model reduces to
the equivalent π circuit shown in Figure 5.
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CONCLUSION
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REFERENCES
[ 1 ] Arthur R. Bergen & Vijay Vittal : “Power System Analysis” , 2nd edition,
Prentice Hall, Inc., 2000.
[ 2 ] Prabha Kundur : “Power System Stability and control” , The EPRI Power
System Engineering Series, McGraw‐Hill, Inc., 1994.
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THANK YOU
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