TRANSMISSION LINE OPERATION (Steady-­‐State) 8/20/13 Tramsmission Line Opera4on 1 Transmission Line Opera<on Distributed parameters: C: F/m L: H/m R: Ω/m all per unit length 8/20/13 Tramsmission Line Opera4on RDC (T ) = ρ (T )

Ω
A
2 Transmission Line Model i ( x + Δx,t )
LΔx
RΔx
+
SENDING END v ( x + Δx,t )
+
v ( x,t )
CΔx GΔx
−
RECEIVING END −
x + Δx
x
v ( x,t ) = v ( x + Δx,t ) − i ( x,t ) RΔx − LΔx
∂i ( x,t )
∂t
i ( x,t ) = i ( x + Δx,t ) − v ( x + Δx,t ) GΔx − CΔx
8/20/13 i ( x,t )
Tramsmission Line Opera4on ∂v ( x + Δx,t )
∂t
3 Transmission Line Model v ( x,t ) = v ( x + Δx,t ) − i ( x,t ) RΔx − LΔx
lim
Δx→0
v ( x + Δx,t ) − v ( x,t )
Δx
=
∂v ( x,t )
∂x
∂i ( x,t )
∂t
= i ( x,t ) R + L
∂i ( x,t )
∂t
i ( x,t ) = i ( x + Δx,t ) − v ( x + Δx,t ) GΔx − CΔx
lim
Δx→0
8/20/13 i ( x + Δx,t ) − i ( x,t )
Δx
=
∂i ( x,t )
∂x
Tramsmission Line Opera4on ∂v ( x + Δx,t )
= Gv ( x,t ) + C
∂t
∂v ( x,t )
∂t
4 Transmission Line Equa<ons Time Domain: ∂v ( x,t )
∂i ( x,t )
=
Ri
x,t
+
L
( )
∂x
∂t
∂i ( x,t )
∂v ( x,t )
=
Gv
x,t
+
C
( )
∂x
∂t
Frequency Domain: ∂ → jω
∂t
dV ( x )
dx
dI ( x )
dx
8/20/13 = ( R + jω L ) I ( x ) = zI ( x ) , z = R + jω L
= ( G + jω C )V ( x ) = yV ( x ) , y = G + jω C
Tramsmission Line Opera4on 5 Transmission Line Equa<ons – Solu<on dV
d 2V
dI
= zI ⇒ 2 = z
dx
dx
dx
dI
= yV
dx
d 2V
+γ x
−γ x
−
zyV
=
0
⇒
V
x
=
A
e
+
A
e
, γ = zy
(
)
1
2
dx 2
I=
zy
1 dV γ
γ
=
A1e+γ x − A2 e−γ x , =
=
z dx z
z
z
⇒ I ( x) =
(
)
(
)
1
A1e+γ x − A2 e−γ x
Zc
y 1
=
z Zc
Zc is called the Characteris4c Impedance or the Surge Impedance (this terminology usually reserved for R = 0 and G = 0) 8/20/13 Tramsmission Line Opera4on 6 Transmission Line Equa<ons – Solu<on V ( x ) = A1e+γ x + A2 e−γ x , γ = zy
1
z
+γ x
−γ x
I ( x ) = ( A1e − A2 e ) , Z c =
Zc
y
Receiving End Voltage: VR = V (0)
⎡ V
⎢ R
⎢ IR
⎣
⎤ ⎡ 1
1
⎥ = ⎢ −1
−1
⎥ ⎢⎣ Z c −Z c
⎦
⎤ ⎡ A1 ⎤
⎥
⎥⎢
⎥⎦ ⎢⎣ A2 ⎥⎦
⎡
⎡ A ⎤
1 ⎤ ⎡ VR
1 ⎢ Z
1
⎥⎢
⎢
⎥=
−1
⎢ A2 ⎥ 2Z c ⎢ Z c−1 −1 ⎥ ⎢ I R
⎣
⎦
⎣
⎦⎣
−1
c
8/20/13 ⎤ 1 ⎡ 1 Z ⎤⎡ V
c
⎥= ⎢
⎥⎢ R
⎥ 2 ⎢ 1 −Z c ⎥ ⎢ I R
⎦
⎣
⎦⎣
Tramsmission Line Opera4on ⎤
⎥
⎥
⎦
VR + Z c I R
2
VR − Z c I R
A2 =
2
A1 =
7 Transmission Line Equa<ons – Solu<on VR + Z c I R + γ x VR − Z c I R − γ x
V ( x) =
e +
e
2
2
V + Z c I R + γ x VR − Z c I R − γ x
I ( x) = R
e −
e
2Z c
2Z c
(
)
(
(
2
+ Z c I R e+ γ x + e− γ x
V ( x) =
VR e + γ x + e − γ x + Z c I R e + γ x − e − γ x
I ( x) =
VR e + γ x − e − γ x
⎡ V ( x)
⎢
⎢ I ( x)
⎣
8/20/13 )
(
2Z c
⎤ ⎡ cosh γ x
Z c sinh γ x
⎥=⎢
⎥ ⎢ Z −1 sinh γ x cosh γ x
⎦ ⎣ c
) =V
)=V
R
R
⎤⎡ V
⎥⎢ R
⎥ ⎢ IR
⎦⎣
cosh γ x + Z c I R sinh γ x
sinh γ x + Z c I R cosh γ x
Zc
⎤ ⎡ A( x ) B ( x )
⎥=⎢
⎥ ⎢ C ( x) D( x)
⎦ ⎣
Tramsmission Line Opera4on ⎤⎡ V
⎥⎢ R
⎥ ⎢ IR
⎦⎣
⎤
⎥
⎥
⎦
8 ABCD Parameters Sending End Voltage: VS = V (  )
⎤ ⎡ V ⎤ ⎡ A(  ) B (  ) ⎤ ⎡ V
⎡ V ⎤ ⎡ cosh γ 
Z
sinh
γ

c
⎥⎢ R
⎢ S ⎥ = ⎢
⎥⎢ R ⎥ = ⎢
−1
⎢
⎢
⎥
I
Z
⎣ S ⎦ ⎣ c sinh γ  cosh γ  ⎥⎦ ⎢⎣ I R ⎥⎦ ⎢⎣ C (  ) D (  ) ⎥⎦ ⎢⎣ I R
ABCD Parameters: IR
IS
+
+
⎡ V ⎤ ⎡
⎡ V
⎤
S
A
B
R
Two-­‐Port ⎢
⎥=⎢
⎥⎢
VS
VR
Network ⎢ IS ⎥ ⎣ C D ⎦ ⎢ I R
⎣
⎦
⎣
−
−
For passive networks AD − BC = 1 8/20/13 Tramsmission Line Opera4on ⎤
⎥
⎥
⎦
⎤
⎥
⎥
⎦
9 Transmission Lines Three kinds of lines: 1. Short lines (< 80 km) 2. Medium lines (80 – 250 km) 3. Long lines (> 250 km) 8/20/13 Tramsmission Line Opera4on 10 Short Length Lines Characterized by the approxima<on I S = I R
IS
Z = z = ( R + jω L ) 
IR
+
VS
+
VR
−
−
Load VR = VS − ZI S ⇒ VS = VR + ZI S = VR + ZI R
IS = I R
⎡ V ⎤ ⎡
⎤ ⎡ VR
S
A
B
⎢
⎥=⎢
⎥⎢
⎢ IS ⎥ ⎣ C D ⎦ ⎢ I R
⎣
⎦
⎣
8/20/13 ⎤ ⎡
⎤ ⎡ VR
1
Z
⎥=⎢
⎥⎢
⎥ ⎣ 0 1 ⎦ ⎢ IR
⎦
⎣
Tramsmission Line Opera4on ⎤
⎥
⎥
⎦
11 Medium Length Lines Characterized by the approxima<on G ≡ 0
Z = z = ( R + jω L ) 
IS
+
VS
IR
+
VR
Y = y = jω C
−
−
Z = z = ( R + jω L ) 
IS
+
VS
−
8/20/13 Y
2
IR
Y
2
Tramsmission Line Opera4on +
VR
−
12 Medium Length Lines I = V Y + VS − VR
S
S
2
Z
KCLs: IS
Z = z = ( R + jω L ) 
+
VS
−
Y
2
IR =
VS − VR
Y
− VR
Z
2
IR
Y
2
+
VR
−
⎛ ZY ⎞
VS = VR ⎜ 1+
+ ZI R
⎟
2 ⎠
⎝
⎛ ZY ⎞
⎛ ZY ⎞
I S = ⎜ 1+
YV
+
I
1+
4 ⎟⎠ R R ⎜⎝
2 ⎟⎠
⎝
⎡ V ⎤ ⎡
⎤ ⎡ VR
S
A
B
⎢
⎥=⎢
⎥⎢
⎢ IS ⎥ ⎣ C D ⎦ ⎢ I R
⎣
⎦
⎣
8/20/13 ⎡
⎤ ⎢
⎥=⎢
⎥ ⎢
⎦ ⎢
⎢⎣
ZY
1+
Z
2
⎛ ZY ⎞
ZY
1+
Y
1+
⎜⎝
4 ⎟⎠
2
Tramsmission Line Opera4on ⎤
⎥⎡
⎥ ⎢ VR
⎥⎢ I
⎥⎣ R
⎥⎦
⎤
⎥
⎥
⎦
13 Long Lines ⎤ ⎡ V ⎤ ⎡ A(  ) B (  ) ⎤ ⎡ V ⎤
⎡ V ⎤ ⎡ cosh γ 
Z
sinh
γ

c
⎥⎢ R ⎥
⎥⎢ R ⎥ = ⎢
⎢ S ⎥=⎢
⎢ I S ⎥ ⎢ Z c−1 sinh γ  cosh γ  ⎥ ⎢ I R ⎥ ⎢ C (  ) D (  ) ⎥ ⎢ I R ⎥
⎣
⎦ ⎣
⎦ ⎣
⎦
⎦⎣
⎦⎣
For computa<onal purposes, circuit models are s<ll o[en used, such as the pi-­‐model: Equate Z′
⎡
Z ′Y ′
Z′
⎢ 1+
⎡ A B ⎤ ⎢
2
Y′
Y′
=
⎢
⎥ ⎢
⎛ Z ′Y ′ ⎞
Z ′Y ′
2
2
C
D
⎣
⎦
1+
Y
1+
′
⎢
⎜
4 ⎟⎠
2
⎢⎣ ⎝
(Note the symmetry) What must Z’ and Y’ equal so that the ABCD matrices are iden<cal? 8/20/13 Tramsmission Line Opera4on 14 ⎤
⎥
⎥
⎥
⎥
⎥⎦
Long Lines ⎡ A B ⎤ ⎡
⎢
⎥=⎢
⎣ C D ⎦ ⎢⎣
⎡
Z ′Y ′
1+
Z′
⎢
cosh γ 
Z c sinh γ  ⎤ ⎢
2
⎥=
Z c−1 sinh γ  cosh γ  ⎥ ⎢ ⎛ 1+ Z ′Y ′ ⎞ Y 1+ Z ′Y ′
⎦ ⎢ ⎜
⎟⎠ ′
4
2
⎝
⎢⎣
B = Z ′ = Z c sinh γ  =
⎤
⎥
⎥
⎥
⎥
⎥⎦
z
sinh γ 
y
Z ′Y ′
2
⎛ Z ′Y ′ ⎞
C = ⎜ 1+
Y′
⎟
4 ⎠
⎝
A = D = 1+
8/20/13 Tramsmission Line Opera4on 15 Long Lines 1
1 z
sinh γ 
Write: Z ′ = z = z Z c sinh γ  = z
sinh γ  = z
z
z y
γ
Similarly: 1+
Z ′Y ′
Y ′ cosh γ  − 1 cosh γ  − 1 γ  cosh γ  − 1
= cosh γ  ⇒ =
=
=
=
sinh γ  z sinh γ 
2
2
Z′
z
γ
⇒
8/20/13 Y ′ y y 2
=
=
2
2
2 y
y
γ
tanh
z
2
γ
y
γ  y 2 1
γ  y tanh 2
tanh =
tanh =
z
2
2  zy
2
2 γ
2
Tramsmission Line Opera4on 16 Long Lines: What must Z’ and Y’ equal so that the ABCD matrices are iden<cal? sinh γ 
Z ′ = z
γ
Y ′ y tanh γ  2
=
2
2 γ 2
Y′
2
8/20/13 Tramsmission Line Opera4on 17 Example: A single-­‐circuit 60-­‐Hz transmission line is 370 km (230) mi) long. The conductors are Rook with flat horizontal spacing and 7.25 m (23.8 [) between conductors. The load on the line is 125 MW at 215 kV with unity power factor. Find the voltage, current, and power at the sending end and the voltage regula<on of the line. Also, determine the wavelength and the velocity of propaga<on of the line. (See Tables that follow) 23.8 \ Deq = 3 23.8 × 23.8 × ( 2 × 23.8) = 30 ft
h]p://www.aluminum.org/AM/Template.cfm?Sec4on=20101&TEMPLATE=/CM/ContentDisplay.cfm&CONTENTID=30772 8/20/13 Tramsmission Line Opera4on 18 Example: ⎛ 1⎞
z = r + j2π fL = r + j f µo ln ( Dm ) + j f µo ln ⎜ ⎟ Ω / m

⎝ Ds ⎠




xd
xa
= 0.1603+ j ( 0.415 + 0.4127 ) = 0.1603+ j0.8277 = 0.8431∠79.04°
10−6
y= j
= j5.105 × 10−6
0.0950 + 0.1009
γ  =  zy = 230 0.8431∠79.04° × 5.105 × 10−6 ∠90° = 0.4772∠84.52°
Zc =
8/20/13 z
= 406.38∠ − 5.48°
y
Tramsmission Line Opera4on 19 8/20/13 Tramsmission Line Opera4on 20 8/20/13 Tramsmission Line Opera4on 21 8/20/13 Tramsmission Line Opera4on 22 Example: 215,000
V
=
= 124,130∠0° to neutral
R
3
125,000
I
=
= 335.7∠0° amps
R
3 × 215,000
⎡ V ⎤ ⎡ cosh γ 
Z c sinh γ  ⎤ ⎡ VR ⎤
S
⎥⎢
⎢
⎥=⎢
⎥
−1
⎢ I S ⎥ ⎢ Z c sinh γ  cosh γ  ⎥ ⎢ I R ⎥
⎣
⎦ ⎣
⎦
⎦⎣
I used Mathcad for the details… 8/20/13 Tramsmission Line Opera4on 23 Example: I used Mathcad for the details… 8/20/13 Tramsmission Line Opera4on 24 Example: Find the equivalent pi-­‐circuit parameters and compare them with results from the exact analysis: Anything within 10% is generally acceptable. 8/20/13 Tramsmission Line Opera4on < 4% error 25 SURGE IMPEDANCE LOADING (SIL) is the power delivered by a lossless line to a resis<ve load equal to the surge impedance Zc =
IS
L
C
+
VS
+
VR
−
−
x=
⎡ V ( x)
⎢
⎢ I ( x)
⎣
8/20/13 IR
Zc =
L
C
x=0
⎤ ⎡ cosh γ x
Z c sinh γ x
⎥=⎢
⎥ ⎢ Z −1 sinh γ x cosh γ x
⎦ ⎣ c
⎤
⎡ cos β x
⎥ = ⎢
⎥ R = 0 ⎢ jZ c−1 sin β x
⎦
⎣
G=0
Tramsmission Line Opera4on jZ c sin β x ⎤ ⎡ VR
⎥⎢
cos β x ⎥ ⎢ I R
⎦⎣
⎤
⎥
⎥
⎦
26 Voltage Profiles – Lossless Line V ( x ) = VR cos β x + jZ c I R sin β x
I ( x ) = jZ c−1VR sin β x + I R cos β x
V − jZ c I R sin β 
V − VR cos β 
V (  ) = VS ⇒ VR = S
, IR = S
cos β 
jZ c sin β 
Surge Impedance Load I R = VR Zc ⇒ V ( x ) = VR exp ⎡⎣ jβ x ⎤⎦ = VS
No Load I R = 0 ⇒ V ( x ) = VRNL cos β x
VS
V
=
0
⇒
V
x
=
Z
I
sin
β
x,
I
=
Short Circuit Load R
SC ( )
c R
R
Z c sin β 
8/20/13 Tramsmission Line Opera4on 27 Voltage Profiles – Lossless Line No-­‐Load VS
SIL Short x=
8/20/13 λ
4
Full-­‐Load x=0
Tramsmission Line Opera4on 28 Steady-­‐State Stability Limit Expression for the real power delivered by a lossless line (derived from the pi-­‐model) VS − VR
Y′
IR =
− VR
IS
IR
Z′
2
Z′
+
+
Y′
Y′
VS
VR
2
2
−
−
Assume that Vr and Vs are held constant and that the angle between them is δ VS e jδ − VR
jω C ′
IR =
− VR
jX ′
2
8/20/13 Tramsmission Line Opera4on 29 Steady-­‐State Stability Limit Complex power delivered to receiving end: ⎛ VS e− jδ − VR
jω C ′ ⎞
*
S R = VR I R = VR ⎜
+ VR
2 ⎟⎠
⎝ − jX ′
VSVR ( cos δ − j sin δ ) − VR 2
ω C ′
= j
+ jVR 2
X′
2
⎛ VSVR
⎞
VSVR
VR 2
2 ω C ′
=
sin δ + j ⎜
cos δ −
+ VR
X′
X′
2 ⎟⎠
⎝ X′
Real Power: P = PR = PS = Re ⎡⎣ S R ⎤⎦ =
8/20/13 VSVR
sin δ
X′
Tramsmission Line Opera4on 30 Steady-­‐State Stability Limit Real Power: P = PR = PS = Re ⎡⎣ S R ⎤⎦ = VSVR sin δ
X′
Real Power VSVR
Pmax =
X′
δ
90°
Pmax is known as the Steady-­‐State Stability Limit of a Lossless Line 8/20/13 Tramsmission Line Opera4on 31 Steady-­‐State Stability Limit The steady-­‐state stability limit is o[en expressed in terms of SIL. Recall Z ′ = Z c sinh γ 
and for a lossless line this becomes Z ′ = jZc sin β  = jX ′
Then VV
VSVR
VV
sin δ
P = S R sin δ =
sin δ = S R
X′
Z c sin β 
Z c sin 2π  λ
Normalizing (per-­‐unit) VS . p.u.VR. p.u SIL
VS VR Vrated 2 sin δ
sin δ
P=
= VS . p.u.VR. p.u SIL
⇒ Pmax =
Vrated Vrated Z c sin 2π  λ
sin 2π  λ
sin 2π  λ
8/20/13 Tramsmission Line Opera4on 32 Steady-­‐State Stability Limit VS . p.u. = VR. p.u = 1
Observa<ons: • Pmax is propor<onal to the square of the line voltage • Pmax decreases with line length 8/20/13 Tramsmission Line Opera4on 33 Maximum Power Flow via ABCD Parameters ⎡
Z ′Y ′
1+
⎢
Recall: ⎡ A B ⎤ ⎡ cosh γ  Zc sinh γ  ⎤ ⎢
2
⎢
⎣ C
⎥ = ⎢ −1
D ⎦ ⎢ Z c sinh γ 
⎣
B = Z ′ = Z c sinh γ  =
cosh γ 
⎥=
⎥ ⎢
⎦ ⎢
⎢⎣
⎤
Z′
⎥
⎥
⎛ Z ′Y ′ ⎞
Z ′Y ′ ⎥
⎜⎝ 1+ 4 ⎟⎠ Y ′ 1+ 2 ⎥
⎥⎦
z
sinh γ 
y
Z ′Y ′
2
⎛ Z ′Y ′ ⎞
C = ⎜ 1+
Y′
⎟
4 ⎠
⎝
A = D = 1+
⎡ V ⎤ ⎡
⎤ ⎡ VR
S
A
B
⎢
⎥=⎢
⎥⎢
⎢ IS ⎥ ⎣ C D ⎦ ⎢ I R
⎣
⎦
⎣
8/20/13 ⎤
V − AVR
⎥ ⇒ VS = AVR + BI R ⇒ I R = S
B
⎥
⎦
Tramsmission Line Opera4on 34 Maximum Power Flow via ABCD Parameters A = cosh γ  = A ∠θ A
Subs<tu<ng: B = Z ′ = Z ′ ∠θ Z
VS = VS ∠δ
VR = VR ∠0°
jθ
jδ
V
e
−
A
e
VR
V
−
AV
R
IR = S
= S
jθ
B
Z
e
′
The complex power delivered to the receiving end is: A
Z
S = PR + jQR = VR I R* =
⇒ PR =
QR =
8/20/13 VS VR
VS VR
Z′
Z′
VS VR e
Z′
cos (θ Z − δ ) −
sin (θ Z − δ ) −
j(θ Z −δ )
A VR
Z′
A VR
Z′
2
A VR e (
2
−
2
j θ Z −θ A )
Z′
cos (θ Z − θ A ) ,
sin (θ Z − θ A )
Tramsmission Line Opera4on 35 Maximum Power Flow via ABCD Parameters For a lossless line: θ A = 0, Z ′ = jX ′ = X ′∠90°
2
VS VR
A VR
VS VR
P
=
cos
90°
−
δ
−
cos
90°
=
sin δ
(
)
(
)
Thus: R
X′
X′
X′
(same as previous Pmax) For a lossy line, 2
V
V
A
V
S
R
R
PR
=
cos (θ Z − δ ) −
cos (θ Z − θ A )
Z′
θZ = δ Z′
2
VS VR
A VR
=
−
cos (θ Z − θ A )
Z′
Z′
The presence of the second term and the fact that Z’ > X’ means that PRmax is less than that of a lossless line. max
8/20/13 Tramsmission Line Opera4on 36 Line Loadability In prac<ce lines are not operated at their theore<cal maximum power. It is operated below this limit to maintain stability during transient disturbances. Loadability is usually limited by the thermal ra<ngs of the conductors or equipment ra<ngs, not by voltage drop as done here. The Steady-­‐State Stability Limit serves more as maximum indicator than as an opera<onal ideal. 8/20/13 Tramsmission Line Opera4on 37 Reac<ve Compensa<on Techniques Reac<ve components are some<mes used on medium and long-­‐
length lines to increase loadability and to maintain voltages near rated values. Two approaches are used. 1. Shunt inductors are installed at selected points along the line from each phase to neutral. Since inductors absorb reac<ve power they serve to reduce overvoltages during light load condi<ons. They also reduce transient overvoltages dues to switching and lightning surges. However, they also reduce line loadability if they are not removed under full-­‐load condi<ons. 2. Series capacitors are used on long lines to increase the line loadability. Capacitor banks are installed in series with each phase conductor at selected points along the line. Their effect is to reduce the net series impedance of the line hence reducing line voltage drops. 8/20/13 Tramsmission Line Opera4on 38 Reac<ve Compensa<on Techniques Series Compensa4on Shunt Compensa4on Transmission Line X′
X′
−
β
−
β
IS
X′
R′
2
2
+
Y′
Y′
Y′
VS −α
2
2
2
−
Circuit Model Examples will help to clarify this… 8/20/13 Tramsmission Line Opera4on IR
−α
Y′
2
+
VR
−
39 Text Example 5.2 – 5.6, 5.9 – 5.10 A three-­‐phase 765-­‐kV, 60-­‐Hz, 300-­‐km line has z = 0.0165 + j0.3303 Ω / km
y = j4.674 × 10−6 S / km
Calculate the exact ABCD parameters and compare the B value with that of the pi-­‐model. 8/21/13 Tramsmission Line Opera4on 40 Text Example 5.2 – 5.6, 5.9 – 5.10 8/21/13 Tramsmission Line Opera4on 41 Text Example 5.2 – 5.6, 5.9 – 5.10 Compare Z and Y with Z’ and Y’ 8/21/13 Tramsmission Line Opera4on 42 Text Example 5.2 – 5.6, 5.9 – 5.10 Neglec<ng line losses, find the steady-­‐state stability limit for the line assuming a 266.a-­‐ohm surge impedance and a 5000-­‐km wavelength. Use VS = VR = 765 kV. Pmax
8/21/13 Tramsmission Line Opera4on VSVR
1
=
Z c sin 2π  λ
43 Text Example 5.2 – 5.6, 5.9 – 5.10 Neglec<ng line losses, find the steady-­‐state stability limit for the line assuming a 266.a-­‐ohm surge impedance and a 5000-­‐km wavelength. Use VS = VR = 765 kV. Pmax =
8/21/13 Tramsmission Line Opera4on VSVR
1
Z c sin 2π  λ
44 Text Example 5.2 – 5.6, 5.9 – 5.10 Determine the theore<cal maximum power that the line can deliver. =
max
θZ = δ
PR
8/21/13 VS VR
Z′
−
A VR
Z′
2
cos (θ Z − θ A )
Tramsmission Line Opera4on 45 Text Example 5.2 – 5.6, 5.9 – 5.10 Suppose the line consists of four conductors each of which can carry a maximum current of 4800 amps. The sending end voltage is held constant at its rated value. Determine: a. The line loadability. Assume that the receiving end voltage is reduce to 0.95 of its rate value and that δ = 35o (maximum angle across the line). b. The full-­‐load current at 0.986 p.f. (leading) based on the above line loadability. c. The exact receiving end voltage for the ful-­‐ load current found in part (b). d. The percent voltage regula<on for the full-­‐load current. 8/21/13 Tramsmission Line Opera4on 46 Text Example 5.2 – 5.6, 5.9 – 5.10 a. The line loadability. Assume that the receiving end voltage is reduce to 0.95 of its rate value and that δ = 35o (maximum 2
angle across the line). V V
AV
PR =
8/21/13 S
Z′
R
cos (θ Z − δ ) −
Tramsmission Line Opera4on R
Z′
cos (θ Z − θ A )
47 Text Example 5.2 – 5.6, 5.9 – 5.10 b. The full-­‐load current at 0.986 p.f. (leading) based on the above line loadability. 8/21/13 Tramsmission Line Opera4on 48 Text Example 5.2 – 5.6, 5.9 – 5.10 c. The exact receiving end voltage for the ful-­‐ load current found in part (b). ⎡ V
⎢ S
⎢ IS
⎣
8/21/13 ⎤ ⎡
⎤ ⎡ VR
A
B
⎥=⎢
⎥⎢
⎥ ⎣ C D ⎦ ⎢ IR
⎦
⎣
⎤
⎥ ⇒ VS = AVR + BI R
⎥
⎦
Tramsmission Line Opera4on 49 Text Example 5.2 – 5.6, 5.9 – 5.10 d. The percent voltage regula<on for the full-­‐load current. VS = AVR + BI R
8/21/13 Tramsmission Line Opera4on 50 Text Example 5.2, 5.3, 5.4, 5.5, 5.6, 5.9, 5.10 Iden<cal shunt reactances (inductors) are connected from each line to neutral at both ends of the line during light load providing 75% compensa<on. The reactances are removed during heavy load condi<ons. Full load is 1.90 kA at unity p.f. at 730 kV. Assuming that the sending end voltage is constant determine: a. The percent voltage regula<on of the uncompensated line. b. The equivalent shunt admiqance and series impedance compensated line. c. Percent voltage regula<on of the compensated line. 8/21/13 Tramsmission Line Opera4on 51 Text Example 5.2, 5.3, 5.4, 5.5, 5.6, 5.9, 5.10 a. The percent voltage regula<on of the uncompensated line. 8/21/13 Tramsmission Line Opera4on 52 Text Example 5.2, 5.3, 5.4, 5.5, 5.6, 5.9, 5.10 b. The equivalent shunt admiqance compensated line. c. Percent voltage regula<on of the compensated line. IS
+
Y′
VS −α
2
−
8/21/13 X′
Y′
2
IR
R′
α = 0.75
Y′
2
Tramsmission Line Opera4on −α
Y′
2
+
VR
−
53 Text Example 5.2, 5.3, 5.4, 5.5, 5.6, 5.9, 5.10 c. No suppose iden<cal series capacitors are installed in each phase at both ends of the line providing 30% compensa<on. Determine the maximum power this line can deliver and compare it with the uncompensated line. Assume VS = VR. IS
+
VS
−
8/21/13 −β
X′
2
Y′
2
X′
R′
β = 0.25
−β
Y′
2
Tramsmission Line Opera4on X′
2
IR
+
VR
−
54 Text Example 5.2, 5.3, 5.4, 5.5, 5.6, 5.9, 5.10 8/21/13 Tramsmission Line Opera4on 55 Text Example 5.2, 5.3, 5.4, 5.5, 5.6, 5.9, 5.10 8/21/13 Tramsmission Line Opera4on 56