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EE3741 L3 Transmission line

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Lecture 3 Transmission line
Subject lecturer: Dr. XU Zhao
Department of Electrical Engineering
Hong Kong Polytechnic University
Email: [email protected]
R
Room:
CF632
Tel: 27666160
Outlines
•
•
•
•
2
Types
Inductance of transmission line
Capacitance of transmission line
Transmission line model
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Types of power lines
• Power lines classified according to voltage class
– Low voltage (LV) lines – 120 V – 600 V
• Lines are insulated conductors,
conductors usually made of aluminium,
aluminium
often extended from local power mounted distribution
transformers to service area of consumer
• Lines may be overhead or underground
– Underground cables found in metropolitan areas with grid
providing dependable service in which some outages will not
cause loss of load
– Medium-voltage (MV) lines – 2.4 kV – 69 kV
• Predominantly radial systems with lines spreading out from
sub-stations to feed power to high-rise buildings, shopping
centres and campuses
– High-voltage (HV) lines - <230 kV
• Lines composed of aerial conductors or underground cables
– Extra-high-voltage (EHV) lines – operate at voltages up to
800 kV
• Used when g
generating
g stations very
y far from load centres
• May be as long as 1000 km
3
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Component of HV
transmission line
• Conductors
Aluminum outer strands
2 layers, 30 conductors
Steell core strands,
St
t d
7 conductors
– Stranded copper conductors or steel-reinforced aluminium
cable (ACSR)
• ACSR usually preferred because result in lighter and more
economical lines
• Insulators
– Serve to support and anchor conductors and insulate them
from ground
• Usually made of porcelain, but glass
and other synthetic insulating materials
may be used
– Must offer high resistivity to surface
leakage currents
– Must be sufficiently thick to prevent
breakdown under high voltage stress
4
Electrical Engineering, HKPU
Single-phase high-voltage cable with solid
dielectric
EE3741 Ass. Prof Zhao Xu
Component of HV transmission line
•Supporting structures
– Keep conductors at safe height from ground and at
adequate distance from each other
• Wooden poles equipped with cross-arms used for
voltages below 70 kV
• For higher voltage two poles used to create H-frame
• For very high voltage lines, steel tower used
– Spacing between conductors must be sufficient to
prevent arc-over under g
p
gusty
y conditions and increased
as distance between tower and line voltages become
higher
5
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Resistance
• DC Resistance at given temperature
RDC,T = ρl/A
l – length of conductor
A – cross-sectional area of conductor
ρ – resistivity of conductor
– Resistivity will depend upon conductor material
6
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Resistance
• If resistance of conductor at certain temperature know, DC
resistance at other temperature given by:
⎛ M + T2 ⎞
⎟⎟ RT 1
RT 2 = ⎜⎜
M + T1 ⎠
⎝
M – temperature constant
• Resistance of nonmagnetic conductors varies with frequency
due to “skin
skin effect”
effect
– Electric current distributions inside conductor not uniform
As frequency increases, current tends to flow nearer to outer
surface of conductor,
conductor decreasing effective cross section
E.g.
RAC = K * RDC
RAC ≈ (1.05
(1 05 ~ 1.10)*
1 10)* RDC @ 60Hz
60H
7
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Inductance
• Series inductance of transmission line consists of two
components
– Internal inductance
• Due to magnetic flux enclosed by conductor
– External inductance
• Due to magnetic flux outside conductor
• Total inductance a combination of both these effects
8
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Internal Inductance
L=λ/I
Current carrying conductor
•Magnetic field intensity around path, radius x,
inside conductor
9
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Internal Inductance
• Flux density around the path inside the conductor
• Flux linkages are equal to fraction of current linked
times flux per meter length. Thus the total flux linkages
inside the conductor
• The flux linkages and the consequent internal
inductance of the conductor
– THE INTERNAL INDUCTANCE IS CONSTANT!, IRRESPECTIVE
OF CONDUCTOR DIAMETER
10
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
External inductance
y to flux between p
• Inductance due only
points at D1 and D2
meters
– Depends
p
comparative
p
distances from current carry
y conductor
11
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Inductance of single phase line
•Consider two wire
transmission system
–Conductor 1 is active
conductor
d t
–Conductor 2 provide
return path for current
• “Neutral” conductor
2- wire transmission line system
12
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Inductance of single phase line
• Inductance of two-wire circuit due to current
flowing
g in conductor 1
• Total
T t l inductance
i d t
can b
be simplified
i
lifi d tto:
r’1=r1e−1/4 : GMR (geometric mean radius) of the conductor
13
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
r’1=rr1e−1/4 : GMR (geometric mean radius) of the conductor
Inductance of single phase line
•r1’ – (GMR) equivalent radius of conductor 1=
0.7788r1
– The impact of self inductance is to reduce the effective
radius of the conductor
• Inductance
d
off conductor
d
2
• The inductance
ind ctance of the complete ci
circuit
c it is then
OR
If r1 = r2
14
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Inductance of Three-Phase Circuit
• Inductance due to current flow in
phase a
p
• Inductance of phase b, c
r’i=rie−1/4 : GMR (geometric mean radius) of the conductor
15
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
For a completely transposed
transmission line, the average
inductance of each conductor over
a complete cycle is the same.
Inductance of Three-Phase Circuit
• If all three conductors have same radius
–GMD – geometric mean distance = (D12D23D31)⅓
• For a three-phase,
p
, equilaterally
q
y spaced
p
transmission
line, phase conductors have equal separation
distances
GMR of the conductor: r ’ = re-1/4
– D12 = D23 = D31 = D
=2×10-7[1/4+ln(D/r)] H/m
If 3-phase line not equilaterally spacedÆ different Φ and L, but small; assume
transposed,
eachHKPU
conductor occupies the original positions of
theAss.
other
16
Electrical i.e.
Engineering,
EE3741
Prof Zhao Xu
16
conductors over equal distances.
Examples
• A single circuit,
circuit fully transposed,
transposed three-phase,
three phase 60Hz transmission line
consists of three conductors arranged as shown. The aluminium
conductor has a diameter of 250mils, find the inductive reactance of the
line p
per kilometre p
per p
phase.
• Note: 1 mil: 1×10-3 in; 1m = 39.36in
17
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Solution
1m = 39.36inches(in)
Diameter: 250mils=0.25in=0.25/39.36=0.006352m
Radius: r = 0.006352/2
0.006352/2=0.003176m
0.003176m
De = (5×5 ×8)1/3=5.848m
For each kilometre of length, the inductance is:
L = 2[1/4 + ln(De/r)] ×10-7 ×103=1.544mH/km
Inductive reactance per km is: XL=2πfL=377×1.544 ×10−3=0.5858Ω
18
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Verify the solution with GMR & GMD
19
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Inductance of Stranded Conductors
• Stranded conductors have 2 or more elements or strands of
wire that are electrically in parallel.
• All strands
t
d are id
identical,
ti l shares
h
currentt equally.
ll
• La =Lb=Lc = 2×10-7 ln(GMD/GMR)
GMR of stranded conductor:
Equivalent radius or GMR (Geometric Mean Radius) of the conductor: r ’ = re-1/4
Dnn
L=LX+LY
20
Electrical Engineering, HKPU
GMR of the kth conducto
G
conductor:: Dkk = rk’ = rke-1/4
EE3741 Ass. Prof Zhao Xu
Inductance of Bundled Conductors
•La =Lb=Lc = 2×10-7 ln(GMD/GMRb)
Two-conductor bundle
GMRb = rb = d rc
Three-conductor bundle
rb = 3 d 2 rc
Four-conductor bundle:
rb = 1.09 4 d 3 rc
rc: GMRc of individual conductor
Bundled Conductors: have 2 or
more conductors belong to the
same phase and are close
t
together
th in
i comparison
i
with
ith the
th
separation distances between the
phases.
2-bundle 3-bundle
21
2-bundle conductor line
4-bundle
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Capacitance
•When
Wh
voltage
lt
applied
li d to
t pair
i off conducting
d ti
plates (separated by non-conducting
medi m) charge
medium)
h ge accumulates
m l te on each
e h side
ide
of plate
–Magnitude
M
it d off charge
h
on each
h side
id off plate
l t equall
–Charges on each side have opposing polarity
•Magnitude
Magnit de of charge
cha ge deposited proportional
p opo tional
to applied voltage
Q = CV
–C – capacitance of line
22
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Capacitance
•Potential difference between conductor in
transmission line cause them to become
charged like capacitors
–Effective capacitance dependent upon size and
separation
i
di
distance off conductors
d
•AC power lines energized by time varying
voltage
–AC voltage causes charge on conductors to vary
• “Charging”
Charging current of line capacitance
–Charging current effects
• Power transmitted and operating power factor
• Voltage drop along line
23
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Electric field around long conductor
• Can consider transmission line as long
g straight
g
conductor having uniform charge throughout length
– Electric field intensity at point P some distance x from
conductor given by:
E = q / (2πεx)
q – change of conductor [coloumbs/m]
permittivity
y of medium surrounding
g conductor=εrε0
ε–p
εr: relative permittivity (dielectric constant) =1 for air
-9
-12 F/mEE3741 Ass. Prof Zhao Xu
Electrical Engineering, HKPU
ε24
0: permittivity of free space = 1/(36π)×10 =8.854×10
Electric field around long conductor
•Instantaneous potential difference between two
point, P1 and P2 around conductor
–Found by considering change in electric field over
radial path between two points
25
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Capacitance of single phase line
• Consider
C
id active
ti
conductor
d t
and return path as two
long lines of charge
• Total voltage between
lines
– Superposition of:
Voltage drop between lines
due to charge on active
conductor
+
Voltage drop between lines
due to charge on return
conductor
26
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Capacitance of single phase line
27
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Capacitance of single phase line
εr: relative permittivity (dielectric constant) =1 for air
ε0: permittivity of free space = 1/(36π)×10-9=8.854×10-12 F/m
28
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Capacitance of single phase line
•Potential difference between each
conductor and ground half the potential
difference between two conductors
–Capacitance to ground / Capacitance to neutral
twice
i capacitance
i
ffrom line
li
- line
li
εr: relative permittivity (dielectric constant) =1 for air
ε0: permittivity of free space = 1/(36π)×10-9=8.854×10-12 F/m
29
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Capacitance of three-phase line
•Capacitance
Capacitance per
phase given by:
a
D3
2
D1
GMD =
(D12D23D31)⅓
=D
provided conductors
h
have
same diameter
di
t
and are equilaterally
spaced
p
1
where
c
D23
b
Bundled line capacitance
30
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Line reactance
• Inductive reactance
– Total
T t l iinductive
d ti
reactance
t
proportional
ti
l to
t li
line llength
th
• Total line inductive reactance found by multiplying inductive
reactance by line length
• Capacitive reactance
– Total capacitive reactance inversely proportional to line
length
• Total line capacitive reactance found by dividing capacitive
reactance by
y line length
g
– Total capacitive susceptance (admittance) proportional to
line length
Xc = Ω⋅m / m = Ω
31
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Transmission line model
• Transmission lines are represented by an equivalent circuit with
parameters on a per-phase basis
– Voltages are expressed as phase-to-neutral
– Currents are expressed for one phase
– The three phase system is reduced to an equivalent single-phase
• All lines are made up of distributed series inductance and
resistance, and shunt capacitance and conductance
– Line parameters: R, L, C, & G
–Capacitance
p
between neighbouring
g
g
conductors, line and ground
– Inductance due to Stranded & Bundled Conductors
• Three types of models
– depend on the length and the voltage level
– short, medium, and long length line models
32
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Transmission Line models
• Classified according to length of line
– Short line – less than 50 miles (80 km)
(or up to 320 km
for some applications depending upon whether line
characteristics can still be represented by lumped
components)
– Medium line – 50 ~ 150 miles (80 ~ 240 km)
– Long line
33
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Short Transmission Lines
• Applicable to lines up to 80.45 km (50 miles) long
• Equivalent circuit of short transmission line
consists of series combination of line resistance
and inductive reactance
– Line capacitance ignored!
34
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Short Transmission Lines
• No shunt branches
– Cu
Current
e a
at se
sending
d g end
e d equal
equa to
o current
cu e at
a receiving
ece
g end
e d
Is = IR
• Voltage at sending end given by:
Vs = VR + IRZ
• Voltage regulation
– Increase
I
in
i receiving
i i
end
d voltage
lt
as load
l d reduced
d
d from
f
full load to no load with sending end voltage held
constant
35
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Two-Port model representation
ABCD two p
port model
Vs = VR + IRZ
Is = IR
36
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Medium Length Lines
• For lines 80 km (50 miles) - 240 km (150 miles) capacitive
reactance between lines and neutral (earth) not
insignificant
• Shunt admittance (usually pure capacitance) included in
equivalent circuit
– Allows formation of nominal π equivalent circuit
• Line impedance
p
still represented
p
as lumped
p
components
p
37
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Medium Length Line
• Sending-end voltage and current expressed in terms
of receiving end voltage and current using “ABCD”
transmission parameters
AD-BC = 1
A&D: dimensionless
B’s unit: (Ω)
C’s unit: siemens
– ABCD parameters governed by line impedance and
admittance
38
• They are a short-cut way of representing line
characteristics
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Medium Length Line
VS=AVR+BIR
IS=CVR+DIR
No load: IR = 0,, VS=AVR
•Voltage Regulation (at specific power
factor)
–At
At no lload,
d receiving-end
i i
d voltage
lt
iis 1/A ti
times
sending end voltage
Sending end power: = PS= 3VSISPFS
Efficient of line: = ((PR//PS))100%
39
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Long transmission line
• For lines greater than 240/250 km (150 miles)
lumped components no longer provide required
accuracy
– Consider parameters distributed evenly throughout line
l: length of line
Z0 = SQRT(z/y): characteristic or surge impedance z: series impedance per unit length, Z=zl
γ=SQRT(zy):
constant
y: shunt admittanceEE3741
per unit
length,
40
Electrical propagation
Engineering, HKPU
Ass. Prof
Zhao Xu Y=yl
40
Long transmission line
•For lines greater than 240/250 km (150
miles) lumped components no longer
provide required accuracy
– Consider parameters distributed evenly throughout line
•Modeling
M d li
off the
th transmission
t
i i
line
li
parameters
– Accuracy obtained by using distributed parameters
– The series impedance per unit length is z
– The shunt admittance per unit length is y
– The
Th distance
di t
from
f
receiving
i i
end
d is
i x
41
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Long transmission line
42
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Long transmission line
43
Electrical Engineering, HKPU
Wave equation for lossy line
EE3741 Ass. Prof Zhao Xu
Long transmission line
• let
d 2 I ( x)
dx 2
= γ 2 I ( x)
α-attenuation/damping
constant
β –phase
p
constant
44
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Two part model
45
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Two part model
46
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Wave propagation
• Substitute
V ( x) = A1eαx e jβx + A2 e −αx e − jβx
• Transform back to time domain
v( x, t ) = 2 Re A1eαx e j (ωt + βx ) + 2 Re A2 e −αx e j (ωt − βx )
-amplitude increases
along positive x direction
-amplitude decreases
along positive x direction
-incident
incident wave -traveling
traveling
towards receiving end
-reflected wave traveling
backward to sending end
Note:
- α >0 for a line with resistance
-Traveling sinusoidal waves in positive x direction
47
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Courtesy by Prof Goran Andersson ETH
48
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Wave propagation
• Wave length: a voltage cycle corresponds to 2 π change of
angular argument βx
λ=2π/β
• If loss neglected g=r=0, and
β = ω LC
C
1
1
λ=
LC
f LC
• Using the equation for per unit length inductance and
capacitance L and C, we can have
1
v≈
= 3 ×108 m / s
u0ε 0
v = λ / T = 2πf / β =
λ≈
49
Electrical Engineering, HKPU
1
= 3 ×108 / 50 = 6km
f u0ε 0
EE3741 Ass. Prof Zhao Xu
Surge impedance
• If loss neglected g=r=0
g=r=0, the characteristic impedance
L
-Surge impedance purely resistive
C(
• Surge
u g impedance
p da
loading
oad g (SIL)
) @ receiving
g end
d
2
3
V
(kVrated ) 2
R
*
SIL = 3VR I R =
=
MW
Zc
Zc
Zc =
• @SIL, voltage and current along lines are constant in magnitude
as sending
di
end
d
V ( x ) = (cos β x + j sin β x )V R = V R ∠ β x
I ( x ) = (cos β x + j sin β x ) I R = I R ∠ β x
• No reflected wave (A2=0) and reactive power in the line, and
reactive loss due to shunt capacitance and series inductance offset
each other
• SIL indicate loading without
itho t reactive
eacti e compensation,
compensation therefore
the efo e
reflecting capacity somehow: [email protected] SIL compensation little,
loading>> SIL-compensation needed
• Zc for overhead line 400-600 ohm, cable: 30-50 ohm, overhead
li
line
lloading
di
can > SIL,
SIL while
hil cable
bl lloading
di
always
l
< SIL (SIL
larger than cable thermal limit)
50
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Wave transient analysis
are reflected wave @receiving end
Let Zc=surge impendence
Voltage reflection coefficient @ receiving end
and SC termination, -1, open circuit termination +1.
Since
current reflection [email protected] end is -ρ
ρR
When wave back to sending end, reflection [email protected]
end
51
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Wave transient analysis-example
• Refer to Page228-230 Power System analysis By
John&William, McGraw-Hill 1994
• A DC source of 120v with negligible resistance is
connected through a switch S to a lossless transmission
line having Zc
Zc=30
30 ohm. The line is terminated in a
resistance of 90 (10) ohm. If S closes at t =0 , plot VR
v.s. time until t=5T, where T is the time for a voltage
wave to travel the length of line.
line
52
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Wave transient analysis-example
Incident wave starts to travel along
line (Note: x from sending end here),
(
) is the unit step
p
where U(vt-x)
function, equal to 1 when vt-x>0,
and 0when vt-x<0
No reflection until wave reach receiving
end with Zc=30,
end,
Zc=30
@ t= T, ρr =(90-30)/(90+30)=1/2
v-=120*1/2=60 V VR=120+60=180 V
@t 2T ρs =(0-30)/(0+30)=-1
@t=2T,
(0 30)/(0+30) 1
v-=-1*60=-60, Vs=120v always
@t=3T, ρr =(90-30)/(90+30)=1/2
v-=1/2*-60=-30,
1/2* 60 30 VR=180-60-30=90
180 60 30 90 V
…
@t=5T, VR=135 v
53
Electrical Engineering, HKPU
Lattice
diagram
EE3741 Ass. Prof Zhao Xu
Wave transient analysis-example
•ZR=10 ohm
•Current lattice diagram can be drawn
with reflection of current is negative
value of voltage one
54
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Wave transient analysis
• Reflection can occurs at the end, and also when a line has a junction
of two different parts, or bifurcation
• E.g.
E g an overhead line connected to cable,
cable the first part terminates at
Zc of the second part cable. The part of wave which continues to
travel and is not reflected back at the junction is called the refracted
wave
• When applies a voltage surge, a same shape voltage surge traveling
backwards as seen in example occurs at end of the lossless line,
• If ZR is not open circuit or SC, reflected wave has reduced
magnitude. When ZR> Zc, the peak terminal voltage > voltage surge
55
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Wave transient analysis
• Terminal equipment tends to overvoltage when surge applies in the
circuit, and therefore protected by surge arrestor (lightening arrestor,
surge divider)
– Conducting at certain v above design rating v
– Limit terminal voltage to its design value
– Noncoducting again when line
line-neutral
neutral voltage drops below rating
• Air gap type, difficult to extinguish current for AC
• Air gap+ nonlinear resistor in series(ohm decreases quick when V rises)
• silicon carbide
carbide, zinc oxide is more popular without need of air gap
56
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Transmission power
•Power transfer by transmission line
VS = VS ∠δ ,VR = VR ∠0
A = A ∠θ A , B = B ∠θ B , I R =
S R = PR + jQ R = 3VR I R = 3
VS VR
*
PR =
QR =
VS ( L − L ) VR ( L − L )
B
VS ( L − L ) VR ( L − L )
B
B
∠(θ B−δ ) − 3
cos(θ B−δ ) −
A VR ( L − L )
sin(θ B−δ ) −
A VR ( L − L )
B
2
∠(θ B−θ A)
cos(θ B−θ A)
2
sin(θ B−θ A)
S R = PR + jQ R = 3VR I R
PS =
57
Electrical Engineering, HKPU
B ∠θ B
2
B
B
A VR
VS ∠δ − A ∠θ A VR ∠0
QS =
A VS ( L − L )
B
IS =
2
cos(θ B−θ A) −
B
A VS ( L − L )
*
2
A ∠θ A VS ∠δ − VR ∠0
B ∠θ B
VS ( L − L ) VR ( L − L )
B
VS ( L − L ) VR ( L − L )
cos(θ B+δ )
sin(θ B−θ A) −EE3741 Ass. Prof Zhao Xu
sin(θ B+δ )
B
Transmission power
• Real
R l and
d reactive
ti
power transmission
t
i i
lloss
PL = PS − PR
QL = QS − QR
• Power transfer by lossless line B = jX ' , θ A = 0, θ B = 900 , A = cos βl
PS = PR =
VS ( L − L ) VR ( L − L )
X
'
sin δ
QR =
VS ( L − L ) VR ( L − L )
X
'
cos δ −
VR ( L − L )
B
2
cos βl
• Power-delta
P
d lt curve: max power transfer
t
f @ 90 degree
d
• Actual stability is much lower [35,45]degree due to
stability
y consideration of generator,
g
, i.e. to withstand
sudden loss of generator or load
58
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Transmission capability
• Power transmission capability is
limited by thermal and stability
limits
• Overloading may cause high
temperature and irreversible
stretching of conductor, i.e. physical
sag of lines due to real power loss
• Thermal limit specified in current
carrying capability Ithemral from
manufacture datasheet
Sthermal = 3VφRated I thermal
59
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Other Limits Affecting Power Transfer
Stability limit
zAngle limits
– while the maximum power transfer occurs when line angle
difference is 90 degrees,
degrees actual limit is substantially less
due to multiple lines in the system and also the generator
stability limits [35, 45]degree
zVoltage stability limits
– as power transfers increases, reactive losses increase as
I2X.
X As reactive power increases the voltage falls
falls, resulting
in a potentially cascading voltage collapse.
60
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Transmission line compensation
• Self reactance and capacitance causing voltage variations and reactive
flow
• Loading variation cause voltage variations along the line
line, e.g.
eg
– @ SIL, voltage profile are flat along line and no reactive flow
– <SIL, voltage rise, line generate VARs
– >SIL,
>SIL voltage drops,
drops line consume VARs
• Shunt v.s. series compensation: Shunt reactors, Shunt capacitor, Series
capacitor, FACTs devices…
61
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Transmission compensation
• Compensate for voltage uneven distribution cause by line
capacitance
• Shunt reactors xLsh @end of long line
• Solving for xLshh
• If requiring
q
g VS=VR
• Substitute
b
Xlsh
• Voltage is uneven along the line, mid point voltage is, why?
62
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
Transmission compensation
• Shunt capacitor for heavy loads with
lagging power factor
• Series capacitor connected in mid of line
– reduce reactance
– Reduce voltage
g drop
p
– improve steady state and transient stability
loading limit for EHV lines at very low costs
comparing to new line costs
PS = PR =
VS ( L − L ) VR ( L − L )
X'
sin δ
is in percentage called percentage compensation
– Drawbacks: special protection to prevent high
current at SC fault occurrence, and SSR
(
(resonant
circuit causing oscillations
ll
when
h
stimulated by a disturbance)
63
Electrical Engineering, HKPU
EE3741 Ass. Prof Zhao Xu
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